text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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If you manage a business that is always striving for a competitive advantage, it would be safe to say that you would have already been taking advantage of the opportunities that social media presents for your company. Your company's probably on Twitter already, right? If not yet, check out this twitter profile setup guide to help you get started.
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Tweet seemingly useless, but interesting information that relates to your products and organization like company history, milestones, partnerships, etc. This can help your fan base know and discover more about your company.
Remember, not everything has to come from you. If your fan mentions you in a Tweet while commending your products/services, use this to your advantage and Retweet it! It's real, unbiased and impartial advertising done by the people you're selling to.
There you have it! If you make use of our ideas and start tweeting like this, you're bound to see a spike in your Twitter account's followers, mentions and retweets. Have any ideas that you feel we missed out on? Feel free to post on the comments section below. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,485 |
package uk.gov.hmrc.ct.ct600.v3
import uk.gov.hmrc.ct.box._
import uk.gov.hmrc.ct.ct600.v3.retriever.CT600BoxRetriever
case class B150(value: Option[Boolean]) extends CtBoxIdentifier("Banks, building societies, insurance companies and other financial concerns") with CtOptionalBoolean with Input with ValidatableBox[CT600BoxRetriever] {
override def validate(boxRetriever: CT600BoxRetriever): Set[CtValidation] = validateBooleanAsMandatory("B150", this)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,949 |
This invention relates to an improved design of a golf putter. More particularly, it relates to a putter having an improved marking enabling a golfer to more accurately line up the putt.
Putting is very important in the game of golf. On a par-72 course, one-half (36) strokes are allotted for putting. Moreover, it does not take great strength or physical ability to be a good putter. For a golfer, one stroke on a green counts just as much as any other stroke in the course.
There are two kinds of putts--long putts and short putts. A long putt may be defined as any putting requiring more than five feet to reach the cup. If the ball lies within about five feet of the hole, it is a short putt. The primary objective of the long putting is to hit the ball so that it ends within approximately three feet of the hole. Although sinking the ball in a long put is pleasant and desirable, the primary goal on long putts should be placing the ball near enough to the hole so that the next stroke will easily make the hole. In addition, for the long putts many variables such as the slope of the ground, the accuracy of the aim, the length to be traveled, and speed of the ball comes into play. The accuracy of the aim, although critically important, only plays a minor role in the overall success of the long putting. On the other hand, the accuracy is the most important factor in short putting. Other variables such as the slope of the ground, speed of the ball, and texture of the turf become less important. A short putt is very important in a game of golf not only because a missed putt will cost a stroke, but the impact it has psychologically to a player. Because of its length, every golfer, pros and amateurs alike, feels compelled to make it. When he does not make it, he thinks he missed an easy shot. Consequently, it tends to destroy a player's confidence and may affect his concentration for the rest of the game.
In a putting situation, the direction of the putt is dictated by the path of the clubhead and the face angle at impact. The path is important and affects direction, but the face angle of the putter at impact is also very important in determining direction. Providing the green within five feet of the hole has no significant slope and texture of the turf is uniform, successful short putting should require the path of the clubhead directly aimed to center of the hole, and club face angle precisely perpendicular to the line between clubhead and the hole. In addition, the center of the ball should be precisely aligned with the marker on top of the putter that generally indicates location of the center of gravity in the toe to heel direction. An infinitesimal deviation from these is the reason for a miss of the short putt. To miss the hole that is four inches wide, from five feet distance the deviation of a face angle from perpendicular to the straight line to the hole must be so small it will not be discernable to the naked eyes. Every player carefully adjusts the face angle of the putter and aligns the ball to the center of gravity marker before he strokes the ball. Nevertheless, a putt is missed because the face of the putter is not truly perpendicular to the direction of the hole, and the ball is off the center of gravity. To achieve a perfect alignment every time, a finely adjusted machine toll is needed. Since hand and eye coordination of a human being is much less precise than a machine, a mistake will occur and a missed putt is the result. The present invention is directed to reducing the small inaccuracies that occur with the prior art putters and automatically compensating for mis-struck putts.
Accordingly, it is a primary object of the present invention to provide an improved golf putter which automatically compensates for misaligned and mis-struck putts in a short putting situation. A typical putter consists of a putter head about four to five inches long with a predetermined weight distribution and a total weight ranging between fifteen and eighteen ounces. The putting face of a putter is horizontally flat and has two to four degrees of loft. There are many putter designs in the market each claiming why it is superior to others. However, for the short putting situation; the most important attribute of a putter has to be how tolerant making up the infinitesimal and almost invisible error in positioning the face angle and the clubhead path a player makes when the player aims and strokes a ball. If there is a putter with a face shaped such a way that the ball is always aimed toward the center of the hole, probability of making the hole increases significantly even when a player makes a small error in aiming the ball toward the hole. Additionally, putter markings are usually taken for granted, consisting of one or several straight lines. A putter with an enhanced marking would assist the golfer with both centering and aiming the putt.
It is desirable to have an alignment on the top of the surface of a putter to aid a player to aim the ball as precisely as possible. The alignment marker will consist of a line perpendicular to the putter face located precisely at the center of gravity of the club head, and a curved horizontal face marker which is an arc of a concentric circle of a golf ball placed abutting or one quarter of an inch in front of the hitting face of the putter. With the horizontal face marker being a concentric circle of the golf ball, the distance between the ball and face marker will be symmetrical and equal along the entire arc of the marker when the ball is aligned properly. However. if the ball is misaligned, the distance between the ball and the marker will be different between the toe-end and the heel-end of the marker, enabling the player to make an adjustment easily. With the alignment marker enabling a player to align the ball more accurately and the concave putter further compensating any misalignment, the improved putter of present invention will enable a player to putt more accurately than a conventional putter.
The putter of the present invention differs from any other conventional or unconventional style putters in the market today. The putter may be precision machined to form a concave horizontal face from the heel to toe of the hitting face.
The curvature of the concave horizontal face may range from an arc of a five to a one-foot radius circle with the center point at the center of the hole. The curvature is defined as the reciprocal of the radius of a circle. Also provided in accordance with the present invention is a marking on the club head, the marking serving to assist with both aiming and centering.
The best way to hit the golf ball in a short putt is like a pendulum. The most golfers cannot do that consistently, but they tend to swing in an arc. Depending on when the putter strikes the ball in that arc, the putter face may be either slightly open or slightly closed at the impact. Because the concave putter always aims a ball toward the center of the hole, slightly open or closed face hit is compensated enough to make the ball to drop into the hole.
Additionally, for right-handed golfers with conventional putters, balls stroked on the inside the sight line toward the heel will travel left of the intended line to the hole. Previous tests show that an average golfer almost always impacts on the toe side of the sight line making the ball to travel right of the intended line to the hole. This is so because the player never looks down at the putter head and ball from directly above, but slightly to the heel side of the sight line. The amount of the offset is very small and almost invisible; but, a small misalignment will result in a missed putt. Since a curved sight line marker enables a golfer to more accurately line up the a putt, the concave putter will compensate the small misalignment and make the ball drop into the hole by automatically aiming the ball toward the center of the hole.
Tests performed on a flat putting green showed that putts stroked with a straight putter resulted in 20 percent traveling on a line left of center and 20 percent traveling right of the center. Of putts stroked with the concave putter, only 10 percent went left of center and 10 percent went right of the center--a statistically significant improvement. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,187 |
\section{Introduction}
Superconductivity in SrTiO$_{3}$, discovered a long time ago
\cite{Schooley,Koonce,Binnig} attracts a renewed experimental
\cite{Lin,Collignon,Lin2,Rischau,Swartz,Rowley} and theoretical
\cite{Rowley,Ruhman,Rosenstein,JSNM2017,DKTM2014,Edge,Wolfle,Gorkov} interest
because strontium titanate exhibits the superconducting transition at very low
carrier densities.
Different physical mechanisms are considered to explain superconductivity in
SrTiO$_{3}$, e.~g., dynamically screened electron-LO-phonon interactions
\cite{Rowley,Ruhman,Rosenstein,JSNM2017,DKTM2014}, interaction of electrons
with quantum ferroelectric fluctuations \cite{Edge}, with the soft TO-phonon
mode \cite{Wolfle} and with local phonons \cite{Gorkov}. All these studies are
in line with experiments. Nevertheless, the physical picture of
superconductivity in SrTiO$_{3}$ is still far from clarity.
Besides physical mechanisms, the key question concerns the method.
Migdal-Eliashberg theory \cite{McMillan} is hardly applicable here, because
the energies of LO phonons in doped SrTiO$_{3}$ are comparable with the Fermi
energy of electrons at typical concentrations relevant for the experiments.
Nevertheless, this standard approach of superconductivity is exploited in
recent works \cite{Ruhman,Gorkov}. A generalization of the BCS theory beyond
the adiabatic regime was derived by Kirzhnits, Maksimov and Khomskii (KMK)
\cite{Kirzhnits} who described the electron-phonon and Coulomb interactions
through the dielectric function.
As a downside of this generalization, the dielectric function method (DFM)
developed by KMK is a weak-coupling approach. The electron-phonon coupling in
strontium titanate is not weak \cite{GLF}, especially at low concentrations.
Consequently, the DFM can be uncontrolled, especially at low densities (where
the electron-phonon interaction is stronger) and not able to obtain
quantitatively accurate results for SrTiO$_{3}$. However it can give a
qualitatively correct picture of the concentration-dependent superconducting
transition temperature \cite{Rowley,Takada,JSNM2017,DKTM2014}. A verification
of the KMK ansatz within the non-adiabatic extension of the Eliashberg theory
\cite{Rosenstein} showed compatibility of these two methods.
The present work does not claim to explain all aspects of superconductivity in
SrTiO$_{3}$, because the picture seems to be more complicated than can what be
described by a unique mechanism for all concentrations. We restrict the
treatment to only the most usual interactions which are definitely present in
SrTiO$_{3}$: the electron-phonon interaction with LO and acoustic phonons, and
the Coulomb repulsion. The DFM with these interactions seems to be applicable
for SrTiO$_{3}$ except at very low densities $n\lesssim10^{18
\operatorname{cm
^{-3}$, where the experiment \cite{Lin} reveals a separate superconducting
dome which correlates with the evolution of the Fermi surface with doping
\cite{Collignon2}.
In Refs. \cite{Rowley,Wolfle}, experimental data on the transition temperature
have been reasonably explained using, respectively, DFM and the Morel-Anderson
theory \cite{Morel} in the effective mass approximation, and including in Ref.
\cite{Wolfle} the deformation potential interaction with the soft TO-phonon
mode. In the present work, we apply the extension of the DFM for a
non-parabolic band structure and incorporate the interaction with acoustic
phonons. It is interesting that the softening of the lowest TO-phonon mode
influences the dielectric function and hence enhances $T_{c}$ even before
introducing the deformational electron -- TO-phonon interaction. The other new
element is a multimode dielectric function different from that used in Ref.
\cite{Takada} and accounting for non-parabolic phonon dispersion. We analyze
also the anomalous isotope effect in SrTiO$_{3}$ \cite{Stucky2016}.
\section{Theoretical description}
In this section, DFM is revisited taking into account non-parabolicity of the
conduction band. In the present study, spin-orbit coupling and
next-nearest-neighbor hopping are neglected. We start from the gap equation
\cite{JSNM2017}, where the band dispersion law $\varepsilon_{\mathbf{k
,\lambda}$ can be, in general, non-parabolic
\begin{align}
\Delta_{\lambda}\left( \mathbf{k}\right) & =-\frac{1}{\left( 2\pi\right)
^{3}}\int d\mathbf{k}^{\prime}~\Delta_{\lambda}\left( \mathbf{k}^{\prime
}\right) \frac{\tanh\frac{\beta\left\vert \varepsilon_{\mathbf{k}^{\prime
},\lambda}\right\vert }{2}}{2\left\vert \varepsilon_{\mathbf{k}^{\prime
},\lambda}\right\vert }\nonumber\\
& \times\frac{2}{\pi}\int_{0}^{\infty}d\Omega\frac{\left\vert \varepsilon
_{\mathbf{k}^{\prime},\lambda}\right\vert +\left\vert \varepsilon
_{\mathbf{k},\lambda}\right\vert }{\Omega^{2}+\left( \left\vert
\varepsilon_{\mathbf{k}^{\prime},\lambda}\right\vert +\left\vert
\varepsilon_{\mathbf{k},\lambda}\right\vert \right) ^{2}}\nonumber\\
& \times V^{R}\left( \mathbf{k}-\mathbf{k}^{\prime},i\Omega\right) .
\label{gapeq
\end{align}
Here $\beta=1/\left( k_{B}T\right) $, $\lambda$ is the index of a subband of
the conduction band, $\Delta_{\lambda}\left( \mathbf{k}\right) $ is the
momentum-dependent gap function, $V^{R}\left( \mathbf{q},i\Omega\right) $ is
the matrix element of the effective electron-electron interaction in the polar
crystal expressed through the total dielectric function of the electron-phonon
system $\varepsilon^{R}\left( \mathbf{q},i\Omega\right) $ in the Matsubara
representation,
\begin{equation}
V^{R}\left( \mathbf{q},i\Omega\right) =\frac{4\pi e^{2}}{q^{2
\varepsilon^{R}\left( \mathbf{q},i\Omega\right) }+V^{ac}\left(
\mathbf{q},i\Omega\right) , \label{poten
\end{equation}
where $V^{ac}\left( \mathbf{q},i\Omega\right) $ is the effective potential
due to the acoustic deformation scattering from Ref. \cite{DKTM2014}. The
acoustic-phonon scattering is not a dominating scattering channel in a
strongly polar crystal, but this contribution is taken into account here for completeness.
Next, we use the density-of-states approximation using the density of states
$\nu_{\lambda}\left( E\right) $ determined by the equation
\begin{equation}
\frac{1}{4\pi^{3}}\int d\mathbf{k}~F_{\lambda}\left( \varepsilon
_{\mathbf{k},\lambda}\right) =\int_{\varepsilon_{\lambda,\min}
^{\varepsilon_{\lambda,\max}}F_{\lambda}\left( E\right) \nu_{\lambda}\left(
E\right) dE, \label{a1
\end{equation}
where $\varepsilon_{\lambda,\max}$ and $\varepsilon_{\lambda,\min}$ are,
respectively, the top and bottom energies in the $\lambda$-th subband, and
$F_{\lambda}\left( E\right) $ is an arbitrary function of the energy.
In the density-of-states approximation, the band energy $\varepsilon
_{\mathbf{k},\lambda}$ is modeled by a spherically symmetric band dispersion
$\varepsilon_{k,\lambda}$. The model band dispersion is determined from the
condition that the \emph{density of states for} $\varepsilon_{k,\lambda}$
\emph{is the same as the exact density of states} for a true energy band
dispersion $\varepsilon_{\mathbf{k},\lambda}$. This condition results in the
equation:
\begin{equation}
\int_{\varepsilon_{\lambda,\min}}^{\varepsilon_{\lambda}\left( k\right)
\nu_{\lambda}\left( E\right) dE=\frac{1}{3\pi^{2}}k^{3}. \label{Ek
\end{equation}
The root of Eq. (\ref{Ek}) determines the model band energy $\varepsilon
_{\lambda}\left( k\right) $ for a given $\nu_{\lambda}\left( E\right) $
corresponding to the true band energy $\varepsilon_{\lambda}\left(
\mathbf{k}\right) $. In this approximation, the gap function depends on the
energy $\varepsilon_{k,\lambda}$, and Eq. (\ref{gapeq}) is transformed to the
equation
\begin{align}
\Delta_{\lambda}\left( \omega\right) & =-\int_{\varepsilon_{\lambda,\min
}-\mu}^{\varepsilon_{\lambda,\max}-\mu}\frac{d\omega^{\prime}}{2\omega
^{\prime}}\tanh\left( \frac{\beta\omega^{\prime}}{2}\right) \nonumber\\
& \times K_{\lambda}\left( \omega,\omega^{\prime}\right) \Delta_{\lambda
}\left( \omega^{\prime}\right) , \label{gapeq2
\end{align}
with the kernel function for non-parabolic bands
\begin{align}
K_{\lambda}\left( \omega,\omega^{\prime}\right) & =\frac{1}{2\pi}\frac
{\nu_{\lambda}\left( \omega^{\prime}+\mu\right) }{k_{\lambda}k_{\lambda
}^{\prime}}\int_{\left\vert k_{\lambda}-k_{\lambda}^{\prime}\right\vert
}^{k_{\lambda}+k_{\lambda}^{\prime}}qdq\nonumber\\
& \times\int_{0}^{\infty}d\Omega\frac{\left\vert \omega^{\prime}\right\vert
+\left\vert \omega\right\vert }{\Omega^{2}+\left( \left\vert \omega^{\prime
}\right\vert +\left\vert \omega\right\vert \right) ^{2}}V^{R}\left(
q,i\Omega\right) , \label{Knp
\end{align}
where the energies $\omega,\omega^{\prime}$ are counted from the chemical
potential $\mu$, (which is close to the Fermi energy). The values of the
momentum $k_{\lambda}\equiv p_{\lambda}\left( \omega\right) $ and
$k_{\lambda}^{\prime}\equiv p_{\lambda}\left( \omega^{\prime}\right) $ are
expressed using the density of states
\begin{equation}
p_{\lambda}\left( \omega\right) =\left( 3\pi^{2}\int_{\varepsilon
_{\lambda,\min}}^{\mu+\omega}\nu_{\lambda}\left( E\right) dE\right) ^{1/3}.
\label{kE
\end{equation}
The total dielectric function entering the effective electron-electron
interaction potential $V^{R}\left( q,i\Omega\right) $ is calculated within
the random phase approximation (RPA). In RPA, the total dielectric function is
a sum of the dielectric function of the lattice and the Lindhard polarization
function $P^{\left( 1\right) }\left( q,i\Omega\right) $ for electrons
\begin{equation}
\varepsilon^{R}\left( q,i\Omega\right) =\varepsilon_{\infty}\prod_{j=1
^{n}\left( \frac{\Omega^{2}+\omega_{L,j}^{2}\left( q\right) }{\Omega
^{2}+\omega_{T,j}^{2}\left( q\right) }\right) -\frac{4\pi e^{2}}{q^{2
}P^{\left( 1\right) }\left( q,i\Omega\right) . \label{TotEps
\end{equation}
The dielectric function of the lattice corresponds to the model of independent
oscillators, where $\omega_{L,j}\left( q\right) $ and $\omega_{T,j}\left(
q\right) $ are, respectively, LO- and TO-phonon frequencies, and
$\varepsilon_{\infty}$ is the high-frequency dielectric constant. Here, we
assume the phonon dispersion law to be isotropic. The TO-mode frequencies have
stronger $q$-dispersion than that for the LO-mode frequencies. Therefore we
suggest only $\omega_{T,j}\left( q\right) $ to be $q$-dependent. Here, the
TO-phonon dispersion is modeled by the expression
\begin{equation}
\omega_{T,j}\left( q\right) =\left[ \omega_{T,j}^{2}+\left( \omega
_{L,j}^{2}-\omega_{T,j}^{2}\right) \sin^{2}\left( qa_{0}/\pi\right)
\right] ^{1/2}, \label{phondisp
\end{equation}
where $a_{0}$ is the lattice constant.
The Lindhard polarization function in the density-of-states approximation
takes the form
\begin{align}
P^{\left( 1\right) }\left( q,i\Omega\right) & =\frac{\pi^{2}}{q
\sum_{\lambda}\int_{0}^{\varepsilon_{\lambda,\max}}dE~\frac{\nu_{\lambda
}\left( E\right) }{k_{\lambda}\left( E\right) }f\left( E-\mu\right)
\nonumber\\
& \times\int_{\varepsilon_{\lambda,a}}^{\varepsilon_{\lambda,b}}dE^{\prime
}~\frac{\nu_{\lambda}\left( E^{\prime}\right) }{k_{\lambda}\left(
E^{\prime}\right) }\frac{E-E^{\prime}}{\Omega^{2}+\left( E-E^{\prime
}\right) ^{2}}. \label{P
\end{align}
with the Fermi distribution functions $f\left( \varepsilon-\mu\right) $. The
integration bounds are given by
\begin{align}
\varepsilon_{\lambda,a} & =\min\left( \varepsilon_{\lambda}\left(
\left\vert p_{\lambda}\left( E\right) -q\right\vert \right) ,\varepsilon
_{\lambda,\max}\right) ,\nonumber\\
\varepsilon_{\lambda,b} & =\min\left( \varepsilon_{\lambda}\left(
p_{\lambda}\left( E\right) +q\right) ,\varepsilon_{\lambda,\max}\right) .
\end{align}
Expanding (\ref{P}) in powers of the momentum up to $q^{2}$, we arrive at the
expansion $\left( 4\pi e^{2}/q^{2}\right) P^{\left( 1\right) }\left(
\mathbf{q},i\Omega\right) =-\omega_{p}^{2}/\Omega^{2}+O\left( q^{2}\right)
$ with the plasma frequency
\begin{equation}
\omega_{p}^{2}=\sum_{\lambda}4\pi e^{2}n_{\lambda}\frac{k_{F,\lambda}}{\pi
^{2}\nu_{\lambda}\left( \mu\right) }. \label{plasmf
\end{equation}
where $k_{F,\lambda}=\left( 3\pi^{2}n_{\lambda}\right) ^{1/3}$, and
$n_{\lambda}$ is the population of the $\lambda$-th subband. This
long-wavelength limit allows us to introduce the effective mass parameter
\begin{equation}
m_{\lambda}\equiv\pi^{2}\nu_{\lambda}\left( \mu\right) /k_{F,\lambda}
\label{mp
\end{equation}
expressed through the density of states at the Fermi energy.
The long-wavelength expansion of $P^{\left( 1\right) }\left( \mathbf{q
,i\Omega\right) $ gives a leading contribution to the effective potential.
Therefore the polarization function $P^{\left( 1\right) }\left(
\mathbf{q},i\Omega\right) $ calculated using the parabolic band approximation
with the effective mass values given by (\ref{mp}), instead of those for the
bottom of the conduction band, provides a good approximation for the effective
interaction potential $V^{R}\left( q,i\Omega\right) $ in non-parabolic
bands. Thus the major effect of the band non-parabolicity comes from the
density of states entering the kernel function (\ref{Knp}) rather than from
the interaction potential.
\section{Concentration dependent critical temperatures}
In this section, we describe the calculated critical temperatures and the
isotope effect in $n$-doped strontium titanate comparing the results with
experimental data and discuss the relation of the present study to other
works. The LO and TO phonon energies are extracted from the results of the
density functional theory calculations using the Vienna ab initio simulation
package (VASP) \cite{Kresse1993,Kresse1996}. All calculations were based on
projector augmented wave pseudopotentials within the density functional
theory. The PBEsol exchange-correlation functional \cite{PBEsol} and a plane
wave energy cutoff of 600 eV were used with a $12\times12\times12$
Monkhorst-Pack $k$-point mesh. The atomic positions were optimized until the
forces were smaller than $0.01$~eV/\AA , and the phonon frequencies were
computed using density functional perturbation theory with an energy accuracy
of $10^{-8}$~eV.
The calculated and measured (shown in brackets) phonon frequencies are shown
in Table \ref{Tab1}. We keep only those LO/TO pairs of frequencies which give
a factor different from in the dielectric function and may be therefore
observed. Even with this reduced set, the DFT calculation gives more
frequencies than measured experimentally, because a significant ratio
$\omega_{L,j}/\omega_{T,j}$ is necessary for detection
\begin{table}[h] \centering
\caption{Energies of polar optical phonons in tetragonal SrTiO$_3$ at the Brillouin zone center
\begin{tabular}
[c]{|l|c|c|}\hline
No. of the branch & $\hbar\omega_{L,j}$ (meV) & $\hbar\omega_{T,j}$
(meV)\\\hline
1 & 103.62 (98.7 \cite{Gervais93}) & 66.983 (67.6 \cite{VDM2008})\\\hline
2 & 61.521 (58.4 \cite{Gervais93}) & 59.990\\\hline
3 & 54.221 & 52.522\\\hline
4 & 31.637 & 21.40 (21.8 \cite{VDM2008})\\\hline
5 & 18.305 (21.2 \cite{Gervais93}) & 17.767\\\hline
6 & 13.716 & 12.504 (11.0 \cite{VDM2008})\\\hline
\end{tabular}
\label{Tab1
\end{table
The lowest-energy TO phonon mode, as measured in Ref. \cite{VDM2008}, exhibits
concentration-dependent softening. We use interpolation to these experimental
data for the lowest TO-phonon energy instead of the calculated value from
Table 1 which does not account for softening. The high-frequency dielectric
constant of SrTiO$_{3}$ is chosen according to Refs. \cite{Kamaras,Ruhman},
$\varepsilon_{\infty}=5.1$. The acoustic deformation potential is used
according to Ref. \cite{Janotti}, $D=4$ eV. In order to see the relative
effect of the interaction with acoustic phonons we add also the transition
temperature with $D=0$
\begin{figure}[tbh
\centering
\includegraphics[
height=2.4033in,
width=3.2067in
{Figure1.eps
\caption{\emph{Solid and dashed curves}: critical temperature in $n$-doped
SrTiO$_{3}$ as a function of the carrier density calculated within DFM using
the density-of-states approximation for non-parabolic bands, with and without
the acoustic-phonon contribution, respectively. \emph{Dotted curve}: the same
assuming the extreme softening of the lowest-energy TO-phonon mode. The
calculated $T_{c}$ is compared with experimental data of Refs.
\cite{Schooley,Koonce,Binnig,Lin,Collignon} (symbols). \emph{Dot-dashed
curve}: the result \cite{Ruhman} of the Eliashberg theory.
\end{figure}
The dispersion of the conduction band is described by the tight-binding
analytic fit for the band Hamiltonian using the expressions and notations of
Ref. \cite{JSNM2017}. We apply the values of the diagonal matrix elements
$t_{\delta},t_{\pi}$ from the tight-binding fit to the calculation performed
using the GW method \cite{GW2018}: $t_{\delta}=54.2$ meV, $t_{\pi}=490.9$ meV.
The conduction band splitting is a rather sensitive parameter, because it is
relatively small with respect to diagonal matrix elements. As discussed in
Ref. \cite{Ruhman}, the experimentally relevant values of the interband
splitting are $\delta\varepsilon_{2}\approx2$ meV and $\delta\varepsilon
_{3}\approx8$ meV. These values are obtained when using splitting parameters
$\xi\approx4.615$ meV and $d\approx0.931$ meV.
It is observed in Ref. \cite{Swartz} that the polaronic effect \cite{Varenna}
can substantially influence parameters of the superconducting state and
critical temperatures. It is taken into account, scaling the bare-electron
band energies $\varepsilon_{\mathbf{k},\lambda}$ by $\varepsilon
_{\mathbf{k},\lambda}^{\left( pol\right) }=\left( m_{\lambda}/m_{\lambda
}^{\ast}\right) \varepsilon_{\mathbf{k},\lambda}$, where the effective mass
parameter $m_{\lambda}$ is determined by (\ref{mp}). In the present
calculation, we have used $\alpha_{eff}\approx2.1$ \cite{KDM2010}, yielding
$m_{\lambda}^{\ast}/m_{\lambda}\approx1.\,35$.
In Fig. 1, we plot critical temperature in $n$-doped SrTiO$_{3}$ as a function
of the carrier density calculated using the dielectric function method within
the density-of-states approximation. The calculated critical temperatures are
compared with experimental data of Refs.
\cite{Schooley,Koonce,Binnig,Lin,Collignon} and with the theoretical result
for $T_{c}$ predicted in Ref. \cite{Ruhman} using the Eliashberg theory. As
can be seen from Fig. 1, the calculated transition temperatures demonstrate
good agreement with the experiments for concentrations $n\sim5\times10^{18}$
to $5\times10^{20
\operatorname{cm
^{-3}$. A deviation between the present calculation and experiment does not
exceed the deviations between different experiments. The latter is
significant, because very small $T_{c}$ may relatively strongly fluctuate in
different experimental conditions. The present calculation does not match the
experimental data of Refs. \cite{Schooley,Lin} for low concentrations
$n\lesssim10^{18
\operatorname{cm
^{-3}$, where the calculated transition temperature decreases to small values
with respect to these experimental results. The measured concentration
dependence of $T_{c}$ in this low-density range exhibits a separate dome. The
superconductivity in this low-density regime may be attributed to another
mechanism than the interaction with bulk phonons, for example, local modes
proposed in Ref. \cite{Gorkov}. The incorporation these local modes in the DFM
is beyond the scope of the present work. There are also other pitfalls when
trying to apply DFM at low concentrations: the electron-phonon interaction at
low densities is less screened, so that the weak-coupling approximation may fail.
The non-parabolic dispersion in SrTiO$_{3}$ leads to a rapidly increasing
density of states in the range of energies corresponding to Fermi energies for
densities up to $n\sim10^{20
\operatorname{cm
^{-3}$. This leads to better quantitative agreement of the calculated $T_{c}$
with experimental data than in our preceding work \cite{JSNM2017}. The
contribution of acoustic phonons results in an increase of $T_{c}$ about 25\%
with a shift of the maximum of superconducting dome to higher densities. The
transition temperature is also calculated assuming extremely strong softening
of the lowest TO-phonon mode to a vanishingly small value. The result is not
sensitive to this soft mode frequency when it is very small. The origin of
this softening can be attributed, e.~g., to quantum criticality \cite{Edge}.
In the present work we do not yet add the deformation potential interaction
with the soft TO-phonon mode considered in Ref. \cite{Wolfle}. Nevertheless,
the softening of TO phonons influences $T_{c}$, because their frequencies
enter the dielectric function and therefore enhance the Fr\"{o}hlich
interaction. As can be seen from Fig. 1, the softening leads to a substantial
rise of $T_{c}$ and to a shift of its maximum to a higher concentration
\begin{figure}[tbh
\centering
\includegraphics[
height=2.4232in,
width=3.1401in
{Figure2.eps
\caption{Isotope effect for the critical temperature in $n$-doped SrTiO$_{3}$
calculated using the dielectric function method within the density-of-states
approximation for non-parabolic bands. \emph{Solid curve}: $T_{c}$ without the
isotope substitution. \emph{Dashed curve}: $T_{c}$ with the complete
substitution $^{16}O~\rightarrow~^{18}O$, and assuming proportional
renormalization of all optical phonon frequencies. \emph{Dotted curve}:
$T_{c}$ with the substitution corresponding to the experimental shift of the
TO-phonon frequency. \emph{Dot-dashed curve}: with the extreme softening of
the lowest-energy TO mode.
\end{figure}
Fig. 2 shows the calculated isotope effect in SrTiO$_{3}$ calculated using new
band splitting parameters. The curves in Fig. 2 correspond to the following
conditions: (1) without the isotope substitution; (2) with the complete
substitution $^{16}O~\rightarrow~^{18}O$ without assuming appearance of the
soft mode (the continuum approach \cite{Born} gives $\omega_{L\left(
T\right) ,j}\left( ^{18}O\right) /\omega_{L\left( T\right) ,j}\left(
^{16}O\right) \approx0.943$); (3) with the proportional change of phonon
frequencies corresponding to the experimentally observed shift of the highest
TO-phonon energy in Ref. \cite{Stucky2016}; (4) assuming extremely strong
softening of the lowest TO-phonon mode as described above. The softening of
the lowest-energy TO-phonon mode is in line with the observed increase of
$T_{c}$ due to the isotopic substitution $^{16}O~\rightarrow~^{18}O$.
The softening leads to an increase of the effective Fr\"{o}hlich coupling
constant [which in the single-mode picture is proportional to $\left(
1/\varepsilon_{0}-1/\varepsilon_{\infty}\right) $]. When $\varepsilon
_{0}\rightarrow\infty$, there is an enhancement of $T_{c}$ (see discussions in
\cite{GLF,Rowley}). Also the phonon dispersion favors an increase of the
transition temperature for relatively high concentrations $n\gtrsim10^{19}$
cm$^{-3}$, where the plasmon contribution is not negligible \cite{Takada}. The
plasmon contribution enters $T_{c}$ in a non-additive way, because the
dielectric function within RPA (\ref{TotEps}) leads to a plasmon-phonon
mixing. It is worth noting also that RPA goes beyond the frequently used
plasmon-pole approximation, which results in damping of plasmon-phonon excitations.
\section{Conclusions}
In the present work, we have found a tractable extension of the dielectric
function method for superconductivity to a non-parabolic band structure,
considering the case of SrTiO$_{3}$. The DFM essentially includes retarded
interactions and can be valid in a non-adiabatic regime, restricted by a
weak-coupling approximation.
The density-of-states approach can represent an interest for practical use,
because, first, it allows us to obtain an easily tractable gap equation.
Second, it opens a way to calculate parameters of the superconducting state
when the band dispersion is not precisely given, and only the density of
states is known, e. g., from first-principle calculations or from experimental data.
The dielectric function method is capable to interpret superconductivity in
strontium titanate without adjustment of material parameters, taking them from
experimental data and microscopic calculations. All phonon branches existing
in strontium titanate, as well as the phonon dispersions, are used for the
numerical calculations. Both a non-parabolic band energy and the dispersion of
phonons are essential for quantitative comparison with experiments. The
dielectric function method predicts the sign of the observed unusual isotope
effect in SrTiO$_{3}$ and a rise of the transition temperature due to
softening of the lowest-energy TO-phonon mode. The resulting critical
temperatures are in agreement with experiments in a range of concentrations
$n\sim5\times10^{18}$ to $5\times10^{20}$ cm$^{-3}$. The applicability of DFM
at very low densities remains an open question. An inclusion of other
interactions (e. g., deformation potential interaction with the soft TO-phonon
mode \cite{Wolfle} and the local phonons \cite{Gorkov}) may improve results
for transition temperatures.
The important feature of DFM is a prediction of superconductivity even in the
case when the effective attraction between electrons does not exceed the
Coulomb repulsion in the normal state. As a consequence, the obtained gap
function exhibits a sign reversal near the Fermi surface, being in line with
ab initio microscopic approaches \cite{Sanna}. This makes it interesting to
treat DFM using the electron and phonon densities of states obtained from
first principles.
\begin{acknowledgments}
This work has been supported by the joint FWO-FWF project POLOX (Grant No. I 2460-N36).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,748 |
<?php
namespace Digbang\Security\Mappings;
use LaravelDoctrine\Fluent\EntityMapping;
use LaravelDoctrine\Fluent\Fluent;
abstract class CustomTableMapping extends EntityMapping
{
private $table;
/**
* Set the custom table name.
*
* @param string $table
*/
public function setTable($table)
{
$this->table = $table;
}
/**
* Get the custom table name, or null if it was not customized.
*
* @return string|null
*/
public function getTable()
{
return $this->table;
}
/**
* Load the object's metadata through the Metadata Builder object.
*
* @param Fluent $builder
*/
public function map(Fluent $builder)
{
if ($this->table) {
$builder->table($this->table);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,647 |
Alison Park is Head of the Society and Social Change Team at NatCen Social Research.
My home was very literary: my dad was an academic, my mum worked in publishing. So perhaps unsurprisingly my earliest vision of a dream job was to be an author.
When did you turn to social research?
I discovered Sociology at A level and really enjoyed it – getting to understand what society really looked like, how it worked and recognising that there are different ways of thinking about the social world. So I took a Social Science degree at what was then Bristol Poly (now UWE). On the side while a student I got involved in various market research projects, the most (sadly) memorable being one that involved standing for hours in Stroud and Gloucester shopping centres trying to persuade people to tell me about what newspapers they read. Then I did an M.Phil. in Sociology at Nuffield College, Oxfordwhere for my dissertation I analysed and wrote up data from a survey of academic careers being overseen by A H Halsey. I focused on the experience of female academics and their career progression, which was really interesting as well as good practice when it came to doing social research analysis. By then I was pretty clear that I wanted to work in research.
I went to NatCen (then known as Social and Community Planning Research, or SCPR) as a junior researcher, assisting on a range of projects, mainly to do with housing. In the mid 1990s I first got involved with the British Social Attitudes Survey, an annual study that began in 1983. It had, and still has, everything I like about social research – interesting topics, rigorous data collection and analysis, a commitment to report the results accessibly, and good media coverage. I have been involved with it ever since.
Perhaps not necessarily the best, but the most challenging professional moment was when, for the 2001 national election, I appeared on the BBC Election Night panel, alongside Anthony King and Andrew Marr with David Dimbleby chairing. They were all seasoned veterans. For me, as a novice, it was both terrifying and exciting, aware that it was going out live (no opportunity for second thoughts or redrafts!). It was also a brilliant insight into what underpins that level of detailed news coverage: so much time! so many staff! such expense!
Do you have a social hero/heroine?
It has to be Roger Jowell who sadly died at the end of last year. Having pursued my career at NatCen he was a strong influence on me and a dear friend. He was forward thinking, charismatic, caring. He leaves behind a great legacy, not just NatCen itself but also the British Attitudes Survey and the more recent European Social Survey.
I believe that there is great value in comparative research for illuminating what is unique in particular contexts. So I like Marshall McLuhan's saying: 'We don't know who discovered water, but we know it wasn't the fish.' Think about it!
Go for it! At its best it is fulfilling, creative, practical – and it develops your ability with words and numbers. But it can be frustrating when you feel that you are not having an impact and your work vanishes without trace. You must work at this as a necessary research skill. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,568 |
\section{Introduction
\section{Introduction}
Trapped ultracold atomic gases serve as an ideal system to model many body systems and to investigate fundamental questions of quantum mechanics.
A paradigm phenomenon are the so called Josephson oscillations, which were demonstrated in a double well potential
for many atoms \cite{Albietz05,Schumm05,Hofferberth06,sewel}; more recently, even the tunneling of individual atoms through a barrier was demonstrated \cite{Kierig08}. A key ingredient
for modelling new systems are novel ways of trapping cold atoms. A standard technique for trapping atoms of a single species is to use either static magnetic
fields \cite{Folman,Wiemann,Fortagh,Dalfovo} or a superposition of static and oscillating magnetic fields leading to the so-called radio-frequency dressed
adiabatic potentials \cite{Lesanovsky06,Hofferberth06b,Lesanovsky06b}. Different species can be trapped in an optical trap making use of the so-called light
shift \cite{Grimm}. The potential results from the intensity maximum (minimum) of a laser that is red (blue) detuned with respect to an atomic transition
frequency. By superpositions of different laser beams and intensity configurations one can create, e.g., optical lattices or double well potentials.
In a more recent approach, two lasers in a Raman configuration were used to trap atoms. Such a setup allows for example the creation of optical lattices
with a reduced lattice spacing compared to standard optical lattices \cite{Zhang05}.
A combination of optical and magnetic fields for creating atom traps has been already described in several works. Even in one of the first BEC
experiments a superposition of a far detuned laser and a magnetic trap was used to trap the atoms \cite{Davis}. In Ref.\ \cite{Deutsch97} Deutsch \textit{et al.} investigated the combination of a constant magnetic field and a state dependent optical lattice that allows for the creation of a lattice of double well potentials. In more recent approaches, the superposition of radio frequency-fields and magnetic fields \cite{Lesanovsky06,Fernholz} or optical lattices \cite{Lundblad08} were used for trapping atoms in tunable potentials. In this paper, we derive the potential for an atom exposed simultaneously to an inhomogeneous magnetic field in a Ioffe-Pritchard trap like configuration and two lasers in a Raman configuration.
In this case, the non-trivial combination of the magnetic and the laser fields cannot longer be reduced to a potential resulting from an effective magnetic field. A direct consequence of this fact is that the potential surfaces for different hyperfine components of an atom do not only differ by a global factor but can be
substantially different. Thus, it is possible to confine the different components of a multi-component BEC in traps of different frequencies or different types, e.g., one component in a double well trap and another one in a single well trap located at the barrier of the double well. Moreover, due to the availability of additional parameters, flexibility is gained in shaping the potentials compared to conventional traps. For example, one can smoothly convert a single well potential into a double well potential or drive a double well potential by varying an offset magnetic field. Furthermore, it is possible to rotate the potential around one axis by changing the phase between the Raman lasers.
The paper is structured as follows. In Section II, we derive the effective Hamiltonian. In Section III our numerical results are presented. Specifically, we provide an
overview of the potential surfaces of the different components and investigate the transition of a double well to a single well potential as well as the rotation
of a double well potential for one component in detail. In Section IV we derive a semi-analytical expression for the potential surfaces; in Sec.~V
loss mechanisms are discussed. Finally, in Sec.~ VI we summarize our results and mention directions for possible future studies.
\section{Analytical Considerations}
\subsection{Hamiltonian and Setup}
The Hamiltonian of an (alkali) atom simultaneously exposed to magnetic and laser fields reads
\begin{equation}\label{hini}
H=-\frac{\hbar^2}{2M}\mathbf{\nabla_{\mathbf{R}}}^2 + H^{\text{e}}(\mathbf{r})+V^{\text{IP}}(\mathbf{r},\mathbf{R}) +V^{\text{AF}}(\mathbf{r},\mathbf{R}).
\end{equation}
Here, $\mathbf{R}$ denotes the center of mass coordinate of the atom and $\mathbf{r}$ the coordinate of the valence electron relative to the center of
mass position; $M$ is the total mass of the atom. $H^\text{e}(\mathbf{r})$ accounts for the field-free electronic structure of the atom; for the scope of
this work, we use $^{87}$Rb as a paradigm. $V^\text{AF}(\mathbf{r,R})$ and $V^\text{IP}(\mathbf{r,R})$ denote the contributions of the Raman lasers and
the magnetic field, respectively.
In order to solve the coupled Schr\"odinger equation associated with Hamiltonian (\ref{hini}), we employ a Born-Oppenheimer separation of the center of
mass motion and the electronic degrees of freedom. We are thereby led to an effective electronic Hamiltonian that parametrically depends on the center of
mass position,
\begin{equation}
H^{\text{eff}}(\mathbf{R})= H^{\text{e}}(\mathbf{r})+V^{\text{IP}}(\mathbf{r};\mathbf{R}) +V^{\text{AF}}(\mathbf{r};\mathbf{R}).
\label{effective Hamiltonian}
\end{equation}
Its solutions $V_\kappa(\mathbf R)$ serve as adiabatic potential energy surfaces for the center of mass motion of the atom; each of these (trapping)
potentials is then associated with a given internal state $\kappa$ of the atom.
Regarding the magnetic field configuration, we consider the setup of a Ioffe-Pritchard trap \cite{Pritchard} which is given by a two-dimensional quadrupole field in the $x_1,x_2$-plane together with a perpendicular offset (Ioffe-) field in the $x_3$-direction; it can be parameterized as $\mathbf B(\mathbf x)=Gx_1\mathbf{e}_1-Gx_2\mathbf{e}_2+B_I\mathbf{e}_3$. $G$ denotes the magnetic gradient of the two-dimensional quadrupole field and $B_I$ the constant offset field oriented along the $x_3$-axis. The quadratic term $\mathbf B_q\propto (x_3^2-\rho^2/2)\mathbf e_3$
that usually arises for a Ioffe-Pritchard configuration can be exactly zeroed by geometry, which we are considering in the following. In actual experimental
setups, $\mathbf B_q$ provides a weak confinement also in the $x_3$-direction. Omitting $\mathbf B_q$, the magnitude of the magnetic field at a certain
position $\mathbf x$ in space is given by $|\mathbf B(\mathbf x)|=\sqrt{B^2+G^2\rho^2}$, which yields a linear asymptote
$|\mathbf B(\mathbf x)|\rightarrow G\rho$ for large values of the coordinates ($\rho=\sqrt{x_1^2+x_2^2}\gg B/G$) and a harmonic behavior
$|\mathbf B(\mathbf x)|\approx B+\frac{1}{2}\frac{G^2}{B}\rho^2$ close to the origin ($\rho\ll B/G$).
The magnetic field interaction within the Born-Oppenheimer approximation reads
\begin{equation}
V^\text{IP}(\mathbf{r;R})=g_F\mu_B \mathbf{F}\cdot \mathbf{B(R)}.
\end{equation}
\begin{figure}
\includegraphics[width=8.5cm]{./fig1.eps}
\caption{State linkage diagram of the unperturbed atom. The solid arrows denote the transitions allowed for $\sigma^+$ and $\sigma^-$ light, whereas the dashed arrows indicate allowed transitions with $\pi$ polarized light. The energy gap between the $5P_{1/2},F=1$ and $5P_{1/2},F=2$ manifold is $\Delta_{\text{hfs}}= 2\pi \hbar \times 0.8$ GHz \cite{Arimondo}.}
\label{fig: fig1}
\end{figure}
The Raman configuration of the excitation lasers is depicted in Fig.~\ref{fig: fig1}. It consists of two oppositely circular polarized lasers that are
close to resonance with the $D_1$ transition line, i.e., being blue-detuned by $\Delta$ with respect to the transition from the $5S_{1/2},F=1$ ground
state manifold to the $5P_{1/2},F=1$ excited state of $^{87}$Rb. The propagation direction of the lasers is chosen to coincide with the direction of the constant
Ioffe Field $B_I$. The overall setup is shown in Fig.\ \ref{fig: fig2}
\begin{figure}
\includegraphics[width=8.5cm]{./fig2.eps}
\caption{(a) Setup showing the propagation direction of the laser beams (large arrows) and the configuration of the magnetic trap. The Helmholtz coils generate the homogeneous Ioffe field oriented along the $Z$ direction and the Ioffe bars the quadrupole field in the $X-Y$ plane, shown in subfigure (b).}
\label{fig: fig2}
\end{figure}
Within the dipole approximation, the potential of an atom exposed to the laser fields is given by
\begin{equation}
V^{\text{AF}}(\mathbf{r};\mathbf{R})=-e\mathbf{r}\cdot\mathbf{E}(\mathbf{R},t),
\end{equation}
$\mathbf{E}(\mathbf{R},t)$ denoting the electric field of the lasers. The latter can be expressed as
\begin{equation}
\mathbf{E}(\mathbf{R},t)=\frac{1}{2}\sum_{i=1}^2 \left[\mathbf{E}_i(\mathbf{R},t)+\mathbf E^{\star}_i(\mathbf{R},t)\right]
\end{equation}
with $\mathbf{E}_i(\mathbf{R},t)=\boldsymbol{\epsilon}_i\varepsilon_i(\mathbf{R})e^{-i(\mathbf{k}_i\mathbf{R}-\omega_it+\phi_i(t))}$ being the electric
field associated with the $i$th laser. The amplitudes $\varepsilon_i(\mathbf{R})$ of the electric fields are spatially dependent in order to account for the
focussing and shape of the laser beams. The factors $\boldsymbol{\epsilon}_i$ are the unit polarization vectors given by
$\boldsymbol{\epsilon}_1=\frac{1}{\sqrt{2}}(\mathbf{e}_1+i\mathbf{e}_2)$ ($\sigma^+$ light) and
$\boldsymbol{\epsilon}_2=\frac{1}{\sqrt{2}}(\mathbf{e}_1-i\mathbf{e}_2)$ ($\sigma^-$ light), respectively.
$\phi_i(t)$ take into account the phases of the lasers, which additionally can depend on time.
Expanding the atom field interaction in the basis $|\alpha\rangle$, defined as the eigenfunctions of the field free atom, leads to
\begin{align}
V^{\text{AF}}(\mathbf{r};\mathbf{R})={}&\frac{1}{2}\sum_{i=1}^2\sum_{\alpha,\gamma}(\omega^{(+)}_{i,\alpha \gamma}
+\omega^{(-)}_{i,\alpha \gamma})|\gamma\rangle\langle\alpha|+\text{h.c.}
\end{align}
with $\omega^{(+)}_{i,\alpha \gamma}=e\langle\alpha|\mathbf{E}_i(t)\mathbf{r}|\gamma\rangle$ and
$\omega^{(-)}_{i,\alpha \gamma}=e\langle\alpha|\mathbf{E}^{\star}_i(t)\mathbf{r}|\gamma\rangle$.
\subsection{Rotating Wave Approximation}
We employ the rotating wave approximation \cite{scully} in order to remove the time-dependence of Hamiltonian (\ref{effective Hamiltonian}) that arises due to the laser interaction. For reasons of simplicity we assume that both lasers have the same frequency $\omega_i\equiv\omega$ and the
same profile $\varepsilon_i(\mathbf{R})\equiv\varepsilon(\mathbf{R})$. Since the magnetic field interaction term is block diagonal, i.e., does not mix states with different
total angular momenta, $V^{\text{IP}}(\mathbf{r;R})$ is not affected by the transformation into the rotated frame. The transformed Hamiltonian reads
\begin{equation}\label{eq: Hamiltonian RWA}
H_{\text{RWA}} = V^{\text{IP}}+V^{\text{e}}_{\text{RWA}}+V^{\text{AF}}_{\text{RWA}},
\end{equation}
where
\begin{align}
V^{\text{e}}_{\text{RWA}}&=\sum_\alpha E_\alpha |\alpha\rangle\langle\alpha | + \sum_l (E_l-\hbar \omega) |l \rangle\langle l| \\
V^{\text{AF}}_{\text{RWA}}&=\frac{1}{2}\sum_{i=1}^2 \sum_{\alpha,l} \varepsilon(\mathbf{R}) e^{-i(\mathbf{k}_i\mathbf{R}+\phi_i(t))} \langle l | \boldsymbol{\epsilon}_i \mathbf r|\alpha\rangle |l\rangle \langle \alpha| +\text{h.c.}
\end{align}
(for clarity, we omit the arguments $\mathbf r$ and $\mathbf R$ of the potentials in the following).
Here, $\alpha$ labels the different states of the ground state manifold and $l$ labels the excited states. Since the lasers are close to resonance
to the $D_1$ transition line, we can restrict our basis to states close to the ground state and the first excited state. Using as a basis all hyperfine states
of the $5S$ and $5P$ manifolds leads to an effective $32 \times 32$ matrix that will be diagonalized.
\subsection{Van Vleck Perturbation Theory}
In the last subsection we derived the Hamiltonian expanded in the eigenfunctions of the unperturbed atom. One can use
\emph{van Vleck perturbation theory} \cite{Shavitt} to adiabatically eliminate the excited $5P$ levels that serve as intermediate states for the Raman
transitions. In this manner, the Hamiltonian can be reduced to an operator acting only on the ground state manifold
(i.e., all states $|\alpha\rangle=|5S_{1/2},F=1,m_F\rangle$ with $m_F\in\{0,\pm1\}$),
\begin{equation}\label{eq: Hamiltonian VV}
H_{\text{VV}}=V^{\text{IP}}+\sum_\alpha E_\alpha |\alpha\rangle\langle\alpha |+\sum_{\alpha,\beta} \mathcal W_{\beta \alpha}|\beta\rangle\langle\alpha|,
\end{equation}
with
\begin{eqnarray}\label{eq:dressWba}
\mathcal W_{\beta\alpha}&=&\frac{1}{2}\sum_l\mathcal V_{\beta l}\mathcal V_{l\alpha}\Big(\frac{1}{E_\alpha-E_l}+\frac{1}{E_\beta-E_l}\Big)\,,
\end{eqnarray}
being the effective interaction within the ground state manifold. Here, the index $l$ labels the excited states, which have been eliminated. For a detailed
derivation of Eq.~(\ref{eq:dressWba}), we refer the reader to the appendix of Ref.~\cite{mayle10}. Employing
\begin{equation}
\mathcal V_{l\alpha}=\frac{1}{2}\sum_{i=1}^2\sum_{\alpha,l} \varepsilon(\mathbf{R}) e^{-i(\mathbf{k}_i\mathbf{R}+\phi_i(t))} \langle l | \boldsymbol{\epsilon}_i \mathbf r|\alpha\rangle.
\end{equation}
for the block off-diagonal matrix elements of $V_\text{RWA}^\text{AF}$ yields the effective interaction $W$ within the ground state manifold
whose matrix representation correspondingly reads
\begin{align}\label{Weffective}
W_{\beta \alpha}={}&\frac{1}{8}\varepsilon(\mathbf{R})^2 \sum_{i,i'=1}^2\sum_l e^{i[\phi_i(t)-\phi_{i'}(t)]}\langle \beta | \boldsymbol{\epsilon}_i \mathbf r|\ l\rangle \langle l | \boldsymbol{\epsilon}_{i'} \mathbf r|\alpha\rangle\nonumber\\
&\times(\frac{1}{E_\alpha-E_l+\hbar \omega}+\frac{1}{E_\beta-E_l+\hbar \omega}).
\end{align}
If one restricts the sum over the intermediate states to the $5P_{1/2}$ hyperfine sublevels, which represent a good approximation, one obtains the compact form
\begin{equation}\label{Wmatrix}
\mathcal W=\frac{|\langle 5S_{1/2}||er||5P_{1/2}\rangle|^2}{72 c\epsilon_0 \Delta}
\begin{pmatrix}
A & 0 & C \\
0 & B & 0 \\
C^{\star} & 0 & A
\end{pmatrix}
\end{equation}
for the atom laser interaction, with
\begin{align}
A &= \left(1+\frac{7}{1-\frac{\Delta_{\text{hfs}}}{\Delta}}\right)I(\mathbf{R}),\\
B &= 2\left(1+\frac{3}{1-\frac{\Delta_{\text{hfs}}}{\Delta}}\right)I(\mathbf R),\\
C &= -\left(1-\frac{1}{1-\frac{\Delta_{\text{hfs}}}{\Delta}} \right)e^{i \Delta\phi(\mathbf{t})}I(\mathbf R),
\end{align}
$\Delta=E_{5S_{1/2},F=1}-E_{5P_{1/2},F=1}-\hbar \omega$ being the detuning of the transition lasers and $\Delta \Phi(t)=\phi_1(t)-\phi_2(t)$ their phase
difference. $I(\mathbf{R})=c\epsilon_0|\varepsilon(\mathbf R)|^2/2$ denotes the intensity of the laser, $\epsilon_0$ being the dielectric constant and $c$ the speed of
light. The reduced matrix element in Eq.~(\ref{Wmatrix}) can be deduced from the measured lifetime of the excited state which yields in our case
$\langle 5S_{1/2}\Vert er\Vert5P_{1/2}\rangle=2.9919 \,ea_0$ \cite{loudon,volz96}.
The individual contributions of the matrix (\ref{Wmatrix}) can be interpreted as follows. The diagonal elements stem from the light shift potential of the
lasers, i.e., the off-resonant coupling of a $m_F$ component of the ground state to an excited state. The off-diagonal elements arise due to the coupling of the $m_F=1$ ($m_F=-1$) component via an intermediate (excited) state to the $m_F=-1$ ($m_F=1$)
component. These off-diagonal matrix elements are not present in radio frequency traps. The specific form of the matrix occurs since the laser light is circularly polarized. Other polarizations would lead to a coupling of different states, i.e., different off-diagonal entries of the matrix $\mathcal W$.
Furthermore, we should note at this point that the matrix is expanded in the field-free basis of the atom. In this basis, the contribution of the magnetic
field interaction $V^\text{IP}$ for the ground state manifold is represented by the spin matrices for total spin $F=1$, giving rise to off-diagonal matrix
elements as well. As a result, the combined action of the magnetic and laser fields leads to the mutual coupling of all magnetic sublevels of the
$5S_{1/2},F=1$ manifold, represented by a fully occupied matrix. In Section \ref{Semi-analytical} we tackle this issue by performing a principal axis
transformation that diagonalizes the magnetic field interaction and thus provides us with a more suitable basis for the interpretation of the underlying physics.
\section{Numerical Results}\label{sec:numerical}
We restrict our investigations to both lasers having a Gaussian profile with a width $\sigma$ in the $X,Y$ plane
and assume that they have a constant intensity in the propagation direction $Z$,
\begin{equation}
I(\mathbf R)=I_0 \exp(-\frac{X^2+Y^2}{\sigma^2}).
\end{equation}
Since the magnetic field interaction term is independent of $Z$ as well, we find a total potential that is constant in the $Z$-direction.
In order to provide a confinement in the $Z$-direction one can, e.g., utilize an additional laser or make use of the defocusing of the laser beams.
If not stated otherwise, we fix the detuning of the lasers to $\Delta=-\Delta_{\text{hfs}}/2$, i.e., right in the middle between the $5P_{1/2},F=1$
and the $5P_{1/2},F=2$ excited states as depicted in Fig.~\ref{fig: fig1}, and the intensity to $I_0=10$ W/m$^2$.
\subsection{Overview Over All Components}
\begin{figure}
\includegraphics[width=8cm]{./fig3.eps}
\caption{Potential curves for $Y=0$ (solid line) and for $X=0$ (dashed line). Two components show an attractive potential in two dimensions whereas one component is exposed to a repulsive potential. The individual potential surfaces are well separated.}
\label{fig: fig3}
\end{figure}
Diagonalizing the Hamiltonian matrix (\ref{eq: Hamiltonian VV}) leads to the adiabatic potential surfaces $V_\kappa(\mathbf R)$ for the center of mass motion as a function of the center of mass coordinate $\mathbf R$.
As expected from the magnetic field interaction, we find one trapped and one anti-trapped component, according to the quantum numbers $m_F=+1$ and $m_F=-1$, respectively. Interestingly, the $m_F=0$ component -- that is untrapped in a pure Ioffe-Pritchard field -- now shows an attractive potential with a double-well structure along the $X$-axis.
Note that the quantum number $m_F$ is only valid in a rotated frame of reference that we are going to introduce in Sec.~\ref{Semi-analytical}.
Nevertheless, we continue to use $m_F$ as a label for the different states in the laboratory frame in order to avoid confusion. Since the different potential surfaces are separated, one can uniquely assign to each state one particular surface.
\begin{figure}
\includegraphics[width=7.5cm]{./fig4.eps}
\caption{Difference of the potential differences for adjacent components for $X=0$ (solid line) and $Y=0$ (dashed line). The deviation between the transition frequencies of adjacent potential surfaces is non-zero and depends on the center of mass position of the atom.}
\label{fig: fig4}
\end{figure}
For investigating the energy spectrum further, let us define the radio transition frequencies $\delta_{-1}(\mathbf R)=V_{m_F=-1}(\mathbf R)-V_{m_F=0}(\mathbf R)$ and $\delta_{1}(\mathbf R)=V_{m_F=0}(\mathbf R)-V_{m_F=1}(\mathbf R)$ between the $m_F=0$ and the $m_F=\mp1$ states, respectively. Figure \ref{fig: fig4} provides a measure for the deviation of both transition frequencies by showing the difference $\delta_{-1}(\mathbf R)-\delta_{1}(\mathbf R)$ for $X=0$ (solid line) and $Y=0$ (dashed line). The fact that the transition frequencies do not coincide, i.e., the difference being non-zero, can be used to mutually couple two components without coupling to the third component. This allows for example a transfer of atoms from the $m_F=-1$ component to the $m_F=0$ component without coupling to the untrapped $m_F=1$ component. Moreover, the radio transition frequencies depend on the center of mass position $\mathbf R$ and thus on the absolute value of the potentials. Therefore, one may couple energy-selectively one component to another one, i.e., couple atoms of the $m_F=0$ or $m_F=-1$ component at a certain position to the untrapped $m_F=1$ component. Such a scheme can be used to evaporatively cool the $m_F=0$ or the $m_F=-1$ component \cite{Alzar06}.
\subsection{$m_F=0$ Component}
\begin{figure*}
\includegraphics[width=17cm]{./fig5.eps}
\caption{Contour plot of the potential surface for the $m_F=0$ component for $\Delta \Phi=0$, $\sigma=10$ $\mu$m, $G=0.1 $G/$\mu$m, and
(a) $B_I=10$ G, (b) $B_I=1$ G, (c) $B_I=0.5$ G, and (d) $B_I=0.1$ G. The grey coded values of the potential are given in nK. By modulating the magnitude of the Ioffe field one can transform the potential smoothly from a single well to a double well potential.}
\label{fig: fig5}
\end{figure*}
We start our detailed investigations of the individual components with the potential surface for the $m_F=0$ component. Note that this component is not confined in a pure magnetic trap. Figure \ref{fig: fig5} shows the contour plot of its trapping potential for $\Delta \Phi=0$, $\sigma=10$ $\mu$m, $G=0.1 $G/$\mu$m, and
(a) $B_I=10$ G, (b) $B_I=1$ G, (c) $B_I=0.5$ G, and (d) $B_I=0.1$ G. The shape of the potential correspondingly changes from (a) close to being rotationally symmetric to (b) a cigar shaped potential in the $X$-direction, and finally to (c,d) a double well in the $X$-direction.
Thus, one can change the shape of the potential from a single- to a double well potential by changing the magnitude of the Ioffe field.
Alternatively, one may also drive the double or single well potential by modulating the magnitude of the Ioffe field.
The parameters of the double well trap can be tuned in different ways. The height of the barrier can be tuned by changing the ratio $\xi=G/B$. This is shown in detail in Fig. \ref{fig: fig8} in Sec.~\ref{Semi-analytical}.
The position of the minima can be controlled by changing the width of the lasers. This leads at the same time
to a change of the height of the barrier and consequently to a change in the number of trapped states within each well.
For a more detailed discussion of the properties of the double well potential, we refer the reader to Sec.~\ref{Semi-analytical} where a semi-analytical expression for the potential surface is derived. By displacing the center of the laser beams in the $X$-direction with respect to the Ioffe Pritchard trap, one can create in addition a tilted double well potential.
\begin{figure}
\includegraphics[width=8.5cm]{./fig6.eps}
\caption{Contour plot of the potential surface for the $m_F=0$ component for $G=0.1$ G/$\mu$m, $\sigma=10$ $\mu$m, $B_I=0.1$ G, and for (a) $\Delta \Phi=0$, (b) $\Delta \Phi=\pi/2$, (c) $\Delta \Phi=\pi$, and (d) $\Delta \Phi=3\pi/2$. A phase difference of $\Delta \Phi$ between the Raman lasers leads to rotation of $\Delta \Phi/2$ of the potential surface around the $Z$-axis.}
\label{fig: fig6}
\end{figure}
Figure \ref{fig: fig6} shows the effects of a phase difference between the two excitation lasers on the potential surface. A phase difference $\Delta \Phi$
leads to a rotation of the whole potential surface about the $Z$-axis by $\Delta \Phi/2$. For a zero phase difference one can add the
electric fields of the lasers, resulting in an effective electric field which is polarized linearly along the $X$-axis, whereas a phase difference of
$\pi$ leads to an effective electric field which is polarized linearly along the $Y$-axis. In general, a phase difference of $\Delta \Phi$ leads to a
rotation of the polarization vector of the total electric field by $\Delta \Phi/2$.
The sensitivity of the alignment of the double well potential on the orientation of the polarization vector seems at first glance surprising, given the azimuthal symmetry of a pure Ioffe-Pritchard trap. It is rooted in the spatially varying quantization axis of the Ioffe-Pritchard trap.
This issue is analyzed further in Sec.~\ref{Semi-analytical}, to which we refer the reader at this point.
\begin{figure}
\includegraphics[width=5cm]{./fig7.eps}
\caption{Potential surface for the case of a single $\sigma^-$ polarized laser for $\sigma=10\mu$ m, $G=0.1$ G/$\mu$m, and $B_I=0.1$ G. The potential is ring shaped.}
\label{fig: fig7}
\end{figure}
Figure \ref{fig: fig7} shows the potential surface for the same setup as in Fig.~\ref{fig: fig6} but for the case of a single $\sigma^-$-polarized laser instead of a pair of $\sigma^+ / \sigma^-$ polarized lasers. The potential is rotationally symmetric with a local maximum at the origin. Therefore it can be used as a ring shaped trap. The absolute value of the potential scales with the laser intensity. Hence, one can increase the height of the barrier by increasing the laser intensity. The position of the local minimum can be varied by changing the ratio $B/G$ or the width of the laser $\sigma$.
\subsection{$m_F=-1$ component}
The potential of the $m_F=-1$ component is dominated by the contribution of the magnetic field and therefore resembles the attractive potential of a pure Ioffe-Pritchard trap, namely a single well with a minimum at the origin. Without lasers, the confinement in the $X$- and $Y$-directions is equal, leading to an isotropic potential. The contributions of the lasers break this symmetry, giving rise to a slightly ellipsoidal potential. The value of the eccentricity depends on the intensity of the lasers. A phase difference between the lasers leads to a rotation of the (anisotropic) potential surface about the $Z$-axis.
\section{Semi-analytical Interpretation of the Potential} \label{Semi-analytical}
\subsection{Principal Axis Transformation}
In the previous sections we investigated the system in the laboratory frame of reference where the quantization axis for the atom is determined by the direction of the constant Ioffe field. However, because of the inhomogeneity of the magnetic field, a more adequate description of our system is to define the quantization axis along the \emph{local} magnetic field vector $\mathbf{B(R)}$. In this chapter, we tackle this issue by introducing the spatially dependent unitary transformation
\begin{equation}
U_r=\exp(-i\alpha F_x)\exp(-i\beta F_y)
\end{equation}
that rotates the local magnetic field vector into the $z$-direction of the laboratory frame of reference with the total spin vector $\mathbf{F}=\mathbf{L}+\mathbf{S}+\mathbf{I}$ consisting of the sum of the electronic orbital angular momentum vector $\mathbf{L}$, the electronic spin vector $\mathbf{S}$, and the nuclear spin vector $\mathbf{I}$. The corresponding rotation angles are defined by
$\sin\alpha\!=\!-GY/\sqrt{B^2+G^2(X^2+Y^2)}$,
$\sin\beta\!=\!GX/\sqrt{B^2+G^2X^2}$,
$\cos\alpha\!=\!\sqrt{B^2+G^2X^2}/\sqrt{B^2+G^2(X^2+Y^2)}$, and
$\cos\beta\!=\!B/\sqrt{B^2+G^2X^2}$.
This rotation diagonalizes the magnetic field contribution in Hamiltonian (\ref{effective Hamiltonian}), giving rise to
\begin{equation}
U_rV^\text{IP}U_r^\dagger=g_F\mu_B F_z |\mathbf{B(R)}|.
\end{equation}
Note that in the rotated frame of reference without lasers $m_F$ remains a good quantum number even in the presence of the inhomogeneous Ioffe-Pritchard field. In absence of the Raman lasers, the trapping potentials correspondingly read $V_\kappa=g_F\mu_B m_F |\mathbf{B(R)}|$.
In order to solve the time-dependent Schr\"odinger equation associated with Hamiltonian (\ref{effective Hamiltonian}), the Hamiltonian for the atom in the Ioffe-Pritchard trap and the laser interaction must be expressed in the same frame of reference. Hence, the unitary transformation $U_r$ must be applied to $V^\text{AF}$ as well. We find
\begin{equation}
U_r\mathbf rU_r^\dagger=
\left(\begin{array}{c}
x\cos\beta + y\sin\alpha\sin\beta - z\cos\alpha\sin\beta\\
y\cos\alpha + z\sin\alpha\\
x\sin\beta - y\sin\alpha\cos\beta + z\cos\alpha\cos\beta
\end{array}\right)\,.
\end{equation}
Therefore, the $\sigma^+$ and $\sigma^-$ laser transitions that are depicted in Fig.~\ref{fig: fig1} become
\begin{align}
\boldsymbol{\epsilon}_\pm\cdot U_r\mathbf rU_r^\dagger={}&\frac{1}{\sqrt{2}}\big[x\cos\beta+y\sin\alpha\sin\beta\nonumber\\
&\quad\,\,-z\cos\alpha\sin\beta
\pm i(y\cos\alpha+z\sin\alpha)\big]\,.\label{eq:dressUeps}
\end{align}
Equation (\ref{eq:dressUeps}) can be rewritten in terms of the polarization vectors $\tilde{\boldsymbol{\epsilon}}_\pm$ and $\tilde{\boldsymbol{\epsilon}}_0$ defined in the rotated frame of reference, showing that in a Ioffe-Pritchard trap contributions of all polarizations emerge away from the trap center \cite{mayle09,mayle10}. This changes drastically the simple transition scheme caused by the $\sigma^+$ and $\sigma^-$ light as depicted in Fig.~\ref{fig: fig1}, leading to a spatially dependent coupling between the involved ground- and excited states.
\subsection{Effective potential for the $m_F=0$ component}\label{sec: effective potential}
For our setup as depicted in Fig.~\ref{fig: fig1} and zero relative phase $\Delta \Phi=0$, the operator describing the interaction between the laser field and the atom reads in the rotated frame of reference
\begin{align}
\hat O(X,Y)\equiv{}&(\boldsymbol{\epsilon}_+ + \boldsymbol{\epsilon}_-)\cdot U_r\mathbf rU_r^\dagger\nonumber\\
={}&\sqrt{2}x\cos\beta + \sqrt{2}y\sin \alpha \sin \beta - \sqrt{2}z\cos \alpha \sin \beta.
\end{align}
Note that the dependence of $\hat O(X,Y)$ on the center of mass coordinates $X$ and $Y$ stems from the dependence of the rotation angles $\alpha$ and $\beta$ on these coordinates.
One finds the following limits. At the origin, the transformation is given by unity, providing $\hat O(0,0)=\sqrt{2}x$, i.e., $\pi$-polarized light
in the $x$-direction. For $X=0$, $\hat O(X,Y)$ is invariant under $U_r$ which results in $\hat O(0,Y)=\hat O(0,0)=\sqrt{2}x$.
For $Y=0$ and $X\rightarrow\infty$ the gradient field dominates and the operator corresponds to the operator of $\pi$-polarized light in the $z$-direction $\hat O(\infty,0)=\sqrt{2}z$.
A non-zero relative phase between the lasers leads to a different polarization of the total electric field in the laboratory frame, giving rise to a rotation of the operators in the rotated frame. This explains the rotation of the potential as seen in Fig.~\ref{fig: fig6}. Since it is straightforward to generalize our results to non-zero relative phases, we will restrict our analytical considerations to a zero relative phase in the following.
Within van Vleck perturbation theory as introduced in Sec.~\ref{effective Hamiltonian}, the effective interaction in the rotated frame of reference becomes
\begin{align}
\mathcal W_{\beta \alpha}={}&\frac{1}{8}\varepsilon(\mathbf{R})^2\sum_{i,i'=1}^2\sum_l \langle \beta | \boldsymbol{\epsilon}_i U_r\mathbf rU_r^\dagger|l\rangle \langle l | \boldsymbol{\epsilon}_i U_r\mathbf rU_r^\dagger|\alpha\rangle\nonumber\\
&\quad\times(\frac{1}{E_\alpha-E_l+\hbar \omega}+\frac{1}{E_\beta-E_l+\hbar \omega}),
\end{align}
cf.\ Eq.~\ref{Weffective}. Note, that we assumed once more that the shapes and frequencies of both lasers are identical.
We are interested in the regime of large enough magnetic fields (adjustable by the homogeneous Ioffe field component $B_I$) where the Zeeman splitting overcomes the light shifts of the laser fields, i.e., $|\mathcal W_{\beta \alpha}|\ll g_F\mu_B |\mathbf{B(R)}|$. Hence, we can approximate the fully occupied effective interaction matrix $\mathcal V^\text{IP}+\mathcal W$ by omitting the off-diagonal matrix elements $\mathcal W_{\beta \alpha},\alpha\neq\beta$, that couple the Zeeman-splitted $m_F$ components. This procedure leaves us with the diagonal matrix elements of the effective interaction,
\begin{align}
\mathcal W_{\alpha \alpha}={}&\frac{1}{2}\varepsilon(\mathbf{R})^2\sum_l\Big[
\cos^2\beta|\langle \alpha | x|l\rangle|^2\nonumber\\
&\quad+\sin^2\alpha\sin^2\beta|\langle \alpha | y| l\rangle|^2\nonumber\\
&\quad+\cos^2\alpha\sin^2\beta|\langle \alpha | z| l\rangle|^2\Big]/(E_\alpha-E_l+\hbar\omega).
\end{align}
Employing $|\langle \alpha | x| l\rangle|^2=|\langle \alpha | y| l\rangle|^2$ and performing the sum over all intermediate states $|l\rangle$ eventually yields for the $m_F=0$ component the effective potential
\begin{align}\label{eq: V_eff}
V_\text{eff}=V_0+ \frac{X^2}{\xi^2+X^2+Y^2}(V_\infty-V_0),
\end{align}
where
\begin{align}
V_0={}&-\frac{I(\mathbf R)}{36c\epsilon_0}\left(\frac{1}{\Delta}+\frac{3}{\Delta+\Delta_{\text{hfs}}}\right)|\langle 5S_{1/2}||er||5P_{1/2}\rangle|^2,\\
V_\infty={}&-\frac{I(\mathbf R)}{9c\epsilon_0}\frac{|\langle 5S_{1/2}||er||5P_{1/2}\rangle|^2}{\Delta+\Delta_{\text{hfs}}}
\end{align}
are the light shifts at the origin and in the limit $Y=0,X\rightarrow\infty$, respectively. $\xi=B/G$ is a length scale characterizing the particular configuration of the Ioffe-Pritchard trap.
Numerical comparison of the effective potential (\ref{eq: V_eff}) with the corresponding eigenvalue of the full problem (\ref{eq: Hamiltonian RWA}) shows a excellent agreement for small laser intensities. We observe a maximal relative deviation of less then $1$\textperthousand{} for $I=10$ W/m$^2$. For larger intensities, the agreement gets worse since the off-diagonal matrix elements $\mathcal W_{\beta\alpha}$ increase in magnitude. However, even for $I=100$ W/m$^2$ the deviation is less than $5$\textperthousand.
\subsection{Discussion of the Effective Potential}\label{sec: discussion of effective potential}
The analytical prediction of the effective potential $V_\text{eff}$ for the $m_F=0$ component allows us to deduce its basic properties by a simple analysis of Eq.~(\ref{eq: V_eff}). The effects of the magnetic field creating the double well potential described in the previous section, are included in the second term in Eq. ~(\ref{eq: V_eff}). In order to obtain the double well structure this term needs to be negative. This is the case for laser light which is red detuned with respect to the $F=2$ state and blue detuned with respect to the $F=1$ state, i.e, $0>\Delta>-\Delta_{\text{hfs}}$. An interesting case occurs for $\Delta=-\Delta_{\text{hfs}}/4$. Then the offset potential $V_0$ vanishes and the height of the barrier is equal to the maximal depth of the potential. For $0>\Delta>-\Delta_{\text{hfs}}/4$ the barrier-height is larger than the depth of the wells with respect to the continuum and for $\Delta_{\text{hfs}}<\Delta<-\Delta_{\text{hfs}}/4$ the height of the barrier is smaller than the depth of the wells with respect to the continuum.\\
In the following, we restrict our investigations again to the case $\Delta=-\Delta_{\text{hfs}}/2$. Since the double well only occurs for $0>\Delta>-\Delta_{\text{hfs}}$, this choice leads to a maximal detuning with respect to the $F=1$ and $F=2$ states. As expected from the numerical solutions provided in Sec.~(\ref{sec:numerical}), $V_\text{eff}$ shows a double well structure that is centered at the origin. The positions $(X_0,Y_0)$ of the two local minima are given by
\begin{equation}\label{xmin}
X_{0}=\pm \frac{\xi}{2}\sqrt{\sqrt{1+8\frac{\sigma^2}{\xi^2}}-3}
\end{equation}
and $Y_0=0$. It is obvious from Eq.~(\ref{xmin}) that the double well only exists if the discriminant is positive, giving rise to the condition $\xi<\sigma$.
For $\xi>\sigma$ one finds a single well potential, whereas for decreasing $\xi< \sigma$ the double well starts to build up. Starting in the limit $\xi\rightarrow0$ from a double well with a barrier with finite height but width going to zero, the distance between the minima increases with $\xi$ up to a local maximum $\Delta X^{\text{max}}_0=2(\sqrt{2}-1)\sigma$ at $\xi_{\text{max}}=\sqrt{3\sqrt{2}-4}\sigma$ and decreases for even larger $\xi$ up to $\xi_{\text{cr}}=\sigma$ where the minima vanish, thus transforming the energy surface to a single well potential. This behavior is illustrated in Fig.~\ref{fig: pots}(a) where a contour plot of the effective potential along the $X$-axis as function of $\xi$ is shown for a fixed width $\sigma=10\,\mu$m of the lasers; the dashed line indicates the positions of the minima.
Figure \ref{fig: pots}(b) shows a similar contour plot of the effective potential along the $X$-axis, but now as a function of $\sigma$ for fixed $\xi=1\,\mu$m. In this case, the shape of the barrier close to the origin is approximately conserved but the position of the minima increases with increasing $\sigma$.
\begin{figure}
\includegraphics[width=8.5cm]{./fig8.eps}
\caption{(a) Contour plot of the effective potential for $Y=0$ for fixed $\sigma=10\,\mu$m as a function of $\xi$. With increasing $\xi$ the barrier gets lower and broader. (b) Same potential for fixed $\xi=1\,\mu$m and as a function of $\sigma$. For $\sigma>\xi$, the shape of the barrier close to the origin is almost conserved. However, the position of the minima increases with increasing $\sigma$. In both subfigures, the dashed line indicates the positions of the minima.}
\label{fig: pots}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{./fig9.eps}
\caption{Dependence of the height of the barrier $\Delta V $ on $\xi$ for different $\sigma$. For increasing $\xi$ the height of the barrier decreases monotonically.}
\label{fig: fig8}
\end{figure}
The height of the barrier is given by $\Delta V = V_\text{eff}(0,0)-V_\text{eff}(X_0,0)$ and is directly proportional to the intensity of the lasers. Figure \ref{fig: fig8} shows the dependence of $\Delta V$ on the parameter $\xi$ for different values of $\sigma$ and fixed laser intensity $I=10$ W/m$^2$. The behavior of $\Delta V$ for different $\sigma$ is qualitatively very similar: the height of the barrier decreases with increasing $\xi$ monotonically. For fixed $\xi$, a more narrow laser entails a more shallow double well. Since the range of $\xi$ is determined by the condition $\xi<\sigma$, a more narrow laser necessitates furthermore tighter magnetic traps, i.e., smaller $\xi$.
Note that the positions of the minima do not depend on the intensity of the lasers. Thus, by increasing the laser intensities one can increase the height of the barrier without changing the position of the minima. In this way, the number of trapped states in each well can be controlled.
\subsection{Effective Potential for a Single Laser}
One can use the same semi-analytical procedure as above in order to predict the effective potential for a single laser. One finds
\begin{equation}\label{eq: V^1_eff}
V^1_{\text{eff}}=\frac{1}{2}V_0+ \frac{1}{8}\left(1-\frac{\xi^2}{\xi^2+R^2}\right)V_\infty
\end{equation}
where $R^2=X^2+Y^2$. The comparison of the effective potential (\ref{eq: V^1_eff}) and the corresponding numerical solutions shows again an excellent agreement. As opposed to the Raman setup involving two lasers, in the case of a single laser the resulting trapping potential for the $m_F=0$ component is rotationally symmetric. The ring-shaped minimum is located at
\begin{equation}
R_{0}=\sqrt{-\frac{5}{6}\xi^2+ \frac{1}{6} \xi^2\sqrt{1+12\sigma^2\xi^2}}
\end{equation}
and exists for $\sigma>\sqrt{2}\xi$. Starting at a barrier with arbitrarily small width with maximal height for $\xi\rightarrow0$, the barrier at the origin gets lower as $\xi$ increases and eventually vanishes as $\xi=\sigma/\sqrt{2}$. The distance of the minimum to the origin increases with $\xi$ up to the local maximum at $\xi_{\text{max}}=\sqrt{5\sqrt{3/2}-6} \sigma$ with a value $R_{\text{max}}=(\sqrt{3}-\sqrt{2})\sigma$. Then it decreases again and becomes zero at $\xi_{\text{cr}}=1/\sqrt{2}\sigma$ thereby transforming the potential into a single well. Note that the potential is not a simple superposition of a magnetic single well potential and a repulsive potential created by the laser. The spatial structure is a direct consequence of the spatially dependent light shift potential of the laser, giving rise to a barrier which is smaller than the width of the laser.
\section{Loss Mechanisms}
In the previous sections we have shown that for appropriate parameters the discussed combination of external fields leads to a confinement for two components of the ground state manifold.
Let us discuss in this section possible loss mechanism for these potentials.
\subsection{Lifetime of the Intermediate State}
One loss channel results from the coupling of the ground manifold of states to the excited $5P_{1/2}$ states. Despite the fact that the Raman lasers are detuned with respect to the excited state, there is a finite probability to excite the atom to this state due to the width of the state and the width of the lasers. The excited atom can subsequently decay spontaneously to the untrapped ground state. The resulting lifetime of the dressed state can be estimated by applying perturbation theory, leading to $\tau_\text{eff}=\tau (\Delta/ \omega_{\text{cp}})^2$ with $\tau$ being the lifetime of the unperturbed excited state and $\omega_{\text{cp}}$ the coupling matrix element of the ground state to the excited state. For our parameters ($\Delta=-\Delta_{\text{hfs}}/2$, $I_1=I_2=10$ W/m$^2$, and $\tau=27$ ns) we get an effective lifetime of $\tau_\text{eff}\sim 27$ ms. The latter can be increased by decreasing the intensities of the lasers. However, one has to bear in mind that this will reduce the depth of the trapping potentials as well.
\subsection{Inelastic Collisions}
Another loss mechanism occurs if more than one atom is loaded into the potential due to the mutual interaction of the atoms.
This mechanism can be estimated on a mean field level, incorporating an effective coupling coefficient that determines the interaction between the atoms. In the field-free case one gets a population transfer from one component to another due to interaction when there is an overlap between the wave functions of two components.
For our setup, however, one obtains new dressed states that are superpositions of the field-free states. Hence, one obtains a state changing contribution due to interaction even if only one dressed state is occupied. However, for the above discussed parameters this additional term can be neglected.
\section{Conclusions and Physical Applications}
We investigated the trapping potentials for $^{87}$Rb ground state atoms simultaneously
exposed to a magnetic trap in a Ioffe-Pritchard like configuration and an
optical trap in a Raman setup. The Raman lasers were detuned between the two
excited $5P_{1/2}$ hyperfine states, the $F=1$ and the $F=2$ state.
By varying the offset field of the Ioffe-Pritchard trap, we demonstrated that the trapping potential of the $m_F=0$ component can be tuned from a rotationally symmetric single well to a double well trap; in the intermediate regime, one finds a cigar shaped trapping potential.
By applying a phase difference between the two Raman lasers, the resulting trapping potentials can be rotated about the propagation direction of the laser beams.
A semi-analytical formula for the potential surfaces has been derived. All relevant properties of the double well potential have been determined analytically as a function of the various trap parameters.
For a single excitation laser, the proposed scheme results in a ring-shaped trap for the $m_F=0$ component.
In order to exploit the unique features of the above discussed potentials, one can think of various experiments. For example, one might trap the $m_F=0$ component in a
double well potential which gives for the $m_F=-1$ component a single well potential located at the center of the barrier. For an asymmetric occupation of the wells one can then observe tunneling of atoms trapped in the double well potential through the atoms trapped in the single well potential. The oscillation frequency of the tunneling can thus be investigated as a function of the occupation number of the second component located at the barrier, which is reminiscent of a single atom transistor \cite{Micheli}. We performed corresponding numerical simulations of the Spinor Gross Pitaevskii equation and observed indeed an increase of the oscillation period with increasing occupation number of the second component.
In a similar setup as mentioned above
(one component is exposed to a a double well potential with a narrow barrier and one component to single well potential centered at the position of the barrier) one might trap atoms in the component exposed to the single well potential and then transfer all atoms by an rf-pulse in the ``double-well'' component that feels a sharp potential maximum at this point. Depending on the energy of the atoms the condensate wave function would consequently either split into two parts or the whole wave function would move into one direction, which could be used as a test of the validity of the Gross Pitaevskii equation \cite{Streltsov07}. Moreover, the possibility to rotate an anisotropic single well trap (which arises for the $m_F=-1$ component) opens up a new possibility to study superfluids under rotation \cite{rev_fetter,PGK:MPLB:04,review}. The possibility to rotate the double well potential allows one to create an effective ring potential by rotating the potential fast enough so that the atoms feel a time averaged potential \cite{Lesanovsky07}. In this way the transition of a double well to a ring-shaped potential can be investigated. Moreover, one can exploit the feature that one can drive the potential surface of the $m_F=0$ component by modulating the Ioffe field strength to investigate non-equilibrium driven systems \cite{lenz08,lenz09}.
As an extension to the present work, it would be interesting to extend the studies to different propagation directions of the laser beams with respect to the orientation of the magnetic trap. Furthermore, one can vary the shape of one or both laser beams by using excited modes.
| {
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Category: FuckOnCam Web Cams
Prejudice, personal Stress, and psychological state in Lesbian, Gay, and Bisexual Populations
15th February 2021 15th February 2021
A difference between individual and team resources is frequently perhaps maybe perhaps not addressed within the coping literature. You will need to differentiate between resources that work on the lagevel that is individuale.g., character), by which members of minority groups differ, and resources that work on a group degree and so are open to all minority people (Branscombe & Ellemers, 1998).
Like many people who deal with basic anxiety, LGB individuals utilize a selection of individual coping mechanisms, resilience, and hardiness to withstand stressful experiences (Antonovsky, 1987; Masten, 2001; Ouellette, 1993). And also to such coping that is personal team degree social structural facets may have psychological state advantages (Peterson, Folkman, & Bakeman, 1996). Jones et al. (1984) described two functions of coping achieved through minority team affiliations: to permit stigmatized persons to see social surroundings for which they're not stigmatized by other people and also to offer help for negative assessment associated with stigmatized minority team. Personal assessment concept indicates another mechanism that is plausible minority coping (Pettigrew, 1967). People in stigmatized teams who possess a strong feeling of community cohesiveness evaluate on their own in comparison to other people who are just like them in place of with people in the principal tradition. The in team might provide a reappraisal for the stressful condition, yielding it less harmful to emotional well being. Continue reading "Prejudice, personal Stress, and psychological state in Lesbian, Gay, and Bisexual Populations" | {
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\section{\label{sec:level1}Introduction}
In spinelectronics, the exchange bias effect is used to pin a ferromagnetic
electrode to an antiferromagnetic layer\citep{newmagneticanisotropy, ebias}. This is crucial in GMR or TMR devices to allow for distinct stable resistance states\citep{spintronics}. For several years, the search for new antiferromagnetic materials has been going on in order to find rare-earth free alternatives for commonly used MnIr\citep{mnir} or MnPt\citep{mnpt, mining}. Recently, we found that antiferromagnetic MnN is a promising candidate\citep{meinert, zilske}: Polycrystalline MnN/CoFe bilayer systems show exchange bias of up to $1800$\,Oe at room temperature. This maximum exchange bias is observed for thicknesses of $t_{\text{MnN}}=30$\,nm and $t_{\text{CoFe}}=1.6$\,nm after annealing and field cooling from $T_{\text{A}}=325\,^{\circ}$C. Significantly higher annealing temperatures lead to a decrease of exchange bias due to nitrogen diffusion. For integration into spinelectronic devices, an increased thermal stability would be desirable.
MnN crystallizes in the $\Theta-$phase of the Mn-N phase diagram\citep{theta}, in a tetragonal variant of the NaCl structure with $a = 4.256$\,\AA{} and $c = 4.189$\,\AA{} at room temperature\citep{Suzuki}. The exact lattice constants depend on the nitrogen content in the lattice. With decreasing nitrogen content, decreasing lattice constants are observed{\citep{Suzuki, Leineweber}. The magnetic moments of MnN are coupled parallel in the $ab$ planes and alternate along the $c$ direction but the spin orientation also depends on the nitrogen content in the lattice\citep{Leineweber, Suzuki2}. The N\'eel temperature of MnN is about $390\,^{\circ}$C\citep{neel}.
\begin{figure*}
\includegraphics[width=0.85\textwidth]{annealing.pdf}
\caption{Dependence of exchange bias (a) and ratio $H_{\text{eb}}/H_{\text{c}}$ (b) on the annealing temperature measured for samples with different thicknesses of MnN. The crossing of the N\'eel temperature of MnN is indicated with dashed lines.}
\end{figure*}
\section{\label{sec:level1}Experimental}
Ta ($10$\,nm) / MnN ($t_{\text{MnN}}$) / CoFe ($1.6$\,nm) / $\text{TaO}_{\text{x}}$ ($2.5$\,nm) stacks were prepared on thermally oxidized Si substrates via DC magnetron sputtering at room temperature. The MnN film was sputtered reactively from an elemental Mn target with a gas ratio of $50\,\%$ Ar to $50\,\%$ $\text{N}_2$, following the same procedure as described in our previous article\citep{meinert}. Post-annealing for 15 minutes and subsequent field cooling in a magnetic field of $H_{\text{fc}}=6.5$\,kOe parallel to the film plane was performed in a vacuum furnace with pressure below $5\cdot10^{-6}$\,mbar to activate exchange bias. Magnetic characterization of the samples was performed using the longitudinal magneto-optical Kerr effect (MOKE) at room temperature. For annealing series, samples were successively annealed and measurements were taken in between the single steps. Structural characterization was performed via X-ray diffraction measurements with a Philips X'Pert Pro MPD, equipped with a Cu source and Bragg-Brentano optics. To investigate nitrogen diffusion inside the stacks, Auger depth profiling with a scanning Auger microscope PHI660 was used. The samples were continuously rotated during sputtering with a $500$\,eV Ar$^+$ ion beam to achieve optimum depth resolution. The measured Auger intensities are defined as peak to peak heights of the differential spectrum of the different components. Target factor analysis\citep{aes2} was used for separating the different chemical states of nitrogen in $\text{TaN}_{\text{x}}$ and $\text{MnN}_{\text{x}}$.
\section{\label{sec:level1}Results and Discussion}
\begin{figure}
\includegraphics[width=0.5\textwidth]{blocking.pdf}
\caption{Results of the reversed field cooling experiments performed on samples with $t_{\text{MnN}}=42,48$\,nm and $t_{\text{CoFe}}=1.6$\,nm: a) exchange bias, b) unblocked ratio UBR and c) derivative of the unblocked ratio in dependence on the temperature of reversed field cooling.}
\end{figure}
Figure 1 shows an exemplary hysteresis loop detected for a sample with $t_{\text{MnN}}=48$\,nm after annealing at $T_{\text{A}}=500\,^{\circ}$C. Very high exchange bias $H_{\text{eb}}$ accompanied by a reasonably small coercive field $H_{\text{c}}$ can be observed. At zero field, the CoFe layer is almost saturated. We can estimate the maximum effective interfacial exchange energy $J_\mathrm{eff} = t_\mathrm{CoFe} M_\mathrm{CoFe} \mu_0 H_\mathrm{eb}\approx0.76$\, erg/cm$^2$ using the saturation magnetization of $M_{\text{CoFe}}\approx 1700$\, emu/cm$^3$ for our Co$_{70}$Fe$_{30}$ composition\citep{cofe}.\\
In Figure 2a), the detailed dependence of exchange bias on the annealing temperature is shown for different MnN thicknesses. Additionally, the dependence of the ratio of exchange bias and coercive field is displayed in Figure 2b) as it provides information about the possible use in GMR or TMR stacks where $H_{\text{eb}}/H_{\text{c}}>1$ is required. In both graphs, it can clearly be seen that the thermal stability of the samples increases with increasing MnN thickness. With a thickness of $t_{\text{MnN}}=32$\,nm, the exchange bias is stable at a maximum value around $1400$\,Oe up to $T_{\text{A}}=400\,^{\circ}$C before it decreases. In contrast, samples with thicker MnN show an increase of exchange bias after annealing at higher temperatures. This is especially observable for the samples with $t_{\text{MnN}}=42,48$\,nm. Both yield exchange bias values of more than $2500$\,Oe with the maximum of $2785$\,Oe reached after annealing at $T_{\text{A}}=525\,^{\circ}$C for $t_{\text{MnN}}=48$\,nm. The highest ratio of $H_{\text{eb}}/H_{\text{c}}=5.4$ is found for the same parameters.
In our recent article\citep{meinert}, we reported that the thermal stability of MnN crucially depends on the nitrogen content in the MnN lattice. Samples that were prepared with a higher amount of nitrogen ($55\,\%$) during the reactive sputtering showed an increase of exchange bias after annealing at $T_{\text{A}}=400\,^{\circ}$C already for thicknesses of $t_{\text{MnN}}=30$\,nm. As thicker MnN films have a larger nitrogen reservoir, the increase of exchange bias after annealing at high temperatures is now observable for the lower amount of nitrogen ($50\,\%$) during reactive sputtering. Hence, we conclude that the effect of large exchange bias after
high-temperature annealing is related to the amount of nitrogen in the MnN film.\\
To obtain information about the influence of high-temperature annealing on the blocking temperature distribution, reversed field cooling experiments\citep{blocking} were performed on samples with $t_{\text{MnN}}=42,48$\,nm. The samples were initially field cooled from $T_{\text{set}}=525\,^{\circ}$C and $550\,^{\circ}$C for $t_{\text{MnN}}=42$\,nm and 48\,nm, respectively, and their hysteresis loops were measured to detect $H_{\text{eb}}(\text{RT})$. After that, they were successively field cooled in a reversed field from $T_{\text{rev}}=50, 75,..., 525\,(, 550)\,^{\circ}$C. In Figure 3a), the dependence of exchange bias on the reversal temperature is shown. The zero of this curve marks the median blocking temperature $<T_{\text{B}}>$ of the antiferromagnetic grains that are still blocked at room temperature. For both MnN thicknesses it lies around $180\,^{\circ}$C. The unblocked ratio (UBR), i.e. the area fraction of unblocked grains, is obtained via
\begin{equation}
\text{UBR}(T_{\text{rev}})= 100\% \cdot \frac {H_{\text{eb}}(\text{RT})-H_{\text{eb}}(T_{\text{rev}})} {2H_{\text{eb}}(\text{RT})},
\end{equation}
representing the cumulative distribution function of the blocking temperature. By taking the derivative of this function, the blocking temperature distribution can be obtained. It is shown in Figure 3c) and has a maximum around $170\,^{\circ}$C. In the course of our previous investigations\citep{meinert} we already performed field cooling experiments on similar samples with $t_{\text{MnN}}=48$\,nm, but initially field cooled them from a lower temperature $T_{\text{set}}=325\,^{\circ}$C. As additionally shown in Figure 3, they yield a median $<T_{\text{B}}>$ of $160\,^{\circ}$C with a maximum of the corresponding blocking temperature distribution around $125\,^{\circ}$C. Comparing this to our new results, the high-temperature annealing has a positive influence on the thermal stability of the MnN/CoFe system and seems to increase either the grain volume or the magnetocrystalline anisotropy energy.\\
In order to identify what causes the giant increase of exchange bias after high-temperature annealing, changes in the crystal structure were investigated for a sample with $t_{\text{MnN}}=42$\,nm. X-ray diffraction scans were detected between each annealing step. In Figure 4, the corresponding diffraction patterns directly after preparation and after annealing at $T_{\text{A}}=325\,^{\circ}$C and $T_{\text{A}}=525\,^{\circ}$C are shown, confirming a growth in (001) direction. After annealing, the (002) and (004) peaks of MnN are shifted towards higher angles, indicating smaller lattice constants due to nitrogen diffusion. Next to that, the peak intensities increase and the peaks become narrower. This indicates growth of the crystal grains and a relaxation of strain. Both can have a beneficial influence on the exchange bias. Larger crystal grains are also in line with the observed enhancement of the blocking temperature\citep{grady}. After annealing at $T_{\text{A}}=525\,^{\circ}$C, an additional peak around $34^{\circ}$ arises that is not related to any phase of Mn-N. Most likely it is attributed to the formation of $\text{TaN}_{\text{x}}$ in the Ta buffer layer caused by the nitrogen diffusion. The inset in Figure 4 displays the detailed evolution of the lattice constant $c$ in dependence on the annealing temperature. Starting at $c=4.23$\,\AA, it decreases monotonously with increasing annealing temperature up to $T_{\text{A}}=475\,^{\circ}$C where it saturates around $c=4.188$\,\AA, close to the value that was determined by Suzuki et al.\citep{Suzuki}.\\
To verify the temperature dependent nitrogen diffusion that is suggested by the X-ray diffraction results, Auger electron depth profiling was performed on a stack with $t_{\text{MnN}}=48$\,nm. Measurements were taken directly after preparation and after annealing at $T_{\text{A}}=325\,^{\circ}$C and $T_{\text{A}}=550\,^{\circ}$C. Figure 5a) shows the nitrogen concentration detected in the Mn and the Ta buffer layer for those three settings. Obviously, the nitrogen concentration in the Ta layer increases with each annealing process. After annealing at $T_{\text{A}}=550\,^{\circ}$C, an almost homogeneous concentration of nitrogen can be found in the whole Ta film. In Figure 5b), the evolution of the Auger peak of nitrogen for different depths is displayed after annealing at $T_{\text{A}}=550\,^{\circ}$C. The color change from blue to pink relates to increasing sputter time/ depth, i.e. the transition from Mn to Ta. A chemical shift of the nitrogen Auger peak is clearly visible. This can be observed in a similar extent after annealing at $T_{\text{A}}=325\,^{\circ}$C. What also strikes is that the nitrogen concentration inside the Mn layer does not decrease with increasing nitrogen concentration in the Ta layer. A strong preferential sputtering of nitrogen is known for other transition-metal nitrides\citep{surface}, which can change the apparent composition of the nitride film. Our results suggest, that this effect is also present for Mn-N and that the same sputter equilibrium ratio of Mn and N is reached for the sample in the as prepared state as well as after annealing. This can mask the expected reduction of the nitrogen concentration in the Mn-N if the nitrogen concentration in the Ta increases. Nonetheless, the depth profiles verify strong nitrogen diffusion caused by the high-temperature annealing. This coincides with the appearance of the additional peak in the XRD scan related to $\text{TaN}_{\text{x}}$ after annealing at high temperatures as well as the decrease of the lattice constant $c$ with increasing annealing temperature. However, this strong diffusion of nitrogen does not worsen but significantly increase the exchange bias.
\begin{figure*}
\includegraphics[width=0.82\textwidth]{xrd.pdf}
\caption{X-ray diffraction spectrum of a stack with $t_{\text{MnN}}=42$\,nm before and after annealing at $T_{\text{A}}=325\,^{\circ}$C and $T_{\text{A}}=525\,^{\circ}$C. The inset shows the dependence of the lattice parameter $c$ of MnN on the annealing temperature for the same sample.}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.84\textwidth]{aes.pdf}
\caption{a) Depth profile displaying the nitrogen concentration in a stack with $t_{\text{MnN}}=48$\,nm detected with Auger electron spectroscopy directly after preparation and after annealing at $T_{\text{A}}=325\,^{\circ}$C and $T_{\text{A}}=550\,^{\circ}$C. b) Normalized KLL-Auger transition of nitrogen in differential spectrum for different depths after annealing at $T_{\text{A}}=550\,^{\circ}$C. Color transition from blue to pink corresponds to increasing depth.}
\end{figure*}
\section{\label{sec:level1}Conclusion}
We found that MnN/CoFe bilayers show an enhanced exchange bias after annealing at temperatures $T_{\text{A}} > 400\,^{\circ}$C. Thus, maximum exchange bias of $2785$\,Oe is achieved accompanied by an increased blocking temperature. This behavior is only observable for samples with $t_{\text{MnN}}>32$\,nm. Even though Auger depth profiling confirmed strong nitrogen diffusion into the Ta buffer layer after annealing, thick MnN films seem to have a large nitrogen reservoir that allows for crystallographic or magnetic modifications. They could cause the strong increase of exchange bias and their identification is the subject of further investigations. Furthermore, depositing MnN on a buffer layer that does not bind nitrogen as strongly as Ta would make high-temperature annealing on thinner MnN films possible. The resulting increase of anisotropy could lead to a reduction of the critical thickness of MnN.
\begin{acknowledgments}
We thank the Ministerium f\"ur Innovation, Wissenschaft und Forschung des Landes Nordrhein-Westfalen (MIWF NRW) for financial support. We further thank G. Reiss for making available laboratory equipment.
\end{acknowledgments}
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Q: Removing new lines from output text using pdfbox Using pdfbox I can get text from PDF. The text goes like this:
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I was trying to remove \n using these methods, but my text file is always with new lines like above.
How can I get text in a one giant string without lines and print it on screen?
A: PDFTextStripper stripper = new PDFTextStripper();
stripper.setLineSeparator(" ");
A: "Phrase 1((?:.|\r?\n)*?)Phrase 2" should capture everything between "Phrase 1" and "Phrase 2", including new lines.
See this for an explaination: http://regex101.com/r/vH9hV1
Alternatively, you can just use the "dotall" flag which makes . match everything including new lines: http://regex101.com/r/aE9dP6
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{"url":"https:\/\/www.physicsforums.com\/threads\/orbit-dynamics.520531\/","text":"# Orbit Dynamics\n\n1. Aug 10, 2011\n\n### dorai007\n\nHi guys,\n\ni cant seem to find the answer .\n\nA satellite in earth orbit has a perigee velocity of 8 km=s and period of 2 hours. From this information,\ndetermine all the orbit parameters that you can. From those parameters, determine its altitude at perigee\n--------------------------------------------------------\n\nmy vp=8km\/sec\nT=7200sec\nVp=sqrt(GM\/r)---is this right and is r =rp?\n\nhow do i calculate my semi major axis? is it from kepler's 3rd law? im stuck at finding rp and eccenctricity and ive been going at this for hours without proper examples in books\/online.\n\n2. Aug 10, 2011\n\n### D H\n\nStaff Emeritus\nThat equation is valid only for circular orbits. The vis viva equation provides a more general answer:\n\n$$\\frac{v^2}{GM} = \\frac 2 r - \\frac 1 a$$\nCorrect. So show some work so we can help you out a bit.\n\n3. Aug 10, 2011\n\n### dorai007\n\nbut if its vis viva will my eccentricity value be more than 0.2..thats the value im getting, this is how im doing my calculations\n\n1) equation 1 : mag(h)=mag(rp)*mag(v)\nequation 2: mag(rp)= [mag(h)^2\/GM]\/(1+ecos(v) - trajectory eqn\nequation 3: T= 2phi\/sqrt(GM) * (mag(h))^3\/2\n\ni am to solve 3 unknowns mag(h),rp and e by simultaneous eqns by substituting all the equations into one another.\n\nive done it a couple of times, either im getting a negative value for e(-0.1) which shldnt be the case.\n\nBut my qn here is. if v=8000m\/sec what is mag(v)? and in this case will a=mag(h) ?\n\n4. Aug 10, 2011\n\n### D H\n\nStaff Emeritus\nThen you are doing it wrong. The eccentricity is less than 0.2. Show your work.\n\nYour equation 1 is only valid for circular orbits, and for elliptical orbits at perifocus and apofocus. (Since the given data point is perigee, this equation is okay here.) Your equation 3 however is only valid for circular orbits.\n\nTry finding a formula that relates semi-major axis (rather than specific angular momentum) to the period.\n\nThe magnitude of the velocity vector is of course 8000m\/s. As far as a=mag(h), no. Look at the units. Specific angular momentum(h) has units of length2\/time. Semi-major axis (a) has units of length.","date":"2018-12-12 09:19:02","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\": 0.7176188826560974, \"perplexity\": 2470.35916701582}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376823817.62\/warc\/CC-MAIN-20181212091014-20181212112514-00520.warc.gz\"}"} | null | null |
\section{#1}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\makeatletter
\@addtoreset{equation}{section}
\makeatother
\date{empty}
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\begin{document}
\begin{titlepage}
\null
\begin{flushright}
April, 2015
\end{flushright}
\vskip 1.5cm
\begin{center}
{\Large \bf Classifying BPS States in
Supersymmetric Gauge Theories
Coupled to Higher Derivative Chiral Models
}
\vskip 1.5cm
\normalsize
\renewcommand\thefootnote{\alph{footnote}}
{\large
Muneto Nitta$^{\dagger}$\footnote{nitta(at)phys-h.keio.ac.jp}
and Shin Sasaki$^\ddagger$\footnote{shin-s(at)kitasato-u.ac.jp}
}
\vskip 0.7cm
{\it
$^\dagger$
Department of Physics, and Research and Education Center for Natural Sciences, \\
\vskip -0.2cm
Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan
\vskip 0.1cm
$^\ddagger$
Department of Physics, Kitasato University \\
\vskip -0.2cm
Sagamihara 252-0373, Japan
}
\vskip 0.5cm
\begin{abstract}
We study $\mathcal{N} = 1$ supersymmetric gauge theories coupled with
higher derivative chiral models
in four dimensions
in the off-shell superfield formalism.
We solve the equation of motion for the auxiliary fields and find
two distinct on-shell structures of the Lagrangian
that we call the canonical and non-canonical branches
characterized by zero and non-zero auxiliary fields, respectively.
We classify BPS states of the models in Minkowski and Euclidean spaces.
In Minkowski space, we find Abelian and non-Abelian vortices,
vortex-lumps (or gauged lumps with fractional lump charges)
as 1/2 BPS states in the canonical branch and
higher derivative generalization of vortices and
vortex-(BPS)baby Skyrmions
(or gauged BPS baby Skyrmions with fractional baby Skyrme charges)
as 1/4 BPS states in the non-canonical branch.
In four-dimensional Euclidean space, we find
Yang-Mills instantons trapped inside a non-Abelian vortex,
intersecting vortices, and intersecting
vortex-(BPS)baby Skyrmions as 1/4 BPS states
in the canonical branch but no BPS states in the non-canonical branch
other than those in the Minkowski space.
\end{abstract}
\end{center}
\end{titlepage}
\newpage
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\newpage
\section{Introduction}
Low-energy effective theories play an important role
in the study of non-perturbative effects of quantum field theory,
such as the chiral Lagrangian of QCD \cite{Leutwyler:1993iq}.
In certain supersymmetric gauge theories, low-energy effective theories are
determined exactly offering full quantum spectra of
Bogomol'nyi-Prasado-Sommerfield (BPS) states
\cite{Seiberg:1994rs}.
BPS states
preserve a part of supersymmetry,
belonging to so-called short multiplets
of supersymmetry algebra,
and consequently they are stable against quantum corrections
perturbatively and non-perturbatively
\cite{Witten:1978mh}.
The low-energy effective field theories
are constructed by a derivative expansion
and are usually complemented by
higher derivative corrections,
as in the chiral perturbation theory
\cite{Leutwyler:1993iq}.
Recently, in our previous paper,
BPS states in the supersymmetric chiral models with
higher derivative terms have been classified
in ${\cal N}=1$ supersymmetric theories in four dimensions
\cite{Nitta:2014pwa}.
The purpose of this paper
is to classify BPS states in
${\cal N}=1$ supersymmetric gauge theories
coupled with higher derivative chiral models
in four-dimensional Minkowski and Euclidean spaces.
Higher derivative corrections to supersymmetric field theories have
a long history because of the auxiliary field problem.
The auxiliary fields $F$ in the off-shell superfield formalism of higher derivative models
are generically acted on by space-time derivatives
and consequently cannot be eliminated algebraically
to obtain on-shell actions.
Supersymmetric higher derivative terms
free from the auxiliary field problem
have been studied individually in various contexts:
the Wess-Zumino-Witten term
\cite{Nemeschansky:1984cd,Gates:1995fx,Gates:2000rp,Nitta:2001rh},
low-energy effective action
\cite{Buchbinder:1994iw, Buchbinder:1994xq, Matone:1996bj, Bellisai:1997ck,
Gomes:2009ev, Banin:2006db, AnDuGh, Gama:2011ws, Kuzenko:2014ypa},
${\mathbb C}P^1$ (Faddeev-Skyrme) model
\cite{BeNeSc,Fr},
Dirac-Born-Infeld (DBI) action
\cite{RoTs,SaYaYo},
$k$-field theory
\cite{AdQuSaGuWe3,AdQuSaGuWe2},
low-energy effective action on BPS solitons \cite{Eto:2012qda},
BPS baby Skyrme model
\cite{Adam:2011hj,AdQuSaGuWe,Nitta:2014pwa,Bolognesi:2014ova},
and nonlinear realizations of
Nambu-Goldstone fields \cite{Nitta:2014fca}.
In the framework of supergravity,
higher derivative terms
\cite{KhLeOv,KoLeOv2,KhLeOv2,KoLeOv,FaKe}
have been applied to
ghost condensations \cite{KhLeOv,KoLeOv2}
and the Galileon inflation models \cite{KhLeOv2}.
Among those, the four derivative term first found in Ref.~\cite{Buchbinder:1994iw}, that can be constructed from
a $(2,2)$ K\"ahler tensor,
was rediscovered in Refs.~\cite{KhLeOv,KoLeOv2}
and has recently been used in various contexts.
By using a K\"ahler tensor containing
space-time derivatives,
one can construct higher derivative terms with
arbitrary number of space-time derivatives
\cite{Nitta:2014fca}.
In our previous paper \cite{Nitta:2014pwa},
the auxiliary field equations
were found to admit at least two distinct solutions
that we called
canonical and non-canonical branches
with $F=0$ and $F\neq 0$, respectively.
In particular,
BPS baby Skyrmions (compactons)
\cite{Adam:2011hj,AdQuSaGuWe}
have been found to be 1/4 BPS states
in the non-canonical branch,
while BPS lumps are 1/2 BPS states
in the canonical branch \cite{Eto:2012qda},
although both of them saturate the same Bogomol'nyi bound.
In the former, the on-shell Lagrangian contains
no usual kinetic term and consists of only a four derivative term,
while in the latter, higher derivative corrections disappear
in solutions and energy.
BPS baby Skyrmions as compactons are currently
paid much attention \cite{Adam:2009px,Adam:2012pm}.
In this paper,
we classify BPS states in ${\cal N}=1$ supersymmetric gauge theories
coupled with higher derivative chiral models
in four-dimensional Minkowski and Euclidean spaces.
Here, we concentrate on the cases where superpotentials are absent for simplicity.
As in the previous cases without gauge fields,
we find canonical and non-canonical branches
corresponding to solutions $F=0$ and $F\neq 0$
of auxiliary field equations, respectively.
We find that
1/2 BPS states that exist
in theories without higher derivative terms
remain 1/2 BPS in the canonical branch
and that corresponding BPS states in the non-canonical branch
are 1/4 BPS states.
On the other hand, we also find that
1/4 BPS states that exist
in theories without higher derivative terms
remain 1/4 BPS in the canonical branch
but there are no corresponding BPS states in the non-canonical branch.
More precisely,
we find that 1/2 BPS equations in the canonical branch
do not receive higher derivative corrections
for an Abrikosov-Nielsen-Olesen (ANO) vortex
\cite{Abrikosov:1956sx}
at the critical (BPS) coupling,
a non-Abelian vortex \cite{Hanany:2003hp},
lumps \cite{Polyakov:1975yp},
vortex-lumps
(gauged lumps with fractional lump charges) \cite{Schroers:1995he,Nitta:2011um}.
We then show that higher derivative generalization of vortices
(that we may call compact vortices)
and vortex-baby Skyrmions
(or gauged baby Skyrmions with fractional baby Skyrme charges)
are 1/4 BPS states
in the non-canonical branch.
In four-dimensional Euclidean space, we find 1/2 BPS Yang-Mills instantons,
1/4 BPS Yang-Mills instantons trapped inside a non-Abelian vortex,
and 1/4 BPS intersecting vortices with instanton charges in the canonical branch.
These configurations were known in supersymmetric theories with
eight supercharges without higher derivative terms
in $4+1$ or $5+1$ dimensions \cite{Hanany:2004ea,Eto:2004rz,Eto:2006pg,Fujimori:2008ee},
and so what we confirm here is that they are still 1/4 BPS states
in theories with four supercharges
in Euclidean four dimensions and that
higher derivative terms are canceled out
in the BPS equations and energy bound.
Further, as new configurations, we find 1/4 BPS vortex-lump string intersections with Yang-Mills instanton charges.
We find no BPS states in the non-canonical branch other than those in Minkowski space.
This paper is organized as follows.
In Sec.~\ref{sec:hdcm}, we give supersymmetric Lagrangian
in the superfield formalism.
The first subsection is devoted to a review for
higher derivative chiral models of chiral multiplets
without coupling to gauge fields.
In the second subsection, we introduce vector multiplets
and coupling of vector and chiral multiplets.
In Sec.~\ref{sec:BPS-Minkowski} we classify BPS states
in four-dimensional Minkowski space.
In Sec.~\ref{sec:BPS-Euclidean}
BPS states in four-dimensional Euclidean space are discussed.
Sec.~\ref{sec:summary} is devoted to a summary and discussion.
Notations and conventions are summarized in Appendix.~\ref{sec:notation}.
Explicit supersymmetry variations of fermions in Euclidean space are
found in Appendix.~\ref{sec:SUSY_variation}
\section{Higher derivative chiral model}\label{sec:hdcm}
In this section we introduce the four-dimensional $\mathcal{N} = 1$ supersymmetric higher
derivative chiral model \cite{KhLeOv, Nitta:2014pwa} and its coupling to
the vector multiplet.
The supersymmetric higher derivative chiral model consists of chiral
superfields $\Phi^i \ (i = 1, \ldots, n)$ with an arbitrary K\"ahler potential $K$,
superpotential $W$ and a symmetric $(2,2)$ K\"ahler tensor
$\Lambda_{ik\bar{j} \bar{l}}$. The tensor $\Lambda_{ik\bar{j}
\bar{l}}$ is an arbitrary function of $\Phi^i, \Phi^{\dagger \bar{j}}$
and its space-time derivatives.
Among other things, the purely bosonic part of the model never contains
the space-time derivatives of the auxiliary fields $F^i$.
Then all the auxiliary fields are integrated out by the algebraic equation of
motion and one finds explicit on-shell Lagrangians.
When global symmetries in the model are gauged, the higher derivative term couples to
the vector multiplet.
In the following, we provide the explicit Lagrangian of the non-gauged higher
derivative chiral model and its coupling to the vector multiplet (gauged
model).
\subsection{Higher Derivative Chiral Models without Gauge Coupling}
We first start from the non-gauged $\mathcal{N} = 1$ supersymmetric
higher derivative model with chiral superfields $\Phi^i$.
We employ the Wess-Bagger convention \cite{Wess:1992cp} in this paper
and detailed conventions and notations are summarized in Appendix \ref{sec:notation}.
The component expansion of the chiral superfield in the chiral base
$y^m = x^m + i \theta \sigma^m \bar{\theta}$ is
\begin{align}
\Phi^i = \varphi^i (y) + \theta \psi^i (y) + \theta^2 F^i (y).
\end{align}
Here $\varphi^i$ is the complex scalar field, $\psi^i$ is the Weyl
fermion, and $F^i$ is the auxiliary complex scalar field.
The Lagrangian of the non-gauged higher derivative chiral model is given by
\begin{align}
\mathcal{L} = & \
\ \int \! d^4 \theta \ K (\Phi^i, \Phi^{\dagger \bar{j}})
+ \frac{1}{16} \int \! d^4 \theta \ \Lambda_{i \bar{j} k \bar{l}} (\Phi,
\Phi^{\dagger})
D^{\alpha} \Phi^i
D_{\alpha} \Phi^k \bar{D}_{\dot{\alpha}} \Phi^{\dagger \bar{j}}
\bar{D}^{\dot{\alpha}} \Phi^{\dagger \bar{l}}
+ \left(\int \! d^2 \theta \ W(\Phi^i) + ({\rm h.c.})\right)
\label{eq:ungauged_HD_Lagrangian}
\end{align}
where $K$ is the K\"ahler potential, $\Lambda_{ik\bar{j}\bar{l}}$ is a
symmetric $(2,2)$ K\"ahler tensor and $W$ is the superpotential.
The fourth derivative part in the Lagrangian is evaluated as
\begin{align}
D^{\alpha} \Phi^i D_{\alpha} \Phi^k \bar{D}_{\dot{\alpha}}
\Phi^{\dagger \bar{j}} \bar{D}^{\dot{\alpha}}
\Phi^{\dagger \bar{l}}
=& \ 16 \theta^2 \bar{\theta}^2
\left[
\frac{}{}
(\partial_m \varphi^i \partial^m \varphi^k) (\partial_n
\bar{\varphi}^{\bar{j}} \partial^n \bar{\varphi}^{\bar{l}})
\right.
\notag \\
&
\left.
- \frac{1}{2}
\left(
\partial_m \varphi^i F^k + F^i \partial_m \varphi^k
\right)
\left(
\partial^m \bar{\varphi}^{\bar{j}} \bar{F}^{\bar{l}}
+ \bar{F}^{\bar{j}} \partial^m \bar{\varphi}^{\bar{l}}
\right)
+ F^i \bar{F}^{\bar{j}} F^k \bar{F}^{\bar{l}}
\right] + I_f.
\label{eq:fourth_deri}
\end{align}
Here $I_f$ stands for terms that contain fermions.
Since the purely bosonic part in Eq.~\eqref{eq:fourth_deri} saturates the
Grassmann coordinate, only the lowest components in $\Lambda_{ik \bar{j}
\bar{l}}$ contribute to the bosonic part of the Lagrangian.
Then, the bosonic part of the Lagrangian is
\begin{eqnarray}
\mathcal{L}_b &=&
g_{i \bar{j}}
\left(
- \partial_m \varphi^i \partial^m \bar{\varphi}^{\bar{j}} + F^i \bar{F}^{\bar{j}}
\right)
+ \frac{\partial W}{\partial \varphi^i} F^i + \frac{\partial
\bar{W}}{\partial \bar{\varphi}^{\bar{j}}} \bar{F}^{\bar{j}}
\nonumber \\
& & +
\Lambda_{ik\bar{j}\bar{l}} (\varphi, \bar{\varphi})
\left\{
(\partial_m \varphi^i \partial^m \varphi^k) (\partial_n \bar{\varphi}^{\bar{j}}
\partial^n \bar{\varphi}^{\bar{l}}) - 2 \partial_m \varphi^i F^k \partial^m
\bar{\varphi}^{\bar{j}} \bar{F}^{\bar{l}} + F^i \bar{F}^{\bar{j}} F^k \bar{F}^{\bar{l}}
\right\}.
\label{eq:off-shell_Lagrangian}
\end{eqnarray}
Here $g_{i\bar{j}} = \frac{\partial^2 K}{\partial \varphi^i \partial
\bar{\varphi}^{\bar{j}}} > 0$ is the K\"ahler metric.
In order to find the on-shell Lagrangian, we integrate out the auxiliary
fields $F^i$.
Since the Lagrangian does not contain space-time derivatives of the auxiliary
fields $F^i$, one can solve the equation of motion for $F^i$ and find the
explicit form of the purely bosonic part of the on-shell Lagrangian\footnote{
There are space-time derivatives of the auxiliary fields $F^i$ in the
fermion term $I_f$. Solutions to $F^i$ that include fermions are
obtained order by order of the fermions.
Since we are interested in the classical configurations of fields,
these fermionic contributions are irrelevant in this paper.
}.
The equation of motion for the auxiliary fields is
\begin{align}
g_{i \bar{j}}
F^i
- 2 \partial_m \varphi^i F^k \Lambda_{ik\bar{j}\bar{l}} \partial^m
\bar{\varphi}^{\bar{l}} + 2 \Lambda_{ik\bar{j}\bar{l}} F^i F^k
\bar{F}^{\bar{l}} + \frac{\partial \bar{W}}{\partial
\bar{\varphi}^{\bar{j}}} = 0.
\label{eq:auxiliary_eom}
\end{align}
As we have advertised, the equation \eqref{eq:auxiliary_eom} is an algebraic equation and it can be
solved in principle. There are distinct on-shell branches associated with different
solutions to the equation \eqref{eq:auxiliary_eom}.
In general, there are two classes of solutions.
The first class has smooth limit $\Lambda_{ik\bar{j}\bar{l}} \to 0$ to the ordinary ({\it i.e.}~without higher
derivative terms) theory. For this class of solutions, higher derivative terms are
introduced as perturbations to the ordinary (with second space-time
derivatives) theory in the on-shell Lagrangian.
We call this case the canonical (perturbative) branch.
On the other hand, the second class of solutions does not have
a smooth limit $\Lambda_{ik\bar{j} \bar{l}} \to 0$
to the ordinary theory. For this class of
solutions, the higher derivative terms enter into the on-shell
Lagrangian non-perturbatively.
We call this case the non-canonical (non-perturbative) branch.
In Ref.~\cite{Nitta:2014pwa}, we studied on-shell structures of the
Lagrangian \eqref{eq:off-shell_Lagrangian} for the single chiral superfield
model. When $W \not= 0$, the equation of motion for the auxiliary field
becomes that of the cubic power of $F$,
and the solutions can be obtained by Cardano's method \cite{SaYaYo}.
The explicit solutions are quite non-linear in
$K$, $\Lambda$, $W$, and $\partial_m \varphi$. Therefore, the on-shell Lagrangian
becomes a highly complicated function of the scalar field $\varphi$.
In the following, we consider models with $W = 0$ and show the explicit
on-shell Lagrangians in the canonical and non-canonical branches.
\paragraph{Canonical branch}
It is apparent that $F^i = 0$ is always a solution to the equation
\eqref{eq:auxiliary_eom}.
In this case, the bosonic part of the on-shell Lagrangian is
\begin{align}
\mathcal{L}_b = - g_{i \bar{j}} \partial_m \varphi^i \partial^m
\bar{\varphi}^{\bar{j}}
+ \Lambda_{ik \bar{j}\bar{l}} (\varphi, \bar{\varphi})
(\partial_m \varphi^i \partial^m \varphi^k) (\partial_n
\bar{\varphi}^{\bar{j}} \partial^n \bar{\varphi}^{\bar{l}}).
\end{align}
The tensor $\Lambda_{ik \bar{j} \bar{l}}$ determines higher derivative
terms in the Lagrangian.
Since $\Lambda_{ik \bar{j} \bar{l}}$ is an arbitrary function
of $\varphi, \bar{\varphi}$, one can construct arbitrary higher
derivative terms for $n=1$ models.
For example, the scalar part of the $\mathcal{N} = 1$
supersymmetric Dirac-Born-Infeld action \cite{RoTs} is obtained by the
single chiral superfield model with a flat K\"ahler potential and
\begin{align}
\Lambda = \frac{1}{1 + A + \sqrt{(1 + A^2) - B}}, \quad
A = \partial_m \Phi \partial^m \Phi^{\dagger}, \quad
B = \partial_m \Phi \partial^m \Phi \partial_n \Phi^{\dagger} \partial^n
\Phi^{\dagger}.
\label{eq:Lambda_DBI}
\end{align}
The supersymmetric Faddeev-Skyrme model is obtained by the
$\mathbb{C}P^1$ Fubuni-Study metric $K_{\varphi \bar{\varphi}} = \frac{1}{(1 +
|\varphi|^2)^2}$ and \cite{Nitta:2014pwa}
\begin{align}
\Lambda = (\partial_m \Phi \partial^m \Phi \partial_n
\Phi^{\dagger} \partial^n \Phi^{\dagger})^{-1} \frac{1}{(1 + \Phi
\Phi^{\dagger})^4}
\left[
(\partial_m \Phi^{\dagger} \partial^m \Phi)^2 - \partial_m \Phi
\partial^m \Phi \partial_n \Phi^{\dagger} \partial^n \Phi^{\dagger}
\right]. \label{eq:FSmodel}
\end{align}
This does not contain
an additional term other than Fadeev-Skyrme term,
in contrast to Refs.~\cite{BeNeSc,Fr} that contain an additional term.
The other examples include a supersymmetric completion of the Galileon inflation
model \cite{KhLeOv}, the ghost condensation \cite{KoLeOv2}
and the effective action of the supersymmetric Wess-Zumino model and QCD
\cite{Kuzenko:2014ypa, Gates:1995fx}.
\paragraph{Non-canonical branch}
Although it is not easy to find explicit solutions $F^i \not= 0$ for
the $n > 1$ case, one finds the solution for a single chiral
superfield model \cite{Nitta:2014pwa}:
\begin{align}
F = e^{i\eta} \sqrt{- \frac{K_{\varphi \bar{\varphi}}}{2\Lambda} +
\partial_m \varphi \partial^m \bar{\varphi}},
\label{eq:non-canonical_sol}
\end{align}
where $\eta$ is a phase factor and $K_{\varphi \bar{\varphi}} =
\frac{\partial^2 K}{\partial \varphi \partial \bar{\varphi}}$.
Then the bosonic part of the on-shell Lagrangian in the non-canonical branch is
\begin{align}
\mathcal{L}_b =& \ \Lambda |\partial_m \varphi \partial^m \varphi|^2 -
\Lambda (\partial_m \varphi \partial^m \bar{\varphi})^2 -
\frac{K^2_{\varphi \bar{\varphi}}}{4 \Lambda}.
\label{eq:neutral_non-cano}
\end{align}
In this case, the ordinary canonical (second space-time derivative)
kinetic term cancels out and
the on-shell Lagrangian contains higher derivative terms only.
An example is the BPS baby Skyrme model \cite{AdQuSaGuWe},
where $\Lambda$ is given by
\begin{align}
\Lambda = \frac{1}{(1 + \Phi \Phi^{\dagger})^4}.
\label{eq:Lambda_bS}
\end{align}
The K\"ahler metric is the Fubini-Study metric of $\mathbb{C}P^1$.
A few comments are in order for the non-canonical branch.
First, since $F \bar{F} \ge 0$, the fields satisfy the constraint
\begin{align}
\partial_m \varphi \partial^m \bar{\varphi} - \frac{K_{\varphi
\bar{\varphi}}}{2 \Lambda} \ge 0.
\label{eq:noncanonical_constraint}
\end{align}
Second, the last term in Eq.~\eqref{eq:neutral_non-cano} is considered as
the scalar potential when $\Lambda$ does not contain space-time derivative term.
One can introduce an arbitrary scalar potential without the superpotential
$W$ or the D-term potential in the non-canonical branch.
This is an alternative way to introduce the scalar potential in
supersymmetric models \cite{KoLeOv}.
\subsection{Gauged higher-derivative chiral models}
In this subsection we study couplings of the gauge field to the
higher derivative chiral models.
We consider the higher derivative model of the type
\eqref{eq:ungauged_HD_Lagrangian} where some global symmetries are assumed.
Let us consider the chiral superfields $\Phi^{ia} \ (a =1, \ldots,
\mathrm{dim} G)$ belonging to the fundamental representation of global
symmetry group $G$ with an additional flavor index $i$.\footnote{
It is straightforward to generalize the result in this subsection to
other representations. Therefore we consider the fundamental
representation of $G$ for the chiral superfield $\Phi^a$ throughout this
paper.}
Then the fourth derivative term which preserves the global
symmetry $G$ is
\begin{align}
\frac{1}{16} \int \! d^4 \theta \ \Lambda_{ik\bar{j} \bar{l}, ab}
{}^{cd} D^{\alpha} \Phi^{ia} D_{\alpha} \Phi^{k b}
\bar{D}_{\dot{\alpha}} \Phi^{\dagger \bar{j}}_c \bar{D}^{\dot{\alpha}}
\Phi^{\dagger \bar{l}}_d,
\label{eq:ungauged_global_4th_derivative}
\end{align}
where the K\"ahler tensor $\Lambda_{ik\bar{j}\bar{l},ab} {}^{cd}$
has indices of the (anti)fundamental representation of $G$.
The gauge field is introduced by the $\mathcal{N} = 1$ vector superfield
$V$ with gauge group $G$.
The generators $T^{\hat{a}} \ (\hat{a} = 0, 1, \ldots, \text{dim}
\mathcal{G} - 1)$ of the gauge algebra $\mathcal{G}$ are normalized as
$\mathrm{Tr} [T^{\hat{a}} T^{\hat{b}}] = k \delta^{\hat{a} \hat{b}} \ (k>0)$.
The component expansion of $V = V^{\hat{a}} T^{\hat{a}}$ in the Wess-Zumino gauge is
\begin{align}
V = - (\theta \sigma^m \bar{\theta}) A_m (x) + i \theta^2 \bar{\theta}
\bar{\lambda} (x) - i \bar{\theta}^2 \theta \lambda (x) + \frac{1}{2} \theta^2
\bar{\theta}^2 D (x).
\label{eq:Vector_component}
\end{align}
Here, $A_m$ is the gauge field, $\lambda_{\alpha},
\bar{\lambda}_{\dot{\alpha}}$ are the gauginos, and $D$ is the auxiliary real scalar field.
All the fields belong to the adjoint representation of $G$.
The coupling of the gauge field to the higher derivative terms is
introduced by gauge covariantizing the supercovariant derivatives in
Eq.~\eqref{eq:ungauged_global_4th_derivative}.
The gauge covariantized supercovariant derivative is defined
by
\begin{align}
\mathcal{D}_{\alpha} \Phi^{ia} = D_{\alpha} \Phi^{ia} + (\Gamma_{\alpha})^a {}_b \Phi^{ib}.
\end{align}
Here $\Gamma_{\alpha}$ is the gauge connection defined by
\begin{align}
\Gamma_{\alpha} = e^{-2gV} D_{\alpha} e^{2gV},
\end{align}
where $g$ is the gauge coupling constant.
The gauge transformations of the superfields are
\begin{align}
\Phi^i \to e^{-i \Theta} \Phi^i, \qquad e^{2gV} \to e^{- i
\Theta^{\dagger}} e^{2gV} e^{i \Theta},
\end{align}
where $\Theta = \Theta^{\hat{a}} (x,\theta,\bar\theta)T^{\hat{a}}$ is
a gauge parameter chiral superfield.
Then the quantities $\mathcal{D}_{\alpha} \Phi^i,
\bar{\mathcal{D}}_{\dot{\alpha}} \Phi^{\dagger \bar{i}}$ are transformed covariantly
under the gauge transformation:
\begin{align}
\mathcal{D}_{\alpha} \Phi^i \to e^{- i \Theta} \mathcal{D}_{\alpha} \Phi^i,
\qquad
\bar{\mathcal{D}}_{\dot{\alpha}} \Phi^{\dagger \bar{i}} \to
\bar{\mathcal{D}}_{\dot{\alpha}} \Phi^{\dagger \bar{i}} e^{i \Theta^{\dagger}}.
\end{align}
We note that the K\"ahler tensor $\Lambda_{ik\bar{j}\bar{l},ab} {}^{cd}$
becomes a function of $\Phi,\Phi^{\dagger}$ and $V$ in general.
Now we look for the concrete realizations of the
gauge invariant generalization of the higher
derivative term \eqref{eq:ungauged_global_4th_derivative}.
We find a manifestly gauge invariant generalization of
\eqref{eq:ungauged_global_4th_derivative} is given by
\begin{align}
- \frac{1}{16}
\int \! d^4 \theta \ \Lambda_{ik \bar{j} \bar{l}}
(\Phi,\Phi^{\dagger}, V) (\bar{\mathcal{D}}_{\dot{\alpha}}
\Phi^{\dagger \bar{j}} e^{2gV} \mathcal{D}^{\alpha} \Phi^i)
(\bar{\mathcal{D}}^{\dot{\alpha}} \Phi^{\dagger \bar{l}} e^{2gV}
\mathcal{D}_{\alpha} \Phi^{k}),
\label{eq:4th_deri_gauge2}
\end{align}
where the K\"ahler tensor is
\begin{align}
\Lambda_{ik\bar{j}\bar{l} ab} {}^{cd} = \Lambda_{ik\bar{j} \bar{l}} (\Phi, \Phi^{\dagger}, V) (e^{2gV})^c
{}_a (e^{2gV})^d {}_b
\end{align}
and $\Lambda_{ik\bar{j}\bar{l}}$ is a gauge invariant $(2,2)$ K\"ahler
tensor which is a function of $\Phi,\Phi^{\dagger}, V$.
The component expansion of the fourth derivative term
\eqref{eq:4th_deri_gauge2} is
\begin{align}
& - \frac{1}{16} (\bar{\mathcal{D}}_{\dot{\alpha}} \Phi^{\dagger \bar{j}} e^{2gV} \mathcal{D}^{\alpha} \Phi^i)
(\bar{\mathcal{D}}^{\dot{\alpha}} \Phi^{\dagger \bar{l}} e^{2gV}
\mathcal{D}_{\alpha} \Phi^k)
\notag \\
=& \ \theta^2 \bar{\theta}^2
\left[
\frac{}{}
(D^m \bar{\varphi}_a^{\bar{j}} D^n \varphi^{ia}) (D_m \bar{\varphi}_b^{\bar{l}} D_n \varphi^{kb})
- \frac{1}{2} (D_m \varphi^{ia} F^{kb} + F^{ia} D_m \varphi^{kb}) (D^m
\bar{\varphi}_a^{\bar{j}} \bar{F}_b^{\bar{l}} + \bar{F}_a^{\bar{j}} D^m \bar{\varphi}_b^{\bar{l}}
)
\right.
\notag \\
& \ \qquad \qquad
\left.
\frac{}{}
+ F^{ia} \bar{F}^{\bar{j}}_a F^{kb} \bar{F}^{\bar{l}}_b
\right] + I'_f,
\label{eq:4th_deri_gauge2_component}
\end{align}
where $I_f'$ is terms that contain fermions.
Again, there are no auxiliary fields with space-time derivatives in the
purely bosonic terms.
Since the bosonic terms in $\bar{\mathcal{D}}_{\dot{\alpha}}
\Phi^{\dagger} \mathcal{D}^{\alpha} \Phi
\bar{\mathcal{D}}^{\dot{\alpha}} \Phi^{\dagger} \mathcal{D}_{\alpha}
\Phi$ already saturate the Grassmann coordinate, the factor $e^{2gV}$ does not
contribute to the purely bosonic sector of the
Lagrangian. However, the factor $e^{2gV}$ is necessary for the gauge
invariance of the higher derivative terms and this indeed contributes to the fermionic
part $I'_f$ in Eq.~\eqref{eq:4th_deri_gauge2_component}.
We also note that the lowest components in $\Lambda_{ik\bar{j}\bar{l}}$ come from the
chiral superfields only. This is because the lowest component in the
vector superfield $V$ contains the Grassmann coordinate $\theta$ in the Wess-Zumino gauge
\eqref{eq:Vector_component}.
In Ref.~\cite{AdQuSaGuWe}, a three-dimensional analogue of the gauge invariant higher derivative model for a $U(1)$ gauge group was discussed.
Introducing the ordinary kinetic terms for $\Phi^{ia}$ and the gauge field,
the total Lagrangian we consider is
\begin{align}
\mathcal{L} =& \ \int \! d^4 \theta \ K(\Phi^{\dagger}, \Phi, V) -
\frac{1}{16} \int \! d^4 \theta \ \Lambda_{ik\bar{j}\bar{l}} (\Phi, \Phi^{\dagger}, V)
(\bar{\mathcal{D}}_{\dot{\alpha}} \Phi^{\dagger \bar{j}} e^{2gV}
\mathcal{D}^{\alpha} \Phi^i) (\bar{\mathcal{D}}^{\dot{\alpha}}
\Phi^{\dagger \bar{l}} e^{2gV} \mathcal{D}_{\alpha} \Phi^k)
\notag \\
& + \frac{1}{16 k g^2} \mathrm{Tr}
\left[
\int \! d^2 \theta \ W^{\alpha} W_{\alpha} + ({\rm h.c.})
\right]
- 2 \kappa g \int \! d^4 \theta \ \mathrm{Tr} V.
\label{eq:gauged_model}
\end{align}
Here we have introduced the Fayet-Iliopoulos parameter $\kappa$
for the purpose of later discussions.
The field strength of the vector superfield $V$ is defined by
\begin{align}
W_{\alpha} = - \frac{1}{4} \bar{D}^2 (e^{-2gV} D_{\alpha} e^{2gV}).
\end{align}
Throughout this paper, we consider the gauge invariant K\"ahler potential
of the form $K (\Phi^{\dagger}, \Phi, V) = \frac{1}{2} (K(\Phi^{\dagger} e^{2gV},
\Phi) + K(\Phi^{\dagger}, e^{2gV} \Phi))$ and general gauge group $G$ if
not mentioned.
Then, the bosonic component of the Lagrangian \eqref{eq:gauged_model} is
\begin{align}
\mathcal{L}_{b} =& \
- \frac{\partial^2 K}{\partial \bar{\varphi}_a^{\bar{j}} \partial \varphi^{ib}}
D_m \bar{\varphi}_a^{\bar{j}} D^m \varphi^{ib}
- \frac{\partial^2 K}{\partial \bar{\varphi}_a^{\bar{j}} \partial \varphi^{ib}}
\bar{F}_a^{\bar{j}} F^{ib} +
\frac{g}{2} D^{\hat{a}}
\left(
\bar{\varphi}_c^{\bar{j}} (T^{\hat{a}})^c {}_d \frac{\partial K}{\partial \bar{\varphi}_d^{\bar{j}}}
+
\frac{\partial K}{\partial \varphi^{ic}} (T^{\hat{a}})^c
{}_d \varphi^{id} - 2 \kappa \delta^{\hat{a}} {}_0
\right)
\notag \\
& \ + \frac{1}{k} \mathrm{Tr}
\left[
- \frac{1}{4} F_{mn} F^{mn} + \frac{1}{2} D^2
\right]
\notag \\
& \ + \Lambda_{ik\bar{j}\bar{l}} (\varphi, \bar{\varphi})
\left[
\frac{}{}
(D^m \bar{\varphi}_a^{\bar{j}} D^n \varphi^{ia}) (D_m \bar{\varphi}_b^{\bar{l}} D_n \varphi^{kb})
\right.
\notag \\
& \ \qquad \qquad \qquad
\left.
\frac{}{}
- \frac{1}{2} (D_m \varphi^{ia} F^{kb} + F^{ia} D_m \varphi^{kb}) (D^m
\bar{\varphi}_a^{\bar{j}} \bar{F}_b^{\bar{l}} + \bar{F}_a^{\bar{j}} D^m \bar{\varphi}_b^{\bar{l}}
)
+ F^{ia} \bar{F}^{\bar{j}}_a F^{kb} \bar{F}^{\bar{l}}_b
\right],
\end{align}
where we have assigned the $U(1)$ generator to $T^0$.
The gauge field strength is
\begin{align}
F_{mn} = \partial_m A_n - \partial_n A_m + i g [A_m, A_n].
\end{align}
The equation of motion for the auxiliary field $D$ is\footnote{
We never introduce higher derivative terms of the vector
superfield $V$. Therefore the equation of motion for $D$ is always
linear and can be solved trivially.}
\begin{align}
D^{\hat{a}} + \frac{g}{2}
\left(
\bar{\varphi}_c^{\bar{j}} (T^{\hat{a}})^c {}_d \frac{\partial K}{\partial
\bar{\varphi}_d^{\bar{j}}}
+ \frac{\partial K}{\partial \varphi^{ic}} (T^{\hat{a}})^c {}_d \varphi^{id}
\right)
- g \kappa \delta^{\hat{a}} {}_0 = 0.
\end{align}
The equation of motion for $\bar{F}_a^{\bar{j}}$ is
\begin{align}
\frac{\partial^2 K}{\partial \bar{\varphi}_a^{\bar{j}} \partial
\varphi^{ib}} F^{ib}
- \Lambda_{ik\bar{j}\bar{l}} (\varphi, \bar{\varphi})
\left[
D_m \varphi^{ib} D^m \bar{\varphi}_b^{\bar{j}} F^{ka} + D_m \varphi^{ia}
D^m \bar{\varphi}^{\bar{l}}_b F^{kb} - 2 F^{ia} F^{kb} \bar{F}^{\bar{l}}_b
\right]
= 0.
\label{eq:auxiliary_eom_gauged}
\end{align}
As in the case of the non-gauged chiral superfield models,
there are two on-shell branches associated with solutions to the equation
\eqref{eq:auxiliary_eom_gauged}.
\paragraph{Canonical branch}
We first consider the canonical branch.
One finds that $F^{ia} = 0$ is always a solution.
Then, the on-shell Lagrangian in the canonical branch is
\begin{align}
\mathcal{L}_{b} =& \
- \frac{\partial^2 K}{\partial \bar{\varphi}_a^{\bar{j}} \partial \varphi^{ib}}
D_m \bar{\varphi}^{\bar{j}}_a D^m \varphi^{ib}
+ \Lambda_{ik\bar{j}\bar{l}} (\varphi, \bar{\varphi}) (D^m \bar{\varphi}_a^{\bar{j}} D^n \varphi^{ia})
(D_m \bar{\varphi}_b^{\bar{l}} D_n \varphi^{kb})
\notag \\
& \ - \frac{g^2}{2}
\left(
\frac{1}{2} \bar{\varphi}_c^{\bar{j}} (T^{\hat{a}})^c {}_d \frac{\partial K}{\partial
\bar{\varphi}_d^{\bar{j}}}
+ \frac{1}{2} \frac{\partial K}{\partial \varphi^{ic}} (T^{\hat{a}})^c {}_d \varphi^{id}
- \kappa \delta^{\hat{a}} {}_0
\right)^2 - \frac{1}{4k} \mathrm{Tr} F_{mn} F^{mn}.
\label{eq:canonical_Lag}
\end{align}
The vacuum of the model is determined by the D-term condition
\begin{align}
\bar{\varphi}_c^{\bar{i}} (T^{\hat{a}})^c {}_d \varphi^{id} - \kappa \delta^a {}_0 = 0.
\end{align}
We stress that $\Lambda_{ik\bar{j}\bar{l}}$
does not contain
the space-time derivatives on $\Phi$ ($\Phi^\dagger$),
unlike the non-gauged cases
for which the space-time derivative can act on
$\Phi$ ($\Phi^\dagger$) in $\Lambda_{ik\bar{j}\bar{l}}$.
This is because the gauge covariant derivative of a chiral superfield $D_m
\Phi^{ia}$ does not provide supersymmetric couplings of the gauge
field. From now on, we therefore consider the tensor $\Lambda_{ik\bar{j}\bar{l}}$
which never contains the space-time derivatives of the superfields.
\paragraph{Non-canonical branch}
It is not so easy to find a $F^{ia} \not= 0$ solution even
for the single chiral superfield model.
However, we find that a $F^a \not= 0$ solution can be explicitly written
down for single chiral superfield models with a $U(1)$ gauge group as
\begin{align}
F^0 =
e^{i \eta} \sqrt{
- \frac{K_{\varphi \bar{\varphi}}}{2 \Lambda} + D_m \varphi D^m
\bar{\varphi}
},
\label{eq:non-canonical_sol_gauged}
\end{align}
where $\eta$ is a phase factor.
The solution in Eq.~\eqref{eq:non-canonical_sol_gauged} is just the gauge
covariantized counterpart of that in Eq.~\eqref{eq:non-canonical_sol}.
The fields satisfy the gauge covariantized constraint \eqref{eq:noncanonical_constraint}.
\begin{align}
|F^0|^2 =
- \frac{K_{\varphi \bar{\varphi}}}{2 \Lambda} + D_m \varphi D^m
\bar{\varphi} \ge 0.
\label{eq:noncanonical_constraint_gauged}
\end{align}
Then the bosonic part of the on-shell Lagrangian in the non-canonical
branch is
\begin{align}
\mathcal{L}_{b} =& - \frac{1}{4} F_{mn} F^{mn}
- \frac{g^2}{2}
\left(
\frac{1}{2} \bar{\varphi} \frac{\partial K}{\partial \bar{\varphi}}
+ \frac{1}{2} \frac{\partial K}{\partial \varphi} \varphi
- \kappa
\right)^2
\notag \\
& \ + \Lambda (|D_m \varphi D^m \varphi|^2 - (D_m \varphi D^m \bar{\varphi})^2)
- \frac{(K_{\varphi \bar{\varphi}})^2}{4 \Lambda},
\label{eq:non-canonical_Lag}
\end{align}
where $F_{mn} = \partial_m A_n - \partial_n A_m$ is the field strength
of the $U(1)$ gauge field.
An example of the Lagrangian \eqref{eq:non-canonical_Lag} is a
supersymmetric generalization of the gauged BPS baby Skyrme model
\cite{Adam:2012pm} whose potential term is determined by the K\"ahler
potential $K$ through the D-term and the term $K^2_{\varphi
\bar{\varphi}}/\Lambda$.
In this case, the explicit function $\Lambda$ is given
in Eq.~\eqref{eq:Lambda_bS}.
\section{BPS states in Minkowski space}\label{sec:BPS-Minkowski}
In this section, we investigate BPS configurations of the model
\eqref{eq:gauged_model} in four-dimensional Minkowski space.
BPS configurations in supersymmetric theories preserve parts of
supersymmetry. BPS equations are obtained from the condition that the
on-shell supersymmetry transformation of the fermions in the model
vanishes, $\delta^{\text{on}}_{\xi} \psi_{\alpha} =
\delta^{\text{on}}_{\xi} \lambda_{\alpha} = 0$.
Here $\delta^{\text{on}}_{\xi}$ ($\delta^{\text{off}}_{\xi}$) is the on-shell
(off-shell) supersymmetry transformation by the parameters
$\xi_{\alpha}$, $\bar{\xi}^{\dot{\alpha}}$.
The off-shell supersymmetry variation of the fermions $\psi$, $\lambda$ is
\begin{align}
\delta^{\text{off}}_{\xi} \psi_{\alpha}^{ia} =& \ \sqrt{2} i
(\sigma^m)_{\alpha \dot{\alpha}} \bar{\xi}^{\dot{\alpha}} D_m
\varphi^{ia} + \sqrt{2} \xi_{\alpha} F^{ia}, \\
\delta^{\text{off}}_{\xi} \lambda_{\alpha} =& \
i \xi_{\alpha} D + (\sigma^{mn})_{\alpha} {}^{\beta} \xi_{\beta}
F_{mn}.
\end{align}
The on-shell supersymmetry transformations are obtained by substituting
the solutions of the auxiliary fields equations into $F$ and $D$.
Therefore they have distinct structures in the canonical and
non-canonical branches.
In Ref.~\cite{Nitta:2014pwa}, we studied BPS equations in the non-gauged higher
derivative models given in Eq.~\eqref{eq:off-shell_Lagrangian} where no gauge fields are present.
We derived the 1/2 BPS domain wall and lump equations in the canonical branch.
These equations are the same for the ordinary (without higher derivative
term) theory. We calculated the BPS bound of the on-shell action
associated with these configurations. Then we found that the BPS bound is
given by the ordinary tension of the domain wall and the lump
(topological) charge, respectively.
Namely, higher derivative effects are totally
canceled in the 1/2 BPS domain wall and lump.
In the non-canonical branch, we found 1/4 BPS configurations for the
domain wall junctions and lump type solitons.
The equation for the domain wall junction receives higher derivative
contributions while the associated BPS bound of the Lagrangian is expressed by the ordinary domain
wall tension and the junction charge.
For the lump type soliton, it is considered as a compacton which is a
soliton with a compact support. Indeed, when the K\"ahler potential $K$ and
$\Lambda$ are chosen appropriately, the 1/4 BPS equation in Ref.~\cite{Nitta:2014pwa} have compacton type solutions \cite{AdQuSaGuWe}.
In the following subsections, we proceed with the analysis of the BPS
configurations for the gauged higher derivative chiral models given in Eq.~\eqref{eq:gauged_model}.
For the ordinary $\mathcal{N} = 1$ supersymmetric gauge theory with fundamental
matters in Minkowski space, there are BPS vortices which are codimension
two solitons.
We study codimension-two vortex configurations
in the canonical and non-canonical branches of the model \eqref{eq:gauged_model}.
\subsection{Canonical branch}
We start from the flat K\"ahler potential $K = \Phi^{\dagger \bar{i}} e^{2gV}
\Phi^i$ and look for the vortex configurations.
The static ansatz for the vortex is given by
\begin{align}
\varphi^{ia} = \varphi^{ia} (x^1,x^2), \qquad F_{12} \not= 0,
\end{align}
where the other components of $F_{mn}$ all vanish.
In the canonical branch, we have the solution $F^{ia} = 0$.
Then, the on-shell supersymmetry variations of the fermions are
\begin{align}
\delta \psi^i =& \ \sqrt{2}i
\left(
\begin{array}{cc}
(D_1 - i D_2) \varphi^i \bar{\xi}^{\dot{2}} \\
(D_1 + i D_2) \varphi^i \bar{\xi}^{\dot{1}}
\end{array}
\right) = 0, \\
\delta \lambda =& \
- i
\left(
\begin{array}{c}
\xi_1 F_{12} - \xi_1 D\\
- \xi_2 F_{12} - \xi_2 D
\end{array}
\right) = 0,
\end{align}
where $D^{\hat{a}} = - g
\left(
\bar{\varphi}_c^{\bar{i}} (T^{\hat{a}})^c {}_d \varphi^{id}
- \kappa \delta^{\hat{a}} {}_0
\right)$.
The vortex configuration is obtained by imposing the following
projection condition on the supersymmetry parameter:
\begin{align}
\frac{1}{2} (\sigma^1 + i \sigma^2) \bar{\xi} = 0.
\label{eq:half_projection}
\end{align}
This is equivalent to the condition $\bar{\xi}^{\dot{2}} = \xi_1 = 0$ so
that the projection \eqref{eq:half_projection} leaves a half of
$\mathcal{N} = 1$ supersymmetry.
Therefore, we obtain the following BPS equations:
\begin{align}
\bar{D}_z \varphi^{ia} = 0, \qquad F_{12}^{\hat{a}} - g
\left(
\bar{\varphi}_c^{\bar{i}} (T^{\hat{a}})^c {}_d \varphi^{id} - \kappa \delta^{\hat{a}} {}_0
\right) = 0.
\label{eq:canonical_half_BPS_vortex}
\end{align}
Here we have defined $z \equiv \frac{1}{2} (x^1 + i x^2)$ and
$D_z \equiv D_1 - i D_2$, $\bar{D}_z \equiv D_1 + i D_2$.
This is just the ordinary 1/2 BPS Abelian (ANO) or non-Abelian vortex equation
\cite{Hanany:2003hp}.
Now we calculate the Lagrangian bound\footnote{When the Lagrangian \eqref{eq:canonical_Lag}
contains higher order time derivatives of fields, the positive energy Hamiltonian
is not defined in general \cite{Ostrogradski}. Therefore, we calculate the Lagrangian bound,
rather than the energy bound, for the BPS configurations.}
associated with the BPS equations \eqref{eq:canonical_half_BPS_vortex}. Using the first
condition in Eq.~\eqref{eq:canonical_half_BPS_vortex}, we find the higher
derivative terms vanish:
\begin{align}
& \Lambda_{ik\bar{j}\bar{l}}
(D^m \bar{\varphi}_a^{\bar{j}} D^n \varphi^{ia}) (D_m \bar{\varphi}_b^{\bar{l}} D_n
\varphi^{kb})
\notag \\
=& \ \frac{1}{4} \Lambda_{ik\bar{j}\bar{l}}
\left(
D_z \varphi^{ia} \bar{D}_z \varphi^{kb} + \bar{D}_z \varphi^{ia} D_z \varphi^{kb}
\right)
\left(
D_z \bar{\varphi}_a^{\bar{j}} \bar{D}_z \bar{\varphi}_b^{\bar{l}} + \bar{D}_z
\bar{\varphi}_a^{\bar{j}} D_z \bar{\varphi}_b^{\bar{l}}
\right)
\notag \\
=& \ 0.
\end{align}
Then, by using the first and the second equations in
\eqref{eq:canonical_half_BPS_vortex},
we obtain the Lagrangian bound
\begin{align}
\mathcal{L} =& \ \kappa g F^0_{12}.
\label{eq:canonical_vortex_bound}
\end{align}
Here $F^0_{12}$ is the $U(1)$ flux density in the $(x^1,x^2)$-plane.
Integrating it in the $(x^1,x^2)$-plane, we obtain the ordinary vortex
topological charge. Therefore, in the canonical branch,
all the higher derivative corrections to the
1/2 BPS vortex are canceled in both the equations
\eqref{eq:canonical_half_BPS_vortex} and the Lagrangian bound
\eqref{eq:canonical_vortex_bound}.
This is a conceivable result since the BPS nature is determined by the
supersymmetry algebra. The model \eqref{eq:canonical_Lag} includes
higher derivative terms but supersymmetry is manifestly realized.
Then we expect that the BPS structure is protected against higher
derivative corrections.
A typical example is the world-volume theory of D-branes where BPS states in
super Yang-Mills theory linearize the non-Abelian DBI action canceling the higher derivative corrections
\cite{Brecher:1998tv}.
While the higher derivative corrections exist in
the non-Abelian vortex effective theory,
the higher derivative effects are canceled in the BPS equation and energy of ${\mathbb C}P^{N-1}$ lumps
inside a non-Abelian vortex
\cite{Eto:2012qda}.
We also comment that this is the same conclusion discussed in
the domain wall and lump in the non-gauged chiral models \cite{Nitta:2014pwa}.
We next consider the general gauge invariant K\"ahler potential of the
form $K(\Phi^{\dagger}, \Phi, V) = \frac{1}{2} (K(\Phi^{\dagger}
e^{2gV}, \Phi) + K(\Phi^{\dagger}, e^{2gV} \Phi))$.
The BPS equations for the 1/2 BPS projection condition \eqref{eq:half_projection}
are
\begin{align}
\bar{D}_z \varphi^{ia} = 0, \qquad F_{12}^{\hat{a}} - \frac{g}{2}
\left(
\bar{\varphi}_c^{\bar{j}} (T^{\hat{a}})^c {}_d \frac{\partial K}{\partial \bar{\varphi}_d^{\bar{j}}}
+
\frac{\partial K}{\partial \varphi^{ic}} (T^{\hat{a}})^c {}_d \varphi^{id}
- \kappa \delta^{\hat{a}} {}_0
\right) = 0.
\label{eq:canonical_half_BPS_general_K}
\end{align}
By using the first condition in
\eqref{eq:canonical_half_BPS_general_K}, we find that the higher derivative terms
vanish. Then, the Lagrangian bound associated with the BPS condition
\eqref{eq:canonical_half_BPS_general_K} is
\begin{align}
\mathcal{L} =& \ - \frac{1}{2} \frac{\partial^2 K}{\partial
\bar{\varphi}_a^{\bar{j}} \partial \varphi^{ib}}
\bar{D}_z \bar{\varphi}_a^{\bar{j}} D_z \varphi^{ib} -
\frac{g^2}{2}
\left(
\frac{1}{2} \bar{\varphi}_c^{\bar{j}} (T^{\hat{a}})^c {}_d \frac{\partial K}{\partial
\bar{\varphi}_d^{\bar{j}}}
+ \frac{1}{2} \frac{\partial K}{\partial \varphi^{ic}} (T^{\hat{a}})^c {}_d
\varphi^{id} - \kappa \delta^{\hat{a}} {}_0
\right)^2
- \frac{1}{2} (F^{\hat{a}}_{12})^2
\notag \\
=& \
- \varepsilon^{st} \partial_s \mathcal{N}_t + \kappa g
F^0_{12},
\label{eq:canonical_bound2}
\end{align}
where we have defined the following quantity
\begin{align}
\mathcal{N}_s =
\frac{i}{2}
\left(
\frac{\partial K}{\partial \bar{\varphi}_a^{\bar{j}}} D_s \bar{\varphi}_a^{\bar{j}} -
\frac{\partial K}{\partial \varphi^{ia}} D_s \varphi^{ia}
\right), \qquad (s,t = 1,2).
\end{align}
The first term in Eq.~\eqref{eq:canonical_bound2} is the gauge covariant
generalization of the lump charge density.
Then the Lagrangian bound is given by the sum of the lump and the
vortex charge densities.
The BPS configurations whose energy bound is given by
Eq.~\eqref{eq:canonical_bound2} have been studied in the gauged non-linear
sigma models where higher derivative corrections are absent \cite{Schroers:1995he,Nitta:2011um}.
In there, the configurations admit fractional lump charges.
Once again, we find that all the higher derivative effects are canceled on the 1/2
BPS states \eqref{eq:canonical_half_BPS_general_K}.
\subsection{Non-canonical branch}
We next consider BPS equations in the non-canonical branch.
The Lagrangian is given by \eqref{eq:non-canonical_Lag} where the gauge
group is $U(1)$ and $K = \Phi^{\dagger} e^{2gV} \Phi$.
The non-zero solution of the auxiliary field $F^0$ is given in Eq.~\eqref{eq:non-canonical_sol_gauged}.
The supersymmetry variation of the fermions is
\begin{align}
\delta \psi =& \ \sqrt{2}
\left(
\begin{array}{cc}
i (D_1 - i D_2) \varphi \bar{\xi}^{\dot{2}} + \xi_1 F^0
\\
i (D_1 + i D_2) \varphi \bar{\xi}^{\dot{1}} + \xi_2 F^0
\end{array}
\right) = 0, \\
\delta \lambda =& \
- i
\left(
\begin{array}{c}
\xi_1 F_{12} - \xi_1 D \\
- \xi_2 F_{12} - \xi_2 D
\end{array}
\right)
= 0.
\end{align}
Since the auxiliary field $F^0$ is non-zero in the non-canonical branch,
the 1/2 BPS projection \eqref{eq:half_projection} gives the equations
\eqref{eq:canonical_half_BPS_vortex} and the following additional condition:
\begin{align}
F^0 = e^{i \eta}
\sqrt{
- \frac{1}{2 \Lambda} + D_m \varphi D^m
\bar{\varphi} } = 0.
\label{eq:noncanonical_F_condition}
\end{align}
Solutions that satisfy the ordinary vortex equations
\eqref{eq:canonical_half_BPS_vortex} do not satisfy the condition in Eq.~\eqref{eq:noncanonical_F_condition} for general $\Lambda$.\footnote{However, when $\Lambda$ is chosen appropriately, it is
possible that the ordinary vortex solution satisfies the condition
\eqref{eq:noncanonical_F_condition}.
}
We therefore look for another BPS condition.
A natural candidate is the gauge covariantized generalization of the BPS lumps
in the non-canonical branch.
Following the BPS lumps studied in Ref.~\cite{Nitta:2014pwa}, we consider the 1/4 BPS projection conditions,
\begin{align}
\frac{1}{2} (\sigma^1 + i \sigma^2)_{\alpha \dot{\alpha}}
\bar{\xi}^{\dot{\alpha}} = 0, \qquad
\frac{1}{2} (\sigma^1 - i \sigma^2)_{\alpha \dot{\alpha}}
\bar{\xi}^{\dot{\alpha}} = i \xi_{\alpha}.
\end{align}
Then, from the variation of the fermions, we find
a set of 1/4 BPS equations:
\begin{align}
\bar{D}_z \varphi =- i e^{ i \eta}
\sqrt{
- \frac{1}{2 \Lambda} + \frac{1}{2}
(D_z \bar{\varphi} \bar{D}_z \varphi + \bar{D}_z \bar{\varphi} D_z \varphi)
}, \qquad
F^0_{12} - g
(\bar{\varphi} \varphi - \kappa) = 0.
\label{eq:gauged_compacton_flat}
\end{align}
\eqref{eq:noncanonical_constraint_gauged}.
The first equation is the gauge covariantized generalization of the compacton-type equation while the second equation is that for the ANO vortex.
We call solutions to these equations as higher derivative vortices.
These equations may admit a vortex with a compact
support for the scalar fields (that we may call a compact vortex).
See Ref.~\cite{Adam:2008rf} for a vortex with a compact support
which are non-BPS in non-supersymmetric theories.
We then calculate the Lagrangian bound associated with the BPS condition
\eqref{eq:gauged_compacton}.
Using the first condition in Eq.~\eqref{eq:gauged_compacton}, we obtain the following relation,
\begin{align}
\Lambda
\left\{
(D_m \varphi D^m \varphi) (D_n \bar{\varphi} D^n \bar{\varphi}) - (D_m
\varphi D^m \bar{\varphi})^2
\right\} =
- \frac{1}{4} \Lambda
\left(
\bar{D}_z \varphi D_z \bar{\varphi} - D_z \varphi \bar{D}_z \bar{\varphi}
\right)^2 = - \frac{1}{4\Lambda}.
\end{align}
By using this relation and the second equation in Eq.~\eqref{eq:gauged_compacton_flat},
we calculate the BPS bound of the Lagrangian as
\begin{align}
\mathcal{L} = \kappa g F^0_{12}.
\end{align}
This is the topological vortex charge density.
Therefore the equations \eqref{eq:gauged_compacton_flat} correspond to
the higher derivative generalization of the ANO vortex rather than the compacton.
We comment that the higher derivative terms cancel out in the Lagrangian
bound even in the non-canonical branch. However, the BPS equation
\eqref{eq:gauged_compacton} receives higher derivative corrections.
The situation is quite similar to the 1/4 BPS domain wall junction and
the compacton in the non-gauged model \cite{Nitta:2014pwa}.
In there, there are higher derivative corrections to the BPS equations.
However, the bounds for the BPS states do not receive higher derivative corrections.
Now we consider the general gauge invariant K\"ahler potential.
A set of 1/4 BPS equations is obtained as
\begin{align}
\bar{D}_z \varphi =- i e^{ i \eta}
\sqrt{
- \frac{K_{\varphi \bar{\varphi}}}{2 \Lambda} + \frac{1}{2}
(D_z \bar{\varphi} \bar{D}_z \varphi + \bar{D}_z \bar{\varphi} D_z \varphi)
}, \qquad
F^0_{12} - \frac{g}{2}
\left(\bar{\varphi} \frac{\partial K}{\partial \bar{\varphi}}
+ \frac{\partial K}{\partial \varphi} \varphi
- \kappa\right) = 0.
\label{eq:gauged_compacton}
\end{align}
Using the first condition in Eq.~\eqref{eq:gauged_compacton}, we find that
the higher derivative terms cancel out in the Lagrangian bound.
The result is
\begin{align}
\mathcal{L} = - \varepsilon^{st} \partial_s \mathcal{N}_t + \kappa g
F^0_{12}, \quad (s,t = 1,2),
\end{align}
where
\begin{align}
\mathcal{N}_s = \frac{i}{2} (K_{\bar{\varphi}} D_s \bar{\varphi} -
K_{\varphi} D_s \varphi).
\end{align}
This is precisely the sum of the lump and the vortex charges.
We therefore expect that the equations \eqref{eq:gauged_compacton}
describe composite states of the higher derivative ANO vortex and
the BPS baby Skyrmions,
or simply gauged BPS baby Skyrmions.
Solutions should carry fractional baby Skyrmion charges
as for the vortex-lumps in the canonical branch.
BPS states in Minkowski space are summarized in Table.~\ref{tb:HDBPS_Minkowski}
\begin{table}[tb]
\begin{center}
\begin{tabular}{|l||l|l|l|}
\hline
& SYM + SUSY NLSM & canonical & non-canonical \\
\hline \hline
L type & 1/2 BPS lump & 1/2 BPS lump & 1/4 BPS baby-Skyrmion \\
\hline
V type & 1/2 BPS vortex & 1/2 BPS vortex & 1/4 BPS HD vortex \\
\hline
VL type & 1/2 BPS vortex-lump & 1/2 BPS vortex-lump & 1/4 BPS vortex-baby Skyrmion \\
\hline
\end{tabular}
\caption{BPS states in the gauged higher derivative (HD) chiral model
and super Yang-Mills with gauged non-linear sigma model (SUSY NLSM).
Theories are defined in Minkowski space. The BPS states are classified
into the lump (L) type, the vortex (V) type and the vortex-lump (VL) type.}
\label{tb:HDBPS_Minkowski}
\end{center}
\end{table}
\section{BPS states in Euclidean space}\label{sec:BPS-Euclidean}
In four-dimensional Euclidean space, one can consider codimension-four
objects. Typical examples are the Yang-Mills instantons and the instantons
trapped inside (intersecting) vortices.
In this section, we study codimension-four BPS configurations of the higher derivative model
\eqref{eq:gauged_model} in Euclidean space.
The off-shell supersymmetry variations of the fermions in Euclidean space
is
\begin{align}
\delta_{\xi} \psi_{\alpha}^i =& \ \sqrt{2} i (\sigma^m_{\text{E}})_{\alpha
\dot{\alpha}} \bar{\xi}^{\dot{\alpha}} D_m \varphi^i + \sqrt{2}
\xi_{\alpha} F^i, \\
\delta_{\xi} \lambda_{\alpha} =& \ i \xi_{\alpha} D +
(\sigma^{mn}_{\text{E}})_{\alpha} {}^{\beta} \xi_{\beta} F_{mn},
\end{align}
where $m = 1,2,3,4$ and the sigma matrices in the Euclidean space are defined by
\begin{align}
(\sigma_{\text{E}}^{m})_{\alpha \dot{\alpha}} = (i \vec{\tau}, \mathbf{1}), \quad
(\bar{\sigma}_{\text{E}}^m)^{\dot{\alpha} \alpha} = (- i \vec{\tau}, \mathbf{1}).
\end{align}
Here, $\vec{\tau}$ are the Pauli matrices.
The explicit supersymmetry variation of the fermions are found in Appendix
\ref{sec:SUSY_variation}.
We note that in Euclidean space, $\xi^{\alpha}$ and
$\bar{\xi}^{\dot{\alpha}}$ are
independent from each other and they are not complex conjugate anymore.
Then it is possible to consider BPS
projections that drop a chiral half of $\mathcal{N} = 1$ supersymmetry $\xi^{\alpha} =
0$, $\bar{\xi}^{\dot{\alpha}} \not= 0$.
Indeed, the standard Yang-Mills
instantons exist in our model \eqref{eq:gauged_model},
that
preserve the (anti)chiral half of supersymmetry and
are 1/2 BPS configurations.
Since BPS states with codimensions less than four in Euclidean space are the same as those
in Minkowski space,
discussed in the previous section,
we focus on codimension-four BPS states
in the higher
derivative model in the following subsections.
\subsection{Canonical branch}
We start from the Lagrangian \eqref{eq:gauged_model} where the K\"ahler
potential is flat.
We consider the 1/4 BPS projection condition\footnote{The other combinations, for example,
$\xi_2 \not= 0, \xi_1 = \bar{\xi}^{\dot{1}} = \bar{\xi}^{\dot{2}} = 0$
and so on give essentially the same form of the BPS equations.}
\begin{align}
\bar{\xi}^{\dot{1}} \not= 0, \quad \bar{\xi}^{\dot{2}} = \xi_1 = \xi_2 =
0.
\label{eq:qBPS_euclidean}
\end{align}
Then from the supersymmetry variation of the fermions,
we obtain the following set of 1/4 BPS equations in the canonical branch:
\begin{align}
& \bar{D}_z \varphi^i = \bar{D}_w
\varphi^i = 0, \quad F^{\hat{a}}_{12} - F^{\hat{a}}_{34} = g (\bar{\varphi}_c^{\bar{i}} (T^{\hat{a}})^c
{}_d \varphi^{id} - \delta^{\hat{a}} {}_0 \kappa) ,
\notag \\
& F^{\hat{a}}_{13} + F^{\hat{a}}_{24} = F^{\hat{a}}_{14} - F^{\hat{a}}_{23} = 0,
\label{eq:vvi_canonical}
\end{align}
where we have defined complex coordinates and derivatives
with respect to them by
\begin{align}
& z \equiv \frac{1}{2} (x^1 + i x^2), \qquad w \equiv \frac{1}{2} (x^4 + i x^3),
\notag \\
& D_z \equiv D_1 - i D_2, \qquad D_w \equiv D_4 - i D_3.
\end{align}
Using the condition $\bar{D}_z \varphi^i = \bar{D}_w \varphi^i = 0$, we find
that the higher derivative terms vanish for the BPS configuration
\eqref{eq:vvi_canonical},
\begin{align}
& \Lambda_{ik\bar{j}\bar{l}}
(D_m \bar{\varphi}_a^{\bar{j}} D^m \bar{\varphi}_b^{\bar{l}})
(D_n \varphi^{ib} D^n \varphi^{kb})
\notag \\
=& \ \frac{1}{4} \Lambda_{ik\bar{j}\bar{l}}
\left(
D_z \varphi^{ia} \bar{D}_z \varphi^{kb} + \bar{D}_z \varphi^{ia} D_z \varphi^{kb} +
D_w \varphi^{ia} \bar{D}_w \varphi^{kb} + \bar{D}_w \varphi^{ia} D_w \varphi^{kb}
\right)
\notag \\
& \qquad \times
\left(D_z \bar{\varphi}_a^{\bar{j}} \bar{D}_z \bar{\varphi}_b^{\bar{l}} +
\bar{D}_z \bar{\varphi}_a^{\bar{j}} D_z \bar{\varphi}_b^{\bar{l}} + D_w \bar{\varphi}_a^{\bar{j}}
\bar{D}_w \bar{\varphi}_b^{\bar{l}} + \bar{D}_w \bar{\varphi}_a^{\bar{j}} D_w \bar{\varphi}_b^{\bar{l}}
\right)
\notag \\
=& \ 0.
\end{align}
Then the BPS bound of the Lagrangian associated with the configuration \eqref{eq:vvi_canonical} is
\begin{align}
\mathcal{L}_{\text{E}} = - \kappa g (F^0_{12} - F^0_{34}) + \frac{1}{4k}
\mathrm{Tr} [F_{mn} \tilde{F}^{mn}],
\label{eq:EucLagrangian_bound_canonical}
\end{align}
where $\tilde{F}_{mn} = \frac{1}{2} \varepsilon_{mnpq} F^{pq}$ is the
Hodge dual of the gauge field strength $F_{mn}$.
We note that the sign of the Lagrangian in
Euclidean space is flipped from that in Minkowski space.
The first and the second terms in
\eqref{eq:EucLagrangian_bound_canonical} correspond to the vortex charge densities
in the $(x^1,x^2)$ and $(x^3,x^4)$-planes, respectively. The last term is the instanton charge density.
Therefore solutions to Eq.~\eqref{eq:vvi_canonical}
are the Yang-Mills instantons trapped inside
intersecting vortices.
A set of these equations were first found in Refs.~\cite{Hanany:2004ea,Eto:2004rz,Eto:2006pg,Fujimori:2008ee} for
supersymmetric theories with eight supercharges
without higher derivative terms,
and configurations were shown to be 1/4 BPS states
\cite{Eto:2004rz}.
Solutions can be constructed in terms of the moduli matrix
\cite{Eto:2006pg} and are mathematically
characterized in terms of
amoeba and tropical geometry
\cite{Fujimori:2008ee}.
We next consider the general gauge invariant K\"ahler potential.
In this case, a set of 1/4 BPS equations that we obtain is
\begin{align}
& \bar{D}_z \varphi^i = \bar{D}_w \varphi^i = 0,
\quad F^{\hat{a}}_{12} - F^{\hat{a}}_{34} =
\frac{g}{2}
\left(
\bar{\varphi}_c^{\bar{j}} (T^{\hat{a}})^c {}_d \frac{\partial K}{\partial \bar{\varphi}_d^{\bar{j}}}
+ \frac{\partial K}{\partial \varphi^{ic}} (T^{\hat{a}})^c {}_d \varphi^{id} - \kappa
\delta^{\hat{a}} {}_0 \right), \notag \\
& F^{\hat{a}}_{13} + F^{\hat{a}}_{24} = F^{\hat{a}}_{14} - F^{\hat{a}}_{23} = 0.
\label{eq:qBPS_canonical_generalK}
\end{align}
Using Eqs.~\eqref{eq:qBPS_canonical_generalK},
the BPS bound of the Lagrangian can be evaluated as
\begin{align}
\mathcal{L}_{\text{E}} = \varepsilon^{st} \partial_s \mathcal{N}_t -
\varepsilon^{s' t'} \partial_{s'} \mathcal{N}_{t'} - \kappa g (F^0_{12}
- F^0_{34}) + \frac{1}{4k} \mathrm{Tr} [F_{mn} \tilde{F}^{mn}],
\end{align}
where $s,t = 1,2$ and $s',t' = 3,4$.
The first and the second terms correspond to the gauge covariantized
extension of the lump charge densities in the $(x^1,x^2)$ and
$(x^3,x^4)$-planes, respectively.
The third and the fourth terms are vortex charge densities in the
$(x^1,x^2)$ and $(x^3,x^4)$-planes, respectively, and the last term is the Yang-Mills
instanton charge density.
Note that when the gauge field vanishes, the configuration corresponds to
the intersecting topological vortex-lumps in the $(x^1,x^2)$- and
$(x^3,x^4)$-planes.
\subsection{Non-canonical branch}
Finally, we consider the non-canonical branch where the gauge group is
$U(1)$.
The 1/4 BPS configurations in the two-dimensional subspaces are
constructed by the same ways discussed in the Minkowski case.
We now look for codimension-four BPS states.
Since the solution of the auxiliary field is not zero in the non-canonical branch,
the 1/4 BPS projection \eqref{eq:qBPS_euclidean} gives the BPS equations
\eqref{eq:vvi_canonical} and the additional condition $F^0 = 0$
\eqref{eq:noncanonical_F_condition}.
As in the case of the Minkowski space,
the solutions to the equations \eqref{eq:vvi_canonical} do not
satisfy the condition \eqref{eq:noncanonical_F_condition} for general
$\Lambda$. Therefore the 1/4 BPS configurations associated with the
projection \eqref{eq:qBPS_euclidean} do not exist in the non-canonical
branch. BPS states in Euclidean space are summarized in Table.~\ref{tb:HDBPS_Euclidean}
\begin{table}[tb]
\begin{center}
\begin{tabular}{|l||l|l|l|}
\hline
& SYM + SUSY NLSM & canonical & non-canonical \\
\hline \hline
L type & 1/2 BPS $\text{L}_{12}$ & 1/2 BPS $\text{L}_{12}$ & 1/4 BPS
$\text{bS}_{12}$ \\
\hline
V type & 1/2 BPS $\text{V}_{12}$ & 1/2 BPS $\text{V}_{12}$ & 1/4 BPS
$\text{HDV}_{12}$ \\
\hline
VL type & 1/2 BPS $\text{VL}_{12}$ & 1/2 BPS $\text{VL}_{12}$ & 1/4 BPS
$\text{HDVbS}_{12}$ \\
\hline
V-V-I type & 1/4 BPS $\text{V}_{12}$-$\text{V}_{34}$-I &
1/4 BPS $\text{V}_{12}$-$\text{V}_{34}$-I & no \\
\hline
VL-VL-I type & 1/4 BPS $\text{VL}_{12}$-$\text{VL}_{34}$-I & 1/4 BPS $\text{VL}_{12}$-$\text{VL}_{34}$-I & no \\
\hline
L-L type & 1/4 BPS $\text{L}_{12}$-$\text{L}_{34}$ & 1/4 BPS $\text{L}_{12}$-$\text{L}_{34}$ & no \\
\hline
\end{tabular}
\caption{BPS states in the gauged higher derivative (HD) chiral model
and super Yang-Mills with gauged non-linear sigma model. Theories are
defined in Euclidean space.
Here L,V,I,VL,HDV, bS, and HDVbS stand for lumps, vortices, instantons,
vortex-lumps, higher derivative vortices, BPS baby Skyrmions, and
higher derivative vortex-BPS baby Skyrmions, respectively.
The subscript stands for subspaces that the soliton is defined.
}
\label{tb:HDBPS_Euclidean}
\end{center}
\end{table}
\section{Summary and discussion} \label{sec:summary}
In this paper,
we have classified BPS states in ${\cal N}=1$ supersymmetric gauge theories
coupled with higher derivative chiral models
in four Minkowski and Euclidean dimensions.
We have found canonical and non-canonical branches
corresponding to solutions $F=0$ and $F\neq 0$
of auxiliary field equations, respectively.
1/2 BPS states
in theories without higher derivative terms
remain 1/2 BPS in the canonical branch
and the corresponding BPS states in the non-canonical branch
are 1/4 BPS states.
1/4 BPS states
in theories without higher derivative terms
remain 1/4 BPS in the canonical branch
but there are no corresponding BPS states in the non-canonical branch.
We have obtained
1/2 BPS equations for
an ANO vortex,
a non-Abelian vortex, a lump,
and a vortex-lump in the canonical branch,
and 1/4 BPS higher derivative generalization of the ANO vortices
in the non-canonical branch.
In four Euclidean dimensions, we have obtained
the 1/4 BPS Yang-Mills instantons trapped inside a non-Abelian vortex,
and 1/4 BPS intersecting vortices or
vortex-lump intersections with instanton charges
in the canonical branch and
no codimension-four BPS states in the non-canonical branch.
While we have given
the superfield Lagrangian of gauged multi-component chiral models,
we have been able to obtain on-shell Lagrangian only for the cases of
a single component because of difficulty solving the equations of motion
for the auxiliary fields for the multi-component cases.
Obtaining on-shell Lagrangians for gauged or non-gauged
multi-component chiral models,
in particular in the presence of an isometry large enough,
remains a future problem.
Our method will give a simple way to construct
higher derivative non-linear sigma models
on K\"ahler manifolds
by gauging chiral fields with
flat target spaces for which
auxiliary field equations of motions are easy to solve.
In the strong gauge coupling limit,
vector superfields do not have gauge kinetic terms
becoming auxiliary superfields, and can be eliminated
by their equations of motion.
This procedure is known as the K\"ahler quotients,
see Ref.~\cite{Higashijima:1999ki} for constructions of
hermitian symmetric spaces.
Thus, it will be possible to construct
higher-derivative non-linear sigma models
on hermitian symmetric spaces,
as a generalization of the Faddeev-Skyrme
${\mathbb C}P^1$ model.
In this paper, we have not introduced superpotentials
while we introduced them for non-gauged chiral models
in our previous paper \cite{Nitta:2014pwa}.
In the presence of a superpotential, there are more
varieties of BPS topological solitons such as
domain walls
\cite{Dvali:1996xe}
in $U(N)$ gauge theories \cite{Isozumi:2004jc},
domain wall junctions
\cite{GiTo,Oda:1999az}
or networks \cite{Eto:2005cp}, and
vortices ending on or stretched between domain walls
\cite{Gauntlett:2000de,Isozumi:2004vg}.
In these cases, the auxiliary field equation can be solved
at most perturbatively even for a single component,
as was so for non-gauged chiral models \cite{Nitta:2014pwa}.
We also comment that in our gauged model, $\Lambda_{ik\bar{j}\bar{l}}$
does not contain space-time derivatives of the chiral superfields,
unlike the non-gauged cases for which it is possible as for
the supersymmetric Dirac-Born-Infeld action in
Eq.~\eqref{eq:Lambda_DBI} and
the supersymmetric Faddeev-Skyrme model in Eq.~(\ref{eq:FSmodel}).
A simple gauge covariant generalization of the form
\eqref{eq:Lambda_DBI} or (\ref{eq:FSmodel}) does not provide supersymmetric interactions of the vector superfield.
It is interesting to introduce the gauge covariant derivatives of $\Phi$
in a supersymmetric way in the K\"ahler tensor $\Lambda_{ik\bar{j}\bar{l}}$,
in order to construct a gauged Dirac-Born-Infeld action
\cite{Sasaki:2009ij} or a gauged Faddeev-Skyrme model.
In Ref.~\cite{Eto:2005sw},
1/2, 1/4, and 1/8 BPS states
were classified in
${\cal N}=2$ supersymmetric field theories
without higher derivative terms.
Extension to ${\cal N}=2$ supersymmetric field theories
with higher derivative terms should be an interesting future problem.
In particular,
1/4 BPS states in the canonical branch
may have 1/8 BPS state counterparts
in the non-canonical branch.
While off-shell supersymmetry for eight supercharges
is a hard task because one needs harmonic superfield
or projective superfield formalisms,
partially off-shell supersymmetry that
BPS solitons preserve can be used
to construct an effective theory of BPS solitons \cite{Eto:2006uw}.
Extension to supergravity is also interesting
for application to cosmology such as the ghost condensations
and the Galileon inflation models in supersymmetric theories
along the line in Refs.~\cite{KhLeOv}--\cite{FaKe}.
\subsection*{Acknowledgments}
The work of M.\ N.\ is supported in part by Grant-in-Aid for
Scientific Research (No. 25400268)
from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
The work of S.~S. is supported in part by Kitasato University Research Grant for Young
Researchers.
\begin{appendix}
\section{Notation and conventions}\label{sec:notation}
We use the convention in the textbook of
Wess and Bagger \cite{Wess:1992cp}.
The component expansion of the $\mathcal{N} = 1$ chiral superfield in
the $x$-basis is
\begin{equation}
\Phi (x, \theta, \bar{\theta}) = \varphi
+ i \theta \sigma^m \bar{\theta} \partial_m \varphi + \frac{1}{4}
\theta^2 \bar{\theta}^2 \Box \varphi + \theta^2 F,
\end{equation}
where
only the bosonic components are presented.
The supercovariant derivatives are defined as
\begin{eqnarray}
D_{\alpha} = \frac{\partial}{\partial \theta^{\alpha}} + i
(\sigma^m)_{\alpha \dot{\alpha}} \bar{\theta}^{\dot{\alpha}}
\partial_m, \quad
\bar{D}_{\dot{\alpha}} = - \frac{\partial}{\partial
\bar{\theta}^{\dot{\alpha}}} - i \theta^{\alpha} (\sigma^m)_{\alpha
\dot{\alpha}} \partial_m.
\end{eqnarray}
The sigma matrices are $\sigma^m = (\mathbf{1}, \vec{\tau})$.
Here $\vec{\tau} = (\tau^1, \tau^2, \tau^3)$ are Pauli matrices.
The bosonic components of the supercovariant derivatives of $\Phi^i$ are
\begin{align}
D^{\alpha} \Phi^i D_{\alpha} \Phi^j =& \
- 4 \bar{\theta}^2 \partial_m \varphi^i \partial^m \varphi^j
+ 4 i (\theta \sigma^m \bar{\theta}) (\partial_m \varphi^i F^j + F^i
\partial_m \varphi^j)
- 4 \theta^2 F^i F^j
\notag \\
& \ + 2 \theta^2 \bar{\theta}^2
\left(
\Box \varphi^i F^j + F^i \Box \varphi^j - \partial_m \varphi^i
\partial^m F^j - \partial_m F^i \partial^m \varphi^j
\right), \\
\bar{D}_{\dot{\alpha}} \Phi^{\dagger\bar{i}} \bar{D}^{\dot{\alpha}}
\Phi^{\dagger\bar{j}} =& \
- 4 \theta^2 \partial_m \bar{\varphi}^{\bar{i}} \partial^m
\bar{\varphi}^{\bar{j}}
- 4 i (\theta \sigma^m \bar{\theta}) (\partial_m \bar{\varphi}^{\bar{i}}
\bar{F}^{\bar{j}} + \bar{F}^{\bar{i}} \partial_m
\bar{\varphi}^{\bar{j}})
+ 4 \bar{\theta}^2 \bar{F}^{\bar{i}} \bar{F}^{\bar{j}}
\notag \\
& \
+ 2 \theta^2 \bar{\theta}^2
\left(
\bar{F}^{\bar{i}} \Box \bar{\varphi}^{\bar{j}} + \Box
\bar{\varphi}^{\bar{i}} \bar{F}^{\bar{j}}
- \partial_m \bar{\varphi}^{\bar{i}} \partial^m \bar{F}^{\bar{j}}
- \partial_m \bar{F}^{\bar{i}} \partial^m \bar{\varphi}^{\bar{j}}
\right),
\\
D^{\alpha} \Phi^i D_{\alpha} \Phi^k \bar{D}_{\dot{\alpha}}
\Phi^{\dagger\bar{j}} \bar{D}^{\dot{\alpha}}
\Phi^{\dagger\bar{l}}
=& \ 16 \theta^2 \bar{\theta}^2
\left[
\frac{}{}
(\partial_m \varphi^i \partial^m \varphi^k) (\partial_m
\bar{\varphi}^{\bar{j}} \partial^m \bar{\varphi}^{\bar{l}})
\right.
\notag \\
&
\left.
- \frac{1}{2}
\left(
\partial_m \varphi^i F^k + F^i \partial_m \varphi^k
\right)
\left(
\partial^n \bar{\varphi}^{\bar{j}} \bar{F}^{\bar{l}}
+ \bar{F}^{\bar{j}} \partial^n \bar{\varphi}^{\bar{l}}
\right)
+ F^i \bar{F}^{\bar{j}} F^k \bar{F}^{\bar{l}}
\right].
\end{align}
When the supercovariant derivative is gauged, we obtain
\begin{align}
\mathcal{D}_{\alpha} \Phi =& \
2 i (\sigma^m)_{\alpha \dot{\alpha}} \bar{\theta}^{\dot{\alpha}} D_m
\varphi + 2 \theta_{\alpha} F + 2 \theta_{\alpha} \bar{\theta}^2 (\Box
\varphi + g D \varphi) - \frac{1}{2} (\sigma^m)_{\alpha \dot{\alpha}}
(\bar{\sigma}^n)^{\dot{\alpha}} {}_{\dot{\beta}} \theta^{\beta}
\bar{\theta}^2 (\partial_m \partial_n \varphi
\notag \\
& \
- 2 i g \partial_m A_n
\varphi) + i \theta^2 (\sigma^m)_{\alpha \dot{\alpha}}
\bar{\theta}^{\dot{\alpha}} \partial_m F.
\end{align}
Using this expression, we obtain Eq.~\eqref{eq:4th_deri_gauge2_component}.
\section{Supersymmetry variation of fermions}\label{sec:SUSY_variation}
The explicit supersymmetry variation of the fermions in the Euclidean
space is given by
\begin{align}
\delta_{\xi} \psi_{\alpha}^i =& \ \sqrt{2}i
\left(
\begin{array}{c}
(\partial_4 + i \partial_3) \varphi^i \bar{\xi}^{\dot{1}} + i
(\partial_1 - i \partial_2) \varphi^i \bar{\xi}^{\dot{2}} - i \xi_1
F^i \\
(\partial_4 - i \partial_3) \varphi^i \bar{\xi}^{\dot{2}} + i
(\partial_1 + i \partial_2) \varphi^i \bar{\xi}^{\dot{1}} - i \xi_2 F^i
\end{array}
\right), \\
\delta_{\xi} \bar{\psi}^{\dot{\alpha}i} =& \ \sqrt{2} i
\left(
\begin{array}{c}
(\partial_4 - i \partial_3) \bar{\varphi}^i \xi_1 - i (\partial_1 - i
\partial_2) \bar{\varphi}^i \xi_2 - i \bar{\xi}^{\dot{1}} \bar{F}^i \\
(\partial_4 + i \partial_3) \bar{\varphi}^i \xi_2 - i (\partial_1 + i
\partial_2) \bar{\varphi}^i \xi_1 - i \bar{\xi}^{\dot{2}} \bar{F}^i
\end{array}
\right).
\end{align}
\begin{align}
\delta_{\xi} \lambda_{\alpha} =& \
\left(
\begin{array}{c}
i \xi_1 D + i \xi_1 (F_{12} + F_{34})
- \xi_2 (F_{13} - i F_{14} - i F_{23} - F_{24}) \\
i \xi_2 D - i \xi_2 (F_{12} + F_{34})
+ \xi_1 (F_{13} + i F_{14} + i F_{23} - F_{24})
\end{array}
\right), \\
\delta_{\xi} \bar{\lambda}^{\dot{\alpha}} =& \
\left(
\begin{array}{c}
- i \bar{\xi}^{\dot{1}} D - i \bar{\xi}^{\dot{1}} (F_{12} -
F_{34}) + \bar{\xi}^{\dot{2}} (F_{13} + i F_{14} - i F_{23}
+ F_{24} ) \\
- i \bar{\xi}^{\dot{2}} D + i \bar{\xi}^{\dot{2}} (F_{12} -
F_{34}) - \bar{\xi}^{\dot{1}} (F_{13} - i F_{14} + i
F_{23} + F_{24})
\end{array}
\right).
\end{align}
\end{appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,947 |
{"url":"http:\/\/dictionnaire.sensagent.leparisien.fr\/RAY%20OPTICS\/en-en\/","text":"Publicit\u00e9 \u25bc\n\nd\u00e9finition - RAY OPTICS\n\nvoir la d\u00e9finition de Wikipedia\n\nWikipedia - voir aussi\n\nPublicit\u00e9 \u25bc\n\nWikipedia\n\nRay (optics)\n\nIn optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing.[1] This allows even very complex optical systems to be analyzed mathematically or simulated by computer. Ray tracing uses approximate solutions to Maxwell's equations that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's wavelength. Ray theory does not describe phenomena such as interference and diffraction, which require wave theory (involving the phase of the wave).\n\nDefinition\n\nA light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore collinear with the wave vector). Light rays bend at the interface between two dissimilar media and may be curved in a medium in which the refractive index changes. Geometric optics describes how rays propagate through an optical system.\n\nA slightly more rigorous definition of a light ray follows from Fermat's principle, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.[2]\n\nSpecial rays\n\nThere are many special rays that are used in optical modelling to analyze an optical system. These are defined and described below, grouped by the type of system they are used to model.\n\nInteraction with surfaces\n\nDiagram of rays at a surface, where $\\theta_\\mathrm i$ is the angle of incidence, $\\theta_\\mathrm r$ is the angle of reflection, and $\\theta_\\mathrm R$ is the angle of refraction.\n\u2022 An incident ray is a ray of light that strikes a surface. The angle between this ray and the perpendicular or normal to the surface is the angle of incidence.\n\u2022 The reflected ray corresponding to a given incident ray, is the ray that represents the light reflected by the surface. The angle between the surface normal and the reflected ray is known as the angle of reflection. The Law of Reflection says that for a specular (non-scattering) surface, the angle of reflection always equals the angle of incidence.\n\u2022 The refracted ray or transmitted ray corresponding to a given incident ray represents the light that is transmitted through the surface. The angle between this ray and the normal is known as the angle of refraction, and it is given by Snell's Law. Conservation of energy requires that the power in the incident ray must equal the sum of the power in the transmitted ray, the power in the reflected ray, and any power absorbed at the surface.\n\u2022 If the material is birefringent, the refracted ray may split into ordinary and extraordinary rays, which experience different indexes of refraction when passing through the birefringent material.\n\nOptical systems\n\n\u2022 A meridional ray or tangential ray is a ray that is confined to the plane containing the system's optical axis and the object point from which the ray originated.[3]\n\u2022 A skew ray is a ray that does not propagate in a plane that contains both the object point and the optical axis. Such rays do not cross the optical axis anywhere, and are not parallel to it.[3]\n\u2022 The marginal ray (sometimes known as an a ray or a marginal axial ray) in an optical system is the meridional ray that starts at the point where the object crosses the optical axis, and touches the edge of the aperture stop of the system.[4][5] This ray is useful, because it crosses the optical axis again at the locations where an image will be formed. The distance of the marginal ray from the optical axis at the locations of the entrance pupil and exit pupil defines the sizes of each pupil (since the pupils are images of the aperture stop).\n\u2022 The principal ray or chief ray (sometimes known as the b ray) in an optical system is the meridional ray that starts at the edge of the object, and passes through the center of the aperture stop.[4][6] This ray crosses the optical axis at the locations of the pupils. As such chief rays are equivalent to the rays in a pinhole camera. The distance between the chief ray and the optical axis at an image location defines the size of the image. The marginal and chief rays together define the Lagrange invariant, which characterizes the throughput or etendue of the optical system.[7] Some authors define a \"principal ray\" for each object point. The principal ray starting at a point on the edge of the object may then be called the marginal principal ray.[5]\n\u2022 A sagittal ray or transverse ray from an off-axis object point is a ray that propagates in the plane that is perpendicular to the meridional plane and contains the principal ray.[3] Saggital rays intersect the pupil along a line that is perpendicular to the meridional plane for the ray's object point and passes through the optical axis. If the axis direction is defined to be the z axis, and the meridional plane is the y-z plane, saggital rays intersect the pupil at yp=0. The principal ray is both sagittal and meridional.[3] All other sagittal rays are skew rays.\n\u2022 A paraxial ray is a ray that makes a small angle to the optical axis of the system, and lies close to the axis throughout the system.[8] Such rays can be modeled reasonably well by using the paraxial approximation. When discussing ray tracing this definition is often reversed: a \"paraxial ray\" is then a ray that is modeled using the paraxial approximation, not necessarily a ray that remains close to the axis.[9][10]\n\u2022 A finite ray or real ray is a ray that is traced without making the paraxial approximation.[10][11]\n\u2022 A parabasal ray is a ray that propagates close to some defined \"base ray\" rather than the optical axis.[12] This is more appropriate than the paraxial model in systems that lack symmetry about the optical axis. In computer modeling, parabasal rays are \"real rays\", that is rays that are treated without making the paraxial approximation. Parabasal rays about the optical axis are sometimes used to calculate first-order properties of optical systems.[13]\n\nFiber optics\n\n\u2022 A meridional ray is a ray that passes through the axis of an optical fiber.\n\u2022 A skew ray is a ray that travels in a non-planar zig-zag path and never crosses the axis of an optical fiber.\n\u2022 A guided ray, bound ray, or trapped ray is a ray in a multi-mode optical fiber, which is confined by the core. For step index fiber, light entering the fiber will be guided if it makes an angle with the fiber axis that is less than the fiber's acceptance angle.\n\u2022 A leaky ray or tunneling ray is a ray in an optical fiber that geometric optics predicts would totally reflect at the boundary between the core and the cladding, but which suffers loss due to the curved core boundary.\n\nReferences\n\n1. ^ Moore, Ken (25 July 2005). \"What is a ray?\". ZEMAX Users' Knowledge Base. Retrieved 30 May 2008.\n2. ^ Arthur Schuster, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 online.\n3. ^ a b c d Stewart, James E. (1996). Optical Principles and Technology for Engineers. CRC. p.\u00a057. ISBN\u00a0978-0-8247-9705-8.\n4. ^ a b Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN\u00a00-8194-5294-7.\u00a0, p. 25[1].\n5. ^ a b Riedl, Max J. (2001). Optical Design Fundamentals for Infrared Systems. Tutorial texts in optical engineering. 48. SPIE. p.\u00a01. ISBN\u00a0978-0-8194-4051-8.\n6. ^ Malacara, Daniel and Zacarias (2003). Handbook of Optical Design (2nd ed.). CRC. p.\u00a025. ISBN\u00a0978-0-8247-4613-1.\n7. ^ Greivenkamp (2004), p. 28[2].\n8. ^ Greivenkamp (2004), pp. 19\u201320[3].\n9. ^ Nicholson, Mark (21 July 2005). \"Understanding Paraxial Ray-Tracing\". ZEMAX Users' Knowledge Base. Retrieved 17 August 2009.\n10. ^ a b Atchison, David A.; Smith, George (2000). \"A1: Paraxial optics\". Optics of the Human Eye. Elsevier Health Sciences. p.\u00a0237. ISBN\u00a0978-0-7506-3775-6.\n11. ^ Welford, W. T. (1986). \"4: Finite Raytracing\". Aberrations of Optical Systems. Adam Hilger series on optics and optoelectronics. CRC Press. p.\u00a050. ISBN\u00a0978-0-85274-564-9.\n12. ^ Buchdahl, H. A. (1993). An Introduction to Hamiltonian Optics. Dover. p.\u00a026. ISBN\u00a0978-0-486-67597-8.\n13. ^ Nicholson, Mark (21 July 2005). \"Understanding Paraxial Ray-Tracing\". ZEMAX Users' Knowledge Base. p.\u00a02. Retrieved 17 August 2009.\n\nContenu de sensagent\n\n\u2022 d\u00e9finitions\n\u2022 synonymes\n\u2022 antonymes\n\u2022 encyclop\u00e9die\n\n\u2022 definition\n\u2022 synonym\n\nPublicit\u00e9 \u25bc\n\ndictionnaire et traducteur pour sites web\n\nAlexandria\n\nUne fen\u00eatre (pop-into) d'information (contenu principal de Sensagent) est invoqu\u00e9e un double-clic sur n'importe quel mot de votre page web. LA fen\u00eatre fournit des explications et des traductions contextuelles, c'est-\u00e0-dire sans obliger votre visiteur \u00e0 quitter votre page web !\n\nEssayer ici,\u00a0t\u00e9l\u00e9charger le code;\n\nSolution commerce \u00e9lectronique\n\nAugmenter le contenu de votre site\n\nAjouter de nouveaux contenus Add \u00e0 votre site depuis Sensagent par XML.\n\nParcourir les produits et les annonces\n\nObtenir des informations en XML pour filtrer le meilleur contenu.\n\nIndexer des images et d\u00e9finir des m\u00e9ta-donn\u00e9es\n\nFixer la signification de chaque m\u00e9ta-donn\u00e9e (multilingue).\n\nRenseignements suite \u00e0 un email de description de votre projet.\n\nJeux de lettres\n\nLes jeux de lettre fran\u00e7ais sont :\n\u25cb\u00a0\u00a0\u00a0Anagrammes\n\u25cb\u00a0\u00a0\u00a0jokers, mots-crois\u00e9s\n\u25cb\u00a0\u00a0\u00a0Lettris\n\u25cb\u00a0\u00a0\u00a0Boggle.\n\nLettris\n\nLettris est un jeu de lettres gravitationnelles proche de Tetris. Chaque lettre qui appara\u00eet descend ; il faut placer les lettres de telle mani\u00e8re que des mots se forment (gauche, droit, haut et bas) et que de la place soit lib\u00e9r\u00e9e.\n\nboggle\n\nIl s'agit en 3 minutes de trouver le plus grand nombre de mots possibles de trois lettres et plus dans une grille de 16 lettres. Il est aussi possible de jouer avec la grille de 25 cases. Les lettres doivent \u00eatre adjacentes et les mots les plus longs sont les meilleurs. Participer au concours et enregistrer votre nom dans la liste de meilleurs joueurs ! Jouer\n\nDictionnaire de la langue fran\u00e7aise\nPrincipales R\u00e9f\u00e9rences\n\nLa plupart des d\u00e9finitions du fran\u00e7ais sont propos\u00e9es par SenseGates et comportent un approfondissement avec Littr\u00e9 et plusieurs auteurs techniques sp\u00e9cialis\u00e9s.\nLe dictionnaire des synonymes est surtout d\u00e9riv\u00e9 du dictionnaire int\u00e9gral (TID).\nL'encyclop\u00e9die fran\u00e7aise b\u00e9n\u00e9ficie de la licence Wikipedia (GNU).\n\nChanger la langue cible pour obtenir des traductions.\nAstuce: parcourir les champs s\u00e9mantiques du dictionnaire analogique en plusieurs langues pour mieux apprendre avec sensagent.\n\n5511 visiteurs en ligne\n\ncalcul\u00e9 en 0,047s\n\nJe voudrais signaler :\nsection :\nune faute d'orthographe ou de grammaire\nun contenu abusif (raciste, pornographique, diffamatoire)\nune violation de copyright\nune erreur\nun manque\nautre\nmerci de pr\u00e9ciser :\nallemand anglais arabe bulgare chinois cor\u00e9en croate danois espagnol esp\u00e9ranto estonien finnois fran\u00e7ais grec h\u00e9breu hindi hongrois islandais indon\u00e9sien italien japonais letton lituanien malgache n\u00e9erlandais norv\u00e9gien persan polonais portugais roumain russe serbe slovaque slov\u00e8ne su\u00e9dois tch\u00e8que thai turc vietnamien\nallemand anglais arabe bulgare chinois cor\u00e9en croate danois espagnol esp\u00e9ranto estonien finnois fran\u00e7ais grec h\u00e9breu hindi hongrois islandais indon\u00e9sien italien japonais letton lituanien malgache n\u00e9erlandais norv\u00e9gien persan polonais portugais roumain russe serbe slovaque slov\u00e8ne su\u00e9dois tch\u00e8que thai turc vietnamien","date":"2021-12-07 19:32:10","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\": 0, \"img_math\": 3, \"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\": 0.5495713949203491, \"perplexity\": 4191.742976283751}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363405.77\/warc\/CC-MAIN-20211207170825-20211207200825-00343.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec.introduction}
Internet of Things (IoT) represents a paradigm in the world of interconnected devices in which sensors, actuators, smartphones communicate with each other using appropriate wireless technologies. In such an environment, users can receive feedback about the physical world that surrounds them, allowing the interaction with it and exchange such data with the digital world~\cite{Mahdavinejad18MLI}. IoT applications are used today in various aspects of the industry, such as wireless sensor networks, data mining, assisted living, etc. giving rise to the concept of Smart City. Application of IoT can be found in every aspect of everyday lives, ranging from the fields of automation, industrial manufacturing, logistics, business/process management, intelligent transportation of people and goods or environmental monitoring~\cite{Atzori10IoTS, Barile18Real-Time}.
IoT denotes a concept in which a large number of interconnected devices create novel applications. Despite the industry forecasts of 50 billion interconnected devices, only 9 billion IoT devices have been materialized until 2020\footnote{\url{https://www.vanillaplus.com/2020/01/16/50049-missing-41bn-iot-devices-biggest-prediction-miss-history/}}. For the IoT to come to life in full, it is necessary to bring the functionality closer to the end-user through a flexible platform where information on the status of "things" being monitored can be obtained and the user is informed if changes occur. Likewise, flexibility must be manifested through the integration with future services where Machine Learning (ML) in IoT applications allows for different estimations and predictions based on data coming from sensor devices.
For example, IoT applications are especially suited for living environments such as agricultural, where the irrigation plays an extremely important role~\cite{Balducci18MLA}. Existing solutions include the use of battery-operated sensing devices, while sensor data is transmitted using the appropriate wireless technology. Recent advancements in Low Power Wide Area Network (LPWAN) gave rise to radio technologies, e.g. LoRa (LoRaWAN), NB-IoT and Sigfox, that are suitable for sporadic transmissions of small sensor data packets over large distances, making them ideal for livestock farming, flood monitoring and/or smart irrigation systems~\cite{Brewster17IoT, Perkovic20Meeting}. In LPWAN architecture application-specific end nodes reach their network server via a gateway. From there, the data are routed to the respective application server. Such architecture gave rise to the commercial LPWAN service providers, e.g. The Things Network (TTN)\footnote{\url{https://www.thethingsnetwork.org/}} and LORIOT\footnote{\url{https://www.loriot.io/}}, that employ functionalities of LoRaWAN and/or application server. Consequently, application specific platforms that allow a user to visualize and possibly control sensor device status have been developed. For example, myDevices Cayenne\footnote{\url{https://www.thethingsnetwork.org/docs/applications/cayenne/}} and Ubidots\footnote{\url{https://ubidots.com/}} provide a service to visualize real-time and historical data sent over The Things Network, such as temperature monitoring, occupancy, predictive maintenance, etc. Libelium, on the other hand, has developed a cloud platform for monitoring a plethora of sensing devices provided by Libelium company\footnote{\url{https://cloud.libelium.com/}}.
All these systems as introduced previously enable users the possibility to insert new proprietary sensor devices. Similarly, as in systems described above, the proposed system allows users to collect the sensor data from TTN cloud and store it in a time-series Influx database. Such a feature is given by default in the majority of services as it allows users to visualize the data using appropriate dashboards. In addition to related services, the proposed solution allows the user to send commands to the sensor data, such as the wake-up period, time synchronization, etc. A special feature of the collected data is a ML service that realizes a novel approach in the big data analysis, allowing users to work on various case studies aimed at solving the domain specific problems. Hence, in this paper an IoT Wallet is introduced, whereas a case study is the soil moisture prediction from a signal strength of an underground LoRa beacon. Using the double prediction of the feed-forward neural network and the long short-term memory (LSTM) network, highly accurate prediction of the soil moisture is achieved solely based on the publicly available data acquired from the State Hydrometeorological Institute and characteristics of the signal received from LoRa end devices.
\section{LoRa-based soil moisture Sensor}
\label{sec:lora}
Using signal strength measurements from a beacon device with the sensor data from other devices, such as air humidity, temperature, and pressure, soil moisture is estimated using related ML techniques\cite{Dujic20MLS}. Hence, soil moisture sensor devices could be replaced with a simple underground LoRa beacon end-device. In this paper, LoRa technology is employed for transmitting communication data from sensor devices to the base station (e.g. soil moisture, air temperature, and humidity). LoRa as a representative of LPWAN allows battery-operated sensors to communicate low throughput data over long distances, making it suitable for applications in scenarios such as agriculture monitoring~\cite{LiedmannHW18}.
\subsection{Data Collection from LoRa-based sensor device}\label{sec:implementation}
Namely, soil moisture, along with signal strength measurements (RSSI and SNR) were collected from humidity sensor device that was buried 14 cm below the ground level.
Besides, air humidity and temperature were collected from another LoRa-based sensor device that was placed 3 m over the ground in constant shade. The core of both sensor devices is Arduino Pro Mini (ATmega328P) that operates at 3.5V. For LoRaWAN communication, an RFM95W module that uses SX1276 chip was used that operates at 868 MHz. In our implementation spring antenna was vertically oriented with $+14$dB transmission power. Also, TPL5110 module was used to preserve energy during inactive periods, where the module simply cuts off power during the inactive period, reducing the overall consumption of the sensor device. The TPL5110 timer was set to power up Arduino every 10 minutes. For the measurement of soil moisture, an I2C soil moisture sensor was employed with capacitive sensing\footnote{\url{https://www.whiteboxes.ch/shop/i2c-soil-moisture-sensor/}}, while for the measurement of air humidity and temperature, SHT-10 mesh protected sensor device was used. Besides, publicly available data was such as air pressures were acquired from the State Hydrometeorological Institute for the city where our system is deployed.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\linewidth]{figs/Soil_moisture_sensor.pdf}
\caption{Implementation of LoRa-based sensor devices.}
\label{fig:implementation}
\end{figure}
As a LoRaWAN gateway, an indoor Raspberry Pi-based gateway device was employed that forwards messages to The Things Network (TTN) cloud infrastructure. Our gateway uses RAK831 concentrator with Procom CXL 900-6LW-NB, 8 dBi gain, 868 MHz, vertically polarized, omnidirectional antenna placed around 60 meters from soil moisture sensor device (Figure~\ref{fig:implementation}).
\begin{figure}[t]
\centering
\includegraphics[width=0.73\linewidth]{figs/RSSI_sensor_2.png}
\includegraphics[width=0.73\linewidth]{figs/SNR_sensor_2.png}
\includegraphics[width=0.73\linewidth]{figs/Humidity_sensor_2.png}
\includegraphics[width=0.73\linewidth]{figs/Humidity_temp_sensor_2.png}
\includegraphics[width=0.73\linewidth]{figs/Pressure_sensor_2.png}
\caption{Snapshot of RSSI and SNR signal captured on LoRaWAN gateway from soil moisture sensor, along with I2C sensor measures of soil moisture. Another LoRaWAN sensor sends information about air humidity and temperature, while State Hydrometeorological Institute gives information regarding air pressure.}
\label{fig:measurements}
\end{figure}
Once the message arrives at the gateway, it is forwarded to the TTN Network and Application server. Furthermore, TTN allows message forwarding from their infrastructure to our dedicated IoT Wallet using MQTT protocol, which is described more in detail in Section~\ref{sec.softarch}. Figure~\ref{fig:measurements} shows a snapshot of soil moisture along with RSSI and SNR values captured on the gateway. Besides, air humidity and temperature from the second LoRa-based sensor devices are collected, along with air pressure data collected from State Hydrometeorological Institute. As can be seen, when the soil moisture increases, both RSSI and SNR signal values drop, showing the tight bound between these values. In addition, in order to improve the prediction of soil moisture, information on air temperature, humidity and pressure are collected as well.
\begin{figure*}[htbp]
\includegraphics[width=\textwidth]{figs/system-overview.pdf}
\caption{High-level overview of the current implementation of the solution.}
\label{fig.system}
\end{figure*}
\section{IoT Wallet Architecture - Overview and Functionalities}
\label{sec.softarch}
The Things Network (TTN) microservice is set up to listen for the LoRa sensor data using MQTT protocol, as depicted in Figure~\ref{fig.system}. Once the signal arrives, the TTN microservice captures and stores the LoRa uplink data into the Influx database. Also, the TTN microservice is capable of listening for LoRa-based downlink messages, which are redirected from the client application to the TTN cloud. The purpose of downlink is to control sensor's behavior within the application itself.
Immediately after the data are stored in the Influx database, TTN microservice sends sensor data to the ML-based forecasting microservice. The data are processed through ML model which forecast the soil moisture values for the following moment or the sequence of following moments, more details are presented in the Section \ref{sec.ml}.
The push notification microservice sends notifications to the user. Notifications are triggered if sensor value is greater than, less than or equal to the value that the user has specified. There is also an option for cumulative notifications for the specific time period. In this case, the user enters a cumulative period, the operator and the value which will trigger notification.
Lastly, the API microservice manages the complete logic of the application. The API microservice has GraphQL server running and listening for all traffic on the pre-defined port. The client application communicates with the API microservice and serves the data to the end user. Main API endpoints are the user, sensors, the sensor type, downlink control etc. Client application is built within the Angular framework. Ionic framework is set up as a top layer of the Angular application and it enables the cross platform deployment. This opens the possibility of serving the application for the web, Android and iOS, simultaneously.
\section{Machine Learning in IoT (Wallet)}
\label{sec.ml}
Data mining, big data analysis and ML are paradigms of particular importance for the current state of IoT and the Industrial IoT.
From smart traffic, health, environment and agriculture all to control, security and forecasting of different natural phenomena, these services can be enhanced and optimized by analyzing the smart data generated and collected in their respective domains \cite{Mahdavinejad_ML_2018}.
The case-study presented in the following subsection is the soil moisture prediction from the signal strength indicator.
\subsection{Case-study: soil moisture Prediction from Signal Strength}
\label{sec.casestudy}
In this work, the idea is to replace expensive capacitive-based soil moisture sensors, most often unreliable and in need of human intervention in terms of calibration and battery change, with simple LoRa beacon end-devices. LoRa beacon lights up sporadically and sends a short signal to the base station where channel information is recorded and the soil moisture is predicted, assuming a nonlinear dependence of the received signal strength by means of RSSI and SNR and the soil moisture.
We incorporate learning by two approaches: training a simple yet powerful deep neural network to extract the current knowledge from the data, and training a LSTM network which allows the historical, long-term dependencies to be captured.
In the first approach, a deep feed-forward neural network with $2$ hidden layers, containing $128$ and $64$ units, respectively, is trained on $23592$ data points. The data have been collected, captured and stored on every $10^{th}$ minute from the mid February 2020 to mid August 2020. Each input consisted of $5$ features, which may be separated into the two logical sets. The first set is closely reflecting the nature of the signal (RSSI and SNR) and the second set is describing current atmospheric conditions (air temperature, air humidity and air pressure) of the area where the LoRa sensors are located. The neural network is trained for $500$ epochs using the batch size of 32. Optimization of the Mean Square Logarithmic Error (MSLE) loss function is performed using ADAM optimization technique introduced in \cite{kingma_adam_2014} with learning rate initially set to $0.0001$. The weight matrix initialization is performed using a well-known Xavier initialization scheme \cite{glorot_bengio_2010}. Each unit is activated using Exponential Linear Unit (ELU) activation function. Before the training itself, the data are split into the training, validation and test set. Additional pre-processing of the data include double normalization. The first step is normalizing the training data by subtracting the mean value. This provides a more general approach to learning and opens the door for applying the same (or minimally re-trained) model to other LoRa sensors in significantly different conditions. The second step is scaling the training data between 0 and 1 so that the order of magnitude of the values that the model is fed with and the weight of the model itself are the same. This allows avoiding the issue of the exploding gradients and creates stable and secure training conditions. In Figure \ref{fig.nn}, the inference on the test set is shown where the mean absolute error (MAE) is $2.0617$.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{figs/nn.pdf}
\caption{The actual data are shown in blue, while the predicted data using a feed-forward neural network are shown in red on the training set (curves to the left of the first black vertical line), validation set (curves between the two black vertical lines) and test set (curves to the right of the second black vertical line), respectively.}
\label{fig.nn}
\end{figure}
In order to capture long-term dependencies in the data, the LSTM network is built and trained \footnote{The first version of the LSTM network is introduced in \cite{Hochreiter_lstm_1997} but since then is upgraded and adjusted to the modern software capabilities \cite{olah_lstm_2015}, which exploit the full potential of the automatic differentiation.}. Because of its recurrent nature, the LSTM network and its units are perfectly adapted to the time-series data (a series of data points indexed in time order and stored at successive equally spaced points in time). The LSTM unit architecture curates the long-term dependency problem with its three-gate configuration allowing information to persist and capturing underlying non-linear pattern in the data, otherwise imperceptible to \textit{vanilla} neural networks. Here, the network consists of a single LSTM layer and a single dense layer with 12 and 20 units, respectively. Training is performed throughout $500$ epochs using the same data and the same input features as for the previous neural network, but here the last $6$ steps are considered in each training step. Scaling and normalization also stayed the same. All hyper-parameters but the activation function stayed the same. The activation function is set by default to Hyperbolic Tangent (\textit{tanh}) function using the LSTM layer deployed in NVIDIA CUDA Deep Neural Network library (cuDNN) adapted for TensorFlow framework \cite{tensorflow_2015}. In Figure \ref{fig.lstm}, the inference on the test set is shown where the MAE is $1.9946$.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{figs/lstm.pdf}
\caption{The actual data are shown in blue, while the predicted data using the LSTM network are shown in red on the training set (curves to the left of the first black vertical line), validation set (curves between the two black vertical lines) and test set (curves to the right of the second black vertical line), respectively.}
\label{fig.lstm}
\end{figure}
Learning curves for both previously outlined models - the feed-forward neural network and the LSTM network, are shown in Figure \ref{fig.loss}. A slight overfitting in the case of the LSTM network is visible on the validation curve by the end of the training. However, due to the fact that the overall validation loss is lower than the training loss, overfitting can be overlooked. This phenomenon may happen usually when the training data is more difficult to train on and learn patterns on, while less change occurs in the validation set.
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{figs/loss.pdf}
\caption{Learning curves by means of the loss change over training epochs of the feed-forward neural network and the LSTM network are shown using blue and red colored full line and blue and red colored dashed line, respectively.}
\label{fig.loss}
\end{figure}
Using the double prediction of the feed-forward neural network and the LSTM network, respectively, in the form of an ensemble of learning, we are able to achieve reliable and accurate prediction of the soil moisture solely on the basis of the publicly available data acquired from the State Hydrometeorological Institute and characteristics of the signal received from LoRa end devices.
\section{Conclusion}
\label{sec.conclusion}
In this paper architecture of IoT Wallet system is introduced. It allows users to manually register the sensor device to TTN service, as well as to visualize the data stored in the database. Besides, depending on the user type, it is allowed to control the sensors, and communicate data towards them. A special feature of the proposed IoT Wallet is a ML feature that allows users to perform predictions from collected sensed data. As a proof-of-concept, soil moisture is predicted from signal strength from an underground LoRa beacon device. As shown in this paper, using the feed-forward neural network and the LSTM network, an accurate prediction of the soil moisture can be estimated based on the signal characteristics of LoRa device along with collected publicly available data from overground air humidity and temperature device along with air pressure collected from Hydrometeorological Institute.
\section*{Acknowledgment}
This Technology Transfer Experiment has received funding from the European Union's Horizon 2020 research and innovation programme under the TETRAMAX grant agreement no 761349.
\bibliographystyle{IEEEtran}{}
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Friday Night Flies – Pink Squirrel. Pink Squirrel is an Attractor pattern. When all else fails through down some Pink. Fish it on a dry line like a Czech style Nymph. Catch a bunch of fish and call it a Day! | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/www.omnimaga.org\/axe-language\/windows-xp-in-axe\/msg402125\/","text":"Author Topic: Windows XP in Axe \u00a0(Read 6379 times)\n\n0 Members and 1 Guest are viewing this topic.\n\nSiphonic_Sugar\n\n\u2022 LV3 Member (Next: 100)\n\u2022 Posts: 51\n\u2022 Rating: +0\/-0\nWindows XP in Axe\n\u00ab on: August 12, 2015, 08:12:30 pm \u00bb\nSo lately, I've been thinking about making another version of Windows XP but this time, unlike every single other version that most people know of, this one will be made in axe. I already have the mouse thing set up...\nCode: [Select]\n.WINDOWXPClrDrawClrHome32->Y48->X[E0C0A00000000000]->Pic1Repeat getkey(15)Pt-On(X,Y,Pic1Line(0,0,95,0\u00a0 \u00a0 \u00a0 \u00a0 \\\\Lines for a borderLine(0,63,95,63\u00a0 \u00a0 \u00a0\\\\Line(0,0,63,0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\\\Line(95,0,95,63\u00a0 \u00a0 \u00a0 \\\\If getkey(1)If Y(doesn't equal)61Y+1->YEndEndIf getkey(2)If X(doesn't equal)1X-1->XEndEndIf getkey(3)If X(doesn't equal)93X+1->XEndEndIf getkey(4)If Y(doesn't equal)1Y-1->YEndEndClrDrawDispGraphEndBut the problem is, I don't know how to make it so the program knows if you have the cursor in the area of the START menu (for example) and if you clicked on it. To tell you the truth, I think that is the main thing that I need help with besides optimization.\n\nSiphonicSugar\n\u00ab Last Edit: August 12, 2015, 10:56:37 pm by Eeems \u00bb\nI'm on Codewalr.us now too, you should check it out.\n<a href=\"https:\/\/codewalr.us\"><img src=\"https:\/\/codewalr.us\/other\/cwaffiliate.png\" \/><\/a>\n\nSorunome\n\n\u2022 Fox Fox Fox Fox Fox Fox Fox!\n\u2022 Support Staff\n\u2022 LV13 Extreme Addict (Next: 9001)\n\u2022 Posts: 7920\n\u2022 Rating: +374\/-13\n\u2022 Derpy Hooves\nRe: Windows XP in Axe\n\u00ab Reply #1 on: August 13, 2015, 04:49:34 am \u00bb\nFor the start menu checking thing you have to check if the cursor is in a certain area, so something like\nIf X>minimumx and (X < maximumx) and (Y > minimumy) and (Y < maximumy)\n THE GAMEAlso, check out my websiteIf OmnomIRC is screwed up, blame me!Click here to give me an internet!\n\nSiphonic_Sugar\n\n\u2022 LV3 Member (Next: 100)\n\u2022 Posts: 51\n\u2022 Rating: +0\/-0\nRe: Windows XP in Axe\n\u00ab Reply #2 on: August 13, 2015, 04:12:41 pm \u00bb\nHim...I never thought about that... Is there a way to put a getkey phrase on the same line as something else if connected by and?\n\nIf getkey(15) and Y(doesn't equal)1\n\nThis never works.\nI'm on Codewalr.us now too, you should check it out.\n<a href=\"https:\/\/codewalr.us\"><img src=\"https:\/\/codewalr.us\/other\/cwaffiliate.png\" \/><\/a>\n\nRuner112\n\n\u2022 Project Author\n\u2022 LV11 Super Veteran (Next: 3000)\n\u2022 Posts: 2289\n\u2022 Rating: +639\/-31\nRe: Windows XP in Axe\n\u00ab Reply #3 on: August 13, 2015, 04:16:07 pm \u00bb\nThat's because Axe's order of operations is simply left-to-right. So that's being evaluated like (getKey(15) and Y)\u22601. Try adding parentheses, like getKey(15) and (Y\u22601), or switching the order, like Y\u22601 and getKey(15).\n\nHaobo\n\n\u2022 LV2 Member (Next: 40)\n\u2022 Posts: 27\n\u2022 Rating: +4\/-0\nRe: Windows XP in Axe\n\u00ab Reply #4 on: August 13, 2015, 06:18:01 pm \u00bb\nsome quick help:\nCode: [Select]\nClrDrawClrHomeya don't need both if in the end, you're only going to use DispGraph every millisecond, ya can remove ClrHome\n\nCode: [Select]\nLine(0,0,95,0\u00a0 \u00a0 \u00a0 \u00a0 \\\\Lines for a borderLine(0,63,95,63\u00a0 \u00a0 \u00a0\\\\Line(0,0,63,0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\\\\Line(95,0,95,63\u00a0 \u00a0 \u00a0 \\\\try the rect command, makes it easier to write (and the recti command)\n\nCode: [Select]\nY+1->Y...X-1->Xthey can be Y++ and X--\n\nHere's a big optimization of your code that you might want to do yourself or not, but anyway, it's not too hard to figure out after you experiment a lot. (but I'll leave it up to you if ya wanna use it)\nSpoiler For Spoiler:\nmin(max(getKey(3)-getKey(2)+X+1,1),96)-1->X\nmin(max(getKey(1)-getKey(4)+Y+1,1),64)-1->Y\n\nthis can replace the whole code after the line part\nand check this out for some other optimizations\nhttps:\/\/www.omnimaga.org\/axe-language\/the-optimization-compilation\/\nProjects:\nStar Cats\nFive Nights at Freddy's\nPhoenix Wright\n\nSiphonic_Sugar\n\n\u2022 LV3 Member (Next: 100)\n\u2022 Posts: 51\n\u2022 Rating: +0\/-0\nRe: Windows XP in Axe\n\u00ab Reply #5 on: August 13, 2015, 06:22:59 pm \u00bb\nOkay, so, obviously when I get a little farther into making this, when you click (2nd) in the bottom left corner, it won't say START (that was just added to see if clicking works), and I'll make an actual start menu pop up.\n\nCode: [Select]\n.WINDOWXPFix 532->Y48->XClrHome[E0C0A00000000000]->Pic10->ARepeat getKey(15)Lbl 1ClrDrawIf A=1Text(2,56,\"STARTEndPt-On(X,Y,Pic1If getKey(1)Y+1->YEndIf getKey(2)X-1->XEndIf getKey(3)0+1->XEndIf getKey(4)Y-1->YEndIf getKey(54)Goto 2EndDispGraphEndLbl 2If Y>54 and X<251->AEndGoto 1\nWhy won't this work with the lbls? When I click on the bottom left corner, the mouse stops working correctly and it won't let me press CLEAR to end the program. Oh and Haubo, thanks for the optimization help. I didn't add that yet because I saw your comment after I posted again.\n\n\u00ab Last Edit: August 13, 2015, 06:42:31 pm by Siphonic_Sugar \u00bb\nI'm on Codewalr.us now too, you should check it out.\n<a href=\"https:\/\/codewalr.us\"><img src=\"https:\/\/codewalr.us\/other\/cwaffiliate.png\" \/><\/a>\n\nHaobo\n\n\u2022 LV2 Member (Next: 40)\n\u2022 Posts: 27\n\u2022 Rating: +4\/-0\nRe: Windows XP in Axe\n\u00ab Reply #6 on: August 13, 2015, 06:40:30 pm \u00bb\n\nTry not to use any goto's in your program unless you know what you're doing with them. Goto's don't goto a certain part of the code, they also take everything with them, including the stack. It will break the program if not used correctly. What you want to do is create subprograms. (not like you really need them here anyway )\nTry this: (I'll change minimally)\nCode: [Select]\n.WINDOWXPFix 532->Y48->XClrHome[E0C0A00000000000]->Pic10->ARepeat getKey(15)ClrDrawPt-On(X,Y,Pic1If getKey(1)Y+1->YEndIf getKey(2)X-1->XEndIf getKey(3)X+1->XEndIf getKey(4)Y-1->YEndIf Y>54?X<25?getKey(54)Text(2,56,\"STARTEndDispGraphEnd\nAlso, I think you should avoid using and's in the code too for that purpose. and doesn't actually compare it that way, check out the command list for more details. For this, use ?\nProjects:\nStar Cats\nFive Nights at Freddy's\nPhoenix Wright\n\nSiphonic_Sugar\n\n\u2022 LV3 Member (Next: 100)\n\u2022 Posts: 51\n\u2022 Rating: +0\/-0\nRe: Windows XP in Axe\n\u00ab Reply #7 on: August 13, 2015, 06:44:38 pm \u00bb\nSure, seems like a good idea. Thanks again for optimization.\nI'm on Codewalr.us now too, you should check it out.\n<a href=\"https:\/\/codewalr.us\"><img src=\"https:\/\/codewalr.us\/other\/cwaffiliate.png\" \/><\/a>\n\nc4ooo\n\n\u2022 Posts: 252\n\u2022 Rating: +10\/-1\n\u2022 The impossible chemical compound.\nRe: Windows XP in Axe\n\u00ab Reply #8 on: August 13, 2015, 07:01:36 pm \u00bb\nHere's a big optimization of your code that you might want to do yourself or not, but anyway, it's not too hard to figure out after you experiment a lot. (but I'll leave it up to you if ya wanna use it)\nCode: [Select]\nmin(max(getKey(3)-getKey(2)+X+1,1),96)-1->Xmin(max(getKey(1)-getKey(4)+Y+1,1),64)-1->Y\nGrrrr I was about to post the same thing myself. That's why I was asking irc about max()^r. to bad you 'd me.\nBut as a side note are you sure it won't be better to remove those -1s and change max() to max()^r? I think that will be smaller.\u00a0 (but i am not sure)\n-German Kuznetsov\nThe impossible chemical compound.\n\nSiphonic_Sugar\n\n\u2022 LV3 Member (Next: 100)\n\u2022 Posts: 51\n\u2022 Rating: +0\/-0\nRe: Windows XP in Axe\n\u00ab Reply #9 on: August 13, 2015, 07:50:39 pm \u00bb\nHmm, where is the recti command?\nI'm on Codewalr.us now too, you should check it out.\n<a href=\"https:\/\/codewalr.us\"><img src=\"https:\/\/codewalr.us\/other\/cwaffiliate.png\" \/><\/a>\n\nc4ooo\n\n\u2022 Posts: 252\n\u2022 Rating: +10\/-1\n\u2022 The impossible chemical compound.\nRe: Windows XP in Axe\n\u00ab Reply #10 on: August 13, 2015, 08:15:12 pm \u00bb\nHmm, where is the recti command?\n[2nd][matrix(x^-1)]->math tab-> option 'B'.\n-German Kuznetsov\nThe impossible chemical compound.\n\nSiphonic_Sugar\n\n\u2022 LV3 Member (Next: 100)\n\u2022 Posts: 51\n\u2022 Rating: +0\/-0\nRe: Windows XP in Axe\n\u00ab Reply #11 on: August 13, 2015, 08:35:50 pm \u00bb\nOkay. Thank you for your help!\n\nHey, I'm going to set aside this Windows XP project for a minute so I can ask this. I'm thinking about learning 68K basic and maybe a little more on Lua but what calculator do you guys think that I should get next?\n\n1) TI-Nspire CX\n\n2) TI-92 Plus\n\n3) TI-84 Plus CE\n\u00ab Last Edit: August 13, 2015, 08:40:41 pm by Siphonic_Sugar \u00bb\nI'm on Codewalr.us now too, you should check it out.\n<a href=\"https:\/\/codewalr.us\"><img src=\"https:\/\/codewalr.us\/other\/cwaffiliate.png\" \/><\/a>\n\nUnicorn\n\n\u2022 LV1 Newcomer (Next: 20)\n\u2022 Posts: 16\n\u2022 Rating: +0\/-0\n\u2022 I am rainbows\nRe: Windows XP in Axe\n\u00ab Reply #12 on: August 13, 2015, 08:51:05 pm \u00bb\nI'm going to say CE, because its new, and we need more games\/programs for it out and about I might be a biased because of my CSE\n\nSiphonic_Sugar\n\n\u2022 LV3 Member (Next: 100)\n\u2022 Posts: 51\n\u2022 Rating: +0\/-0\nRe: Windows XP in Axe\n\u00ab Reply #13 on: August 13, 2015, 08:55:08 pm \u00bb\nHaha, I hate the CSE. The CE is almost 8 times faster!\nI'm on Codewalr.us now too, you should check it out.\n<a href=\"https:\/\/codewalr.us\"><img src=\"https:\/\/codewalr.us\/other\/cwaffiliate.png\" \/><\/a>\n\nIvoah\n\n\u2022 LV6 Super Member (Next: 500)\n\u2022 Posts: 336\n\u2022 Rating: +3\/-0\nRe: Windows XP in Axe\n\u00ab Reply #14 on: August 13, 2015, 08:58:35 pm \u00bb\nNspire CX (CAS if you can get it) It can run linux so you can really put any programming language on it. It also has Lua built in.\nhttp:\/\/codinghobbit.no-ip.org\nMy Calcs:\nTI-86 (now broken) $2 TI SR-56 -$0\nTI-Nspire CX CAS - $152 TI-84+ Silver Edition -$56\nTI-84+ Silver Edition - $0 TI-85 -$0\nTI-73 Explorer VS - $10 ViewScreen -$3","date":"2021-04-21 18:10:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.29403579235076904, \"perplexity\": 12959.28144231378}, \"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-2021-17\/segments\/1618039546945.85\/warc\/CC-MAIN-20210421161025-20210421191025-00155.warc.gz\"}"} | null | null |
Q: Detecting dual monitors in Silverlight Is there a way to detect if the user has 1 monitor or 2? And if so, is it possible to determine which is left and which is right and open a browser window on that specific monitor?
I'm looking for either a Silverlight solution to this problem or a Javascript / DOM solution or an HTML5 solution.
This is an in-browser app, by the way
Thanks
A: Javascript, Silverlight and Flash are bounded by the sandbox within the browser where they are running. To do what you require, would most likely mean breaking out of that sandbox which would be a huge security concern and is not allowed.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,259 |
.class synthetic Landroid/net/MobileDataStateTracker$2;
.super Ljava/lang/Object;
.source "MobileDataStateTracker.java"
# annotations
.annotation system Ldalvik/annotation/EnclosingClass;
value = Landroid/net/MobileDataStateTracker;
.end annotation
.annotation system Ldalvik/annotation/InnerClass;
accessFlags = 0x1008
name = null
.end annotation
# static fields
.field static final synthetic $SwitchMap$com$android$internal$telephony$PhoneConstants$DataState:[I
# direct methods
.method static constructor <clinit>()V
.locals 3
.prologue
.line 1560
invoke-static {}, Lcom/android/internal/telephony/PhoneConstants$DataState;->values()[Lcom/android/internal/telephony/PhoneConstants$DataState;
move-result-object v0
array-length v0, v0
new-array v0, v0, [I
sput-object v0, Landroid/net/MobileDataStateTracker$2;->$SwitchMap$com$android$internal$telephony$PhoneConstants$DataState:[I
:try_start_0
sget-object v0, Landroid/net/MobileDataStateTracker$2;->$SwitchMap$com$android$internal$telephony$PhoneConstants$DataState:[I
sget-object v1, Lcom/android/internal/telephony/PhoneConstants$DataState;->DISCONNECTED:Lcom/android/internal/telephony/PhoneConstants$DataState;
invoke-virtual {v1}, Lcom/android/internal/telephony/PhoneConstants$DataState;->ordinal()I
move-result v1
const/4 v2, 0x1
aput v2, v0, v1
:try_end_0
.catch Ljava/lang/NoSuchFieldError; {:try_start_0 .. :try_end_0} :catch_3
:goto_0
:try_start_1
sget-object v0, Landroid/net/MobileDataStateTracker$2;->$SwitchMap$com$android$internal$telephony$PhoneConstants$DataState:[I
sget-object v1, Lcom/android/internal/telephony/PhoneConstants$DataState;->CONNECTING:Lcom/android/internal/telephony/PhoneConstants$DataState;
invoke-virtual {v1}, Lcom/android/internal/telephony/PhoneConstants$DataState;->ordinal()I
move-result v1
const/4 v2, 0x2
aput v2, v0, v1
:try_end_1
.catch Ljava/lang/NoSuchFieldError; {:try_start_1 .. :try_end_1} :catch_2
:goto_1
:try_start_2
sget-object v0, Landroid/net/MobileDataStateTracker$2;->$SwitchMap$com$android$internal$telephony$PhoneConstants$DataState:[I
sget-object v1, Lcom/android/internal/telephony/PhoneConstants$DataState;->SUSPENDED:Lcom/android/internal/telephony/PhoneConstants$DataState;
invoke-virtual {v1}, Lcom/android/internal/telephony/PhoneConstants$DataState;->ordinal()I
move-result v1
const/4 v2, 0x3
aput v2, v0, v1
:try_end_2
.catch Ljava/lang/NoSuchFieldError; {:try_start_2 .. :try_end_2} :catch_1
:goto_2
:try_start_3
sget-object v0, Landroid/net/MobileDataStateTracker$2;->$SwitchMap$com$android$internal$telephony$PhoneConstants$DataState:[I
sget-object v1, Lcom/android/internal/telephony/PhoneConstants$DataState;->CONNECTED:Lcom/android/internal/telephony/PhoneConstants$DataState;
invoke-virtual {v1}, Lcom/android/internal/telephony/PhoneConstants$DataState;->ordinal()I
move-result v1
const/4 v2, 0x4
aput v2, v0, v1
:try_end_3
.catch Ljava/lang/NoSuchFieldError; {:try_start_3 .. :try_end_3} :catch_0
:goto_3
return-void
:catch_0
move-exception v0
goto :goto_3
:catch_1
move-exception v0
goto :goto_2
:catch_2
move-exception v0
goto :goto_1
:catch_3
move-exception v0
goto :goto_0
.end method
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,333 |
Een roman van Billie Letts, zie Where the Heart Is (boek)
Drie televisieseries:
Een Amerikaanse soapserie uit 1969, zie Where the Heart Is (Amerikaanse soap).
Een Britse soapserie uit 1997, zie Where the Heart Is (Britse soap).
Een Maleisische dramaserie uit 2008m, zie Where the Heart Is (dramaserie).
een aflevering van de televisieserie ER, zie Where the Heart Is (ER).
Twee films:
Een romantische komedie uit 1990, zie Where the Heart Is (1990)
Een verfilming van bovengenoemde roman uit 2000, zie Where the Heart Is (2000)
Muziek:
Where the Heart Is (Haevn), een nummer van Haevn | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,771 |
Ženská dvouhra Monterrey Open 2019 probíhala v první polovině dubna 2019. Do singlové soutěže monterreyského tenisového turnaje hraného na antuce nastoupilo třicet dva hráček. Obhájkyní titulu byla Muguruzaová.
Vítězkou se stala druhá nasazená Španělka Garbiñe Muguruzaová, jež ve finále zdolala běloruskou turnajovou pětku Viktorii Azarenkovou. Poražená bývalá světová jednička za stavu 1–6 a 1–3 utkání skrečovala pro poranění pravého lýtka. 25letá Muguruzaová si v probíhající sezóně připsala premiérové turnajové vítězství, které představovalo sedmý singlový titul na okruhu WTA Tour a vůbec první obhájený.
Nasazení hráček
Pavouk
Finálová fáze
Horní polovina
Dolní polovina
Odkazy
Reference
Externí odkazy
Monterrey Open
WTA Tour 2019 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,480 |
\section{Introduction}
Large scale poloidal magnetic flux that threads the event horizon
(EH) of a rotating black hole (BH) forms an event horizon
magnetosphere (EHM) that is a viable source of BH driven jets. Since
the BH cannot be a source of plasma, the EHM is charge starved
(lacks a supply of charge that is required to support a frozen-in
magnetosphere, everywhere) and strong analogies with pulsar driven
winds have been made \citep{bla77}. A large distinction between
these two environments is that the neutron star (NS) is a
superconductor and supports magnetic fields with the largest field
strengths in the known Universe ($\sim 10^{8}\rm{G} -
10^{14}\rm{G}$), whereas the BH cannot support its own magnetic
field, since the field must be produced outside of the EH
\citep{pun08}. This distinction has not been considered in depth in
previous treatments of EHM jets. This paper discusses plausible
astrophysical circumstances in which this distinction has a crucial
effect on the physics of the system.
\par The EHM is located within the vortex of the accretion flow and requires
plasma injection in order to maintain a jetted system \citep{bla77}.
In seminal efforts, two viable options for producing the plasma in
the EHM were postulated. The first was drawn directly from pulsar
theory. In the charge starved limit, various types of vacuum gaps
and null (zero density) surfaces can exist in principle. As in
pulsar theory, the semi-vacuum electric field in these gaps can
accelerate leptons to very high energy thereby powering multi-stage
pair creation scenarios that seed the magnetosphere with an ideal
magnetohydrodynamic (MHD) plasma \citep{stu71,che86}. Analogously,
EHM gap models always assume that a background magnetic field is
already present \citep{bes92,hir98,hir16,bro15,pti16,lev00,lev11}.
The other idea, unique to BHs, is that the ambient $\gamma$-ray
field (presumably from the accretion flow) can produce enough
electron-positron pairs to seed the EHM. This study considers these
scenarios in the context of creating (as opposed to perpetuating) an
EHM in realistic astrophysical environments. In particular, in any
BH time evolution problem a causal temporal order of events is
required to establish the initial state. Without this key element as
part of the solution, it is not clear that a physical solution is
attained.
\par In Section 2, the details of the
scenario in which the EHM is created by a slow accumulation of thin,
weak, isolated magnetic flux tubes that are transported to the EH by
an accretion flow is explored. This assumed model of the seeding of
the EHM is the basis of the analysis of the charge starved limit
discussed in this paper. By ``thin, weak, isolated magnetic flux
tubes" it is meant:
\begin{itemize}
\item thin: the dimensions of the flux tubes are very narrow compared to the
dimensions of the disk and BH,
\item isolated: the large scale
poloidal flux that extends above or below the disk consists of a few
strands of flux that extend off to infinity as opposed to closing as
loops back into the disk. They accrete sporadically as opposed to a
nearly uniform, continual deposition of flux tubes into the EHM
\item weak: the field lines are readily deformed by the
surrounding disk atmosphere. The field strength is less than that
which is required to initiate a pair cascade.
\end{itemize}
The EHM solution considered here is evaluated in the charge starved
limit. Without sufficient plasma, it is shown that the accreted
poloidal magnetic flux readily dissipates in the EHM. The
dissipation is rapid relative to the rate that plausible accretion
scenarios can replenish the flux. Thus, a highly magnetized EHM is
not created.
\par The radio galaxy, M87, appears
to be an ideal candidate for the new EHM solution. It has a very low
luminosity accretion flow with arguably too low a photon flux to
support significant pair creation on weak accreting flux tubes in
the EHM. Furthermore, new high resolution Very Long Baseline (VLBI)
86 GHz VLBI observations resolve the jet in M87 on scales much
closer to the central BH than has been accomplished for any other
radio loud active galactic nucleus (AGN) \citep{kim16,had16}. These
images reveal a jet with an unexpected forked topology that seems to
represent a hollow jet (see Section 4). There is no evidence of
significant jet emission along the central spine above the event
horizon in agreement with the new EHM solution to be presented in
this paper. The new EHM solution is particularly relevant in the
context of recent models of hollow jets emanating from the inner
regions of an accretion flow that can describe a very wide range of
plausible broadband spectra (mm wavelengths to UV) of the base of
the jet in M87 on scales $\sim 15 -30\mu\rm{as}$. In addition to
explaining broadband emission from the region that produces the
correlated 230 GHz flux detected by the Event Horizon Telescope
(EHT), the jet base has sufficient power to energize the entire jet
out to kiloparsec scales \citep{pun18}. There is no need to invoke a
powerful invisible spine jet driven by the EHM in order to power the
jet \citep{mos16}. This supports the most direct interpretation of
the 86 GHz VLBI images: the jet is hollow because the EHM jet is
intrinsically weak in accord with the model presented here. Thus
motivated, much of the discussion to follow is focused on the
example of M87.
The paper is organized as follows. Section 2 is a discussion of the
details of the time evolution of weak, isolated flux tubes in the
completely charge starved limit. This section assumes negligible
plasma injection into the EHM in order to describe the new solution
of the EHM that is proposed here. Without plasma injection from the
external environment or a particle creation gap in the weak flux
tubes, currents cannot be maintained. Flux is dissipated, not
accumulated, if it accretes to the EHM. In Appendices A -C, the
details of the dissipation of the poloidal magnetic flux transported
within the charged starved, accreting flux tube is explored by means
of approximate solutions to Maxwell's equations in curved spacetime.
The lack of a reservoir for accreted flux in the EHM indicates a
weak EH driven jet.
\par The second part of the paper focuses on the application of the model to M87. The new model of the
magnetosphere is predicated on the mode of accretion and inefficient
pair creation. It is shown in Appendix D that for any plausible
model there is some minimum field strength below which the posited
accreted flux tubes will not produce a potential difference across
the vacuum gap large enough to initiate a pair cascade. Thus, pair
production in an external $\gamma$ -ray field would be required to
seed the EHM with plasma and would determine the maximum sustainable
magnetic field and jet power in the EHM. In Section 3, the
observational evidence that bounds the $\gamma$ -ray luminosity of
the inner accretion flow, from above, in M87 is discussed. No
$\gamma$ -ray telescope can resolve the inner accretion flow. The
highest resolution observations of the hard photon spectrum are with
the Chandra X-ray telescope. The core flux within 0.67 arcsec of the
nucleus is extracted. This is combined with broadband hard photon
spectra of active galactic nuclei from INTEGRAL in order to give
bounds on the $\gamma$ -ray luminosity from the nucleus. This in
turn implies an upper bound on the maximum sustainable magnetic
field strength in the EHM and the resultant maximum Poynting flux
that can be delivered by an EHM jet in M87. It is concluded that M87
is likely an example of a source with a weak $\gamma$-ray field near
the EH that is incapable of producing enough pairs to support the
currents required for an astrophysically significant EHM. In Section
4, it is noted that the results of Sections 2 and 3 and Appendix C
indicate that M87 is a possible example in which the EHM is so
charge starved that any jet produced in this region will be very
weak. It is shown that HSA (High Sensitivity Array) observations at
86 GHz support the new EHM model. There is a profound nadir of
emissivity along the central spine at the jet base above the
putative EHM that is consistent with this basic consequence of the
new EHM solution. In the following, it is assumed that $ M = 6
\times 10^{9} M_{\odot}$ ($8.4 \times 10^{14} \rm{cm}$ in
geometrized units) appropriate for M87 \citep{geb11}.
\begin{figure*}
\includegraphics[width= 0.5\textwidth]{be.eps}
\includegraphics[width= 0.5\textwidth]{isotube_iso.eps}
\caption{The accretion of weak flux tubes into the EHM occurs in the
SFMHD+MF simulation of \citet{bec09} in the left hand frame. The
right hand frame shows isolated flux tubes in the 3-D radiatively
inefficient simulations of \citep{igu08} and \citep{pun09}. The
strength of the vertical poloidal magnetic field is color coded.
Dark blue is no field and red is a strong field (near equipartition
with the gas pressure). The inner calculational boundary is a circle
of radius 2M. Notice the weaker, green, small patches of vertical
flux in the inner accretion flow. See the text for more details.}
\end{figure*}
-------------------------------------------------------
\section{The Creation of an EHM by Accretion}
This study considers a possible new EHM solution that might occur in
some astrophysical black hole accretion systems. It is predicated on
a particular mode of accretion onto a rotating (Kerr) BH described
by a mass, $M$, and an angular momentum per unit mass, $a$. The
context is the initial seeding of the BH magnetosphere with large
scale poloidal flux. This is the initial state for the time of
evolution of the EHM. The specific details of how an EHM is
established are not known, and the processes involved are on too
small a scale to be observed directly, even if one were to be
observing during the initial stages. Thus any scheme for
establishing the EHM must rely on assumptions. It is known that the
flux must be delivered from the external environment since the Kerr
BH does not support a magnetic field in isolation. A plausible
method of creating a significant EHM is the radiatively inefficient
accretion of weak poloidal magnetic flux from large distances over a
long period of time \citep{igu08,bec09,mck12}. Similar ideas have
been proposed for protostellar systems \citep{ler99}. It has been
suggested that the large scale magnetic flux near a black hole has
its origins in the advection of the weak large scale patchy magnetic
field in the intergalactic medium or from a magnetized stellar wind
or a tidal disruption event of a nearby magnetized star
\citep{mck12}. This is the scenario considered in this model of the
EHM and it is the fundamental assumption of this paper.
\par Note that the charge starved limit and the assumed pair creation in an EHM violates
perfect MHD. There are no existing numerical simulations that can
study this limit. Perfect MHD numerical experiments involving
accreting mass always end with the code crashing before the
charge-starved limit is approached (see \citet{mei01} and references
therein). Thus, numerical simulations artificially insert a non-MHD
mass floor that perpetuates the solution \citep{dev03,mck04}. One
can distinguish these from ideal MHD simulations by denoting them as
SFMHD+MF (single fluid MHD plus mass floor) in the following. A
numerical simulation that utilizes a mass floor is not an acceptable
device if one is considering the time evolution of a charge depleted
system as is the case here. Thus, the dynamics of the charge starved
accretion into the EHM will be described in what follows by
approximate analytic arguments.
The dynamics take place in the background spacetime of a rotating
black hole, the Kerr solution. In Boyer--Lindquist coordinates, the
Kerr metric, $g_{\mu\nu}$, is given by the line element in
geometrized units
\begin{eqnarray}
&&\mathrm{d}s^{2} \equiv g_{\mu\nu}\, \mathrm{d}x^{\mu}\mathrm{d}
x^{\nu}= -\left (1-\frac{2Mr}{\rho^{2}}\right)
\mathrm{d}t^{2}+\rho^{2}\mathrm{d}\theta^{2} \nonumber \\
&& +\left(\frac{\rho^{2}}{\Delta}\right)\mathrm{d}r^{2}
-\frac{4Mra}{\rho^{2}}\sin^{2}\theta\, \mathrm{d}\phi \,
\mathrm{d}t \nonumber \\
&& +\left [(r^{2}+a^{2})+\frac{2Mra^{2}}{\rho^{2}}\sin^{2}
\theta\right ] \sin^{2} \theta \, \mathrm{d}\phi^{2} \; ,
\end{eqnarray}
where $\rho^{2} = r^{2}+a^{2}\cos^{2}\theta$ and
\begin{equation}
\Delta = r^{2}-2Mr+a^{2} \equiv \left(r-r_{{+}})(r-r_{{-}}\right ).
\end{equation}
There are two event horizons given by the roots of the equation
$\Delta=0$. The outer horizon at $r_{{+}}$ is of physical interest
\begin{equation}
r_{{+}}=M+\sqrt{M^{2}-a^{2}} \; .
\end{equation}
In order to simplify the calculations, one can compute quantities in
a hypersurface orthogonal, orthonormal frame. There exists an
orthonormal, Zero Angular Momentum Observers (ZAMO) frame associated
with each coordinate pair, $(r,\, \theta)$. The ZAMOs can be used to
express, locally, the electromagnetic field in terms of electric and
magnetic (observer-dependent) fields. There are three main benefits
of calculating in the ZAMO frames. The orthonormality condition is
beneficial for utilizing many results and techniques from special
relativity. By contrast, the Boyer-Lindquist coordinates are
curvilinear and not even orthogonal. Thus, a physical interpretation
of the covariant and contravariant quantities near the black hole is
far from trivial. Secondly, unlike other orthonormal frames, being
hypersurface orthogonal, the ZAMO frame provide an unambiguous
definition of the electromagnetic field that is integrable
\citep{pun08}. Most importantly, as shown in Appendix B, one can
rotate the poloidal direction to always be along the local poloidal
magnetic field direction. This greatly simplifies the interpretation
of the electromagnetic quantities. Even though calculations are much
clearer in the rotated ZAMO basis, ultimately we need to express the
results in terms of the Boyer-Lindquist coordinates associated with
the stationary observers at asymptotic infinity. Thus, we describe
the transformation between frames. The ZAMO basis vectors are
\begin{eqnarray}
&& \hat{e}_0 = \alpha_{\rm{Z}} ^{-1}\left(\frac{\partial}{\partial t} + \Omega_{\rm{Z}}\frac{\partial}{\partial\phi}\right) \;\nonumber \\
&& \Omega_{\rm{Z}} =\frac{-g_{\phi\, t}}{g_{\phi\phi}} \;,\; \alpha_{\rm{Z}} =\frac{\sqrt{\Delta} \sin{\theta}}{\sqrt{g_{\phi\phi}}}\;, \nonumber \\
&&\hat{e}_\phi=\frac{1}{\sqrt{g_{\phi\phi}}}\frac{\partial}{\partial\phi}\
,\;\hat{e}_r = \left( \frac{\Delta^{1/2}}{\rho} \right)
\frac{\partial}{\partial r} \ , \;\hat{e}_\theta =\left(
\frac{1}{\rho} \right) \frac{\partial}{\partial \theta} \; .
\end{eqnarray}
The lapse function, $\alpha_{\rm{Z}}$, is the gravitational redshift
of the ZAMOs as measured by the stationary observers at asymptotic
infinity (i.e., astronomers on earth). Note that
\begin{eqnarray}
&&\lim_{r \rightarrow \infty} \alpha_{\rm{Z}} = +1 \; ,\\
&&\lim_{r \rightarrow r_{+}} \alpha_{\rm{Z}} = 0 \;.
\end{eqnarray}
Similarly, $\Omega_{\rm{Z}}$, is the angular velocity of the ZAMOs
as viewed by stationary observers at asymptotic infinity.
The basis covectors are
\begin{eqnarray}
&&\omega^{\hat 0}=\alpha_{\rm{Z}}dt\; ,\; \omega^{\hat
r}=\sqrt{g_{rr}}dr\; , \nonumber \\ && \omega^{\hat
\theta}=\sqrt{g_{\theta\theta}}d\theta\;, \;\omega^{\hat
\phi}=\sqrt{g_{\phi\phi}}d\phi \;.
\end{eqnarray}
Boyer-Lindquist evaluated quantities are distinguished from ZAMO
evaluated quantities by the use of a "tilde" on the variables. Both
formalisms will be utilized in the description of the flux
evolution.
\subsection{Weak Isolated Flux Tubes in the EHM} The concept of a weak isolated magnetic flux tube is
introduced by means of SFMHD+MF simulations. In the initial state
there is no large scale poloidal flux that threads the event
horizon. There needs to be a mechanism that can transport large
scale poloidal flux to the EHM. The accretion flow is the natural
place to look for such a source. Attempts to spontaneously create
the flux from the accretion flow itself by means of the
magneto-rotational instability (MRI) proved to be unsuccessful
\citep{bec09}. A simulation requires a net poloidal flux in the
accretion flow in order to build up a significant EHM
\citep{igu08,bec09}. When the simulation starts there is a transient
state when the first flux tubes approach the event horizon. It will
look similar to the t=1500M snapshot from a SFMHD+MF simulation of
\citet{bec09}, depicted in Figure 1. The magnetic flux is clearly
weaker in the EHM than in the disk and a single field line is
separated by a large gap from the magnetic field in the disk. This
is an accreted isolated flux tube created in the early stages of a
SFMHD+MF simulation. All transient early stages of SFMHD+MF
simulations create an EHM by beginning with the arrival of a first
weak flux tube, unless the initial state is unphysical and posits
large amounts of flux in the initial state in the EHM proper or
adjacent to the EHM. This is true even if a saturated magnetosphere
is attained at large times \citep{ mck12}.
\par The right hand frame of Figure 1 shows a different depiction of
isolated flux tubes in the 3-D radiatively inefficient simulation of
\citep{igu08} and \citep{pun09}. This frame is from the on-line
movies of the latter reference. The strength of the vertical
poloidal magnetic field is color coded. Dark blue is no field and
red is a strong field (near equipartition with the gas pressure of
the surrounding accretion flow). Notice that the field accumulates
in isolated patches. Even though it was definitely not the intent of
this simulation, in this image there are small patches of weak field
near the inner boundary (a circle of radius 2M). The greenish-yellow
patches have a magnetic pressure $\sim 2\% -10\%$ of equipartition
with the gas pressure of the surrounding gas. These are examples of
weak isolated flux tubes. It is important to note that in this
simulation they formed as a consequence of the amalgamation of a
steady influx of very weak field from the outer calculation
boundary. The flux reservoir at the outer boundary is axisymmetric,
but the accretion flow is not. The 3-D accretion flow is driven by
the MRI as in the \citet{bec09} simulations. However, these
simulations have a much larger reservoir of flux at the outer
boundary. If there is a large reservoir of poloidal flux,
condensations of vertical flux will naturally occur as a consequence
of the MRI driven turbulence. In general the isolated flux tubes are
more magnetized in other time snapshots. However, these simulations
suggest that weak isolated vertical flux tubes might be natural in
an accretion flow. The patches of vertical magnetic flux near the
black hole should be weaker and more isolated if the reservoir of
flux is a weak patchy intergalactic magnetic field as opposed to a
constant flood of flux as in the simulation in Figure 1.
\subsection{Relevant Assumptions of SFMHD+MF Simulations} In this
paper, the early time behavior of a nascent EHM is analyzed after
abandoning some major assumptions of the SFMHD+MF simulations. In
particular:
\begin{enumerate}
\item The notion of a mass floor is dropped. Physically, this
equates to a black hole accretion system in which there is no
efficient plasma injection mechanism to support the flux in the EHM.
\item There is no large
reservoir of magnetic flux that persistently deposits flux into the
EHM. It is instead assumed that the flux deposits into the EHM on
astronomically large time scales. For example, the jet propagation
speeds indicate a jet lifetime of $>10^{6}$ years for many radio
loud AGN \citep{wil99}. This is $>10^{8}$ light travel times across
the black hole in M87. Even a small fraction of this time scale is
not computer resource efficient for SFMHD+MF simulations, so a more
compact flux source is assumed in those numerical models. However, a
compact source is not a valid assumption if the rate that flux
accretes is dynamically important as in this section and Appendix C.
\item It is also not assumed that the distant flux reservoir is uniform,
but is composed of small distinct patches of isolated flux,
\end{enumerate}
By dropping assumption 1), there will be insufficient plasma to
support MHD. In the low or zero pair creation limit, it is shown in
Appendix C that the magnetic flux will dissipate in the EHM on a
timescale, $t_{\rm{dis}}$. that is estimated. Dropping assumption 3)
allows for an non uniform deposition of flux into the EHM over time.
This naturally produces temporal gaps between episodes in which
isolated patches of accreted flux are deposited in the EHM. The
dynamical timescale to deposit more flux, $t_{\rm{dyn}}$, can exceed
$t_{\rm{dis}}$ allowing the flux tube to dissipate before an
accumulation of flux can occur.
\subsection{Maxwell's Equations Description of a Weak Isolated flux
Tube} Figure 2 shows an idealized isolated, large scale, poloidal
flux tube accretion scenario. There are two components of the
magnetic field in the accretion flow since the system is in rotation
with the plasma, $B^{\phi}$ and $B^{P}$, azimuthal and poloidal
respectively. In the thin flux tube limit (so thin that cross-field
gradients in the current and field are negligible compared to the
gradients at the boundaries), the electromagnetic sources are
approximately surface currents. To quantify this for flux tubes that
emanate from the accretion disk, a cylindrical coordinate system in
flat space is chosen for demonstrative purposes,
$(\rho^{\rm{cyl}},\, \phi, \, z)$. The inner boundary of the flux
tube is $\rho^{\rm{cyl}}_{-}(z)$ and the outer boundary is
$\rho^{\rm{cyl}}_{+}(z)$, where axisymmetry is assumed for
simplicity. The thin flux tube limit is defined for small
$\epsilon>0$ by the conditions,
\begin{eqnarray}
&&\frac{\mid B^{\phi}(\rho^{\rm{cyl}}_{-}(z)-\epsilon, \,z)\mid }{\mid B^{\phi}(\rho^{\rm{cyl}}_{-}(z), \,z)\mid}\ll 1\\
&&\frac{B^{\phi}(\rho^{\rm{cyl}}_{+}(z)+\epsilon, \,z)\mid }{\mid B^{\phi}(\rho^{\rm{cyl}}_{+}(z), \,z)\mid}\ll 1\\
&&\frac{\mid B^{P}(\rho^{\rm{cyl}}_{-}(z)-\epsilon, \,z)\mid }{\mid B^{P}(\rho^{\rm{cyl}}_{-}(z), \,z)\mid}\ll 1\\
&&\frac{\mid B^{P}(\rho^{\rm{cyl}}_{+}(z)+\epsilon, \,z)\mid }{\mid B^{P}(\rho^{\rm{cyl}}_{+}(z), \,z)\mid}\ll 1\\
&&\frac{\mid B^{\phi}(\rho^{\rm{cyl}}_{-}(z), \,z) -
\mid B^{\phi}(\rho^{\rm{cyl}}_{+}(z), \,z)\mid }{\mid B^{\phi}(\rho^{\rm{cyl}}_{-}(z), \,z)\mid}\ll 1\\
&&\frac{\mid B^{P}(\rho^{\rm{cyl}}_{-}(z), \,z) - \mid
B^{P}(\rho^{\rm{cyl}}_{+}(z), \,z)\mid }{\mid
B^{P}(\rho^{\rm{cyl}}_{-}(z), \,z)\mid}\ll 1 \\
&&\frac{\rho^{\rm{cyl}}_{+}(z)-\rho^{\rm{cyl}}_{-}(z)}{\rho^{\rm{cyl}}_{-}(z)}\ll
1
\end{eqnarray}
The fact that the slowly accreting isolated flux tubes have a $B$
field much stronger than that of the plasma on both sides of the
flux tube means that the surface current will change the field
strength from approximately zero to $B$ in Ampere's law at the inner
face of the flux tube. Similarly, the surface current will change
the field from $B$ to near zero at the outer face of the flux tube.
Since the flux tube accretes with the plasma in the disk, it
essentially spirals with the Keplerian velocity with a relatively
slow inward radial drift \citep{sad11}. Thus, to first
approximation, one can ignore displacement current in Amperes's law
for the field inside the axisymmetric flux tube. Let $\mathbf{K}$
designate a surface current. In the approximately cylindrical
configuration. by Ampere's Law and Equations (8) - (14),
\begin{eqnarray}
&&\frac{4\pi}{c}K^{\phi}(\rho^{\rm{cyl}}_{-}(z), \,z) \approx -B^{P}(\rho^{\rm{cyl}}_{-}(z), \, z)\\
&&\frac{4\pi}{c}K^{\phi}(\rho^{\rm{cyl}}_{+}(z), \,z) \approx B^{P}(\rho^{\rm{cyl}}_{+}(z), \, z)\\
&&\frac{4\pi}{c}K^{P}(\rho^{\rm{cyl}}_{-}(z), \,z) \approx \rho^{\rm{cyl}}_{-}(z)B^{\phi}(\rho^{\rm{cyl}}_{-}(z), \, z)\\
&&\frac{4\pi}{c}K^{P}(\rho^{\rm{cyl}}_{+}(z), \,z) \approx -\rho^{\rm{cyl}}_{+}(z)B^{\phi}(\rho^{\rm{cyl}}_{+}(z), \, z)\\
&&\frac{4\pi}{c}K^{\phi}(\rho^{\rm{cyl}}_{-}(z), \,z) \approx - K^{\phi}(\rho^{\rm{cyl}}_{+}(z), \,z)\\
&&\frac{4\pi}{c}K^{P}(\rho^{\rm{cyl}}_{-}(z), \,z) \approx -
K^{P}(\rho^{\rm{cyl}}_{+}(z), \,z)
\end{eqnarray}
The integral of $K^{P}$ over an orthogonal cross-section of either
the inner or outer boundary of the flux tube (the total poloidal
current) is approximately conserved from the disk to asymptotic
infinity in the axisymmetric, magnetically dominated limit and a
conserved value represents electromagnetic angular momentum flux
conservation in the flux tube \citep{pun08}. $K^{\phi}$ is set by
the poloidal magnetic flux conservation condition from the accretion
flow to asymptotic infinity in each flux tube. The corresponding
curved spacetime versions of these surface current equations are
derived in Appendix B in the ZAMO frames.
\begin{eqnarray}
&&\frac{4\pi}{c}K^{\phi}_{\rm{Z}}(r_{\rm{in}}, \, \theta_{\rm{in}}) \approx -B^{P}(r_{\rm{in}}, \, \theta_{\rm{in}})\\
&&\frac{4\pi}{c}K^{\phi}_{\rm{Z}}(r_{\rm{out}}, \,
\theta_{\rm{out}}) \approx B^{P}(r_{\rm{out}}, \,
\theta_{\rm{out}})\\
&&\frac{4\pi}{c}K^{P}_{\rm{Z}}(r_{\rm{in}}, \, \theta_{\rm{in}}) \approx B^{\phi}(r_{\rm{in}}, \, \theta_{\rm{in}})\\
&&\frac{4\pi}{c}K^{P}_{\rm{Z}}(r_{\rm{out}}, \, \theta_{\rm{out}})
\approx -B^{\phi}(r_{\rm{out}}, \, \theta_{\rm{out}})\;.
\end{eqnarray}
These equations are required near the black hole. The
Boyer-Lindquist coordinates, $(r_{\rm{in}}, \, \theta_{\rm{in}})$
indicates a point on the inner boundary of the flux tube and
$(r_{\rm{out}}, \, \theta_{\rm{out}})$ indicates a point on the
outer boundary of the flux tube.
\subsection{The Dynamics of Accreted Weak Isolated Flux
Tubes} During the inflow through the disk, the source of the charges
that create the currents that sustain the magnetic flux is in the
base of the flux tube that is frozen into the accretion flow. Plasma
is shot outward by magneto-centrifugal forces in the rotating flux
tube and dragged inward near the base (accretion) by gravity
\citep{igu08}. The plasma that is shot outward is provided by the
accretion flow before the flux tube enters the EHM.\begin{figure}
\includegraphics[width= 0.5\textwidth]{ft2.eps}
\caption{ The dynamics of the accretion of weak flux into the EHM is
depicted above. The accretion flow in this image is likely much
thinner than what occurs in M87. The concepts that are illustrated
are not a consequence of the accretion flow thickness.}
\end{figure}
\par Figure 2 shows that the dynamic that existed in the flux tube in the accretion disk
persists as it enters the EHM. In particular, plasma is still shot
outward by magneto-centrifugal forces in the rotating flux tube and
still dragged inward (accretion) by gravity \citep{mei01,sem04}. At
the flow division surface, the flow divides into an accretion flow
and an outgoing wind \citep{phi83}. Due to gravitational redshifting
and frame dragging, the plasma in the flux tube near the EH is out
of causal contact with the large scale poloidal flux \citep{pun08}.
In this discussion, it is assumed that there is no external plasma
injection mechanism such as pair production (see Appendix D and
Section 3 for the likelihood of this possibility in M87). Plasma
that is already threaded on the flux tube must provide the outgoing
plasma and the currents supporting the magnetic field.
\footnote{Note that there is no dynamic at the disk-EHM boundary
that naturally changes the MHD flux tube, with its local current
system, into a flux tube in which the source is transferred to a
surface current that resides at the inner surface of the disk.}.
There is a finite amount of plasma in the flux tube and the plasma
quickly becomes tenuous. The plasma starts to drain from the flow
division surface producing a vacuum gap as depicted in Figure 3. The
figure is a schematic diagram that shows the split that occurs in
the distribution of plasma, not the field lines, as the vacuum gap
begins to expand. Initially, the field lines are not severed in the
vacuum gap. However, the poloidal magnetic field is not uniform in
this region. The poloidal field bulges outward and inward as
fringing effects become pronounced. A laboratory example of this
effect would occur if one split a long solenoid in the middle, then
pulled the two halves away from each other along the axis of
symmetry. A non-uniform bulging field occurs in the gap between the
two coils. At later times, the fringing fields associated with the
spreading vacuum gap expand and can approach other fringing field
lines along circles (due to axisymmetry) of X-type reconnection
points. This reconnection process can change the topology of the
poloidal magnetic field.
\begin{figure}
\includegraphics[width= 0.5\textwidth]{M87_3.eps}
\caption{ In the charge-starved limit, a vacuum gap will spread
outward from the flow division surface if there is no substantive
pair injection mechanism as is quite possibly the case in M87. In
this charge starved limit, the surrounding magnetosphere is a
semi-vacuum. The only significant source of electromagnetic fields
is within the flux tube. The very tenuous stray charges have
trajectories that are affected by these fields, but the number
density is too small to provide a source for a significant
perturbation to these fields. Charges can only flow inward across
the charge horizon \citep{kom03,pun04}}
\end{figure}
\par In Appendix C, we discuss a model of an accreted flux tube in
which there are insufficient charges to maintain the source currents
in the EHM - charge starved. The lower portion of the flux tube
contracts toward the black hole by gravity and the outer is slung
out by magneto-centrifugal forces. Evaluating Maxwell's equations as
the inner portion of the flux tube approaches the event horizon
indicates that the large scale poloidal magnetic field in the EHM
will approximately be a decaying magnetic dipole (see Figure C.2).
Since the calculation is very long and involved, we only summarize
the logic and results in the main text.
\begin{enumerate}
\item Equations (21) -(24) are used to describe the current
distribution in the ingoing portion of the severed flux tube as two
nested, coaxial helical surface currents distributions, one in each
hemisphere.
\item In Section C.1, it is shown that due to gravitational redshift
as these helical current flows accrete close to the event horizon
they seem to be frozen in corotation with the horizon, hovering just
above it, as viewed by external observers. Thus, these axisymmetric
electromagnetic source are approximately time stationary to the
external observers (eg. in Boyer-Lindquist coordinates) that would
be affected by the large scale poloidal magnetic field. Therefore,
Laplace's equations can be used to accurately depict the large scale
poloidal magnetic field for these sources at any given
Boyer-Lindquist time, $t$ \citep{pun89}.
\item In Section C.2, it is shown that at late times in the accretion, near the event
horizon, the large scale poloidal magnetic field from the helical
current sources can be approximated as the large scale poloidal
magnetic field due to 4 azimuthal current rings that are located
near the black hole (see Figure C.1).
\item In Section C.3, the large scale poloidal magnetic field produced by the four
current loops is calculated by means of Laplace's equations in
curved spacetime and the results plotted in Figure C.2. The large
scale poloidal magnetic field is approximately a decaying magnetic
dipole.
\item In Section C.4, it is estimated that the flux tube dissipates (magnetic dipole
decays) on a time scale, $t<10M$, after the vacuum gap starts to
spread apart. This time scale is much less than any time scale of
the accretion flow. Thus, for the accretion scenario posited in this
section, the flux will dissipate before more flux can accumulate in
the EHM. A highly magnetized EHM will not form.
\item In Section C.4, based on Figure C.2, it is argued
that surface currents induced in the disk during the field decay do
not prevent accreted, thin, isolated flux tubes in a charge starved
EHM from dissociating. These currents are decaying and are of the
wrong sign to maintain the accreted flux within the EHM.
\end{enumerate}
This suggests that an interesting new dynamic can exist in the EHM
if the EHM is charge starved: no vacuum gap pair cascades and weak
$\gamma$-ray pair production. Thus motivated, standard vacuum gap
pair production in the EHM are considered in Appendix D and
$\gamma$-ray pair production is discussed in the case of M87 in the
next section.
\section{The $\gamma$-Ray Induced Pair Creation in the EHM of M87}
\begin{figure*}
\begin{center}
\includegraphics[width= 0.8\textwidth]{bmax.eps}
\caption{Plots of the upper limits on $L_{C}$ and the maximum
sustainable Poynting flux from the EHM for M87 as a function of
$E_{c}$ for two cases, $L_{x}$ is entirely from accretion or $L_{x}$
from accretion is attenuated by an absorbing screen, $N_{H}$. The
maximum sustainable magnetic field in the EHM assuming that
$a/M=0.9$ is also plotted.}
\end{center}
\end{figure*}
In the weak field limit, proposed in the last section, the EHM will
not be able to sustain pair creation in a vacuum gap (see Appendix D
for more elaboration). Thus, pair creation in an external
$\gamma$-ray field is required in order to provide plasma to the
accreted flux tubes and this will determine the maximum sustainable
magnetic field strength in the EHM. This particle injection
mechanism is considered in the context of the accretion scenario of
Section 2 in the environment of M87.
\par $\gamma$-rays from the jet in M87 are produced relatively far away and beamed away
from the EH and do not contribute to EHM pair production. However,
the $\gamma$-ray field of the accretion flow can produce
electron-positron pairs in the EHM. In this section, the available
data related to the hard photon spectrum of M87 is considered in
order to make as precise as possible any constraints that can be
imposed on the $\gamma$-ray luminosity. The resolution of telescopes
in the $\gamma$-ray band is many orders of magnitude too low to be
of any use. However, the low energy region of the hard photon
spectrum can be resolved to within 0.67 arcsec by Chandra. This
information is used in consort with what is known about the hard
photon spectra of other AGN (in particular, the cutoff energy) in
order to constrain the $\gamma$-ray luminosity in M87. Even though
it will be concluded that the Chandra flux is likely from the jet
itself, this detection still provides a useful and non-arbitrary
bound on the hard photon spectrum from the accretion flow.
\par The number density of created pairs from a background
$\gamma$-ray field can be estimated by balancing the infall
(free-fall) rate with the pair creation rate \citep{phi83}
\begin{equation}
n \sim \left( \frac{m_p}{m_e} \right) \left(
\frac{L_{_C}}{L_{_{Edd}}} \right)^2 10^{13} M^{-1}_{8} \,
\mbox{cm}^{-3} \; ,
\end{equation}
\noindent where $L_{_C}$ is the luminosity of $\gamma$-rays $> 1$
MeV from the accretion flow, $L_{_{Edd}}$ is the Eddington
luminosity and $M_{8}$ is the mass of the black hole in units of
$10^{8}M_{\odot}$. If the pair creation process can produce a charge
density in excess of the Goldreich-Julian density, $ \rho_{_{G-J}}$,
then the growth of the electric field in the vacuum gap can be
quenched and the surface current flow sustained on the flux tube
\citep{gol69}. One can estimate $ \rho_{_{G-J}}$ near the EH
\begin{eqnarray}
&& \rho_{_{G-J}}\sim \frac{\Omega_{F} B}{2 \pi ce} \sim \sigma
\frac{10}{M_{8}}\left(\frac{B}{10^{4}\,\mathrm{G}}
\right)\mathrm{cm}^{-3} \;, \\
&& \Omega_{F} \equiv \sigma \Omega_{H} \;,
\end{eqnarray}
\noindent where $\Omega_{F}$ and $\Omega_{H}$ are the angular
velocity of the magnetic field and the event horizon angular
velocity as viewed from asymptotic infinity, respectively. For a
given $\gamma$-ray field, the condition, $n_{e}>\rho_{_{G-J}}$,
determines the maximum sustainable $B$ field in a thin accreting
magnetic flux tube in the EHM.
\par $L_{_C}$ in M87 is constrained by revisiting the estimate of the accretion flow X-ray
luminosity, $L_{x}$, from \citet{har09}, with a smaller extraction
region (correcting for the PSF outside the region) of 0.67 arcsec
(versus 1 arcsec) to avoid contamination from the knot, HST-1, in
the Chandra data \citep{har03}. No detectable X-ray excess above a
single unabsorbed power law flux density was observed: $\alpha_{x} =
1.1$, $L_{E} \propto E^{-\alpha_{x}}$, where $E$ is photon energy
and $L_{x}= 2.9 \times 10^{40}\rm{erg/s}$ from 2-10 keV. The
nucleus is a continuation of the large scale X-ray jet with similar
values of $L_{x}$ and $\alpha_{x}$ to those of the knots in the jet
\citep{wil02}. Mid-IR and optical studies conclude that there is no
hidden strong accretion source, but just a synchrotron nuclear
source in M87 \citep{why04,chi99}. Broadband correlations amongst
the nuclear synchrotron and X-ray fluxes in many Fanaroff-Riley I
(FRI) radio galaxies such as M87 also imply a jet origin for X-rays
\citep{har00,har09}.
\par An upper bound for $L_{_C}$ due to accretion can be estimated in two ways from the Chandra data.
First, consider the limiting scenario (although it is unlikely
considering the discussion above) that the Chandra nuclear flux is
from the accretion flow. This estimate is performed in order to
establish the most conservative limit on the upper bound on
$L_{_C}$, Secondly, it is assumed that the accretion X-ray source is
hidden by an attenuating column of neutral hydrogen,
$10^{22}\rm{cm}^{-2}< N_{H}< 10^{23}\rm{cm}^{-2}$ and $\alpha_{x} =
0.7$ \citep{har09}. Note that there is no evidence of such a large
$N_{H}$ in M87. In this case, an intrinsic $L_{x}< 1.9 \times
10^{39}\rm{erg/s}$ from 2-10 keV with 90\% confidence is estimated.
These are ``worst case," not necessarily likely, scenarios for
producing upper bounds on $L_{_C}$.
\par The wideband $L_{x}(\rm{wb})$ from accretion in AGN and Galactic compact objects
is typically approximated by a cutoff power law, $L_{x}(\rm{wb})
\propto E^{-\alpha_{x}} \rm{e}^{-E/E_{c}}$, where $E_{c}$ is the
cutoff energy \citep{mal14}. It is assumed that the spectral index,
$\alpha_{x}$, is constant from keV to MeV energies in the following
calculations. However, the upper bounds that are computed below are
valid as long as the power law does not flatten at higher energies.
Figure 4 contains plots of three upper bounds as functions of
$E_{c}$ for both scenarios: $L_{_C}$, the associated maximum
sustainable Poynting flux from the EHM and the maximum sustainable
value of $B$ from Equations (25)-(27). The $B$ plot assumes the
seminal value of $\sigma=0.5$ from \citet{bla77} and $a/M=0.9$. The
range of $E_{c}$ appropriate to the putative accretion source of
$L_{x}(\rm{wb})$ is motivated by INTEGRAL observations indicating an
average $E_{c} =125$ keV for type I AGN and radio loud AGN in which
$L_{x}(\rm{wb})$ is not of blazar (jet) origin \citep{mal14}. The
MHD Poynting flux in the magnetically dominated limit is
\begin{equation}
\int{S^{P}\mathrm{d}A_{_{\perp}}} =
k\frac{\Omega_{F}^{2}\Phi^{2}}{2\pi^{2} c} \approx
\frac{\Omega_{F}^{2} (4\pi B(r_{+}^2))^{2}}{2\pi^{2} c}\;,
\end{equation}
where $\Phi$ is the total magnetic flux enclosed within the jet
(through the EH), $\mathrm{d}A_{_{\perp}}$ is the cross-sectional
area element (surface area element of EH) and $k$ is a geometrical
factor that equals 1 for a uniform highly collimated jet
\citep{pun08}. Using the fact that $\Omega_{H}= a/(2Mr_{+})$ and
Equations (25) - (28), the upper bound on the approximate Poynting
flux is independent of BH spin and the jet model for $\sigma$ over a
wide range: $0.4< a < 0.95$ and $0.1< \sigma <1$.
\par Figure 4 shows that the Chandra data likely imply a $\gamma$-ray accretion source in M87 that is
insufficient to support even a 1G field in a charge-starved EHM.
Furthermore, the largest upper bounds on Poynting flux are more than
three to four orders of magnitude less than the estimated jet power
of $\sim 10^{43} \rm{ergs/s} - 10^{44} \rm{ergs/s}$.
\citep{mcn11,sta06}.
\begin{figure*}
\begin{center}
\includegraphics[width= 0.70\textwidth]{map.ps}
\includegraphics[width= 0.70\textwidth]{M87cs1.eps}
\caption{The central flux nadir of the jet near its base is apparent
in the 86 GHz HSA image from Hada et al. (2016) (restored with a
0".0001 beam) in the top frame. The bottom frame plots surface
brightness cross-sections from the image above. The central flux
nadir is resolved within 56 M (0".0002) of the EH. The central flux
nadir surface brightness is $\sim 4\% - 8\%$ of the average surface
brightness on the limbs, 56 M - 112 M from the EH and $\sim 20 -
30\%$ of the average surface brightness of the limbs at 140 M.}
\end{center}
\end{figure*}
\section{Evidence for a Hollow Jet in M87}
\par This section considers possible evidence in support of
the posited model of the EHM for the particular case of M87. New
data reductions from high sensitivity 86 GHz VLBI are provided that
indicate a much larger deficit of luminosity along the jet spine at
the base of the jet in M87 than has been previously demonstrated at
lower resolution. This result is combined with lower resolution data
in order to examine the details of the new EHM model and previous
explanations of limb brightening.
\par The fundamental testable consequence of this model of the EHM is
the existence of a jet with a base that is wider than the EH (i.e.,
driven from the accretion flow) that will have a dearth of intrinsic
emissivity along its central spine, above the EH. The jet in M87 is
likely optically thin since the flux density, $F_{\nu} \propto
\nu^{-0.8}$ \citep{had16}. If the jet is hollow to first order,
lines of sight (LOS) that are nearly parallel (or anti--parallel) to
the tangent to the circumference of the jet will intersect larger
column densities of optically thin plasma than a LOS through the
middle of the jet. Thus, one expects a limb brightened appearance in
two places, one where the LOS is parallel and one where it is
anti-parallel to the tangent of the jet circumference if the jet is
hollow to first order as predicted by the new EHM model. In order to
test this prediction, consider the HSA image at 86 GHz in Figure 5
\citep{had16}. The flux nadir along the center of the jet is
resolved within 56 M (0.2 mas) of the EH and is not transient,
occurring in multiple epochs \citep{kim16}. The flux nadir from 0.2
mas to 0.4 mas can be described quantitatively in terms of the integrated
flux density. The total 86 GHz flux density of the central flux
nadir in the region 0.2 mas - 0.4 mas from the black hole is
$\approx 6\%$ of the flux density of the surrounding outer sheath
jet (hollow jet).
\par Patches of enhanced surface brightness are clearly detected in
the central void (``the spine") at 0.5 mas in Figure 5 and beyond 1.5
mas in multiple epochs with increasing prominence downstream
\citep{mer16}. In particular, the velocity field of the inner jet in
M87 has been mapped by means of a 43 GHz VLBA wavelet analysis
\citep{mer16}. Even though, the data is from 2007, seven years before
the HSA observations, the components line up reasonably well with the
ridges seen in the 86 GHz image between 0.5 mas and 1.5 mas. The 43
GHz wavelet analysis is consistent with new 22 GHz VLBI data from 2014
\citep{had17}. Within 1.5 mas, the wavelet based apparent velocities
are similar to the values obtained by \citet{had16} for the HSA
observation, $\sim~0.1c -0.4 c$, quite subluminal. The 43-GHz analysis
also provides valuable evidence of the dynamics of the spine beyond
1.5 mas from the core. The apparent velocity, $v_{\rm{app}}$, of the
individual components of the spine, at the smallest displacements from
the core for which the signal to noise of the spine is sufficient for
such estimates (1.5 mas - 2 mas from the core), is $v_{\rm{app}/}/c =
1.33 \pm 0.63 $ and $v_{\rm{app}/}/c = 1.16 \pm 0.77 $ for the
surrounding limbs \citep{mer16}. The similarity of the velocity field
for the spine and the limbs suggests that the spine is gradually being
filled by plasma that originates in the surrounding sheath and slowly
spreads inward towards the central axis, as would be expected in the
model in which the spine is empty at the jet base. In other words, the
$v_{\rm{app}}$ distribution and increased spine prominence downstream
is well explained in terms of a weak EHM jet surrounded by a hollow
jet that slowly fills in the relative void with kinematically similar
plasma as it propagates.
We consider a few possible alternative models for the observations.
\subsection{Bifurcating Obstacle} There could be an obstacle $\leq 120M$ downstream
from the black hole. When the jet collides with this obstacle, it
would bifurcate, rendering the central parts of the jet empty
without it being intrinsically so. However, the jet has the hollow
morphology in multiple epochs \citep{kim16}. So there needs to be a
quasi-stationary feature hovering $\leq 120M$ above the black hole.
We know of no physical mechanism that could create such a
quasi-equilibrium above the black hole.
\subsection{Doppler Suppression} The central spine could be
of similar emissivity to the observed sheath, but have a much higher
speed, so that Doppler suppression reduces the observed spine
surface brightness. Given the Doppler factor for the approaching jet
${\cal D} = 1/(\Gamma[1-\beta\cos\theta)]$, where $\Gamma$ is the
bulk Lorentz factor, it can easily be shown that Doppler suppression
takes place for angles to the line of sight $\theta >
\cos^{-1}[(\Gamma
-1)/(\Gamma^2-1)^{1/2}]$. For example, the bulk Lorentz factors
($\Gamma \sim 10$--$50$) often implied by observations of
superluminal motion in blazars, \citet{lis16}, Doppler suppression
will take place unless the angle to the line of sight is smaller
than a critical angle in the range $\sim 25^\circ - \sim 10^\circ$.
While this model cannot be ruled out in principle, we regard the
observed similar apparent velocities in the sheath and spine region
as evidence against it; in such a model we might expect to see
higher apparent speeds in the center of the jet.
\subsection{Ghost Jet} {The central spine could have the same speed as the
observed sheath, but have a low emissivity because the energy
density of the particles is low \citep{mos16}, forming a `ghost
jet'.} However, it is not obvious that the Poynting flux core can be
protected from an infusion of high energy particles, if it is
surrounded by an energetic outflow of protonic material from the
surrounding disk/corona accretion system. There are three
significant sources of high energy particles.
\par First, the accretion vortex in numerical simulations of
radiatively inefficient accreting systems is not the ordered
force-free environment envisioned in theoretical treatments when the
putative Poynting jet does exist in the EHM. In the simulations of
\citet{kro05}, it was found that the EHM and the jet base are very
unsteady and the accretion vortex appeared to be a cauldron of
strong MHD waves rather than what would be expected of a force-free
structure (even though the energy density of the particles is much
less than the energy density of the electromagnetic field). This
appears to be the case in the simulations of \citet{tch12}, as well,
based on the supporting online movies in which the field lines in
the vortex whip around chaotically. As these strong MHD waves crash
against the bounding sheath jet, fast magnetosonic shocks are
created. Even though, in this magnetically dominated limit, these
shocks are not highly effective at accelerating plasma to high
energy (see \citet{ken84}), there would be many such shocks. This
would be expected to imbue the Poynting flux core with a back-flow
of particles from the high energy tail of the plasma that is
energized at the shock front.
\par Secondly, it is difficult to keep the sheath plasma from mixing
into the jet, if it is there. Near the base of the jet, it was found
in 3-D numerical simulations that the corona/jet interface is
unsteady with large fingers of hot gas being injected into the
Poynting jet on scales of $\sim 20M - 30 M$ from the BH
\citep{pun07}. To accurately model such mixing of corona and jet gas
requires an accurate numerical scheme. For example, codes like HARM
which is used in \citet{tch12,mck12} do not utilize the contact
discontinuity in their Riemann solver. The absence of the contact
discontinuity tends to numerically dissipates effects associated
with abrupt density gradients \citep{pun16}. Furthermore, a recent
study of \citet{how17} showed that the typical numerical resistivity
in MHD simulations is large enough that mixing modes such as the
Kelvin-Helmholtz instability (associated with a strong magnetic
coronal loop) are highly suppressed. Thus, it is an open question
how much the corona and sheath will seed a putative strong Poynting
jet core with high energy plasma.
\par Thirdly, the chaotic behavior in the accretion vortex and the
large toroidal twisting of the field lines is not conducive to
maintaining an ordered, untangled field. Field tangling is often
called braiding in solar physics. Braided fields are believed to
release the extra energy of tangling as they relax to a more
simplified state by reconnection \citep{wil10}. Reconnection of the
braided fields in the jet can also provide high energy plasma to the
jet and the fields are strongest near its base \citep{wil10,bla15}.
\par Based on the fact that the putative ghost jet would support a pair
cascade of high energy particles in the accretion vortex,
\citet{bro15}, and the three plausible sources of high energy plasma
described above, it is not at all clear that the energy density of
the jet can be maintained low enough to keep it invisible or
extremely weak at mm wavelengths. Thus, the study of alternative
scenarios that require fewer assumptions, such as a weak EHM jet, is
worthwhile.
\section{Conclusion} This paper considers an EHM that is built up by the
accumulation of accreted weak, isolated strands of magnetic flux
over a long period of time. In the absence of a significant
background photon field, an analysis based on Maxwell's equations in
curved spacetime that was developed in Appendices A - C indicates
that the magnetic flux will readily dissipate in the EHM instead of
accumulate in the EHM. In this accretion scenario, the resultant
weak field that can be sustained in the EHM is determined by the
pair creation rate in the $\gamma$ -ray field of the accretion
disk/corona. In Section 3, evidence that M87 appears to have a weak
$\gamma$ -ray accretion source was presented based on the Chandra
X-ray spectrum of the nucleus and the high energy cutoffs of other
AGN derived form INTEGRAL observations. The derived upper bounds on
the $\gamma$ -ray luminosity renders the EHM of M87 ineffectual for
jet launching. In Section 4, it is shown that 86 GHz HSA
observations reveal a bizarre forked jet 50M - 400M from the black
hole. This is a manifestation of the weak central spine of the jet
above the EH that is expected as a consequence of the new solution
of the EHM. Many other FRI and some FRII radio galaxies also appear
to have weak accretion X-ray emission and likely weak $\gamma$ -ray
emission as well \citep{har09}. Thus, a weak or absent EHM might be
common to radio galaxies with radiatively inefficient accretion such
as M87. It is tempting to speculate that jet bases with a forked
morphology might occur in other radiatively inefficient radio
galaxies.
\par The EHM solution is consistent with recent hollow jet models from
the inner accretion flow of M87 \citep{pun18}. The models are able
to fit an extremely wide range of plausible spectra of broadband
emission emanating from $15-30 \mu\rm{as}$ scales including the 230
GHz correlated flux detected by the EHT. For high spin black holes,
$a/M=0.99$, the jet transports $10^{43} -10^{44} \rm{ergs/sec}$ if
the poloidal magnetic field is 8 - 15 G in the inner accretion flow.
Thus, these models can supply the entire jet power of M87 that has
been estimated from the analysis of large scale features
\citep{mcn11,sta06}. The accord with constraints based on broadband
spectra and jet power is achieved with a magnetic field strength
that is consistent with assumption 1) of Section 2. In particular,
based on Appendix D, $ 8 - 15\rm{G}$ is $ \ll$ than the $ \sim 225
\rm{G}$ that would be required for a self-sustaining pair creation
mechanism on an accreted flux tube in the EHM in the absence of a
significant ambient soft photon flux. Thus, the key assumption of
the EHM solution presented here, a weak accreted magnetic field, is
a property of a wide range of high spin BH, hollow jets models of
M87 that have both a plausible mm wavelength to UV spectrum and a
jet power of $10^{43} -10^{44} \rm{ergs/sec}$.
\par The EHM solution described in this article could be used to
argue that a steady accretion of weak axisymmetric flux would also
dissipate in a charge starved EHM. But, more importantly, the flux
dissipation does not depend on the assumption of axisymmetry. Even
for non-axisymmetric flux tubes, as in the right hand frame of
Figure 1, the charges will drain off without a plasma source in the
EHM and the flux will be dissipated. Even though an axisymmetric
disk was used in the models of the broadband luminosity of the jet
in \citet{pun18}, this is not necessary to drive the jet from the
inner accretion flow. In the quasar jet launching study of
\citet{pun14}, the jets are considered to originate in isolated flux
tubes (magnetic islands), as in the right hand frame of Figure 1,
within the innermost accretion flow. In this case, the jet Poynting
flux is altered slightly from our Equation (28). Instead of the jet
power from the inner disk scaling as $(B^{P})^{2}$ as in Equation
(28), it scales as $(fB^{P})^{2}$, where $f$ is the filling fraction
of the disk threaded by isolated flux tubes with a vertical field
strength, $B^{P}$. It should be noted that in general (more
realistically) there would be a bivariate distribution of field
strengths and filling fractions. In the example of M87, as noted
above, for $a/M=0.99$ the broadband spectrum and jet power was fit
in \citet{pun18} with an inner accretion disk field strength of 8 -
15 G. For a filling factor, $f \sim 50\%$, this corresponds to $
B^{P} \sim 15 -30 \rm{G}$ in order to reproduce the jet power.
\par The EHM solution described in this article provides an alternative to assuming a
powerful invisible (or highly under-luminous) ghost jet along the
central spine on sub-mas scales that is also posited to be the
primary power source for the large scale jet on kpc scales. Being
under-luminous, by assumption, a powerful jet cannot be directly
verified by any observation on sub-mas or mas scales. It can only be
ascertained indirectly with deductive reasoning or it must dissipate
violently farther out in the jet, thereby revealing its intrinsic
power. Evidence of this second alternative, would be a spine that
far out shines the limbs over an extended region. Putative spine
emission on larger scales falls far short of satisfying this
requirement \citep{had18}. The heretofore only posited deductive
argument is that a powerful spine is required to energize regions of
enhanced emission such as the knot HST-1 nearly 1 arcsec from the BH
\citep{sta06,mer16}. However, in this context, it was shown in
\citet{pun18} that a hollow jet from the inner accretion flow not
only explains a multitude of plausible spectra of broadband emission
emanating from $15-30 \mu\rm{as}$ scales, but also supports $\sim
10^{44} \rm{ergs/s}$ of jet power. Thus, a powerful ghost jet is not
required to power the large scale jet (including energizing the knot
HST-1). This renders deductive arguments that the ghost jet must be
powerful in order to meet global energy requirements untenable. In
summary, a powerful ghost jet is not indicated directly by any
observation nor is it required to explain any of the observations.
\par By contrast, there are two very extreme properties in M87 that are observed near
the nucleus. Both are fundamental elements of the new EHM solution.
There is the extreme central flux nadir in the base of the jet near
the event horizon. There is also the extraordinarily weak high
energy luminosity of the accreting gas given the large central black
hole mass. The EHM solution presented here implies that these two
extreme circumstances might not be coincidental in M87. If the new
EHM solution applies to M87 then a luminous jet should extend back
towards its source in the inner accretion disk as in the hollow jet
models \citep{pun18}. The detection of a luminous forward jet on
scales $< 30 \mu\rm{as}$ by future EHT imaging would be direct
evidence of a powerful hollow jet connecting the accretion flow to
kpc scales and the compatible new EHM solution. This is in contrast
to models of ghost jets surrounded by a luminous sheath that predict
no strong forward jet emission at 230 GHz - 370 GHz on scales $< 40
\mu\rm{as}$ \citep{dex12,mos16}. Future EHT imaging might be able to
discriminate between these two models.
\begin{acknowledgements}
We would like to thank Robert Antonucci for many valuable
comments. This paper also benefitted from the insightful review of an anonymous
referee.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,881 |
{"url":"https:\/\/blog.r-project.org\/2022\/10\/20\/concordances\/","text":"# Concordances\n\nOne of the strengths of R is its ability to help in producing documents. Sweave and knitr can work with .Rnw files, evaluating and automatically inserting the results of R code to produce a LaTeX document in a .tex file. We call this \u201cpreprocessing\u201d, since the later steps were originally designed with the assumption that the .tex file was directly edited by the user and then processed to produce PDF or other output formats. R Markdown (using knitr) does the same for documents written in the Markdown language.\n\nA difficulty with preprocessors is that errors arising in the later steps will produce error messages that refer to the intermediate files: for example, LaTeX errors will refer to the .tex file rather than the .Rnw file that is the true source. Errors in the HTML code generated from help files are reported by the HTML Tidy utility according to their line in the .html file, not the .Rd or .R file which the user originally wrote.\n\nConcordances address this issue. A concordance is a mapping between lines in the intermediate file and lines in the input file. If an error is reported at \"file:line\" by LaTeX or HTML Tidy, the concordance allows that location to be translated into the corresponding location in the .Rnw or .Rd file. I added concordances to Sweave many years ago, and wrote the patchDVI package to use them with previewers and to translate LaTeX error messages. (See the details in the history below.) With the upcoming release 4.3.0 of R, concordances have been extended to help files. Messages from HTML Tidy will be reported with both the .html file location and the .Rd file location.\n\nFor example, the file hello.Rd could contain this code:\n\n\\name{hello}\n\\alias{hello}\n\\title{Hello, World!}\n\\usage{\nhello()\n}\n\\description{\nPrints 'Hello, world!'.\n\n\\out{<foobar>}\n}\n\nThe second last line inserts the literal text <foobar> into the output. This is not a legal HTML token, and HTML Tidy will complain. With the new changes, the complaint will be shown as\n\n* checking HTML version of manual ... NOTE\nFound the following HTML validation problems:\nhello.html:25:1 (hello.Rd:10): Error: <foobar> is not recognized!\nhello.html:25:1 (hello.Rd:10): Warning: discarding unexpected <foobar>\n\nThis indicates that the bad token was spotted by HTML Tidy in column 1 of line 25 of the hello.html file, and that line originated from line 10 of hello.Rd. There may also be an error reported in producing the PDF version of the manual; at present those are not automatically translated by R, but as shown below, the location can be found manually.\n\n## Concordance code\n\nThe concordance code is mainly intended for internal use, but it is being made available to package writers. One package that might be able to use it is roxygen2; among other things, it creates help files from .R source files. The new code would allow it to embed its own concordance in the .Rd file so that HTML Tidy would report a reference to the true source in the .R file. (There are some difficult issues in producing that concordance due to Pandoc limitations, so this might not happen soon.)\n\n### Some details about the new code\n\nThere\u2019s a new class named \"Rconcordance\", and three related functions exported by the tools package. The \"Rconcordance\" objects are simple lists with three fields:\n\n\u2022 offset: If only part of the output file is related to the input file, the initial offset lines can be skipped.\n\u2022 srcLine: This is a vector of line numbers from the original source file corresponding to a range of lines of the output file starting at line offset + 1.\n\u2022 srcFile: In simple cases, this is a single filename for the source file; in more complicated cases, it can be a vector of filenames of the same length as srcLine, possibly giving a different source file for each of those lines. There is a print method for the class:\nlibrary(tools)\nconcordance <- structure(list(offset = 5,\nsrcLine = 20:30,\nsrcFile = \"myHelpfile.Rd\"),\nclass = \"Rconcordance\")\nconcordance\n## srcFile srcLine\n## 6 myHelpfile.Rd 20\n## 7 myHelpfile.Rd 21\n## 8 myHelpfile.Rd 22\n## 9 myHelpfile.Rd 23\n## 10 myHelpfile.Rd 24\n## 11 myHelpfile.Rd 25\n## 12 myHelpfile.Rd 26\n## 13 myHelpfile.Rd 27\n## 14 myHelpfile.Rd 28\n## 15 myHelpfile.Rd 29\n## 16 myHelpfile.Rd 30\n\nThe row labels are the output line numbers, the columns give the source filename and line corresponding to each.\n\nThe as.character method for \"Rconcordance\" objects converts them into one or more fairly compact strings, suitable for inclusion into a final document. For example,\n\nconc_as_char <- as.character(concordance)\nconc_as_char\n## [1] \"concordance::myHelpfile.Rd:ofs 5:20 10 1\"\n\nThe as.Rconcordance function is a generic function, with a default method defined. That method looks for strings like the one above in its input, and combines all of them into a single concordance object. For example:\n\nnewconcordance <- as.Rconcordance(conc_as_char)\nnewconcordance\n## srcFile srcLine\n## 6 myHelpfile.Rd 20\n## 7 myHelpfile.Rd 21\n## 8 myHelpfile.Rd 22\n## 9 myHelpfile.Rd 23\n## 10 myHelpfile.Rd 24\n## 11 myHelpfile.Rd 25\n## 12 myHelpfile.Rd 26\n## 13 myHelpfile.Rd 27\n## 14 myHelpfile.Rd 28\n## 15 myHelpfile.Rd 29\n## 16 myHelpfile.Rd 30\n\nFinally, the tools::matchConcordance function does the translation of locations in intermediate files to locations in the source file. For example, when proofreading the HTML help files, you may have noticed \u201cHello, world!\u201d on lines 1, 19 and 23 of the hello.html file and decided to change it, but because your actual help file was so large, this isn\u2019t the trivial problem it would be with my example. So what you could do is the following:\n\n1. Run tools::Rd2HTML(\"hello.Rd\", concordance = TRUE). This will print the HTML source for the help page, ending with\n<!-- concordance::hello.Rd:3 19 0 1 4 1 0 3 1 2 0 1 -6 1 0 1 1 3 0 1 7 1 0 1 1 5 0 -->\n1. Convert that string to a concordance object using\nconcordance <- tools::as.Rconcordance(\"<!-- concordance::hello.Rd:3 19 0 1 4 1 0 3 1 2 0 1 -6 1 0 1 1 3 0 1 7 1 0 1 1 5 0 -->\")\n1. Find the source corresponding to lines 1, 19 and 23 using\ntools::matchConcordance(c(1, 19, 23), concordance)\n## srcFile srcLine\n## [1,] \"hello.Rd\" \"3\"\n## [2,] \"hello.Rd\" \"3\"\n## [3,] \"hello.Rd\" \"8\"\n\nThe first two arose from the \\title{} specification, and the third one came from a line of text in the \\description section.\n\n## Ancient History\n\nMany years ago I used Sweave for writing papers, presentations, exams, etc. It took .Rnw files as input, and produced .tex files as output. I would run those files through latex to get .dvi files which I could preview, print, or convert to PDF for distribution.\n\nPreviewers existed in those days that let you click on a particular word in the preview, and they\u2019d tell your text editor to jump to the corresponding location in the .tex file. That was kind of nice, but also kind of irritating: I then had to figure out the right location in the .Rnw file to make my edits, or make the edits in the .tex file and be frustrated when they got wiped out by Sweave on the next run!\n\nMy first solution to this problem was to get Sweave in R 2.5.0 to keep a record of the correspondence between the lines of the .Rnw file and the .tex file it produced, which I called the \u201cconcordance\u201d. Given a line in the .tex file, it was then possible to find the corresponding line in the .Rnw file. By embedding this record in the latex output, this could be made automatic. I wrote the patchDVI package to modify the links in the .dvi file so that the previewer would automatically jump to the right place in the right file. Happiness!\n\nOver the years there were lots of developments. I started using pdflatex which skipped the .dvi stage, but supported synctex, so I added support for that into Sweave and patchDVI. knitr arrived to improve on Sweave, and included concordance support. I switched text editors and previewers several times, writing new scripts each time to connect things.\n\nUnfortunately, R Markdown is processed by Pandoc, and as far as I know, Pandoc doesn\u2019t support any way to relate input lines to output lines. I\u2019d love to be corrected if I\u2019m wrong about that! So concordances don\u2019t work with R Markdown or other processors like Quarto that rely on Pandoc. I believe roxygen2 uses Pandoc for processing some help files, so it will also be difficult.","date":"2023-03-22 16:02:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.498459130525589, \"perplexity\": 2489.185255832976}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943845.78\/warc\/CC-MAIN-20230322145537-20230322175537-00592.warc.gz\"}"} | null | null |
\section{Introduction}
In series theory, there is well known Cauchy convergence test which
is used to determine convergence of a given series,but Cauchy
convergence test usually is not practical in most applications,so
there come out various convergence test such as D'Alembert
convergence test ,integral convergence test and so on. In
Diophantine approximation theory, we are in totally different
situation that we already have necessary and sufficient condition
to determine if a given real number is an irrational number or a
transcendental number such as well known Roth theorem but seems to
be lack of practical test just as various convenient test in series
theory.
\newline
The purpose of this paper is to propose some sufficient conditions
for convenient use in determining if a given real number is an
irrational number or a transcendental number and also give out
various interesting examples to illustrate how to apply these
conditions,particularly, we will explain an example coming from
complex analytic dynamics in detail. At the end of this paper, we
propose a conjecture about rational approximation of any irrational
number.
\begin{theorem}
Assume that series $\sum_{n=1}^{\infty}c_{n},
c_{n}=\frac{a_{n}}{b_{n}}\neq 0 ,(n=1,2,\cdots)$ are rational
numbers and satisfy following conditions:
\newline
(1) $$b_n |b_{n+1},(n=1,2,\cdots)$$
\newline
(2)$$lim_{n\rightarrow\infty}a_n\frac{c_{n+1}}{c_{n}}=0$$
\newline
(3)for any natural number $ N$, $\sum_{n=N}^{\infty}c_{n}\neq 0$
\newline
then the series$\sum_{n=N}^{\infty}c_{n}$ converges to an
irrational number.
\end{theorem}
\begin{proof} First of all, since condition (2) implies
$lim_{n\rightarrow\infty}\frac{c_{n+1}}{c_{n}}=0$, the series
$\sum_{n=1}^{\infty}c_{n}$ is convergent and set the convergent
result to be $\theta$. We will use indirect method to show $\theta$
is an irrational number. Suppose that $\theta=\frac{s}{r}$ is a
rational number. By (1), when $k\leq n$,$b_k |b_{n}$, we have
\begin{eqnarray}
A_n =
rb_{n}(\frac{s}{r}-\frac{a_1}{b_1}-\cdots-\frac{a_n}{a_n})\nonumber
\\ =rb_{n}(\frac{a_{n+1}}{b_{n+1}}+\frac{a_{n+2}}{b_{n+2}}+\cdots
)\nonumber \\=rb_{n}(c_{n+1}+c_{n+2}+\cdots)\neq 0\nonumber
\end{eqnarray}
and $A_n$ is an integer number, we notice
that$$A_n=rb_{n}c_{n}\frac{c_{n+1}}{c_{n}}(1+\frac{c_{n+2}}{c_{n+1}}+\frac{c_{n+3}}{c_{n+1}}+\cdots
)$$ and $lim_{n\rightarrow\infty}\frac{c_{n+1}}{c_{n}}=0$, thus
there exists $N_1$, when $n\geq N_1$ we have
$$\frac{|c_{n+1}|}{|c_{n}|}<\frac{1}{2},\frac{|c_{n+2}|}{|c_{n+1}|}<\frac{1}{2},\cdots$$,
so
$$\frac{|c_{n+3}|}{|c_{n+1}|}=\frac{|c_{n+3}|}{|c_{n+2}|}\frac{|c_{n+2}|}{|c_{n+1}|}<(\frac{1}{2})^2$$
$$\frac{|c_{n+4}|}{|c_{n+1}|}=\frac{|c_{n+4}|}{|c_{n+3}|}\frac{|c_{n+3}|}{|c_{n+2}|}\frac{|c_{n+2}|}{|c_{n+1}|}<(\frac{1}{2})^3$$
Therefore,$$|A_n|\leq |r||a_{n}|
\frac{c_{|n+1}|}{c_{|n|}}(1+\frac{c_{|n+2|}}{c_{|n+1|}}+\frac{c_{|n+3|}}{c_{|n+1|}}+\cdots
)$$
$$<|r||a_{n}|
\frac{c_{|n+1}|}{c_{|n|}}(1+\frac{1}{2}+\frac{1}{2^2}+\cdots )$$
$$=2|r||a_{n}|
\frac{c_{|n+1}|}{c_{|n|}}$$ By (2),for $\frac{1}{2|r|}>0$,there
exists $N_2$,when $n\geq N_2$,$|a_{n}|
\frac{c_{|n+1}|}{c_{|n|}}<\frac{1}{2|r|}$. Set $N=max(N_1,N_2)$,when
$n\geq N$,we get $|A_n|<2|r||a_{n}|
\frac{c_{|n+1}|}{c_{|n|}}<2r\frac{1}{2r}=1$ which contradicts the
fact that $A_n$ is an integer and $A_n\neq 0$. That follows the
theorem.
\end{proof}
When the series just contains positive terms, condition (3) is
naturally satisfied, we have
\begin{theorem}
Assume that series $\sum_{n=1}^{\infty}c_{n},
c_{n}=\frac{a_{n}}{b_{n}}> 0 ,(n=1,2,\cdots)$ are rational numbers
and satisfy following conditions:
\newline
(1) $$b_n
|b_{n+1},(n=1,2,\cdots)$$
\newline
(2) $$lim_{n\rightarrow\infty}a_n\frac{c_{n+1}}{c_{n}}=0$$ then the
series$\sum_{n=1}^{\infty}c_{n}$ converges to an irrational number.
\end{theorem}
{\bf Remark.} In the above theorem, the condition
$lim_{n\rightarrow\infty}a_n\frac{c_{n+1}}{c_{n}}=0$ is not
sufficient because $\sum_{n=0}^{\infty}\frac{n+1}{n!}=2e$ is an
irrational number and
$a_n\frac{c_{n+1}}{c_{n}}=\frac{n+2}{n+1}\rightarrow 1(n\rightarrow
\infty)$
\bigskip
{\bf Example 1.}
$$e=\sum_{n=0}^{\infty}\frac{1}{n!}=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots
+\frac{1}{n!}+\cdots$$ is an irrational number. Because by the
theorem 2,
$$lim_{n\rightarrow\infty}a_n\frac{c_{n+1}}{c_{n}}=lim_{n\rightarrow\infty}\frac{n!}{(n+1)!}=lim_{n\rightarrow\infty}\frac{1}{n+1}=0$$
Where $a_{n}=1$
{\bf Example 2.}
$$\theta=\sum_{n=1}^{\infty}\frac{n^4}{(n!)^5}=1+\frac{2^4}{(2!)^5}+\cdots+\frac{n^4}{(n!)^5}+\cdots
$$ is an irrational number, because by the theorem 2,$$a_n\frac{c_{n+1}}{c_{n}}=a_{n+1}\frac{b_{n}}{b_{n+1}}=\frac{1}{n+1}\rightarrow 0
(n\rightarrow \infty)$$ where $a_n=n^4, b_n=(n!)^5$.
Let's look at a more complicated example as follows:
{\bf Example 3.} Suppose that $r\geq 1$ is an integer, then
$sin\frac{1}{r}$ is an irrational number.
Since
$$sin\frac{1}{r}=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{(2n-1)!r^{2n-1}}=\sum_{n=1}^{\infty}c_n=\sum_{n=1}^{\infty}\frac{a_n}{b_n}$$
where $a_n=(-1)^{n-1},b_n=(2n-1)!r^{2n-1}$, then
$$lim_{n\rightarrow\infty}a_n\frac{c_{n+1}}{c_{n}}=lim_{n\rightarrow\infty}a_n\frac{b_{n}}{b_{n+1}}=lim_{n\rightarrow\infty}
\frac{(-1)^{n}(2n-1)!r^{2n-1}}{(2n+1)!r^{2n+1}}=0$$.
The remaining
is to verify $\sum_{n=N}^{\infty}c_n\neq 0$
$$\sum_{n=N}^{\infty}c_n=\frac{(-1)^{N-1}}{(2N-1)!r^{2N-1}}+\frac{(-1)^{N}}{(2N+1)!r^{2N+1}}+\frac{(-1)^{N+1}}{(2N+3)!r^{2N+3}}+\cdots$$
When $N$ is odd
$$\sum_{n=N}^{\infty}c_n=(\frac{1}{(2N-1)!r^{2N-1}}-\frac{1}{(2N+1)!r^{2N+1}})+(\frac{1}{(2N+3)!r^{2N+3}}-\frac{1}{(2N+5)!r^{2N+5}})+\cdots$$
$$=\frac{(2N+1)!r^{2}-(2N-1)!}{(2N-1)!(2N+1)!r^{2N+1}}+\frac{(2N+5)!r^{2}-(2N+3)!}{(2N+3)!(2N+5)!r^{2N+5}}+\cdots >0$$
Similarly,when $N$ is even
$$\sum_{n=N}^{\infty}c_n=-(\frac{1}{(2N-1)!r^{2N-1}}-\frac{1}{(2N+1)!r^{2N+1}})-(\frac{1}{(2N+3)!r^{2N+3}}-\frac{1}{(2N+5)!r^{2N+5}})-\cdots<0$$
Thus for any natural number $N,\sum_{n=N}^{\infty}c_n\neq 0$,by the
theorem 1,$sin\frac{1}{r}$ is an irrational number.
Since the sum of
two irrational numbers is not necessarily an irrational number, the
following theorem is interesting.
\begin{theorem}
Assume that $\alpha=\sum_{n=1}^{\infty}c_n$,where
$c_{n}=\frac{a_n}{b_n}>0$ and $\beta=\sum_{n=1}^{\infty}c_n^{'}$,
where$ c_n^{'}=\frac{a_n^{'}}{b_n^{'}}>0$ and above two number are
irrational numbers determined by the theorem 2 and satisfy the
following conditions:
$lim_{n\rightarrow\infty}\frac{a_{n+1}b_{n}b_{n}^{'}}{b_{n+1}}=0$
and
$lim_{n\rightarrow\infty}\frac{a_{n+1}^{'}b_{n}^{'}b_{n}}{b_{n+1}^{'}}=0$
then $\alpha +\beta$ is also an irrational number.
\end{theorem}
\begin{proof}
Let $\gamma=\alpha+\beta=\sum_{n=1}^{\infty}d_n$ and
$d_n=c_n+c_n^{'}=\frac{a_n}{b_n}+\frac{a_n^{'}}{b_n^{'}}=\frac{a_{n}b_n^{'}+b_{n}a_n^{'}}{b_{n}b_n^{'}}=\frac{\tilde{a_n
}}{\tilde{b_n}}$ where $\tilde{a_n }=a_{n}b_n^{'}+b_{n}a_n^{'},
\tilde{b_n}=b_{n}b_n^{'}$ then
$$lim_{n\rightarrow\infty}\tilde{a_n}\frac{d_{n+1}}{d_{n}}=lim_{n\rightarrow\infty}\tilde{a_{n+1}}\frac{\tilde{b_{n}}}{\tilde{b_{n+1}} }$$
$$=lim_{n\rightarrow\infty}(a_{n+1}b_{n+1}^{'}+b_{n+1}a_{n+1}^{'})\frac{b_{n}b_n^{'}}{b_{n+1}b_{n+1}^{'}}
=lim_{n\rightarrow\infty}\frac{a_{n+1}b_{n}b_{n}^{'}}{b_{n+1}}+lim_{n\rightarrow\infty}\frac{a_{n+1}^{'}b_{n}^{'}b_{n}}{b_{n+1}^{'}}=0$$
In additional,we notice that $\tilde{b_n}|\tilde{b_{n+1}}$, so by
the theorem 2 we get $\alpha+\beta$ is an irrational number
\end{proof}
\newline
{\bf Example 4.}Let
$\alpha=\sum_{n=1}^{\infty}\frac{1}{2^{n!}},\beta=\sum_{n=1}^{\infty}\frac{1}{3^{n!}}$,we
can use above theorem to verify that $\alpha+\beta$ is an irrational
number as follows: First of all, $\alpha,\beta$ are irrational
numbers because of theorem 2,secondly, let
$b_n=2^{n!},b_n^{'}=3^{n!}$,then
$$\frac{b_{n}^{'}b_{n}}{b_{n+1}^{'}}=\frac{3^{n!}}{2^{nn!}}\rightarrow 0,(n\rightarrow \infty)$$
$$\frac{b_{n}^{'}b_{n}}{b_{n+1}^{'}}=\frac{2^{n!}}{3^{nn!}}\rightarrow 0,(n\rightarrow \infty)$$
so $\alpha+\beta$ is an irrational number.
Essentially, we can replace condition 1 of theorem 2 by a more
general condition as follows:
\begin{theorem}
Let series $\sum_{n=1}^{\infty}\frac{a_n}{b_n}$ where
$\frac{a_n}{b_n}>0$ are rational numbers and
$lim_{n\rightarrow\infty}\frac{a_{n+1}}{b_{n+1}}[b_1,b_2,\cdots
b_n]=0$ where $[b_1,b_2,\cdots b_n]$ denotes least common multiple
of $b_1,\cdots b_n$ then $\sum_{n=1}^{\infty}\frac{a_n}{b_n}$
converges an irrational number
\end{theorem}
\begin{proof}
Let $c_n=\frac{a_n}{b_n}$ and
$$c_n=\frac{a_{n}\frac{[b_1,b_2,\cdots b_n]}{b_n}}{[b_1,b_2,\cdots
b_n]}=\frac{\tilde{a_n}}{\tilde{b_n}}=\tilde{c_n}$$ where
$\tilde{a_n}=\frac{[b_1,b_2,\cdots
b_n]}{b_n},\tilde{b_n}=[b_1,b_2,\cdots b_n]$ and
$$lim_{n\rightarrow\infty}\tilde{a_n}\frac{\tilde{c_{n+1}}}{\tilde{c_{n}}}=
lim_{n\rightarrow\infty}\frac{\tilde{a_{n+1}}}{\tilde{b_{n+1}}}\tilde{b_n}=
lim_{n\rightarrow\infty}\frac{a_{n+1}}{b_{n+1}}[b_1,b_2,\cdots
b_n]=0$$ Obviously,$\tilde{b_n}|\tilde{b_{n+1}},n=1,2\cdots$ and the
series $\sum_{n=1}^{\infty}\tilde{c_{n}}$ satisfies conditions of
theorem 2 ,and
$$\sum_{n=1}^{\infty}\frac{a_n}{b_n}=\sum_{n=1}^{\infty}c_{n}=\sum_{n=1}^{\infty}\tilde{c_n}$$then we
finish the proof.
\end{proof}
\newline
{\bf Example 5.}
$$\theta=\frac{1}{p_{2^{2^{1!}}}}+\frac{1}{p_{2^{2^{2!}}}}+\cdots+\frac{1}{p_{2^{2^{n!}}}}+\cdots$$
is a irrational number, where $p_{2^{2^{n!}}}$ is $2^{2^{n!}}$-th
prime number. Let's show it as follows:
\newline
We need the famous result\cite{Hua}: let $p_n$ is $n$-th prime
number, there exists two positive numbers such that
$c_{1}n\ln{n}<p_n<c_{2}n\ln{n}$,then
$$\frac{a_{n+1}}{b_{n+1}}[b_1,b_2,\cdots
b_n]=\frac{p_{2^{2^{1!}}}p_{2^{2^{2!}}}\cdots
p_{2^{2^{n!}}}}{p_{2^{2^{(n+1)!}}}}$$
$$<\frac{c_{2}2^{2^{1!}}\ln{2^{2^{1!}}}c_{2}2^{2^{2!}}\ln{2^{2^{2!}}}\cdots c_{2}2^{2^{n!}}\ln{2^{2^{n!}}}}
{c_{1}2^{2^{(n+1)!}}\ln{2^{2^{(n+1)!}}}}$$
$$=\frac{2^{2^{1!}+2^{2!}+\cdots+2^{n!}+n\log_{2}C}}{2^{2^{(n+1)!}}}
\frac{2^{1!+2!+\cdots+n!}}{2^{(n+1)!}}k(\ln2)^{n-1}$$ Where
$k=\frac{1}{c_1},c=c_{2}$ Since
$$1!+2!+\cdots+n!\leq nn!<(n+1)!$$, $\frac{2^{1!+2!+\cdots+n!}}{2^{(n+1)!}}<1 .$
Also, there exists $N$ such that when $n\geq N$ $\log_{}2C<2^{n!}$.
Therefore
$$2^{2^{1!}+2^{2!}+\cdots+2^{n!}+n\log_{2}C}\leq n2^{n!}+n2^{n!} \leq 2n2^{n!}\leq 2^{(n+1)!}$$
we get
$$\frac{2^{2^{1!}+2^{2!}+\cdots+2^{n!}+n\log_{2}C}}{2^{2^{(n+1)!}}}\leq 1$$
Thus when $n\geq N$
$$lim_{n\rightarrow\infty}\frac{a_{n+1}}{b_{n+1}}[b_1,b_2,\cdots
b_n]\leq k(\ln2)^{n-1}\rightarrow 0(n\rightarrow \infty)$$ ,by
theorem 4,$\theta $ is an irrational number
\newline
The following theorem shows that condition 2 of theorem 2 is also
necessary in some special case.
\begin{theorem}
Assume that the sequence $c_{n} =\frac{1}{a^{P_{m}(n)}}$ ,where
$a\geq 2$ is an integer and
$P_{m}(x)=b_{0}x^{m}+b_{1}x^{m-1}+\cdots+b_{m-1}x+b_m$ is an
polynomial with positive integer coefficients,then series
$\sum_{n=1}^{\infty}c_n$ converges to an irrational number if and
only if $lim_{n\rightarrow\infty}\frac{c_{n+1}}{c_{n}}=0$
\end{theorem}
\begin{proof}
Obviously, $P_{m}(n)<P_{m}(n+1)$,so
$a^{P_{m}(n)}|a^{P_{m}(n+1)},n=1,2,\cdots$ which satisfies condition
1 of theorem 2 and
\begin{eqnarray}
P_{m}(n+1)-P_{m}(n) &=& b_{0}(n+1)^{m}+\cdots
+b_{m}-(b_{0}^{m}+\cdots+b_{m}
+b_m)\nonumber\\
&=&b_{0}(n^{m}+mn^{m-1}+\cdots)+\cdots +b_m-(b_{0}^{m}+\cdots+b_m
+b_m)\nonumber\\
&= &mb_{0}n^{m-1}+l_{1}n^{m-2}+l_{2}n^{m-3}+\cdots
\end{eqnarray}
then
$$\frac{c_{n+1}}{c_n}=\frac{a^{P_{m}(n)}}{a^{P_{m}(n+1)}}
=\frac{1}{a^{P_{m}(n+1)-P_{m}(n)}}=\frac{1}{a^{mb_{0}n^{m-1}+l_{1}n^{m-2}+
\cdots}}$$ Thus
\begin{eqnarray}
lim_{n\rightarrow\infty}\frac{c_{n+1}}{c_{n}}=\left \{
\begin{array}{ll}
0 & m\geq 2 \\
a^{b_0} & m=1
\end{array}
\right.
\end{eqnarray}
By theorem 2, when $m\geq 2$ series $\sum_{m=1}^{\infty}c_n$
converges to an irrational number . and
$lim_{n\rightarrow\infty}\frac{c_{n+1}}{c_{n}}\neq 0$ means $m=1$
then
$$\sum_{n=1}^{\infty}c_n=\sum_{n=1}^{\infty}\frac{1}{a^{b_{0}n+b_{1}}}=\frac{1}{a^{b_1}(a^{b_0}-1)}$$
is a rational number.
\end{proof}
\begin{theorem}
Let $\theta=\sum_{n=1}^{\infty}\frac{a_n}{b_n}$,where
$\frac{a_n}{b_n}>0 (n=1,2\cdots)$ are rational number, and assume
that $ f(b_n)$ is a function of $b_n$ and $ f(b_n)>0$ ,furthermore
if the following conditions are satisfied:
\newline
(1)$b_1<b_2<\cdots$ and $b_n|b_{n+1},n=1,2,\cdots$ \newline
(2)$f(b_n)>0$ and $\frac{f(b_{n+1})}{f(b_n)}<\frac{1}{2}$ ($n$ is
big enough)\newline (3)$\frac{f(b_{n})}{b_{n+1}}a_{n+1}<\frac{1}{2}$
($n$ is big enough)\newline (4)$\frac{b_n}{f(b_n)}\rightarrow 0
(n\rightarrow \infty)$
\newline
then
\newline
(1)$\theta$ is an irrational number
\newline (2) When $n$ is big enough, there exists infinite fractions
$\frac{c_n}{b_n}$ such that $|\theta -
\frac{c_n}{b_n}|<\frac{1}{f(b_n)}$
\end{theorem}
\begin{proof}
According condition 3,when $n$ is big enough, we have $\frac
{f(b_{n})}{b_{n+1}a}a_{n+1}<\frac{1}{2}$ or equivalently
$\frac{a_{n+1}b_n}{b_{n+1}}<\frac{b_n}{2f(b_n)}$,so
$$lim_{n\rightarrow\infty}a_n\frac{c_{n+1}}{c_{n}}=lim_{n\rightarrow\infty}\frac{a_{n+1}b_n}{b_{n+1}}=0$$
In the last step we use condition 4.By theorem 2, $\theta$ is an
irrational number.
\newline Let's prove the second part.
$$|\theta-\frac{a_1}{b_1}-\frac{a_2}{b_2}-\cdots-\frac{a_n}{b_n}|=\frac{a_{n+1}}{b_{n+1}}+\frac{a_{n+2}}{b_{n+2}}+\frac{a_{n+3}}{b_{n+3}}+\cdots$$
$$=\frac{1}{f(b_n)}(\frac{f(b_{n})a_{n+1}}{b_{n+1}}+\frac{f(b_{n})a_{n+2}}{b_{n+2}}+\frac{f(b_{n})a_{n+3}}{b_{n+3}}+\cdots)$$
$$=\frac{1}{f(b_n)}(\frac{f(b_{n})a_{n+1}}{b_{n+1}}+\frac{f(b_{n})}{f(b_{n+1})}\frac{f(b_{n+1})a_{n+2}}{f(b_{n+2})}+\frac{f(b_{n})}{f(b_{n+1})}\frac{f(b_{n+1})}{f(b_{n+2})}\frac{f(b_{n+2})a_{n+3}}{f(b_{n+3})}+\cdots)$$
$$<\frac{1}{f(b_n)}(\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots)=\frac{1}{f(b_n)}$$
We use condition 2 and 3 in the last two steps.
Let
$c_{n}=b_{n}(\frac{a_1}{b_1}+\frac{a_2}{b_2}+\cdots+\frac{a_n}{b_n})$,since
$\frac{a_n}{b_n}>0,(n=1,2,\cdots)$,
$\frac{c_n}{b_n}<\frac{c_{n+1}}{b_{n+1}}$ Thus there are infinite
number of $\frac{c_n}{b_n}$ satisfy
$$|\theta-\frac{c_n}{b_n}|=|\theta-\frac{a_1}{b_1}-\frac{a_2}{b_2}-\cdots-\frac{a_n}{b_n}|<\frac{1}{f(b_n)}$$
(when $n$ is big enough )That proves the theorem
\end{proof}
\newline
{\bf Remark.} In above theorem, condition 2 and 3 can be replaced by
$lim_{n\rightarrow\infty}\frac{f(b_n)}{f(b_{n+1})}=l<\frac{1}{2}$
and
$lim_{n\rightarrow\infty}\frac{f(b_{n})}{b_{n+1}}a_{n+1}=k<\frac{1}{2}$
Using theorem 6 and two following known results, we can get two
useful theorems , one is about how to determine a given number is
transcendental number, the other is about complex analytic dynamics.
{\bf Theorem(K.Roth).} Let $\theta$ be a $n\geq 2$ degree algebraic
number, then for any given $\epsilon > 0$, there exists only finite
positive integer pairs $x,y$ such that $|\theta-\frac{x}{y}|<
\frac{1}{y^{2+\epsilon}}$
{\bf Theorem (H.Cremer)\cite{Cremer}} If irrational number $\theta$
satisfies the condition that there exists infinite positive integers
such that $|\theta-\frac{n}{m}|\leq \frac{1}{m^{d^{m}-1}}$,
indifferent fixed point $z=0$ of polynomial
$f(z)=z^{d}+\cdots+e^{2\pi i \alpha}$ belongs to Julia set. First of
all, we use theorem 6 and Roth theorem to derive following theorem:
\begin{theorem}
Let $\theta=\sum_{n=1}^{\infty}\frac{a_n}{b_n}$,where
$\frac{a_n}{b_n}>0 (n=1,2\cdots)$ are rational numbers which satisfy
\newline
(1)$b_1<b_2<\cdots$ and $b_n|b_{n+1},n=1,2,\cdots$
\newline
(2) for some $\epsilon >0,$
$\frac{a_{n+1}b_{n}^{2+\epsilon}}{b_{n+1}}<\frac{1}{2}$ ($n$ is big
enough)
\newline
then $\theta=\sum_{n=1}^{\infty}\frac{a_n}{b_n}$ is a transcendental
number.
\end{theorem}
\begin{proof}
In the theorem 6,let's take $f(b_n)=b_{n}^{2+\epsilon}$,then it's easy to
verify $f(b_n),n=1,2,\cdots$ satisfy condition 1 and 3 of theorem
6, we only need to check condition 2 and 4.
\newline
Since $\epsilon>0$ and $a_n,b_n$ are positive integers, we have
$$\frac{f(b_{n})}{f(b_{n+1})}=\frac{b_{n}^{2+\epsilon}}{b_{n+1}^{2+\epsilon}}< \frac{b_{n}^{2+\epsilon}}{b_{n+1}}<\frac{a_{n+1}b_{n}^{2+\epsilon}}{b_{n+1}}<\frac{1}{2}$$ ,thus we pass condition 2 of theorem 6.
Also because $b_1<b_2<\cdots$ and
$\epsilon>0$,$$\frac{b_n}{f(b_n)}=\frac{b_n}{b_{n}^{2+\epsilon}}=\frac{1}{b_{n}^{1+\epsilon}}\rightarrow
0(n\rightarrow \infty)$$, then we finish checking all conditions of
theorem 6 get satisfied. By theorem 6,
$\theta$ is an irrational
number, or equivalently, it's not a first order algebraic number,by
the conclusion 2, when $n$ is big enough, there exists infinite
fractions $\frac{c_n}{b_n}$ satisfy $|\theta-\frac{c_n}{b_n}|<
\frac{1}{{b_n}^{2+\epsilon}}$,by Roth theorem we get $\theta$ is a
transcendental number.
\end{proof}
\newline
{\bf Example 6.} $\sum_{m=1}^{\infty}\frac{1}{10^{m!}}$ is an
transcendental number.
\newline
Because by taking $a_{n}=1,n=1,2,\cdots, b_{n}=10^{n!}$ and
$\epsilon=1$,
$\frac{a_{n+1}b_{n}^{2+\epsilon}}{b_{n+1}}=\frac{(10^{n!})^3}{10^{{n+1}!}}=\frac{1}{10^{n!(n-2)}}\rightarrow
0 (n\rightarrow \infty)$,by theorem 7,
$\sum_{m=1}^{\infty}\frac{1}{10^{m!}}$ is an transcendental number.
\newline
{\bf Example 7.} $\sum_{n=1}^{\infty}\frac{3^n}{2^{3^n}}$ is an
transcendental number.
\newline
Because by taking $a_{n}=3^n,n=1,2,\cdots, b_{n}=2^{3^n}$ and
$\epsilon=\frac{2}{3}$,
$\frac{a_{n+1}b_{n}^{2+\epsilon}}{b_{n+1}}=9\frac{3^{n-1}}{2^{3^{n-1}}}
\rightarrow 0 (n\rightarrow \infty)$,by theorem 7,
$\sum_{n=1}^{\infty}\frac{3^n}{2^{3^n}}$is an transcendental
number.
\begin{theorem}
Let $\theta=\sum_{n=1}^{\infty}\frac{a_n}{b_n}$,where
$\frac{a_n}{b_n}>0 (n=1,2\cdots)$ are rational numbers which satisfy
\newline
(1)$b_1<b_2<\cdots$ and $b_n|b_{n+1},n=1,2,\cdots$
\newline (2)
$\frac{a_{n+1}b_{n}^{d^{b_n}-1}}{b_{n+1}}<\frac{1}{2}$ ($d\geq 2$ is
an integer and $n$ is big enough)
\newline
Then indifferent fixed point $z=0$ of polynomial
$f(z)=z^{d}+\cdots+e^{2\pi i \alpha}$ belongs to Julia set
\end{theorem}
\begin{proof}
Set $f(b_n)=b_{n}^{d^{b_n}-1}$,it's easy to verify $f(b_n)$ satisfy
condition 1 and 3 of theorem 6. We only need to verify condition 2
and 4. Since $b_n|b_{n+1},(n=1,2,\cdots)$ and $b_{n+1}\geq 2b_{n}$
,we have
$$\frac{f(b_{n})}{f(b_{n+1})}=\frac{b_{n}^{d^{b_n}-1}}{b_{n+1}^{d^{b_{n+1}}-1}}\leq
\frac{(\frac{1}{2}b_{n+1})^{d^{\frac{1}{2}b_{n+1}}-1}}{b_{n+1}^{d^{b_{n+1}}-1}}=
\frac{(\frac{1}{2})^{d^{\frac{1}{2}b_{n+1}}-1}b_{n+1}^{d^{\frac{1}{2}b_{n+1}}-1}}{b_{n+1}^{d^{b_{n+1}}-1}}$$
$$<(\frac{1}{2})^{d^{\frac{1}{2}b_{n+1}}-1}<\frac{1}{2}$$
thus condition (2) is satisfied. Let's verify condition (4), Since
when $n\geq 2$,$b_n \geq 2$ and $d\geq 2$,$d^{b_n}-1\geq 3$, thus
when $n\geq 2$
$$\frac{b_n}{f(b_n)}=\frac{b_n}{b_{n}^{d^{b_n}-1}}\leq \frac{1}{b_n^2 }\rightarrow 0(n\rightarrow \infty)$$
By theorem 6,$\theta$ is an irrational number and there exists
infinite fractions $\frac{c_n}{b_n}$ such that
$|\theta-\frac{c_n}{b_n}|<\frac{1}{b_{n}^{d^{b_n}-1}}$ ($n$ is big
enough), then by Cremer theorem, we get our result
\end{proof}
\newline
{\bf Example 8.} In order to illustrate this example, we need some
notation to describe a special series so called "nth exponential
floor"as follows:
\newline
Set $[a_n,a_{n-1},\cdots, a_1]_{n} =f_n$ where $f_n$
is defined inductively by $f_{k+1}=(a_{k+1})^{f_k},k=1,2\cdots, n-1$
and $f_{1}=a_1$
For any positive integer $d\geq2$, let $b_{n}=[d,\cdots,d,nd]_{2n}$
and $\theta=\sum_{n=1}^{\infty}\frac{1}{b_n}$, we will
show that indifferent fixed point $z=0$ of polynomial $g(z)=Z^{d}+\cdots+e^{2\pi
i\theta}z$ belongs to Julia set.
Let's check $b_n$ satisfy conditions of theorem 8,
condition 1 is obvious and by noticing
$[d,\cdots,d,nd]_{2n}=d^{[d,\cdots,d,nd]_{2n-1}}$,we have
$$\frac{a_{n+1}b_{n}^{d^{b_n}-1}}{b_{n+1}}=\frac{([d,\cdots,d,nd]_{2n})^{[d,\cdots,d,nd]_{2n+1}-1}}{[d,\cdots,d,(n+1)d]_{2(n+1)}}$$
$$<\frac{([d,\cdots,d,nd]_{2n})^{[d,\cdots,d,nd]_{2(n+1)}}}{[d,\cdots,d,(n+1)d]_{2(n+1)}}$$
$$=\frac{d^{([d,\cdots,d,nd]_{2n-1})([d,\cdots,d,nd]_{2n+1})}}{[d,\cdots,d,(n+1)d]_{2(n+1)}}
=\frac{d^{d^{[d,\cdots,d,nd]_{2n-2}+[d,\cdots,d,nd]_{2n}}}}{[d,\cdots,d,(n+1)d]_{2(n+1)}}$$
$$=\frac{d^{d^{[d,\cdots,d,nd]_{2n-2}+[d,\cdots,d,nd]_{2n}}}}{d^{d^{[d,\cdots,d,(n+1)d]_{2n}}}}$$
We notice that $\frac{[d,\cdots,d,nd]_{2n-2}+[d,\cdots,d,nd]_{2n}}{[d,\cdots,d,(n+1)d]_{2(n)}}\rightarrow 0(n\rightarrow\infty)$
That means when $n$ is big enough,$\frac{a_{n+1}b_{n}^{d^{b_n}-1}}{b_{n+1}}<\frac{1}{2}$ and we finish checking this example satisfies all
conditions of theorem 8 and thus indifferent fixed point of $g(z)$
belongs to Julia set.
\newline
At the end of paper, we are going to propose a conjecture which
relates to theorem 2. To do this, we need following definition
firstly.
\begin{definition}
Let $\alpha$ be an irrational number ,if $\alpha$ satisfies
following conditions:
\newline
(1) $\alpha=\sum_{n=1}^{\infty}c_{n}, $ where
$c_{n}=\frac{a_{n}}{b_{n}} ,(n=1,2,\cdots)$ and $a_n,b_n$ are
positive integers.
\newline
(2) $$b_n |b_{n+1},(n=1,2,\cdots)$$
\newline
(3) $$lim_{n\rightarrow\infty}a_n\frac{c_{n+1}}{c_{n}}=0$$
We call the irrational number $\alpha$ has $E$ rational
approximation.
\end{definition}
{\bf Conjecture.}Every positive irrational number has $E$ rational
approximation.
{\bf Remark.} The positive answer of above conjecture will give an
explicit character of any positive irrational number.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,011 |
namespace headless {
namespace switches {
// Instructs headless_shell to print document.body.innerHTML to stdout.
const char kDumpDom[] = "dump-dom";
// Uses a specified proxy server, overrides system settings. This switch only
// affects HTTP and HTTPS requests.
const char kProxyServer[] = "proxy-server";
// Use the given address instead of the default loopback for accepting remote
// debugging connections. Should be used together with --remote-debugging-port.
// Note that the remote debugging protocol does not perform any authentication,
// so exposing it too widely can be a security risk.
const char kRemoteDebuggingAddress[] = "remote-debugging-address";
// Runs a read-eval-print loop that allows the user to evaluate Javascript
// expressions.
const char kRepl[] = "repl";
} // namespace switches
} // namespace headless
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,898 |
Q: transitive rails associations and magical count In Rails, to automatically count associations, you do:
class Script
has_many :chapters
end
class Chapter
belongs_to :script
end
and you add a chapters_count column into the Script model.
Now, what if you want to count the number of paragraphs in a Script without having a script_id key in the paragraph model ?
class Script
has_many :chapters
has_many :paragraphs # not complete
end
class Chapter
has_many :paragraphs
belongs_to :script
end
class Paragraph
belongs_to :chapter
end
How do you automatically associate script to paragraph and count them using the automatic count of Rails ?
A: You're on the right track. But first you've got to address a small error. Rails won't update a counter cache unless you instruct it to.
class Chapter
belongs_to :script, :counter_cache => true
end
Will automatically update @script.chapter_count before creation and after destruction of all associated Chapters.
Unfortunately things aren't so simply when dealing :through relationships. You will need to update the associated script's paragraph counter through callbacks in the Paragraph model.
N.B.: The following assumes you want to keep a paragraph counter in Chapter as well.
Start by applying the same theory to the Chapter model, and a paragraphs count column to the Script table.
class PrepareForCounterCache < ActiveRecord::Migration
def self.up
add_column :scripts, :paragraphs_count, :integer, :default => 0
add_column :chapters, :paragraphs_count, :integer, :default => 0
Chapter.reset_column_information
Script.reset_column_information
Chapter.find(:all).each do |c|
paragraphs_count = c.paragraphs.length
Chapter.update_counters c.id, :paragraphs_count => paragraphs_count
Script.update_counters c.script_id, :paragraphs_count => paragraphs_count
end
end
def self.down
remove_column :scripts, :paragraphs_count
remove_column :chapters, :paragraphs_count
end
end
Now to set up the relationships:
class Script
has_many: chapters
has_many: paragraphs, :through => :chapters
end
class Chapter
has_many: paragraphs
belongs_to :script, :counter_cache => true
end
class Paragraph
belongs_to :chapter, :counter_cache => true
end
All that's left is to tell Paragraph to update the paragraph counters in script as a callback.
class Paragraph < ActiveRecord::Base
belongs_to :chapter, :counter_cache => true
before_save :increment_script_paragraph_count
after_destroy, :decrement_script_paragraph_count
protected
def increment_script_paragraph_count
Script.update_counters chapter.script_id, :paragaraphs_count => 1
end
def decrement_script_paragraph_count
Script.update_counters chapter.script_id, :paragaraphs_count => -1
end
end
A: The quick and simple way, without using a cache is to do:
class Script
has_many :chapters
has_many :paragraphs, :through => :chapters
end
script = Script.find(1)
puts script.paragraphs.size #get the count
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,105 |
Home » Azerbaijan significantly improves its position in the National Cybersecurity Index
Azerbaijan significantly improves its position in the National Cybersecurity Index
BAKU, Azerbaijan, January 25. Azerbaijan improved its ranking in the National Cybersecurity Index by 34 positions to 52nd place, the Association of Cybersecurity Organizations of Azerbaijan (AKTA) told Trend.
According to the association, successes in the field of digital development and cybersecurity have allowed Azerbaijan to improve its position in the international index of national cybersecurity.
"The National Cybersecurity Index ranks and assesses 46 digital environment indicators from 161 countries. Our country improved its ranking by 34 positions – from 86th to 52nd," the association said.
Azerbaijan's previous score was 37.66, while it rose to 59.74 in the latest report.
It was noted that the National Cybersecurity Index, developed by the Estonian Academy of E-Governance Foundation, is a global index that, in real time, measures the readiness of countries to ensure cybersecurity and manage cyber incidents.
President Ilham Aliyev has repeatedly stated that the development of cybersecurity in all areas, as well as the strengthening of cyber protection of crucial information resources, is one of the main priorities of the country. The Head of State gave appropriate instructions on the way forward for the development of this sphere.
Azerbaijan's national cybersecurity and information strategy is being developed by a number of departments and organizations, including the Ministry of Digital Development and Transport, the Electronic Security Service , the State Security Service and a number of other institutions.
analytic Azerbaijan news Caspian News Central Asia Company News economic news economy financial news Georgia news IEC news Kazakhstan news latest news news from iran oil and gas news Politics the caspian sea Turkey news Turkmenistan news Uzbekistan news world news
January 26, 2023 Cybersecurity
EC-Council Announces 2022 Academia Award-Winning Cybersecurity Institutions in North America | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 376 |
Q: How do i update content of infowindow in loop with setInterval I am having problem in updating the infowindow content everytime my loop runs with the specified setInterval.Every 1 second the content of the infowindow will change each marker array...but I am having problem If I put the closure in the if statement. after 5 minuets the browser will crash because of adding the listener in every 1 second.what I want is to update only the content of each marker.how would I do that.?
Thank you in advance.
var map;
var marker;
var markerarray =[];
setInterval(function(){
$.ajax({
type: "post",
url: "vehiclecordinates.php",
success: function(data){
coordinates = data.latlng;
vehiclename = data.vehiclename;
heading = data.heading;
devname = data.vehiclename;
thedate = data.trackdate;
for (var i = 0; i < coordinates.length; i++) {
newcoordinate = new google.maps.LatLng(coordinates[i].split(",")[0],coordinates[i].split(",")[1]);
if (markerarray[vehiclename[i]] && markerarray[vehiclename[i]].setPosition){
markerarray[vehiclename[i]].icon.rotation = parseInt(heading[i]);
markerarray[vehiclename[i]].setPosition(newcoordinate);
var con = '<div style="font: 11px arial,tahoma,helvetica,sans-serif;">Vehicle Name:' + devname + '<br/>' +'Date='+thedate+'</div>';
}else {
marker = new MarkerWithLabel({
map:map,
labelClass: "mylabels",
labelStyle: {opacity: 1.0},
labelContent: '<div>'+ vehiclename[i]+'</div>',
icon:{
path: google.maps.SymbolPath.CIRCLE,
scale:.7,
strokeColor: 'white',
strokeWeight: .10,
fillOpacity: 1,
fillColor: '#404040',
offset: '5%',
rotation: parseInt(heading[i]),
anchor: new google.maps.Point(10, 50)
}
});
marker.setPosition(newcoordinate);
markerarray[vehiclename[i]] = marker;
var con = '<div style="font: 11px arial,tahoma,helvetica,sans-serif;">Vehicle Name:' + devname + '<br/>' +'Date='+thedate+'</div>';
google.maps.event.addListener( marker, 'click', (function (marker, con, infowindow) {
return function () {
infowindow.setContent(con);
infowindow.open(map, marker);
};
})( marker, con, infowindow));
}
}
}
});
},1000);
A: You need to name your anonymus clickEventListener function.
*
*Then use javacript code to remove click event.
*If you use jquery to add the eventlistener then you can easily unbind() the function, or search for if it exists then don't add.
*You can set a bool: clickSet=true, that you set after eventlistener once called.
Edit:
Code:
function whatever(marker, con, infowindow) {
return function () {
infowindow.setContent(con);
infowindow.open(map, marker);
};
};
var map;
var marker;
var markerarray =[];
setInterval(function(){
$.ajax({
type: "post",
url: "vehiclecordinates.php",
success: function(data){
coordinates = data.latlng;
vehiclename = data.vehiclename;
heading = data.heading;
devname = data.vehiclename;
thedate = data.trackdate;
for (var i = 0; i < coordinates.length; i++) {
newcoordinate = new google.maps.LatLng(coordinates[i].split(",")[0],coordinates[i].split(",")[1]);
if (markerarray[vehiclename[i]] && markerarray[vehiclename[i]].setPosition){
markerarray[vehiclename[i]].icon.rotation = parseInt(heading[i]);
markerarray[vehiclename[i]].setPosition(newcoordinate);
var con = '<div style="font: 11px arial,tahoma,helvetica,sans-serif;">Vehicle Name:' + devname + '<br/>' +'Date='+thedate+'</div>';
}else {
marker = new MarkerWithLabel({
map:map,
labelClass: "mylabels",
labelStyle: {opacity: 1.0},
labelContent: '<div>'+ vehiclename[i]+'</div>',
icon:{
path: google.maps.SymbolPath.CIRCLE,
scale:.7,
strokeColor: 'white',
strokeWeight: .10,
fillOpacity: 1,
fillColor: '#404040',
offset: '5%',
rotation: parseInt(heading[i]),
anchor: new google.maps.Point(10, 50)
}
});
marker.setPosition(newcoordinate);
markerarray[vehiclename[i]] = marker;
var con = '<div style="font: 11px arial,tahoma,helvetica,sans-serif;">Vehicle Name:' + devname + '<br/>' +'Date='+thedate+'</div>';
google.maps.event.addListener( marker, 'click', whatever);
}
}
}
});
},1000);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,101 |
Q: Type of conditional expression cannot be determined because there is no implicit conversion between 'Entities.KRAParameterInfo' and 'string' lstInitializeGoal = (from itemEmployees in itemsEmployees.Cast<SPListItem>().AsEnumerable()
select new Business.Entities.InitializeGoal
{
UserId = null != itemEmployees[Business.Enums.Employees.LoginName.ToString()]
? Convert.ToString(itemEmployees[Business.Enums.Employees.LoginName.ToString()]).Split(';')[0]
: string.Empty,
EmployeeName = null != itemEmployees[Business.Enums.Employees.LoginName.ToString()]
? Convert.ToString(itemEmployees[Business.Enums.Employees.LoginName.ToString()]).Split('#')[1]
: string.Empty,
LeadUserId = null != itemEmployees[Business.Enums.Employees.ReportingTo.ToString()]
? Convert.ToString(itemEmployees[Business.Enums.Employees.ReportingTo.ToString()]).Split(';')[0]
: string.Empty,
ReportingTo = null != itemEmployees[Business.Enums.Employees.ReportingTo.ToString()]
? Convert.ToString(itemEmployees[Business.Enums.Employees.ReportingTo.ToString()]).Split('#')[1]
: string.Empty,
Designation = null != itemEmployees[Business.Enums.Employees.Designation.ToString()]
? Convert.ToString(itemEmployees[Business.Enums.Employees.Designation.ToString()]).Split('#')[1]
: string.Empty,
WorkStatus = null != itemEmployees[Business.Enums.Employees.WorkStatus.ToString()]
? Convert.ToString(itemEmployees[Business.Enums.Employees.WorkStatus.ToString()])
: string.Empty,
GoalRequired = null != itemEmployees[Business.Enums.Employees.GoalRequired.ToString()]
? Convert.ToString(itemEmployees[Business.Enums.Employees.GoalRequired.ToString()])
: string.Empty,
GoalStatus = "Pending",
KRAParameter = null != itemEmployees[Business.Enums.Employees.Designation.ToString()]
? kraParameterColl.FirstOrDefault(tempKRAParameter =>
tempKRAParameter.Designation == Convert.ToString(itemEmployees[Business.Enums.Employees.Designation.ToString()]).Split('#')[1])
: string.Empty,
IsEnabled = true
}).ToList();
A: My guess is that it's the second-to-last assignment in that code, where you appear to have two different types on either side of the : in your ternary condition:
KRAParameter = (null != itemEmployees[Business.Enums.Employees.Designation.ToString()])
? kraParameterColl.FirstOrDefault(tempKRAParameter =>
tempKRAParameter.Designation == Convert.ToString(
itemEmployees[Business.Enums.Employees.Designation.ToString()]).Split('#')[1])
: string.Empty
Looking specifically at the types, the conditional expression looks something like:
KRAParameter = (condition) ? KRAParameterInfo : string;
The error is saying that it can't determine the type to assign because there is no common type conversion between a KRAParameterInfo and a string.
A possible fix would be to use the default of the type (which is null for classes):
KRAParameter = (null != itemEmployees[Business.Enums.Employees.Designation.ToString()])
? kraParameterColl.FirstOrDefault(tempKRAParameter =>
tempKRAParameter.Designation == Convert.ToString(
itemEmployees[Business.Enums.Employees.Designation.ToString()]).Split('#')[1])
: default(KRAParameterInfo)
Or explicitly convert the first part to a string:
KRAParameter = (null != itemEmployees[Business.Enums.Employees.Designation.ToString()])
? Convert.ToString(kraParameterColl.FirstOrDefault(tempKRAParameter =>
tempKRAParameter.Designation == Convert.ToString(
itemEmployees[Business.Enums.Employees.Designation.ToString()]).Split('#')[1]))
: string.Empty
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,369 |
{"url":"https:\/\/www.physicsforums.com\/threads\/does-every-quantum-field-have-a-non-zero-groundstate.595855\/","text":"# Does every quantum field have a non-zero groundstate?\n\n1. Apr 12, 2012\n\n### a dull boy\n\nDoes every quantum field have a non-zero groundstate? I understand that the energy ground state is non-zero, due to the uncertainty principle virtual particles pop in and out of the vaccum. But is this true for any quantum field (higgs field, for example)?\n\nThanks, Mark\n\n2. Apr 12, 2012\n\n### Jano L.\n\nProbably you refer to the fact that the lowest possible energy of harmonic oscillator, defined in a standard way as the first eigenvalue of the operator\n\n$$H=\\frac{p^2}{2m}+\\frac{1}{2}m\\omega^2 x^2$$\n\nis $\\hbar \\omega \/2$, which is positive.\n\nHowever, it is important to realize that energy can be defined up to arbitrary constant, so the value of the energy alone does not tell anything important.\n\nThe really important thing is that the position and momentum have nonzero statistical scatters, which imply particle is still moving when the wave function is that adjoined to the lowest eigenvalue of energy.\n\nI do not know much about what happens in quantum theory of field, but I have heard there is not even ground-state solution known (vacuum).\n\n3. Apr 16, 2012\n\n### a dull boy\n\nThanks very much Jano, that help. -Mark\n\n4. Apr 16, 2012\n\n### Lapidus\n\nYes, that is true for any quantum field. When you measure very accurately the vacuum for a given field, you will measure non-zero magnitudes for this field. But then also, a split moment later the magnitude and\/ or the direction of the field will be completely different.\n\nHow large the magnitudes and the fluctuations are, depends on how accurate you measure. When you measure at very short distances and very short time intervalls, the magnitudes and fluctuation will be higher.\n\nFor the most fields the fluctuations cancel each other out and give the field a vanishing vacuum expectation value (VEV). The values vary from one point to another and one time to the next, resulting in a zero VEV.\n\nFor one field that this is assumed not true and a non-zero VEV is postulated, and that is the Higgs field.\n\nAlso very important, note the difference between the magnitude of a field and the energy or energy density of a field. The energy density depends only on the field strength, not on its direction. So while the field fluctuates back and forth its energy density does not average out to zero. Quite to contrary, if you zoom in into smaller space and time intervals, the energy becomes infinte! Normally, physicist stop at the Planck scale, saying at that point some new and yet unknown physics must come into play. But even though this prevents the energy of the vacuum from becoming infinite, when we include all the vacuum fluctuation (of all kind of fields that exist!) down to planck scale, we still get a crazy huge number for the energy density of the vacuum! 10^88 tons per cubic centimeter!\n\nOf course, this not what we observe. And that what makes it one of the biggest puzzle in physics.\n\nLast edited: Apr 16, 2012\n5. Apr 17, 2012\n\n### Demystifier\n\nBosonic fields have positive groundstate energy, while fermionic fields have negative groundstate energy. Supersymmetric theories have equal number of bosonic and fermionic fields, so that their groundstate energies cancel up.\n\n6. Apr 17, 2012\n\n### a dull boy\n\nThanks Lapidus, that helps me think a little more on what spontaneous symmetry breaking means.\n\nThanks Demystifier - can I build on your answer? I think it must be related to this statement \"The wavefunction is said to be symmetric (no sign change) under boson interchange and antisymmetric (sign changes) under fermion interchange. This feature of the wavefunction is known as the Pauli principle.\"\n\nCan you tie the energy groundstates for fermions vs. bosons to this statement?\n\nThanks, Mark\n\n7. Apr 17, 2012\n\n### A. Neumaier\n\nThe vacuum _is_ the ground state of every quantum field theory. It is completely inert; nothing can pop in and out of the vacuum = ground state. (One needs energy provided by fields to create particles. The presence of a field means that one is no longer in the ground state.)\n\n8. Apr 18, 2012\n\n### Demystifier\n\nYes, these two things are definitely related, but a detailed explanation would require more formalism (which can be found in many textbooks on quantum field theory.)\n\n9. Apr 18, 2012\n\n### Lapidus\n\nAre you saying there is no such thing as vacuum energy? That the vacuum is completely empty, devoid of anything? When vacuum is completely inert, what does that imply for the cosmological constant problem?\n\n10. Apr 19, 2012\n\n### Demystifier\n\nNo, he is (correctly) saying that vacuum is inert, that no changes happen in vacuum. Being inert is not the same as having no energy.\n\n11. Apr 19, 2012\n\n### Lapidus\n\nHow do you know? When I look\/ measure for fields in the vacuum, the vacuum is a buzzling sea of fluctuations.\n\n12. Apr 19, 2012\n\n### Lapidus\n\nAlso, how can something have energy but does not change in quantum physics?\n\nThe particle in the groundstate of the harmonic oscillator does not sit still like in classical physics.\n\n13. Apr 19, 2012\n\n### martinbn\n\nA pedantic remark, you don't mean non-zero as the ground state is always a non-zero vector.\n\n14. Apr 19, 2012\n\n### Demystifier\n\nI calculate it from the theory.\n\nI guess you calculate it from the theory as well, since I think there is no such experiment performed in practice. But even the theory says that, to measure something, you need a measuring apparatus. And clearly, the presence of the measuring apparatus implies that the total state is not the vacuum.\n\n15. Apr 19, 2012\n\n### Demystifier\n\nThe wave function of the ground-state (or any energy eigenstate) has a trivial time dependence proportional to exp(-iEt), so the probability density of any observable is time-independent. In other words, nothing observable changes.\n\nTo have any observable change at all in quantum physics, the system must be in a state which is not an energy-eigenstate.\n\n16. Apr 19, 2012\n\n### A. Neumaier\n\nYes. Everything you can compute about the vacuum is constant.\nNothing, as currently there is no accepted quantum field theory featuring the cosmological constant.\n\n17. Apr 19, 2012\n\n### A. Neumaier\n\nHow do you look at or measure fields in the vacuum? If you can, you don't have a vacuum.\n\nHowever, nobody ever has measured or seen a buzzling sea of fluctuations. This is just fashionalbe visualization imagery for laymen.\n\n18. Apr 19, 2012\n\n### A. Neumaier\n\nIt only changes an immaterial phase without any influence on measurements.\n\n19. Apr 19, 2012\n\n### Demystifier\n\nHow about the correlation function <0| phi(x) phi(y) |0> ?\n\n20. Apr 19, 2012\n\n### A. Neumaier\n\nIt is translation invariant; changing x and y by the same 4-vector doesn't change the correlation function. Thus what is observable about it is time-independent. (Relative times have no simple meaning in quantum field theory.)\n\nLast edited: Apr 19, 2012","date":"2018-09-21 22:14:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6106886863708496, \"perplexity\": 936.4991040607324}, \"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-2018-39\/segments\/1537267157569.48\/warc\/CC-MAIN-20180921210113-20180921230513-00245.warc.gz\"}"} | null | null |
Q: What is difference between Titan and Neo4j graph database? I had worked on relational database; but now want to learn about graph database. I came to know that these two are graph database. What is difference between these two databases. What should we prefer among them?
A: One approach is to simply try to choose one database over the other. For example, you might quickly search around to find that Titan has been forked to JanusGraph where it is more actively maintained. In your research you may find that there are other open source graph databases as well like OrientDb, ChronoGraph, or Sqlg as well as commercial alternatives like Microsoft's CosmosDb, DSE Graph or IBM Graph. How do you decide now?
There is a graph framework that ties together all of these graphs including Neo4j/Titan (and more than those listed here): Apache TinkerPop. TinkerPop provides an abstraction over different graph databases and graph processors allowing the same code to be used with different configurable backends. This pattern is quite similar to the one you find in SQL with JDBC which helps make your code vendor agnostic.
You can try all of the different supported graph databases before you make a choice and you can do this type of prototyping/benchmarking fairly quickly with the Gremlin Console. You will be able to make self-informed choice as to what is the best way to go for your project.
It occurs to me as I come to the end of this post that I haven't directly answered your question. If you are just getting started and are just interested in learning about graph databases, then I likely wouldn't recommend starting with Titan/JanusGraph as it requires a bit of configuration to get started (schemas, backend selection, etc). Start with TinkerGraph or Neo4j using the Gremlin Console to try out some simple graph traversals and go from there.
A: Titan was originally backed by Aurelius, which was bought by DataStax in 2015. This move was designed to give DataStax a jump-start into the Graph DB world, as they now offer their own "DSE Graph" enterprise product. Titan was since been forked (as previously mentioned) into JanusGraph.
The nice thing about Titan/Janus (IMO) is that it is "pluggable" with other existing back-end and search technologies. So it will "play nice" with things like Cassandra, HBase, Hadoop, Solr, and ElasticSearch.
The drawback is that the community support is tough. The Titan project has been effectively killed, and Janus scores a whopping 0.23 on DBEngines. That makes it the 16th most-popular Graph DB (231st overall), which is pretty low.
Neo4j is backed by Neo Technology, and is regarded as the front-runner in the Graph DB community (score of 38.52 right now, 1st graph DB and 21st overall). It is open source, but controlled by Neo Technologies so they can dictate a difference in feature set between open source and enterprise.
The nice thing about Neo4j is that they have a lot of tutorials and learning aids built right-in to the Neo4j Browser, which is a nice, user-friendly web interface. Their documentation is top-notch, easy to read and search through, and they have a pretty good following here on Stack Overflow.
Neo4j Browser screenshot:
The drawback of Neo4j, is that some features (like clustering) are only available in the enterprise version. But if you work for a big company who doesn't mind shelling-out $ for an enterprise license, that may not be a big deal.
Consistency: Titan/Janus is a part of the "eventual consistency" crowd, while Neo4j aims to be strong-consistent (especially in a causal clustering scenario). Although consistency can be tuned with configuration in both, with Titan/Janus that can be dependent on your choice of pluggable backend (ex: typically strong-consistent with HBase, while eventually consistent with Cassandra).
Recommendation:
If you're just starting to learn graph databases and modeling, you can't go wrong with Neo4j. Simply download/install the community edition, run it, and execute :play movies as your first command (tutorial that walks you through loading, modeling, and querying movie relationships).
If you have some experience with graph, and you don't mind troubleshooting/googling to figure out things (like how to set the max frame size for Thrift), then you could probably do some really cool things with Titan.
Try each out, and see which one works for you.
A: Neo4j uses native graph technology.
*
*Native graph technology ensures that data is stored efficiently by writing nodes and relationships close to each other.
*It optimizes the graph DB.
*With native graph technology, processing becomes faster because it uses index-free
adjancey. That means each node directly references its adjacent nodes.
Titan (Now JanusGraph) uses non-native graph technology.
*
*In non-native we use different storage backends like Cassandra, HBase
*With non-native processing becomes slowers compared to native because database uses
many types of indexs to link nodes together.
A: There are far more than two graph databases - there are dozens. That being said, there are two with real market share: Neo4j and Titan/JanusGraph. But there are dozens of other graph datases, each with interesting strengths for different specific application spaces. That being said, I wouldn't dig into all of the niche players to start with - learning the basic idea of graph databases can be done with one of the two lead players.
Neo4j is the most mature, with the most nicely packaged install and documentation, tons of reference code, and support from a wide range of partners.
Titan/JanusGraph is the next most popular, as it's free/open source and has very strong support (e.g. IBM, Google, Hortonworks, AWS, ...). There's a recent complexity in that the leaders of the Titan project were acquired, freezing the Titan project. But the community forked the project into JanusGraph. So while JanusGraph is a new project, it's literally the same Titan code, with even broader industry support than Titan had.
Related to the two is the language used to work with the graphs. Neo4j uses its proprietary language, Cypher, while nearly everyone else uses Gremlin, and the TinkerPop open source tool set (which is a part of the Apache set of open source projects). Nearly all graph databases, including Neo4j, support Gremlin and TinkerPop. So, for example, you can use either Cypher or Gremlin to query Neo4j, though Neo (and some other proprietary graph database vendors) support Gremlin as a second-class citizen, so to speak. For example, you can connect to Neo using Gremlin from the (external) Gremlin console, but you can't use Gremlin in the (very nice) Neo4j console.
Note that there are many graph databases that support Gremlin other than Titan/JanusGraph. One new entrant that's very interesting is Microsoft's Azure Cosmos DB, which is a managed graph database that's "cheap and easy" if you use Azure already. And there are several vendors that provide managed JanusGraph.
For personal learningk I'd say that Neo4j is the easiest to set up and learn - you download and run it, and open a web browser onto their web-based console, which only takes a few minutes. That being said, if you're comfortable on a command line JanusGraph only took a half hour to install and get running for me, so it's not too hard.
For learning the concepts Neo4j is great. Neo4j's query language, Cypher, and JanusGraph's query language, Gremlin, are semantically identical, just spelled differently, so you'll learn the concepts either way.
For building a real system, either could work (and there are many successful following both approaches).
For which you choose, you'll want to think about whether you want to be strategically tied to a single vendor (Neo4j) or in a broader standards-based community. There's comfort level in picking the market leader with the most mature product - Neo4j. And there's a comfort level in picking open standards with strong industry support - JanusGraph. So IMO there's no "wrong" answer - people using either one are happy and successful. But since you have to pick, you'll need to think about which you're more comfortable with long-term.
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THE PROCRASTINATION EQUATION
How to Stop Putting Things Off
and Start Getting Stuff Done
PIERS STEEL, PhD
"To my brother Toby. He knew that the clock is always ticking."
## Contents
Cover
Title Page
Chapter One - Portrait of a Procrastinator
Chapter Two - The Procrastination Equation
Chapter Three - Wired for Procrastination
Chapter Four - Procrastinations
Chapter Five - The Personal Price of Procrastination
Chapter Six - The Economic Cost of Procrastination
Chapter Seven - Optimizing Optimism
Chapter Eight - Love It or Leave It
Chapter Nine - In Good Time
Chapter Ten - Making it Work
Postscript - Procrastination's Chapter 11
Notes
Index
Acknowledgments
About the Author
Author's Note
Copyright
About the Publisher
[Chapter One
Portrait of a Procrastinator](contents.xhtml#ch_01)
_Never put off till tomorrow,_
_what you can do the day after tomorrow._
**M** **ARK** **T** **WAIN**
This book is about every promise you made to yourself but broke. It is about every goal you set but let slide, never finding the motivation. It is about diets postponed, late-night scrambles to finish projects, and disappointed looks from the people who depend on you—or from the one you see in the mirror. It is about being the slacker in your family and the straggler in your circle of friends. It is about that menacing cloud of uncompleted chores, from the late bill payments to the clutter that fills your home. It is about that doctor's appointment you have been putting off and the finances still in disarray. It is about dawdling, delay, opportunity lost, and more. Much more. This book is also about the _other_ side, the moments of action when procrastination gives way to crystal clarity and attention, work is devoured without hesitation, and giving up never even occurs to you. It is about personal transformation, about unencumbered desire free of internal competition, and the guiltless leisure you can enjoy when your daily tasks are done. This book is about potential, wasted and fulfilled; about dreams that fade into obscurity and dreams we can make come true. Best of all, this book is about shifting the rest of your life away from putting it off to getting it done.
The pivot point that tips us away from accomplishing what we want and need to do is procrastination. It isn't a question of laziness, although the two are easily confused. Unlike the truly slothful, procrastinators want to do what they need to do—and usually do get around to it, but not without a lot of struggle. I will show that this dillydallying is in part hereditary, and that we are hardwired to delay. Our tendency to put things off took a hundred million years to form and is now almost etched into our being. But research shows that, despite its ingrained nature, we can modify our habits and change this behavior. Procrastinators who understand the processes behind their inaction can master them and become less stressed about their deadlines and more able to meet them.
This book tells procrastination's story. It stretches from Memphis of ancient Egypt to modern New York City, from the cancer ward to the stock market floor. I hope to enlighten you about why we procrastinate, what comes of procrastination, and what strategies we can employ to do something about it. We will start off simply, establishing what procrastination is, helping you decide whether you are a procrastinator, and if so, how you likely experience a bout of procrastination. If you are a procrastinator—and the odds are good that you are—you are part of a very large community indeed. It is time we all got to know each other a little bit better.
WHAT PROCRASTINATION IS AND ISN'T
There is so much confusion about procrastination that it is best to lay our subject bare on the dissecting table and start immediately separating the dilly from the dally. By procrastinating you are not just delaying, though delay is an integral part of what you are doing. Procrastination comes from the Latin _pro,_ which means "forward, forth, or in favor of," and _crastinus,_ which means "of tomorrow." But procrastination means so much more than its literal meaning. Prudence, patience, and prioritizing all have elements of delay, yet none means the same as procrastination. Since its first appearance in the English language in the sixteenth century, procrastination has identified not just any delay but an _irrational_ one—that is, when we voluntarily put off tasks despite believing ourselves to be worse off for doing so. When we procrastinate, we know we are acting against our own best interests.
Still, you will find people mischaracterizing wise delays as procrastination. Seeing a co-worker stretched out in his office chair, arms crossed behind his head, relaxed, you ask what he is up to and get a cheerful response of "Me? I'm procrastinating!" But he isn't. He is happily putting off a report because he knows there is a good chance that the project is going to be cancelled later this week, and if it isn't, well, he can still definitely write it at the last minute anyway. This is smart. In this scenario, it is the person who compulsively has to finish everything as soon as possible who is irrational, tackling work even when it is destined to become irrelevant. The obsessive who completes every task at the first opportunity can be just as dysfunctional as the procrastinator who leaves everything to the last moment. Neither one is scheduling time intelligently.
Consequently, it isn't procrastination if you fail to arrive at a party far earlier than everyone else or if you don't get to the airport for your flight three hours in advance. By delaying a little bit, you save awkward moments with your host, who is likely still getting things ready, and you will be spared uncomfortable hours at your gate waiting for your plane to take off. Neither is it procrastination to respond to emergencies by dropping (and putting off) everything else. Insisting that you should finish mowing the front lawn before attending to your house, which has just caught fire, isn't smart. Sure, you didn't put off trimming the grass, but the charred ruin of your home is too high a price to pay. Alternatively, flexibly adapting your schedule to respond to the pressing needs of a spouse or a child will likely save you from ruining your family. Not everything can happen at once; it is in your choice of what to do now and what to delay that procrastination happens, not in delay itself.
YOU THE PROCRASTINATOR
Now that we understand what procrastination is, do you practice it? Where do you land in the ranks of procrastination? Are you a garden-variety dillydallier or are you hardcore with "tomorrow" tattooed across your back? There are some entertaining methods that may reveal your propensity to procrastinate. To begin, check your handwriting. If it is sluggish and disjointed, it may indicate you are likewise. Alternatively, look to the stars . . . well, really the planets. Astrologers note that when Mercury is in retrograde or in opposition to Jupiter, procrastination tends to be on the uptick. Or try a tarot card reading. The "Two of Swords" often indicates you are split with a dilemma and procrastinating on your decision. Personally, I prefer a more scientific approach.
You can go to my website, www.procrastinus.com, for a comprehensive test that I've administered to tens of thousands of subjects, and compare your level of irrational delay with those of individuals around the world. However, if time is pressing and you wish not to delay, you might try the shorter quiz provided below. Complete the mini-version here by circling your response to each of these nine items and then calculating the total. Note that questions 2, 5, and 8 are scored in the opposite direction from the other items:
**Stands For:**
1. VERY SELDOM OR NOT TRUE OF ME
2. SELDOM TRUE OF ME
3. SOMETIMES TRUE OF ME
4. OFTEN TRUE OF ME
5. VERY OFTEN TRUE OR TRUE OF ME
* * *
1. I delay tasks beyond what is reasonable.
**1 2 3 4 5**
2. I do everything when I believe it needs to be done.
**5 4 3 2 1**
3. I often regret not getting to tasks sooner.
**1 2 3 4 5**
4. There are aspects of my life that I put off, though I know I shouldn't.
**1 2 3 4 5**
5. If there is something I should do, I get to it before attending to lesser tasks.
**5 4 3 2 1**
6. I put things off so long that my well-being or efficiency unnecessarily suffers.
**1 2 3 4 5**
7. At the end of the day, I know I could have spent the time better.
**1 2 3 4 5**
8. I spend my time wisely.
**5 4 3 2 1**
9. When I should be doing one thing, I will do another.
**1 2 3 4 5**
* * *
**TOTAL SCORE _______**
**SCORE**
19 or less
**COMPARED TO EVERYONE ELSE**
You are in the bottom 10%
Your mantra is "first-things-first"
**SCORE**
20-23
**COMPARED TO EVERYONE ELSE**
You are in the bottom 10-25%
**SCORE**
24-31
**COMPARED TO EVERYONE ELSE**
You are in the middle 50%
Average procrastinator
**SCORE**
32-36
**COMPARED TO EVERYONE ELSE**
You are in the top 10-25%
**SCORE**
37 or more
**COMPARED TO EVERYONE ELSE**
You are in the top 10%
Tomorrow is your middle name
Where did you end up? Are you legendary for leaving things to the last minute or do you only put off exercising and taxes, like almost everyone else?
PROCRASTINATION POLKA
The higher you scored on that procrastination test, the greater the chance that you are procrastinating right now. Certain other tasks should be occupying your attention—which sadly means you have better things to do than reading this book. These tasks are likely unpleasant, possibly administrative and boring, and perhaps difficult to visualize as being successfully accomplished. Let me make a few guesses about what is on your plate:
• Is your laundry basket overflowing?
• Are there dirty dishes in the sink?
• Do your smoke detectors need new batteries?
• How about your car battery? What is the air pressure in your tires and how long has it been since the last oil change?
• Isn't there a ticket to book, a room to reserve, a bag to pack, a passport to renew?
• Have you informed your boss about your vacation plans?
• Have you bought a gift for that upcoming birthday?
• Have you filled out your time sheets, performance reviews, and expense reports?
• Did you hold that difficult conversation with the employee whose work is not up to par?
• Have you scheduled the meeting you are dreading?
• What about the big project your boss gave you? Are you making progress?
• Did you make it to the gym this week?
• Have you called your mom?
How does that list strike you? You can add to it, of course. Even if I didn't score a direct hit, you were likely procrastinating somewhere else, pushing a task into the future. On its own, each of these postponed tasks has few repercussions. Together, they can culminate in misery by nibbling away at your life. The major project, the one with the hard deadline, is the mother of all such concerns; it can keep you awake at night and make it difficult to accomplish any of the other tasks on your list. At one time or another, we have _all_ felt motivationally marooned and unable to get around to the report, the research, the writing, the presentation to prep, or the exam to ace.
There is a common pattern to all procrastination and it goes something like this. At the start of a big project, time is abundant. You wallow in its elastic embrace. You make a few passes at getting down to it, but nothing makes you feel wholeheartedly engaged. If the job can be forgotten, you'll forget it. Then the day arrives when you really intend to get down to work; but suddenly it's just something you don't feel like doing. You can't get traction. Every time you try to wrap your mind around it, something distracts you, defeating your attempts at progress. So you forward your task to a date with more hours, only to find that every tomorrow seems to have the same twenty-four. At the end of each of these days, you face the disquieting mystery of where it went. This goes on for a while.
Eventually, time's limited nature reveals itself. Hours, once tossed carelessly away, become increasingly limited and precious. That very pressure makes it hard to get started. You want to get going on the big project but instead you take on peripheral chores. You clean your office or clean up your e-mail; you exercise; you shop and cook. Part of you knows this isn't what you should be doing, and so you say to yourself, "I am doing this; at least I am preparing by doing something." Eventually, it is too late in the day to really get started, so you may as well go to bed. And the cycle of avoidance starts again with the dawn.
Sometimes, to quell your anxiety, you give in to total diversion. You take a moment to check your e-mail or the sports scores. From there, why not respond to a few messages or watch a few minutes of TV? Soon these temptations have seduced you. The task still waggles itself in the periphery of your vision, but you don't want to look it in the eye—it will have you if you look—so you burrow deeper into your distractions. You write long passionate comments on online forums, troll for news tidbits, or manically switch TV channels at the first ebb of interest. Pleasure turns to powerlessness as you become unable to extract yourself.
As the deadline approaches, you make the diversions more intense so that they will sufficiently distract you. Banishing anything that reminds you of the dreaded thing, you shun calendars and timepieces. In a willful distortion of reality, you shift your plans from what you once could solidly accomplish to what is minimally possible. When you should be working harder than ever, you are sleeping in, daydreaming of alternative worlds, of winning the lottery, of being anywhere but here. As anxiety mounts, you want immediate relief, escape, rewards—anything that gives you the illusion of safe harbor. If friends or relatives or co-workers try to separate you from your diversions, you meet them with an annoyed: "Just a minute! I'LL DO IT AFTER THIS!" Unfortunately, "this" never ends. Secretly, you are full of self-recrimination and self-doubt, envious of those who simply get things done.
Energy builds until finally a threshold is crossed and something clicks. You start working. Some inner mind has quietly boiled the task down to its essence, as there are no more moments to spare. You wade into the work, making ruthless decisions and astonishing progress. In place of that menacing cloudiness, a glittering clarity comes over you. There is purity to your work, fueled by the real urgency of now or never. For a lucky few, this surge of efficiency will enable them to get the project done. For others, this initial rush wanes before the cursed thing is completed. After too many hours of sleepless concentration, brains shut down. Caffeine and sugar only offer an unsatisfying buzz. Tick, tock . . . the time has run out. You limp across the finish line with insufficient preparation, giving the world your second best.
This is so common as to be unremarkable—except to the person who has suffered through the experience and knows the performance was not up to par. The relief at getting a job done doesn't always make up for doing a sloppy job. Even if you managed to perform brilliantly, the achievement is tainted with a whiff of what might have been. And this kind of procrastination has likely cast a cloud on an evening out, a party, or a vacation, which you couldn't fully enjoy because half of your mind was elsewhere, obsessing about what you were avoiding. You resolve that this will never happen again; the cost of procrastination is too great.
The trouble with such resolutions is that procrastination is a habit that tends to endure. Instead of dealing with our delays, we excuse ourselves from them—self-deception and procrastination often go hand-in-hand. Exploiting the thin line between _couldn't_ and _wouldn't,_ we exaggerate the difficulties we faced and come up with justifications: a bad chest cold, an allergic reaction that caused sleepiness, a friend's crisis that demanded our attention. Or we deflect responsibility entirely by saying, "Gee whiz, who knew?" If you couldn't have anticipated the situation, then you can't be blamed. For example, how would you respond to the following questions regarding your last bout of procrastination?
• Did you know the task was going to take so long?
• Did you realize that the consequences of being late were so dire?
• Could you have expected that last-minute emergency?
The honest answers are likely yes, yup, and definitely, but it's difficult to answer honestly, isn't it? And that is the problem.
Some procrastinators will even try to frame their self-destructive inaction as a thoughtful choice. For example, is it wrong to put off your career to pursue more family time? It depends on who you are. Some people relish the work-focused model of success, resenting time taken away from the job, and so they may miss out on family dinners and school plays. Others prosper in the home and community, enjoying the relationships nurtured there, at the expense of tasks at work. To the casual observer, it isn't easy to tell which choice is procrastination and which is a purposeful decision. Only the procrastinator knows for sure.
In the back of their minds, many procrastinators hope they won't need excuses. They bank on Lady Luck. Sometimes it works. Frank Lloyd Wright drew his architectural masterpiece, Fallingwater, in the three hours before his patron, Edgar Kaufmann, came to see the sketches. Tom Wolfe cranked out in a midnight panic forty-nine pages of almost unedited prose for an _Esquire_ magazine piece on California's hotrod and custom car culture. Byron Dobell, his editor, simply removed "Dear Byron" from the top of Wolfe's memo and printed it under the title "There Goes (Varoom! Varoom!) That Kandy-Kolored Tangerine-Flake Streamline Baby," and a new style of journalism was born. But I don't need to tell you how rare such outcomes are. By your own standards, if you thought delay was a good idea in the first place, you wouldn't be procrastinating.
THE PROCRASTINATOR'S PROFILE
If it makes you feel any better, procrastination puts us in good company. It's as common as morning coffee. Across scores of surveys, about 95 percent of people admit to procrastinating, with about a quarter of these indicating that it is a chronic, defining characteristic. "To stop procrastinating" is at any time among the world's top reported goals. Procrastination is so prevalent that it has its own brand of humor. Possibly the best excuse for missing a deadline came from Dorothy Parker. When asked by _The New Yorker_ 's editor, Harold Ross, for a piece that was late, she woefully explained, using her dark and sorrowful eyes to full effect, "Somebody was using the pencil." And, of course, there is the most infamous of all procrastination jokes. Don't you know it? I will tell you later.
No occupational category seems immune from procrastination, but writers seem especially prone. Agatha Christie was guilty of it and Margaret Atwood admitted she often spends "the morning procrastinating and worrying, and then plunges into the manuscript in a frenzy of anxiety around 3:00 p.m." Newscasters can also suffer from it; witness Ted Koppel's quip: "My parents and teachers used to be exasperated by the fact I would wait until the last minute, and now people are fascinated by it." Procrastinators come from every letter of the occupational alphabet, from astronauts to Episcopalian priests and from X-ray technicians to zookeepers. Unfortunately, whatever the job, procrastinators are more likely to be unemployed or working part-time compared to their non-procrastinating counterparts. Procrastinators can be of either sex, though the Y chromosome has a slight edge. A group of a hundred hardened procrastinators would likely be composed of 54 men and 46 women, leaving 8 unmatched males vying for a female dalliance. You see, procrastinators tend to be available . . . sort of. They are more likely to be single than married but also more likely to be separated than divorced. They put off ending as well as beginning commitment. Age also determines procrastination. As we progress from grade school through to the retirement home and the closer we come to life's final deadline, the less we put off. Those who have matured physically are, unsurprisingly, more mature in character.
This demographic exploration, though interesting, isn't as useful as identifying procrastinators by their psychological profile. There is indeed a core trait explaining why we put off, but it might not be what you have heard. It is commonly thought that we delay because we are perfectionists, anxious about living up to sky-high standards. This perfectionist theory of procrastination sounds good and even feels good. Perfectionism can be a desirable trait, as shown by the canned response to the interview question, "What is your biggest weakness?" When Bill Rancic was asked that question just before winning the first season of Donald Trump's _The Apprentice,_ he replied, "I'm too much of a perfectionist; it's a flaw," prompting his interviewer to interject, "Being a perfectionist is a good thing; it means you keep striving." But the perfectionism-procrastination theory doesn't pan out. Based on tens of thousands of participants—it's actually the best-researched topic in the entire procrastination field—perfectionism produces a negligible amount of procrastination. When the counseling psychologist Robert Slaney developed the Almost Perfect Scale to measure perfectionism, he found that "perfectionists were _less_ likely to procrastinate than non-perfectionists, a result that contradicted the anecdotal literature." My research backs him up: neat, orderly, and efficient perfectionists don't tend to dillydally.
How, then, did we come to believe that perfectionism causes procrastination? Here is what happened. Perfectionists who procrastinate are more likely to seek help from therapists, so of course they turn up in clinical research about procrastination in greater numbers. Non-perfectionist procrastinators (and for that matter, non-procrastinating perfectionists) are less likely to seek professional help. Perfectionists are more motivated to do something about their failings because they are more likely to feel worse about whatever they are putting off. Consequently, it is not perfectionism that is the problem but the _discrepancy_ between perfectionist standards and performance. If you are a perfectionist and are suffering from high standards that are unachievable, you might want to do something about that too, but you will need an additional book: this one is about procrastination.
What is really the main source of procrastination? Thirty years of research and hundreds of studies have isolated several personality traits that predict procrastination, but one trait stands above the rest. The Achilles Heel of procrastination turns out to be _impulsiveness;_ that is, living impatiently in the moment and wanting it all now. Showing self-control or delaying gratification is difficult for those of us who are impulsive. We just don't have much ability to endure short-term pain for long-term gain. Impulsiveness also determines how we respond to task anxiety. For those of us who are less impulsive, anxiety is often an internal cue that gets us to start a project early, but for those who are more impulsive it is a different story: anxiety over a deadline will lead straight to procrastination. The impulsive try to avoid an anxiety-provoking task temporarily or block it from their awareness, a tactic that makes perfect sense if you're thinking short term. In addition, impulsiveness leads procrastinators to be disorganized and distractible or, as my colleague Henri Schouwenburg puts it, to suffer from "weak impulse control, lack of persistence, lack of work discipline, lack of time management skill, and the inability to work methodically." In other words, impulsive people find it difficult to plan work ahead of time and even after they start, they are easily distracted. Procrastination inevitably follows.
LOOKING FORWARD
So there it is. Procrastination is pervasive. Almost as common as gravity and with an equal downward pull, it is with us from the overfull kitchen garbage can in the morning to the nearly empty tube of toothpaste at night. In the next chapter, I'll let you in on the research that has helped me understand why we delay things irrationally and why procrastination is so widespread. I'll reveal and explain the Procrastination Equation, a formula that shows the dynamics of this way of behaving, and then I'll tell you about the amazing opportunity I had to study this phenomenon in the real world. Subsequent chapters will describe the different elements that are at play in our minds and hearts, and then we'll look at the price of procrastination in our lives and in society at large. There's always a good side to the kind of research I present—within the causes we can also find the cures. So the last part of the book will offer ways in which individuals, bosses, teachers, and parents can improve their own motivation and motivate others, in the hope that procrastination will be less of a scourge. The final chapter pushes you to put these proven practices into your own life. The advice here is evidence-based, as scientifically vetted and pharmaceutically pure as it gets; it's the good stuff from behind the counter, so don't overdo it.
[Chapter Two
The Procrastination Equation](contents.xhtml#ch_02)
THE RESULT OF EIGHT HUNDRED STUDIES PLUS ONE
_My own behavior baffles me. For I find myself doing what_
_I really hate, and not doing what I really want to do!_
ST. PAUL
Rejection is wearing thin on Eddie during his first sales job. He attentively attended each sales seminar, read all the recommended books, and dutifully repeats the positive affirmations "I can do it! I am a winner!" each morning in the mirror. Still, after another day without a sale he is looking at his phone with dread. As he picks it up and cold calls another prospect, the only response he anticipates hearing is yet another "I am in a meeting" or "click" as they hang up halfway through his introduction. Indeed, he is brushed off once again. "What is the point?" Eddie asks himself. Demoralized, he organizes his desk, fills out all the paperwork to update his benefits package, and surfs the Internet to get insights about competitors' products. He puts off his phone calls until later—the dregs of the day when most of his potential clients are leaving for home. His boss checks in on him and recognizes the signs. Eddie's decision to delay is the beginning of the end of his sales career.
* * *
Valerie's face is as blank as her computer screen. She stares at it, knowing that words should be there, words written by her, but nothing appears. Not even a letter. "Why? Why?" she wonders. It is not like she hasn't done pieces like this before, but for some reason this assignment on municipal politics due tomorrow is mind numbing. "Write," she thinks, "Press your fingers into the keyboard." In response, "asdfkh" appears on the screen. Better than nothing. Convincing herself she needs a short break from interminable boredom, Valerie starts texting her friends who direct her to a nifty website spoofing popular bands. After watching a few music videos, she finds a satire site on television shows and texts that link back to her friends. Soon, Val's virtual group is trying to one-up each other to find the funnier and cleverer clip. Hours go by, then it dawns on her that it is near the end of day and she feels even less inspired than when she first chose to take that "short break." She dives into the writing but the end product reflects the effort and time put into it. It's crap.
* * *
The vacation plans are set! Tom is for once ahead of the ball, booking time off in advance to fly to the Dominican Republic. Thanks to his foresight he even paid for his flight on points from his frequent-flyer program. The only detail left is to reserve a room at a hotel, but that can be done at anytime. But what can be done at anytime is often done at no time. As the months slip by, Tom pushes the task forward to each sub-sequent week or forgets about it altogether. There is always something more pressing to attend to, like his favorite television show. Finally, as he thinks about what to pack, he realizes that there are no more weeks to push the task forward into and that he has left it far too late. He goes online and, finding little available, makes a hurried and haphazard reservation. When his plane later sets down in the Dominican, he hopes that his hotel is as beautiful as the island. It isn't. It's too far from the beach, his room is decorated with dead mosquitoes, adjoins a disgusting bathroom, and the hotel dining gives him food poisoning.
* * *
Eddie, Valerie, and Tom are all procrastinators but they are not identical. Just as a car can stop running because of an empty gas tank, a blown tire, or a dead battery, there are a multitude of causes for procrastination—even if the outward behavior is the same. Eddie, Valerie, and Tom all procrastinated for different underlying reasons and each one represents a facet of the Procrastination Equation, the mathematical formula I derived that describes irrational delay. Understanding why Eddie, Valerie, and Tom put off their respective tasks is the essence of this book. To this end, we are going to do a little more assessment. In the last chapter, we established the degree to which you procrastinate. In this chapter, we are going to find out why you put things off. Are you an Eddie, a Valerie, a Tom or some hybrid of all three? Take this test by circling your response to each of the following 24 items and find out:
**Stands For:**
1. VERY SELDOM OR NOT TRUE OF ME
2. SELDOM TRUE OF ME
3. SOMETIMES TRUE OF ME
4. OFTEN TRUE OF ME
5. VERY OFTEN TRUE OR TRUE OF ME
* * *
1. When I put in the hours, I am successful.
**1 2 3 4 5**
2. Uninteresting work defeats me.
**1 2 3 4 5**
3. I get into jams because I will get entranced by some temporarily delightful activity.
**1 2 3 4 5**
4. When I apply myself, I see the results.
**1 2 3 4 5**
5. I wish my job was enjoyable.
**1 2 3 4 5**
6. I take on new tasks that seem fun at first without thinking through the repercussions.
**1 2 3 4 5**
7. If I try hard enough, I will succeed.
**1 2 3 4 5**
8. My work activities seem pointless.
**1 2 3 4 5**
9. When a temptation is right before me, the craving can be intense.
**1 2 3 4 5**
10. I am confident that my efforts will be rewarded.
**1 2 3 4 5**
11. Work bores me.
**1 2 3 4 5**
12. My actions and words satisfy my short-term pleasures rather than my long-term goals.
**1 2 3 4 5**
13. I am persistent and resourceful.
**1 2 3 4 5**
14. I lack enthusiasm to follow through with my responsibilities.
**1 2 3 4 5**
15. When an attractive diversion comes my way, I am easily swayed.
**1 2 3 4 5**
16. Whatever problems come my way, I will eventually rise above them.
**1 2 3 4 5**
17. When a task is tedious, again and again I find myself pleasantly daydreaming rather than focusing.
**1 2 3 4 5**
18. I have a hard time postponing pleasurable opportunities as they crop up.
**1 2 3 4 5**
19. I can overcome difficulties with the necessary effort.
**1 2 3 4 5**
20. I don't find my work enjoyable.
**1 2 3 4 5**
21. I choose smaller but more immediate pleasures over those larger but more delayed.
**1 2 3 4 5**
22. Winning is within my control.
**1 2 3 4 5**
23. If an activity is boring, my mind slips off into other diversions.
**1 2 3 4 5**
24. It takes a lot for me to delay gratification.
**1 2 3 4 5**
* * *
To score, add up your answers to each of the following questions:
Eddie's Scale = 1 + 4 + 7 + 10 + 13 + 16 +19 + 22 =
Valerie's Scale = 2 + 5 + 8 + 11 + 14 + 17 + 20 + 23 =
Tom's Scale = 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24 =
If you scored 24 or _lower_ for _Eddie's Scale,_ you have some similarities with his situation. On the other hand, if you scored 24 or _higher_ for _Valerie's Scale_ or _Tom's Scale,_ you really should give them a call as you have a lot in common. You see, Eddie, Valerie, and Tom represent respectively the three basic elements of motivation: Expectancy, Value, and Time. Once you grasp their situations, you will understand the components of the Procrastination Equation. After this, we will look at how each of these pieces fits together with the others to form the overall formula. Yes, there will be math, but don't balk. A version of this principle was illustrated within just two glossy pages of _Yes! The Science Magazine for Kids._ If twelve-year-olds can get it, so will you.
LOW EXPECTANCY EDDIE
Eddie's story is regrettably common in sales. Rejection is part and parcel of the job, and most sales people receive an ungodly number of "no's" before they get a "yes," especially at the beginning of their careers. Many aspiring salespeople, like Eddie, succumb to this steady stream of rebuffs and find themselves lacking the motivation to perform; it takes especially resilient people to rise above relentless negativity. What is sapping Eddie's motivation and causing his procrastination? It is _Expectancy_ —what he _expects_ will happen. After a series of attempts that all resulted in failure, he began to expect failure even before he started. High expectancy forms the core of self-confidence and optimism: but if you start believing your goals aren't achievable, you stop effectively pursuing them. Consequently, if during your self-assessment you _disagreed_ with statements like "I am confident that my efforts will be rewarded" or "Winning is within my control," you are like low-expectancy Eddie.
The results from thirty-nine procrastination studies consisting of almost seven thousand people indicate that while some procrastination stems from overconfidence, the opposite is far more common. Procrastinators are typically less confident, especially about the tasks they are putting off. If you are procrastinating about schoolwork, you likely consider the assignment difficult. If you are procrastinating about getting healthier, by starting an exercise program, for example, or by eating better, odds are that you question your ability to follow through. And, if you are unemployed, you are likely procrastinating on your job search because you are discouraged about your chances of getting hired.
The seminal work of Martin Seligman, one of the leaders of the positive psychology movement, demonstrates the connection between lack of self-confidence or optimism and procrastination. If you love dogs, as I do, please try to forgive Dr. Seligman; he experimented by jolting canines with electricity.2a The gist of what he did was to yoke together two sets of dogs, and zap them at random intervals. Both sets received the same electric shocks and for the same duration, but the first group could press a lever that terminated the shocks for all the dogs. The second group had no control and was entirely dependent on their counterparts for ending the agony. Seligman then changed the setup; he tested both sets of dogs again but this time in a shuttle-box divided by a low barrier. One side of the box became electrified and _all_ the dogs had the possibility of escaping simply by jumping over the partition. The first group of dogs, who previously had control of the lever, readily learned to jump over the barrier. The second group had also learned something from their previous experience. When the box was electrified, they didn't jump, but lay down and took the shock. Like low-expectancy Eddie, these dogs had learned that what they did made no difference; they had learned they were helpless.
Learned helplessness is connected to quickly giving up, whether in a complacent acceptance of a prolonged sickness or in a lackluster school performance. Learned helplessness also helps explain why putting off decisions more than usual is one of the symptoms of depression. The underlying cause is reduced self-confidence, which makes it difficult to invest in any demanding work. On balance, a degree of learned helplessness is common. Many of us have been in situations where our world was seemingly not set up for our success. For low-expectancy Eddie, it was his sales job; for someone else it may have been a harsh upbringing in which family or classmates enforced rigid roles. Restraining beliefs can become internalized and be carried within us long after we leave the home or schoolyard where they started. Our learned self-perception becomes a self-fulfilling prophecy—by expecting to fail, we make failure a certainty. We never dig in and really try, and the end result is more procrastination.
VALERIE WITHOUT VALUE
How do you feel about what you are putting off right now? As you reflect on the question, you will be channeling Valerie. Like her, with her sluggish attempts at writing on municipal politics, we all tend to put off whatever we dislike. Consequently, that chore you are currently deferring is probably something you don't especially enjoy. The technical term for this measure of enjoyment is _Value_ and the less of it a task has for you, the harder it is for you to get started on it. We have no problem initiating lengthy conversations with friends over a few drinks and a decadent dessert, but most of us delay starting on our taxes or cleaning out our basement. Similarly, the top reason that students give for essay procrastination is that they "really dislike writing term papers."2b Although the fact that we are less likely to promptly pursue an unpleasant task may seem pretty obvious, the scientific field lacks your insight. Scientists have committed over a dozen studies involving a good two thousand respondents to reach the same conclusion. Well, at least now we are sure.
To the degree that some tasks are universally burdensome, they reveal some touchstones of procrastination. Since everyone wants to put off whatever they detest, it is no surprise that we commonly avoid cleaning up, or organizing, or seeing our doctor or dentist. Since many find exercising an imposition, 70 percent of us rarely use our long-term gym memberships. Similarly, many find Christmas shopping stressful, thus helping to keep Christmas Eve the busiest shopping day of the year. On the other hand, to the degree that individuals consider certain chores uniquely unpleasant, the exact bundle of procrastinating tasks will differ from person to person. Depending on the nature of their dillydallying owners, some households contain kitchen counters cluttered with dishes, while others have medicine cabinets stuffed with long-defunct prescriptions. Some have fridges needing to be filled with food, while others have dining room tables needing to be filled with friends.
Given the connection between what is pleasurable and what is promptly pursued, it makes sense, then, that chronic procrastinators tend to detest life's allotment of responsibilities. Their jobs, their chores, their duties are all irksome, and they avoid tackling these tasks as long as possible. If you agreed with statements like "Work bores me" or "I lack enthusiasm to follow through with my responsibilities," the absence of pleasurable value is likely a source of your procrastination. Laundry makes you listless, cooking makes you crabby, and washing dishes and paying the bills are hardships rather than innocuous incidentals. You have tremendous difficulty keeping your attention on the mundane. For you, boredom signals irrelevance and your mind slides off to something else. This very characteristic has provided me with quite a challenge in writing this book. I am painfully aware of your fickle nature and your unforgiving attention span—meaning that I'd better keep a lively pace at all times. In other words, ever onward.
TIME-SENSITIVE TOM
While Eddie's Expectancy and Valerie's Value are contributing factors to procrastination, Tom's reason is at its core. Tom had to book a hotel room but couldn't find the motivation until just before the deadline, letting himself get distracted every time he made an intention to act. When he finally did do something, he knew he should have acted earlier and he suffered for his tardiness. In all likelihood, if you procrastinate, you feel some kinship with Tom and have admitted that you too "get into jams" because you are "entranced by some temporarily delightful activity" or that you "choose smaller but more immediate pleasures over those larger but more delayed." The biggest factor in determining what you pursue is not the associated rewards or the certainty of receiving them, but their timing. You value rewards that can be realized quickly far more highly than rewards that require you to wait; simply, you are impulsive.
As I mentioned in the last chapter, scientific evidence of the connection between impulsiveness and procrastination is unequivocal. Scores of studies based on many thousands of people have established that impulsiveness and the related personality traits of low conscientiousness, low self-control, and high distractibility are at the core of procrastination. I myself have collected personality profiles from more than twenty thousand people to take a closer look. And I found confirmation that of these traits, impulsiveness shares the strongest bond with procrastination. This isn't surprising if you look at specific aspects of impulsiveness: intense cravings, a lack of caution and reserve, and an inability to see tasks through. Though all have their role in why we put things off, the last of these is almost equivalent to procrastination in itself: not seeing tasks through means agreeing to statements like "I'm not good about pacing myself so as to get things done on time." People who act without thinking, who are unable to keep their feelings under control, who act on impulse, are also people who procrastinate.
The influence of time itself also contributes to the connection between impulsiveness and procrastination. We tend to see tomorrow's goals and concerns abstractly—that is, in broad and indistinct terms—but to see today's immediate goals and concerns concretely—that is, with lots of detail on the particulars of who, what, where, and when. Actions or goals framed in abstract terms, like "engaging in self-development," are less likely to be immediately pursued than goals framed in concrete terms, like "reading this book." Similarly, the broad goal of "exercising" is less motivating than "running for an hour," and "getting a promotion" is harder to act on than the more immediate goal of "writing this report." Since we consistently frame long-term goals abstractly, the result is that we are more likely to postpone them, at least until they become short-term goals and we start thinking about them concretely. Psychologists Nira Liberman and Yaacov Trope have recently specialized in the scientific study of this phenomenon, but the basics aren't that new. David Hume wrote about the same thing over 250 years ago in his book _A Treatise of Human Nature._ 12
Right now, if you like, you can experience the influence of time on whether you view events concretely or abstractly. Let's plan a shopping trip for the distant future, say next year. Take a moment and imagine yourself twelve months from now. What would you buy? Do you have a clear picture or is the vision cloudy and smudged? Now imagine the money currently warming your pocket. If you had to spend it today, this very moment, where exactly would that money go? Likely, what you plan to buy a year from now seems generic, as vanilla as "nice shoes" or "good sports equipment." Such goals are ghosts, ethereal and with no handles to grab onto. Today's spending plans, however, are likely concrete and meaty, something you can sink your teeth into. Instead of "shoes," they are Manolo Blahnik's "Sizzle," the python sandal that will make you the envy of every shoe fetishist. Rather than just "sporting goods," you are obsessing over a TaylorMade Quad r7 425 TP Driver, the one with the oversized titanium sweet spot, used by the pros on the PGA tour. As you contrast these concrete and abstract options, the differences in their ability to excite you should be palpable. This is procrastination's dark heart. It is largely because we view the present in concrete terms and the future abstractly that we procrastinate.
PUTTING THE PIECES TOGETHER
Eddie's, Valerie's, and Tom's situations—that is Expectancy, Value, and Time—are the basic components of procrastination. Decrease the certainty or the size of a task's reward—its expectancy or its value—and you are unlikely to pursue its completion with any vigor. Increase the delay for the task's reward and our susceptibility to delay—impulsiveness—and motivation also dips. Understanding procrastination at this component level isn't bad, but we can do better.2c
The first step is to figure out how Expectancy and Value fit together. To this end, we can tap into an entire family of formulations called Expectancy Theories, the most famous being _Expected Utility Theory._ You might not have heard of it by that name but you are more familiar with it than you know. Expected Utility Theory forms the basis of mainstream economics; every successful gambler adheres to its rule. It proposes that people make their decisions by multiplying expectancy and value together. That is:
EXPECTANCY × VALUE
Here is how it works. Imagine there are two piles of money in front of you. The one on the right I will definitely give you—very nice of me—but the one on the left I probably won't. If you could ask for only one pile, which would it be? My bet is that you would take the sure thing, revealing how _expectancy_ affects your decisions. Expectancy, as you might expect, refers to probability or chance. We prefer _more_ likely to _less_ likely rewards. However, what if I told you the sure thing on the right was a much smaller pile of money than the riskier one on the left? This is actually a pretty common situation, like choosing whether to put your money into riskless but low rate government bonds or to speculate on the stock market. To make sense of your options, now you have to incorporate _value_ into your decision making in order to judge how much bigger the pile needs to be to inspire you to take more risk. As I vary the size of a pile and the probability of you receiving it, your preferences will flip from right to left and vice versa. The formula "Expectancy × Value" does a fair job of predicting which pile you would end up choosing. Multiplying the two together, you go with whatever pile has the highest outcome. Economists try to understand all of human behavior using just this equation. From their viewpoint, every choice you make—from pouring milk on your cereal to wiping your child's nose—is based on how much pleasure you will receive and the degree of certainty that you will receive it. Unfortunately, they are wrong.
You can't rely on "Expectancy × Value" alone to describe human nature. For starters, the theory is considered an expression of rational decision making, meaning that it doesn't leave room for any form of irrational behavior. No matter what you do, from eating an ice cream cone to getting hooked on heroin, it is all reasonable from an economist's perspective. Consequently, their theory also excludes the possibility of procrastination—irrational delays—and since I am currently writing a book on the topic and you are currently reading one, let's consider this a weakness. The economic model of human nature isn't so much incorrect as just incomplete. Consistently, we do respond to incentives (i.e., value) to the extent we believe (i.e., expect) that they are obtainable, but that isn't the entire picture. There is a third factor—time.
Economists need to update how their model of human nature deals with time, and I'm not the only one saying so. Back in 1991, in a lecture aptly titled "Procrastination and Obedience," the Nobel Prize-winning economist George Akerlof spoke to the American Economic Association about how his field would be better off if it considered how we irrationally find present costs more salient than future costs. In the following year, George Loewenstein, an economics professor from Yale, co-authored _Choice Over Time,_ which reviews how economics can best include time. Since then, behavioral economics, a sub-speciality of economics that integrates time, has opened up, with researchers such as Ted O'Donoghue and Matthew Rabin studying procrastination specifically. These behavioral economists are simply using observations of the world to refine their model, which is like using feedback from your eyes to keep the car on the road. Sounds very, you know . . . rational.
The theory of time that these behavioral economists are most attracted to is from the psychological field of behaviorism. Behaviorists developed a little equation called the _Matching Law,_ which proved pretty good at predicting the average behavior of mice and men. Here it is in one of its simplest forms:
Since the product of Expectancy × Value is divided by Delay, the greater the delay, the less your motivation.
How important is the inclusion of time? Let me invent my own game show called _Now Deal or Then Deal._ You are a contestant and have won $1,000. It is put into your hands in ten crisp $100 bills, a short stack you stuff into your pocket. However, I also have a certified check—guaranteed money—postdated to one year from now. Here is the dilemma. What is the minimum amount that I have to put on that check to get you to dig into your pocket, hand back all those hundreds, trade me for the check, and wait 365 days to cash it? I have run this little thought experiment with hundreds of people in my classes, and most say that they would wait a year for between $2,000 and $3,000, especially if I ask for an immediate, gut decision. Unless you have been taught a reasonable rate of return and given time to mentally calculate it, thereby preventing yourself from reacting emotionally, it is likely that these responses are not so far off from your own. The more money you require to make the swap, the more sensitive you are to delay; that is, the more impulsive you are. Unfortunately, this sensitivity to delay is still missing from the equation.
Impulsiveness provides the last missing piece of the puzzle, updating the basic Matching Law. Impulsiveness provides a more sophisticated understanding of time by letting the effects of delay grow and shrink. The more impulsive you are, the more sensitive you will be to delay and the more you will discount the future—and, in the game of _Now Deal or Then Deal,_ the more cash you'll require to endure waiting. Without impulsiveness, there wouldn't be such a thing as chronic procrastination. Popping this into our equation, we have:
And there it is: the Procrastination Equation—inspired by the common elements that determine when we procrastinate, and crafted together from the most deeply researched elements of social sciences' strongest motivational theories.2d The Procrastination Equation accounts for every major finding for procrastination. As the deadline for any task gets pushed further into the future, Delay increases and our motivation to tackle the task decreases. Impulsiveness multiplies the effects of Delay, and so impulsive people feel the effects of time far less acutely, at least at first. Consequences have to be on their doorstep before they start paying attention to them—unless they are particularly large. And what makes consequences large? Expectancy and Value. The bigger the payoff and the greater the likelihood of receiving it, the sooner it will capture your attention. The Procrastination Equation also explains one of the most pernicious aspects of procrastination: the intention-action gap.
Studies show that procrastinators usually make the same plans to get to work as their more diligent counterparts. Where they differ is in acting upon their plans. Unfortunately, what was a heartfelt intention to work next week or next weekend seems a lot less important when the moment of truth actually comes around. Instead of buckling down to work, the procrastinator's intentions buckle. Unsurprisingly, one of the most common laments of procrastinators is, "No matter how much I try, I still put things off!" This complaint illustrates an intention-action gap: you truly don't want to slack off tomorrow but you constantly find yourself slacking off when tomorrow comes. This is exactly what the Procrastination Equation predicts and here's why.
Let's create an intention. Two weeks from now, you will have a choice between staying up late and honing a budget proposal for work, due the next day, or meeting your friends for drinks at the bar. At the moment, you value polishing your proposal far more than seeing your friends, as the former could lead to a sizable pay raise while the latter will only be a fun get-together. You wisely intend to work on the proposal that night, but will you stand fast? Flash forward two weeks to the very night the choice must be enacted, and life suddenly switches from the abstract to the concrete. It isn't just friends, it is Eddie, Valerie, and Tom. These are your best friends; they are texting you to come down to the bar; Eddie is so funny; Tom owes you a drink and you owe Valerie a drink; and maybe you can bounce some ideas off them. Besides, you deserve a break because you've worked so hard. So you give in, and once you are there, you forget about going back to work. Instead, you pledge to get up early tomorrow morning because "your mind will be fresher then." The culprit for your intention-action gap is time. When you headed down to the bar, it probably took you just 15 minutes to get there, a minuscule delay compared to the deadline for tomorrow's task, which is orders of magnitude off into the future—specifically 96 times greater (i.e., 24 hours divided by 15 minutes). As per the Procrastination Equation, that difference causes an almost hundredfold increase in the relative effects of delay. Indeed, there's no time like the present, and it's no wonder your intentions fell through.
THE PROCRASTINATION EQUATION IN ACTION
To see all the pieces of the Procrastination Equation in action at once, it would be tempting to try swapping in your own scores on impulsiveness, expectancy, and value and checking out the results. Unfortunately, it isn't that easy. To accurately apply the equation to a specific individual we would need a controlled laboratory experiment. In the lab we can put everything into an exact and measurable metric by artificially simplifying your choices, having you push a bar, or run a maze to receive a food pellet, for example.
To demonstrate how the Procrastination Equation operates in a realistic setting, a better way is to apply it to the prototypical procrastinator. And nobody—nobody at all—procrastinates like college students, who spend, on average, a third of their days putting work off. Procrastination is by far students' top problem, with over 70 percent reporting that it causes frequent disruption and fewer than 4 percent indicating that it is rarely a problem. Part of the reason that colleges are filled with procrastinators is that their inhabitants are young and therefore more impulsive. However, the campus environment must shoulder most of the blame. Colleges have created a perfect storm of delay by merging two separate systems that contribute to procrastination, each devastating in its own right.
The first system is the essay. The more unpleasant you make a task—the lower its value—the less likely people will be to pursue it. Unfortunately, writing causes dread, even revulsion for almost everyone. But welcome to the club. Writing is hard. George Orwell, author of the classics _Nineteen Eighty-Four_ and _Animal Farm,_ had this to say: "Writing a book is a horrible, exhausting struggle, like a long bout of some painful illness. One would never undertake such a thing if one was not driven by some demon that one can neither resist nor understand." Gene Fowler, who wrote about twenty books or screenplays, was equally despairing: "Writing is easy, all you do is sit staring at a blank sheet of paper until the drops of blood form on your forehead." To write this very book, I have been leaning on William Zinsser's _On Writing Well: The Classic Guide to Writing Nonfiction._ Sure enough, on page 87, Zinsser confesses, "I don't like to write."
Added to the cruelty of assigned writing is the capriciousness of grading—low expectancy. Any essay that is re-marked by another professor may shift remarkably in grade—a B+ could become an A+ if you are lucky, or a C+ if you are not. This is not because the marker is sloppy; it is because measuring performance is inherently hard. Just look at the variation in judges' scores at Olympic events or among reviewers critiquing films. From the students' perspective, such discrepancies mean there is no guarantee that their hard work will be recognized. Quite possibly, it won't be.
The final aspect of the essay system that contributes to student procrastination is the distant due date—high delay. There are often no intermediary steps—you just hand the paper in when you are finished. At first the due date seems months and months away, but that is just the start of a slippery slope. You blink and it has become weeks and weeks, then days and days, and then hours and hours, until suddenly you are considering Plan B. Approximately 70 percent of all reasons given for missing a deadline or bombing an exam are excuses, because the real reason—procrastination—is unacceptable.2e As students themselves report, their top strategy is to pore over the instructions with a lawyer's eye for any detail that could possibly be misinterpreted, later claiming, "I didn't understand the instructions."
There it is; university essays hit each key variable of the equation. Essays are grueling (low value), their results are very uncertain (low expectancy), and they have a single distant deadline (high delay). And if essays are hard in and of themselves, there are few harder places to do them than a college dorm. This leads us to the second system in that perfect storm: the place where this essay is supposed to be written.
College dorms are infernos of procrastination because the enticements—the alternatives to studying—are white hot. Superior in every aspect to essay writing, these pleasures are reliable, immediate, and intense. Consider campus clubs alone. At the university where I earned my PhD, there are about a thousand of them, catering to every recreational, political, athletic, or spiritual need, ranging from Knitting for Peace to the Infectious Disease Interest Group. These clubs will give you a new set of friends, with whom you will want to socialize—likely in one of the dozens of coffee shops and pubs a short walk from any place on campus. They'll also entice you to go to one of a dozen events occurring every week, from poetry readings to tailgate parties. With all the camaraderie, alcohol, sex, and—headiest of all temptations—the freedom to enjoy them all, university can lure us into the unregulated state of bliss where the liberties of adulthood are combined with only a minority of the responsibilities. From the moment students step into the classroom, inevitable conflicts are set in motion. Even Tenzin Gyatso, better known as the 14th Dalai Lama, reported of his student days, "Only in the face of a difficult challenge or an urgent deadline would I study and work without laziness."
We can graph this dilemma using Eddie, Valerie, and Tom when they were back in their university days. They hang together, as they have a lot in common and all of them like to socialize rather than work. Still, there are differences among them. Valerie knows she isn't especially bright but she has two cardinal virtues—she is levelheaded and responsible. Though she isn't competitive, she sees the future pretty clearly, and can imagine one day graduating from college and getting her dream job. Tom is more ambitious and more confident of his abilities than either of his classmates but he is also the most impulsive. His cockiness and spontaneity arouse mixed feelings of envy and hate in many people who know him. Eddie, on the other hand, lacks desire as well as self-confidence. He was pressured by his family to go to college and he is unsure whether he can survive much less thrive academically. In fact, he doesn't really care. At least he is comfortable being a slacker.
One mid-September morning, Eddie, Valerie, and Tom walk into my Introduction to Motivation class, where they find that a final essay is due three months later, on December 15. The graph on the facing page charts their likely levels of motivation and when each of them will start working. Their common motivation to socialize, represented by the dotted line, starts off strong in the semester and tapers off toward the end, partly in response to a lack of opportunity and ever-increasing guilt. Valerie, being the least impulsive, is the first to start working, on November 29 (the smooth, unadorned graph line). It takes another week before Eddie or Tom bears down—a significant gap.
In terms of the Procrastination Equation, although Tom is more confident (high expectancy) and competitive (high value) than Eddie, his impulsiveness means that most of his motivation is reserved until the end (the graph line with squares). Valerie's motivation flows more steadily, like water from a tap, while Tom's gushes like a fire hose when eventually turned on. Even though Tom starts working the same day as Eddie the slacker (the graph line with triangles), Tom's motivation in the final moments should enable him to outstrip the others' best efforts.
MY OWN RESEARCH
Although Eddie, Valerie, and Tom are fictional, they are composite characters based on the thousands of students I have taught. As I stressed, there is no better venue for finding procrastinators than universities. Harnessing all this wasted motivation for science is the trick. It was great luck that as a graduate student I worked with Dr. Thomas Brothen. Thomas taught an introductory psychology course at the University of Minnesota's General College, an institution designed specifically to increase the diversity of the university. Significantly, the class was administered through a Computerized Personalized System of Instruction, a nifty arrangement that allows students to pro-gress through a course at their own pace but is well known for creating high levels of procrastination. In fact, procrastination is such a problem that students are repeatedly warned throughout the course about the dangers of delay. And here is the beautiful part. Its being computerized meant that every stitch of work that the students completed for the course had a time-date stamp exact to the second. You truly can't find a superior setting for studying procrastination.
Before the General College was closed, Thomas and I managed to follow and assess a few hundred students with his wired classroom. We even got around to publishing some of the results. Here are the basics of what we found. Observed procrastination and confessions of procrastination were closely linked, confirming that we were using the right venue. Also, procrastinators tended to be the lowest performers in the course and were more likely to drop out, confirming that they were worse off for putting off. Now these problems didn't occur because procrastinators are intrinsically lazy; they were making the same intentions to work as everybody else. They just had trouble following through with their intentions at the beginning of the course. Toward the end, a different story emerged. Procrastinators actually started logging more hours than they had intended, with one student completing 75 percent of the course in the final week alone. They also weren't procrastinating because of anxiety. The real reasons for inaction were the following: impulsiveness, hating the work, proximity to temptation, and failing to plan. And most significantly, each of these findings directly follows from the Procrastination Equation.
The ability of the Procrastination Equation to formulate these and other results forms the backbone of this book. I have already talked in depth about the connection between the intention-action gap and impulsiveness. Similarly, putting off work because it is unpleasant simply illustrates the effect of value on procrastination. Proximity to temptation highlights the effect of time. Students who said that if they chose not to study "they could immediately be doing more enjoyable activities" or that in their study location there were "a lot of opportunities to socialize, to play, or to watch TV" procrastinated more, a lot more. Remember that Eddie, Valerie, and Tom needed their motivation to write to exceed their motivation to socialize before they could get down to work. But the more readily available temptations become, the stronger they become and the longer they will dominate choices, necessarily creating procrastination. The findings from our study, such as procrastinators' failure to properly plan or to create efficient study schedules, also pointed to ways of combating procrastination. Proper planning allows you to transform distant deadlines into daily ones, letting your impulsiveness work for instead of against you. We will talk more about how to plan properly and the rest of these issues as you go through the book. But one last thing about this study.
There is an epiphany I want to share that occurred to me when I graphed the work pace of the class. Would their work pace replicate the curve that the Procrastination Equation predicted, starting off slow and then spiking toward the end like a shark's fin? Would it follow the pattern that Eddie, Valerie, and Tom's experience suggested? I couldn't expect an exact match, as the equation couldn't take into account weekends or the midterm-break lull, but I was hoping for something close. My findings are what you see on the next page. The dotted line is a hypothetical steady work pace, the dark line is what we observed, and the gray line is what the Procrastination Equation predicts. Notice which lines match together almost perfectly.
LOOKING FORWARD
To some, a mathematical model of procrastination is threatening; it reduces humankind to a robotic formula. I am sympathetic. We are all more complicated and nuanced than any equation could capture, and the subtle details of each person's procrastination are personal. Exactly when your self-confidence peaks, what you find deathly dull, and where your vices lie all combine to determine your individual procrastination profile. The Procrastination Equation isn't seeking to form a comprehensive depiction of who you are but to create a succinct snapshot that can explain a lot with a little.
The Procrastination Equation attempts economically to describe the underlying neurobiology that creates procrastination. I will tell you right now: the biology and the math won't match exactly. A road map of a city, for example, no matter how recent or detailed, can't represent every corner and crevasse of reality; it skips over details like architectural styles or fire hydrant placement. Judiciously focusing on streets and highways allows the map to emphasize navigation. If this big picture doesn't satisfy you and you want all the details, don't fret. The next chapter will give you what you are looking for.
[Chapter Three
Wired for Procrastination](contents.xhtml#ch_03)
PUTTING OFF IS HUMAN NATURE
_Think of all the years passed by in which you said to yourself "I'll do it tomorrow," and how the gods have again and again granted you periods of grace of which you have not availed yourself. It is time to realize that you are a member of the Universe, that you are born of Nature itself, and to know that a limit has been set to your time._
MARCUS AURELIUS
Every day, we experience our souls as being split. Who hasn't struggled between a reasonable intention and an opposing pleasurable impulse? As the dessert cart pulls into view, commitment starts to crumble in the heat of the internal battle of "I want to eat that cake, but I don't _want_ to want to eat cake." Have you ever skipped exercising, knowing that you would later regret it? Have you ever scratched an itch, knowing that you just made it worse? You are not alone; it's a permanent part of the human condition. Thousands of years ago, Plato described this internal clash as a chariot being pulled by two horses, one of reason, well-bred and behaved, and the other of brute passion, ill-bred and reckless. At times, the horses pull together and at other times they pull apart. Thousands of years later, Sigmund Freud continued Plato's equestrian analogy by comparing us to a horse and rider. The horse is desire and drive personified; the rider represents reason and common sense. This division has been rediscovered by dozens of other investigators, each with their own angle, emphasis, and terminology for the same divided self: emotions versus reason, automatic versus controlled, doer versus planner, experiential versus rational, hot versus cold, impulsive versus reflective, intuitive versus reasoned, or visceral versus cognitive. Understanding how the architecture of the brain enables this division is the secret to understanding the biological basis of procrastination.
The brain has been considered the last frontier of human science because its workings have been so difficult to investigate. Emerson Pugh, a Carnegie Mellon University physics professor, concluded that, "If the human mind was simple enough to understand, we'd be too simple to understand it." He is right. And the Procrastination Equation is only a model of how you might behave. Though I like to think of it as a supermodel, it is still merely an approximation of how motivation works. Our brains aren't actually doing these calculations any more than a falling stone is calculating its mass times its acceleration to determine with what force it will hit the ground. Rather, the equation summarizes a more complex underlying process, the interplay between the limbic system and the prefrontal cortex. This is where we must turn for a more fundamental understanding of procrastination.
Recent advances in brain science have allowed us to pull the curtain aside and see our own minds in operation. The basic methodology isn't that hard to describe. You place participants in your choice of brain scanner, likely a functional magnetic resonance imager (fMRI), which detects subtle changes in magnetic signals associated with blood flow and neural processing (i.e., thinking). Once the participant is strapped in, you then ask questions carefully designed to engage aspects of decision making and observe which parts of the brain light up. For example, if we had J. Wellington Wimpy as a subject, we could ask him, "If I gave you a hamburger today, how much would you pay me on Tuesday?" Sure enough, what then comes up on the electronic monitors are not one but two internal messages, which science has blandly come to call System 1 and System 2.
Asking a thirsty person a question such as what drink she would like _now_ primarily activates System 1, the limbic system. This is the beast of the brain ("the horse"), the origin of pleasure and of fear, of reward and of arousal. Questions about future benefits, however, activate System 2, the prefrontal cortex ("the rider"). Though studies are still refining the exact subsection of the prefrontal cortex that is involved, the consensus is that this is willpower's throne. The prefrontal cortex is often described as the _executive_ function, appropriately evocative of CEOs making strategic company plans. Without it, long-term pursuits or considerations become almost impossible, as it is—literally—what keeps our goals in mind. This prefrontal cortex is the place from which planning arises. The more active it becomes, the more patient we can be. It allows us to imagine different outcomes and, with help from our speedy and definitive limbic system, helps us to choose what to do. This interplay of instinct and reason has enabled the human race to create the world in which we live. But it also has created procrastination.
You see, this decision-making arrangement is not the most elegant. It's often described as a haphazard _kluge,_ the clumsy outcome of an evolutionary process. Because the limbic system evolved first, it is very similar across species. It makes decisions effortlessly, spurring action through instinct. Its purview is the here and now, the immediate and the concrete. Our more recently evolved prefrontal cortex is more flexible in its decision making, but also slower and more effortful. It is best at big-picture thinking, abstract concepts, and distant goals. When the limbic system is aroused by immediate sensations of sight, smell, sound, or touch, an increase in impulsive behavior results, and the "now" dominates. Future goals espoused by the prefrontal cortex are cast aside and we find ourselves seduced into diversions—despite knowing what we should be doing, we simply don't want to do it. Also, because the limbic system runs automatically at an incredibly fast rate and is thus less accessible to consciousness, desires can often come over us inexplicably and unexpectedly. People feel helpless to stop intense cravings and they display little insight into their ensuing actions other than, "I felt like it."
In essence, procrastination occurs when the limbic system vetoes the long-term plans of the prefrontal cortex in favor of the more immediately realizable; and the limbic system, aside from being the quicker of the two and in charge of our first impulse, is often the stronger. When near events get this evaluative boost from our limbic system, their vividness increases and our attention shifts to their immediate and highly valued consumable aspects (what we can see, smell, hear, touch, and taste). Deadlines are often put off until they are close or concrete enough to get a hint of that limbic system zing, whereupon both parts of our brain are finally shouting in agreement, "Get to work! Time is running out!"
OF BABES AND BEASTS
Procrastination increases whenever our more recently acquired prefrontal cortex is compromised. The less potent the prefrontal cortex, the less patient we become. Those with brain damage can provide particularly vivid examples of this, Phineas Gage being the most famous. Gage was a shrewd, responsible, hardworking, methodical railway foreman who, in a workplace accident in 1848, had over three feet of iron rod blown through the top of his skull and the front of his brain. He recovered, incredibly, but he became a man of the moment: impatient, vacillating, profane, inconsiderate, uninhibited, and uncontrollable. The iron rod had severed the connection between Gage's limbic system and his prefrontal cortex. The planning part of the brain needs the fast and accurate input from the limbic system to understand the world, and that's what Gage lost. A more modern example is Mary J. who was completely transformed within a year by a brain tumor that debilitated her prefrontal cortex. Before the tumor, she was a quiet teetotaling Baptist, on the dean's list at an Ivy League university, and engaged to be married. Until the tumor was surgically removed, she was angry and extremely promiscuous, failing school, drinking hard, and using drugs. Her executive function was disabled and she became all impulse, ruled by whatever temptation was put before her.
There is a way people can experience Phineas' or Mary's predicament, and happily it doesn't involve a nail gun. We can temporarily lesion the prefrontal cortex with transcranial magnetic stimulation, which uses electromagnetic induction to briefly knock out focused sections of the brain. Alternatively, taking alcohol, amphetamines, or cocaine either hypercharges the limbic system or hinders the prefrontal cortex's ability to perform, creating actions that "seemed like a good idea at the time" but later prompt regrets. Or, the prefrontal cortex can simply become exhausted through sleeplessness, stress, or resisting other temptations; by fighting one enticement, we often become more susceptible to another. Finally, if you are a teenager, you might not need to go to any of these extremes, since your prefrontal lobes are still receiving their final touches. Combining the effects of youth, stress, and alcohol together, the most impulsive and uninhibited place on this planet is a group of teenagers celebrating the finish of a willpower-depleting stretch of studying with a weeklong drink-fest. Indeed, Phineas Gage would fit right in during Spring Break in Cancún, with wet T-shirt contests, drinking games, and random hook-ups. If you don't diminish the prefrontal cortex, you can't have _Girls Gone Wild._
If you can't make it to Spring Break to see the limbic system dominate action, there are good alternatives closer to home. In fact, they are likely in your home. Do you have a pet or a child? Both are heavy on the limbic system, making owning a pet the neurobiological equivalent of raising a child. We are their external prefrontal cortices. We have to be the ones providing patience and doing our best to coax it out of those who don't have much of it or who are still developing it.
THE NOW OF BABES
There is a rhyming biological heuristic that goes "ontogeny recapitulates phylogeny." It means that the way we develop within our lives roughly reenacts the course that human evolution has taken over millions of years. When in the womb, more or less, we morph from fish to reptile before eventually emerging as a mammal. But the process isn't done yet. The last aspect of us to evolve is the prefrontal cortex, which continues to develop after birth. For those who have children, and as I write this, I have two still in diapers, we don't need a biology degree to know that infants aren't born with the ability to plan ahead and put their immediate needs on hold for the benefit of some future goal. Just try asking for patience from a hungry baby or a little one with a full diaper and my point will be made. They are merciless in their need.
As children develop, their prefrontal lobes grow too and eventually they achieve the ability to put things off just a bit. You can't ask a baby to put off a feeding, but eventually you may ask a toddler to say "Please" before getting a treat. It takes the development of the prefrontal cortex for this modicum of control to appear—which happens all too slowly for my taste. Year-old children have almost no executive control, instantly batting down any pile of blocks or grabbing your eyeglasses, but just one year later, brief moments of patience become possible, say twenty seconds. By the age of three, children are routinely waiting a full minute and by four they are piling their blocks high, putting off the blast until they can enjoy the big burst of noise when their soaring towers tumble.
At the age of four, children can play "Simon Says." This is a significant advance, because the game is all about self-control, about inhibiting the immediate impulse from the limbic system so that the prefrontal cortex can mull over whether Simon has actually said "Simon Says" before they respond. How well this acquired ability transfers into kindergarten is another matter, because kindergarten requires sitting when you want to run, listening when you want to shout, and taking turns when you want it all to yourself. Fortunately, between the ages of four and seven there is a burst of development in children's executive function. They are progressively better able to make plans for tomorrow, to pay prolonged attention to more than the television set, and to shut out distracting events other than parents calling them to come in for dinner.
The normal maturation of the prefrontal cortex is assisted by endless hours of patient teaching by parents trying to get their little ones to put off their needs for just a moment without tears or the stomping of feet. Unwearyingly insisting that gifts can be unwrapped only at Christmas and then only your own, that dessert comes after dinner, or that toys must be shared with others coaxes a little more from the prefrontal cortex and a little less from the limbic system. Unfortunately for parents, their role as their children's external prefrontal cortices is a long one. It can last until about the age of nineteen or twenty, when the biological basis of self-control is finally fully in place. Until then, parents can only herd their teenagers away from all the vices that impulsiveness ensures youth will find especially tempting: risky sex, excessive alcohol, petty crime, reckless driving, and, of course, procrastination. The younger you are, the more you seek instant gratification, from socializing late into the night and then facing tomorrow's exam half asleep to dillydallying so long you have to pack your bags in a flurry and almost miss your plane. Though the young act as if they will live forever, they really are living just for today.
The novelist Elizabeth Stone has written that having a child "is to decide forever to have your heart go walking around outside your body," but our role as walking prefrontal lobes comes to an end at this point. As adults, our children no longer need us for guidance and any mental inequalities between us go into a long lull, perhaps broken briefly by the arrival of grandchildren if they are forthcoming. We can expect apologies from our kids as they try to raise a few of their own and learn firsthand the vigilance required to be a parent. And then, long later and hopefully not at all, our roles may change entirely. As we grow older so do our brains, losing the snap they had in earlier years, especially the prefrontal cortex—following the last in, first out rule. Though some avoid this fate, remaining razor sharp into their final years, others get it worse, assisted by the senility of frontotemporal dementia that affected my grandmother Eileen. I am well aware that I too might encounter a second childhood and once again be as vulnerable as my two young sons are now. Indeed, we'd better raise our kids well, as their love might be the only thing that stands between us and a world that views us as prey made easy through old age and a compromised mind.
BIRD BRAINS
Animals might be our fellow procrastinators. After all, we share many other "human" personality traits with dozens of other species, from rhesus monkeys to octopi. Wild great tits, for example, exhibit varying degrees of aggressiveness and risk taking, traits that enable greater environmental exploration.3a The bolder birds expose themselves to more danger but also reap the gains of better nesting places, food sources, and choice of mates. For another example, just ask any dog or cat owner if their pets have a unique personality; the owners will rightly insist that their furry friends differ in terms of affection, anxiousness, aggressiveness, and curiosity. Significantly, this list of shared traits includes impulsiveness, the cornerstone of procrastination. But this doesn't necessarily translate into procrastination itself.
Whether they are meowing, barking, or chirping, animals are clearly limbic-heavy in their decision making. But that's only half the story. You need some prefrontal cortex or its equivalent to procrastinate, for without it you can't make plans that you later irrationally put off. Do animals have this mental capacity? Apparently some do, showing the ability to anticipate and plan for the future, especially regarding food. Scrub jays can anticipate being denied breakfast tomorrow and will cache food to snack on later. Rats seem to have some sense of time, being able to recall where and when feeding events occur. Chimpanzees can wait up to eight minutes to exchange a small cookie for a large one, showing slightly more patience than a young human child. Male chimps will invest in future mating opportunities by sharing meat with a female, with the hope of being favored when she comes into heat. Also, consider Santino, a particularly farsighted chimpanzee from the Swedish Furuvik Zoo. He will spend his morning collecting stones to hurl at annoying zoo visitors in the afternoon. In combination with impulsiveness, all the pieces for procrastination are there: animals can make plans for the future and, what's more, they can impulsively put them off, despite expecting to be worse off for it.
James Mazur, a Harvard-trained psychologist, has directly demonstrated procrastination in animals. He trained pigeons to two different work schedules and then gave them a choice of which to pursue. Both schedules delivered a tasty treat at the same time, but the first started with a little work followed by a long delay, while the second started with the long delay and ended with _a lot_ more work, up to four times as much. Essentially, the birds had to choose between doing a little hard work now (followed by rest and recreation) and taking it easy immediately (followed by a lot of hard work). The pigeons proved to be procrastinators, putting off their work despite having to do more of it to obtain their reward in the end. Like a twisted version of a Cole Porter song, birds delay doing it and even chimpanzees in the zoo delay doing it. Since most animals, including pigeons, have the capacity for procrastination, procrastination is pretty well confirmed as a fundamental part of our motivational firmament. The last time we all went to the same family reunion was over 286 million years ago during the Carboniferous period, before the time of the dinosaur.
Inevitably, then, having an animal as a pet is largely an exercise in dealing with this limbic-heavy decision making. Dogs, for example, naturally act in the moment and grab food that isn't theirs, chase stray animals across busy streets, and bark or whimper incessantly by the door waiting for you to open it. It would be easier in the short run to let the dog be, but patience and long-range thinking on our part can make all the difference for a life with any four-legged friend. This is what expert dog trainers stress, like Cesar Millan, the dog whisperer, or Andrea Arden, author of _Dog-Friendly Dog Training:_ the primary responsibility of an owner is "to convince your dog that waiting for something—which is typically not a natural instinct for dogs—is the best option." The big trick is convincing owners to do this in the first place. Teaching impulse control uses a lot of our prefrontal cortex, a resource we often don't have a surplus of to begin with.
EVOLUTIONARY PROCRASTINATION
By all appearances, from the evidence of brain science to animal studies, the capacity to procrastinate is engrained in us. It's even in our genetic code: several studies indicate that about half of most people's lack of self-discipline has a genetic origin. This makes sense, given that DNA allows adaptive genetic mutations to be passed down through subsequent generations, a process known as "descent with modification." Without a genetic component, the ability to procrastinate couldn't easily be passed on.
We evolved to be procrastinators, but why? Procrastination is an irrational delay, whereby we voluntarily put off tasks until later despite expecting to be worse off for the decision. By definition, procrastination is harmful and should have been culled long ago from our gene pool rather than filling it to the brim. Are we the butt of some cosmic joke? Maybe. But there is another possibility to consider. Some traits occur as by-products of other once-more-adaptive processes. For example, belly buttons are a by-product of being born, and though they can be pretty, they don't have any pressing purpose in themselves. Since procrastinators are above all else impulsive, the evolutionary explanation for impulsiveness is the one to focus on. Procrastination is a by-product.
Essentially, impulsiveness is about living for the moment. Long-term desires and tomorrow's deadlines are ignored until they become imminent—until the future becomes the now. Though today impulsiveness isn't usually a helpful trait, evolution operates through hindsight; that is, it custom fits us to the environment we _were_ in, with no anticipation or prediction. This is known as _ecological rationality,_ in that what is rational depends upon the environment you operate in. It is like getting a tailored suit for your wedding day. You look magnificent in it, but try it on again twenty years later and it pinches in all the wrong places. Likewise, procrastination may be steeped within our existence because having an impulsive mindset made a lot of sense when we were hunter-gatherers. When our ancestors needed to do the basic four "F"s of survival—feeding, fighting, fleeing, and mating—it would aid their cause if they also wanted to. Let's briefly consider the last and first of these four: what we have for dinner and who we seek to spend the evening with afterward.
FAST FOOD
From our teeth, which chew it up, to our intestines, which digest it, food has played a major role in our evolution. We have evolved to love the taste of fats and sugars because, in a world where starvation and predation were constant concerns, stocking up on high-caloric foods was once an adaptive preference. When the food supply was sporadic, we would have to gorge when the going was good, focusing on energy foods rich in sugar and fat. There were no Neanderthals on self-imposed diets. Consequently, for most of human history, being "overweight" has been considered beautiful, affluent, and enviable. The exigencies of eating may explain how we all became so impulsive and, consequently, procrastinators.
Let's consider two types of primates, common marmosets and cotton-top tamarins, which are almost identical except in their choice of food. Marmosets are gummivores, which scratch tree bark and then sip on the sap that flows. Tamarins are insectivores; they pounce on bugs whenever they can find them. Marmosets show a lot more self-control than tamarins, as they are selected for it. Sap takes a while to flow, demanding patience, whereas the hunt for jumping and scurrying bugs requires immediate action. For animals in general, the fine tuning of impulsiveness to their food source is called _optimal foraging._ 37 We are optimized to get the most calories in the shortest time; consequently, the longer it takes to kill, eat, and digest, the less impulsive a species typically becomes. In short, we develop the self-control we need to ensure our next meal.3b
Being omnivores and at the top of the food chain, humans are superstars of self-control. We have the patience to kill and eat almost anything that lives. Birds' ability to delay gratification, in comparison, hardly registers; even a ten-second wait is remarkable. Similarly, ten minutes of patience is an eternity for a chimp. For all our superior self-control, though, in today's whirlwind, we don't have enough. We have been favored with enough patience for a world without grocery stores or refrigerators, enough for hunting animals or gathering berries. Yet, we have a relatively small window compared to what we currently need. Procrastination results from a disconnect in our genetic inheritance, as we now pursue projects and plans that require weeks, months, and years to complete, timelines for which we are motivationally mismatched. In the forest, a bird in the hand might be worth two in the bush, but in the city, the discount rate is far more slender; invest in a bird today and tomorrow you are lucky to earn a chicken wing's worth of interest.
JUST SAY YES
Now on to the second example, the one you've been waiting for—sex. Evolution is steeped in sex, as those who succeed breed. Since procrastinators' impulsive nature is ingrained in their DNA, it can be passed on to their offspring and, if it lets them have more kids, the trait quickly becomes common. Just consider my family. The males on my mother's side tend to have children later in life. My great grand-dad was Owen Owen, who people in the UK might remember from his string of similarly named though now-defunct department stores. Since Owen Owen was born in 1847 and I had my son Elias in 2007, each generation of my family tree is spaced forty years apart. If we were in a stork race with another family that started a new generation every twenty years (thereby reproducing twice as fast), by now there could easily be over eighty of them for every one of us. Getting an early start on baby making makes a big difference.
Sure enough, procrastinators' impulsiveness has been linked to an early start for parenthood through teen pregnancy as well as sexual promiscuity. The one thing that procrastinators don't tend to put off is "getting some." No wonder. The fun part of copulation comes immediately, while the harder part of child raising . . . well, that's almost a year away. This state of sexual affairs also helps explain why men tend to be more impulsive and procrastinate more than women. Reproduction strategies favor a quality versus quantity split—that is, raising a few kids well or having lots in the hope some of them work out. Since it is easier for men to invest less in their offspring, they definitely lean toward the quantity option. As Geoffrey Miller, author of _The Mating Mind,_ wrote: "Men are more motivated to have short-term sexual flings with multiple partners than women are." Women tend to favor the quality strategy, taking a longer-term and more responsible perspective. She waits patiently for Mr. Right while he impulsively wants Ms. Right Away.
Sex also ensures a range of impulse-driven procrastination in the populace; some will procrastinate a little and others a lot. If it was always advantageous to get pregnant as soon as possible, the world would be like the Mike Judge movie _Idiocracy._ In that film, everyone who was smart and cautious held off having kids, and the intelligent were out-bred by the clueless and carefree. There is, however, no optimal level of impulsiveness to maximize the number of your descendants. Much depends on the resources available to raise children, for as costs increase, it is better to have smaller families. Other tradeoffs occur when there is an excess of men pursuing the "quantity" reproduction strategy. If too many men are focused on short-term sexual encounters, they swamp the singles bars and strain the goodwill of the available women. In this scenario, committed family men are a rarity and thus more valued. Men demonstrating loyalty would find themselves vigorously pursued, able to pick the prettiest and most compatible of spouses.
A BRIEF HISTORY OF PROCRASTINATION
This evolutionary explanation of procrastination directly demonstrates why procrastination is so widespread. No matter which country or language you are reading this book in, there is a name for irrationally putting things off, from Hawaii's _napa_ to Scotland's _maffling._ Everywhere we have looked for procrastination, we've found it—easily. Today's age of procrastination was inevitable the moment we walked out of the trees into the savanna, learned to make fire, and began trading among tribes. Procrastination grew alongside civilization.
The history of procrastination likely began around nine thousand years ago, sprouting along with the invention of agriculture. Planting crops in the spring to reap them in the fall was our first artificial deadline; it was a task that civilization and survival required but not one we had evolved to perform. This is why all the earliest written records of procrastination deal with farming. Four thousand years ago, ancient Egyptians chiseled at least eight hieroglyphs to indicate delay, but one in particular also indicates neglect or forgetfulness. Translated as procrastination, this hieroglyph is most often associated with agricultural tasks, especially those connected with the yearly cycle of the river Nile, as it overflowed its banks and fertilized the floodplains. Similarly, the ancient Greeks struggled with procrastination, as recounted by Hesiod. Living around 700 B.C., Hesiod was one of the greatest poets of Greek literature, rivaled only by Homer. In Hesiod's epic 800-line poem, _Work and Days,_ he exhorts: "Do not put your work off till tomorrow and the day after; for a sluggish worker does not fill his barn, nor one who puts off his work: industry makes work go well, but a man who puts off work is always at hand-grips with ruin." This warning was especially important because the Greeks were in the midst of a financial crisis of such proportions that many Greek farmers put up not only their farms but also their families as collateral. Procrastination led not only to a poor credit rating, but also to seeing your sons and daughters become the exclusive property of your richer neighbors.
By 440 B.C., procrastination was spilling over from farming into fighting. Thucydides, the father of scientific history, wrote about it in the _History of the Peloponnesian War,_ which chronicled the conflict between Athens and Sparta. This account, still studied in military colleges, discusses various aspects of personalities and strategies. Thucydides clearly considered procrastination to be the most wicked of character traits, useful only to delay the commencement of a war so as to lay a better groundwork for winning it. Another notable Greek reference to this trait is found in the work of the philosopher Aristotle, who wrote much of his _Nicomachean Ethics_ on the weakness of the will, what the Greeks called _akrasia._ Specifically, Aristotle discusses a form of _akrasia_ called _malakia,_ which is not doing something that you know you should (clearly, procrastination).3c
Moving a few centuries forward, we see procrastination entering politics. Marcus Tullius Cicero was a major political player around 44 B.C. His position put him in conflict with Marcus Antonius, better known as Mark Antony, Cleopatra's lover. In a speech denouncing Mark Antony, Cicero declares: _In rebus gerendis tarditas et procrastinatio odiosae sunt_ ("in the conduct of almost every affair slowness and procrastination are hateful"). Perhaps because of Cicero's advice, or perhaps because Cicero made thirteen other speeches denouncing him, Mark Antony delayed little in killing him.
Then, for a millennium and a half, procrastination made inroads into religion and it is referenced in the texts of every major faith. For example, in the earliest written Buddhist scriptures from the Pali Canon, the monk Utthana Sutta concludes that "Procrastination is moral defilement." Moving forward seven centuries, the Indian Buddhist Shantideva is still on message, writing in _The Way of the Boddhisattva,_ "Death will be so quick to swoop on you; Gather merit till that moment comes!" By the sixteenth century, procrastination starts appearing in English texts without translation. The playwright Robert Greene, for example, wrote in 1584, "You shall find that delay breeds danger, and that procrastination in perils is but the mother of mishap."
Finally, when the Industrial Revolution came into its own, so did procrastination. In 1751, Samuel Johnson wrote a piece for the weekly periodical _The Rambler,_ describing procrastination as "one of the general weaknesses, which, in spite of the instruction of moralists, and the remonstrances of reason, prevail to a greater or less degree in every mind."3d Four years later, Dr. Johnson enshrined the word within his influential English dictionary and, ever since, the term has remained in common use. If procrastination is indeed a core characteristic of humankind, it is acting just as you would expect: it's maintaining itself as a reoccurring theme in our history books, right from the beginning of the written word.
LOOKING FORWARD
I'd like to end this chapter on the evolution of procrastination with the story of Adam and Eve. They lived in the Garden of Eden, naked and unashamed, fitting in perfectly with nature. Then, in humankind's first act of disobedience, Adam and Eve bit an apple from the tree of knowledge, and they were cast out by God, forced to survive through agriculture. Though biblical in origin, this story maps perfectly onto the story of evolution as well.
In the environment where we evolved, we drank when thirsty, ate when hungry, and worked when motivated. Our urges and what was urgent were the same. When we started to anticipate the future, to plan for it, we put ourselves out of step with our own temperament, and had to act not as nature intended. We are all hardwired with a time horizon that is appropriate for a more ancient and uncertain world, a world where food quickly rots, weather suddenly shifts, and property rights have yet to be invented. The result is that we deal with long-term concerns and opportunities with a mind that is more naturally responsive to the present. With paradise lost and civilization found, we must forever struggle with procrastination.
Bottom line: procrastination is not our fault, but we have to deal with it nonetheless. We encounter procrastination across almost all of life's domains, from the boardroom to the bedroom, shifting from major to minor. Is it your home life, your finances, or your health that suffers the most from procrastination? Is it your e-mail or TV habit sucking away your productivity? Odds are, not only is the amount of your procrastination increasing, but so is the number of places you are doing it. But I am getting ahead of myself—that's the subject of the next chapter.
[Chapter Four
ProcrastiNations](contents.xhtml#ch_04)
HOW MODERN LIFE ENSURES DISTRACTION
_Over the bleached bones and jumbled residues of numerous civilizations are written the pathetic words: Too Late._
MARTIN LUTHER KING JR.
Our love affair with the present moment is at the root of procrastination. The fact that we tend to be more impetuous than reasonable is an evolutionary heirloom handed down through a thousand generations. But we can't blame our neurobiology entirely. Every distraction the modern world offers also exacerbates the mismatch between who we are and what we need to be. This chapter diagnoses the growing divergence between our plans and our impulses. To better write it, I reacquainted myself with an old distraction, purposefully re-infecting myself with what had afflicted me for so long as a student—video games. Gaming's capacity to absorb and dominate my attention is remarkable. There have been days, which stretched into nights, when I tore myself away from the screen only long enough to cram fast food into my mouth and to take care of my bodily needs. Eventually, I winnowed down every scheduled responsibility or meeting to its barest essentials in order to minimize the time it took away from gaming. My girlfriend viewed it as my mistress. My motto was Just One More Turn.
So, for the purposes of this book, I decided to explore Conquer Club, an online version of Risk©. I hadn't played the board game since my college days, rolling dice with a few friends and beers, so I found its nostalgic aspect attractive. Also, Conquer Club's free version only allowed you to play four games at a time, so I believed it would be difficult for it to get out of control. Since you are playing against people around the world, moves are done at all hours and the game progresses at unexpected moments. Consequently, I found myself checking the website fairly often, even when it wasn't my turn. Suddenly there it was—procrastination, in the familiar form of gaming when I should be working. I could feel the hooks sinking deeper into me and I knew I was taking an extended walk along a sharp precipice—you know, the one you are going to slip over sooner or later (and secretly you are full of anticipation and excitement waiting for the stumble).
My, how quickly addictions reassert themselves. One Friday night, at the end of a long unrewarding work week, with sick kids, and after a trivial tiff with my wife, I felt life owed me more. Perhaps it was a bad idea to then upgrade to the premium version of Conquer Club and play twenty-five games simultaneously.4a Periodic checks on the progress of the game became my life's punctuation marks, the periods capping off any of my tasks. At any break in the day's flow, I would peek into my games and see what battles had been fought in my absence or perhaps (joy, oh joy!) take my turn. Conquer Club continued to draw me back for weeks after that fateful Friday night. I checked on my games before and after the drives to work and home. It was the last thing I did before I went to sleep, the first thing I did when I woke up; and I dreamt about it in between. Oh, the sacrifices I make for science! But don't worry about me. Being both a victim and a detective of the fine art of distraction has its advantages. I know how to wind down this obsession, which I will do just after recapturing Kamchatka. While I wait for my turn, let's talk about why you, along with everyone you know, have likely experienced similar problems.
FULL-IMPULSE DRIVE
One of the elements that made me a slave to Conquer Club corresponds perfectly to the first and strongest findings from my research program: proximity to temptation is one of the deadliest determinants of procrastination. Since every computer offers an opportunity to play, it is hard to keep temptation at bay. The second element is the virulence of the temptation; the more enticing the distraction, the less work we do. Conquer Club followed what is known as a "variable reinforcement schedule"—that is the reward (i.e., reinforcement) occurred at unpredictable times. For over fifty years, ever since B.F. Skinner and C.B. Fester's 1957 magnum opus _Schedules of Reinforcement,_ we have known that such variable patterns of reinforcement are very addictive. Skinner found that from pigeons to primates, we all work much harder for rewards when they are unpredictable but _instantaneous_ when they arrive. You can see the power of variable reinforcement in gambling. Slot machines are fine-tuned for addiction because they have these schedules of winning hardwired into them. Every time a grandparent spends the grandkids' inheritance on these one-armed bandits, you can give a nod to the wonderful power of motivational psychology. Unfortunately, as my Conquer Club example confirms, the Internet has given rise to a variety of similarly structured diversions. Paradoxically, while the Internet has made it easier for us to work, it has also created a series of behavioral traps that make it harder for us to work at all. In case it helps, I put this in graph form. It shows what comes between our wanting to accomplish a task and our ability to actually complete it.
The graph's two horizontal dashed bars represent temptations, the lower bar being a small temptation (something nice) and the higher bar being a large one (something great). The solid line that eventually swoops up is the work curve, showing that, as we have seen, most of one's motivation is reserved until just before the deadline. This is a _fixed interval schedule,_ meaning that there is a fixed deadline before your work is assessed and you get "paid."4b On the other hand, variable reinforcement schedules (the horizontal dashed bars representing small and large temptations) exert a constant state of motivation, typically much higher than fixed schedules. The motivation to play is always there and doesn't go away. The more attractive we make the temptation (making it large instead of small), the higher its bar moves and the longer it takes for the competing work line to become the dominant choice. So, we can see that when the allure of temptations rises, so does procrastination.
RAISING THE BAR
In award-winning research, Vas Taras, a professor from the University of North Carolina, and I assembled a database that tracked changes in the world's culture over the last forty years. It required piecing together hundreds of studies from social scientists of all stripes who used dozens of different scales. What we found was that as countries "modernize," they start to converge around a set of values typical of Western free market economies. One major finding was that the world has become more individualistic: people look after themselves with less concern for others. Another was that modernization brings with it procrastination. As our economies have grown over the last few decades, we have experienced a fivefold increase in chronic procrastination. In the 1970s, 4 to 5 percent of people surveyed indicated that they considered procrastination a key personal characteristic. Today, that figure is between 20 and 25 percent, the logical consequence of filling our lives with ever more enticing temptations.
Consider how the world was transformed during the last century. In 1911, William Bagley, writing in _The Craftmanship of Teaching,_ described the "hammock on the porch," the "fascinating novel," and the "happy company of friends" as the "seductive siren call of change and diversion, that evil spirit of procrastination!" Bagley's temptations, though real, were relatively pedestrian compared to what was to come. Also in 1911, the first film studio opened in Hollywood, and the next few decades saw the rise of multi-million-dollar productions and multi-millionaire movie stars, along with their scandals; both Charlie Chaplin and Errol Flynn—the comic tramp and the romantic swashbuckler—seemed to like their women a little bit younger than the law allowed. The spectacle of Cecil B. DeMille's _The Ten Commandments_ hooked viewing audiences, and by the 1930s, the popular press was referring to movies as a common form of procrastination. Still, you had to leave your home or office to see the silver screen. But not for long. The end of the Second World War coincided with the development of television, and the number of Americans with TVs leaped from 9 percent to 65 percent between 1950 and 1955. During popular show times, streets would empty and stores would shut so that everyone could tune into episodes of _I Love Lucy._ By 1962, with television sets now in 90 percent of American homes, _Popular Science_ published a book digest, "How to Gain an Extra Hour Every Day," connecting television watching to procrastination.
In the mid-1970s, a new temptation emerged on the scene. I was eight years old when Pong, the first successful video game, was introduced to our household. My father hooked up one end of the game box to our black-and-white TV and the other end to two "paddles" that were nothing more than knobs on cords. You turned your knob and a small bar moved vertically on either the left or right side of the screen, depending on which paddle you had. If a moving electronic ball hit your paddle, it was deflected back to the other side of the screen for your opponent to return. That's it, but it was magic and I loved it. Sure enough, by 1983, psychology texts were reporting video gaming in the list of typical procrastination behaviors.
Having looked at some historical baselines for comparison, you should be able to see why procrastination has risen to today's levels. While the pleasure derived from working has remained fairly constant over the decades, the power of distractions only seems to increase. The temptation bar in Skinner's graph is raised ever higher, while the work curve remains the same. Let's reconsider today's video games, which make Pong laughable by comparison. Unfathomably more advanced, these games are the product of untold millions of programming hours, and tax the capacity of even the most advanced computer systems. They beat hands-down anything that Bagley wrote about. Many people play anywhere, anytime—it is not uncommon for students to engage in head-to-head online games during university lectures. Furthermore, as good as these games are, they are getting better. With each evolving iteration of Grand Theft Auto, Guitar Hero, or World of Warcraft, choosing _not_ to procrastinate becomes harder. The graphics, the story, the action, the console—all of them advance. In the battle for your attention, it is as if work is still fighting with bows and arrows while gaming has upgraded to auto-cannons, sniper rifles, and grenade launchers. Consequently, it is becoming increasingly common for people of all ages to become consumed by games, and intervention centers are proliferating to treat video game addiction. In Korea, for example, about 10 percent of young people show advanced signs of addiction, developing up to seventeen-hour-a-day habits. In response, the government has sponsored 240 counselling centers or hospital programs. There are even resources for particular games, such as www.WoWdetox.com, which is dedicated to World of Warcraft players as well as their spouses, typically called Warcraft widows.
What is more shocking is that there are worse creatures than video games for inciting procrastination. The worst isn't getting a bite to eat or napping, though they remain popular choices. The king of distraction—and there is only one—is television. Since its halcyon years in the 1950s, television has continued to perfect itself, gaining all the features it needs to win the competition for our time. The magic of the remote allows us to change channels without moving. The advent of cable and satellite has ensured that there is always at least one available channel that reliably caters to our tastes. And with multiple television sets throughout the house—more TVs than people according to Nielsen Media Research—we can watch our shows anywhere we like. If our interest in a particular program flags even momentarily—zappp!—we are off to other worlds in this 500-channel universe. So attractive is television that we are often guilty of over-consumption, feeling TV'd out and wishing at day's end that we'd watched a little less.
Most Americans spend about half of their leisure hours in television's glow. Other nations aren't far behind. According to the latest national census data, Americans watch an average of 4.7 hours per day, beating out Canadians, who watch 3.3 hours per day. The average Thai spends 2.9 hours in front of the tube; a Brit, 2.6 hours, and a Finn, 2.1 hours. Reading, for comparison's sake, clocks in at an international average of 24 minutes a day. This means, of course, that you have likely been plugging through this book for about three months now.
Worse still, as with video games, TV is getting more and more attractive. Not only is the hardware becoming sleeker, and higher-tech, but your options of what to watch are also stepping up. Season box sets are commonplace, as are digital video recorders (DVRs). These DVRs allow you to record multiple programs simultaneously, store hundreds of hours, keep track of what you have watched, and help you find desirable episodes. Watching scheduled TV seems primitive today. The future promises even more. As television continues to evolve, viewing options become almost endless. For example, the technology already exists to download any movie in less than a second. When a fraction of that power becomes available to the ordinary household, we can expect our TV viewing to rise correspondingly. Any movie, any show, any clip, can be seen by anyone, almost anywhere, all at an insanely crisp resolution. Inevitably, as television pumps up, it muscles out the rest of life. It is already happening. For every country with data, the amount of television watching has increased. In just eight years, from 2000 to 2008, TV watching in the United States went from 4.1 to 4.7 hours, a 15 percent increase. Since time is finite, everything else must suffer—and suffer it does. It's not just chores that we put off in favor of tele-vision—it's eating with the family or connecting with friends.
I have painted a pretty bleak picture, but it could be worse. Actually, the worst is still ahead. It doesn't quite take up as much time as television yet, but it shows a lot more potential. It is the Internet, which has all the allure of video games, television, and more on a single platform. About 80 percent of students already report that Internet activities are particularly problematic for them. No wonder. There are websites and blogs that cater to every fetish and interest, videos to download, music to acquire, and text-messages to respond to. The newest twist in the Internet procrastination saga are the hundreds of social networking sites, like Facebook, Bebo, MySpace, and Twitter. Less than a year from its inception in early 2004, Facebook was reported in _The New York Times_ as a key enabler of dawdling, with students hitting the refresh key on their screens hundreds of times a day to check for updates. This habit is not dissimilar to Skinner's variable reinforcement research on rats and pigeons, who also pressed their keys hundreds of times for an eventual but unpredictable reward pellet. Since researching Facebook in person would likely endanger the hours I needed to write this book, I decided to locate an expert, someone already intimately entangled and familiar with the site's details. As if to underscore Facebook's prevalence in the university population, it took me less than five minutes to track down such an authority, specifically a graduate student. She "Facebooks" about ninety minutes a day and has even set up a Dogbook page for her pug, Schmeebs. I sat down beside her computer so she could give me the tour:
• "The first thing," she explains, "is that this gives you a way to connect to all your friends and controls what type of contact you want from them. For example, I am interested in other people's pictures, so this section provides previews of their photographs from their Facebook pages."
º I comment that it seems to be a large section of the screen.
• "Oh, well I have quite a few Facebook friends."
º How many?
• "Let me see," she responds. "Here it is, 603."
º That does seem like a lot. Do you really have 603 friends?
• "No, no, a lot of them are just acquaintances and you can treat them differently. You control security and access to what you get from them and what they can post on your website. There is a wall here, where my friends can post comments, see."
º I see.
• "Some get priority. My friend Jen is one of three people who I get text messages from every time she updates her Facebook site . . ."
º Which is?
• " . . . about twice a day."
º How long do you wait until checking what the update is about?
• "Well, the text message is usually incomplete, so I have to go online to read the whole thing."
º So, immediately.
• "Exactly. Right away."
º Even if you're at a movie, at dinner, or with family?
• "Of course," she responds, "though if it is with family, I will sneak off first."
º Very polite of you. What else can you do?
• "So, so much. You can 'poke' people just to tell them you are thinking of them, you can give them virtual gifts . . ."
º Why?
• "Because you can. Some gifts are free or corporate-sponsored, others you have to pay for. Here is a bunch of booze-related ones. I am not sure why I have these. You can also use Facebook to send invitations to events."
º So this enables you to meet and interact with people more?
• "No, not really. It has replaced a lot of my socializing.
But, I do feel I have gotten to know a lot of my closer friends better. You post funny quotes you have heard, videos of trips, whatever tidbits interest you. Oh look! Chelsea's dog added Schmeebs as a friend!"
My expert then leads me to the myriad procrastination-dedicated Facebook groups, such as: "AP-Advanced procrastination" (over 18,000 members); "I'm majoring in Napping and Facebook with a minor in procrastination" (over 30,000 members); "I was doing homework and then I ended up on Facebook" (over 900,000 members). There are also over 600,000 members of a Fan Page dedicated solely to "Procrastination" itself. Joining these groups makes a statement about your identity, and provides a wide assortment of suggested diversions, as well as an opportunity to talk about them. Ironically, a recurring discussion among members was focused on how they could limit or quit Facebook itself (e.g., "let parents change your password and only tell you after exams"). Not to be perceived as a Facebook prude, I agree that the site is attractive and intriguing, and that it has useful applications, especially networking. Napoleon Hill, an achievement guru of the last century, considered networking a key element of success. On the other hand, Facebook is a tremendous distraction and it is this and not the networking itself that dominates. A true sign of the addictive aspects of Facebook is that half of those who quit reactivate their accounts. They can't keep away.
HOW WE GOT HERE
The rise of procrastination is hard to avoid, given its deep roots in our brain's neurobiology. The limbic system focuses on the now while the prefrontal cortex deals with longer-term concerns. In other words, when building a fire, the limbic system is eyeing that can of gasoline while the prefrontal cortex says branches and logs would provide slow, steady heat. The first wants the immediate million-dollar check, while the second favors a weekly five grand for life. Though both the limbic system and the prefrontal cortex come together to reach a final decision, their duet ensures the rise of procrastination. Here's an example to show you what I mean.
Let's take two snack food vendors, _Nutrity Nuggets_ and _Tasty Tempts._ _Nutrity Nuggets_ provides healthy fare, which appeals to our long-term and more abstract goals of a slender waist and improved physical well-being. This is brain food . . . well, at least its prefrontal portion. _Tasty Tempts_ provides sugar and fat in a dozen deep-fried combinations, immediately delicious and sure to tickle the fancy of our limbic system. Set up across from each other at the mall, which of these two stores will sell more snacks? You don't need a degree in marketing to conclude that sugar's moment on the lips wins out over its lifetime on the hips. _Nutrity Nuggets_ will be most people's choice of tomorrow, what they _intend_ to eat, while _Tasty Tempts_ is the choice for today, what they _are_ eating. Furthermore, as sure as the high price of movie theater popcorn, _Tasty Tempts_ delivers a much larger profit margin because impulsive purchasing curtails comparison shopping. _Nutrity Nuggets_ may well go under while _Tasty Tempts_ becomes an international franchise. Businesses respond to our dominant desires, so there is no coercion or conspiracy here, just the invisible hand of the market building a limbic system wonderland. With the ubiquitous overemphasis on the immediate and the material, on the instant and the consumable, people are seduced into putting off long-term but ultimately more satisfying goals involving career achievement, volunteering in the community, raising a family or following a spiritual path. Materialism and consumerism are merely emergent properties of our neurobiology given free rein in a free market.
The process of seduction all starts with the sophisticated science of market research. I should know; I met my wife, Julie, while she was getting an advanced degree in this area. Market research has many different applications, some of which are benevolent. For example, my wife's university adviser conducted research on how to craft warning labels on cigarettes so that people _don't_ buy a pack. But, like most of life, market research usually concentrates on where the money is, and from children's television to political parties, marketers are adjusting products to our tastes or even creating desire for them. In doing so, they invariably appeal to our limbic system by creating temptations. The food industry, in particular, has used market research to the hilt, finding out what is most delicious to consumers and how best to package it. In his book _The End of Overeating,_ Dr. David Kessler, a commissioner of the U.S. Food and Drug Administration under two presidents and the former dean of the Yale School of Medicine, investigates the incredible vigor and purpose that food producers commit to getting us to eat more of their cheap and nutritionally challenged products. The amount of engineering that goes into creating visually appealing, flavorful food that scrumptiously crunches, melts, or rolls in your mouth is astounding, rivaling only by that put into designing your flat screen TV or Blu-Ray player. With exquisite fine-tuning of the proportions of sugar, fat, and salt in a recipe, for example, dishes can be created that provide no satiation—we always have room for one more bite.
Once a product has been devised, the emphasis on the limbic system continues during its presentation. Advertisements, which comprise about 1 to 2 percent of most economies, typically accentuate the most concrete and salient aspects of any merchandise. When you next walk through a grocery store, note how prominently the look or taste of any product is displayed; compare that to the extra effort required to find its nutritional content or its relative cost, both of which would appeal more to your prefrontal cortex. Finally, the virulence of temptations is truly maximized if desired products are immediately accessible; availability encourages the impulse purchase. Since this principle of instant availability considerably strengthens the role of the limbic system in our decision making, we see a lot of it. "Buy-now, pay-later" sales strategies stress the present moment, and sales gurus like Zig Ziglar emphasize: "If the decision is yes, then you . . . could be enjoying the benefits NOW!" As David Mesla, a weight management scientist at the Unilever Health Institute, notes: "Every single day and every single place you go, those foods are there, those foods are readily available for you to engage in. There is constant, constant opportunity." Universal proximity is exactly the goal—to shave enough seconds off the mechanisms of delivery that all products can be purchased as impulsively as the candy by the checkout counter. Once this happens, the world becomes an inescapable cage of temptation and if your willpower ever lapses, even for just a second, that's all the time they need to get you. But wait, there's more.
Aside from making their products and presentations more alluring to your limbic system, marketers also make concerted efforts to push that pesky prefrontal cortex aside. Habits and rituals, in particular, bypass the prefrontal cortex during decision making, and so great efforts are made to cultivate them in consumers. Many of our purchases are triggered rather than chosen, just as the addictive aroma of KFC's eleven special herbs and spices is designed to create a sudden craving for deep-fried chicken. Regardless of our original intentions, once cued, our actions can be "emotionally hijacked" and we end up automatically eating fast food or ordering that specialty coffee. We are all vulnerable to this, me included; I am guilty of consistently buying overpriced movie theater popcorn, which science confirms is less of a conscious choice than a ritualistic act cued by entry to the theater. Research on exactly how vulnerable we are is the specialty of Brian Wansink, a professor of consumer behavior at Cornell University. His studies have estab-lished that our eating choices are indeed mostly routines that are weakly based on hunger and more strongly based on context, such as the size of plate or portion, or how visible the treat is. For a study that won him the Ig Nobel Prize, he infamously devised a bottomless soup bowl that surreptitiously refilled itself as people ate from it. Though they reported not feeling any fuller than those who had a single bowl of soup, those who ate from the ever-full bowl consumed almost twice as much—76 percent more. Such habits now make up about 45 percent of our daily actions, and increasing this percentage by providing easy options and clear cues is big business.
The extent to which we can become wedded and welded to our habits is best revealed by the way we rely on our personal digital assistants (PDAs), like Apple's iPhone or Research in Motion's BlackBerry. People text everywhere, even while driving. This is a salient example of a pleasurable impulse overriding good judgment, for, as common sense concludes, and my own as well as others' research shows, any cell phone use while driving (hands free or handheld) slows one's reaction time dangerously. Such are the addictive qualities of PDAs that the World College Dictionary voted "CrackBerry" as the word of the year in 2006. They are so embedded in people's lives that at times our brains, in a testament to their neural plasticity, adopt the devices as part of our bodies. When it isn't around, people feel something like phantom limb syndrome (sometimes termed "fauxcellarm"). Others report the more familiar problem of repetitive motion disorder, like "BlackBerry Thumb," which is recognized by the American Physical Therapy Association as an official workplace injury. And what are people frenetically doing with their PDAs so much as to wear out their joints and ligaments? With tens of thousands of applications available, the company comScore undertook to categorize the top twenty-five applications downloaded into the iPhone. The only one that wasn't entertainment, a game, or a social networking site was Flashlight—a utility that turns your iPhone into a light source.
SUPPORTING VOICES
So here's the situation. Procrastination isn't just battling a hundred million years of evolution. It is battling a hundred million years of evolution that are being actively exploited at every turn by the very fabric of our society. In 1958, Aldous Huxley, in his book _Brave New World Revisited,_ warned: "All the resources of psychology and the social sciences are mobilized" with the aim of controlling people by finding "the best ways to take advantage of their ignorance and to exploit their irrationality." In 1985, with the rise of the video game, the influential cultural critic Neil Postman specifically built on Huxley's work in _Amusing Ourselves to Death,_ where he points out that "the rationalists who are ever on the alert to oppose tyranny 'failed to take into account man's almost infinite appetite for distractions.'" And we have Avner Offer, professor of economic history at Oxford University, reviewing in 2006 how current Internet consumption contributes to many of the world's ills, in _The Challenge of Affluence._ 31 In sum, the free market is geared toward providing increasingly irresistible temptations that distract us from our greater goals.4c
Are we right, Huxley, Postman, Offer, and I? Well, look around you. Exactly how many recreational or entertainment pursuits do you have readily available? A typical household can hold hundreds, from widescreen TVs to Internet portals. Never before in our history have there been as many temptations, as succulently devised, as readily available, and as adeptly marketed. Adam and Eve only had to deal with a juicy apple purveyed by a serpent. Nowadays, our apple is caramel coated and chocolate dipped, marketed with a multi-million dollar advertising campaign in a blitz of commercials, pop-ups, and inserts. Inevitably, as our lives drown in these diversions, our procrastination is on the rise.
LOOKING FORWARD
There is no turning our backs on modern life. The free market, in one form or another, will continue and the pace of invention will only accelerate. We will benefit from many of these innovations, but not all. The exploitation of the limbic system is baked into capitalism and you can't stop it without making the entire wonderful wealth-generating machinery grind to a halt. Someone will always create a product that provides short-term pleasure along with considerable but deferred pain simply because we will buy it. Consequently, dealing with constant temptation and its potential for creating procrastination is and will continue to be part of living in this world. Good thing you are reading this book, then. Learning better ways to cope with temptation and procrastination is what we will be doing together in later chapters; we will make the Procrastination Equation work for us, one variable at a time. But first, let's whet your appetite by acknowledging what procrastination costs you and society at large. A single incident of procrastination can be petty, but once you see the ultimate toll, I think you will find that it is an opponent worth fighting. I surveyed four thousand people to find out where they procrastinated most. The next chapter reveals what they told me and the personal price they paid for putting things off.
[Chapter Five
The Personal Price of Procrastination](contents.xhtml#ch_05)
WHAT WE MISS, WHAT WE LOSE, AND WHAT WE SUFFER
_We have left undone those things which we ought to have done; and we have done those things which we ought not to have done._
THE BOOK OF COMMON PRAYER
For enduring fame and sheer depth of procrastination, Samuel Taylor Coleridge stands alone. One of the great poets of the nineteenth-century romantic era, Coleridge might have been its greatest, but that title is more often given to his one-time and more diligent friend, William Wordsworth. Coleridge's tragic weakness was procrastination. He put off his work and his obligations, at times for decades. The poems for which he is best remembered, and which are still regularly studied in English literature classes, all show traces of procrastination. _Kubla Khan_ and _Christabel_ were both eventually published as fragments—unfinished works—nearly twenty years after he began them, and _The Rime of the Ancient Mariner,_ though completed, was five years late to press.
Everyone—his family, friends and even he himself—recognized Coleridge's procrastination. His nephew and editor, Henry, wrote that his uncle was "the victim of a procrastinating habit," and Coleridge himself describes his procrastination as "a deep and wide disease in my moral Nature . . . Love of Liberty, Pleasure of Spontaneity, these all express, not explain, the fact." However, it was his close friend Thomas de Quincey who provided the best account, having shared with Coleridge not only a proclivity to procrastinate but also a severe drug addiction—Quincey's autobiography is aptly titled _Confessions of an English Opium-Eater._ As Quincey wrote:
I now gathered that procrastination in excess was, or had become, a marked feature in Coleridge's daily life. Nobody who knew him ever thought of depending on any appointment he might make. Spite of his uniformly honourable intentions, nobody attached any weight to his assurances _in re future_ [in regard to the future]. Those who asked him to dinner, or any other party, as a matter of course sent a carriage for him, and went personally or by proxy to fetch him; and as to letters, unless the address was in some female hand that commanded his affectionate esteem, he tossed them all into one general _dead-letter bureau,_ and rarely, I believe, opened them at all.
Coleridge's excuses for lateness have themselves become legendary. His correspondences consist frequently of apologies, at times even an extended run of them; witness his letters to a Mr. Cottle, a publisher who bought the copyright to a book of his poems—sadly, in advance. Deserving special mention is the "Person from Porlock," who Coleridge claimed irrevocably interrupted his recollection of the opium-induced dream that served as the basis of his poem _Kubla Khan._ The poem runs only 54 lines, instead of the intended 200 to 300. As judged by Robert Pinsky, an American poet of our time, the "Person from Porlock" is the most famous of fibs from a long line of writers who are "better at making excuses or self-indictments than at getting things written."
But what did this procrastination reap for Coleridge? As Molly Lefebure describes his situation in her book _A Bondage of Opium,_ "his existence became a never-ending squalor of procrastination, excuses, lies, debts, degradation, failure." Financial problems pervaded his life, and most of his projects, despite elaborate planning, were barely begun or finished. His health was terrible, exacerbated by his opium addiction, for which he delayed medical treatment for an entire decade. His enjoyment of work dissolved in the stress of unmet deadlines—"My happiest moments for composition are broken in upon by the reflection that I must make haste." He lost rare friends, such as Wordsworth, and his marriage ended in separation because of it.
Coleridge's woes clearly illustrate that procrastination is capable of damaging all aspects of our lives. However, only the most confirmed of procrastinators will experience anything that approaches Coleridge's sad life. Most of us procrastinate significantly in only a few of life's domains. To find out about the procrastination habits of ordinary people living in our own time, I put up a survey on my website and four thousand people answered it. I asked them to tell me how much they procrastinated in each of twelve major life domains and to rank what was most problematic for them. The table on the next page reflects the results. The first column is the domain, and the second column records people's average level of procrastination, with a score of 2 indicating _seldom,_ 3 indicating _sometimes,_ and 4 indicating _often._ The third column indicates the percentage of people who chose that life domain as a "top three problem."
As you look at this table, pay attention to the domains where the numbers in the last two columns are both high: these are the trouble spots.
Procrastination causes us grief at school, at work, and in our private lives, particularly in relation to health. Eighty-nine percent of people believe they have major problems in _at least_ one of these three life domains alone, with 9 percent approaching Coleridge's levels by reporting all three. There is also a pattern to people's procrastination; most of these life domains cluster or hang together in groups. For example, many of the people who reported procrastinating about their financial situation also reported putting off education and other activities that might improve their careers (domains 2, 3, and 6). If you are suffering in one of these three areas, you're likely to feel that the other two are also problematic. This "Success" cluster of concerns is overall the most prevalent in terms of procrastination.
A second cluster focuses on "Self-Development," as people who put off their health issues (domain 1) also tend to put off spiritual quests, leisure activities, and self-improvement programs (domains 7, 10, and 11). This is the broadest cluster, as it is also connected to your social life, being part of your community, or pursuing a romance (domains 4 and 5). A final cluster could be labeled "Intimacy" by virtue of grouping together close friends, family, and parenting (domains 8, 9, and 12). This is the least problematic of the lot, especially in regard to parenting. Happily, very few respondents report putting off raising their kids—there is an immediacy to caring for children that trumps everything else.
Whether your procrastination lies in the Success, Self-Development, or Intimacy cluster determines the price you pay for procrastination, as these three areas translate into three major costs: your Wealth, Health, and Happiness. Naturally, those who put off the Success cluster and its career or financial aspects will be less wealthy. Those who procrastinate on Self-Development will experience poorer health, both of body and spirit. And though happiness is affected by the previous two clusters, Success and Self-Development, it has the strongest ties to Intimacy. In a different meta-analysis that I conducted, based on close to twelve hundred studies, I established that the biggest predictors of happiness are traits leading to fulfilling interpersonal relationships; great wealth and good health mean less without someone to share them with. Wherever your procrastination lies, the more you do it, the greater the cost. Just take a look.
FINANCIAL PROCRASTINATION
The most common excuse I hear from people who procrastinate at work is that they are more creative under pressure. I can see how it might appear this way. If all your work occurs just before a deadline, that is when all your insights will happen. Unfortunately, these insights will be relatively feeble and few compared to the insights of those who got an earlier start, since under tight timelines and high pressure people's creativity universally crumbles. The bleary-eyed 3:00 a.m. crowd scrambling to finish a project will usually come up with routine, unremarkable solutions. Innovative ideas are typically built on the bedrock of preparation, which includes a laborious mastery of your topic area followed by a lengthy incubation period.
Other procrastinators try to justify their delays by indicating that they work most efficiently closer to the deadline. This time, they are partly right. You do have more motivation just before the clock strikes twelve and you cross the deadline. But what the procrastinator is creatively arguing here is not whether we work hardest at the eleventh hour (which we do) but that working _earlier_ actually harms our performance. In other words, this procrastinator says that working today _and_ tomorrow is worse than working only tomorrow—a clumsy lie.
No matter what index of success we examine, procrastinators tend to perform worse than non-procrastinators. There is some variation based on whether we look at education, career, or income, but not in a way you are going to like: as procrastination moves from school to job to measures of overall wealth, the worse its effects. The results (which I summarize in my article "The Nature of Procrastination") look like this. For high school and college students, only about 40 percent of procrastinators have above-average grades, while 60 percent are below. If you are one of the lucky 40 percent, you should recognize that though procrastination is a handicap, you are probably compensating for it with other attributes, like a brilliant mind. Don't fall into the trap of thinking that this vice is actually helping you out. Not that I should judge . . . I put off studying for too many finals and tried to erase my late start by doing a series of all-nighters, a strategy that ended with me dozing peacefully through the last half of a French exam. The consequence of that impromptu nap has me dreading foreign languages to this day. The funny thing is that almost everyone else has a similar story to share.
Students spend roughly a third of their waking hours on diversions they themselves describe as procrastination. On average, students engage in over eight hours of leisure activities on the two days prior to exams, and their inability to effectively manage their time is a self-reported top concern as well as a reason for dropping a course. Worse, this trend doesn't abate when the stakes get raised. It is a major reason why most potential PhDs leave school before graduation, and the three little letters they get after their name are ABD (all but dissertation). ABDs are so common that cartoonist Jorge Cham, for example, makes a living by writing _PhD Comics,_ a strip dedicated to chronicling PhD students' procrastination. Incredibly, after graduate students have gotten into a competitive academic program, done all their course work, perhaps even gathered their dissertation data, and need only to write it up and defend their thesis, at least half never complete the process despite the immense investment of time and the significant rewards for completion (on average, a 30 percent increase in salary). Procrastination is the primary culprit.
Moving on to the field of career success, we find that the impact of procrastination intensifies a little bit. As judged by their peers, 63 percent of procrastinators are in the below-average, unsuccessful group. From the get-go, procrastinators have trouble getting going, and they put off the job hunt. When unemployed, they stay that way for longer. Once employed, most will find that work life is less forgiving than college or high school. The stakes are higher, so it is harder to get your boss or your clients to give extensions. For example, Michael Mocniak, general counsel at Calgon Carbon, got fired from his job for putting off processing his invoices—$1.4 million worth. Furthermore, on-the-job projects can be larger and much harder to complete at the last minute; work is less predictable, so you could find the final hours before a deadline suddenly double-booked. Still, there is that 37 percent of procrastinators whose wealth is at least above average despite their character flaw. If over the breakfast table you call the CEO mom and the Chairman of the Board dad, you are likely going to be wealthy no matter what your personal vices are. Other chronic procrastinators might end up in a career—and there are a few—where it is very difficult to procrastinate—careers with built-in daily goals, like sales or journalism. With everything due today, the leeway to procrastinate is exceedingly slim.
Finally, when we talk about overall financial success, the procrastination numbers again step up. By their own admission, only 29 percent of procrastinators consider themselves successful, with the remainder describing themselves as below average. The reasons for this are legion: procrastination's harmful touch extends into dozens of nooks and crannies that affect your bank balance. For example, the U.S. government gets at least an extra $500 million each year thanks to tax procrastination. A typical procrastinator's mistake is failing to sign the forms in the last-minute rush, making them invalid and subject to a late penalty. But procrastination hits our savings and spending in many other ways.
Savings speak to what Albert Einstein called the eighth wonder of the world—compound interest. The money you save not only earns interest, but the interest earns interest, like your children having grandchildren. Such is its power that if you put aside $5,000 each year between the ages of 20 and 30, you would retire richer than if you started putting that five grand aside _every_ year from the age of 30 on. Alternatively, consider the native Indians who sold Manhattan Island for about $16 worth of beads. Had they taken the money and invested it, with compound interest they could pretty much have bought back the entire island today and everything on it—from the Christmas trimmings at Rockefeller Center to the boardroom leather chairs at Trump Tower. Too bad procrastinators rarely act on their intention to sock money away for retirement or even a rainy day. If they were characters in one of Aesop's cautionary tales, they'd play the grasshopper instead of the ant. All that compound interest, all those potential investment dividends, lost and almost impossible to regain. As a paper in _The Financial Services Review_ concludes, "We find the levels of contributions required for individuals who start saving late are so high it is questionable whether they are affordable for anyone not on a high income."
What's more, procrastinators tend to be credit card revolvers; that is, they have hefty rotating unpaid balances on their monthly statements. When you combine all the credit cards in a household, the total often exceeds $10,000. Thanks to something called "universal default," most procrastinators are probably being charged the maximum rate for those balances, close to 29 percent per year, or 32 percent if it's on your Sears card and up to 113 percent if you live in Mexico. Universal default means that if you are late paying one bill, such as the phone or electric bill, the credit card company can jack up their rates. Make one mistake, anywhere, and they've got you (goodbye 0 percent introductory annual rate). Here compound interest again rears its head, but this time it's ugly. How are the credit card companies making record profits? At the dallying hands of procrastinators. They affectionately call revolvers the "sweet spot" of their industry.
Of all the scientific studies that show how procrastination is dangerous for your financial health, here is one I found particularly revealing. It deals with MBA students, so I've nicknamed them "Leaders of Tomorrow." The study demonstrates the self-defeating choices of procrastinators by examining University of Chicago MBAs, who often pride themselves on being cutthroat competitors. After playing a series of games in which they could win up to $300, study subjects were given a choice about how they could receive their winnings. They could either get a check now or wait two weeks and get an even larger sum. Here is why procrastinators tend to be poorer: even though most of them demanded to be paid now, they didn't cash their checks until, on average, four weeks later. In other words, it took them _twice_ as long to get to the bank as they would have had to wait for the larger reward. This brutal mixture of procrastination and impatience is common: two-thirds of the students wanted their money up front.
If none of this has resonated with you yet, I will add one final example, _the_ final example: your last will and testament. Way back in 1848, Lewis Judson noted that procrastinators not only borrow excessively but they put off their estate planning too: "Most men postpone making their wills until on a sick bed, and often then, until too weak to make them clearly and the lawyers take more of the estate than the heirs." In the ensuing one and a half centuries, nothing much has changed; right now, I bet your will is almost certainly either out of date or completely undone.5a Though you will be dead when the ramifications of this particular procrastination play out, it is probably the ugliest possible legacy to leave for your friends and family. Dying intestate—without a legal will—is common, happening to around three quarters of the population. George Gershwin (American composer), Richie Valens (rock pioneer), Howard Hughes (reclusive billionaire), Keith Moon (drummer for the Who), and Barry White (the smooth bass soul singer) all died intestate. Abraham Lincoln and Martin Luther King, Jr., despite both rallying against procrastination themselves and receiving a stream of death threats, died intestate too.5b What dying intestate entails depends on the jurisdiction where you live. Your whole estate may go to the government, to a hated sibling, or perhaps to your ex-spouse from whom you are separated but not yet divorced (and his or her new partner). Nothing may go to your soul mate, your best friend, or your favorite charity, and all your family heirlooms could be sold at bargain prices. The law favors descendants over ancestors so, if you do have children, they could get your estate all at once at the supremely responsible age of eighteen with no strings attached.
MEDICAL PROCRASTINATION
Despite being an important medical procedure, the thought of a colonoscopy makes most people squeamish. Even a description, which I'm about to provide, often provokes discomfort. The first step in a colonoscopy is to clean out your innards as much as possible. Typically, this involves drinking a gallon of a very strong laxative until what you pass resembles what you put in. You may also need an enema, which requires you to take in another few quarts, except from the other end. After this cleaning, upside and down, you are ready for the physician. You will go to the hospital, put on a gown, and then be sedated. You certainly don't want to tense up when asked to lie on your left side and then receive a half-inch colonoscope through your rectum. A little air is usually injected to help inflate your bowel and allow a good look around. The doctor looks through the colonoscope and into you for about thirty minutes and your buttocks will feel greasy for a while afterward, but that's about it.
You should start getting these colonoscopies pretty regularly after the age of fifty, if not earlier, but a surprising number of people put it off, including oncologists. Even my father-in-law, who ran a large health sciences center and should have known better, unduly delayed his. It does sound unpleasant, but the downside of delaying a colonoscopy is potential death from colorectal cancer, the second most deadly form of cancer, right behind lung cancer. But unlike lung cancer, colorectal cancer is very treatable and preventable if you catch it early. It comes in stages, from 0 to 4, the survival rate plummeting with each successive stage. The number one reason for failing to get screened is procrastination. Putting off a recommended colonoscopy because of fear, discomfort, or embarrassment is a widespread problem even among the most capable. Katie Couric, while she was co-anchor of the _Today_ show, lost her husband to it. My father lost his second wife. By the time she finally went to see her doctor, a colonoscopy wasn't needed because you could already feel the cancer through the wall of her stomach. After seeing her vibrancy slowly fade away in my father's care over her last year, I can confidently say that this is as serious and tragic as it gets. However, the story of colonoscopies is not unusual for medicine. For many diseases, infections, growths, and general ailments, early detection and treatment is always better, and yet people consistently delay. Given this lead up, I'm sure you won't be surprised to learn that procrastinators tend to be among the least healthy of people.
To rub salt into the wound, not only are procrastinators less likely to pursue treatments but they are more likely to indulge in the very behaviors that create the need for treatments in the first place. Procrastinators are health risks because their impulsive nature makes them susceptible to vices, attracting them to short-term pleasures despite their long-term pains. On the other hand, they are less predisposed to virtues—that is, short-term pains with long-term rewards. For example, do you floss? Though you know you should and often plan to, if you are a procrastinator, you very likely don't. Exploring the effects of this oversight, I asked my dentist about the worst case he had seen. He recalled one patient with more tartar than tooth, tartar so thick that it formed a solid wall, obscuring the teeth. He offered to show me a picture; I wisely declined. Here are a few other misbehaviors that affect procrastinators' health.
If you are a major procrastinator, you likely have some cigarettes on you. At least, they are tobacco-based rather than cannabis (but you probably have had those too at some point). And what goes better with a cigarette than a drink, one that has a little alcoholic bite to it? Better not have too many—you don't want to pass out while smoking, because you haven't checked your smoke detector or changed its batteries in quite a while. And that wasn't a salad you had for dinner, not with all those calories. Well, if you got it at a drive-through, what do you expect? This brings up your driving. Have you noticed that most people are scared when you are behind the wheel? Don't get angry with me, though you do tend to get angry quite a bit. Don't you?
In short, smoking, excessive alcohol use, drug abuse, recklessness, overeating, risky driving, and fighting, not to mention promiscuous sex, are all activities that procrastinators tend to do a little more of rather than a little less. They all tap into procrastinators' impulsiveness, making gratification the one thing they don't delay. If you currently partake in even half of these vices, you are not exactly a poster child for a healthy lifestyle. Odds are, your choices will catch up with you.
RELIGIOUS PROCRASTINATION
Despite being born in the fourth century, St. Augustine is interesting enough to this day that a musician, specifically Sting, has written a song about him. Prior to his conversion, St. Augustine was a follower of what was then the world's most popular religion, Manichaeism, and he knew the pleasures of the flesh way better than you would expect of any saint. Though Manichaeism was against procreational sex—which partly explains why it died out—it found recreational sex more forgivable, an option that St. Augustine and his multiple mistresses indulged in enthusiastically. Their libidinous lifestyle more than explains how St. Augustine became the patron saint of beer or, at least, of brewers; it became his defining temptation. After converting to Christianity in a.d. 386, he had trouble turning his back on a woman's embrace, his most famous quote being, "Please lord, make me chaste, just not today!" He kept putting off celibacy, feeling utterly defeated by his procrastination.5c Then one day, in his garden in Milan, he heard God directing him through a child's voice to "take up and read." He grabbed the Bible, which opened to this precise passage from St. Paul's Epistle to the Romans " . . . not in orgies and drunkenness, not in promiscuity and licentiousness, not in rivalry and jealousy. But put on the Lord Jesus Christ, and make no provision for the desires of the flesh." With such a direct message, he redoubled his efforts for a holier life.
St. Augustine's plight is a common one.5d The world's great religions are tough on procrastination, universally viewing it as a detour from the path of salvation and enlightenment. Their disapproval makes sense, because putting off good acts in order to sin will put you in spiritual jeopardy. Here are some samples that show how.
Hinduism, to start with, is defined by the _Mahabharata,_ especially a section called the _Bhagavad Gita,_ a religious text preached by the god-figure Krishna. In it Krishna declares, "Undisciplined, vulgar, stubborn, wicked, malicious, lazy, depressed, and procrastinating; such an agent is called a Taamasika agent," unworthy of rebirth. In Islam, postponement of good deeds is primarily what the Arabic word for procrastination, _taswif,_ refers to. Similarly, _The Pillars of Islam,_ the foundational book on Islamic law, has much to say on procrastination, none of it good. The same is true of Buddhism, despite its often being taken to be the world's undemanding and unobtrusive feel-good religion. From the Pali Canon, the earliest written Buddhist scriptures dating from about the first century B.C. until today, the message has been consistent and clear. As the American-born lama, Surya Das, says: "We have to stop procrastinating, pretending that we have forever to do what we want to do and be what we long to be." But the religion in which procrastination appears to be the biggest problem, judging from the number of times it is mentioned, is Christianity. Sermons aplenty preach against procrastination, mainly because the faith emphasizes repentance.5e People may lead a sinful, selfish life, but can seek forgiveness on their deathbed and still be redeemed, cramming for the finals so to speak.
Procrastination is a universal theme in all these religions because we cannot predict when we will die; thus, the best time to repent, to act morally, to commit ourselves to doing good is now. A parable from _The Mahabharata,_ Hinduism's epic narrative, demonstrates this reasoning. The hero, Yudhisthira, promises to donate some money to a beggar _tomorrow._ His younger brother Bhima hears of this and runs out to ring the court's victory bells. "Why," asked Yudhisthira, "did you ring the bells?" Bhima replies, "To have made such as promise, you must have victory over life. Otherwise, who knows what tomorrow will bring?" Similarly, Sayyiduna Ali Murtadha, the fourth Caliph of Islam, wrote, "Everyone who is taken by death asks for more time, while everyone who still has time makes excuses for procrastination." If our clock suddenly stops, our souls may be damned if we put off good deeds, meditative practice, and requests for forgiveness.
The universal holy war, then, isn't against forces of darkness but against forces of nature, our own human nature. Religions are all battling procrastination among their believers and converts because whatever promised lands or promised rewards they offer will most likely be granted in the _distant_ future. Inevitably, everlasting salvation is being deeply discounted against a backdrop of sins that provide pleasures immediately. The world may be spiritually divided by how we view God or the good, but when it comes to damnation, procrastination leaves no doubt that all religions have a lot in common.
THE PURSUIT OF HAPPINESS
If procrastinators tend to be less wealthy and healthy than the doers among us, it is likely that they are going to be less happy too. And they are. This is partly on account of the stress of procrastination, which frequently gives rise to guilt. It is not unusual for procrastinators to suffer more for putting off the work than they would have suffered by actually doing the work itself. Consequently, when they finally tackle the task, they are often relieved, admitting, "This isn't as bad as I thought." Rita Emmett, in her _Procrastinator's Handbook,_ considers this a law, which she codifies under her own name, as Emmett's Law: "The dread of doing a task uses up more time and energy than doing the task itself."
The online procrastination discussion boards often serve as confessionals of delay-induced torment. Here are half a dozen examples culled from two online forums, _Procrastinators Anony-mous_ and _Procrastination Support:_
•I've been very successful in many ways and managed to accomplish a lot in my life. But the process is miserable—I procrastinate, feel terribly guilty, get depressed, do work marathons, promise to change, and then start procrastinating again. I'm now at a point professionally where I've procrastinated so much on so many things that the work has really piled on and I'm fearful and unclear about how to dig myself out of the hole I'm in.
• The semester started two weeks ago and so far it has gone well. I was doing every assignment early and had so much free time but since then I have reverted to my old self. I fear for the worst and I have about two months until mid-term when my marks are due. I know I'm not as bad a student as shown in my report cards but I can't seem to get my work schedule in gear.
• Whenever I told people I was a horrible procrastinator, they would usually laugh, and then say they were too. But they seemed fine; their lives weren't on the brink of destruction because of their procrastination, like mine was. Can any of you please help me out?
• I really just want to DO WHAT I'M SUPPOSED TO WHEN I'M SUPPOSED TO DO IT! Whether I intrinsically want to or not, like NORMAL PEOPLE do. It hurts me so much that I cannot simply do that.
• And I'm so ashamed of even needing to resort to something like this. What kind of person am I that I have such a lack of self-control? I have fought and fought and fought over the years . . . I feel like it's a dying battle.
• This habit isn't funny, but I've always pretended it was. Really, though, it's pretty tragic. It takes me months to respond to e-mails, costing me personally, socially, and financially . . . the only thing I really ever finish is dessert.
Unfortunately for these procrastinators, guilt and poor performance won't be the entire story. When it comes to gratification, procrastinators stress immediacy. Like the spoiled rich girl Veruca Salt from Charlie and the Chocolate Factory, they don't care how, they want it now. Immediate gratification often comes at the cost of larger, later rewards, so consequently, procrastination is like running up a charge on your emotional credit card. You don't have to pay it now, but when the bill is finally due there will be compound interest. We fritter away the days with the small pleasures of television and computer games, of Internet surfing and Sudoku puzzles and end up with nothing to show for it. This is a recipe for regret.
In the short term, we regret what we do, but in the long term, we regret what we don't get done. Inaction causes us the greater suffering. Not to have done, not to have tried, to have put it off—this is part of the human condition, so we all suffer from it to some degree. You almost certainly have or will have regrets in at least one of these three life areas: Success, Self-Development, and Intimacy. Looking back on our lives, it is common to feel that we should have gone for that degree or tried harder in class, that we should have mustered up the courage and risked rejection for that date, or made time for that phone call to Mom. We are haunted by the ghosts of our own lost possible selves—what we might have been: could've, should've, but didn't.
I am no exception to procrastination's rule of regret. My brother Toby suffered from sarcoidosis, the same debilitating disorder that took the life of comedian Bernie Mac. When my family had to make the decision to take Toby off his ventilator and wait with him until he took his last breath, I was crushed with knowing what a fool I had been with my time. I regret putting off trips to see his plays. I regret not making it to the hospital sooner to see him. I regret the littlest things, like not taking more time to watch a bad movie on TV with him while eating take-out. He was the smartest, funniest person I have ever known and I took it all for granted. In keeping with life's synchronicity, soon after my brother's funeral I found a poem in the newspaper written by Mary Jean Iron. I clipped it to remind myself of my carelessness. It is still there, in my desk drawer, waiting for this moment:
_Normal day, let me be aware of the treasure you are._
_Let me learn from you, love you, bless you before you depart._
_Let me not pass you by in quest of some rare and perfect tomorrow._
_Let me hold you while I may, for it may not always be so._
_One day I shall dig my nails into the earth, or bury my face in the pillow,_
_Or stretch myself taut or raise my hands to the sky_
_And want more than all the world, your return._
Put down this book and get going. Don't hesitate: call your mother, start writing that essay, ask out that special person you have had your eye on. Now is the moment you have been waiting for.
LOOKING FORWARD
Have you really put this book down? I didn't think so, but don't worry. I know it is not that simple. Interventions are still coming—you will hit them when you reach chapter seven. Right now, I want to continue focusing on the price of procrastination. In the next chapter, we look at the economic cost of procrastination to society. When we calculate the final figure, it will probably be larger than even your most outlandish guess.
[Chapter Six
The Economic Cost of Procrastination](contents.xhtml#ch_06)
HOW BUSINESSES AND NATIONS LOSE
_Momentary passions and immediate interests have a more active and imperious control over human conduct than general or remote considerations of policy, utility or justice._
ALEXANDER HAMILTON
When exploring procrastination, no other country provides as many good examples as the United States. Almost two-thirds of all procrastination research is done with American citizens, and no wonder, given what it costs them. Here's how to calculate it. First, how many workers are there in a country? For the United States, the figure is over 130 million, but we will round down for ease of calculation. Second, what is the annual average wage for those workers? Estimates can reach over $50,000, but we will be conservative and go with the lower figure of $40,000. Finally, how many hours do people work each year? The Organisation for Economic Co-operation and Development provides that figure: Americans clock in 1,703 hours, or slightly more than 212 eight-hour workdays, each year. Finally, we have to determine how many hours each day people procrastinate. Two companies, America Online and Salary.Com, partnered together to survey the work habits of more than ten thousand people; the result was over two hours of procrastination in every eight-hour day, not including lunch and scheduled breaks. Once again, we will round the estimate for ease of calculation, this time downward to an even two hours.
Keep in mind as we calculate the final figure that I've used conservative estimates at every step. We have 130 million people who spend about two hours out of every eight at work procrastinating, or 414 hours per year. Each hour is worth at least $23.49 (i.e., $40,000 divided by 1,703 hours), though if their employers are making a profit, they are worth more than that. At a minimum, then, procrastination is costing organizations about $9,724 per employee each year ($23.49 times 414). Multiply that by the total number of employees in the United States, and you get $1,264,1200,000,000. In other words, a conservative estimate of the cost of procrastination for just one country in just one year is over a trillion dollars. This number may seem surprisingly large, but not if you are an economist. Gary Becker, who won the Nobel Prize for economics, concludes, "Indeed, in a modern economy, human capital [the work people do] is by far the most important form of capital in creating wealth and growth." With a quarter of each person's work day spent dithering, procrastination is going to be costly.
Still, if this trillion-dollar figure makes you balk, fine. Revise any part of these calculations downward to what _you_ think is reasonable. Cut the number of procrastination hours in half, pay everyone minimum wage, but pretty much anything times 130 million is still going to be a hefty sum. Myself, I think the true costs of procrastination are far more than a trillion dollars. Procrastination during the business day is only part of the picture. Our ability to save money or make timely political decisions is also affected by procrastination, and the costs there should be over a trillion dollars too. And here is how it is happening.
TIME IS MONEY
The more we procrastinate at work, the more it costs us. Unfortunately, it's not just entry level workers who procrastinate but their managers and CEOs as well. Consider the Young Presidents' Organization, a club of corporate heads under forty-five who run companies worth more than ten million in revenue. In a survey of 950 of its members, the most troublesome problem reported was "facing up to a task which was, for various reasons, personally distasteful." As my own research program shows, organizational teams, work groups, and task forces procrastinate. The graph on the next page charts the average work pace of business groups over the course of their projects (the solid line) along with a hypothetical steady work pace (the dashed line). In both form and content, it parallels the graph from chapter 2 that featured student procrastination. As can be seen, both students and business groups demonstrate the same shape of curve, whereby people start off slow and then pick up the pace.6a
ORGANIZATIONAL TEAM PROCRASTINATION
How has procrastination wormed its way into every inch of the business world? For the most part, by way of the same device that tempts students from their studies—the Internet. Dubbed e-breaking or cyberslacking, surfing the Net is the most serious of employees' time-wasting activities. About one in four people admit to playing online games on the job. In fact, gaming websites report a sharp drop in traffic at exactly 5:00 p.m., the end of most people's work day. Similarly, "video snacking," when people surf for and trade clips of all types, is a huge distraction. Though video use tends to spike during the lunch hour, it is prevalent at all times, and is expected to soon account for half of all Internet traffic. As summarized by Miguel Monteverde, executive director of AOL Video, "Based on the traffic I'm seeing, our nation's productivity is in question." Interestingly enough, this trend extends to pornographic sites as well, which get 70 percent of their traffic from the nine-to-five crowd. Finally, of course, there is social networking. The company Talkswitch provides a perfect example; it recognized it had a problem when it discovered that all sixty-five of its employees were using Facebook—simultaneously.
To cope with this tsunami of procrastination, most companies ban inappropriate Internet use, but it is difficult to enforce. Employees rearrange their computer screens so they can't be easily seen from the doorway, giving them time to hit a "Boss Key" that quickly opens a legitimate application. There are also several applications that mask illicit activities, such as one that allows Internet browsing within a Microsoft Word shell, making it difficult to detect dillydallying ways. Especially notable is the website "Can't You See I'm Busy," which makes it hard to detect games hidden within graphs and charts. In response, two-thirds of companies firewall their servers, fettering people's Internet access to various degrees. WebSense, which ironically makes software that filters the Internet, automatically monitors employees' Internet use and cuts off their access when they reach two hours of personal surfing. Other organizations enforce wide-ranging, perpetual restrictions on gambling, pornography, video sharing, and social networking sites alike.
Banishing people from games and Internet sites does not eliminate polymorphic procrastination because it can manifest itself in so many ways. Solitaire is pre-loaded on most Windows platforms, making it the top computer game of all time, even favored by former president George W. Bush. Memory keys often have games embedded on their chips, as do personal digital assistants (PDAs), which provide unrestricted Internet access. You can also go old school and avoid the computer completely. The ritual start of many a working day involves the diversion of the news. When I visit my sister, we scramble to be the first to get to the Sudoku in the morning paper. In the White House, Bill Clinton completed the _New York Times_ crossword puzzle daily.
Procrastination isn't fuelled by games alone. As Robert Benchley quipped, "Anyone can do any amount of work providing it isn't the work he is supposed to be doing at that moment." We procrastinate on important tasks by doing the unimportant. For many of us, this means e-mail, which now takes up 40 percent of work life. With every notifying "ding," workers instantly redirect their attention to reading the latest in an endless stream of electronic missives. Only a small seam of this e-mail bonanza is useful; the rest is junk. Though this deluge of electronic debris is partly composed of spam—unsolicited bulk e-mail—our greatest threat is the enemy behind the lines. Coined _friendly spam,_ much of the junk we receive is created by our friends and co-workers who carelessly mass e-mail us about every social event, virus hoax, urban myth, trivia tidbit, or arcane corporate policy change. Since all these e-mails have the potential to be useful, they must be read to conclude they aren't. And then there are e-mail's peripheral effects. In a study of Microsoft workers, people took an average of fifteen minutes to re-focus on their core tasks after answering an e-mail interruption. Combine this with the finding that information workers check their e-mail accounts over fifty times a day, over and above the seventy-seven times they text message, and theoretically no work should ever get done. More realistically, the business research firm Basex puts the interruption and recovery time at a little over a quarter of the work day (about two hours), which is consistent with studies on multi-tasking that conclude that switching attention is extremely detrimental to performance. In short, despite the veneer of activity that e-mail checking provides, there is not much light for all that heat.
SAVING FOR LATER YEARS TOO LATE
Procrastination doesn't just diminish our wealth by decreasing our productive hours. It also reduces the benefit we gain from our productivity itself. Our wealth is determined not only by the money we make but also by the money we save. Saving is a tried-and-true path to riches, as every dollar you put aside starts to reap the miracle of compound interest. Furthermore, since the dollars you save are invested, savings can help the nation as a whole, spurring economic expansion. When adopted, a policy of savings can be hugely successful. Since 2004, average Singaporeans, for example, have been wealthier than the average American largely because they save more. Unfortunately, when procrastination overtakes a society, saving becomes the exception and borrowing becomes the rule, a trend that can easily lead to financial ruin. Just consider your retirement savings account.
Aside from your plans to win the lottery, retirement rests on a three-legged stool. The first leg is the government, which, due to a bad habit of spending more than it receives, won't always be able to deliver on what little it promises. In the United States by the year 2040, for example, people can hope to receive only about two-thirds of their scheduled Social Security benefits, and on the heels of the 2008 global financial crisis, this percentage will probably decrease. The second leg is represented by businesses, which can put money aside for you as a form of compensation, typically in the form of a Defined Contribution plan.6b In such a plan, you decide how much, or rather how little, of your paycheck to contribute, and most allocations are matched by the company. The third leg is you, your decision to initiate and open independent retirement accounts. This is the most dependable option—except, of course, that it still depends on you.
By becoming a society of procrastinators, we have caused the retirement stool to be increasingly wobbly, as most people are socking away less. People are neither starting their own retirement accounts nor contributing to company plans, despite the fact that allocation matching is the equivalent of getting free money. When they leave work, their financial backsides rest on a stool supported by a single stubby peg derived from the government's forced savings program. Again, procrastination proves to be particularly poignant in the United States. In 2005, after decades of decline from originally double-digit rates, American household savings finally went into the negative. In other words, instead of saving today's money for the future, people were going into greater debt by spending tomorrow's money today—on average about half a percent more than they earned. To do this, not only did they borrow against their homes, in the form of mortgages, but about one in five borrowed against funds they had already set aside for retirement, putting themselves further behind. Worst of all, some of this financing was arranged through "liar loans," which initially seem affordable but eventually create financial ruin. Variable mortgages entice homeowners to buy well beyond their means, while "pay day" advances provide the desperate with temporary respite but leave them much worse off. They end up repaying each loan many times over; the interest rates of "check cashing" shops often exceed 500 percent a year. These are financial products that procrastinators are prone to fall for, products with short-term benefits but exceedingly high long-term costs.
The experts share the consensus that this situation is not ideal. At least something should be put aside for retirement; ideally, you should be saving 10 to 20 percent of your salary or higher if you are already in your forties. Even before the 2008 global financial crisis, which alone lowered pension accounts by at least a fifth, an increasing number of Americans believed they were not putting enough aside for their old age. And they are right. When retirement comes, more than four out of five Americans will find they haven't saved enough for their needs and by then it will be far too late to do anything about it.
Retirement procrastination transforms the golden years into grim and gray poverty. It means living on skid row or with the kids, if you had them and if they'll have you. To prevent this from happening, governments have employed a few tricks. Tax breaks for contributing to registered saving plans are a good start, but to make the most of them, these breaks need to be accompanied by a definite deadline: that's what procrastinators respond to. Stipulating that retirement contributions must be put aside by tax time is an effective strategy, as it breaks down long-term retirement savings into a series of yearly goals. Still, on its own, it hasn't proved to be sufficient, and so governments around the world are exploring another technique: automatic enrollment. Applying the same negative-option marketing ploy used by mail order book clubs, employers can now _automatically enroll_ their employees in pension programs with default investing options. Employees are free to withdraw or adjust their investment strategy at any time, but procrastinators will typically delay this decision, too. The result is a huge bump in enrollment. Another neat trick comes from the trademarked _Save More Tomorrow_ plan, developed by the behavioral economists Richard Thaler and Schlomo Benartzi. Rather than automatic enrollment, they use a strategy that exploits procrastinators' tendency to discount the future: employees can choose _now_ to save _later._ 6c That is, they must decide this year whether to start saving next year, and just as in automatic enrollment plans, once they have filled out the paperwork that commits them to saving, they will put off filing more paperwork to reverse their decision.
POLITICAL PROCRASTINATION
Governments, like people, have a bad habit of spending more than they receive. As I write this book, central government debts around the world are reaching commanding heights, often exceeding half the wealth their respective countries annually generate. By the time you are reading this book, it will be even worse. The United States, for example, will likely have finally hit the 100 percent mark, the point where it owes everything it makes in a year (that is, its total GDP). In dollar terms, that's an eye-popping $16 trillion. How did we get so deeply in debt? Governments display the same intention-action gap that defines all procrastinators: they form intentions to stop spending but change their minds when the moment to act is upon them. The United States has repeatedly tried to curb its own spending by legislating a borrowing limit—essentially reining in the government credit card. Unfortunately, this is akin to an alcoholic locking the door to the liquor cabinet but leaving the key in the hole. Politicians simply vote away their previous debt resolution and install a new higher limit, a process they have repeated _hundreds_ of times.
Governments are perpetually focused on quick fixes that solve the issues of the moment; the urgent displaces the important. This isn't a new insight. The American founding fathers understood this early on. I opened this chapter with a quotation from Alexander Hamilton, "Father of the Constitution," featured on every American ten-dollar bill. Similarly, James Madison, "Father of the Bill of Rights," wrote, "Procrastination in the beginning and precipitation toward the conclusion is the characteristic of such [legislative] bodies." And regarding the threat of debt specifically, here is a revealing quotation from George Washington: "Indeed, whatever is unfinished of our system of public credit, cannot be benefited by procrastination; and, as far as may be practicable, we ought to place that credit on grounds which cannot be disturbed, and to prevent that progressive accumulation of debt which must ultimately endanger all governments."
The American founding fathers were right; just take a look at the graph on the next page, which is similar to the two you have already seen, on student procrastination and the dillydallying of organizational teams. This one shows the average length of time it took the U.S. Congress to pass bills over the years from 1947 to 2000. For every session in fifty, Congress passed the bulk of its bills toward the end of the session.
CONGRESSIONAL PROCRASTINATION
Though some bills are delayed due to political maneouvering, a large part of the delay should be due to procrastination. Furthermore, one can determine which groups are the worst procrastinators by comparing the surface area between the two lines—that is, between the steady work pace (the dotted line) and the actual work pace (the solid line). The more they are procrastinating, the greater the surface area. And Congress soundly beats out even the average college student when it comes to putting things off.
The result of all this procrastination is more than simply delay dealing with the national debt. All long-term national goals and challenges tend to be put off as well, no matter how threatening. The outcome of America's War of Independence was partly determined by procrastination. In a key battle, George Washington crossed the Delaware to destroy a Hessian garrison: Colonel Rahl, head of the garrison, actually had prior warning of the invasion but decided not to read the report until later, after a card game he never had the chance to finish playing. Winston Churchill and Dwight D. Eisenhower, both wartime leaders, explicitly struggled with procrastination in their own governments, which put off preparing for war with Germany and, later, the Cold War with Russia.
Today, the most pressing issue facing all governments is environmental depletion and destruction. We are in the midst of several ongoing ecological disasters, all projected to peak at the same time: 2050. That may seem far away, but environmental issues are like supertankers. They take so long to stop that they must be tackled decades in advance; by the time they are in your face, they can't change course. Across the board, governments are putting off the issue until it is too late. To begin with, the soil beneath our feet is eroding and depleting. With about 40 percent of agricultural land already damaged or infertile, what will happen in 2050 when the little remaining arable land must feed over nine billion people? It is also doubtful whether there will be enough fresh water to grow the necessary crops; the projection is that 75 percent of countries will be experiencing extreme water shortages by that same date. The sea tells an almost identical story. Approximately 40 percent of oceans are already fouled and overfished, with species disappearing around the world. But it won't get _really_ bad until 2050, when the last of the wild fisheries are projected to collapse.
Interestingly—if that's the right word—these environmental disasters make the debate over global warming almost superfluous. With so many catastrophes projected, the consensus is grim. Even the futurist Freeman Dyson, who doubts global warming, concludes, "We live on a shrinking and vulnerable planet which our lack of foresight is rapidly turning into a slum." However, if climate projections hold true, we can expect about a three-degree increase in temperature by 2050. No matter what country you are in, there won't be any place that will truly benefit from this change. Entire ecosystems, like the Amazon rainforest, are expected to collapse, about a third of all animals and plants will become extinct, and billions of famine refugees will fight to determine who starves to death first. Since many of us will be around in 2050, it is worth taking a private moment to envision what this tomorrow will mean to you.
Government bodies have been alerted to this possible future for a long time. In 1992, 1,700 of the world's leading scientists, including most Nobel Prize winners, signed the "World Scientists' Warning to Humanity," which stated in the most explicit terms: "A great change in our stewardship of the earth and the life on it is required, if vast human misery is to be avoided and our global home on this planet is not to be irretrievably mutilated." For even longer, we have known what to do about it. Unfortunately, we are procrastinating about translating this knowledge into action. We could have avoided all these environmental issues if we had acted early. We can still mitigate them if we act now. The problem isn't informational or technological; it is motivational.
Still, be thankful that government procrastination isn't worse. Since the founding fathers of America were among the first to acknowledge the problem of procrastination, they did try to reduce its effects. Recognizing that what is expeditious can too easily prevail over what is wise, they tried to put temptation at a distance through _bicameralism:_ legislation must pass through two houses or chambers. Using the exact terminology of "hot" and "cold" cognition favored by today's scientists, George Washington explained to Thomas Jefferson why they needed a senate as well as a house of representatives.
"Why do you pour coffee into your saucer?" Washington asked.
"To cool it," Jefferson replied.
"Even so," Washington said. "We pour legislation into the Senatorial saucer to cool it."
Aside from Washington advocating a scandalous breach of etiquette (as "to pour tea or coffee into a saucer . . . are acts of awkwardness never seen in polite society"), it is a solid strategy that has been widely adopted. What can be initiated immediately will hold much, much greater sway over tomorrow's better options. By purposefully building in delays, such as a sena-torial house of sober second thought, the Constitution reduces the effects of time. Since it takes longer to pass all legislation, bicameralism focuses decision making on factors other than whether an aim is immediately obtainable. In other words, the added delay of a second house ensures that everything is going to take a while.
LOOKING FORWARD
We live in a world where our impulsive nature is only appreciated by those seeking to exploit it. But this is beginning to change. The field of behavioral economics, which recognizes our capacity for irrationality, is being incorporated into governmental public policy. Recently, the Gallup Organization hosted the inaugural Global Behavioral Economics Forum. Events like this have started to draw the attention of economic and political leaders from all shades of the political spectrum; both British Conservative leader David Cameron and U.S. President Barack Obama are exploring behavioral economic solutions. Phrases from Obama's inaugural address highlighting this need for change appropriately resonate, especially our need "to confront problems, not to pass them on to future presidents and future generations." Some of this thinking has already been translated into action, such as legislation making it easier for businesses to automatically enroll workers in retirement savings plans. Still, much more needs to be done.
As individuals and as a society, we pay a hefty price for our procrastination and have done so since the beginning of history. But we can bring millennia of dillydallying to an end today. A good start is to continue reading—the rest of the book is dedicated to actionable intelligence that puts putting off in its place. No matter what your procrastination profile—whether you lack confidence, hate your work, or are ruled by impulsiveness—there are proven steps you can take. And though we may have wished for this advice to have been available earlier in our lives, as we all know, working ahead of time is not really in our nature, is it? Perhaps we're now ready.
[Chapter Seven
Optimizing Optimism](contents.xhtml#ch_07)
BALANCING UNDER- AND OVER-CONFIDENCE
_A positive attitude may not solve all your problems, but it will annoy enough people to make it worth the effort._
HERM ALBRIGHT
Iremember few darker days of the soul than those I spent hunting for a job during a harsh economy. Job hunting is humbling—and humiliating—and it tests you to the very core. As rejections and months of unemployment add up, a gnawing uncertainty makes you doubt who you are. When bills mount so does the pressure to settle for less, to take that job you swore was beneath you. But then, when you finally lower yourself to apply, you find that even that possibility is out of reach. Here is where the value of faith comes in, whether in yourself or in a God with a plan. Against all facts and experience, you have to believe that the next interview, the next lead, or the next day will bring a different answer. Belief in oneself separates the successful person from the procrastinator; without such confidence, the couch beckons, the television distracts, and dreams of the future become what could have been. Many procrastinators doubt their ability to succeed and as a result, stop making the effort. Once effort disappears, failure is inevitable.
Beliefs are powerful because they form or directly affect _expectancy,_ making them a motivational keystone of the Procrastination Equation. As you become less optimistic or less confident in your ability to achieve, your motivation also ebbs: the more uncertain you are of success, the harder it is to keep focused. This self-doubt is usually associated with novel and difficult tasks, but it can also become a chronic condition: expectation of failure. Poor self-perception then becomes a self-fulfilling prophecy—by expecting to fail, we make failure a certainty because we never dig in and make an intensive effort. Since beliefs can create reality, we need a healthy dose of optimism to motivate us toward success.
On the other hand, too much optimism can also lead to procrastination. Remember Aesop's fable about the race between the Tortoise and the Hare? The far faster hare was so certain of his victory that he took a nap halfway through the race. The tortoise, moving slowly and steadily, overtook his slumbering competitor and won. As Michael Scheier and Charles Carver, psychologists who have spent their lives studying optimism, write: "It may be possible to be too optimistic, or to be optimistic in unproductive ways. For example, unbridled optimism may cause people to sit and wait for good things to happen, thereby decreasing the chance of success."
Over-optimism is particularly prevalent when we estimate the time a task will take. It's called "the planning fallacy." Most people are not very good at predicting the length of time required for completing even commonplace tasks. For estimating the time it will take to shop for Christmas presents, to make a phone call, to write an essay, the rule is "longer than you think." I myself am making edits to this very chapter far closer to my publisher's deadline than I'd like. We can't really help ourselves; it's a built-in bias of memory. To estimate how long future events take, we recall how long they took in the past. Our retrospection automatically abbreviates this time, and edits out much of the effort and obstacles. Unfortunately, this exacerbates the negative effects of procrastination. If you are leaving something to the last minute, there is actually far less time than that.
We need to find a balance between gloomy pessimism and Pollyanna optimism. Jeffrey Vancouver, a psychologist at Ohio University who specializes in the study of motivation, has succeeded in locating optimism's sweet spot. He found that, in a sense, we are motivational misers who constantly fine-tune our effort levels so that we strive just enough for success and use the prospect of failure as an indicator that we should up our game.7a Look at the figure on the next page. The vertical axis is motivation and the horizontal axis is optimism (that is, how difficult we perceive the task to be). Sensibly, we want the greatest reward for the least effort. Along the horizontal axis moving right, we start off with impossible tasks, too difficult to pursue. Why concentrate our resources where we will reap no reward? As tasks become easier and our optimism increases, we reach a tipping point. Motivation suddenly peaks: we believe that a win is possible, even though it will require considerable effort. As our optimism rises even further, our motivation falls, this time slowly. Eventually, we end up at the far right of the figure with tasks we believe we can easily perform. We're not motivated to accomplish these tasks because we deem them literally effortless. Most procrastinators are on the left of this chart, underestimating their ability, but a few are on the far right, believing that they are better than they really are.
Since most procrastinators tend to be less confident than non-procrastinators, we will start off by focusing on how to increase optimism, as it plays a central role in expectancy. Then we will consider overly confident procrastinators and learn how to gently deflate their overblown expectations.
REALISTIC OPTIMISM
A little optimism helps us persist when it comes to tackling difficult tasks. "Next time," you might optimistically think, "it will happen for me." Such a belief will keep you going much longer than a more realistic "Success is going to take about two or three dozen more tries." But it isn't obvious how to achieve this sunny disposition. Slogans and aphorisms such as "Be positive!" tend to be as ineffective as they are popular; they work best for people who are already optimistic and can actually make matters worse for people who aren't. But don't despair. After more than fifty years of research into developing effective options for improving optimism, researchers have identified three major proven techniques: Success Spirals, Vicarious Victory, and Wish Fulfillment.
SUCCESS SPIRALS
Whatever sport you are passionate about, from football to table tennis, your favorite athletic icon likely embodies the principle of success spirals. I am a fan of mixed martial arts, which I first started watching in the mid-1990s when I took up tae kwon do with a friend. Although I quickly sustained a knee injury, which stopped my martial arts practice in its tracks, I kept watching. I became fascinated by Royce Gracie and Matt Hughes, seemingly unbeatable fighters who once dominated the sport with their respective contributions of Brazilian jujitsu or wrestling skills and conditioning. Each victory, though, was a lesson to their competitors; eventually, these champions' abilities were countered or copied, and they fell. A titleholder of five years ago would likely be hard pressed to remain a contender today. One of the few champions who managed to endure is Georges St. Pierre. Remarkably, he attributes his present success to an old failure—he was knocked out by Matt Sera. As St. Pierre puts it, "I think that loss was the best thing that ever happened to me, and skill-wise I'm way better than I used to be before." In a rematch between the two the following year, the referee stopped the fight when Sera was unable to defend himself from St. Pierre's attacks.
What makes Georges St. Pierre such a resilient combatant is his history of overcoming adversity, which includes a hardscrabble Montreal childhood. His persistence enabled him to transform initial failure into success, which in turn gave him the confidence to continue fighting and to improve in the future. This is an example of a success spiral: if we set ourselves an ongoing series of challenging but ultimately achievable goals, we maximize our motivation and make the achievement meaningful, reflecting our capabilities. Each hard-won victory gives a new sense of self and a desire to strive for more. It is similar to the way Polynesian explorers colonized the South Pacific. From their home port they saw in the distance signs of a new island—a new goal—reachable if they made the proper provisions. Setting sail, they eventually made land, only to see another distant island from their new vantage point. Every step forward is enabled by the step just taken.
For those who suffer from chronic discouragement and expect only failure, success spirals offer a way out. Initiating them is the trick, as everyday living doesn't easily provide a structured and confidence-building series of accomplishments. However, great opportunities are available: wilderness classes and adventure education. Much like tribe members in a season of _Survivor,_ participants from management trainees to juvenile delinquents go on outings where they are challenged to overcome extremely difficult tasks with the help of inspirational guides. Outward Bound is the longest-running and most popular of these wilderness programs. In small groups, participants complete demanding expeditions on land or sea that can involve rafting, sailing, rock climbing, caving, orienteering, or horseback riding. Problem solving and personal responsibility are built in; individuals have to make key decisions, both before (what to pack?) and during (which way and how?). As a hundred studies have concluded, these wilderness programs improve self-concept, particularly self-confidence.
One of the keys to the power of such programs is that participants leave with a vivid success experience they can hold on to—there's nothing vague about crossing a river or climbing a mountain or figuring out how to deal with the unexpected. Personal stories of triumph can bolster people's spirits for years to come. "I did it!" translates into "I can do it." In follow-up assessments, wilderness program participants report that their self-confidence kept growing; having accomplished in the wild tasks they thought they couldn't possibly do, they set higher goals for themselves at home. This is the essence of a success spiral: accomplishment creates confidence, which creates effort resulting in more accomplishment.
Parents can start these success spirals in their children. Structured extracurricular activities that provide a circle of encouragement and a venue for achievement can increase a child's academic achievement and self-esteem as well as reduce drug use, delinquency, and dropping out. In particular, scouting provides an almost textbook recipe for creating tangible challenges that promote feelings of confidence. With the motto "learning by doing," the Scouts reward a progressive series of tasks with proficiency badges that recognize each accomplishment, culminating in the coveted super-scout Baden-Powell Award. Building a fire, setting up a tent, camping out, and cooking a meal for the group are all accomplishments kids can tell their parents about and—more importantly—remember themselves. Such success stories gradually build into a narrative that helps a child face the next challenge.7b
Here is a personal example of a success spiral in action. A close friend of mine has a son with self-confidence and anxiety problems: since he doesn't expect to succeed, he gives up quickly. So, his parents enrolled him in martial arts at a very strict tae kwon do dojo. It took the boy several attempts to get his yellow belt, but eventually he did. This turned out to be the pivotal experience that changed the course of his life, and it wasn't because he became better at fighting. Every time he was tempted to give up in other areas of his life, especially school, his parents reminded him of how he had to persevere to get that yellow belt and how good it felt to receive it in the end. Having overcome obstacles in the past, he now routinely strives to overcome any new ones that arrive.
As adults, you might not have the time to try Outward Bound or share my passion for martial arts, and you are definitely too old for the Scouts. No worries; there are plenty of other options to create a success spiral. The secret is to start small and pay attention to incremental improvement, breaking down large and intimating tasks into manageable bits. Like the old adage about how to eat an elephant—one bite at a time—you carve difficult projects into a series of doable steps, purposefully planning in some early accomplishments. If you don't feel up to writing a whole report, find a small portion you do feel capable of. Could you do the headings? Perhaps there are a few apt quotes to set aside? How about finding a few similar pieces to inspire you or to provide direction for organization? If you can't run a mile, then run a block. Stop when you've done that and next time try two blocks. Keep note of your progress, and watch how quickly you get to a mile. Nobody has to know about your small successes; keep them as your own happy secret and let them encourage you. The trick is taking the time to acknowledge incremental change, perhaps by recording your performance in a daily log.
Remember, there is always a path toward progress, no matter how small the increments. The better you are able to recognize subtle advances toward your goal, the more likely your confidence will continue to grow. Success breeds success.
To help you put this into practice, throughout this chapter and the next two, I've included sections called _Action Points._ These sections give you pointers about how to put what you have read directly into action, easily and without delay. Here is the first.
**_1. Action Points for Success Spirals:_** Think of an area of your life of real interest and then strive to improve just a little beyond your present skill set. As your confidence builds, you can also try exploring life outside your comfort zone. Consider this list (and add to it):
• Volunteer for more responsibility, either at work or in your community. If it involves hard physical work, like building houses for the homeless, all the better. Those sore muscles will remind you of your effort and your success.
• Travel to a place you've always wanted to go but thought you never would. Give yourself bonus points if you don't speak the local language.
• Try an adventure course such as white-water rafting, mountain climbing, bungee jumping, or skydiving.
• Learn a new skill. Sign up for a class in cooking, kickboxing, photography, or music. As you advance, pay attention to the small improvements in your skills and recognize them as victories.
• Challenge yourself by pushing an old hobby to a new level. If you are a runner, train for a race; join an amateur sports league; or tackle the harder solos in _Guitar Hero._
• Break down the tasks that daunt you into smaller and smaller pieces. Keep formal track of your progress. Count your successes.
Vicarious Victory
When I was a child, zoos were made up of cages, not habitats, and animals were truly captive. My father once took me to see the elephants. A mother elephant and child were on display side by side, both of their right hind legs secured to the ground. A large and heavy chain limited the baby, but the mother only had a slender rope. "Why Daddy?" I asked. "Shouldn't the big chain be around the big elephant?" No, he explained to me, the younger elephant needs the bigger chain because it is still struggling to become free. Eventually, it will accept that the chain won't break and, like the mother, it will stop trying. Once the baby elephant believes that it can't escape, the flimsy rope will be as effective as any cage.
Though I told it in the first person, this is a motivational story I've heard many times. Its implication is that we have untold strength, but that we were broken and tamed at some point and we don't realize how easily our potential could be regained if only we tried. I find it almost impossible not to be stirred by it, longing to break my own metaphorical ropes. There are many other motivational stories with this capacity to give us vicarious victory—from King Henry's "St. Crispin's Day" speech to Winston Churchill's "We Shall Fight on the Beaches." The most powerful of these are biographies of successful people that you can relate to.
Consider the effect one such story had on entrepreneur Kaaydah Schatten. Despite being raised in profound poverty by alcoholic parents, today she is a multi-millionaire and international franchise owner, a transformation she partly attributes to early inspiration. At a young age, Schatten read the life story of Catherine the Great and, seeing a common thread with her own heritage—Kaaydah is of a royal line, being the hereditary chieftain of the Quakiutl tribe—she adopted Catherine as a role model. To reap a similar benefit, perhaps you too can find the right story, another's life history that resonates with your own and speaks to your potential.
But people with extremely low self-confidence may need something stronger than inspirational stories to help them take their first step. Pessimists tend to put down any personal victories with a stream of negative self-talk: "Anyone could have done that," "It was all luck," or "It won't happen again." They need active forms of encouragement to believe that their success is due to their own effort: that when they try, good things happen. We normally absorb encouragement of this kind through social support, peer groups, and role models. From adolescence on, our peer group is a determining factor of our own development. Hang with the wrong crowd and they can hold us back. Hang with the right crowd and their successes can inspire us to think, "If they can do it, I can too!" Attitudes are catching, so you would be smart to hang out with groups of upbeat people. The social group we associate with helps cement our own view of what is possible and what we ourselves should strive to be. Giving up or continuing to strive—both are contagious.
A few groups seem particularly well structured for fostering a positive spirit. Service clubs like the Elks, Masons, Rotarians, or Shriners have millions of members worldwide, all bent on doing good work for their communities, but your options don't end there. My wife goes to a local Calgary group, the Famous Five, which holds women's leadership luncheons. I'm indebted to Toastmasters, a club that promotes public speaking and is endlessly encouraging and welcoming. You can even start a group yourself. Benjamin Franklin, for example, labeled his friends the Junto or the Leather Apron Club. Every Friday night, they would have a few beers at a pub and discuss how they could help their community.
**_2. Action Points for Vicarious Victory:_** Seek inspiration from stories or, better yet, from social groups. It is easier to believe in yourself if you are surrounded by others who believe in themselves—or you. Here are some suggestions:
• Watch inspirational movies. Here are a few I've seen: _Men of Honor,_ _My Left Foot,_ _Apollo 13,_ _Invictus,_ and _Hotel Rwanda._
• Read inspirational biographies or autobiographies. The most effective will resonate with your own background, so use the bookstore staff to help you find an appropriate book. For example, if you are a chef, read _Humble Pie_ by Gordon Ramsey, in which he speaks of his hard upbringing.
• Listen to inspirational speakers. Great athletes, heroes, and entrepreneurs regularly speak about their experiences. Seek them out.
• Join a community, service, or professional association. By hanging out with people who are trying to better themselves or the world around them, you will be infused with optimism.
• Start your own support group. As long as it contains a circle of mutually encouraging friends, it can be your running clique, your religious study group, or in the case of Ben Franklin, your drinking buddies.
WISH FULFILLMENT
Professional athletes often use visualization to achieve their goals. Before going to sleep every night, they imagine the perfect golf swing or triple axel landing. The detailed mental recreation of a performance engages mirror neurons that engrave the act in your brain almost as deeply as if you were actually practicing it. Visualization can also combat procrastination through the technique of _mental contrasting._
The expert on mental contrasting is Gabriele Oettingen from New York University, who has made this technique a cornerstone of her psychology career. Begin by imagining what you want to achieve. If it is a car, visualize yourself behind the wheel, cruising for all to see. If it is a job, see yourself in that dream career. Have you got a good mental picture? Good.
Now here's the all-important second step. Contrast where you want to be with where you are now. Visualize that dinged-up rust-bucket you drive or your dumbed-down joke of a job with its paltry paycheck. The result will be that your present situation becomes framed as an obstacle standing in the way of your dreams. Mental contrasting doesn't create optimism but it maximizes optimism's motivational benefits, creating energy and effort as well as jumpstarting planning. People who practice mental contrasting almost immediately start pursuing their dreams, putting a crimp in procrastination.
What happens if you forgo the second step and just focus on the positive fantasies alone? _Creative visualization_ advocates just that. It involves creating vivid and compelling pictures of your heart's desire, with the aim of drawing this vision toward you. But Oettingen, who has researched this for twenty years, finds that such fantasies tend to have the _opposite_ effect than advertised; they sap motivational energy.7c The only wealth created by creative visualization is a rich fantasy life. Whether the task is preparing for exams, getting a job, recovering from surgery, smoking less, dating an attractive stranger, or improving personal relationships, she found that the worst-performing group used positive fantasies alone. You are better off not using the technique at all.
**_3. Action Points for Wish Fulfillment:_** Fans of creative visualization don't have to stop what they are doing; they just need to add to it. Keep with the affirmations, the personal mission statements, but afterward reflect on where you really are. Here is a step-by-step walkthrough to make wish fulfillment work for you:
• Sit down in a quiet place and clear your mind. Think about the life you want for yourself.
• Break off a manageable piece of this future by focusing on just one aspect you desire. It may be a relationship, a job, a home, or a healthy body.
• Elaborate on all that makes this mental picture attractive to you. You can use a daily diary, create a collage of images, or just spend some quiet time concentrating on it.
• Then _mentally contrast_ this future with where you are now. Focus on the gap. Put the same emphasis on vividly reflecting on this discrepancy as you did on imagining your idealized future.
• If, after mentally contrasting, you remain optimistic about realizing this ideal future, you will find more motivation to pursue your goal. Procrastination will disappear as you start actively closing the gap between where you are now and where you want to be. You know what to do and have the drive to do it.
Fantasy Land
Overconfidence is just as problematic as under-confidence. Forty-one days before the start of the Iraq war, the U.S. defense secretary, Donald Rumsfeld, estimated that it could "last six days, six weeks. I doubt six months." The allied troops would surely be greeted as liberators. The cost? It was supposed to be fifty to sixty billion dollars, not almost a trillion. Regrettably, military overconfidence leading to lengthy and unprofitable wars is quite common.
In the business world, overconfidence creates a host of similar problems: mergers aren't usually finished within time or on budget. Overconfidence, for example, contributed to the Concorde fiasco; despite mounting evidence that it wouldn't be profitable, Air France and British Airways continued to pursue its development. Entrepreneurs often exemplify this point, reflecting Jeffrey Vancouver's observation that optimism has a sweet spot (see the graph on page 118). Confidence is definitely needed to start a business, and entrepreneurs tend to have more of it than the rest of us. Just as the graph predicts, however, overly confident entrepreneurs tend to fail. When confidence becomes supersized and unearned, it fuels procrastination because the overconfident tend to discount serious problems and subsequently delay responding to them.
Certain philosophies, such as the philosophy of Pangloss, a character created by Voltaire to epitomize naive and unrestrained optimism, exacerbate the problem of overconfidence. Over the last few centuries, unbounded positive belief has formed the basis of several success systems, such as Phineas Quimby's _New Thought Movement_ or Norman Vincent Peale's _Power of Positive Thinking._ 27 The best modern example of Panglossian thinking is _The Secret,_ a book (and movie) developed by Australian television executive Rhonda Byrne. According to Byrne, thoughts have magnetic energy that draws like to like by a Law of Attraction—think positive and the positive will come toward you. There are millions of followers of this philosophy but I am not one of them. The Law of Attraction separates positive belief from action, leaving belief free-floating and unconnected. It changes the story of the _Little Engine That Could_ from "I think I can" to "I think it will." That's a big difference.7d
To prevent ourselves from falling into over-optimism, we need a teaspoon of pessimism. As Freud put it, we need to activate the _reality principle:_ to confront the reality of the situation when we are seeking the best way to achieve our goals. Invoking the reality principle is a sign that we have outgrown our childish and impulsive ways and can acknowledge the price we must realistically pay for our dreams. This entails imagining what could go wrong and how you would prevent or mitigate potential pitfalls. Neil Armstrong, the first man on the moon, used this principle during his lunar escapades. "Well," he would say, "I think we tried very hard not to be overconfident, because when you get overconfident, that's when something snaps up and bites you."
In business, this reality check is a standard step of _crisis management._ Adages to this effect are well worn: "If you fail to plan, you plan to fail," or "An ounce of prevention is worth a pound of cure." We can apply this principle to procrastination in two ways: Plan for the Worst, Hope for the Best, and Accept That You're Addicted to Delay.
PLAN FOR THE WORST, HOPE FOR THE BEST
Very few succeed in major life reforms on the first try; most of us need multiple attempts. Take New Year's resolutions, for example: it often takes five attempts before vows last for more than six months. I myself sweated out several attempts to quit smoking before I successfully put cigarettes aside. For more serious alcohol or drug problems, the same need for repetition applies. Whatever you do, don't wallpaper over this painful and repetitive process; wishful thinking will only increase your procrastination.
Psychologists Janet Polivy and Peter Herman describe such dysfunctional over-optimism as the _False Hope Syndrome._ Overconfidence about the size, speed, and ease of major life changes is associated with lower success rates. If people have unrealistic, supersized expectations, they discount modest achieve-ments. They _only_ lost ten pounds. They smoked at a party. They skipped the gym for a week. They see these as "failures" and they lose momentum—they are more likely to give up and feel worse than before they made the resolution to change. This disillusionment is common, as the self-help industry instills incredibly high expectations and promises. If you are in the vast majority who don't transform as quickly as advertised, you feel that your failing is personal rather than a failure of the program.
Success requires balancing optimism with realism: it will be a hard slog and there will likely be lapses, but you can get back on track. When I quit smoking, I paid attention to two variables: how many cigarettes I smoked during a binge and the length of time between binges. As long as the first number went down and the second went up, I was getting somewhere. Rather than believing you can entirely and easily beat the problem of procrastination, believe that you can beat it down. Instead of aiming to never procrastinate, aim to start just a little bit earlier on more and more projects. Modest gains can have significant outcomes. I have some students who start studying for exams only forty-eight hours beforehand, but if they started one day earlier, they'd increase their cramming time by 50 percent. As the author Louis L'Amour counsels: "Victory is won not in miles but in inches. Win a little now, hold your ground, and later, win a little more."
**_4. Action Points for Plan for the Worst, Hope for the Best:_** Life won't always go your way. Rather than expecting perfection, anticipate difficulties and setbacks. When they inevitably occur, you won't be as easily derailed. Here is how to inject some healthy pessimism into your plans.
• Determine what could go wrong to distract you on the way to your goal. Reflect honestly on your past experiences and seek counsel from others who have gone through similar difficulties. For example, take a look at the online forums about procrastination.
• Make a list of the ways you habitually procrastinate, and post it where you work.
• Avoid these pre-identified risky situations. For example, if text messaging is your problem, turn off your cell phone or PDA before you get to work.
• Develop a disaster recovery plan ahead of time. If you stumble and start skipping the gym, what emergency cord can you pull? Do you have a friend you can go to for a pep talk? Can you hire a personal trainer to get you back on track?
• If you find your motivation derailed, use your recovery plan. Focus on reducing the depth and duration of your motivational lapse.
ACCEPT THAT YOU'RE ADDICTED TO DELAY
When procrastination gets really serious, baby steps might not work. You may need a heavy-duty technique, one that you borrow from the Alcoholics Anonymous' twelve-step recovery program. AA's first step is: "We admit we are powerless over alcohol." Many find this admission a strange start to sobriety, since it runs counter to any notions of optimism and increases the chance that once you've had one drink, you will abandon all self-control and go on a bender. Still, acknowledging powerlessness over alcoholism or procrastination can paradoxically lead to the elimination of both.
Indeed, it is possible to improve self-control by embracing your pessimism. How does this work? Well, truly acknowledging that any single failure of willpower inevitably leads to the collapse of all your self-control gives you far more motivation than believing that occasional lapses can be safely contained. Abstinence is a preferable antidote to rationalizing every slip and indulgence. Since one drink, a lone candy bar, or a solitary cigarette is in itself inconsequential, we can trick ourselves into downplaying their significance. If we indulge in thinking one additional day of delay is always all right, then tomorrow's day of action never comes. Maury Silver and John Sabini, who researched procrastination in the 1970s, describe this problem in terms of the prototypical student procrastinator:
Now suppose you had to decide what to do for just the next five minutes—either work on the paper or play one game of pinball. The paper can wait for one game—there is little long-term cost. In the short run, five minutes of pinball is far more pleasurable than five minutes of paper writing, and after all, how much of a paper can you do in five minutes? Pinball is the obvious choice. The game is over so you must decide about the next five minutes. The situation is only trivially changed, so you will reach the same result. Once you've taken the possibility of pinball seriously and fragmented your night into five-minute intervals, you may be doomed to play until you run out of money, the machine breaks, someone meaner than you wants to play. The trouble is, even five minutes has a real cost to the paper. Because a single game of pinball is brief, it is particularly seductive.
When it is time to decide whether to work or procrastinate, there is no shortage of excuses for giving in to temptation. Conditions will be better tomorrow, so I will start then; I'll work better after I get something to eat; it will be easier if I clean up first; I'll start after I finish this level, finish watching this show, finish this e-mail; this party/episode/diversion is especially good, so it would be unfair if I missed it; I deserve a break because I've been working so hard already; others are procrastinating, so why shouldn't I; it's just this once so it doesn't count; there's still plenty of time; and finally, it's already too late to make a difference so there's no need to start now. These are all justifications after the fact. Their only purpose is to assuage your anxiety and guilt.
There is only one surefire way to stop you from justifying your way into procrastination. Follow the Victorian era's greatest maxim: "Never suffer an exception to occur."7e This is the same advice given by Alcoholic Anonymous. You buttress your commitment to early starts by believing that any slip will be catastrophic, that the initial step toward procrastinating is merely the first link in an endless chain. The specifics of tomorrow will be much the same as today: you will be tempted to incur a small but cumulative cost to gain a moderate immediate pleasure. If you decide to delay even once, your decision will be replicated daily and the consequences will grow. It is as these verses from Goethe's masterpiece, _Faust,_ indicate. Asking for more time can be a deal with the devil:
_Lose this day loitering—'twill be the same story
To-morrow—and the next more dilatory;
Each indecision brings its own delays,
And days are lost lamenting o'er lost day,s
Are you in earnest? seize this very minute—
Boldness has genius, power and magic in it.
Only engage, and then the mind grows heated—
Begin it, and then the work will be completed!_
**_5. Action Points for Accepting That You're Addicted to Delay:_** If you find yourself chronically procrastinating, consistently able to fool yourself into extended delays by finding moment-by-moment excuses, this may be the technique you have been looking for. Procrastination has a very deep hold on you, and to defeat it you need to accept this humbling fact.
• Take a moment to reflect upon how many times you have talked yourself out of your plans and into trouble. Start keeping a daily log to track your procrastination habits.
• Acknowledge that your biggest worry is your own weak will, that you _will_ try to fool yourself into thinking "just this once."
• Accept that the first delay allows you to justify all the subsequent ones. By doing so, you will be far less likely to take that first step.
LOOKING FORWARD
This chapter is primarily for the low-expectancy Eddies, those who need just a little more confidence to reach their potential. Back in chapter 2, Eddie stopped believing in his ability to make the sale and because of this self-doubt, his failure became inevitable. If he had attended more to his progress, he could have initiated a success spiral. If he had supplemented this momentum by attending a sales support group, creating a little vicarious victory, he could have maintained a career in selling. You might have also stopped believing in your ability to advance your career, your personal life, or your health. You make plans to change but no longer truly believe in your ability to pull them off. Take a moment to review your results from the self-assessment quiz in chapter 2. If you scored 24 or below on the Expectancy scale, maybe you, like Eddie, should take a closer look at the techniques presented here.
On the other hand, there are a few of you who are overconfident, and you may be equally at risk. Confidence or optimism turns out to be a lot like vitamin A: too little of it will lead to blindness but too much of it can kill you. The trick is to find the sweet spot between being pessimistic and being happy-go-lucky, a place where you have faith in your ability to succeed but not so much faith that you fail to put in the effort. Whether you need your positive expectations fired up or dampened down, you are in luck. All the techniques shown here are rock-solid and scientifically sound. They will start working immediately and you will get better with practice. Believe me.
[Chapter Eight
Love It or Leave It](contents.xhtml#ch_08)
FINDING RELEVANCE IN WORK
_If time flies when you're having fun, it hits the afterburners when you don't think you're having enough._
JEF MALLETT
To warm up my students for a class on motivation, we play a game called _My Job Is Worse Than Your Job._ Since misery loves company, it is a lot of fun. We try to find the worst past employment experience in our group and then deconstruct the job to determine why it was so terrible. The room echoes with groans of sympathy as the students talk about summers spent shoveling pig manure or their scorching, exhausting, and mosquito-plagued months spent tree planting. But invariably, the jobs voted "worst" aren't the physically demanding ones; the worst are the mind-numbingly boring ones. For example, one bright young man used to spend his potential straightening cardboard boxes when they occasionally became misaligned crossing conveyor belts. I was once a lifeguard at a waterslide park, assigned to watch the same few meters of splash zone for time without end.
These sorts of jobs turn us into clock-watchers who wait for each agonizing minute to pass. Since every aspect of the job has been mapped out, we are left with little to say about when or how to do the work, little chance for initiative or innovation. We must repeat the same actions endlessly. Are we doing a good job? Nobody really knows except when we slip up. Movies such as _Modern Times_ and _Office Space,_ in which the protagonists escape such employment purgatory, become cult classics. More recently, the award-winning television show _The Office_ has been a success in half a dozen versions around the world. Part of the show's charm is its ability to demonstrate how humanity manages to rise above the squelch of meaningless work. Factory and office jobs, however, weren't always like this.
We owe the "modern" workplace largely to Frederick Winslow Taylor, the originator of scientific management. Before he came along, the majority of work was skilled and somewhat immune to direct management, performed by craftsmen who learned their trade through years of apprenticeship and specialization. Managers couldn't easily supervise such artisans when they had little idea of how they did their jobs and it wasn't in the workers' interest to tell them. Taylor's breakthrough was to fragment work into more easily managed elements—simple routine tasks lacking autonomy. When his system, Taylorism, was first implemented, back in the late nineteenth and early twentieth century, it was considered an abomination that lobotomized the human spirit, robbing work of its meaning and pleasure. People hated it so much that its introduction at the U.S. government arsenal at Watertown, Massachusetts, spurred a strike that led to a special investigation by the U.S. House of Representatives. The congressional committee concluded that man should naturally resent "the introduction of any system which deals with him the same way as a beast of burden or an inanimate machine," and took action to prevent the adoption of Taylorism in governmental facilities. When the industrialist Henry Ford implemented a similar system, employee turnover at his automobile factories increased almost tenfold; workers stayed barely a month before leaving. Taylorism, however, had an ace in the hole: it was efficient and it was profitable. Though Ford eventually had to double wages to fill his factory jobs, the improvement in efficiency allowed him to simultaneously increase his workers' pay and cut the cost of his Model T cars by almost half. In the end, Taylorism helped to give rise to cheap goods and a wealthy middle class that could purchase them. Assembly line work, on the other hand, still often sucks.
The tasks we hate are among those we tend to postpone. Because Taylor's system leads to standardized, repetitive, and rigidly controlled tasks, hating work can be a chronic state of being, an inevitable result of jobs being designed around mechanistic instead of motivational models. What can we do about this? We might dream of returning to a time when what we desired to do and what we needed to get done were the same, but that's not realistic. Even if you are your own boss and can dictate your own terms, you still need to accomplish some tasks that are no fun, and these are exactly the jobs that people put off. Perhaps it's time to think about tricking yourself into getting them done. As the title of this chapter goes, you love it or leave it—until later that is.
GAMES AND GOALS
Whoever we are, we are likely to put off doing whatever we find excruciatingly dull. Boredom signals that what we are doing is irrelevant, and so the mind slides off the task. It makes sense, then, that procrastinators are much more likely than non-procrastinators to perceive life's daily tasks as drudgery. Of all the boring tasks that fill the world, the one that tops most people's hate list is routine paperwork. The busywork—filling in timesheets, submitting expense reports, and supplying the data that companies and governments endlessly require—seems pointless, even when it isn't. Remember Michael Mocniak, that general counsel who got fired for putting off completing $1.4 million worth of invoices? Fortunately, however, boredom isn't inherently part of any job—anything can be made more interesting simply by the way we treat it. Tom Sawyer, for example, managed to get the village boys to _pay him_ for the privilege of whitewashing his Aunt Polly's picket fence. How? By insisting that they couldn't help and making them envy an unenviable chore. Here are a few effective techniques for turning leaden tasks into golden ones.
To relieve task boredom, try making things more difficult for yourself. (Don't overdo it though—when tasks are too difficult, frustration can take hold.) Finding the balance between the difficulty of your task and your ability to do it is a key component for creating _flow,_ a state of total engagement. Flow states don't happen naturally, since many jobs are structured around an unvarying level of difficulty, whereas most workers' ability increases with practice. When work is new and its difficulty exceeds your ability, anxiety rises as you fumble to perform. Then, as you improve, the work can become engaging, but this motivational fit is momentary. When you have gained true mastery, boredom becomes the rule; you have done it all before. To prevent this descent into dullness, game playing becomes a common strategy. Set your own standards, create your own feedback, and try to beat your score. Can you do it in half the time? How about one-handed? Eyes closed? The comedy group Broken Lizard devised a film, _Super Troopers,_ around this theme: five Vermont state troopers find ways of weaving games and shenanigans into their work, making their days passable.8a One elderly potato-chip factory worker kept her days full by collecting unusual chip deviations that resembled famous people. Competitive swimmers keep boredom at bay by imagining sharks in the pool water.
By the way, I can't help but notice that you are continuing to read this book despite the shelves of other books to choose from. I'd guess that procrastination is a problem you or someone in your family experiences, and that as a consequence you are finding these pages personally pertinent and interesting. You could put the book aside, but relevance keeps you reading. This is equally true for other actions and tasks: the risk of procrastination diminishes when tasks are relevant, instrumentally connected to topics and goals of personal significance. Actions that don't fit self-determined and self-defined goals are _amotivational._ 11 They are imposed upon us and we reluctantly comply. At my university, we have many managers voluntarily coming to school each evening after working a full day at the office in order to earn their MBAs. I imagine their motivational chain of objectives goes something like this:
• They read the book so they can prepare for a test.
• They prepare for the test so they can ace the course.
• They ace the course so they can get the grades.
• They get the grades so they can receive an MBA.
• They get the MBA so they can get a promotion.
• They get the promotion so they can make more money and enjoy their work.
All the sub goals in this hierarchy are predicated on the last—getting a promotion so as to enjoy more interesting work. You need a string of future goals that you find intrinsically motivating to hook your present responsibilities onto. Break this motivational chain at any point and you leave it anchorless; goal commitment is negligible and, like a balloon, attention floats away with every waft.
The relevance factor is a major reason why procrastination decreases with age. As we mature, we increasingly connect the dots, seeing reasons for what we once thought was pointless. If you lack large resonating goals—life tasks—then your purpose now is to find them. It is a big world and you need to experience at least some of it. In the meantime, I will give you a generic goal that will inject any task with more meaning. Frame what you are putting off as a test of your willpower, and, as a buttressing side bet, tell your friends of your intention to start early. The goal of staying true to yourself and portraying your consistency to others will increase the pleasure of sticking with the task and resisting tempting alternatives. For example, Barack Obama's public announcement that he intended to quit smoking helped him put cigarettes aside, with only the occasional lapse.
To further maximize your intrinsic motivation, frame your long-term goals in terms of the success you want to achieve—an _approach_ goal—rather than the failure you want to prevent—an _avoidance_ goal. People who generate positive long-term goals go on to procrastinate less and perform better. Advice like "Don't fall!" to the precariously balanced or "Don't forget the lyrics!" to singers increases the likelihood of the very outcomes they profess to prevent. Consequently, "I really want this book to get good reviews" is better than "I hope not to be openly mocked for my writing." Thinking "I want her to like me" is better than "I don't want to be rejected again." Almost any goal can be flipped from avoidance to approach, from what you don't want to happen to what you desire. Just look at the following table:
AVOIDANCE GOALS ARE . . .
1. Not staying home
2. Not being tired
3. Not staying in a dead-end job
4. Not struggling with bills
5. Not leaving the glass empty
6. Not beginning late
APPROACH GOALS ARE . . .
Exploring the World
Having energy
Finding your calling
Making more money
Filling the glass up
Starting early
On which side of that table do you usually reside? Do you focus on not eating treats when dieting (an avoidance goal) or on eating healthy meals (an approach goal)? Do you think about not procrastinating (an avoidance goal) or about starting earlier (an approach goal)? I thought so. So the lesson is: Stop making avoidance goals
**_1. Action Points for Games and Goals:_** It is said, at least by people quoting Shakespeare, that there is nothing good or bad in this world but thinking makes it so. The Bard is exaggerating a bit but he is essentially right. Frame your tasks appropriately; the way you view them significantly determines their value.
• Avoid boredom by making tasks more challenging. Games can be handy here, with the rules limited only by your imagination and common sense. For example, when you are competing against your colleagues, almost any task can become a race to finish first or to get the most work done. In competing against yourself, you could also try to finish the task in fewer hours.
• Connect tasks to your long-term goals, to what you find intrinsically motivating. For example, if you are a social person, you could frame cleaning your house as "Providing an inviting home for family and friends."
• Frame your goals in terms of what you want to achieve rather than what you are trying to avoid. For example, think "I want to succeed" instead of "I don't want to fail."
ENERGY CRISIS
When I moved to Minnesota to work on my PhD, my wife, Julie, and I managed to snag a dream apartment: a converted warehouse loft. Rent was low—a key feature on a student income—and the location was close to both my university and her workplace. Even better, only a wide golden field separated us from the Mississippi River. Nothing, however, is all good. That field was full of ragweed, which triggered my hay fever. My allergies had never been bad enough to warrant medication, but after my third box of tissues, I quickly opted for an over-the-counter allergy drug. Suddenly, I couldn't get out of bed in the morning without repeated sharp prods from my wife. Work became incredibly difficult, like running through deep powder snow. What was wrong with me? Was I depressed? Overwhelmed? Finally, I read the back of the medication box: "May cause drowsiness." Later I learned that most allergy medications contain antihistamines, which have the same active ingredient as the sleep-aid _Nytol._ I was taking the equivalent of sleeping pills and no wonder I was having trouble tackling tasks.
Whether tiredness is drug-induced or not, being too tired is the number one reason given for procrastinating; 28 percent of people claim, "Didn't have enough energy to begin the task" as the cause. When you are tired at the end of day, after your job has already got the best part of you, cleaning out the garage is the last thing you are going to do. Fatigue increases task-aversion, saps interest, and makes the difficult excruciating. Whether it is an exhausted muscle or an exhausted mind, you can feel the burn of being burnt out. When you're tired, it becomes even harder to force yourself to tackle jobs you dislike. Burnout saps your willpower because the exercise of will—self-control and self-motivation—takes energy. Whenever you have to suppress a competing impulse, you exhaust your energy stores and willpower. If you have to stop yourself from eating that cookie, you deplete your willpower. If you suppress an emotion, like laughter or anger, you deplete your willpower. If you are coping with stress, your willpower depletes. This decrease in self-control occurs after you make difficult choices, one reason why clothes shopping can be an exhausting ordeal if you lack an innate fashion sense. Those bizarre outfits that languish in your closet were likely purchased toward the end of a shopping trip.
To some extent, we should accept that we don't have infinite mental energy and acknowledge our motivational limitations along with our physical ones. Everyone understands why you can't run back-to-back marathons but it's not so obvious that equivalent internal struggles can be just as onerous. Perhaps we have trouble with procrastination because we demand too much of ourselves in a day, and it's possible that pursuing a less stressful, slower paced life would help us get energized. Regrettably, we don't always have a choice. So what can we do when our "get-up-and-go" has "got-up-and-gone"?
Recognizing that our energy reserves are limited, we can strategically refuel and allocate them. You don't want to ever completely exhaust yourself; when you are sapped, you are likely to give in to your impulses. That is why dieters shouldn't let themselves get hungry, because they are likely to satiate themselves with the simple carbohydrate and fat combinations that saturate our world. Ironically, sweet treats will restore willpower just long enough for you to regret the indulgence. So shield yourself from distractions by using moments of strength to enact other longer-lasting self-control techniques, especially distancing yourself from temptations. This is the beauty of offices. Once purged of temptations, an office can become a temple of productivity, a place where following up on your intentions to work takes a lot less willpower.
Facing the challenges of writing a report at the end of the day, when you're already wiped, isn't the best idea either. You want to tackle it when you have the most zip, and when that occurs depends upon your circadian rhythm. Some of us are morning larks, relentlessly chipper and active early in the morning, filling gyms in the pre-dawn hours. Others are night owls, slow starters whose energy levels peak later in the day. Night owls are more likely to be procrastinators, with a chronobiology best suited for after-hours endeavors; forcing themselves into an unnatural schedule, they gulp down caffeine in the morning in order to wake up, and alcohol in the evening to wind down.
Whatever your rhythm, schedule that report writing to start a few hours after you wake up; it's when your mind operates at maximum efficiency, a period that lasts about four hours. If you woke at seven in the morning, for instance, your peak performance likely occurs between ten and two, not really that wide a window. But if you clear your desk, turn off your e-mail, and shut your door for those hours, you can get an amazing amount of work done. You can extend this efficiency phase with a brief nap, twenty minutes or so, but if you're in an office environment, that's usually not possible. Still, a quick walk around the block can also refresh you around lunchtime. In any event, it's smart to start shifting toward less creative, more routine work in the late afternoon: you are losing IQ points by the hour. When you finally get home, the only decision you might be able to effectively make is whether to wind down with a glass of wine or a pint of beer. The good news is that the timing is perfect; twelve hours after waking is when your liver best metabolizes alcohol.
Finally, a typical pattern that many of us fall into when stressed is to cut back on exercise and sleep and make up for them with diet and stimulants, usually sugar, caffeine, and nicotine. In the short-term, this can be an effective energy strategy, but in the long-term, it will leave you worse off. Not only do stimulants lose their effectiveness with repeated use, but they can make exercise and sleeping even more difficult to achieve. As quality of concentration is gradually swapped for quantity of effort, you work longer hours while producing less, eventually working late into the night when you should be sleeping. These are bad energy habits.
You probably already know what you should be doing to solve these problems. Committing to a regular schedule of exercise has been shown to decrease procrastination. Since many people in North America aren't getting a good night's sleep, I also recommend you start learning about sleep hygiene, which prevents people from polluting their bedrooms with the stress of the day, maintaining it instead as a sanctuary for escape. Sleep hygiene is the only thing that worked for my wife, who comes from a family of chronic insomniacs.
**_2. Action Points for Energy Crisis:_** Being too tired is the top reason for procrastination. Your energy stores are both a limited and a renewable resource, so actively replenish them and allocate your efforts wisely.
• Reserve your morning and mid-day peak performance hours for your most difficult tasks.
• Don't let yourself get hungry. Graze on small nutritious snacks as needed.
• Make time for exercise several days each week.
• Make sleep predictable, going to bed at the same time each night with a regular wind-down routine.
• Respect your own limitations. If after all this, you still are too tired to tackle your responsibilities, try to cut back on your commitments or get help completing them.
YOU SHOULD SEE THE TASK I'M AVOIDING
The sun sets and long shadows disappear into the darkness. Eyes dilate to adjust, but still the blackness obscures: uncertainty shrouds us and anything could emerge. Vulnerable now to the limitless unknown, we feel a suffocating fear. With night comes the time of monsters. Pull the blankets over your head and don't say a word: this is about survival . . . at least it used to be. Like three-quarters of kids, I grew up afraid of the dark, a dread largely passed on from my ancestors. When nighttime was truly dangerous, that fear of ghouls and ghosts kept children quiet, stationary, and safe. Imaginary fears were an adaptive part of any culture. The Northern Inuit teach their children of the Qallupilluit, which kidnaps children who walk too close to cracks in the ice, while the Japanese have the Kappa, water creatures that eat urchins.8c Maybe we can conjure our own monster to scare off procrastination as well.
The technique of productive procrastination might employ such a monster. It is a well-established ploy, advocated by no less than Sir Francis Bacon, the seventeenth-century philosopher and statesman. He proposed that we try to "set affection against affection, and to master one by another; even as we use to hunt beast with beast." We see productive procrastination in action when people spend precious hours sharpening pencils, scrubbing stoves, or cleaning bedrooms as an imminent deadline towers over them. Though by all outward appearances they seem suddenly afflicted by obsessive-compulsive disorder, such procrastination isn't entirely a waste of their time. Things are getting done—though not quite the right things. Psychoanalysts would consider it an example of displacement, whereby we shift impulses into a related but less threatening outlet, like picking a fight with a friend after being upbraided by our boss. Behavioral psychologists would point out that we are willing to pursue any vile task as long as it allows us to avoid something worse.
Productive procrastination isn't perfect—it reduces the cost of dillydallying but doesn't eliminate it. Rather than doing nothing useful while avoiding the big project, you are at least taking care of minutiae, "robbing Peter to pay Paul." It isn't as constructive as tackling the real work, but it does clear your plate and puts you in a much better position to dig in when you're ready. Sooner or later, though, you will have to face that monster you have been avoiding.
**_3. Action Points for You Should See the Task I'm Avoiding:_** Don't let the perfect—never procrastinating—get in the way of the good—productively procrastinating. Meet your procrastination impulse halfway. By engaging in productive procrastination, you put off one task only to spur yourself toward tackling another.
• Identify a target task that you ideally should be doing now but have been putting off.
• Identify tangent tasks that also should be done and are _relatively_ more enjoyable than your target task. You might be putting these off too.
• Accept the trade-off of avoiding the target task by tackling the tangent tasks. When you eventually get to the target task, you will be in a better position to complete it.
DOUBLE OR NOTHING
We are all too familiar with guilty pleasures. You know, the ones you indulge in after a long day doing things for others, after the kids are fed and in bed, the dishes done, and you finally get an hour to yourself. You slip off your work clothes, step into a robe, pour yourself a drink, and watch . . . oh yes, reality TV. Ah, the sweet cerebral abyss of spoon-fed entertainment. We all have the ability to self-reward, whether it be with a trashy book, a bowl of ice cream, or a luxury purchase. So let's put this talent to good use.
A principal problem with procrastinators is that they tend not to reward themselves after completing a task, often failing to appreciate their own hard work. They give themselves no whispered kind word or planned treat after a task well done. Too bad, as such rewards are the easiest to implement and personalize. The specifics of soothing self-talk or a deserved indulgence will differ from person to person, but the effects remain the same. Whether your catchphrase is a silent "Atta boy!" or a "You go girl!" a little internal self-praise is a costless incentive for overcoming a challenging task. Similarly, whether it is a fine meal or a full vacation, a self-administered reward can pull us through the drudgery of work toward a project's completion. Even better, they offer motivational dividends, realized during subsequent endeavors.
This technique is called learned industriousness: people can learn to love their work. You see, the enjoyable emotions generated by self-praise and other rewards tend to creep backward into the effort itself. That is, activities take on the attributes of their goals and can become rewarding in themselves. Money is the principal example of this phenomenon, having been instilled with value by virtue of what it can later buy. Hard work, by virtue of the achievement it can later generate, can be similarly infused, making such effort rewarding in the moment. Consequently, successful people find themselves in a virtuous circle: the anticipated rewards from winning help make the work more enjoyable, and that enjoyment helps them to win. With the future flavoring the present, they savor victory long before it is realized. It is a very nice arrangement, but the trick is in how to get it started. It may take a number of effort-reward cycles before the effort itself takes on the taste of the later reward.
While waiting for learned industriousness to kick in, you can enhance the pleasure of work in a more direct way: blend bitter medicine with sweet honey. Try to find a compatible pairing between a long-term interest and a short-term impulse. If you combine an unpleasant task with one you find more enjoyable, the mixture may be enough to get you going. Getting together with a workout partner can spur you to exercise. Treating yourself to a specialty coffee can help you focus on your time sheets or your budget. But this method has its risks as well. Engaging a partner to help you finish a report or prep for an exam, for example, can degenerate into an evening-long bull-session with little learning to show for it. Still, the principle is sound. In the Adam Sandler movie _Billy Madison,_ the title character has to redo his entire schooling, twelve grades in twenty-four weeks, to receive a sizable inheritance. In desperation, he engages an attractive tutor, who for every correct answer he gives, removes an article of her clothing.
**_4. Action Points for Double or Nothing:_** Take the time to recognize and reward your progress. Though success itself will eventually make effort enjoyable, right now you can artificially graft a little pleasure onto most tasks.
• Make a list of rewards you can administer to yourself, such as self-praise, frivolous purchases, or a night out.
• Promise yourself these rewards upon completion of the task you have been avoiding.
• Consider ways of making tasks more enjoyable, such as listening to music, sipping a specialty coffee, or working with a friend.
• Make sure that what makes the work more enjoyable, like partnering, doesn't override the work itself.
LET YOUR PASSION BE YOUR VOCATION
Perfect work exists, tasks people would do even in the absence of a paycheck. One example is gold farming. Gold farmers are professional video game players who have become experts in massive multiplayer online role-playing games (MMORPG) like _World of Warcraft,_ _RuneScape,_ or _Star Wars Galaxies._ With their honed skills and long hours of play—at times, eighteen hours a day—they gain virtual gold and rare items that they then sell to other players for real cash. As documented by Ge Jin, a University of California PhD student and independent filmmaker, these professional gamers blur the line between work and play in a constructive way. Jin admits he was "shocked by the positive spirit there, the farmers are passionate about what they do, and there is indeed camaraderie between them." Most telling is what many gold farmers do in their spare time—they continue to play.
Apart from the problem of who would buy all this make-believe money, gold farming isn't and can't be for everyone. Still, it captures the Holy Grail of job design, marrying high performance with job satisfaction. And it illustrates that finding work you want to do is a major step toward avoiding procrastination. Being intrinsically motivated by your job means you are rewarded simply by doing it; no need to delay gratification here. This combination can make work almost addictive; motivation shoots upward stratospherically, souping up creativity, learning, and persistence. Speaking for myself, I love learning about motivation and I willingly work hard at it. Finding work you love is tricky, but let's try.
Finding your perfect job is at least as difficult as finding your soul mate. With almost 50 percent of marriages in our culture ending in divorce, the challenge is a tough one. With love, we seek the person who complements us; with work, we seek the job that could become a calling. In either case, a satisfying match is known as congruence, and it can be darn difficult to accomplish. The best predictor of love is familiarity brought about by physical proximity,8d a good recipe insofar as it keeps down travel costs while dating. At work too, we gravitate toward the best of our known options, not the best of all possible jobs. Expanding our world and improving our career choices is not a simple matter. We need to better understand ourselves and what different jobs can offer us, and then find a way to link the two.
For most people, finding themselves and their calling is an ongoing struggle. If we all went with our first impulse, the working world would be primarily composed of firefighters and ballerinas. If we followed the dreams of our teenage years, we would mostly be professional athletes, fashion designers, or rap stars. Ask college students, and many of them want a career in film. On the other hand, choosing sensibly to be a doctor or a lawyer doesn't always pan out either; these were the initial career paths of Graham Chapman and John Cleese before they created Monty Python's Flying Circus. Most of us have to search for a calling while we are already working, deepening the commitment to a current and perhaps inappropriate career path. We may need the help of a matchmaker or, as we call them in the world of work, a counseling or vocational psychologist. These professionals assess your personality as it relates to work, typically relying on an assessment tool that divides interests into six themes: realistic (doing), investigative (thinking), artistic (creating), social (helping), enterprising (persuading), and conventional (organizing). Jobs are profiled too, with firefighting falling under "realistic" and ballerina under "artistic." Vocational counselors will point you toward a variety of job choices, though it is up to you to at least try "dating" them. The assessment on the next page is my own, completed when I was seventeen years old. One profession clearly dominates my profile, one that requires a combination of investigative and artistic interests—a professor. I did not take these findings seriously at the time, but drifted about for a decade before finally coming to the same conclusion. Blame my strong need for autonomy.
As in seeking love, there is more involved in finding your calling than identifying what you desire. Though a certain job could be the one for you, your feelings may not be reciprocated. Some jobs are out of our league because they are already being pursued by an excessive number of applicants. Supply and demand is harsh, and there may not be a demand for what you supply. Fortunately, there are plenty of other jobs that you could like just as much. The O*NET program in the United States catalogues nearly a thousand jobs, identifying those that are in demand by employers and that fit your profile.8e
After accounting for your personality and for the job market, you will also need to consider your abilities. Can you do what the job requires? Firefighters and ballerinas need to be athletically gifted, ranking in the top positions of the physical fitness category. If you want to be a rocket scientist or a brain surgeon, you'd better be blazingly smart. Linking individual abilities, such as stamina or mental capability, with the world of work isn't easy. For example, I can tell you that if you are five feet tall, you shouldn't foster aspirations for a career in the NBA. But most of the time it isn't obvious whether you are following a dream or pursuing a lost cause. Just be aware that you want to find work that you not only love but have the capacity to excel at too.
**_5. Action Points for Let Your Passion Be Your Vocation:_** Not everyone has job mobility. Some are tied down by obligations and economic constraints and have to make choices based on security or availability. If you have the gift of choice, don't blow it! For the next little while, finding a compatible fit between what you do and who you are should be an ongoing occupation.
• Look at careers involving activities you love or like doing.
• Filter out all the occupations for which you don't possess (or aren't willing to learn) the necessary skills or abilities.
• Rank the remaining careers by what is in demand. The harder the economic times, the fewer your choices will be.
• If you need help answering any of these questions, find a reputable career service for employment advice.8f
• Start job hunting!
LOOKING FORWARD
In chapter 2, Valerie Without Value hated to write and put off her municipal politics assignment for so long that what she produced was second-rate. Instead of working, she indulged in the far more pleasurable acts of texting her friends and binging on video snacks. Hers is a regrettably common story, especially among writers.8g To stop procrastinating, Valerie needs to find a way to heighten the value of her work. Connecting it to her greater career goals would be a good start. By identifying the type of writing she wants to do and framing the present task as a stepping-stone toward this goal, she should enact strategy elements of both _Let Your Passion Be Your Vocation_ and _Games and Goals._ Also, she could have started earlier in the day, when she had the most energy, instead of toward the end, when her willpower was weakest (see _Energy Crisis_ ). And at the very least, she could have tried _Double or Nothing_ and used that municipal politics piece to motivate her to get other work done, procrastinating productively rather than cyberslacking.
If you scored 24 or above on Valerie's scale about value from chapter 2, you probably can relate to her life, though your problem might lie elsewhere than in writing. If so, reviewing the techniques in this chapter would be a good idea, as there is indeed some wiggle room in the world to find work that suits us better and to fashion this work into something we love (or at least like). Let's transmute those motivationally inert and tire-some tasks into golden goals that engage you. Just think, it might even be fun!
[Chapter Nine
In Good Time](contents.xhtml#ch_09)
MANAGING SHORT-TERM IMPULSES AND LONG-TERM GOALS
_He that has not a mastery over his inclinations, he that knows not how to resist the importunity of present pleasure or pain, for the sake of what reason tells him is fit to be done, wants the true principle of virtue and industry, and is in danger never to be good for anything._
JOHN LOCKE
Impulsiveness is the last cause of procrastination we will address, despite its overwhelming desire to be first in all things. "Now, now, I want it now" is its mantra. If we have an inner child, this is it, and it wants that candy right away. Impulsiveness runs through every vice that involves weakness of the will. Not only does impulsiveness form the core of procrastination but it is strongly connected to dysfunctional relationships, lousy leadership, suicide, substance abuse, and violence. In their groundbreaking book _A General Theory of Crime,_ criminologists Michael Gottfredson and Travis Hirschi argue that most misdeeds and misdemeanors are due to impulsiveness alone. What inevitably happens when vices give more immediate satisfaction than virtues? The most impulsive person will be the most corruptible.
Consequently, impulsiveness stands at procrastination's center-field, and has a much more intense relationship with procrastination than with any other personality trait. Whereas low self-confidence (expectancy) and propensity for boredom (value) have definite roles in creating procrastination, they are not in the same league as impulsiveness. Impulsiveness multiplies the effect of delay, making it a major determinant of the Procrastination Equation's outcome. A person with twice the average level of impulsiveness as a typical person will generally let the deadline become twice as close before starting to work. Unfortunately, if you are impulsive, you will always be somewhat susceptible to putting life off. Though you will experience a modest decrease in impulsiveness as you age and not all situations will trigger impulsive action, you can't escape your fate. Impulsiveness is not something you have, but something you are.
So what can we do about a chronic lack of self-control? Civilization has been chewing over this problem for thousands of years, figuring out how to tone down the limbic system and pump up the prefrontal cortex. Since every generation has to rediscover these solutions in their own words, it is time for us to revisit and reframe a little ancient wisdom. Let's go back to the beginning of the Greek empire, its legendary poet Homer, and his epic _The Odyssey._
COMMIT NOW TO BONDAGE, SATIATION, AND POISON
Known as Odysseus or Ulysses, this King of Ithaca reigned more than three thousand years ago, but is widely remembered to this day. In the battle to retrieve the beautiful Helen, it was Ulysses who thought up the famous Trojan horse, a giant wooden statue in which forty Greeks were hidden. Since the phrase "Beware of Greeks bearing gifts" was still hours away from being coined, Troy accepted the peace offering, only to have Ulysses and his men descend from the horse's belly behind their lines. For us, the most important of Ulysses' stories happens afterward on his sea voyage return. In a poorly planned itinerary, he fights dozens of monsters—the Cyclops, giants, drugged-out hippies known as lotus-eaters—but most important of all, the Sirens. These beautiful women, despite being perpetually naked and available, are unattached for good reason. They sing and their voices are so pure and captivating that they are irresistible; enthralled by their melody, you will want nothing but to listen and will blissfully starve, die, and rot. What do you do? Fortunately, on one of his previous stops, Ulysses had met the goddess Circe, who gave him some handy advice: fill his men's ears with wax to make them deaf and bind himself to the ship's mast so he could hear the irresistible song but couldn't act upon his urges. The bondage worked and Ulysses traveled on.
How does this apply to us? Consider Ulysses' situation in terms of the Procrastination Equation in the chart on the following page. On the vertical axis, we have Ulysses' desire, showing that he always acts on what he wants most. On the horizontal axis, there is the time dimension, starting off on the left with the way he feels right now and then moving to the right, tracking the way his desires change over time, especially as he approaches the Sirens and then Ithaca. Initially, he wants to go home to Ithaca, surprise his wife, Penelope, after his twenty-year absence, and slaughter all the suitors vying for her hand—as represented by the dashed line. He is noticeably less enthusiastic about dying at the hands of the Sirens, as represented by the solid line. However, his preference reverses when he reaches the island of the Sirens, where briefly the solid line peaks above the dashed line. If he hadn't taken Circe's advice and protected himself and his crew, they would have all stayed and died on the island. This is exactly what the Procrastination Equation predicts. As you get closer to a temptation, your desire for it peaks, allowing the temptation to trump later but better options. This probably happens to you all the time.
Right now, I'm sure you have no shortage of long-term goals: you want to lose ten pounds, stop smoking, get out more, or work harder. Maybe you want to start saving money for retirement or just for a trip. Standing between us and our aspirations are our Sirens. Instead of beautiful bare-breasted babes, they are the dessert cart, the television, or the amazing videogame. We wake in the morning with a clear desire to hit the gym in the afternoon only to succumb to the succubi of the immediately available. We want to diet but when some apple-crumble cake wafts under our nose, our willpower crumbles too. But if you can anticipate these powerful temptations, you can act in advance to ward them off. You can use the concept of _precommitment._ 6
Because he heeded the warning about the Sirens, Ulysses acted before the urge was upon him, precommitting now to prevent himself from later weakness. Because he followed Circe's advice, Ulysses lived to sail another day. Unfortunately, we don't have our own goddess to warn us of our Sirens; it is notoriously difficult to anticipate our own temptations in the moment. Using economic terminology, _sophisticates_ acknowledge their self-control problems, while _naïfs_ are caught unaware by sudden shifts in their inclinations. Most of us are _naïfs,_ unable to fully anticipate how we will feel when cravings leap upon us. In biological terms, our prefrontal cortex and limbic system just don't get each other, so we tend to underestimate the power of our own arousal—the heat of the moment—whether it is hunger, anger, or sexual excitement. And we forget the degree of regret we will feel after acting on these urges. Looking groggily into the mirror the morning after, we are mystified by exactly what our limbic system was thinking the night before.
Though we might be slow learners in regard to the power of our temptations, we do eventually learn. Give it some thought. When you are procrastinating, what are you doing? Do a few specific distractions come to mind? Can you name your Sirens? If so, let's start precommitting. Keeping true to your goals can be a limited time offer, so here's how to act now.
THROW AWAY THE KEY
A common military strategy to prevent your ships from being captured is to destroy them yourself, but such destruction has another purpose. The Spanish conquistador Hernán Cortés scuttled his own ships after landing in Mexico by filling them with water, even though the enemy was not yet in sight. Similarly, William the Conqueror burned a few of his boats symbolically and had the rest dismantled when he made landfall in England. In both cases, these men profited from their decision and went on to establish new dominions. Cortés destroyed the Aztec Empire and took their ruler Montezuma hostage. William's conquest of England ensured that the native-born nobility were replaced by those of Norman origin for centuries to come. By eliminating the means of retreat, they left their troops no option but to win, a strategy that dates back thousands of years. Sun Tzu summarizes it in his sixth-century text, _The Art of War:_ "Throw the troops into a position from which there is no escape, and even faced with death they will not flee. For if prepared to die, what can they not achieve? Then officers and men together put forth their utmost effort."
Applying this principle to procrastination, we can also shield our long-term goals from immediate temptations. Our ships in this case are our alternatives, which we try to eliminate. Herman Melville reportedly had his wife chain him to his desk while he wrote _Moby-Dick._ To keep writing, Victor Hugo had his servant strip him naked in his study and not return with his clothes until the appointed hour. Knowing that I will devour half the Halloween candy ahead of time, I don't buy it until hours before the trick-or-treating starts and take leftovers to the office for my colleagues the day after. Smokers, attempting to quit, give their packs away, telling friends not to lend them cigarettes. Revelers going out to the bar leave their credit cards at home and bring limited cash so they don't break their budgets.9a
Unfortunately, as with so many of the strategies we have already encountered, precommitment can be difficult to enact, especially on your own. Ulysses had his crew to tie him up to the mast, but we usually find ourselves without sailors at our command. Technology is beginning to fill this gap. A few years ago, I was interviewed for an article in _Newsday_ celebrating the fiftieth anniversary of the snooze button. The snooze button is the devil's device, a procrastination-enabling technology that lets you easily put off your original goal of waking up, in order to grab a few more minutes of low-quality slumber. To counteract this temptation, people hide their alarm clocks across the bedroom, or make use of Clocky, a clock on wheels that, after you hit the snooze button once, bolts off your nightstand and beeps and flashes like a robot in distress. A number of similar applications have been developed for the computer. Google has the "Take a break" button, which disables your e-mail for fifteen minutes. Another feature is Mail Goggles, which prevents late-night drunken e-mailing by requiring you first to solve simple math problems after 10:00 p.m. Others are being constantly developed, including a wide selection of add-ons for the Internet browser Firefox ( _MeeTimer,_ _LeechBlock_ ); for Apple users, there is the _Freedom_ program, which will block your access to the Internet for up to eight hours. Unfortunately, most elaborate commercial time-control software, such as _Chronager,_ is based on the idea of parental control instead of self-control; once you have the system of checks and balances set up, you will need a friend to surreptitiously change your password and keep the new one a secret.
Despite their usefulness, however, such precommitments aren't entirely effective. Most of these examples merely make succumbing to temptation difficult but still not impossible. The crux of the problem is that the same cunning you employed to set them up is now turned against you; indeed, you are your own worst enemy. You can always run to the store to buy another treat, reformat your computer to get around nanny software, and throw pillows to suffocate Clocky. Samuel Coleridge hired thugs to prevent him from frequenting opium dens, only to fire them when the urge came upon him once again. In _Trainspotting,_ Ewan McGregor's character nailed himself into a room so he could quit his heroin habit, only to extract himself later with the same determination. More realistically, the mechanism at work here is delaying—not preventing—your access to temptations. As the delay lengthens, with luck the desire for the temptation is reduced in strength. A bowl of ice cream might beckon if placed within arm's reach, but its voice is muffled when shut inside the freezer. Naturally, the greater the desire for the vice, the greater the distance required to silence it.
SATIATION
Have you ever gone to the grocery store hungry? Bad idea. You likely wheeled down the aisles, filling your cart with indulgences that weren't on your list. Unpacking the bags at home, you loaded your cupboard and freezer with goodies that took you weeks to plow through and added pounds to your midriff. Really, all you needed was a small treat but in your state of deprivation you impulsively bought yourself a sizable feast. The pearl of wisdom, aside from "Never grocery shop on an empty stomach," is that more basic concerns must be attended to before concentration can be applied elsewhere. Abraham Maslow, the father of humanistic psychology, based his theory of self-actualization on this insight, positing that we have a hierarchy of needs whereby basic, more visceral desires, like food and safety, must be attended to first.
To precommit using satiation, we try to meet our needs in a safe and managed manner before they intensify and take control. If your appetite becomes too extreme, you will gorge yourself in seeking to satisfy it. Two common precommitment strategies are having a glass of water and garden salad at the beginning of a meal and grazing on small healthy snacks throughout the day.9b A rather fun way of encouraging fidelity is to make love before your partner leaves for a prolonged trip, endorsed by no less than St. Paul the Apostle.9c Smokers use the nicotine patch to reduce their cravings while heroin users take methadone. A broader use of this strategy is to schedule your recreational activities in your calendar first. Then pencil in your chores. Called an "unschedule," it can breathe energy back into life's grind. In all these cases, the idea is to let off a little steam before our boilers burst.
TRY POISON
Even though registration deadlines are posted months ahead and reinforced with early bird discounts, the crush of applications for anything from training courses to 10K races typically occurs just before the deadline. No surprises here. Presenting at a conference in New York a few years ago, I met Victor Vroom, an expert in leadership and motivation. Crossing Times Square with him, I noticed that neither of us had managed to secure rooms in the main hotel because we had both registered too late. Procrastinators, however, are paradoxically not always the last to sign up; sometimes they are the first. In an effort to precommit, they sign long-term health club contracts, buy season subscriptions to the symphony, or request home delivery of highbrow films from DVD movie clubs far in advance. By acting now, they hope to irrevocably force their future selves to do what their present selves are unwilling to pursue, even if it means poisoning other alternatives.
A one-time common form of this precommitment device was the Christmas Club. Invented by the Carlisle Trust Company in 1909, banks offered low-interest savings accounts that penalized you for early withdrawal. Despite today's easier access to credit, variations on Christmas Clubs still exist. Why would anyone use them? Because they want to be under the threat of punishment: without the looming penalty, they fear they will withdraw and spend their money prematurely, and have nothing but good intentions to leave under the tree. The same principle can be useful in preventing weight gain. Weight Watchers is an international company designed to punish people for putting on the pounds. It provides assistance and advice for getting to and maintaining a target weight. Once you are firmly established at your ideal size, you receive a free Lifetime membership. But there is a catch. You must weigh in once a month and if you are more than two pounds over, the membership fees are reinstated until you again shed the pounds. I've also heard of a Danish chain of gyms that offers membership free of charge as long as you show up once a week. Fail to exercise regularly and you have to pay.
With the help of a merciless friend or perhaps an agreeable enemy, you too can raise the stakes on any venture. Just make a painfully large bet that you will lose only if you put off striving toward the goal you want to attain. Economists John Romalis and Dean Karlin, for instance, opted for their own enhanced version of Weight Watchers. In their pact to stay trim, either of them can call an impromptu weigh-in, with the fine for weight gain being $10,000. Karlin later teamed up with a different economics professor, Ian Ayres, to create _stickK.com,_ a website to help others devise their own precommitment contracts. A similar but earlier effort is the website "Covenant Eyes," founded by Ronald DeHass. To curtail pornography consumption, it tracks and e-mails all your Internet visits to the "accountability partner" of your choice. It could be a friend, a spouse, or perhaps a pastor. For a technological solution in the same cast as Clocky, there is the alarm clock SnuzNLuz. Every time you press the snooze button, it donates ten or more dollars to your most detested charity; a little extra sleep comes at the cost of assisting groups that represent the antithesis of your political position, sexual orientation, or environmental stance.
Like all precommitment methods, these devices aren't foolproof. To begin with, they are inflexible, so you can't change your mind even for legitimate reasons. Where would Ulysses be if his ship started sinking or was attacked by pirates with him still bound to the mast? You might desperately need the money you've tied up in Christmas Clubs or fall ill and be unable to use long-term gym memberships. On the other hand, if you don't make them strong enough, disincentives can be circumvented. In keeping with the saying, "Those who flee temptation generally leave a forwarding address," be careful that your future self isn't smarter or more determined than your present version. If there is a will—and there most definitely _is_ a will—then there'd better not be a way. Adults who nail-bite will coat their fingers with the same bitter ointment used to discourage children from sucking their thumbs, only to endure the taste or find inventive ways to wash it off. Similarly, in Mordecai Richler's novel _Joshua Then and Now,_ Joshua Shapiro helps his friend Seymour overcome a precommitment strategy by swapping underwear with him: Seymour was wearing "black satin panties with a delicate lace trim" to prevent his adulterous ways. After all, what type of woman would want to sleep with you after she found out that you clad your manliness in lacey undergarments? Well, I guess it depends on what crowd you hang with, but that's beside the point.
**_1. Action Points for Commit Now to Bondage, Satiation, and Poison:_** Staying true to your goals can be a time-limited offer, requiring you to act before temptation overcomes you. You first need to identify your temptations, what distracts you when you should be working. If you need assistance, ask your family and friends. They likely know. After identifying your temptations, you have three options about what to do about them. Take your pick.
• _Bondage:_ Put these temptations out of reach or at least far away. For example, erase your video games or disconnect your Internet connection. Remove the battery from your PDA or unplug your television set.
• _Satiation:_ Satisfy your needs before they get too intense and distract you from your work. Ironically, you can often work harder if you first schedule in some time for leisure.
• _Poison:_ Add disincentives to your temptations to make them sufficiently unattractive. For example, a monetary bet with someone else that you won't give in to your temptation can be applied to almost anything.
MAKING PAYING ATTENTION PAY
About the time I was born, the award-winning psychologist Walter Mischel started experimenting on children using marshmallows to test the power of their will. In a series of studies, he would offer the kids a marshmallow, but tell them if they could wait a little while, they would get two marshmallows. Some waited a little, others a lot, with the average being about five minutes. The children's ability to delay gratification and get the larger but later treat proved critical as they grew up. The self-control they exhibited as kids predicted everything from their Scholastic Aptitude Test (SAT) scores to their adult social skills. Character is indeed destiny. Subsequently, Mischel tried to change the destiny of a new cohort of children by improving their strategies for dealing with temptations, usually tripling their self-control, getting them to wait three times as long. What was his magic? He simply showed them how to pay attention.
Mischel's approach to conquering inattention will seem very familiar. As I do for the Procrastination Equation, Mischel emphasizes our mind's dual nature; procrastination arises from the interplay between our limbic system and our prefrontal cortex. To master attentional control as a means of increasing our self-control, we must first go from the inside out, to change what we see and how we see the world. Second, we go from the outside in, to remove or reinforce external cues, changing the world we see.
INSIDE OUT: PAY ATTENTION PLEASE!
It is time to play a game called "The Unlikely Beast." It will take precisely a minute. Take out your watch to time yourself and for the entire minute don't think of a _pink elephant._ No pink elephants, not even one. Got it? Since you probably didn't think of any pink elephants today, this should be pretty easy. If you can make it an additional sixty seconds without thinking of pink elephants, you win. Are you ready? Go!
* * *
INSERT SIXTY SECONDS HERE
* * *
Did you win? I doubt it. According to Daniel Wegner, who wrote the book on thought suppression, the game is rigged against you. To make sure you aren't thinking about pink elephants, you have to keep some notion of them in mind, otherwise you can't watch out for violations. Ironically, by actively suppressing thoughts, you help to maintain them. This mechanism forms the basis of Freudian slips; trying to repress a trauma or a temptation seems to cause the dreaded idea to surface. For the few of you who did suppress the beast successfully for a whole sixty seconds, did you notice the post-suppression rebound? Your mind, in a sigh of release, probably indulged in a series of pink elephant fantasies as soon as the time was up. Despite its disastrous track record, thought suppression is a popular technique used to combat—ineffectively—everything from homosexual urges to racial stereotypes. If you find yourself pestered with an intrusive temptation, whether it be for an illicit lover or a new television show, you can find better ways to stop thinking about it. Here's what works.
Instead of avoiding thinking about your temptation, you can mentally distance yourself from it by framing your temptation in terms of its abstract and symbolic features. For example, Mischel had children delay eating pretzels by having them focus on the snack's shape and color ("the pretzels are long and thin like little logs") rather than on their taste and texture. Similarly, anthropologist Terrence Deacon managed to get chimpanzees to make food choices more strategically by using a form of symbolic representation called lexigrams. The chimps were to choose between two portions of fruit, kiwis and strawberries, and received the fruit _they didn't select._ Only chimps who learned the lexigram equivalents of kiwis and strawberries (respectively a black square with a blue "Ki" versus a red square with two horizontal white lines) were able to enact the winning strategy of pointing to the less desirable fruit option and, in return, receive the more desirable one. As Deacon concluded, seeing the world in symbols tips the balance away from the stimuli-driven limbic system toward the abstraction-loving prefrontal cortex, enabling us to make better choices.9d To take advantage of this quirk, we need to keep our thoughts as airy and formless as possible, as if seeing temptations from a great distance. As the seventeenth-century Japanese swordsman Miyamoto Musashi wrote in the _Book of Five Rings:_ "Perception is strong and sight weak. In strategy it is important to see distant things as if they were close and to take a distanced view of close things."
Your second line of defense is to run a "smear campaign" on whatever features your limbic system finds desirable. You can ascribe negative qualities and consequences to every temptation to counteract its enticing features. Those pretzels, for example, could be stale or sneezed upon. The more such disgusting possibilities you generate, the more unpleasant the indulgence will seem. Furthermore, by imagining some really horrific outcomes, you engage in something called _covert sensitization._ 35 This technique is to pair your temptation with an undesirable image, hopefully infusing the former with the latter. Here is a generic one I developed specifically for procrastination:
I want you to imagine you've just put off a major project, one that you thought you still had plenty of time for. You are doing other less important work, surfing the Internet, watching TV at home—procrastinating. Finally, the moment comes when you can't really put it off any longer and, though it will be stressful, you should be able to handle it—except you just came down with a throbbing headache. Given all the extra time you had to take on the project, you can't use this as an excuse without looking lazy and incompetent. You start working on it, but the headache gets worse and worse, like a knife twisting behind your eyes. You are producing nothing of value despite the excruciating pain as you try to work. As your eyes almost tear up with agony, you take some pain medication only to find that it makes you sleepy, and indeed you do sleep. When you wake up, it is morning and you are late for work. Rushing to get there, you find that your boss has decided to gather all of your colleagues in the boardroom for you to present your project. The president of your company stops by and decides to listen in too. Being late, you are rushed to the front of the podium and everyone waits for you to get started. As you try to explain you have accomplished nothing because of a headache, you stumble over your words and look like an utter fool. There is a long silence broken only by a few sniggers, with your colleagues looking away, embarrassed to be associated with you. Afterward, your boss explains that she was thinking of promoting you but now she will have to fire you instead—what you have done is inexcusable. One of the people at the meeting recorded your "presentation" with her cell phone and posted it on YouTube, where people everywhere mock you. No one in your industry will even give you an interview and your career is ruined.
Feel free to change this scenario to fit your situation, tailoring it to your specific distractions. Joshua Shapiro's friend Seymour, for example, might have had better luck with fidelity by focusing on negative possibilities, like getting a stranger pregnant, catching a disease, or destroying his marriage. For yourself, just remember that when you leave tasks to the last moment, you can get sick, competing emergencies do happen, and work almost always takes longer than you thought. As for the dire outcomes that result from your procrastination, imagine the worst. The consulting company Opera Solutions lost a million-dollar contest by submitting their solution twenty minutes too late. Elisha Gray lost credit for inventing the telephone to Alexander Graham Bell by submitting his idea to the patent office a day late. Delay makes bad things happen. Why not to you?
Attentional control and covert sensitization aren't perfect techniques, though. They require effort, and will eventually exhaust your energy stores—you can't avert your eyes forever. As Mischel's work showed, children's ability to delay gratification was increased, but remained limited. Still, some delay may be enough for your purposes. Many temptations are time sensitive, like dessert at the end of dinner; if you can avoid them for an hour or so, the desire to indulge will disappear. It isn't perfect, but it is better. If you are looking for more long-lasting solutions, read on.
OUTSIDE IN: NOW YOU SEE IT, NOW YOU DON'T
Here is a trick that will give you an extra month of efficiency each year. It is easy to implement, immediately effective, and doesn't cost a cent. First, go to your e-mail program. Second, disable all the audio alerts and mailbox pop-ups. In Microsoft Outlook, they are buried pretty deep under "Advanced E-Mail Options," but the controls are definitely there. Just unclick every-thing under "When new items arrive in my Inbox." That's it, there is no third step. Banishing e-mail notifications will make you about 10 percent more efficient and over a year that translates into one more month of productivity.9e The best work happens when you engage deeply in a single task. Every time you stop your flow, you have to once again decide to work and then it takes time to become fully re-engaged. Unfortunately, we are conditioned to answer e-mail instantly, responding to the tell-tale "ding" like Pavlov's dogs. Unless you have a pressing reason, check your e-mail at your convenience, during natural breaks in your productivity.
What we are doing here by changing our e-mail settings is regaining _stimulus control._ Part of our decision making occurs subconsciously, in our limbic system. This is not the brightest part of our minds; it takes much of its lead from environmental cues—that is, from the _stimuli_ of sight, smell, sound or touch. A provocative image pops up and we think of sex, a tasty smell wafts our way and we become hungry, or we hear a snippet of a song and start humming the tune. These associative cues cause our mind to wander and we forget the original task. With just a little nudge, our imagination slips down the rabbit-hole and we find ourselves mulling over some more personally relevant issue, like what's for lunch. We have been distracted.
These distracting cues are powerful and pervasive, and are actively pumped into our world. John Bargh, head of Yale's Automaticity in Cognition, Motivation and Emotion (ACME) Lab, has spent decades showing how little it takes to influence our minds. We can be prepared—primed—for almost anything, all without being aware of it. A slight dimming of the lights increases our fearfulness. Hold a hot cup of coffee and warm feelings infuse us, causing us to be more charitable. Putting Hershey's chocolate kisses on a secretary's desk in a clear rather than opaque bowl, thereby making them more visible but not more available, increases snacking at the office by 46 percent. The power of cues is such that they can create cravings that leap upon us—"If you speak of the Devil, so he will appear." Addicts often feel an overwhelming urge to relapse when they encounter a strong drug cue, such as a neighborhood hangout or a former fellow user.
Big business has been aggressively trying to direct these cognitive cues, deluging us with over a thousand advertisements each day. To take back control of our environment, essentially we need to run our own personal advertising department. As it is, our workplaces and schools are motivationally toxic, polluted with distractions. We need to make them sanctuaries of performance, taking advantage of the "out of sight, out of mind" adage to purge our offices and classrooms of irrelevant cues. At the beginning of this section, I asked you to turn off all your e-mail alerts. I also told you about how Ulysses had his crew seal their ears with wax to avoid hearing the Sirens. Both of these examples draw upon the same principle of eliminating external cues. You need to identify your distractions and cleanse their accompanying cues from your life. I bet you have more than a few Internet sites hot-linked on your computer for easy access. Start by deleting those. While you are at it, get rid of any quick-launch icons for games, or better yet, erase the games completely. At home, hide the remote control for the TV or close the doors of the television cabinet if you have one. Now for the really hard part.
A messy workspace, cluttered and disorganized, is a minefield of distractions. For every minute you hunt for a misplaced report or book, the likelihood increases that some tangential tidbit will entrance you. Everything extraneous on your desk distracts and detracts, making it harder to find and focus on your primary purpose. But here is the catch-22: the number one activity that people postpone is "cleaning out closets, drawers, and other cluttered spaces." Procrastinators are more likely to leave clutter, which in turn, increases their procrastination. You need some help. You can combat clutter with some of the other procrastination-fighting techniques in this book—the structured or productive procrastination we looked at in chapter 8 is particularly relevant for you. The most motivating time to de-clutter your life always seems to be before another pressing deadline. Alternatively, look outside these pages for help. Just search online under the word "clutter" to find books on how to organize your life. You can also call in organizing experts; it's no more unusual than hiring a personal trainer to jumpstart your exercise program.9f
Once you have banished the signs of temptation, the other half of this stimulus-control strategy is filling the void. External reminders of our goals are important, but instead of motivational posters featuring generic catch phrases, your reminders need to be personally relevant. They need to speak to _you._ What do you strongly associate with the target task? If there is a quotation you find particularly inspiring, have your screensaver produce it whenever you idle. If you are slow at paying bills or taxes, place them prominently on your kitchen or coffee table, where you can't ignore them. Even writing a list is a good reminder, especially on a sticky note posted to the side of your computer screen. All these cues solidify into an unwaveringly effective concentrative strategy, focusing your attention toward your goal.
To emphasize how effective this concentrative strategy can be, consider the boost it can give to your household energy efficiency. The problem with energy consumption is that it is distant and vague, only realized in a monthly bill long after the kilowatts have been killed. If we made a very small change and put your electricity meter on the _inside_ rather than the _outside_ of your house, this visible and constant reminder of your energy cost would coordinate your limbic system with your prefrontal cortex, sparking you to turn off unneeded lights and replace the remainder with efficient fluorescents. Mark Martinez from Southern California Edison, for example, had his customers use an _Ambient Orb_ that glowed red when electricity was expensive. Within weeks, peak hour consumption voluntarily reduced by 40 percent; and other similar experiments have indicated about a 10 percent savings in monthly utility bills.
For work, stimulus cues don't have to be store-bought. Anything associated with a task can spur you to complete it: time of day, preceding activity, and colleagues all can be transformed into work triggers. Most usefully, you can make your place of work itself a cue, so that focus comes automatically as soon as you sit down. This strategy requires dedicating your environment exclusively to labor. To do this, work in your office until your motivation leaves you and goofing off becomes irresistible. At this point, do your web surfing, your social networking, your game playing _somewhere else._ This may require you to get a second computer, one for play, but when the added productivity kicks in, the purchase will pay for itself. If you keep work and play in discrete domains, associations will build and attention will become effortless—your environment will be doing all the heavy motivational lifting. Three studies have investigated the effectiveness of this technique with students, and found that the use of dedicated work areas decreased procrastination significantly within weeks. Similar applications, such as using separate banking accounts to prevent impulsive spending, can be almost instantaneously effective. Without this segregation between work and play, you get conflicting cues every time you sit down at your desk, one indicating that you should research your report and the other egging you on to check your Facebook page.
To sharpen role boundaries between clashing life domains, typically family and work, we need to keep the demarcation lines pristine. If you can't afford a separate computer, then at least create a second profile that requires you to log out of your workplace identity before you slip into your lazier alter ego. If you find your BlackBerry allows the office to pollute family time, get a stripped-down second cell phone to use when you punch out. You might also include a transition ritual to help you move from one domain to another, such as winding down with the radio during your commute or changing out of your "work clothes" when you arrive home. If you need to work at home, have a separate office, no matter how small or symbolic. These environmental cues will fence off distracting temptations, allowing you to truly _be_ in each place.
**_2. Action Points for Making Paying Attention Pay:_** Distractions are a major enabler of procrastination, so learning how to effectively handle them is a must. Your options are to denigrate, eliminate, or replace cues that remind you of your temptations.
• Sully tempting alternatives by using covert sensitization, imagining disgusting ways they may be tainted, or envision possible disastrous outcomes from procrastinating. The more vividly you can imagine the contamination or the catastrophe, the more effective this technique will be.
• When confronted with distracting temptations, focus on their most abstract aspects. Triple chocolate cheesecake, for example, can be construed as another fat and sugar combination.
• Entirely eliminate cues that remind you of distracting alternatives where possible. Keeping your workplace clear of clutter will help you accomplish this.
• Once you have purged your workplace of distracting cues, replace them with meaningful messages or pictures that remind you of why you are working. For some, a desk photo of loved ones can be an effective reminder.
• Foster these work cues by compartmentalizing your place of work and play, keeping them as separate as possible.
SCORING GOALS
Inch by inch, life's a cinch; yard by yard, life is hard. How powerful is this mantra? Joe Simpson, in one of mountaineering's greatest survival stories, used it to save his life. Left for dead at the bottom of a crevasse in an isolated Peruvian mountain with a shattered shinbone, he had three days to pull himself to a base camp through five miles of truly treacherous glacier field or be really dead. He was already utterly exhausted from an arduous marathon of an ascent, with no food and only a little water, so this journey should have been impossible, except for one critical survival tool: his wristwatch. With it, he set goals. Setting the alarm for twenty minutes at a time, he made for a nearby rock or drift—he was elated when he reached it in time and he despaired when he didn't. Battling exhaustion, pain, and eventually delirium, he repeated this process hundreds of times and reached the perimeter of the base camp just hours before his friends' intended departure.
Simpson's story, recounted in his book _Touching the Void,_ highlights the power of goal setting. As Mark Twain wrote: "The secret of getting ahead is getting started. The secret of getting started is breaking your complex overwhelming tasks into small manageable tasks, and then starting on the first one." Further notions about how to construct goals to maximize their motivational benefits, however, are shrouded in confusion. Despite thousands of scientific studies on how best to set goals, little of this know-how has permeated into the mainstream. Since the mid-eighties, over five hundred books have stressed S.M.A.R.T. goals, an acronym that has both too many and too few letters. S.M.A.R.T. stands for: Specific, Measurable, Attainable, Realistic, and Time-Anchored. There are too many letters, in that _Specific_ is redundant with both _Measurable_ and _Time-Anchored_ while _Attainable_ is redundant with _Realistic._ 57 There are too few in that it is still missing major concepts. Let me tell you what you actually need to know.
We have already touched on some of what makes a goal good. In chapter 7, we mentioned that making goals challenging is more inspiring than making them attainable. Easy goals are attainable. You know what happens after obtaining your easy goal? The same thing that happens after you cross the finish line of any race: you stop. In chapter 8, we focused on making goals meaningful by linking them to personally relevant aspirations. If you see how present tasks lead to future rewards, you will value them more highly. In this chapter, we will put the finishing touches on goal setting by putting time back on your side.
THE FINISH LINE IS JUST AHEAD
Almost invariably, reporters contact me about their piece on procrastination mere hours before it is due. _Slate_ magazine, for example, which did a special issue on procrastination, confessed: it was "originally planned for the week of May 5. Seriously. We'd planned to publish that Monday morning, but there was one problem: only a handful of our writers had managed to get their work in on time." My theory is that the fourth estate is full of unrepentant procrastinators, drawn there because it is one of the few places they fit. Every day the job itself generates a specific and proximal deadline: so many words on this topic by this hour _or else!_ This is exactly the type of goal that procrastinators excel at meeting. To get motivated, they need a clear and close finish line. Their action curve follows directly from the Procrastination Equation; as delay shrinks, motivation peaks.
To apply this principle to your life, you need a concrete and exact notion of what needs to be done because vague and abstract goals (such as "Do your best!") rarely lead to anything excellent. The level of detail required differs from person to person but you should be able to sense when you've got enough. Goals should have a corporeal rather than an ethereal feel—you should be able to sink your teeth into them. "Complete my Last Will and Testament before flying on the 15th" is an achievable goal. "Get my finances together," not so much.
After creating a specific finish line, schedule it soon. You may need to break up a long-term project into a series of smaller steps. Consider the following chart, which represents most work situations. In the background, there is always a buzz of temptation and though it will have its peaks and valleys, on average, we can represent it by a straight horizontal dashed line. Until our desire for work exceeds this constant, we won't be working. Typically, we allow the environment to set our goals for us and it is pictured by a single goal: the deadline. The triangle line represents a person with no self-set goals, whose motivation is mostly reserved until just before the deadline. What to do? How about artificially moving the deadline closer? The unadorned solid line represents a person who has broken down the task into two earlier subgoals, allowing work motivation to crest above the temptation line sooner. As can be seen, the sum of the parts can be greater than the whole, as the person who sets subgoals works for twice as long as the person who doesn't.
There are no hard rules for how specific and how proximal your goals must be to be effective. Your success depends on how impulsive you are, how unappealing you find the task, and what temptations you are battling. But keep in mind that too-frequent goals can be cumbersome. Daily goals typically provide a good balance; they are both effective and practical. Still, many find that the hard outer shell of a chore, the first few minutes, remains the initial obstacle. How many times have you put off a task only to realize it wasn't so bad once you got started? Cleaning, exercising, and even writing are often difficult at first. It is a bit like swimming in the lake by my in-laws' cabin, just north-east of Winnipeg (the coldest city in the world with a population greater than 600,000). The water is deliciously invigorating but, for most, the initial temperature shock is an effective barrier against reaping the subsequent reward. By focusing solely on the initial jump off the dock, I can plunge in and, after a few intense seconds, enjoy myself. An extremely short-term or mini-goal, then, is excellent for busting through such motivational surface tension. Ten-minute goals are an application of this technique, such as the ten-minute clean-up around the house. Consequently, if you have trouble writing, just sit down and type a few words. If you don't want to exercise, at least get your workout clothes on and drive to the gym. Once you have completed your mini-goal, re-evaluate how you feel and see if you are willing to immediately commit to a longer stretch. Having broken through that motivational surface tension and immersed yourself in the project, you, like most, will opt to continue.
Your final choice is how to structure your goals. Do you prefer _inputs,_ the time invested, or _outputs,_ what is produced? For exercise, are you going to run for an hour or for five miles? Both are good options. A modest but regular schedule, if it really is regular, produces wonders. B.F. Skinner thought "fifteen minutes of writing] a day, every day, adds up to about a book every year," though most professional writers aim to do far more than a quarter of an hour. Others go by the word count; science fiction author Robert Sawyer, for one, writes two thousand words each day, including his blog. Ernest Hemingway combined both inputs and outputs, writing for six hours or producing about five hundred words, a useful strategy. If you have a fruitful day and hit your output quota early, be it words or widgets, reward yourself and go fishing; if the productivity doesn't come, the input or time requirement ensures that something is produced. To help keep you honest about your productivity, try using free software like _ManicTime_ or _RescueTime._[ 62 They are nifty applications that automatically track your computer work habits, allowing you to easily monitor your activities. How much time are you spending on e-mail? How about web surfing? How much do you actually spend on work? This kind of reality check will make you aware of your productivity and I'll personally vouch that it is useful for winding down an Internet gaming habit.
FULL AUTO
Occasionally, on my ride home from work, I am charged with stopping off at the grocery store to pick up milk or diapers. This side trip entails taking the earlier exit off the highway, which I invariably drive right past. I then need to negotiate a laborious series of travel corrections to get to where I should already be. The problem is that I've done my commute so many times that I'm on autopilot. We have dozens of these automatic routines in our lives, which we can perform even when dead tired. In a mindless blur, we eat breakfast, brush our teeth, and tie our shoes. Despite their zombie-like quality, these routines have power we can tap—the force of habit.
Both the strength and weakness of routines lies in their lack of flexibility. Their weakness is that once we fall into a habit, we tend to follow through even when a change of pace would be beneficial. We go to the same restaurants, order the same food, watch the same shows, without really considering possibly better options. On the other hand, routines are easy to maintain and can be undertaken even when we're exhausted. By intentionally adopting a routine, we can pursue long-term goals even when our wills are weary and temptations abound. We push forward oblivious to other choices, choices that might mean stopping, resting, doing otherwise. The fewer moments of choice there are, the less likely you will be to procrastinate. That is, if you have the right kind of habits. Routines are like Don Quixote's windmills; they can raise you up to the heavens or drop you down into the mud. Though we have our share of bad habits—reflexively turning on the TV or finishing a bag of potato chips—we can create good ones. We can turn exercising, cleaning, or working into at least semi-automatic routines. Scientific study confirms the benefit of this effort; procrastinators perform as well as anyone else when the work is routine.
Building a routine requires activating many of the same precepts as stimulus cues. You want _predictability._ Devise rituals of performance, keeping as many of the environmental variables as stable as possible, especially time and place. Exercise programs, for example, should take place at regularly scheduled times, leaving little guesswork about where and what the fitness activities will be. Like clockwork, every Tuesday afternoon at 5:00, you go and lift weights, and every Thursday morning at 6:00, you go running. Take whatever you have been putting off and specify where and how you intend to implement it. For instance, make a vow: "When breakfast is finished on Saturday morning, I will clean out the storage room." This seems so easy and simple that it couldn't work, but it does. When you make an explicit intention to act, the desired behavior just happens. The expert on the psychology of intentions, Peter Gollwitzer, finds that forming intentions almost doubles the chances that you will follow through with almost any activity. The effectiveness of explicit intentions has been scientifically confirmed on everything from cervical screening to testicular self-examinations and from recycling to writing a research report over the holidays. In terms of ease and power, this is as good as it gets. Making an intention is a remarkably accessible back door into your brain; it programs your limbic system to effortlessly act on cue as you see fit. Intentions can even be used to implement other self-regulatory techniques, especially when expressed in an "If . . . then" format. If you have energy issues, make the intention of " _If_ I get tired, _then_ I will persevere." If you are easily distracted, it would be " _If_ I lose focus, _then_ I will move my attention back to the task." And of course, " _If_ I am pursuing a goal, _then_ I will use implementation intentions."
Be warned that when trying to start your routine, you will invent a ceaseless onslaught of excuses not to follow through. You will get sick, go on vacation, have extra work, fall behind elsewhere, and find it ever so convenient to let your schedule slip. Defend fiercely against these slippages! Routines get stronger with repetition, so every time you slack off, you weaken your habit and it becomes even harder to follow through the next time. If you protect your routine, eventually it will protect you. At the start, your regimen will need constant nursing. Some temporary professional assistance can be a good investment; after all, you are investing in yourself. Personal trainers to run you through your paces or professional organizers to help you clean up can help launch you in the right direction. To draft your last will and testament, hire an estate planner or a wills and estates lawyer. They provide as much motivational help as legal expertise, structuring the process to maximize your follow-through. But hired help can't do it on their own, nor can this or any other book. In the end, the responsibility lies where it has always been—with you.
**_3. Action Points for Scoring Goals:_** This is really saving the best for last. Goal setting—proper goal setting—is the _smartest_ thing you can do to battle procrastination. Though every other technique discussed so far has its place, goal setting alone may be all you need. Along with making your goals challenging (chapter 7) and meaningful (chapter 8), follow these remaining steps. Regardless of what other books say, this is what's proven to maximize your motivation.
• Frame your goals in specific terms so that you know precisely when you have to achieve them. What exactly do you have to do? And when do you have to do it by? Instead of "Do my expense report" it should be "Gather all my receipts, itemize them and record them by lunchtime tomorrow."
• Break down long-term goals into a series of short-term objectives. For particularly daunting tasks, begin with a mini-goal to break the motivational surface tension. For example, a goal of tackling just the first few pages of any required reading can often be enough to get you to finish the entire text.
• Organize your goals into routines that occur regularly at the same time and place. Predictability is your pal, so open your schedule and pencil in reoccurring tasks. Better yet, use an indelible pen.
LOOKING FORWARD
If only time-sensitive Tom could have read this chapter! He put off booking his hotel and subsequently had a vacation to forget instead of one to remember. He probably didn't even need all the techniques in this chapter to have changed his fate. Perhaps it would have been enough to set a specific deadline for himself, say, next Thursday night, and frame his intention to act in explicit terms, as in: "Immediately after dinner I will research hotels in the area and book a room." For good measure, he could have imagined some worst-case scenarios, such as: if he continued to procrastinate, then his room would be far away from the beach and in desperate need of redecoration. Those of you who scored 24 and above on the impulsiveness self-assessment scale in chapter 2 should pay special attention to the techniques here, but almost everyone would benefit from them as well. Though some of us are more impulsive than others, we all can make regrettably impulsive choices.
The fundamental challenge in implementing these steps is that attempts to increase self-control require some self-control to begin with. The obstacle is similar to strength training; in order to initiate the process, we need to be able to lift at least the lightest of the available weights. As for procrastination, the worse it is, the harder it becomes to remedy. The very motivational deficits that create your procrastination also hamper your attempts at change. If you are unable to delay gratification, for example, methods to increase your patience must initially be immediately rewarding in themselves. Otherwise, advice becomes useless shouting from the sidelines, annoyingly extolling you to "do first things first." If you could simply do that, you wouldn't need the advice in the first place. Fortunately, most of these techniques are easy to adopt, like turning off your e-mail ding or making those explicit intentions to act. These immediate successes will give you the confidence and the self-control to increase your efforts, all of which will become even easier with practice. From here on out, life becomes better, not harder.
[Chapter Ten
Making it Work](contents.xhtml#ch_10)
PUTTING THE PIECES INTO PRACTICE
_Do or do not do. There is no try._
MASTER YODA
Before I get into this chapter, I want to thank you for persevering. People who procrastinate tend to get distracted and turn to other things. So since you have reached chapter 10—and I am assuming you haven't skipped ahead to the end—you deserve a little praise. After all, the tendency to put off has such a deep resonance in our beings that it is more remarkable when we _don't_ procrastinate than when we do. Having read through the book, you have a good grasp of the underpinnings of procrastination, how it emerges from our brain's architecture, the ways in which the modern world makes it worse, and what you can do about diminishing it. There is just one last step to putting procrastination in its place. You need to believe what you read.
I can't really blame you if you are a little suspicious. If you are familiar with self-help books, you have certainly earned some cynicism. There is so much misinformation in the field of motivation—so many promises that don't deliver—that "What if someone wrote a self-help book that actually worked?" is the premise of Will Ferguson's international bestselling novel _HappinessTM._ Satirizing the self-help industry, Ferguson invents the character Tupak Soiree, who writes _What I Learned on the Mountain,_ a tome that genuinely helps you lose weight, make money, be happy, and have great sex.10a Now I can't promise the last of these, but _The Procrastination Equation_ is about making the rest of _What I Learned on the Mountain_ a reality. Every technique in this book is based on the bedrock of scientific study, so it had better work. Just flip ahead a few more pages and look at the research I have laid out in the Endnotes.
_The Procrastination Equation,_ just like _What I Learned on the Mountain,_ is still only an inconsequential book if the techniques stay locked inside its covers. In Ferguson's novel, the challenge was just getting people to read it. For a while, _What I Learned on the Mountain_ 's potential effectiveness was derailed, as you might guess, by procrastination. As Edwin, the book's editor, concludes: "I forgot about procrastinators. Don't you see? All those people out there who purchased the book or were given it as a gift and still haven't got around to reading it." For my book, the requirements are a little steeper, but as you can see, you have already pretty much finished it. To make what you are reading effective, you also need to take its contents seriously. You need to adopt these techniques into your life and start seeing your decision making in terms of that interplay between your limbic system and your prefrontal cortex. To lift the ideas off these pages and into your life, we are going to take one parting look at Eddie, Valerie, and Tom and imagine how they are getting along. You'll see that they are using all these techniques in combination, and thriving because of it. And if you can see yourself doing the same, then you will be able to get your act together, and you will soon be putting procrastination behind you as well.
EDDIE AND VALERIE
After Eddie lost his sales job, he was depressed for a long time—that is, until he met Valerie. She always found a way of putting a smile on his face and it was natural that the two got married. Now in their thirties, with two full-time jobs and a lovable toddler named Constance, they have a wonderful life. But they are always on the run, and lately the demands have been getting worse.
Valerie is often on crushing deadlines, and her home responsibilities take second place when she is in a crunch. She knows how lucky she is to have a job at the local newspaper, but there have been cuts, and she is now doing the work of two people, maybe more. The pressure to meet all her deadlines is serious—this isn't about career advancement, it's about staying employed. Eddie has to travel for his job in marketing, which means that he leaves before dawn and is away for days, leaving Valerie in a lurch. When Constance gets sick, all hell breaks loose. She keeps them up at night, and somebody has to stay home with her. When the washing machine breaks down, somebody has to wait for the repairman. Valerie and Eddie feel as if they haven't had enough sleep in years. And they are right. They know how lucky they are to have two jobs and their little girl, but they are stressed beyond words.
Valerie and Eddie shuttle between work and home like mechanical dolls, always late, grabbing a kiss or a donut on their way out. When they are at home, they worry about the work they are not doing, and so they often go to the computer after the baby is asleep, working through exhaustion late into the night. If the baby is sick, the one who goes to work frets about how she is, and when she is well, they are both checking her out on the webcam at daycare—spending precious work minutes monitoring her well-being. They can hardly handle paying the bills and getting to the pediatrician's office for checkups and shots. They e-mail each other dozens of times a day, and Eddie has to control himself from texting Valerie from the car on the way to his next meeting.
Eddie promised himself he would clean out the garage last summer, but it's October, and the junk remains. Valerie has lost control of her vegetable garden, which she started as an altruistic family project but which has devolved into a sad collection of wilted greenery. They are considering canceling their joint membership in the gym—they are both too tired to work out at the end of the day, and mornings reach a level of chaos that drives them both nuts—dressing the baby, exchanging directives about multiple tasks, suddenly full diapers and fussy moments . . . you can fill in the blanks.
This is actually the best-case scenario. It could easily be worse. They face no sudden illness, no job loss, no financial straits, and no tragedy. But Eddie and Valerie's lives are out of control and they are facing the conflicts that every working couple with kids has to deal with. Recently, Valerie began to feel that she is never in the right place—at work, she thinks she should be home; and at home, she worries about all the work she should be doing. She is feeling frayed and tattered, and is starting to hate her life. Looking for some cheering up, she calls her sister, who listens sympathetically, and then offers a little advice: "There's this book I've been reading that has a few ideas that might help. Do you want to borrow it?"
Like all such offered books, this one was gratefully accepted but put aside. That is until one stressful sleepless night, when Valerie in desperation decided to crack it open. After skimming through the pages, she noticed the research behind it. "Well now," she thought. "This stuff has really been battle-tested. Let's see what I can find for Eddie and me." Taking some paper and a pencil, she slowed down and made some notes about what she might be able to use.
The next night when Eddie shuffled home, Valerie sat him down and told him flat out, "I'm not happy. Things are going to have to change."
Eddie sighed and, revealing his low expectancy, said, "I'm not happy either, but this is just the way life is. We can't change it."
"You always say that and you're usually wrong," Valerie replied. "I think there are steps we can take to make our life better. My sister lent me a book and it's based on scientific research. I hear it has helped a lot of people and we could use some help ourselves. I think we should at least try some of the ideas. For starters, we just need to lay down a few goals."
Eddie was too exhausted to argue with her, so he played along. "I have a goal," he said with a small smile, "I want to be happy."
"They have to be _specific_ goals," Valerie said patiently. "They have to be concrete and doable, something we can get excited about."
"How about I want to be happy today?" Eddie suggested.
Valerie thumbed her way to the relevant page of the book." We start by making some goals about the minimum changes we need to make to stay sane. I need to see my friends more often. I haven't seen them properly since Constance's baby shower and talking this over with them always makes me feel like my problems are more manageable."
Slumping into a chair, Eddie sullenly replied, "And my goal is to hit the gym every weeknight."
Valerie kept on message. "Get realistic. I think you can spare me one evening every other week. In return, I'm willing to cover you every Saturday morning if you want to exercise."
"That would be nice," admitted Eddie. "But I don't think I am up for handling an evening with Constance on my own."
Valerie pointed out that he often bathed Constance and put her to bed. "I want you to imagine hitting that gym, Eddie, how good your muscles are going to feel afterward. Also, imagine how much happier I'll be around here if I get some time with my friends. Can you picture that? Take a second and bask in its glow. Great! Now open your eyes and come back to reality. Does that give you the motivation?"
"All right," Eddie conceded, warming to the idea. "Let's do it."
With a little mental contrasting to spur them on, Valerie and Eddie's goal-setting techniques and "unschedule" (scheduling in realistic leisure time first) do indeed work. Valerie is seeing her friends, and after sharing her problems and hearing others deal effectively with their own issues, she is gaining a little more perspective. She is reassured that Constance will grow up and the economy will get better. It is amazing what a little social support (see _Vicarious Victory_ ) can do for a person. Eddie himself is glad to get to the gym once in a while. The exercise takes away a lot of his stress. He sleeps a little better and has more energy to tackle the rest of his life (see _Energy Crisis_ ). Still, a few weeks later, Eddie suddenly announces he has to work late and tells Valerie she has to cancel her plans. When he finally gets home, Valerie is not pleased.
Eddie pleads his case, "Look, I'm sorry you missed your night out but I had work to do and that takes precedence."
"Night out?" snapped Valerie. "It's more than a night out. I need that time with my friends. I wouldn't mind if you had to leave on one of your road trips for work but you e-mailed me fifteen times today while you were at the office."
"I thought you liked those texts!" retorted Eddie.
Composing herself, Valerie replied: "Here's what I like. I like face-to-face time with you and with my friends. For every minute you take to text me or send off an e-mail, that's ten minutes less we have at home. It takes ten minutes at least for you to get your mind back into your work after taking a break."
This surprised Eddie, but he wasn't going to give up his text-ing without a fight. "That may be so, but you text too. Besides, I can't work like a machine at the office. I need my breaks."
"And why are you tired?" asked Valerie.
"Well, it's impossible to get to bed early with all the evening work . . ." Then Eddie paused, making the connection. "Oh! Yeah, that might work."
"If we stop texting during business hours, stop Internet surfing, stop mindlessly checking our e-mails, that'll make at least two extra hours each day for the both of us. Hours we can use for sleep."
"My mind will zonk out from so much concentration," said Eddie.
"The book has a few ideas about how to make it work. Start with this. Create a second computer profile for yourself with a different background and layout. Log out of your regular work persona and into this play persona whenever you need a rest. If you aren't willing to take the minute to do it, you don't need the break. Here, I got you a present to help you commit."
"I like presents. What is it?"
Valerie pulled a silver-framed photo from her purse. "A framed picture of Constance and me. Every time you think of slacking off, this will remind you of why we're both pushing ourselves so hard. Remember, this is about us spending more time together as a family. Promise me you'll do this?"
"OK. I'll do it if you do," said Eddie.
And it works, of course. By ridding their workplace of their major temptations (see _Making Paying Attention Pay_ ), they have become more productive in the time they are at work and more relaxed when at home. They are starting to wind down for bed and are getting a better night's sleep, so that they can perform even better (see _Energy Crisis_ ). To help them get to where they need to be and remind them what this is all about, Eddie keeps that framed photo of his family on his desk (see _Games and Goals_ ), especially since it reminds him of what he really wants to do—spend more time at home, not texting at the office (setting approach goals, not avoidance goals). It didn't hurt that Valerie raised the stakes by extracting a little verbal precommitment from Eddie. In the end, they have a little more time than either expected, with both of them hitting the gym at least once or twice a week. Sicknesses, surprises, and other obligations still push them out of their routine, but now they are learning how to push back. They know they are fighting for a life that works. Eventually, Eddie even has the time to do some light reading, which he never used to have the energy for.
After putting Constance to bed, Eddie poured Valerie and himself a cup of tea and plopped into his comfy chair. "I've been looking through this book of yours," he said, "and I see where your ideas come from."
Picking up her own cup, Valerie replied, "Well, the secret was in actually following through with them not just reading the book."
"You're right," said Eddie, "but I have a suggestion of my own."
"Go on. I'm listening," said Valerie.
"Here's a technique called Let Your Passion Be Your Vocation."
Her eyes widening in horror, Valerie gasped, "You're not thinking of leaving work to be a golf pro!"
"No, no, no, I'm not thinking that at all. Well maybe a little, but no," teased Eddie. "But how about this? Getting home earlier is reminding me of how much I used to love to cook. Remember those romantic meals I made for you when we first starting dating? Well, you don't mind cleaning up as much as I do. So, I'll tell you what: I'll do all the cooking if you do the cleaning."
Sweetening the arrangement, Valerie added, "If you throw in grocery shopping too, you've got a deal."
"If cleaning includes laundry, I'll shake on it," said Eddie.
"Done and done."
A sensible pair, they have now allocated the tasks of child-rearing and housekeeping according to their differing tastes and talents. So Eddie does the cooking and shopping for groceries. He goes to the supermarket on Saturday or Sunday and stocks up for the week. This is easy for him because he loves shopping and the peace and quiet of chopping. Valerie, who never cared much about food, watches the baby when Eddie is doing the cooking. She cleans up after him, and she does the never-ending batches of laundry. Constance goes to daycare during the week, and they trade off taking her there early in the morning and picking her up after work. Life is getting better. Not insanely better. Not perfect. Just noticeably better. Valerie and Eddie are beginning to live life in harmony with who they are and what motivates them.
TIME-SENSITIVE TOM
On his journey back home from his disastrous vacation in the Dominican Republic, time-sensitive Tom was delayed at the airport for most of the day. It was hurricane season, which he had not thought about when he planned the trip. Sitting in the lounge, Tom reflected on his life. He was never much of a student, and constantly struggled with deadlines. But he knew that his friends at the fraternity were always glad to see him. An upbeat kind of guy, Tom always had a word of encouragement for the freshmen who were having trouble adjusting to college and being away from home for the first time; he enjoyed helping out. How did he get stuck in such a terrible rut? Without anything else to do, for hours he reflected on how much his procrastination had detracted from his own success, aspirations, and happiness. He thought about how it had affected not only his work life but also his home life. He realized that even if his vacation hadn't been such a mess, much of his leisure time would still have been focused on all the work waiting for him back at the office. He desperately yearned for that childhood feeling of unfettered time and guiltless play unpolluted by pressing obligations. His mind primed, he couldn't help but notice a title in one of the airport bookstores, a book that promised help. After buying it, he read it in its entirety during his wait and then on his flight home. Excited about the book's possibilities, he couldn't wait to put the techniques to use—this time his impulsiveness worked for him rather than against him.
On his first day back at work, Tom purged his office of temptations. He loaded software to keep track of his productivity, and he started setting specific, timely, and challenging goals. The results were immediate. Instead of being constantly behind, Tom found extra time to help others with their projects. "All the better," he thought; he always enjoyed talking and helping the people he worked with. Happy with the results, on a whim he went hardcore and used precommitment, promising to his boss that if he didn't get his next report finished in seven days, they could keep his upcoming year-end bonus. This got his boss's attention. When he handed the report in a day earlier than promised, people were amazed. What had happened to Tom in the Dominican Republic, they wondered. Over time Tom's interest in helping his colleagues and his fidelity to deadlines made his superiors think that he was showing leadership potential, and so they promoted him.
As the excitement of the promotion started to fade, Tom shared the news with his older brother Tim. After a few congratulatory drinks, Tom confessed it wasn't all good, "What did I get myself into? What do I know about leadership? I'm not a leader. I just barely learned how to get myself in shape. You know about these things. You took that leadership course back in college. What should I do?"10b
Tim laughed, "Well, I guess it's too late to say 'don't panic.' But you have a right to be worried. No one who knew you a year ago would have expected you to be doing so well."
"Thanks for taking the pressure off, Tim," Tom replied sarcastically. "I guess you forgot all that leadership material anyway."
Rising to the bait, Tim put down his drink and focused. "Sorry. You're right; you do need to know this stuff. Leadership is important and not just for your organization's success. Most employees rate their relationship with their boss as their top concern. If you screw up, it can make your employees more miserable than if you took away a huge chunk of their paycheck. You now have the power to crush a considerable number of people's spirits."
"And that's why I'm talking to you," said Tom.
"Well, I'm happy to help," Tim replied. "I've been thumbing through that book you lent me and most of the basic leadership techniques are already laid out—you just need to apply them to other people, just the way you did when you applied them to yourself. You can practice leadership along with self-leadership."
"Good, because I am not planning to go back to college," said Tom. "Let's get down to it."
Tim looked up at the ceiling, trying to remember the details. "There are two basic leadership styles: _transformational,_ a people-oriented approach, and _transactional,_ a task-oriented approach. Since you're a people person, Tom, start using your people skills—go transformational!"
"So buddy up to them?" asked Tom.
"Nope," said Tim. "The first thing to do is to focus on creating confidence. What you need is an early success, to help them build faith in you and their ability to succeed under you. It's a basic principle, that you create achievable goals to recognize and celebrate. Later, this will help give everyone the confidence to persevere and hit the harder milestones."
"Ah, create a success spiral!" exclaimed Tom, making the connection.
"Exactly!" said Tim. "I knew a teacher who did this. She built confidence in us by starting off the semester with a few simple quizzes before proceeding into more difficult assignments. I really had a crush on her. One time after class, I remember . . ."
" . . . you're going off topic," interrupted Tom.
"Where was I?" said Tim, finishing his drink. "Well, you can also use the vicarious victory principle by setting the tone. Confidently and clearly articulate a vision of where you want to be, exude optimism, provide pep-talks and in general be the role model. It's textbook."
"Me? Be the role model? What are you thinking?" Tom complained.
"Heavy is the crown . . . Of course, you could always quit or just take their money and wait for them to fire you. To me, that sounds a little bit like stealing, but I guess you have your own moral compass . . ."
Tim looked expectantly at Tom, letting the point linger.
"All right, all right, I'll do it," said Tom. "I was just thinking it through."
So on Tom's first day in charge, he gathered his staff together and gave them a prepared speech about what he intended to accomplish. He told them that though there were areas of excellence in what they had been doing, they were taking too long to finish financial reports despite logging tons of overtime. He then set that first achievable goal. "For starters," he enthusiastically told them, "I want us to cut the average time we take to compile our reports by a day this month. I think we can do it. In fact, I know we can do it." And Tom did know; it was a pretty easy goal. Still, he stayed on message at their weekly meeting, realizing that enthusiasm can be contagious. And at the end of the month, he found that indeed they did cut their production time by a day, precisely one day. "That's a start," he thought to himself, "but really we need to cut our production time by a week." He phoned Tim about his success and his situation.
"Well, that's great news," said Tim. "It's one thing for you to ask for advice but for you to actually follow through is impressive."
"Well, it was good advice to begin with," said Tom, "but enough of this love-fest. I'm not sure the team will keep this up despite the fact that they could easily do much more. What else have you got for me?"
Thinking about it, Tim replied, "Let's think about the value variable. What can you give them that they value? How can you reward them?"
"Do you mean pay them more?" asked Tom.
"Can you do that?"
"Well, no," admitted Tom. "Not unless I want to drain my own bank account."
"Then don't bring it up," said Tim, "but no worries. Money does talk, but it's not the only speaker in this conversation. There's something out there that most people value more than cash—recognition. Simply be aware when they do something right and recognize it in a timely manner—not next month or next week but that day. A person's pride can feed off a sincere 'awesome' or 'job well done' for a long time, while a cup emblazoned with the company logo or even a certified check doesn't provide the same bang for your buck."
"That's an awesome point, Tim."
"Thanks," Tim said warmly, oblivious to the immediate use of the strategy he had just recommended.
"I really like it," said Tom. "If it gets me out of my own office a little more, that's great. I like one-on-one conversations more than those weekly meetings anyway."
"You're lucky. Many managers are promoted solely on technical skills and find the interpersonal part of the job difficult. Since you are so good at it, start using the Games and Goals strategy too. You know the story of the bricklayer, right?"
"Umm, remind me," said Tom, not willing to admit he hadn't heard of it at all.
"It's short. When two bricklayers were asked what they were doing, the first bricklayer replied, 'Building a wall.' The second took his time and, after some thoughtful reflection, responded, 'Building a cathedral.' You want to instill the bigger picture, why what they do matters, because if you do . . ."
" . . . all my dreams will come true," said Tom. "I see what you're doing—giving me the bigger picture. Got it. Timely recognition and frame the picture—communicate why what they do is important."
Tom allocated an hour a day to walk around the office, checking in on people to see how they were doing. If they impressed him, he told them so and sometimes did a little bit more; when one of his employees did a brilliant job presenting, he spontaneously offered to buy her lunch that day. Explaining the significance of the work was a little bit harder. He found that his employees needed the bigger picture to be framed in different ways. For some, how it would help their career made sense; for others, it was positioning an assignment as a symbol of responsibility; and for still others, it was about how their work affected their colleagues. Finding the right frame for the right person was a bit of a puzzle, but he got it correct more often than not. With one difficult employee, he explained it this way: "When you are done with your piece, it passes on to Suzanne. If you are late, she has to stay here late, which means scrabbling to find someone to pick up her kids from daycare, feed them, and put them into bed. You finish early, you make Suzanne's life easy. You finish late, you make her life hell." He never had trouble with that employee again. For good measure, he also tried to respect his employees' chronobiology and energy levels by instituting some _flextime._ Looking up some research, he found that just as students improved by a letter grade when allowed to sleep in by an hour, corporations that enabled flextime, allowing their employees to show up later but stay later in return, saw a nice bump in work performance.
One night after work, Tim picked up Tom for dinner at a favorite restaurant. After they were seated and had ordered their food, Tim asked, "How's the leadership thing going?"
"Great," bragged Tom. "This transformer leadership is a snap."
"That's transformational leadership," said Tim. "Transformer is a type of robot, like Megatron or Optimus Prime."
Tom had been joking, but he corrected himself, "That's right, transformational and transactional." Then, he quickly changed the topic. "Speaking of which, you never told me about transactional leadership."
"Well, most people tend to favor one style or the other," said Tim, "but the best leaders have a combination of both. Transactional leaders excel at making plans, assigning tasks, _and_ goal setting."
"Aah! This makes so much sense," said Tom. "I never knew what a pain my procrastination was to other people until I had to deal with procrastinators myself. Goal setting worked for me and it'll work for them too."
"Yup. That's what transactional leaders do. They divide distant deadlines into a series of short-term, specific, and realistic goals for their employees. Of course, too many goals and you become a micro-manager, also known as a control freak."
"I'm in no danger of becoming that. But still, how many goals do I need to set?" asked Tom.
"There's no firm answer on that," admitted Tim. "Essentially, people work hardest as the clock runs out, so you want to set as many deadlines as practical. At least have regular meetings, where you review people's progress and set new milestones. Keep in mind that some are already self-motivated and don't need much, while others need a lot."
"Yeah, I'm thinking of a few people who could benefit from minute-to-minute goals," said Tom.
"Just don't do what I have seen your company do," said Tom, as he began mimicking a pompous corporate voice: "'We want to raise revenue by 20% by year end!' That never works. I don't even know why they bother."
"I know what you mean. It's so distant and abstract, nobody can get motivated by it. Also, I don't know if anyone thinks it's even realistically achievable, especially with the downturn in the economy."
While looking over Tom's shoulder to see if their food would be arriving, Tim said, "Last year, your company made the goal too easy. When it's too easy, people do what they do when they cross any finish line—they coast, leaving extra performance on the table."
"Just like my initial goal," said Tom, "where everyone beat my schedule by _exactly_ one day. I thought it was suspicious. I guess it's time for me to set the bar a little higher."
"When you do—and if I know you, you are going to love this—try partying," said Tim.
"You've got my attention. Go on."
"Don't forget to have a party at the end when you do accomplish it," said Tim. "People remember two things about a task: its best moment and the end moment. A party at the end will make it seem all worthwhile."
"I get it. Like good food at the end of a long conversation," said Tom, noticing that the waitress had finally arrived with their order.
Tom started to incorporate effective goal setting. When asking what his employees were up to, he kept pushing them to make concrete, short-term, and challenging goals. When he met them later, he had them give him updates on their progress. Some were naturals at this and used the opportunity to brag, which was fine with Tom—he was paying them with recognition. Others needed to be coaxed. Finally, he set a big group goal: they were going to cut production time by a week this month and if they could do it, which he told them he was sure they could, they were going to cut out early the following Friday for a party. Babysitting for parents and cab rides home for everyone would be on the company's tab. For the rest of the month, his team worked with a purpose and met the goal. The party was fantastic—as much a reward for Tom as for his team. He loved parties. In fact, whenever "his crew," as he started to call them, looked as if they weren't going to meet a goal, he doubled his efforts to make sure they met it and won the blow out. "Next time," he thought, "I'll put some money in the budget for a white-water rafting extravaganza. I can probably expense it as a team-building exercise anyway. And a prize for whoever gets the most reports out this month too."
Just when Tom was starting to get comfortable in this role as leader and manager, word came down from higher up. Unlike most other department heads, Tom was getting his budgets in on target and did his performance appraisals ahead of time. His performance was exceptional and his department was consistently the most satisfied and the most productive in his workgroup. Inevitably, he was to be promoted once again. The secret to Tom's success was simply learning that what motivated other people was pretty much the same as what motivated him. To follow in his footsteps and become a better leader, you need to do the same. Good leadership is a skill that the world eagerly, even desperately, wants you to possess.
A WORD OF WARNING
Eddie, Valerie, and Tom benefited from enacting the principles of the Procrastination Equation, repeatedly hitting the three key components of expectancy, value, and time. When you put into practice the suggestions put forth here, you will benefit too. Just don't overdo it. While procrastination can lead to an inauthentic life, in which long-term dreams sour inside you, so can our efforts to completely eliminate procrastination. A genuine and autonomous individual seeks a life endorsed by the whole self, not just a fragment of it. Trying to squelch your impulsive side entirely is ultimately self-defeating; the wants and appetites that propel a life depend upon being attended to. Overregulation—seeking the perfect over the real—isn't healthy and won't make you happy. You are going to have to find a balance.
Just as the Procrastination Equation's techniques can work too well, so could the techniques in Will Ferguson's fictional self-help book. In his novel, after people read _What I Learned on the Mountain,_ they did become blissful, contented, kind, and vice-free. They replaced their cigarettes and alcohol addictions with hugs and self-acceptance and swapped their oversized cheeseburgers with sensibly sized ones made from tofu. But all this virtue came at a cost: though everyone was equally content, they were also equally bland, interchangeable, and forgettable. Their personalities were whitewashed by their yearning to overcome all their flaws, and along with their vices so went desserts, fashion sense, and desire.
Procrastination represents a single swing of the pendulum, an emotional short-sightedness that sees only the present. As the pendulum swings to the other side, rational far-sightedness can become equally troublesome; we tend to focus only on the future. When asked about their past regrets, workaholic employees wished they had occasionally goofed off, and exceptionally industrious students regretted studying through Spring Break.10c Consequently, optimal self-control involves not the denial of emotions but a respect for them. Not all indulgent delays are irrational. You need to have moments of expression, when you can laugh freely with friends, or let yourself go to be indulged and pampered. Using the words of W. H. Davies, a vagabond Welsh poet of my mother's youth: "What is this life if, full of care, we have no time to stand and stare." To be idle, frivolous, spontaneous, and whimsical—these qualities deserve a place in our lives too.
LOOKING FORWARD
Nine thousand years ago, procrastination didn't exist. Back then, if we worked when motivated, slept when sleepy, and acted on other urges as they came upon us, we did so more or less adaptively. In that golden age, our compulsions fit our daily demands like jigsaw puzzle pieces. We were designed for that world, life before the invention of agriculture. Fast forward nine thousand years and that same human nature has equipped us with inclinations that are ill-suited to the everyday. We have to-do lists filled with diets, early wake-ups, and exercise schedules, among a host of other ugly and motivationally indigestible ordeals. Almost every aspect of our lives reflects this maddening mismatch between our desires and our responsibilities, as we overemphasize the present and sacrifice the future. We overindulge in the immediate pleasures of fats, sugars, and television, putting off dieting and exercise. We let loose anger and rage, putting off needed reflection and reconciliation. We have predilections toward the easy pleasure of promiscuity, risking long-term relationships and reproductive health for the forbidden but immediate. Each of these examples reflects a nature that was once adaptive but is no longer, a nature that outrageously values the now more than the later. The story, however, doesn't need to end here.
As _The Procrastination Equation_ stresses, irrationally putting off is a tendency, not an inevitability. If we can accept our internal state of affairs, we can counter it. Instead of believing we have the temperament of divine beings, we can reconcile ourselves with our humanity—to the fact that we are flawed and compromised creatures—and act accordingly. We can make procrastination an irrational delay of yesterday, what we all once did or didn't do, but only if we acknowledge our own limitations and adopt advice consistent with this understanding. To put it all into practice, you don't need to ask permission. There will be no handwritten invitation. To live your life as you always wanted to, to be the person you always wanted to be, you know what to do. You are holding all the answers in your hands. Now do it.
[Postscript
Procrastination's Chapter 11](contents.xhtml#post_01)
The beauty of procrastination is its ubiquity; tracking its scent leads into dozens of scientific fields. If you duplicate my path, you will start with psychology, where the bulk of the work has been done, but you will quickly find yourself in economics, which is becoming a dominant force on the topic. You will take a stroll through the applied issues, like retirement or debt procrastination, perhaps taking a peek into the legal implications, such as suggested bankruptcy laws. From economics, you would naturally wander into neuroeconomics and become interested in the neurobiology of procrastination, a detour that, of course, would give you a chance to look at the basis of all biological study, evolution. You would learn that procrastination is a common and consistent human trait, one we share with species across the animal kingdom. Then, instead of considering where we came from, you might reverse your perspective and see where we are going, getting into societal issues, especially long-term concerns like environmental degradation. If you start wondering why the government doesn't do more, you will discover that they and other organizations have procrastination problems of their own.
Having been studied in so many disciplines, procrastination has become a Rosetta Stone, where the same phenomenon is translated into a dozen tongues. This pool of resources allows us not only to translate findings from different fields, such as from economics into psychology, but also to form a common language of human behavior, an Esperanto of the social sciences. It's an important accomplishment. As Christopher Green concludes, writing in psychology's premier journal, "it [integration] would doubtless be considered the greatest scientific victory in the history of the discipline," one that can rescue psychology from the realms of a "would-be science." And if you can integrate psychology with economics, sociology, and biology too, even better. This was actually my original purpose in creating the Procrastination Equation—to help integrate the social sciences.
Unfortunately for procrastination, its pervasiveness makes it an obvious target. Having a common basic model, one that each discipline can adopt and customize, could be incredibly dangerous to our familiar enemy. Integration enables exponentially more progress in all disciplines. This understanding has already permitted the physical sciences to provide an endless stream of game-changing advances, from the laptop I am using to write this book to the nuclear energy that powers the electrical lines. By working from a common model of reality, the physical sciences share and pass knowledge across specialties and research foci. Similarly, such synergy could supercharge the social sciences. Herbert Gintis, an economics professor emeritus from the University of Massachusetts who has long argued for integration, concludes: "The true power of each discipline's contribution to knowledge will only appear when suitably qualified and deepened by the contribution of the others." You see, it is all connected, all of it, as we are all studying the same thing: people's decision making and behavior. As one area informs the other, our fight against procrastination necessarily gives insights into reducing obesity, building better retirement savings programs, and much, much more.
Once disciplinary integration comes about, we will have gone a long way toward truly mastering our own minds. As it currently stands, we as a society can do better. Consider that the top two ways that people procrastinate are through their televisions and through their computers—about a quarter of their waking hours in some parts of the world. People seeking help to curb their addictions freely acknowledge that they use these temptations to excess. Because TV watching has been associated with the rise of obesity and the erosion of the family, huge efforts have been put forth to reduce our consumption. Nothing has been truly effective; the hours used and incidence of abuse rise yearly. If we adopt a more integrated viewpoint, using some of the principles from _The Procrastination Equation,_ we can change this. We only need to apply the principles of self-control to our own technology.
When I watch too much TV, I blame my digital video recorder (DVR). It makes it easy for me to find a show I like and watch it when I want. Naturally, the easier it is to find good programming and the faster it can be accessed, the more I will make use of it. You will too. Though DVRs are part of the problem, they are going to be part of the solution too, as they are also the perfect platform to enable self-control techniques. Self-control improves when we receive accurate feedback about our behavior, which we can then use as reminding cues and to help us set goals (see _Scoring Goals_ and _Making Attention Pay_ ). An add-on for a DVR could be a prominent digital display that reflects how much TV you have watched today or this week. As you see the hours visibly rise while you watch, so will the desire to turn off the set. The DVR could even track your long-term viewing, calculating when and what you are watching.
Also, DVRs could permit precommitment. Devices are available to enable parents to limit the viewing habits of their kids, but there are few options for parents themselves. With a DVR, a series of precommitment measures could be incorporated. The first few could just be devices for enabling delays. A long code, for example, could be laboriously inputted before viewing. Alternatively, it could lock you out for a few minutes, or perhaps require confirmation multiple times, giving you a chance to have second thoughts. As delay lengthens and impulsive choices become impossible, you should be able to make more rational use of your viewing time. If this isn't enough, you could lock yourself out temporarily, perhaps only allowing viewing within given time periods or up to a total number of hours each day. Best of all, whichever of these options we as viewers want to activate, if any, the choice—the intention—is ultimately ours.
For Internet procrastination, similar solutions are already on the market. Attentional control programs like _RescueTime,_ which let you see exactly what you have been doing with your day, are freely available. As an added feature, _RescueTime_ also assists in goal setting and permits the creation of comparison work groups, thereby activating the _Vicarious Victory_ principle. Cyberly seeing others hard at work should inspire, or at least spark, your competitive spirit. Furthermore, _RescueTime_ allows you to voluntarily block your own access to the Internet for chosen periods of time, permitting a precommitment strategy that eliminates distractions. If this could be complemented with a sophisticated and difficult-to-subvert nannyware program—like _Chronager,_ except self-administered—it is hard to imagine a more effective self-control platform. Right now, the pieces are all there; we just have to bundle them together.
These tools for rationalizing television and computer use could be easy to build and implement. Though not yet fully developed, they have almost coalesced. When they are finally built, the market is virtually everyone, but definitely the chronically procrastinating quarter of the population. These tools would have society-wide effects and an observable impact on national GDP; if they cut procrastination even by half, that would amount to trillions of extra productivity each year worldwide. With further advances in integration, more such tools that address our own weak wills should become commonplace, designed into our society's fabric. And ironically, for all this, we can partly thank procrastination. Fittingly for an irrational self-defeating delay, by making possible the groundwork for integration, procrastination may have contributed to its own defeat.
Notes
Author's Note
And everyone knew it. Here is an excerpt from a letter my late brother sent to my uncle: "Have you heard from Piers about his research? He has forged himself into an expert on procrastination, publishing numerous articles on the subject and being interviewed on national radio and in the press. I get a chuckle, as Piers was the worst procrastinator during his high school and college years."
Even the philosophers have been fascinated by procrastination, able to sit and watch it for hours:
Andreou, C. (2007). Understanding procrastination. Journal for the Theory of Social Behavior, 37(2), 183–193.
Gosling, J. (1990). Weakness of the will. New York: Routledge.
Silver, M. (1974). Procrastination. Centerpoint, 1(1), 49–54.
Sorensen, R. (2006). Originless sin: Rational dilemmas for satisficers. The Philosophical Quarterly, 56(223), 213–223.
Katz, I., de Deyn, P., Mintzer, J., Greenspan, A., Zhu, Y., & Brodaty, H. (2007). The efficacy and safety of risperidone in the treatment of psychosis of Alzheimer's disease and mixed dementia: A meta-analysis of 4 placebo-controlled clinical trials. International Journal of Geriatric Psychiatry, 22(5), 475–484.
Lee, J., Seto, D., & Bielory, L. (2008). Meta-analysis of clinical trials of probiotics for prevention and treatment of pediatric atopic dermatitis. The Journal of Allergy and Clinical Immunology, 121(1), 116–121.
Bowen, F., Rostami, M., & Steel, P. (2010). Meta-analysis of organizational innovation and performance. Journal of Business Research.
Caird, J., Willness, C. R., Steel, P., & Scialfa, C. (2008). A meta-analysis of the effects of cell phones on driver performance. Accident Analysis & Prevention, 40(4), 1282–1293.
Peloza, J., & Steel, P. (2005). The price elasticities of charitable contributions: A meta-analysis. Journal of Public Policy & Marketing, 24(2), 260–272.
Taras, V., Kirkman, B. L., & Steel, P. (2010). Examining the impact of Culture's Consequences: A three-decade, multi-level, meta-analytic review of Hofstede's cultural value dimensions. Journal of Applied Psychology, 95 (3), 405–439.
Steel, P. & Kammeyer-Mueller, J. (2002). Comparing meta-analytic moderator search techniques under realistic conditions. Journal of Applied Psychology, 87(1), 96–111.
Steel, P., & Kammeyer-Mueller, J. (2008). Bayesian variance estimation for meta-analysis: Quantifying our uncertainty. Organizational Research Methods, 11(1), 54–78.
Steel, P., & Kammeyer-Mueller, J. (2009). Using a meta-analytic perspective to enhance Job Component Validation. Personnel Psychology, 62, 533–552.
Steel, P., & Ones, D. (2002). Personality and happiness: A national level of analysis. Journal of Personality and Social Psychology, 83(3), 767–781.
Steel, P., & Taras, V. (2010). Culture as a consequence: A multilevel multivariate meta-analysis of the effects of individual and country characteristics on work-related cultural values. Journal of International Management.
Steel, P., Schmidt, J., & Shultz, J. (2008). Refining the relationship between personality and subjective well-being. Psychological Bulletin, 134(1), 138–161.
Chapter One
Astrologers also cluster the twelve signs of the zodiac into quadruplicities, with the Gemini, Virgo, Sagittarius, and Pisces group being particularly relevant. Quoting Bertrand Russell, this "quadruplicity drinks procrastination," as both Sagittarius and Pisces are inebriated with the trait. If you found that sentence awkward, I have a confession. In truth, Bertrand Russell actually intended these words to be an example of a sentence with correct grammar, but whose meaning is nonsensical. However, I was inspired to make sense out of nonsense by the Harvard linguist Yuen Ren Chao, who was Russell's interpreter when he went to China in the 1920s. He did the same for "Colorless green ideas sleep furiously." On the other hand, this type of wordplay doesn't increase your popularity.
Gendler, T. S. (2007). Self-deception as a pretense. Philosophical Perspectives, 21(1), 231–258.
Gosling, J. (1990). Weakness of the will. New York: Routledge.
Martin, M. (1986). Self-deception and morality. Lawrence, KS: University Press of Kansas.
Steel, P. (2007). The nature of procrastination: A meta-analytic and theoretical review of quintessential self-regulatory failure. Psychological Bulletin, 133(1), 65–94.
See www.43things.com, a website that has helped millions of people create their life lists.
Horn, S. (2001). ConZentrate: Get focused and pay attention—when life is filled with pressures, distractions, and multiple priorities. New York: Saint Martin's Press.
As part of my research, I documented the professions of 20,000 self-professed procrastinators. For example, it even afflicts pageant contestants; Sara Hoots, former Miss Hooters winner, revealed in her audition video: "My worst trait is procrastination." However, astronaut and zookeeper weren't among the ones I recorded. For those instances, we have confessions from Slate's "Procrasti-Nation: Workers of the world, slack off!"
Gröpel, P., & Steel, P. (2008). A mega-trial investigation of goal-setting, interest enhancement, and energy on procrastination. Personality and Individual Differences, 45, 406–411.
Silverman, I. (2003). Gender Differences in Delay of Gratification: A Meta-Analysis. Sex Roles, 49(9), 451–463.
Take your pick:
Burka, J. B., & Yuen, L. M. (1983). Procrastination: Why you do it, what to do about it. Reading, MA: Addison-Wesley.
Fiore, N. (1989). The now habit: A strategic program for overcoming procrastination and enjoying guilt-free play. New York: Penguin Putnam, Inc.
Knaus, W. (2002). The procrastination workbook: Your personalized program for breaking free from the patterns that hold you back. Oakland, CA: New Harbinger Publications, Inc.
Peterson, K. E. (1996). The tomorrow trap: Unlocking the secrets of the procrastination-protection syndrome. Deerfield Beach, FL: Health Communications, Inc.
McGarvey, J. (1996). The almost perfect definition. Research/Penn State, 17(3). Retrieved from http://www.rps.psu.edu/sep96/almost.html.
In addition to my "Nature of Procrastination" article, see also:
Canter, D. (2008). Self-appraisals, perfectionism, and academics in college undergraduates. Unpublished PhD, Virginia Commonwealth University, Richmond, VA.
Yao, M. (2009). An exploration of multidimensional perfectionism, academic self-efficacy, procrastination frequency, and Asian American cultural values in Asian American university students. Unpublished PhD, Ohio State University, Columbus, Ohio.
Pullen, F. J. (2003). Perfectionism, procrastination, and other self-reported barriers to completing the doctoral dissertation. Unpublished PhD, The University of Iowa, Iowa City, IA.
Schouwenburg, H. C. (2004). Academic procrastination: Theoretical notions, measurement, and research. In H. C. Schouwenburg, C. H. Lay, T. A. Pychyl, & J. R. Ferrari (Eds.), Counseling the procrastinator in academic settings (pp. 3–17). Washington, DC: American Psychological Association.
Arce, E., & Santisteban, C. (2006). Impulsivity: A review. Psicothema, 18(2), 213–220.
Bembenutty, H., & Karabenick, S. A. (2004). Inherent association between academic delay of gratification, future time perspective, and self-regulated learning. _Educational Psychology Review, 16_ (1), 35–57.
Enticott, P., & Ogloff, J. (2006). Elucidation of impulsivity. _Australian Psychologist, 41_ (1), 3–14.
Whiteside, S., & Lynam, D. (2001). The Five Factor Model and impulsivity: Using a structural model of personality to understand impulsivity. _Personality and Individual Differences, 30_ (4), 669–689.
Bui, N. H. (2007). Effect of evaluation threat on procrastination behavior. Journal of Social Psychology, 147(3), 197–209.
Schouwenburg, H. C. (2004). Academic procrastination: Theoretical notions, measurement, and research. In H. C. Schouwenburg, C. H. Lay, T. A. Pychyl, & J. R. Ferrari (Eds.), Counseling the procrastinator in academic settings (pp. 3–17). Washington, DC: American Psychological Association.
Chapter Two
Overmier, J. B., & Seligman, M. E. P. (1967). Effects of inescapable shock upon subsequent escape and avoidance responding. Journal of Comparative and Physiological Psychology, 63, 28–33.
Seligman, M., & Csikszentmihalyi, M. (2000). Positive psychology: An introduction. American Psychologist, 55, 5–14.
Seligman, M. E. P., & Maier, S. F. (1967). Failure to escape traumatic shock. Journal of Experimental Psychology, 74, 1–9.
When I was taught about learned helplessness, my instructor related to me a story about a captured cricket. If you put a cricket in a jar, give it some food and water, and punch in some small air holes for breathing, it will try to escape, launching itself up and hitting its head against the lid. Come back in a few days and take the lid off. The cricket will jump but just stop short of where the top used to be. It can escape at any time, but no longer engages in the behavior that will realize its freedom—the cage is now inside its head.
Beck, A. T., & Beck, R. W. (1972). Screening depressed patients in family practice: A rapid technique. Postgraduate Medicine, 52, 81–85.
Sadly, procrastination itself can even be a cause of the deviation-amplifying loop known as a depression spiral. That is, depression may lead to procrastination, which can cause guilt and self-loathing that deepens the depression, which completes the cycle by causing more procrastination. Such a hollowing out of existence can be further exacerbated if the activities one is putting off are of community and accomplishment, both of which help avoid depression from the start.
Thase, M. E. (1995). Cognitive behavior therapy. In I. D. Glick (Ed.), Treating depression (pp. 33–70). San Francisco: Jossey-Bass, Inc.
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McCrea, S., Liberman, N., Trope, Y., & Sherman, S. (2008). Construal level and procrastination. Psychological Science, 19(12), 1308–1314.
Here is Hume reflecting on how the nearby and concrete always seems to supersede the long-term and abstract: "In reflecting on any action which I am to perform a twelvemonth hence, I always resolve to prefer the greater good, whether at that time it will be more contiguous or remote; nor does any difference in that particular make a difference in my present intentions and resolutions. My distance from the final determination makes all those minute differences vanish, nor am I affected by anything but the general and more discernible qualities of good and evil. But on my nearer approach, those circumstances which I at first overlooked begin to appear, and have an influence on my conduct and affections. A new inclination to the present good springs up, and makes it difficult for me to adhere inflexibly to my first purpose and resolution. This natural infirmity I may very much regret, and I may endeavour, by all possible means, to free myself from it."
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The graph includes two-thirds of the courses students, excluding those who dropped the course or had finished the entire course work more than four days before the deadline and couldn't be potentially procrastinating in this part of their lives. See also, the following articles that find procrastination almost perfectly fits a hyperbolic curve.
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Chapter Three
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As William James, the godfather of psychology, puts it when discussing the economic theory of behavior: "Not one man in a billion, when taking his dinner, ever thinks of utility. He eats because the food tastes good and makes him want more."
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As Adam Gifford puts it:
Evolution cannot discard existing designs and start over from scratch, it can only build the new on top of the old—the old higher biology-based time preference mechanisms are still built into the human brain. These mechanisms must be overridden in decision making by the inhibition process, which is significantly enhanced in humans by language. It is this divergence between the cultural and biological rates of time preference that creates a potential internal nature versus nurture conflict leading to self-control problems [like procrastination]. The higher level prefrontal working memory system allows the agent to consider possible events in the extended future and to discount those events at a rate appropriate to the individual's current environment. The lower level [limbic system] does not have access to events not yet experienced, and as a result, ignores these purely abstract events; it also incorporates the high level discount rate similar to that used by non-human primates and some other mammals that is a product of natural selection.
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Bechara, A. (2005). Decision making, impulse control and loss of willpower to resist drugs: A neurocognitive perspective. Nature Neuroscience, 8, 1458–1463.
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Gifford, A. (2002). Emotion and self-control. Journal of Economic Behavior & Organization, 49, 113–130.
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Reyna, V. F., & Farley, F. (2006). Risk and rationality in adolescent decision making: Implications for theory, practice, and public policy. Psychological Science in the Public Interest 7(1), 1–44.
Rosso, I. M., Young, A. D., Femia, L. A. & Yurgelun-Todd, D. A. (2004). Cognitive and emotional components of frontal lobe functioning in childhood and adolescence. Annals of the New York Academy of Sciences, 1021, 355–362.
Wood, J. N., & Grafman, J. (2003). Human prefrontal cortex: Processing and representational perspectives. Nature Reviews, 4, 139–147.
Yurgelun-Todd, D. A. & Killgore, W. D. S. (2006) Fear-related activity in the prefrontal cortex increases with age during adolescence: A preliminary fMRI study. Neuroscience Letters, 406, 194–199.
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Indeed, the reason why pigeons can procrastinate is that they do have a prefrontal cortex counterpart, the nidopallium caudolaterale.
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As Cesar Millan stressed, to instill discipline in your pet, you need to have discipline in yourself. "Exercise, discipline, and affection." Too often the middle ingredient is left out.
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There is also a Wikipedia page: http://en.wikipedia.org/wiki/Owen_Owen
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To investigate this topic, I tried to locate a 1971 book by Paul T. Ringenbach, Procrastination through the Ages: A Definitive History. Ringenbach is a U.S. Air Force Officer with a PhD from the University of Connecticut. His work was described as "an interesting survey" by the late Albert Ellis on the very first page of his book Overcoming Procrastination, making it a must-have for anyone interested in the topic. After spending weeks hunting with a pack of librarians, I finally found some correspondence with Gil Campbell from Filter Press, the book's publisher, buried in the appendix of an old 1982 doctoral thesis by Margaret Aitken. The letter indicated that Procrastination through the Ages was never actually written. Colonel Ringenbach was asked to write it, but kept putting it off for so long that it metamorphosized into an elaborate prank, with Campbell telling everyone for fifteen years that it was coming out imminently. I tracked Colonel Ringenbach to his Texan address, where after a series of e-mails and phone calls, I extracted a full confession.
By the speed of my response, I guess you realize that procrastination is alive and well. Procrastination through the Ages: A Definitive History first appeared in Books in Print in the 1971–1972 edition. How it came about was that Gil Campbell of the Filter Press was also the Acquisitions chief at the US Air Force Academy when I first met him . . . He asked me to do a short piece for him on "Black Cowboys" that he could publish. After a time with no progress, he suggested I write a book on procrastination because I was so good about it. Months rolled on with no progress, so finally he said give me a title; I want to include it in my next catalogue. I gave him the title and he did not print it in the catalogue on purpose but included it as a loose insert on colored paper with the excuse to the readers that he hadn't got around to including it in the actual text, but here it is anyway. At this point he put it in Books in Print with a date not set, price not set notation. After all we surmised, how could one ever complete a book on procrastination? It continued in Books in Print about 15 years until Gil took it out because he was tiring of the continuing inquiries that he always sent along dutifully to me to reply.
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Handily available in book form as well, called the Phillipics.
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Chapter Four
Though officially not being "associated with Risk or Hasbro in any way."
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Applebome, P. (2004, December 1, 2004). On campus, hanging out by logging on. New York Times.
Aspan, M. (February 13, 2008). Quitting Facebook gets easier. New York Times.
Kessler, D. A. (2009). The end of overeating: Taking control of the insatiable American appetite. New York: Rodale.
Offer, A. (2006). The challenge of affluence: Self-control and well-being in the United States and Britain since 1950. New York: Oxford University Press.
Dittmar, H. (2005). Compulsive buying—a growing concern? An examination of gender, age, and endorsement of materialistic values as predictors. British Journal of Psychology, 96, 467–491.
LaRose, R., & Eastin, M. S. (2002). Is online buying out of control? Electronic commerce and consumer self-regulation. Journal of Broadcasting and Electronic Media, 46(4), 549–564.
Percoco, M. (2009). Estimating individual rates of discount: A meta-analysis. Applied Economics Letters, 6(12), 1235–1239.
Verplanken, B., & Herabadi, A. (2001). Individual differences in impulse buying tendency: Feeling and no thinking. European Journal of Personality, 15, 71–83.
Youn, S., & Faber, R. J. (2000). Impulse buying: Its relation to personality traits and cues. Advances in Consumer Research, 27, 179–185.
Baumeister, R. F. (2002). Yielding to temptation: Self-control failure, impulsive purchasing, and consumer behavior. Journal of Consumer Research, 28, 670–676.
Baumeister, R., Sparks, E., Stillman, T., & Vohs, K. (2008). Free will in consumer behavior: Rational choice and self-control. Journal of Consumer Psychology, 18, 4–13.
LaRose, R., & Eastin, M. S. (2002). Is online buying out of control? Electronic commerce and consumer self-regulation. Journal of Broadcasting and Electronic Media, 46(4), 549–564.
Lynch, J. G., & Zauberman, G. (2006). When do you want it? Time, decisions, and public policy. Journal of Public Policy & Marketing, 25(1), 67–78.
Ziglar, Z. (1991). Ziglar on selling. New York: Thomas Nelson.
Kessler, D. A. (2009). The end of overeating: Taking control of the insatiable American appetite. New York: Rodale.
Duhigg, C. (July 13, 2008). Warning: Habits may be good for you. New York Times.
Ji, M., & Wood, W. (2007). Purchase and consumption habits: Not necessarily what you intend. Journal of Consumer Psychology, 17(4), 261–276.
Wood, W., & Neal, D. T. (2007). A new look at habits and the habit-goal interface. Psychological Review, 114(4), 843–863.
Wansink, B. (2006). Mindless eating: Why we eat more than we think. New York: Bantam-Dell.
Ariely, D., Loewenstein, G., & Prelec, D. (2006). Tom Sawyer and the construction of value. Journal of Economic Behavior & Organization, 60(1), 1–10.
Lindstrom, M. (2005). BRAND sense: Build powerful brands through touch, taste, smell, sight, and sound. New York: Free Press.
Ramanathan, S., & Menon, G. (2006). Time-varying effects of chronic hedonic goals on impulsive behavior. Journal of Marketing Research, 43(4), 628–641.
Wood, W., & Neal, D. T. (2007). A new look at habits and the habit-goal interface. Psychological Review, 114(4), 843–863.
Caird, J., Willness, C. R., Steel, P., & Scialfa, C. (2008). A meta-analysis of the effects of cell phones on driver performance. Accident Analysis & Prevention, 40(4), 1282–1293.
Strictly speaking, there were a few other categories but all in the same genre, such as Shazam for "Music" or the Virtual Zippo Lighter for "Lifestyle."
Huxley, A. (2004). Brave New World and Brave New World Revisited. New York: HarperCollins.
Postman, N. (1985). Amusing ourselves to death: Public discourse in the age of show business. New York: Penguin Group.
Offer, A. (2006). The challenge of affluence: Self-control and well-being in the United States and Britain since 1950. New York: Oxford University Press.
Novotney, A. (July/August, 2008). What'$ behind American con$umeri$m? Monitor on Psychology, 39(7), 40–42.
Vyse, S. (2008). Going broke: Why Americans can't hold on to their money. New York: Oxford University Press.
Davenport, T., & Beck, J. (2001). The Attention Economy: Understanding the new currency of business. Harvard Business School Press.
Shenk, D. (1997). Data smog: Surviving the information glut. New York: HarperCollins.
Chapter Five
Ferrari, J. R., Barnes, K. L., & Steel, P. (2009). Life regrets by avoidant and arousal procrastinators: Why put off today what you will regret tomorrow? Journal of Individual Differences, 30(3), 163–168.
Roese, N. J., & Summerville, A. (2005). What we regret most . . . and why. Personality and Social Psychology Bulletin, 31(9), 1273–1285.
Steel, P., Schmidt, J., & Shultz, J. (2008). Refining the relationship between personality and subjective well-being. Psychological Bulletin, 134(1), 138–161.
Baer, M., & Oldham, G. R. (2006). The curvilinear relation between experienced creative time pressure and creativity: Moderating effects of openness to experience and support for creativity. Journal of Applied Psychology, 91, 963–970.
Amabile, T. M., Hadley, C. N., & Kramer, S. J. (2002). Creativity under the gun. Harvard Business Review, 80(8), 52–61.
Steel, P. (2007). The nature of procrastination: A meta-analytic and theoretical review of quintessential self-regulatory failure. Psychological Bulletin, 133(1), 65–94.
Pychyl, T. A., Lee, J. M., Thibodeau, R., & Blunt, A. (2000). Five days of emotion: An experience-sampling study of undergraduate student procrastination. Journal of Social Behavior & Personality, 15(5), 239–254.
Patry, D. A., Blanchard, C. M., & Mask, L. (2007). Measuring university students' regulatory leisure coping styles: planned breathers or avoidance? Leisure Sciences, 29(3), 247–265.
Bernold, L. E. (2007). Preparedness of engineering freshman to inquiry-based learning. Journal of Professional Issues in Engineering Education and Practice, 133, 99–106.
Doherty, W. (2006). An analysis of multiple factors affecting retention in Web-based community college courses. The Internet and Higher Education, 9(4), 245–255.
Finck, J., & DeLine, A. (2008). Do students listen to advice from their experienced peers? College Teaching Methods & Styles Journal, 4(9), 19–26.
Laven, A. V. (2007). Freshmen college student mental health and their resource usage. Unpublished EdD dissertation, University of California, Los Angeles, CA.
Moore, B. (2006). Goal conflicts, self-regulation, and course completion: A comparison of Web-based learners to traditional classroom learners. Unpublished PhD dissertation, University of South Florida, Tampa, FL.
Bair, C. R., & Haworth, J. G. (2004). Doctoral student attrition and persistence: A meta-synthesis of research. Higher education: Handbook of theory and research, 19, 481–534.
Green, G. D. (1981). Dissertation procrastination. Unpublished PhD dissertation, University of Washington, Seattle, WA.
Muszynski, S. Y., & Akamatsu, T. J. (1991). Delay in completion of doctoral dissertations in clinical psychology. Professional Psychology—Research & Practice, 22(2), 119–123.
Mariano, C. M. (1993). A study of Ed.D.s, Ph.D.s and ABDs in educational administration (dissertation completion, Ed.D. candidates, Ph.D. candidates). Unpublished EdD dissertation, Boston College, Boston, MA.
Pullen, F. J. (2003). Perfectionism, procrastination, and other self-reported barriers to completing the doctoral dissertation. Unpublished PhD dissertation, University of Iowa, New Haven, IA.
Based on the average salary difference between those with a Master's and those with a Doctorate education.
Lacey, J. & Crosby, O. (2005). Job outlook for college graduates. Occupational Outlook Quarterly, 48(4), 15–27.
Lay, C. H., & Brokenshire, R. (1997). Conscientiousness, procrastination, and person-task characteristics in job searching by unemployed adults. Current Psychology: Developmental, Learning, Personality, Social, 16(1), 83–96.
Senecal, C., & Guay, F. (2000). Procrastination in job-seeking: An analysis of motivational processes and feelings of hopelessness. Journal of Social Behavior & Personality, 15(5), 267–282.
Nawrocki, J. (2006, June 15, 2006). When you're a GC, procrastination doesn't work. Corporate Counsel, Retrieved from http://www.law.com/jsp/ihc/PubArticleIHC.jsp?id=1150275918375
Angeletos, G.-M., Laibson, D., Repetto, A., Tobacman, J., & Weinberg, S. (2001). The hyperbolic consumption model: Cali-bration, simulation, and empirical evaluation. Journal of Economic Perspectives, 15(3), 47–68.
Bankston, J. (2001). IRS experts blame procrastination for simple oversights on tax returns. The Augusta Chronicle, GA. Knight Ridder/Tribune Business News.
Kasper, G. (2004). Tax procrastination: Survey finds 29% have yet to begin taxes [Electronic Version] from http://www.prweb.com/releases/2004/03/prweb114250.htm.
Weinstein, G. (2004). The procrastinator's guide to taxes made easy. New York: Penguin Group.
(2006). Compound interest, Manhattan & the Indians. Retrieved from: http://www.savingadvice.com/blog/2006/01/15/10341_compound-interest-manhattan-the-indians.html
Byrne, A., Blake, D., Cairns, A., & Dowd, K. (2006). There's no time like the present: The cost of delaying retirement saving Financial Services Review, 15(3), 213–231.
Lazarus, D. (April 24, 2009). Obama scolds card issuers, and their silence speaks volumes. Los Angeles Times. Retrieved from http://www.latimes.com/business/la-fi-lazarus24–2009apr24,0,6516756.column
Heidhues, P., & Koszegi, B. (2008). Exploiting naivete about self-control in the credit market. University of California, Berkeley.
Shui, H., & Ausubel, L. M. (2005). Time inconsistency in the credit card market. University of Maryland.
Spinella, M., Yang, B., & Lester, D. (2004). Prefrontal system dysfunction and credit card debt. International Journal of Neuroscience, 114, 1323–1332.
Frontline (2008). The secret history of the credit card. Retrieved from: http://www.pbs.org/wgbh/pages/frontline/shows/credit/view/
Reuben, E., Sapienza, P., & Zingales, L. (2008). Procrastination and impatience: NBER Working Paper.
Judson, L. C. (1848). The moral probe: Or one hundred and two common sense essays on the nature of men and things, interspersed with scraps of science and history. New York: Published by the author.
Matlin, E. (2004). Procrastinator's guide to wills and estate planning. New York: Penguin.
Like procrastination, this is more common than you think. The American Dental Association indicates that only 12 percent of Americans floss daily and about half don't floss at all.
Harrison, H. C. (2005). The three-contingency model of self-management. Unpublished PhD dissertation, Western Michigan University, Kalamazoo, MI.
Arce, E., & Santisteban, C. (2006). Impulsivity: A review. Psicothema, 18(2), 213–220.
Bickel, W. K., Yi, R., Kowal, B. P., & Gatchalian, K. M. (2008). Cigarette smokers discount past and future rewards symmetrically and more than controls: Is discounting a measure of impulsivity? Drug and Alcohol Dependence, 96, 256–262.
Carver, C. S. (2005). Impulse and constraint: Perspectives from personality psychology, convergence with theory in other areas, and potential for integration. Personality and Social Psychology Review, 9(4), 312–333.
Chamberlain, S., & Sahakian, B. (2007). The neuropsychiatry of impulsivity. Current Opinion in Psychiatry, 20(3), 255.
Enticott, P., & Ogloff, J. (2006). Elucidation of impulsivity. Australian Psychologist, 41(1), 3–14.
Schmidt, C. (2003). Impulsivity. In E. F. Coccaro (Ed.), Aggression: Psychiatric assessment and treatment (pp. 75–87). New York: Informa Health Care.
Sirois, F. M. (2004). Procrastination and intentions to perform health behaviors: The role of self-efficacy and the consideration of future consequences. Personality & Individual Differences, 37(1), 115–128.
Sirois, F. M., & Pychyl, T. A. (2002). Academic procrastination: Costs to health and well-being. Paper presented at the American Psychological Association, Chicago.
Soble, A. G. (2002). Correcting some misconceptions about St. Augustine's sex life. Journal of the History of Sexuality, 11(4), 545–569.
Bland, E. (2008). An appraisal of psychological & religious perspectives of self-control. Journal of Religion and Health, 47(1), 4–16.
McCullough, M. E., & Willoughby, B. L. B. (2009). Religion, self-regulation, and self-control: Associations, explanations, and implications. Psychological Bulletin, 135(1), 69–93.
Alternatively, you can open the Panchatantra section of the Mahabharata to read Vishnu Sharma's words, "The man who is tardy acting where utmost speed is called for, rouses the ire of the gods who would set up obstacles in his way; you can bank on that." Also by Sharma, "Time drinks up the essence of every great and noble action, which ought to be performed, and is delayed in the execution." Gandhi, M. K., Strohmeier, J., & Nagler, M. N. (2000). The Bhagavad Gita according to Gandhi. Berkeley, CA: Berkeley Hills Books.
Cosan, M. E. (1996). Ramadhan and Taqwa training. (H. H. Erkaya, Trans.). Retrieved from http://gumushkhanawidargah.8m.com/books/ramadhan/
Similarly, the Islamic scholar Dr. Umar Sulaiman al-Ashqar titles an entire section of his book "Satan hinders the slave from acting by means of procrastination and laziness." As he notes, some of the earliest religious advice underscores the seriousness of procrastination: "Beware of procrastinating. It is the greatest of the soldiers of Satan." Al-Nu'man, A. (2002). The pillars of Islam. (A. Fyzeem, Trans., revised, and annotated by I. Poonawala). New Delhi: Oxford University Press. (Original work published 960).
al-Ashqar, U. S. (1998). World of the Jinn and Devils. (J. Zarabozo, Trans.). Al-Basheer Publications.
Olcott, H. S. (1887). Golden rules of Buddhism. London: Theosophical Publishing House.
Also, we have Tenzin Gyatso, the fourteenth Dalai Lama: "You must not procrastinate. Rather, you should make preparations so that even if you did die tonight, you would have no regrets." Das, S. (2000). Awakening to the sacred: Creating a spiritual life from scratch. London: Bantam.
Giloviqh, T., & Medvec, V. H. (1995). The experience of regret: What, when, and why. Psychological Review, 102(2), 379–395.
Roese, N. J., & Summerville, A. (2005). What we regret . . . and why. Personality and Social Psychology Bulletin, 31(9), 1273–1285.
King, L. A., & Hicks, J. A. (2007). Whatever happened to "What might have been"?: Regrets, Happiness, and Maturity. American Psychologist, 62(7), 625–636.
Chapter Six
Hayden, A. (2003). International work-time trends: The emerging gap in hours. Just Labour, 2, 23–35.
Wasow, B. (2004). Comparing European and U.S. Living Standards (The Century Foundation). Accessed at: http://www.tcf.org/list.asp?type=NC&pubid=596.
Malachowski, D. (2005). Wasted time at work costing companies billions. from http://salary.com
This is in line with other estimates that also put the cost of procrastination at over $9,000 per employee. D'Abate, C., & Eddy, E. (2007). Engaging in personal business on the job: Extending the presenteeism construct. Human Resource Development Quarterly, 18(3), 361–383.
Wheelan, C. (2002). Naked economics: Undressing the dismal science. New York. W. W. Norton.
Critchfield, T., & Kollins, S. (2001). Temporal discounting: Basic research and the analysis of socially important behavior. Journal of Applied Behavior Analysis, 34(1), 101–122.
Spencer, L. (1955). 10 problems that worry presidents. Harvard Business Review, 33, 75–83.
Steel, P. & König, C. J. (2006). Integrating theories of motivation. Academy of Management Review, 31, 889–913.
Lavoie, J. A. A., & Pychyl, T. A. (2001). Cyberslacking and the procrastination superhighway: A web-based survey of online procrastination, attitudes, and emotion. Social Science Computer Review, 19(4), 431–444.
Johnson, P. R., & Indvik, J. (2003). The organizational benefits of reducing cyberslacking in the workplace. Proceedings of the Academy of Organizational Culture, Communications and Conflict, 7(2), 53–59.
Malachowski, D. (2005). Wasted time at work costing companies billions. from http://salary.com
Villano, M. (September 30, 2007). It's only a game, but it's played at work. New York Times.
Lawler, R. (Monday, June 16, 2008). Cisco sees a zettaflood of IP traffic—driven by video. Contentinople, from http://www.contentinople.com/author.asp?section_id=450&doc_id=156555
Stelter, B. (January 5, 2008). Noontime web video revitalizes lunch at desk. New York Times.
Kelly, E. P. (Spring, 2001). Electronic monitoring of employees in the workplace. National Forum. Retrieved from: http://findarticles.com/p/articles/mi_qa3651/is_200104/ai_n8939300
Ladurantaye, S. (April 2, 2008). Corporate crackdown targets employee surfing: Home e-mail accounts, instant messaging, gaming and video-watching websites . . . they're all on the hit list as employers increasingly restrict what content they permit employees to access. Globe & Mail.
This corporate "big brother" mentality can become incredibly annoying when you have a legitimate reason to access these sites. My colleague Allen Ponak is a professional labor arbitrator, whose job requires him to mediate a variety of union-management disputes, including when an employee is caught downloading porn to his computer. Part of his job—and I am led to believe he is paid for this—is to examine the content of these sites.
American Management Association (2005). Electronic monitoring & surveillance survey. New York: Author.
Levin, J. (May 14, 2008). Solitaire-y confinement: Why we can't stop playing a computerized card game. Slate.
Phillips, J. G., & Reddie, L. (2007). Decisional style and self-reported Email use in the workplace. Computers in Human Behavior, 23(5), 2414–2428.
Song, M., Halsey, V., & Burress, T. (2007). The hamster revolution: How to manage your email before it manages you. San Francisco: Berrett-Koehler Publishers.
Thatcher, A., Wretschko, G., & Fridjhon, P. (2008). Online flow experiences, problematic Internet use and Internet procrastination. Computers in Human Behavior, 24, 2236–2254.
Iqbal, S. T., & Horvitz, E. (2007). Conversations amidst computing: A study of interruptions and recovery of task activity. Proceeds of User Modeling, 350–354.
Richtel, M. (June 14, 2008). Lost in E-mail, tech firms face self-made beast. New York Times.
Alboher, M. (June 10, 2008). Attention must be paid. New York Times.
Monsell, S. (2003). Task switching. TRENDS in Cognitive Sciences, 7(3), 134–140.
Rubinstein, J. S., Meyer, D. E., & Evans, J. E. (2001). Executive control of cognitive processes in task switching. Journal of Experimental Psychology: Human Perception and Performance, 27(4), 763–797.
Akerlof, G., & Shiller, R. (2009). Animal spirits: How human psychology drives the economy, and why it matters for global capitalism. Princeton, NJ: Princeton University Press.
Dunleavy, M. P. (December 2, 2006). Plan to retire but leave out Social Security. New York Times.
As Avner Offer describes it, "the long-term pattern is that the overall capacity for saving has declined quite substantially since the 1960s, suggesting a declining capacity for prudence."
Offer, A. (2006). The challenge of affluence: Self-control and well-being in the United States and Britain since 1950. New York: Oxford University Press.
Weber, E. (2004). Who's afraid of a poor old-age? Risk perception in risk management decisions. In O. Mitchell & S. Utkus (Eds.), Pension design and structure: New lessons from behavioral finance (pp. 53–66). New York: Oxford University Press.
Transamerica Center for Retirement Studies (2008). The attitudes of American workers and their employers regarding retirement security and benefits. Ninth Annual Transamerica Retirement Survey. Available at: http://www.transamericacenter.org/resources/Building-ConfidencePresentation%20TCRS%201002–0208.pdf
Brooks, D. (2009). Usury country. Harper's, 318 (1907), 41–48.
Byrne, A., Blake, D., Cairns, A., & Dowd, K. (2006). There's no time like the present: The cost of delaying retirement saving. Financial Services Review, 15(3), 213–231.
Notably, the economist Matthew Rabin, one of the authors of Procrastination in Preparing for Retirement, has included himself among those who aren't saving enough.
O'Donoghue, T., & Rabin, M. (1999). Procrastination in preparing for retirement. In H. J. Aaron (Ed.), Behavioral dimensions of retirement economics (pp. 125–156). New York: Brookings Institution Press.
Transamerica Center for Retirement Studies (2008). The attitudes of American workers and their employers regarding retirement security and benefits. Ninth Annual Transamerica Retirement Survey. Available at: http://www.transamericacenter.org/resources/Building-ConfidencePresentation%20TCRS%201002–0208.pdf
Organisation of Economic Cooperation and Development (December 2008). Pension Markets in Focus OECD Newsletter, 5, 1–20.
Byrne, A., Blake, D., Cairns, A., & Dowd, K. (2006). There's no time like the present: the cost of delaying retirement saving. Financial Services Review, 15(3), 213–231.
Hewitt Associates (July, 2008). Hewitt study reveals widening gap between retirement needs and employee saving behaviors. Retrieved: http://www.businesswire.com/portal/site/google/?ndmViewId=news_view&newsId=20080701005267&newsLang=enVenti, S. (2006). Choice, Behavior and Retirement Saving. In G. Clark, A. Munnell & M. Orszag (Eds.), Oxford Handbook of Pensions and Retirement Income (Vol. 1, pp. 21–30). Oxford: Oxford University Press.
O'Donoghue, T., & Rabin, M. (1999). Procrastination in preparing for retirement. In H. J. Aaron (Ed.), Behavioral dimensions of retirement economics (pp. 125–156). New York: Brookings Institution Press.
Armour, P., & Daly, M. (2008). Retirement savings and decision errors: Lessons from behavioral economics. FRBSF Economic Letter, 16, 1–3.
Legorano, G. (2009). Automatic enrollment gains ground for DC plans. Global Pensions from http://www.globalpensions.com/global-pensions/news/1557589/automatic-enrollment-gains-ground-dc-plans
Mitchell, O., & Utkus, S. (2003). Lessons from behavioral finance for retirement plan design. The Wharton School: University of Pennsylvania.
Turner, J. (2006). Designing 401 (k) plans that encourage retirement savings: Lessons from behavioral finance. Benefits Quarterly, 22(4), 1–19.
Choi, J., Laibson, D., & Madrian, B. (2004). Plan design and 401 (k) savings outcomes. National Tax Journal, 57(2), 275–298.
Thaler, R., & Benartzi, S. (2004). Save More Tomorrow(tm): Using behavioral economics to increase employee saving. Journal of Political Economy, 112(S1), 164–187.
The debt ceiling was deemed "a meaningless strait jacket" by Robinson as early as 1959.
Austin, D. (2008). The debt limit: History and recent increases. Congressional Research Service.
Robinson, M. A. (1959). The national debt ceiling: An experiment in fiscal policy. Washington, D. C.: Brookings Institute.
Critchfield, T. S., Haley, R., Sabo, B., Colbert, J., & Macropoulis, G. (2003). A half century of scalloping in the work habits of the United States Congress. Journal of Applied Behavior Analysis, 36, 465–486.
Weisberg, P., & Waldrop, P. (1972). Fixed-interval work habits of Congress. Journal of Applied Behavior Analysis, 5(1), 93–97. Also, special thanks to Tom Critchfield for personally providing me with the data.
America's history is particularly steeped in procrastination. During their civil war, procrastination by the General Longstreet cost the South the war when his delays prevented him securing the key positions of Little Round Top and Cemetery Ridge during the Battle of Gettysburg. Meanwhile, Abraham Lincoln struggled with the procrastination of General George Brinton McClellan, which ensured that the war dragged on three extra years. Regarding the procrastination that cost Colonel Rahl his life but in return put America a step closer to independence: there are a few words about this event by British ambassador Nolbert Quayle, "Only a few minutes' delay cost him [Colonel Rahl] his life, his honor, and the liberty of his soldiers. Earth's history is strewn with the wrecks of half-finished plans and unexecuted resolutions. 'Tomorrow' is the excuse of the lazy and refuge of the incompetent." Unfortunately for Quayle, the only record I could find of his existence is this quotation itself.
The policy of appeasing Hitler is often characterized as delay that gave the Führer greater time to prepare for battle. Winston Churchill is best known for capturing this sentiment, saying three years prior to Germany's invasion of Poland: "The era of procrastination, of half-measures, of soothing and baffling expedients, of delays, is coming to a close. In its place we are entering a period of consequences . . . We cannot avoid this period; we are in it now." In the aftermath of the war, Dwight D. Eisenhower, former Supreme Commander of the Allied forces in Europe and thirty-fourth president of the United States, found procrastination still undefeated. The Soviets were preparing themselves for a nuclear confrontation, with little being done to prevent it in Western Europe. Eisenhower's key concern was that the North Atlantic Treaty Organization (NATO) was still just a paper invention, unfunded and militarily toothless. In a speech that Churchill considered to be the greatest he ever heard, at least by an American, we have Eisenhower saying: "The project faces the deadly danger of procrastination, timid measures, slow steps and cautious stages. Granted that the bars of tradition and habit are numerous and stout, the greatest bar to this, or any human enterprise, lie in the minds of men themselves. The negative is always the easy side, since it holds that nothing should be done. The negative is happy in lethargy, contemplating almost with complacent satisfaction, the difficulties of any other course."
Andreou, C. (2007). Environmental preservation and second-order procrastination. Philosophy & Public Affairs, 35(3), 233–248.
Caney, S. (2008). Climate change, human rights and intergenerational equity. Oxford: Magdalen College.
Hepburn, C. (2003). Hyperbolic discounting and resource collapse, Discussion-Paper No. 159. Department of Economics, University of Oxford.
Read, D. (2001). Intrapersonal dilemmas. Human Relations, 54(8), 1093–1117.
Hurni, H., Herweg, K., Portner, B., & Liniger, H. (2008). Soil erosion and conservation in global agriculture. In A. Braimoh & P. L. G. Vlek (Eds.), Land Use and Soil Resources (pp. 41–72). New York: Springer.
Montgomery, D. (2007). Soil erosion and agricultural sustainability. _Proceedings of the National Academy of Sciences, 104_ (33), 13268–13272.
Sample, I. (August 31, 2007). Global food crisis looms as climate change and population growth strip fertile land. _The Guardian_.
Hightower, M. & Pierce, S. A. (2008) The energy challenge. Nature, 452, 285–286.
Editorial. (March 9, 2008). Oceans at risk. New York Times.
Worm, B., Barbier, E., Beaumont, N., Duffy, J., Folke, C., Halpern, B., Jackson, J., Lotze, H., Micheli, F., & Palumbi, S. (2006). Impacts of Biodiversity Loss on Ocean Ecosystem Services. Science, 314, 787–790.
Simpson, J. (November 26, 2008). Fishing the fish stocks to extinction. Globe and Mail.
Lynas, M. (2007). Six degrees: Our future on a hotter planet. New York: HarperCollins.
Spratt, D., & Sutton, P. (2008). Climate Code Red: The case for emergency action. Melbourne: Scribe Publications.
Bamberg, S. (2003). How does environmental concern influence specific environmentally related behaviors? A new answer to an old question. Journal of Environmental Psychology, 23(1), 21–32.
Orr, D. W. (2004). The nature of design: Ecology, culture, and human intention. New York. Oxford University Press.
Farrand, M. (Ed.) (1966). Records of the federal convention (Vol. 3). New Haven, CT: Yale University Press.
Actually, drinking tea from a saucer only became a social faux pas after Washington's and Jefferson's time. Back then, it was quite fashionable to drink tea from a saucer, with accompanying "cup plates" to allow tea drinkers to park their mugs while saucer sipping. Frost, S. (1869). Frost's laws and by-laws of American society. New York: Dick & Fitzgerald.
Titus, S. Tea: A Brief History. http://www.memorialhall.mass.edu/-classroom/curriculum_12th/unit3/lesson8/bkgdessay.html
Cumming, L. (2008). To guide the human puppet: Behavioural economics, public policy and public service contracting: Serco Institute.
Chapter Seven
Booth, D., & James, R. (2008). A literature review of self-efficacy and effective job search. Journal of Occupational Psychology, Employment and Disability, 10(1), 27–42.
Lay, C. H., & Brokenshire, R. (1997). Conscientiousness, procrastination, and person-task characteristics in job searching by unemployed adults. Current Psychology: Developmental, Learning, Personality, Social, 16(1), 83–96.
Senecal, C., & Guay, F. (2000). Procrastination in job-seeking: An analysis of motivational processes and feelings of hopelessness. Journal of Social Behavior & Personality, 15(5), 267–282.
Sigall, H., Kruglanski, A., & Fyock, J. (2000). Wishful thinking and procrastination. Journal of Social Behavior & Personality, 15(5), 283–296.
Scheier, M. F., & Carver, C. S. (1993). On the power of positive thinking: The benefits of being optimistic. Current Directions in Psychological Science, 2(1), 26–30.
Despite the fact that we all tend to underestimate the time it takes, procrastinators tend to be worse at this.
Buehler, R., Griffin, D., & Ross, M. (1994). Exploring the "planning fallacy": Why people underestimate their task completion times. Journal of Personality and Social Psychology, 67, 366–381.
Kahneman, D., & Tversky, A. (1979). Intuitive prediction: Biases and corrective procedures. TIMS Studies in Management Sciences, 12, 313–327.
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Sigall, H., Kruglanski, A., & Fyock, J. (2000). Wishful thinking and procrastination. Journal of Social Behavior & Personality, 15(5), 283–296.
Vancouver, J., More, K., & Yoder, R. (2008). Self-efficacy and resource allocation: Support for a nonmonotonic, discontinuous model. Journal of Applied Psychology, 93(1), 35–47.
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Kruger, J., & Dunning, D. (1999). Unskilled and unaware of it: How difficulties in recognizing one's own incompetence lead to inflated self-assessments. Journal of Personality and Social Psychology, 77(6), 1121–1134.
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Most poignantly illustrative of the potential danger of aphorisms is the story of Felix Powell, a British Staff Sergeant who wrote the music of the morale-building marching song "Pack Up Your Troubles in Your Old Kit Bag and Smile, Smile, Smile," one of the most optimistic songs ever written. Dressed in the uniform of the Peacehaven Home Guard, Powell shot himself in the heart with a rifle, committing suicide. Indeed, positive self-statements can make matters worse for those with low self-esteem.
In business academia, they call success spirals an "efficacy-performance deviation amplifying loop."
Lindsley, D., Brass, D. J., & Thomas, J. B. (1995). Efficacy-performance spirals: A multilevel perspective. Academy of Management Review, 20(3), 645–678.
For early stages of a complex venture, it is often best to have process or learning goals rather than product or outcome goals. That is, the goals are acquiring or refining new skills or steps (the process) rather than winning or getting the highest score (the product). Not only will confidence be maximized but, in the end, higher performance results.
Schunk, D., & Meece, J. (2006). Self-efficacy development in adolescences. In F. Pajares & T. Urdan (Eds.), Self-efficacy beliefs of adolescents (pp. 71–96). Greenwich CT: Information Age.
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Hans, T. A. (2000). A meta-analysis of the effects of adventure programming on locus of control. Journal of Contemporary Psychotherapy, 30(1), 33–60.
Hattie, J., Marsh, H. W., Neil, J. T., & Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. _Review of Educational Research, 67_ (1), 43–87.
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World Organization of the Scout Movement (1998). Scouting: An educational system. Geneva, Switzerland: World Scout Bureau.
Gestdottir, S., & Lemer, R. M. (2007). Intentional self-regulation and positive youth development in early adolescence: Findings from the 4-H study of positive youth development. Developmental Psychology, 43(2), 508–521.
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Early efforts to combat procrastination often focused on just this one step, using cognitive therapy to challenge people's self-limiting beliefs. It was notably used by the late Albert Ellis, whose approach is being continued by his co-author, William Knaus.
Ellis, A., & Knaus, W. J. (1977). Overcoming procrastination: Or how to think and act rationally in spite of life's inevitable hassles. Institute for Rational Living.
Schunk, D., & Meece, J. (2006). Self-efficacy development in adolescences. In F. Pajares & T. Urdan (Eds.), Self-efficacy beliefs of adolescents (pp. 71–96). Greenwich CT: Information Age.
This includes the leaders we follow as much as the spouses we choose (e.g., "Behind every great man/woman, there is a great woman/man"). Aside from role models and comparison groups being key determinants of self-efficacy, what others believe (that is, normative beliefs and subjective norms) play a major role in forming an intention to act.
Aarts, H., Dijksterhuis, A., & Dik, G. (2008). Goal contagion: Inferring goals from others' actions—and what it leads to. In J. Y. Shah & W. L. Gardner (Eds.), _Handbook of motivation_ (pp. 265–280). New York: Guilford Press.
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van Knippenberg, D., van Knippenberg, B., De Cremer, D., & Hogg, M. (2004). Leadership, self, and identity: A review and research agenda. _The Leadership Quarterly, 15_ (6), 825–856.
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Oettingen, G., Mayer, D., Thorpe, J. S., Janetzke, H., & Lorenz, S. (2005). Turning fantasies about positive and negative futures into self-improvement goals. Motivation and Emotion, 29(4), 236–266.
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Pham, L. B., & Taylor, S. E. (1999). From thought to action: Effects of process- versus outcome-based mental simulations on performance. Personality and Social Psychology Bulletin, 25, 250–260.
It may also be a bad idea to promote a pattern of thinking that puts a person at increased risk for a wide variety of mental illnesses. In compensation, though, those who are extremely fantasy-prone can enjoy imagined food as much as the real and can imagine themselves to orgasm without physical stimulation.
Levin, R., & Spei, E. (2004). Relationship of purported measures of pathological and nonpathological dissociation to self-reported psychological distress and fantasy immersion. Assessment, 11(2), 160–168.
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Schneider, S. L. (2001). In search of realistic optimism. Meaning, knowledge, and warm fuzziness. American Psychologist, 56(3), 250–263.
Waldo, T. G., & Merritt, R. D. (2000). Fantasy proneness, dissociation, and DSM-IV axis II symptomatology. Journal of Abnormal Psychology, 109(3), 555–558.
Johnson, D. D. P. (2004). Overconfidence and war: The havoc and glory of positive illusions. Cambridge, MA: Harvard University Press.
Armor, D., & Taylor, S. (2002). When predictions fail: The dilemma of unrealistic optimism. In T. Gilovich, D. Griffin & D. Kahneman (Eds.), Heuristics and biases: The psychology of intuitive judgment (pp. 334–347). New York: Cambridge University Press.
Asterbro, T., Jeffrey, S., & Adomdza, G. K. (2007). Inventor perseverance after being told to quit: The role of cognitive biases. _Journal of Behavioral Decision Making, 20_ (3), 253–272.
Lovallo, D., & Kahneman, D. (2003). Delusions of success. How optimism undermines executives' decisions. _Harvard Business Review, 81_ (7), 56–63.
Moore, D., & Healy, P. (2007). _The trouble with overconfidence._ Unpublished manuscript, Carnegie-Mellon University, Pittsburgh.
Baker, W., & O'Malley, M. (2008). Leading with kindness: How good people consistently get superior results. New York: AMACOM/American Management Association.
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Hmieleski, K., & Baron, R. (2009). Entrepreneurs' optimism and new venture performance: A social cognitive perspective. Academy of Management Journal, 52(3), 473–488.
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Sigall, H., Kruglanski, A., & Fyock, J. (2000). Wishful thinking and procrastination. _Journal of Social Behavior & Personality, 15_(5), 283–296.
Though having several critics, such as the influential psychologist Albert Ellis, and accused of being a confidence (con) man, Peale's popularity still remains strong.
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Barbara Held, a psychology professor at Bowdoin College, describes it this way: "The positive attitude has—in some of its manifestations—become tyrannical, in that Americans have come to live not only with a historically/culturally grounded inclination for optimism but with the expectation, with the demand, that they maintain a positive attitude at all times and at all costs."
De Raeve, L. (1997). Positive thinking and moral oppression in cancer care. European Journal of Cancer Care, 6(4), 249–256.
Ehrenreich, B. (2009). Bright-sided: How the relentless promotion of positive thinking has undermined America. New York: Metropolitan Books.
Fineman, S. (2006). On being positive: Concerns and counterpoints. The Academy of Management Review, 31(2), 270–291.
Gilovich, T. (2005). The perceived likelihood of events that "tempt fate." Paper presented at the Annual Meeting of the Society of Personality and Social Psychology, New Orleans.
Held, B. (2002). The tyranny of the positive attitude in America: Observation and speculation. Journal of Clinical Psychology, 58(9), 965–991.
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Jones, F., Harris, P., Waller, H., & Coggins, A. (2005). Adherence to an exercise prescription scheme: The role of expectations, self-efficacy, stage of change and psychological well-being. British Journal of Health Psychology, 10, 359–378.
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Polivy, J., & Herman, C. P. (2002). If at first you don't succeed: False hopes of self-change. American Psychologist, 57(9), 677–689.
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Aspinwall, L. G., & Taylor, S. E. (1997). A stitch in time: Self-regulation and proactive coping. Psychological Bulletin, 121, 417–436.
Baumeister, R. F., Heatherton, T. F., & Tice, D. M. (1994). Losing control: How and why people fail at self-regulation. San Diego, CA: Academic Press, Inc.
Klassen, R. M., Krawchuk, L. L., & Rajani, S. (2008). Academic procrastination of undergraduates: Low self-efficacy to self-regulate predicts higher levels of procrastination. Contemporary Educational Psychology, 33(4), 915–931.
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Known as the abstinence violation effect.
Larimer, M. E., Palmer, R. S., & Marlatt, G. A. (1999). Relapse prevention: An overview of Marlatt's cognitive-behavioral model. Alcohol Research & Health, 23(2), 151–160.
Howard Rachlin gives a similar account under the rubric of "restructuring" and Jeong-Yoo Kim considers the same phenomenon from an economic perspective. Another pair of economists, Benabou and Tirole, discuss how it is best to assume that you don't have the self-control to resist possible addictions, even if there is a good chance you could use without risk. Interestingly, Buddhists actually use an enhanced form of this technique by believing bad choices (i.e., karma) will not only negatively impact your future self but also your future reincarnations.
Ainslie, G. (1992). Picoeconomics: The strategic interaction of successive motivational states within the person. New York: Cambridge University Press.
Ainslie, G. (2001). Breakdown of the will. New York: Cambridge University Press.
Benabou, R., & Tirole, J. (2004). Willpower and personal rules. Journal of Political Economy, 112(4), 848–886.
Kim, J.-Y. (2006). Hyperbolic discounting and the repeated self-control problem. Journal of Economic Psychology, 27(3), 344–359.
Rachlin, H. (2000). The science of self-control. Cambridge, MA: Harvard University Press.
Gosling, J. (1990). Weakness of the will. New York: Routledge.
Silver, M., & Sabini, J. (1981). Procrastinating. Journal for the Theory of Social Behavior, 11(2), 207–221.
Chapter Eight
Fried, Y., & Ferris, G. R. (1987). The validity of the Job Characteristics Model: A review and meta-analysis. Personnel Psychology, 40(2), 287–322.
Hackman, J. R., & Oldham, G. R. (1976). Motivation through the design of work: Test of a theory. Organizational Behavior and Human Performance, 16, 250–279.
Humphrey, S., Nahrgang, J., & Morgeson, F. (2007). Integrating motivational, social, and contextual work design features: A meta-analytic summary and theoretical extension of the work design literature. Journal of Applied Psychology, 92(5), 1332–1356.
Others were involved, such as Frank and Lillian Gilbreth who pioneered time and motion studies. The Gilbreths' work and life were chronicled in a book, Cheaper by the Dozen, written by two of their twelve children (Frank Jr. and Ernestine). Lillian was arguably the first of my kind—an Industrial/Organizational Psychologist—getting a PhD in management psychology (as well as receiving twenty-two other honorary degrees). The book became a film in 1950, not to be confused with the 2003 feature by the same name. This later version, starring Steve Martin and Bonnie Hunt, has some changes. Instead of Industrial/Organizational Psychology, this adaptation centers around a football coach because apparently there just aren't enough movies produced each year featuring football.
Kanigel, R. (1997). The one best way: Frederick Winslow Taylor and the enigma of efficiency. New York: Viking Penguin.
Furthermore, the harder employees worked, the less they were paid for each unit they produced. This is the typical outcome of most piece-rate systems, whereby you get paid for what your produce. Paradoxically, it is an inherent temptation for managers to reduce incentives as employees provide the very performance they were trying to incent. Known as the rachet effect, only a very few companies, like Lincoln Electric, have the discipline to avoid it and make the piece-rate system work.
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Science studies the malleable nature of value under the term "psychophysics," with research emphasizing, as here, that value is constructed (that is, dependent on how it is presented) and relative (i.e., dependent on what it is being compared to).
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Csíkszentmihályi, M. (1990). Flow: The psychology of optimal experience. New York: Harper and Row.
Johnny Carson of "The Tonight Show" invited her as a guest and pretended to eat her prized Elvis Presley chip. CNN (January 24, 2005). Your Johnny Carson memories. Retrieved from: http://www.cnn.com/2005/SHOWBIZ/TV/01/23/your.memories/index.html
Miller, R. B., & Brickman, S. J. (2004). A model of future-oriented motivation and self-regulation. Educational Psychology Review, 16(1), 9–33.
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Wolters, C. A. (2003). Understanding procrastination from a self-regulated learning perspective. Journal of Educational Psychology, 95(1), 179–187.
Ryan, R. M., & Deci, E. L. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. American Psychologist, 55(1), 68–78.
Lonergan, J. M., & Maher, K. J. (2000). The relationship between job characteristics and workplace procrastination as moderated by locus of control. Journal of Social Behavior & Personality, 15(5), 213–224.
Miller, R. B., & Brickman, S. J. (2004). A model of future-oriented motivation and self-regulation. Educational Psychology Review, 16(1), 9–33.
Shah, J., & Kruglanski, A. (2000). The structure and substance of intrinsic motivation. In C. Sansone & J. M. Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The search for optimal motivation and performance (pp. 106–130). San Diego, CA: Academic Press.
I like Franklin Jones' quote to this effect: "Nothing makes it easier to resist temptation than a proper bringing-up, a sound set of values—and witnesses."
Becker, H. (1960). Notes on the concept of commitment. American Journal of Sociology, 66(1), 32–40.
Magen, E., & Gross, J. J. (2007). Harnessing the need for immediate gratification: Cognitive reconstrual modulates the reward value of temptations. Emotion, 7(2), 415–428.
Powell, D., & Meyer, J. (2004). Side-bet theory and the three-component model of organizational commitment. Journal of Vocational Behavior, 65(1), 157–177.
Newman, T. (December 20, 2008). Barack Obama, I quit smoking—all the time. Newsday. Retrieved from http://www.newsday.com/news/opinion/ny-opnew205971623dec20,0,6796122.story.
Elliot, A., & Friedman, R. (2006). Approach-avoidance: A central characteristic of personal goals. In B. R. Little, K. Salmela-Aro & S. D. Phillips (Eds.), Personal project pursuit: Goals, action, and human flourishing (pp. 97–118). Mahwah, NJ: Lawrence Erlbaum Associates.
Howell, A. J., & Watson, D. C. (2007). Procrastination: Associations with achievement goal orientation and learning strategies. Personality and Individual Differences, 43(1), 167–178.
Mogilner, C., Aaker, J., & Pennington, G. (2007). Time will tell: The distant appeal of promotion and imminent appeal of prevention. Journal of Consumer Research, 34(5), 670–681.
Polivy, J., & Herman, C. P. (2002). If at first you don't succeed: False hopes of self-change. American Psychologist, 57(9), 677–689.
Schneider, S. L. (2001). In search of realistic optimism. Meaning, knowledge, and warm fuzziness. American Psychologist, 56(3), 250–263.
Wolters, C. A. (2003). Understanding procrastination from a self-regulated learning perspective. Journal of Educational Psychology, 95(1), 179–187.
Wolters, C. A. (2004). Advancing achievement goal theory: Using goal structures and goal orientations to predict students' motivation, cognition, and achievement. Journal of Educational Psychology, 96(2), 236–250.
Valkyrie, K. T. (2006). Self-regulated learning: An examination of motivational, cognitive, resource management, metacognitive components and academic outcomes with open admissions community college students. Unpublished PhD dissertation, University of Houston, Houston, TX.
Also, you can further upgrade your approach goals by making them about mastery. Mastery is viewing life as a prolonged opportunity to improve, to live to your potential. Each challenge, won or lost, is another step toward consummate skill. Mastery goals much more reliably produce the intrinsic motivation you are looking for. Similarly, those who are already at the top can eke out a little extra motivation by framing their approach goals in terms of prevention; that is, achievement will prevent them from losing their desirable position. Goals that emphasize protecting and maintaining standing and success will help you start a little earlier than everyone else.
Freitas, A. L., Liberman, N., Salovey, P., & Higgins, E. T. (2002). When to begin? Regulatory focus and initiating goal pursuit. Personality and Social Psychology Bulletin, 28(1), 121–130.
Molden, D. C., Lee, A. Y., & Higgins, E. T. (2007). Motivations for promotion and prevention. In W. L. G. James Y. Shah (Ed.), Handbook of motivation science (pp. 169–187). New York: Guilford Press.
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Pennington, G. L., & Roese, N. J. (2003). Regulatory focus and temporal distance. Journal of Experimental Social Psychology, 39, 563–576.
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Gröpel, P., & Steel, P. (2008). A mega-trial investigation of goal setting, interest enhancement, and energy on procrastination. Personality and Individual Differences, 45, 406–411.
Reduced energy is another reason why, aside from reduced self-confidence as per the last chapter, depression is connected to procrastination.
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A Kids in the Hall comedy sketch called "Chocolate" depicts this back and forth between wanting to diet and to wanting to eat chocolate. After a few bites, our protagonist throws away his chocolate bar, only to change his mind over and over.
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Though Jim Horne, from the University of Loughborough's Sleep Research Centre, contends we are actually sleeping better now than in most of history.
Horne, J. (18 October, 2008). Time to wake up to the facts about sleep. New Scientist, 2678, 36–38.
Mooallem, J. (November 18, 2007). The sleep-industrial complex. The New York Times.
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Muris, P., Merckelbach, H., Ollendick, T., King, N., & Bogie, N. (2001). Children's nighttime fears: Parent-child ratings of frequency, content, origins, coping behaviors and severity. Behaviour Research and Therapy, 39(1), 13–28.
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Bettelheim, B. (1977). The uses of enchantment: The meaning and importance of fairy tales. New York: Knopf.
Ferrari, J. R., & McCown, W. (1994). Procrastination tendencies among obsessive-compulsives and their relatives. Journal of Clinical Psychology, 50(2), 162–167.
Rachman, S. (1993). Obsessions, responsibility and guilt. Behaviour Research & Therapy, 31(2), 149–154.
Kaplan, A., & Hollander, E. (2004). Comorbidity in compulsive hoarding: a case report. CNS Spectrums, 9(1), 71–73.
Benton, T. H. (2005). Productive procrastination. The Chronicle of Higher Education, 52(1).
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Febbraro, G., & Clum, G. (1998). Meta-analytic investigation of the effectiveness of self-regulatory components in the treatment of adult problem behaviors. Clinical Psychology Review, 18(2), 143–161.
Ferrari, J. R., & Emmons, R. A. (1995). Methods of procrastination and their relation to self-control and self-reinforcement: An exploratory study. Journal of Social Behavior & Personality, 10(1), 135–142.
Eisenberger, R. (1992). Learned industriousness. Psychological Review, 99, 248–267.
Renninger, K. (2000). Individual interest and its implications for understanding intrinsic motivation. In C. Sansone & J. M. Harackiewicz (Eds.), Intrinsic and extrinsic motivation: The search for optimal motivation and performance (pp. 373–404). San Diego, CA: Academic Press.
Stromer, R., McComas, J. J., & Rehfeldt, R. A. (2000). Designing interventions that include delayed reinforcement: Implications of recent laboratory research. Journal of Applied Behavior Analysis, 33, 359–371.
Technically known as impulse pairing or fusion.
Ainslie, G. (1992). Picoeconomics: The strategic interaction of successive motivational states within the person. New York: Cambridge University Press.
Murray, H. A. (1938). Explorations in personality. New York: Oxford University Press.
Though this is the common term, some practitioners find it derogatory, dismissing the skill it can involve.
Dibbell, J. (June 17, 2007). The life of the Chinese gold farmer. The New York Times Magazine.
Jin, G. (2006). Chinese gold farmers in the game world [Electronic Version]. Consumers, Commodities & Consumption 7. Retrieved from https://netfiles.uiuc.edu/dtcook/www/CCCnewsletter/7–2/jin.htm.
Jin, G. (2008). Gold farmers. Retrieved from http://chinesegoldfarmers.com/Index.html
Akerman, D. S., & Gross, B. L. (2007). I can start that JME manuscript next week, can't I? The task characteristics behind why faculty procrastinate. Journal of Marketing Education, 29(2), 97–110.
Sansone, C., & Harackiewicz, J. (2000). Intrinsic and extrinsic motivation: The search for optimal motivation and performance. San Diego, CA: Academic Press.
Bordens, K., & Horowitz, I. (2001). Social psychology. Mahwah, NJ: Lawrence Erlbaum Associates.
Moreland, R. L., & Beach, S. R. (1992). Exposure effects in the classroom: The development of affinity among students. _Journal of Experimental Social Psychology, 28_ (3), 255–276.
Fouad, N. (2007). Work and vocational psychology: Theory, research, and applications. Annual Review of Psychology, 58, 543–564.
If you want to know your own profile, there are a variety of free versions available online. Just conduct an Internet search using the term "RIASEC."
Lubinski, D., & Benbow, C. P. (2000). States of excellence. American Psychologist, 55(1), 137–150.
It is possible to select a job for you with considerably more accuracy than currently available, directing you to areas of work that you would love as well as those in which you would excel. Unfortunately, despite being designed, proven, and patented, such a system has yet to be built. Sorry for this, but I have been busy writing a book. The patent number is US 20080027771. Interested parties should contact University Technologies International (tech@uti.ca).
Scherbaum, C. A. (2005). Synthetic validity: Past, present, and future. Personnel Psychology, 58(2), 481–515.
Steel, P. D., Huffcutt, A. I., & Kammeyer-Mueller, J. (2006). From the work one knows the worker: A systematic review of the challenges, solutions, and steps to creating synthetic validity. International Journal of Selection and Assessment, 14(1), 16–36.
Steel, P., & Kammeyer-Mueller, J. (2009). Using a meta-analytic perspective to enhance Job Component Validation. Personnel Psychology, 62(3), 533–552.
Tullier, L. (2000). The complete idiot's guide to overcoming procrastination. Indianapolis, IN: Alpha Books.
Chapter Nine
Akerlof, G. A. (1991). Procrastination and obedience. American Economic Review, 81, 1–19.
Arneklev, B., Elis, L., & Medlicott, S. (2006). Testing the General Theory of Crime: Comparing the effects of "imprudent behavior" and an attitudinal indicator of "low self-control." _Western Criminology Review, 7_ (3), 41–55.
Carver, C. S. (2005). Impulse and constraint: Perspectives from personality psychology, convergence with theory in other areas, and potential for integration. _Personality and Social Psychology Review, 9_ (4), 312–333.
Glomb, T., Steel, P., & Arvey, R. (2002). Office sneers, snipes, and stab wounds: Antecedents, consequences, and implications of workplace violence and aggression. In R. G. Lord, R. J. Klimoski, & R. Kanfer (Eds.), _Emotions in the workplace: Understanding the structure and role of emotions in organizational behavior_ (pp. 227–259). San Francisco, CA: Jossey-Bass.
Gottfredson, M. R., & Hirschi, T. (1990). _A General Theory of Crime._ Stanford, CA: Stanford University Press.
Hirschi, T. (2004). Self-control and crime. In R. F. Baumeister & K. D. Vohs (Eds.), _Handbook of self-regulation: Research, theory, and applications_ (pp. 537–552). New York: Guilford Press.
Schmidt, C. (2003). Impulsivity. In E. F. Coccaro (Ed.), _Aggression: Psychiatric assessment and treatment_ (pp. 75–87). New York: Informa Health Care.
Roberts, B. W., Walton, K. E., & Viechtbauer, W. (2006). Patterns of mean-level change in personality traits across the life course: A meta-analysis of longitudinal studies. Psychological Bulletin, 132, 1–25.
Funder, D. C. (2001). Personality. Annual Review of Psychology, 52, 197–221.
Ainslie, G. (1975). Specious reward: A behavioral theory of impulsiveness and impulse control. Psychological Bulletin, 82(4), 463–496.
Ariely, D., & Wertenbroch, K. (2002). Procrastination, deadlines, and performance: Self-control by precommitment. Psychological Science, 13(3), 219–224.
Funk, I. K. (1895). The complete preacher: Sermons preached by some of the most prominent clergymen in this and other countries, and in the various denominations. University of Michigan: Funk & Wagnalls.
Sally, D. (2000). I, too, sail past: Odysseus and the logic of self-control. Kyklos, 53, 173–200.
Stanford, W. (1954). The Ulysses theme: A study in the adaptability of a traditional hero. Ann Arbor, MI: University of Michigan Press.
Strotz, R. (1956). Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies, 23(3), 165–180.
Precommitment is a term devised by Thomas Schelling, the Nobel Prize winning economist. Known for influencing fields from strategic bargaining to global warming, Schelling was also particularly good at dreaming up precommitment examples.
Schelling, T. (1984). Choice and consequence: Perspectives of an errant economist. Cambridge, MA: Harvard University Press.
Schelling, T. C. (1992). Self-command: A new discipline. In G. Loewenstein & J. Elster (Eds.), Choice over time (pp. 167–176). New York: Russell Sage Foundation.
O'Donoghue, T., & Rabin, M. (2008). Procrastination on long-term projects. Journal of Economic Behavior & Organization, 66, 161–175.
This lack of self-awareness is known as a "projection bias," whereby we project our present desires onto our future selves.
Loewenstein, G., & Angner, E. (2003). Predicting and indulging changing preferences. In R. F. Baumeister, G. Loewenstein & D. Read (Eds.), Time and decision: Economic and psychological perspectives on intertemporal choice (pp. 351–391). New York: Russell Sage Foundation.
That he burnt his ships is a myth, probably caused by a mistranslation or confusing the story with William the Conqueror. Regardless, it still makes for a good example.
Reynolds, W. (1959). The burning ships of Hernán Cortés. Hispania, 42(3), 317–324.
Ibeji, M. (2001). 1066: BBC History. Retrieved from: http://www.bbc.co.uk/history/british/normans/1066_01.shtml
Working at my university office, I can't really use this nudist technique without people starting petitions and protests. However, it might confine me to my home, which is also a great technique. In his book on precommitment Thomas Schelling cites the Times Literary Supplement for January 22, 1982, in which George Steiner interviews the Hungarian radical Georg Lukacs: "When I first called on him, in the winter of 1957–8, in a house still pockmarked with shell bursts and grenade splinters, I stood speechless before the armada of his printed works, as it crowded the bookshelves. Lukacs seized on my puerile wonder and blazed out of his chair in a motion at once vulnerable and amused: 'You want to know how one gets work done? It's easy. House arrest, Steiner, house arrest!'"
Schelling, T. (1984). _Choice and consequence: Perspectives of an errant economist._ Cambridge, MA: Harvard University Press.
Wallace, I. (1977). Self-control techniques of famous novelists. _Journal of Applied Behavior Analysis, 10_ (3), 515–525.
Weir, W. (January 12, 2006). Wake up! You snooze, you lose—Multiple hits on the snooze alarm may be hazardous to your sleep and motivation. Newsday.
Richtel, M. (June 14, 2008). Lost in E-mail, tech firms face self-made beast. New York Times.
Williams, A. (October 19, 2008). Drunk, and dangerous, at the keyboard. New York Times.
It is based on the Irvine Welsh book of the same name, but I have only seen the movie.
To underscore the importance of this wisdom, there are dozens of other sayings to this effect. For instance, George Eliot noted: "No man can be wise on an empty stomach"; Albert Einstein thought: "An empty stomach is not a good political adviser"; and William Cowper concluded: "No man can be a patriot on an empty stomach." My favorite, though, is number 214 of the Ferengi Rules of Acquisition.
Not always though. As Maslow wrote: "We have spoken so far as if this hierarchy was a fixed order, but actually it is not nearly so rigid as we may have implied. It is true that most of the people with whom we have worked have seemed to have these basic needs in about the order that has been indicated. However, there have been a number of exceptions . . ."
Maslow, A. H. (1954). Motivation and personality. New York: Harper.
Cantor, N., & Blanton, H. (1996). Effortful pursuit of personal goals in daily life. In P. M. Gollwitzer & J. A. Bargh (Eds.), The psychology of action: Linking cognition and motivation to behavior (pp. 338–359). New York: Guilford Press.
Fiore, N. (1989). The now habit: A strategic program for overcoming procrastination and enjoying guilt-free play. New York: Penguin Putnam, Inc.
Schneider, F. W., & Green, J. E. (1977). The need for affiliation and sex as moderators of the relationship between need for achievement and academic performance. Journal of School Psychology, 15, 269–277.
Su, X. (2007). A model of consumer inertia with applications to dynamic pricing. Berkeley: University of California.
This form of precommitment is also known as counteractive control, contingency management, and side bets.
Loewenstein, G., & Angner, E. (2003). Predicting and indulging changing preferences. In R. F. Baumeister, G. Loewenstein, & D. Read (Eds.), Time and decision: Economic and psychological perspectives on intertemporal choice (pp. 351–391). New York: Russell Sage Foundation.
Milkman, K. L., Rogers, T., & Bazerman, M. (2008). Highbrow films gather dust: A study of dynamic inconsistency and online DVD rentals. Boston: Harvard Business School.
Moeller, F., Barratt, E., Dougherty, D., Schmitz, J., & Swann, A. (2001). Psychiatric aspects of impulsivity. American Journal of Psychiatry, 158(11), 1783–1793.
Read, D., Loewenstein, G., & Kalyanaraman, S. (1999). Mixing virtue and vice: Combining the immediacy effect and the diversification heuristic. Journal of Behavioral Decision Making 12, 257–273.
Strotz, R. (1956). Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies, 23(3), 165–180.
Trope, Y., & Fishbach, A. (2000). Counteractive self-control in overcoming temptation. Journal of Personality and Social Psychology, 79(4), 493–506.
Surowiecki, J. (Feb. 14, 2006). Bitter money and Christmas Clubs. Forbes.
Ashraf, N., Karlin, D., & Yin, W. (2008). Female empowerment: Impact of a commitment savings product in the Philippines. Boston: Jameel Poverty Action Lab. Retrieved from: http://www.povertyactionlab.org/papers/ashraf_karlan_yin_female_empowerment_0308.pdf
Retrieved from http://www.marginalrevolution.com/marginalrevolution/2008/09/markets-in-self.html
Here is one more example. To stop addicts from relapsing, a Denver cocaine addiction center encourages self-inflicted blackmail. Patients write an incriminating letter to the authorities, revealing their misdeeds and urging the strongest punitive response. If these patients then fail to pass a random series of drug tests, those letters are delivered.
Schelling, T. C. (1992). Self-command: A new discipline. In G. Loewenstein & J. Elster (Eds.), Choice over time (pp. 167–176). New York: Russell Sage Foundation.
Thaler, R., & Sunstein, C. (2008). Nudge. New Haven, CT: Yale University Press.
Lane Olinghouse.
Allen, K. (1996). Chronic nailbiting: A controlled comparison of competing response and mild aversion treatments. Behavior Research and Therapy, 34(3), 269–272.
As Seymour originally bragged, "You are looking at a man who developed a foolproof system for fidelity." Richler, M. (1980). Joshua then and now. Toronto, ON: McClelland & Stewart.
Mischel, W., & Ayduk, O. (2004). Willpower in a cognitive-affective processing system. In I. Baumeister & K. Vohs (Eds.), Handbook of self-regulation: Research, theory, and applications (pp. 99–129). New York: Guilford Press.
See also: Caspi, A., Roberts, B., & Shiner, R. (2005). Personality development: Stability and change. Annual Review of Psychology, 56, 453–484.
Lee, P., Lan, W., Wang, C., & Chiu, H. (2008). Helping young children to delay gratification. Early Childhood Education Journal, 35(6), 557–564.
The average is over one violation per minute under attempts of active suppression. If you did make it to the minute mark, see if you can go an additional sixty seconds. It becomes much tougher.
Wenzlaff, R., & Wegner, D. (2000). Thought suppression. Annual Reviews in Psychology, 51(1), 59–91.
Wegner, D. (1994). White bears and other unwanted thoughts: Suppression, obsession, and the psychology of mental control. New York: The Guilford Press.
Of note, this is an inherent problem with any Panglossian approach that advocates you not think any negative thoughts. Such advice is doomed to fail by its very design.
Alternatively, the twentieth-century cultural critic Ernst Cassirer observed: "Physical reality seems to recede in proportion as man's symbolic activity advances."
Mischel, W., & Baker, N. (1975). Cognitive appraisals and transformations in delay behavior. Journal of Personality and Social Psychology, 31, 254–261.
Deacon, T. W. (1997). The symbolic species. New York: W. W. Norton & Company.
Gifford, A. (2002). Emotion and self-control. Journal of Economic Behavior & Organization, 49, 113–130.
Gifford, A. (2009). Rationality and intertemporal choice. Journal of Bioeconomics, 11(3), 223–248.
Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124–1131.
Kearney, A. (2006). A primer of covert sensitization. Cognitive and Behavioral Practice, 13(2), 167–175.
My example was actually rather mild compared to the ones Joseph Cautela, one of the originators of the technique, develops. Here Joseph describes its use in avoiding desserts:
I want you to imagine you've just had your main meal and you are about to eat your dessert, which is apple pie. As you are about to reach for the fork, you get a funny feeling in the pit of your stomach. You start to feel queasy, nauseous, and sick all over. As you touch the fork, you can feel food particles inching up in your throat. You're just about to vomit. As you put the fork into the pie, the food comes up into your mouth. You try to keep your mouth closed because you are afraid that you'll spit the food out all over the place. You bring the piece of pie to your mouth. As you are about to open your mouth, you puke; you vomit all over your hands, the fork, over the pie. It goes all over the table, over other people's food. Your eyes are watering. Snot, mucus are all over your mouth and nose . . .
Cautela goes on (and on) but I think that is all the description you or I can probably stomach. I have no reason to hate apple pie and I would like to keep it that way. But it was effective, wasn't it?
Cautela, J. R. (1972). Covert sensitization scenes: A compilation of typical scenes used in the application of covert sensitization to a variety of maladaptive behaviors. Chestnut Hill, MA: Boston College.
Lohr, S. (September 22, 2009). A $1 million research bargain for Netflix, and maybe a model for others. New York Times, B1.
Of note, mindfulness meditation may be a relevant way of increasing your attention control, but this has yet to be proven. As described by Jon Kabat-Zinn, a molecular biologist who pioneered the practice in the West, "meditation means cultivating a non-judging attitude toward what comes up in the mind . . . to witness whatever comes up . . . and to recognize it without condemning it or pursuing it." Consequently, even if the impulse to pursue a temptation does arise, the decision to act upon this impulse is not automatic. If mindfulness meditation does prove helpful, however, I am still skeptical about its practical value. It can take a long time to effectively master and in the meantime you will find it really, really boring. This makes it exactly the type of practice that boredom-sensitive procrastinators are going to put off. In other words, if you have the patience to foster mindfulness, you probably don't need the added self-control in the first place.
Brown, K., Ryan, R., & Creswell, J. (2007). Mindfulness: Theoretical foundations and evidence for its salutary effects. Psychological Inquiry, 18(4), 211–237.
Kabat-Zinn, J. (1994). Wherever you go there you are: Mindfulness meditation in everyday life. New York: Hyperion.
Masicampo, E. J., & Baumeister, R. F. (2007). Relating mindfulness and self-regulatory processes. Psychological Inquiry, 18(4), 255–258.
Kavanagh, D. J., Andrade, J., & May, J. (2005). Imaginary relish and exquisite torture: The elaborated intrusion theory of desire. Psychological Review, 112(2), 446–467.
Smallwood, J., & Schooler, J. (2006). The restless mind. Psychological Bulletin, 132(6), 946–958.
Bargh, J. A., & Chartrand, T. L. (1999). The unbearable automa-ticity of being. American Psychologist, 54(7), 462–479.
Bargh, J. A., & Ferguson, M. J. (2000). Beyond behaviorism: On the automaticity of higher mental processes. Psychological Bulletin, 126(6), 925–945.
Bargh, J. (2006). What have we been priming all these years? On the development, mechanisms, and ecology of nonconscious social behavior. European Journal of Social Psychology, 36(2), 147–168.
Carey, B. (July 31, 2007). Who's minding the mind? New York Times.
Wansink, B. (2004). Environmental factors that increase the food intake and consumption volume of unknowing consumers. Annual Review of Nutrition, 24, 455–479.
Childress, A., Hole, A., Ehrman, R., Robbins, S., McLellan, A., & O'Brien, C. (1993). Cue reactivity and cue reactivity interventions in drug dependence. In L. S. Onken, J. D. Blaine & J. J. Boren (Eds.), Behavioral treatments for drug abuse and dependence (pp. 73–96). Rockville, MD: National Institute on Drug Abuse.
Lustig, C., Hasher, L., & Tonev, S. T. (2001). Inhibitory control over the present and the past. European Journal of Cognitive Psychology, 13(1), 107–122.
Tullier, M. (2000). The complete idiot's guide to overcoming procrastination. Indianapolis, IN: Alpha Books.
Especially see the work of psychologist Fuschia Sirios, whose work on household safety behaviors emphasizes reducing clutter, such as putting away "hazardous tools after they are used" or keeping "stairs and walkways at home free of clutter and other tripping hazards."
Sirois, F. M. (2007). "I'll look after my health, later": A replication and extension of the procrastination-health model with community-dwelling adults. Personality and Individual Differences, 43(1), 15–26.
Lay, C. H., & Schouwenburg, H. C. (1993). Trait procrastination, time management, and academic behavior. Journal of Social Behavior & Personality, 8(4), 647–662.
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Again, even pigeons are capable of using this type of attentional control.
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There are lots of do-it-yourself kits that provide precisely this, like the _Contr014,_ _Kill A Watt,_ _Wattson Energy Meter_ or the _Owl_ (aka, _The Electrisave_ ); they should pay for themselves within months. Also, the hypermiler car subculture is an early adopter of this insight. With an arsenal of tricks, a few not for the faint of heart, like coasting in the draft of an eighteen-wheeler or the "death turn," they eke out incredible gas mileage just by the way they drive. But what hypermilers rave about most is a mini-computer called the _Scan Gauge,_ which plugs into any car built after 1995. Velcroed prominently on your dashboard, it provides instantaneous feedback on a choice of critical outcomes like cost per mile and cost per trip, not just miles per gallon (though that is a nice start). Suddenly, driving cost-consciously and environmentally becomes upfront and second nature. Once the abstract notion of reduced gas consumption, which appeals to our prefrontal cortex, becomes more immediate, tangible, and vivid, so that it appeals to our limbic system, we will freely use less gas. For example, I have seen my thrifty mother-in-law drive for thirty minutes to a fabric store just to return one _extremely_ low-cost item. Once you calculate in gas costs, the round-trip cost her money, but still she drove. Travel costs are vaguely known, while that purchase was right there in her hands. If she drove a different kind of car, one which calculated her travel costs automatically on the dashboard, I doubt she would have made the journey. This type of technology should increase our mileage by 25 percent just by reducing idling, speeding, and unnecessary acceleration. If we could tie in the automatic "tire pressure monitoring system," indicating how much your underinflated tires are costing you, driving efficiency could increase over 3 percent. Incorporating an "air filter monitoring system" and gas mileage potentially jumps another 10 percent. Given that cars produce the bulk of greenhouse gases, this implementation alone could easily meet the targeted reductions for the Kyoto Protocol, the international environmental treaty.
Gaffney, D. (January/February 2007). This guy can get 59 MPG in a plain old Accord. Beat that, punk. _Mother Jones_.
Grunwald, M. (August, 2008). The tire-gauge solution: No joke. _Time._
Jones, T. Y. (June, 2008). Hypermilers: Breaking the 100-MPG barrier. _Edmunds Inside Line._
Thompson, C. (2007). Clive Thompson thinks: Desktop orb could reform energy hogs. Wired, 15.08.
Lohr, S. (January 10, 2008). Digital tools help users save energy, study finds. New York Times.
Minosi, A., Martinola, A., Mankan, S., Balzarini, F., Kostadinov, A., & Prevostini, A. (2003). Intelligent, low-power and low-cost measurement system for energy consumption. Paper presented at the International Symposium on Virtual Environments, Human-Computer Interfaces, and Measurement Systems, Lugano, Switzerland
Aarts, H., Dijksterhuis, A., & Dik, G. (2008). Goal contagion: Inferring goals from others' actions—and what it leads to. In J. Y. Shah & W. L. Gardner (Eds.), Handbook of motivation (pp. 265–280). New York: Guilford Press.
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Lopez, F., & Wambach, C. (1982). Effects of paradoxical and self-control directives in counseling. Journal of Counseling Psychology, 29(2), 115–124.
Mulry, G., Fleming, R., & Gottschalk, A. C. (1994). Psychological reactance and brief treatment of academic procrastination. Journal of College Student Psychotherapy, 9(1), 41–56.
Ziesat, H. A., Rosenthal, T. L., & White, G. M. (1978). Behavioral self-control in treating procrastination of studying. Psychological Reports, 42, 59–69.
Economists actually refer to a version of stimulus cuing as mental accounting, which deals with how easily we categorize the world into discrete domains. This tendency also helps to explain the success of Christmas Clubs.
Thaler, R. (1999). Mental accounting matters. Journal of Behavioral Decision Making, 12, 183–206.
Surowiecki, J. (February 14, 2006). Bitter money and Christmas clubs. Forbes.
Ashforth, B. E., Kreiner, G. E., & Fugate, M. (2000). All in a day's work: Boundaries and micro role transitions. The Academy of Management Review, 25(3), 472–491.
Locke, E., & Latham, G. (2002). Building a practically useful theory of goal setting and task motivation: A 35-year odyssey. American Psychologist, 57(9), 705–717.
For example, as the management training group RapidBi documents on their website (http://www.rapidbi.com/created/WriteSMARTobjectives.html), the S.M.A.R.T. acronym has dozens of variations. However, people already invariably add a time frame when giving examples of specific goals. For example, RapidBi suggests that people should indicate "When do I want this to be completed?" when creating specific goals. Similarly, a typical definition for attainable goals is that they be "realistic." Take a look at almost any book or example on the topic.
Tayntor, C. B. (2001). Incorporating six sigma concepts into systems analysis. In P. Tinnirello (Ed.), New directions in project management (pp. 161–172). Boca Raton, FL: CRC Press LLC. http://www.topachievement.com/smart.html
Prendergast, C. (1999). The provision of incentives in firms. Journal of Economic Literature, 37, 7–63.
Schlinger, H. D., Derenne, A., & Baron, A. (2008). What 50 years of research tell us about pausing under ratio schedules of reinforcement. The Behavior Analyst, 31, 39–40.
Hall, P. A., & Fong, G. T. (2003). The effects of a brief time perspective intervention for increasing physical activity among young adults. Psychology and Health, 18(6), 685–706.
Miller, R. B., & Brickman, S. J. (2004). A model of future-oriented motivation and self-regulation. Educational Psychology Review, 16(1), 9–33.
Engber, D. (May 15, 2008). The unfinished stories: All the stuff we never got around to including in the special issue. Retrieved from http://www.slate.com/id/2191420/
Amabile, T. (2001). Beyond talent: John Irving and the passionate craft of creativity. American Psychologist, 56(4), 333–336.
Wallace, I. (1977). Self-control techniques of famous novelists. Journal of Applied Behavior Analysis, 10(3), 515–525.
http://www.rescuetime.com/dashboard; http://manictime.com/
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McCrea, S., Liberman, N., Trope, Y., & Sherman, S. (2008). Construal level and procrastination. Psychological Science, 19(12), 1308–1314.
Wood, W., & Neal, D. T. (2007). A new look at habits and the habit-goal interface. Psychological Review 114(4), 843–863.
Psychologist Peter Gollwitzer calls this process action planning and the resulting plans implementation intentions. Gallo, I. S., & Gollwitzer, P. M. (2007). Implementation intentions: A look back at fifteen years of progress. Psicothema, 19(1), 37–42.
Gollwitzer, P., & Sheeran, P. (2006). Implementation intentions and goal achievement: A meta-analysis of effects and processes. Advances in Experimental Social Psychology, 38, 69–119.
Gollwitzer, P. M. (1999). Implementation intentions: Strong effects of simple plans. American Psychologist, 54(7), 493–503.
Owens, S., Bowman, C., & Dill, C. (2008). Overcoming procrastination: The effect of implementation intentions. Journal of Applied Social Psychology, 38(2), 366–384.
Oaten, M., & Cheng, K. (2006). Improved self-control: The benefits of a regular program of academic study. Basic & Applied Social Psychology, 28(1), 1–16.
Oaten, M., & Cheng, K. (2007). Improvements in self-control from financial monitoring. Journal of Economic Psychology, 28(4), 487–501.
And more than a few proverbs:
"I say that habit's but a long practice, friend, and this becomes men's nature in the end."—Aristotle
"Habit, if not resisted, soon becomes necessity."—St. Augustine
"The chains of habit are generally too small to be felt until they are too strong to be broken."—Samuel Johnson
"Habit is a cable; we weave a thread each day, and at last we cannot break it."—Horace Mann
"Man becomes a slave to his constantly repeated acts. What he at first chooses, at last compels."—Orison Swett Marden
"Habits are at first cobwebs, then cables."—Chinese Proverb
Wood, W., Tam, L., & Witt, M. (2005). Changing circumstances, disrupting habits. Journal of Personality and Social Psychology, 88(6), 918–933.
Grant, A. (2003). The impact of life coaching on goal attainment metacognition and mental health. Social Behavior and Personality, 31(3), 253–263.
Matlin, E. (2004). The procrastinator's guide to wills and estate planning. New York: Penguin Group.
Chapter Ten
Frincke, J. (2008). Job satisfaction. Alexandria, VA: Society for Human Resource Management.
Kaiser, R., Hogan, R., & Craig, S. (2008). Leadership and the fate of organizations. American Psychologist, 63(2), 96–110.
Sousa-Poza, A., & Sousa-Poza, A. A. (2000). Well-being at work: A cross-national analysis of the levels and determinants of job satisfaction. Journal of Socio-Economics, 29(6), 517–538.
Bass, B. M. (1998). Transformational leadership: Industry, military, and educational impact. Mahwah, NJ: Erlbaum.
Eagly, A., Johannesen-Schmidt, M., & van Engen, M. (2003). Transformational, transactional, and laissez-faire leadership styles: A meta-analysis comparing women and men. Psychological Bulletin, 129(4), 569–591.
Yukl, G. (2006). Leadership in organizations (6th ed.). Upper Saddle River, NJ: Prentice Hall.
Baltes, B., Briggs, T., Huff, J., Wright, J., & Neuman, G. (1999). Flexible and compressed workweek schedules: A meta-analysis of their effects on work-related criteria. Journal of Applied Psychology, 84(4), 496–513.
Tom was truly exceptional. In survey after survey and study after study, about three quarters of employees report that the worst aspect of their job is their immediate supervisor, and about two thirds of supervisors would be considered incompetent by any objective standards.
Hogan, R., & Kaiser, R. (2005). What we know about leadership. Review of General Psychology, 9(2), 169–180.
Milgram, N. A. (1991). Procrastination. In R. Dulbecco (Ed.), Encyclopedia of human biology (Vol. 6, pp. 149–155). New York: Academic Press.
Ainslie, G. (2001). Breakdown of will. Cambridge University Press.
Ryan, R. M., & Deci, E. L. (2006). Self-regulation and the problem of human autonomy: Does psychology need choice, self-determination, and will? Journal of Personality & Social Psychology, 74(6), 1557–1586.
Vohs, K. D., & Baumeister, R. F. (2007). Can satisfaction reinforce wanting? In J. Y. Shah & W. L. Gardner (Eds.), Handbook of motivation science (pp. 373–389). New York: Guilford Press.
Kivetz, R., & Keinan, A. (2006). Repenting hyperopia: An analysis of self-control regrets. Journal of Consumer Research, 33, 273–282.
Tangney, J., Baumeister, R., & Boone, A. (2004). High self-control predicts good adjustment, less pathology, better grades, and interpersonal success. Journal of Personality, 72(2), 271–324.
Postscript
Carver, C. S. (2005). Impulse and constraint: Perspectives from personality psychology, convergence with theory in other areas, and potential for integration. Personality and Social Psychology Review, 9(4), 312–333.
Cervone, D., Shadel, W. G., Smith, R. E., & Fiori, M. (2006). Self-regulation: Reminders and suggestions from personality science. Applied Psychology: An International Review, 55(3), 333–385.
Mesoudi, A., Whiten, A., & Laland, K. (2006). Towards a unified science of cultural evolution. Behavioral and Brain Sciences, 29(4), 329–347.
Tooby, J., & Cosmides, L. (2007). Evolutionary psychology, ecological rationality, and the unification of the behavioral sciences. Behavioral and Brain Sciences, 30(1), 42–43.
Green, C. D. (1992). Is unified positivism the answer to psychology's disunity? American Psychologist, 47, 1057–1058.
Staats, A. W. (1999). Unifying psychology requires new infrastructure, theory, method, and a research agenda. Review of General Psychology, 3(1), 3–13.
Stanovich, K. E. (2007). The psychology of decision making in a unified behavioral science. Behavioral and Brain Sciences, 30(1), 41–42.
It is why one of my key articles is titled Integrating Theories of Motivation.
Steel, P. & König, C. J. (2006). Integrating theories of motivation. Academy of Management Review, 31, 889–913.
Wilson, E. (1998). Consilience: The unity of knowledge. New York: Knopf.
Gintis, H. (2004). Towards the unity of the human behavioral sciences. Politics, Philosophy & Economics, 3(1), 37–57.
Akerlof, G. A. (1991). Procrastination and obedience. American Economic Review, 81(2), 1–19.
Glimcher, P., & Rustichini, A. (2004). Neuroeconomics: The consilience of brain and decision. Science, 306, 447–452.
Kubey, R., & Csikszentmihalyi, M. (2002). Television addiction is no mere metaphor. Scientific American, 286(2), 62–68.
Young, K. (1998). Internet addiction: The emergence of a new clinical disorder. Cyberpsychology and Behavior, 1, 237–244.
Hancox, R., & Poulton, R. (2006). Watching television is associated with childhood obesity: but is it clinically important? International Journal of Obesity, 30, 171–175.
Vandewater, E., Bickham, D., & Lee, J. (2006). Time well spent? Relating television use to children's free-time activities. Pediatrics, 117(2), 181–191.
Hall, L., Johansson, P., & Léon, D. d. (2002). The future of self-control: Distributed motivation and computer-mediated extrospection. Lund: Lund University.
2a An ethical choice that Dr. Seligman struggled with, as he recounts in his book Learned Optimism. He discontinued this experimental method as soon as he had obtained the data he needed.
2b As determined by four separate surveys of the Procrastination Assessment Scale—Students, which assesses twenty-six possible reasons for procrastinating.
2c In fact, I wrote an article called "Integrating Theories of Motivation," dedicated to doing better. Regularly assigned reading for university students around the world, the paper acknowledges that there are a hundred years of motivational science to draw upon conducted by an army of researchers. Let's not let this go to waste.
2d The strict equation also includes the addition of a small constant at the bottom, typically the number "1," as in "Impulsiveness × Delay + 1." This constant's principal purpose is to prevent the equation from skyrocketing to infinity if impulsiveness or delay ever reaches zero.
2e The most infamous excuse is to claim the death of a grandparent. Mortality of grandparents increases several hundredfold during final exams, a statistic that if taken seriously suggests that seeing the grandkids being tested is extremely stressful for the elderly.
3a The Latin name for Great Tits is Parus major; despite the suggestive name, all but begging to be made fun of, they are the best studied bird in the world.
3b Gary Marcus, a New York University psychologist and author of Kluge: The Haphazard Construction of the Human Mind, concludes that "over hundreds of millions of years, evolution selected strongly for creatures that lived largely in the moment."
3c In Greek today, malakia has a somewhat fouler meaning, possibly best translated as "wanker."
3d Naturally, Dr. Johnson procrastinated writing that very article until the last possible moment, composing it in Sir Joshua Reynolds' parlor while the errand boy waited outside to bring it to press. "Typical," as his friend Hester Piozzi remembered, given that "numberless are the instances of his writing under immediate pressure of importunity or distress."
4a It was definitely a bad idea.
4b Alternatively, you will get almost the same type of slope with a fixed ratio schedule, which occurs when there is a set amount of work to be done before reaping your reward. For example, piece-rate factory workers who get paid for every one hundred units produced tend to work a little harder as they approach that hundredth mark and then they take a breather. In the professional literature, this is known as "break and run," the pattern of taking a break after completing a work block before accelerating once again toward the next finish line.
4c Similarly, and at about the same time, the psychologist Stuart Vyse reports in Going Broke: Why Americans Can't Hold On to Their Money, "Any time the urge strikes, we now have the capability to act on it impulsively, and that creates a much greater challenge for us than was ever the case before. It's only natural that we are having trouble with debt."
5a Not me, but only because I wrote this book. With two kids and a wife, I completed my will within a few days of writing this sentence.
5b Abraham Lincoln: "The leading rule for the lawyer, as for the man of every other calling, is diligence. Leave nothing for to-morrow which can be done to-day."
Martin Luther King: "How soon not now, becomes never."
5c In his own words: "And when Thou didst on all sides show me that what Thou said was true, I, though convinced of its truth, only repeated my dull and drowsy words, 'Right away, one minute, leave me but a little.' But 'right away' wasn't ever right now, and my 'little while' went on for a long while . . ."
5d For another example, see St. Gabriel Possenti, who consistently swore whenever he became seriously ill that he would enter a religious order, only to have his resolve disappear when he healed. It took several bouts of illness before he kept his word, whereupon he contracted tuberculosis and died a few years later.
5e Such as Johnathan Edwards' eighteenth-century classic Procrastination, or The Sin and Folly of Depending on Future Time. Similarly, you have Reverend Edward Irving's, "Procrastination is the kidnapper of souls and the recruiting-officer of hell;" and Reverend Aughey's, "Procrastination has populated hell. All the doomed and damned from Christian lands are victims of this pernicious and destructive wile of the devil. It is foolish to procrastinate."
6a Group-level procrastination is actually common enough to receive an official name; many business academics call it punctuated equilibrium.
6b These Defined Contribution plans are usually supported by the government and go by a variety of names, depending on the country you live in. For example, Americans have their 401(k)s, while Canadians have RRSPs. The UK has Pension Provisions, while France has Special Retirement Plans.
6c Furthermore, the amount put aside doesn't rely upon their present wages but draws solely on the extra money gained from assumed future salary raises. This is a nifty ploy, based on the "wage illusion." Wage increases typically keep salaries level with inflation, so you aren't really any richer. Still, a raise often feels like "extra money," instead of drawing on exactly what you are making now.
7a Sports teams constantly struggle against this trend, as it is natural to feel that last year's victory ensures the next season's success. As Bill Russell, winner of the NBA's most valuable player award five times over, notes, "It's much harder to keep a championship than to win one . . . there's a temptation to believe that the last championship will somehow win the next one automatically."
7b We could also cite the International Farm Youth Exchange or 4-H clubs (i.e., Head, Heart, Hands, and Health). With a similar slogan of "learn by doing," they also aid in youth development. Having branched out considerably from their agricultural beginnings, they actively prepare students to excel across a variety of specialties, especially the sciences. Ask any alumni of any 4-H club what they thought of it; overwhelmingly they will testify that it was a major contribution to their self-confidence.
7c Sigmund Freud much earlier drew a similar conclusion. Fantasy is primarily a process whereby we form an image of our desire and receive gratification from it alone. This is much like addiction to Internet pornography, where pixels take the place of people.
7d Little of this is new. Benjamin Franklin wrote about the need for hard work in The Way to Wealth, over 150 years before Wallace Wattles' The Science of Getting Rich, the book that inspired The Secret. Even if you adopt the premise that magical thinking works, it is traditionally thought to operate contrary to the way professed by The Secret. Magnets actually attract their counter, that is positive attracts negative. Consequently, boasting about or predicting a positive result means it is less likely to come true; we jinx the outcome by tempting fate. It is why we knock on or touch wood after reporting good luck or health, in an effort to avoid the curse and allow the good luck to continue.
7e This is from William James' 1890 textbook, Principles of Psychology. James is actually summarizing a recommendation made four years earlier by Alexander Bain: "It is necessary, above all things, never to lose a battle. Every gain on the wrong side undoes the effect of many conquests on the right." For that matter, what James considers the second greatest Victorian maxim is also relevant: "Seize the very first possible opportunity to act on every resolution you make.
8a For those who have seen the movie, "Meow."
8b Oops! Make approach goals!
8c Which they start feasting on first through the anus. It makes for a nice bedtime story.
8d As Sir Peter Ustinov concluded, "Contrary to popular belief, I do not believe that friends are necessarily the people you like best; they are merely the people who got there first."
8e Go to: http://online.onetcenter.org/find/descriptor/browse/Interests/. If you check it out, look up my profession, Industrial-Organizational Psychologist. You will see that in addition to researching motivation, we also counsel workers about their careers.
8f One suggestion is Career Vision, which focuses on both job success and satisfaction: http://www.careervision.org/
8g For example, Douglas Adams, the bestselling author of The Hitchhiker's Guide to the Galaxy, had a legendary ability to avoid writing. As he quipped: "I love deadlines. I like the whooshing sound they make as they fly by."
9a Another great example is from Tony Wilson in the movie 24 Hour Party People. Tony was a Manchester music mogul and aficionado of punk rock. Despite his success, he never retained much money. His explanation is pure precommitment: "I have protected myself from ever having to sell out by having nothing to sell out."
9b Gastric surgery or stomach stapling is a more drastic form of satiation precommitment in that it reduces the amount of food needed to feel sufficiently suffonsified. That there is a non-negligible chance of dying during the procedure emphasizes the desperate measures people are willing to undertake to combat their desires.
9c Do not withhold yourselves from each other unless you agree to do so just for a set time, in order to devote yourselves to prayer. Then you should come together again so that Satan does not tempt you through your lack of self-control." (1 Corinthians 7:5)
9d Or, in Deacon's own words, chimps need the assistance of symbolic representation, for without it "being completely focused on what they want, they seem unable to stand back from the situation, so to speak, and subjugate their desire to the pragmatic context."
9e At least a month. See chapter 5.
9f Here are two national associations: http://www.napo.net/about_napo/; http://www.organizersincanada.com/.
10a More or less. I don't want to spoil his plot.
10b Aside from being referenced in dozens of college textbooks, the Procrastination Equation is also used during managerial training programs. For example, the company Intulogy bases motivational training for managers around the Procrastination Equation and it works. As one of their clients testifies, "When you first told me that you wanted to introduce yet another motivation theory, I thought it was a big waste of my time. Yet, it worked in class. Then, I spent all summer thinking about the theory. I have realized how much it applies to everything in life. It's incredibly powerful."
10c As psychologists Walter Mischel and Ozlem Ayduk wrote: "An excess of will can certainly be as self-defeating as its absence. Postponing gratification can be an unwise and even stifling joyless choice, but unless people develop the competencies to sustain delay and continue to exercise their will when they want and need to do so, the choice itself is lost."
Index
The pagination of this electronic edition does not match the edition from which it was created. To locate a specific passage, please use the search feature of your e-book reader.
**_Boldface numerals denote graphs and charts._**
ABD (all but dissertation), 87
abstinence, 134
abstinence violation effect, 267n32
academic dishonesty, 34
academic procrastination. _See_ students
accountability partners, 171
Adams, Douglas, 159n
adaptability, 4, 212, 213
adaptive genetic mutation, 52
adventure education, 121-22, 125
advertising, 75, 79, 179
age
and impulsiveness, 33, 162
and prefrontal cortex, 49-50
procrastination determinant, 12
and relevancy factor, 144
agriculture
invention, 57
sustainability, 112
Akerlof, George, 29
al-Ashqar, Umar Sulaiman, 249n26
alcohol, 46-47, 49, 93, 132, 134, 136, 149
Alcoholics Anonymous (AA), 134, 136
Almost Perfect Scale, 12-13
Ambient Orb, 181
America Online, 101
American Dental Association, 248n22
American Economic Association, 29
American Physical Therapy Association, 77
_Amusing Ourselves to Death_ (Postman), 78
Ancient Egypt, 57
Ancient Greece, 57-59
anger, 46, 93, 165, 213
_Animal Farm_ (Orwell), 33
animals
and impulsiveness, 54, 55
limbic system, 50
optimal foraging, 54
planning ability, 51
procrastination in, 51-52
anxiety, 8, 13, 38, 123, 135, 142
AOL Video, 103
AP-Advanced procrastination (group), 73
Apple Inc., 77
_Apprentice, The_ (TV), 12
approach goals, 144-45, 200, 271-72n16
Arden, Andrea, 52
Aristotle, 58
Armstrong, Neil, 131-32
arousal
biological origin, 44
power of, 165.
_See also_ temptation
_Art of War, The_ (Sun Tzu), 166
associative cues, 178-82
astrology, 4, 222n1
attentional control, 173-74
Atwood, Margaret, 11
automatic enrollment plans, 108, 109, 115
Automaticity in Cognition, Motivation and Emotion (ACME), 179
avoidance cycle, 7
avoidance goals, 144-45, 200
Ayduk, Ozlem, 212n
Ayres, Ian, 171
Bacon, Sir Francis, 151
Baden-Powell Award, 122
Bagley, William, 67, 68
bank accounts, separate, 182
Bargh, John, 179
Basex (co.), 105
beauty pageant contestants, 223n6
beauty, perception of, 54
Bebo (social networking site), 71
Becker, Gary, 101
behavioral economics, 29, 108-9, 230n3
behaviorism, 29, 151
beliefs, 117
Bell, Alexander Graham, 177
Benartzi, Schlomo, 109
Benchley, Robert, 105
_Bhagavad Gita_ , 95
bicameralism, 113-14
bill paying, 24, 84, 89, 98, 117, 146, 181, 182, 196
bills, congressional, 110-11, 111
birds, 55
BlackBerry, 77, 182
_Bondage of Opium_ , A (Lefebure), 82
_Book of Five Rings_ (Musashi), 175
books, inspirational, 127
boredom, 24, 140-43, 144, 162
"Boss Key," 104
brain
damage, 46
science, advances, 43-44
structure, xii, 43. _See also_ limbic system; prefrontal cortex
tumors, 46
_Brave New World Revisited_ (Huxley), 78
break and run pattern, 66n
Broken Lizard (comedy group), 143
Brothen, Thomas, 37
Buddhism, 59, 267n33
burnout, 147
Bush, George W., 104
Byrne, Rhonda, 131
cable TV, 69
Calgon Carbon, 87
Cameron, David, 114
Campbell, Gil, 239-40n44
"Can't You See I'm Busy" (website), 104
career domain, 83
career success, 87-88
Career Vision, 159n
Carlisle Trust Company, 170
Carver, Charles, 117
Catherine the Great, 126
Cautela, Joseph, 283n36
cell phones, 77, 134, 177, 183
CEOs, 102
_Challenge of Affluence, The_ (Offer), 78
Cham, Jorge, 87
Chaplin, Charlie, 67
Chapman, Graham, 156
check-cashing shops, 107
children
delayed gratification experiment, 173, 174-75, 177
neurological evolution, 47-49
success spirals, 122-23.
_See also_ parenting; teenagers
chimpanzees, 51, 52, 175, 175n
_Choice Over Time_ (Loewenstein), 29
_Christabel_ (Coleridge), 80
Christianity, 94, 95, 95n
Christie, Agatha, 11
Christmas Clubs, 170, 171, 288n54
Christmas shopping, 23, 118
_Chronager_ (software), 168-69
Churchill, Winston, 112
Cicero, Marcus Tullius, 58-59
circadian rhythm, 148-49
clarity, 1, 9
Cleese, John, 156
Cleopatra, 59
climate change, 112-13
Clinton, Bill, 105
Clocky, 167, 168, 171
clutter, 23, 180, 285n46. _See also_ de-cluttering; dedicated work space
Cold War, 112, 256n7
Coleridge, Samuel Taylor, 80-82, 84, 168
colonoscopy, 91-92
colorectal cancer, 92
commercials, 79
community domain, 83
companies
associative cues, 179
crisis management, 132
employee-employer relationship, 203-10
overconfident, 130
retirement plans, 106, 106n. _See also_ automatic enrollment plans
comparison shopping, 74
compound interest, 88, 89, 106
compulsiveness, 3
comScore, 77-78
concentrative strategies, 180-81
confessional procrastination, 97-98
_Confessions of an English Opium-Eater_ (Quincey), 81
confidence, 20, 22, 117, 121, 122-24, 123n, 162, 204
congressional procrastination, 110-11, 111
congruence, 155
Conquer Club (board game), 63-64, 65
Constitution (U.S.), 110, 114
consumerism, 74-77
Cortés, Hernán, 166
cotton-top tamarins, 54
counselling psychologists, 156
Couric, Katie, 92
Covenant Eyes (website), 171
covert sensitization, 177, 283n36
"CrackBerry," 77
_Craftsmanship of Teaching, The_ (Bagley), 67
cravings
aspect of procrastination, 25
biological basis, 45, 165
and cues, 180
precommitment strategy, 170.
_See also_ temptation
creative visualization, 128-29
creativity, 85, 155
credit card revolvers, 89
crime, 49, 162
cues. _See_ associative cues
cyberslacking, 103, 159
Dalai Lama, 35, 249-50n30
Davies, W. H., 212
daydreaming, 8
de-cluttering, 180, 190, 196
Deacon, Terence, 175, 175n
deadlines
avoidance cycle, 7
and brain, 45
and career success, 195, 203
dividing into short-term goals, 208-9
eleventh hour, 4, 9, 45, 53, 85-87
excuses for missing. _See_ excuses and planning, 39
registration, 169-70
and stress, 2, 82
students (graph), 228-29n17.
_See also_ essay writing; students; writers
death, 95-96, 98-99
debt, 78n, 82, 107, 215. _See also_ government procrastination and debt
decision making. _See_ limbic system
dedicated work space, 181-82, 200, 202
Defined Contribution plans, 106-7, 106n
DeHass, Ronald, 171
DeMille, Cecil B., 67
dental health, 92-93, 248n22
depression, 22, 226n4
descent with modification, 52
desk photos, 183, 199-200
diet, 21, 76-77, 83, 213
diets, 138, 149. _See also_ fats; sugar
digital video recorders (DVRs), 70, 216-17
disaster recovery plan, 134
disciplinary integration, 214-18
displacement, 151
divided self, 42-43
Dobell, Byron, 10-11
doctors' appointments. _See_ medical procrastination
_Dog-Friendly Dog Training_ (Arden), 52
Dog Whisperer, 52, 237n32
domestic tasks, 7, 23, 24, 146, 147, 151, 180, 187, 189, 201
driving
efficiency, 286-87n49
reckless, 49, 77, 93
dropouts, 38, 86, 122
drug dependence, 46-47, 81, 82, 93, 122, 132, 179, 281n23
Dyson, Freeman, 112
e-breaking, 103
e-mail, 105, 167, 178, 199
ecological rationality, 53
education domain, 83
Edwards, Jonathan, 95n
effort-reward cycle, 152-53
Einstein, Albert, 88
Eisenhower, Dwight D., 112, 255-56n36
Ellis, Albert, 239n44, 261n15, 264n27
emergencies, 4
Emmett, Rita, 96-97
Emmett's Law, 97
emotions, respect for, 212
employees
electronic monitoring of, 104
and flextime, 207-8
and illicit Internet use, 103-4, 250n3
and online games, 104-5
recognition. _See_ praise; success, celebrating
retirement accounts, 107-9, 107n
and work pace, 102-3, 103.
_See also_ CEOs; e-mail; managers; organ-izational team procrastination
_End of Overeating, The_ (Kessler), 75
energy (physical), 47, 147-50, 190
energy consumption, 181, 286-87n49
environmental cues. _See_ associative cues
environmental degradation
and governmental procrastination, 112-13
and interdisciplinarity concept, 214
projections, 112
_Esquire_ magazine, 10-11
essay writing, 23, 23n, 33-35, 136
estate planning, 90-91, 190-91
evolutionary procrastination, 52, 54-57, 231-32n7
excuses, 9-10, 34, 34n, 81-82, 85, 135, 190
executive function, 44-45, 46, 49
exercise, 21, 23, 83, 149, 150, 187, 189, 190, 200, 213
Expectancy Theories, 27-28
expectancy, low, 20-22, 34, 35, 137, 197
expectancy-value theory, 26-32
expectations, unrealistic, 132-33
Expected Utility Theory, 27-29
experts, hiring, 190-91
4-H clubs, 123n
Facebook, 71-73, 104, 182
factory jobs, 66n, 140-41
failure, expectation of. _See_ self-fulfilling prophecy
faith, 117, 138, 204
Fallingwater (house), 10
False Hope Syndrome, 132
family domain, 83
Famous Five, 126
fantasies, 128-29
fatigue. _See_ energy (physical)
fats, 54, 75, 148, 212
_Faust_ (Goethe), 134
fear(s)
biological origin, 44
and cues, 179
imaginary, 150-51
and medical procrastination, 92
Ferguson, Will, 194, 211
fidelity precommitment, 169
financial procrastination
and career success, 87-88
and credit cards, 89
and education, 86-87
and estate planning, 90-91
and MBAs, 89-90
and savings/spending, 88-89
and taxation, 88
Firefox (browser), 167
fisheries, 113
fixed interval schedule, 65, 66
_Flashlight_ (iPhone), 78
flexibility, 4, 188-89
flextime, 207-8
flow, 142-43, 178
Flynn, Erroll, 67
food
and evolutionary procrastination, 54-55
global supply, 112
and market research, 75-76
Ford, Henry, 141
forgetfulness, 57
Fowler, Gene, 33
Franklin, Benjamin, 127, 131n
free market economy, 66, 74, 78
_Freedom_ (program), 167
Freud, Sigmund, 131
Freudian slip, 174
friendly spam, 105
friends (domain), 83
frontotemporal dementia, 50
functional magnetic resonance imager (fMRI), 44
Furuvik Zoo (Sweden), 51
Gage, Phineas, 46
gambling, 64-65
game playing, and boredom, 143, 144
gaming, 62-64, 98, 188. _See also_ video games
gastric surgery, 169n
Ge Jin, 155
General College, University of Minnesota, 37-38
_General Theory of Crime, A_ (Gottfredson/Hirschi), 162
Gershwin, George, 90
Gilbreth, Frank, 268n2
Gilbreth, Lillian, 268n2
Gintis, Herbert, 215
Global Behavioral Economic Forum, 114
global financial crisis (2008), 106, 108
global food supply, 112
global warming, 112-13
goals
abstract, 25-26, 32, 185, 209, 227-28n12
attainable, 121, 204, 209, 288n57
breaking down, 125, 191, 208
challenging, 184, 202
concrete, 25-26, 32, 185, 192, 197, 198, 202, 227-28n12
daily, 88, 187
easy, 184, 209
inputs/outputs, 187-88
life tasks, 144
mastery, 271-72n16
meaningful, 184-85
motivational chain, 143-44
process/learning, 260n9
structuring, 187-88
too-frequent, 187
top, 11.
See also approach goals; avoidance
goals; mini-goals; S.M.A.R.T.
goals; sub-goals
Going Broke: Why Americans Can't Hold On to Their Money (Vyse), 78n
gold farming, 154-55
Gollwitzer, Peter, 190
Google, 167
Gottfredson, Michael, 162
government savings programs, 106, 107
governmental procrastination
consequences, 111-12
and debt, 110-11
and environmental issues, 113
and founding fathers (U.S.), 110, 113-14
and passage of bills (U.S.), 110-11, 111
Gracie, Royce, 120
Grand Theft Auto (game), 68
gratification, postponing, 212, 212n. _See also_ instant gratification
Green, Christopher, 215
Greene, Robert, 59
grocery shopping, 74, 168-69, 201
guilt, 96, 98, 135, 226n4
_Guitar Hero,_ 68, 125
gummivores, 54
gyms, 23, 170-71, 196, 197, 198
habit. _See_ routine/habit
Hamilton, Alexander, 110
handwriting, 4
happiness, 85, 97-99, 197, 202
_Happiness(tm)_ (Ferguson), 194, 211
health domain, 83, 84, 85. _See also_ medical procrastination
Hemingway, Ernest, 188
Herman, Peter, 132
Hesiod, 57-58
hierarchy of needs, 169, 280n16
Hill, Napoleon, 73
Hinduism, 95, 96
Hirschi, Travis, 162
_History of the Peloponnesian War_ (Thucydides), 58
_Hitchhiker's Guide to the Galaxy, The_ (Adams), 159n
hobbies, 83, 125, 201
home offices, 182
Homer, 58, 162-64
Hoots, Sara, 222n6
housework. _See_ domestic tasks
"How to Gain an Extra Hour Every Day," 67
Hughes, Howard, 90
Hughes, Matt, 120
Hugo, Victor, 166
human nature, economic model, 27-29
humanistic psychology, 169
Hume, David, 26, 227-28n12
humour, 11
hunter-gatherers, 53
Huxley, Aldous, 78
I Love Lucy, 67
_Idiocracy_ (film), 56
"If...then" scenarios, 190
Ig Nobel Prize, 77
implementation intentions, 191
impulse spending, 74, 76, 78n, 182
impulsiveness
and age, 33, 162
and animals, 54, 55
biological origin, 45, 46
evolutionary explanation, 53-54
gender comparisons, 56-57
—procrastination connection, 13-14, 25-26, 30, 31, 161-62
teenagers, 49 and time, 30, 36, 37
independent retirement accounts, 107
individualism, 66
indulgences, 152, 154, 212
Industrial-Organizational Psychologists, x, 158n, 268n2
Industrial Revolution, 59-60
insectivores, 54
instant gratification, 13, 48, 49, 55, 74, 93, 98, 161, 162, 173, 177, 192
instinct/reason interplay, 44, 46
"Integrating Theories of Motivation" (Steel), 27n
intention-action gap, 38, 109
intentions, 189-91, 261n17
Internet
blocking access to, 104, 168, 217
and electronic monitoring, 104
online discussion boards, 97-98
online games, 103
rationalizing use, 217-18
time-control software, 167.
_See also_ e-mail; employees; students
interruption and recovery, 105
intestacy, 90-91
intimacy, 83-84, 98, 201
iPhone, 77, 78
Iraq war, 130
Iron, Mary Jean, 99
irrational delay. _See_ procrastination
Irving, Edward, 95n
Islam, 95, 96
James, William, 136n, 230n3
Jefferson, Thomas, 113-14, 257n44
job hunting, 21, 116-17. _See also_ career success
job mobility, 158
jobs
ability to perform, 158, 276n41
assessment, 156, 157
database, 158
factory, 66n, 140-41
finding perfect, 155-59
market, 157-58
office, 140
sales, 15-16, 20, 88
Johnson, Samuel, 59-60, 59n
jokes, 11
_Joshua Then and Now_ (Richler), 172, 177
journalism, deadlines, 88
Judge, Mike, 56
Jupiter, 4
Kabat-Zinn, 284n38
Karlin, Dean, 171
Kaufman, Edgar, 10
Kessler, David, 75
KFC, 76
King, Martin Luther, Jr., 90, 90n
_Kluge: The Haphazard Construction of the Human Mind_ (Marcus), 54
Koppel, Ted, 11
_Kubla Khan_ (Coleridge), 80, 81
L'Amour, Louis, 133
labor statistics, 100-1
Law of Attraction, 131
laziness, 2, 38
leadership, 203-10
learned helplessness, 21-22, 225-26n2
learned industriousness, 153
_LeechBlock_ , 167
Lefebure, Molly, 82
leisure (domain), 83
liar loans, 107
Liberman, Nira, 26
life domains survey, 83-85, 84
limbic system
animals, 47, 50
and brain damage, 46
children, 47
evolution, 45, 231-32n7
and intentionality, 190
and market research, 74-76
role, 44, 73-74
and stimulus control, 178-79
Lincoln, Abraham, 90, 90n
love, 155-56
Loewenstein, George, 29
Lukacs, Georg, 279n11
lung cancer, 92
Mac, Bernie, 98
Madison, James, 110
magical thinking, 131, 131n
_Mahabharata_ , 95, 96, 249n26
Mail Goggles, 167
managers, 102, 143-44, 206
Manichaeism, 93-94
_ManicTime_ (software), 188
Marcus, Gary, 54n
Mark Antony, 59
market research, 74-77
marmosets, 54
marriage, 155
martial arts, 120-21, 123, 125
Martinez, Mark, 175
Maslow, Abraham, 169
massive multiplayer online role-playing games (MMORPG), 154-55
Matching Law, 29-30
materialism, 74. _See also_ consumerism
_Mating Mind, The_ (Miller), 56
maturity, 12, 144
Mazur, James, 51-52
MBAs, 143-44
McGregor, Ewan, 168
medical procrastination, 23, 91-93, 196
_MeeTimer_ , 167
Melville, Herman, 166
mental accounting, 288n54
mental contrasting, 129-30
Mercury, 4
Mesla, David, 76
meta-analysis, x-xi
Microsoft, 104, 105
Microsoft Outlook, 178
military overconfidence, 130
Millan, Cesar, 52, 237n32
Miller, Geoffrey, 56
mindfulness meditation, 284n38
mini-goals, 187
Mischel, Walter, 173, 174-75, 177, 212n
_Moby-Dick_ (Melville), 166
Mocniak, Michael, 87, 142
_Modern Times_ (film), 140
modernization, 66-67, 212-13
Monteverde, Miguel, 103
Moon, Keith, 90
mortgages, 107
motivation
* beliefs/expectancy, 117
chain, 143-44
fixed interval schedule, 65, 66
fixed ratio schedule, 66n
* optimism, 118-19, 119
variable reinforcement, 65, 65, 66
motivational psychology, 65
movie stars, 67
movies
development, 67
inspirational, 127
multi-tasking, 105
Musashi, Miyamoto, 175
MySpace, 71
NATO, 256n36
"Nature of Procrastination, The" (Steel), xi, 86
negative self-talk, 126
neglect, 57
networking, 73. _See also_ social networking sites
neurobiology. _See_ brain; limbic system; prefrontal cortex
New Thought Movement, 131
_New York Times, The_ , 71, 105
New Year's resolutions, 132
_New Yorker, The_ , 11
_Nichomachean Ethics_ , 58
_Nineteen Eighty-Four_ (Orwell), 33
Nobel Prize, 101, 113
O*NET program, 158
O'Donoghue, Ted, 29
Obama, Barack, 114-15, 144
obesity, 216
occupations, 11-12, 223n6
_Odyssey, The_ (Homer), 162-64, 164, 165, 167, 171
Oettingen, Gabriele, 128, 129
Offer, Avner, 78, 252n23
_Office Space_ , 140
_Office, The_ (TV), 140
_On Writing Well: The Classic Guide to Writing Nonfiction_ (Zinsser), 34
Opera Solutions, 177
optimal foraging, 54
optimism, 21-22, 21n, 117-19, 119, 205, 265n28
Organisation for Economic Co-operation and Development, 100
organizational team procrastination, 102-5, 103
Orwell, George, 33
Outward Bound, 121-22
_Overcoming Procrastination_ (Ellis), 239n44
overconfidence, 21, 130-32, 137, 138
overeating, 93, 212-13
overoptimism, 117-18
overregulation, 211
Owen, Owen, 55
packaging, 74
Pali Canon, 59, 95
Panglossian thinking, 130-31, 282n31
paperwork, 142
parental controls, 168, 217
parenting, 47-50, 83, 84, 122-23
Parker, Dorothy, 11
partnering, 153-54
patience, 3, 44, 47, 48, 52, 55
pay day advances, 107
Peale, Norman Vincent, 131, 246n27
perfectionism, 12-13
performance
documenting, 124, 125
and flextime, 207-8
mental re-creations. _See_ mental contrasting
peak hours, 148-49, 150
and perfectionism, 13
rituals of, 189-90
and switching attention, 105
persistence, 14, 121, 155
"Person from Porlock" excuse, 81, 82
personal digital assistants (PDAs), 77-78, 104, 133
personal trainers, 134, 180, 190, 191
pessimism, 20, 118, 121, 126, 131, 132-33
pets, 47, 50, 52
_PhD Comics_ , 87
PhDs, 86-87
piece-rate systems, 140, 269n3
_Pillars of Islam, The_ , 95
Pinsky, Robert, 82
Piozzi, Hester, 59n
planning, biological origin, 44
planning fallacy, 118
pleasurable vs. unpleasant tasks. _See_ value
pleasure, biological origin, 44
political activism, 83
political procrastination, 58-59, 109-14
Polivy, Janet, 132
polymorphic procrastination, 102-5, 103
Ponak, Allen, 251n14
pop-ups, 178
_Popular Science_ , 67
porn sites, 103-4, 171, 251n14
positive psychology movement, 21-22
positive self-statements, 120, 126, 259n7
Postman, Neil, 78
poverty, in retirement, 108
Powell, Felix, 259n7
_Power of Positive Thinking, The_ (Peale), 131
powerlessness, 8, 134-37. _See also_ learned helplessness
praise, 204, 207, 210
precommitment, 165-172, 200, 202-3, 217, 278n6
predation, 53, 54
predictability, 181
prefrontal cortex
animals, 50-51
evolution, 231-32n7
and exhaustion, 47
and impulse control, 49-50, 52
and market research, 75, 76
maturation, 47-50
role, 73-74
pregnancy, 55-57
pride, 206
_Principles of Psychology_ (James), 136n
procrastination
biological basis, 40, 43-57
cluster pattern, 83
defined, 3-4, 53
economic cost, 58, 100-15, 250n3
history, 57-60
hours, 101-2
* impulsiveness connection, 13-14, 25-26, 30, 31, 161-62
life domains survey, 83-85, 84
log, 137
misdiagnosis, xii
pattern, 7-9
personal price, 80-99
quiz, 4-6
reasons for, 7, 15-17, 38. _See also_ excuses
statistics, 57, 100-1
top goal, 11
ubiquity, 214, 215
"Procrastination" (fan page), 73
Procrastination Assessment Scale—Students, 23, 23n
Procrastination Equation, 15-40
elements, 27-31, 31n
time/motivation graph, 37, 39
work pace graph, 39, 40
Procrastination Support (group), 97
_Procrastination Through the Ages: A Definitive History_ , 239n44
_Procrastinator's Handbook_ (Emmett), 97
procrastinators—profile, 4, 25, 40
demographics, 11-12
psychological, 12-14. _See also_
self-assessment quiz
productive procrastination, 151-52, 180
professional organizers, 180, 190
projection bias, 278n8
promiscuity, 46, 49, 56, 93, 94, 213
prudence, 3
psychophysics, 269n6
Pugh, Emerson, 43
Quimby, Phineas, 131
Quincey, Thomas de, 81
Rabin, Matthew, 29
rachet effect, 269n3
rage, 213
Rahl, Colonel, 111-12
_Rambler, The_ (periodical), 59
Ramsey, Gordon, 127
reading, 69-70
realistic optimism
success spirals, 120-25, 123n
vicarious victory, 125-28
wish fulfillment, 128-30
reality principle, 131
acknowledging powerlessness, 134-37
healthy pessimism, 132-34
reflection, 213
registered savings plans, 108
registration, late, 169-70
regret, 98-99, 165
rejection, 15, 20, 116
relevancy, 4, 143, 144, 184-85
religious procrastination, 93-96
remote control, 69
repetitive motion disorder, 77
reproduction, 55-57
_RescueTime_ (software), 188, 217
Research in Motion, 77
retirement savings plans, 107-9, 106n
rewards
biological origin, 44
and expectancy, 27-28
timing, 24-25. _See also_ instant gratification
Reynolds, Sir Joshua, 59n
Richler, Mordecai, 172
_Rime of the Ancient Mariner_ (Coleridge), 80
Ringenbach, Paul T., 239n44
Risk (board game), 63
risk-taking behavior, 49
role models, 126, 205
Romalis, John, 171
romance domain, 83
Ross, Harold, 11
routine/habit, 76-77, 188-91
Rumsfeld, Donald, 130
_RuneScape_ , 154
Russia, 112, 255-56n36
Sabini, John, 134
Salary.Com, 101
sales jobs, 15-16, 20, 88
satiation precommitment, 168-69, 172
satellite TV, 69
Save More Tomorrow plan, 108-9
savings, 88-89, 106-9
Sawyer, Robert, 188
Sayyiduna Ali Murtadha, 96
_Scan Gauge_ , 286-87n49
Schatten, Kaaydah, 126
Scheier, Michael, 117
Schelling, Thomas, 278n6
Scholastic Aptitude Test (SAT), 173
Schouwenburg, Henri, 14
_Scientific American_ , xi
scientific history, 58
scouting, 122-23
Second World War, 68, 255n36
_Secret, The_ (Byrne), 131, 131n
self-actualization theory, 169, 280n16
self-assessment quiz, 18-20
self-confidence. _See_ confidence
self-control, 13, 14, 25, 47-49, 54, 134, 147, 162, 192, 212, 216
self-deception, 9-10
self-development, 83-84, 98
self-doubt, 8, 117, 137
self-efficacy. _See_ intentions
self-esteem, 122, 257-259n7
self-fulfilling prophecy, 20, 22
self-help industry, 132-33, 193-94
self-loathing, 226n4
self-perception, learned, 22
self-praise, 152-53
self-recrimination, 8
Seligman, Martin, 21-22, 21n
Sera, Matt, 120-21
service clubs, 127
sex, 55-57
Shantideva, 55
shopping, 118, 147. _See also_ Christmas shopping; grocery shopping; impulse spending
Silver, Maury, 134
Singapore, wealth, 106
Skinner, B.F., 64, 68, 71
Slaney, Robert, 12-13
_Slate_ magazine, 185
sleep, 149-50
S.M.A.R.T. goals, 184, 288n57
smoking, 75, 93, 132, 133, 134, 144, 169
snooze button, 167, 171
SnuzNLuz, 171
social networking sites, 71-72, 104
social support, 126, 127-28, 197, 198, 199
soil depletion, 112
Solitaire, 104
Southern California Edison, 181
speakers, inspirational, 127
species survival, 112, 113
spirituality, 83, 84, 117
St. Augustine, 93-94, 94n
St. Gabriel Possenti, 94n
St. Paul the Apostle, 169
St. Pierre, Georges, 120-21
Star Wars Galaxies, 154
starvation, 54
Steel, Toby, 98-99
stickK.com (website), 171
stimulants, 149
stimulus control, 178-83
stimulus cues, 189
Stone, Elizabeth, 49
stress, 82, 96, 147, 149
students
campus clubs, 35
campus environment, 33, 35
grades, 34, 86
and Internet use, 70-73
and justifying procrastination, 134-35
Procrastination Assessment Scale, 23, 23n
Procrastination Equation applied to, 33-37, 37
SATs, 173
time/motivation graph, 37, 228-29n17
time management, 87
work pace graph, 40.
_See also_ academic dishonesty; dropouts; essay writing; MBAs; PhDs
sub-goals, 186, 186. _See also_ mini-goals
success, celebrating, 209, 210
success cluster, 83-84, 98
success spiral, 120-25, 123n, 204, 259n8
Sudoku, 98, 105
sugar, 9, 54, 74, 75, 149, 213
Sun Tzu, 166
_Super Troopers_ (film), 143
Surya Das, 95
_24 Hour Party People_ (film), 167n
Talkswitch, 104
tangent tasks. _See_ productive procrastination
Taras, Vas, 66
tarot cards, 4
tax deductions, retirement plans, 108
tax procrastination, 88, 181
Taylor, Frederick Winslow, 140-41
Taylorism, 140-41
teenagers, 47, 49, 56, 69
television, 79, 98, 180, 213, 216-17
development, 67
international comparisons, 69-70
viewing options, 70
temperature, increases. _See_ climate change
temptation
biological origin, 46
covert sensitization, 175-77
delaying access to, 168
distancing from, 175
proximity to, 38-39, 64-66, 65, 75-76, 163-64, 164, 168, 172
resistance-susceptibility, 47
time sensitive, 177
_Ten Commandments, The_ (film), 67
text messaging, 77, 105, 133, 196, 199, 200
"There Goes (Varoom! Varoom!)
That Kandy-Kolored Tangerine-Flake
Streamline Baby" (Wolfe), 11
thought suppression, 173-74
Thaler, Richard, 109
Thucydides, 58
time
and Expectancy Utility Theory, 28-30
and irrational delay, 3-4
motivation element, 24-26
software, 188.
_See also_ intention-action gap
time and motion studies, 140-41, 268n2
Toastmasters, 127
_Today_ show, 92
_Trainspotting_ , 168, 279n14
transactional leadership, 204, 208
transcranial magnetic stimulation, 46-47
transformational leadership, 204-8
_Treatise of Human Nature_ , A (Hume), 26
Trope, Yaacov, 26
Trump, Donald, 12
Twitter, 71
unemployment, 11-12, 87, 116
Unilever Health Institute, 76
United States
congressional procrastination, 110-11, 111
cost of procrastination, 101
founding fathers, 110, 113-14
government debt, 109-10
household savings, 107
labor stats, 100-1
retirement planning, 106, 108
wealth comparisons, 106
universal default, 89
"Unlikely Beast" game, 173-74
unschedule, 169, 198
U.S. Food and Drug Administration, 75
Ustinov, Sir Peter, 155n
Utthana Sutta (monk), 59
Valens, Richie, 90
value
and expectancy. _See_ expectancy-value theory
malleable nature, 269n6
motivation element, 22-24
Vancouver, Jeffrey, 118, 130
variable reinforcement, 64-65, 65
vicarious victory, 125-28, 217
video games, 62-63, 78, 103, 104, 154-55, 180
video snacking, 103
virtuous circle, 143
vision, 205
Visnu Sharma, 249n26
visualization, 128, 143. _See also_ creative visualization; mental contrasting
vocational psychologists, 156
Voltaire, 130
volunteerism, 74, 83, 124
Vroom, Victor, 170
Vyse, Stuart, 78n
Wansink, Brian, 76-77
War of Independence (U.S.), 111
Washington, George, 110, 111-12, 113-14, 257n44
water shortages, 112
_Way of the Boddhisattva, The_ (Shantideva), 59
Wegner, Daniel, 174
Weight Watchers, 170, 171
White, Barry, 90
wild great tits, 50
wilderness programs, 121-22, 125
William the Conqueror, 166
willpower, 44, 58, 76, 134, 137, 147, 148
wills, 90-91, 190
winding-down routines, 149, 150, 182
wish fulfillment, 128-30
wish-fulfilling prophecy, 117
Wolfe, Tom, 10-11
Wordsworth, William, 80, 82
_Work and Days_ (Hesiod), 58
work hour stats, 100-1
work/life balance, 182
work/play segregation, 182, 195-200
"World Scientists' Warning to Humanity," 113
_World of Warcraft_ , 68, 69, 154
Wright, Frank Lloyd, 10
writers, 10-11, 33-34, 59n, 80-82, 159, 159n, 166, 168, 185, 249-50n44
Young Presidents Organization, 102
Ziglar, Zig, 76
Zinsser, William, 34
Acknowledgments
This book all started with a phone call from an immensely talented and likable literary agent, Sally Harding. After seeing my research covered in the press, she insisted long before anyone else that I was the person to write the book on procrastination. Who was I to argue? The Cooke Agency was wise to merge their agency with hers, and she equally so to form a partnership with them. With Dean Cooke, Suzanne Brandreth, and Mary Hu, they make a fine crew that can steer a book through any waters, foreign or domestic.
My thanks also go to Louise Dennys at the Knopf Random House Canada Group, who saw the potential of this book, and to the extraordinarily erudite Anne Collins, who wields a golden pen. She is an editor's editor, and becoming publisher at Knopf Random Canada was inevitable. Anne improved every page here. I am indebted to Nancy Miller, who championed the book early on and then to Jonathan Burnham at HarperCollins US, who ensured that it had a home. Also, I am grateful to my editor there, Sally Kim, for stubbornly insisting that what I thought was good enough should be better. Talented and thoughtful, she even gave me her own umbrella when I got caught in a New York rainstorm. Special thanks to Jane Isay for bringing the manuscript home by providing finishing editorial touches and making sure the narrative flowed. With her extensive experience and her familiarity with psychodynamics, psychology, and neurobiology, we made a good team. Lastly, the lovely Jane McWhinney gave the final polish, making sure each sentence gleamed. Like raising a child, writing a book takes a village, and I am thankful to have had so many gifted people in my corner.
Early in my academic career at the University of Minnesota, I was lucky to have Dr. Deniz Ones teach me meta-analysis and Dr. Thomas Brothen initiate my lifelong fascination with procrastination. At the University of Calgary, where I currently reside, much appreciation goes to my colleague and friend Dr. Daphne Taras, who fought to make sure I received my sabbatical to write this book and who provided, or credibly feigned, interest in the manuscript development. Though I wished the sabbatical had been longer, those uninterrupted months proved invaluable. I also appreciate the efforts of her son, Matthew Taras, for confirming historical facts. Further appreciation goes to my sisters, Anita and Marion, for reading earlier drafts and to my father-in-law, John Horne, a consulting economist, for his critical eye.
For everything else, and everything in general, I thank my wife, Julie. The conditions for writing this book, like so much of life, were not ideal and yet here it is. Teaching, researching, and running a department aren't easy for a parent of a toddler and a newborn. With both of our families in other cities, it seemed ridiculous to think I could also take on writing a book, but we did it anyway. My wife and I traded off sleeping on different nights, tag-teamed the children, and I absolutely relied on her support and faith. Though the motivational principles contained within this book proved invaluable, her reserves of strength are the platform on which this book was built. And, through all of it, I learned that she is a very gifted copy editor with a most discerning eye. The reader, as am I, should be very happy we are married.
About the Author
PIERS STEEL, PhD, is the world's leading researcher and speaker on the science of motivation and procrastination. He studied and taught at the business and psychology schools of the University of Minnesota before moving to the University of Calgary's Haskayne School of Business, where he is a professor of human resources and organizational dynamics. He has been studying procrastination and its effects for more than ten years, and has spent the decades before that practicing it. Dr. Steel's award-winning research has appeared in magazines ranging from _Psychology Today_ and _New Scientist_ to _Good Housekeeping_ and _Profit._ His work has been reported in the _Los Angeles Times_ , the _Wall Street Journal_ , the _New York Times_ , and _USA Today._ Winner of the Killam Emerging Research Leader Award, he lives in Calgary, Alberta, with his wife and two sons.
Visit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.
Author's Note
Procrastination has been my life's work—both as a researcher and as a practitioner. With research so often being "me-search," this isn't accidental. Scientists often intimately know the subjects they study—they are problems they themselves face. It's true that I have sympathy for the procrastinator's plight because it is one I shared for many years. Nowadays my work has received international acclaim, I have coached national college champions in business school competitions, and awards for teaching and research hang on my office wall. But for most of my life, I felt potential languishing inside me mingled with frustration because I couldn't sustain any of my many attempts to improve. Encountering people who were naturally more capable of getting things done simply reminded me of my own deficiencies, curdled my spirit, and raised considerable misplaced resentment. Luckily, I was attracted to a profession whose very purpose was to identify the key enablers of change, which I then systematically put into practice in my own life one by one.
My PhD is in Industrial/Organizational Psychology, the scientific study of our actions and minds in the workplace. Psychology applied to work focuses on how to improve people's performance, well-being, and, quite appropriately, motivation or lack thereof. Unfortunately, many of the techniques of this discipline aren't well known, buried in the depths of obscure journals and written in scholarly language comprehensible only to the initiated. For procrastination, the problem gets even more complex. This subject has attracted the attention of all the social sciences and inspired research around the world. With over eight hundred scientific articles on the topic from fields spanning economics to neuroscience, in languages ranging from German to Chinese, the challenge is to find and make sense of them all. And this is where I come in. I found two ways to study procrastination. The first was by doing my own research, which you will read about presently. That gave me the basis for a theory of how and why we put things off. But then I needed to deal with the panoply of disciplines that have studied procrastination and published results in so many different journals and books. I was lucky enough to stumble upon meta-analysis, a recently developed scientific technique, and adapt it for my research.
Meta-analysis mathematically distills the results from thousands of studies to their core consensus. At a basic level, meta-analysis is what lets science progress. By enabling a synthesis of knowledge, it reveals the underlying truths we seek. It is very powerful, it has applications in every field, and it increasingly provides the information we need to run the world. The medical treatment you get from your doctor, for example, is likely based on the results of meta-analyses, from asthma to Alzheimer's. It is a discipline I have mastered: I have created some of its basic techniques, I teach it to others, and I have developed software for it. I like to think of it as something I am good at. It was natural, then, to meta-analyze the body of research on procrastination, given that there was no other way to put together all the findings. I have to say that the field of procrastination proved to be daunting, as almost every possible scientific methodology and technique has been thrown at it. Researchers have run laboratory experiments, read through personal diaries, twiddled with neurotransmitters, and dissected DNA. They have monitored every setting, from airports to shopping malls; they have wired entire classrooms to track every student's twitch and shudder; and they have studied procrastinators from every background, including pigeons, vermin, and members of the U.S. Congress. Making them all fit coherently together was like being a conductor of a madhouse orchestra. The strings, woodwinds, brass, and percussion are all playing the same tune but not in the same room, in the same rhythm, or in the same key. Turning that noise into music is what this book is about.
What I found will surprise you and challenge the status quo. Some of my work has already been published, such as my article "The Nature of Procrastination," which appeared in _Psychological Bulletin,_ the social sciences' most respected journal. Some of it has already been reported in hundreds of media venues around the world, from India to Ireland and from _Scientific American_ to _Good Housekeeping_ and _The Wall Street Journal._ But most of what I found is presented here for the first time. Within these pages, you will find out that we've been misdiagnosing procrastination for decades, attributing it to a trait associated with less procrastination, not more. The real reasons for procrastination are partly genetic and can be traced to the fundamental structure of our brains, which is why procrastination is seen in every culture and throughout history. The environment, however, isn't blameless; it may not be responsible for procrastination's existence, but it is responsible for its intensity—modern life has elevated procrastination into a pandemic. And guess what? All these findings follow from the application of a simple mathematical formula I devised—the Procrastination Equation.
Because I was able to tease out the fundamentals of the dynamic that makes us procrastinate, I have also been able to figure out strategies that we can use throughout our lives—school, work, or personal—to combat our innate tendency to put things off. A tall order? You bet. That's why it has taken me so many years to write this book. I hope the hours you spend reading it will reward you with a new way of thinking about how to spend—and waste—your time.
Copyright
THE PROCRASTINATION EQUATION. Copyright © 2011 by Piers Steel. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.
FIRST U.S. EDITION
Library of Congress Cataloging-in-Publication Data has been applied for.
ISBN: 978-0-06-170361-4
EPub Edition © 2011 ISBN: 9780062035257
11 12 13 14 15 OFF/RRD 10 9 8 7 6 5 4 3 2 1
About the Publisher
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{"url":"https:\/\/neos-guide.org\/guide\/types\/","text":"# Optimization Problem Types\n\nAs noted in the Introduction to Optimization, an important step in the optimization process is classifying your optimization model, since algorithms for solving optimization problems are tailored to a particular type of problem. Here we provide some guidance to help you classify your optimization model; for the various optimization problem types, we provide a linked page with some basic information, links to algorithms and software, and online and print resources.\n\n\u2022 Convex Optimization versus Nonconvex Optimization\n\u2022 Continuous Optimization versus Discrete Optimization\nSome models only make sense if the variables take on values from a discrete set, often a subset of integers, whereas other models contain variables that can take on any real value. Models with discrete variables are discrete optimization problems; models with continuous variables are continuous optimization problems. Continuous optimization problems tend to be easier to solve than discrete optimization problems; the smoothness of the functions means that the objective function and constraint function values at a point $$x$$ can be used to deduce information about points in a neighborhood of $$x$$. However, improvements in algorithms coupled with advancements in computing technology have dramatically increased the size and complexity of discrete optimization problems that can be solved efficiently. Continuous optimization algorithms are important in discrete optimization because many discrete optimization algorithms generate a sequence of continuous subproblems.\n\u2022 Unconstrained Optimization versus Constrained Optimization\nAnother important distinction is between problems in which there are no constraints on the variables and problems in which there are constraints on the variables. Unconstrained optimization problems arise directly in many practical applications; they also arise in the reformulation of constrained optimization problems in which the constraints are replaced by a penalty term in the objective function. Constrained optimization problems arise from applications in which there are explicit constraints on the variables. The constraints on the variables can vary widely from simple bounds to systems of equalities and inequalities that model complex relationships among the variables. Constrained optimization problems can be furthered classified according to the nature of the constraints (e.g., linear, nonlinear, convex) and the smoothness of the functions (e.g., differentiable or nondifferentiable).\n\u2022 Deterministic Optimization versus Stochastic Optimization\nIn deterministic optimization, it is assumed that the data for the given problem are known accurately. However, for many actual problems, the data cannot be known accurately for a variety of reasons. The first reason is due to simple measurement error. The second and more fundamental reason is that some data represent information about the future (e. g., product demand or price for a future time period) and simply cannot be known with certainty. In optimization under uncertainty, or stochastic optimization, the uncertainty is incorporated into the model. Robust optimization techniques can be used when the parameters are known only within certain bounds; the goal is to find a solution that is feasible for all data and optimal in some sense. Stochastic optimization models take advantage of the fact that probability distributions governing the data are known or can be estimated; the goal is to find some policy that is feasible for all (or almost all) the possible data instances and optimizes the expected performance of the model.\n##### Continuous Optimization\n\nIn continuous optimization, the variables in the model are allowed to take on any value within a range of values, usually real numbers. This property of the variables is in contrast to discrete optimization, in which some or all of the variables may be binary (restricted to the values 0 and 1), integer (for which only integer values are allowed), or more abstract objects drawn from sets with finitely many elements.\n\nAn important distinction in continuous optimization is between problems in which there are no constraints on the variables and problems in which there are constraints on the variables. Unconstrained optimization problems arise directly in many practical applications; they also arise in the reformulation of constrained optimization problems in which the constraints are replaced by a penalty term in the objective function. Constrained optimization problems arise from applications in which there are explicit constraints on the variables. There are many subfields of constrained optimization for which specific algorithms are available.\n\n##### Discrete Optimization\n\nIn discrete optimization, some or all of the variables in a model are required to belong to a discrete set; this is in contrast to continuous optimization in which the variables are allowed to take on any value within a range of values. Here, we consider two branches of discrete optimization. In integer programming, the discrete set is a subset of integers. In combinatorial optimization, the discrete set is a set of objects, or combinatorial structures, such as assignments, combinations, routes, schedules, or sequences.\n\n##### Unconstrained Optimization\n\nUnconstrained optimization problems consider the problem of minimizing an objective function that depends on real variables with no restrictions on their values. Mathematically, let $$x \\in \\mathcal{R}^n$$ be a real vector with $$n \\geq 1$$ components and let $$f : \\mathcal{R}^n \\rightarrow \\mathcal{R}$$ be a smooth function. Then, the unconstrained optimization problem is $\\mbox{min}_x \\; f(x).$\n\nUnconstrained optimization problems arise directly in some applications but they also arise indirectly from reformulations of constrained optimization problems. Often it is practical to replace the constraints of an optimization problem with penalized terms in the objective function and to solve the problem as an unconstrained problem.\n\n##### Constrained Optimization\n\nConstrained optimization problems consider the problem of optimizing an objective function subject to constraints on the variables. In general terms,\n$\\begin{array}{lllll} \\mbox{minimize} & f(x) & & & \\\\ \\mbox{subject to} & c_i(x) & = & 0 & \\forall i \\in \\mathcal{E} \\\\ & c_i(x) & \\leq & 0 & \\forall i \\in \\mathcal{I} \\end{array}$ where $$f$$ and the functions $$c_i(x) \\,$$ are all smooth, real-valued functions on a subset of $$R^n \\,$$ and $$\\mathcal{E}$$ and $$\\mathcal{I}$$ are index sets for equality and inequality constraints, respectively. The feasible set is the set of points $$x$$ that satisfy the constraints.\n\n##### Optimization Under Uncertainty\n\nThe optimization problem types described in the Continuous Optimization section and the Discrete Optimization section implicitly assume that the data for the given problem are known accurately. For many actual problems, however, the problem data cannot be known accurately for a variety of reasons. The first reason is due to simple measurement error. The second and more fundamental reason is that some data represent information about the future (e. g., product demand or price for a future time period) and simply cannot be known with certainty.\n\nStochastic Programming and Robust Optimization are the most popular frameworks for explicitly incorporating uncertainty. Stochastic programming uses random variables with specified probability distributions to characterize the uncertainty and optimizes the expected value of the objective function. Robust optimization uses set membership to characterize the uncertainty and optimizes a worst possible case of the problem.","date":"2022-08-12 06:36:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6390866637229919, \"perplexity\": 217.82507326315474}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571584.72\/warc\/CC-MAIN-20220812045352-20220812075352-00104.warc.gz\"}"} | null | null |
The Toronto Fringe Festival
720 Bathurst St
Mon - Fri 10am - 5pm
The Fringe: Theatre Al Fresco
The "fringe movement" is a worldwide network of indie theatre festivals. As the name suggests, the movement is about celebrating under-represented voices and those on the margins of the performing arts world. As a result, the shows are a result of a lottery system of producers and writers who submitted their plays. It's a festival where anyone can put on any show, without having to pass through a jury – where theatre students can mount their first production outside of school, where emerging artists can get their big break, and where established artists can test out new work. Popular shows like da Kink in my Hair and The Drowsy Chaperone got their start at Fringe. Whether you're watching a show (or number of shows) in the back alley behind the famed Honest Ed's, or smaller theatres in Toronto like the Helen Gardiner Phelan Playhouse, you'll be in for a cheap treat.
By Natalie Taylor , AFAR Local Expert | {
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const rule = require('../../../lib/rules/no-observers');
const RuleTester = require('eslint').RuleTester;
// ------------------------------------------------------------------------------
// Tests
// ------------------------------------------------------------------------------
const eslintTester = new RuleTester();
eslintTester.run('no-observers', rule, {
valid: [
{
code: 'export default Controller.extend();',
parserOptions: { ecmaVersion: 6, sourceType: 'module' },
},
{
code: 'export default Controller.extend({actions: {},});',
parserOptions: { ecmaVersion: 6, sourceType: 'module' },
},
],
invalid: [
{
code: 'Ember.observer("text", function() {});',
parserOptions: { ecmaVersion: 6, sourceType: 'module' },
output: null,
errors: [{
message: 'Don\'t use observers if possible',
}],
},
],
});
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Program crashed, when Searching or Exporting to a text file.
New option in Export: "With caption for each node"
Intrinsic outliner can now be exported to a plain text file.
-- In the plain text format, only one kind of links (one of https://, http://, file:///) is created, even though the page has other kinds of links.
-- When the [Enter] key is pressed in the async mode, the contents of the node are not displayed.
-- After browsing http://... pages, when you select a node from the combo box, it is not displayed.
-- The file's extention name is corruped when an image is dropped from the Explorer.
-- When you click on a link to an image (jpg, gif, png), a dialog to save the file appears.
Microsoft DLL files (MSVCR120.DLL, MSVCP120.DLL) are now removed from the installer package.
- A large number is displayed, when the node size is 0, on the node properties dialog.
- New nodes sometimes get Read Only.
Bug fix: The Group pane will not be resized even though the whole window is resized.
Bug fix: When you create a new node and soon delete it, Foliaro clashes.
MSVCR120.DLL, MSVCP120.DLL are included in the setup package.
Bug fix: Unable to open a second Foliaro file from Explorer.
Bug fix: Unable to export image files imported in August 2015(?) or later.
Save the modification date/time when importing image files from the dialog of the Imported Images.
When exporting image files, the modification date/time will be set to the files.
(1)New links do not work. (2)When a link is edited, its destination URL is not actually corrected yet. In both cases, please click the Reload button.
Bug fix: Unable to hoist in the Intrinsic outliner mode.
Bug fix: Sometimes, unable to open from Menu→File→Favourite file.
When you first open a file from the menu of the button "Favourite/Recently opened files", and then try to open another file from the menu File→Favourite files, it won't open.
Messages of the upcoming expiration day will show up from 5 days before the expiration.
Zooming by the mouse wheel while pressing the Ctrl key.
Save the X scroll position, too, when closed.
When the content is displayed again, the previous (X, Y) position will be restored.
When "Remove format" is selected from the drop-down menu of [¶ ] in the edit toolbar, the attributes "name" and "id" will be removed, too.
If the image does not exist, the ALT string will be displayed with colour.
In the list of images, the date-time property "mtime", if any, will be displayed.
Save tabs with external URL when closing the file, in order to resume the tabs when reopend.
To enable this feature, go to the Tools menu -> Settings -> Preference dialog.
Jump to a specific location within the node (Supporting Anchor #name).
You can directly jump to a specific location within the node, scrolling the page so that the location will be seen. Put an anchor name after # when setting a link.
Modified the menu shown on the icon "Recently opened files" in the main toolbar.
Submenu "Favourite files" has been added to the top of the menu.
Select a zooming % predefined in the Zoom menu (View menu->Zoom). You can define the zooming percentages in the settings.ini file.
New menu "Paste text only" in the Edit menu.
Alternatively, you may press the Ctrl+Shift+V keys.
Modified "Remove Format" (on the "Paragraphs..." icon in the Edit toolbar).
The attributes of 'class' and 'style' will be removed from all the HTML tags except <br> <span> <b> <i> <u>.
Added "IFrame.html" in the template folder.
The registree will be shown for several seconds on the Tip bar (beneath the Contents pane).
Added the Smart Paste icon on the Bookmark dialog.
The Image dialog has a button "Show more..." / "Show less" for detailed settings.
When Foliaro is run in the 64bit-CPU environment, it would not finish completely even if all the Foliaro windows are closed; so Foliaro cannot start again unless you finish it using the Task Manager.
[serious] Crash when exporting to a text file or during search through the whole document.
Spaces were removed in the String textbox on the Link dialog.
You can customize the icons on the toolbars by editing the settings.ini file.
In the Windows menu, a newly opened window is placed below the old ones.
■ As for tables, each row is output in a line, and Tab characters will be put between columns.
■ A horizontal ruler is output as --------- .
The output text of intrinsic outliner is not pretty yet. This will be improved later.
You might often enter link info with copy & paste. To make it easy, a Paste button and a Clear button have been added.
* You can enter the URL and string(text) with one click, if you've copied a portion of an HTML page.
* If you copy a link on an HTML page and then show the Link Dialog, the URL and the string (text) are entered automatically.
While searching the contents pane, a tip will show up when the search string was not found.
Some styles -- .bg_blue, .bg_red, .bg_green -- are added to the Master styles.
The Lifetime Days box is enabled only when Specify Lifetime is selected.
Select a tab: "Properties" to edit or "Preview" to view the appearance.
You can preview the style settings while editing properties.
When selecting a style, the "Preview" tab is selected by default.
Foliaro switches to the last selected tab, when opening a file, which was closed with multiple tabs.
Improved domain names in the password list for sites.
(Ex.) www.domain.jp → the domain name "domain.jp" is shown.
Bug fix: crash when the URL is http://localhost:port .
Fixed the problem: Crash when accessing the local HTTP server which is running on the local PC.
The pasted image is imported to the Foliaro file. The default image type is BMP(bitmap). You can change the default type to BMP or JPG by the option "Default image type when pasted" (Tools->Settings->Preferences).
You can directly import the image when you select "Copy image" on your browser. Set the option "Pasting image from browser" to "import" (Tools->Settings->Preferences).
You can copy an image to the clipboard.
Press the right mouse-button on an image, and select "Copy image" from the context menu.
Formerly the pasted HTML source was enclosed by <div>...</div>. Now the <div> tag is removed, meaning, only the HTML source is pasted.
When you select an image from the File Dialog or the Imported Images Dialog, the file name is put in the ALT Text box.
Importing images has been improved (internally).
Domain names are listed on the left side in the alphabetical order.
Improved removing formats(style and class attributes) from paragraphs, blocks, headings, etc.
Added the Search Range option on the Search dialog.
Selectable from: (1)Current article, (2)Entire document, (3)Group, (4)Current node and its descendants.
Added the option Case Sensitive on the Replace dialog.
When the "Replace with" is empty, the strings of "Search for" are deleted.
Added the "Paste plain text" button on the HTML Edit dialog.
Available with the existing "Paste HTML source" button.
Fixed the bug (character encoding) when pasting plain text.
An empty line (<br>) is inserted when a new table is created.
In the earlier versions, when a table is inserted at the end of the article, no text can be entered below the table.
"Remove Format" in the "Paragraphs, Headings..." menu has been improved.
Now the format (the class and style attributes) in the child elements are also removed. When a fragment of HTML is pasted from a browser, unneeded classes and long style definitions are added, making the size larger. This command in the "Paragraphs, Headings..." removes long styles to make the size smaller.
There is a similar icon "Remove Format"; this cannot remove the style attributes in the paragraph/headings (<div> <p> etc.) themselves.
Added the ALT Text property on the Image dialog.
This text will be shown for an incorrect URL.
Selecting an alignment of the image has been changed to using combo-box instead of radio buttons.
The scroll position of each node, that you've viewed so far, are saved when the file is closed. (the option "Save window states on close" must be checked ON). When you open the file again, the previous scroll positions will be restored.
For advanced edit, you can paste HTML source, which has been copied with a browser. The ordinary Ctrl+V operation pastes only the plain text.
When the position of the window that is about to open is outside the display screen, the window will be moved to the upper left corner, so that you can actually view it. Chances are that the window's position gets outside the display screen, when you open a file that was created with another PC, and the current PC's display is smaller than that PC's.
The color of the Navigator toolbar will be changed when in the async mode, to distinguish the mode apparently.
You can change the colors of the toolbars and tabs, although, currently no settings-dialog is provided. You have to edit the settings.ini file directly yourself. The settings-dialog will be provided in a later version if there are requests.
Focus will be set on the Article pane, when you open a file, select a tab, etc. so that the key operation gets easier.
Added an attribute "line-height:1.3em" for #editText in the customstyle.css file (the same as #text) Applied to the nodes in Plain Text format.
Show the DDE error message. An error at launch. Normally not shown.
The saved data size becomes smaller (about minus 30 bytes per topic). When a node of Intrinsic Outliner is displayed, the data is automatically converted to the new format. You can view the node of new Outliner with an earlier version of Foliaro, but some operations are impossible.
Improved event procedures for Intrinsic Outliner and links.
Resolved the problem unable to set the username/password in some sites.
In some sites, when the mouse pointer hovers over a link, the page scrolls back to the top. This is caused by the size change of the Article pane when the tip appears.If this option is checked, the Tip pane stays on, making the Article size unchanged.
When you press a key (F1, F5, etc) or select another node after no interaction with Foliaro for 5 minutes or more, Foliaro will respond with delay of 2..3 seconds. This option enables quicker response in such a situation. See Preferences for more details.
This inserts a space between topics in the Intrinsic Outliner. You can adjust the space by changing the value (2px). This attribute does not exist in the existing files created by earlier versions; you have to add the attribute yourself.
-- Pasted relative paths: when a relative path is pasted to another file, it is converted to a path relative to that file. If the target file is new, the path is converted to absolute.
* Fixed bugs: "../../../" (or "../../") was prepended to the URL of a link / image.
* Support path names in Unicode.
-- Internal images are imported into the destination file.
-- Links to the internal nodes become invalid.
-- Relative paths to the images are converted to paths relative to the destination file.
-- NOTE: Relative paths in the link URLs are left unchanged!
When you move a topic to the right, it becomes a child of the above topic, which, however, is not expanded from the collapsed state.
Use relative paths for links, images, and bookmarks by checking "Make the path relative" on each dialog. The path becomes relative to the current Foliaro file. Useful, for instance, when you store images in sub-folders. When exporting, you can choose relative or absolute -- leaving the paths relative or converting them into absolute paths.
You can edit text files that is bookmarked in the Navigator. Click on the "Editable" checkbox to edit on the Bookmark dialog.
Text files are transformed into HTML format to be displayed; that is, long lines are wrapped at the right end of the article pane.
When moving a topic with Ctrl+Up/Down keys, it will be moved one by one topic upward/downward. For instance, when you move down a topic toward an expanded topic, it becomes the first child of the expanded topic below. On the contrary, when you move up a topic toward an expanded topic, it becomes the last child of the expanded topic. When the first or last child of a topic is moved up/down, it becomes a sibling of the (old) parent. But if you move a topic toward a collapsed topic, it is moved below/above the collapsed topic (does not become a child).
In addition to the default search engine, you can set up more search engines. Useful for searching web dictionaries. The search menu will be shown when you press the right mouse-button while selecting a word in the article. Set up by clicking Menu Tools→Settings→"Search Engine settings for selected text"
"Favorite files" and "Search Engine settings" are now on separate dialogs. Click the links on the Preference panel.
When you created a new group and you selected it, the URL combo-box showed like "/doc/1/2/3" instead of its caption.
Markers in Intrinsic outliners were not shown in the exported HTML.
An error occurred when exporting to a text file, if the default character set is selected.
Improved the creation and license check of Portable Foliaro.
Moved "Layout groups horizontally" and "Reset layout" to Windows menu.
When the right mouse button is clicked while a text is selected, you can "Search web for the selected text"
You can install Foliaro on a USB stick.
Folders for the files containing cookies etc. are changed. If you update Foliaro, old cookies are not carried to the new folder.
Added "Zoom" to the View menu.
On the calendar dialog, "Add time" is not checked (OFF) by default.
When a file was opened from the Recently Opened Files, and it was not found, the file will be removed from the list.
Added a style ".underline_red" to the customstyle.css file.
Retaining the selection state of bookmark nodes has been improved.
This is the improved feature of Focus in the earlier version.
You can click on each item in the Help to carry out the operation. A Help button is displayed at the top right corner in the Article pane. When the mouse hovers over the button, the Help is shown or hidden. To hide the Help button, press the ctrl+H keys.
When the Ctrl+Shift+O keys are pressed on the top text, the line is split there, and a new topic is created. On the other hand, when the Ctrl+J keys are pressed, the first topic is joined with the top text.
This feature is useful when you create topics from the existing article.
The Ctrl+T is not available any longer; instead, use the Ctrl+Q.
Data size has got smaller.
When you get back to the old page (node), it scrolls to the position you previously viewed.
Note, scroll positions of external pages are not retained. When the file is closed, all the scroll positions are reset to the top.
You can select a window out of the currently open windows.
Added the Cancel button on the Save Changes dialog when closing the file.
Tidier border style in the Article pane when in the Edit mode.
Menu name changed: Tools⇒"Options" to "Settings"
Separate Style file for exported HTML documents.
Flexible cell width of a table: defining with the min-width property.
Couldn't create a new topic when an existing node is set to the Intrinsic Outliner.
Discard the document if no file name is entered when closing a new document (in case the option "Save without prompt" is checked).
The document's style was not applied when previewing the HTML on the Insert Object/Template/HTML dialog.
In the preview above, images (imported) were not displayed.
Crashed when trying to show the link to non-existing node in a read-only window.
Crash when a new document created while another file is open.
* when moving a body with Left/Right arrow keys.
* a new node is put in a wrong position when created on a body topic.
* all the topics are expanded when a topic is moved by Ctrl+Right arrow key.
Bug when Ctrl+S keys are pressed in the Ariticle pane.
F5 : Toggle the focus on Navigator/Article.
Use tabs in the Options dialog.
Check for Update: displays HTML.
Enalbed to open a group in a new tab.
Fixed bug: dirty display of link URL.
Fixed bug: Select Node X in Group A→Move the Node X to another group /or delete the Node X→The Root Group is always shown /or the app crashes.
Bug fixed: Did Not Save even though the option "Save without prompt" is checked.
Reading / saving template files. Templates (HTML snippets) can be read from the Insert Object/Template/HTML in the Edit Toolbar.
Added "Check for update" in the Help menu, enabling to check for the latest program; other announcements, such as the scheduled date, are shown as well.
Shows the URL at the bottom of the Article pane when the mouse pointer hovers over a link.
Incorporated RegNow's Affiliate Tracking Code.
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Terminology: changed "Custom Style" to "Style"
Now the Root node can be pasted or dropped to another file.
The URL to a local HTML file is now file:///... instead of /file/..., and can be shown in a browser.
The bookmark to a local HTML file can be shown in a browser.
On exit, delete the temporary files in the Temp folder.
Shows the URL contents same as of the current tab in the new window (when the icon clicked) (the old version showed the contents of the current node).
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Specifies the data folder for the Key file, when registering the licence.
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Editing captions: supports Drag-drop and Copy-paste.
Added some icons in the Navigator's context menu.
Added the Edit button on the Custom Style Selection dialog.
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Paste only plain text when HTML has been dropped on the node of Plain Text type.
Bug fix: when the indentation becomes 0, the property style="" still remains.
Bug fix: abort when duplicating images to another file.
Bug fix: '+' character changes to a space in the link URL.
Includes Custom Styles; displays imported images; auto-modifies URLs in imported images and node-to-node links; converts plain text node to HTML; choosing with or without Navigator; showing the output file.
When nodes are pasted(dropped) in a different file, copies the imported images referred to by the nodes. Auto-modifies URLs of imported images and node-to-node links.
Creates the root and its child node on New File.
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On the Image List dialog, the list is sorted by clicking on the column-label.
Bug fix: the message "Save changes?" is shown, when you've edited nothing, but only changing nodes with multiple tabs. | {
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} | 3,284 |
\section{Introduction}
\qquad The volume of date generated by many existing and forthcoming astronomical instruments is simply too large for traditional analysis techniques. Two extreme cases are the Large Synoptic Survey Telescope (LSST; \citealt{LSST14}) and the Gaia mission \citep{Gaia15}.
Optimal use of modern astronomical instrumentation requires open and efficient access to the resulting observations. Such access is provided by large and well-organised databases, (e.g. the Hubble Space Telescope (HST), Gaia, or the Sloan Digital Sky Survey archives). As happens with data reduction, the exploitation of these vast data sets cannot be made using traditional tools (see e.g. the discussion by \citealt{Bailer-Jones02}). Classification is the first step in any automated analysis. It can be used to identify and discard noisy data or to group like objects to follow a common interpretation pipeline. It is certainly needed when exploring new types of data and it is also an invaluable tool to identify rare objects, usually the most telling from a scientific point of view.
Numerous works have been done that explore the performance of the automatic MK classification of spectra (see e.g. \citealt{Bailer-Jones1998, Singh, Bailer-Jones2001, Rodriguez2004, Giridhar, Manteiga2009, Navarro2012}). The main approach followed in these works was to apply supervised learning training using labelled data. The unsupervised approach was also applied in works like \cite{vanderplas}, \cite{Daniel2011} and \cite{itamar}. In this work we focus on an unsupervised approach that does not aim to reproduce the MK classification.
Among all unsupervised classification methods, $K$-means (e.g. \citealt{macqueen67}, \citealt{everitt92}, \citealt{50years}) is a flexible clustering algorithm that has being extensively used in the literature. We have already employed $K$-means in several applications, including the identification of similar targets to average and reduce noise \citep{sanchez09}, the classification of one million galaxy spectra representative of the local universe \citep{sanchez10},
a systematic search for rare extremely metal-poor galaxies \citep{Morales-Luis11, sanchez16}, and the classification of the large stellar spectra data set available from the Sloan Digital Sky Survey, in particular data from the Sloan Extension for Galactic Understanding and Exploration (SEGUE; \citealt{sanchez13}). In this work we show the virtues and limitations of $K$-means in this context, making a first step in the search for alternatives. This work is also the first to perform classification on APOGEE.
In this paper, we turn our attention to high-resolution stellar spectroscopy, and in particular to the Apache Point Galactic Evolution Experiment (APOGEE), part of the Sloan Digital Sky Survey (\citealt{eisenstein01}; \citealt{blanton17}). We examine whether or not the massive APOGEE data set is amenable to a sensible unsupervised classification scheme based on $K$-means. Section 2 describes the APOGEE spectroscopic data in detail, including the APOGEE Stellar Parameters and Chemical Abundances Pipeline (ASPCAP; \citealt{garcia-perez16}). Section 3 is devoted to the details of the classification algorithm, and Section 4 describes its application to the APOGEE data, preceded by numerical experiments based on simulated data. Section 5 discusses the main results, and Section 6 summarises the conclusions.
\section{Data set}
\qquad APOGEE makes use of a novel fibre-fed high-resolution $H-$band spectrograph to obtain simultaneously up to 300 stellar spectra \citep{wilson10, wilson12}. The APOGEE spectrograph is usually coupled to the Sloan Foundation 2.5-m telescope at the Apache Point Observatory, but has also been linked to the New Mexico State University 1-m telescope at the same location. The project has already obtained spectra for more than 300,000 stars in the Milky Way, focusing on red giants and therefore covering a broad range of galactocentric distances. Working in the near-IR, between 1.5 and 1.7 $\mu$m, APOGEE can access regions of the Galaxy heavily obscured by dust, such as the mid-plane of the Galaxy, or the bulge and the Galactic bar near the centre \citep{majewski16}.
APOGEE spectra are processed by a custom-made data pipeline that extracts the spectra, calibrates them, and corrects telluric absorption and sky emission lines before measuring radial velocities \citep{Nidever15}. The pipeline ASPCAP performs an automated analysis based on model atmospheres, delivering atmospheric parameters and chemical abundances for the majority of the observed stars.\footnote{Approximately 93 per cent of the spectra in APOGEE DR12 have uncalibrated atmospheric parameters, [M/H], [$\alpha$/M], [N/M]\, and [C/M]\, determined. The calibrated values are defined to $\approx$ 63 per cent of the spectra.} The atmospheric model grid boundaries in effective temperature are 3500 and 8000 K, in $\log g $ the boundaries are 0 and 5, and in [M/H]\, they are -2.5 and 0.5 dex. More details about the grid can be found in Table 2 of \citealt{Holtzman15}.
The APOGEE pipelines are in constant evolution and the data set continues to grow. In this work, we have adopted the data made publicly available in DR12\footnote{The catalogue is available at \href{http://data.sdss3.org/sas/dr12/apogee/spectro/redux/r5/allStar-v603.fits}{allstar file}.}, the final data release from SDSS-III (\citealt{Alam15}; \citealt{Holtzman15}). This data set includes over 150,000 stars observed between 2011 and 2014. The resolving power of the APOGEE data is $R\equiv \lambda/\delta \lambda\simeq 22,500$, and the typical signal-to-noise ratio exceeds 100 per half a resolution element. In addition, we used quality and target flags\footnote{The flags were extracted from the objects \textsc{targflags}, \textsc{starflags}, \textsc{andflags} and \textsc{aspcapflags}.
A complete description can be found in \href{http://www.sdss.org/dr13/algorithms/bitmasks/}{bitmasks documentation}.}, and the uncalibrated parameters derived by ASPCAP\footnote{This parameters are accessible through the objects \textsc{fparam} and \textsc{felem}, see \href{https://data.sdss.org/datamodel/files/APOGEE_REDUX/APRED_VERS/APSTAR_VERS/ASPCAP_VERS/RESULTS_VERS/allStar.html}{data model documentation}.} in order to evaluate the result of the classification (Section \ref{sec:description}). Besides sky coordinates and atmospheric parameters (temperature, surface gravity, and micro turbulence), the data set includes metallicities, $\alpha$-element abundance, and individual chemical abundances for 15 elements.\footnote{Al, Ca, C, Fe, K, Mg, Mn, Na, Ni, N, O, Si, S, Ti, and V.} As described in \citealt{Holtzman15}, the DR12 results were calibrated using star clusters' data in order to eliminate abundance trends with temperature and systematic differences with the literature. Since calibrated parameters are not available for all stars in DR12, we chose to use the uncalibrated parameters and chemical abundances. This choice should not affect the interpretation of our results; we are not interested in absolute values for each object, but in relative differences among spectra with intrinsically different shapes. In addition, using the uncalibrated data we can arguably better understand ASPCAP.
\section{Classification algorithm}
\label{sec:algorithm}
\qquad Cluster analysis aims to organise a collection of objects into classes based on a similarity criterion, such that objects in the same class are more alike than objects in different classes. There is a numerous set of cluster algorithms available in the literature (e.g. \citealt{everitt92}), but in general, all involve the following main steps: (1) Feature selection, the identification of the features that better represent the objects in the data set; (2) choosing a feature proximity indicator, the figure of merit that optimally defines the similarity between objects in the data set; (3) establishing the grouping criterion - meaning the clustering algorithm itself, and (4) cluster validation, an evaluation of the output quality.
In the feature selection phase, we excluded all pixels potentially affected by sky emission and telluric absorption. Standard $K$-means algorithms are designed such that all input objects must have the same dimensions, and therefore we have to consider the same pixels in all spectra. For the vast majority of APOGEE observations 35 fibres are devoted to observe warm stars, measuring telluric absorption, 35 fibres to observe the sky, pointing them to blank regions in the sky, and 230 fibres to acquire science spectra. To determine which are the pixels more affected by sky emission and telluric absorption, we have taken the average of the normalised sky, and the telluric spectra for all fields in APOGEE DR12, and used them to identify and exclude in our analysis all pixels for which the mean sky count is above 1 per cent of the maximum mean normalised sky count. We have also excluded all pixels at which the mean normalised telluric spectrum falls more than five per cent below the continuum.
Figure \ref{fig:Sky_Tel} shows the mean sky and telluric spectra, the cuts applied, and the regions excluded from the spectra used in the $K$-means classification. In this figure we have displaced vertically the mean sky spectrum for clarity. Since stars have different heliocentric velocities, the spectra were corrected for Doppler shifts, and therefore they are affected by sky emission and telluric absorption at different wavelengths for different stars. This can be seen in Figure \ref{fig:Sky_Tel} from the width of the mean normalised telluric lines and sky emissions lines. From the $8575$ original wavelength pixels, we kept $4838$ pixels, or $56$ per cent of the APOGEE spectral coverage. All the spectra were also normalised using a fourth degree polynomial regression for each of the three chips in the APOGEE spectrograph. We have also removed values in the normalised flux higher than 1.02 (i.e. two per cent above the pseudo continuum level), setting the flux value to 1.02, avoiding any remaining problem with sky emission lines.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{./img/Sky_Tell.png}
\caption{\label{fig:Sky_Tel} Mean sky normalised emissions (blue line) and telluric absorption (red line) spectra for the 153,847 spectra in the sample. Mean sky normalised emissions fluxes are displaced by one unit to help visualisation. Black lines define the cut applied to each spectrum. Grey shades highlight the areas excluded from the $K$-means classification.}
\end{figure}
The chosen feature proximity metric was the Euclidean distance. That is the most straightforward possibility, since the objects to be classified are normalised spectra, which can be regarded as data points in an N-dimensional space. It also has the advantage of being easily interpreted and having a low computational cost.
The grouping criterion is the way one assigns each object to a certain cluster and is how groups are designed. For example, groups can be selected in a single partition, that is to say, all clusters are simple partitions, hierarchically equivalent samples, otherwise they would be hierarchical clusters that have a structure with clusters and sub-clusters. Furthermore, clustering is said to be hard if it assigns each object to a single cluster, in opposition to soft clustering where the objects are assigned as having a non-zero probability of belonging to more than one cluster.
In this work we explore the use of $K$-means \citep{macqueen67}, a partitional hard clustering algorithm. It is one of the most popular clustering algorithms, mainly because it is easy to implement and its computational cost scales linearly with the number of objects to be classified. The fundamental steps in $K$-means are (1) to choose the number of clusters $K$; (2) define $K$ initial cluster centres; (3) assign each object in the sample to the closest cluster; (4) recompute cluster centres as the centroid of the objects assigned to each cluster; (5) repeat steps 3 and 4 until a convergence criterion is met. Usually the convergence criterion is either a decrease of the within-cluster variance under a threshold or a minimal re-assignation between two consecutive iterations. Here we adopt the criterion of having less than one per cent of re-assignation between two consecutive iterations.
Initialisation also can be done in different ways. The simplest is to randomly choose objects in the entire sample, but if the data set has an over-abundance of a particular kind of object, the clusters would over-sample those objects. In order to avoid this, we initialise in an iterative fashion; we carry out a couple of $K$-means iterations with $K = 10$, randomly choose an object in the most abundant cluster as initial centre, discard all objects in this cluster and repeat the process until the desired number of initial cluster centres is reached. During the process, if more than 95 per cent of the objects are discarded, we select the remaining cluster centres randomly in the whole sample. In this work we have translated the algorithm presented by \citealt{sanchez10} from IDL\footnote{\href{http://www.harrisgeospatial.com/ProductsandSolutions/GeospatialProducts/IDL.aspx}{http://www.harrisgeospatial.com/ProductsandSolutions/GeospatialProducts/}} to Python \footnote{\href{www.python.org}{www.python.org}}. Besides serial and parallel performance optimisation, no major modifications were made. Using Python we achieved a simpler and faster code, which also has the advantage of being available in an open source platform.
We have compared our results with the results using scipy\footnote{\href{www.scipy.org}{www.scipy.org}} and scikit learn\footnote{\href{http://scikit-learn.org}{http://scikit-learn.org}} algorithms. The results are qualitatively equivalent. The advantage of using our own code is that we are coherent with previous works in the literature \citep{sanchez09, sanchez10, Morales-Luis11, sanchez13, sanchez16}.
A major drawback in any clustering classification is that the algorithm will always return partitions regardless of the existence of clusters or not. In addition, the algorithm does not guarantee convergence to a global solution. Moreover, many implementations require choosing the number of clusters. In order to overcome these problems, or even just to find out how serious they are, we apply cluster validation techniques. We are interested in verifying whether the data have intrinsic clusters, whether there is an optimal number of clusters, and whether the clusters derived in flux space exist in parameters' space.
\subsection{Choosing the number of clusters}
\label{subsec:find_K}
Choosing the optimal number of clusters is a critical step in $K$-means classification. There is no universal criterion to do it, although many heuristic criteria have being developed over the last fifty years \citep{tibshirani2001estimating}. In an attempt to select the most suitable criteria for our problem we built a testbed data set with 6900 synthetic spectra spread over 69 well-defined clusters in surface gravity ($\log g $, in cgs units), temperature ($T_\mathrm{eff}$), $\alpha$ abundance ([$\alpha$/M]), and metallicity ([M/H]), as shown in Figure \ref{fig:synt_parms}. The brackets, for two given elements X and Y, [X/Y] is defined as: $$
[\mathrm{X}/\mathrm{Y}]= \log_{10}{\left({\frac{N_{\mathrm{X}}}{N_{\mathrm{Y}}}}\right)_{\mathrm{star}}}-\log_{10}{\left({\frac{N_{\mathrm{X}}}{N_{\mathrm{Y}}}}\right)_{\odot}},
$$ where $N_{X}$ and $N_{Y}$ are the number of X and Y nuclei per unit volume, respectively. Metallicity is a measure of all the chemical elements heavier than He, assuming they vary in the same proportions with respect to the solar values. Analogously, [$\alpha$/M]\, is a measure of all $\alpha$-elements (O, Ne, Mg, Si, S, Ar, Ca, and Ti) assuming they vary in union. The centres of the clusters were chosen based on the most dense regions in the HR diagram of the empirical data set with parameters from DR12. The parameters for each spectrum were randomly chosen around each cluster centre, following a normal distribution with $\sigma_{T_\mathrm{eff}} = 50 $K and $\sigma_{\log g} = \sigma_{\mathrm{[M/H]}} = \sigma_{[\alpha/\mathrm{M}]} = 0.05$. The synthetic spectra were built using the code \textsc{FERRE}\footnote{\href{https://github.com/callendeprieto/ferre}{https://github.com/callendeprieto/ferre}.}, interpolating in a grid of theoretical models (\citealt{ferreII, ferreIII, zamora2015}). We use a model grid with seven parameters per spectrum, microturbulence velocity $(\xi_\mathrm{v})$, carbon abundance ([C/M]), nitrogen abundance ([N/M]), mean $\alpha-$elements abundance ([$\alpha$/M]), metallicity ([M/H]), surface gravity ($\log g $), and effective temperature ($T_\mathrm{eff}$). But the parameters $\xi_\mathrm{v},\, \mathrm{[C/M]},\, \mathrm{[N/M]}$ were fixed to the mean values\footnote{$\langle \xi_\mathrm{v} \rangle = 0.169\,km\,s^{-1},\, \langle \mathrm{[C/M]} \rangle = 0.122\,,\, \langle [N/M] \rangle = 0.227.$} of the stars in the DR12 sample for all spectra. In order to explore the best-case scenario we have not added any noise to the spectra.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{./img/synt_parms.png}
\caption{\label{fig:synt_parms} Atmospheric parameters for the synthetic data set. Left panel shows effective temperature and surface gravity for the synthetic spectra. The right top panel presents the projection of the clusters in the $T_\mathrm{eff} - \mathrm{[M/H]}$ plane, while the right bottom panel shows the plane $T_\mathrm{eff} - [\alpha/\mathrm{M}]$.}
\end{figure}
We applied $K$-means to the simulated data set ten times, with $K$ varying from 5 to 100. We then applied four different statistical criteria trying to recover the optimal number of clusters, knowing that the actual number is 69. We tried the KL index \citep{DIFF_ref}, the gap statistic \citep{tibshirani2001estimating}, the CH index \citep{CH_ref}, and the silhouette index \citep{silhouette}.
These indexes were selected for being the most widely and successfully used in the literature. None of the chosen criteria was able to identify the right number of clusters, with the CH index being the only one capable of giving consistent results over different initializations, finding $K = 9 \pm 1.8$, far from the true value of 69. The other methods found a $\sigma_{K} > 12$ over the ten different runs, while randomly selecting ten numbers in this range would result in $\sigma_{K} \approx 25$. A possible explanation for this failure is that, despite the clusters being well-defined in parameters' space, the classification is made in flux space, where the separation between classes seems to be more subtle.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/mean_std_synt.png}
\caption{\label{fig:mean_std_synt} Variation of median standard deviation as function of the number of clusters $K$ in the synthetic data set. The solid black lines represent the median standard deviation of the classes in a run. The solid horizontal grey lines show the median standard deviation at $K = 69$. Dashed black lines show the input standard deviation. The top left panel refers to $\alpha$ abundances [$\alpha$/M], the top right to metallicity [M/H], bottom left to logarithmic surface gravity $\log g $\,, and the bottom right to effective temperature $T_\mathrm{eff}$.}
\end{figure}
In the absence of better criteria, we have chosen the numbers of clusters based on the within-class standard deviation of the atmospheric parameters and chemical abundances. Figure \ref{fig:mean_std_synt} shows the variation of the median $\sigma$ values for each of the four main input parameters. We use the notation $\widehat{X}$ meaning the median of $X$. It is important to use medians instead of means in order to avoid the predominance of the fewer classes which gather faulty and unusual spectra. Especially when we start to work with the observed data set, with classes having few spectra ($< 30$) and a large dispersion in atmospheric parameters and chemical abundances. We see a decrease in $\widehat{\sigma}_{X}$ as $K$ grows for all quantities. This means that dividing the spectra in flux space into more classes results also in finer partitions in atmospheric parameters and abundances spaces. Therefore, we can choose $K$ based on a threshold value for $\sigma$. The extreme case would be to increase $K$ until having one star per class, reaching the minimum variation. However, since the computational cost scales with $K$ and we also lose generality when increasing $K$, we should choose $K$ making a compromise between accuracy, agility, and generality.
We know the $\widehat{\sigma}_{X}$ values and $K$ for the synthetic data set. Therefore we can verify how much we can trust the variance for the choice of $K$. Figure \ref{fig:mean_std_synt} shows that when $K = 69$ we have exactly the input metallicity dispersion; $\widehat{\sigma}_{\log(g)}$ is highly above the input level, while $\widehat{\sigma}_{\alpha}$ and $\widehat{\sigma}_{Teff}$ are both below the input level.
The figure also shows the slope ($|\partial \widehat{\sigma}_{X}/\partial K|$) of the curves decreases rapidly for $K \gtrsim 50$. Therefore, increasing $K$ does not produce a significant change in $\widehat{\sigma}_{T_\mathrm{eff}}$ and $\widehat{\sigma}_{\log g}$ for $K \gtrsim 50$. The plots also reveal a different sensitivity for each parameter.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/mean_std_real.png}
\caption{\label{fig:mean_std_real} Variation of median standard deviation as function of the number of clusters $K$ in the real data set. Top panel refers to effective temperature $T_\mathrm{eff}$, while the bottom panel to the variation of median standard deviation for the other 20 parameters available in DR12 as indicated in the legend box.}
\end{figure}
The actual APOGEE data set behaves in a similar way. Figure \ref{fig:mean_std_real} shows how $K$ affects the median of the standard deviation of $T_\mathrm{eff}$, $\log g $, [M/H]\, and the abundances of carbon, nitrogen, and $\alpha$-elements with respect to metallicity, and the same for the abundances of the chemical elements Al, Ca, C, Fe, K, Mg, Mn, Na, Ni, N, O, Si, S, Ti, and V. From these plots we have chosen $K = 50$ as the number of clusters to be used throughout the paper, since beyond that value increasing $K$ does not reduce significantly the within-cluster parameters' dispersion.
\begin{figure*}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width= 0.92\textwidth]{./img/coincidence_real.png}
\caption{\label{fig:mean_confusion} Top panel shows the mean coincidence matrix comparing the chosen classification with the other 99 performed classifications. Elements on the mean diagonal represent the coincidence ratio of a class and can be interpreted as the stability of the class. The elements in the diagonal are labelled with their corresponding class number and highlighted in green if the class has a coincidence ratio above 75 per cent or in red if the class has coincidence ratio below 25 per cent. Elements off the diagonal can be interpreted as the confusion rate between two classes. We highlight confusion rates above 25 per cent with white stars. The bottom panel presents a histogram of the coincidence ratios corresponding to the diagonal of the coincidence matrix. A green dashed line marks the 75 per cent level, while a red line marks the 25 per cent level. }
\end{minipage}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/d_rms_distribution.png}
\caption{\label{fig:chi2} Two-dimensional histogram comparing distances from each spectra to its best fit with the distance from each spectra to its class centroid. Each pixel in the image is colour-coded according to the number of spectra in that region, as indicated by the colour bar.}
\end{figure}
\begin{figure*}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=0.98\textwidth]{./img/Groups_MH_Teff_new.png}
\caption{\label{fig:Groups_MH_Teff} Contour diagrams in the $T_\mathrm{eff} - \mathrm{[M/H]}$ plane. Different colours are used to distinguish different classes. Each class is represented by four colour shades; from dark to light, the shades enclose 15, 30, 45, and 68.3 per cent of the data points in the class. The groups are separated into three panels minimising the superposition of classes. Panel $(a)$ shows groups 0, 1, 4, 5, and two classes of group 7, panel $(b)$ groups 2, 3, and three classes of group 7 and panel $(c)$ shows group 6. In these panels each class is flagged with a floating label in the form G\textsc{x}C\textsc{xx}, C referring to class and G to its group. Class 21 is represented as a scatter plot, since it is too concentrated to present visible contours on this scale.}
\end{minipage}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/MH_Teff_G8.png}
\caption{\label{fig:MH_Teff_G8} Scatter plot of $T_\mathrm{eff}$\, against [M/H]\, for classes in group 8. The classes are identified as shown in the legend. The stars in this group are scattered throughout the plane.}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.98\textwidth]{./img/Groups_LOGG_Teff.png}
\caption{\label{fig:Groups_LOGG_Teff} Contour diagram for the groups in the $T_\mathrm{eff} - \log g$ plane. Each group is represented by a different colour. Colour shades enclose, from dark to light, 15, 30, 45, and 68.3 per cent of the objects in each group.}
\end{figure*}
\begin{figure*}
\begin{minipage}{\textwidth}
\centering
\includegraphics[angle=0, width=0.97\textwidth] {./img/spectra_pile_windows_final.png}
\caption{\label{fig:spectra_pile} Mean spectra of the classes in the wavelength range from 16178 to 16222 \AA, where the differences among classes are particularly enhanced. Top to bottom: Panels show the mean spectra for classes belonging to groups from 0 to 7. Each mean spectrum is drawn with the same colours used in Figure \ref{fig:Groups_MH_Teff}. In all panels we plot the spectral windows used in ASPCAP to determine the chemical abundances of stars. Each set of element windows is colour-coded as indicated in the legend of the first panel.}
\end{minipage}
\end{figure*}
\subsection{Repeatability of the classification}
\label{subsec:repeat}
\qquad The randomized initialization of $K$-means implies that different runs generate slightly different results. In order to evaluate the repeatability of the process we define a coincidence index $\varepsilon$, which measures the ratio of coincidence between two different classifications based on the number of spectra in equivalent classes, as described in \citealt{sanchez10}. We note that the label assigned to a class can vary over the classifications, even when the class remains with essentially the same objects. Therefore, when comparing two different classifications, we first need to cross identify the classes. For example, let $X$ be a set of $N$ objects, $X = \{\vec{x}_0, \vec{x}_1,..., \vec{x}_N\}$ classified in $K$ clusters, with two different initializations. Each initialization generates a set of clusters, say $\Omega = \{\omega_0, \omega_1,..., \omega_K\}$ in one classification and $\Gamma = \{\gamma_0, \gamma_1,..., \gamma_K\}$ in a second classification. In each classification we label the classes ensuring that the number of objects in the $i$th class ($n_{i}$) follows the rule $n_{i} \geq n_{i+1}$. To build a comparison between clusters we define a coincidence matrix $ \mathbf{A}_{K,K}$, with the elements $a_{i,j}$ being the number of objects in cluster $\omega_i$ that are also in cluster $\gamma_j$.
\begin{equation}
\centering
a_{i,j} = \sum_{\iota \epsilon \omega_i} \delta_{\iota}^{j}\textrm{, where } \delta_{\iota}^{j} = \begin{cases} 1, & \textrm{if $\vec{x}_\iota$ is in cluster $\gamma_j$} \\ 0, & \textrm{if it is not.} \end{cases}
\end{equation}
Thus, we match the $j$th cluster in $\Gamma$ to the cluster in $\Omega$ having the maximum number of coincidences with it, $j_{match} = argmax\{a_{0, j}, a_{1, j},..., a_{i, j}\}$, always ensuring no cluster in $\Omega$ is assigned to more than one cluster in $\Gamma$. Then we use the matches to transform the matrix $\mathbf{A}$ into $\mathbf{A}'$ permuting its columns to have their largest numbers in the diagonal. The elements of the diagonal of $\mathbf{A}'$ ($a'_{i,j}$, with $i = j$) are counts of the number of agreement between the two classifications, while the other elements ($a'_{i,j}$, with $i \neq j$) are counts of the number of confusions between the two classifications. The trace of $\mathbf{A}'$ divided by the total number of classified objects gives an estimate of the mean overall coincidence rate between the two classifications, $ \bar{\varepsilon}_{total} = \textrm{Tr} \left\{\mathbf{A}'\right\}/N $. By defining the mean normalised coincidence matrix between a chosen classification ($\bar{\mathbf{A}}'_{chosen}$) and a set of $\eta$ classifications with the same $K$ as $\bar{\mathbf{A}}'_{chosen}$, the diagonal elements will give the mean coincidence ratio of each class over the $\eta$ classifications, which is a measure of how stable the classes in the chosen classification are. Likewise, the elements out of the diagonal measure the mean confusion ratio between different classes.
\subsubsection{Synthetic data set}
We performed a series of classifications for the synthetic data set varying the number of clusters from $K = 5$ to $100$. For each value of $K$ we initialized the classification with ten different random seeds, the same ten seeds for all values of $K$. In order to avoid some possible bias caused by choosing a particular reference, the coincidence ratio was measured for every pair of classifications having the same $K$. For the expected number of clusters in the synthetic data set ($K=69$) the mean coincidence ratio is $\bar{\varepsilon}(K=69) = 74.7 \pm 6.2$ per cent. The mean coincidence ratio computed for all runs with $K = 5$ to $100$ for the synthetic data set is $75.1 \pm 8.4$ per cent.
\subsubsection{DR12 data set}
\qquad Under equivalent conditions, that is, comparing all combinations of the ten classifications per value of $K$, with $K$ from 5 to 100, the DR12 data set had a mean coincidence ratio of $\bar{\varepsilon} = 77.9 \pm 7.8$ per cent.
When we consider only the $K=50$, for which we performed 100 classifications with different random initialization, and using the chosen classification (see Sec. \ref{subsec:chosen}) as reference, the mean coincidence ratio is found to be $\bar{\varepsilon}(K=50) = 79.6 \pm 2.6 $ per cent.
To understand what a mean coincidence ratio of 79.6 per cent means, we measured the mean difference between the matching classes over the 100 classifications, and compared this with the mean within the cluster dispersion of the chosen classification (see Appendix \ref{sec:app0} for more details). We found that the variations of the class centroid over the 100 classifications amount to $6.4 \pm 3.3$ per cent of the average mean internal dispersion of its corresponding class in the chosen classification. That is to say, even for runs with different classifications, for about 25 per cent of the spectra (coincidence of 75 per cent) the main classes end up having their centres displaced by about 6 per cent of the internal dispersion of its class in the $4838$-dimensional flux space. As we show in Section \ref{subsec:chosen}, the confusion occurs mainly between classes sharing borders in the space $T_\mathrm{eff} - \log g - \mathrm{[M/H]}$. Except for some outlier classes, the shapes of the classes are very similar over different classifications.
\subsection{Chosen classification}
\label{subsec:chosen}
\qquad After running $K$-means a hundred times with $K = 50$, we chose the classification with the lowest sum of squared error (SSE). As we are working with the Euclidean metric, the SSE is computed as
\begin{equation}
\centering
\mathrm{SSE} = \sum_{i=1}^{K} \sum_{\iota \epsilon \omega_i}||\vec{x_{\iota}} -\vec{\mu_i}||^2, \textrm{ where } \vec{\mu}_i = \frac{1}{n_i} \sum_{\iota \epsilon \omega_i} \vec{x}_\iota,
\end{equation}
where $x_\iota$ is the $\iota$th spectrum in cluster $\omega_i$ and $\mu_i$ the centroid of the class $i$. The chosen run has an SSE 9 per cent smaller than the average SSE over all classifications. As mentioned in \ref{subsec:repeat}, the coincidence ratio is measured by the number of spectra sharing the same class over two distinct classifications. Comparing the chosen classification with the other 99 runs, the average coincidence ratio is $ 79.6 \pm 2.6 $ per cent, which can be considered a high repeatability rate. Also the mean variation of the centres of the most popular classes, containing 99 per cent of the objects, is $\approx 2.4 $ per cent of the mean within-cluster variation of the classes in the chosen classification. Again, this is a comparison between the standard deviation of the centroids over the 100 classification with the internal standard deviation of the main classes in the chosen classification. In this case the number falls from 6.4 to 2.4\% because we are only taking into account the classes containing 99\% of the spectra in the sample, classes from 0 to 31.
In Figure \ref{fig:mean_confusion} we plot $\mathbf{A}'_{chosen}$, comparing the chosen classification with the other 99 classifications. The elements of $\mathbf{A}'_{chosen}$ are represented by a colour scale in a 2D histogram; the bottom panel in this figure shows a histogram with the main diagonal values of $\mathbf{A}'_{chosen}$. This plot will be useful in Section \ref{sec:description}, where we will describe each group of classes and comment on the stability of each class. From now on, we will refer to the elements in the main diagonal of $\mathbf{A}'$ as coincidence rates and to its other elements as the confusion rates.
In Figure \ref{fig:chi2} we show a comparison between the root mean squared distances for each spectrum in the sample to its best fit spectrum with respect to the centroid of its assigned class. The plot shows the centroids are n average approximately five times closer to the spectra than its best fit. These higher distances between the spectra and the models are due to systematic differences between synthetic spectra based on model atmospheres and real spectra.\footnote{This can also be seen in panel F of the summarised plots in the appendix. For instance, \href{https://garciadias.github.io/APOGEE/group0/class2/index.html}{Class 02} present these systematic differences near 16205 \AA\, and 16215 \AA.}
Table \ref{tab:sens} shows a comparison between the standard deviation within clusters ($\widehat{\sigma}$) and the overall standard deviation ($\sigma_{random}$), corresponding to clusters randomly built. For example, $T_\mathrm{eff}$\, and $\log g $\, have a $\widehat{\sigma}$ about 3.6 and 4.2 times smaller than their corresponding $\sigma_{random}$, respectively. This means that the algorithm is especially sensitive to $T_\mathrm{eff}$\, and $\log g $.
In Table \ref{tab:sens} we also highlight the parameters that present $\widehat{\sigma}$ at least two times smaller than its $\sigma_{random}$. They are $T_\mathrm{eff}$, $\log g $, [M/H], [Ca/H], [C/H], [Mg/H], [N/H], [Si/H], [S/H]\, and [Ti/H]. Since these are the most sensitive parameters to $K$-means, we will focus mainly on them in order to interpret the classes in the next section.
\begin{table}
\centering
\caption{\label{tab:sens} Comparison of the internal median standard deviation (third column) with the overall standard deviation (second column) for each parameter. The fourth column displays the ratio of these quantities. We highlight the parameters that have internal median standard deviation at least two times smaller than the overall standard deviation.}
\begin{tabular}{cccc}
\centering
Parameter& $\sigma_{random}$&$\widehat{\sigma}^{K=50}$&$\sigma_{random}/\widehat{\sigma}^{K=50}$ \\ \hline
$T_\mathrm{eff}$\, (K) & 553 & 152 & \textbf{3.6}\\
$\log g$ & 1.17 & 0.28 & \textbf{4.2}\\
$\mathrm{[M/H]}$ & 0.35 & 0.17 & \textbf{2.1}\\
$\mathrm{[C/M]}$ & 0.12 & 0.11 & 1.1 \\
$\mathrm{[N/M]}$ & 0.18 & 0.12 & 1.5 \\
$\mathrm{[\alpha/M]}$ & 0.10 & 0.08 & 1.3 \\
$\mathrm{[Al/H]}$ & 0.13 & 0.10 & 1.3 \\
$\mathrm{[Ca/H]}$ & 0.48 & 0.22 & \textbf{2.2}\\
$\mathrm{[C/H]}$ & 0.31 & 0.15 & \textbf{2.1}\\
$\mathrm{[Fe/H]}$ & 0.38 & 0.23 & 1.7 \\
$\mathrm{[K/H]}$ & 0.12 & 0.10 & 1.2 \\
$\mathrm{[Mg/H]}$ & 0.75 & 0.35 & \textbf{2.1}\\
$\mathrm{[Mn/H]}$ & 0.15 & 0.09 & 1.6 \\
$\mathrm{[Na/H]}$ & 0.15 & 0.10 & 1.5 \\
$\mathrm{[Ni/H]}$ & 0.29 & 0.18 & 1.6 \\
$\mathrm{[N/H]}$ & 0.32 & 0.16 & \textbf{2.0}\\
$\mathrm{[O/H]}$ & 0.39 & 0.21 & 1.9 \\
$\mathrm{[Si/H]}$ & 1.00 & 0.43 & \textbf{2.3}\\
$\mathrm{[S/H]}$ & 0.77 & 0.35 & \textbf{2.2}\\
$\mathrm{[Ti/H]}$ & 0.71 & 0.33 & \textbf{2.1}\\
$\mathrm{[V/H]}$ & 0.36 & 0.19 & 1.9
\end{tabular}
\end{table}
\section{Results}
\label{sec:description}
\qquad After visual inspection we divided all the classes into nine groups sharing similar properties. Here we describe in detail each group, giving a summary of their classes' mean properties. In Figure \ref{fig:Groups_MH_Teff} we present contour plots in $T_\mathrm{eff} - \mathrm{[M/H]}$ space. We highlight regions enclosing progressively 15, 30, 45, and 68.3 per cent of the stars in each class, with the colour shades varying from strong to light respectively. Class 21 is too concentrated to have its contours seen at this scale, so it is represented by purple dots in the figure. In some cases the separation of the contours is too tight and only three contours are visible. The figure is divided into three panels, aiming to minimise the superposition of classes. We use different colours to help identifying borders between classes. Some classes have the same colour, but there is no overlap between classes with the same colour. Classes are identified with labels.
For the labels we use the abbreviations $G$ for group and $C$ for its associated classes. Classes in group 8 have few objects, which are sparsely distributed in the $T_\mathrm{eff} - \mathrm{[M/H]}$\, plane, making this plot very noisy and hard to read; for these objects we present a scatter plot in Figure \ref{fig:MH_Teff_G8}. Figure \ref{fig:Groups_LOGG_Teff} shows the main distribution of the groups in the $T_\mathrm{eff} - \log g$ plane. Besides the differences found in the $T_\mathrm{eff} - \mathrm{[M/H]}$ space, we also found some other particularities in the classes and groups, some of them based in the spatial distribution (RA - DEC), global chemical abundances, or spectral fluxes.
In Figure \ref{fig:spectra_pile} we present the mean spectra, in a limited spectral window, for all classes in groups 0 to 7. Each panel in this figure shows the mean spectrum of the classes in each group colour-coded as in Figure \ref{fig:Groups_MH_Teff}. In order to offer the highest contrast between the classes' mean spectra, we chose the spectral coverage which maximises the cumulative variance over the first 32 classes in a 150-pixels-long window. The grey shades in the background of these plots highlight the masked pixels (those discarded from the classification, as discussed in Section \ref{sec:algorithm}). Besides the description presented in this section, we include complementary material with detailed plots for many of the DR12 available features in the supplementary online material described in the Appendix. Table \ref{tab:desc} gives a short description for each class and provides links to the online material. With these figures the reader can find more details about the atmospheric parameters, spatial distributions, and chemical abundances for each class presented.
Tables \ref{tab:quantiles0} and \ref{tab:quantiles1} present the median values for the atmospheric parameters and all the individual chemical elements in each class. The error bars presented in the tables, as well as those shown in the next sections, were calculated by taking the interval around the median, which encloses 68.3 per cent of the points in each class.
\subsection[G0: Metal-rich RC/warm RGB]{Metal-rich RC/warm RGB - Group 0 (Classes 2, 4, 6, 8 and 9)}
\label{subsec:G0}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/Group_0_LOGG_TEFF.png}
\caption{\label{fig:G0_LOGG} $T_\mathrm{eff} - \log g$ distribution for classes in group 0. The same rules and colours from Figure \ref{fig:Groups_MH_Teff} were applied to contours here. Top and right panels show histograms of the distributions of $T_\mathrm{eff}$\, and $\log g $, respectively. The histogram line colours match the colours of the contours.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width= 0.48\textwidth]{./img/projection_G0.png}
\caption{\label{fig:projection_G0} Mollweide's projection of the Galactic coordinates distribution of classes 8 and 9. Yellow and blue contours enclose 68.3 per cent of the stars in classes 8 and 9, respectively. Yellow squares represent stars in class 8 and blue squares represent stars in class 9 out of the regions containing 68.3 per cent of the points. The contour shades follow the same rule as in Figure \ref{fig:Groups_MH_Teff}.}
\end{figure}
\qquad From the distribution of $\log g $\, and $T_\mathrm{eff}$\, values in Figure \ref{fig:G0_LOGG} one can spot this group among the red clump (RC) stars and at the warmest end of the red giant branch (RGB) \citep{binney}. Comparing these classes with \citealt{bovy14}'s catalogue of red clump stars, we found that 31, 26, 26, 1, and 21 per cent of the stars in classes 2, 4, 6, 8, and 9, respectively, belong to the red clump. The classes increase in metallicity in the sense $-0.07\pm^{0.10}_{0.11} = \widehat{\mathrm{[M/H]}}_{c2} < \widehat{\mathrm{[M/H]}}_{c8} < \widehat{\mathrm{[M/H]}}_{c4} < \widehat{\mathrm{[M/H]}}_{c6} < \widehat{\mathrm{[M/H]}}_{c9} = 0.30\pm^{0.09}_{0.12}$. As metallicity increases, the position of the RC moves towards cooler regions in the plane $T_\mathrm{eff}$ - $\log g $, as shown in Figure \ref{fig:G0_LOGG}. Chemical abundances for individual elements also vary inside this group; for example, [Si/H]\, varies as follows: $-0.22\pm^{0.20}_{0.30} = \widehat{\mathrm{[Si/H]}}_{c2} < \widehat{\mathrm{[Si/H]}}_{c8} < \widehat{\mathrm{[Si/H]}}_{c4} < \widehat{\mathrm{[Si/H]}}_{c6} < \widehat{\mathrm{[Si/H]}}_{c9} = 0.26 \pm^{0.18}_{0.19}$. This group is similar to group 5 in terms of atmospheric parameters, but classes here are more metal rich.
For this group there is some confusion among classes, as shown in Figure \ref{fig:mean_confusion}. About 30 per cent of the spectra belonging to class 4 in the chosen classification are assigned to class 2 in other classifications.
Classes 2 and 8 are similar in chemical abundances, but differ in $\log g $. Besides metallicity differences, classes 8 and 9 also differ in their spatial distribution over the Galactic plane, as shown in Figure \ref{fig:projection_G0}. While stars in class 8, with lower [M/H], lie preferentially at higher galactic longitudes, stars in class 9, which are cooler and more metal rich, are mainly towards the galactic centre. In general the fittings for class 9 are poor, the spectral lines are deeper than the chosen models. Classes 2, 4, and 6 follow approximately the same spatial distribution of the APOGEE sample.
In the top panel of Figure \ref{fig:spectra_pile} we have a comparison of the mean spectra for all the classes in this group. For group 0, we see that their mean spectra are very similar in shape, but with different line strengths ($s$). The intensity of lines grows in the sense $s_{c8} < s_{c2} < s_{c4} < s_{c6} < s_{c9}$, following their median temperatures.
Together, these classes include $\approx 27$ per cent of the spectra in DR12.
\subsection[G1: Metal poor cool RGB]{Metal poor cool RGB - Group 1 (Classes 7, 14, 19, 25, 26 and 28)}
\label{subsec:G1}
\qquad As shown in Figure \ref{fig:G1_LOGG}, the classes in group 1 are composed of cooler stars in the RGB ($3500 \lesssim \widehat{T_\mathrm{eff}} \lesssim 4200 $ K and $0.79 \lesssim \widehat{\log g} \lesssim 2.03 $ ) \citep{binney}. All classes are mainly formed of low latitude stars, composed of a mixture of thin and thick disk population, except for class 28 which is mainly projected towards the Galactic centre and with high $\alpha$ abundances, $\widehat{[\alpha/\mathrm{M}]} = 0.24\pm^{0.04}_{0.11}$. All of them are classes composed of stars in the RGB, but with increasing metallicities,
\footnote{$ -0.81\pm^{0.19}_{0.33} = \widehat{\mathrm{[M/H]}}_{c28} < \widehat{\mathrm{[M/H]}}_{c26} < \widehat{\mathrm{[M/H]}}_{c19} < \widehat{\mathrm{[M/H]}}_{c25} < \widehat{\mathrm{[M/H]}}_{c7} < \widehat{\mathrm{[M/H]}}_{c14} = -0.09 \pm 0.13.$}
surface gravities, \footnote{ $ 0.79\pm^{0.25}_{0.37} = \widehat{\log g}_{c25} < \widehat{\log g}_{c19} < \widehat{\log g}_{c26} < \widehat{\log g}_{c28} < \widehat{\log g}_{c14} < \widehat{\log g}_{c7} = 2.03 \pm 0.22.$}
and temperatures.\footnote{$ 3561\pm^{84}_{60} = \widehat{T_\mathrm{eff}}_{c25} < \widehat{T_\mathrm{eff}}_{c19} < \widehat{T_\mathrm{eff}}_{c26} < \widehat{T_\mathrm{eff}}_{c28} < \widehat{T_\mathrm{eff}}_{c14} < \widehat{T_\mathrm{eff}}_{c7} = 4236\pm^{97}_{100}$ K.}
Concerning the stability of the classes, class 25 is very stable, having a mean coincidence ratio of 82 per cent. As shown in Figures \ref{fig:Groups_MH_Teff} and \ref{fig:G1_LOGG}, this class consists of giant stars at the tip of the RGB. Confusion higher than 10 per cent occurs between classes inside the group. The highest confusion rates are 12 per cent and 16 per cent between class 7 and classes 14 and 28, respectively, 16 per cent between classes 14 and 26, 16 per cent between classes 19 and 26, and 30 per cent between classes 26 and 28. Again, classes overlapping in the 3D space $T_\mathrm{eff} - \log g - \mathrm{[M/H]}$ present the highest degrees of confusion. Between classes in this group and other classes out of the group, the confusion rate is above 5 per cent only between class 14 and class 22 (10 per cent).
Tables \ref{tab:quantiles0} and \ref{tab:quantiles1} show the classes in this group are selecting stars within narrow distributions of the parameters, including the abundances. They typically have $\bar{\sigma}_{T_\mathrm{eff}} \approx 100$ K, $\bar{\sigma}_{\log g} \approx 0.30,$ and, for example, in class 14, the within class dispersion of the parameter can reach $\bar{\sigma}_{X} \leq 0.1$ for [$\alpha$/M], [N/M], [C/M], [Na/H], [Mn/H]\, and [K/H].
Class 28 is particularly spread in $\widehat{\mathrm{[C/M]}} = -0.09 \pm^{0.15}_{0.30}$, $\widehat{\mathrm{[Fe/H]}} = -1.14 \pm^{0.42}_{0.73}$ , and $\widehat{\mathrm{[Al/H]}} = -0.10 \pm^{0.16}_{0.31}$. In Figure \ref{fig:spectra_pile}, second panel from top to bottom, we see the mean spectra of the stars in this group. As in group 0, we see very similar spectral shapes, but with different line strengths.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/Group_1_LOGG_TEFF.png}
\caption{\label{fig:G1_LOGG} The distribution in $T_\mathrm{eff} - \log g$ for classes in group 1. The same rules and colours from Figure \ref{fig:Groups_MH_Teff} were applied to the contours here. Top and right panels show histograms of the distributions of $T_\mathrm{eff}$\, and $\log g $, respectively. The colours of the histogram match the colours of the contours.}
\end{figure}
\subsection[G2: Warm Stars]{Warm stars - Group 2 (Classes 3, 11, and 13)}
\label{subsec:G2}
\qquad This group assembles the warmest stars in DR12. The sample includes 15,233 spectra flagged as telluric standards, warm objects ideal for characterising the telluric lines that plague the IR, of which 67 per cent are in class 3, 16 per cent in class 11, and 12 per cent in class 13. According to target-selection flags, 96 per cent of the 10,628 objects in class 3 are telluric standards, while classes 11 and 13 have up to 50 per cent of stars of this kind. The differences between the classes in this group are mainly found in $T_\mathrm{eff}$\, and [M/H], as seen in panel b of Figure \ref{fig:Groups_MH_Teff}; class 3 is the warmest, containing A and B type stars, according to a match with the SIMBAD catalogue \citep{simbad}, while classes 11 and 13 are RGB stars, cooler and richer in metals compared with class 3 (see Table \ref{tab:quantiles0}). The third panel in Figure \ref{fig:spectra_pile} shows the differences between the mean spectra of the classes in group 2. The mean spectrum of class 3 is almost featureless, while the mean spectrum in class 13 has the strongest lines in the group. Moreover, there is a difference in their spatial distribution; while class 3 mainly occupies low latitudes, classes 11 and 13 are found primarily out of the Galactic plane and towards the Galactic centre.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/Group_2_3_LOGG_TEFF.png}
\caption{\label{fig:Group_2_3_LOGG_TEFF} Scatter plot for $T_\mathrm{eff} - \log g$ distributions of the classes in groups 2 and 3. Top and right panels show histograms of the distributions of $T_\mathrm{eff}$\, and $\log g $, respectively. To aid visualisation, both panels are split into two plots with different scales. The histogram line colours match the colours of the scatter plot.}
\end{figure}
As we would expect, since they are easily distinguishable even by eye, classes in this group are among the most stable classes in the classification, with mean coincidence rates of 94 per cent, 73 per cent, and 80 per cent for classes 3, 11, and 13, respectively. As class 11 is cooler than classes 3 and 13, it has the highest mean confusion rate with other classes (for example it has about 10 per cent mean confusion with classes 5 and 24). Classes 5 and 24 are among the most metal-poor in the classification, emphasising the role that the degeneracy between $T_\mathrm{eff}$\, and [M/H]\, plays in the determination of the stellar parameters.
All the chemical elements have very wide distributions except for [K/H]\, in class 11. Nevertheless, the atmospheric parameters of the stars in this group are out of the DR12 model grid, and thus it should be seen as a failure of the model fittings, as suggested by the ASPCAP flag \textit{star warn} found in $\approx$ 35 per cent of the objects in this class.
\subsection[G3: Fast rotators]{Fast rotators - Group 3 (Classes 27 and 29)}
\label{subsec:G3}
\qquad This group is formed by fast rotating stars. For both classes the ASPCAP models poorly fit their spectra. As a consequence of this, some artefacts are observed in their abundances, for example, the abundances of [C/M], [$\alpha$/M], [Al/H], [K/H], [Na/H]\, and [Si/H]\, are not continuous; they appear in clumps, having gaps of at least 0.2 in abundance between them.
In terms of atmospheric parameters, this group is very close to group 6 (dwarfs), but their spectra are remarkably different. The spectra of group 3 have fewer, shallower, and broader lines than those found in group 6, as can be seen in the fourth and seventh panel in Figure \ref{fig:spectra_pile}. This shows that the algorithm is sensitive to rotation, since it is able to split the stars affected by $\log g $\, line broadening from those affected by rotational line broadening. On the other hand, ASPCAP determines that the great majority of the stars in this group have $\log g $\, greater than 4.9 (see Figure \ref{fig:Group_2_3_LOGG_TEFF}), but since the rate of stars flagged with a fast rotation warning are 81 per cent and 93 per cent for classes 27 and 29, respectively, we cannot trust these determinations. The rate of stars flagged with a rotation warning in the entire DR12 data set is 7 per cent.
Class 29 is the most unstable of the classes, excluding the outliers (see Section \ref{subsec:G8}). It has a confusion rate of 62.8 per cent with class 27, which means that for some classifications class 29 dissolves mainly in classes 13, 23, 27 and 29. Class 27 is more stable, with 63 per cent of coincidence, having some degree of confusion with class 10 (13 per cent), which has the shallower lines in group 6.
About one quarter of the stars in class 27 and about half of the stars in class 29 are either young embedded cluster members or known calibration cluster members. Statistically we expect fast rotating stars to be younger than those that rotate more slowly \citep{van13}. In addition, the great majority of stars form in star clusters, dispersing latter on, and thus the fastest rotating stars are expected to be in young embedded clusters.
\subsection[G4: Metal-rich cool RGB]{Metal-rich cool RGB - Group 4 (Classes 16, 18 and 22)}
\label{subsec:G4}
\qquad Group 4 classes include metal rich stars covering the RGB with effective temperatures from 3620 to 4140 K, and with metallicities from 0.17 to 0.22 in the order $\widehat{\mathrm{[M/H]}}_{c16} < \widehat{\mathrm{[M/H]}}_{c18} < \widehat{\mathrm{[M/H]}}_{c22}$. Some stars in this group are near the edge of the model grid, at $\mathrm{[Fe/H]} = 0.50$ (36 per cent in class 16, 26 per cent in class 18, and 24 per cent in class 22). That also happens in $T_\mathrm{eff}$\, for class 16, which has 43 per cent of the stars cooler than 3600 K.
The stars in these classes are very concentrated in the Galactic disk, with [$\alpha$/M]\, close to the solar value. As shown in Figure \ref{fig:projection_G4}, the spatial distribution of class 16 is more concentrated towards the Galactic centre than classes 18 and 22.
Classes 16 and 18 are very stable, with a coincidence rate of 91 per cent and 80 per cent, respectively. Class 22 is much less stable having a coincidence rate of 29 per cent. The highest degree of confusion for class 22 occurs with class 9 (38 per cent), but classes 14 and 18 also contaminate class 22. Those three classes, 9, 14, and 18, share borders with class 22 in the space $T_\mathrm{eff} - \mathrm{[M/H]}$, as shown in Figure \ref{fig:Groups_MH_Teff}, and also with superposition in $\log g $, as can be seen by comparing Figures \ref{fig:G1_LOGG} and \ref{fig:G4_LOGG}. Once again, we see that the overlap in the space $T_\mathrm{eff} - \mathrm{[M/H]} - \log g$ is the main cause of confusion between classes.
The abundance distributions for these classes are narrow, as reflected in Tables \ref{tab:quantiles0} and \ref{tab:quantiles1}.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/projection_G4.png}
\caption{\label{fig:projection_G4} Galactic coordinates in Mollweide's projection for objects in classes 22 (orange triangles and contours), 18 (purple triangles and contours), and 16 (grey circles and contours), all belonging to group 4. The contour shades follow the same rule as in Figure \ref{fig:Groups_MH_Teff}.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/Group_4_LOGG_TEFF.png}
\caption{\label{fig:G4_LOGG} $T_\mathrm{eff} - \log g$ distribution for classes in group 4. The same rules and colours from Figure \ref{fig:Groups_MH_Teff} were applied to contours here. Top and right panels show histograms of the distributions of $T_\mathrm{eff}$\, and $\log g $, respectively. The colours of the histograms match the colours of the contours.}
\end{figure}
\subsection[G5: Metal-poor RC/warm RGB]{Metal-poor RC/warm RGB - Group 5 (Classes 0, 1 and 5)}
\label{subsec:G5}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/corner_G5C05.png}
\caption{\label{fig:class_5_corner} Properties of class 5 (group 5), which contains 9,144 stars ($N_\star$). The panels in the uppermost diagonal contain histograms for $T_\mathrm{eff}$, [M/H]\, and [$\alpha$/M], from left to right, respectively. In these plots vertical black dashed lines show the median value and the limits enclosing 68.3 per cent of the data points around the median value. The green histograms correspond to the objects in class 5 and the grey histogram shows the distribution of the whole group 5. As indicated by labels in the axes, the other three panels show 2D histograms for $T_\mathrm{eff} - \mathrm{[M/H]}$, $T_\mathrm{eff} - [\alpha/\mathrm{M}]$ and $[\alpha/\mathrm{M}] - \mathrm{[M/H]}$. From outside to inside the contours enclose 68.3, 45, 30, and 15 per cent of the objects in the class.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/Group_5_LOGG_TEFF.png}
\caption{\label{fig:Group_5_LOGG_TEFF} $T_\mathrm{eff} - \log g$ distribution for classes in group 5. The same rules and colours from Figure \ref{fig:Groups_MH_Teff} were applied to contours here. Top and right panels show histograms of the distributions of $T_\mathrm{eff}$\, and $\log g $, respectively. The colours of the histograms match the colours of the contours.}
\end{figure}
\qquad Just like group 0, this group is made of classes that include stars from the RC and the warmest end of the RGB. For classes 0, 1, and 5 the ratios of red clump stars are 30, 31, and 16 per cent according to a comparison with \citealt{bovy14}. In comparison with group 0, this group is more metal-poor, with $ - 0.45 \lesssim \widehat{\mathrm{[M/H]}} \lesssim -0.22 $. The group lacks stars in the direction of the Galactic centre, being homogeneously distributed in all other directions. Relative to group 0, group 5 is more dense in regions with Galactic latitudes higher than 30 degrees. All three classes are a mixture of thin and thick disk populations, but class 5 is more populated by high [$\alpha$/M]\, stars than other classes in the group, as shown in Figure \ref{fig:class_5_corner}.
As shown in Figure \ref{fig:Group_5_LOGG_TEFF}, class 5 almost completely overlaps with classes 0 and 1 in $T_\mathrm{eff} - \log g$ space. The median temperatures of class 0 stars are about 150 K warmer than class 1 stars. Class 5 is particularly broad in $T_\mathrm{eff}$\, and $\log g $, covering temperatures from 4125 to 7170 K, with a median value of $\widehat{T_\mathrm{eff}} = 4942 \pm^{584}_{202}$ K and $\log g = 3.16 \pm^{+1.04}_{-0.38}$. Figure \ref{fig:class_5_corner} shows the distribution of the stellar parameters in the planes $T_\mathrm{eff} - \mathrm{[M/H]}$, $T_\mathrm{eff} - [\alpha/\mathrm{M}]$ and $[\alpha/\mathrm{M}] - \mathrm{[M/H]}$. The dispersion there is likely to be an artefact due to the degeneracy between $T_\mathrm{eff}$\, and [M/H]\, in the ASPCAP parameter determination pipeline. Also the class is broadly spread in $\widehat{\mathrm{[Si/H]}} = -1.38 \pm^{0.96}_{1.38}$, which may also be an artefact of ASPCAP. In this range of atmospheric parameters the pipeline is probably confusing warmer temperatures with lower metallicities, as discussed in \cite{Holtzman15}.
\subsection[G6: Dwarfs stars]{Dwarfs stars - Group 6 (Classes 10, 12, 15, 17 and 20)}
\label{subsec:G6}
\qquad With $\log g $\, ranging from 4.23 to 4.35, group 6 has only dwarf stars. The classes differ because of their different temperatures and abundance patterns. Figure \ref{fig:Group_6_LOGG_TEFF} shows the distribution of $\log g $\, and $T_\mathrm{eff}$\, for this group.
Class 12 is over-abundant in Mg ($\widehat{\mathrm{[Mg/H]}} = +0.38 \pm^{0.32}_{0.28}$ ), and classes 15 and 20 have low [$\alpha$/M], especially in [Ca/H]\, and [O/H]. Some bimodality is found for [Al/H]\, and [K/H]\, for classes 15 and 20. However, 99 per cent of the objects in the group have their chemical abundances flagged with a warning and are not reliable, so this strange behaviour is likely to be an artefact of ASPCAP.
In Figure \ref{fig:spectra_pile} we see the FeI line around 16210 \AA\, is blended with the CN and CO lines near it for classes 15 and 20. In other regions of the spectra, blends like this are present. This is caused by the enhancement of molecular lines at low $T_\mathrm{eff}$\, values.
Class 20 presents two separate blobs of [$\alpha$/M]\, abundances, one around solar values and the other around $\widehat{[\alpha/\mathrm{M}]} = -0.3$, but almost 70 per cent of the stars in this class are flagged with the star warning, so abundance determination for these stars is not reliable.
The abundance distributions of these classes are very narrow, as shown in Tables \ref{tab:quantiles0} and \ref{tab:quantiles1}.
The classes here are relatively stable. Class 17 is the most unstable (50 per cent of mean coincidence rate), but has a significant degree of confusion only with classes 10, 12, and 15. Class 20 is the most stable in the group with a mean coincidence rate of 81 per cent. Other significant confusion rates are found only between classes inside the group, showing that the classes are stable as a group.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/Group_6_LOGG_TEFF.png}
\caption{\label{fig:Group_6_LOGG_TEFF} $T_\mathrm{eff} - \log g$ distribution for classes in group 6. The same rules and colours from Figure \ref{fig:Groups_MH_Teff} were applied to contours here. Top and right panels show histograms of the distributions of $T_\mathrm{eff}$\, and $\log g $, respectively. The histogram line colours match the colours of the contours.}
\end{figure}
\subsection[G7: Sparse classes]{Sparse classes - Group 7 (Classes 21, 23, 24, 30 and 31)}
\label{subsec:G7}
\qquad This group is formed by the most peculiar classes, with a number of objects corresponding to at least 0.5 per cent of the whole DR12 sample. The group is very diverse, so in this case we describe each class individually. All classes that represent less than 0.5 per cent of the sample are treated as outliers and are discussed in Section \ref{subsec:G8}. Figure \ref{fig:Group_7_LOGG_TEFF} shows the $T_\mathrm{eff}$ - $\log g $\, distribution for the group.
\subsubsection{M-giants/Bulge - Class 21}
Ninety-seven per cent of the stars in class 21 are at the edge of the model grid in $T_\mathrm{eff}$. That is to say, their temperatures are likely to be lower than the minimum $T_\mathrm{eff}$\, of the models in the spectral library. The class presents other anomalies; except for [C/M], [N/M], [$\alpha$/M], [Al/H], [K/H], [Mn/H]\, and [Na/H], all other abundances are also at the edge of the model grid. Lacking sufficiently cool spectra, ASPCAP probably tries to change the abundances until reaching its limits. For these stars, the problem has been corrected in DR13 \citep{DR13}.
This class is the most stable class with a coincidence rate of 95 per cent. Figure \ref{fig:spectra_pile}, bottom panel, shows that the mean spectra of this class looks totally different from the other classes, with very strong molecular bands, so $K$-means easily identifies these spectra as a class.
Spatially, the stars are concentrated at low latitude, specially towards the galactic centre, as shown in Figure \ref{fig:projection_G7}. This class also gathers 23 per cent of the bulge targets in DR12, according to its target flags.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/Group_7_LOGG_TEFF.png}
\caption{\label{fig:Group_7_LOGG_TEFF} Scatter plot for $T_\mathrm{eff}$\, versus $\log g $\, of the classes in group 7. Top and right panels show histograms of the distributions of $T_\mathrm{eff}$\, and $\log g $, respectively. Top panel is divided in two plots with different scales. The colours of the lines in the histograms match the colours of the scatter plot, as indicated in the legends.}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/projection_G7.png}
\caption{\label{fig:projection_G7} Galactic coordinates distribution of classes 21 (purple triangles), 23 (orange triangles) and 31 (blue circles).}
\end{figure}
\subsubsection{Metal-poor M dwarfs - Class 23}
This class is dominated by metal-poor ($\widehat{\mathrm{[M/H]}} \approx -0.54$ ) M dwarfs. The distribution of [$\alpha$/M]\, is divided into four clumps, showing there is some problem with the determination of these abundances, since very similar spectra correspond to differences of 0.25 in [$\alpha$/M]. The mean spectrum is similar to that of class 20, but with cooler stars; here more than 60 per cent of the stars are at the minimum $T_\mathrm{eff} = 3500$ K. This similarity in their spectra causes a mean confusion rate with class 20 of 12 per cent. However, class 23 is quite stable, with a mean coincidence rate of 87 per cent.
Similar to what happened to class 21, this class has many anomalies in its parameters, gaps in chemical abundances, and a high concentration at the borders of the model grid. This can also be related to limitations in ASPCAP.
As shown in Figure \ref{fig:projection_G7}, there seems to be no anisotropy in this class. It approximately follows the spatial distribution of APOGEE.
\subsubsection{K-giants from the Halo - Class 24}
This is a very metal-poor class with stars lying over the whole RGB, $\widehat{T_\mathrm{eff}} = 4583 \pm^{322}_{330}$ K and $\widehat{\log g} = 2.22 \pm^{0.60}_{0.54}$, as shown in Figure \ref{fig:Group_7_LOGG_TEFF}. With a median metallicity of $\widehat{\mathrm{[M/H]}} = -1.20 \pm^{0.22}_{0.25}$ it is one of the most metal-poor classes in the classification, certainly the most well-defined class among the metal-poor ones. This class is also $\alpha$ enhanced, with $\widehat{[\alpha/\mathrm{M}]} = 0.24 \pm 0.07$. We find that 593 out of 2388 ($\approx 25$ per cent) of these objects are globular cluster members used in APOGEE's calibration. Its spacial distribution is more dense in Galactic latitudes above 30$^o$.
Class 24 has a very low stability, having a coincidence rate of 18 per cent. Its stars are classified as class 11 members 59 per cent of the time.
\subsubsection{M31 GCs - Class 30}
In APOGEE DR12, 236 integrated spectra of Globular Clusters (GCs) in M31 were observed; each of these spectra appears as duplicate in the dataset. In order to remove the contamination from the unresolved M31 stellar population in these spectra, 141 background spectra near to the clusters were obtained \citep{Zasowski13}. Altogether they add up to 613 spectra in the region of M31. This class has the largest number of objects in this region, 171, with 33 background spectra and 69 duplicated GCs spectra. In general the spectra present high absorption in the continuum, as shown for the mean behaviour by the yellow line in the bottom panel of Figure \ref{fig:spectra_pile}. Its spectra are poorly fitted by the ASPCAP, and their wide chemical abundances and atmospheric parameters distributions (see Tables \ref{tab:quantiles0} and \ref{tab:quantiles1}) should not be trusted since they are all flagged with ASPCAP warnings. \citealt{Sakari16} have determined the abundance for 25 of the GCs in DR12 (eight are in this class) and we refer to their work as a better source of chemical abundances for these objects. This group also has 62 stars in embedded clusters, two member candidates of the GC Palomar 1, six bulge giants, and many metal-poor RGB stars. The class has 562 spectra, from which 93 per cent are flagged with star warnings, so the ASPCAP values cannot be trusted.
\subsubsection{M31 GCs/high persistence - Class 31}
Class 31 also has some spectra in the region of M31 (84 out of 613), from which 20 are background spectra and 64 are duplicated spectra of 32 clusters. In this class the spectra seem to be less affected by continuum absorption. As shown by the light blue circles in Figure \ref{fig:projection_G7}, this class has a peculiar spacial distribution, being more dense in $ 60^o \leq l \leq 90^o $ and $0^o \leq b \leq 45^o $. Further investigation is needed to determine why the stars in that direction have these characteristics. In this class there are 88 calibration cluster members and 38 spectra that overlap with the Kepler mission sample. Comparing the position of the stars of this class in Figure \ref{fig:projection_G7} with Figure 2 in \citealt{Zasowski13} one sees the position of these objects match the locus of observation targets of the halo population, the Kepler mission, and some of the calibration cluster. Thirty-five per cent (170) of the spectra in this class are flagged with a warning.
Thirty one per cent of the stars in this class are flagged as \textit{high persistence} observations. Persistence refers to the latent image of a previous exposure appearing in subsequent images, due to a slow release of an appreciable fraction of accumulated charge in the previous exposure over the subsequent ones. It affects the bluest chip particularly \citep{Nidever15}. The intensity of the persistence effect depends on the brightness of the spectra and their history of previous observations. In DR12, a flag is used to inform the relevance of the persistence effect on each spectra \citep{Holtzman15}. Some of the affected spectra by persistence present an obvious excess/deficit of flux in the blue chip. This behaviour is flagged as a \textit{positive/negative jump in blue chip}.
\subsection[G8: Outliers]{Group 8 and outliers}
\label{subsec:G8}
\qquad Ninety-nine per cent of the stars in APOGEE are in the classes presented in sections \ref{subsec:G0} to \ref{subsec:G7}. We briefly discuss the remaining 1 per cent. In addition, we also investigate the outliers of the main classes, that is, those spectra in classes from 0 to 31 for which the distance to the class mean spectrum is larger than 3-$\sigma$. Figure \ref{fig:count_G8} shows the number of spectra in the classes of group 8. Figure \ref{fig:projection_G8} shows the spatial distribution of these classes. In this figure the classes are represented by different symbols and colours. Figure \ref{fig:outliers_32-38} shows the spectra in classes from 32 to 38 in the same wavelength window as Figure \ref{fig:spectra_pile}; we plot the spectra as semi-transparent black lines to highlight the locations where the spectra are more similar to each other. In Figure \ref{fig:outliers_32-38} the mean spectrum of each class is drawn as a white dashed line.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{./img/hist_outliers_count.png}
\caption{\label{fig:count_G8} Number of objects in outlier classes.}
\end{figure}
\begin{figure*}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth]{./img/projection_G8.png}
\caption{\label{fig:projection_G8} Galactic coordinates distribution of targets in group 8.}
\end{minipage}
\end{figure*}
\subsubsection{Bulge giants - Class 32}
\qquad This class has 269 spectra, from which 71 are of supergiant stars in the bulge, 33 bulge giants, and 44 spectra in the region of M31 (20 of the background and 24 of 12 duplicated GCs). Forty-one per cent of the spectra in this class are flagged as having a negative jump in blue chip, 19 per cent of them as having high persistence, and 99 per cent of them are flagged as \textit{star bad}, assigned if there is warning about any of the following issues: $T_\mathrm{eff}$, $\log g $, model fitting $\chi^2$, rotation, S/N (signal-to-noise ratio), and if the difference between photometric and spectroscopic temperature is greater than 500K.
\begin{figure*}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{./img/spectra_pile_outliers_32-38.png}
\caption{\label{fig:outliers_32-38} Spectra of the objects in classes from 32 to 38. Each spectra is plotted as a semi-transparent line, in a way that the darkest regions represent the most dense regions in this flux window. The wavelength coverage here is the same of figure \ref{fig:spectra_pile}.}
\end{minipage}
\end{figure*}
\subsubsection{M31 GCs/high persistence - Class 33} Class 33 has 116 spectra in the region of M31, 18 background spectra, and 98 spectra from 49 GCs. There are 39 spectra flagged as emission line stars in DR12, eight of which are in this class. Figure \ref{fig:outliers_32-38} second panel from the top shows all 232 spectra overlapped. Emission lines are not visible in this figure because all spectra were truncated at 1.02 of the normalised flux. In spite of this constraint, the algorithm is able to identify emission lines since they affect the form of the continuum around them. Ninety-five per cent of the 232 spectra in this class are flagged as \textit{star bad}, 56 per cent are flagged with the rotation warning, and 33 per cent of these spectra are flagged as high persistence spectra. In Figure \ref{fig:outliers_32-38}, second panel from the top, we see there is no clear resemblance between the spectra in the class. The pixels with lower dispersion seem to be emission-dominated by lines, suggesting the spectra are either actually emission line stars or have some problem with the sky subtraction.
\subsubsection{Bad pixels - Class 34}
\qquad Seventy six per cent of the 170 stars in this class are flagged as high persistence observations. In Figure \ref{fig:outliers_32-38}, third panel from the top, we see they are mainly giant stars whose spectra have sequences of bad pixels, as those seen between 16,205 and 16,220 \AA.
\subsubsection{M31 GCs/high persistence - Class 35}
\qquad Class 35 has 88 spectra in the region of M31, 38 background spectra, and 50 spectra from 25 duplicated GCs. These 88 spectra are 70 per cent of the 126 spectra in the class. There are 99.2 per cent of the objects in this class flagged as \textit{star bad}, and 94 per cent of them have signal to noise lower than 30. As we see in Figure \ref{fig:outliers_32-38}, central panel, all the spectra are very noisy.
\subsubsection{1m Telescope - Class 36}
\qquad There is 817 spectra observed with the 1m telescope in DR12, and 93 of these are in this class. With 123 spectra, it corresponds to 76 per cent of the spectra in the class. Apart from a few cases, the spectra seem to contain sequences of a few bad pixels like the ones seen in class 34, Fig. \ref{fig:outliers_32-38}, but in different regions of the spectrum.
\subsubsection{Emission line stars/M31 GCs - Class 37}
\qquad This class has 13 emission line stars. There are also 11 spectra in the M31 region, one spectrum from the background, and ten spectra of five GCs. There are six objects identified by SIMBAD as galaxies.
\subsubsection{Negative flux - Class 38} This class has 36 spectra, from which eight are embedded cluster members, four are Sagittarius dwarf galaxy members, and one is an integrated spectra of the Pal1 GC. Eighty three per cent of the spectra in the class have pixels with negative counts.
\subsubsection{Classes from 39 to 49}
\qquad Except for class 42, all classes here have extreme negative flux values in some pixels. These negative counts imply high Euclidean distances between these spectra and those restricted to positive fluxes. Therefore they are segregated within these classes. Here we give a brief description of these objects.
\begin{itemize}
\item \textbf{Class 39:} Three noisy spectra, one of them flagged as an embedded cluster member.
\item \textbf{Class 40:} Two duplicated spectra of a globular cluster in M31 and one spectrum of the background in the M31 region.
\item \textbf{Class 42:} Two stars with a very similar pattern of sequences of pixels with flux equal to zero.
\item \textbf{Class 43:} One spectrum of the Pal1 globular cluster. This spectrum has deep asymmetric lines.
\item \textbf{Class 44:} One noisy spectrum with negative spikes.
\item \textbf{Class 45:} One background spectrum in the region of M31.
\item \textbf{Class 46:} One stellar spectrum with broad absorption lines.
\item \textbf{Class 47:} One spectrum with great negative spikes.
\item \textbf{Class 48:} One spectrum with high persistence and a positive jump in the blue chip.
\item \textbf{Class 49:} One noisy spectrum with wide absorption lines.
\end{itemize}
\subsubsection{Outliers in classes.} For the first 32 classes, we define the outliers as the spectra with a distance outside of the 3$\sigma$ interval around the mean spectrum of their class. It corresponds, on average, to $1.7 \pm 0.6 $ per cent of the objects in the classes. Exploring their target flags, we notice some phenomena as having high persistence, a positive jump in blue chip, emission lines, sequences of bad pixels, and many stars with signal to noise below 70.
\section{Summary and conclusion}
\subsection{Main results}
\qquad We performed an automated unsupervised classification of 153,847 APOGEE spectra included in SDSS DR12, using $K$-means. We classified the spectra into 50 classes, which were afterwards sorted manually into nine major groups. By construction, each class collects spectra that are very similar. The resulting classes and groups are interpreted using the physical parameters inferred by the APOGEE Stellar Parameters and Chemical Abundances Pipeline (ASPCAP). We found that classes were divided mainly according to their $T_\mathrm{eff}$, $\log g $\, and [M/H], and less strongly by other characteristics, such as elemental abundances or the quality of the spectra. Groups from 0 to 7 include 32 classes containing 99.3 per cent of the spectra in DR12. The identified groups can be described as follows:
\begin{itemize}
\item \textbf{Group 0:} Includes five classes dominated by red clump (RC) stars and the warmest end of the red giant branch (RGB) with different chemical abundances;
\item \textbf{Group 1:} Composed of six classes with stars from the RGB, cooler than those in group 0, and mainly separated from each other by their chemical abundances;
\item \textbf{Group 2:} Made up of three classes mainly populated by warm dwarfs, warm subgiant stars, and some A- and B-type stars used for telluric correction;
\item \textbf{Group 3:} Composed of two classes with fast rotating stars. Due to the strong line broadening, they are among the most poorly-fitted spectra in the survey;
\item \textbf{Group 4:} Has two classes covering almost the same range of $T_\mathrm{eff}$\, and $\log g $\, as group 1, RGB stars, but with higher metallicities;
\item \textbf{Group 5:} Contains three classes formed by stars from the RC and the warm end of RGB, with stellar populations from both the thin and thick disk;
\item \textbf{Group 6:} Formed of five classes composed of dwarf stars over a wide range of temperatures;
\item \textbf{Group 7:} Including five classes with peculiar stars;
\item \textbf{Group 8:} Collects 18 classes with all the outliers of the classification, less than 1 per cent of the spectra in SDSS DR12.
\end{itemize}
\subsection{Uses of the classification}
\quad As with any classification, this work can be used to provide an overview of the APOGEE DR12 data set, which simplifies the visualisation and highlights some features of the survey. For example, we can easily see that class 3, composed of very warm stars with almost featureless spectra, has an unexpectedly well-behaved distribution of values for [C/M], [N/M], [$\alpha$/M], [Mn/H]\, and [Na/H]. It also easily identifies strange behaviours such as the bimodality in [K/H]\, for class 15, the gaps in metallicity found in class 11, and the similarity in parameters of stars with very different spectra, as is the case for classes 20 and 27.
We provide extensive online and appendix material in order to encourage the search for features that may be interesting for specific purposes. For example, the catalogue provides a set of standard spectral templates that could be applied in stellar populations synthesis for galaxies. The mean spectrum (centroids) of the classes are arguably more reliable templates than the traditional synthetic models of standard MK type stars. However, the application should be restricted to those classes with a high number of members and low internal dispersion. Moreover, calibration of the atmospheric parameters and abundances is required, since the ones presented here are based on uncalibrated parameters.
The centroids of the classes are also useful to find substantial differences between the spectra and their best fit model found by ASPCAP. Since the classes are a collection of very similar spectra, the comparison between the class' mean and the mean of their best fit model can underline systematic differences between spectra and models. This comparison will be implemented soon and made available in a future publication.
Some classes have a different spatial distribution without an obvious reason, for example, classes in group 2 differ in their spatial distribution, something unexpected since the main difference among them is the $T_\mathrm{eff}$\, of their member stars. Class 31 has an especially peculiar distribution, occupying mainly the region with $60^\mathrm{o} \leq l \leq 90^\mathrm{o} $ and $0^o \leq b \leq 45^o $. The reason is unclear. Further investigations must be carried out to find out the cause of this spatial segregations. Other spatial distributions are less surprising, for example, classes in group 4 are concentrated in the disk. This is to be expected, since their metallicity and [$\alpha$/M]\, distributions match those expected for red giants that are part of the thin disk population. Classes 24 and 28, formed by metal-poor stars with high $\alpha$-element abundances, corresponding to the halo population, are expected to be out of the galactic disk, as we found. Class 21 can be interpreted as the population of the bulge, with high $\alpha$-element abundances and high metallicity, and is also expected to have a preferential spatial distribution like the one observed. These are the most evident examples of spatial segregation, but others can be found among the classes.
Finally, the extensive online and appendix material we provide can be used to explore deeper aspects of DR12 APOGEE. We encourage the use of Tables \ref{tab:desc} and \ref{tab:assign} for the reader to explore the results of the classification.
\subsection{Additional issues}
\quad In this work we face the problem of determining the optimal number of clusters for the $K$-means classification. In our case, none of the standard criteria provided a reliable answer. That is probably a consequence of the continuous nature of the dataset. In general, there are no sharp changes in the spectral properties of the stars. Indexes like CH and KL are mathematically proven to work in data sets with well separated clusters, but perform poorly in overlapping clusters or continuous distributions. In this case, $K$-means provides a way of artificially dividing a continuous space into meaningful slices, maximising the similarity among objects in the same class. Thus, the number of classes can be tuned according to the degree of within-class compactness we are interested in, as shown in Section \ref{subsec:find_K}.
Another consequence of applying $K$-means to a continuous data set is a significant observed degree of confusion between classes sharing borders in the space $T_\mathrm{eff} - \log g - \mathrm{[M/H]}$. However, these issues are not restricted to $K$-means. Any analysis tool, independently of whether it is supervised or not, will face the intrinsic degeneracy of these quantities in the stellar spectra. Soft clustering algorithms such as fuzzy $K$-means or density based algorithms such as Gaussian mixture models or DBSCAN could provide a more natural way to deal with this kind of problem, but would not solve the overlap of the classes in the space of parameters.
We have shown how the random seed used by the algorithm affects its solution. Although there is no unique solution, the variations are negligible compared to the internal dispersion of the classes. In addition, we show how the centroids of the classes are much closer to the spectra in the class than their corresponding best fit models. This suggests that $K$-means can be used to identify the systematic deficiencies of the modelling adopted in the determination of physical parameters and abundances with ASPCAP, and improve the agreement with the data.
Although the within-class dispersions in the parameter space are larger than the typical uncertainties derived from this kind of data, $K$-means provides good insight into the general characteristics of the spectra in the data set. In this sense, $K$-means is not the optimal algorithm to be used for parameter determination, but can be useful in an early analysis of the data, helping to design solutions and map the general behaviour of the data set.
$K$-means essentially performs hyperspherical cuts in the N-dimensional space. Future works in unsupervised spectral classification should address the issues presented in this section and search for algorithms that can more generically divide the space taking into account its density distribution. Also a soft clustering approach can arguably produce a more reliable classification. However, more complex algorithms are also more computationally expensive, therefore any further application has to address the scalability problem.
\subsection{Conclusions}
\quad As exemplified in this work, $K$-means provides an easy way to divide complex problems into smaller pieces, which are simpler to solve. The version of ASPCAP used in DR12 was designed to work optimally on K and early-M giant stars. For dwarfs, warmer ($T_{\rm eff} >$ 6000 K), cooler ($T_{\rm eff} <$ 3800 K), or metal-poor stars ([M/H] $< -1$), the results are less accurate. Prior to a model-atmospheres spectral analysis, $K$-means can provide guidance on the most natural groups in the data set. This can be very useful to design a pipeline that treats differently the distinct groups of objects, which is necessary for groups such as 2, 6, 7, and 8, for example.
\citealt{NFLT} puts forward what is known as the 'no free lunch' theorem for machine learning. That is to say, there is no best machine learning algorithm; it is always a matter of which one is better suited to the specific features of a given problem. Knowing the problem, we can only presume which kind of algorithm is most suitable for solving it, but finding the best solution always requires testing some algorithms and tuning their parameters. This work adds to previous applications of $K$-means \citep{sanchez09, sanchez10, Morales-Luis11, sanchez13, sanchez16} consolidating a guideline for the use of this algorithm in the analysis of spectroscopic data, and providing a new perspective for the APOGEE data.
In this work we made a serious effort to organise the spectra into classes and groups according to the similarity within their spectra. This classification is completely independent of any atmospheric and spectroscopic model. It provides a useful way to explore the data in APOGEE, since it allows a quick identification of the main different types of objects in the survey.
\input{Quant0.tex}
\input{Quant1.tex}
| {
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} | 2,036 |
{"url":"https:\/\/hackage.haskell.org\/package\/free-5.0.2\/docs\/Control-Monad-Free.html","text":"free-5.0.2: Monads for free\n\nCopyright (C) 2008-2015 Edward Kmett BSD-style (see the file LICENSE) Edward Kmett provisional MPTCs, fundeps Safe Haskell2010\n\nControl.Monad.Free\n\nDescription\n\nMonads for free\n\nSynopsis\n\n# Documentation\n\nclass Monad m => MonadFree f m | m -> f where Source #\n\nMonads provide substitution (fmap) and renormalization (join):\n\nm >>= f = join (fmap f m)\n\nA free Monad is one that does no work during the normalization step beyond simply grafting the two monadic values together.\n\n[] is not a free Monad (in this sense) because join [[a]] smashes the lists flat.\n\nOn the other hand, consider:\n\ndata Tree a = Bin (Tree a) (Tree a) | Tip a\n\ninstance Monad Tree where\nreturn = Tip\nTip a >>= f = f a\nBin l r >>= f = Bin (l >>= f) (r >>= f)\n\n\nThis Monad is the free Monad of Pair:\n\ndata Pair a = Pair a a\n\n\nAnd we could make an instance of MonadFree for it directly:\n\ninstance MonadFree Pair Tree where\nwrap (Pair l r) = Bin l r\n\n\nOr we could choose to program with Free Pair instead of Tree and thereby avoid having to define our own Monad instance.\n\nMoreover, Control.Monad.Free.Church provides a MonadFree instance that can improve the asymptotic complexity of code that constructs free monads by effectively reassociating the use of (>>=). You may also want to take a look at the kan-extensions package (http:\/\/hackage.haskell.org\/package\/kan-extensions).\n\nSee Free for a more formal definition of the free Monad for a Functor.\n\nMethods\n\nwrap :: f (m a) -> m a Source #\n\nAdd a layer.\n\nwrap (fmap f x) \u2261 wrap (fmap return x) >>= f\n\n\nwrap :: (m ~ t n, MonadTrans t, MonadFree f n, Functor f) => f (m a) -> m a Source #\n\nAdd a layer.\n\nwrap (fmap f x) \u2261 wrap (fmap return x) >>= f\n\n\nInstances\n\n (Functor f, MonadFree f m) => MonadFree f (ListT m) Source # Methodswrap :: f (ListT m a) -> ListT m a Source # (Functor f, MonadFree f m) => MonadFree f (MaybeT m) Source # Methodswrap :: f (MaybeT m a) -> MaybeT m a Source # Applicative f => MonadFree f (Free f) Source # Methodswrap :: f (Free f a) -> Free f a Source # Functor f => MonadFree f (Free f) Source # Methodswrap :: f (Free f a) -> Free f a Source # Functor f => MonadFree f (F f) Source # Methodswrap :: f (F f a) -> F f a Source # Monad m => MonadFree Identity (IterT m) Source # Methodswrap :: Identity (IterT m a) -> IterT m a Source # (Functor f, MonadFree f m) => MonadFree f (ExceptT e m) Source # Methodswrap :: f (ExceptT e m a) -> ExceptT e m a Source # (Functor f, MonadFree f m, Error e) => MonadFree f (ErrorT e m) Source # Methodswrap :: f (ErrorT e m a) -> ErrorT e m a Source # (Functor f, MonadFree f m) => MonadFree f (IdentityT * m) Source # Methodswrap :: f (IdentityT * m a) -> IdentityT * m a Source # (Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) Source # Methodswrap :: f (WriterT w m a) -> WriterT w m a Source # (Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) Source # Methodswrap :: f (WriterT w m a) -> WriterT w m a Source # (Functor f, MonadFree f m) => MonadFree f (StateT s m) Source # Methodswrap :: f (StateT s m a) -> StateT s m a Source # (Functor f, MonadFree f m) => MonadFree f (StateT s m) Source # Methodswrap :: f (StateT s m a) -> StateT s m a Source # (Applicative f, Applicative m, Monad m) => MonadFree f (FreeT f m) Source # Methodswrap :: f (FreeT f m a) -> FreeT f m a Source # (Functor f, Monad m) => MonadFree f (FreeT f m) Source # Methodswrap :: f (FreeT f m a) -> FreeT f m a Source # MonadFree f (FT f m) Source # Methodswrap :: f (FT f m a) -> FT f m a Source # (Functor f, MonadFree f m) => MonadFree f (ContT * r m) Source # Methodswrap :: f (ContT * r m a) -> ContT * r m a Source # (Functor f, MonadFree f m) => MonadFree f (ReaderT * e m) Source # Methodswrap :: f (ReaderT * e m a) -> ReaderT * e m a Source # (Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) Source # Methodswrap :: f (RWST r w s m a) -> RWST r w s m a Source # (Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) Source # Methodswrap :: f (RWST r w s m a) -> RWST r w s m a Source #\n\ndata Free f a Source #\n\nThe Free Monad for a Functor f.\n\nFormally\n\nA Monad n is a free Monad for f if every monad homomorphism from n to another monad m is equivalent to a natural transformation from f to m.\n\nWhy Free?\n\nEvery \"free\" functor is left adjoint to some \"forgetful\" functor.\n\nIf we define a forgetful functor U from the category of monads to the category of functors that just forgets the Monad, leaving only the Functor. i.e.\n\nU (M,return,join) = M\n\nthen Free is the left adjoint to U.\n\nBeing Free being left adjoint to U means that there is an isomorphism between\n\nFree f -> m in the category of monads and f -> U m in the category of functors.\n\nMorphisms in the category of monads are Monad homomorphisms (natural transformations that respect return and join).\n\nMorphisms in the category of functors are Functor homomorphisms (natural transformations).\n\nGiven this isomorphism, every monad homomorphism from Free f to m is equivalent to a natural transformation from f to m\n\nShowing that this isomorphism holds is left as an exercise.\n\nIn practice, you can just view a Free f a as many layers of f wrapped around values of type a, where (>>=) performs substitution and grafts new layers of f in for each of the free variables.\n\nThis can be very useful for modeling domain specific languages, trees, or other constructs.\n\nThis instance of MonadFree is fairly naive about the encoding. For more efficient free monad implementation see Control.Monad.Free.Church, in particular note the improve combinator. You may also want to take a look at the kan-extensions package (http:\/\/hackage.haskell.org\/package\/kan-extensions).\n\nA number of common monads arise as free monads,\n\n\u2022 Given data Empty a, Free Empty is isomorphic to the Identity monad.\n\u2022 Free Maybe can be used to model a partiality monad where each layer represents running the computation for a while longer.\n\nConstructors\n\n Pure a Free (f (Free f a))\n\nInstances\n\n Source # This is not a true monad transformer. It is only a monad transformer \"up to retract\". Methodslift :: Monad m => m a -> Free m a # (Functor m, MonadWriter e m) => MonadWriter e (Free m) Source # Methodswriter :: (a, e) -> Free m a #tell :: e -> Free m () #listen :: Free m a -> Free m (a, e) #pass :: Free m (a, e -> e) -> Free m a # (Functor m, MonadState s m) => MonadState s (Free m) Source # Methodsget :: Free m s #put :: s -> Free m () #state :: (s -> (a, s)) -> Free m a # (Functor m, MonadReader e m) => MonadReader e (Free m) Source # Methodsask :: Free m e #local :: (e -> e) -> Free m a -> Free m a #reader :: (e -> a) -> Free m a # (Functor m, MonadError e m) => MonadError e (Free m) Source # MethodsthrowError :: e -> Free m a #catchError :: Free m a -> (e -> Free m a) -> Free m a # Functor f => MonadFree f (Free f) Source # Methodswrap :: f (Free f a) -> Free f a Source # Functor f => Monad (Free f) Source # Methods(>>=) :: Free f a -> (a -> Free f b) -> Free f b #(>>) :: Free f a -> Free f b -> Free f b #return :: a -> Free f a #fail :: String -> Free f a # Functor f => Functor (Free f) Source # Methodsfmap :: (a -> b) -> Free f a -> Free f b #(<\\$) :: a -> Free f b -> Free f a # Functor f => MonadFix (Free f) Source # Methodsmfix :: (a -> Free f a) -> Free f a # Functor f => Applicative (Free f) Source # Methodspure :: a -> Free f a #(<*>) :: Free f (a -> b) -> Free f a -> Free f b #liftA2 :: (a -> b -> c) -> Free f a -> Free f b -> Free f c #(*>) :: Free f a -> Free f b -> Free f b #(<*) :: Free f a -> Free f b -> Free f a # Foldable f => Foldable (Free f) Source # Methodsfold :: Monoid m => Free f m -> m #foldMap :: Monoid m => (a -> m) -> Free f a -> m #foldr :: (a -> b -> b) -> b -> Free f a -> b #foldr' :: (a -> b -> b) -> b -> Free f a -> b #foldl :: (b -> a -> b) -> b -> Free f a -> b #foldl' :: (b -> a -> b) -> b -> Free f a -> b #foldr1 :: (a -> a -> a) -> Free f a -> a #foldl1 :: (a -> a -> a) -> Free f a -> a #toList :: Free f a -> [a] #null :: Free f a -> Bool #length :: Free f a -> Int #elem :: Eq a => a -> Free f a -> Bool #maximum :: Ord a => Free f a -> a #minimum :: Ord a => Free f a -> a #sum :: Num a => Free f a -> a #product :: Num a => Free f a -> a # Traversable f => Traversable (Free f) Source # Methodstraverse :: Applicative f => (a -> f b) -> Free f a -> f (Free f b) #sequenceA :: Applicative f => Free f (f a) -> f (Free f a) #mapM :: Monad m => (a -> m b) -> Free f a -> m (Free f b) #sequence :: Monad m => Free f (m a) -> m (Free f a) # Eq1 f => Eq1 (Free f) Source # MethodsliftEq :: (a -> b -> Bool) -> Free f a -> Free f b -> Bool # Ord1 f => Ord1 (Free f) Source # MethodsliftCompare :: (a -> b -> Ordering) -> Free f a -> Free f b -> Ordering # Read1 f => Read1 (Free f) Source # MethodsliftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Free f a) #liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Free f a] #liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Free f a) #liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Free f a] # Show1 f => Show1 (Free f) Source # MethodsliftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Free f a -> ShowS #liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Free f a] -> ShowS # Alternative v => Alternative (Free v) Source # This violates the Alternative laws, handle with care. Methodsempty :: Free v a #(<|>) :: Free v a -> Free v a -> Free v a #some :: Free v a -> Free v [a] #many :: Free v a -> Free v [a] # (Functor v, MonadPlus v) => MonadPlus (Free v) Source # This violates the MonadPlus laws, handle with care. Methodsmzero :: Free v a #mplus :: Free v a -> Free v a -> Free v a # (Functor m, MonadCont m) => MonadCont (Free m) Source # MethodscallCC :: ((a -> Free m b) -> Free m a) -> Free m a # Traversable1 f => Traversable1 (Free f) Source # Methodstraverse1 :: Apply f => (a -> f b) -> Free f a -> f (Free f b) #sequence1 :: Apply f => Free f (f b) -> f (Free f b) # Foldable1 f => Foldable1 (Free f) Source # Methodsfold1 :: Semigroup m => Free f m -> m #foldMap1 :: Semigroup m => (a -> m) -> Free f a -> m #toNonEmpty :: Free f a -> NonEmpty a # Functor f => Apply (Free f) Source # Methods(<.>) :: Free f (a -> b) -> Free f a -> Free f b #(.>) :: Free f a -> Free f b -> Free f b #(<.) :: Free f a -> Free f b -> Free f a #liftF2 :: (a -> b -> c) -> Free f a -> Free f b -> Free f c # Functor f => Bind (Free f) Source # Methods(>>-) :: Free f a -> (a -> Free f b) -> Free f b #join :: Free f (Free f a) -> Free f a # Functor f => Generic1 * (Free f) Source # Associated Typestype Rep1 (Free f) (f :: Free f -> *) :: k -> * # Methodsfrom1 :: f a -> Rep1 (Free f) f a #to1 :: Rep1 (Free f) f a -> f a # (Eq1 f, Eq a) => Eq (Free f a) Source # Methods(==) :: Free f a -> Free f a -> Bool #(\/=) :: Free f a -> Free f a -> Bool # (Typeable (* -> *) f, Data (f (Free f a)), Data a) => Data (Free f a) Source # Methodsgfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Free f a -> c (Free f a) #gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Free f a) #toConstr :: Free f a -> Constr #dataTypeOf :: Free f a -> DataType #dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Free f a)) #dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Free f a)) #gmapT :: (forall b. Data b => b -> b) -> Free f a -> Free f a #gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Free f a -> r #gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Free f a -> r #gmapQ :: (forall d. Data d => d -> u) -> Free f a -> [u] #gmapQi :: Int -> (forall d. Data d => d -> u) -> Free f a -> u #gmapM :: Monad m => (forall d. Data d => d -> m d) -> Free f a -> m (Free f a) #gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Free f a -> m (Free f a) #gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Free f a -> m (Free f a) # (Ord1 f, Ord a) => Ord (Free f a) Source # Methodscompare :: Free f a -> Free f a -> Ordering #(<) :: Free f a -> Free f a -> Bool #(<=) :: Free f a -> Free f a -> Bool #(>) :: Free f a -> Free f a -> Bool #(>=) :: Free f a -> Free f a -> Bool #max :: Free f a -> Free f a -> Free f a #min :: Free f a -> Free f a -> Free f a # (Read1 f, Read a) => Read (Free f a) Source # MethodsreadsPrec :: Int -> ReadS (Free f a) #readList :: ReadS [Free f a] #readPrec :: ReadPrec (Free f a) #readListPrec :: ReadPrec [Free f a] # (Show1 f, Show a) => Show (Free f a) Source # MethodsshowsPrec :: Int -> Free f a -> ShowS #show :: Free f a -> String #showList :: [Free f a] -> ShowS # Generic (Free f a) Source # Associated Typestype Rep (Free f a) :: * -> * # Methodsfrom :: Free f a -> Rep (Free f a) x #to :: Rep (Free f a) x -> Free f a # type Rep1 * (Free f) Source # type Rep1 * (Free f) = D1 * (MetaData \"Free\" \"Control.Monad.Free\" \"free-5.0.2-LFMuTEkYorP4u8Pz3JwyRq\" False) ((:+:) * (C1 * (MetaCons \"Pure\" PrefixI False) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1)) (C1 * (MetaCons \"Free\" PrefixI False) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) ((:.:) * * f (Rec1 * (Free f)))))) type Rep (Free f a) Source # type Rep (Free f a) = D1 * (MetaData \"Free\" \"Control.Monad.Free\" \"free-5.0.2-LFMuTEkYorP4u8Pz3JwyRq\" False) ((:+:) * (C1 * (MetaCons \"Pure\" PrefixI False) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * a))) (C1 * (MetaCons \"Free\" PrefixI False) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 * (f (Free f a))))))\n\nretract :: Monad f => Free f a -> f a Source #\n\nretract is the left inverse of lift and liftF\n\nretract . lift = id\nretract . liftF = id\n\n\nliftF :: (Functor f, MonadFree f m) => f a -> m a Source #\n\nA version of lift that can be used with just a Functor for f.\n\niter :: Functor f => (f a -> a) -> Free f a -> a Source #\n\nTear down a Free Monad using iteration.\n\niterA :: (Applicative p, Functor f) => (f (p a) -> p a) -> Free f a -> p a Source #\n\nLike iter for applicative values.\n\niterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a Source #\n\nLike iter for monadic values.\n\nhoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g b Source #\n\nLift a natural transformation from f to g into a natural transformation from FreeT f to FreeT g.\n\nfoldFree :: Monad m => (forall x. f x -> m x) -> Free f a -> m a Source #\n\nThe very definition of a free monad is that given a natural transformation you get a monad homomorphism.\n\ntoFreeT :: (Functor f, Monad m) => Free f a -> FreeT f m a Source #\n\nConvert a Free monad from Control.Monad.Free to a FreeT monad from Control.Monad.Trans.Free.\n\ncutoff :: Functor f => Integer -> Free f a -> Free f (Maybe a) Source #\n\nCuts off a tree of computations at a given depth. If the depth is 0 or less, no computation nor monadic effects will take place.\n\nSome examples (n \u2265 0):\n\ncutoff 0 _ == return Nothing\ncutoff (n+1) . return == return . Just\ncutoff (n+1) . lift == lift . liftM Just\ncutoff (n+1) . wrap == wrap . fmap (cutoff n)\n\nCalling 'retract . cutoff n' is always terminating, provided each of the steps in the iteration is terminating.\n\nunfold :: Functor f => (b -> Either a (f b)) -> b -> Free f a Source #\n\nUnfold a free monad from a seed.\n\nunfoldM :: (Traversable f, Applicative m, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a) Source #\n\nUnfold a free monad from a seed, monadically.\n\n_Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a)) Source #\n\nThis is Prism' (Free f a) a in disguise\n\n>>> preview _Pure (Pure 3)\nJust 3\n\n>>> review _Pure 3 :: Free Maybe Int\nPure 3\n\n\n_Free :: forall f m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (f (Free f a))) -> p (Free f a) (m (Free f a)) Source #\n\nThis is Prism' (Free f a) (f (Free f a)) in disguise\n\n>>> preview _Free (review _Free (Just (Pure 3)))\nJust (Just (Pure 3))\n\n>>> review _Free (Just (Pure 3))\nFree (Just (Pure 3))","date":"2019-02-22 01:41:00","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\": 0.18368567526340485, \"perplexity\": 12451.263462572799}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247511573.67\/warc\/CC-MAIN-20190221233437-20190222015437-00402.warc.gz\"}"} | null | null |
Q: Wave equation equivalence I have to prove that the wave equation $\partial_{tt} \zeta = c^2 \partial_{xx} \zeta$ is equivalent to $$\begin{cases}\partial_tv&=c\partial_xw\\\partial_tw&=c\partial_xv\end{cases}$$ with the variable change $v=\partial_t\zeta$, $w=c\partial_x\zeta$.
If you subsitute the variable change in the wave equation you get the first equivalence, but how about the second? I do not know if I am missing something about the variable change.
Any recommendations?
A: $$\partial_{tt}\zeta = \partial_t(\partial_t\zeta) = \partial_t v = c \partial_x w
= c \partial_x (c \partial_x \zeta) = c^2 \partial_{xx} \zeta$$
| {
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Some algebraically defined bimodules}
\label{sec:AlgDA}
Having defined the holomorphic objects of study, we now turn to their
computation. In~\cite{Bordered2}, we associated bimodules to
crossings, maxima, and minima. Their description did not involve
pseudo-holomorphic curves: rather, they were defined via explicit,
algebraic descriptions. These bimodules were defined over the algebra
$\Alg$, and some versions were defined over a related algebra
$\DuAlg$. In this section, we adapt these constructions to the curved
framework. These modified bimodules will play a central role in an
explicit computation of the holomorphic objects
(e.g. Theorem~\ref{thm:MainTheorem} and Theorem~\ref{thm:ComputeD}
below).
\subsection{ Algebraically defined, curved bimodules}
\label{subsec:FormalModules}
Consider the $DA$ bimodules from~\cite{Bordered2} associated to a
positive crossings, a negative crossing, a maximum, and a minimum,
$\lsup{\Alg_2}\Pos^i_{\Alg_1}$, $\lsup{\Alg_2}\Neg^i_{\Alg_1}$,
$\lsup{\Alg_2}\Max^c_{\Alg_1}$, and $\lsup{\Alg_2}\Min^c_{\Alg_1}$.
(We suppress here the parameters (matching and strand numbers) of the algebras
appearing in these bimodules.) By slightly modifying the construction, we obtain corresponding
curved bimodules over $\cBlg$, which are related to the original bimodule as follows:
\begin{prop}
\label{prop:CurvedDABimodules}
There are curved bimodules associated to crossings, maxima, and
minima over $\Blg$, $\lsup{\cBlg_2}\Pos^i_{\cBlg_1}$,
$\lsup{\cBlg_2}\Neg^i_{\cBlg_1}$, $\lsup{\cBlg_2}\Max^c_{\cBlg_1}$, and
$\lsup{\cBlg_2}\Min^c_{\cBlg_1}$, which are related to the corresponding
bimodules over $\Alg$ defined in~\cite{Bordered2}
by the following relations:
\begin{align*}
\lsup{\Alg_2}T_{\cBlg_2} \DT \lsup{\cBlg_2}\Pos^i_{\cBlg_1} &\simeq
\lsup{\Alg_2}\Pos^i_{\Alg_1}\DT \lsup{\Alg_1}T_{\cBlg_1} \\
\lsup{\Alg_2}T_{\cBlg_2} \DT \lsup{\cBlg_2}\Neg^i_{\cBlg_1} &\simeq
\lsup{\Alg_2}\Neg^i_{\Alg_1}\DT \lsup{\Alg_1}T_{\cBlg_1} \\
\lsup{\Alg_2}T_{\cBlg_2} \DT \lsup{\cBlg_2}\Max^c_{\cBlg_1} &\simeq
\lsup{\Alg_2}\Max^c_{\Alg_1}\DT \lsup{\Alg_1}T_{\cBlg_1} \\
\lsup{\Alg_2}T_{\cBlg_2} \DT \lsup{\cBlg_2}\Min^c_{\cBlg_1} &\simeq
\lsup{\Alg_2}\Min^c_{\Alg_1}\DT \lsup{\Alg_1}T_{\cBlg_1},
\end{align*}
where $\lsup{\Alg_i}T_{\cBlg_i}$ is the $\Blg$-to-$\Alg$ transformer
from Definition~\ref{def:Transformer}.
\end{prop}
\begin{proof}
This follows from the fact that the bimodules defined over $\Alg$
are ``standard'' in the sense
of~\cite[Section~\ref{BK2:sec:Algebras}]{Bordered2}, which we recall
presently. First, recall that the algebras $\Alg_i$ are equipped
with preferred elements $C_i$, and each is equipped with a
subalgebra $\Blg_i\subset \Alg_i$, with the property that $d C_i\in
\Blg_i$; indeed, $d C_i$ is the curvature $\mu_0^{\Blg_i}$
associated to the matching. A {\em standard sequence} is a sequence
$a_1,\dots,a_{\ell-1}$ of elements of $\Alg_1$ with the property
that each $a_i$ is either equal to $C_1$ or it is an element of
$\Blg_1\subset \Alg_1$.
\begin{defn}
\label{def:Standard}
A type $DA$ bimodule $\lsup{\Alg_2}X_{\Alg_1}$ is called {\em standard} if
the following conditions hold:
\begin{enumerate}[label=(DA-\arabic*),ref=(DA-\arabic*)]
\item
\label{prop:Adapted}
The bimodule $X$ is finite-dimensional (over $\Field$),
and it has a a $\mathbb Q} \newcommand{\R}{\mathbb R$-grading $\Delta$ and a further grading $\Agr$, as follows.
Think of $\Alg_i$ as associated to a union of points $Y_i$,
and fix a cobordism $W_1$ from $Y_1$ to $Y_2$.
The weights of a
a homogenous algebra $a$ element can be viewed as an element
of $H^0(Y_i)$; taking the coboundary then induces an element
$\Agr(a)$ in $H^1(W,\partial W)$.
These various gradings are related by the formulas:
\begin{align*}
\Delta(\delta^1_{\ell}(\mathbf x,a_1,\dots,a_{\ell-1}))&=
\Delta(\mathbf x)+\Delta(a_1)+\dots+\Delta(a_{\ell-1})-\ell+2 \\
\Agr(\delta^1_{\ell}(\mathbf x,a_1,\dots,a_{\ell-1}))&=
\Agr(\mathbf x)+\Agr(a_1)+\dots+\Agr(a_{\ell}));
\end{align*}
(This is the condition that $X$ is {\em adapted to $W_1$}
in the sense
of~\cite[Definition~\ref{BK2:def:AdaptedDA}]{Bordered2}.)
\item
The one-manifold $W$ is compatible with the
matching $\Matching_1$ used in the algebra $\Alg_1$, in the
sense that
$W\cup W(\Matching_1)$ has no closed components.
\item\label{prop:LandsInB} For any standard sequence of elements $a_1,\dots,a_{\ell-1}$
with at least some $a_i\in\Blg_1$,
$\delta^1_{\ell}(\mathbf x,a_1,\dots,a_{\ell-1})\in\Blg_2\otimes X$.
\item
\label{prop:CStandard}
For any $\mathbf x\in X$,
\[
C^2\otimes \mathbf x +
\sum_{\ell=0}^{\infty}\delta^1_{1+\ell}(\mathbf x,\overbrace{C^1,\dots,C^1}^{\ell})\in
\Blg_2\otimes X.\]
\end{enumerate}
\end{defn}
If $\lsup{\Alg_2}X_{\Alg_1}$ is standard in the above sense, then
the actions on
$\lsup{\Alg_2}X_{\Alg_1}\DT \lsup{\Alg_1}T_{\cBlg_1}$ are sums of
actions on $X$ where the input is a standard sequence in the
above sense. It follows from Property~\ref{prop:LandsInB} that
the output lies in $\Blg_2$, except in the special case of the
$\delta^1_1$ action on the tensor product, in which case
Proposition~\ref{prop:CStandard} expresses the output in
$\Blg_2\otimes X$ plus $C^2\otimes X$. Dropping the term in
$C^2\otimes X$, we obtain actions defining
$\lsup{\cBlg_2}X_{\cBlg_1}$. As the notation suggests, the operations so defined give a curved
DA bimodule. Indeed, the curved bimodule relations can be seen as direct
consequence of the ordinary bimodule relations for
$\lsup{\Alg_2}X_{\Alg_1}$: in our construction of
$\lsup{\Blg_2}X_{\Blg_1}$, we have dropped the term $C^2\otimes \mathbf x$,
but its contribution to the $\Ainfty$ relation in $\Blg_2\otimes X$
is precisely $(\partial C^2) \otimes \mathbf x=\mu_0^{\Blg_2}\otimes \mathbf x$;
moreover, the terms involving for $\mu_0^{\Blg_1}$ account for the
terms in the $\Ainfty$ relation containing $\partial C^1$.
By construction,
\[ \lsup{\Alg_2}T_{\cBlg_2}\DT \lsup{\cBlg_2}X_{\cBlg_1}
=\lsup{\Alg_2}X_{\Alg_1}\DT \lsup{\Alg_1}T_{\cBlg_1}:\]
The proposition now follows since all the bimodules listed above
are standard. (See~\cite[Proposition~\ref{BK2:prop:PosExt} and
Theorems~\ref{BK2:thm:MaxDA} and~\ref{BK2:thm:MinDual}]{Bordered2}.)
\end{proof}
(Note that the actions on $\lsup{\cBlg_2} X_{\cBlg_1}$ are precisely
the actions $\gamma_k$ defined
in~\cite[Section~\ref{BK2:sec:Fast}]{Bordered2}.)
\subsection{Bimodules over $\DuAlg$}
Recall that in~\cite{Bordered2}, we considered an algebra $\DuAlg$ that
was dual to $\Alg$. Sometimes, it is convenient to work with bimodules over this algebra
(and its quotient, described in Subsection~\ref{subsec:nDuAlg}).
It is natural to consider idempotents in $\DuAlg$
that are complementary to those in $\Alg$. In the present paper, when
we write $\cBlg$, we understand $\Blg(2n,n)$, with curvature specified by some matching
$\Matching$. Correspondingly, when we
write $\DuAlg$ without decoration, we understand $\DuAlg(2n,n+1,\Matching)$
from~\cite{Bordered2}.
In~\cite[Section~\ref{BK2:sec:DuAlg}]{Bordered2}, we constructed
bimodules associated to crossings over the algebras $\DuAlg$, denoted
$\lsup{\DuAlg_1}\Pos^i_{\DuAlg_2}$ and $\lsup{\DuAlg_1}\Neg^i_{\DuAlg_2}$, which
become homotopy equivalent to the type $DA$ bimodule over $\Alg$,
after tensoring with the canonical $DD$ bimodule. (See
Proposition~\ref{prop:DualityRelationship} below). Before giving the
precise statement, we describe a similar bimodule for a local minimum,
as well. (This latter bimodule was not needed in~\cite{Bordered2}; rather, it
was sufficient to have only its corresponding type $DD$ module.)
\subsubsection{The local minimum $\lsup{\DuAlg_1}\Min^c_{\DuAlg_2}$}
\label{subsec:DDmin}
Let $\lsup{\Blg_2}\Min^c_{\Blg_1}$ denote the DA bimodule for a
minimum from~\cite{Bordered2}. We define now the corresponding $DA$
bimodule $\lsup{\DuAlg_1}\Min^c_{\DuAlg_2}$ (i.e. where the input
algebra $\DuAlg_2$ has $2$ fewer strands than the output algebra
$\DuAlg_1$). This discussion is is very similar to the construction
of the corresponding bimodule for a local maximum (over $\Alg$)
from~\cite[Section~\ref{BK2:sec:Max}]{Bordered2}; see
also~\cite[Section~\ref{BK1:sec:Crit}]{BorderedKnots}.
Let $\phi_c\colon \{1,\dots,2n\}\to \{1,\dots,2n+2\}$ be the map
\begin{equation}
\label{eq:DefInsert}
\phi_c(j)=\left\{\begin{array}{ll}
j &{\text{if $j< c$}} \\
j+2 &{\text{if $j\geq c$.}}
\end{array}\right.
\end{equation}
Let
\begin{equation}
\label{eq:MaxAlgebras}
\DuAlg_1=\DuAlg(n+1,\Matching_1)
\qquad{\text{and}}\qquad
\DuAlg_2=\Alg(n,\Matching_2),
\end{equation}
where $\Matching_2$ is obtained from $\Matching_1$ by the property
that:
\begin{align}
\{\phi_c(i),\phi_c(j)\}\in\Matching_1&\Rightarrow
\{i,j\}\in\Matching_2; \nonumber \\
\{c,\phi_c(s)\},\{c+1,\phi_c(t)\}\in\Matching_1&\Rightarrow\{s,t\}\in\Matching_2. \label{eq:M1andM2}
\end{align}
We call an idempotent state $\mathbf y$ for $\DuAlg_1$ an {\em allowed idempotent state for $\DuAlg_1$} if
\begin{equation}
\label{eq:AllowedIdempotents}
|\mathbf y\cap\{c-1,c,c+1\}|\leq 2\qquad\text{and}\qquad c\in\mathbf y.
\end{equation}
There is a map $\psi'$ from
allowed idempotent states $\mathbf y$ for $\DuAlg_1$ to idempotent states for $\Blg_2(n)$,
where $\mathbf x=\psi'(\mathbf y)\subset \{0,\dots,2n\}$ is characterized by
\begin{equation}
\label{eq:SpecifyPsi}
|\mathbf y\cap \{c-1,c,c+1\}| +
|\mathbf x\cap \{c-1\}| =2~\qquad{\text{and}}~\qquad \phi_c(\mathbf x)\cap \mathbf y=\emptyset.
\end{equation}
Similarly, we can define $\psi$ from idempotent states for $\DuAlg_1$ to idempotent states for $\DuAlg_2$ by
\begin{equation}
\label{eq:SpecifyPsiPrimed}
\psi(\mathbf y)=\{0,\dots,2n\}\setminus\psi'(\mathbf y).
\end{equation}
We have the following:
\begin{lemma}
\label{lem:ConstructDeltaTwo}
If $\mathbf x$ is an allowed idempotent state (for $\DuAlg_1$) and $\mathbf y$ is
an idempotent state for $\DuAlg_2$ so that $\psi(\mathbf x)$ and $\mathbf y$ are
close enough (i.e. $\Idemp{\psi(\mathbf x)}\cdot \Blg_1\cdot \mathbf y\neq 0$),
then there is an allowed idempotent state $\mathbf z$ (for $\DuAlg_1$) with
$\psi(\mathbf z)=\mathbf y$ so that there is a map
\[ \Phi_{\mathbf x}\colon
\Idemp{\psi(\mathbf x)}\cdot \DuAlg_2\cdot \Idemp{\mathbf y}\to
\Idemp{\mathbf x}\cdot \DuAlg_1\cdot \Idemp{\mathbf z}\]
with the following properties:
\begin{itemize}
\item $\Phi_\mathbf x$ maps the portion of $\Idemp{\psi(\mathbf x)}\cdot \Blg_2\cdot \Idemp{\mathbf y}$ with weights
$(v_1,\dots,v_{2n})$ surjectively onto the portion of
$\Idemp{\mathbf x}\cdot \Blg_1\cdot\Idemp{\mathbf z}$ with
$w_{\phi_c(i)}=v_i$ and $w_{c}=w_{c+1}=0$,
\item $\Phi_{\mathbf x}$ further satisfies the relations
\begin{align*}
\Phi_{\mathbf x}(U_i\cdot a)&=U_{\phi_c(i)} \cdot \Phi_{\mathbf x}(a) \\
\Phi_{\mathbf x}(E_i\cdot a)&=
\left\{\begin{array}{ll}
E_{\phi_c(i)}\cdot \Phi_{\mathbf x}(a) & {\text{if $i\neq t$}} \\
(E_{\phi_c(t)}+E_{c}\llbracket E_{\phi_c(t)},E_{c+1}\rrbracket)\cdot \Phi_\mathbf x(a) &{\text{if $i=t$}}
\end{array}\right.
\end{align*}
for any $i\in 1,\dots,2n$ and $a\in \Idemp{\psi(\mathbf x)}\cdot \DuAlg_2\cdot \Idemp{\mathbf y}$.
\end{itemize}
Moreover, the state $\mathbf z$ is uniquely characterized by the existence of
such a map $\Phi_\mathbf x$.
\end{lemma}
\begin{proof}
This is a straightforward adaptation
of~\cite[Lemma~\ref{BK2:lem:ConstructDeltaTwo}]{Bordered2}; the only
novelty is that here we are using the algebra elements $E_i$
extending $\Blg$ to $\DuAlg$ rather than the algebra elements $C_p$
extending $\Blg$ to $\Alg$ as in that lemma.
\end{proof}
\begin{defn}
Let $\lsup{\DuAlg_1}\Min^c_{\DuAlg_2}$ be the vector space generated
by elements ${\mathbf Q}_\mathbf y$ generated by
allowed idempotent states $\mathbf y$ for $\DuAlg_1$. Endow this with
the structure of a
$\IdempRing(\DuAlg_1)-\IdempRing(\DuAlg_2)$ bimodule
by
\[ \Idemp{\psi(\mathbf y)}\cdot {\mathbf Q}_\mathbf y\cdot \Idemp{\mathbf y}={\mathbf Q}_\mathbf y.
\]
Let
\[ \delta^1_1\colon \lsup{\DuAlg_1}\Max^c_{\DuAlg_2}
\to \DuAlg_1\otimes \lsup{\DuAlg_1}\Max^c_{\DuAlg_2} \]
be the map specified by
\[ \delta^1_1({\mathbf Q}_{\mathbf y})= \Idemp{\mathbf y}\cdot \left(R_{c+1} R_{c} +
L_{c} L_{c+1} + U_c E_{c+1}\right)\otimes \sum_{\mathbf z} {\mathbf
Q}_{\mathbf z}.\] where the sum is taken over all allowed idempotents
$\mathbf z$ for $\DuAlg_2$.
Let
\[\delta^1_2\colon \lsup{\DuAlg_1}\Max^c_{\DuAlg_2}\otimes
\DuAlg_2\to \DuAlg_1\otimes \lsup{\DuAlg_1}\Max^c_{\DuAlg_2} \]
be the map characterized by the property that if
$a=\Idemp{\psi(\mathbf x)}\cdot a\cdot \Idemp{\mathbf y}\in\DuAlg_1$ is a non-zero
algebra element,
then
$\delta^1_2({\mathbf Q}_\mathbf x\cdot a)=\Phi_{\mathbf x}\cdot {\mathbf Q}_{\mathbf z}$,
where $\mathbf z$ is as in Lemma~\ref{lem:ConstructDeltaTwo}.
\end{defn}
\begin{prop}
The above specified actions $\delta^1_1$ and $\delta^1_2$ (and
$\delta^1_\ell=0$ for all $\ell>2$) give
$\lsup{\DuAlg_2}\Min^c_{\DuAlg_1}$ the structure of a $DA$
bimodule, equipped with a $\Delta$-grading and a grading by $H^1(W,\partial)$,
where $W$ is the one-manifold specified in the diagram.
\end{prop}
\begin{proof}
The proof is
straightforward. (Compare~\cite[Theorem~\ref{BK2:thm:MaxDA}]{Bordered2}.)
The differential of the term $E_{c}\llbracket E_{c+1},E_t\rrbracket$
in $\Phi_\mathbf x(E_s)$, which is $U_{c} \cdot \llbracket
E_{c+1},E_t\rrbracket$, cancels against the anti-commutator of
the other term in
$\Phi_{\mathbf x}(E_t)$, $E_{\phi_c(t)}$,
with the term $\delta^1_1(\mathbf x)=U_c\cdot E_{c+1}$.
\end{proof}
\subsubsection{$DA$ bimodules over $\DuAlg$}
The bimodule over $\DuAlg$ associated to a local minimum defined above
and the bimodules over $\DuAlg$ associated to crossings
in~\cite[Section~\ref{BK2:sec:DualCross}]{Bordered2} are related to
the corresponding bimodules over $\Alg$.
In a little more detail, given an integer $n$, a matching $\Matching$
on $\{1,\dots,2n\}$, and an integer $i\in 1,\dots,2n-1$, there are
associated algebras which we abbreviate $\Alg_1$, $\DuAlg_1$,
$\Alg_2$, and $\DuAlg_2$, and imodules $\lsup{\Alg_2}\Pos^i_{\Alg_1}$
and
$\lsup{\Alg_2}\Neg^i_{\Alg_1}$. In~\cite[Section~\ref{BK2:sec:DualCross}]{Bordered2},
we also associated bimodules $\lsup{\DuAlg_1}\Pos_{\DuAlg_2}$
and $\lsup{\DuAlg_1}\Neg_{\DuAlg_2}$.
\begin{prop}
\label{prop:DualityRelationship}
For algebras $\Alg_j$ and $\DuAlg_j$ as above, we have relations
\begin{align*}
\lsup{\Alg_2}\Pos^i_{\Alg_1}\DT \lsup{\Alg_1,\DuAlg_1}\CanonDD &\simeq
\lsup{\DuAlg_1}\Pos^i_{\DuAlg_2}\DT \lsup{\DuAlg_2,\Alg_1}\CanonDD \\
\lsup{\Alg_2}\Neg^i_{\Alg_1}\DT \lsup{\Alg_1,\DuAlg_2}\CanonDD &\simeq
\lsup{\DuAlg_1}\Neg^i_{\DuAlg_2}\DT \lsup{\DuAlg_2,\Alg_2}\CanonDD
\end{align*}
Similarly, now using algebras $\DuAlg_j$ as in Equation~\ref{eq:MaxAlgebras}
(and corresponding algebras $\Alg_j$), we have that
\begin{align*}
\lsup{\Alg_2}\Min_{\Alg_1}\DT \lsup{\Alg_1,\DuAlg_1}\CanonDD &\simeq
\lsup{\DuAlg_1}\Min_{\DuAlg_2}\DT \lsup{\DuAlg_2,\Alg_2}\CanonDD
\end{align*}
\end{prop}
\begin{proof}
Note that the $DA$ bimodules over $\DuAlg$ are all bounded, so the
above tensor products all make sense. The relation involving the
positive crossing
is~\cite[Proposition~\ref{BK2:prop:PosExt}]{Bordered2}. The
relationship with the negative crossing follow formally, since both
negative crossing bimodules are constructed as ``opposite modules''
to the positive crossing bimodules;
see~\cite[Definition~\ref{BK2:def:NegCrossing} and
Subsection~\ref{BK2:subsec:DuAlgNegCross}]{Bordered2}.
For the relation involving the minimum, note that
in~\cite[Section~\ref{BK2:subsec:DDmin}]{Bordered2},
we defined the type $DD$ bimodule associated to a minimum,
denoted $\lsup{\Alg,\DuAlg}\Min$.
In Theorem~\ref{BK2:thm:MinDual}, it is verified that
\[\lsup{\Alg_2}\Min^c_{\Alg_1}\DT \lsup{\Alg_1,\DuAlg_1}\CanonDD
\simeq \lsup{\Alg_2,\DuAlg_1}\Min.\]
The similar verification that
\[\lsup{\Alg_2,\DuAlg_1}\Min\simeq
\lsup{\DuAlg_1}\Min_{\DuAlg_2}\DT \lsup{\DuAlg_2,\Alg_2}\CanonDD\]
is a straightforward exercise in the definitions.
\end{proof}
\begin{rem}
Observe that for all $X\in \Min^c$,
\begin{equation}
\label{eq:IdealToIdeal}
\delta^1_2(X,\llbracket E_i,E_j\rrbracket)=
\left\{\begin{array}{ll}
\llbracket E_{\phi_c(i),\phi_c(j)}\rrbracket\otimes X &{\text{if $\{i,j\}\in M_2$,
$\{i,j\}\neq \{s,t\}$}} \\
\llbracket E_{\phi_c(s)},E_{c}\rrbracket\cdot
\llbracket E_{\phi_c(t)},E_{c+1}\rrbracket\otimes X
&{\text{if $\{i,j\}=\{s,t\}$.}}
\end{array}\right.
\end{equation}
\end{rem}
\subsection{A new algebra $\nDuAlg$}
\label{subsec:nDuAlg}
We will find it convenient to work in a quotient of $\DuAlg$.
Specifically,
let $\nDuAlg$ be the quotient of $\DuAlg$ by the relations $\llbracket E_i,
E_j\rrbracket = 1$ for all $\{i,j\}\in\Matching$. There is a quotient
map $q\colon \DuAlg\to\nDuAlg$.
Recall that $\DuAlg$ is equipped with a $\Delta$-grading defined by
\[ \Delta(a)=\#(\text{$E_j$~that divide $a$})-\sum_{i} \weight_i(a).\]
Since $\llbracket E_i,E_j\rrbracket$ is homogeneous
with $\Delta(\llbracket E_i,E_j\rrbracket)=0$, it follows that the
$\Delta$ grading descends to $\nDuAlg$.
The weight vector, which was a grading on $\DuAlg$, no longer descends
to $\nDuAlg$. However, there is an Alexander grading on $\DuAlg$,
induced by the matching and the orientation on its associated
one-manifold, defined by
\[ \AlexGr(a)=\bigoplus_{\{i,j\}\in\Matching} \weight_i(a)-\weight_j(a), \]
where $W$ represents an arc oriented from $i$ to $j$.
This grading descends to $\nDuAlg$.
Evidently, the set of algebra
elements in $\nDuAlg$ with a fixed $\Delta$-grading is
finite-dimensional (unlike for $\DuAlg$). It follows that bimodules
over $\nDuAlg$ automatically satisfy the boundedness properties
required to form their tensor products, as
in~\cite[Proposition~\ref{BK2:prop:AdaptedTensorProd}]{Bordered2}.
Specifically, a bimodule is called {\em adapted $W$} if it has a grading
by $\mathbb Q} \newcommand{\R}{\mathbb R$ and $H^1(W)$, as in Property~\ref{prop:Adapted} from
Definition~\ref{def:Standard} (using the algebras
$\nDuAlg$ in place of $\Alg$, equipped with the above Alexander grading).
\begin{prop}
\label{prop:AdaptedTensorProdnDuAlg}
Choose $W_1\colon Y_1\to Y_2$, $W_2\colon Y_2\to Y_3$, $\nDuAlg_1$,
$\nDuAlg_2$, and $\nDuAlg_3$ as above. Suppose moreover that $W_1\cup
W_2$ has no closed components, i.e. it is a disjoint union of
finitely many intervals joining $Y_1$ to $Y_3$. Given any two
bimodules $\lsup{\nDuAlg_2}X^1_{\nDuAlg_1}$ and
$\lsup{\nDuAlg_3}X^2_{\nDuAlg_2}$ adapted to $W_1$ and $W_2$ respectively,
we can form their tensor product
$\lsup{\nDuAlg_3}X^2_{\nDuAlg_2}\DT~\lsup{\nDuAlg_2}X^1_{\nDuAlg_1}$ (i.e. only
finitely many terms in the infinite sums in its definition
are non-zero); and moreover, it is a bimodule that is adapted to
$W=W_1\cup W_2$.
\end{prop}
\begin{proof}
This follows exactly as
in~\cite[Proposition~\ref{BK2:prop:AdaptedTensorProd}]{Bordered2},
in view of the fact that the set of algebra elements in $\nDuAlg$
with a fixed $\Delta$-grading is finite-dimensional. (The same
property holds for the algebras $\Alg$ considered in that
proposition.)
\end{proof}
We construct bimodules over $\nDuAlg$ that are related to bimodules over
$\DuAlg$ in the following:
\begin{lemma}
\label{lem:BimodulesOvernDuAlg}
There are $DA$ bimodules
$\lsup{\nDuAlg_1}\Pos^i_{\nDuAlg_2}$,
$\lsup{\nDuAlg_1}\Neg^i_{\nDuAlg_2}$,
$\lsup{\nDuAlg_1}\Min^c_{\nDuAlg_2}$
related to the above bimodules by the relations
\begin{align}
\lsup{\nDuAlg_1}
q_{\DuAlg_1} \DT~ \lsup{\DuAlg_1}\Min^c_{\DuAlg_2} &\simeq
\lsup{\nDuAlg_1}\Min^c_{\nDuAlg_2}\DT~ \lsup{\nDuAlg_2}q_{\DuAlg_2} \label{eq:MinnDuAlg}\\
\lsup{\nDuAlg_1}q_{\DuAlg_1} \DT~ \lsup{\DuAlg_1}\Pos^i_{\DuAlg_2} &\simeq
\lsup{\nDuAlg_1}\Pos_{\nDuAlg_2}\DT~ \lsup{\nDuAlg_2}q_{\DuAlg_2} \label{eq:PosnDuaAlg}\\
\lsup{\nDuAlg_1}q_{\DuAlg_1} \DT~ \lsup{\DuAlg_1}\Neg^i_{\DuAlg_2} &\simeq
\lsup{\nDuAlg_1}\Neg_{\nDuAlg_2}\DT~ \lsup{\nDuAlg_2}q_{\DuAlg_2}
\end{align}
\end{lemma}
\begin{proof}
Equation~\eqref{eq:IdealToIdeal} ensures that
all the operations where some element of the
input algebra lies in the ideal in $\DuAlg$ whose quotient is
$\nDuAlg$ have the property that the output is contained in the corresponding
ideal in the output algebra.
It follows at once that there is an induced module
$\lsup{\nDuAlg_1}\Min^c_{\nDuAlg_2}$ satsisfying
Equation~\eqref{eq:MinnDuAlg}.
The bimodule associated to a positive crossing is constructed in
Section~\cite[Section~\ref{BK2:sec:DuAlg}]{Bordered2}. It is constructed
so that all operations commute with multiplication by
$\llbracket E_i,E_j\rrbracket$. Specifically, there are relations
\begin{align*}
\delta^1_2(X,\llbracket E_i,E_j\rrbracket\cdot a)&=\llbracket
E_{\tau(i)},E_{\tau(j)}\rrbracket\cdot \delta^1_2(X,a)\nonumber\\
\delta^1_3(X,\llbracket E_i,E_j\rrbracket\cdot a_1,a_2)&=
\delta^1_3(X,a_1,\llbracket E_i,E_j\rrbracket\cdot a_2)
=
\llbracket E_\tau(i),E_\tau(j)\rrbracket\cdot
\delta^1_3(X,a_1,a_2),
\end{align*}
where $\tau\colon \{1,\dots,2n\}\to \{1,\dots,2n\}$ is the
transposition that switches the two adjacent elements, corresponding
to the strands that enter the crossing. (Note that $\delta^1_k=0$
for $k>3$.) It follows at once that there is a bimodule
$\lsup{\nDuAlg_1}\Pos_{\nDuAlg_2}$ satisfying
Equation~\eqref{eq:PosnDuaAlg}.
The corresponding construction for the negative crossing follow
similarly. (Compare~\cite[Section~\ref{BK2:subsec:DuAlgNegCross}]{Bordered2}.)
\end{proof}
We wish to state an analogue of
Proposition~\ref{prop:DualityRelationship}, using the algebras $\nDuAlg$. To this end, we will
define the canonical type $DD$ bimodule
$\lsup{\cBlg,\nDuAlg}\CanonDD$.
Generators of $\lsup{\cBlg,\nDuAlg}\CanonDD$, as a vector space,
correspond to $n$-element subsets $\mathbf x\subset\{0,\dots,2n\}$, i.e.
$I$-states for $\Blg(2n,n)$. Let $\gen_\mathbf x$ be the generator corresponding to $\mathbf x$. The left $\IdempRing(2n,n)\otimes \IdempRing(2n,n+1)$-module
structure is specified by
\[ (\Idemp{\mathbf x}\otimes \Idemp{\{0,\dots,2n\}\setminus \mathbf x}) \cdot \gen_\mathbf x = \gen_x.\]
The differential is specified by the element
\[
A = \sum_{i=1}^{2n} \left(L_i\otimes R_i + R_i\otimes L_i\right) + \sum_{i=1}^{2n}
U_i\otimes E_i \in
\Blg\otimes \nDuAlg,\]
\[ \delta^1 \colon \CanonDD \to \Alg\otimes\nDuAlg \otimes
\CanonDD.\] by $\delta^1(v)=A\otimes v$.
That is, $\lsup{\cBlg,\nDuAlg}\CanonDD$ is obtained from the description
of $\lsup{\Alg,\DuAlg}\CanonDD$ by dropping the terms involving $C_{\{i,j\}}$,
and then taking the quotient to go from $\DuAlg$ to $\nDuAlg$, or, more formally,
\[ \lsup{\Alg}T_{\cBlg}\DT \lsup{\cBlg,\nDuAlg}\CanonDD=
\lsup{\nDuAlg_1}q_{\DuAlg}\DT \lsup{\DuAlg,\Alg}\CanonDD.\]
For future reference, we describe analogous type $DD$ bimodules for
a local minimum and for a crossing.
\subsubsection{Algebraically defined curved $DD$-bimodule of a local minimum}
Fix some positive integers $n$ and $c$,
with $1\leq c\leq 2n+1$, and a matching $\Matching_1$ on $\{1,\dots,2n+2\}$.
Let
\[
\nDuAlg_1=\nDuAlg(n+1,\Matching_1)
\qquad{\text{and}}\qquad
\cBlg_2=\cBlg(n,\Matching_2),
\]
where $\Matching_2$ is induced from $\Matching_1$ as in Equation~\ref{eq:M1andM2}.
Define $\lsup{\cBlg_2,\nDuAlg_1}\Min_c=\Min$, as follows.
We call an idempotent state $\mathbf y$ for $\Blg_1$ an {\em allowed idempotent state for $\Blg_1$} if
\[ |\mathbf y\cap\{c-1,c+1\}|\neq\emptyset\qquad\text{and}\qquad c\not\in\mathbf y.\]
Note that the allowed idempotent states $\mathbf y$ for $\Blg_1$
are those that are of the form $\{0,\dots,2n+2\}\setminus \mathbf y'$, where
$\mathbf y'$ is an allowed idempotent state for $\DuAlg_1$, in the sense of Equation~\eqref{eq:AllowedIdempotents}.
As a vector space, $\Min$ is spanned by vectors that are in
one-to-one correspondence with allowed idempotent states for $\nDuAlg_1$;
if $\mathbf y$ is an allowed idempotent state for $\nDuAlg_1$, let ${\mathbf P}_\mathbf y$ be its corresponding generator.
The bimodule structure, over the rings of idempotents $\IdempRing(\Blg_2)\otimes \IdempRing(\nDuAlg_1)$
is specified by the relation
\[ (\Idemp{\psi(\mathbf y)}\otimes \Idemp{\{0,\dots,2n+2\}\setminus \mathbf y})\cdot {\mathbf P}_{\mathbf y}= {\mathbf P}_\mathbf y,\]
where $\psi$ is as in Equation~\eqref{eq:SpecifyPsi}.
The differential is specified by the the following element in $\Blg_2\otimes\nDuAlg_1$:
\begin{align}
A''&=(1\otimes L_{c} L_{c+1}) +
(1\otimes R_{c+1} R_{c})
+ \sum_{j=1}^{2n} R_{j} \otimes L_{\phi(j)} + L_{j} \otimes R_{\phi_c(j)} + U_{j}\otimes E_{\phi_c(j)}
\label{eq:defDDmin}\\
&
+ 1\otimes E_{c} U_{c+1}
+ U_{s}\otimes E_{c+1}.
\nonumber
\end{align}
(Note that in~\cite[Section~\ref{BK2:subsec:DDmin}]{Bordered2}, we described a $DD$ bimodule
$\lsup{\nDuAlg_1,\Alg_2}\Min_c$ associated to a local minimum. The above description
is obtained by elminiating all the terms involving $C_{i,j}$, and specializing from $\DuAlg$
to $\nDuAlg$.)
The bimodule can be described in a little more detail, in terms of the classification of
idempotents for $\nDuAlg_1$ into three types, labelled $\XX$, $\YY$, and $\ZZ$:
\begin{itemize}
\item $\mathbf y$ is of type $\XX$ if $\mathbf y\cap \{c-1,c,c+1\}=\{c,c+1\}$,
\item $\mathbf y$ is of type $\YY$ if $\mathbf y\cap \{c-1,c,c+1\}=\{c-1,c\}$,
\item $\mathbf y$ is of type $\ZZ$ if $\mathbf y\cap \{c-1,c,c+1\}=\{c-1,c+1\}$.
\end{itemize}
There is a corresponding classification of the generators ${\mathbf
P}_{\mathbf y}$ into $\XX$, $\YY$, and $\ZZ$.
With respect to this decomposition,
terms in the differential are of the following four types:
\begin{enumerate}[label=(P-\arabic*),ref=(P-\arabic*)]
\item
\label{type:OutsideLR}
$R_{j}\otimes L_{\phi_c(j)}$
and $L_{c}\otimes R_{\phi_c(j)}$ for all $j\in \{1,\dots,2n\}\setminus \{c-1,c\}$; these connect
generators of the same type.
\item
\label{type:UC}
$U_{j}\otimes E_{\phi_c(j)}$
\item
\label{type:UC2}
$1\otimes E_{c} U_{c+1}$
\item
\label{type:Curved}
$U_s \otimes E_{c+1}$
\item
\label{type:InsideCup}
Terms in the diagram below that connect generators
of different types.
\begin{equation}
\label{eq:CritDiag}
\begin{tikzpicture}[scale=1.5
\node at (-1.5,0) (X) {$\XX$} ;
\node at (1.5,0) (Y) {$\YY$} ;
\node at (0,-2) (Z) {$\ZZ$} ;
\draw[<-] (X) [bend right=7] to node[below,sloped] {\tiny{$1\otimes R_{c+1} R_{c}$}} (Y) ;
\draw[<-] (Y) [bend right=7] to node[above,sloped] {\tiny{$1\otimes L_{c} L_{c+1}$}} (X) ;
\draw[<-] (X) [bend right=7] to node[below,sloped] {\tiny{$R_{c-1}\otimes L_{c-1}$}} (Z) ;
\draw[<-] (Z) [bend right=7] to node[above,sloped] {\tiny{$L_{c-1}\otimes R_{c-1}$}} (X) ;
\draw[<-] (Z) [bend right=7] to node[below,sloped] {\tiny{$R_{c} \otimes L_{c+2}$}} (Y) ;
\draw[<-] (Y) [bend right=7] to node[above,sloped] {\tiny{$L_{c}\otimes R_{c+2}$}} (Z) ;
\end{tikzpicture}
\end{equation}
\end{enumerate}
With the understanding that
if $c=1$, then the terms containing $L_{c-1}$ or $R_{c-1}$ are missing;
similarly, if $c=2n$, the terms containing $R_{c+2}$ and $L_{c+2}$ are missing.
\subsubsection{Algebraically defined curved bimodule of a positive crossing}
\label{subsec:AlgPosCross}
We construct the bimodle $\lsup{\cBlg_2,\nDuAlg_1}\Pos_i$.
Fix $i$ with $1\leq i\leq 2n-1$,
fix a matching $\Matching$ on $\{1,\dots,2n\}$.
Let $\tau=\tau_i\colon \{1,\dots,2n\}\to \{1,\dots,2n\}$
be the transposition that
switches $i$ and $i+1$, and let
$\tau(\Matching)$ be the induced matching; i.e.
$\{j,k\}\in\Matching$ iff $\{\tau(j),\tau(k)\}\in\tau(\Matching)$
Let
\begin{equation}
\label{eq:DefA1A2}
\nDuAlg_1=\nDuAlg(n,\tau_i(\Matching))
\qquad{\text{and}}\qquad
\cBlg_2=
\cBlg(n,\Matching).
\end{equation}
\begin{figure}[ht]
\input{PosCrossDD.pstex_t}
\caption{\label{fig:PosCrossDD} {\bf{Positive crossing $DD$ bimodule generators.}}
The four generator types are pictured to the right.}
\end{figure}
As an $\IdempRing(\Blg_1)-\IdempRing(\nDuAlg_2)$-bimodule, $\Pos_i$ is the submodule
of $\IdempRing(\Blg_1)\otimes_{\Field}\IdempRing(\DuAlg_2)$ generated by elements
$\Idemp{\mathbf x}\otimes \Idemp{\mathbf y}$ where $\mathbf x\cap\mathbf y=\emptyset$ or
\begin{equation}
\label{eq:PosGens}
\mathbf x\cap\mathbf y=\{i\}\qquad{\text{and}}\qquad \{0,\dots,2n\}\setminus (\mathbf x\cup \mathbf y) =\{i-1\}~\text{or}~\{i+1\}.
\end{equation}
Generators can be classified into four types,
$\North$, $\South$, $\West$, and $\East$: for generators of type $\North$
the subsets $\mathbf x$ and $\mathbf y$ are complementary subsets of $\{0,\dots,2n\}$
and $i\in \mathbf x$;
for generators of type $\South$,
$\mathbf x$ and $\mathbf y$ are complementary subsets of $\{0,\dots,2n\}$ with $i\in \mathbf y$;
for generators of type $\West$, $i-1\not\in \mathbf x$ and $i-1\not\in \mathbf y$,
and $\mathbf x\cap\mathbf y=\{i\}$;
for generators of type $\East$, $i+1\not\in \mathbf x$ and $i+1\not\in \mathbf y$,
and $\mathbf x\cap\mathbf y=\{i\}$.
The differential has the following types of terms:
\begin{enumerate}[label=(P-\arabic*),ref=(P-\arabic*)]
\item
\label{type:OutsideLRP}
$R_j\otimes L_j$
and $L_j\otimes R_j$ for all $j\in \{1,\dots,2n\}\setminus \{i,i+1\}$; these connect
generators of the same type.
\item
\label{type:UCP}
$U_{j}\otimes E_{\tau(j)}$
for all $j=1,\dots,2n$
\item
\label{type:InsideP}
Terms in the diagram below that connect generators
of different types:
\begin{equation}
\label{eq:PositiveCrossing}
\begin{tikzpicture}[scale=1.8]
\node at (0,3) (N) {$\North$} ;
\node at (-2,2) (W) {$\West$} ;
\node at (2,2) (E) {$\East$} ;
\node at (0,1) (S) {$\South$} ;
\draw[->] (S) [bend left=7] to node[below,sloped] {\tiny{$R_i\otimes U_{i+1}+L_{i+1}\otimes R_{i+1}R_i$}} (W) ;
\draw[->] (W) [bend left=7] to node[above,sloped] {\tiny{$L_{i}\otimes 1$}} (S) ;
\draw[->] (E)[bend right=7] to node[above,sloped] {\tiny{$R_{i+1}\otimes 1$}} (S) ;
\draw[->] (S)[bend right=7] to node[below,sloped] {\tiny{$L_{i+1}\otimes U_i + R_i \otimes L_{i} L_{i+1}$}} (E) ;
\draw[->] (W)[bend right=7] to node[below,sloped] {\tiny{$1\otimes L_i$}} (N) ;
\draw[->] (N)[bend right=7] to node[above,sloped] {\tiny{$U_{i+1}\otimes R_i + R_{i+1} R_i \otimes L_{i+1}$}} (W) ;
\draw[->] (E)[bend left=7] to node[below,sloped]{\tiny{$1\otimes R_{i+1}$}} (N) ;
\draw[->] (N)[bend left=7] to node[above,sloped]{\tiny{$U_{i}\otimes L_{i+1} + L_{i} L_{i+1}\otimes R_i$}} (E) ;
\end{tikzpicture}
\end{equation}
\end{enumerate}
Note that for a generator of type $\East$, the terms of
Type~\ref{type:OutsideLRP} with $j=i+2$ vanish; while for one of type
$\West$, the terms of Type~\ref{type:OutsideLRP} with $j=i-1$ vanish.
There is a $\mathbb Q} \newcommand{\R}{\mathbb R$-grading on $\Pos_i$ defined by
\begin{equation}
\label{eq:DeltaGradingPos}
\Delta(\North)=\Delta(\South)=\Delta(\West)-\OneHalf=\Delta(\East)-\OneHalf,
\end{equation}
which is homological in the sense that if $(a\otimes b)\otimes Y$
appears with non-zero coefficient in $\delta^1(X)$, then
\[ \Delta(X)-1=\Delta(Y)+\Delta(a)+\Delta(b)
=\Delta(Y)-\weight(a)-\weight(b)+\# \text{($E$ in $b$)}.\]
There is an additional $\MGradingSet=\mathbb Q} \newcommand{\R}{\mathbb R^{2n}$-valued grading on
$\Pos_i$.
The grading set can be thought of as the vector space
spanned by the strands in the picture.)
Fix a standard basis
$e_1,\dots,e_{2n}$ for $\mathbb Q} \newcommand{\R}{\mathbb R^{2n}$ (where we can think of $e_k$ as
labelling the strand whose output is the $k^{th}$ spot from the left,
in the algebra corresponding to $\Blg$), and set
\begin{equation}
\label{eq:AlgGradeCrossing}
\begin{array}{llll}
\gr(\North)=\frac{e_i +e_{i+1}}{4} &
\gr(\West)=\frac{e_{i}-e_{i+1}}{4} &
\gr(\East)=\frac{-e_{i}+e_{i+1}}{4} &
\gr(\South)=\frac{-e_{i}-e_{i+1}}{4}.
\end{array}
\end{equation}
This induces an Alexander grading on the bimodule, in the following
sense. Let $\Agr^{\Matching}$ denote the Alexander grading on the
algebra from Subsection~\ref{subsec:Gradings}.
Now, if
$(a\otimes b)\otimes Y$
appears with non-zero multiplicity in $\delta^1(X)$,
then
\[
\Pi(\gr(X))=\Agr^{\Matching}(a)-\Agr^{\tau_i(\Matching)}(b)+\Pi(\gr(Y)),\]
where $\Pi\colon \mathbb Q} \newcommand{\R}{\mathbb R^{2n}\to \mathbb Q} \newcommand{\R}{\mathbb R^{n}$ is the quotient by the
matching.
(More abstractly, if $W_1$ is the $1$-manifold with $n$ components
associated to the matching, and $W_2$ denotes $1$-manifold with $2n$
components appearing in the picture of the crossing, the map $\Pi$ is
induced by the inclusion map $H^1(W_2,\partial W_2)\to H^1(W_2\cup
W_1,\partial W_2\cup W_1)\cong H^1(W_1,\partial W_1)$.)
The definition is almost the same as the $DD$ bimodule
$\lsup{\Alg_2,\DuAlg_1}\Pos_i$ associated to a crossing
from~\cite[Section~\ref{BK2:subsec:DDcross}]{Bordered2}, except that the
terms involving $C_{\{i,j\}}$ are dropped; i.e. by its construction, it
is clear that
\[ \lsup{\nDuAlg_1}q_{\DuAlg_1} \DT\lsup{\Alg_2,\DuAlg_1}\Pos_i
= \lsup{\Alg_2}T_{\cBlg_2}\DT \lsup{\cBlg_2,\nDuAlg_1}\Pos_i. \]
\subsubsection{Algebraically defined curved bimodule of a negative crossing}
Taking opposite modules, we can form
\[{\overline{\lsup{\Alg_1,\DuAlg_2}
\Pos_i}}=\overline{\Pos}_i^{\Alg_1,\DuAlg_2}
=\lsup{\Alg_1^{\opp},(\DuAlg_2)^{\opp}}{\overline{\Pos}}_i.\] Combining
this with the identification $\Opposite$ of $\Alg_1$ and $\DuAlg_2$
with their opposites (an identification that switches the roles of $R_i$ and $L_i$), we arrive at a type $DD$ bimodule, denoted
$\lsup{\Alg_1,\DuAlg_2}\Neg_i$.
Concretely, we reverse all the arrows in Figure~\ref{eq:PositiveCrossing} and replace algebra elements $L_j$ and $R_j$
with $R_j$ and $L_j$ respectively (leaving $U_j$ unmodified).
Note that this is related to the $DD$ bimodule associated to a crossing
$\lsup{\DuAlg,\Alg}\Pos$ from~\cite[Section~\ref{BK2:subsec:DDcross}]{Bordered2}
by the relation
\[ \lsup{\Alg}T_{\cBlg}\DT \lsup{\cBlg,\nDuAlg}\Pos
= \lsup{\nDuAlg_1}q_{\DuAlg} \DT \lsup{\DuAlg,\Alg}\Pos.\]
\subsection{Curved $DA$ bimodules over $\Blg$ and those over $\nDuAlg$}
We have the following analogue of Proposition~\ref{prop:DualityRelationship}:
\begin{prop}
\label{prop:BimodulesOvernDuAlg}
There is the following relationship between the curved bimodules
associated to a minimum $\lsup{\cBlg_2}\Min^c_{\cBlg_1}$,
$\lsup{\cBlg_2}\Pos^i_{\cBlg_1}$, and $\lsup{\cBlg_2}\Neg^i_{\cBlg_1}$
(from Proposition~\ref{prop:CurvedDABimodules})
and the corresponding $DA$ bimodules over $\nDuAlg$ (from Lemma~\ref{lem:BimodulesOvernDuAlg}):
\begin{align*}
\lsup{\cBlg_2}\Min^c_{\cBlg_1}\DT \lsup{\cBlg_1,\nDuAlg_1}\CanonDD &\simeq
\lsup{\nDuAlg_1}\Min^c_{\nDuAlg_2}\DT \lsup{\nDuAlg_2,\cBlg_2}\CanonDD \\
\lsup{\cBlg_2}\Pos^i_{\cBlg_1}\DT \lsup{\cBlg_1,\nDuAlg_1}\CanonDD &\simeq
\lsup{\nDuAlg_1}\Pos^i_{\nDuAlg_2}\DT \lsup{\nDuAlg_2,\cBlg_2}\CanonDD \\
\lsup{\cBlg_2}\Neg^i_{\cBlg_1}\DT \lsup{\cBlg_1,\nDuAlg_2}\CanonDD &\simeq
\lsup{\nDuAlg_1}\Neg^i_{\nDuAlg_2}\DT \lsup{\nDuAlg_2,\cBlg_2}\CanonDD
\end{align*}
\end{prop}
\begin{proof}
It is straightforward to see that
$\lsup{\cBlg_1}\Min^c_{\cBlg_2}\DT \lsup{\cBlg_2,\nDuAlg_2}\CanonDD
=\lsup{\cBlg_1,\nDuAlg_2}\Min_c$, where the latter is the type $DD$ bimodule
from Subsection~\ref{subsec:DDmin}.
The verification:
$\lsup{\nDuAlg_1}\Min^c_{\nDuAlg_2}\DT \lsup{\nDuAlg_2,\cBlg_2}\CanonDD
=\lsup{\cBlg_1,\nDuAlg_2}\Min_c$
follows similarly.
The identification
$\lsup{\cBlg_2,\nDuAlg}\Pos_i=~\lsup{\cBlg_2}\Pos^i_{\cBlg_1}\DT
\lsup{\cBlg_1,\nDuAlg_1}\CanonDD$
is similarly straightforward.
A homotopy equivalence
$\lsup{\nDuAlg_1}\Pos^i_{\nDuAlg_2}\DT \lsup{\nDuAlg_2,\cBlg_2}\CanonDD\simeq
\lsup{\nDuAlg_1,\cBlg_2}\Pos_i$
is given exactly as in the proof
of~\cite[Lemma~\ref{BK2:lem:DuAlgDualCross}]{Bordered2}
(where it is shown that
$\lsup{\DuAlg_1}\Pos^i_{\DuAlg_2}\DT \lsup{\DuAlg_2,\Alg_2}\CanonDD\simeq
\lsup{\DuAlg,\Alg}\Pos_i$).
The relation with a negative crossing follows formally, since $\lsup{\cBlg,\nDuAlg}\CanonDD$
is preserved by the symmetry induced by taking the algebra to its opposite.
\end{proof}
\subsection{Restricting idempotents}
Consider $\Blg(n)$, and let
\begin{equation}
\label{eq:DefCidemp}
\Cidemp(n)=\sum_{\{\mathbf x\big|\mathbf x\cap\{0,2n\}=\emptyset\}}\Idemp{\mathbf x}.
\end{equation}
Recall that $\Clg(n)$ is the subalgebra of $\Blg(n)$ determined by
$\Clg(n)=\Cidemp(n)\cdot\Blg(n)\cdot\Cidemp(n)$. We can think of the
inclusion map $i\colon \Clg(n)\to \Blg(n)$ as a curved type $DA$ bimodule
$\lsup{\cBlg(n)}i_{\cClg(n)}$. As usual, we drop $n$ (in the notation
for $\cClg$, $\cBlg$, and $\Cidemp$) when it
clear from the context.
\begin{lemma}
\label{lem:RestrictIdempotents}
There are $DA$ bimodules
$\lsup{\cClg_2}\Pos^i_{\cClg_1}$,
$\lsup{\cClg_2}\Neg_{\cClg_1}$,
$\lsup{\cClg_2}\Min_{\cClg_1}$
related to the above bimodules by the relations
\begin{align*}
\lsup{\cBlg_2}i_{\cClg_2} \DT~ \lsup{\cClg_2}\Pos_{\cClg_1} &\simeq~
\lsup{\cBlg_2}\Pos_{\cBlg_1}\DT~ \lsup{\cBlg_1}i_{\cClg_1} \\
\lsup{\cBlg_2}i_{\cClg_2} \DT~ \lsup{\cClg_2}\Neg_{\cClg_1} &\simeq~
\lsup{\cBlg_2}\Neg_{\cBlg_1}\DT~ \lsup{\cBlg_1}i_{\cClg_1} \\
\lsup{\cBlg_2}i_{\cClg_2} \DT~ \lsup{\cClg_2}\Min_{\cClg_1} &\simeq~
\lsup{\cBlg_2}\Min_{\cBlg_1}\DT~ \lsup{\cBlg_1}i_{\cClg_1}
\end{align*}
\end{lemma}
\begin{proof}
It suffices to notice that all three bimodules
$\lsup{\cBlg_2}X_{\cBlg_1}=\lsup{\cBlg_2}\Min_{\cBlg_1}$,
$\lsup{\cBlg_2}\Pos^i_{\cBlg_1}$, and $\lsup{\cBlg_2}\Neg^i_{\cBlg_1}$,
and have the property that if
$X\cdot \Cidemp=\Cidemp\cdot X$, with $\Cidemp$ as in Equation~\eqref{eq:DefIota}.
(Note that $\Cidemp$
depends on the underlying algebra; in particular for $\Min$, the left and
right ``$\Cidemp$'' differ.) This in turn it follows from properties of
these three objects as (ordinary) bimodules over the idempotent
algebras;
c.f.~\cite[Proposition~\ref{BK2:prop:RestrictIdempotent}]{Bordered2}.
\end{proof}
\section{Algebra}
\label{sec:Algebra}
We recall the algebraic preliminaries used in this paper, starting in
Section~\ref{subsec:Framework} with an algebraic framework familiar
from bordered Floer homology~\cite{InvPair}, applied to the case of
algebras with curvature. In Subsection~\ref{subsec:BorderedAlgebras},
we recall the bordered algebras from~\cite{BorderedKnots}
and~\cite{Bordered2}, and explain how to fit them into this framework.
\subsection{Algebraic framework}
\label{subsec:Framework}
For the purposes of the present paper, we find it convenient to work
with smaller algebras than those from~\cite{Bordered2}, but equipped
with further algebraic structure, in the form of a {\em curvature}
element. Such curved algebras have long played a role in Floer
homology; see for example~\cite{FOOO}. They have also played a role in
categorified knot invariants; see for example~\cite{KhovanovRozansky}.
Using curved algebras allows us to work with somewhat smaller
algebras, and the curve counting involves fewer types of
pseudo-holomorphic curves; see also~\cite{TorusMod}.
The following is a straightforward adaptation of the algebraic
material familiar in bordered Floer homology (see
especially~\cite[Chapter~2]{InvPair} and~\cite[Chapter~2]{Bimodules})
to the curved setting.
Fix a ground ring $R$. A {\em curved algebra} is a graded
$R$-bimodule $B$, equipped with an associative multiplication
\[ \mu_2\colon B\otimes_{R}B \to B \]
(with unit)
and a preferred central element
$\mu_0\in B$. We denote the triple $(B,\mu_0,\mu_2)$ by $\cBlg$.
A {\em curved module} over $\cBlg$ is a right $R$-module $M$,
equipped with right module maps $m_1\colon M \to M$
and $m_2\colon M\otimes_{R} B \to M$
satisfying the structure relations
\begin{align}
\label{eq:CurvedDiff}
m_1\circ m_1 + m_2\circ (\Id_M \otimes \mu_0) &= 0 \\
m_1\circ m_2 + m_2\circ (m_1\otimes \Id_{B})&= 0 \\
m_2\circ (m_2\otimes \Id_{B})=m_2\circ (\Id_M\otimes \mu_2)
\end{align}
This has a generalization to the $\Ainf$ context, as follows.
A (curved) $\Ainf$ module $M_{\cBlg}$ consists of a right $R$-module
$M$, equipped with right $R$-module homomorphisms
\[ m_i\colon M\otimes_R \overbrace{B\otimes_R\dots \otimes_R
B}^{i-1}\to B,\] satisfying a sequence of $\Ainf$ relations, indexed
by an integer $k\geq 0$, an element $x\in M$,
and a (possibly empty) sequence $b_1,\dots,b_k$ of algebra
elements:
\begin{align*}
0 &=\sum_{j=0}^k m_{k-j+1}(m_{j+1}(x,b_1,\dots,b_j),b_{j+1},\dots,b_k) \\
&+ \sum_{i=1}^{k-1} m_{k}(x,b_1,\dots,b_{i-1},\mu_2(b_i,b_{i+1}),b_{i+2},\dots,b_k) \\
&+ \sum_{i=1}^{k+1}
m_{k+2}(x,b_1,\dots,b_{i-1},\mu_0,b_{i},\dots,b_{k}).
\end{align*}
For example, when $k=0$, the above relation gives Equation~\eqref{eq:CurvedDiff}.
A {\em (curved) type $D$ structure} is a left $R$-module $X$ equipped
with a left $R$-module homomorphism
$\delta^1\colon X \to B\otimes_{R} X$,
satisfying the structure relation
\[ 0 = \mu_0\otimes \Id_{X} + (\mu_2\otimes \Id_X)\circ(\Id_B\otimes
\delta^1)\circ \delta^1,\] thought of as maps $X\to B\otimes_{R}
X$. We will denote this data by $\lsup{\cBlg}X$. (This notion is
equivalent to the notion of the {\em matrix factorizations} considered
in~\cite{KhovanovRozansky}.)
The definition of the tensor product $M_{\cBlg}\DT \lsup{\cBlg}X$ is
the same as in the uncurved context. The verification that this is a
chain complex is the same as in the uncurved case (see for
example~\cite[Section~2.4]{InvPair}), except that the curved type $D$
structure relations give rise to extra terms which are cancelled by
the extra terms in the curved $\Ainf$ relation.
Bimodules also can be generalized to the curved case in a
straightforward way. For example, let $(B_1,\mu^{B_1}_0)$ and
$(B_2,\mu^{B_2}_0)$ be two curved algebras over ground rings $R_1$ and $R_2$ respectively. A {\em curved type $DA$
bimodule} is a $R_2$-$R_1$-bimodule $X$, equipped
with operations
\[ \delta^1_{\ell+1}\colon X \otimes_{R_1} \overbrace{B_1\otimes_{R_1}\dots\otimes_{R_1}\otimes B_1}^{\ell}\to B_2\otimes_{R_2} X,\]
satisfying the structural relations for any $x\in X$,
\begin{equation}
\label{eq:CurvedDAbimoduleRelation1}
0 =(\mu^{B_2}_2\otimes\Id_X)\circ (\Id_{B_2}\otimes \delta^1_1)\circ \delta^1_1(x) \\
+ \delta^1_{2}(x,\mu_0^{B_1})
+ \mu_0^{B_2} \otimes x;
\end{equation}
and, for each sequence $b_1,\dots,b_k$ in $B_1$ (with $k\geq 1$),
\begin{align}
0 &=\sum_{j=0}^k \mu_2^{B_2}\circ(\Id\otimes \delta^1_{k-j+1})\circ
(\delta^1_{j+1}(x,b_1,\dots,b_j),b_{j+1},\dots,b_k)
\label{eq:CurvedDAbimoduleRelation2}
\\
&+ \sum_{i=1}^{k-1} \delta^1_{k}(x,b_1,\dots,b_{i-1},\mu_2^{B_1}(b_i,b_{i+1}),b_{i+2},\dots,b_k) \nonumber \\
&+ \sum_{i=1}^{k+1}
\delta^1_{k+2}(x,b_1,\dots,b_{i-1},\mu_0^{B_1},b_{i},\dots,b_{k}). \nonumber
\end{align}
(Compare~\cite{Bimodules} in the uncurved case.)
Note that the curvature on $B_2$ appears in Equation~\eqref{eq:CurvedDAbimoduleRelation1}, but not Equation~\eqref{eq:CurvedDAbimoduleRelation2}.
\begin{example}
Fix curved algebras $(B_1,\mu_0^{B_1})$ and $(B_2,\mu_0^{B_2})$ over
$R$, and suppose that $\phi\colon B_1\to B_2$ is an
$R$-algebra homomorphism with the property that
$\phi(\mu_0^{B_1})=\mu_0^{B_2}$. Then $\phi$ induces a type $DA$
bimodule $\lsup{B_2}[\phi]_{B_1}$, whose underlying
$R$-bimodule is isomorphic to
$\lsub{R}R_{R}$, and whose
operations are specified by
$\delta^1_2(x,b)=\phi(b)\otimes x$ (where $x$ corresponds to
$\One\in\lsub{R}R_{R}$) and $\delta^1_k\equiv 0$
for $k\neq 2$.
\end{example}
The above notions have obvious generalizations to the case where $B$
is a differential graded algebra, i.e. it is equipped with a
differential $\mu_1$ (which satisfies a Leibniz rule) so that
$\mu_1(\mu_0)=0$.
In practice, we will often have another DGA $\nDuAlg$ over $R$,
and will consider various bimodules over $\cBlg$ and $\nDuAlg$. For
example, thinking of $\nDuAlg\otimes\cBlg$ as curved, with curvature
$\One\otimes \mu_0$, we define a type $DD$ bimodule $X$ over $\nDuAlg$ and
$\cBlg$ to be a curved type $D$ structure over $\nDuAlg\otimes\cBlg$.
\subsection{The bordered algebras}
\label{subsec:BorderedAlgebras}
In~\cite{BorderedKnots}, to each pair of integers $(m,k)$ with
$0\leq k\leq m+1$,
we associated an algebra
$\Blg(m,k)$. We recall the construction presently.
$\Blg(m,k)$ is constructed as the quotient of a larger algebra,
$\BlgZ(m,k)$, associated to $(m,k)$. The base ring of $\BlgZ(m,k)$ is the polynomial
algebra $\Field[U_1,\dots,U_m]$. Idempotents correspond to $k$-element
subsets $\mathbf x$ of $\{0,\dots,m\}$ called {\em idempotent states}. We think of
these as generators of a ring of idempotents $\IdempRing(m,k)$.
Given idempotents states $\mathbf x,\mathbf y$, the $\Field[U_1,\dots,U_m]$-module
$\Idemp{\mathbf x}\cdot\BlgZ(m,k)\cdot\Idemp{\mathbf y}$ is identified with
$\Field[U_1,\dots,U_m]$, given with a preferred generator $\gamma_{\mathbf x,\mathbf y}$.
To specify the product, we proceed as follows. Each idempotent state
$\mathbf x$ has a {\em weight vector} $v^\mathbf x\in \Z^m$, with components $i=1,\dots,m$
given by
$v^\mathbf x_i=\#\{x\in \mathbf x\big|x\geq i\}$.
The multiplication is specified by
\[ \gamma_{\mathbf x,\mathbf y}\cdot \gamma_{\mathbf y,\mathbf z}=U_1^{n_1}\cdots U_m^{n_m} \cdot \gamma_{\mathbf x,\mathbf z},\] where
\[ n_i = \frac{1}{2}(|v_i^\mathbf x-v_i^\mathbf y|+|v_i^\mathbf y-v_i^\mathbf z|-|v_i^\mathbf x-v_i^\mathbf z|).\]
For $i=1,\dots,m$, let $L_i$ be the sum of $\gamma_{\mathbf x,\mathbf y}$, taken
over all pairs of idempotent states $\mathbf x, \mathbf y$ so that there is some integer $s$
with $x_s= i$ and $y_s=i-1$ and $x_t=y_t$ for all $t\neq
s$. Similarly, let $R_i$ denote the sum of all the $\gamma_{\mathbf y,\mathbf x}$
taken over the same pairs of idempotent states as
above.
Under the identification $\Idemp{\mathbf x}\cdot \BlgZ(m,k)\cdot
\Idemp{\mathbf y}\cong \Field[U_1,\dots,U_m]$, the elements that correspond to monomials
in the $U_1,\dots,U_m$ are called {\em pure algebra elements in $\BlgZ(m,k)$}.
These elements are specified by their idempotents $\mathbf x$ and $\mathbf y$, and their
{\em relative weight vector} $w(b)\in
\mathbb Q} \newcommand{\R}{\mathbb R^m$, which in turn is uniquely characterized by
\[
\weight_i(\gamma_{\mathbf x,\mathbf y})=\OneHalf|v_i^\mathbf x-v_i^\mathbf y| \qquad
\weight_i(U_j\cdot b) = \weight_i(b)+\left\{\begin{array}{ll}
0 &{\text{if $i\neq j$}} \\
1 &{\text{if $i=j$.}}
\end{array}\right.
\]
Let ${\mathcal J}\subset \BlgZ(m,k)$ be the two-sided ideal generated by
$L_{i+1}\cdot L_i$, $R_{i}\cdot R_{i+1}$ and, for all choices of
$\mathbf x=\{x_1,...,x_k\}$ with $\mathbf x\cap \{j-1,j\}=\emptyset$,
the element
$\Idemp{\mathbf x}\cdot U_j$.
Then,
\[ \Blg(m,k)=\BlgZ(m,k)/{\mathcal J}.\]
The pure algebra elements in $\Blg(m,k)$ are those elements that
are images of pure algebra elements in $\BlgZ(m,k)$ under the above quotient map.
Idempotents $\mathbf x=x_1<\dots<x_k$ and $\mathbf y=y_1<\dots<y_k$ are said to be {\em too far}
if for some $t\in \{1,\dots,k\}$, $|x_t-y_t|\geq 2$. If $\mathbf x$ and $\mathbf y$ are too far,
then $\Idemp{\mathbf x}\cdot\BlgZ(m,k)\cdot\Idemp{\mathbf y}\in {\mathcal J}$; i.e.
$\Idemp{\mathbf x}\cdot\Blg(m,k)\cdot\Idemp{\mathbf y}=0$.
We restate here the concrete description of the ideal ${\mathcal J}$ given in~\cite{BorderedKnots}:
\begin{prop}\cite[Proposition~3.7]{BorderedKnots}
\label{prop:Ideal}
Suppose that $b=\Idemp{\mathbf x}\cdot b\cdot \Idemp{\mathbf y}$ is a pure algebra element in
${\mathcal J}$. Then, either $\mathbf x$ and $\mathbf y$ are too far, or there is
a pair of integers $i<j$ so that
\begin{itemize}
\item $i,j\in\{0,\dots,m\}\setminus \mathbf x\cap\mathbf y$
\item for all $i<t<j$, $t\in\mathbf x\cap\mathbf y$
\item $\weight_t(b)\geq 1$ for all $t=i+1,\dots,j$
\item $\#(x\in \mathbf x\big| x\leq i)=\#(y\in \mathbf y\big| y\leq i)$.
\end{itemize}
\end{prop}
We specialize to the case of $\Blg(n)=\Blg(2n,n)$,
which we think of as an algebra over
the idempotent ring $\IdempRing(n)=\IdempRing(2n,n)$.
We will typically consider a subalgebra
$\Clg(n)\subset \Blg(2n,n)$
given by
\[ \Clg(n)= \left(\sum_{\{\mathbf x\mid \mathbf x\cap \{0,
2n\}=\emptyset\}} \Idemp{\mathbf x}\right)\cdot \Blg(2n,n)\cdot
\left(\sum_{\{\mathbf x\mid \mathbf x\cap \{0, 2n\}=\emptyset\}}
\Idemp{\mathbf x}\right).\] In particular, the elements $L_1$, $R_1$,
$L_{2n}$, $R_{2n}$ are not in this subalgebra; but $U_1$ and $U_{2n}$ are.
Let $\RestrictIdempRing(n)\subset \Clg(n)$ denote the subring
spanned by the idempotents $\Idemp{\mathbf x}$ where $\mathbf x\cap \{0,2n\}=\emptyset$.
Thus,
\[ \Clg(n)=\RestrictIdempRing(n)\cdot\Blg(n)\cdot\RestrictIdempRing(n).\]
Sometimes, we also consider
\[ \ClgZ(n)=\RestrictIdempRing(n)\cdot\BlgZ(n)\cdot\RestrictIdempRing(n).\]
A {\em matching} is a partition of $\{1,\dots,2n\}$ into $2$-element subsets.
The matching $\Matching$ specifies a central algebra element
in $\Blg(n)$,
\begin{equation}
\label{eq:CurvatureOfB}
\mu_0^{\Matching}=\sum_{\{i,j\}\in\Matching} U_i U_j,
\end{equation}
which we
think of as specifying a curvature for
$\cBlg(n)=(\Blg(n),\mu^{\Matching}_0)$ or for
\begin{equation}
\label{eq:DefcClg}
\cClg(n)=(\Clg(n),\mu^{\Matching}_0).
\end{equation}
We
compare this with the algebraic set-up from~\cite{Bordered2}. In that paper, we
defined an algebra $\Alg(n,\Matching)$ containing $\Blg(n)$, with
new variables $C_{i,j}$ for each $\{i,j\}\in\Matching$ satisfying
$d C_{i,j}=U_i U_j$ and
$C_{i,j}^2=0$.
\begin{defn}
\label{def:Transformer}
The {\em $\Blg$-to-$\Alg$ transformer} is the
type $DA$ bimodule $\lsup{\Alg}T_{\cBlg}$
which, as a bimodule over $\IdempRing(2n,n)$, is identified with
$\IdempRing(2n,n)$, and with operations specified by
\begin{align*}
\delta^1_1(\One)&=\sum_{(i,j)\in\Matching} C_{i,j}\otimes \One \\
\delta^1_2(\One,b)&= b\otimes \One \\
\delta^1_\ell(\One,b_1,\dots,b_{\ell-1})&= 0 \quad\text{for $\ell>2$}.
\end{align*}
\end{defn}
Thus, a curved type $D$ structure over $\cBlg$
naturally gives rise to a type $D$ structure over $\Alg(n,\Matching)$,
$\lsup{\cBlg}X\to \lsup{\Alg}T_{\cBlg}\DT~\lsup{\cBlg}X$.
\subsection{Gradings}
\label{subsec:Gradings}
Our algebras are equipped with two types of gradings: an Alexander
grading, with values in some Abelian group, which is preserved by the
algebra operations; and a homological grading, with values in $\Z$, so
that $\mu_i$ shifts by $i-2$ (and in particular, the element $\mu_0$ has
Alexander grading zero and homological grading $-2$).
The weight function induces a grading on the algebra $\Blg(n)$ with
values in $(\OneHalf\Z)^{2n}\subset \mathbb Q} \newcommand{\R}{\mathbb R^{2n}$.
Choose for each $\{i,j\}\in\Matching$ a preferred ordering $(i,j)$ of the integers $i$ and $j$.
There is an induced {\em Alexander vector}
$\AlexGr\colon \Matching\to \mathbb Q} \newcommand{\R}{\mathbb R$
defined by
\begin{equation}
\label{eq:DefAgrAlg}
{\mathbf A}_{\{i,j\}}(a)=\weight_i(a)-\weight_j(a),
\end{equation}
where $(i,j)$ is the ordering on $\{i,j\}$.
Of course, this can be thought of as a grading with values in $\mathbb Q} \newcommand{\R}{\mathbb R^n$.
Since $\AlexGr(\mu_0)=0$, the Alexander function induces a
well-defined (Alexander-type) $\mathbb Q} \newcommand{\R}{\mathbb R^n$-grading on the curved algebra $\cBlg$.
Furthermore, there is an induced $\mathbb Q} \newcommand{\R}{\mathbb R$-valued Alexander grading specified by
the function on homogeneous algebra elements
$A=\sum_{\{i,j\}\in\Matching} \AlexGr_{\{i,j\}}$.
Sometimes, when we wish to distinguish this from Alexander gradings on modules,
we write $\Agr^{\Matching}$.
More abstractly, we can think of the matching as giving rise to a
one-manifold $W=W(\Matching)$, consisting of $n$ arcs and boundary the
points $Y$ in $\{1,\dots,2n\}$ (i.e. each pair $\{i,j\}\in\Matching$
determines an arc connecting $i$ and $j$). The weight of a given
algebra element gives an element of $H^0(Y)$; and the Alexander
grading can be thought of as an element of the cokernel $H^0(W)\to
H^0(Y)$, which is identified with $H^1(W,\partial W)\cong \mathbb Q} \newcommand{\R}{\mathbb R^n$. A
choice of isomorphism above is equivalent to an orientation on $W$.
We have also a homological $\Delta$-grading, determined by
\begin{equation}
\label{eq:DefDeltaAlg}
\Delta(a)=-\sum_{i} \weight_i(a)
\end{equation}
if $a\in \Blg$. Note that $\Delta(\mu_0)=-2$, as required.
\subsection{Adapted bimodules}
We follow the algebraic set-up from~\cite[Section~2]{Bordered2} with
slight modifications.
We can think of $\Blg(n)$ as an algebra associated to the
zero-manifold $Y_2$, which consists of $2n$ points. A matching on
$Y_2$, which we think of as a one-manifold $W_2$ with $\partial
W_2=Y_2$, specifies a curvature $\mu_0\in\Blg(Y_2)$. There is an
induced grading on $\Agr_{W_2}$ on $\Blg(Y_2)$ by
$H^1(W_2,Y_2)$, for which $\Agr_{W_2}(\mu_0)=0$.
Thus, we think of $\cBlg(Y_2,W_2)$ as graded by $\Agr_{W_2}$.
Fix cobordism $W_1$ from $Y_2$ to $Y_1$, and let $W=W_1\cup_{Y_2}
W_2$. If $X$ is a $H^1(W_1,\partial W_1)$-graded vector space, then
$X\otimes \cBlg(Y_2,W_2)$ inherits a grading by $H^1(W,\partial W)$,
using the natural map
\begin{equation}
\label{eq:MayerVietoris}
H^1(W_1,\partial W_1)\oplus H^1(W_2,\partial
W_2)\to H^1(W,\partial W).
\end{equation}
\begin{defn}
Suppose that $\cBlg_2$ is an algebra graded by $H^1(W_2,Y_2)$.
Fix a cobordism $W_1\colon Y_2\to Y_1$. A curved type $DA$
bimodule $\lsup{\cBlg_1}X_{\cBlg_2}$ is called {\em adapted to $W_1$} if it
is equipped with the following additional data:
\begin{itemize}
\item a grading of $X$ by $H^1(W_1,\partial W_1)$, satisfying the
following compatibility condition: if
$a_1,\dots,a_{\ell-1}$ are $H^2(W_2,\partial W_2)$-homogenous elements,
and $\mathbf x$ is an $H^1(W_1,\partial W_1)$-homogenous element, then
$\delta^1_{\ell}(\mathbf x,a_1,\dots,a_{\ell-1})$ is
$H^1(W,\partial W)$-homogeneous, where $W=W_1\cup W_2$,
with
grading given by
\[\gr(x) + \Agr(a_1)+\dots + \Agr(a_\ell),\]
viewed as an
element of $H^1(W,\partial W)$ using the Mayer-Vietoris maps
from Equation~\eqref{eq:MayerVietoris}).
\item
a grading of $X$ by $\mathbb Q} \newcommand{\R}{\mathbb R$, so that if $\mathbf x$, $a_1,\dots,a_{\ell-1}$
are homogeneous, then $\delta^1_{\ell}(\mathbf x,a_1,\dots,a_{\ell-1})$
is homogeneous of degree
\[ \Delta_X(\mathbf x) + \Delta(a_1)+\dots+\Delta(a_{\ell-1})-\ell+2.\]
\item $X$ is a finite-dimensional $\Field$-vector space.
\end{itemize}
\end{defn}
By contrast, recall that the adapted bimodules in the uncurved case
(\cite[Section~2]{Bordered2}) were graded by $H^1(W_1,\partial)$,
rather than $H^1(W_1\cup W_2,\partial)$. This causes no additional
difficulties. In particular, we have the following straightforward
modification
of~\cite[Proposition~\ref{BK1:prop:AdaptedTensorProducts}]{BorderedKnots}:
\begin{prop}
Let $W_3$ be an oriented one-manifold with $Y_3=\partial W_3$. Fix
also $W_2\colon Y_3\to Y_2$ and $W_1\colon Y_2\to Y_1$. If
$\lsup{\cBlg_2}Y_{\cBlg_3}$ is adapted to $W_2$ and
$\lsup{\cBlg_1}X_{\cBlg_2}$ is adapted to $W_1$,
and $W_1\cup W_2$ has no closed components; then we can form
their tensor product $\lsup{\cBlg_1}X_{\cBlg_2}\DT \lsup{\cBlg_2}
Y_{\cBlg_3}$ to get a curved DA bimodule $\lsup{\cBlg_1}(X\DT
Y)_{\cBlg_3}$ adapted to $W_1\cup W_2$. \qed
\end{prop}
\section{Comparing the knot invariants}
\label{sec:Comparison}
An {\em acceptable knot diagram} $\Diag$ is a diagram for
an oriented knot whose local maxima are
at the same level, all of whose other events occur at different
heights, and whose global minimum is the marked edge; thus, an
acceptable knot diagram is obtained from an acceptable upper knot
diagram in the sense of Section~\ref{sec:computeD} by adding a single
global minimum.
The methods of this paper now give the following explicit computation of knot Floer homology:
\begin{thm}
\label{thm:ComputeHFK}
Let $\Diag$ be an acceptable knot diagram, obtained by adding a
global minimum to $\DiagUp$, and let $\Hup$ be upper diagram
associated to $\DiagUp$. Then, there is a homotopy equivalence
\[\lsup{\Ring}\CFKsimp(\orK)\simeq\lsup{\Field[U,V]}[\Psi]_{\cClg(1)}\DT~\lsup{\cClg(1)}\DmodAlg(\Hup),\]
where
$\Psi\colon \Clg(1)\to \Field[U,V]$ is the isomorphism from Equation~\eqref{eq:IsoClg1}.
\end{thm}
\begin{proof}
To compute $\lsup{\Ring}\CFKsimp(\orK)$,
we use
By the pairing theorem (Theorem~\ref{thm:PairAwithD})
\[ \lsup{\Ring}\CFKsimp(\orK)=
\lsup{\Ring}\Amod(\Hdown)_{\Clg(1)}\DT \lsup{\Clg(1)}\Dmod(\Hup).\]
The result now follows from Lemma~\ref{lem:GlobalMinimum} with
Theorem~\ref{thm:ComputeD}.
\end{proof}
\begin{rem}
The application of Theorem~\ref{thm:PairAwithD}
in the above proof in fact uses the
special case where $n=1$, where we glue to the standard lower diagram.
This is a particularly simple special case of the pairing theorem;
the more interesting applications of the pairing theorem
are contained in the use of Theorem~\ref{thm:ComputeD}.
\end{rem}
The above description of knot Floer homology is not exactly the
construction formulated in~\cite{Bordered2}, but it is close enough
that the proof of Theorem~\ref{thm:MainTheorem} follows quickly:
\begin{proof}[Proof of Theorem~\ref{thm:MainTheorem}]
In~\cite{Bordered2}, we defined $\Hwz(\orK)$ as the homology of a
complex $\Cwz(\Diag)$ associated to a diagram. The complex
$\Cwz(\Diag)$ is constructed similarly to $\DmodAlg(\DiagUp)$
described above: the diagram $\Diag$ is broken into pieces, and to
each elementary piece we associate bimodules, and $\Cwz(\Diag)$ is
obtained by tensoring together these pieces. Specifically, to an upper diagram
$\DiagUp$, tensoring together local bimodules
defines an algebraically defined type $D$ structure
$\lsup{\Alg(n,\Matching)}\PartInv(\DiagUp)$, where $2n$ is the
number of strands out of $\DiagUp$ and $\Matching$ is the matching
induced by $\DiagUp$; and if $\Diag$ is obtained by adding a global
minimum to $\DiagUp$, then
$\Cwz(\Diag)=\lsup{\Field[U,V]}\TerMin_{\Alg(1)}\DT\lsup{\Alg(1)}\PartInv(\DiagUp)$,
where $\lsup{\Field[U,V]}\TerMin_{\Alg(1)}$ is the a bimodule
described in~\cite[Section~\ref{BK2:subsec:ConstructInvariant}]{Bordered2}.
We claim that
\begin{equation}
\label{eq:IdentifyPartialInvariants}
\lsup{\Alg}T_{\cBlg}\DT~ \lsup{\cBlg}i_{\cClg}\DT~ \lsup{\cClg}\DmodAlg(\DiagUp)\simeq
\lsup{\Alg}\PartInv(\DiagUp),
\end{equation}
where $\Alg=\Alg(n,\Matching)$, $\Blg=\Blg(n,\Matching)$,
$\Clg=\Clg(n,\Matching)$ for $\Matching$ as determined by $\DiagUp$.
This is seen by induction on the number of events in $\DiagUp$. For
the basic case, where $\DiagUp$ contains only maxima, recall
from~\cite{Bordered2} that $\lsup{\Alg}\PartInv(\DiagUp)$ is
generated by a single element $\mathbf x$, and $\delta^1(\mathbf x)=C\otimes \mathbf x$,
where $C=\sum_{i=1}^{n} C_{\{2i-1,2i\}}$; i.e. there is an
identification of $\lsup{\Alg}\PartInv(\DiagUp)$ with
\[ \lsup{\Alg}T_{\cBlg}\DT~ \lsup{\cBlg}i_{\cClg}\DT~ \lsup{\cClg}k =
\lsup{\Alg}T_{\cBlg}\DT~ \lsup{\cBlg}i_{\cClg}\DT~ \lsup{\cClg}\DmodAlg(\DiagUp).\]
For the inductive step, when we add the bimodule associated to one
more standard piece, we use Proposition~\ref{prop:CurvedDABimodules}
and Lemma~\ref{lem:RestrictIdempotents}.
Next, we consider $\lsup{\Ring}\TerMin_{\Alg(1)}$, used in the definition of $\Cwz(\Diag)$.
The bimodule $\TerMin$ has three generators $\XX$,
$\YY$, and $\ZZ$, with
\[ \XX \cdot \Idemp{\{0\}}=\XX, \qquad
\YY \cdot \Idemp{\{1\}}=\YY, \qquad
\ZZ \cdot \Idemp{\{2\}}=\ZZ.\]
When $1$ is oriented upwards, $\TerMin$ is the $DA$ bimodule with $\delta^1_k=0$ for $k\neq 2$, and
all $\delta^1_2$ are determined by
\[
\begin{array}{lll}
\delta^1_2(\YY,L_1)= u\otimes \XX, & \delta^1_2(\XX,R_1)= u\otimes \YY, \\
\delta^1_2(\YY,R_2)= v\otimes \ZZ, & \delta^1_2(\ZZ,L_2)= v\otimes \YY, \\
\delta^1_2(\XX,C_{\{1,2\}})=
\delta^1_2(\YY,C_{\{1,2\}})=
\delta^1_2(\ZZ,C_{\{1,2\}})=0.
\end{array}
\]
(When $1$ is oriented downwards, we define the actions as above, exchanging the roles of $u$ and $v$.)
It follows immediately from this description that
\begin{equation}
\label{eq:DescribeTerMin}
\lsup{\Ring}\TerMin_{\Alg(1)}\DT~ \lsup{\Alg(1)}T_{\cBlg(1)}\DT~\lsup{\cBlg(1)}i_{\cClg(1)}=\lsup{\Ring}[\Psi]_{\cClg(1)},
\end{equation}
where $\Psi$ is as in Equation~\eqref{eq:IsoClg1}.
Thus, combining the definition of $\Cwz$ with Equations~\eqref{eq:IdentifyPartialInvariants},
\eqref{eq:DescribeTerMin}, and Theorem~\ref{thm:ComputeHFK}, we find that
\begin{align*}
\Cwz(\Diag)&=\lsup{\Ring}\TerMin_{\Alg(1)}\DT~\lsup{\Alg(1)}\PartInv(\Hup) \\
&= \lsup{\Ring}\TerMin_{\Alg(1)}\DT~ \lsup{\Alg(1)}T_{\cBlg(1)}\DT~ \lsup{\cBlg(1)}i_{\cClg(1)}\DT~ \lsup{\cClg(1)}\DmodAlg(\DiagUp) \\
&= \lsup{\Ring}[\Psi]_{\cClg(1)}\DT~ \lsup{\cClg(1)}\DmodAlg(\Hup) \\
&= \lsup{\Ring}\CFKsimp(\orK),
\end{align*}
as required.
\end{proof}
\section{Computing the $DD$ bimodules of standard middle diagrams}
\label{sec:ComputeDDmods}
Given an extended middle diagram $\HmidExt$, we can think of the bimodule
\[\lsup{\cBlgout}\DAmodExt(\HmidExt)_{\cBlgin}\DT
~\lsup{\cBlgin,\nDuAlgin}\CanonDD\]
as the type $DD$ bimodule
associated to $\HmidExt$.
Our aim here is to compute these bimodules
for some standard middle diagrams, expressed in terms of the
algebraically defined $DD$ bimodules from Section~\ref{sec:AlgDA}.
\begin{thm}
\label{thm:ComputeDDmods}
For each number $n$ of strands equipped with some matching
$\MatchIn$, there are extended middle Heegaard diagrams for local
minima (provided that $\{c,c+1\}\not\in\MatchIn$), positive, and
negative crossings, denoted $\Hmin{c}$, $\Hpos{i}$ $\Hneg{i}$; and
whose associated $DA$ bimodules
$\lsup{\cBlg_2}\DAmodExt(\Hmin{c})_{\cBlg_1}$,
$\lsup{\cBlg_2}\DAmodExt(\Hpos{i})_{\cBlg_1}$,
$\lsup{\cBlg_2}\DAmodExt(\Hneg{i})_{\cBlg_1}$
are related to the
bimodules from Lemma~\ref{lem:BimodulesOvernDuAlg} by:
\begin{align}
\lsup{\nDuAlg_1}\Min^c_{\nDuAlg_2} \DT~ \lsup{\nDuAlg_2,\cBlg_2}\CanonDD &\simeq
\lsup{\cBlg_2}\DAmodExt(\Hmin{c})_{\cBlg_1} \DT~ \lsup{\cBlg_1,\nDuAlg_1}\CanonDD \label{eq:ComputeMinDD} \\
\lsup{\nDuAlg_1}\Pos^i_{\nDuAlg_2} \DT~ \lsup{\nDuAlg_2,\cBlg_2}\CanonDD &\simeq
\lsup{\cBlg_2}\DAmodExt(\Hpos{i})_{\cBlg_1} \DT~ \lsup{\cBlg_1,\nDuAlg_1}\CanonDD \label{eq:ComputePosDD}\\
\lsup{\nDuAlg_2}\Neg_{\nDuAlg_1} \DT~ \lsup{\nDuAlg_1,\cBlg_1}\CanonDD &\simeq
\lsup{\cBlg_2}\DAmodExt(\Hneg{i})_{\cBlg_1} \DT~ \lsup{\cBlg_1,\nDuAlg_1}\CanonDD. \label{eq:ComputeNegDD}
\end{align}
\end{thm}
We split the verification of Theorem~\ref{thm:ComputeDDmods} into
parts (Subsections~\ref{subsec:LocalMin}, \ref{subsec:Pos}, and
\ref{subsec:Neg}), starting first with a warm-up
(Subsection~\ref{subsec:CanonDD}).
\subsection{The canonical $DD$ bimodule}
\label{subsec:CanonDD}
Although this is not logically needed for the subsequent computations,
we compute the bimodule associated to the middle diagram for the identity cobordism, as follows:
\begin{prop}
\label{prop:ComputeIdDD}
For the (extended middle) Heegaard diagram $\Hid$ for the identity cobordism
from Figure~\ref{fig:IdDiag},
stabilized as in Section~\ref{sec:ExtendDA}, we have that
\[ \lsup{\cBlg}\DAmod(\Hid)_{\cBlg}\DT \lsup{\cBlg,\nDuAlg}\CanonDD
\simeq \lsup{\cBlg,\nDuAlg}\CanonDD, \]
\end{prop}
\begin{proof}
The stabilized identity diagram Figure~\ref{fig:IdDiag} is redrawn in
Figure~\ref{fig:IdDiag2}.
\begin{figure}[h]
\centering
\input{IdDiagram2.pstex_t}
\caption{{\bf Extended middle diagram of the identity, again.}}
\label{fig:IdDiag2}
\end{figure}
We claim that for the diagram,
\[\delta^1_2(\One,L_i)=L_i\otimes\One \qquad
\delta^1_2(\One,R_i)=R_i\otimes \One,\qquad
\delta^1_2(\One,U_i)=U_i\otimes \One.\]
The holomorphic disks counting the first two kinds of actions
are polygons (in fact, in cases where $1<i<2n$, they are rectangles).
For example, the rectangles counting the operations
\[
\delta^1_2(\Idemp{\{2,3,4\}},L_2)=L_2\otimes \Idemp{\{1,3,4\}} \qquad
\delta^1_2(\Idemp{\{1,3,4\}},R_2)=R_2\otimes \Idemp{\{2,3,4\}}
\]
are shown in the top line of Figure~\ref{fig:IdDomains}.
The holomorphic disk representing the action
$\delta^1_2(\One,U_i)=U_i\otimes \One$ is a (twice punctured) annulus,
which is the region bounded by $\beta_{i-1}$ and $\beta_{i}$ (labelled
so that $\beta_0$ and $\beta_{2n}$ encircle the two ``middle''
boundary components $\Zmid_0$ and $\Zmid_1$); but the precise
combinatorics are dictated by the idempotent type of the input. When
only one of $i-1$ or $i$ is present in the in-coming generator, there
is a cut from the generator to $\Zin_i$, and another cut along
$\alphaout_{i-1}\cup\alphaout_{i}$, which might or might not cut through
$\Zout_i$. In both cases, this annulus always has an odd number of
pseudo-holomorphic representatives that can be realized as branched
double-covers of the disk, and the output algebra element is always
$U_i$.
When both $i-1$ and $i$ are present in the in-coming generator, there
are two cuts going in towards $\Zin_i$; one goes exactly until
$\Zin_i$ and the other not quite. The total number of such ends is odd.
To see this, we consider the one-dimensional moduli space
with the given homotopy class, and which contains the Reeb orbit.
This has one end, which is a boundary degeneration. The other ends of this moduli space occur when a cut goes exactly out till $\Zin_i$; thus there is an odd number of such ends.
It follows that the generators of $\lsup{\cBlg}\DA(\Hid)_{\cBlg}\DT
\lsup{\cBlg,\nDuAlg}\CanonDD$ correspond to complementary idempotents,
and the terms $L_i\otimes R_i$, $R_i\otimes L_i$, and $U_i\otimes E_i$
appear with non-zero coefficient in the differential.
The $DD$ bimodule inherits a $\Delta$-grading, defined by by
\begin{equation}
\label{eq:DeltaGradenAlg}
\Delta ((a\otimes b)\otimes \mathbf x)=\# \text{($E$ in $b$)}-\weight(a)-\weight(b).
\end{equation}
By the Alexander grading,
the coefficients $(a\otimes b)\mathbf x$ in $\delta^1(\mathbf y)$
satisfy the relation that $\weight(a)=\weight(b)$.
Thus, $L_i\otimes R_i$, $R_i\otimes L_i$, and $U_i\otimes E_i$ are all the elements
with grading $-1$, and all other algebra elements have
$\leq -2$, so they cannot appear in the differential.
It follows that
\[ \lsup{\cBlg}\DA(\Hid)_{\cBlg}\DT
\lsup{\cBlg,\nDuAlg}\CanonDD=\lsup{\cBlg,\nDuAlg}\CanonDD.\]
\end{proof}
\begin{figure}[h]
\centering
\input{IdDomains.pstex_t}
\caption{{\bf Some holomorphic disks.}
Redrawing the picture from Figure~\ref{fig:IdDiag}.}
\label{fig:IdDomains}
\end{figure}
\subsection{A local minimum}
\label{subsec:LocalMin}
Take the standard middle diagram for the local minimum occuring between
strands $c$ and $c+1$, and consider its stabilized middle diagram $\Hmin{c}$,
as pictured in Figure~\ref{fig:MinHDiag}. Equip the strands with a
matching $\Matching$ with $\{c,c+1\}\not\in\Matching$ (i.e., so that
$\Matching$ is compatible with $\Hmid$.)
\begin{figure}[h]
\centering
\input{MinimumDoms.pstex_t}
\caption{{\bf Heegaard diagram for a minimum, stabilized.} Compare Figure~\ref{fig:MinimumHeeg}. We have labelled two
domains.}
\label{fig:MinHDiag}
\end{figure}
We now prove the following:
\begin{prop}
\label{prop:MinimumComputation}
For each $n$, there is a Heegaard diagram for
which Equation~\eqref{eq:ComputeMinDD} holds.
\end{prop}
\begin{proof}
It is clear that the Heegaard states are in one-to-one
correspondence with allowed idempotents for $\mathbf y$; the
correspondence is simply given by $\mathbf x\mapsto \alphain(\mathbf x)$.
We verify
actions that connect different terms of the following form:
\[ \begin{tikzpicture}[scale=1.5
\node at (-1.5,0) (X) {$\XX$} ;
\node at (1.5,0) (Y) {$\YY$} ;
\node at (0,-2) (Z) {$\ZZ$} ;
\draw[->] (X) [bend right=7] to node[below,sloped] {\tiny{$1\otimes (R_{c}, R_{c+1})$}} (Y) ;
\draw[->] (Y) [bend right=7] to node[above,sloped] {\tiny{$1\otimes (L_{c+1}, L_{c})$}} (X) ;
\draw[->] (Y) [bend right=7] to node[above,sloped] {\tiny{$L_{c-1}\otimes L_{c-1}$}} (Z) ;
\draw[->] (Z) [bend right=7] to node[below,sloped] {\tiny{$R_{c-1}\otimes R_{c-1}$}} (Y) ;
\draw[->] (Z) [bend right=7] to node[above,sloped] {\tiny{$R_c\otimes R_{c+2}$}} (X) ;
\draw[->] (X) [bend right=7] to node[below,sloped] {\tiny{$L_c\otimes L_{c+2}$}} (Z) ;
\draw[->] (X) [loop above] to node[above,sloped]{\tiny{$1\otimes (R_{c},U_{c+1},L_{c})+ U_t\otimes U_c$}} (X);
\draw[->] (Y) [loop above] to node[above,sloped]{\tiny{$1\otimes (L_{c+1},U_{c},R_{c+1}) + q \cdot U_s\otimes U_{c+1}$}} (Y);
\draw[->] (Z) [loop below] to node[below,sloped]{\tiny{$1\otimes (R_{c},U_{c+1},L_{c})+U_t\otimes U_c$}} (Z);
\end{tikzpicture}
\]
(Note, though, that there are other actions as well.)
Here, $q=0$ or $1$ (though we shall see afterwards that it must be $1$).
We also prove that $\delta^1_2(\XX,U_{c+1})=0=\delta^1_2(\YY,U_c)=\delta^1_2(\ZZ,U_{c+1})$.
To set up the computation, we introduce some notation.
Let $\{\phi_c(s),c\}\in
\Matching_1$ and $\{c+1,\phi_c(t)\}\in \Matching_2$, and orient the
strand through the minimum as indicated in Figure~\ref{fig:OrientMin};
i.e. $c$ corresponds to an even orbit, $c+1$ to an odd orbit,
and
$\tau(c)=\tau(c+1)=t$. (See Figure~\ref{fig:OrientMin}.)
The six arrows that connect different generator types are represented by
polygons. For example, the verification $\XX$ appears with non-zero
multiplicity in $\delta^1_3(\YY,L_{c+1},L_c)$ can be seen by considering
the quadrialteral (denoted ${\mathcal D}_1$ in
Figure~\ref{fig:MinHDiag}), and noting that it represents a unique
rigid holomorphic flow-line. Traversing the $\alpha$-curves,
$L_{c+1}$ occurs before $L_c$.
\begin{figure}[h]
\centering
\input{OrientMin.pstex_t}
\caption{{\bf Orienting a local minimum.}}
\label{fig:OrientMin}
\end{figure}
Next we consider other terms from $\XX$ to $\XX$. Consider
holomorphic curves with shadow the annulus, ${\mathcal D}_1 +
{\mathcal D}_2$. If we cut exactly to $\Zin_c$, there is a
holomorphic disk containing $e_{c+1}$ in its interior; if we cut
further, exactly to $\Zin_{c+1}$, the holomorphic disk is
interpreted as a $\delta^1_4$. Thus, this homotopy class gives two
terms
\[ \begin{CD}
\XX@>{U_{t}\otimes U_{c}}>{{\mathcal D}_1+{\mathcal D}_2}> \XX
\end{CD}\qquad{\text{and}}\qquad \begin{CD}
\XX@>{1\otimes (R_{c},U_{c+1},L_{c})}>{{\mathcal D}_1+{\mathcal D}_2}> \XX.
\end{CD}
\]
Observe that there are no other holomorphic representatives for this
homology class: the order of the algebra elements $R_c$, $U_{c+1}$, and
$L_c$ is pre-determined by the order they appear on the
$\alpha$-arc.
Moreover, $\delta^1_2(\XX,U_{c+1})=0$.
Consider next $\YY$. In this case, if we cut exactly to
$\Zin_{c+1}$, there is a holomorphic disk containing $e_c$ in its
interior. But this homotopy class alone does not contribute to the
differential, since $e_c$ is an ``even'' Reeb orbit.
For this to count in an action for the DA bimodule, there must also be some $v_{s}$ occuring
at the same time. When $s$ is distinct from $c-1$ and $c+2$,
it is easy to find a corresponding action
\[ \begin{CD}
\YY@>{U_{s}\otimes U_{c+1}}>{{\mathcal D}_1+{\mathcal D}_2 + {\mathcal D}_s}> \YY
\end{CD}, \]
for a choice of ${\mathcal D}_s$ which is an annular domain containing $\Zin_{\phi_c(s)}$ and $\Zout_s$ in its interior.
Cutting further, we obtain the action
\[
\begin{CD}
\YY@>{1\otimes (L_{c+1}, U_c, R_{c+1})}>{{\mathcal D}_1+{\mathcal D}_2}> \YY,
\end{CD}
\]
(which is obvious for any choice of $s$).
Moreover, $\delta^1_2(\YY,U_{c})=0$.
Tensor with the identity type $DD$ bimodule, we obtain terms
\[\begin{CD}
\XX @>{U_{t}\otimes E_{c} + 1\otimes (U_c E_{c+1})}>> \XX \\
\YY @>{q \cdot (U_{s}\otimes E_{c}) + 1\otimes (E_{c} U_{c+1})}>>\YY.
\end{CD}\]
So far, we have verified that $q=1$ only under hypotheses on $s$; but
$q=1$ is forced from algebraic considerations, and the existing other terms, as follows.
We refer to the structural equation for a $DD$ bimodule simply as $\delta\circ\delta=0$:
this includes both terms that multiply terms in $\delta^1$ with other such terms,
and terms that differentiate terms in $\delta^1$.
The term in $\delta\circ \delta$ arising from anti-commuting the terms
$\YY\mapsto (U_{s}\otimes E_{\phi_c(s)})\otimes \YY$ and the above
$\YY\mapsto((1\otimes (E_{c} U_{c+1}))\otimes\YY$ gives a term
$\YY\to (U_{s}\otimes U_{c+1})\otimes \YY$. Note that $(1\otimes U_{c+1})\otimes \YY\neq 0$,
so we need a term in $\delta\circ\delta$ to cancel this term $(U_{s}\otimes U_{c+1})\otimes \YY$. In fact,
the only
possible term that can cancel $U_{s}\otimes U_{c+1}$ is the differential of $U_{s}\otimes E_{c+1}$.
Starting at $\ZZ$, the space of almost-complex structures has a
chamber structure: pseudo-holomorphic flows correspond to annuli
obtained by cutting the annulus $A=\cald_1\cup\cald_2$ in along
$\alphain_{c-1}$ (from the left) and $\alphain_{c+1}$ (from the
right). To determine which $\Ainfty$ operations these induce
involves understanding whether the cut from the left reaches the
boundary punctures before the cuts on the right.
\begin{figure}[h]
\centering
\input{MinimumChamber.pstex_t}
\caption{{\bf Stretch along the dotted curve.}}
\label{fig:MinimumChamber}
\end{figure}
Choose a complex structure where the length of the curve
$(\alphain_{c-1}\cup\alphain_{c})\cap (\cald_1\cup\cald_2)$ is much
shorter than $(\alphain_{c+1}\cup\alphain_{c})\cap
(\cald_1\cup\cald_2)$. (This choice
is equivalent to stretching sufficiently much normal to the dotted
curve in Figure~\ref{fig:MinimumChamber}.)
Thus, for the pseudo-holomorphic flowlines,
the cuts from the left reach all the punctures before the cuts from
the right. In this
case, the annulus supports exactly two non-trivial operations
\[\begin{CD}
\ZZ@>{U_{t}\otimes U_{c}}>{{\mathcal D}_1+{\mathcal D}_2}> \ZZ
\end{CD}\qquad{\text{and}}\qquad \begin{CD}
\ZZ@>{1\otimes (R_{c},U_{c+1},L_{c})}>{{\mathcal D}_1+{\mathcal D}_2}> \ZZ
\end{CD}
\]
(which looks like the operations from $\XX$ to itself.)
Thus,
as in the case of generators of type $\XX$, we obtain terms
in the $DD$ bimodule of the form.
\[ \begin{CD}
\ZZ @>{U_{t}\otimes E_{c} + 1\otimes (U_c E_{c+1})}>> \ZZ
\end{CD}
\]
Having verified some of the terms in the type $DD$ bimodule
$\lsup{\Blg_2}\DAmod(\Hmin{c})_{\Blg_1}\DT \lsup{\Blg_1,\nDuAlg_1}\CanonDD$,
algebra also forces some additional terms
\[\begin{CD}
\XX @>{1\otimes L_{c} L_{c+1}E_{c} E_{c+1}}>> \YY \\
\YY @>{1\otimes R_{c+1}R_c E_{c} E_{c+1}}>> \XX.
\end{CD}\] (The first are needed cancel terms in $\delta\circ
\delta$ arising from composing the terms from $\XX$ to $\YY$
labelled by $1\otimes L_{c} L_{c+1}$, with terms from $\XX$ to $\XX$
labelled $1\otimes U_c E_{c+1}$ or (on the other side) terms from
$\YY$ to $\YY$ labelled $1\otimes E_{c} U_{c+1}$. The second terms
follow similarly.)
Maslov index considerations allow the following
additional types of terms in the differential: $L_i\otimes R_i E_c
E_{c+1}$, $R_i\otimes L_i E_c E_{c+1}$, and $U_i\otimes E_i E_c
E_{c+1}$. But these would contribute terms to $\delta\circ\delta$
that cannot be cancelled. Consider the case where the generator is
of type $\XX$. Then, $d(U_i\otimes E_i E_c E_{c+1})$ contains the
non-zero term $U_i \otimes E_i E_c U_{c+1}$, which can be factored
as $(U_i\otimes E_i)\cdot (1\otimes E_{c} U_{c+1})$, but $1\otimes
E_c U_{c+1}$ does not connect two generators of type $\XX$. The same
argument for type $\YY$ idempotents, using the other non-zero term
$U_i\otimes E_i U_c E_{c+1}$, excludes the existence of terms of the
form $U_i\otimes E_i E_c U_{c+1}$. Similar considerations apply to
the other types of terms listed above.
A $DD$ isomomorphism from
\[\lsup{\Blg_2}\DAmod(\Hmin{c})_{\Blg_1}\DT \lsup{\Blg_1,\nDuAlg_1}\CanonDD \to \lsup{\cBlg_2,\nDuAlg_1}\Min \]
is now defined by
\[ h^1(\XX)=(1+E_c E_{c+1})\otimes \XX, \qquad h^1(\YY)=\YY,\qquad h^1(\ZZ)= (1+E_c E_{c+1})\otimes \ZZ.\]
\end{proof}
\subsection{A positive crossing}
\label{subsec:Pos}
Consider a positive crossing between the $i^{th}$ and $(i+1)^{st}$ strands.
A Heegaard diagram for this crossing (stabilized) is
shown in Figure~\ref{fig:PosCrossDiag}. Our aim here is to verify
Theorem~\ref{thm:ComputeDDmods} for this diagram.
There are two
cases, according to whether or not $i$ and $i+1$ are matched. We
consider these two cases separately.
\begin{figure}[h]
\centering
\input{PosCrossDiag.pstex_t}
\caption{{\bf Positive crossing diagram.}
Stabilize Figure~\ref{fig:CrossDiag}.}
\label{fig:PosCrossDiag}
\end{figure}
\begin{figure}
\begin{tikzpicture}[scale=1.85]
\node at (0,4) (N) {${\mathbf N}$} ;
\node at (-2,2.5) (W) {${\mathbf W}$} ;
\node at (2,2.5) (E) {${\mathbf E}$} ;
\node at (0,-1) (S) {${\mathbf S}$} ;
\draw[->] (S) [bend left=10] to node[below,sloped] {\tiny{$R_{i} U_{i+1} \otimes E_{i} E_{i+1}+L_{i+1}\otimes R_{i+1}R_{i}
+ R_i U_\beta\otimes 1$}} (W) ;
\draw[->] (W) [bend left=10] to node[above,sloped] {\tiny{$L_{i}\otimes 1$}} (S) ;
\draw[->] (E) [bend right=10] to node[above,sloped] {\tiny{$R_{i+1}\otimes 1$}} (S) ;
\draw[->] (S)[bend right=10] to node[below,sloped] {\tiny{$R_{i} \otimes L_{i} L_{i+1} + L_{i+1} U_{i}\otimes E_{i+1}
E_{i}+L_{i+1}U_\alpha\otimes 1$}} (E) ;
\draw[->] (W)[bend right=10]to node[below,sloped] {\tiny{$1\otimes L_{i}$}} (N) ;
\draw[->] (N)[bend right=10] to node[above,sloped] {\tiny{$U_{i+1}\otimes R_{i} + R_{i+1} R_{i} \otimes L_{i+1}$
}} (W) ;
\draw[->] (E)[bend left=10]to node[below,sloped]{\tiny{$1\otimes R_{i+1}$}} (N) ;
\draw[->] (N)[bend left=10] to node[above,sloped]{\tiny{$U_{i}\otimes L_{i+1} + L_{i} L_{i+1}\otimes R_{i}$}}
(E) ;
\draw[->] (N) [loop above] to node[above]{\tiny{$U_{i}\otimes E_{i+1} + U_{i+1}\otimes E_{i}$}} (N);
\draw[->] (W) [loop left] to node[above,sloped]{\tiny{$U_{i+1}\otimes E_{i}$}} (W);
\draw[->] (E) [loop right] to node[above,sloped]{\tiny{$U_{i}\otimes E_{i+1}$}} (E);
\draw[->] (E) [bend right=5] to node[above,pos=.3] {\tiny{$R_{i+1} R_{i} \otimes E_{i+1}$}} (W) ;
\draw[->] (W) [bend right=5] to node[below,pos=.3] {\tiny{$L_{i} L_{i+1}\otimes E_{i}$}} (E) ;
\draw[->] (S) to node[below,sloped,pos=.3] {\tiny{$L_{i+1}\otimes E_{i} R_{i+1} + R_{i}\otimes L_{i} E_{i+1}$}} (N) ;
\end{tikzpicture}
\caption{\label{fig:PosDD} {\bf $DD$ bimodule of a positive crossing.}}
\end{figure}
\begin{lemma}
\label{lem:PosDDchar}
Suppose that $i$ and $i+1$ are not matched; indeed, $\{\alpha,i\},
\{i+1,\beta\}\in M_1$.
There is a unique type $DD$ bimodule over $\lsup{\Blg_2,\nDuAlg_1}M$
with the following properties:
\begin{itemize}
\item The generators of $M$ over the idempotent ring
are of the four types $\North$, $\South$, $\East$, and $\West$
as in the bimodule $\lsup{\cBlg_2,\DuAlg}\Pos_i$
described in Section~\ref{subsec:AlgPosCross}
\item The bimodule $M$ has a $\Delta$ grading is as specified in Equation~\eqref{eq:DeltaGradingPos}
\item The bimodule $M$ has an Alexander grading is as specified in
Equation~\eqref{eq:AlgGradeCrossing}.
\item
The $\North$ coefficient of $\delta^1_1(\North)$ is $U_i\otimes
E_{i+1}+U_{i+1}\otimes E_i + \sum_{j\not\in\{i,i+1\}} U_j\otimes
E_j$.
\item
The $\West$ coefficient of $\delta^1_1(\West)$ is
$U_{i+1}\otimes E_i+ \sum_{j\not\in\{i,i+1\}} U_j\otimes E_j$.
\item
The $\East$ coefficient of $\delta^1_1(\East)$ is
$U_{i}\otimes E_{i+1}+ \sum_{j\not\in\{i,i+1\}} U_j\otimes E_j$.
\item
The $\South$ coefficient of $\delta^1_1(\South)$ is
$\sum_{j\not\in\{i,i+1\}} U_j\otimes E_j$.
\item
The $\East$ coeficient of $\delta^1(\West)$ is $R_{i+1}R_i\otimes E_{i+1}$.
\item
The $\West$ coeficient of $\delta^1(\West)$ is $L_i L_{i+1}\otimes E_i$.
\end{itemize}
Moreover, there is an isomorphism
$\lsup{\cBlg_2,\nDuAlg_1}M\cong~\lsup{\cBlg_2,\nDuAlg_1}\Pos_i$,
where the right-hand-side bimodule is the one described in
Section~\ref{subsec:AlgPosCross}.
\end{lemma}
\begin{proof}
Consider the $\North$ coefficient of $\delta^1\circ \delta^1(\North)$
There is a term of $U_{i+1}\otimes U_i$ coming from
differentiating the self-arrows.
Gradings now ensure the term can arise
only from a term of $(1\otimes L_i)\otimes \North$ in $\delta^1(\West)$
and $(U_{i+1}\otimes R_i)\otimes \West$ in $\delta^1(\North)$.
The coefficient of $R_i \otimes L_i E_{i+1}$ on the arrow from $\South$ to
$\North$ is forced from the $\North$
component of $\delta^1\circ \delta^1(\West)$.
The term of $R_i U_{i+1}\otimes E_{i+1} E_i$ on the arrow from
$\South$ to $\West$ is forced from the $\West$ coefficient of
$\delta^1 \circ \delta^1(\West)$. (Note that degree considerations
alone allow also for terms $R_i U_{i+1}\otimes E_{i} E_{i+1}$ and
$R_i U_{i+1}\otimes 1$; but the $DD$ structure relations and our
hypotheses about the existing coefficients eliminates these
possibilities.)
Symmetric arguments give $L_{i+1}\otimes R_{i+1} E_i$ on the arrow
from $\South$ to north, $L_{i+1} U_i\otimes E_i E_{i+1}$ on the
arrow from $\South$ to $\East$.
Now, considering the $\South$ coefficient fo $\delta^1\circ \delta^1(\North)$
ensures the terms $R_{i+1} R_i \otimes L_{i+1}$ on the arrow from
$\North$ to $\West$ and $L_i L_{i+1}\otimes R_i$ on the arrow from $\North$
to $\East$.
The curvature term in $\delta^1\circ \delta^1(\West)$ (of $U_i
U_\beta\otimes 1$) ensures the arrow from $\South$ to $\East$
labelled by $R_i U_\beta$.
This constructs all the actions in $\lsup{\cBlg_2,\nDuAlg}M$.
The isomorphism is provided by the map
\[ h^1\colon M \to \Pos_i \]
defined by
\[ h^1(X) = \left\{\begin{array}{ll}
\South+ (L_2\otimes E_1)\cdot \East + (R_1\otimes E_2)\cdot
\West & {\text{if $X=\South$}} \\
X &{\text{otherwise.}}
\end{array}
\right.\]
\end{proof}
\begin{prop}
\label{prop:PosDDUnmatched}
If $i$ and $i+1$ are unmatched, then Equation~\eqref{eq:ComputePosDD}
holds.
\end{prop}
\begin{proof}
It is straightforward to see that Lemma~\ref{lem:PosDDchar}
computes $\lsup{\nDuAlg_1}\Pos^i_{\nDuAlg_2} \DT~
\lsup{\nDuAlg_2,\cBlg_2}\CanonDD$.
It remains to verify that there is a complex structure for which
$\lsup{\cBlg_2}\DAmodExt(\Hpos{i})_{\cBlg_1} \DT~ \lsup{\cBlg_1,\nDuAlg_1}\CanonDD$
is also computed by Lemma~\ref{lem:PosDDchar}.
We consider the actions on
$\lsup{\cBlg_2}\DAmodExt(\Hpos{i})_{\cBlg_1}$ which could pair to give the actions
required by the lemma.
\begin{figure}[h]
\centering
\input{PosCrossDoms.pstex_t}
\caption{{\bf Labeling domains in the positive crossing diagram.} }
\label{fig:PosCrossDoms}
\end{figure}
Consider labels shown in Figure~\ref{fig:PosCrossDoms}. We have polygons that exhibit
actions
For $X\in\{\North,\West\}$
\[\begin{CD}
X@>{U_{i+1}\otimes U_i}>{\cald_2+\cald_4+\cald_6}> X
\end{CD}.\]
For $X'\in\{\South,\East\}$, there is no action
\[\begin{CD}
Y@>{U_{i+1}\otimes U_i}>{\cald_2+\cald_4+\cald_6}> Y
\end{CD},\]
as can be seen from the geometry of that bigon.
The following action is given by a polygon, and hence
it exists for all choices of almost-complex structure:
\[ \begin{CD}
\West@>{L_i L_{i+1}\otimes U_i}>{\cald_1+\cald_2+\cald_4}>\East
\end{CD}\]
Consider possible actions
\begin{equation}
\label{eq:ExcludeThis}
\begin{CD}
Y'@>{U_{i}\otimes U_{i+1}}>{\cald_1+\cald_3+\cald_5}> Y'.
\end{CD}
\end{equation}
for $Y'\in\{\South,\West\}$.
We can choose a complex structure
so that neither exists, but
\begin{equation}
\label{eq:IncludeThis}
\begin{CD}
\East@>{R_{i+1} R_i\otimes E_{i+1}}>{\cald_1+\cald_3+\cald_5}> \West
\end{CD}
\end{equation}
exists. This can be seen as follows.
\begin{figure}[h]
\centering
\input{StretchPosDoms.pstex_t}
\caption{{\bf Stretch normal to the black curve.} }
\label{fig:StretchPosDoms}
\end{figure}
First, we stretch normal to the black curve from
Figure~\ref{fig:StretchPosDoms}. In the limit, this curve degenerates
to the point $p$; this is indicated in Figure~\ref{fig:DegenPosDoms}
(once we identify the two curves labelled $p$ to points).
\begin{figure}[h]
\centering
\input{DegenPosDoms.pstex_t}
\caption{{\bf Degenerating the black curve from
Figure~\ref{fig:StretchPosDoms}.} }
\label{fig:DegenPosDoms}
\end{figure}
The domain $\cald_3+\cald_5$ can be thought of as the shadow of a
pseudo-holomorphic flow from $\South$ to $\West$ with Maslov index
$0$. As the domain is degenerated, the domain decomposes into two
regions as shown in Figure~\ref{fig:DegenPosDoms}. One is
$\cald_3'+\cald_5'$, which is now an index one flow from $\South$ to
$\West$ on the left; and the remaining region which we denote here
${\mathcal D}''$, thought of as connecting a generator to itself, and
following the Reeb chord covering $\Zin_{i+1}$ once. Both flows
contain Reeb orbits at $p$ in their interior.
Let $s_1=s\circ \ev_p(\ModFlow^{\cald_3'+\cald_5'}(\South,\West))$ and
$s_2=s\circ\ev_p(\ModFlow^{\cald''}(x,x;\rho))$,
where $\rho$ is the length one Reeb orbit that covers $\Zin_{i+1}$ once.
We will choose our complex structure so that $s_1<s_2$.
Consider the maps
\begin{equation}
\label{eq:SegmentOne}
s\circ \ev_p\colon \ModFlow^{\cald_1+\cald_3'+\cald_5'}(Y',Y')\to
[0,1] \qquad{\text{and}}\qquad s\circ \ev_p\colon
\ModFlow^{\cald_3'+\cald_5'+\cald_6}(\East,\West)\to [0,1]
\end{equation} We
claim that these two moduli spaces map degree one to $[0,s_1]$ and
$[s_1,1]$ respectively. For example, consider the first of these maps
$Y'=\South$. The moduli space
$\ModFlow^{\cald_1+\cald_3'+\cald_5'}(\South,\South)$ has two ends:
one is a $\beta$-boundary degeneration, where $s\circ \ev_p$
converges to $0$; and the other is a broken flowline, juxtaposing the
flow with shadow $\cald_1$ from $\South$ to $\West$, followed by the
flowline with shadow $\cald_3'+\cald_5'$ from $\West$ to $\South$.
Here, $s\circ \ev_p$ converges to $s_1$.
When $Y'=\West$, the order of the two parts of the broken flowline are reversed.
It follows that the first map from Equation~\eqref{eq:SegmentOne}
has degeree one onto $[0,s_1]$. The analysis of the second map
from Equation~\eqref{eq:SegmentOne} follows similarly (except in that
case, rather than a boundary degeneration, the
flow degenerates to an index one flowline containing
a Reeb chord along its $\alpha$-boundary).
By the usual stretching arguments, the degree of the first map at
$s_2$ counts the number of representatives of
Equation~\eqref{eq:ExcludeThis} (for either choice of $Y'$), while
the degree of the second map counts the number of representatives of
Equation~\eqref{eq:IncludeThis}. Thus, for our choice of $s_1<s_2$,
the actions from Equation~\eqref{eq:ExcludeThis} are excluded
and the one from Equation~\eqref{eq:IncludeThis} is included.
We can now apply Lemma~\ref{lem:PosDDchar} to
$\lsup{\cBlg_2}\DAmodExt(\Hpos{i})_{\cBlg_1} \DT~
\lsup{\cBlg_1,\nDuAlg_1}\CanonDD$
\end{proof}
\begin{prop}
\label{prop:PosDDMatched}
Equation~\eqref{eq:ComputePosDD} holds when $i$ and $i+1$ are matched.
\end{prop}
\begin{proof}
First note that
if $i$ and $i+1$ are matched, then the $DD$ bimodule of a positive crossing exhibited in
Figure~\ref{fig:PosDD}, with $\alpha=i+1$ and $\beta=i$, is uniquely characterized by
the properties listed in the statement of Lemma~\ref{lem:PosDDchar}. The proof follows exactly
as in the proof of Lemma~\ref{lem:PosDDchar}. In the present application, note that
we expect a curvature term of $U_i U_{i+1}\otimes 1$ in $\delta\circ \delta$.
With this said, the proof of Proposition~\ref{prop:PosDDUnmatched} applies.
\end{proof}
\subsection{A negative crossing}
\label{subsec:Neg}
\begin{prop}
\label{prop:ComputeNegDD}
For the extended middle diagram for a negative crossing,
Equation~\eqref{eq:ComputeNegDD} holds.
\end{prop}
\begin{proof}
Observe that there is a symmetry (which is reflection through a
vertical axis in Figure~\ref{fig:PosCrossDiag}) taking the positive
crossing diagram to the negative crossing diagram (shown in
Figure~\ref{fig:NegCrossDiag}) This switches $\West$ and $\East$, $i$
and $2n-i$, and reverses the orientation of the Heegaard diagram,
\begin{figure}[h]
\centering
\input{NegCrossDiag.pstex_t}
\caption{{\bf The (extended) negative crossing diagram.} }
\label{fig:NegCrossDiag}
\end{figure}
Thus, the differential in the type $DD$ bimodule associated to
$\Hneg{i}$ is computed by taking the differential the type $DD$
bimodule $\Hpos{2n-i}$, reversing the arrows, switching $\West$ and
$\East$, and switching strand $j$ with strand $2n-j$. This
operation also transforms $\lsup{\cBlg_2,\nDuAlg_1}\Pos_i$
to the $DD$ bimodule for $\Neg_{2n-i}$.
With these remarks, the proposition follows immediately from
Proposition~\ref{prop:PosDDUnmatched} or~\ref{prop:PosDDMatched}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:ComputeDDmods}]
The proof follows immediately
from Propositions~\ref{prop:MinimumComputation}, \ref{prop:PosDDUnmatched}, \ref{prop:PosDDMatched}.
and~\ref{prop:ComputeNegDD}.
\end{proof}
\section{Computing the type $D$ structures of an upper diagram}
\label{sec:computeD}
Suppose that $\DiagUp$ is an upper knot diagram (thought of as a
diagram in an upper half space). After Reidemeister moves and
isotopies, we can arrange for all the local maxima of $\DiagUp$ to be
global maxima, and for each of the local minima and crossings to occur
at different heights. We can construct an upper Heegaard diagram for
each upper slice of an {\em acceptable} diagram, as follows. We start from
the standard upper diagram corresponding to all the local maxima, and
then glue on the standard middle diagrams for the remaining crossings
and local mimuma. In this manner we associate the {\em canonical upper
diagram} $\Hup(\DiagUp)$ associated to the acceptable diagram
$\DiagUp$.
Given an acceptable upper knot diagram $\DiagUp$ with $n$ strands at
the bottom and matching $\Matching$, there is an associated {\em
algebraically defined type $D$ structure}
$\lsup{\cBlg}\DmodAlg(\DiagUp)$, where $\cBlg=\cBlg(n,\Matching)$,
obtained by starting with $\lsup{\cClg(n,\Matching)}k$ for the global
maxima, and then tensoring with the bimodules
$\lsup{\cClg_2}\Pos^i_{\cClg_1}$, $\lsup{\cClg_2}\Neg^i_{\cClg_1}$,
$\lsup{\cClg_2}\Min^c_{\cClg_1}$ as dictated by the local slices of
$\DiagUp$. (This is very closely related to the algebraic description
appearing in~\cite{Bordered2}; see the proof of
Theorem~\ref{thm:MainTheorem} below.)
\begin{thm}
\label{thm:ComputeD}
Let $\DiagUp$ be any acceptable knot diagram $\DiagUp$, and
let $\Hup$ be its associated upper Heegaard diagram.
Then there is an identification
\begin{equation}
\label{eq:ComputeD}
\lsup{\cClg(n,\Matching)}\Dmod(\Hup)\simeq~\lsup{\cClg(n,\Matching)}\DmodAlg(\DiagUp).
\end{equation}
\end{thm}
We will prove the above theorem at the end of this section. The
hypothesis that $\DiagUp$ is acceptable is removed in
Section~\ref{sec:Further} (see especially
Theorem~\ref{thm:ComputeD2}); though it is not logically necessary for
the computation of knot Floer homology (Theorem~\ref{thm:MainTheorem},
proved in Section~\ref{sec:Comparison}).
\begin{defn}
\label{def:Relevant}
For $\Clg=\Clg(n,\Matching)$,
a type $D$ structure $\lsup{\cClg}X$ is called {\em relevant} if
there is an $\Ainfty$ module $Z_{\nDuAlg}$ over $\nDuAlg=\nDuAlg(n,\Matching)$
with the property that
\[
\lsup{\cBlg}i_{\cClg}\DT~ \lsup{\cClg}X=Z_{\nDuAlg}\DT~\lsup{\nDuAlg,\cBlg}\CanonDD.\]
\end{defn}
Consider the type $D$ structure $\lsup{\cClg}k$ from Section~\ref{subsec:ExampleTypeDs} (which is identified
in Lemma~\ref{lem:StandardTypeD} with the
type $D$ structure of the standard upper diagram).
We shall see that $\lsup{\cClg}k$ is relevant, in the sense of
Definition~\ref{def:Relevant}. Since $\lsup{\cClg}k$ is a one-dimensional vector space, so is $Z_{\nDuAlg}$. The
$\Ainfty$-actions on this module are fairly
complicated. Letting if $\mathbf x$ be the generator,
we have that
\[ \mathbf x\cdot E_{2i-1} \cdot E_{2i} + \mathbf x \cdot E_{2i}\cdot E_{2i-1} =\mathbf x;\]
i.e. exactly one of the two following equations holds:
\[ \mathbf x\cdot E_{2i-1}\cdot E_{2i}=\mathbf x\qquad {\text{or}}\qquad
\mathbf x\cdot E_{2i}\cdot E_{2i-1}=\mathbf x.\]
Suppose that the first equation holds. Then, applying the $\Ainfty$
relation with $\mathbf x$, $E_{2i-1}$ and $E_{2i}$, we can conclude that
exactly one of
\[
m_3(\mathbf x,E_{2i-1}, U_{2i})=\mathbf x\qquad
{\text{or}}\qquad
m_3(\mathbf x,U_{2i-1}, E_{2i})=\mathbf x\]
holds. The $\Ainfty$ relations then imply many further actions. We
organize this construction in the following:
\begin{lemma}
\label{lem:kIsRelevant}
The type $D$ structure $\lsup{\cClg}k$ is relevant.
\end{lemma}
\begin{proof}
Let $\mathbf x_1$ denote the idempotent state for $\nDuAlg$ $\mathbf x_1$ that
consists of all odd numbers between $1$ and $2n-1$, together with
$0$. Let $\BigMod$ be the right $\Blg(2n,n+1)$-module consisting of $2^{n}$
copies of $\Idemp{\mathbf x_1}\cdot (L_1 L_2\backslash \Blg(2n,n+1))$, indexed by
subsets $S$ of $\{1,\dots,n\}$; i.e.
\[ \BigMod = \bigoplus_{S\subset\{1,\dots,n\}} \BigMod_S,\]
and there are preferred identifications of $\BigMod_S\cong \BigMod_T$ for all
$S, T\subset \{1,\dots,n\}$.
We endow $\BigMod$ with a differential, as follows.
Given $i=1,\dots,n$, if $i\not\in S$, let
\[\partial_i \colon \BigMod_S \to \BigMod_{S\cup\{i\}} \]
be multiplication by $U_{2i-1}$
(composed with the preferred identification $\BigMod_S\cong \BigMod_{S\cup\{i\}}$) and
\[\partial_i \colon \BigMod_{S\cup\{i\}} \to \BigMod_{S}; \]
be multiplication by $U_{2i}$.
Note that $\partial = \sum_{i=1}^n \partial_i \colon \BigMod \to \BigMod$
satisfies $\partial\circ\partial =0$. (This follows from the fact
that $\partial^2$ is multiplication by $z=\sum_{i=1}^{n} U_{2i-1}U_{2i}$,
and $\Idemp{\mathbf x_1}\cdot z = \Idemp{\mathbf x_1}\cdot U_1 U_2 = \Idemp{\mathbf x_1}
L_1 L_2 \cdot R_2 R_1$.
We can extend the action on $\BigMod$ by $\Blg(2n,n+1)$ to all of
$\nDuAlg$, as follows.
For fixed $S$ with $i\not\in S$, let
\[m_2(\cdot,E_{2i-1})\colon \BigMod_{S} \to
\BigMod_{S\cup\{i\}}\qquad\text{and}\qquad
m_2(\cdot,E_{2i})\colon \BigMod_{S \cup \{i\}} \to \BigMod_{S}
\]
be the zero map, while
\[m_2(\cdot,E_{2i})\colon \BigMod_{S} \to
\BigMod_{S\cup\{i\}}\qquad\text{and}\qquad
m_2(\cdot,E_{2i-1})\colon \BigMod_{S \cup \{i\}} \to \BigMod_{S}
\]
are the standard identifications. On each $\BigMod_{S}$ multiplication by
exactly one of $E_{2i-1}\cdot E_{2i}$ or $E_{2i}\cdot E_{2i-1}$ is
non-zero, and the non-zero one is the preferred identification. It
follows that the action by $\DuAlg$ descends to an action by
$\nDuAlg$. Moreover, if $\{j,k\}$ are unmatched, then $E_{j}\cdot
E_{k}$ acts the same as $E_{k} E_{j}$. It follows that there is an
induced action of $\nDuAlg$ on $\BigMod$.
We also claim that the homology of $\BigMod$ is one-dimensional, generated
by
$P_{\{\}} \left(\prod_{i=1}^{n-1} R_{2i}\right)$. We see this as
follows.
Let $Q_j\subset \BigMod$ be the vector subspace generated by elements of
the form $b \cdot P_S$ with the property that
\[ b= \prod_{i=1}^{j} R_{2i}\cdot c \] with $w_i(c)=0$ for
$i=1,\dots,2j-1$ and $\{1,\dots,j\}\cap S=\emptyset$. In
particular, $Q_0=\BigMod\supset Q_1\supset\dots\supset Q_{n}$, and $Q_{n}$
is the one-dimensional vector space spanned by $P_{\{\}}\cdot
(\prod_{i=1}^{n-1}R_{2i})$. Observe that $Q_j$ is a subcomplex.
Define
\[ H(a\cdot P_S) =\left\{\begin{array}{ll}
b\cdot (E_{2i-1} \cdot P_S) & \text{if $a=U_{2i-1} \cdot b$
and $w_j(b)<1$ for all $j<2i-1$} \\
b\cdot (E_{2i} \cdot P_S) & \text{if $a=U_{2i} \cdot b$
and $w_j(b)<1$ for all $j<2i$}
\end{array}\right.\]
We claim that $\Id + \partial\circ H + H\circ \partial$ maps $Q_j$
to $Q_{j+1}$. Thus, iterating the chain map $F=\Id + \partial\circ
H + H\circ \partial$ $n$ times, we obtain a chain homotopy contraction of
$\BigMod$ onto the one-dimensional subcomplex spanned by $P_{\{\}}\cdot
(\prod_{i=1}^{n-1} R_{2i})$, as claimed.
The homological perturbation lemma now endows the one-dimensional
vector space $Z_{\nDuAlg}=H(\BigMod,\partial)$, with an $\Ainfty$ action
by $\nDuAlg$.
We claim that $Z_{\nDuAlg}\DT
\lsup{\nDuAlg,\cBlg}\CanonDD$ has trivial differential.
This follows from the $\Delta$-gradings: the algebra output in
$\nDuAlg$ of each element is either $0$ (if it is some $E_i$)
$-1/2$ (if it is some $L_i$ or $R_i$). But such sequences of algebra
elements act trivially on $Z_{\nDuAlg}$, for if
\[ \Delta(x)=m_{\ell+1}(x,a_1,\dots,a_{\ell})=\Delta(x) + \ell-1 +
\Delta(a_1)+\dots+\Delta(a_\ell) \geq \Delta(x)+ \frac{\ell}{2}-1
\] so $\ell=1$. The case where $\ell=1$ is excluded since
$m_2(\mathbf x,E_i)=0$. The case where $\ell=2$ is also excluded by
direct consideration.
Moreover,
\[
\Idemp{\{0,2,\dots,2i,\dots,2n\}}\cdot \left(Z_{\nDuAlg}\DT\lsup{\nDuAlg,\cBlg}\CanonDD\right)
=Z_{\nDuAlg}\DT\lsup{\nDuAlg,\cBlg}\CanonDD. \]
The lemma follows.
\end{proof}
\begin{prop}
\label{prop:InductiveStep}
Let $\cClg_1=\cClg(n,\Matching_1)$
Suppose $\lsup{\Clg_1}X$ is relevant, then
\begin{align}
\lsup{\cClg_2}\Min^c_{\cClg_1} \DT~ \lsup{\cClg_1}X &\simeq
\lsup{\cClg_2}\DAmodExt(\Hmin{c})_{\cClg_1} \DT~ \lsup{\cClg_1}X \label{eq:MinRel}\\
\lsup{\cClg_2}\Pos^i_{\cClg_1} \DT~ \lsup{\cClg_1}X &\simeq
\lsup{\cClg_2}\DAmodExt(\Hpos{i})_{\cClg_1} \DT~ \lsup{\cClg_1}X \label{eq:PosRel}\\
\lsup{\cClg_2}\Neg_{\cClg_1} \DT~ \lsup{\cClg_1}X &\simeq
\lsup{\cClg_2}\DAmodExt(\Hneg{i})_{\cClg_1} \DT~ \lsup{\cClg_1}X; \label{eq:NegRel}
\end{align}
and moreover all the type $D$ structures appearing on the left
are relevant.
\end{prop}
\begin{proof}
Consider Equation~\eqref{eq:MinRel}.
Combining Lemma~\ref{lem:RestrictIdempotents},
the relevance of $\lsup{\Blg_1}X$, and the associativity of tensor product,
we have that
\begin{align*}
\lsup{\cBlg_2}i_{\cClg_2}\DT(\lsup{\cClg_2}\Min^c_{\cClg_1}\DT
\lsup{\cClg_1}X) &\simeq
\lsup{\cBlg_2}\Min^c_{\cBlg_1}\DT\lsup{\cBlg_2}i_{\cClg_2}\DT
\lsup{\cClg_1}X \\
&\simeq
\lsup{\cBlg_2}\Min^c_{\cBlg_1}
\DT
(Z_{\nDuAlg_1}\DT\lsup{\nDuAlg_1,\cBlg_1}\CanonDD) \\
&\simeq
Z_{\nDuAlg_1}
\DT
(\lsup{\cBlg_2}\Min^c_{\cBlg_1}\DT\lsup{\cBlg_1,\nDuAlg_1}\CanonDD).
\end{align*}
By Proposition~\ref{prop:BimodulesOvernDuAlg},
\[
Z_{\nDuAlg_1}\DT
(\lsup{\cBlg_2}\Min^c_{\cBlg_1}\DT\lsup{\nDuAlg_1,\cBlg_1}\CanonDD)
\simeq
(Z_{\nDuAlg_1}\DT
\lsup{\nDuAlg_1}\Min^c_{\nDuAlg_2})\DT\lsup{\nDuAlg_2,\cBlg_2}\CanonDD;
\]
so
$\lsup{\cClg_2}\Min^c_{\cClg_1}\DT\lsup{\cClg_1}X$ is relevant.
Applying Theorem~\ref{thm:ComputeDDmods} and Proposition~\ref{prop:ExtendDAPrecise},
\begin{align*}
Z_{\nDuAlg_1}\DT
(\lsup{\cBlg_2}\Min^c_{\cBlg_1}\DT\lsup{\nDuAlg_1,\cBlg_1}\CanonDD)
&\simeq
Z_{\nDuAlg_1}\DT
(\lsup{\cBlg_2}\DAmodExt(\Hmin{c})_{\cBlg_1}
\DT \lsup{\cBlg_1,\nDuAlg_1}\CanonDD) \\
&\simeq
\lsup{\cBlg_2}\DAmodExt(\Hmin{c})_{\cBlg_1}\DT
(Z_{\nDuAlg_1}
\DT \lsup{\nDuAlg_1,\cBlg_1}\CanonDD) \\
&\simeq \lsup{\cBlg_2}\DAmod(\Hmin{c})_{\cBlg_1}\DT~ \lsup{\cBlg_1}i_{\cClg_1} \DT ~\lsup{\cClg_1}X \\
&\simeq \lsup{\cBlg_1}i_{\cClg_1} \DT \lsup{\cClg_2}\DAmod(\Hmin{c})_{\cClg_1}\DT ~ \lsup{\cClg_1}X.
\end{align*}
It follows that
\begin{equation}
\label{eq:MinRel1}
\lsup{\cBlg_2}i_{\cClg_2}\DT~ \lsup{\cClg_2}\Min^c_{\cClg_1}\DT~ \lsup{\cClg_1}X
\simeq \lsup{\cBlg_2}i_{\cClg_2}\DT~
\lsup{\cClg_2}\DAmodExt(\Hmin{c})_{\cClg_1} \DT~ \lsup{\cClg_1}X.
\end{equation}
Equation~\eqref{eq:MinRel} follows from Equation~\eqref{eq:MinRel1} and
the following observation: if $\lsup{\cClg_2}P$ and $\lsup{\cClg}Q$ are any two type $D$ structures, then
\[ \mathrm{Mor}^{\cClg_2}(\lsup{\cClg_2}P,\lsup{\cClg_2}Q)=
\mathrm{Mor}^{\cBlg_2}(\lsup{\cBlg_2}i_{\cClg_2}\DT\lsup{\cClg_2}P,\lsup{\cBlg_2}i_{\cClg_2}\DT\lsup{\cClg}Q),\]
since
\[ \lsub{\cClg}i^{\cBlg}\DT\lsub{\cBlg}\Blg_{\cBlg}\DT \lsup{\cBlg}i_{\cClg}=
{\iota}\cdot \Blg \cdot {\iota}=\cClg\]
(with $\iota$ as in Equation~\eqref{eq:DefIota});
so $\lsup{\cClg_2}P\simeq \lsup{\cClg_2}Q$ iff
$\lsup{\cBlg_2}i_{\cClg_2}\DT\lsup{\cClg_2}P\simeq
\lsup{\cBlg_2}i_{\cClg_2}\DT\lsup{\cClg_2}Q$.
Equations~\eqref{eq:PosRel} and~\eqref{eq:NegRel} follows similarly.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:ComputeD}]
We prove Theorem~\ref{thm:ComputeD}, together with the statement
that $\lsup{\Clg}\Dmod(\Hup)$ is homotopy equivalent to a relevant
type $D$ structure, by induction on the sum of the number of
crossings and local minima. When this sum is zero, $\Hup$ is the
standard upper diagram, and the theorem follows from
Lemma~\ref{lem:StandardTypeD}. Moreover, this module is relevant by
Lemma~\ref{lem:kIsRelevant}.
The inductive step is now provided by
Proposition~\ref{prop:InductiveStep} and
Theorem~\ref{thm:PairDAwithD}.
\end{proof}
\begin{remark}
With a little extra work, one can show that the type $D$ structure
$\Dmod(\Hup)$ is independent of the choice of upper Heegaard diagram $\Hup$.
\end{remark}
\section{More holomorphic curves}
\label{sec:CurvesA}
We describe here the pseudo-holomorphic curves used in the
constructions of the type $A$ modules for a given lower diagram
$\Hdown$. The moduli spaces we consider here are similar to the ones
considered in Section~\ref{sec:CurvesD}, except that now the limiting
values as $t\goesto \pm \infty$ are lower (rather than upper) Heegaard
states; moreover, the partitions we consider here
$\Partition=(\rhos_1,\dots,\rhos_\ell)$ of the east and interior
punctures need not be simple
(c.f. Definition~\ref{def:DecoratedSource}): each term in the
partition $\rhos_i$ is some (non-empty) subset of chords and orbits.
More formally:
\begin{defn}
\label{def:ConstraintPacket}
a {\em constraint packet} $\rhos$ is a pair consisting
of a set of Reeb orbits, denoted $\orbits(\rhos)$; and a set of Reeb
chords, denoted $\chords(\rhos)$.
\end{defn}
Thus, in our curve counting, for
the partition $(\rhos_1,\dots,\rhos_{\ell})$, each term $\rhos_i$ is a
constraint packet. Sometimes, we find it convenient to generalize
this slightly: a {\em generlized constraint packet}, consisting of
{\em multi-sets} of Reeb orbits and chords (i.e. the same chord or
orbit can occur with positive multiplicity).
The differential and, indeed, all the module actions (of sequences of
algebra elements on the module) will count pseudo-holomorphic curves
with constraint packets specified by algebra elements; and these
constraint packets will have a rather special form
(cf. Definition~\ref{def:CompatiblePacket}). In particular, each
constraint packet in the algebra action definition will contain at
most one Reeb orbit.
We will prove an $\Ainfty$ relation for modules, which will involve
analyzing ends of one-dimensional moduli spaces. Two story building
degenerations correspond to terms in the $\Ainfty$ relation involving
two applications of the module actions. In~\cite{InvPair}, these
moduli spaces have another kind of end, called {\em join curve ends},
which correspond to the differential in the bordered algebra. By
contrast, in the present case, join curve ends of the various moduli
spaces which we consider cancel in pairs. (One could construct a
larger algebra than the one considered here, equipped with a
differential, so that the join curve ends of moduli spaces correspond
to terms in the differential of the algebra. The present algebra can
be thought of as the homology of this larger algebra.) See
Figure~\ref{fig:JoinEnd} for an illustration. As~\cite{InvPair},
there are also {\em collision ends}, which correspond to
multiplication in the algebra. There is another new kind of end,
corresponding to the formation of $\beta$-boundary degenerations. Some
of these cancel against orbit curve ends; the remaining orbit ends are
accounted for by the curvature of our algebra.
Our goal here is to formulate the moduli spaces we consider precisely,
and to describe the ends of one-dimensional moduli spaces. The
algebraic interpretations of the counts of these ends described above
will be given in detail in Section~\ref{sec:TypeA}.
\subsection{Pseudo-holomorphic flows in lower diagrams}
We will need to name Reeb chords for lower diagrams as shown in Figure~\ref{fig:ChordNamesA}; i.e. switched from the conventions from
the type $D$ side; c.f. Figure~\ref{fig:ChordNames}). This will be
useful for the gluing of diagrams. As a point of comparison: the chord
labelled $L_i$ on the $D$ side has initial point on $\alpha_{i-1}$ and
terminal point on $\alpha_{i}$; while the chord labelled $L_i$ on the
$A$ side has initial point on $\alpha_i$ and terminal point on
$\alpha_{i-1}$.
\begin{figure}[h]
\centering
\input{ChordNamesA.pstex_t}
\caption{{\bf Names of Reeb chords for lower diagrams.} The Reeb
chord $L_i$ is indicated by the oriented half circle.}
\label{fig:ChordNamesA}
\end{figure}
Strong boundary monotonicity is a closed condition, in the following sense:
\begin{lemma}
\label{lem:StrongMonotoneClosed}
Suppose that $u$ is a pseudo-holomorphic flowline which appears in
the Gromov limit of a sequence of strongly boundary boundary
monotone curves. Then $u$, is strongly boundary monotone, as well.
\end{lemma}
\begin{proof}
This follows exactly as in~\cite[Lemma~5.55]{InvPair}: strong
boundary monotonicity can be phrased in terms of the monotonicity of
the function $t\circ u$ restricted to each boundary component.
Monotone functions can limit to constant functions, but in that
case, there is an $\alpha$-boundary degeneration component, which in
turn is ruled out by Condition~\ref{prop:CurvesNotToCoverA}.
\end{proof}
Given a lower diagram $(\Sigma,\alphas,\betas,w,z)$,
the operations on the type $A$ module are defined by counting
$J$-holomorphic curves
\[ u\colon (S,\partial S)\to (\Sigma\times[0,1]\times \R,
(\alphas\times\{1\}\times \R)\cup(\betas\cup\{0\}\times \R),\]
subject to the constraints~\ref{property:First}-\ref{prop:FiniteEnergy}
and~\ref{prop:WeakBoundaryMonotone}, with the understanding that presently, we set
\begin{equation}
\label{eq:dUp}
d=g+n.
\end{equation}
We will make the following
further hypothesis:
\begin{enumerate}[label=(A${\mathcal M}$-\arabic*),ref=(A${\mathcal M}$-\arabic*)]
\setcounter{enumi}{\value{bean}}
\item \label{prop:CurvesNotToCoverA}
At least one of $n_\wpt(u)$ or $n_\zpt(u)$ vanishes.
\end{enumerate}
The strong boundary monotonicity condition on such a map $u$ looks
exactly as it did earlier
(c.f. Condition~\ref{property:StronglyBoundaryMonotone}). As in
Section~\ref{sec:CurvesD}, this condition can be formulated in terms
of combinatorial data. Specifically, given a lower generator $\mathbf x$ and
a sequence of sets of Reeb chords and orbits
$(\rhos_1,\dots,\rhos_\ell)$, define the terminal $\alpha$-set and
strong boundary monotonicity of $(\mathbf x,\rhos_1,\dots,\rhos_\ell)$ as in
Definition~\ref{def:StronglyBoundaryMonotone}, with the understanding
that now $\mathbf x$ is a lower, rather than upper, Heegaard state.
Given lower Heegaard states $\mathbf x,\mathbf y$, a marked source $\Source$ a
strongly boundary monotone sequence
$\rhos_1,\dots,\rhos_\ell=\vec{\rhos}$, we can form moduli spaces of
such pseudo-holomorphic flows, denoted
$\UnparModFlow(\mathbf x,\mathbf y,\Source,\rhos_1,\dots,\rhos_\ell)$
or simply $\UnparModFlow(\mathbf x,\mathbf y,\Source;\vec{\rhos})$.
Lemma~\ref{lem:SBB} has the following straightforward adaptation:
\begin{lemma}
\label{lem:SBA}
If $(\mathbf x,\vec{\rhos})$ is strongly boundary monotone and $u\in
\ModFlow^B(\mathbf x,\mathbf y,\Source,\vec{\rhos})$, then $u$ is strongly boundary
monotone. Conversely, if $u\in\ModFlow^B(\mathbf x,\mathbf y,\Source,\vec{\rhos})$
is strongly boundary monotone, then $(\mathbf x,\vec{\rhos})$ is strongly
boundary monotone.\qed
\end{lemma}
Let $\rhos$ be a constraint packet consisting entirely of chords (i.e.
it contains no orbits. Think of each $\rho\in \rhos$ as a path in
$[0,1]\times Z$. Let $\inv(\rhos)$ denote the minimal number of
crossings between the various chords. Given any point $p\in Z$, let
$m([\rhos],p)$ denote the multiplicity with which the Reeb chords in
$\rhos$ cover $p$, with the convention that if $p=\alpha_i\cap Z_i$ or
$\alpha_{i-1}\cap Z_i$, then $m([\rhos],p)$ is the average of
$m([\rhos],p')$ over the two nearby choices $p'\in Z$ on both sides of
$p$. For example, if $p=\alpha_i\cap Z_i$ or $\alpha_{i-1}\cap Z_i$,
then $m(L_i,p)=1/2$.
Let
\begin{equation}
\label{eq:DefIota}
\iota(\rhos)=\inv(\rhos)-m([\rhos],\rhos^-).
\end{equation}
(Note that $m([\rhos],\rhos^-)=m([\rhos],\rhos^+)$.)
\begin{example}
\label{eq:IotaOfChord}
If $\rho$ is a single Reeb chord, then
$\iota(\{\rho\})=-\weight(\rho)$.
\end{example}
\begin{example}
\label{ex:JoinCurveEnd}
Fix integers $a,b\geq 0$, $i\in \{2,\dots,2n-1\}$, and
let $\rho_1=(L_i R_i)^\alpha L_i$,
$\rho_2=(R_i L_i)^\beta R_i$, and $\rhos=\{\{\rho_1\},\{\rho_2\}\}$.
Then,
\begin{align*}
\inv(\rho_1,\rho_2)&=|\alpha-\beta| \\
m([\{\rho_1,\rho_2\}],\{\rho_1,\rho_2\}^-)&=
2(\alpha+\beta+1).
\end{align*}
Note that $\iota(\{(L_i R_i)^{\alpha+\beta+1}\})=-\alpha-\beta-1$, so
\[ \iota(\rhos)-\iota(\{(L_i R_i)^{\alpha+\beta+1}\})
= |\alpha-\beta|-\alpha-\beta-1\leq -1 ,\]
with equality iff $a=0$ or $b=0$.
\end{example}
\begin{example}
\label{ex:CollisionEnd}
Fix integers $a,b\geq 1$, $i\in\{2,\dots,2n-1\}$;
let $\rho_1=(L_i R_i)^a$, $\rho_2=(R_i L_i)^b$,
and $\rhos=\{\rho_1,\rho_2\}$.
Then
\begin{align*}
\inv(\{\rho_1\},\{\rho_2\})&= |a-b| \\
m([\{\rho_1,\rho_2\}],\{\rho_1,\rho_2\}^-)&= 2a+2b.
\end{align*}
Note that $\iota(\rho_1)=-a$, $\iota(\rho_2)=-b$; so
\[ \iota(\{\rho_1,\rho_2\})
= |a-b|-2a-2b <-a-b
=\iota(\{\rho_1\})+\iota(\{\rho_2\}).\]
\end{example}
\subsection{The chamber structure on $\Tb$}
We will be interested in some further structure on $\Tb$ induced by
the boundary degenerations, as defined in
Definition~\ref{def:BoundaryDegenerations} (using $d$ now
as in Equation~\eqref{eq:dUp}).
For each $\{j,k\}\in\Mdown$, there is a corresponding component of
$\Bjk$ of $\Sigma_0\setminus \betas$, which contains the two
boundary components $Z_j$ and $Z_k$; or equivalently, component
$\Bjk\subset \Sigma\setminus\betas$ which contains
the two punctures corresponding to $Z_j$ and $Z_k$.
Let $t\colon {\mathbb H}\cong [0,\infty)\times \R \to \R$ denote the
projection to the second factor. Given
$w\in\ModDeg_J^\Bjk$, we have two punctures $q_1$ and $q_2$, labelled
by orbits $j$ and $k$ respectively. Let $\delta(w)=t\circ
w(q_1)-t\circ w(q_2)$. Note that $\delta(w)$ is not invariant under
conformal automorphisms of ${\mathbb H}$; but the sign of $\delta(w)$
is.
\begin{lemma}
\label{lem:Walls}
For generic $J$, the subspace of $d$-dimensional torus $\Tb$
\[ \Wall^{\orb_j=\orb_k}=\{\mathbf x\in\Tb\big| \exists v\in\UnparModDeg^{\Bjk}(\mathbf x)~\text{so that}~\delta(v)=0\}\]
is the image under a smooth map of a smooth manifold of dimension $d-1$.
\end{lemma}
\begin{proof}
The map $\delta\colon \ModDeg^{\Bjk}(\mathbf x)\to \R$ is a smooth
map. By transversality arguments, for generic $J$, $0$ is a regular
value, so $\delta^{-1}(0)$ is a submanifold. Similarly, if we take the quotient
by the automorphism of group of ${\mathbb H}$, the quotient of $\delta^{-1}(0)$
is a codimension one submanifold of $\UnparModDeg^{\Bjk}(\mathbf x)$.
Now, $\Wall^{\orb_j=\orb_k}$ is the image of this submanifold under the evaluation map $\evB$ from Definition~\ref{def:evB}.
\end{proof}
We have the following analogue of
Lemma~\ref{lem:BoundaryDegenerationsDegree1} for lower diagrams:
\begin{lem}
\label{lem:BoundaryDegenerationsDegree1A}
The evaluation map $\evB\colon \ModDeg_J^{{\mathcal B}_{\{r,s\}}}\to \Tb$ has odd degree.
\end{lem}
It follows that the complement of $\Wall^{\orb_j=\orb_k}$ in $\Tb$
consists of two (disjoint) chambers:
\begin{align*}
\Chamber^{\orb_j>\orb_k}&=\{\mathbf x\in\Tb\big| \#(w\in\ModDeg^{\Bjk}(\mathbf x)~\text{so that}~\delta(w)>0)\equiv 1 \pmod{2}\} \\
\Chamber^{\orb_j<\orb_k}&=\{\mathbf x\in\Tb\big| \#(w\in\ModDeg^{\Bjk}(\mathbf x)~\text{so that}~\delta(w)<0)\equiv 1 \pmod{2}\}
\end{align*}
\subsection{Smooth moduli spaces}
\begin{defn}
\label{def:IndexTypeA}
Let $\mathbf x$ and $\mathbf y$ be lower states,
suppose that $(B,\rhos_1,\dots,\rhos_\ell)=(B,\vec{\rhos})$ is strongly boundary monotone.
Define
\begin{align}
\chiEmb(B,\vec{\rhos})&= d+ e(B) - n_\mathbf x(B)-n_\mathbf y(B) -\sum_{i=1}^{\ell} \Big(\iota(\chords(\rhos_i))+\weight(\chords(\rhos_i))\Big)
\label{eq:ChiEmbA} \\
\ind(B,\mathbf x,\mathbf y;\vec{\rhos}) &= e(B)+n_\mathbf x(B)+n_\mathbf y(B)+\ell \label{eq:EmbIndA}
\\
& \qquad -\weight(\vec{\rhos})+\iota(\chords(\vec{\rhos}))+\sum_{o\in\orb(\vec\rhos)} (1-\weight(o)), \nonumber
\end{align}
where
\[ \iota(\vec{\rhos})=\sum_{i=1}^{\ell} \iota(\chords(\rhos_i))
\qquad{\text{and}} \qquad
\weight(\rhos)=\sum_{i=1}^{\ell} \weight(\rhos_i),
\]
\end{defn}
\begin{rem}
Note that in the special case where each packet in $\vec{\rhos}$
contains a single chord (so we write $\vec{\rhos}=\vec{\rho}$),
Example~\ref{eq:IotaOfChord} shows that $\iota(\vec{\rho})=-\weight(\vec{\rho})$;
so the above definition of the embedded index is consistent with
Equation~\eqref{eq:ChiEmb}.
\end{rem}
We have the following analogue of Proposition~\ref{prop:ExpectedDimension}
\begin{prop}
\label{prop:ExpectedDimensionA}
Suppose that $(\mathbf x,\vec\rhos)$ is strongly boundary monotone.
If $\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec\rhos)$ is represented by some
pseudo-holomorphic $u$, then $\chi(\Source)=\chiEmb(B)$ if and only
if $u$ is embedded. In this case, the expected dimension of the
moduli space is computed by $\ind(B,\mathbf x,\mathbf y;\vec\rhos)$. Moreover, if a
strongly monotone moduli space $\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec\rhos)$
has a non-embedded holomorphic representative, then its expected
dimension $\leq \ind(B,\mathbf x,\mathbf y;\vec\rhos)-2$.
\end{prop}
\begin{proof}
To deduce Equation~\eqref{eq:ChiEmbA}, we apply the proof of
Proposition~\ref{prop:ExpectedDimension}. As in that proof, we
compare the intersection number $u\cap
\tau_R(u)=n_\mathbf x(B)+n_\mathbf y-\frac{d}{2}$ with $u\cap
\tau_\epsilon(u)$. In that argument, we used the fact that
$\tau_\epsilon(u)=b_\Sigma$. In the present case, however, there are additional
intersection points from $u\cap \tau_\epsilon(u)$ that come from the
double points at the boundary (arising from the constraint packets).
Thus,
\[u \cap \tau_\epsilon(u)=b_\Sigma + \sum_{i=1}^\ell N(\chords(\rhos_i)),\] where
$N(\rhos_i)$ is the number of intersection points of
$u\cap \tau_\epsilon(u)$ that come from the constraint packet $\rhos_i$.
We will show that
\begin{equation}
\label{eq:ComputeCorrection}
N(\chords(\rhos))=-\iota(\chords(\rhos))-\weight(\chords(\rhos)).
\end{equation}
To see this, note that contributions arise only for pairs of chords
in the packet that are contained in some fixed boundary component
$\Zout_j$. Suppose then that there are exactly two chords $\rho_1$
and $\rho_2$ in $\rhos$ that are contained in $\Zout_j$. Suppose
that the length of $\rho_1$ is $a$ and the length of $\rho_2$ is
$b$. (Here, we normalize so that the whole boundary has length $1$;
so in particular $L_i$ has length $\OneHalf$.) By boundary
monotonicity, $a+b$ is an integer. In our local model, the surface
has a component where $f$ is modelled on $\tau\mapsto \tau^{2a}$ and another
modeled on $\tau\mapsto c\cdot \tau^{2b}$, for $\tau\in\C$ with
$\Real(\tau)\geq 0$, and some $c\in \R^{<0}$. To count
double points, we can halve the number of double points on the maps
$\tau\mapsto \tau^{2a}$ and $\tau\mapsto c \cdot \tau^{2b}$ for
$\tau\in \C$. Counting double points there is equivalent to counting
the intersection number of the quadratic function $(z-\tau^{2a})(z-c
\tau^{2b})$ with the discriminant locus, which in turn is equivalent
to the order of vanishing of the function
\[ (\tau^{2a}+c \tau^{2b})^2-4c \tau^{2(a+b)}=(\tau^{2a}-c \tau^{2b})^2=\min(4a,4b),\] which, by Examples~\ref{ex:JoinCurveEnd}
and~\ref{ex:CollisionEnd}, verifies
Equation~\eqref{eq:ComputeCorrection}. (Note that the when $a$ has
fractional length, we are using Example~\ref{ex:JoinCurveEnd} with
$a=\alpha+\OneHalf$, $b=\beta+\OneHalf$.)
Having verified Equation~\eqref{eq:ComputeCorrection},
Equation~\eqref{eq:ChiEmbA} follows at once.
Deducing the index from the Euler characteristic as in
proof of Proposition~\ref{prop:ExpectedDimension}, noting that
\[ \ind=2e+b_{\CDisk}-\sum_{i=1}^{\ell}(|\rhos_i|-1)+\sum_{\rho\in\chords(\rhos_i)}(1-2\weight(\rho))+\sum_{o\in\orbits(\rhos_i)}(2-2\weight(o)).
\]
\end{proof}
\begin{example}
Consider the shadow in Figure~\ref{fig:DoublepointFree}. This
shadow occurs for four different boundary monotone moduli spaces:
$\UnparModFlow^B(\mathbf x,\mathbf y,(\{\orb_i\};\Source_1))$, where $\Source_1$ is a
disk with four boundary punctures, and a single orbit puncture;
$\UnparModFlow^B(\mathbf x,\mathbf y,(\{L_iR_i\};\Source_2))$, where $\Source_2$ is a
disk with five boundary punctures (one of which is an East infinity
boundary puncture, labelled by $L_iR_i$);
$\UnparModFlow^B(\mathbf x,\mathbf y,(\{R_iL_i\};\Source_3))$, similar to the above
moduli space, and finally,
$\UnparModFlow^B(\mathbf x,\mathbf y,(\{L_i,R_i\};\Source_4))$, where $\Source_4$ is a
disjoint union of two three-punctured disks (one of which has an
East infinity puncture labelled by $L_i$ and the other of which has
a puncture labelled by $R_i$). The dimensions of these moduli
spaces are $2$, $1$, $1$, and $0$ respectively.
\begin{figure}[h]
\centering
\input{DoublepointFree.pstex_t}
\caption{{\bf Moduli spaces with given shadow.}
\label{fig:DoublepointFree}}
\end{figure}
\end{example}
There are evaluation maps $\evB_i\colon
\ModFlow(\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_\ell)\to \Tb$, obtained as
follows. Suppose that the punctures at level $\rhos_i$ are mapped via
$t$ to $\tau\in \R$; then
\[ \evB_i(v)=\pi_{\Sigma}^{-1}(\{(0,\tau)\}).\]
These maps descend to give maps
\begin{equation}
\label{eq:EvaluationAtStage}
\evB_i\colon
\UnparModFlow(\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_\ell)\to \Tb.
\end{equation}
The following is a mild elaboration on Theorem~\ref{thm:GeneralPosition}
(see~\cite[Proposition~5.6]{InvPair}):
\begin{thm}
\label{thm:GeneralPositionA}
Choose a generic $\{J_t\}$. Suppose that $(\mathbf x,\vec{\rhos})$ is
boundary monotone. If $\ind(B,\mathbf x,\mathbf y;\vec{\rhos})\leq 2$, then the
moduli space $\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec{\rhos})$ is a smooth
manifold of dimension given by $\ind(B,\mathbf x,\mathbf y;\vec{\rhos})$. Moreover,
$\evB_i$ are transverse to all of the walls
$\Wall^{\orb_j=\orb_k}$ for all $\{r,s\}\in\Mdown$; in particular, all the
disks $v$ appearing in the zero-dimensional moduli spaces
$\UnparModFlow^B(\mathbf x,\mathbf y,\Source,\vec{\rhos})$ with
$\ind(B,\mathbf x,\mathbf y;\vec{\rhos})=1$ have
$\evB_i\in \Chamber^{\orb_j>\orb_k}$ or $\Chamber^{\orb_j<\orb_k}$.
\end{thm}
\begin{proof}
For generic $\{J_t\}$, the moduli spaces $\ModFlow^B(\mathbf x,\mathbf y;\Source)$
are transversely cut out by the $\dbar$ operator. The $t$
evaluations of the various punctures give a map
from this moduli
space to $\R^{P}$, where the set $P$ corresponds with the
interior and $\east$-punctures of $\Source$.
This map $\ev_P\colon \ModFlow^B(\mathbf x,\mathbf y;\Source)\to \R^P$
is a submersion.
The moduli space $\ModFlow^B(\mathbf x,\mathbf y;\Source;\vec\rhos)$
can be thought of as the preimage under this evaluation map of a
suitable diagonal $\Delta_P$ in $\R^P$. (For example, if
some subset $\{p_1,\dots,p_k\}$
of punctures are assigned to the same constraint packet,
then the corresponding diagonal in $\R^P$ consists of
those $P$-tuples whose components at $p_1,\dots,p_k$ coincide.)
It follows readily that
$\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec{\rhos})$ is a smooth
manifold of dimension given by $\ind(B,\mathbf x,\mathbf y;\vec{\rhos})$.
At each puncture $p$, we also have a corresponding evaluation map
$\ev_p^\beta\colon \ModFlow^B(\mathbf x,\mathbf y;\Source)$ defined by
$\ev_p^\beta(v)=\pi_\Sigma^{-1}(\{(0,t(v(p)))\})$;
taking the product over each puncture gives a submersion
\[ \ev_P^\beta\colon \ModFlow^B(\mathbf x,\mathbf y;\Source)\to (\Tb)^{P}.\]
It follows that for generic $\{J_t\}$, the evaluations
are transverse to the diagonals in $\R^P$ and the walls
$\Wall^{\orb_j=\orb_k}$ from Lemma~\ref{lem:Walls};
in particular, for generic $\{J_t\}$,
$\ev^\beta_i$ on
$\UnparModFlow(\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_\ell)$
(which is obtained by evaluating $\ev^\beta_p$ at
any puncture $p$ belonging to the $i^{th}$ packet)
is transverse to the codimension one walls
$\Wall^{\orb_j=\orb_k}$.
(Compare~\cite[Proposition~5.6]{InvPair};
see also~\cite[Section~3.4]{McDuffSalamon}.)
\end{proof}
If a holomorphic curve $v$ represents a point in
$\UnparModFlow^B(\mathbf x,\mathbf y;\Source;\vec{\rhos})$ with
$\evB_i\in \Chamber^{\orb_j>\orb_k}$ or
$\Chamber^{\orb_j<\orb_k}$, we write $v\in \Chamber^{\orb_j>\orb_k}_i$ or
$\Chamber^{\orb_j<\orb_k}_i$.
\subsection{Ends of one-dimensional moduli spaces}
We set up some preliminaries used in the description of the ends of one-dimensional moduli spaces.
\begin{defn}
\label{def:OrbitMarking}
Observe that there are two {\em special} Reeb orbits which are not
matched with any other Reeb orbit. If $\orb_j$ is one of these
orbits, recall that the corresponding component of
$\Sigma\setminus\betas$, which we denote ${\mathcal B}_{\{j\}}$,
contains one of the two basepoints $\wpt$ or $\zpt$. We call the other
orbits {\em non-special}. An {\em orbit marking} is a partition of
the orbits into two types, the {\em even} ones and the
{\em odd} ones, so that the following conditions hold:
\begin{itemize}
\item each even one is matched with an odd one in
$\Mdown$.
\item there is exactly one even special orbit and one odd one.
\end{itemize}
\end{defn}
\begin{defn}
\label{def:Allowed}
Fix an orbit marking. A constraint packet is called {\em allowed by
the orbit marking}, or simply {\em allowed}, if it satisfies the
following properties:
\begin{enumerate}[label=($\rhos$-\arabic*),ref=($\rhos$-\arabic*)]
\item Each of the chords appearing in $\rhos$ are disjoint from one another.
\item \label{rhos:OneOrbit}
It contains at most one orbit, and that orbit is
simple. If it contains no orbits, it is called {\em orbitless}.
\item If $\rhos$ contains an even type orbit, then it contains exactly one
Reeb chord, as well; and the chord is disjoint from the orbit.
\item If $\rhos$ contains an odd type orbit, then it contains
no Reeb chords.
\end{enumerate}
\end{defn}
Given two constraint packets $\rhos_1$ and
$\rhos_2$, a {\em contained collision} is a new (possibly generalized)
constraint packet
$\sigmas$, where
$\orbits(\sigmas)=\orbits(\rhos_1)\cup\orbits(\rhos_2)$ as multi-sets
(i.e. it might consist of the same orbit with multiplicity $2$), and
$\chords(\sigmas)$ is the union of the following three sets:
\begin{itemize}
\item those Reeb chords in $\rhos_1$ that cannot be prepended onto any Reeb chord in $\rhos_2$
\item those Reeb chords in $\rhos_2$ that cannot be appended to any Reeb chord in $\rhos_1$
\item the joins $\rho_1\uplus\rho_2$
of all possible pairs of joinable (i.e.
``strongly composable'') Reeb chords
$\rho_1\in \chords(\rhos_1)$ and $\rho_2\in \chords(\rhos_2)$.
\end{itemize}
The contained collision is called {\em visible} if no orbit in
$\orbits(\sigmas)$ is contained in both $\orbits(\rhos_1)$ and
$\orbits(\rhos_2)$. The collision is called {\em strongly composable}
if whenever the chords $\rho_1\in \chords(\rhos_1)$ and $\rho_2\in
\chords(\rhos_2)$ are weakly composable (as in
Definition~\ref{def:ComposableChords}), they are in fact strongly
composable.
\begin{rem}
A collision between two constraint packets $\rhos_1$ and $\rhos_2$
might be merely a a generalized constraint packet. For example, both
$\rhos_1$ and $\rhos_2$ may contain the same orbit, and their
collision can contain the same orbit with multiplicity two. When the
collision is contained and visible this does not occur: the
multi-set of orbits in the collision is in fact a set. (Indeed, if
the collision occurs in a boundary monotone sequence, it is also
easy to see that the multi-set of chords is also a set.)
\end{rem}
\begin{defn}
\label{def:BoundaryDegCollision}
Suppose that $\rhos_1$ and $\rhos_2$ are allowed constraint packets
(in the sense of Definition~\ref{def:Allowed}),
which also have the property that there are $\{j,k\}\in\Mdown$ so that
$\orb_j\in\orbits(\rhos_1)$ and $\orb_k\in\orbits(\rhos_2)$. In this
case, $\chords(\rhos_1)\cup\chords(\rhos_2)$ consists of a single
chord, which we denote $\sigma$. We say that the constraint packet is
$\{\sigma\}$ (i.e. with the two orbits removed) is the {\em boundary
degeneration collision} of $\rhos_1$ and $\rhos_2$.
\end{defn}
With these remarks in place, we state the following analogue
of~\cite[Theorem~5.61]{InvPair}, which will be used in
Section~\ref{sec:TypeA} in the verification of the $\Ainf$ relation:
\begin{thm}
\label{thm:AEnds}
Let $\Hdown$ be a
lower diagram and $\Mdown$ the induced relation among
$\{1,\dots,2n\}$. Choose also an orbit marking
as in Definition~\ref{def:OrbitMarking}.
Fix a lower Heegaard state $\mathbf x$ and a sequence of
constraint packets $\vec{\rhos}$ with the following properties:
\begin{itemize}
\item $(\mathbf x,\vec{\rhos})$ is strongly
boundary monotone.
\item Each constraint packet $\rhos_i$ is allowed,
in the sense of Definition~\ref{def:Allowed}
\end{itemize}
Let $\mathbf y$ be a
lower Heegaard state, and $B\in\pi_2(\mathbf x,\mathbf y)$, whose local multiplicity
vanishes either at $\wpt$ or $\zpt$ (or both). Choose $\Source$ and
$\vec{P}$ so that $[\vec{P}]=(\rhos_1,\dots,\rhos_\ell)$ and
so that $\chi(\Source)=\chiEmb(B)$;
and suppose that $\ind(B,\mathbf x,\mathbf y;\vec{\rhos})=2$, and abbreviate
$\UnparModFlow=\UnparModFlow^B(\mathbf x,\mathbf y;\Source;{\vec{\rhos}})$. The total
number of ends of $\UnparModFlow$ of the following types are even in
number:
\begin{enumerate}[label=(AE-\arabic*),ref=(AE-\arabic*)]
\item \label{endA:2Story}
Two-story ends, which are of the form
\[ \UnparModFlow(\mathbf x,\w;\Source_1;\rhos_1,\dots,\rhos_i)\times
\UnparModFlow(\w,\mathbf y;\Source_2;\rhos_{i+1},\dots,\rhos_{\ell}),\]
taken over all lower Heegaard states $\w$ and choices of $\Source_1$ and $\Source_2$ so that
$\Source_1\natural \Source_2=\Source$, and $B_1\natural B_2=B$.
\item
\label{endA:Orbit}
Orbit curve ends,
of the form $\UnparModFlow^B(\mathbf x,\mathbf y,\Source';\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_{\ell})$,
where $\orbits(\sigmas)=\orbits(\rhos_i)\setminus\{\orb_r\}$, $\chords(\sigmas)=\chords(\rhos_i)\cup\{\longchord_r\}$
where $\longchord_r$ is a Reeb chord that covers the boundary component $Z_r$ with multiplicity $1$.
\item
\label{endA:ContainedCollisions}
Contained collision ends for two consecutive packets $\rhos_i$ and
$\rhos_{i+1}$, which correspond to points in
$\UnparModFlow^{B'}(\mathbf x,\mathbf y,\Source;\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+2},\dots,\dots,\rhos_\ell)$
with the following properties:
\begin{enumerate}[label=(C-\arabic*),ref=(C-\arabic*)]
\item The collision is visible.
\item The packets $\rhos_i$ and $\rhos_{i+1}$ are strongly composable.
\item The packet $\sigmas$ is a contained collision of $\rhos_i$
and $\rhos_{i+1}$
\item \label{c:Disjoint}
The chords in $\sigmas$ are disjoint from one
another.
\end{enumerate}
\item
\label{endA:Join}
Join ends, of the form
$\UnparModFlow^B(\mathbf x,\mathbf y,\Source';\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_{\ell})$,
$\orbits(\sigmas)=\orbits(\rhos_i)$, and the following conditions hold:
\begin{enumerate}[label=(J-\arabic*),ref=(J-\arabic*)]
\item $(\mathbf x,\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_{\ell})$
is strongly boundary monotone.
\item There is some $\rho\in \chords(\rhos_i)$ with the property
that $\rho=\rho_1\uplus \rho_2$, and
$\chords(\sigmas)=(\chords(\rhos_i)\setminus \{\rho\})\cup
\{\rho_1,\rho_2\}$.
\item \label{OneIsShort} In the above decomposition, at least one of $\rho_1$ and
$\rho_2$ covers only half of a boundary component.
\end{enumerate}
\item \label{endA:BoundaryDegeneration}
Boundary degeneration collisions $\sigmas$
between two consecutive packets $\rhos_i$ and $\rhos_{i+1}$;
when
$\orb_j\in\orbits(\rhos_i)$, $\orb_k\in\orbits(\rhos_{i+1})$
and $\{j,k\}\in\Mdown$; these correspond to points in
$\ModFlow^{B'}(\mathbf x,\mathbf y,\Source';\rhos_1,\dots,
\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_\ell)$
in the chamber $\Chamber^{\orb_j<\orb_k}$.
The homology class $B'$
is obtained from $B$ by removing
a copy of $\Brs$.
\item \label{endA:SpecialBoundaryDegeneration}
Special boundary degeneration ends,
when $\rhos_i$ contains a special Reeb orbit $\orb_k$.
When $\sigmas=\rhos_i\setminus\{\orb_k\}$ is non-empty,
these are identified with
\[ \UnparModFlow^{B'}(\mathbf x,\mathbf y,\Source';\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+2},\dots,
\rhos_\ell)\] for $B=B'+{\mathcal B}_{\{k\}}$;
when $\rhos_i=\{\orb_k\}$, then $\ell=1$, $\mathbf x=\mathbf y$, $B={\mathcal B}_{\{k\}}$, and the end is unique.
\end{enumerate}
\end{thm}
\begin{rem}
In the above statement, some of the sources $\Source'$ are different
from the original source $\Source$. We have not spelled out the
precise relationship between $\Source'$ and $\Source$; it is clear
from the context.
\end{rem}
\begin{rem}
The packets $\sigmas$ that appear in the join curve ends are not
{\em allowed} in the sense of Definition~\ref{def:Allowed}; moreover,
packets appearing in contained collision ends need not be allowed.
\end{rem}
Let
\[ \UnparModFlow^B(\mathbf x,\mathbf y,\vec{P})=
\bigcup_{\Source}\UnparModFlow^B(\mathbf x,\mathbf y,\Source;\vec{P}).\]
See Figure~\ref{fig:JoinEnd} for a picture of a join curve end.
\begin{figure}[h]
\centering
\input{JoinEnd.pstex_t}
\caption{{\bf Join curve end.}
The moduli space corresponding to the shaded homotopy class, from $\mathbf x=\{x_1,x_2,x_3,x_4\}$
to $\mathbf y=\{y_1,y_2,y_3,y_4\}$ with the Reeb chord $L_i R_i$ has a join curve end as the cut parameter $a\mapsto 0$.
(There is another end which is a two-story building, corresponding to a flow from $\{x_1,x_2,x_3,x_4\}$ to
$\{y_1,t,x_3,x_4\}$, followed by the flow from $\{y_1,t,x_3,x_4\}$ to $\{y_1,y_2,y_3,y_4\}$ that crosses
$L_i R_i$).)}
\label{fig:JoinEnd}
\end{figure}
\begin{example}
Suppose that $\mathbf x$ and $\mathbf y$ are generators. Let $c$ denote the number of
points in $\UnparModFlow^B(\mathbf x,\mathbf y,(\{L_i, R_i\}))$.
Let $a$ be the number of two-story ends of
$\UnparModFlow^B(\mathbf x,\mathbf y,(\{L_i R_i\}))$; and $b$ be the number of
two-story ends of $\UnparModFlow^B(\mathbf x,\mathbf y,(\{R_i L_i\}))$.
The above theorem applied to $\UnparModFlow^B(\mathbf x,\mathbf y,(\{L_i R_i\}))$
implies that $a+c\equiv 0\pmod{2}$; and applied to
$\UnparModFlow^B(\mathbf x,\mathbf y,(\{R_i L_i\}))$ gives $b+c\equiv 0\pmod{2}$.
\end{example}
\begin{example}
Suppose that $\mathbf x$ and $\mathbf y$ are generators.
The above theorem implies that the number of two-story ends
of $\UnparModFlow^B(\mathbf x,\mathbf y,\{L_i R_i L_i\})$ is even; in particular,
there are no join curve ends because the constraint packet
$\{L_i, R_i L_i\}$ is not
part of a boundary monotone sequence.
\end{example}
\begin{example}
Suppose that $\mathbf x$ and $\mathbf y$ are generators.
Let $c_1$ denote the number of points in
$\UnparModFlow(\mathbf x,\mathbf y,\{L_i,R_i L_i R_i\})$,
$c_2$ denote the number of points in
$\UnparModFlow(\mathbf x,\mathbf y,\{R_i,L_i R_i L_i\})$,
$a$ denote the number of two-story ends of
$\UnparModFlow(\mathbf x,\mathbf y,\{L_i R_i L_i R_i\})$
and $b$ the number of two-story ends of
$\UnparModFlow(\mathbf x,\mathbf y,\{R_i L_i R_i L_i \})$.
The above theorem implies that
\begin{align*}
a+c_1+c_2& \equiv 0\pmod{2} \\
b+c_1+c_2& \equiv 0\pmod{2}.
\end{align*}
\end{example}
\begin{example}
The above theorem shows that there is an even number of two-story
ends of $\UnparModFlow(\mathbf x,\mathbf y,(\{L_{i+1}\},\{L_i\}))$, since the collision
between $L_{i+1}$ and $L_{i}$ is weakly, but not strongly, composable.
(Note that there are three types of such two-story ends.)
\end{example}
\begin{example}
Consider the moduli space
$\UnparModFlow(\mathbf x,\mathbf y,\{L_i R_i\},\{ L_i R_i\})$. Let $a$ denote the number of two-story ends
(again, of three possible types); $b$ denote the number of join curve ends
(of two types, corresponding to the sequence
$(\{L_i,R_i\},\{L_i R_i\})$ or the sequence
$(\{L_i R_i\},\{L_i, R_i\})$);
and $c$ be the number of points in
$\UnparModFlow(\mathbf x,\mathbf y,\{L_i R_i L_i R_i\})$.
Then, $a+b+c\equiv 0\pmod{2}$.
\end{example}
\begin{example}
Let $a$ be the number of two-story ends of
the moduli space $\UnparModFlow^B(\mathbf x,\mathbf y,\{\orb_i,R_{i+1}\})$,
and $b_1$ denote the number of points in $\UnparModFlow^B(\mathbf x,\mathbf y,\{L_i R_i, R_{i+1}\})$,
and $b_2$ denote the number of points in $\UnparModFlow^B(\mathbf x,\mathbf y,\{R_i L_i, R_{i+1}\})$.
Then,
$a+b_1+b_2\equiv 0\pmod{2}$.
\end{example}
\subsection{Curves at East infinity}
We recall (with very minor adaptation) the material
from~\cite[Section~5.3]{InvPair}. Let $Z=\bigcup_{i=1}^{2n} Z_i$ be
the boundary of $\Sigma_0$. Let ${\mathbf a}=Z\cap
\bigcup_{i=1}^{2n-1}\alpha_i$. We consider moduli spaces of
holomorphic curves in $\R \times Z \times [0,1] \times \R$. The ends
of the first $\R$ factor are called east and west infinity; the ends
of the second $\R$ factor are called $\pm\infty$. There is an
$\R\times \R$-action on $\R \times Z \times [0,1] \times \R$,
projection maps $\pi_{\R\times Z}$ (onto the first two factors), $s$
(to $[0,1]$) and $t$ (to the last $\R$ factor). Fix a split complex
structure $J$ on $\R\times Z\times [0,1]\times \R$.
\begin{defn}
An {\em east source} $\EastSource$ is:
\begin{itemize}
\item a smooth two-manifold $T$ with boundary
and punctures
\item a labeling of each puncture of $T$ by $\east$ or $\west$
\item a labeling of each $\west$ or $\east$ puncture $q$ by a Reeb orbit, if
the $q$ is in the interior of $T$, and a labeling of $q$ by a
chord in $(Z,{\mathbf a})$ if the puncture is on the boundary of
$T$.
\end{itemize}
\end{defn}
Given a east source $\EastSource$, we consider maps:
\[ v\colon (T,\partial T)\to (\R\times Z \times [0,1]\times \R,
\R \times {\mathbf a}\times \{1\}\times \R) \]
satisfying:
\begin{enumerate}[label=(E-\arabic*),ref=(E-\arabic*)]
\item \label{EC1} $v$ is $(j,J)$-holomorphic with respect to some almost-complex
structure $j$ on $T$.
\item $v$ is proper.
\item $(s,t)\colon v \to [0,1]\times \R$ is constant.
\item At each $\west$ puncture $q$ of $T$ labeled by $\rho$ (a chord or orbit),
$\lim_{z\goesto q}\pi_\Sigma\circ v(z)$ is
$\rho\subset \{-\infty\}\times Z$.
\item\label{ECn} At each $\east$ puncture $q$ of $T$ labeled by $\rho$ (a chord or orbit),
$\lim_{z\goesto q}\pi_\Sigma\circ v(z)$ is
$\rho\subset \{+\infty\}\times Z$.
\end{enumerate}
Note that if $T$ has non-empty boundary, then $s\circ v=1$.
\begin{defn}
Let $\ModEast(\EastSource)$ denote the moduli space of
holomorphic maps from $T$ satisfying Properties~\ref{EC1}-\ref{ECn} above.
\end{defn}
For each puncture $q$ in $\EastSource$, there is a corresponding
evaluation $\ev_q\colon \ModEast(\EastSource)\to \R$ which computes
the $t\circ v$ on the component of $T$ containing $q$. There are
evaluation maps
\begin{align*}\ev_\west&=\prod_{q\in\West(\EastSource)} \ev_q \colon
\ModEast(\EastSource)\to ([0,1]\times \R)^{|\West(\EastSource)|} \\
\ev_\east&=\prod_{q\in\East(\EastSource)} \ev_q \colon
\ModEast(\EastSource)\to ([0,1]\times \R)^{|\East(\EastSource)|}.
\end{align*}
Consider the $t$-projection of the evaluation map; e.g. $t\circ
\ev_{\west}\to \R^{|\West(\EastSource)|}$ Given an east source
$\EastSource$, and ordered partitions $P_\west$ and $P_\east$, we let
$\ModEast(\EastSource;P_\west,P_\east)\subset \ModEast(\EastSource)$
be the subspaces obtained by cutting down by the partial diagonals
associated
to $t\circ \ev_\west$ and $t\circ \ev_\east$.
The following is~\cite[Proposition~5.14]{InvPair}; compare~\cite[Section~3.3]{McDuffSalamon}:
\begin{prop}
If $\EastSource$ has the property that all of the components of $T$ are topological disks,
then $\ModEast(\EastSource)$ is transversely cut out by the $\dbar$ equation for any
split complex structure on $\R\times Z\times [0,1]\times \R$.
\end{prop}
Let $\UnparModEast(\EastSource)=\ModEast(\EastSource)/\R\times \R$.
The following particular components are illustrated in Figure~\ref{fig:EastInfinity}.
A {\em trivial component} is a component of $\EastSource$ with exactly
two punctures, one $\east$ and one $\west$, both labelled by the same Reeb chord.
The holomorphic map to $\R\times Z$ is invariant under translation by $\R$.
A {\em join component} is a component of $\EastSource$ which is a
topological disk with two west boundary punctures and one east
boundary puncture. Labeling, in counterclockwise order $(e,\rho_e)$,
$(w,\rho_1)$, and $(w,\rho_2)$. There is a holomorphic map from such
a component if and only if $\rho_e=\rho_2\uplus\rho_1$; if it exists,
it is unique up to translation. The puncture $(w,\rho_1)$ is called
the {\em top puncture} and $(w,\rho_2)$ is called the {\em bottom
puncture}. A {\em join curve} is a curve that consists of one join
component and a collection of trivial components.
An {\em split component} is defined similarly, only now there the
punctures in counterclockwise order, are $(w,\rho_w)$, $(e,\rho_1)$
and $(e,\rho_2)$. Again, there is a holomorphic map if and only if
$\rho_w=\rho_1\uplus\rho_2$. The puncture $(e,\rho_2)$ is called the
{\em top puncture} and $(e,\rho_1)$ is called the {\em bottom
puncture}.
\begin{remark}
Note that there are join and split curves that cover the cylinder with
arbitrarily large multiplicity; on the left of
Figure~\ref{fig:EastInfinity}, we have illustrated a split curve
that covers the cylinder with multiplicity one, and these are the
split curves that will occur in our considerations for type $D$
structures. When considering type $A$ modules, though, we will be
forced to consider join and split curve ends that cover the boundary
cylinder with higher multiplicity.
\end{remark}
An {\em orbit component} is a disk with a single boundary puncture,
labelled $(w,\rho)$, and a single orbit puncture $(e,\orb)$ in its
interior, so that $\orb$ is a simple orbit. There is a holomorphic
map from such a component if and only if $\rho$ is one of the two
chords that covers the boundary component containing $\orb$ with
multiplicity one.
\begin{rem}
There are other components with disk sources one might consider.
For example, the ``shuffle curves'' from~\cite[Section~5.3]{InvPair}
have a natural analogue, which one might call ``orbit-shuffle
curves''. These have a west puncture that is an orbit, another west
puncture labelled by a chord $\rho$ on the same boundary component
as $\orb$, and a single east puncture labelled by $\rho\uplus u$,
where $u$ is one of the two curves that covers the boundary once.
The map to $\R\times Z$ has a single branch point in the
interior. These curves, however, will not enter our
considerations. This is because in an allowed constraint packet (in
the sense of Definition~\ref{def:Allowed}), the orbits are disjoint
from the chords. These other curves would enter if we were to try to
define the theory over $\Alg$ from~\cite{Bordered2}, which we can
avoid by some algebraic considerations; see
Section~\ref{sec:Comparison}.
\end{rem}
\begin{figure}[h]
\centering
\input{EastInfinityCurves.pstex_t}
\caption{{\bf Curves at East infinity.} The boundary on the left is
glued to the source, and the boundary to the right is the ``east
infinity'' portion. We have illustrated from left to right: a
split curve, an orbit curve, and an orbit shuffle curve. The join
curve is obtained by reflecting the leftmost picture through a vertical axis.}
\label{fig:EastInfinity}
\end{figure}
\subsection{Curves at West infinity}
\label{subsec:CurvesAtWest}
Unlike in~\cite{InvPair}, generalized flowlines can degenerate also
at west infinity. In formulating this degeneration, we need
to generalize slightly the
notion of a decorated source and pre-flowline, as follows:
\begin{defn}
A {\em decorated source with West punctures} is a decorated source
as in Definition~\ref{def:DecoratedSource} equipped with a further
set of boundary punctures that are labelled $\west$. A {\em pre-flowline
with west punctures} is a map
\[ u\colon (\Source,\partial\Source)\to
(\Sigma\times[0,1]\times\R,(\alphas\times\{1\}\times \R)\cup(\betas\times\{0\}\times\R))\]
where $\Source$ is equipped with boundary punctures $\West(\Source)$
with the following properties:
\begin{itemize}
\item for each west puncture $q$,
$\lim_{z\goesto q} u(z)$ converges to a point in
$\Sigma\times 0\times \R$;
i.e. if we fill in the west punctures to form a source curve
$\Source'$
(without West punctures), $u$ extends
uniquely to a $u'\colon \Source'
\to \Sigma\times [0,1]\times \R$
\item The extension $u'$ obtained as above is a pre-flowline
in the sense of Definition~\ref{def:GenFlow}.
\item The set of west punctures in $\Source$
comes with an ordered partition into $d$-tuples $q_1,\dots,q_d$,
so that
\[ \pi_{\CDisk}\circ u(q_1)=\dots=\pi_{\CDisk}\circ u(q_d).\]
We call this the {\em disk partition} of the punctures at West infinity.
\end{itemize}
A pseudo-holomorphic flowline with West punctures is a pre-flowline
with West punctures which, when filled in, give a pseudo-holomorphic
flowline in the sense of Definition~\ref{def:HolFlow}.
\end{defn}
Obviously, when $\West(\Source)=\emptyset$, the above definition agrees
with the usual definition of a source curve and pre-flowline
(Definitions~\ref{def:DecoratedSource} and~\ref{def:GenFlow} respectively).
For a pre-flowline $u$, let $[\West(\Source)]$ denote the set of
$d$-tuples in the disk partition; equivalently, it is the set of
equivalence classes of West punctures on $\Source$, modulo the equivalence
relation $q\sim q'$ if $t\circ u(q)=t\circ u(q')$. In particular,
\[ |[\West(\Source)]|=|\West(\Source)|/d.\]
Fix a pre-flowline $u$ with west punctures, and ${\mathbf
q}=\{q_1,\dots,q_d\}\in[\West(\Source)]$. There is a corresponding
evaluation
\[ \evB_{\mathbf q}(u)=\pi_\Sigma(u(q_1))\times \dots\times \pi_\Sigma(u(q_d)).\]
By boundary monotonicity,
$\evB_{\mathbf q}(u)\in \beta_1\times \dots\times \beta_g$
Taking the product over $[\West(\Source)]$, we obtain a map
\[ \evB_\east\in (\Tb)^{[\West(\Source)]}\]
We consider moduli spaces of holomorphic curves in $\Sigma\times
{\mathbb H}$, generalizing the boundary degenerations of
Definition~\ref{def:BoundaryDegenerations}.
\begin{defn}
A {\em boundary degeneration with West punctures}
is a map
\[ w\colon
(\wSource,\partial\wSource)\to(\Sigma\times\HH,\betas\times\R)\]
whose $d$ punctures over the point at infinity in $\HH$ are called
{\em east punctures}; and equipped with additional punctures in $\partial\wSource$, called {\em west punctures} with the following properties:
\begin{itemize}
\item for each west puncture $q$, $\lim_{z\goesto q} u(z)$ converges
to a point in $\Sigma\times \R\subset \Sigma\times \HH$; i.e. if
we fill in in the west punctures to form a source curve $\wSource'$,
then $\west$ extends uniquely to a continuous map
\[ w'\colon (\wSource',\partial\wSource')\to
(\Sigma\times\HH,\Sigma\times \R).\]
\item The extension $w'$ obtained as above is a boundary
degeneration as in Definition~\ref{def:BoundaryDegenerations}.
\item The set of west punctures in $\wSource$ comes with a partition
into $d$-tuples
$q_1,\dots,q_d$ so that
\[\pi_\HH\circ u(q_1)=\dots=\pi_\HH\circ u(q_d).\]
\end{itemize}
\end{defn}
\begin{defn}
A {\em boundary degeneration level} is a finite union of boundary
degenerations with West punctures. We can think of its source curve
$\WestSource$ (which has possibly many components) as marked with
a set of East punctures $\East(\WestSource)$ and west punctures $\West(\WestSource)$;
punctures of each type come in $d$-tuples (again, referred to as the disk partition).
\end{defn}
For a boundary degeneration level, let $[\East(\WestSource)]$ resp.
$[\West(\WestSource)]$ and
denote
the set of $d$-tuples in the disk partition of $\East(\WestSource)$
resp. $\West(\WestSource)$.
For a boundary degeneration level, we have, as before, evaluations
\[ \evB_\east(w)\in (\Tb)^{[\East(\WestSource)]}
\qquad \text{and}\qquad
\evB_\west(w)\in (\Tb)^{[\West(\WestSource)]}.\]
\subsection{Compactness}
As in~\cite[Section~5.4]{InvPair}, we use the Eliashberg-Givental-Hofer compactness~\cite{EGH}.
\begin{defn}
\label{def:HolomorphicStory}
A {\em holomorphic story} is the following data:
\begin{itemize}
\item a sequence $(w_{\ell},\dots,w_{1},u,v_1,\dots,v_k)$ for some
$\ell\geq 0$, $k\geq 0$ where $u$ is a pseudo-holomorphic flowline
with West punctures, with source $\Source$; $\{w_i\}_{i=1}^{\ell}$
is a sequence of boundary degeneration levels,
where $w_i$ has source $\WestSource_i$; $\{v_i\}_{i=1}^{k}$ is a
sequence of curves at East infinity.
\item One-to-one correspondences between the following objects:
$\West(\Source)$ and $\East(\WestSource_{1})$;
$\West(\WestSource_i)$ and $\East(\WestSource_{i+1})$
for $i=1,\dots,\ell-1$ (which respect the disk partition);
$\East(\Source)$ and $\West(\EastSource_1)$;
$\East(\EastSource_i)$ and $\West(\EastSource)_{i+1}$ (for
$i=1,\dots,k-1$),
\end{itemize}
so that the following conditions hold:
\begin{itemize}
\item $u\in\ModFlow^B(\mathbf x,\mathbf y;\Source)$
\item $v_i\in\ModEast(\EastSource_i)$
\item $w_{i}\in\ModWest(\WestSource_{i})$
\item $\ev_\east(u)=\ev_\west(v_1)$ in
$\R^{\East(\Source)}/\R\cong \R^{\West(\EastSource_1)}/\R$
(here, and in the next few conditions, we use
the isomorphism of product spaces
induced by the one-to-one correspondence between punctures).
\item $\ev_\east(v_{i})=\ev_\west(v_{i+1})$ in
$\R^{\East(\EastSource_{i})}/\R\cong
\R^{\West(\EastSource_{i+1})}/\R$
for $i=1,\dots,k-1$.
\item $\evB_\west(u)=\evB_\east(w_{1})$ in
$(\Tb)^{[\West(\Source)]}\cong(\Tb)^{[\East(w_1)]}$
\item $\evB_\west(v_w)=\evB_\east(w_{i+1})$ in
$(\Tb)^{[\West(\WestSource_i)]}\cong
(\Tb)^{[\East(\WestSource_{i+1})]}$
for $i=1,\dots,\ell-1$.
\end{itemize}
A holomorphic story with $\{k,\ell\}=\{0,1\}$ is called a {\em
simple holomorphic comb}. A {\em holomorphic comb of height
$N$} is a sequence
$(w_{j,\ell_j},\dots,w_{j,1},u_j,v_{j,1},\dots,v_{j,k_j})$ for
$j=1,\dots,N$ of holomorphic stories with $u_j$ a stable curve in
$\ModFlow^{B_j}(\mathbf x_j,\mathbf x_{j+1};\Source_j)$ for some sequence of
generalized generators $\mathbf x_1,\dots,\mathbf x_{N+1}$.
\end{defn}
\begin{rem}
In the above statement, we require the curves $u_j$ to be stable. This excludes
where all the components of the source $\Source_j$ are disks with exactly two punctures:
one at $+\infty$, another at $-\infty$, and $\pi_{\Sigma}\circ u_j$ is a constant map
(to $\mathbf x_j=\mathbf x_{j+1}$).
\end{rem}
As in~\cite[Section~5.4]{InvPair} (following~\cite{EGH}), the space of
holomorphic combs can be used to construct a compactification
of $\UnparModFlow(\mathbf x,\mathbf y;\Source)$, denoted
$\ClosedModFlow(\mathbf x,\mathbf y;\Source)$; and similarly
a compactification
$\ClosedModFlow(\mathbf x,\mathbf y;\Source;\vec{\rhos})$
of $\UnparModFlow(\mathbf x,\mathbf y;\vec{\rhos})$.
This is a compactification in the following sense:
\begin{prop}
\label{prop:Compactness}
If $\{U_n\}$ is a sequence of holomorphic combs in a fixed homology
class, then $\{U_n\}$ has a subsequence which converges to a (possibly)
nodal holomorphic curve in the same homology class.
\end{prop}
\begin{proof}
See~\cite[Proposition~5.24]{InvPair} for the compactness result at
east infinity. The west infinity curves are extracted by a more
standard Gromov compactification, rescaling from the $\CDisk$ factor
to $\HH$, as in the proof of~\cite[Lemma 3.82]{LipshitzCyl}.
\end{proof}
Let $([0,1]\times\R)^{\vec\rhos}$ denote the space of functions
from all of the punctures on the source to $[0,1]\times \R$.
Note that if two punctures $p$ and $q$
correspond to the same level $\rhos_i$, then
their $t$ values coincide.
The evaluation maps
\[
\ev\colon \UnparModFlow^B(\mathbf x,\mathbf y,\vec{\rhos})\to ([0,1]\times
\R)^{\vec\rhos}/\R \] extend continuously to the space of holomorphic
stories, where now we evaluate on all east-most punctures.
Each curve at west infinity $w$ has a shadow $B$, which determines the
total multiplicity of the Reeb orbits. When the shadow is $\Bjk$,
there are exactly two Reeb orbits, $\orb_j$ and $\orb_k$; in this
case, we call the curve at west infinity a {\em simple boundary
degeneration component}. There are also two special orbits that are
not matched, with corresponding shadows ${\mathcal B}_{\{j\}}$
containing no other orbit. We also call the corresponding boundary
degeneration components simple.
A {\em simple boundary degeneration} is a
curve all of whose components are either trivial or simple west
components.
When $w$ is a simple boundary degeneration, and $\ev(w)\in
\Chamber^{\orb_k>\orb_j}$, we call $\orb_k$ the {\em top orbit} and
$\orb_j$ the {\em bottom orbit}; see
Figure~\ref{fig:ExtractBoundaryDegeneration}.
\begin{figure}[h]
\centering
\input{ScaleOut.pstex_t}
\caption{{\bf Extracting a boundary degeneration.}
As the two orbits on the left come together, we can rescale
(and rotate) to construct a boundary degeneration as pictured on the right.}
\label{fig:ExtractBoundaryDegeneration}
\end{figure}
\subsection{Gluing}
The ends of two-dimensional moduli spaces stated in
Theorem~\ref{thm:AEnds} are modeled by certain gluing results, which
we collect here; compare~\cite[Section~5.5]{InvPair}.
The following is~\cite[Proposition~5.39]{InvPair}. To state it, use
the following terminology from there. If $(u,v)$ is a simple
holomorphic comb, with $v\in\ModEast(\EastSource)$,
a {\em smeared neighborhood} of $(u,v)$ in
$\ClosedModFlow^{B}(\mathbf x,\mathbf y,\Source,P)$ is an open neighborhood of
\[\{ (u,v')|v'\in\ModEast(\EastSource),
(u,v')\in\ClosedModFlow^B(\mathbf x,\mathbf y;\Source;P)\}.\] Also, given
$p,q\in\EastSource$, let $\compactev_{p,q}\colon
\ClosedModFlow(\mathbf x,\mathbf y;\Source)\to [-\infty,\infty]$ be the
continuous extension of $\ev_{p,q}$.
\begin{prop}
Suppose that $(u,v)$ is a simple holomorphic comb in
${\ClosedModFlow}^B(\mathbf x,\mathbf y;\Source, \Partition)$. Assume
that $v$ is a split curve and there are two parts $P_1$ and $P_2$
such that for each split component of $\EastSource$, its bottom
puncture belongs to $P_1$ and its top puncture belongs to
$P_2$. Assume that $\ind(B,\mathbf x,\mathbf y;\Source,P)=2$. Let
$q_1\in \Partition_2$ and $q_2\in \Partition_1$ be the top and
bottom punctures, respectively, on one of the split components of
$\EastSource$. Then for generic $J$, there is a smeared neighborhood
$U$ of $(u,v)$ in
${\ClosedModFlow}^B(\mathbf x,\mathbf y;\Source, \Partition)$ so that
$\compactev_{q_1,q_2}\colon U \to \R_+$ is proper near $0$ and of
odd degree near $(u,v)$.
\end{prop}
The above is proved in~\cite[Proposition~5.39]{InvPair}.
The following result is also used in~\cite{InvPair} (see especially
the proof of ~\cite[Theorem~5.61]{InvPair}):
\begin{prop}
\label{prop:JoinCurve}
Let $\Source$ be a source curve equipped with some ordered partition
$\Partition$, and $\Partition'$ be the ordered partition
where consecutive packets $P_j$ and $P_{j+1}$ in $P$ collide.
Suppose that $\ind(B,\mathbf x,\mathbf y;\Source',\Partition')=2$.
Suppose that $(u,v)$ is a simple holomorphic comb in
$\ClosedModFlow^B(\mathbf x,\mathbf y;\Source',\Partition')$, so that $u\in
\ModFlow^B(\mathbf x,\mathbf y;\Source,\Partition)$ is a smooth point and $v$
is a join curve.
Let $\Delta_P\subset \R^{\Partition}$ denote the diagonal
that specifies the collision of levels $j$ and $j+1$, and suppose that
$u$ is a smooth, isolated point in $\ModFlow(\mathbf x,\mathbf y;\Source,\Partition)\times_{\ev_{P}}\Delta_P$.
Then, there is a smeared neighborhood $U$ of $(u,v)\in\ClosedModFlow^B(\mathbf x,\mathbf y,\Source';\Partition')$
with the property that $U\times_{\ev_{P'}}\Delta_{P'}$
is homeomorphic to $[0,1)$.
\end{prop}
\begin{proof}
Note first that since the domain of $v$ is a planar surface, $v$
represents a smooth point in its moduli space
$\ModEast$. (See~\cite[Proposition~5.16]{InvPair}.) The hypothesis
that $u$ is a smooth point includes the statement that the
evaluation map is transverse to the diagonal where the two chords to
be joined are mapped to the same $t$-position. Thus, $u$ and $v$ have neighborhoods
$U_u$ and $U_v$, so that $(u,v)$ is a
transverse intersection point of $\ev\colon U_u\to \R^{\East(\Source)}$ and
$\ev\colon U_v\to \R^{\West(\EastSource)}$. Thus, gluing gives an
identification of a neighborhood of $(u,v)$ that is identified with
$U_u\times_{\ev} U_v\times [0,1)$. In fact, the image of $\ev(U_v)$
is the diagonal $\Delta_P$ (that determines the portion of
$\ModFlow^B(\mathbf x,\mathbf y,\Source;P')$ where the two consecutive packets
$P_j$ and $P_{j+1}$ collide). Thus, gluing identifies a
neighborhood of $(u,v)$ with $(U_u\times_{\ev_P} \Delta_P)\times
[0,1)$. By assumption, $U_u\times_{\ev_P}\Delta_P$ consists of the point $u$.
\end{proof}
In a similar vein, we have the following analogue for orbit curves.
To state it, fix some $q_0\in \Source$, and let $p\in \Source$ be a puncture marked by an orbit.
The map
\begin{equation}
\label{eq:EvAtP}
s\circ \ev_{\{p\}} \colon \ModFlow(\mathbf x,\mathbf y;\Source)\to (0,1)
\end{equation}
specified by $s\circ \ev_p(u)=s\circ u(q)$, extends continuously to a map
\[ {\overline {s\circ \ev_{\{p\}}}}\colon \ClosedModFlow(\mathbf x,\mathbf y;\Source) \to [0,1].\]
\begin{prop}
\label{prop:OrbitCurve}
Suppose that $(u_0,v)$ is a simple holomorphic comb in
${\ClosedModFlow}^B(\mathbf x,\mathbf y;\Source)$, and assume that $v$ is an orbit
curve with orbit $\orb_j$. Let $\Source_0$ denote the source for $u_0$;
it has a distinguished boundary puncture $p_0$ labelled by some
length one chord $\longchord_j$; and $\Source$ is obtained by
replacing this boundary puncture with an interior puncture $p$ labelled by
$\orb_j$. Choose also the following data:
\begin{itemize}
\item a set of punctures $P$ on $\Source$; and let $P_0$ be
the corresponding collection of punctures on $\Source_0$
(i.e. with $p$ replaced by $p_0$)
\item a smooth manifold $M$
\item a smooth map $\phi\colon M\to \R^{P}\cong \R^{P_0}$
\item a point $u_0\times m_0\in \ModFlow(\mathbf x,\mathbf y;\Source_0)\times_{\ev_{P_0}} M$.
\end{itemize}
Suppose that:
\begin{itemize}
\item
$\phi$ is transverse to
$\ev_{P}\colon \ModFlow^B(\mathbf x,\mathbf y;\Source_0)\to \R^{P_0}=\R^{P}$
\item $\phi$ is transverse to
$\ev_{P}\colon
\ModFlow^B(\mathbf x,\mathbf y;\Source)\to \R^{P}$
\item the fibered product
$\ModFlow^B(\mathbf x,\mathbf y;\Source_0)\times_{\ev_{P_0}} M$ is
two-dimensional.
\item
$u_0\times m_0$ is an (isolated) point in
$\ModFlow^B(\mathbf x,\u;\Source_0)\times_{\ev_{P_0}} M$.
\end{itemize}
Then, there is a smeared
neighborhood $U$ of $(u_0,v)$ in $\ClosedModFlow^B(\mathbf x,\mathbf y;\Source)$
so that
$U\times_{\compactev_{P_0}} M$ is homeomorphic to
$u_0,\times m_0 \times (0,1]=(0,1]$. Moreover,
\[\begin{tikzpicture}
\node at (0,0) (X) {$u_0\times m_0\times (0,1] \subset
\ClosedModFlow^B(\mathbf x,\mathbf y;\Source)\times_{\compactev_{P_0}} M$};
\node at (5,0) (Y) {$[0,1]$};
\draw[->] (X) to node[above]{\begin{tiny} $\overline{s\circ \ev}_{\{p\}}$\end{tiny}} (Y);
\end{tikzpicture}\]
is proper and of odd degree near
$1$.
\end{prop}
\begin{proof}
This proof is similar to the proof of
Proposition~\ref{prop:JoinCurve}. Again,
the moduli space $\ModEast$ for an orbit curve is smooth. In this
case, the evaluation map
\[ \ev\colon \ModEast(\EastSource,P)\to \R^{P} \] is a
diffeomorphism near $v$, so gluing gives a smeared neighborhood of
$(u_0,v)$ in $\overline{\ModFlow}(\mathbf x,\mathbf y;\Source)$
which is homeomorphic to $U_0\times (0,1]$, where $U_0$ is a neighborhood
of $u_0$ in $\ModFlow(\mathbf x,\mathbf y;\Source_0)$.
Similarly,
$(U_0\times (0,1])\times_{\ev_{P_0}}M$ is identified with
$(U_0\times_{\ev_{P_0}} M)\times (0,1]$.
Since $p$ is an interior puncture, it follows that
$s\circ \ev_{p}(u\times m\times t)<0$ for $t<1$;
it is straightforward also to see that
$s\circ \compactev_{p}(u\times m\times 1)=1$. The degree statement follows.
\end{proof}
The following is similar to Proposition~\ref{prop:OrbitCurve}. To
state it, we generalize the notion of smeared neighborhood in the
following straightforward manner to simple combs $(w,u)$ where $w$ is
a curve at West infinity.
Let $(w,u)$ be a simple holomorphic curve with $w\in\ModWest(\WestSource)$,
and $u$ a pseudo-holomorphic flowline with west punctures.
A {\em smeared neighborhood} of $(w,u)$ in is an open neighborhood
in $\ClosedModFlow^B(\mathbf x,\mathbf y,\Source)$ of the set
\[ \{(w,u')\in\ModWest(\WestSource),(w,u)\in\ClosedModFlow(\mathbf x,\mathbf y,\Source)\}\]
In this neighborhood, $u$ is to be thought of as a curve with West punctures
(over the $t$-value where it meets $w$).
\begin{prop}
\label{prop:BoundaryDegenerationNbd}
Suppose that $(w,u_0)$ is a simple holomorphic comb in
${\ClosedModFlow}^B(\mathbf x,\mathbf y;\Source)$, and assume that $w$ is a simple
boundary degeneration with some puncture $p$ labelled by an orbit
$\orb_j$. Let ${\mathbf q}$ denote the $d$-tuple of west punctures in $u_0$.
Fix also the following data:
\begin{itemize}
\item a smooth manifold $M$
\item a smooth map $\phi\colon M\to \R^{P_0}$, where $P_0$ denotes the
set of punctures which do not appear in the boundary degeneration
component
\item an auxiliary puncture $q_0\in P_0$ with $t\circ u_0(q_0)=t \circ u_0(q)$
for all $q\in {\mathbf q}$.
\end{itemize}
Assume that:
\begin{enumerate}[label=(W-\arabic*),ref=(W-\arabic*)]
\item $\phi$ is transverse to
\[ \ev_{P_0}\colon \ModFlow^{B_0}(\mathbf x,\mathbf y;\Source_0)\to \R^{P_0}, \]
where $\Brs+B_0=B$ and $\Brs$ is the shadow of $w$.
\item
$\phi\times 0 \colon M \to \R^{P_0}\times \R$ is transverse to
\[ \ev_{P_0} \times (\ev_{p_1}-\ev_{q_0}) \colon \ModFlow^{B}(\mathbf x,\mathbf y;\Source)\to \R^{P_0}\times \R\]
\item
the fibered product
$\ModFlow^{B}(\mathbf x,\mathbf y;\Source)\times_{\ev_{P_0}\times
\ev_{p_1}-\ev_{q_0}} (M\times 0)$ is two-dimensional,
\item
$(w\times u_0)\times m_0$ is
a point in $\ClosedModFlow(\mathbf x,\mathbf y;\Source)\times_{\ev_{P_0}}M$.
\end{enumerate}
Then there is a smeared neighborhood $U$ of $(w,u_0)$ in
$\ClosedModFlow^B(\mathbf x,\mathbf y;\Source)$ with the following properties:
\begin{itemize}
\item The map ${\overline{s\circ \ev_{p}}}\colon U\times_{\ev_{P_0}} M \to [0,1]$
is proper near $0$.
\item The above map has odd degree near $(w,u_0)$.
\end{itemize}
\end{prop}
\begin{proof}
Each simple boundary degeneration $w$ is transversely cut out in the
moduli space $\UnparModWest^{\Bjk}(\mathbf x;\WestSource)$. We are assuming
that $u$ is also transversely cut out in
$\UnparModFlow^{B_0}(\mathbf x,\mathbf y;\Source;P_2)$. It follows from standard
gluing results (see~\cite[Proposition~5.31]{InvPair}; see
also~\cite[Section~5.3]{Bourgeois}) that for sufficiently small open
neighborhoods $U_w$ and $U_{u_0}$ of $w$ and $u_0$, there is an open
neighborhood in
$\ClosedModFlow^B(\mathbf x,\mathbf y;\WestSource\natural\Source_0;P)$ which is
homeomorphic to $(U_w\times_{\evB} U_{u_0})\times [0,1)\times
(-1,1)$. Here, the interval $[0,1)$ parameterizes the gluing
parameter, and the interval $(-1,1)$ parameterizes the $t$ value of
the $d$-tuple of west punctures over $u_0$. Since $\evB\colon
U_w\to {\mathbb T}_{\betas}$ has odd degree
(Lemma~\ref{lem:BoundaryDegenerationsDegree1}), this neighborhood is
identified with an odd number of copies of $U_{u_0}\times
[0,1)\times (-1,1)$. Choosing $U_{u_0}$ sufficiently small that
$U_{u_0}\times_{\ev_{P_0}} M$ consists of the single point
$(u_0,m_0)$, we have that the smeared neighborhood $U$ of $(w,u_0)$
has the property that is $U\times_{\ev_{P_0}} M$ is identified with
an odd number of copies of $[0,1)\times (-1,1)$.
Pick one such component ${\overline {\mathcal M}}$, and
let ${\mathcal M}$ denote its interior.
Consider the
maps
\[ F_1=s\circ \ev_{p} \colon {\mathcal M}\to (0,1)
\qquad{\text{and}}\qquad F_2 =\ev_{p}-\ev_{q_0}\colon {\mathcal M}\to \R. \]
The maps $F_1$ and $F_2$ extend
continuously to give maps
${\overline F}_i \colon {\overline{\mathcal M}} \to \R$ for $i=1,2$;
and in fact $F_1\colon {\overline{\mathcal M}}\to [0,1)$ is proper near $0$.
Clearly, the restriction of ${\overline F}_2$ to
$0\times (-1,1)$ has odd degree near
$0$. Moreover, since $p$ is an interior puncture, $F_1(s,t)>0$ for
all $s>0$ and $t\in (-1,1)$. Also, ${\overline F}_1(0,t)=0$. It
follows that $F\colon [0,1)\times (-1,1)\to \R\times \R$ has odd
degree. Clearly, the neighborhood of
\[ {\mathcal M}\times_{\ev_{P_0}\times
(\ev_{p}-\ev_{q_0})} (M\times 0) \] is given as $F_2^{-1}(0)$, and
the restriction of ${s\circ\ev_p}$ to this set is $F_1$. It follows
that the degree of $s\circ \ev_p$ to $U\times_{\ev_{P_0}} M$ agrees
with an odd multiple of the degree of $F_1|_{F_2^{-1}(0)}$, which
agrees with the degree of $F$, which we have shown to be odd.
\end{proof}
In the above proposition, we measured the distance of the boundary
degeneration by measuring $s\circ u(p)$, where $p$ is the puncture
which goes into the boundary degeneration. In the next very similar
proposition, we measure the distance to the boundary by $t\circ
u(p_1)-t\circ u(p_2)$, where now $\{p_1,p_2\}$ are the two punctures
which go into the boundary degeneration.
\begin{prop}
\label{prop:WestInftyEnd}
Suppose that $(w,u_0)$ is a simple holomorphic comb in
${\ClosedModFlow}^B(\mathbf x,\mathbf y;\Source)$, and assume that $w$ is a simple
boundary degeneration with exactly two punctures $p_1$ and $p_2$,
labelled by orbits
$\orb_j$ and $\orb_k$ respectively. Fix also the following data:
\begin{itemize}
\item a smooth manifold $M$
\item a smooth map $\phi\colon M\to \R^{P_0}$, where $P_0$ denotes the
set of punctures which do not appear in the boundary degeneration
component
\item an auxiliary puncture $q_0\in P_0$ with $t\circ u(q_0)=t \circ u_0(q)$
for all $q\in {\mathbf q}$.
\end{itemize}
Assume that:
\begin{itemize}
\item $\phi$ is transverse to
\[ \ev_{P_0}\colon \ModFlow^{B_0}(\mathbf x,\mathbf y;\Source_0)\to \R^{P_0}, \]
where $\Brs+B_0=B$ and $\Brs$ is the shadow of $w$.
\item $\phi$ is transverse to
\[ \ev_{P_0}\times (\ev_{p_1}-\ev_{q_0})\colon
\ModFlow^{B}(\mathbf x,\mathbf y;\Source)\to \R^{P_0}\times \R, \]
where $\Brs+B_0=B$ and $\Brs$ is the shadow of $w$.
\item
the fibered product
$\ModFlow^{B}(\mathbf x,\mathbf y;\Source)\times_{\ev_{P_0}\times
\ev_{p_1}-\ev_{q_0}} (M\times 0)$ is two-dimensional,
\item
$(w\times u_0)\times m_0$ is
a point in $\ClosedModFlow(\mathbf x,\mathbf y;\Source)\times_{\ev_{P_0}}M$.
\item The map $\evB_{\mathbf q}\colon \ModFlow(\mathbf x,\mathbf y;\Source)\to \Tb$
is transverse to the wall $\Wall^{\orb_j=\orb_k}$.
\end{itemize}
Then there is a smeared neighborhood $U$ of $(w,u_0)$ in
$\ClosedModFlow^B(\mathbf x,\mathbf y;\Source)$ with the following properties:
\begin{itemize}
\item Let $f\colon U\times_{\ev_{P_0}\times (\ev_{p_1}-q_0)} (M\times 0)
\to \R$ be defined by
$u\mapsto t\circ u(p_1)-t\circ u(p_2)$ is proper,
sending the portion of $U$ in the interior $\ModFlow^B(\mathbf x,\mathbf y;\Source)$
to $\R\setminus 0$, and mapping ideal combs to $0$.
\item
if $\evB_{\mathbf q}(u_0)$ is in the $\Chamber^{\orb_j>\orb_k}$, then
$f$ has odd degree near $(w,u_0)$
for $\epsilon>0$ and even degree for $\epsilon<0$;
if $\evB_{\mathbf q}(u_0)$ is in $\Chamber^{\orb_j<\orb_k}$
$f$ has even degree for $\epsilon>0$ and odd degree for $\epsilon<0$.
\end{itemize}
\end{prop}
\begin{proof}
As in the proof of Proposition~\ref{prop:BoundaryDegenerationNbd},
we can choose a smeared neighborhood $U$ so that
$U\times_{\ev_{P_0}} M$ is identified with an odd number of copies
of $[0,1)\times (-1,1)$, indexed by simple boundary degenerations
$\{w_i\}$. Since $\evB_{\mathbf q}$ is not on a wall, it follows
that that for each such boundary degeneration, $t\circ w_i(p_1)-t
\circ w_i(p_2)$ is non-zero. Thus, we can give each component of
$U\times_{\ev_{P_0}} M$ an {\em associated sign}, which is positive
or negative, according to the sign of the difference $t\circ
w_i(p_1)-t\circ w_i(p_2)$.
Choosing $U$ sufficiently small, we can ensure
that $t\circ u(p_1)-t\circ u(p_2)$ is non-zero; and in fact,
on each component ${\mathcal M}$, the sign of $t \circ u(p_1)-t \circ(p_2)$
is determined by the associated sign of the component.
Pick any component ${\overline {\mathcal M}}$ with positive sign,
and let ${\mathcal M}$ denote its interior.
Consider the maps
\[ F_1=t \circ \ev_{p_1} - t\circ \ev_{p_2}={\mathcal M}\to \R^{>0} \qquad{\text{and}}\qquad F_2
=\ev_{p_1}-\ev_{q_0}\colon {\mathcal M}\to \R. \] The maps $F_1$ and
$F_2$ extend continuously to ${\overline{\mathcal M}}$
to give maps ${\overline F}_1 \colon {\overline{\mathcal M}} \to \R^{\geq 0}$ and
${\overline F}_2\colon {\overline{\mathcal M}}\to \R$.
Clearly, the restriction of ${\overline F}_2$ to
\[ 0\times
(-1,1)=\partial {\overline{\mathcal M}}\]
has odd degree near
$0$. Moreover, $F_1(s,t)>0$ for all $s>0$ and $t\in (-1,1)$,
with ${\overline F}(0,t)=0$. It follows that
${\overline F}\colon {\overline{\mathcal M}}\to [0,1)\times (-1,1)$
has odd degree; and this is the same as the degree of the restriction
$F_1|_{F_2^{-1}(0)}$.
If the component ${\overline{\mathcal M}}$ has negative associated sign,
the map $F_1=t\circ \ev_{p_1}-t\circ \ev_{p_2}$ maps to
$\R^{<0}$, and indeed
the above argument shows that
\[ {\overline F}\colon {\overline{\mathcal M}}\to \R^{\leq 0}\times \R \]
has odd degree over the origin.
Now, if $\evB_{\mathbf q}(u_0)\in \Chamber^{\orb_j>\orb_k}$, there
is an odd number of positive associated components and an even
number of negative associated ones; whereas if $\evB_{\mathbf
q}(u_0)\in \Chamber^{\orb_j<\orb_k}$, there is an odd number of
negative associated components and an even number of positive ones.
The degree statement now follows.
\end{proof}
\subsection{Ends of one-dimensional moduli spaces}
\begin{proof}[Proof of Theorem~\ref{thm:AEnds}]
Consider the limiting comb in the end of the one-dimensional moduli space
${\widehat{\mathcal M}}$, whose existence is guaranteed by
Gromov compactness (cf. Proposition~\ref{prop:Compactness}).
This limit contains no $\alpha$-boundary degenerations, because such
a degeneration covers both basepoints $\wpt$ and $\zpt$.
Suppose next that a $\beta$-boundary degeneration occurs in the
limit. By the dimension formula, the boundary degeneration must be
simple (otherwise, the remaining curve has codimension greater than
one). There are two cases, according to whether or not the boundary
degeneration is special.
Suppose first that boundary degeneration is not special. Then there
are two different orbits $\orb_j$ and $\orb_k$ on the boundary
degeneration, and, by the hypotheses on allowed constraint packets,
it follows that the boundary degeneration occurs as two packets
collide. Moreover, since $\orb_j$ and $\orb_k$ are matched in
$\Mdown$, and our packets are allowed, one of the two orbits is
even, and so it is constrained to lie at the same level as a
chord. Thus, the collision is a ``boundary degeneration collision''
in the sense of Definition~\ref{def:BoundaryDegCollision}. Moreover,
the hypotheses of Proposition~\ref{prop:BoundaryDegenerationNbd} are
satisfied, using for $M$ the diagonal which specifies the chord
packets. That proposition now ensures that there is an odd number of
such ends, and they are all in the chamber as specified
in~\ref{endA:BoundaryDegeneration}.
Suppose that the boundary degeneration is special. There are two
subcases, in the notation
of~\ref{endA:SpecialBoundaryDegeneration}, according to whether or not
$\sigmas=\rhos_i\setminus\{o_k\}$ is empty.
If $\sigmas$ is non-empty, we can
apply Proposition~\ref{prop:WestInftyEnd}, to find an odd number of
ends. If $\sigmas$ is empty, the dimension formula ensures that the
limiting curve lies in a moduli space with expected dimension
zero. This moduli space is empty, unless it corresponds to the
constant flowline. That is the case where $\ell=1$, $\mathbf x=\mathbf y$, and
$B={\mathcal B}_{\{k\}}$. These ends occur with an odd multiplicity,
where we glue the constant flowline at $\mathbf x=\mathbf y$ to the odd number of
simple boundary degenerations $w$ with $\evB(w)=\mathbf x$,
according to Lemma~\ref{lem:BoundaryDegenerationsDegree1}.
This completes the cases where $\beta$-boundary
degenerations occur.
The classification of the remining ends follows very similarly to the proof
of~\cite[Theorem~5.61]{InvPair}.
In the present case, we have the possibility of orbit curves forming.
Each such end appears with odd multiplicity by Proposition~\ref{prop:OrbitCurve}.
The fact that the chords in the collision are disjoint from one
another (Condition~\ref{c:Disjoint}) follows from the dimension
formula: combining the index formula the computation from
Example~\ref{ex:CollisionEnd}, it follows that boundary-monotone
collisions between non-disjoint constraint packets occur in
codimension greater than one.
As in~\cite{InvPair}, boundary monotonicity and the non-existence
of $\alpha$-boundary degenerations implies that in every collision,
the joined chords are strongly composable.
We argue that collisions that are not visible occur with even
multiplicity. To see this, observe that if the limiting curve has
two punctures $q_1$ and $q_2$, both of which are marked by the same
orbit $\orb_i$, and $t(u(q_1))=t(u(q_2))$, then that curve can be
obtained as a limit of curves in
$\ModFlow(\mathbf x,\mathbf y,\Source,\rhos_1,\dots,\rhos_\ell)$, so that $q_1$
belongs to the packet $\rhos_i$ (and $q_2$ to $\rhos_{i+1}$); or it
can be obtained as a limit of curves in the same moduli space but
where $q_2$ belongs to $\rhos_{i+1}$. Thus, these two ends cancel.
For the join curve ends as in Case~\ref{endA:Join}, the
decomposition must satisfy Property~\ref{OneIsShort}; for other
decompositions occur in larger codimension than one. (See the
computation from Example~\ref{ex:JoinCurveEnd}.)
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:DEnds}]
This follows as in the proof of Theorem~\ref{thm:AEnds}.
The key difference is that in the present case, orbits are never constrained
to lie in the same level as other Reeb chords;
and indeed if $j$ and $k$ are matched in $\Mup$,
i.e. both appear in a boundary degeneration, only one of $\orb_j$ or $\orb_k$
is allowed to appear in a constraint packet. It follows that the only boundary
degenerations that can appear are special ones. The dimension formula once
again ensures that the remaining curve, in this case, is a constant.
\end{proof}
\section{Holomorphic curves used for type $D$ structures}
\label{sec:CurvesD}
We will describe now the holomorphic curves that go into the
construction of the type $D$ structure associated to an upper diagram,
returning to the type $A$ structure in Section~\ref{sec:CurvesA}. The
curves counted in the present work are similar to the curves counted in~\cite{InvPair}. Since our context here is slightly
different, we recall material from~\cite{InvPair}, with an emphasis on
the differences.
Fix some upper
diagram \[\Hup=(\Sigma_0,Z_1,\dots,Z_{2n},\{\alpha_1,\dots,\alpha_{2n-1}\},\{\alpha^c_1,\dots,\alpha^c_{g}\},
\{\beta_1,\dots,\beta_{g+n-1}\}).\]
Filling in the boundary of $\Sigma_0$ as explained in Section~\ref{sec:FillSurface},
we get $\Sigma_0\subset \Sigma\subset \cSigma$.
We will use Lipshitz's reformulation
of Heegaard Floer homology~\cite{LipshitzCyl}, where the
pseudo-holomorphic curve counting takes place in
$\Sigma\times[0,1]\times \R$. To this end, we will use the class of
almost-complex structures appearing there (see
also~\cite[Definition~5.1]{InvPair}), which we recall presently.
There are two projection maps
\[
\pi_{\Sigma}\colon \Sigma\times [0,1]\times \R \to \Sigma\qquad\text{and}\qquad
\pi_{\CDisk}\colon \Sigma\times [0,1]\times \R \to [0,1]\times \R.
\]
The last projection map $\pi_{\CDisk}$ can be further decomposed into
its components
\[ s\colon \Sigma\times [0,1]\times \R \to [0,1] \qquad{\text{and}}\qquad
t\colon \Sigma\times[0,1]\times \R\to \R. \]
\begin{defn}
\label{def:AdmissibleAlmostCx}
An almost complex structure $J$ on ${\overline\Sigma}\times [0,1]\times \R$ is
called {\em admissible} if
\begin{itemize}
\item The projection $\pi_\CDisk$ is
$J$-holomorphic.
\item
$J$ preserves the
subspace $\ker(d_p \pi_\Sigma)\subset T_p(\Sigma\times [0,1]\times \R)$.
\item The $\R$-action is $J$-holomorphic.
\item The
complex structure is split in some $\R$-invariant neighborhood of
\[\{p_1,\dots,p_{2n}\}\times [0,1]\times \R,\] where the $p_i$ are the punctures.
\end{itemize}
\end{defn}
We will consider $J$-holomorphic curves
\[ u\colon (S,\partial S)\to (\Sigma\times[0,1]\times \R,
(\alphas\times\{1\}\times \R)\cup(\betas\cup\{0\}\times \R)),\] with
certain asymptotic conditions. To state those, we view $\Sigma\times
[0,1]\times \R$ as having three kinds of infinities, $\Sigma\times
[0,1]\times\{+\infty\}$, $\Sigma\times [0,1]\times\{-\infty\}$, and
$\{p_i\}\times[0,1]\times \R$; the first two of these are
referred to as $+\infty$ and $-\infty$ respectively.
Let
\begin{equation}
\label{eq:DefineMultUpper}
d=g+n-1.
\end{equation}
The asymptotics
of the holomorphic curves we consider are as follows:
\begin{itemize}
\item At $\pm\infty$, $u$ is asymptotic to a $d$-tuple of
chords of the form $x_i\times[0,1]\times \{\pm \infty\}$,
where $\mathbf x=\{x_i\}_{i=1}^{d}$ is an upper Heegaard state.
\item For boundary punctures $p_i$,
at $\{p_i\}\times [0,1]\times \R$, the curve $u$ is
asymptotic to a collection of Reeb chords
$\rho_i\times 1 \times t_i$; where $\rho_i$ is a Reeb chord in
$\partial\Sigma_0=Z_1\cup\dots\cup Z_{2n}$ with endpoints on
${\mathbf a}=\alphas\cap Z$. These ends are called {\em east
infinity} boundaries of $u$, and $t_i$ is called its {\em
height}.
\item For interior punctures $p_i$,
at $\{p_i\}\times [0,1]\times \R$, the curve $u$ is
asymptotic to a collection of Reeb orbits $\{\orb_i\}\times
s_i\times t_i$ for $0<s_i<1$, where $\orb_i$ is the simple Reeb orbit
corresponding to the puncture $p_i$. These ends are called {\em
middle infinity}, and the values $t_i$ are also called their
{\em height}.
\end{itemize}
We give the details in this section.
\subsection{Naming Reeb chords}
\label{subsec:NamingChords}
A Reeb chord is an arc $\rho$ in some boundary component $Z_i$, with
endpoints on the intersection points between $Z_i$ and
$\alpha_{i-1}\cup\alpha_i$. As such, it has an initial point $\rho^-$
and a terminal point $\rho^+$.
When describing Reeb chords, we will use the following notation. Each
circle boundary component $Z_i$ with $i=2,\dots,2n-1$ meets two
$\alpha$-curves $\alpha_{i-1}$ and $\alpha_{i}$, and so the boundary
circle is divided into two Reeb chords. Label $L_i$ the chord that
goes from $\alpha_{i-1}$ to $\alpha_{i}$ with respect to the boundary
orientation of the circle, and $R_i$ the one which goes from
$\alpha_{i}$ to $\alpha_{i-1}$; see Figure~\ref{fig:ChordNames}. All
Reeb chords on $Z_i$ can thus be represented as words in the $L_i$ and
$R_i$ that alternate between the two letters. In particular, the two
Reeb chords that cover the circle once can be written as $L_i R_i$ and
$R_i L_i$; moreover, $L_i R_i$ starts and ends at $\alpha_{i-1}$ and
$R_i L_i$ starts and ends at $\alpha_{i}$.
The boundary component $Z_1$ meets only one $\alpha$-arc, $\alpha_1$;
and hence all Reeb chords are multiples of the same Reeb chord from
$\alpha_1$ to itself. For consistency with the above, we label this
basic Reeb chord, that covers $Z_1$ once, $R_1 L_1$ (although
independently, $R_1$ and $L_1$ do not make sense); similarly, we label
the Reeb chord that covers $Z_{2n}$ once $L_{2n} R_{2n}$.
\begin{figure}[h]
\centering
\input{ChordNames.pstex_t}
\caption{{\bf Names of Reeb chords.}
The Reeb chord $R_i$ is indicated by the oriented half circle.}
\label{fig:ChordNames}
\end{figure}
\subsection{Pre-flowlines}
We start with a more precise formulation of the asymptotic conditions for our holomorphic curves.
\begin{defn}
\label{def:DecoratedSource}
A {\em decorated source} $\Source$ is the following collection of data:
\begin{enumerate}
\item a smooth oriented surface $S$ with boundary and punctures
(some of which can be on the boundary)
\item a labeling of each boundary puncture of $S$ by one of $+$, $-$, or $\east$
\item a labeling of each $\east$ puncture on $S$ by a Reeb chord
\item a labeling of each interior puncture by a Reeb orbit.
\end{enumerate}
Let $\East(\Source)\subset S$ denote the set of punctures marked $\east$;
$\IntPunct(\Source)\subset S$ denote the set of interior punctures,
and $\AllPunct(\Source)=\East(\Source)\cup\IntPunct(\Source)$.
\end{defn}
\begin{remark}
In Section~\ref{subsec:CurvesAtWest}, our decorated sources will
also include boundary punctures marked by $w$. Unlike the $\east$
punctures, such punctures are not labelled by Reeb chords.
\end{remark}
We recall what it means for a map to be asymptotic to a given Reeb
chord $\rho$ at $q$. Suppose that $S$ is a decorated source with a
puncture $q$ on its boundary, thought of as a point in $\cS$.
$f\colon S \to \Sigma$ is a smooth map with a continuous extension
${\overline f}\colon \cS\to \cSigma$, so that $f(q)=z_i$. Let
\[ \CDisk^+=\{z=x+iy\in\C\mid |z|\leq 1, y\geq 0\}\] and let
$\phi\colon \CDisk^+ \to {\overline\Source}$ be a holomorphic
parameterization around the puncture $q$, i.e. so that
$\phi(0)=q$. If the chord $\rho$ is supported in $Z_i$, consider the
identification of the corresponding puncture in $\Sigma$,
\[ \psi\colon S^1 \times [0,\infty) \to \Sigma, \] with image
$Z_i\times [0,\infty)\subset \Sigma$.
We say that {\em $u$ is asymptotic to $\rho$ at the
puncture $q$} if the family of function $[0,1]\to S^1$ indexed
by $r\in (0,1]$ specified by
\[ \theta\mapsto \pi_{S^1} \circ \psi^{-1}\circ \pi_{\Sigma} \circ u\circ
\phi(r e^{\pi i\theta}) \] converges to $\rho$ as
${\mathcal C}^{\infty}$ functions from $[0,1]$ to $S^1$, as $r\goesto 0$.
This definition has a straightforward adaptation to
interior punctures $q$ in $\Sigma$, where $\orb$ is some Reeb orbit.
In that case, we choose a parameterization around $q$ by the punctured disk
$\{z\in \C\setminus\{0\}\big||z|\leq 1\}$ about $q$, and
we require that the ${\mathcal C}^{\infty}$ functions
from $S^1\to S^1$ indexed by $r\in (0,1]$ defined by
\[ \theta\mapsto \pi_{S^1} \circ \psi^{-1}\circ \pi_{\Sigma} \circ u\circ
\phi(r e^{2 \pi i\theta}) \]
converge to the given Reeb orbit as $r\goesto 0$.
A {\em generalized upper Heegaard state} is a $d$-element subset of
points $\mathbf x=\{x_i\}_{i=1}^d$ in $\Sigma_0$, each of which is contained
in the intersection of the various $\alpha$-and $\beta$-curves,
distributed so that each $\beta$-circle contains exactly one point in
$\mathbf x$, each $\alpha$-circle contains exactly one some point in $\mathbf x$,
and no more than one point lies on any given $\alpha$-circle. Note
that a generalized upper Heegaard state can have more than one element
on the same $\alpha$-arc.
Analogous to~\cite[Section~5.2]{InvPair},
given a decorated source $\Source$, we consider maps as follows:
\begin{defn}
\label{def:GenFlow}
A {\em pre-flowline} is a map
\[ u\colon (\Source,\partial\Source)\to
(\Sigma\times[0,1]\times\R,(\alphas\times\{1\}\times \R)\cup(\betas\times\{0\}\times\R))\]
subject to the constraints:
\begin{enumerate}[label=(${\mathcal M}$-\arabic*),ref=(${\mathcal M}$-\arabic*)]
\item
\label{property:First}
$u\colon \Source\to \Sigma\times[0,1]\times \R$ is proper.
\item
\label{property:ProperTwoa}
$u$ extends to a proper map ${\overline u}\colon
{\bSource}' \to {\overline{\Sigma}}\times[0,1]\times \R$,
where $\bSource'$ is obtained from $\Source$ by filling in the interior
and the $\east$
punctures
(so $\bSource\subset \bSource'\subset \cS$)
\item
\label{prop:BrCover}
$\pi_{\CDisk}\circ u$ is a $d$-fold branched cover
(with $d$ as in Equation~\eqref{eq:DefineMultUpper}).
\item At each $-$-puncture $q$ of $\Source$,
$\lim_{z\goesto q}(t\circ u)(z)=-\infty$.
\item At each $+$-puncture $q$ of $\Source$,
$\lim_{z\goesto q}(t\circ u)(z)=+\infty$.
\item
At each $\east$-puncture $q$ of $\Source$,
$\lim_{z\to q}(\pi_{\Sigma}\circ u)(z)$ is the Reeb
chord $\rho$ labeling $q$.
The same holds for
middle infinity punctures $q$, with limits
to the corresponding Reeb orbit.
\item \label{prop:FiniteEnergy}
There are a generalized upper Heegaard states $\mathbf x$ and $\mathbf y$
with the property that as $t\goesto - \infty$,
$\pi_{\Sigma}\circ u$ is asympotic to $\mathbf x$ and
as $t\goesto +\infty$, $\pi_{\Sigma}\circ u$ is asymptotic
to $\mathbf y$.
\setcounter{bean}{\value{enumi}}
\item
\label{prop:WeakBoundaryMonotone}
For each $t\in \R$ and $i=1,\dots,d$,
$u^{-1}(\beta_i\times\{0\}\times\{t\})$ consists of exactly one point.
Similarly, for each $t\in \R$ and each $i=1,\dots,g$,
$u^{-1}(\alpha_i^c\times\{1\}\times\{t\})$ consists of exactly one point.
\end{enumerate}
\end{defn}
By the definition of branched covers of manifolds with boundary, Condition~\ref{prop:BrCover}
ensures that $\pi_{\CDisk}\circ u$ (and indeed $t\circ u$) has no critical points over
$\partial [0,1]\times \R$.
\begin{defn}
Let $\mathbf x$ and $\mathbf y$ be generalized upper Heegaard states, and $u$ a pre-flowline
that connects them.
The {\em Reeb asymptotics} of $u$ is the ordered partition of Reeb chords
$\vec{P}=(P_1,\dots,P_{\ell})$ appearing in the asymptotics of $u$,
ordered by the value of $t\circ u$.
\end{defn}
Property~\ref{prop:WeakBoundaryMonotone} is called {\em weak boundary
monotonicity}. We will often consider curves satisfying the
following additional condition, called {\em strong boundary
monotonicity}:
\begin{enumerate}[label=(${\mathcal M}$-\arabic*s),ref=(${\mathcal M}$-\arabic*s)]
\setcounter{enumi}{\value{bean}}
\item
\label{property:StronglyBoundaryMonotone}
For each $t\in \R$ and $i=1,\dots 2n-1$,
$u^{-1}(\alpha_i\times\{1\}\times \{t\})$ consists of at most
one point.
\setcounter{bean}{\value{enumi}}
\end{enumerate}
If $u$ satisfies this stronger condition, then $u$ is asymptotic to
upper Heegaard states $\mathbf x$ and $\mathbf y$ over $-\infty$ and $+\infty$ respectively.
\begin{rem}
\label{rem:FlowsAsDisks}
Pre-flows can be thought of as Whitney disks in
$\Sym^{g+n-1}(\cSigma)$, mapping one of the boundary arcs into the
smooth torus $\beta_1\times\dots\times \beta_{g+n-1}$, and the other
boundary arc $a$ into the singular space
$\alpha^c_1\times\dots\times \alpha^c_g\times \Sym^{n-1}(I)$, where
$I={\overline\alpha}_1\cup\dots\cup{\overline\alpha}_{2n-1}$. The
strong monotonicity condition guarantees that the arc $a$ in fact
maps into a smooth part of
$\alpha_1^c\times\dots\times\alpha_g^c\times \Sym^{n-1}(I)$.
\end{rem}
\subsection{On boundary monotonicity}
\label{subsec:BoundaryMonotone}
If $\rhos=\{\rho_1,\dots,\rho_m, \orb_1,\dots,\orb_k\}$ is a set of
Reeb chords and orbits, let $\rhos^-=\{\rho_1^-,\dots,\rho_m^-\}$ be
the multi-set (i.e. set with repeated entries) of initial points of
the Reeb chords, and $\rhos^+=\{\rho_1^+,\dots,\rho_m^+\}$ be the
multi-set of terminal points.
Strong boundary monotonicity can be
formulated in terms of the initial generalized Heegaard state $\mathbf x$ and
the Reeb asymptotics.
\begin{defn}
\label{def:StronglyBoundaryMonotone}
Let $\mathbf x$ be a generalized upper Heegaard state and
$(\rhos_1,\dots,\rhos_\ell)$, a sequence of sets of Reeb chords and
orbits. We formulate {\em strong boundary monotonicity} of
$(\mathbf x,\rhos_1,\dots,\rhos_\ell)$ inductively in $\ell$; at the same
time, we also define the {\em terminal $\alpha$-set} of a strongly
boundary monotone sequence
$\alpha(\mathbf x,\rhos_1,\dots,\rhos_\ell)\subset \{1,\dots,2n-1\}$, as
follows. When $\ell=0$, $(\mathbf x)$ is called strongly boundary monotone
if $\mathbf x$ is an upper Heegaard state; and its terminal $\alpha$-set is
defined to be the set of $i=1,\dots,2n-1$ so that $\mathbf x\cap
\alpha_i\neq \emptyset$. (Note that this definition of $\alpha(\mathbf x)$
coincides with the earlier definition given in
Definition~\ref{def:UpperState}.) For $\ell\geq 1$, we say that
$(\mathbf x,\rhos_1,\dots,\rhos_{\ell})$ is strongly boundary monotone if
all of the following conditions hold:
\begin{itemize}
\item The sequence $(\mathbf x,\rhos_1,\dots,\rhos_{\ell-1})$ is strongly boundary monotone.
\item No two points in $\rhos_{\ell}^-$
lies on the same $\alpha$-arc, and no two points in $\rhos_{\ell}^+$
lies on the same $\alpha$-arc.
\item Letting $A_-$ resp. $A_+\subset \{1,\dots,2n-1\}$
consist of all $i$ so that $\rhos_\ell^-\cap \alpha_i\neq \emptyset$ resp.
$\rhos_\ell^+\cap \alpha_i\neq \emptyset$, we require that
\[ A_-\subset \alpha(\mathbf x,\rhos_1,\dots,\rhos_{\ell-1}).\]
\item
The set
\[\alpha(\mathbf x,\rhos_1,\dots,\rhos_{\ell})=A_+\cup\Big(\alpha(\mathbf x,\rhos_1,\dots,\rhos_{\ell-1})\setminus A_-\Big)\]
consists of $n-1$ elements.
\end{itemize}
\end{defn}
Note that Condition~\ref{prop:BrCover} for a pseudo-holomorphic
flowline follows from the other conditions, as follows. It is clear
that $\pi_{\CDisk}\circ u$ is a pseudo-holomorphic map from $\Source$
to $[0,1]\times \R$. Since $\Source$ has positive and negative
punctures,
$t\circ u$ is not constant, so $\pi_{\CDisk}\circ u$ is a branched cover.
The degree of the branching is determined by Property~\ref{prop:FiniteEnergy}.
The following is a variant of~\cite[Lemma~5.53]{InvPair}:
\begin{lemma}
\label{lem:SBB}
Suppose that $u$ is a weakly boundary monotone
flowline from $\mathbf x$ to $\mathbf y$ with asymptotics specified by $\vec{\rhos}$.
Then, $(\mathbf x,\vec{\rhos})$ is
strongly boundary monotone if and only if $u$ is strongly boundary monotone.
\end{lemma}
\begin{proof}
Fix $\tau\in\R$, so $(t\circ u)^{-1}(\tau)$ contains none of the punctures of $\Source$, and
let
\[ \alpha(u,\tau)=\{i\in\{1,\dots,2n-1\}\big| u^{-1}(\alpha_i\times
\{1\}\times \{\tau\})\neq \emptyset\}.\] Let $q$ be some puncture on
$\Source$ labelled by $\rho$, a Reeb chord with $\rho^-$ on
$\alpha_i$ and $\rho^+$ on $\alpha_j$. Since $t\circ u$ is strictly
monotone on the arc through $q$ (in view of
Property~\ref{prop:BrCover}), it follows that for all sufficiently
small $\epsilon>0$, $i\in\alpha(u,t(q)-\epsilon)$ and
$j\in\alpha(u,t(q)+\epsilon)$. In fact, by continuity (and
induction on $\ell$), we see that
$\alpha(\mathbf x,\rhos_1,\dots,\rhos_{\ell-1})=\alpha(u,\tau)$ for all
$\tau$ with $t_{\ell-1}<\tau<t_\ell$, where $t_i$ denotes the
$t$-value of the punctures labelled by $\rhos_i$, and $t_0=-\infty$.
It follows easily that the two
formulations of boundary monotonicity coincide:
strong boundary monotonicity on $u$ is a condition on
$\alpha(u,\tau)$ and strong boundary monotonicity of $(\mathbf x,\vec{\rhos})$
is the corresponding condition on the $\alpha(\mathbf x,\rhos_1,\dots,\rhos_\ell)$.
\end{proof}
The following result will allow us to restrict attention to moduli
spaces containing only strongly boundary monotone sequences:
\begin{prop}
\label{prop:SBD}
Suppose that $\mathbf x$ and $\mathbf y$ are upper Heegaard states. If $u$ is a weakly,
but not strongly boundary monotone pre-flowline representing
$\phi\in\doms(\mathbf x,\mathbf y)$, then $b_0(\phi)$ is in the
ideal ${\mathcal J}\subset \BlgZ$; i.e. its image in
$\Blg$ vanishes.
\end{prop}
\begin{proof}
Let $X\subset \R$ denote the set of points $\tau\in\R$ for which
$u^{-1}(\alpha_i\times\{1\}\times \{\tau\})$ consists of more than one
point for some $i$. The set $X$ is bounded below since $u$ is asymptotic to
an upper Heegaard state $\mathbf x$ as $t\goesto -\infty$. Thus, it has an infimum
$\tau_0$. There must be some puncture $p\in
\partial\cS$ asymptotic to a Reeb chord $\rho$ that ends on
$\alpha_i$, with $t(u(p))=\tau_0$, and another point
$q\in\partial{\cS}$ with
$\pi_{\Sigma}(u(q))\in{\overline\alpha}_i$ and $t(u(q))=\tau_0$. The
initial point of $\rho$ cannot be on $\alpha_i$, for that would
violate boundary monotonicity for the portion of the curve
in
values $<\tau_0$.
Given a pre-flowline $u$ and generic $\tau<\tau_0$, we construct certain pure
algebra elements $b_\tau,c_\tau\in \BlgZ$ with $\Iup(\mathbf x)\cdot
b_{\tau} =b_{\tau}$, and
$b_0(\phi) = b_\tau\cdot c_\tau$.
The algebra element $b_\tau$ for any $\tau<\tau_0$ is specified by
its initial idempotent $\Iup(\mathbf x)$, and its weight, which is given
by the sum of the weights of all the Reeb chords and orbits in
$(t\circ u)^{-1}((-\infty,\tau))$. Note that $b_\tau =
\Iup(\mathbf x)\cdot b_\tau\cdot\Idemp{\mathbf x_\tau}$, where
\[ \mathbf x_\tau= \{1,\dots,2n-1\}\setminus \{i\mid \pi_{\Sigma}(u(1,\tau))\cap \alpha_i\neq \emptyset \}, \]
and $\Idemp{\mathbf x_\tau}$ denotes its corresponding idempotent.
Assume that the initial point of $\rho$ is on $\alpha_{i-1}$, so
that $\rho$ is of the form $L_i (R_i L_i)^k$. (The case where the
initial point of $\rho$ is on $\alpha_{i+1}$ will follow
similarly.) We could write
$b_0(\phi)=b_{\tau_0-\epsilon}\cdot c_{\tau_0-\epsilon}$, where $\Iup(\mathbf x)\cdot
b_{\tau_0-\epsilon} \cdot \Idemp{\mathbf x_{\tau_0-\epsilon}}=b_{\tau_0-\epsilon}$ and
$i-1,i\not\in\Idemp{\mathbf x_\tau}$.
There are two cases. Either $\weight_i(c_{\tau_0-\epsilon})\geq 1$, in
which case $c_{\tau_0-\epsilon}=U_i\cdot c'$ for some algebra
element $c'$; so $c_{\tau_0-\epsilon}\in{\mathcal J}$. If
$\weight_i(c_{\tau_0-\epsilon})=1/2$, then $c_{\tau_0-\epsilon}$ moves one
of its coordinates from $\geq i+1$ to $\leq i-1$, so once again
$c_{\tau_0-\epsilon}\in{\mathcal J}$.
\end{proof}
\subsection{Pseudo-holomorphic flows}
\begin{defn}
\label{def:HolFlow}
A {\em pseudo-holomorphic flowline} is a
pre-flow satisfying the following further hypothesis:
\begin{enumerate}[label=(${\mathcal M}$-\arabic*h),ref=(${\mathcal M}$-\arabic*h)]
\setcounter{enumi}{\value{bean}}
\item
\label{property:Holomorphic}
The map $u$ is $(j,J)$-holomorphic with respect to some fixed admissible
almost-complex structure $J$ (Definition~\ref{def:AdmissibleAlmostCx})
and complex structure $j$ on $\Source$.
\end{enumerate}
\end{defn}
Recall that $\cSigma$ is equipped with $2n$ points $z_1, \dots,
z_{2n}$. If $u$ is a pseudo-holomorphic flow, then
$f=\pi_{\Sigma}\circ u$ is a local branched cover over $z_i$, with
branching specified by the Reeb chords.
Generalized pseudo-holomorphic flowlines can be collected into
homology classes. Specifically, if $u$ is a pre-flowline from $\mathbf x$ to
$\mathbf y$, then the projection to $\Sigma$ induces a two-chain from $\mathbf x$ to
$\mathbf y$, in the sense of Definition~\ref{def:TwoChainFromXtoY}, obtained
from assembling the local multiplicities of $\pi_{\Sigma}\circ u$. We
call the two-chain so obtained $\Shadow(u)$, the {\em shadow} of $u$.
Fix an admissible almost-complex structure $J$.
We will consider moduli spaces $\ModFlow^B(\mathbf x,\mathbf y;\Source;\vec{P})$ of
curves from a decorated source
asymptotic to $\mathbf x$ and $\mathbf y$ at
$-\infty$ and $+\infty$ respectively, with given shadow
$B\in \doms(\mathbf x,\mathbf y)$, and respecting the partition ${\vec{P}}$.
We will typically take
the quotient of these moduli spaces by the natural $\R$ action, to get
moduli spaces
\[ \UnparModFlow^B(\mathbf x,\mathbf y;\Source;\vec{P})=\ModFlow^B(\mathbf x,\mathbf y;\Source;\vec{P})/\R.\]
\begin{example}
Consider the top picture in Figure~\ref{fig:AlgebraRelations},
showing a shaded domain $B$ connecting
upper states $\mathbf x$ and $\mathbf y$. (We have illustrated only two components of each
Heegaard state; assume for all $i>2$, $x_i=y_i$.) The two holomorphic
disks crossing $L_i$ and $L_{i+1}$ can be translated relative to one other
to obtain a one-parameter family of holomorphic curves in
$\ModFlow^B(\mathbf x,\mathbf y,(\{L_i\},\{L_{i+1}\}))$ (which are not boundary monotone) and
$\ModFlow^B(\mathbf x,\mathbf y,(\{L_{i+1}\},\{L_i\}))$ (which are boundary monotone),
and a single curve in $\ModFlow^B(\mathbf x,\mathbf y,(\{L_i,L_{i+1}\}))$.
Note that $b_0(B)=\Iup(\mathbf x)\cdot L_{i+1} L_{i}\in{\mathcal J}$.
\end{example}
\begin{example}
Consider the second line in Figure~\ref{fig:AlgebraRelations}. Here,
the shaded domain supports holomorphic curves in six different
moduli spaces, $\ModFlow^B(\mathbf x,\mathbf y,\vec{\rhos})$,
with partitions $\vec{\rhos}=(\{L_i\},\{R_i\})$,
$(\{R_i\},\{L_i\}))$, $(\{L_i\cdot R_i\})$,
$(\{R_i\cdot L_i\})$, $(\orb_i)$, and $(\{R_i,L_i\})$.
The first two are not boundary
monotone, and the four are. (The first five are two-dimensional
moduli spaces and the last one is one-dimensional.) Now,
$b_0(B)=\Iup(\mathbf x)\cdot U_i \cdot \Iup(\mathbf y)$ (noting that $\Iup(\mathbf x)=\Iup(\mathbf y)$,
and $i-1,i\not\in\{1,\dots,2n-1\}\setminus \alphas(\mathbf x)$).
\end{example}
\begin{figure}[h]
\centering
\input{AlgebraRelations.pstex_t}
\caption{{\bf Some moduli spaces.}}
\label{fig:AlgebraRelations}
\end{figure}
\subsection{The expected dimension of the moduli spaces}
\begin{defn}
A {\em Reeb sequence} $\vec{\rho}=(\rho_1,\dots,\rho_{\ell})$ is an ordered
sequence of Reeb orbits and chords.
\end{defn}
A Reeb sequence $\vec{\rho}=(\rho_1,\dots,\rho_{\ell})$ gives rise to
a partition of Reeb chords, where each term consists of one
element sets, $\vec{\rhos}=(\{\rho_1\},\dots,\{\rho_{\ell}\})$. We
call such a partition {\em simple}. We will use interchangeably a Reeb
sequence with its associated simple partition; e.g. we say that
$(\mathbf x,\vec{\rho})$ is boundary monotone if $\mathbf x$, together with the
simple partition associated to $\vec{\rho}$ is. Similarly, given a
Reeb sequence $\vec{\rho}$, when we write
$\ModFlow^B(\mathbf x,\mathbf y,\Source,\vec{\rho})$, we mean the moduli space with
associated simple partition.
Fix $(B,{\vec\rho})$ with $B\in\doms(\mathbf x,\mathbf y)$, and $\vec\rho$ is a
sequence of Reeb chords and orbits. We say that
$(B,{\vec\rho})$ is {\em compatible} if the sum of the weights of ${\vec\rho}$
agree with the local multiplicities of $B$ around the boundary, and
$(B,{\vec\rho})$ is strongly boundary monotone.
\begin{defn}Let $|\orb(\vec\rho)|$ be the number of
Reeb orbits appearing in
$\vec\rho$, and $|\chords(\vec\rho)|$ be the number of chords.
If $(B,{\vec\rho})$ is compatible, we can define the {\em embedded
Euler characteristic}, the {\em embedded index}, and the {\em
embedded moduli space}:
\begin{align}
\chiEmb(B)&= d + e(B)-n_\mathbf x(B)-n_\mathbf y(B) \label{eq:ChiEmb} \\
\ind(B,\mathbf x,\mathbf y;\vec{\rho})&=e(B)+n_\mathbf x(B)+n_\mathbf y(B)
+ 2|\orb(\vec\rho)|+|\chords(\vec\rho)|
-2 \weight_\partial(B) \label{eq:IndEmb}\\
\ModFlow^B(\mathbf x,\mathbf y,\vec{\rho})&=
\bigcup_{\chi(\Source)=\chiEmb(B)}
\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec{\rho}),
\label{eq:EmbedMod}
\end{align}
where $\weight_\partial(B)$ is the total weight of $B$ at the boundary; i.e.
\[\weight_\partial(B)=\sum_{\rho_i}\weight(\rho_i).\]
\end{defn}
Note also that $\ind(B,\mathbf x,\mathbf y;\vec\rho)=\Mgr(B)$, in cases where each Reeb
chord in $\vec\rho$ has length $1/2$ and each orbit has length $1$.
\begin{rem}
\label{rem:EulerMeasures}
When comparing the above formulas with, for
example,~\cite[Section~5.7.1]{InvPair}, bear in mind that
there, the Euler measure of $B$ is defined in terms of
the Heegaard surface with boundary ($\Sigma_0$); whereas here we
think of it as the Euler measure in $\overline\Sigma$ instead.
\end{rem}
The following is a straightforward adaptation
of~\cite[Proposition~5.29]{InvPair}:
\begin{prop}
\label{prop:ExpectedDimension}
If $\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec\rho)$ is represented by some
pseudo-holomorphic $u$, then $\chi(\Source)=\chiEmb(B)$ if and only
if $u$ is embedded. In this case, the expected dimension of the
moduli space is computed by $\ind(B,\mathbf x,\mathbf y;\vec\rho)$. Moreover, if a
strongly monotone moduli space $\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec\rho)$
has a non-embedded holomorphic representative, then its expected
dimension is $\leq \ind(B,\mathbf x,\mathbf y;\vec\rho)-2$.
\end{prop}
\begin{proof}
The proof is as in~\cite[Proposition~5.29]{InvPair}, which in turn follows~\cite{LipshitzCyl}.
Suppose that $u$ is a weakly boundary monotone pre-flow.
Let $b_\Sigma$ be the
ramification number of $\pi_\Sigma\circ u$, defined so that each
interior branch point contributes $1$; each boundary branched points
contribute $1/2$. We think of $\cS$ as a manifold with corners, one
for each $\pm\infty$ puncture (but the Reeb orbit punctures fill in
to give ordinary boundary). Let $e(\cS)$ denote the corresponding
Euler measure.
By the
Riemann-Hurwitz formula,
\begin{equation}
\label{eq:RiemannHurwitz}
\chi(\cS)=e(\cS)+\frac{d}{2}=
e(B)+\frac{d}{2}-b_\Sigma.
\end{equation} Let $\tau_R(u)$ denote a copy
of $u$ translated by $R$ units in the $\R$-direction.
Since $u$ is
embedded, for small $\epsilon$, $u$ and $\tau_\epsilon(u)$ intersect
only near branch points of $f=\pi_\Sigma\circ u$;
and since both are
pseudo-holomorphic, their algebraic intersection number is precisely
$b_\Sigma$. When $R$ is large,
\[u\cdot
\tau_R(u)=n_\mathbf x(B)+n_\mathbf y(B)-\frac{d}{2}.\] As in~\cite{LipshitzCyl},
the intersection number $u\cap\tau_t(u)$ is independent of $t$, so
in particular
\begin{equation}
\label{eq:InParticular}
u \cdot \tau_R(u)=u\cdot \tau_\epsilon(u).
\end{equation}
(In~\cite{InvPair}, the intersection number is not independent of
$t$; rather, there are possible correction terms when Reeb chords
are slid past one another. This contribution takes the form of a
linking number near the boundary which, in the present context
vanishes.)
Thus, Equation~\eqref{eq:InParticular} shows that
\[ b_\Sigma = n_\mathbf x(B)+n_\mathbf y(B)-\frac{d}{2}.\] Substituting this back
into Equation~\eqref{eq:RiemannHurwitz} shows that
\[ \chi(\cS)=d+e(B)-n_\mathbf x(B)-n_\mathbf y(B) = \chiEmb(B),\]
in the case where $u$ is embedded.
When $u$ is pseudo-holomorphic,
but not embedded, it has $s>0$ (positive) double points
(and no negative double-points). By boundary monotonicity, do
not occur on the boundary. In this case
$u\cdot\tau_\epsilon(u)=b_\Sigma + 2 s$,
Equation~\eqref{eq:InParticular} shows that
$b_\Sigma+ 2s = n_\mathbf x(B)+n_\mathbf y(B)-\frac{d}{2}$, and so
\[ \chi(\cS)=\chiEmb(B)+2s>\chiEmb(\cS).\]
Suppose once again that $u$ is embedded. Thinking of
of $\cS$ as a branched cover of the disk with branching $b_\CDisk$,
we have that
\[ b_{\CDisk}=d-\chi(\cS)=n_\mathbf x(u)+n_\mathbf y(u)-e(B).\]
From the point of view of the symmetric product, since $u$ is embedded,
$b_{\CDisk}$ is the
intersection number of the disk corresponding to $u$ with the diagonal locus
in the symmetric product.
In the case where the sequence $\vec{\rho}$ is empty, $\ind(u)$ is
computed by a Maslov index, which, according a result of
Rasmussen~\cite{RasmussenThesis}, equals $2e(\Shadow(u))+b_{\CDisk}$.
Thus, in this case where $\ell=0$,
\[ \ind(u)= e(B)+n_\mathbf x(B)+n_\mathbf y(B).\]
In general, each Reeb chord and orbit gives a correction to the
above formula for the index. If a Reeb chord has weight $w$,
its correction is $1-2w$. This can be seen by looking, for example,
at a model computation as shown on the right in
Figure~\ref{fig:IndexModel}. In this example, we have arranged for
the source to be a $\Source$ is a disk with a single Reeb chord with
weight $w$; $B$ has $e(B)=2w$; and the moduli space of
pseudo-holomorphic representatives (modulo $\R$) is rigid, and hence
has index $1$. For the Reeb orbit with weight $w$, a similar rigid
solution can be found with $e=\frac{w+1}{2}$, $n_\mathbf x(B)+n_\mathbf y(B)=\frac{w-1}{2}$.
\begin{figure}[h]
\centering
\input{IndexModel.pstex_t}
\caption{{\bf Model computations for the index.} }
\label{fig:IndexModel}
\end{figure}
Finally, when $u$ is not embedded, and has $s$ double points, the
intersection number with the diagonal corresponds to $b_\CDisk+2s$, so
$\ind(u)=\ind(B,\mathbf x,\mathbf y;\rhos)-2s$.
\end{proof}
\subsection{Boundary degenerations}
We formalize the notion of $\beta$-boundary degenerations.
\begin{defn}
\label{def:BoundaryDegenerations}
Let $\HH\subset \C$ (the ``lower half plane'') consist of
$\{x+iy\big| x,y\in \R, y\leq 0\}$, so $\partial\HH=\R$. A {\em
boundary degeneration} consists of the following data.
\begin{itemize}
\item a smooth, oriented surface $\wSource$ with boundary and punctures,
exactly $d$ of which are on the boundary.
\item a labelling of the interior punctures of $\wSource$ by Reeb orbits
\item a complex structure structure $\jSource$ on $\wSource$.
\item
a smooth map
$w\colon (\wSource,\partial \wSource)\to (\Sigma\times \HH,\betas\times\R\times\{0\})$.
\item a constant $\tau\in\R$.
\end{itemize}
These data are required to satisfy the following conditions:
\begin{enumerate}
\item The map $w\colon \wSource\to\Sigma\times\HH$ is proper, and it extends to
a proper map
${\overline w}\colon
{\wSource}' \to {\overline{\Sigma}}\times\HH$,
where $\wSource'$ is obtained from $\wSource$ by filling in the $\east$ punctures
(so $\wSource\subset \wSource'\subset \overline{\wSource}$).
\item The map $\pi_\HH\circ w$ is a $d$-fold branched cover.
\item At each interior puncture of $\wSource$, $\pi_\Sigma\circ w$ is asymptotic
to the Reeb orbits which labels the puncture.
\item For each $t\in \R=\partial\HH$ and $i=1,\dots,d$, $w^{-1}(\beta_i\times
\{t\})$ consists
of exactly one point.
\item The map $w$ is holomorphic, with respect to $\jSource$ on
the domain and the split complex structure
$j\times j_\HH$ on the range.
\end{enumerate}
\end{defn}
The map $w\colon \wSource\to \Sigma\times \HH$
extends to a map
\[ {\overline{w}}\colon {\overline\wSource}\to \cSigma\times {\overline\HH}.\]
\begin{defn}
\label{def:evB}
Think of the boundary punctures in $\wSource$ as a $d$-tuple of points
in ${\overline\wSource}$. The images of these points under
${\overline{\pi_\HH\circ w}}$ gives a point, denoted $\evB(w)$, in
$\beta_1\times\dots\times\beta_d=\Tb$.
\end{defn}
The boundary degeneration induces a map $u\colon
(\wSource,\partial\wSource)\to (\Sigma\times [0,1]\times
\R,\betas\times \R)$, so that $\pi_\Sigma\circ u=\pi_\Sigma \circ w$,
$s\circ u = 0$, and $t\circ u\equiv \tau$.
Such a boundary degeneration $w$ has a shadow which is a two-chain $B$
which is a formal linear combination of the components of
$\Sigma\setminus\betas$.
For $\{r,s\}\in\Matching$, there is a two-chain $\Brs$ corresponding
to the component of $\Sigma\setminus \betas$ containing $z_r$ and
$z_s$. Let $\UnparModDeg_j^{\Brs}$ denote the moduli space of
boundary degenerations with shadow $\Brs$ as above modulo the (real
two-dimensional) group of automorphisms of $\HH$.
\begin{prop}
\label{prop:SmoothBoundaryDeg}
For a generic complex structure $j$ on $\Sigma$, the moduli
space $\ModDeg_j^{\Brs}$ of boundary degeneration
is a smooth manifold of dimension $d=g+n-1$.
\end{prop}
\begin{proof}
This follows from the fact that the corresponding moduli space is
somewhere injective near the boundary; see~\cite[Proposition~3.9 and
Lemma~3.10]{LipshitzCyl}; see also~\cite[Proposition~3.14]{HolDisk}
and~\cite{OhBoundary}.
\end{proof}
There is an evaluation map $\evB\colon\ModDeg_j^{{\mathcal
B}_{\{r,s\}}} \to \Tb$. Given $\mathbf x\in\Tb$, let
\[ \ModDeg_j^{\Brs}(\mathbf x)=(\evB)^{-1}(\mathbf x).\]
The following result will be important:
\begin{lem}
\label{lem:BoundaryDegenerationsDegree1}
The evaluation map $\evB\colon \ModDeg_j^{{\mathcal B}_{\{r,s\}}}\to \Tb$ has odd degree.
\end{lem}
\begin{proof}
Consider first the case where $g=0$. In this case, $\evB$ is clearly a
homeomorphism: the boundary degeneration consist of $d-1$ constant
disks, and one disk that maps to $\Brs$ with degree one. We can
move the constants around
on at $d-1$ dimensional portion of $\Tb$ and reparameterize the remaining component
to obtain the claimed homeomorphism. In cases where $g>0$, gluing spheres gives the desired
degree statement~\cite[Section~12]{LipshitzCyl}; see also~\cite[Section~10]{HolDisk}.
\end{proof}
\subsection{Regular moduli spaces of embedded curves}
\begin{defn}
\label{def:Typical}
A Reeb sequence $(\rho_1,\dots,\rho_\ell)$ is called {\em{typical}}
if each chord $\rho_i$ appearing in the sequence covers half of some
boundary circle (i.e. it is one of $L_i$ or $R_i$), and each Reeb
orbit covers some boundary circle exactly once.
\end{defn}
\begin{thm}
\label{thm:GeneralPosition}
Choose a generic $J$. Let $B$ be a shadow with $\Mgr(B)\leq 2$ and
$\vec{\rho}$ is a typical Reeb sequence. Then, the moduli spaces
$\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec{\rho})$ is a smooth manifold of
dimension given by $\Mgr(B)$. Moreover, if $\Mgr(B)\leq 1$ and
$\vec{\rho}$ is not typical, then
$\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec{\rho})$ is empty.
\end{thm}
\begin{proof}
Consider first the moduli space $\ModFlow^B(\mathbf x,\mathbf y,\Source)$, where
the order of the punctures is left unspecified. Standard arguments
show that, for generic $J$, the corresponding moduli space is a
manifold transversely cut out by the $\dbar$ operator;
see~\cite[Proposition~5.6]{InvPair}. Moreover, the evaluation map
at the punctures gives a map from the moduli space to $\R^{E}$,
where here $E=E(\Source)$ denotes the number of east punctures of
$\Source$. There is a dense set of $J$ for which the evaluation map
is transverse to the various diagonals in $\R^{E}$, so that their
preimages give submanifolds of $\ModFlow^B(\mathbf x,\mathbf y,\Source)$; the
moduli spaces $\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec{\rho})$ are the
complement of these submanifolds.
This transversality argument shows that
$\ModFlow^B(\mathbf x,\mathbf y,\Source;\vec\rho)$ is a smooth manifold with
dimension (as computed in
Proposition~\ref{prop:ExpectedDimension}) given by
\begin{align*}
\ind(B,\mathbf x,\mathbf y;\vec{\rho})&=e(B)+n_\mathbf x(B)+n_\mathbf y(B)
+ 2|\orb(\vec\rho)|+|\chords(\vec\rho)|
-2 \sum_{\rho_i}\weight(\rho_i)\\
&= \Mgr(B)-\sum_{\orb\in \orb(\vec\rho)} (2\weight(\orb)-2) -
\sum_{\rho\in \chords(\vec\rho)} (2\weight(\rho)-1) \\
& \leq \Mgr(B),
\end{align*}
with equality exactly when each orbit has length one and each chord has length $1/2$.
\end{proof}
\begin{rem}
The above general position can be seen from the point of view of the
symmetric product as follows. As in Remark~\ref{rem:FlowsAsDisks},
we think of our pseudo-holomorphic curves as giving
pseudo-holomorphic disks in $\Sym^d(\overline\Sigma)$. The subspace
$\alpha_1^c\times\dots\times \alpha_g^c\times \Sym^{n-1}(I)$ is
equipped with codimension one walls of the form $\{p_i\}\times\Sym^{n-2}(I)$,
with $p_i\in\cSigma$.
Typical sequences arise for holomorphic disks that are transverse to
these walls; chords or orbits with larger weight occur when
the curves have higher order contact with the submanifolds.
\end{rem}
\subsection{Ends of one-dimensional moduli spaces}
We will consider ends of one-dimensional moduli spaces
$\UnparModFlow^B(\mathbf x,\mathbf y,\Source;\vec{P})$. These will include ends that
consist of two-story buildings. Another kind of end consists of the
formation of an ``orbit curve'' at east infinity. This occurs when
some Reeb orbit constraint $\orb_i$ in $\vec{P}$ slides off to $s=1$.
(See Figure~\ref{fig:OrbitEnd}.) Other ends occur when two consecutive
parts in $\vec{P}$ collide. We call these {\em collision ends}.
In formulating these ends, we use the following terminology from~\cite{InvPair}:
\begin{defn}
\label{def:ComposableChords}
An ordered pair of Reeb chords $\rho$ and $\sigma$ are called {\em
weakly composable} if $\rho^+$ and $\sigma^-$ are contained on the same $\alpha$-arc. Moreover, if $\rho$ and $\sigma$ weakly
composable, they are further called {\em strongly composable} if
$\rho^+=\sigma^-$. If $\rho$ and $\sigma$
are strongly composable we can join them to get a new Reeb chord
$\rho\uplus\sigma$.
\end{defn}
Thus, $L_i$ and $L_{i+1}$ are weakly but not strongly composible,
while $L_i$ and $R_i$ are strongly composable.
\begin{defn}
A collision end is called {\em invisible} if $\rho_i$ and
$\rho_{i+1}$ are the same Reeb orbit. Otherwise, it is called {\em visible}.
\end{defn}
\begin{thm}
\label{thm:DEnds}
Let $\Hup$ be an upper diagram and $\Matching$ its associated
matching. Fix upper Heegaard states $\mathbf x$ and $\mathbf y$ and a typical
Reeb sequence $\vec{\rho}$. Suppose moreover that $(\mathbf x,\vec{\rho})$
is strongly boundary monotone, and
$B\in\doms(\mathbf x,\mathbf y)$ has vanishing local multiplicity somewhere.
Fix $\Source$ and $\vec{\rho}$ so that
$\ind(B,\mathbf x,\mathbf y;\Source,\vec{\rho})=2$. Let
$\UnparModFlow=\UnparModFlow^B(\mathbf x,\mathbf y;\Source;{\vec{\rho}})$. The total
number of ends of $\UnparModFlow$ of the following types are even in
number:
\begin{enumerate}[label=(DE-\arabic*),ref=(DE-\arabic*)]
\item
\label{typeDE:2StoryEnd}
Two-story ends, which are of the form
\[ \UnparModFlow^{B_1}(\mathbf x,\w;\Source_1;\rho_1,\dots,\rho_i)\times
\UnparModFlow^{B_2}(\w,\mathbf y;\Source_2;\rho_{i+1},\dots,\rho_{\ell}),\]
taken over all upper Heegaard states $\w$ and choices of $\Source_1$ and $\Source_2$ so that
$\Source_1\natural \Source_2=\Source$, and $B_1\natural B_2=B$.
\item
\label{typeDE:OrbitEnd}Orbit curve ends,
of the form $\UnparModFlow^B(\mathbf x,\mathbf y,\Source;\rho_1,\dots,\rho_{i-1},\longchord_j,\rho_{i+1},\dots,\rho_{\ell})$,
where some Reeb orbit component $\rho_i=\orb_j$ slides off to $s=1$
and is replaced by a Reeb chord $\longchord_j$ that
covers $Z_j$ with multiplicity $1$. (When $j\neq 1$ or $2n$,
there are two possible choices: $\longchord_j=R_j L_j$ or $L_j R_j$.)
\item
\label{typeDE:CollisionWithOrbit} Visible collision ends where at least one of $\rho_i$ or $\rho_{i+1}$ is a Reeb orbit.
\item
\label{typeDE:ChordChord}
Collision ends where $\rho_i$ and $\rho_{i+1}$ are Reeb chords,
where one of the two conditions are satisfied:
$\rho_i$ and $\rho_{i+1}$ are not weakly composable, or they are strongly composable.
\item
\label{typeDE:BoundaryDegeneration}
An end consisting of a boundary degeneration that meets a constant
flowline. In this special case, $\vec{\rho}$ consists of exactly two
constraints, which are matched Reeb orbits, and $\mathbf x=\mathbf y$. Moreover,
if $\{r,s\}\in\Matching$, then the number of boundary degeneration
ends of the union
\[ \ModFlow(\mathbf x,\mathbf x,\{e_r\},\{e_s\})\cup\ModFlow(\mathbf x,\mathbf x,\{e_s\},\{e_r\})\]
is odd.
\end{enumerate}
\end{thm}
\begin{figure}[h]
\centering
\input{OrbitEnd.pstex_t}
\caption{{\bf Orbit curve end.} Consider the moduli space
$\UnparModFlow^B(x,y,\{\orb_i\})$, where $B$ is shaded on the left.
This one-dimensional moduli space has an end which is a two-story
building, and another which is an orbit curve end with $\longchord_i=L_i R_i$.}
\label{fig:OrbitEnd}
\end{figure}
The above result is proved in Section~\ref{sec:CurvesA}.
\section{Bimodules}
\label{sec:Bimodules}
In this section, we describe how to associate a type DA bimodule to a
middle Heegaard diagram (cf. Definition~\ref{def:MiddleDiagram}),
together with a matching on the incoming boundary components. Loosely
speaking, the incoming boundary is treated as type $A$, and the
outgoing as type $D$. (See~\cite{Bimodules} for the corresponding
construction in bordered Floer homology.)
In more detail, fix a middle diagram
\begin{align*} \Hmid=(\Sigma_0,(\Zin_1,&\dots,\Zin_{2m}),(\Zout_1,\dots,\Zout_{2n}),
\{\alphain_1,\dots,\alphain_{2m-1}\},
\{\alphaout_1,\dots,\alphaout_{2n-1}\},\\
&\{\alpha^c_1,\dots,\alpha^c_{g}\},
\{\beta_1,\dots,\beta_{g+m+n-1}\}),
\end{align*}
and let $\MatchIn$ be a matching on $\{1,\dots,2m\}$, thought of as indexing the
components of $\Zin$. The Heegaard diagram induces a matching $\Mmid$
on all the boundary components of $\Sigma_0$.<
Together, $\MatchIn$ and $\Mmid$ given an equivalence
relation on the components of $\partial\Sigma$.
\begin{defn}
\label{def:CompatibleDA}
We say that $\MatchIn$ is
{\em compatible} with $\Hmid$ if every equivalence class has some
component of $\Zout$ in it.
\end{defn}
Form $\Wmid=W(\Hmid)$ as in Definition~\ref{def:AssociatedW}, and
$\Win=W(\MatchIn)$. The compatibility condition is equivalent to the
condition that the one-manifold $W=\Wmid\cup\Win$ has no closed
components.
\begin{defn}
Let $\Hmid$ be a middle diagram, equipped with a matching $\MatchIn$ on the
incoming boundary components.
The {\em full incoming} algebra
$\ClginBig(\Hmid)$ and {\em full outgoing algebra}
$\ClgoutBig(\Hmid)$ are defined by
\[\Clgin(\Hmid)=\bigoplus_{k=0}^{2m-1}\Clg(2m,k);
\qquad \Clgout(\Hmid)=\bigoplus_{k=0}^{2n-1}\Clg(2n,k).\]
We will be primarily interested in the $k=n$, summands,
which we call the {\em incoming algebra} and the {\em outgoing algebra}
respectively:
\[\Clgin(\Hmid)=\Clg(2m,m);
\qquad \Clgout(\Hmid)=\Clg(2n,n).\]
\end{defn}
Each middle Heegaard state $\mathbf x$ determines two subsets
\[ \alphain(\mathbf x)\subset \{1,\dots,2m\}\qquad
{\text{resp.}}\qquad
\alphaout(\mathbf x)\subset \{1,\dots,2n\}\]
consisting of those $i\in\{1,\dots,2m\}$ resp. $\{1,\dots,2n\}$
with $\mathbf x\cap \alphain_i\neq \emptyset$ resp
$\mathbf x\cap\alphaout_i\neq\emptyset$.
Each middle Heegaard state $\mathbf x$ has an {\em idempotent type}
$k=|\alphain(\mathbf x)|$.
Let $\DAmodBig(\Hmid)$ be the $\Field$-vector space
spanned by
the middle Heegaard states of $\Hmid$.
Let
\[ \IdempIn(\mathbf x)=\Idemp{\alphain(\mathbf x)}
\qquad{\text{and}}\qquad
\IdempOut(\mathbf x)=\Idemp{\{1,\dots,2n-1\}\setminus \alphaout(\mathbf x)}.\]
An
$\IdempRing(2n,k-m+n)-\IdempRing(2m,k)$-bimodule structure is specified by
\[ \IdempOut(\mathbf x)\cdot \mathbf x
\cdot \IdempIn(\mathbf x)=\mathbf x.\]
There is a splitting
\[ \DAmodBig(\Hmid)=\bigoplus_{k\in\Z}
~~~{\lsup{\IdempRing(2n,k-m+n)}\DAmod(\Hmid)}_{\IdempRing(2m,k)},\]
where $k$ is the idempotent type of $\mathbf x$. We will be primarily
interested in the summand where $k=m$,
\[ \lsup{\IdempRing(2n,n)}\DAmod(\Hmid)_{\IdempRing(2m,m)}.\]
Our goal here is to endow $\DAmod(\Hmid)$ with the structure of a type $DA$ bimodule structure
$\DAmod(\Hmid,\MatchIn)=\lsup{\Clgout}\DAmod(\Hmid)_{\Clgin}$, where
$\Clgin=\Clgin(\Hmid)$ and $\Clgout=\Clgout(\Hmid)$.
To equip $\DAmod(\Hmid)$ with the structure of a $DA$ bimodule, we
choose further an orientation $\orW$ on $W=\Wmid\cup\Win$. Each boundary
component $\Zin_i$ or $\Zout_j$ of $\Hmid$ corresponds to some point
on $W$.
This data specifies an orbit marking, in the following sense:
\begin{defn}
\label{def:OrbitMarkingDA}
A {\em special orbit} in $\Hmid$ covers an orbit in $\Zin_i$ that is matched
with $\Zout_j$.
An {\em orbit marking} in a middle diagram $\Hmid$ is a partition of the
simple orbits of $\Zin$ so that:
\begin{itemize}
\item $\MatchIn$ matches even with odd orbits,
\item if components of $\Zin$ are matched by $\Mmid$, then one one
is even and the other is odd.
\end{itemize}
Let $\OmegaInEv\subset \{1,\dots,2m\}$ denote the even boundary components of $\Zin$;
and $\OmegaInOdd\subset \{1,\dots,2m\}$ denote the odd ones.
\end{defn}
An orientation on $W$ is equivalent to an orbit marking: each segment
of $W$ in $\Wmid$ is oriented from even to odd, and each segment in $\Win$
is oriented from odd to even.
The orientation on $W$ also induces a pair of functions
\[ \sigma\colon \{1,\dots,2m\}\to \{1,\dots,2n\}, \qquad \tau\colon
\{1,\dots,2m\}\to \{1,\dots,2n\},\] the {\em starting} and {\em
terminal} points respectively. Namely, $\sigma(p)=i$ and
$\tau(p)=j$ if $\Zin_p$ is contained on the oriented interval in $W$
starting at $\Zout_i$, and $\tau(p)=j$ if $\Zin_p$ is contained on the
oriented interval terminating in $\Zout_j$.
See Figure~\ref{fig:OrbitMarkingDA} for an example.
\begin{figure}[h]
\centering
\input{OrbitMarkingDA.pstex_t}
\caption{{\bf Orbit markings in a middle diagram.}
The even orbits (1,3,6) are colored white and the odd ones (2,4,5) are colored black.
Moreover,
$\sigma(1)=\sigma(2)=\sigma(3)=\sigma(4)=2$;
$\tau(1)=\tau(2)=\tau(3)=\tau(4)=1$;
$\sigma(5)=\sigma(6)=3$;
$\tau(5)=\tau(6)=4$.}
\label{fig:OrbitMarkingDA}
\end{figure}
The $DA$ bimodule $\DAmod(\Hmid)$ depends on the incoming matching,
but we typically suppress data from the notation; in fact, even the
curvature element in the incoming algebra $\Clgin$ depends on this
choice. The bimodule also depends on an orientation of $W$, which we
also suppress.
\subsection{Type $DA$ bimodules}
\label{subsec:DAmodConstruction}
We adapt the definitions from Section~\ref{subsec:AlgebraicConstraints}
in the following straightforwarde manner.
Suppose that $\rhos$ is a set of Reeb chords for $\Hmid$
that is algebraic in the sense of Definition~\ref{def:AlgebraicPacket},
and that is supported entirely on $\Zin$. For each $\rho\in \rhos$,
$\alpha(\rho^+)$ resp.
$\alpha(\rho^-)$ be the curve $\alphain_i$ with $\rho^+\in\alphain_i$
resp. $\rho^-\in\alphain_i$.
Let \[ I^-(\rhos)=\sum_{\{{\mathbf{s}}\big| \{\alphain(\rho_1^-),\dots,\alphain(\rho_j^-)\}\subset
\{\alphain_i\}_{i\in{\mathbf{s}}}\}} I_{\mathbf{s}}
\qquad\text{and}\qquad
I^+(\rhos)=\sum_{\{{\mathbf{s}}\big| \{\alphain(\rho_1^+),\dots,\alphain(\rho_j^+)\}\subset
\{\alphain_i\}_{i\in{\mathbf{s}}}\}} I_{\mathbf{s}}
\]
Then, $\bIn_0(\rhos)$ be the algebra element $a_0\in\BlgZ(2m,m)$ with
$a=I^- \cdot a\cdot I^+$ and whose weight $w_i(a)$ is the average local
multiplicity at $\Zin_i$ for $i=1,\dots,2m$.
Let $\bIn(\rhos)$ be the image of $\bIn_0(\rhos)$
in $\Clgin$.
The definition of constraint packets has the following immediate
generalization to middle diagrams:
\begin{defn}
\label{def:CompatiblePacketDA}
Fix a Heegaard state $\mathbf x$ and a sequence $\vec{a}=(a_1,\dots,a_\ell)$ of pure
algebra elements of $\Clgin(\Hmid)$.
A sequence of constraint packets $\rhos_1,\dots,\rhos_k$ is called
\em{$(\mathbf x,\vec{a})$-compatible} if there is a sequence
$1\leq k_1<\dots<k_\ell\leq k$ so that the following conditions hold:
\begin{itemize}
\item the constraint packets $\rhos_{k_i}$ consist of chords
in $\Zin$, and they are algebraic, in the sense
of Definition~\ref{def:AlgebraicPacket},
\item $\IdempIn(\mathbf x)\cdot \bIn(\rhos_{k_1})\otimes\dots\otimes \bIn(\rhos_{k_\ell})=
{\mathbf I}(\mathbf x)\cdot a_1\otimes\dots\otimes a_{\ell}$,
as elements of $\DAmod(\Hdown)\otimes \Clgin^{\otimes \ell}$
\item
for each $t\not\in \{k_1,\dots,k_\ell\}$,
the constraint packet $\rhos_t$ is one
of the following types:
\begin{enumerate}[label=($DA\rhos$-\arabic*),ref=($DA\rhos$-\arabic*)]
\item
\label{eq:OddOrbit}
it consists of the orbit $\{\orb_i\}$,
where $\orb_i$ is {\em odd} for the $\orW$-induced orbit marking.
\item
\label{eq:EvenOrbit}
it is of the form
$\{\orb_i,\longchord_j\}$, where $\orb_i$ is even
$\{i,j\}\in\Matching$, and
$\longchord_j$ is one of the two Reeb chords
that covers $\Zin_j$ with multiplicity one.
\item
\label{eq:OutOrbit}
it is of the form $\{\orb_k\}$, where $\orb_k$ is the simple Reeb orbit
around some component in $\Zout$
\item
\label{eq:OutChord} it is of the form $\{\rho_j\}$, where $\rho_j$ is a Reeb chord
of length $1/2$ supported in some $\Zout$.
\end{enumerate}
\end{itemize}
Let $\llbracket \mathbf x,a_1,\dots,a_\ell\rrbracket$
be the set of all sequences of constraint packets $\rhos_1,\dots,\rhos_k$
that are $(\mathbf x,\vec{a})$-compatible.
\end{defn}
Given $(\rhos_1,\dots,\rhos_k)\in\llbracket
\mathbf x,a_1,\dots,a_{\ell}\rrbracket$, we can consider
$\pi_2(\mathbf x,\rhos_1,\dots,\rhos_k,\mathbf y)$, the space of homology
classes of maps with asymptotics at $\Zin$
specified by the given Reeb chords.
We define the index $\ind(B,\mathbf x,\mathbf y;\vec{\rhos})$ exactly as in the case
of type A modules (Definition~\ref{def:IndexTypeA}):
\begin{align*}
\ind(B,\mathbf x,\mathbf y;\vec{\rhos}) &= e(B)+n_\mathbf x(B)+n_\mathbf y(B)+\ell \label{eq:EmbIndA} \\
& \qquad -\weight(\vec{\rhos})+\iota(\chords(\vec{\rhos}))+\sum_{o\in\orb(\vec\rhos)} (1-\weight(o)), \nonumber
\end{align*}
\begin{lemma}
\label{lem:MgrAndAgr}
Given $B\in\doms(\mathbf x,\mathbf y)$,
the quantity
\[ \Mgr(B)=e(B)+n_\mathbf x(B)+n_\mathbf y(B)-\weight_\partial(B) \] is indendent
of $B$ (depending only on $\mathbf x$ and $\mathbf y$).
Also, if $\{j,k\}\in\Mmid$, oriented in $\Wmid$ from $k$ to $j$, then the integer
\[ \Agr_{\{j,k\}}=\weight_{j}(B)-\weight_{k}(B),\]
is independent of the choice of
$B$ (depending only on $\mathbf x$ and $\mathbf y$).
\end{lemma}
\begin{proof}
Clearly, $\{j,k\}\in\Mmid$ if and only if $i$ and $j$ are contained
in the same component $\Bjk$ of
\[ \Sigma_0\setminus(\beta_1\cup\dots\cup\beta_{g+m+n-1}).\]
Now,
\[ \pi_2(\mathbf x,\mathbf y)\cong \bigoplus_{\{j,k\}} \Z\cdot \Bjk.\]
The lemma follows from the following:
\begin{align*}
e(\Bjk)+n_{\mathbf x}(\Bjk)+n_{\mathbf y}(\Bjk)&=2 \\
\weight_j(\Bjk)&=\weight_k(\Bjk)=1.
\end{align*}
\end{proof}
We can think of $\Agr(B)$ as defining an element ${\mathbb
A}(B)\in H^1(\Wmid,\partial \Wmid)\cong \Z^{m+n}$, the {\em Alexander grading}.
Now, the Alexander grading, with
values in $H^1(\Wmid,\partial \Wmid)$; and the $\delta$-grading $\gr$, with
values in $\mathbb Q} \newcommand{\R}{\mathbb R$, are determined up to overall constants by
\begin{align*}
\Agr(\mathbf x)-\Agr(\mathbf y)&=\Agr(B) \\
\gr(\mathbf x)-\gr(\mathbf y)&=e(B)+n_\mathbf x(B)+n_\mathbf y(B)-\weight_\partial(B)
\end{align*}
Given $B\in\pi_2(\mathbf x,\rhos_1,\dots,\rhos_h,\mathbf y)$,
let $\bOut_0(B)=a\in\BlgZ(2n,n)$ be the algebra element
with $\Iup(\mathbf x)\cdot a=a$ and whose weight
at $i\in\{1,\dots,2n\}$ agrees with the local multiplicity of
$B$ at $\Zout_i$.
Let $\bOut(B)$ denote the image of $\bOut_0(B)$ in $\Clgout$.
Given, an $(\mathbf x,\vec{a})$-compatible sequence of constraint packets
$\rhos_1,\dots,\rhos_h$, we define a corresponding monomial in the
variables $U_1,\dots,U_{2n}$, denoted $\gamma(\rhos_1,\dots,\rhos_h)$, to
be product over all packets of Type~\ref{eq:OddOrbit} of the element
$U_{\tau(i)}$.
Let
\begin{equation}
\label{eq:bOut}
\bOut(B,\rhos_1,\dots,\rhos_k)=\gamma(\rhos_1,\dots,\rhos_h)\cdot
\bOut(B).
\end{equation}
Let $\ModFlow(\mathbf x,\mathbf y,\rhos_1,\dots\rhos_h)$ denote the moduli space of
flowlines as in Definition~\ref{def:GenFlow}, with the understanding
that now the asymptotics at $\pm \infty$ go to middle Heegaard states
$\mathbf x$ and $\mathbf y$. Note also that for middle diagrams, the number of
$\beta$-curves is given by $d=g+m+n-1$.
Define
\begin{align}
\label{eq:DefDA-Action}
\delta^1_{\ell+1}&(\mathbf x,a_1,\dots,a_\ell)\\
&=
\sum_{
\left\{\begin{tiny}
\begin{array}{r}
\mathbf y\in\States \\
(\rhos_1,\dots,\rhos_k)\in \llbracket \mathbf x,a_1,\dots,a_\ell\rrbracket \\
B\in\pi_2(\mathbf x,\rhos_1,\dots,\rhos_h,\mathbf y)
\end{array}
\end{tiny}\Big| \ind(B,\rhos_1,\dots,\rhos_h)=1\right\}}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\#\UnparModFlow(\mathbf x,\mathbf y,\rhos_1,\dots\rhos_h)\cdot \bOut(B,\rhos_1,\dots,\rhos_h)\otimes \mathbf y.
\nonumber
\end{align}
Lemma~\ref{lem:CompatWithMgr} has the following straightforward adaptation:
\begin{lemma}
\label{lem:CompatWithMgrDA}
Fix $\mathbf x,\mathbf y\in\States$, a sequence of pure algebra elements
$\vec{a}=(a_1,\dots,a_\ell)$, an $(\mathbf x,\vec{a})$-compatible sequence
of constraint packets $\rhos_1,\dots,\rhos_h$, and
$B\in\doms(\mathbf x,\mathbf y)$. If there is a pre-flowline $u$ whose shadow is
$B$ and whose packet sequence is $(\rhos_1,\dots,\rhos_\ell)$,
then
\[ \gr(\mathbf x)+\ell-\sum_{i=1}^{\ell}\weight(a_i)=\gr(\mathbf y)-\weight_{\Zout}(\bOut(B))+
\ind(B,\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_h).\]
\end{lemma}
\begin{proof}
We have that
\[ \gr(\mathbf x)-\gr(\mathbf y)=e(B)+n_\mathbf x(B)+n_\mathbf y(B)-\weight_\partial(B), \]
and
\[
\ind(B)=e(B)+n_\mathbf x(B)+n_\mathbf y(B)+h-\weight_\partial(B)+\sum\iota(\chords(\rhos_i)).
\]
Taking the difference, we find that
\[ \gr(\mathbf x)-\gr(\mathbf y)-\ind(B,\mathbf x,\mathbf y,\vec{\rhos})
=-h-\sum\iota(\chords(\rhos_i)).\]
Now, if $\rhos_i$ is an algebraic packet, then
$\iota(\chords(\rhos_i))=-\weight(\rhos_i)$;
if it is of Type~\ref{eq:OddOrbit},
$\iota(\chords(\rhos_i))=0$;
if it is of Type~\ref{eq:EvenOrbit},
$\iota(\chords(\rhos_i))=-1$;
if it is of Type~\ref{eq:OutOrbit},
the contribution is $0$;
if it is of Type~\ref{eq:OutChord},
we get $-1/2$.
It follows that
\begin{align*}
\gr(\mathbf x)-\gr(\mathbf y)-\ind(B,\mathbf x,\mathbf y,\vec{\rhos})
&=-\ell+\left(\sum_{i=1}^{\ell}\weight_{\Zin}(a_i)\right) \\
& \qquad
-\#(\text{odd orbits coming in})-\weight_{\Zout}(B) \\
&=-\ell+\left(\sum_{i=1}^{\ell}\weight_{\Zin}(a_i)\right)
-\weight(\bOut(B,\vec{\rhos_i})).
\end{align*}
\end{proof}
\begin{lemma}
Given $\mathbf x$ and a sequence of algebra elements $(a_1,\dots,a_\ell)$,
there are only a finite number of non-negative homology classes
$B\in\pi_2(\mathbf x,\mathbf y,\vec{\rhos})$ where $\rhos_1,\dots,\rhos_h$ are
$(\mathbf x,\vec{a})$-compatible and with
$\ind(B,\mathbf x,\mathbf y,\vec{\rhos})=1$.
\end{lemma}
\begin{proof}
According to Lemma~\ref{lem:MgrAndAgr}, $(\mathbf x,\vec{a})$, $\mathbf y$, and
$\ind(B,\mathbf x,\mathbf y,\vec{\rhos})=1$ determines the total weight of $b$.
The lemma also shows that
$\{i,j\}\in\Mmid$, then $\weight_i(B)-\weight_j(B)$ is independent
of $B$ (depending only on $\mathbf x$ and $\mathbf y$).
Suppose next that
$\{i,j\}\in\MatchIn$, and $i$ is an odd orbit, then if
$c_{i,j}=\weight_i(a_1\otimes\dots \otimes a_\ell)-
\weight_j(a_1\otimes\dots\otimes a_\ell)$, then clearly
\[ c_{i,j}\leq \weight_i(B)-\weight_j(B)\leq c_{i,j}+\weight_{\tau(i)}(b).\]
Finally,
if $i\in\Zout$ is $W$-initial, then
$\weight_i(B)=\weight_i(b)$.
Since every in-coming boundary component is equivalent (using the
equivalence relation generated by $\Mmid$ and $\MatchIn$) to an
$W$-initial out boundary component, we have universal upper bounds
on the weights of $B$ at all of its boundary points (which we also
assumed to be non-negative). Since the map $B\mapsto
\bigoplus_{i}\weight_i(B)$ gives an injection of $\pi_2(\mathbf x,\mathbf y)$
into $\Z^{m+n}$, the lemma follows.
\end{proof}
Theorem~\ref{thm:AEnds} has the following straightforward adaptation
to middle diagrams.
\begin{remark}
We will need the following theorem in the case where the in-coming
sequence of packets are compatible with some sequence of algebra
elements $(a_1,\dots,a_\ell)$. To underscore the similarity with
Theorem~\ref{thm:AEnds}, we have stated it under slightly weaker
hypotheses. (Compare Definition~\ref{def:Allowed}.)
\end{remark}
\begin{thm}
\label{thm:DAEnds} Choose a middle diagram $\Hmid$, and fix a
compatible matching $\Mup$. Choose also an orbit marking
(Definition~\ref{def:OrbitMarkingDA}). Fix a lower Heegaard state $\mathbf x$
and a sequence of constraint packets $\vec{\rhos}$ with the
following properties:
\begin{itemize}
\item $(\mathbf x,\vec{\rhos})$ is strongly
boundary monotone.
\item The chords appearing in each packet $\rhos_i$ are disjoint from one another
\item Each packet contains at most one orbit, and that orbit is simple.
\item If a packet contains an even (in-coming) orbit, then it
contains exactly one other Reeb chord, as well; and that chord
is disjoint from the orbit.
\item If the packet contains an orbit which is not even,
then it contains no other chord.
\end{itemize}
Let $\mathbf y$ be a
lower Heegaard state, and $B\in\pi_2(\mathbf x,\mathbf y)$, whose local multiplicity
vanishes somewhere. Choose $\Source$ and
$\vec{P}$ so that $[\vec{P}]=(\rhos_1,\dots,\rhos_\ell)$ and
so that the $\chi(\Source)=\chiEmb(B)$;
and suppose that $\ind(B,\mathbf x,\mathbf y;\vec{\rhos})=2$, and abbreviate
$\UnparModFlow=\UnparModFlow^B(\mathbf x,\mathbf y;\Source;{\vec{P}})$. The total
number of ends of $\UnparModFlow$ of the following types are even in
number:
\begin{enumerate}[label=(DAE-\arabic*),ref=(DAE-\arabic*)]
\item \label{endDA:2Story}
Two-story ends, which are of the form
\[ \UnparModFlow(\mathbf x,\w;\Source_1;\rhos_1,\dots,\rhos_i)\times
\UnparModFlow(\w,\mathbf y;\Source_2;\rhos_{i+1},\dots,\rhos_{\ell}),\]
taken over all lower Heegaard states $\w$ and choices of $\Source_1$ and $\Source_2$ so that
$\Source_1\natural \Source_2=\Source$, and $B_1\natural B_2=B$.
\item
\label{endDA:Orbit}
Orbit curve ends, of the form
$\UnparModFlow^B(\mathbf x,\mathbf y,\Source';\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_{\ell})$,
where $\orbits(\sigmas)=\orbits(\rhos_i)\setminus\{\orb_r\}$,
$\chords(\sigmas)=\chords(\rhos_i)\cup\{\longchord_r\}$ where
$\longchord_r$ is a Reeb chord that covers the boundary component
$Z_r$ with multiplicity $1$.
\item
\label{endDA:ContainedCollisions}
Contained collision ends for two consecutive packets $\rhos_i$ and
$\rhos_{i+1}$, which correspond to points in
$\UnparModFlow^{B'}(\mathbf x,\mathbf y,\Source;\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+2},\dots,\dots,\rhos_\ell)$
with the following properties:
\begin{itemize}
\item The collision is visible.
\item The packets $\rhos_i$ and $\rhos_{i+1}$ are strongly composable.
\item The packet $\sigmas$ is a contained collision of $\rhos_i$
and $\rhos_{i+1}$
\item
The chords in $\sigmas$ are disjoint from one
another.
\end{itemize}
\item
\label{endDA:Join}
Join ends, of the form
$\UnparModFlow^B(\mathbf x,\mathbf y,\Source';\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_{\ell})$,
$\orbits(\sigmas)=\orbits(\rhos_i)$, and the following conditions hold:
\begin{itemize}
\item $(\mathbf x,\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_{\ell})$
is strongly boundary monotone.
\item There is some $\rho\in \chords(\rhos_i)$ with the property
that $\rho=\rho_1\uplus \rho_2$, and
$\chords(\sigmas)=(\chords(\rhos_i)\setminus \{\rho\})\cup
\{\rho_1,\rho_2\}$.
\item In the above decomposition, at least one of $\rho_1$ and
$\rho_2$ covers only half of a boundary component.
\end{itemize}
\item \label{endDA:BoundaryDegeneration}
Boundary degeneration collisions $\sigmas$
between two consecutive packets $\rhos_i$ and $\rhos_{i+1}$;
when
$\orb_j\in\orbits(\rhos_i)$, $\orb_k\in\orbits(\rhos_{i+1})$
and $\{j,k\}\in\Mmid$.
When $\sigmas=\rhos_i\setminus \{\orb_j,\orb_k\}$
is non-empty,
these correspond to points in
\[ \ModFlow^{B'}(\mathbf x,\mathbf y,\Source';\rhos_1,\dots,
\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_\ell)\]
in the chamber $\Chamber^{\orb_j<\orb_k}$, where
the homology class $B'$
is obtained from $B$ by removing
a copy of $\Bjk$. When $\sigmas=\emptyset$, then $\ell=1$,
$\mathbf x=\mathbf y$, $B'=0$, and the end is unique.
\end{enumerate}
\end{thm}
\begin{proof}
The proof is exactly as in the proof of Theorem~\ref{thm:AEnds}.
The key difference is that in a middle diagram, we do not treat
special boundary degenerations separately. (Theorem~\ref{thm:AEnds}
would have a had a similar statement if we had treated the marked
points $\wpt$ and $\zpt$ as orbits around the punctures.)
\end{proof}
\begin{prop}
\label{prop:DAmid}
Let $\Hmid$ be a middle diagram that is compatible with a given
matching $M$ on its incoming boundary. Choose an orientation on
$W=W(\Hmid)\cup W(M)$. The $\IdempRing(2n)-\IdempRing(2m)$-bimodule
$\DAmod(\Hmid)$, equipped the operations
\[\delta^1_{\ell+1}\colon \DAmod(\Hmid)\otimes\Clgin^{\otimes \ell}\to \Clgout\otimes\DAmod(\Hmid)\]
defined above endows $\DAmod(\Hmid)$ with the structure of
a curved $\Clg(n,\Mout)-\Clg(m,M)$ $DA$ bimodule (cf. Equations~\eqref{eq:CurvedDAbimoduleRelation1} and~\eqref{eq:CurvedDAbimoduleRelation2}),
where $\Mout$ is the matching on $\{1,\dots,2n\}$ induced by
$\Mmid$ and $M$.
\end{prop}
\begin{proof}
This is a combination of Propositions~\ref{prop:CurvedTypeD} and
Proposition~\ref{prop:CurvedTypeA}.
In more detail, look at ends of one-dimensional moduli spaces. Note
that the moduli spaces that contribute to the outgoing algebra
element cannot cover all of the outgoing boundary with positive
weight, so we can restrict attention to homology classes $B$ that do
not cover all of $\Sigma$; i.e. Theorem~\ref{thm:DAEnds} applies.
Consider first the $\Ainfty$ relation with no incoming algebra
elements.
When $\mathbf x\in\Chamber^{\orb>\orb'}$, there is a corresponding end of
$\ModFlow(\mathbf x,\mathbf x,\{\orb\},\{\orb'\})$. In turn, that moduli space
contributes to $\mu_2\circ(\Id\otimes \delta^1)\circ \delta^1$ only
in two cases:
\begin{itemize}
\item when both $\orb$ and $\orb'$ are Reeb orbits on
the out-going boundary, or
\item one of the two is a Reeb orbit on the
out-going boundary and the other is an odd orbit on the in-coming
boundary.
\end{itemize}
Each equivalence class of orbits contains exactly one pair of orbits
which can be paired in a simple boundary degeneration as above: the
boundary degeneration is the unique simple boundary degeneration that
contains the orbit corresponding to the endpoint of the given
$W$-equivalence class. Explicitly, when the $W$-equivalence class
contains no in-coming boundary components, then $\orb$ and
$\orb'$ are the two (out-going) orbits in the equivalence
class; otherwise, if $\orb$ is outgoing and it is paired with
an in-coming $\orb'$, then that $\orb'$ must be the
last odd orbit in the $W$-equivalence class (under the ordering induced
by its orientation).
By switching the order of $\orb$ and $\orb'$ if needed (since
$\mathbf x\in\Chamber^{\orb>\orb'}$ or $\Chamber^{\orb'>\orb}$), we can conclude
each equivalence class of orbits contributes $U_j U_k$,
where $\Zout_j$ and $\Zout_k$ are the two out-going boundary components
in the equivalnce class.
Thus, these boundary degenerations to
$\mu_2\circ(\Id\otimes\delta^1)\circ\delta^1(\mathbf x)$ gives a term of
the form $\mu_0^{\Clgout}\otimes \mathbf x$.
As in the proof of Proposition~\ref{prop:CurvedTypeA}, there are orbit
curve ends which can be identified with
$\delta^1(\mathbf x,\mu^{\Clgin}_0)$.
All other contributions cancel in pairs as in the proof of
Proposition~\ref{prop:CurvedTypeA}, verifying the weighted type $DA$
structure equation with no algebra inputs,
Equation~\eqref{eq:CurvedDAbimoduleRelation1}.
Note that we do have collision ends when packets on $\Zin$ collide
with packets on $\Zout$; but once again, these cancel with ends
where the two packets are permuted.
\end{proof}
\subsection{Another pairing theorem}
Let $\Hup$ be an upper diagram and $\Hmid$ be a middle diagram, so
that $\partial\Hup$ is identified with $\partial^{\vee}\Hmid$. Then,
we can form a new upper diagram $\Hmid\#\Hup$. Evidently, there is a
one-to-one correspondence between pairs of states $\mathbf x$ and $\mathbf y$, where
$\mathbf x$ is an partial Heegaard state for $\Hmid$ and $\mathbf y$ is an upper
Heegaard state for $\Hup$, and
$\alpha(\mathbf x)=\{1,\dots,2n\}\setminus\alphaout(\mathbf y)$.
We have the following analogue of the pairing theorem
Theorem~\ref{thm:PairAwithD}; compare also~\cite[Theorem~11]{Bimodules}:
\begin{thm}
\label{thm:PairDAwithD}
Let $\Hmid$ and $\Hup$ be as above.
Let $\Clg_1=\Clg(\Hup)=\Clgin(\Hmid)$;
$\Clg_2=\Clgout(\Hmid)$.
Under the above hypotheses, there is a quasi-isomorphism
of curved type $D$ structures
\[\lsup{\Clg_2}\Dmod(\Hmid\#\Hup)
\simeq
\lsup{\Clg_2}\DAmod(\Hmid)_{\Clg_1}\DT \lsup{\Clg_1}\Dmod(\Hup).\]
\end{thm}
\subsection{Proof of the DA bimodule pairing theorem}
The proof of Theorem~\ref{thm:PairDAwithD} is very similar to the
proof of Theorem~\ref{thm:PairAwithD}. The key algebraic difference is
that in the present context, there is a curvature in the result; and
analytically, the curvature is produced by boundary
degenerations. We give the details presently.
Let $\Hup_2$ be an upper diagram with $2m$ outgoing boundary
components and $\Hmid$ be a middle diagram with $2m$ incoming boundary
components and $2n$ outgoing ones. Fix an identification between
$\Zout(\Hup_1)$ and $\Zin(\Hmid)$, and use this to form the upper
$\Hup=\Hmid\#\Hup_2$.
Definition~\ref{def:MatchingPair} has the following straightforward
generalization:
\begin{defn}
Let $\Hup=\Hmid\#\Hup_2$. States $\mathbf x_1\in\States(\Hmid)$,
$\mathbf x_2\in\States(\Hup)$ are called {\em matching states}
if $\alphain(\mathbf x_1)$ is the complement of
$\alpha(\mathbf x_2)$ (i.e. $\Idown(\mathbf x_1)=\Iup(\mathbf x_2)$).
\end{defn}
There is a one-to-one correspondence between pairs of matching states
$\mathbf x_1$ and $\mathbf x_2$, and upper states for $\Hup$; and hence there is a
one-to-one correpsondence between generators of
$\DAmod(\Hmid)\DT\Dmod(\Hup_2)$ and $\Dmod(\Hup)$.
\begin{defn}
Suppose that $(\mathbf x_1,\mathbf y_1)$ and $(\mathbf x_2,\mathbf y_2)$ are matching states,
and $B_1\in \doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$.
We say that $B_1$ and $B_2$ are {\em matching domains}
if the local multiplicities of $B_2$
around each component of $\Zout_i(\Hup_2)$ coincide with
the local multiplicities around each component of
$\Zin_i(\Hmid)$.
\end{defn}
For $\mathbf x=\mathbf x_1\#\mathbf x_2$ and $\mathbf y=\mathbf y_1\#\mathbf y_2$, there is a one-to-one
correspondence between $B\in\doms(\mathbf x,\mathbf y)$ and matching domains
$B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf x_2)$. In that case,
we write $B=B_1\# B_2$.
Definition~\ref{def:MatchedPair} has the following generalization:
\begin{defn}
\label{def:MatchedPairDA}
Suppose $n>1$.
Fix two pairs $(\mathbf x_1,\mathbf x_2)$ and $(\mathbf y_1,\mathbf y_2)$ of matching states,
i.e. where $\mathbf x_1,\mathbf y_1\in\States(\Hmid)$,
$\mathbf x_2,\mathbf y_2\in\States(\Hup_2)$, so that $\mathbf x=\mathbf x_1\#\mathbf x_2$ and
$\mathbf y=\mathbf y_1\#\mathbf y_2$ are Heegaard states for $\HD$.
A {\em matched pair} consists of the following data
\begin{itemize}
\item a holomorphic curve $u_1$ in $\Hmid$ with source $\Source_1$ representing homology class $B_1\in\doms(\mathbf x_1,\mathbf y_1)$
\item a holomorphic curve $u_2$ in $\Hup_2$ with source $\Source_2$ representing homology class $B_2\in\doms(\mathbf x_2,\mathbf y_2)$
\item a bijection $\psi\colon \AllPunct(\Source_2)\to \AllPunctIn(\Source_1)$,
where $\AllPunctIn(\Source_1)\subset\AllPunct(\Source_1)$ denotes the set
of punctures on $\Source_1$ that are labelled by chords and orbits
supported on $\Zin$,
\end{itemize}
with the following properties:
\begin{itemize}
\item For each $q\in \AllPunct(\Source_2)$ is marked with a Reeb orbit
or chord, the corresponding puncture $\psi(q)\in \AllPunct(\Source_1)$
is marked with the matching Reeb orbit or chord
(in the sense of Definition~\ref{def:MatchingChords}).
\item For each $q\in\AllPunct(\Source_2)$,
\[ (s\circ u_1(\psi(q)),t\circ u_1(\psi(q)))=(s\circ u_2(q),t\circ u_2(q)).\]
\end{itemize}
If $(u_1,u_2)$ is a matched pair with homology class
$B_1$ and $B_2$, then $B_1$ and $B_2$ are matching domains.
Let $\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)$
denote the modul space of matched pairs with shadow $B=B_1\# B_2$.
\end{defn}
We have the following analogue of Lemma~\ref{lem:BoundaryMonotone}:
\begin{lemma}
\label{lem:BoundaryMonotoneDA}
Let $(\mathbf x_1,\mathbf x_2)$ and $(\mathbf y_1,\mathbf y_2)$ be two pairs of matching states,
and fix a matched pair $(u_1,u_2)$ connecting $\mathbf x_1\#\mathbf x_2$ to
$\mathbf y_1\#\mathbf y_2$, with $u_1\in\ModFlow(\mathbf x_1,\mathbf y_1,\rhos_1,\dots,\rhos_m)$
and $u_2\in\ModFlow(\mathbf x_2,\mathbf y_2,\rhos_1',\dots,\rhos_m')$ (where the
chords in $\rhos_i$ all match, in the sense of
Definition~\ref{def:MatchingChords} with chords in $\rhos_i'$).
Then, both $u_1$ and
$u_2$ are strongly boundary monotone; moreover,
$\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_\ell)$ is complementary to
$\alpha(\mathbf x_2,\rhos_1',\dots,\rhos_\ell')$ for all $\ell=0,\dots,m$.
\end{lemma}
Given a holomorphic curve $u_1$ for a middle diagram, let $\cin_1$
denote the number of chords in $u_1$ on $\Zin(\Hmid)$ $\oin_1$ denote
the number of orbits in $u_1$ on $\Zin(\Hmid)$, and $\win_1$ denote
the total weight of $u_1$ at $\Zin(\Hmid)$.
If $(u_1,u_2)$ is a
matched pair of curves, if $c_2$, $o_2$, and $w_2$ denote the number
of chords, orbits, and total weight at $\Zout(\Hup_2)$ respectively,
then $\cin_1=c_2$, $\oin_1=o_2$, $\win_1=\weight_2$.
The space of matched pairs lies in a moduli space whose expected dimension is
given by
\begin{equation}
\label{eq:DefIndMatchedPairDA}
\ind(B_1,\Source_1;B_2,\Source_2)=
\ind(B_1,\Source_1)+\ind(B_2,\Source_2)-c_2-2o_2.
\end{equation}
\begin{defn}
The {\em embedded index} of a matched pair is defined by the formula
\begin{align}
\label{eq:DefEmbInd}
\ind(B_1;B_2)&=e(B_1)+n_{\mathbf x_1}(B_1)+n_{\mathbf y_1}(B_1)
+ e(B_2)+n_{\mathbf x_2}(B_2)+n_{\mathbf y_2}(B_2)-2\weight_{\Zin}(B_1) \nonumber \\
&= e(B_1\natural B_2)+n_{\mathbf x}(B_1\natural B_2)+n_{\mathbf y}(B_1\natural B_2).
\end{align}
\end{defn}
Lemma~\ref{lem:TransversalityDA} has the following analogue:
\begin{lemma}
\label{lem:TransversalityDA}
Fix $B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ so that
$\weight_i(B_1)=\weight_i(B_2)$ for $i=1,\dots,2m$,
and the local multiplicity of $B_1$ vanishes somewhere.
For generic admissible almost complex
structures on $\Sigma_i\times [0,1]\times \R$, and
$\ind(B_1,\Source_1;B_2,\Source_2)\leq 2$, the moduli space of
matched pairs
\[ \ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)\] is
transversely cut out by the $\dbar$-equation and the evaluation map;
in particular, this moduli space is a manifold whose dimension is
given by Equation~\eqref{eq:DefIndMatchedPairDA}.
\end{lemma}
Proposition~\ref{prop:EmbeddedModuliSpaces} has the following analogue:
\begin{prop}
\label{prop:EmbeddedModuliSpacesDA}
Fix $\mathbf x=\mathbf x_1\#\mathbf x_2$ and $\mathbf y=\mathbf y_1\#\mathbf y_2$, and decompose
$B\in\doms(\mathbf x,\mathbf y)$ as $B=B_1\natural B_2$, with
$B_i\in\doms(\mathbf x_i,\mathbf y_i)$. Fix source curves $\Source_1$ and
$\Source_2$ together with a one-to-one correspondence $\psi\colon
\AllPunct(\Source_2)\to\AllPunct(\Source_1)$ which is consistent with the chord
and orbit labels, so we can form $\Source=\Source_1\natural_\psi\Source_2$.
Suppose that $\ModFlow^B(\mathbf x,\mathbf y;\Source)$
(i.e. the moduli space for curves in $\HD$) and
$\ModFlow^{B_i}(\mathbf x_i,\mathbf y_i;\Source_i)$
(which are moduli spaces for curves in $\HD_i$)
are non-empty for $i=1,2$.
Then,
$\ind(B_1,\Source_1;B_2,\Source_2)\leq \ind(B)$
if and only if
$\chi(\Source_i)=\chiEmb(B_i)$ for $i=1,2$;
and all the chords in $\Source_1$ have weight $1/2$ and all
the orbits in $\Source_1$ have weight $1$.
\end{prop}
In the present case, the Gromov compactification can contain
$\beta$-boundary degenerations. Nonetheless, excluding the
$\alpha$-boundary degenerations and closed curve components work as
before, as in the following. Compare Lemmas~\ref{lem:NoBoundaryDegenerations}
and~\ref{lem:NoClosedCurves} respectively:
\begin{lemma}
\label{lem:NoBoundaryDegenerationsDA}
Fix $B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ with so
that
\begin{itemize}
\item the local multiplicities of $B_1$ along $\Zin(\HD_1)$ agree
with the local multiplicities of $\Zout(\HD_2)$
\item $B_1$ has
vanishing local multiplicity somewhere.
\end{itemize}
Then, curves in the Gromov
compactification of
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2)$ contain no
$\alpha$-boundary degenerations.
\end{lemma}
\begin{lemma}
\label{lem:NoClosedCurvesDA}
Suppose that $m>1$. Fix matching domains $B_1\in\doms(\mathbf x_1,\mathbf y_1)$
and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$, so that $B_1$ has a vanishing local multiplicity
somewhere, and so that
$\ind(B_1\natural B_2)\leq 2$. Curves in the Gromov
compactification of
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2)$ contain no
closed components.
\end{lemma}
\begin{lemma}
\label{lem:NoBoundaryDegenerations2DA}
Fix matching domains
$B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ so that
$w_i(B_1)=w_i(B_2)$ so that
$\ind(B_1\natural B_2)\leq 2$. Then, curves in the Gromov
compactification of
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2)$ contain no
$\beta$-boundary degenerations, except in the special case where
$B=\Bjk$ for some matched pair $j$ and $k$.
\end{lemma}
\begin{proof}
The proof of Lemma~\ref{lem:NoBoundaryDegenerations} actuall shows that
if there is $\beta$-boundary degeneration (on either side), then
in fact, it forms along with a sequence of boundary degenerations
that contain all the orbits in a single component of $W$.
A straightforward computation shows that the remaining components
have index $0$, and therefore, if it has a pseudo-holomorphic representative,
the remainder must correspond to a constant map. It follows that
$B=\Bjk$.
\end{proof}
\subsubsection{Type $D$ structures of matched curves}
\begin{defn}
\label{def:MatchTypeD}
Let $\Hup=\Hmid\cup \Hup_2$ be a decomposition of an upper diagram.
Let $X$ denote the vector spacegenerated by
upper states for $\Hup$, equipped with an operator
\[ \delta^1_{\natural}\colon X \to \Clg\otimes X \]
characterized by
\[ \delta^1_{\natural}(\mathbf x)=\sum_{\{B=B_1\# B_2\big| ind(B)=1\}}
\#\ModMatched^B(\mathbf x,\mathbf y)\cdot \bOut(B)\otimes \mathbf y.\]
\end{defn}
The analogue of Theorem~\ref{thm:NeckStretch} is the following:
\begin{thm}
\label{thm:NeckStretchDA}
Provided $m>1$.
$(X,\delta^1_{\natural})$ is a curved type $D$ structure, which is
homotopy equivalent to $\Dmod(\Hup)$.
\end{thm}
\begin{proof}
The stretching argument from Theorem~\ref{thm:NeckStretchDA}
proves that $\delta^1_{\natural}$ agrees with $\delta^1$
for a suitable complex structure on $\Hup$.
It follows that $\delta^1_{\natural}$ is a curved type $D$ structure.
\end{proof}
\begin{rem}
The above argument shows that boundary degenerations which are
allowed in Lemma~\ref{lem:NoBoundaryDegenerations2DA} indeed do
occur; and their algebraic count is $1$.
\end{rem}
\subsubsection{Self-matched curves}
We adapt Definition~\ref{def:SelfMatched} to middle diagrams as follows:
\begin{defn}
\label{def:SelfMatchedDA}
Let $\Source_1$ be a decorated source for $\Hmid$.
according to whether the punctures are marked by chords or orbits on
$\Zout$ or $\Zin$ respectively. Further partition
the punctures of $\Source_1$ labelled by chords and orbits in $\Zin$,
as
\[ \EastIn(\Source_1)\cup \OmegaInEv(\Source_1)\cup
\OmegaInOdd(\Source_1),\] where $\EastIn(\Source_1)$ consists of
boundary punctures, $\OmegaInEv(\Source_1)$ consists of interior
punctures marked by even Reeb orbits, and $\OmegaInOdd(\Source_1)$
consists of interior punctures marked by odd Reeb orbits. A {\em
self-marked source} is a decorated source, together with an
injection $\phi\colon \OmegaInEv(\Source_1)\to
\EastIn(\Source_1)$ with the property that if
$p\in\OmegaInEv(\Source_1)$ is marked by some orbit $\orb_j$, then
$\phi(p)$ is marked by a length one chord that covers the boundary
component $Z_k$ so that $\{j,k\}\in \Mup$. A {\em self-matched
curve} $u$ is an element
$u\in\ModFlow^{B_1}(\mathbf x,\mathbf y;\Source_1,\phi)$, subject to the following
additional constraints: for each puncture
$p\in\OmegaInEv(\Source_1)$,
\begin{equation}
\label{eq:SelfMatchingDA}
t\circ u(p)=t\circ u(\phi(p)).
\end{equation}
Let
$\ModMatchedChanged^{B_1,B_2}(\mathbf x,\mathbf y;\Source_1,\Source_2,\phi,\psi)$
denote the moduli space self-matched curve pairs.
\end{defn}
Let $X$ be the $\Field$-vector space generated by matching states
$\mathbf x_1$ and $\mathbf x_2$.
We use the self-matched moduli spaces to construct
a map
\[ \DChanged\colon X \to \Blg\otimes X \]
defined as in Definition~\ref{def:MatchTypeD},
only using $\ModMatchedChanged^B(\mathbf x,\mathbf y)$ instead of
$\ModMatched^B(\mathbf x,\mathbf y)$.
\begin{proof}[Proof of Theorem~\ref{thm:PairDAwithD}]
The proof is very similar to the proof of
Theorem~\ref{thm:PairAwithD}. Theorem~\ref{thm:NeckStretchDA}
provides the $D$-module isomorphism
$\Dmod(\Hup)\cong(X,\delta^1_{\natural})$. Suppose $m>1$.
We wish to define a sequence of intermediate complexes Specifically,
in the definitions of the intermediate complexes, we chose a
particular ordering on $\{1,\dots,m\}$ in which to deform the
matching condition; specifically, choose a one-to-one correspondence
$f\colon \{1,\dots,2m\}\to\{1,\dots,2m\}$ so that if
$n_1,\dots,n_{2k}$ are the orbits in $\Zin$ on a given component of
$W$, arranged in the opposite to the order the appear in $W$ (with
its orientation), then
$\{f(n_2),f(n_1),f(n_3),\dots,f(n_{2k}),f(n_{2k-1})\}$ is a sequence
of consecutive integers. See Figure~\ref{fig:LabelOrbitDA} for an
example.
We can now define the sequence of intermediate type $D$ structures
$(X,\delta^1_\ell\colon X\to \Clg\otimes X)$ using operators
$\delta^1_\ell$ that count $\ell$-self-matched curve pairs, adapting
Definition~\ref{def:IntermediateComplexes}.
\begin{figure}[h]
\centering
\input{LabelOrbitDA.pstex_t}
\caption{{\bf Labeling the orbits for a middle diagram.}
For this picture, we have listed a valid numbering $f$.}
\label{fig:LabelOrbitDA}
\end{figure}
Isomorphisms are constructed as in
Proposition~\ref{prop:Intermediates}. Finally, the identification of
$(X,\DChanged)$ with the tensor product is achieved by time dilation.
The case where $m=1$ is handled separately, using special matched
pairs, as in Subsection~\ref{subsec:Nequals1}.
\end{proof}
\section{Gradings on upper Heegaard states}
\label{sec:Shadows}
Throughout this section, we will fix an upper diagram $\Hup$ with $2n$ boundary circles
\[\Hup=(\Sigma_0,Z_1,\dots,Z_{2n},\{\alpha_1,\dots,\alpha_{2n-1}\},\{\alpha^c_1,\dots,\alpha^c_{g}\},
\{\beta_1,\dots,\beta_{g+n-1}\})\] throughout. Let $\Matching$ be the
matching on $\{1,\dots,2n\}$ induced by $\Hup$.
In Section~\ref{sec:TypeD}, we will explain how to associate a type
$D$ structure to $\Hup$, over the curved algebra $\cClg$ from
Equation~\eqref{eq:DefcClg}. This structure has a differential, which
is defined by counting pseudo-holomorphic curves. Here, we explain
the data needed to specify gradings on these structures.
\subsection{Preliminaries: filling in the Heegaard surface}
\label{sec:FillSurface}
Before proceeding to the main material in this section, we introduce
some notation which will be used throughout the paper.
Recall that the Heegaard surface $\Sigma_0$ for $\Hup$ is an oriented, connected
two-manifold of genus $g$ with boundary
$\partial{\Sigma_0}=Z=Z_1\cup\dots\cup Z_{2n}$. By attaching
infinite cylinders $Z\times [0,\infty)$ to $\Sigma_0$, we obtain an oriented
two-manifold $\Sigma$ with punctures $p_1,\dots,p_{2n}$. Filling in
these punctures, we obtain a compact surface, denoted $\cSigma$.
Extend $\alphas$ in $\Sigma$, by attaching two rays in each
cyclinder $Z_i\times [0,\infty)$ for $i=2,\dots,2n-1$ and a single ray
inside each of $Z_1\times [0,\infty)$ and $Z_{2n}\times[0,\infty)$.
In the filled surface $\cSigma$
the union of $\alpha$-arcs completes to
form a single closed interval. Let ${\overline\alphas}\subset \overline\Sigma$
denote the subspace which is the union of the above defined interval and the union of curves $\{\alpha_i^c\}_{i=1}^g$.
\subsection{Gradings}
To each upper Heegaard state $\mathbf x$
for $\Hup$, there is an associated idempotent in
$\RestrictIdempRing(n)$ (the ring generated by the idempotent states in $\Clg(n)\subset
\Blg(n)$), defined by the formula
\[ \Iup(\mathbf x)=\Idemp{\{1,\dots,2n-1\}\setminus\alpha(\mathbf x)},\]
where $\alpha(\mathbf x)$ is defined as in Definition~\ref{def:UpperState}.
The complement of ${\overline\alphas}\cup\betas$ inside $\cSigma$ can be written
as a disjoint union of connected open sets called {\em elementary
domains}.
\begin{defn}
\label{def:TwoChainFromXtoY}
Given upper states $\mathbf x$ and $\mathbf y$, a {\em two-chain from $\mathbf x$ to $\mathbf y$}
is a formal integral combination $\phi$ of the elementary domains in
$\cSigma$, with the following property. If $\partial_{\alpha}(\phi)$
resp. $\partial_\beta(\phi)$ denotes the portion of the boundary of
$\phi$ contained in $\alphas$ resp. $\betas$, we require that
$\partial (\partial_{\alpha}(\phi))=\mathbf y-\mathbf x$ (and hence $\partial
(\partial_{\beta}(\phi))=\mathbf x-\mathbf y$). Let $\doms(\mathbf x,\mathbf y)$ denote the space
of two-chains from $\mathbf x$ to $\mathbf y$.
\end{defn}
Given $\phi\in\doms(\mathbf x,\mathbf y)$ and $\psi\in\doms(\mathbf y,\mathbf z)$,
their sum can be viewed as an element $\phi*\psi\in\doms(\mathbf x,\mathbf z)$.
\begin{defn}
Let $\phi\in\doms(\mathbf x,\mathbf y)$. Define $b_0(\phi)$ to be the element
$b\in \BlgZ(2n,n)$ characterized by the following two properties that
\begin{itemize}
\item
$\Iup(\mathbf x)\cdot b\cdot \Iup(\mathbf y) = b$; and
\item for all $i=1,\dots,2n$,
$\weight_i(b)$ is the average of the local multiplicities of $\phi$ in the
two elementary domains adjacent to $Z_i$.
\end{itemize}
We will also write $\weight(b)=\sum_{i=1}^{2n}\weight_i(b)$.
\end{defn}
\begin{lemma}
\label{lem:MultUnderJuxtaposition}
$b_0(\phi*\psi)=b_0(\phi)\cdot b_0(\psi)$,
where the right hand side is multiplication in $\BlgZ(2n,n)$.
\end{lemma}
\begin{proof}
This is clear from the additivity of the local multiplicities
under juxtaposition.
\end{proof}
Each elementary domain ${\mathcal D}_i$ in $\cSigma$ has an {\em Euler
measure}, which is the integral of $1/2\pi$ times the curvature of a
metric for which the boundary consists of geodesics meeting at
$90^\circ$ angles along the corners.
For $\phi\in\doms(\mathbf x,\mathbf y)$, we define its {\em point measure} $P(\phi)$
to be the sum $n_{\mathbf x}(\phi)+n_{\mathbf y}(\phi)$, so that each elementary
domain ${\mathcal D}$ contributes $1/4$ times the number of components
of $\mathbf x$ and $\mathbf y$ contained as corners of ${\mathcal D}$. The
{\em Maslov grading of $\phi$} is defined by the formula:
\begin{equation}
\label{eq:DefineMgr}
\Mgr(\phi)=e(\phi)+P(\phi).
\end{equation}
\begin{lemma}
\label{lem:GradingsWellDefined}
If $\doms(\mathbf x,\mathbf y)$ is non-empty, then for any $\phi\in\doms(\mathbf x,\mathbf y)$,
the integers $\Mgr(\phi)-\weight(b_0(\phi))$ and
$\weight_i(\phi)-\weight_j(\phi)$ (with $\{i,j\}\in \Matching$) are independent of the
choice of $\phi$.
\end{lemma}
\begin{proof}
Before verifying the independence of the choice of $\phi$, we start
by verifying that $\Mgr(\phi)$, which is evidently a rational
number, is in fact an integer. This could be seen either by the
interpretation of $\Mgr(\phi)$ as a Maslov index, but instead we
recall here a more elementary argument. Since
$\Mgr(\phi+\Sigma)=\Mgr(\phi)+2$, it suffices to verify the
integrality of $\Mgr(\phi)$ for positive $\phi$. If $\phi$ is
positive, it is elementary to construct a surface with corners $F$
(cf.~\cite[Lemma~2.17]{HolDisk}) at $\mathbf x$ and $\mathbf y$; this surface is
equipped with a branched covering to $\Sigma$ with branching at the
intersection points of the $\alpha$- and the
$\beta$-circles. Suppose for simplicity that each elementary domain
is topologically a disk, so that the $\alpha$ and $\beta$-arcs and
circles give $\Sigma$ the structure of a $CW$ complex. The surface
$F$ has a $CW$ complex structure, obtained by pulling back this $CW$
complex structure. Consider the function $f$ on subcomplexes of $F$
that associates to each $2$-cell $1$, to each edge $-1/2$, and to
each vertex $1/4$. Clearly, $f(\cald)=e(\cald)$ for each elementary
domain. Moreover, since each interior edge is contained in two
domains, and each interior vertex is contained in four elementary
domains, it follows that
\[ e(F)=\chi(F)+ \OneHalf \#(e\subset \partial F)+ \sum_{p}
\left(\frac{N_p}{4}-1\right)=\chi(F)+ \sum_{p}
\left(\frac{N_p-2}{4}\right),\] where $N_p$ denotes the number of
elementary domains that meet at a $0$-cell in $F$. The integrality of
$\Mgr(\phi)$ follows from the observation that at each corner point
$p$, $\frac{N_p-2}{4}+n_{p\cap \mathbf x}(\phi)+n_{p\cap \mathbf y}(\phi)$ is an integer.
This argument can be easily adapted also to the case where
the elementary domains are not disks.
If $\phi,\phi'\in\doms(\mathbf x,\mathbf y)$ then $\phi-\phi'$ can be written as a
formal sum of components ${\mathcal D}$ of
$\cSigma\setminus\betas$. This is true since
$\alpha_1^c,\dots,\alpha_g^c$ are linearly independent in
$H_1(\cSigma)$ and the intersection of their span with the
span of $\beta_1,\dots,\beta_{g+n-1}$ is trivial (by
Condition~\ref{UD:NoPerDom}). Each of those latter components has
$e({\mathcal D})=1$ and $P({\mathcal D})=1$, contributing
$2$ to $\Mgr(\phi)$; similarly, the addition of ${\mathcal D}$
contributes $2$ to $\weight(b_0(\phi))$. To complete the lemma, note
that if $i$ and $j$ are matched, then $\weight_i({\mathcal
D})=\weight_j({\mathcal D})$ for any ${\mathcal D}$ of
$\cSigma\setminus\betas$.
\end{proof}
Given $\phi\in\doms(\mathbf x,\mathbf y)$, let $\bOut(\phi)\in\Idemp{\mathbf x}\cdot
\Blg\cdot \Idemp{\mathbf y}$ denote the image of $b_0(\phi)$ under the
quotient map $\BlgZ(n)\to\Blg(n)$. Clearly,
$\bOut(\phi)\in\Clg(n)\subset\Blg(n)$.
\begin{prop}
\label{prop:DefineBigradingD}
There is a function $\Mgr\colon \States(\Hup)\to \Z$
that is uniquely characterized up to an overall constant by the property that
\begin{equation}
\label{eq:DefMgr}
\Mgr(\mathbf x)-\Mgr(\mathbf y)=\Mgr(\phi)-\weight(b_0(\phi)),
\end{equation}
for any $\phi\in\doms(\mathbf x,\mathbf y)$. Similarly, given an orientation for
the one-manifold $W$ specified by the matching $\Matching(\Hup)$,
there is another function $\Agr\colon \States(\Hup)\to \OneHalf\Z^n$
with components $\Agr_{\{i,j\}}$ cooresponding to each
$\{i,j\}\in\Matching$, characterized by the following condition,
uniquely up to overall translation by some vector in
$\OneHalf\Z^n\subset\mathbb Q} \newcommand{\R}{\mathbb R^n$:
\begin{equation}
\label{eq:DefAgr}
\Agr_{\{i,j\}}(\mathbf x)-\Agr_{\{i,j\}}(\mathbf y)=
\weight_i(b_0(\phi))-\weight_j(b_0(\phi)),
\end{equation}
for any choice of $\phi\in\doms(\mathbf x,\mathbf y)$;
i.e. $\Agr(\mathbf x)-\Agr(\mathbf y)=\Agr(b_0(\phi))$, where the right-hand-side uses the $\mathbb Q} \newcommand{\R}{\mathbb R^n$-valued Alexander vector grading on the algebra.
\end{prop}
\begin{proof}
Fix $\mathbf x$ and $\mathbf y$. Condition~\ref{UD:NoHone} ensures that for any
two upper states $\mathbf x$ and $\mathbf y$, there is some $\phi\in\doms(\mathbf x,\mathbf y)$.
Thus, according to Lemma~\ref{lem:GradingsWellDefined}, given $\mathbf x$
and $\mathbf y$, the right-hand-side of Equation~\eqref{eq:DefMgr} is
well-defined. To see that it can be written as $\Mgr(\mathbf x)-\Mgr(\mathbf y)$,
it suffices to observe that the right hand side of
Equation~\eqref{eq:DefMgr} is additive under juxtaposition. This is
mostly straightforward; see~\cite[Theorem 3.3]{SarkarWhitney} for an
elementary proof of the additivity of $\Mgr$ under juxtapositions.
The corresponding statement for $\Agr$ follows similarly.
\end{proof}
\begin{rem}
\label{rem:OurDomains}
We could have chosen to work instead with elementary domains
${\mathcal D}_i^0$, which are the closures of the components of
$\Sigma_0\setminus\alphas\cup\betas$.
Clearly each elementary domain
${\mathcal D}_i$ in $\cSigma$ is obtained
from some elementary domain ${\mathcal D}^0_i$ in $\Sigma_0$ by
attaching half disks to each boundary component of ${\mathcal D}_i$
obtained as $Z_j\cap {\mathcal D}_i$ with $j\neq 1$ or $2n$; and
attaching disks along the boundary components $Z_1\cap{\mathcal
D}_i$ and $Z_{2n}\cap{\mathcal D}_i$.
Thus, the Euler measure of
each elementary doman in ${\overline \Sigma}$ equals the Euler
measure of the corresponding domain in $\Sigma_0$ plus $1/2$ for
each boundary component induced by $Z_j$ with $j\neq 1$ or $2n$ and
$1$ for the boundary components coming from $Z_1$ or $Z_{2n}$.
We could work with $\doms_0(\mathbf x,\mathbf y)$, which are domains in $\Sigma_0$.
The Euler measure on elementary domains can be extended linearly to obtain an Euler measure
of any $\phi_0\in\doms_0(\mathbf x,\mathbf y)$.
If $\phi_0\in\doms_0(\mathbf x,\mathbf y)$, and $\phi\in \doms(\mathbf x,\mathbf y)$ is
the corresponding domain in $\cSigma$, then
\[ e(\phi)=e(\phi_0)+\weight(b_0(\phi)).\]
With these conventions, then,
Lemma~\ref{lem:GradingsWellDefined} says that
and
\[ \Mgr(\phi)-\weight(b_0(\phi))=e(\phi_0)+P(\phi_0)\]
is independent of the choice of $\phi_0\in\doms_0(\mathbf x,\mathbf y)$.
\end{rem}
\section{Extending the $DA$ bimodules}
\label{sec:ExtendDA}
As defined so far, our type DA bimodules are curved bimodules over
$\Clg$. Our aim here is to extend these modules to modules over
$\Blg$. Specifically, if $\Hmid$ is a middle diagram, let
$\Blgin=\Blgin(\Hmid)=\bigoplus_{k=0}^{2m-1}\Blg(2m,k)$,
$\Blgout=\Blgout(\Hmid)=\bigoplus_{k=0}^{2n-1}\Blg(2n,k)$, so
that
\begin{align*}
\Clgin&=
\left(\sum_{\{\mathbf x\mid \mathbf x\cap \{0,
2m\}=\emptyset\}} \Idemp{\mathbf x}\right)\cdot \Blgin\cdot
\left(\sum_{\{\mathbf x\mid \mathbf x\cap \{0, 2m\}=\emptyset\}}
\Idemp{\mathbf x}\right) \\ \\
\Clgout&=
\left(\sum_{\{\mathbf x\mid \mathbf x\cap \{0,
2m\}=\emptyset\}} \Idemp{\mathbf x}\right)\cdot \Blgout\cdot
\left(\sum_{\{\mathbf x\mid \mathbf x\cap \{0, 2m\}=\emptyset\}}
\Idemp{\mathbf x}\right).
\end{align*}
In particular, there are inclusion maps
${\iota}\colon \Clgin\to \Blgin$ and ${\iota}\colon \Clgout\to \Blgout$.
In the next statement we will use the corresponding bimodules
$\lsup{\Blgin}[\iota]_{\Clgin}$ and
$\lsup{\Blgout}[\iota]_{\Clgout}$
\begin{prop}
\label{prop:ExtendDA}
Let $\Hmid$ be a middle diagram, and $\lsup{\Clgout}\DAmod(\Hmid)_{\Clgin}$
be the corresponding type $DA$ bimodule.
Then, there is a type $DA$ bimodule
$\lsup{\Blgout}\DAmod_{\Blgin}$, with the property that
\[ \lsup{\Blgout}[\iota]_{\Clgout}~
\DT \lsup{\Clgout}\DAmod(\Hmid)_{\Clgin}=
\lsup{\Blgout}\DAmodExt_{\Blgin}\DT~\lsup{\Blgin}[\iota]_{\Clgin}.\]
\end{prop}
The bimodule over $\Blg$ in the statement of
Proposition~\ref{prop:ExtendDA} is constructed by extending the notion of
middle diagrams, and defining their associated type $DA$ bimodules, as
we do presently. The proof of the above theorem is given in
Section~\ref{sec:DestabilizationTheorem}.
\subsection{Extended middle diagrams}
We define now relevant generalization of middle diagrams used
in the construction of the module from Proposition~\ref{prop:ExtendDA}.
\begin{defn}
An {\em extended middle diagram} is a Heegaard diagram obtained
from an upper diagram with $2m+2n+2$ boundary components
$\{Z_0,\dots,Z_{2m+2n+1}\}$, and adding a new arc $\alpha_{2m+2n+1}$ connecting $Z_0$ and
$Z_{2m+2n+1}$ ,
and adding two new beta circles, so that the $\beta$ circles now are labelled
$\{\beta_i\}_{i=1}^{g+m+n+1}$.
We think of $2m$ of these components as
``incoming'' boundary, writing $\Zin_i=Z_i$ for
$i=1,\dots,2m$; $2n$ ``outgoing'' boundary components, writing
$\Zout_i=Z_{2m+2n+2-i}$ for $i=1,\dots,2n$; and two ``middle'' boundary
components $\Zmid_0=Z_0$ and $\Zmid_1=Z_{2m+1}$. We also let
$\alphain_i=\alpha_i$ for $i=0,\dots,2m$;
$\alphaout_i=\alpha_{2m+2n+1-i}$ for $i=0,\dots,2n$.
We write this data
\begin{align*} \HmidExt=(\Sigma_0,\{\Zin_i\}_{i=1}^{2m},
\{\Zout_i\}_{i=1}^{2n},
\{\Zmid_0,\Zmid_1\},
\{\alphain_i\}_{i=0}^{2m},
\{\alphaout_i\}_{i=0}^{2n},
\{\alpha^c_i\}_{i=1}^g,
\{\beta_i\}_{i=1}^{g+m+n+1}\}).
\end{align*}
\end{defn}
Note that $\alphain_0$ connects $\Zin_1$ to
$\Zmid_0$, $\alphaout_0$ connects $\Zmid_0$ to $\Zout_1$,
$\alphain_{2m}$ connects $\Zin_{2m}$ to $\Zmid_1$, and
$\alphaout_{2n}$ connects $\Zmid_1$ to $\Zout_{2n}$.
\begin{example}
\label{ex:HmidEx}
Let
\[\Hmid=(\Sigma_0,\{\Zin_i\}_{i=1}^{2m}, \{\Zout_i\}_{i=1}^{2n},
\{\alphain_i\}_{i=1}^{2m-1}, \{\alphaout_i\}_{i=1}^{2n-1},
\{\alpha^c_i\}_{i=1}^g, \{\beta_i\}_{i=1}^{g+m+n-1})\] be a middle
diagram. We can form an extended middle diagram as follows.
Connect $\Zin_1$ and $\Zout_1$ by a path that crosses only
$\beta$-circles. Remove a disk centered at a point on the path from
$\Sigma_0$ to obtain a new boundary component $\Zmid_0$, and
introduce new arcs $\alphain_0$ connecting $\Zin_1$ to $\Zmid_0$ and
$\alphaout_0$ connecting $\Zmid_0$ to $\alphaout_1$. Add also a new
small $\beta$-circle $\beta_{0}$ in a collar neighborhood of
$\Zmid_0$. Connect $\Zin_{2m}$ and $\Zout_{2n}$ by another path,
remove another disk to obtain the boundary component $\Zmid_1$, and
introduce arcs $\alphain_{2m}$ connecting this to $\Zin_{2m}$ and
$\alphaout_{2n}$ connecting it to $\Zout_{2n}$, and introduce a
circle $\beta_{g+m+n}$ which encricles $\Zmid_1$, intersecting
$\alphain_{2m}$ and $\alphaout_{2n}$ in one point apiece. In this
manner, we obtain a new extended middle diagram, $\HmidExt$, called
a {\em stabilization} of $\Hmid$.
\end{example}
See Figure~\ref{fig:ExtMidDiag} for an example.
\begin{figure}[h]
\centering
\input{ExtMidDiag.pstex_t}
\caption{{\bf Extended middle diagram of local minimum.}
\label{fig:ExtMidDiag}}
\end{figure}
\begin{defn}
An {\em extended partial Heegaard state} is a $g+m+n+1$-tuple of
points, where one lies on each $\beta$-circle, one lies on each
$\alpha$-circle, and no more than one lies on each $\alpha$-arc.
Let $\alphain(\mathbf x)\subset \{0,\dots,2m\}$ be the set of
$i\in\{0,\dots,2n\}$ so that $\mathbf x\cap \alphain_i\neq \emptyset$; and
$\alphaout(\mathbf x)\subset\{0,\dots,2n\}$ be the set of
$i\in\{0,\dots,2n\}$ so that $\mathbf x\cap \alphain_i\neq \emptyset$. An
exended middle Heegaard state has an idempotent type
$k=|\alphain(\mathbf x)|$.
\end{defn}
\subsection{$\DA$ bimodules for extended diagrams}
An exteded middle diagram induces a matching on the components of
$\partial\Sigma$; in fact, if $\HmidExt$ is an extension of $\Hmid$,
then the matchings are the same.
\begin{defn}
Let $\HmidExt$ be an extended middle diagram compatible with a matching $M$ on
the incoming boundary components. The {\em incoming algebra}
$\Blgin(\HmidExt)$ and the {\em outgoing algebra}
$\Blgout(\HmidExt)$ are defined by
\[\Blgin(\Hmid)=\bigoplus_{k=0}^{2m-1}\Blg(2m,k);
\qquad \Blgout(\Hmid)=\bigoplus_{k=0}^{2n-1}\Blg(2n,k).\]
\end{defn}
We will define a type $DA$ bimodule $\DAmodExt(\HmidExt)=
\lsup{\Blgout}\DAmodExt(\HmidExt)_{\Blgin}$. As a vector space,
it is spanned by the extended middle Heegaard states of $\HmidExt$.
\[ \Idemp{\{0,\dots,2n\}\setminus \alphaout(\mathbf x)} \cdot \mathbf x
\cdot \Idemp{\alphain(\mathbf x)}=\mathbf x.\]
Again, there is a splitting
\[ \DAmodExt(\HmidExt)=\bigoplus_{k\in\Z}
~~~{\lsup{\IdempRing(2n,k-m+n)}\DAmodExt(\Hmid)}_{\IdempRing(2m,k)},\] where $k$
is the idempotent type of $\mathbf x$. We will be primarily interested in
the summand where $k=m$,
\[ \lsup{\IdempRing(2n,n)}\DAmodExt(\Hmid)_{\IdempRing(2m,m)}.\]
The material from Section~\ref{subsec:DAmodConstruction}
has a straightforward adaptation to extended diagrams,
with the understanding that the pseudo-holomorphic curves in consideration
now have zero multiplicity at the middle boundary $\Zmid_0$ and $\Zmid_1$.
In particular, the definition of $(\mathbf x,\vec{a})$-compatible sequences
given in Definition~\ref{def:CompatiblePacketDA}
can be readily defined when $\mathbf x$ is an extended partial Heegaard state,
and the sequence $\vec{a}=(a_1,\dots,a_{\ell})$ lies in $\Blgin(\HmidExt)$
(rather than $\Clgin(\Hmid)$, as it was there).
Observe given $(B,\rhos_1,\dots,\rhos_h)$, the associated element
$\bOut(B,\rhos_1,\dots,\rhos_h)$ defined as in Equation~\eqref{eq:bOut}
naturally lies in $\Blgout(\HmidExt)$ for extended diagrams, rather than merely
in $\Clgout$.
With these observations in place, Equation~\eqref{eq:DefAction}
induces now a map
\[ \delta^1_{\ell+1}\colon \DAmodExt(\HmidExt)\otimes\Blgin^{\otimes \ell}\to
\Blgout\otimes \DAmodExt(\HmidExt).\]
Proposition~\ref{prop:DAmid} has the following analogue for extended middle diagrams:
\begin{prop}
\label{prop:DAmidEx}
Let $\HmidExt$ be an extended middle diagram that is compatible with a given
matching $M$ on its incoming boundary. Choose an orientation on
$W=W(\Hmid)\cup W(M)$. The $\IdempRing(2n)-\IdempRing(2m)$-bimodule
$\DAmodExt(\HmidExt)$, equipped the operations
$\delta^1_{\ell+1}\colon \DAmodExt(\HmidExt)\otimes\Blgin^{\otimes m}\to \Blgout\otimes\DAmodExt(\HmidExt)$
defined above endows $\DAmodExt(\HmidExt)$ with the structure of
a curved $\Blgin-\Blgout$ type $DA$ bimodule.
\end{prop}
\begin{proof}
Theorem~\ref{thm:DAEnds} has a straightforward adaptation to
extended middle diagrams. The proposition then is an immediate
consequence.
\end{proof}
\subsection{Destabilizing extended diagrams}
\label{sec:DestabilizationTheorem}
Our aim now is to prove the following version of
Proposition~\ref{prop:ExtendDA}:
\begin{prop}
\label{prop:ExtendDAPrecise}
Let $\HmidExt$ be an extension of a middle diagram $\Hmid$, as in
Example~\ref{ex:HmidEx}. The type $DA$ bimodules
associated to $\Hmid$ and $\HmidExt$ are related by the formula
\[ \lsup{\Blgout}[\iota]_{\Clgout}~
\DT \lsup{\Clgout}\DAmod(\Hmid)_{\Clgin}=
\lsup{\Blgout}\DAmodExt_{\Blgin}(\HmidExt)\DT~\lsup{\Blgin}[\iota]_{\Clgin}.\]
\end{prop}
\begin{figure}[h]
\centering
\input{ExtRemove1.pstex_t}
\caption{{\bf Removing $\alphain_0$.}
At the left, we have a portion of an extended middle diagram;
at the right, we have the corresponding half-extended middle diagram. \label{fig:ExtRemove1}}
\end{figure}
This is proved in two steps.
{\bf{Step 1: Remove $\alphain_0$ and $\alphain_m$.}}
From $\Hmid$ we constructed $\HmidExt$. There is another diagram,
$\HmidHalfExt$ which is obtained from $\HmidExt$ by removing
$\alphain_0$ and $\alphain_{2m+1}$. (See Figure~\ref{fig:ExtRemove1}
for an illustration of removing $\alphain_0$.)
We call such diagrams {\em half-extended middle diagrams}.
We can define the associated DA bimodule as before, except that now
the input algebra for a half extended middle diagram is obviously
$\Clgin$, whereas the output algebra is {\em a priori} $\Blgout$.
\begin{lemma}
\label{lem:RemoveAlphaIns}
The DA bimodules for an extended middle diagram and a half-extended
middle diagram are related by
\[ \lsup{\Blgout}\DAmodHalfExt(\HmidHalfExt)_{\Clgin}=~
\lsup{\Blgout}\DAmodExt(\HmidExt)_{\Blgin}\DT~\lsup{\Blgin}[\iota]_{\Clgin}.\]
\end{lemma}
\begin{proof}
Recall that $\beta_0$ and $\beta_{g+m+n}$ denote the two
$\beta$-circles that encircle $\Zmid_0$ and $\Zmid_1$
respectively. Let $x_0=\beta_0\cap\alphaout_0$ and
$x_{2m+1}=\beta_{g+m+n}\cap\alphaout_{2n}$; and
$y_0=\beta_0\cap\alphain_0$ and
$y_{2n+1}=\beta_{g+m+n}\cap\alphaout_{2n+1}$; The generators of
$\lsup{\Blgout}\DAmodExt(\HmidExt)_{\Blgin}\DT~\lsup{\Blgin}[\iota]_{\Clgin}$
are those middle states with $\{x_0,x_{2m+1}\}\subset \mathbf x$, which
in turn are the states generating $\DAmodHalfExt(\HmidHalfExt)$.
Similarly, the holomorphic curves counted in the module actions
for
$\lsup{\Blgout}\DAmodExt(\HmidExt)_{\Blgin}\DT~\lsup{\Blgin}[\iota]_{\Clgin}$
cannot have an $\alpha$-interval which maps to $\alphain_0$ or
$\alphain_{2m+1}$: for the endpoints would have to either be at
Reeb chords, but the incoming algebra does not allow Reeb chords
with boundary on $\alphain_0$ or $\alphain_{2m+1}$; or they would
have to be $\pm \infty$ punctures, but the generators we are
considering contain $x_0$ and $x_{2m+1}$, not the corresponding
$y_0$ or $y_{2n+1}$.
Thus, the holomorphic curves used to define the $\Ainfty$ operations on
$\lsup{\Blgout}\DAmodExt(\HmidExt)_{\Blgin}\DT~\lsup{\Blgin}[\iota]_{\Clgin}$
coincide with the ones used to define
$\lsup{\Blgout}\DAmodExt(\HmidHalfExt)_{\Clgin}$.
\end{proof}
{\bf{Step 2: Remove $\alphaout_0$ and $\alphaout_{2n}$.}}
This is a neck stretching argument, in the spirit of Step 1 in the
pairing theorem.
We start by analyzing moduli spaces that are relevant after the neck
stretching. To this end, let $\Hmid_0$ consist of a punctured disk,
whose puncture we think of as the (filled) middle boundary $\Zmid$,
and whose boundary we label ${\mathcal Z}$, equipped with a single
embedded $\beta$-circle that separates the puncture from the boundary,
and a single $\alpha$-arc, denoted $\alpha$, that runs from the
boundary to the puncture, meeting $\beta$ in a single point, which we
denote $x$. See Figure~\ref{fig:LocalStab}.
\begin{figure}[h]
\centering
\input{LocalStab.pstex_t}
\caption{{\bf The diagram ${\mathcal H}_0$.} We will consider
homology classes with local multiplicity $k$ near the boundary $Z$,
as indicated.}
\label{fig:LocalStab}
\end{figure}
A homology class of flows from $x$ to itself is determined by its
local multiplicity $k$ at the puncture; denote the space
$\ModFlow^{[k]}(x,x)$.
\begin{lemma}
The space $\ModFlow^{[k]}(x,x)$ is $2k$-dimensional.
There is a dense, open subset of $\ModFlow^{k}(x,x)$ consisting of
those curves whose Reeb orbits are simple.
\end{lemma}
\begin{proof}
The dimension is a straightforward application of the dimension formula:
the Euler measure of the region is $k$, and the point measure is $k$.
Smoothness is a consequence of the fact that the domain curve is also planar.
\end{proof}
Consider the evaluation map $\ev\colon \ModFlow^k(x,x)\to
\Sym^k([0,1]\times \R)$, which projects the punctures of the source
curves in $\ModFlow^{k}(x,x)$ to $[0,1]\times \R$.
\begin{lemma}
\label{lem:EvaluationDegreeOne}
The map
$\ev\colon \ModFlow^{[k]}(x,x)\to \Sym^k([0,1]\times \R)$
is a proper map of odd degree.
\end{lemma}
\begin{proof}
Properness can be thought of as a consequence of Gromov's compactness
theorem, since any non-constant holomorphic curves from $x$ to $x$
projects to some symmetric product $\Sym^k([0,1]\times\R)$. It
remains to compute the degree, which we do with a model computation.
Invert the Heegaard surface, so that it is a disk $D$ in the complex
plane, $Z$ corresponds to the origin, $\beta$ corresponds to the
unit circle. We can think of the domain of the holomorphic curve
$D^+$ as the upper half disk (i.e. $z\in D$ whose imaginary part is
non-negative). The holomorphic disks we are considering are
holomorphic maps from $D^+$ to $D$ which carry the real interval
$[-1,1]$ in $\partial D^+$ to the real interval in $D$, the upper half
circle in $\partial D^+$ to the boundary of $D$, and the points $\{\pm
1\}$ to $1$. By the Schwartz reflection principle, these correspond
to holomorphic maps $f\colon D\to D$ so that
\begin{enumerate}[label=(B-\arabic*),ref=(B-\arabic*)]
\item $f(\partial D)\subset \partial D$
\item
$f(D\cap \R)\subset \R$; i.e. ${\overline{f(z)}}=f({\overline z})$.
\end{enumerate}
Consider those $f$ for which $f(D\cap \R)\neq 0$. (These correspond
to those holomorphic curves that have no Reeb chord on their boundary.)
By classical complex analysis holomorphic maps $f$ as above can be uniquely
written in the form
\[ f(z)=\prod_{i=1}^{k}\frac{(z-\alpha_i)(z-{\overline\alpha_i})}
{(1-{\overline\alpha}_iz)\cdot (1-{\alpha}_i z)},\]
where $\{\alpha_1,\dots,\alpha_k\}\in\Sym^k(D^+)$.
This shows that $\ev$ induces a homeomorphism, for suitable choices
of complex structure $\beta$. It follows that the degree is odd in general.
\end{proof}
\begin{prop}
\label{prop:ExtendMainStep}
There is an identification
\[ \lsup{\Blgout}\DAmodHalfExt(\HmidHalfExt)_{\Clgin}
\simeq
\lsup{\Blgout}[\iota]_{\Clgout}\DT~ \lsup{\Clgout}\DAmod(\Hmid)_{\Clgin}\]
\end{prop}
\begin{proof}
Start from $\Dmod(\Hmid)$. We will add first $\Zmid_0$, $\beta_0$,
$\alphaout_0$ to obtain a diagram ${\mathcal H}'$; and then add
$\Zmid_1$, $\beta_{g+m+n}$, and $\alphaout_{2n}$ to obtain $\HmidHalfExt$.
Fix a curve $\gamma_0$ in ${\mathcal H}'$ which is parallel to
$\beta_0$, dividing $\HmidHalfExt$ into two components, one of which
is homeomorphic to $\Hmid_0$, and another looks like $\HmidHalfExt$
with $\beta_0$ removed. Stretch the neck normal to $\gamma_0$, so
that $\HmidHalfExt={\mathcal H}'\cup \Hmid_0$.
The diagram ${\mathcal H}'$ can also be used to define a $DA$ bimodule
of the form $\lsup{\Blgout}\DAmodP_{\Clgin}$.
Let $\ModFlow^B(\mathbf x,\mathbf y;\rhos_1,\dots,\rhos_k)$ be some moduli space of
holomorphic curves which appears in the definition of the module
actions for $\Hmid$, equipped with a puncture point $z_0$ (thought
of as a filling of $\Zmid_0$). We have an analogous
moduli spaces $\ModFlow^B_{{\mathcal H}'}(\mathbf x,\mathbf y;\rhos_1,\dots,\rhos_k)$ which
define the actions on $\lsup{\Blgout}\DAmodP_{\Clgin}$.
Stretching the neck gives a fibered product description:
\[ \ModFlow^B_{{\mathcal H}'}(\mathbf x,\mathbf y;\rhos_1,\dots,\rhos_k)
=\ModFlow^B(\mathbf x,\mathbf y;\rhos_1,\dots,\rhos_k)\times_{\Sym^d([0,1]\times\R)}
\ModFlow^{[d]}(\Hmid_0),\]
where the fibered product is taken over the evaluation at the puncture $z_0$
\[ \ModFlow^B(\mathbf x,\mathbf y;\rhos_1,\dots,\rhos_k)\to
\Sym^d([0,1]\times\R) \] (where here $d$ depends on the homotopy
class of $B$), and the evaluation map from
Lemma~\ref{lem:EvaluationDegreeOne}. Indeed, by
Lemma~\ref{lem:EvaluationDegreeOne}, it follows that
$\DAmod(\Hmid)\cong \DAmodP({\mathcal H}')$.
Adding $\beta_{g+m+n}$, $\Zmid_1$, and $\alphaout_{2n}$ to
${\mathcal H}'$ in the same manner, we obtain an
isomorphism $\DAmodP({\mathcal H}')\cong \DAmodExt(\HmidExt)$.
\end{proof}
We now have all the ingredients to prove the following:
\begin{proof}
[Proof of Proposition~\ref{prop:ExtendDAPrecise}] Combine
Lemma~\ref{lem:RemoveAlphaIns} and
Proposition~\ref{prop:ExtendMainStep}.
\end{proof}
\section{Further remarks on invariance}
\label{sec:Further}
This section is a further elaboration on Theorem~\ref{thm:ComputeD},
wherein we explore the invariance properties of the two sides of
Equation~\eqref{eq:ComputeD}.
Observe that the object appearing on the right hand side of
Equation~\eqref{eq:ComputeD}, $\lsup{\cClg}\DmodAlg(\DiagUp)$, is
associated to an acceptable upper diagram $\DiagUp$.
We explain first how to define $\lsup{\cClg}\DmodAlg(\DiagUp)$ for a
more generic kind of knot diagram.
By analogy with~\cite[Section~\ref{BK1:sec:ConstructionAndInvariance}]{BorderedKnots},
we say that an upper diagram $\DiagUp$ is in {\em bridge position} if
it is drawn on the $y\geq 0$ plane so that:
\begin{enumerate}
\item all the critical points are local
minima or maxima
\item minima, maxima, and crossings all have distinct $y$ coordinates
\item $\DiagUp$ has no closed components
\end{enumerate}
To an upper diagram in bridge position, we can associate an
algebraically defined, curved type $D$ structure
$\lsup{\cClg}\DmodAlg(\DiagUp)$ in the obvious way: we tensor together
the curved DA bimodules associated to the crossings local maxima, and
local minima ($\lsup{\cBlg_2}\Pos^i_{\cBlg_1}$,
$\lsup{\cBlg_2}\Neg^i_{\cBlg_1}$, $\lsup{\cBlg_2}\Max^c_{\cBlg_1}$,
and $\lsup{\cBlg_2}\Min^c_{\cBlg_1}$ from
Proposition~\ref{prop:CurvedDABimodules}) as they appear in $\DiagUp$.
Adapting methods from~\cite[Section~\ref{BK1:sec:ConstructionAndInvariance}]{BorderedKnots}
(see also~\cite[Section~\ref{BK2:sec:Construction}]{Bordered2}), we can show that $\lsup{\cClg}\DmodAlg(\DiagUp)$
depends only on the planar isotopy class of the diagram $\Diag$.
\begin{prop}
\label{prop:CurvedDPlanarIsotopies}
If $\DiagUp$ and $\DiagUp'$ are two upper diagrams that differ
by planar isotopies (fixing $y=0$), then
the associated curved type $D$ structures $\lsup{\cClg}\DmodAlg(\DiagUp)$
and $\lsup{\cClg}\DmodAlg(\DiagUp')$ are homotopy equivalent
(as curved type $D$ structures).
\end{prop}
The above is proved in Section~\ref{subsec:AlgInvar}, after some
algebraic background is set up in Section~\ref{subsec:LocalMaximum}.
With a little more work, we can show that this curved type
$D$-structure is in fact an invariant of the tangle represented by
$\DiagUp$.
\begin{prop}
\label{prop:TangleInvariance}
If $\DiagUp$ and $\DiagUp'$ are two upper diagrams that represent the same tangle in $S^3$, then
the associated curved type $D$ structures $\lsup{\cClg}\DmodAlg(\DiagUp)$
and $\lsup{\cClg}\DmodAlg(\DiagUp')$ are homotopy equivalent.
\end{prop}
To an upper diagram $\DiagUp$ in bridge position, there is an
associated Heegaard diagram $\HD(\DiagUp)$, obtained by stacking the
middle diagrams associated to the crossings and critical points as
they appear in $\DiagUp$.
We have the following variant of Theorem~\ref{thm:ComputeD}:
\begin{thm}
\label{thm:ComputeD2}
Let $\DiagUp$ be an upper diagram in bridge position, and let $\Hup$
be its associated upper Heegaard diagram. Then there is an
identification
\begin{equation}
\label{eq:ComputeD2}
\lsup{\cClg(n,\Matching)}\Dmod(\Hup)\simeq~\lsup{\cClg(n,\Matching)}\DmodAlg(\DiagUp).
\end{equation}
\end{thm}
Combining Theorem~\ref{thm:ComputeD2} with
Proposition~\ref{prop:CurvedDPlanarIsotopies}, we can conclude thet
the homotopy class of the analytically defined
$\lsup{\cClg(n,\Matching)}\Dmod(\Hup)$, which appears to depend on the
choice of Heegaard diagram, in fact is an invariant for tangles. (One
could alternatively prove invariance
$\lsup{\cClg(n,\Matching)}\Dmod(\Hup)$ using more analytical methods
by showing invariance under isotopies, handleslides, and
stablizations, in the spirit of~\cite{HolDisk}.)
Theorem~\ref{thm:ComputeD2} is proved in Section~\ref{subsec:ComputeD2}
\subsection{Local maxima}
\label{subsec:LocalMaximum}
Consider the curved DA bimodule $\lsup{\cBlg_1}\Max^{c}_{\cBlg_2}$
from Proposition~\ref{prop:CurvedDABimodules};
i.e. where
\[\cBlg_1=\Blg(n,\Matching_1)
\qquad {\text{and}} \qquad \cBlg_1=\Blg(n,\Matching_2),\] where
$\Matching_1$ is some matching on $\{1,\dots,2n\}$, and $\Matching_2$
is the matching on $\{1,\dots,2n+2\}$ obtained by adding to
$\phi_c(\Matching_1)$, the additional pair $\{c,c+1\}$, with $\phi_c$
as in Equation~\eqref{eq:DefInsert}.
By construction, $\lsup{\cBlg_2}\Max^{c}_{\cBlg_1}\cdot \iota =
\iota \cdot \lsup{\cBlg_1}\Max^{c}_{\cBlg_2}$
(compare Lemma~\ref{lem:RestrictIdempotents}); i.e.
there is a corresponding bimodule
\[ \lsup{\cBlg_2}i_{\cClg_2} \DT~ \lsup{\cClg_2}\Max^c_{\cClg_1} \simeq~
\lsup{\cBlg_2}\Max^c_{\cBlg_1}\DT~ \lsup{\cBlg_1}i_{\cClg_1} \]
The key algebraic step required to adapt
the invariance proof from~\cite{Bordered2} is the existence of a bimodule
$\lsup{\nDuAlg_1}\Max^c_{\nDuAlg_2}$ that is dual to
$\lsup{\cBlg_2}\Max^c_{\cBlg_1}$, as follows:
\begin{prop}
\label{prop:DualMax}
Given $\cBlg_1$ and $\cBlg_2$ as above,
there is a bimodule
$\lsup{\nDuAlg_1}\Max^c_{\nDuAlg_2}$ with the property that
\[
\lsup{\cBlg_2}\Max^c_{\cBlg_1}\DT \lsup{\cBlg_1,\nDuAlg_1}\CanonDD
\simeq
\lsup{\nDuAlg_1}\Max^c_{\nDuAlg_2}\DT \lsup{\nDuAlg_2,\cBlg_2}\CanonDD. \]
\end{prop}
The proof occupies the rest of this subsection.
The construction of $\lsup{\nDuAlg_1}\Max^c_{\nDuAlg_2}$ is similar to
the constructions of the bimodules associated to local minimum
from~\cite{Bordered2} and~\cite{BorderedKnots}. As
in~\cite[Section~\ref{BK2:sec:Min}]{Bordered2}, we start with the
construction of $\Max^c$ in the case where $c=1$.
A {\em preferred idempotent state} for
$\nDuAlg_2$ is an idempotent state $\mathbf x$ with
\[ \mathbf x\cap \{0,1,2\}\in\{\{0\},\{2\},\{0,2\}\}.\] We have a map $\psi$
from preferred idempotent states $\mathbf x=\{x_1,\dots,x_{k+1}\}$ of $\nDuAlg_2$ to idempotents of
$\nDuAlg_1$ defined byLet $\nDuAlg_1=\DuAlg(n+1,k+1,M_1)$ and
$\nDuAlg_2=\DuAlg(n,k,M_2)$
\[
\psi(\mathbf x) = \left\{
\begin{array}{ll}
\{0,x_3-2,\dots,x_{k+1}-2\} &{\text{if $|\mathbf x\cap\{0,1,2\}|=2$}} \\
\{x_2-2,\dots,x_{k+1}-2\} &{\text{if $|\mathbf x\cap\{0,1,2\}|=1$}} \\
\end{array}\right.
\]
Generators of $\Max^1=\lsup{\nDuAlg_1}\Max^1_{\nDuAlg_2}$ correspond to preferred idempotent
states. Letting $\nMinGen{\mathbf x}$ denote the generator corresponding to the preferred
idempotent state $\mathbf x$, the bimodule action is specified by
\[ \Idemp{\mathbf x}\cdot \nMinGen{\mathbf x}\cdot \Idemp{\psi(\mathbf x)}=\nMinGen{\mathbf x}.\]
The module is constructed via the homological perturbation lemma,
following~\cite[Section~\ref{BK2:subsec:AltConstr}]{Bordered2}.
Fix a matching $M_2$ on $\{1,\dots,2n\}$, and let $M_1$
be the matching $\{1,2\}\cup \phi_1(M_1)$.
Let
\begin{align*} \nBlg_1=\Blg(2n+2,n+2)\qquad{\text{and}}\qquad
\nBlg_2=\Blg(2n,n+1).
\end{align*}
Consider the idempotent in $\nBlg_1$ given by
\[ {\mathbf I}=\sum_{\{\mathbf x\big|x_1=1\}} \Idemp{\mathbf x}.\]
Consider the right $\nBlg$-module
\[ M= (L_1 L_2 \nBlg\backslash {\mathbf I}\cdot\nBlg)\oplus (L_1 L_2
\nBlg\backslash {\mathbf I}\cdot\nBlg),\] which can also be viewed as a left module
over the subalgebra of ${\mathbf I}\cdot \nBlg\cdot {\mathbf I}$ consisting of
elements with $\weight_1=\weight_2=0$, which in turn is identified
with $\nBlg2$. Denote this identification
\[\phi\colon \nBlg_2\to {\mathbf I}\cdot \nBlg\cdot {\mathbf I}.\]
Let $\XX$ and $\YY$ be the generators of the two summands of $M$.
Let
\[ m_{1|1|0}(b_2,\XX\cdot b_1)=\XX \cdot \phi(b_2)\cdot b_1 \qquad
m_{1|1|0}(b_2,\YY\cdot b_1)=\YY \cdot \phi(b_2)\cdot b_1,\]
where
$b_2\in\nBlg_2$ and $b_1\in{ {\mathbf I}\cdot \nBlg}\backslash{L_1 L_2 \cdot \nBlg}$.
Equip $M$ with the differential
\[ m_{0|1|0}(\XX)= \YY\cdot U_2 \qquad m_{0|1|0}(\YY)= \XX\cdot U_1.\]
Think of the right $\nBlg$-action as inducing further operations
\[ m_{0|1|1}(\XX\cdot b_1,b_1')=\XX\cdot (b_1\cdot b_1') \qquad
m_{0|1|1}(\YY\cdot b_1,b_1')=\YY\cdot (b_1\cdot b_1').\] All the
operations described above give $M$ the structure of a $\nBlg_2-\nBlg$
bimodule, written $\lsub{\nBlg_2}M_{\nBlg}$.
As explained in~\cite[Lemma~\ref{BK2:lem:BigDAGens}]{Bordered2}, $M$
can be viewed as arising from a type $DA$ bimodule
$\lsup{\nBlg_2}\Theta_{\nBlg_1}$ via
$\lsub{\nBlg_2}M_{\nBlg_1}=\lsub{\nBlg_2}(\nBlg_2)_{\nBlg_2}\DT~\lsup{\nBlg_2}\Theta_{\nBlg_1}$. (The
key notational difference here is twe have restricted the incoming
algebra from the algebra $\Blg$ of
\cite[Lemma~\ref{BK2:lem:BigDAGens}]{Bordered2} to its subalgenra
$\nBlg_1$ where the idempotent states have $n+2$ elements; and
correspondingly the output algebra has a similar constraint.)
In~\cite{Bordered2}, this DA bimodule $\Theta$ is promoted to a
bimodule over versions of $\Alg$. Instead, we promote here the
incoming algebra to $\nDuAlg_1$ (and the outgoing one to $\nDuAlg_2$).
This is done by introducing the following actions:
\[\begin{array}{rl}
\delta^1_2(\XX,E_1)=1\otimes \YY &
\delta^1_2(\YY,E_2)=1\otimes \XX\\
\delta^1_2(\XX,E_2)=0 &
\delta^1_2(\YY,E_1)=0\\
\delta^1_2(\XX,E_{\phi_c(i)})=E_{i}\otimes \XX &
\delta^1_2(\YY,E_{\phi_c(i)})=E_{i}\otimes \YY \\
\delta^1_2(\XX,E_1 E_2)= 1\otimes \XX &\qquad
\delta^1_2(\YY,E_2 E_1)= 1\otimes \YY \\
\delta^1_2(\XX,E_2 E_1)= 0& \qquad
\delta^1_2(\YY,E_1 E_2)= 0
\end{array}
\]
These actions are illustrated in the following picture:
\begin{equation}
\label{eq:ModuleVersionMin}
\mathcenter{\begin{tikzpicture}[scale=1.5]
\node at (0,0) (X) {$\XX$} ;
\node at (6,0) (Y) {$\YY$} ;
\draw[->] (X) [bend left=15] to node[above,sloped] {\tiny{$U_2 + 1 \otimes E_{1}$}} (Y) ;
\draw[->] (Y) [bend left=15]to node[below,sloped] {\tiny{$U_1+ 1\otimes E_{2}$}} (X);
\draw[->] (X) [loop] to node[above,sloped]{\tiny{$1\otimes E_{1} E_{2}+\sum_{i=1}^{2n} E_i\otimes E_{\phi_c(i)}$}} (X);
\draw[->] (Y) [loop] to node[above,sloped]{\tiny{$1\otimes E_{2}E_{1}+\sum_{i=1}^{2n} E_i\otimes E_{\phi_c(i)}$}} (Y);
\end{tikzpicture}}
\end{equation}
(Here the arrow labels with $U_2$ and $U_1$
alone represent $\delta^1_1$, actions where the outgoing algebra element
is $1$.)
We denote the result by $\lsup{\nDuAlg_2}{\BigMin}_{\nDuAlg_1}$.
\begin{lemma}
\label{lem:BigDAGens2}
The operations described above make
$\lsup{\nDuAlg_2}{\BigMin}_{\nDuAlg_1}$ into a type $DA$ bimodule.
\end{lemma}
\begin{proof}
This is straightforward.
\end{proof}
Consider the map $h^1\colon \BigMin \to \BigMin$
determined on the generating set $\XX\cdot a$ and $\YY\cdot a$ for $a\in \Gamma$) by
\begin{align*}
h^1(\XX a)&=\left\{\begin{array}{ll}
\YY a' &{\text{if there is a $a'\in \Gamma$ with $a=U_1 a'$}} \\
0 &{\text{otherwise}}
\end{array}\right. \\
h^1(\YY a)&=\left\{\begin{array}{ll}
\XX a' &{\text{if there is a pure $a'\in \Gamma$ with $a=U_2 a'$}} \\
0 &{\text{otherwise}}
\end{array}\right.
\end{align*}
Let $Q\subset \BigMin$ be the two-dimensional vector space $\XX L_1$
and $\YY R_2$. There are inclusions $i^1\colon Q \to \BigMin\subset \nDuAlg_2\otimes \BigMin$ and a projection $\pi^1\colon \BigMin\to Q$
\begin{lemma}
For the above operators, we have the identities:
\[ \begin{array}{lllll}
(\pi^1\circ i^1) =\Id_{Q}, & i^1\circ \pi^1=\Id_{\BigMin}+ dh^1, &
h^1\circ h^1=0, & h^1\circ i^1=0, & \pi^1\circ h^1 = 0.
\end{array}\]
\end{lemma}
The homological perturbation lemma, we can gives $\XX L_1 \oplus \YY
R_2$ the structure of a type $DA$ bimodule.
\begin{defn}
Let $\lsup{\nDuAlg_1}\Max^c_{\nDuAlg_2}$ be the $DA$ bimodule
structure on $\XX L_1 \oplus \YY R_2$ induced by the homological
perturbation lemma.
\end{defn}
We can now verify duality:
\begin{lemma}
\label{lem:MaxDual}
The bimodule $\lsup{\nDuAlg_1}\Max^{1}_{\nDuAlg_2}$ as defined above
satisfies the requirements of Proposition~\ref{prop:DualMax} when $c=1$.
\end{lemma}
\begin{proof}
This is a simple computation using the homological perturbation lemma,
exactly as in~\cite[Lemma~\ref{BK2:lem:MinDual}]{Bordered2}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:DualMax}]
By analogy with~\cite[Section~\ref{BK2:subsec:GenMin}]{Bordered2},
we can define $\lsup{\nDuAlg_1}\Max^c_{\nDuAlg_2}$ inductively in $c$,
using the action of the bimodules of crossings; i.e.
\[ \lsup{\nDuAlg_1}\Max^c_{\nDuAlg_2} =
\lsup{\nDuAlg_1}\Max^{c-1}_{\nDuAlg_4}\DT~\lsup{\nDuAlg_4}\Pos^c_{\nDuAlg_3}
~\DT\lsup{\nDuAlg_3}\Pos^{c-1}_{\nDuAlg_2},\] for suitably chosen
$\nDuAlg_3$ and $\nDuAlg_4$. Property~\ref{prop:DualMax} is then
proved by induction, with the basic case given by
Lemma~\ref{lem:MaxDual}, and the inductive step using
Proposition~\ref{prop:BimodulesOvernDuAlg} (for the positive
crossing). (The steps are as in the proof
of~\cite[Theorem~\ref{BK2:thm:MinDual}]{Bordered2}.)
\end{proof}
\subsection{Algebraic invariance}
\label{subsec:AlgInvar}
\begin{proof}[Proof of Proposition~\ref{prop:CurvedDPlanarIsotopies}]
Following terminology
from~\cite[Section~\ref{BK1:subsec:InvarainceUnderBridgeMoves}]{BorderedKnots},
Proving that the invariant $\lsup{\cClg}\DmodAlg(\DiagUp)$ is an isotopy invariant amounts to proving invariance under bridge moves,
which are:
\begin{enumerate}
\item Commutations of distant crossings
\item Trident moves
\item Critical points commute with distant crossings
\item Commuting distant critical points
\item Pair creation and annihilation.
\end{enumerate}
In~\cite{BorderedKnots,Bordered2}, these are verified
as obtained identities between corresponding $DA$
bimodules. To expedite computations, we worked in stead
with $DD$ bimodules.
In the notation of the present paper, the requisite identities for the
$DD$
module versions of commutation moves (for distant crossings and
critical points) are of the form
\[
X^i \DT Y^j \DT \CanonDD \simeq Y^{j'}\DT X^{i'}\DT \CanonDD
\]
where $X$, $Y$ can be either of $\Pos$, $\Neg$, $\Max$ or $\Min$,
$|i-j|>1$ (and the superscripts $i'$ and $j'$ have to be chosen
accordingly), curved modules over $\cBlg$ (as in
Subsection~\ref{subsec:FormalModules}), and
$\CanonDD=\lsup{\cBlg,\nDuAlg}\CanonDD$. Trident moves come from
identities
\begin{align*}
\Min^c\DT \Pos^{c+1}\DT \CanonDD& \simeq
\Pos^{c+1}\DT \Min^c\DT \CanonDD \\
\Pos^{c+1}\DT \Max^c\DT \CanonDD& \simeq
\Min^c\DT \Pos^{c+1} \DT \CanonDD
\end{align*}
Pair creation and annihilation invariance from the identities
\[ \Max^{c+1}\DT \Min^c\simeq \Id \simeq
\Max^c\DT \Min^{c+1}. \]
The verifications from~\cite{Bordered2} (dropping $C_{i,j}$, and
viewing the algebras over $\Blg$ as curved) now appy verbatim to give
the stated identities.
In~\cite{Bordered2}, the type $DD$ identities implied corresponding
corresponding identities of type $DA$ modules, due to a Koszul
duality, which we have not established here. Nonethless, the above
identities do establish corresponding identities where the $DA$
modules act on any ``relevant'' type $D$ structure, in the sense of
Definition~\ref{def:Relevant}. Thus, the needed invariance follows
once we know that $\lsup{\cClg}\DmodAlg(\Diag)$ is relevant.
Relevance
of $\lsup{\Clg}\DmodAlg(\Diag)$ is proved by induction on the number
of crossings and critical points in $\Diag$ (as in the proof of
Proposition~\ref{prop:InductiveStep}), with the inductive step
furnished by the fact that for $X$ is $\Pos$, $\Neg$, $\Max$, or
$\Min$ (curved DA bimodules over $\Blg$), there are corresponding DA
bimodules $X''$ so that
\[ \lsup{\Blg_2}X_{\Blg_1}\DT \lsup{\Blg_1,\nDuAlg_1}\CanonDD \simeq
\lsup{\nDuAlg_1}X_{\nDuAlg_2}\DT \lsup{\Blg_1,\nDuAlg_1}\CanonDD. \]
For $X\in\{\Pos,\Neg,\Min\}$, this was verified in
Proposition~\ref{prop:CurvedDABimodules}, while for $\Max$, it was
Proposition~\ref{prop:DualMax}.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:TangleInvariance}]
In view of Proposition~\ref{prop:CurvedDPlanarIsotopies},
it suffices to verify invariance of $\lsup{\cClg}\DmodAlg(\Diag)$
under the Reidemeister moves.
The corresponding $DD$ module identities for $\Alg$ and $\DuAlg$
Reidemeister $1$ invariance follows from the identity
$\Pos^c\DT\Max^c\DT \CanonDD \simeq \Pos^c\DT \CanonDD$, whereas the
other Reidemeister moves follow from the braid relations
from~\cite[Section~\ref{BK2:sec:BraidRelations}]{Bordered2}. The
identities are verified exactly as they are done there, appealing to
the fact that $\lsup{\cClg}\Dmod(\Diag)$ is relevant, as shown in the proof
of Proposition~\ref{prop:CurvedDPlanarIsotopies} above.
\end{proof}
\subsection{Computing the holomorphically defined invariant}
\label{subsec:ComputeD2}
\begin{proof}[Proof of Theorem~\ref{thm:ComputeD2}]
The proof of Theorem~\ref{thm:ComputeD} applies, once we have shown
the analogue of Theorem~\ref{thm:ComputeDDmods} for a local
maximum; i.e.
\[
\lsup{\cClg_2}\Max^c_{\cClg_1} \DT~ \lsup{\cClg_1,\nDuAlg_1}\CanonDD \simeq
\lsup{\cClg_2}\DAmodExt(\Hmax{c})_{\cClg_1} \DT~
\lsup{\cClg_1,\nDuAlg_1}\CanonDD. \]
This verification is very similar to the proof of Proposition~\ref{prop:MinimumComputation}, and is left to the reader.
\end{proof}
\section{Heegaard diagrams}
\label{sec:Heegs}
\subsection{Upper diagrams}
An {\em upper Heegaard diagram} is the following data:
\begin{itemize}
\item a surface $\Sigma_0$ of genus $g$ and
$2n$ boundary components, labelled $Z_1,\dots,Z_{2n}$,
\item a collection
of disjoint, embedded arcs $\{\alpha_i\}_{i=1}^{2n-1}$, so that
$\alpha_i$ connects $Z_i$ to $Z_{i+1}$,
\item a collection of
disjoint embedded closed curves $\{\alpha^c_i\}_{i=1}^{g}$
(which are also disjoint from $\alpha_1,\dots,\alpha_{2n-1}$),
\item
another collection of embedded, mutually disjoint closed curves
$\{\beta_i\}_{i=1}^{g+n-1}$.
\end{itemize}
We require this data to also satisfy the following properties:
\begin{enumerate}[label=(UD-\arabic*),ref=(UD-\arabic*)]
\item For each $i\in\{1,\dots,2n-1\}$, $j\in \{1,\dots,g\}$, and $k\in\{1,\dots,g+n-1\}$,
$\alpha_i$ and $\alpha_j^c$ curves are transverse to $\beta_k$.
\item
Both sets of $\alpha$-and the
$\beta$-circles consist of homologically linearly
independent curves (in $H_1(\Sigma_0)$).
\item
\label{UD:TwoBoundariesApiece}
The surface
obtained by cutting $\Sigma_0$ along $\beta_1,\dots,\beta_{g+n-1}$,
which has $n$ connected components, is required to contain exactly
two boundary circles in each component.
\item
\label{UD:NoPerDom}
The subspace of $H_1(\Sigma_0;\Z)$ spanned by the curves
$\{\alpha_i^c\}_{i=1}^g$ meets transversely the space spanned by
$\{\beta_i\}_{i=1}^{g+n-1}$.
\item
\label{UD:NoHone}
Letting $\cSigma$ denote the closed surface obtained by filling
in $\partial \Sigma_0$ with $2n$ disks,
the space $H_1(\cSigma;\Z)$ is spanned by the images of the curves
$\{\alpha_i^c\}_{i=1}^g$ and
$\{\beta_i\}_{i=1}^{g+n-1}$.
\end{enumerate}
\begin{rem}
An upper diagram specifies a three-manifold $Y$ whose boundary is a
sphere, with an embedded collection of $n$ arcs.
Condition~\ref{UD:NoPerDom} ensures that $H_2(Y;\Z)=0$.
(Compare~\cite[Proposition~2.15]{HolDisk}.)
When $g=0$, the three-manifold $Y$ is a three-ball.
Also, when $g=0$, Condition~\ref{UD:NoHone} is automatically satisfied.
\end{rem}
Condition~\ref{UD:TwoBoundariesApiece} gives a
matching $\Matching$ on $\{1,\dots,2n\}$ (a partition into two-element subsets), where $\{i,j\}\in \Matching$ if $Z_i$ and $Z_j$
can be connected by a path that does not cross any $\beta_k$.
We sometimes abbreviate the data
\[\Hup=(\Sigma_0,Z_1,\dots,Z_{2n},\{\alpha_1,\dots,\alpha_{2n-1}\},\{\alpha^c_1,\dots,\alpha^c_{g}\},
\{\beta_1,\dots,\beta_{g+n-1}\}),\]
and let $\Matching(\Hup)$ be the induced matching.
\begin{figure}[h]
\centering
\input{UpperHeeg.pstex_t}
\caption{{\bf An upper Heegaard diagram.} The black dots represent an
upper Heegaard state $\mathbf x$ with $\alpha(\mathbf x)=\{2,4,6\}$}
\label{fig:UpperHeeg}
\end{figure}
\begin{defn}
\label{def:UpperState}
An {\em upper Heegaard state} is a subset $\mathbf x$ of $\Sigma_0$
consisting of $g+n-1$ points in the intersection of the various
$\alpha$-and $\beta$-curves, distributed so that each $\beta$-circle
contains exactly one point in $\mathbf x$, each $\alpha$-circle contains
exactly one point in $\mathbf x$, and no more than one point lies on any
given $\alpha$-arc. Each Heegaard state $\mathbf x$ determines a subset
$\alpha(\mathbf x)\subset \{1,\dots,2n-1\}$ with cardinality $n-1$
consisting of those $i\in\{1,\dots,2n-1\}$ with $\mathbf x\cap \alpha_i\neq
\emptyset$.
\end{defn}
\subsection{Lower diagrams}
A {\em lower diagram} $\Hdown$ is an upper diagram, equipped with an
extra pair of basepoints $w$ and $z$ and one
additional $\beta$-circle $\beta_{g+n}$
so that exactly two of the $n+1$ components in $\Sigma_0\setminus\betas$
contains one boundary component of $\Sigma_0$ apiece, and these are marked
by the basepoints $w$ and $z$ (and the remaining $n-1$ components meet
two boundary components of $\Sigma_0$).
Note that a lower diagram also determines an equivalence relation
$\Mdown$ on the boundary circles $Z_1,\dots,Z_{2n}$, where $\{i,j\}\in
\Mdown$ if $Z_i$ and $Z_j$ can be connected by a path that does not
cross any $\beta_k$.
A {\em lower Heegaard state} $\mathbf x$ is a set of $g+n$ points, where one
lies on each $\beta$-circle, one lies on each $\alpha$-circle, and no
more than one lies on each $\alpha$-arc. Each lower Heegaard state $\mathbf x$
determines a subset $\alpha(\mathbf x)$ of $\{1,\dots,2n-1\}$ with
cardinality $n$, once again, consisting of those $i=1,\dots,2n-1$ with
$\mathbf x\cap \alpha_i\neq \emptyset$.
\begin{figure}[h]
\centering
\input{LowerHeeg.pstex_t}
\caption{{\bf A lower Heegaard diagram.} The basepoints $w$ and $z$
are marked; furthermore, a lower state $\mathbf x$ with
$\alpha(\mathbf x)=\{1,3,6,7\}$ is indicated by black dots.}
\label{fig:LowerHeeg}
\end{figure}
\subsection{Gluing upper and lower diagrams}
\label{subsec:Gluing}
\begin{defn}
\label{def:CompatibleMatching}
Let $M_1$ and $M_2$ be two matchings on $\{1,\dots,2n\}$. We say
that $M_1$ and $M_2$ are {\em compatible} if together they generate
an equivalence relation with a single equivalence class.
\end{defn}
It is convenient to phrase this geometrically in the following terms.
\begin{defn}
\label{def:AssociatedW}
Let $M$ be a matching on a finite set $\{1,\dots,2k\}$. There is an
associated one-manifold $W(M)$, whose boundary components correspond
to the points in $\{1,\dots,2k\}$, and whose components correspond to
$\{i,j\}\in M$; that component connects the boundary component $i$ to
the boundary component $j$.
\end{defn}
Suppose $\{1,\dots,2k\}$ is given with a matching $M_1$ and another
matching $M_2$ on a subset $S\subset\{1,\dots,2k\}$. Those two
matchings together induce an equivalence relation $M_3$ on
$\{1,\dots,2k\}\setminus S$. The associated spaces are related by
\[ W(M_3)=W(M_2)\cup_{S} W(M_1).\]
In this language, the compatibility of $M_1$ and $M_2$ is
equivalent to the condition that $W(M_1)\cup_S W(M_2)$ has no closed
components. See Figure~\ref{fig:AssociateW}.
\begin{defn}
\label{def:Compatible}
Suppose that $\Hup$ and $\Hdown$ are upper and lower diagrams with the
same number $2n$ of boundary circles, and genera $g_1$ and $g_2$.
We say that
$\Hup$ and $\Hdown$ are {\em compatible} if the corresponding
matchings $\Matching(\Hup)$ and $\Matching(\Hdown)$ are compatible,
in the sense of Definition~\ref{def:CompatibleMatching}
\end{defn}
\begin{example}
The upper Heegaard diagram $\Hup$ from Figure~\ref{fig:UpperHeeg} determines
the matching on $\{1,\dots,8\}$, $\Mup=\{\{1,3\},\{2,4\},\{5,7\},\{6,8\}\}$;
and the lower diagram $\Hdown$ from Figure~\ref{fig:LowerHeeg}
determines the matching
$\Mdown=\{\{2,7\},\{3,6\},\{4,5\}\}$. together, they determine an
equivalence relation with two equivalence classes,
$\{2,4,5,7\}$ and $\{1,3,6,8\}$;
thus, $\Hup$ and $\Hdown$ are not
compatible, in the sense of Definition~\ref{def:Compatible}.
\begin{figure}[h]
\centering
\input{AssociateW.pstex_t}
\caption{{\bf One-manifold associated to pairs of matchings.}}
\label{fig:AssociateW}
\end{figure}
\end{example}
\begin{prop}
\label{prop:ConstructDiagram}
Suppose that $\Hup$ and $\Hdown$ are compatible upper and lower diagrams
(with underlying surfaces $\SigmaUp$ and $\SigmaDown$ respectively).
Glue $\SigmaUp$ and $\SigmaDown$ together along their boundaries
to get a closed surface $\Sigma$ of genus $g=g_1+g_2+2n-1$. Collect
the $\beta$-circles for $\Hup$ and $\Hdown$ into a $g$-tuple,
thought of now as circles in $\Sigma$; and form a $g$-tuple
of $\alpha$-circles consisting of the $\alpha$-circles in $\Hup$ and $\Hdown$
and further $\alpha$-circles formed by gluing
$\alpha$-arcs in $\Hup$ to $\alpha$-arcs in $\Hdown$.
The result is a doubly-pointed Heegaard diagram.
\end{prop}
\begin{proof}
It is straightforward to see that $\Sigma$ has the stated genus, and
that the $\alpha$-circles are homologically linearly independent.
The compatibility condition guarantees that the complement of the
$\beta$-circles is connected in $\Sigma$, and hence these circles
are also linearly independent.
\end{proof}
We denote the doubly-pointed Heegaard diagram constructed in Proposition~\ref{prop:ConstructDiagram}
$\Hup\#\Hdown$.
\subsection{Examples}
\label{subsec:Examples}
The {\em standard upper diagram} is the diagram $\Hup(n)$ equipped with a
planar surface, $\alpha$-arcs as in Figure~\ref{fig:StandardUpperDiagram}
(with $n=4$),
$\beta$-circles arranged as
in the figure
(after deleting $\beta_4$),
and no $\alpha$-circles.
(That figure also contains two basepoints $\wpt$ and $\zpt$ which should be
disregarded.)
Note that the standard upper diagram
has a single state $\mathbf x$ with $\alpha(\mathbf x)=\{2,\dots,2n-2\}$.
\begin{figure}[h]
\centering
\input{StandardUpperDiagram.pstex_t}
\caption{{\bf Standard Heegaard diagrams $n=4$.} This is the
standard lower diagram; the standard upper diagram is obtained by
removing the basepoints $w$ and $z$ and the circle $\beta_4$.}
\label{fig:StandardUpperDiagram}
\end{figure}
The {\em standard lower diagram} is obtained from the standard upper
diagram by adding an extra $\beta$-circle around $Z_1$, and two adjacent
basepoints $w$ and $z$ as shown
in Figure~\ref{fig:StandardUpperDiagram}.
Let $\Hup$ be any upper diagram drawn with Heegaard surface
$\Sigma_0$, and fix an orientation preserving diffeomorphism
$\phi\colon \Sigma_0 \to \Sigma_0$. There is a new upper Heegaard
diagram
\[ \phi(\Hup)=(\Sigma_0,\alphas,\phi(\betas))\cong (\Sigma_0,\phi^{-1}(\alphas),\betas).\] We have
illustrated this new upper diagram in the case where $\phi$ is a half
twist switching $Z_2$ and $Z_3$ in
Figure~\ref{fig:BraidGroupAction}.
\begin{figure}[h]
\centering
\input{BraidGroupAction.pstex_t}
\caption{{\bf Acting on the standard upper diagram by a half twist switching
$Z_2$ and $Z_3$}.
\label{fig:BraidGroupAction}}
\end{figure}
Let $\Hup$ be an upper diagram on a surface $\Sigma_0$ of genus zero.
Every mapping class of $\C$ punctured at points
$\{p_1,\dots,p_{2n}\}$ can be represented by a diffeomorphism
$\phi_0$ of $\Sigma_0$ to itself. Letting $\phi_0$ act on $\Hup$, we
obtain an action of the mapping class group of
$\C\setminus\{p_1,\dots,p_{2n}\}$ (which in turn is identified with
the quotient of the braid group on $2n$ elements by the cyclic
subgroup generated by a full twist; see~\cite{Birman}) on genus zero
upper diagrams modulo isotopy. The braid group is generated by
consecutive switches $\sigma_i$; their corresponding mapping classes
are represented by half twists: suppose that $\gamma_i$ is a curve
that encloses exactly two puncture points $p_i$ and $p_{i+1}$, the
half twist along $\gamma_i$ is the mapping class $\tau_{i,+}$
supported in the compact component of $\C\setminus\gamma_i$ which
switches $p_i$ and $p_{i+1}$, and whose square is a Dehn twist along
$\gamma_i$.
Gluing the standard upper diagram to $\phi$ applied to the standard
lower diagram induces a Heegaard diagram representing some knot in
$S^3$, provided that the permutation $\sigma$ induced by $\phi$ has
the property that the product of permutations
\[ \sigma\cdot
((1,2),(3,4),\dots,(2n-1,2n))\cdot \sigma^{-1}\cdot
((1,2),(3,4),\dots,(2n-1,2n))\]
has order $n$.
Consider a decorated knot projection, equipped with a height
function, so that the marked edge contains the global minimum for the height
function. Consider the associated Heegaard diagram
(following~\cite{AltKnots}). Slicing the knot projection along a
generic horizontal slice corresponds to decomposing the Heegaard
diagram as the gluing of some upper and lower Heegaard diagrams; see
Figure~\ref{fig:SliceProjection}.
As usual, the Heegaard surface $\Sigma$ is oriented as the boundary
of the $\alpha$-handlebody.
\begin{figure}[h]
\centering
\input{SliceProjection.pstex_t}
\caption{{\bf Slicing the Heegaard diagram corresponding
to a decorated knot projection.}
Cutting the decorated projection on the left along the dotted line
corresponds to slicing the Heegaard diagram along the right along the
circles $Z_1\cup Z_2\cup Z_3\cup Z_4$. }
\label{fig:SliceProjection}
\end{figure}
\subsection{Middle diagrams}
In this paper, we will define algebraic objects using
holomorphic curve counts associated to upper and lower diagrams. Knot
Floer homology then will be expressed as a pairing between the
invariants of the upper and lower diagrams. To compute these
individual objects, it will help to be able to decompose the
knot diagram into further pieces, the {\em middle diagrams} we introduce presently.
\begin{defn}
\label{def:MiddleDiagram}
A {\em middle diagram} is a Heegaard diagram obtained from an upper
diagram with $2m+2n$ boundary components, by deleting the arc
$\alpha_{2m}$ which would have connected $Z_{2m}$ with $Z_{2m+1}$.
Thus, the $\alpha$-arcs are labelled
$\alpha_i$ for
\[ i\in\{1,\dots,2m-1,2m+1,\dots,2m+2n-1\}.\]
Think of some boundary components of $\Sigma_0$
as ``incoming'' boundary and other components as ``outgoing'' boundary.
The incoming boundary components are labelled as
$\Zin_i=Z_i$ for $i=1,\dots,2m$; and the outgoing boundary components
$\Zout_i=Z_{2m+i}$
for $i=1,\dots,2n$; $\alphain_i=\alpha_i$ for $i=1,\dots,2m-1$ and
$\alphaout_i=\alpha_{2m+i}$ for $i=1,\dots,2n-1$. The $\beta$-circles
are labelled $\{\beta_i\}_{i=1}^{g+m+n-1}$. We abbreviate the
resulting data
\begin{align*} \Hmid=(\Sigma_0,(\Zin_1,&\dots,\Zin_{2m}),(\Zout_1,\dots,\Zout_{2n}),
\{\alphain_1,\dots,\alphain_{2m-1}\},
\{\alphaout_1,\dots,\alphaout_{2n-1}\},\\
&\{\alpha^c_1,\dots,\alpha^c_{g}\},
\{\beta_1,\dots,\beta_{g+m+n-1}\}).
\end{align*}
\end{defn}
We will be primarily interested in five model middle diagrams: the
{\em identity diagram} $\Hid=\Hid(n)$ (Figure~\ref{fig:IdDiag}), the
{\em middle diagram of a local maximum} $\Hmax{c,n}$
(Figure~\ref{fig:MaximumHeeg}), the {\em middle diagram of a local
mimum} $\Hmin{c,n}$ (Figure~\ref{fig:MinimumHeeg}), the {\em middle
diagram of a positive crossing} $\Hpos{i,n}$, and the {\em middle
diagram of a negative crossing} $\Hneg{i,n}$ (both in
Figure~\ref{fig:CrossDiag}).
We explain the interpretations of these Heegaard diagrams in terms of
partial knot projection; though this connection will become important
for us only much later (in Section~\ref{sec:ComputeDDmods}). For
$\Hmax{c,n}$, where $2n$ denotes the number of input strands (so there are
$2n+2$ outputs), and $c\in 1,\dots,2n+1$ is chosen so that strands $c$ and
$c+1$ come out of the local maximum. By contrast, for $\Hmin{c,n}$,
$2n$ is the number of output strands and, $c$ is choosen so that
the output strands $c$ and $c+1$ connect to the local maximum. The diagrams for
$\Hpos{i,n}$ and $\Hneg{i,n}$ both represent $2n$ strands, so that the
$i$ and $i+1^{st}$ ones cross. Orienting each strand upwards, the
crossing is positive for $\Hpos{}$ and negative for $\Hneg{}$. (In
practice, the strands in a crossing region will not all be oriented
pointing upward.)
\begin{figure}[h]
\centering
\input{IdDiagram.pstex_t}
\caption{{\bf Middle diagram of the identity $\Hid(n)$.}
We have labelled a middle Heegaard state, with $|\alphain(\mathbf x)|=2$.}
\label{fig:IdDiag}
\end{figure}
\begin{figure}[h]
\centering
\input{Maximum.pstex_t}
\caption{{\bf Middle diagram of a local maximum $\Hmax{c,n}$.} We have drawn here the case when $n=1$ and $c=2$.
\label{fig:MaximumHeeg}}
\end{figure}
\begin{figure}[h]
\centering
\input{Minimum.pstex_t}
\caption{{\bf Middle diagram of a local minimum $\Hmin{c,n}$.} We have drawn here the case where $n=2$ and $c=1$.
\label{fig:MinimumHeeg}}
\end{figure}
\begin{figure}[h]
\centering
\input{CrossingDiagram.pstex_t}
\caption{{\bf Middle diagram of crossings $\Hpos{i,n}$ and $\Hneg{i,n}$.}
$\Hneg{2,2}$ is above and $\Hpos{1,2}$ is below.}
\label{fig:CrossDiag}
\end{figure}
A {\em middle Heegaard state} is a set of $g+m+n-1$ points of $\Sigma_0$ in
the locus where the various $\alpha$- and $\beta$-curves intersect,
distributed so that one lies on each $\beta$-circle, one lies on each
$\alpha$-circle, and no more than one lies on each $\alpha$-arc. For
each middle Heegaard state, let $\alphain(\mathbf x)\subset \{1,\dots,2m-1\}$
consists of those $i$ for which $\mathbf x\cap\alphain_i\neq
\emptyset$; and $\alphaout(\mathbf x)\subset\{1,\dots,2n-1\}$ consist of
those $j$ for which $\mathbf x\cap \alphaout_{j}\neq \emptyset$.
Obviously,
\begin{equation}
\label{eq:alpaInOut}
|\alphain(\mathbf x)|+|\alphaout(\mathbf x)|=m+n-1.
\end{equation}
We will be primarily interested in middle states with $|\alphain(\mathbf x)|=m$.
An upper diagram can be thought of as a middle diagram with $m=0$.
The $\beta$-circles induce an equivalence relation $\Mmid$ on
the components of $\partial\Sigma_0$.
We should say a word about orientation conventions in pictures such
as Figure~\ref{fig:CrossDiag}. According to our
conventions from~\cite{HolDisk}, the Heegaard surface is oriented as
the boundary of the $\alpha$-handlebody. Thus, the Heegaard diagram in
picture in Figure~\ref{fig:CrossDiag} is oriented opposite to the
orientation it inherits as the boundary of a neighborhood of the
crossing apparent in the figure.
\subsection{Gluing middle diagrams to upper diagrams}
Let $\Hup_1$ be an upper diagram with $2m$ outgoing boundary
components, and $\Hmid$ a middle diagram with $2m$ incoming boundary
components and $2n$ outgoing ones. Combine the equivalence relation
$\Matching(\Hup_1)$ with the equivalence relation $\Matching(\Hmid)$
to obtain an equivalence relation on all the boundary components of
$\Hmid$. If every equivalence class contains an outgoing boundary
component (of $\Hmid$), then we say that $\Hup_1$ and $\Hmid$ are {\em
compatible}.
If $\Hup_1$ and $\Hmid$ are compatible, we can glue the boundary of
$\Hup_1$ to the incoming boundary of $\Hmid$ to form a new upper diagram
$\Hup_2=\Hup_1\# \Hmid$. Note that $\Matching(\Hup_2)$ is the matching
on $\{1,\dots,2n\}$, thought of as labelling the outgoing boundary components of $\Hmid$
induced by restricting $\Matching(\Hup_1)\cup\Matching(\Hmid)$ to the outgoing boundary of $\Hmid$.
Each upper state on $\Hup_2$ can be
restricted to give an upper state on $\Hup_1$ and a middle state in
$\Hmid$. By Equation~\eqref{eq:alpaInOut}, for any middle state
obtained in this manner, $|\alphain(\mathbf x)|=m$.
We say that two upper diagrams $\Hup_1$ and $\Hup_2$ are {\em
equivalent} if we can obtain $\Hup_2$ from $\Hup_1$ by isotopies,
handle slides, and stabilizations/destabilizations. Handleslides here
involve two $\beta$ circles, two $\alpha$ circles, or an $\alpha$-arc
slid over an $\alpha$-circle.
Observe that if $\Hid$ is the standard diagram for the identity with
$2n$ outgoing boundary components, then for $\Hup_2=\Hup_1\#\Hid$, we
can slide each new outgoing $\alpha$-arc over its corresponding
newly-formed $\alpha$-circle, and then destabilize that $2n-1$ new
$\beta$-circles with their newly-formed $\alpha$-circles, to get back
the original Heegaard diagram $\Hup_1$; i.e. $\Hup_1$ and $\Hup_2$ are
equivalent.
\subsection{The Heegaard diagram of a knot projection}
Fix the sphere with four boundary circles, and four $\alpha$-arcs
connecting them in a circular configuration. Equip with a further
$\beta$-circle. When the $\beta$-circle meets each of the
four arcs exactly once, we can think of the configuration as specifying a crossing,
where the type of the crossing depends on how the $\beta$-circle meets
the $\alpha$-arcs. We denote the two diagrams $H_+$ and $H_-$; see Figure~\ref{fig:CrossingPieces}.
(As in the case of $\Hpos{}$ and $\Hneg{}$, the sign of the crossing using the usual conventions from knot theory agrees with
the sign of $H_{\pm}$ when both strands are oriented upwards.)
\begin{figure}[h]
\centering \input{CrossingPieces.pstex_t}
\caption{{\bf Crossing pieces.}
\label{fig:CrossingPieces}}
\end{figure}
We can glue $H_+$ or $H_-$to two consecutive output circles $Z_i$ and
$Z_{i+1}$ in an upper diagram $\Hup_1$ to get new upper diagrams
$\Hup_1\cup_{i} H_+$ or $\Hup_1\cup_{i}H_-$. These diagrams be
easily seen to be equivalent to the diagram obtained by gluing
$\Hup_1$ to the middle diagrams $\Hmid_{+,i}$ and $\Hmid_{-,i}$.
We can relate this construction with the action of the mapping class
group on the upper diagram discussed in Section~\ref{subsec:Examples}.
Let $\tau_{i,+}$ be the half twist whose square is a positive Dehn
twist around a curve that encircles $Z_i$ and $Z_{i+1}$. We have seen
that $\Hup\cup_{i} H_{+}$ is equivalent to $\Hup\cup\Hmid_{+,i}$ We
claim these two diagrams are in turn equivalent to
$\tau_{i,+}(\Hup)$. This can be seen by applying three handleslides to
$\Hup\cup_i H_+$, so that the newly-formed $\alpha$-circle and the
$\beta$-circle supported in $H_+$ meet in a single
point. Destabilizing the resulting diagram, we obtain a new diagram
which is diffeomorphic to $\tau_{i,+}(\Hup)$; see
Figure~\ref{fig:Handleslides}. An analogous computation shows the
equivalence between $\Hup\#\Hneg{i,n}$ and $\tau_{i,-}(\Hup)$.
\begin{figure}[h]
\centering
\input{Handleslides.pstex_t}
\caption{{\bf Attaching $H_+$ is equivalent to acting on a diagram by
a half twist.} At the left, we have a portion of an upper
diagram (with two consecutive boundary components separated by an
arc; in general, they are separated by a collection of parallel
arcs), which is glued to $H_+$. Performing three handleslides (the
first two of which are indicated) and destabilizing is
diffeomorphic to the original upper diagram and acting on its
$\alpha$-curves by a half twist.}
\label{fig:Handleslides}
\end{figure}
Suppose that $K$ is a knot projection all of whose local maxima are
global maxima, and whose local minima are global minima. The
crossings correspond to the factorization of some mapping class
$\phi$ as a product of half twists. From the above discussion, it is
clear that the Heegaard diagram of the knot projection is
equivalent to the Heegaard diagram obtained by gluing the standard
upper and lower diagrams $\Hup(n)$ and $\Hdown(n)$, twisted by
$\phi$; i.e. $\phi(\Hup(n))\# \Hdown(n)$.
\section{Introduction}
\label{sec:Intro}
The aim of the present paper is to identify the knot invariant
from~\cite{Bordered2} with a certain specialization of the knot Floer
homology from~\cite{Knots} and~\cite{RasmussenThesis}.
One version of knot Floer homology associates to a knot $K$, a
bigraded module over $\Ring=\Field[U,V]/UV=0$ (where $\Field=\Zmod{2}$
is the field with two elements), which we denote here by
$\HFKsimp(K)$. That module in turn is the homology of a chain
complex, denoted here $\CFKsimp(\HD)$, associated to a doubly-pointed
genus $g$ Heegaard diagram $\HD$, whose differential counts
pseudo-holomorphic disks $u$ in $\Sym^g(\Sigma)$, weighted with the
monomial $U^{n_w(u)} V^{n_z(u)}$. Specializing further to $V=0$ (and
then taking homology), we obtain the knot invariant denoted $\HFKm(K)$
in~\cite{Knots}; and setting $U=V=0$, we obtain the knot invariant
$\HFKa(K)$. (Note that throughout the present paper, we are using
knot Floer homology groups with coefficients mod $2$, though we
suppress this from the notation.) In~\cite{AltKnots}, we described a
Heegaard diagram associated to a knot projection, where the generators
correspond to Kauffman states for the knot diagram; but the
differentials still remained elusive counts of pseudo-holomorphic
disks.
Taking algebraic clues from the bordered Floer homology
of~\cite{InvPair}, in~\cite{BorderedKnots,Bordered2}, we
defined chain complexes whose generators correspond to Kauffman
states and whose differentials are determined by certain explicit
algebraic constructions; we then verified that their homology groups are knot
invariants. Specifically, the constructions from~\cite{Bordered2}
give an oriented knot invariant $\Hwz(\orK)$, that is a bigraded module
over $\Ring$, whose $V=0$ specialization gives the knot invariant $\KHm(-\orK)$
from~\cite{BorderedKnots}.
The aim of the present paper is to prove the following:
\begin{thm}
\label{thm:MainTheorem}
Bigraded knot Floer homology $\HFKsimp(K)$ (with coefficients mod
$2$) is identified with the bordered knot invariant $\Hwz(\orK)$
(again, with coefficients mod $2$) from~\cite{Bordered2}; the
bigraded knot Floer homology $\HFKm(K)$ is identified with the
bordered knot invariant $\KHm(\orK)$ from~\cite{BorderedKnots}; the
bigraded knot Floer homology $\HFKa(K)$ is identified with the
$U=V=0$ specialization of $\Hwz(\orK)$.
\end{thm}
(As the notation suggests, knot Floer homology is independent of the
choice of orientation on $K$; in view of the above theorem,
$\Hwz(\orK)$ is independent of this choice, as well.)
The bordered knot invariants are defined in terms of a decorated knot
projection. We start from the projection, cut it up into elementary
pieces, associate bimodules to those pieces, and then define the
invariant to be a (suitable) tensor product of the bimodules;
compare~\cite{Bimodules, HFa} for the three-dimensional analogue.
In its original formulation, knot Floer homology can be constructed
using any Heegaard diagram representing a knot. We work here with the
``standard Heegaard diagram'' associated to a decorated knot
projection, as used in~\cite{ClassKnots}. After degenerating this
Heegaard diagram, we obtain partial Heegaard diagrams, formalized in
Section~\ref{sec:Heegs}, which come in two kinds: the ``lower
diagrams'' and the ``upper diagrams''. Suitable counts of
pseudo-holomorphic curves in upper diagrams leads to the definition of
the ``type $D$ structure'' associated to an upper diagram
(Section~\ref{sec:TypeD}), and counting pseudo-holomorphic curves in
lower diagrams leads to the definition of the ``type $A$ structure''
associated to the lower diagram (Section~\ref{sec:TypeA}). The
underlying algebras for these modules are closely related to the ones
defined in~\cite{BorderedKnots,Bordered2}. Specifically, we will be
working with curved modules over the algebra $\Blg(2n,n)$
from~\cite{BorderedKnots} (which we abbreviate here by $\Blg(n)$), with a curvature specified by a matching
$\Matching$ (as described in Section~\ref{sec:Algebra}), which in turn
is very similar to working over the algebra
$\Alg(n,\Matching)$ from~\cite{Bordered2}.
A suitable tensor product of the type $A$ and type $D$ structures
computes knot Floer homology, by an analogue of the pairing theorem
from~\cite{InvPair}, proved in Section~\ref{sec:Pairing}. The theory
is generalized to bimodules in Section~\ref{sec:Bimodules}.
With the bordered theory in place, the verification of
Theorem~\ref{thm:MainTheorem}, which is completed in
Section~\ref{sec:Comparison}, rests on some model computations
(see Section~\ref{sec:ComputeDDmods}).
The pseudo-holomorphic curve counting here has a few complications
beyond the material present in~\cite{InvPair}. As in~\cite{TorusMod},
we must account for ``Reeb orbits'' rather than merely Reeb chords.
The present analysis is simpler, though, because our hypotheses on the
partial diagrams exclude boundary degenerations on the
``$\alpha$-side''. Rather, the boundary degenerations that occur here
all happen on the ``$\beta$-side'', and indeed they happen in a
controlled manner that can be accounted for neatly in the algebra.
With this understanding, the proof of the pairing theorem
from~\cite{InvPair} adapts readily.
Theorem~\ref{thm:MainTheorem} can be viewed as providing explicitly
computable models for variants of knot Floer homology. Grid diagrams
also give explicit combinatorial models for the necessary holomorphic
curve counts; see~\cite{MOS,MOST}. While the more algebraic bordered
constructions studied here (and in~\cite{BorderedKnots,Bordered2}) at
present describe a specialized version of these invariants, their
computations are much more efficient; see~\cite{Program}.
\subsection{Organization}
This paper is organized as follows. In Section~\ref{sec:Heegs}, we
formalize the partial Heegaard diagrams used in the bordered
pseudo-holomophic construction. Section~\ref{sec:Algebra} contains
the algebraic background, with a formalization of the framework of
curved algebras and the modules over them, which will be used
throughout. Next we recall bordered algebras defined
in~\cite{BorderedKnots}, and show how they can be fit into the curved
framework. The next few sections deal with upper
diagrams, with the ultimate aim of defining their associated type $D$
structures. In Section~\ref{sec:Shadows}, we formalize the shadows of
domains that connect upper Heegaard states, and use these to define a
grading on the upper Heegaard states. In Section~\ref{sec:CurvesD}, we
formulate the pseudo-holomorphic curves that represent these shadows,
and lay out properties of their moduli spaces. In Section~\ref{sec:TypeD} we define the
type $D$ structure associated to an upper diagram, and verify that it
satisfies the required structural identities, using the results from
Section~\ref{sec:CurvesD}. In Section~\ref{sec:CurvesA}, we
generalize the material from Section~\ref{sec:CurvesD} to the curves
relevant for the type $A$ modules, which are defined in
Section~\ref{sec:TypeA}. In Section~\ref{sec:Pairing}, we prove a
pairing theorem for the modules associated to upper diagrams with
lower diagrams, expressing knot Floer homology in terms of the modules
associated to the two pieces. In Section~\ref{sec:Bimodules}, the
material is adapted to the case of type $DA$ bimodules associated to
``middle diagrams''. A corresponding pairing theorem is established
(Theorem~\ref{thm:PairDAwithD}). In fact, the above material is set up
over a subalgebra of $\Blg(n)$, corresponding to restricting the
idempotent ring. In Section~\ref{sec:ExtendDA}, we extend the $DA$
bimodules over all of $\Blg(n)$.
We would like to understand these bimodules for certain standard
middle diagrams. For this description, we adapt the
algebraically-defined bimodules from~\cite{Bordered2}; to the curved
context. This adaptation is given in
Section~\ref{sec:AlgDA}. Bimodules derived from these are identified
with the bimodules associated to certain basic middle diagrams in
Section~\ref{sec:ComputeDDmods}. Combining these model computations
with the pairing theorem for bimodules, we obtain an algebraic
description of the type $D$ modules associated to Heegaard diagrams
for upper knot diagrams, in Section~\ref{sec:computeD}. This is
readily adapted to a proof of Theorem~\ref{thm:MainTheorem} in
Section~\ref{sec:Comparison}.
\subsection{Acknowledgements}
The authors wish to express their gratitude to Robert Lipshitz and
Dylan Thurston. Many of the ideas here were inspired by bordered Floer
homology as described in~\cite{InvPair}; and moreover the use of Reeb
orbits also figured heavily in joint work with the first author which
will appear soon~\cite{TorusMod}.
We would also like to thank Andy Manion for sharing with us
his work~\cite{Manion1,Manion2}; and to thank Nate Dowlin, Ian Zemke,
and Rumen Zarev for interesting conversations,
\section{The pairing theorem}
\label{sec:Pairing}
Start with a doubly-pointed Heegaard diagram $\HD$ for an oriented
knot $\orK$ in $S^3$, together with a decomposition along a union $Z$
of $2n$ circles along which $\HD$ decomposes as a union of a lower and
an upper diagram, $\HD=\Hdown\cup_Z \Hup$. For example, we could
consider the doubly-pointed Heegaard diagram for a knot projection as
in~\cite{AltKnots}, sliced in two along $2n$ circles corresponding to
a horizontal cut of the diagram, arranged so that the distinguished
edge is in the bottom; see for example
Figure~\ref{fig:SliceProjection}.
Letting $\Mup$ and $\Mdown$ be the matchings
induced by $\Hup$ and $\Hdown$ respectively, the hypothesis
ensures that $\Mup$ is compatible with $\Mdown$, in the sense
of Definition~\ref{def:CompatibleMatching}.
Let $C_\Ring(\HD)$ denote the Heegaard-Floer complex of the doubly-pointed diagram $\HD$:
i.e. if $C$ is the free $\Ring$-module generated by Heegaard states of $\HD$,
equipped with the differential specified by
\begin{equation}
\label{eq:OriginalComplex}
\partial \mathbf x =\sum_{\mathbf y\in\States(\HD)}\sum_{B\in\doms(\mathbf x,\mathbf y)}
\# \UnparModFlow^B(\mathbf x,\mathbf y)\cdot U^{n_\wpt(B)}V^{n_\zpt(B)} \cdot \mathbf y.
\end{equation} The homology
of this complex $C_\Ring(\HD)=(C,\partial)$, thought of as a bigraded $\Ring$-module, is the knot
invariant $\HFKsimp(K)$ discussed in the introduction.
Our aim here is to prove the following pairing theorem, describing
knot Floer homology in terms of type $D$ and type $A$ structures.
\begin{thm}
\label{thm:PairAwithD}
For $\HD=\Hdown\cup_Z\Hup$ as above
there is a quasi-isomorphism of bigraded
chain complexes over $\Ring$
\[\CFKsimp(\HD)\simeq \Amod(\Hdown)\DT\Dmod(\Hup).\]
\end{thm}
The proof will occupy the rest of this section. Like the of the
pairing theorem from~\cite[Chapter~9]{InvPair}, we ``deformation
the diagonal''. The key novelty here is to deform our orbits as well
as chords, as in~\cite{TorusMod}. However, the boundary degenerations
present in the current set-up cause this step to be more intricate.
As usual, the proof begins by inserting a sufficiently long neck, to
identify the Floer complex $\CFKsimp(\HD)$ with another complex which
is isomorphic to $C$ as a $\Ring$-module, and whose differential
$\partial^{(0)}$ counts points in a fibered product of moduli spaces
coming from the two sides. In our case, the fibered product is a
fibered product over $[0,1]\times \R$ (thought of as having
coordinates $(s,t)$) of moduli spaces of pseudo-holomorphic curves in
$\Hdown$ and $\Hup$. This identification is spelled out in
Subsection~\ref{subsec:FiberedProduct}. Note that this identification
works slightly differently when $n=1$. In this outline we will assume
$n>1$; the case where $n=1$ is handled in
Subsection~\ref{subsec:Nequals1}.
In Section~\ref{subsec:MatchedComplex}, we give an independent proof
that $\partial^{(0)}$ is indeed a differential. Although this is not
logically necessary (it follows from the identification with the
Heegaard Floer differential), elements of the proof will be used throughout.
We will compare the fibered product complex with another chain complex,
described in Subsection~\ref{subsec:SelfMatched}, that corresponds to
deforming the $s$-coordinate matchings to the boundary. Again, the
underlying $\Ring$-module is $C$, but the differential $\dChanged$
now, instead of counting points in the fibered product, counts curves
where orbits for the curves in $\Hup$ match with long chords on the
$\Hdown$-side. These moduli spaces allow for some additional orbits on
the type $\Hdown$ side; details are given in
Subsection~\ref{subsec:SelfMatched}.
Interpolating between these two extremes is a sequence of complexes
(Subsection~\ref{subsec:Intermediates}), $\{(C,\partial^{(k)})\}_{k=0}^{2n}$,
where $\partial^{(2n)}=\dChanged$.
Isomorphisms $\Phi_k\colon (C,\partial^{(k)})\to (C,\partial^{(k+1)})$
are constructed in Section~\ref{subsec:Interpolate}, using moduli
spaces in which the $s$-matching on Reeb orbits are deformed by a
parameter $r\in (0,1)$.
Looking at ends of these moduli spaces give the desired relation
\[ \partial^{(k+1)}\circ \Phi_k + \Phi_k \circ \partial^{(k)}=0.\]
By an energy argument, these maps are isomorphisms.
Having fully deformed the $s$-parameter in the matching to obtain
$(C,\dChanged)$, we can now deform the $t$-parameter in the
matching as in~\cite[Chapter~9]{InvPair} (``time dilation''); see
Section~\ref{subsec:TimeDilation}. We now proceed to establish these
steps.
\subsection{The Heegaard Floer differential and matched curves.}
\label{subsec:FiberedProduct}
The discussion here is closely modelled on~\cite[Section~9.1]{InvPair}.
Let $\Sigma_1$ denote the Heegaard surface for $\Hdown$ and $\Sigma_2$ denote the surface for $\Hup$.
\begin{defn}
\label{def:MatchingPair}
Let $\HD=\Hdown\#\Hup$. States $\mathbf x_1\in\States(\Hdown)$,
$\mathbf x_2\in\States(\Hup)$ are called {\em matching states}
if $\alpha(\mathbf x_1)$ is the complement of
$\alpha(\mathbf x_2)$ (i.e. $\Idown(\mathbf x_1)=\Iup(\mathbf x_2)$)
\end{defn}
There is a one-to-one correspondence between pairs of matching states
$\mathbf x_1$ and $\mathbf x_2$ and Heegaard states for $\Hdown\#\Hup$. Thus, the
generators of $\CFKsimp(\HD)$ correspond to the generators of
$\Amod(\Hdown)\otimes\Dmod(\Hup)$, where the latter tensor product is
over the idempotent subalgebra $\RestrictIdempRing(n)\subset \Clg(n)$.
In the Heegaard Floer differential for $\CFKsimp(\HD)$, the
holomorphic disk counting for the homology class $B$ is weighted by
$U^{n_\wpt(B)} V^{n_\zpt(B)}$. Since in $\Ring$ we impose the relation
$UV=0$, it follows that the homology classes with non-zero contribution
have vanishing local multiplicity at $\wpt$ or $\zpt$.
For $x\in \alpha_i$, we write $\alpha(x)=i$. This definition will be used
both for $x\in\Sigma_1$ and $\Sigma_2$.
\begin{defn}
\label{def:MatchingChords}
Let $\rho_1$ be a Reeb chord in $\Hup$ and $\rho_2$ be a Reeb chord
in $\Hdown$. We say that $\rho_1$ and $\rho_2$ are {\em matching
chords} if the following two conditions hold:
\begin{itemize}
\item $\weight(\rho_1)=\weight(\rho_2)$
\item $\alpha(\rho_1^-)\neq \alpha(\rho_2^-)$.
\end{itemize}
\end{defn}
The first condition implies that if $\rho_1$ is a chord on $\Zin_i$,
then $\rho_2$ is a chord in $\Zout_i$. The second condition means that
the initial point of a chord is on $\alpha_{i-1}$ if and only if the
initial point of its matching chord is on $\alpha_i$. It is
equivalent to replace the second condition by the condition
$\alpha(\rho_1^+)\neq \alpha(\rho_2^+)$. Note that
chords in $\Zin_1$ or $\Zin_{2n}$ do not match with any chords in
$\Zout_1$ or $\Zout_{2n}$.
A more succinct description of the matching condition can be given
following our labeling conventions from Figure~\ref{fig:ChordNames}
(for the upper diagram) and~\ref{fig:ChordNamesA} (for the lower
diagram): with these conventions, $\rho_1$ and
$\rho_2$ are matching precisely if their labels are the
same.
\begin{defn}
\label{def:MatchedPair}
Suppose $n>1$.
Fix two pairs $(\mathbf x_1,\mathbf x_2)$ and $(\mathbf y_1,\mathbf y_2)$ of matching states,
i.e. where $\mathbf x_1,\mathbf y_1\in\States(\Hdown)$,
$\mathbf x_2,\mathbf y_2\in\States(\Hup)$, so that $\mathbf x=\mathbf x_1\#\mathbf x_2$ and
$\mathbf y=\mathbf y_1\#\mathbf y_2$ are Heegaard states for $\HD$.
A {\em matched pair} consists of the following data
\begin{itemize}
\item a holomorphic curve $u_1$ in $\Hdown$ with source $\Source_1$
representing homology class $B_1\in\doms(\mathbf x_1,\mathbf y_1)$.
\item a holomorphic curve $u_2$ in $\Hup$ with source $\Source_2$ representing homology class $B_2\in\doms(\mathbf x_2,\mathbf y_2)$
\item a bijection $\psi\colon \AllPunct(\Source_2)\to \AllPunct(\Source_1)$
\end{itemize}
with the following properties:
\begin{itemize}
\item For each $q\in \AllPunct(\Source_2)$ marked with a Reeb orbit
or chord, the corresponding puncture $\psi(q)\in \AllPunct(\Source_1)$
is marked with the matching Reeb orbit or chord.
\item For each $q\in\AllPunct(\Source_2)$,
\[ (s\circ u_1(\psi(q)),t\circ u_1(\psi(q)))=(s\circ u_2(q),t\circ u_2(q)).\]
\end{itemize}
If $B_1$ and $B_2$ induce $B\in \doms(\mathbf x,\mathbf y)$, let $\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)$
denote the moduli space of matched pairs.
\end{defn}
\begin{lemma}
\label{lem:SimpleLemma}
Assume that $\rho_1,\sigma_1$ are Reeb chords on two different
boundary components in $\Sigma_1$ and $\rho_2,\sigma_2$ are the
matching Reeb chords in $\Sigma_2$. The four numbers
$\{\alpha(\rho_1^+), \alpha(\sigma_1^+),\alpha(\rho_2^+),
\alpha(\sigma_2^+)\}$ are distinct if and only if the four numbers
$\{\alpha(\rho_1^-),
\alpha(\sigma_1^-),\alpha(\rho_2^-),\alpha(\sigma_2^-) \}$ are.
\end{lemma}
\begin{proof}
Let $\xi_1$ and $\xi_2$ be matching chords, then clearly
\[\{\alpha(\xi_1^-),\alpha(\xi_2^-)\}=\{\alpha(\xi_1^+),\alpha(\xi_2^+)\}.\]
The lemma follows readily.
\end{proof}
\begin{lemma}
\label{lem:BoundaryMonotone}
Let $(\mathbf x_1,\mathbf x_2)$ and $(\mathbf y_1,\mathbf y_2)$ be two pairs of matching states,
and fix a matched pair $(u_1,u_2)$ connecting $\mathbf x_1\#\mathbf x_2$ to
$\mathbf y_1\#\mathbf y_2$, with $u_1\in\ModFlow(\mathbf x_1,\mathbf y_1,\rhos_1,\dots,\rhos_m)$
and $u_2\in\ModFlow(\mathbf x_2,\mathbf y_2,\rhos_1',\dots,\rhos_m')$ (where the
chords in $\rhos_i$ all match, in the sense of
Definition~\ref{def:MatchingChords} with chords in $\rhos_i'$).
Then, both $u_1$ and
$u_2$ are strongly boundary monotone; moreover,
$\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_\ell)$ is complementary to
$\alpha(\mathbf x_2,\rhos_1',\dots,\rhos_\ell')$ for all $\ell=0,\dots,m$.
\end{lemma}
\begin{proof}
We prove the following by induction on $\ell$:
\begin{enumerate}
\item \label{item:SBM} $(\mathbf x_1,\rhos_1,\dots,\rhos_\ell)$ and
$(\mathbf x_2,\rhos_1',\dots,\rhos_\ell')$ are strongly boundary monotone
\item \label{item:Disjoint} $\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_\ell)$ and
$\alpha(\mathbf x_2,\rhos_1',\dots,\rhos_\ell')$ are disjoint.
\end{enumerate}
The case where $\ell=0$ is simply the hypothesis that $\mathbf x_1$ and
$\mathbf x_2$ are matching states. Given a subset ${\mathfrak s}$,
let \[\alpha({\mathfrak s})=\{k\in 1,\dots,2n-1\big| \alpha_k\cap
{\mathfrak s}\neq \emptyset\}.\] Recall that $\rhos^-_i$ denotes the
set of initial points of all the chords in $\rhos_i$. Let
$A_-=\alpha(\rhos_i^-)$ and $A_+=\alpha(\rhos_i^+)$. Continuity
ensures that $A_-\subset \alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{\ell-1})$
and $B_-\subset \alpha(\mathbf x_2,\rhos'_1,\dots,\rhos'_{\ell-1})$.
Weak boundary monotonicity implies that $|A_-|=|\rhos_i^-|$ and $|B_-|=|{\rhos_i'}^-|$.
By
the induction hypothesis, $A_-$ and $B_-$ are disjoint. From
Lemma~\ref{lem:SimpleLemma}, it follows that
\[ |A_-|=|\rhos_i^-|, |B_-|=|{\rhos_i'}^-|, |A_-\cap B_-|=\emptyset
\Rightarrow
|A_+|=|\rhos_i^+|, |B_+|=|{\rhos_i'}^+|, |A_+\cap B_+|=\emptyset. \]
In particular, we have verified that $A_+$ and $B_+$ are disjoint.
It follows at once that
\[ \alpha(\mathbf x_1,\rhos_1,\dots,\rhos_i)=(\alpha(\mathbf x,\rhos_1,\dots,\rhos_{\ell-1})\setminus A_-)\cup A_+ \]
is disjoint
from
\[ \alpha(\mathbf x_2,\rhos_1',\dots,\rhos_i')=(\alpha(\mathbf x_2,\rhos_1',\dots,\rhos_{\ell-1}')\setminus B_-)\cup B_+, \]
verifying Property~\eqref{item:Disjoint} for the inductive step.
Since $|A_+|=|A_-|$, it follows that
\[
|\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_i)|=|\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{i-1})|-|A_+\cap
(\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{i-1})\setminus A_-)|. \] We show that
$A_+\cap (\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{i-1})\setminus A_-)$ is empty; for if it were
non-empty, there would be some chord $\rho_1\in \rhos_i$ with
$\rho_1^+\neq \rho_1^-$, and $\alpha(\rho_1^+)\in
\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{i-1})$. But for any chord with
$\rho_1^+\neq \rho_1^-$, the matching chord $\rho_2$ satisfies
$\alpha(\rho_1^+)=\alpha(\rho_2^-)$; so $\alpha(\rho_2^-)\in
\alpha(\mathbf x_2,\rhos'_1,\dots,\rhos'_{i-1})$. But $\alpha(\rho_2^-)$ is
contained in both $\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{i-1})$ and
$\alpha(\mathbf x_2,\rhos_1',\dots,\rhos_{i-1}')$, violating an inductive
hypothesis. We conclude that $|A_+\cap
(\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{i-1})\setminus A_-|=0$, so
\[ |\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_i)|=|\alpha(\mathbf x_1,\rhos_1,\dots,\rhos_{i-1})|. \]
A symmetric argument shows that
\[ |\alpha(\mathbf x_2,\rhos'_1,\dots,\rhos'_i)|=|\alpha(\mathbf x_2,\rhos_1',\dots,\rhos_{i-1}')|, \]
verifying Property~\eqref{item:SBM} for the inductive step.
\end{proof}
Recall that
the expected dimensions of the moduli spaces $\ModFlow^{B_i}(\mathbf x_i,\mathbf y_i;\Source_i)$ for $i=1,2$ are given by the indices
\begin{align*}
\ind(B_1,\Source_1)&=g_1+n + 2e(B_1)-\chi(\Source_1) + 2 o_1
+ c_1-2 \weight_1 \\
\ind(B_2,\Source_2)&=g_2+n-1 + 2e(B_2)-\chi(\Source_2) + 2 o_2
+ c_2 -2 \weight_2,
\end{align*}
where $o_i$ denotes the number of interior punctures of $\Source_i$,
$c_i$ denotes the number of boundary punctures of $\Source_i$, and
$\weight_i$ denotes the total weight of $B_i$ at the boundary.
(In comparing this formula with, for example, \cite[Equation~9.8]{InvPair},
note that the convention on the Euler measures here are different from the ones used in~\cite{InvPair}; c.f. Remark~\ref{rem:EulerMeasures}.)
\begin{defn}
The {\em index} of a matched pair
is defined by the formula
\begin{align}
\label{eq:DefIndMatchedPair}
\ind(B_1,\Source_1;B_2,\Source_2)&=
\ind(B_1,\Source_1)+\ind(B_2,\Source_2)-c-2o\\
&=g+2e(B_1)+2e(B_2)-\chi(\Source_1)-\chi(\Source_2)
+ 2o+c-4\weight,
\nonumber
\end{align}
where
$g=g_1+g_2+2n-1$ is the genus of $\Hup\#\Hdown$,
$c=c_1=c_2$,
$o=o_1=o_2$, and $\weight=\weight_1=\weight_2$.
\end{defn}
We can think of the moduli space
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)$ as a
fibered product
\[ \ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)
= \ModFlow^{B_1}(\mathbf x_1,\mathbf y_1,\Source_1)\times_\ev \ModFlow^{B_2}(\mathbf x_2,\mathbf y_2,\Source_2)\]
where $\times_\ev$ denotes the fibered product over the evaluation maps
at the punctures
\[
\ev_1\colon \ModFlow^{B_1}(\mathbf x_1,\mathbf y_1,\Source_1)\to ([0,1]\times
\R)^k \qquad \ev_2\colon \ModFlow^{B_2}(\mathbf x_2,\mathbf y_2,\Source_2)\to
([0,1]\times \R)^k,\] where
$k=|\AllPunct(\Source_1)|=|\AllPunct(\Source_2)|=c+o$. Then, the index of the
moduli space is the expected dimension of the moduli space of
matched pairs inherited from its description as a fibered product
over $([0,1]\times \R^k)$.
The moduli space $\ModMatched(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)$ comes with an $\R$ action
which is free except in the special case where both sides consist of trivial strips.
The quotient is denoted by
\[ \UnparModMatched(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)
= \ModMatched(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)/\R. \]
\begin{lemma}
\label{lem:Transversality}
Fix $B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ so that
$\weight_i(B_1)=\weight_i(B_2)$ for $i=1,\dots,2n$, and at least one of
$n_{\wpt}(B_1)$ or $n_{\zpt}(B_1)$ vanishes.
For generic admissible almost complex
structures on $\Sigma_i\times [0,1]\times \R$, and
$\ind(B_1,\Source_1;B_2,\Source_2)\leq 2$, the moduli space of
matched pairs
\[ \ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)\] is
transversely cut out by the $\dbar$-equation and the evaluation map;
in particular, this moduli space is a manifold whose dimension is
given by Equation~\eqref{eq:DefIndMatchedPair}.
\end{lemma}
\begin{proof}
This is is essentially~\cite[Lemma~9.4]{InvPair}.
\end{proof}
The data $(\Source_1,\Source_2,\psi\colon
\AllPunct(\Source_2)\to\AllPunct(\Source_1))$ of a matched pair can be
used to form a new source curve $\Source_1\natural_{\psi}\Source_2$,
obtained by gluing punctures to punctures (i.e. connected sum for
orbits and boundary connected sum for boundary punctures). If
$B_1\in\doms(\mathbf x_1,\mathbf y_1)$, $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ are represented by
a matched pair, we can construct $B=B_1\natural B_2\in\doms(\mathbf x,\mathbf y)$,
where $\mathbf x=\mathbf x_1\#\mathbf x_2$ and $\mathbf y=\mathbf y_1\#\mathbf y_2$, represented by a (not
necessarily holomorphic) curve with source
$\Source=\Source_1\natural_\psi\Source_2$.
It is elementary to see that
\begin{align}
e(B_1\natural B_2)&=e(B_1)+ e(B_2)-2 \weight \label{eq:EulerMeasures} \\
\chi(\Source_1\natural_\psi \Source_2)&=\chi(\Source_1)+\chi(\Source_2) - 2o-c.
\label{eq:EulerCharacteristics}
\end{align}
It is an easy consequence that
\[ \ind(B_1\natural
B_2,\Source_1\natural_\psi\Source_2)=\ind(B_1,\Source_1;B_2,\Source_2),\]
identifying expected dimension of the moduli space of curves in
$\Source_1\natural_\psi\Source_2$ with expected dimension of the
moduli spaces of matched pairs. Our goal is to refine this to an
identification of moduli spaces.
As in Definition~\ref{eq:EmbedMod}, let
\begin{align*}
\chiEmb(B)&=g+e(B)-n_\mathbf x(B)-n_\mathbf y(B) \\
\ind(B)&=e(B)+n_\mathbf x(B)+n_\mathbf y(B).
\end{align*}
\begin{prop}
\label{prop:EmbeddedModuliSpaces}
Fix $\mathbf x=\mathbf x_1\#\mathbf x_2$ and $\mathbf y=\mathbf y_1\#\mathbf y_2$, and decompose
$B\in\doms(\mathbf x,\mathbf y)$ as $B=B_1\natural B_2$, with
$B_i\in\doms(\mathbf x_i,\mathbf y_i)$. Fix source curves $\Source_1$ and
$\Source_2$ together with a one-to-one correspondence $\psi\colon
\AllPunct(\Source_2)\to\AllPunct(\Source_1)$ which is consistent with the chord
and orbit labels, so we can form $\Source=\Source_1\natural_\psi\Source_2$.
Suppose that $\ModFlow^B(\mathbf x,\mathbf y;\Source)$
(i.e. the moduli space for curves in $\HD$) and
$\ModFlow^{B_i}(\mathbf x_i,\mathbf y_i;\Source_i)$
(which are moduli spaces for curves in $\HD_i$)
are non-empty for $i=1,2$.
Then,
$\chi(\Source)=\chiEmb(B)$ if and only if
$\chi(\Source_i)=\chiEmb(B_i)$ for $i=1,2$;
and all the chords in $\AllPunct(\Source_1)$ have weight $1/2$ and
all the orbits have length $1$.
\end{prop}
\begin{proof}
Let $\weight$ denote the total weight of $B_1$ at the boundary,
$o$ denote the number of orbits in $\Source_1$, and $c$ the number of chords.
Since $g=g_1+g_2+2n-1$, it follows immediately
from Equation~\eqref{eq:EulerMeasures} that
\[ \chiEmb(B_1\natural B_2)=\chiEmb(B_1)+\chiEmb(B_2)-2\weight;\] so using
Equation~\eqref{eq:EulerCharacteristics}, it follows that
\[
\chi(\Source)-\chiEmb(B)
= \left(\chi(\Source_1)-\chiEmb(B_1)\right)
+ \left(\chi(\Source_2)-\chiEmb(B_2)\right)
+ 2\weight-c-2o.\]
Clearly, $2\weight-c-2o\geq 0$, with equality iff each orbit has weight $1$ and each chord has weight $1/2$.
Also, the hypotheses that
$\ModFlow^B(\Source)\neq \emptyset$ and
$\ModFlow^{B_i}(\Source_i)\neq \emptyset$ for $i=1,2$ ensure that
\[ \chi(\Source)\geq \chiEmb(B),\qquad \chi(\Source_i)\geq \chiEmb(B_i) \]
with equality iff
the curves are embedded.
\end{proof}
The Gromov compactification
of the space of matched curves is provided by a space of matched combs,
which we define presently.
Let $(w_{\ell_1},\dots,w_1,u,v_1,\dots,v_k)$ be a story with total
source ${\overline\Source}$. Let $\East({\overline\Source})$ be the
eastmost punctures in ${\overline\Source}$ (i.e. these are the East
punctures on curves that are not matched with west punctures on other
curves at East infinity), $\IntPunct({\overline\Source})$ be all the
orbit-marked punctures in all the components of ${\overline\Source}$,
and
\[ \AllPunct({\overline\Source})=\East({\overline\Source})\cup\IntPunct({\overline\Source}).\]
\begin{defn}
\label{def:dMatched}
Given $\mathbf x_1,\mathbf y_1\in\States(\Hdown)$ and $\mathbf x_2,\mathbf y_2\in\States(\Hup)$,
a {\em matched story from $\mathbf x=\mathbf x_1\#\mathbf x_2$ to $\mathbf y=\mathbf y_1\#\mathbf y_2$} consists of the following data:
\begin{itemize}
\item a
pair of holomorphic stories
\[ {\overline
u}_1=(w_{\ell_1}^1,\dots,w_1^1,u^1,v_1^1,\dots,v_{k_1}^1)
\qquad{\text{and}} \qquad {\overline
u}_2=(w_{\ell_2}^2,\dots,w_1^2,u^2,v_1^2,\dots,v_{k_2}^2), \]
\item
a one-to-one correspondence
\[ \psi\colon \AllPunct({\overline\Source}_2)\to
\AllPunct({\overline\Source}_1) \] so that for all $p\in
\AllPunct({\overline\Source}_2)$, the Reeb chord or orbit marking
$p$ and $\psi(p)$ have matching labels.
\end{itemize}
satisfying the following condition for each $q\in \AllPunct({\overline\Source}_2)$:
\[ (s\circ {\overline u}_2(q),t\circ {\overline u}_2(q))=
(s\circ {\overline u}_1(\psi(q)),t\circ {\overline u}_1(\psi(q))).\]
for all $q\in\AllPunct({\overline \Source}_2)$.
The story is called {\em stable} if there are no unstable east or
west infinity curves on either side, and either $u^1$ or $u^2$ is stable.
\end{defn}
\begin{defn}
A matched comb of height $N$ is a sequence of stable
matched stories running from
$(\mathbf x_1^{j},\mathbf x_2^{j})$ to $(\mathbf x_1^{j+1},\mathbf x_2^{j+1})$
for sequences of lower states $\{\mathbf x_1^j\}_{j=1}^{N+1}$ (for $\Hdown$)
and upper states $\{\mathbf x_2^j\}_{j=1}^{N+1}$ (for $\Hup$).
\end{defn}
In principle, the Gromov compactification could contain more
complicated objects: closed components contained in some story, or
$\alpha$-boundary degenerations. We exclude these possibilities in the
next two lemmas.
\begin{lemma}
\label{lem:NoBoundaryDegenerations}
Fix $B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ so that
$\weight_i(B_1)=\weight_i(B_2)$ for $i=1,\dots,2n$, and at least one of
$n_\wpt(B_1)$ or $n_\zpt(B_1)$ vanishes, and so that $\ind(B_1\natural
B_2)\leq 2$. Then, curves in the Gromov compactification of
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)$ contain no
boundary degenerations.
\end{lemma}
\begin{proof}
Fix a curve $({\overline u}_1,{\overline u}_2)$ denote the matched
comb in the Gromov compactification of matched curve pairs, so that
${\overline u}_1$ denotes the portion in $\Hdown$ and ${\overline
u}_2$ in $\Hup$.
We first prove that neither ${\overline u}_1$ nor ${\overline u}_2$
can contain a $\beta$-boundary degeneration. This is equivalent to
showing that ${\overline u}_i$ does not contain a puncture $p_i$
marked with a Reeb orbit, and with the property that
$s({\overline u}_i(p_i))=0$.
To see this, we prove first that if there is a puncture $p$ marked
by a Reeb orbit in ${\overline u}_i$ with $\pi_\CDisk(p)=(0,\tau)$,
then in fact for each $j=1,\dots,2n$, there are corresponding
punctures $q_1$ and $q_2$ in ${\overline u}_1$ and ${\overline u}_2$
with
\[ \pi_{\CDisk} ({\overline u}_1(q_1))=\pi_{\CDisk}(u_2(q_2))=(0,\tau),\] so that $q_1$ is marked by some orbit that
covers $\Zin_j$ and $q_2$ is marked by some orbit that covers
$\Zout_j$. This follows from the following observations:
\begin{itemize}
\item
The curve ${\overline u}_1$ contains a puncture $q_1$
marked by Reeb orbit that covers $\Zin_j$
with $\pi_{\CDisk}u(q)=(0,\tau)$ if and only ${\overline u}_2$
contains a puncture $q_2$ marked by a Reeb orbits that covers $\Zout_j$
with $\pi_{\CDisk}u(q)=(0,\tau)$. This is immediate from
the matching condition.
\item If ${\overline u}_1$ contains a puncture $q$ marked by a
Reeb orbit that covers $\Zin_j$ with $\pi_{\CDisk}u(q)=(0,\tau)$,
and $\{j,k\}\in\Mdown$,
then there is another puncture $q'$ marked by a Reeb orbit that
covers $\Zin_k$.
This follows at once from the fact that $q$ is contained
in a $\beta$-boundary
degeneration component.
\item If ${\overline u}_2$ contains a puncture $q$ marked
by a Reeb orbit that covers $\Zout_j$ with $\pi_{\CDisk}u(q)=(0,\tau)$,
then it also contains another puncture $q'$
marked by a Reeb orbit that covers $\Zout_k$ with
$\pi_{\CDisk}u(q')=(0,\tau)$, where $\{j,k\}\in\Mup$.
This follows as above.
\end{itemize}
Compatibility of $\Mup$ and $\Mdown$ now allows us to conclude
that statement at the beginning of the paragraph. Moreover, from
that statement it follows that if ${\overline u}_1$ or ${\overline
u}_2$ contains any boundary degeneration, then in fact it
contains some boundary degeneration that contains $\wpt$ and
another boundary degeneration that contains $\zpt$. But this
violates the hypothesis that $n_\wpt(B_1)=0$ or
$n_\zpt(B_1)=0$.
We turn to the possibility of a $\Ta$ boundary degeneration. Since
$B_1$ does not cover both $\wpt$ and
$\zpt$, our homological hypotheses on the type $A$ side ensures that
the limiting curve contains no $\alpha$-boundary degenerations on
the $\Hdown$ side.
It remains to consider the possibility that
an $\alpha$-boundary degenerations that occurs on
the $\Hdown$ side, with projecting to $(1,\tau)$ for some $\tau\in\R$.
The chords and orbits in the boundary degeneration in
${\overline{u}_2}$ can match with chords and orbits on various East
infinity curves in ${{\overline u}_1}$. Nonetheless, the
$(s,t)$-projections of all of these chords and orbits are
$(1,\tau)$, for some fixed $\tau\in\R$, by the matching
condition. Moreover, the weight at each boundary component $Z_i$ of
these chords in ${{\overline u}_1}$ must be at least $1$ (since the
same is true for each boundary degeneration in $\Hdown$). Consider
now the main component $u_1$ of ${{\overline u}_1}$, obtained after
removing East infinity curves. Note that the total weights of the
Reeb chords and orbits of an East infinity curve are the same at
both their Eastern and their Western boundary; it follows that $u_1$
has a chord packet $\rhos$ at $(s,t)=(1,\tau)$ with the property
that $\weight_i(\rhos)\geq 1$ for all $i=1,\dots,2n$. Observe also the
main component cannot have any orbits whose $s$-projection is
$1$. It follows that $\rhos^-$ contains points on all of the
boundary components of $\Sigma_1$ ; i.e. $\rhos^-$ contains at least
$2n$ points. But this violates boundary monotonicity: $\rho^-$ can
contain at most $n$ points.
\end{proof}
\begin{lemma}
\label{lem:NoClosedCurves}
Suppose that $n>1$.
Fix $B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ so that
$\weight_i(B_1)=\weight_i(B_2)$ for $i=1,\dots,2n$, and at least one of
$n_\wpt(B_1)$ or $n_\zpt(B_1)$ vanishes, and so that $\ind(B_1\natural
B_2)\leq 2$. Curves in the Gromov compactification of
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)$ contain no
closed components.
\end{lemma}
\begin{proof}
By our hypotheses on the homology class, homologically non-trivial
closed components cannot occur
on the $\Hdown$ side.
Suppose there is a curve in the Gromov compactification of
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)$ which has a single closed
component in $\Hup$, and it has multiplicity $k>0$. Let $(u_1',u_2')$
be the main components of this limit, so that
$u_1'\in\ModFlow^{B_1}(\mathbf x_1,\mathbf y_1;\Source_1')$ and
$u_2'\in\ModFlow^{B_2-k[\Sigma_2]}(\mathbf x_2,\mathbf y_2;\Source_2')$. We will
show now that $(u_1',u_2')$ lies in a moduli space whose expected
dimension $\ind(u_1',u_2')$ satisfies
\begin{equation}
\label{eq:ConstrainedModSpaceDim}
\ind(u_1',u_2')\leq \ind(B_1\natural B_2)+2-2nk.
\end{equation}
Let $o_i'$ denote the number of orbits in $\Source_i'$,
$c_i'$ denote the number of chords in $\Source_i$,
$\weight$ denote $\weight_{\partial}([B_1])=\weight_{\partial}([B_2])$,
$\weight_i'$ denote $\weight_{\partial}([B_i'])$.
Since there are no boundary degenerations, we conclude that $B_1=B_1'$,
and in particular $\weight_1'=\weight$.
Also, the matching conditions ensure that $c_1'=c_2'$.
If $\delta$ is the number of interior punctures in $\Source_1'$ that match with punctures in the (removed) closed component, then
$o_1'=o_2'+\delta$.
The pair $(u_1',u_2')$ lies in a moduli space whose expected dimension is given by
\begin{align*}
\ind(u_1',u_2')&=\ind(B_1,\Source_1')+\ind(B_2,\Source_2)-2(\delta-1)-2 o_2'-c_2'\\
&=\ind(B_1,\Source_1')+\ind(B_2,\Source_2)-2o_1'-c_1'+2
\end{align*}
To see why, note that the $\delta$ orbits in
$\Source_1'$ are all required to have the same $(s,t)$ projection
(hence the term $2(\delta-1)$); the additional $2o_2'+c_2'$ constraints come
from the orbits and chords in $\Source_2'$, which project to the same
$(s,t)$ values as corresponding orbits and chords in $\Source_1'$.
It is elementary to see that
\begin{align*}
e([\Sigma_2])+n_{\mathbf x_2}([\Sigma_2])+n_{\mathbf y_2}([\Sigma_2])&=
2-2g_2 + 2 (g_2 + n-1)=2n.
\end{align*}
Since $[B_1']=[B_1]$, the index formula gives
\begin{align*}
\ind(u_1',u_2')-\ind(B_1\natural B_2)
&= e(B_1')+n_{\mathbf x_1}(B_1)+n_{\mathbf y_1}(B_1)-e(B_2)-n_{\mathbf x_2}(B_2)-n_{\mathbf y_1}(B_2) \\
&\qquad
-2\weight_1'-2\weight_2'+2\weight
+2o_2'+c_2'+2 \\
&= -2nk -2\weight_2'+2o_2'+c_2'+2 \\
&\leq -2nk+2,
\end{align*}
using the obvious inequality $-2\weight_2'+2o_2'+c_2'\leq 0$;
i.e. Inequality~\ref{eq:ConstrainedModSpaceDim} holds.
Our hypothesis that $\ind(B_1\natural B_2)=2$ ensures that
$(u_1,u_2')$ lives in a moduli space ${\mathcal M}$ with a free $\R$
action (since $u_1'$ contains orbits, so it cannot be constant) and
$\dim({\mathcal M})=4-2nk$. Since $n>1$ and $k\geq 1$, we conclude that this
moduli space is empty.
We have ruled out homologically non-trivial closed components.
Homologically trivial ``ghost'' components are also ruled out
by the dimension formula, as in~\cite[Lemma~5.57]{InvPair}.
\end{proof}
\begin{remark}
When $n=1$, spheres can occur in zero-dimensional moduli spaces, so
that case must be handled separately; see
Section~\ref{subsec:Nequals1}.
\end{remark}
\begin{defn}
Given a shadow $B\in \doms(\mathbf x_1\#\mathbf y_1,\mathbf x_2\#\mathbf y_2)$,
the {\em embedded matched moduli space}
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)$ is the union of all
$\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi)$
taken over all compatible pairs $\Source_1$ and $\Source_2$
with $\chi(\Source_1\natural\Source_2)=\chiEmb(B)$.
Moreover,
\[\UnparModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)=
\ModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)/\R.\]
\end{defn}
\begin{prop}
\label{prop:StretchNeck}
We can find a generic almost-complex structures
on $\Hup$, $\Hdown$, and $\HD=\Hdown\#_Z\Hup$
with the following property. For each $\mathbf x=\mathbf x_1\#\mathbf x_2$
and $\mathbf y=\mathbf y_1\#\mathbf y_2\in\States(\HD)$, $B\in\doms(\mathbf x,\mathbf y)$
and $\ind(B)=1$,
with at least one of $n_{\wpt}(B)$ or $n_{\zpt}(B)$ vanishing,
there is an identification of moduli spaces
of curves in $\HD$ with matched curves:
\[ \UnparModFlow^B(\mathbf x_1\# \mathbf x_2,\mathbf y_1\#\mathbf y_2)\cong
\UnparModMatched^B(\mathbf x_1,\mathbf y_1; \mathbf x_2,\mathbf y_2).\]
\end{prop}
\begin{proof}
With Lemmas~\ref{lem:NoBoundaryDegenerations}
and~\ref{lem:NoClosedCurves} in place, this is a standard gluing
argument; see~\cite[Proposition~9.6]{InvPair}.
\end{proof}
\begin{defn}
\label{def:MatchComplex}
Let $\HD=\Hup\#_Z\Hdown$ be a decomposition of a
doubly-pointed Heegaard diagram for a knot in $S^3$,
and assume that $n>1$.
The {\em chain complex of matched curves} is the pair $(C,\partial^{(0)})$,
where,
as before, $C$ denotes the free $\Ring$-module generated by Heegaard states
for $\HD$ or, equivalently, matching pairs of states (as in Definition~\ref{def:MatchingPair}).
The operator
\[\partial^{(0)}\colon C\to C \]
is the $\Ring$-module endomorphism specified by
\begin{equation}
\label{eq:DefD0}
\partial^{(0)}(\mathbf x_1\#\mathbf x_2)=\sum_{(\mathbf y_1,\mathbf y_2)} \sum_{\{B\mid \ind(B)=1\}}
\#\left(\UnparModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)\right)\cdot
U^{n_\wpt(B)}V^{n_\zpt(B)} \cdot \mathbf y_1\#\mathbf y_2.
\end{equation}
\end{defn}
\subsection{The chain complex of matched curves}
\label{subsec:MatchedComplex}
Although it is not technically necessary (it can be thought of as a
consequence of Proposition~\ref{prop:StretchNeck}, we include here a proof
that $\partial^{(0)}$ induces a differential. The methods appearing in
the proof will appear again in the proof of
Theorem~\ref{thm:PairAwithD}.
The following is a slight adaptation of~\cite[Proposition~9.16]{InvPair}:
\begin{lemma}
\label{lem:dMatchedCompactify}
Suppose that $\ind(B_1,\Source_1;B_2,\Source_2)=2$, then for generic
$J$, the moduli space
$\ModMatched^{B}(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2)$ can be
compactified by adding the following objects:
\begin{enumerate}[label=(ME-\arabic*),ref=(ME-\arabic*)]
\item two-story matched holomorphic curves
(i.e. neither story contains curves at East or
West infinity)
\item
\label{def:JJ}
matched stories $(u^1,v^1)$ and $(u^2,v^2)$
where the only
non-trivial components of $v_1$ and $v_2$ are join curves, and
where $\West(v^1)$ and $\West(v^2)$ are the two distinct length one Reeb chords
that cover the same boundary component,
\item
\label{def:OO} matched stories $(u^1,v^1)$ and $(u^2,v^2)$ where the only
non-trivial components of $v^1$ and $v^2$ are orbit curves, and
$\West(v^1)$ and $\West(v^2)$ are the two distinct length one Reeb chords that
cover the same boundary component.
\end{enumerate}
\end{lemma}
\begin{figure}[h]
\centering
\input{JoinOrbit.pstex_t}
\caption{{\bf Cancellation of join ends with orbit ends.}}
\label{fig:JoinOrbit}
\end{figure}
\begin{proof}
The main components are strongly boundary monotone by the argument
from Lemma~\ref{lem:BoundaryMonotone}. This rules out boundary
double points, as in~\cite[Lemma~5.56]{InvPair}. Boundary
degenerations are ruled out as in
Lemma~\ref{lem:NoBoundaryDegenerations}. Dimension considerations
rule out closed components as in Lemma~\ref{lem:NoClosedCurves}.
We follow proof of~\cite[Proposition~9.16]{InvPair}. Suppose there
is some matched story that appears in the boundary of the moduli
space. We could write that matched story as a pair of stories with a
matching condition on the eastmost punctures;
instead we combine the curves at East infinity
into a single sequence, writing, more symmetrically,
$(u_1,v_1,\dots,v_\ell,u_2)$. Let
$(\Source_1',\EastSource_1,\dots,\EastSource_\ell,\Source_2')$
denote the corresponding sequence of source curves. Let
$\EastSource=\EastSource_1\natural\dots\natural\EastSource_\ell$
(where we glue along all boundary punctures and all interior
punctures, as well). There is an induced partition $P_1$ on the
east punctures of $\Source_1$: two punctures are in the same
partition if they are assigned to same components of
$\EastSource$. There is a similar partition $P_2$ on the east
punctures of $\Source_2$. Combining Equations~\eqref{eq:DefIndMatchedPair}
and~\eqref{eq:EulerMeasures}, we see that
\[ \ind(B_1,\Source_1;B_2,\Source_2)=g-\chi(\Source_1)-\chi(\Source_2)+2e(B)+c+2o,\]
where $c=|\East(\Source_1)|=|\East(\Source_2)|$
and $o=|\IntPunct(\Source_1)|=|\IntPunct(\Source_2)|$.
The limit curves
$(u_1,u_2)$ live in a fibered product $\ModFlow$ whose dimension (after dividing out by the free $\R$ action) is
given by
\begin{align*}
\dim(\ModFlow)&=\ind(B_1,\Source_1',P_1)+\ind(B_2,\Source_2',P_2)-k-1-2o', \\
&= g-\chi(\Source_1')-\chi(\Source_2')+2e(B)+k-1+2o';
\end{align*}
where $k$ is the number of components of $\EastSource$, which agrees
with $|P_1|=|P_2|$; and $o'=|\IntPunct(\Source_1')|=|\IntPunct(\Source_2')|$
is the number of interior punctures in $\Source_1'$. By elementary topology,
\[ \chi(\Source_1)+\chi(\Source_2)-c-2o=\chi(\Source_1')+\chi(\Source_2')+\chi(\EastSource)-c_1-c_2-2o',\]
where $c_i=|\East(\Source_i')|$ for $i=1,2$.
It follows that
\[ \dim(\ModFlow)=\ind(B_1,\Source_1;B_2,\Source_2)+(\chi(\EastSource)-k)+(k-c_1)+(k-c_2)-1.\]
Since $\chi(\EastSource)\leq k$ and $k\leq c_i$ for $i=1,2$, we conclude that
if $\dim(\ModFlow)\geq 0$ and $\ind(B_1,\Source_1;B_2,\Source_2)=2$,
then exactly one of the following three possibilities can occur:
\begin{enumerate}[label=(d-\arabic*),ref=(d-\arabic*)]
\item
\label{case:SplitJoin}
all the components of $\EastSource$ are disks, and
$\{c_1,c_2\}=\{k,k+1\}$
\item
\label{case:Annulus}
$k=c_1=c_2$, exactly one component in $\EastSource$ is an annulus,
and all other components are trivial strips.
\end{enumerate}
Case~\ref{case:SplitJoin} is excluded, as follows. Suppose that
$c_1=k+1$ and $c_2=k$, so that there are two punctures on $\Source_1'$
labelled by two chords $\rho_1$ and $\rho_2$ (for $\Hdown$)
that are matched
by $\EastSource$; and $\rho_1\uplus\rho_2$ is the corresponding
chord in $\Source_2'$. Both chords $\rho_i$ have weight $1/2$, so
clearly $\{\rho_1^-,\rho_2^-\}$ are the two $\alpha$-curves on some
fixed boundary component, while $\rho_1\uplus\rho_2^-$ is also an
$\alpha$-curve on the corresponding boundary component in
$\Hup$. But this violates Lemma~\ref{lem:BoundaryMonotone},
according to which the sets of $\alpha$-occupied curves are complementary.
The case where $c_1=k$ and $c_2=k+1$ is excluded the same way.
Consider next Case~\ref{case:Annulus}. Drop all the trivial strip components,
letting $\EastSource_0$ be the annulus. Dropping all unstable components
from east infinity, we have
$\EastSource_0=\EastSource_1\natural\dots\natural\EastSource_\ell$.
Obviously, we cannot have $\ell=1$: for an orbit at East infinity cannot match
with an interior puncture. Consider next $\ell=2$.
This can occur in two ways. Suppose
$\EastSource=\EastSource_1\natural\EastSource_2$. We have that
\[ 0=\chi(\EastSource)=\chi(\EastSource_1)+\chi(\Source_2)-2o-c;\]
where $o$ resp. $c$ denotes the number of interior resp. boundary
punctures in $\EastSource_1$ (which are matched with corresponding
punctures in $\EastSource_2$). Clearly, this forces $\EastSource_1$
and $\EastSource_2$ to be disks, with either $o=0$ and $c=2$ or
$o=1$ and $c=0$. The first case is Case~\ref{def:JJ}, and the second
is Case~\ref{def:OO}. The same Euler characteristic considerations
show that decompositions with $\ell>2$ have unstable components.
It remains to consider the case where there is more than one story.
In that case, by the additivity of the index, it follows that there are
two stories, both have index $1$, and neither has curves at East infinity.
\end{proof}
\begin{remark}
In~\cite[Proposition~9.16]{InvPair}, Case~\ref{case:Annulus} is
ruled by the existence of a basepoint on the boundary, while
Case~\ref{case:SplitJoin} does occur,
unlike in the above proof, where it is ruled out by combinatorics
of the Reeb chords.
\end{remark}
\begin{lemma}
\label{lem:Join}
Let $(u^1,v^1)$ and $(u^2,v^2)$ be a matched story with
sources $(\Source_1,\EastSource_1)$ and $(\Source_2,\EastSource_2)$
with the property that for $i=1,2$,
the only non-trivial component of $v^i$ is a
join curve forming a length one Reeb chord, then there are arbitrary
small open neighborhoods
$U$ of $(u^1,v^1)\times (u^2,v^2)$
in $\ModFlow^{B_1}(\mathbf x_1,\mathbf y_1,\Source_1\natural\EastSource_1)\times
\ModFlow^{B_2}(\mathbf x_2,\mathbf y_2,\Source_2\natural\EastSource_2)$
so that
$\partial{\overline U}$
meets
$\ModFlow^B(\mathbf x_1,\mathbf y_1,\Source_1\natural\EastSource_1;
\mathbf x_2,\mathbf y_2;\Source_2;\Source_2\natural\EastSource_2)$ in an odd number of points.
The same conclusion holds if for each $i=1,2$, the only non-trivial component
of $v^i$ is an orbit curve.
\end{lemma}
\begin{proof}
If $(u_1,v_1)$ and $(v_2,v_2)$ is a matched story where the
only non-trivial components of $v^1$ and $v^2$ are join curves,
the result follows from Proposition~\ref{prop:JoinCurve}.
When $v^1$ and $v^2$ are orbit curves, the result follows from
Proposition~\ref{prop:OrbitCurve}.
See Figure~\ref{fig:DSquaredZero}.
\end{proof}
\begin{figure}[h]
\centering
\input{DSquaredZero.pstex_t}
\caption{{\bf Cancellation of join ends with orbit ends; an example.}}
\label{fig:DSquaredZero}
\end{figure}
\begin{prop}
\label{prop:dMatchedZqZero}
The endomorphism $\partial^{(0)}$ is a differential.
\end{prop}
\begin{proof}
Fix $\mathbf x_1\#\mathbf x_2, \mathbf y_1\#\mathbf y_2\in\States(\Hdown\#\Hup)$ and $B\in
\pi_2(\mathbf x_1\#\mathbf x_2,\mathbf y_1\#\mathbf y_2)$ with $\ind(B)=2$. Consider the ends
of $\ModFlow^B(\mathbf x_1,\mathbf x_2;\mathbf y_1,\mathbf y_2)$, consisting of embedded,
matched holomorphic curves representing $B$. By
Lemma~\ref{lem:dMatchedCompactify}, ends of these moduli spaces are
either two-story matched curves; of they correspond to
matched combs, with height two, of two types (with join
curves or orbit curves at East infinity). These two types of ends
cancel in pairs according to Lemma~\ref{lem:Join}; so the total
number of two-story ends is even, and those count the $\mathbf y_1\#\mathbf y_2$
coefficient of
$\partial^{(0)}\circ \partial^{(0)}(\mathbf x_1\#\mathbf x_2)$.
\end{proof}
We identify $(C,\partial^{(0)})$ with the Heegaard-Floer chain
complex $\CFKsimp(\HD)$ associated to the doubly-pointed Heegaard
diagram $\HD$, via the following adaptation
of~\cite[Theorem~9.10]{InvPair}:
\begin{thm}
\label{thm:NeckStretch}
Let $\HD$ be a Heegaard diagram representing $K$, equipped with a
decomposition $\HD=\Hdown\cup_Z \Hup$ as a union of an upper and a
lower diagram along $2n$ circles, with $n>1$.
For suitable choices of almost-complex structures $J$ used to define
$\CFKsimp(K)$, there is an isomorphism (of chain complexes)
$\CFKsimp(K)\cong (C,\partial^{(0)})$.
\end{thm}
\begin{proof}
This follows from Lipshitz's reformulation of Heegaard Floer
homology (\cite[Theorem~2]{LipshitzCyl}) and
Proposition~\ref{prop:StretchNeck}.
\end{proof}
\subsection{Self-matched curves}
\label{subsec:SelfMatched}
\begin{defn}
\label{def:SelfMatched}
Let $\Source_1$ be a decorated source. Partition
let $\IntPunctEv(\Source_1)$ resp. $\IntPunctOdd(\Source_1)$
denote the set of interior punctures of $\Source_1$
that are labelled by even resp. odd orbits.
A {\em self-marked source} is a decorated
source, together with an injection $\phi\colon \IntPunctEv(\Source_1)\to
\East(\Source_1)$ with the property that if $p\in\IntPunctEv(\Source_1)$
is marked by some orbit $\orb_j$, then $\phi(p)$ is marked by a length
one chord that covers the boundary component $Z_k$ with
$\{j,k\}\in \Mup$. A {\em self-matched curve} $u$ is an element
$u\in\ModFlow^{B_1}(\mathbf x,\mathbf y;\Source_1,\phi)$, subject to the following additional
constraints: for each puncture $p\in\IntPunctEv(\Source_1)$,
\begin{equation}
\label{eq:SelfMatching}
t\circ u(p)=t\circ u(\phi(p)).
\end{equation}
\end{defn}
Note although ``self-matched curves'' are supported in $\Hdown$, the
matching condition depends on $\Hup$, through its induced matching
$\Mup$.
\begin{defn}
\label{def:SelfMatchedCurvePair}
A {\em self-matched curve pair} consists of the following data:
\begin{itemize}
\item a self-matched source
$(\Source_1,\phi\colon \IntPunctEv(\Source_1)\to \East(\Source_1))$
\item a marked source $\Source_2$
\item an injection $\psi\colon \AllPunct(\Source_2)\to
\East(\Source_1)$, where
$\AllPunct(\Source_2)=\IntPunct(\Source_2)\cup\East(\Source_2)$
\item a pseudo-holomorphic self-matched curve $u_1$ with source
$\Source_1$
in $\Hdown$ and
\item a pseudo-holomorphic curve $u_2$ with source $\Source_2$ in $\Hup$,
\end{itemize}
satisfying the following properties
\begin{enumerate}[label=(smpc-\arabic*),ref=(smcp-\arabic*)]
\item
\label{smcp:1to1}
The map $\psi$ is a one-to-one correspondence between
$\East(\Source_2)$ and $\East(\Source_1)\setminus
\phi(\IntPunctEv(\Source_2))$.
\item If $p\in \IntPunct(\Source_2)$, is labelled by an
orbit $\orb_j$, then $\phi(p)$ is labelled by
a length one Reeb chord that covers the boundary component of $\Zin_j$.
\item If $p\in\East(\Source_2)$, then
the name of the Reeb chord in $\Hup$ marking $p$
is the same as the name of the Reeb chord in $\Hdown$ marking $\psi(p)$.
\item For each puncture $q\in \AllPunct(\Source_2)$:
\begin{equation}
\label{eq:SelfMatchedCurvePair}
t\circ u_1(\psi(q))=t\circ u_2(q).
\end{equation}
\end{enumerate}
Let
$\ModMatchedChanged^{B_1,B_2}(\mathbf x,\mathbf y;\Source_1,\Source_2,\phi,\psi)$
denote the moduli space of self-matched curve pairs.
\end{defn}
Note that the even orbits in $\Source_1$ are constrained by the
self-matching condition; and the odd orbits in $\Source_1$ are not constrained by any additional condition.
Let $(u_1,u_2)$ be a self-matched curve pair with sources $\Source_1$
and $\Source_2$. Let $o_1^+$ resp. $o_1^-$ denote the number of even
resp. odd orbits in $\Source_1$; let $c_i$ denote the number of
boundary punctures in $\Source_i$. For example, note that
$c_1=c_2+o_2+o_1^+$.
Given homology classes $B_1\in\pi_2(\mathbf x_1,\mathbf y_1)$ and
$B_2\in\pi_2(\mathbf x_2,\mathbf y_2)$, we construct a corresponding class
$B=B_1\# B_2\in\pi_2(\mathbf x_1\#\mathbf y_1,\mathbf x_2\#\mathbf y_2)$, as follows. First, we
obtain a homology class $B_2'$ from $B_2$ by summing, for each
puncture in $\Source_1$ labelled by some orbit $\orb_j$ with $f(j)=2k-1$,
all the components of $\Sigma_2\setminus \betas$ that contain boundary
components $\Zout_\ell$ with $f(\ell)\leq 2k$. Similarly, we obtain a homology
class $B_1'$ from $B_1$ by adding, for each puncture in $\Source_1$
labelled by $\orb_j$ with $f(j)=2k-1$, all the components of
$\Sigma_1\setminus\betas$ that contain boundary components $\Zin_\ell$
with
$f(\ell)\leq 2k$.
Let $B_1\# B_2=B_1'\natural B_2'$.
We formalize now the expected dimension of the moduli space of pairs
of curves, with the time constraints coming from
Equations~\eqref{eq:SelfMatching} and~\eqref{eq:SelfMatchedCurvePair}.
\begin{defn}
For a self-matched curve pair
with
$\chi(\Source_i)=\chiEmb(B_i)$, define the {\em index of the self-matched curve pair} by
\[ \ind^{\sharp}(B_1,\Source_1;B_2,\Source_2)
= \ind(B_1,\Source_1)+\ind(B_2,\Source_2)-c_1 \]
\end{defn}
\begin{prop}
For a self-matched curve pair $(u_1,u_2)$, as above the domain
$B_1\sharp B_2$ has the following properties:
\begin{align*}
B_1\sharp B_2&\in \doms(\mathbf x_1\#\mathbf x_2,\mathbf y_1\#\mathbf y_2) \\
n_\wpt(B_1\sharp B_2)&=n_\wpt(B_1)+o^-_1 \\
n_\zpt(B_1\sharp B_2)&=n_\zpt(B_1) \\
\end{align*}
Moreover, if $\chi(\Source_i)=\chiEmb(B_i)$ for $i=1,2$, then
\[ \ind(B_1\sharp B_2)=\ind^{\sharp}(B_1,\Source_1;B_2,\Source_2).\]
\end{prop}
\begin{proof}
This is a straightforward consequence of the fact that if ${\mathcal
D}$ is any of the components in $\Sigma_i\setminus\betas$ then
$\ind(B_i+{\mathcal D})=\ind(B_i)+2$.
\end{proof}
Let $\mathbf x=\mathbf x_1\#\mathbf x_2$ and $\mathbf y=\mathbf y_1\#\mathbf y_2$.
Define
\begin{align*} \ModMatchedChanged^B&(\mathbf x,\mathbf y)\\
&=
\bigcup_{\left\{\begin{tiny}\begin{array}{r}
B_1\in \pi_2(\mathbf x_1,\mathbf y_1) \\B_2\in\pi_2(\mathbf x_2,\mathbf y_2)
\end{array}\end{tiny}\Big|~B=B_1\sharp B_2\right\}}
\bigcup_{\{(\Source_1,\Source_2,\phi,\psi)\big| \ind^{\sharp}(B_1,\Source_1;B_2,\Source_2)=\ind(B)\}}
\ModMatchedChanged^{B_1,B_2}(\mathbf x,\mathbf y,\phi,\psi).
\end{align*}
We use these moduli spaces to construct an endomorphism of $C$, specified by its values
$\mathbf x=\mathbf x_1\#\mathbf x_2$ by
\begin{align} \dChanged &(\mathbf x)
&=\sum_{\mathbf y}
\sum_{\{B\in\pi_2(\mathbf x,\mathbf y)|\ind(B)=1\}}
\#\left(\frac{\ModMatchedChanged^B(\mathbf x,\mathbf y)}{\R}\right)\cdot U^{n_\wpt(B)}V^{n_\zpt(B)}
\cdot \mathbf y.
\label{eq:dChanged}
\end{align}
\begin{prop}
\label{prop:dChangedSqZero}
The endomorphism $\dChanged$ satisfies the identity $\dChanged\circ \dChanged=0$.
\end{prop}
As usual, the proof involves understanding the ends of one-dimensional
moduli spaces of self-matched curve pairs. These ends contain terms
counted in $\dChanged\circ\dChanged$; and all other terms cancel. We
give the proof after some preliminary results; see Figures~\ref{fig:JJandXO}, ~\ref{fig:OXoddWX},
and~\ref{fig:OXevenXW} for pictures.
\begin{figure}[h]
\centering
\input{JJandXO.pstex_t}
\caption{{\bf Ends of Type~\ref{SMCP:XO} cancel ends of type~\ref{SMCP:JJ}.}
This is a combination of Lemmas~\ref{lem:JJ} and~\ref{lem:XO}.}
\label{fig:JJandXO}
\end{figure}
\begin{figure}[h]
\centering
\input{OXoddWX.pstex_t}
\caption{{\bf Ends of Type~\ref{SMCP:OX} with an odd orbit cancel against ends of Type~\ref{SMCP:WX}.}
This cancellation is a combination of Lemmas~\ref{lem:OXodd} and~\ref{lem:WX}.}
\label{fig:OXoddWX}
\end{figure}
\begin{figure}[h]
\centering
\input{OXevenXW.pstex_t}
\caption{{\bf Ends of Type~\ref{SMCP:OX} with an even orbit cancel against ends of Type~\ref{SMCP:XW}.}
This cancellation is a combination of Lemmas~\ref{lem:OXeven} and~\ref{lem:XW}.}
\label{fig:OXevenXW}
\end{figure}
\begin{defn}
If ${\overline\Source}$ is the source of a holomorphic story
$(w_\ell,\dots,w_1,u,v_1,\dots,v_m)$, let
$\IntPunctOdd({\overline\Source})$ resp
$\IntPunctEv({\overline\Source})\subset \IntPunct({\overline\Source})$
denote the subset of punctures in ${\overline\Source}$ marked by
Reeb orbits which are odd resp. even. A {\em self-matched story}
consists of the following data:
\begin{itemize}
\item a holomorphic story ${\overline u}=(w_\ell,\dots,w_1,u,v_1,\dots,v_m)$
\item an injective map $\phi\colon \IntPunctEv({\overline\Source})\to
\East({\overline\Source})$ with the property that if
$p\in {\overline\Source}$ is a puncture marked by some orbit
$\orb_j$, then $\phi(p)$ is marked by some chord $\longchord_k$
that covers $Z_k$ with multiplicity one, so that $\{j,k\}\in
\Mup$.
\end{itemize}
\end{defn}
\begin{defn}
A {\em self-matched story pair}
consists of the following data:
\begin{itemize}
\item a self-matched holomorphic story ${\overline u}_1$ in $\Hdown$,
with source ${\overline\Source}_1$
\item a holomorphic story ${\overline u}_2$ in $\Hup$,
with source ${\overline\Source}_2$
\item a one-to-one correspondence
\[\psi\colon \AllPunct({\overline\Source}_2)\to
\East({\overline\Source}_1)\setminus
\phi(\IntPunct({\overline \Source}_1),\]
\end{itemize}
satisfying the following properties:
\begin{itemize}
\item if $q\in \IntPunct({\overline\Source}_2)$ is marked by an orbit
$\orb_j$ (in $\Hup$), then $\psi(q)$ is marked by one of the two
length $1$ chords that covers $\Zin_j$
\item if $q\in\East({\overline\Source}_2)$,
the Reeb chord in $\Hup$ label on $q$ has the same
name as the Reeb chord in $\Hdown$ that marks
$\psi(q)$
\item for each $q\in \AllPunct(\Source_2)$,
\[ t\circ {\overline u}_1(\psi(q))=
t\circ {\overline u}_2(q).\]
\end{itemize}
\end{defn}
\begin{prop}
\label{prop:SMCP-Ends}
Suppose that $\ind(B_1,\Source_1;B_2,\Source_2)=2$.
Every comb pair appearing in the boundary of the moduli space
$\ModMatchedChanged^B(\mathbf x,\mathbf y;\Source_1,\Source_2)$
is of one of the following types:
\begin{enumerate}[label=(sME-\arabic*),ref=(sME-\arabic*)]
\item
\label{SMCP:2Story}
a two-story self-matched curve pair
\item
\label{SMCP:JoinCurve}
a self-matched story pair of the form $(u_1,v_1),u_2$,
where $v_1$ is a join curve
\item
\label{SMCP:XO}
a self-matched story pair of the form $u_1,(u_2,v_2)$
where $v_2$ is an orbit curve
\item\label{SMCP:OX} a self-matched story pair $(u_1,v_1),u_2$ where
$v_1$ is an orbit curve
\item
\label{SMCP:JJ}
a self-matched story pair of the form $(u_1,v_1),(u_2,v_2)$
where $v_1$ and $v_2$ are join curves, and the corresponding
punctures in $u_1$ and $u_2$ are the two distinct length one Reeb
chords that cover the same boundary component.
\item
\label{SMCP:WX}
a self-matched story pair of the form $(w_1,u_1),u_2$,
where $w_1$ is a simple boundary degeneration
\item
\label{SMCP:XW}
a self-matched story pair of the form $u_1,(w_2,u_2)$,
where $w_2$ is a simple boundary degeneration.
\end{enumerate}
\end{prop}
\begin{proof}
Suppose that we have a sequence of self-matched curve pairs
with fixed sources $\Source_1$ and $\Source_2$, representing
fixed homology classes $B_1$ and $B_2$.
The index of these is computed nby
\begin{align*}
\ind^{\sharp}(\Source_1,\Source_2)&=
\ind(B_1;\Source_1)+\ind(B_2;\Source_2)-o_+-c_2-o_2\\
&= d_1+n_{\mathbf x_1}(B_1)+n_{\mathbf y_1}(B_1) + e(B_1)-2\weight^1_\partial \\
&\qquad
+ d_2+n_{\mathbf x_2}(B_2)+n_{\mathbf y_2}(B_2)
+e(B_2)-2\weight^2_\partial+2o_-+o_++o_2+c_1,
\end{align*}
where $o_+$ resp. $o_-$ denotes the number of punctures
in $\Source_1$ marked by even resp. odd orbits; $c_2$ denotes the number of East
punctures in $\Source_2$ and $o_2$ the number of interior punctures.
Take a Gromov limit, and assume that it does not contain any curves at West infinity,
and that it consists of a singly story.
Consider the main component $(u_1',u_2')$,
with sources $\Source_1'$ and $\Source_2'$. This limit inherits
certain matching conditions, whose expected dimension we will now
express.
Let $\lambda_+$ denote the number of East punctures
that arise as limits of East marked orbits in $\Source_1$.
Let $\lambda_-$ denote the number of East punctures
that are not matched with any other punctures in either $\Source_i'$.
(These punctures arise as limits of interior punctures in $\Source_1$
marked by odd orbits.)
Let $c_2'$ denote the number of East punctures in
$\Source_2'$ and $o_2'$ denote the number of its interior punctures.
Each
interior puncture of $\Source_1'$ marked by an even orbit is
constrained to lie at the same $t$-level as some corresponding East
puncture $\Source_1'$; and each puncture of $\Source_2'$ is
constrained to lie at the same $t$-level as a corresponding East
puncture in $\Source_1'$. Thus, the expected dimension of the moduli
space in which $(u_1',u_2')$ lives is computed by
\begin{align*}
\ind(u_1',u_2')&\leq
\ind(B_1;\Source_1')+\ind(B_2;\Source_2')-o_+'-\lambda_+-c_2'-o_2' \\
&= d_1+n_{\mathbf x_1}(B_1)+n_{\mathbf x_2}(B_1)+e(B_1)-2\weight^1_\partial \\
&\qquad +d_2+ n_{\mathbf x_1}(B_1)+n_{\mathbf x_2}(B_1)+e(B_2)-2\weight^2_\partial+c_1'+o_+'+2o_-'+o_2',
\end{align*}
Thus,
\[ \ind(u_1',u_2')-\ind^\sharp(\Source_1,\Source_2)
\leq (c_1'-\lambda_--\lambda_+-c_1)+(o_+'-o_+)+(2o'_-+\lambda_--2o_-)+(o_2'-o_2).\]
Each quantity in parentheses is clearly non-positive. Since
we assumed that $\ind^\sharp(\Source_1,\Source_2)=2$, at most one of the quantities
can be $-1$ (and the others zero) for $(u_1',u_2')$ to exist.
If $2o_+'+\lambda_--2o_-=-1$, then $\lambda_-=1$. In this case,
an (odd) orbit curve forms in $\Source_1$ (Case~\ref{SMCP:OX}). On the other hand,
if $2o_+'+\lambda_--2o_-=0$, then $\lambda_-=0$.
Also, $o_+'+\lambda_+\leq o_+$, so if $o'_+-o_+=-1$, then
$\lambda_+=0$ or $1$. Assume first that $\lambda_+=0$. In this
case, $\Source_1'$ must contain a multiple even orbit. In that case,
it is also the case that $c_1'<c_1$, which is a
contradiction. Assume next that $\lambda_+=1$. In that case,
$c_1'=c_1+1$. This is the case of an (even) orbit curve appearing in
$\Source_1$ (Case~\ref{SMCP:OX}, again).
If $c_1'-c_1=-1$, there are two possibilities. A join curve can form
on $\Source_1$ without one on $\Source_2$
(Case~\ref{SMCP:JoinCurve}), or it can form on both sides
(Case~\ref{SMCP:JJ}).
If $o_2'-o_2=-1$, we have an orbit curve forming on $\Source_2$ (Case~\ref{SMCP:XO}).
This finishes the cases where curves at West infinity do not form.
Suppose that the limiting curve $(u',v')$ contains weight $2k$
boundary degeneration with $2m$ distinct orbits in it.
Then, $(u',v')$ lies in a moduli space whose index is computed by
\[ \ind(B_1,\Source_1)+\ind(B_1',\Source_1')-(c_1-2k)-(2m-1):\]
there are $c_1-2k$ height constraints coming from chords in
$\Source_1'=\Source_1$ that are not matched with orbits in the boundary
degeneration, and the remaining group of $2m$ chords are required to occur at
the same $t$-value.
\begin{align*}
\ind(B_1,\Source_1')&\leq \ind(B_1,\Source_1) -c_1+c_1'=\ind(B_1,\Source_1)-2(k-m) \\
\ind(B_2',\Source_2')&\leq \ind(B_2,\Source_2) -2k.
\end{align*}
Thus,
\[ \ind(u_1',u_2')\leq \ind^{\sharp}-2k+1.\]
Thus, for the limiting object to be non-empty, we require $k=1$,
i.e. the boundary degeneration is simple; this is
Case~\ref{SMCP:XW}. More generally, each boundary degeneration
level carries codimension at least $1$, so it follows that no more
than one boundary degeneration can occur, and if it does, then there
are no other curves at East infinity.
Suppose next that there is a weight $2k$ boundary degeneration West
infinity on the $\Source_1$-side; let $a$ denote the total weight of
the orbits, and let $m$ denote the number
of chords in $\Source_1'$ that are matched with the boundary degeration.
Suppose that $m>0$.
Then, $(u_1',u_2')$ lives in a moduli space with expected dimension
\[ \ind(u_1',u_2')=\ind(B_1',\Source_1')+\ind(B_2,\Source_2')-c_1'+1,\]
since there are $c_1'-m$ matching comditions coming from the chords
that do not go into the boundary degeneration, and the remaining $m$
chords are required to lie at the same height, imposing $m-1$
further constraints. It is straightforward to see that
\begin{align*}
n_{\mathbf x_1}(B_1')+n_{\mathbf y_1}(B_1')+e(B_1')&=n_{\mathbf x_1}(B_1)+n_{\mathbf y_1}(B_1)+e(B_1)-4k \\
\weight^1_\partial(B_1')&=\weight^1_{\partial}(B_1) -2k\\
c_1'&\leq c_1-a+m \\
o_1'&\leq o_1-a \\
\weight^1_\partial(B_1')&=\weight^1_{\partial}(B)-a
\end{align*}
so $\ind(u_1',u_2')\leq \ind^{\sharp}+ 1-2k$.
Thus, $k>1$ forces $(u_1',u_2')$ to be in an empty moduli space;
the case $k=1$ is allowed, and it is
Case~\ref{SMCP:WX}.
We turn our attention to $m=0$.
In this case,
\[ \ind(u_1',u_2')=\ind(B_1',\Source_1')+\ind(B_2,\Source_2')-c_1',\]
and the boundary degeneration is forced to contain exactly one odd
orbit: it is a special boundary degeneration with weight $2k=2a$.
In this case, computing as above, we find that $\ind(u_1',u_2')\leq \ind^{\sharp}-2k$. The
moduli space is once again empty if $k>1$. When $k=1$, there is a
special case where it is non-empty: when $(u_1',u_2')$ is
constant. This case also falls under Case~\ref{SMCP:WX}.
\end{proof}
Let $(u_1,v_1),(u_2,v_2)$ be a self-matched story pair where $v_1$ and
$v_2$ are join curves, as in Case~\ref{SMCP:JJ}. Then, $(u_1,u_2)$
are self-matched pairs. Similarly, if $u_1,(u_2,v_2)$ is a
self-matched story pair where $v_2$ is an orbit curve as in
Case~\ref{SMCP:OX}, then $(u_1,u_2)$ are also self-matched pairs. In
these cases, we call $(u_1,u_2)$ the {\em trimming} of the
corresponding limit curve.
\begin{lemma}
\label{lem:JJ}
The number of ends of Type~\ref{SMCP:JJ} coincides with the number
of self-matched pairs $(u_1,\Source_1,u_2,\Source_2,\phi,\psi)$
for which $\East(\Source_2)$
consists of one length $1$ chord and all other chords have length
$1/2$.
\end{lemma}
\begin{proof}
Let $\ModMatchedX(\mathbf x,\mathbf y,\Source_1;\Source_2)$ be the moduli space of
self-matched curve pairs, where exactly one of the boundary punctures $q$ of
$\Source_2$ is labelled by a length $1$ chord, which we denote $\longchord$ (and all others
boundary punctures are marked by length $1/2$ chords). This moduli space
embeds in a larger moduli space
${\widetilde\ModMatchedX}$, where we drop the condition
from Equation~\eqref{eq:SelfMatchedCurvePair} for the puncture $q\in \Source_2$ marked
by the length $1$ chord. This moduli space in turn admits an evaluation map
\[\ev_q-\ev_{\psi(q)}\colon \widetilde\ModMatchedX \to \R \]
with $0$ as a regular value, whose preimage is $\ModMatchedX(\mathbf x,\mathbf y,\Source_1;\Source_2)$.
Given $u=(u_1,u_2)\in \ModMatchedX(\mathbf x,\mathbf y;\Source_1,\Source_2)$, let ${\widetilde U}\subset \widetilde\ModMatchedX$
be a neighborhood so that $\ev_q-\ev_{\psi(q)}\colon {\widetilde U}\to (-\epsilon,\epsilon)$ is a diffeomorphism
(i.e. so that $u$ is the preimage of $0$).
Let $\Source_1'=\Source_1\natural_{\psi(q)} \EastSource_1$ and $\Source_2'=\Source_2\natural_q \EastSource_2$.
Thus, $\Source_2$ comes with an extra pair of punctures $\{q_1,q_2\}$ (labelled by
chords which can be joined to form $\longchord$).
Let $\BigModMatched$ be the moduli space like $\ModMatchedChanged(\mathbf x,\mathbf y,\Source_1',\Source_2')$,
except now that at the two distinguished punctures $\{q_1,q_2\}$ in $\Source_2'$ coming from the east infinity
curve, we do not require the time constraint from Equation~\eqref{eq:SelfMatchedCurvePair}.
There is a map $F=(F_1,F_2,F_3)\colon \BigModMatched \to \R^3$, with components
\[
F_1=t\circ \ev_{\psi(q_1)}-t\circ \ev_{q_1}, \qquad
F_2=t\circ \ev_{q_1}-t\circ \ev_{q_2} \qquad
F_3=t\circ \ev_{\psi(q_1)}-t\circ \ev_{\psi(q_2)},
\]
with the property that if $\Delta\subset \R^2$ denotes the diagonal, th
Type~\ref{SMCP:JJ}
end of $\ModMatchedChanged(\mathbf x,\mathbf y,\Source_1',\Source_2')$
at $(u_1,\Source_1,u_2,\Source_2,\phi,\psi)$.
Gluing the east infinity curves to $u_1$ and $u_2$, we get a gluing map
\[ \gamma \colon {\widetilde U}\times (0,\epsilon)\times (0,\epsilon) \to \BigModMatched.\]
The gluing map continuously extends to a map
from ${\widetilde U}\times [0,\epsilon)\times [0,\epsilon)$
to the Gromov compactification of $\BigModMatched$
so that for all $r_1,r_2>0$,
\begin{align*}
\gamma(u_1\times u_2\times \{0\}\times \{0\})&=((u_1,v_1),(u_2,v_2)) \\
F_1\circ \gamma|_{{\widetilde U}\times \{0\}\times \{0\}}&=
{\ev_{\psi(q)}-\ev_{q}}|_{\widetilde U} \\
F_2\circ \gamma({\widetilde U}\times \{r_1\}\times \{0\})&=0 \\
F_2\circ \gamma({\widetilde U}\times \{r_1\}\times \{r_2\})&>0 \\
F_3\circ \gamma({\widetilde U}\times \{0\}\times \{r_2\})&=0 \\
F_3\circ \gamma({\widetilde U}\times \{r_1\}\times \{r_2\})&>0.
\end{align*}
It follows that $(F\circ \gamma)^{-1}(\{0\}\times \Delta)$ has a
single endpoint over the origin, and that is the point
$((u_1,v_1),(u_2,v_2))$, giving the stated correspondence
betwen ends and self-matched curves stated in the lemma.
\end{proof}
\begin{lemma}
\label{lem:XO}
The number of ends of Type~\ref{SMCP:XO} coincides with the number
of self-matched pairs for which $\East(\Source_2)$ consists of one length $1$ chord
and all other chords have length $1/2$.
\end{lemma}
\begin{proof}
Let $\ModMatchedX(\mathbf x,\mathbf y,\Source_1,\Source_2)$, ${\widetilde\ModMatchedX}$, and $U$ be as in the proof of Lemma~\ref{lem:JJ}.
Gluing the orbit curve to $\Source_2$ now gives a gluing map
\[ \gamma\colon U \times (0,\epsilon)\to \ModMatchedChanged(\mathbf x,\mathbf y,\Source_1,\Source_2'),\]
where now $\Source_2'=\Source_2\natural\EastSource$ is obtained by gluing on an orbit curve, which we denote here by $v_2$.
(Note that $\Source_2'$ and $v_2$ mean different objects than they did in the proof of Lemma~\ref{lem:JJ}.)
Gluing extends continuously to the Gromov compactification, giving
\[ \gamma\colon U \times [0,\epsilon)\to \overline\ModMatchedChanged(\mathbf x,\mathbf y,\Source_1,\Source_2'),\]
so that for all $r>0$,
\begin{align*}
\gamma(u_1,\times u_2\times \{0\})&=(u_1,(u_2,v_2)) \\
\ev_{\longchord_i}-\ev_{\orb_i'}|_{\gamma(U\times \{0\})}&=\ev_{\longchord_i}-\ev_{\longchord_i'} \\
s\circ \ev_{\orb_i'}(\gamma(u,0)&=1 \\
s\circ \ev_{\orb_i'}(\gamma(u,r)&<1.
\end{align*}
There is a neighborhood $U'$ of $(u_1,(u_2,v_2))$ in ${\overline\ModMatchedChanged}(\mathbf x,\mathbf y,\Source_1,\Source_2')$ so that
\[ U'\cap\ModMatchedChanged(\mathbf x,\mathbf y;\Source_1,\Source_2')=(\ev_{\orb_i}-\ev_{\orb'_i})^{-1}(0),\]
so that $s\circ \ev_{\orb_i'}\colon U'\to (1-\epsilon,1]$ is a proper map near $1$.
It follows that the ends
$U'\cap\ModMatchedChanged(\mathbf x,\mathbf y;\Source_1,\Source_2')=(\ev_{\orb_i}-\ev_{\orb'_i})^{-1}(0)$
is modelled on the preimage at $s=1$.
\end{proof}
\begin{defn}
\label{def:PartialSMCP}
Let $X$ be a set consisting of one or two Reeb chords. A
{\em{$X$-partially self-matched curve pair }} (or $X$-partial-smcp) is the data $(\Source_1,\phi,\Source_2,\psi)$ as
in Definition~\ref{def:SelfMatchedCurvePair}, except
Condition~\ref{smcp:1to1} is replaced by the following:
\begin{itemize}
\item
$\East(\Source_1)$ is partitioned into three disjoint sets:
\[ \phi(\IntPunctEv(\Source_1))\qquad \psi(\East(\Source_2))\qquad X',\]
where $X'$ is a set with $|X'|=|X|$, and the labels on the punctures of $X'$
are specified in $X$.
\end{itemize}
Moreover, if $|X|=2$, let $X'=\{p_1,p_2\}$. In this case, we also require
\begin{equation}
\label{eq:NearMatchedConstraint}
t\circ u_1(p_1)=t\circ u_1(p_2).
\end{equation}
Let $\NearModMatched{X}{B}(\mathbf x,\mathbf y)$ denote the moduli space of $X$-partial smcps,
with homology class $B$.
\end{defn}
Let $(u_1,v_1),u_2$ be a self-matched story pair where $v_1$ is an
orbit curve; and let $\longchord$ be the chord labelling the boundary
puncture of $v_1$. If the orbit in $v_1$ is odd, then $(u_1, u_2)$ is
a $\{\longchord\}$-partial smcp. If the orbit is even, then
$(u_1,u_2)$ is a $X$-partial smcp, where $X$ consists of
two length one chords
that cover $\Mup$-matched boundary components. Similarly, if
$(w_1,v_1),u_2$ is a self-matched story pair where $w_1$ is a boundary
degeneration, the curves $u_1$ and $u_2$ is a partial smcp with one
remaining chord, and if $u_1,(w_2,u_2)$ is a self-matched story pair
where $w_2$ is a boundary degeneration, then $(u_1,u_2)$ is a
$\{\longchord\}$-partial smcp, where $\longchord$ is a length one chord
that covers some boundary component. In all the above cases, we
call $(u_1,u_2)$ the {\em{trimming}} of the self-matched story pair.
The next few lemmas show that the number of ends are determined by the
trimmings of the limiting configurations.
\begin{lemma}
\label{lem:OXodd}
Suppose that $\orb_j$ is an odd orbit, and let $\longchord_j$ be the
chord of length one that covers the corresponding boundary component
$\Zin_j$. The number of curves in
$\NearModMatched{\{\longchord_j\}}{B}(\mathbf x,\mathbf y;\Source_1,\Source_2)$
has the same parity as the number of ends of
$\ModMatchedChanged^B(\mathbf x,\mathbf y;\Source_1',\Source_2')$ of
Type~\ref{SMCP:OX}, where $v_1$ is an orbit curve that has a
boundary puncture marked by $\orb_j$.
\end{lemma}
\begin{proof}
This follows from the gluing
\[ \gamma \colon
\NearModMatched{\longchord_j}{B}(\mathbf x,\mathbf y;\Source_1',\Source_2) \times (0,\epsilon) \to
\ModMatched^B(\mathbf x,\mathbf y,\Source_1,\Source_2), \]
gluing the orbit curve to $\Source_1'$ at $\longchord_j$,
which parameterizes the end with $s(u(q))\goesto 0$,
where here $q$ denotes the $\orb_j$-marked puncture;
cf. Proposition~\ref{prop:OrbitCurve}.
\end{proof}
The following is a variant of Proposition~\ref{prop:BoundaryDegenerationNbd}:
\begin{lemma}
\label{lem:WX}
Suppose that $\{j,k\}\in\Mup$, where $f(j)$ is odd. Let $\longchord_j$
be a chord of length one that covers an boundary component $\Zin_j$.
The number of curves in
$\NearModMatched{\{\longchord_j\}}{B}(\mathbf x,\mathbf y;\Source_1,\Source_2)$ has
the same parity as the number of ends of
$\ModMatchedChanged^B(\mathbf x,\mathbf y;\Source_1',\Source_2')$ of
Type~\ref{SMCP:WX}, where $w_1$ is a (smooth) simple boundary degeneration
that contains $\orb_j$.
\end{lemma}
\begin{proof}
Let $\Source_1'=\WestSource\natural\Source_1$.
This curve contains three distinguished punctures, $q_1$ and $q_2$ marked by orbits
(coming from $\WestSource$), labelled so that $q_2$ is the even orbit; and a third puncture $q_3$ that
marked by a length $1$ chord, with $q_3=\phi(q_2)$.
(In the notation of the lemma statement, $q_3$ is labelled by $\longchord_j$.)
Consider a moduli space ${\BigModMatched}$ containing $\ModMatchedChanged(\mathbf x,\mathbf y,\Source_1',\Source_2)$,
where we drop the height constraint from Equation~\eqref{eq:SelfMatching} for the orbit $q_2$ and its corresponding
chord $q_3=\phi(q_2)$. Thus, there is an evaluation map
\[ t\circ \ev_{q_1}-t\circ \ev_{q_2}\colon \BigModMatched\to \R \]
so that $0$ is a regular value, and whose preimage is identified with
$\ModMatchedChanged(\mathbf x,\mathbf y,\Source_1',\Source_2)$.
We will consider a map $F=(F_1,F_2)\colon \BigModMatched \to \R^2$ whose components are given by
\[ F_1= t\circ \ev_{q_1}-t\circ \ev_{q_2}\qquad F_2=t\circ \ev_{q_1}-t\circ\ev_{q_3}.\]
This map extends continuously to the Gromov compactification of $\BigModMatched$.
Fix $((u_1,w_1),u_2)$ in the Gromov compactification.
Gluing gives a map
\[ \gamma\colon \ModFlow\times_{\Tb}
\ModWest(\WestSource)\times (-\epsilon,\epsilon)\times (0,\epsilon)\to \BigModMatched.\]
Here, the $(-\epsilon,\epsilon)$ specifies the $t$-coordinate where the gluing
is performed, and $[0,\epsilon)$ represents the gluing scale.
We restrict this to $(u_1,u_2)\times w_1\in \ModFlow\times_{\Tb}\ModDeg$,
and denote the resulting map
\[ \gamma\colon (-\epsilon,\epsilon)\times(0,\epsilon)\to \BigModMatched.\]
Consider the two punctures $q_1$ and $q_2$ in $\WestSource$
Since $w_1$ is generic, $t\circ w_1(q_1)\neq t\circ w_1(q_2)$.
Assume $t \circ w_1(q_1)>t\circ w_1(q_2)$.
The map
\[ (t\circ \ev_{q_1}-t\circ \ev_{q_3}) \circ\gamma(\cdot,0) \colon(-\epsilon,\epsilon)\to (-\epsilon,\epsilon) \]
has odd degree. By our assumption, for all $r>0$,
\[ F_1\circ \gamma(r,t)>0.\]
Clearly, also
\[ F_1(0,t)=0.\]
Thus,
\[ F\circ \gamma\colon (-\epsilon,\epsilon)\times [0,\epsilon) \to \R^{\geq 0}\times \R \]
is proper map of degree $1$ to points in $\R^{\geq 0}\times \R$ near the origin.
The moduli space consists of points in the preimage of $\R^{>0}\times \{0\}$, a smooth one-manifold whose end consists of the preimage
of the origin.
The same argument applies when $t\circ w_1(q_1)<t\circ w_1(q_2)$, with minor modifications.
\end{proof}
\begin{lemma}
\label{lem:OXeven}
Suppose that $\orb_j$ is an even orbit, and let
$\longchord_j$ be a chord of length one that covers a the corresponding boundary component $\Zin_j$,
and $\longchord_k$ be a chord of length one that covers $\Zin_k$, with $\{j,k\}\in \Mup$.
The number of curves in
$\NearModMatched{\{\longchord_j,\longchord_k\}}{B}(\mathbf x,\mathbf y;\Source_1,\Source_2)$
has the same parity as the number of ends of
$\ModMatchedChanged^B(\mathbf x,\mathbf y;\Source_1',\Source_2')$
of Type~\ref{SMCP:OX}, where $v_1$ is an orbit curve with boundary puncture marked by $\longchord_j$.
\end{lemma}
\begin{proof}
Let ${\widetilde\ModMatchedX}$ denote the moduli space containing
$\NearModMatched{\{\longchord_j,\longchord_k\}}{B}(\mathbf x,\mathbf y,\Source_1,\Source_2)$,
of data $(u_1,u_2,\Source_1,\phi,\Source_2,\psi)$ satisfying the
conditions from Definition~\ref{def:PartialSMCP}, except that for
the two distinguished punctures $q_1$ and $q_2$ on $\Source_1$
labelled by $\longchord_j$ and $\longchord_k$ (not contained
contained in $\phi(\IntPunctEv(\Source_1))$ or
$\psi(\East(\Source_2))$), we no longer require that $t\circ
u_1(q_1)=t\circ u_1(q_2)$; i.e. we drop the constraint from
Equation~\eqref{eq:NearMatchedConstraint}.
We have a map
\[t\circ \ev_{q_1}-t\circ \ev_{q_2}\colon {\widetilde\ModMatchedX}\to \R \]
so that $0$ is a regular value, and
\[ (t\circ \ev_{q_1}-t\circ \ev_{q_2})^{-1}(0)= \NearModMatched{\{\longchord_j,\longchord_k\}}{B}(\mathbf x,\mathbf y,\Source_1,\Source_2).\]
Each $(u_1,u_2)\in \NearModMatched{\{\longchord_j,\longchord_k\}}{B}(\mathbf x,\mathbf y,\Source_1,\Source_2)$
neighborhood ${\widetilde U}\in{\widetilde M}$, so that
$F_1\colon {\widetilde U} \to (-\epsilon,\epsilon)$
is a homeomorphism.
Let $\BigModMatched$ be the moduli space like ${\widetilde\ModMatchedX}$,
except now $q_1$ is marked by the orbit $\orb_j$ rather than $\longchord_j$.
In particular,
we have the map
\[ F_1=t\circ\ev_{q_1}-t\circ \ev_{q_2}\colon \BigModMatched\to \R \]
with regular value $0$, so that
\[ F_1^{-1}(0)=\ModMatchedChanged^B(\mathbf x,\mathbf y,\Source_1',\Source_2').\]
Let $F_2=s\circ \ev_{q_1}$, we have a map
\[ F=(F_1,F_2)\colon \BigModMatched\to \R\times \R^{< 1}. \]
Gluing on the orbit curve
at $q_1$ gives a map $\gamma\colon {\widetilde U}\times [0,r) \to
\ModMatchedChanged^B(\mathbf x,\mathbf y,\Source_1',\Source_2')$.
We have that
\[ F_1\circ \gamma \colon {\widetilde U}\times (0)\to (-\epsilon,\epsilon) \]
has degree $1$ near $0$; and for all ${\widetilde u}\in {\widetilde U}$, $r\in (0,\epsilon)$.
\begin{align*}
F_2\circ \gamma({\widetilde u},0)&=1 \\
F_2 \circ \gamma({\widetilde u},r)&<1
\end{align*}
Thus,
\[ F\circ \gamma({\widetilde U}\times [0,\epsilon)) \to \R\times \R^{\leq 1} \]
is proper of degree $1$ for points near $(0,1)$ in $\R\times \R^{\leq 1}$.
It follows at once that the smooth manifold
\[
\ModMatchedChanged^B(\mathbf x,\mathbf y,\Source_1',\Source_2')=F_1^{-1}(\{0\}) \]
has one end over the point $(0,1)\in \R\times \R^{\leq 1}$.
\end{proof}
\begin{lemma}
\label{lem:XW}
Suppose that $\{j,k\}\in\Mup$, and let $\longchord_j$ and
$\longchord_k$ be chords of length one that cover the boundary
components $\Zin_j$ and $\Zin_k$ respectively. The number of curves in
$\NearModMatched{\{\longchord_j,\longchord_k\}}{B}(\mathbf x,\mathbf y;\Source_1,\Source_2)$ has
the same parity as the number of ends of
$\ModMatchedChanged^B(\mathbf x,\mathbf y;\Source_1,\Source_2')$ of
Type~\ref{SMCP:XW}, where $w_2$ is a simple boundary degeneration
that contains $\orb_j$ and $\orb_k$.
\end{lemma}
\begin{proof}
Let ${\widetilde\ModMatchedX}$ be as in Lemma~\ref{lem:OXeven}.
Let $\BigModMatched$ be the moduli space containing the moduli space
of self-matched curve pairs
$\ModMatchedChanged^{B}(\mathbf x,\mathbf y,\Source_1,\Source_2')$ as in
Definition~\ref{def:SelfMatchedCurvePair}, except that now there are
two distinguished (interior) punctures $q_1,q_2$ in $\Source_2$
marked by orbits $\orb_j$ and $\orb_k$ respectively, where we do not
impose the corresponding height constraints (from
Equation~\eqref{eq:SelfMatchedCurvePair}). Note that $u_1$
represents the homology class $B_1$ and $u_2$ represents the
homology class $B_2+{\mathcal D}$, where ${\mathcal D}$ is the
elementary domain $\Sigma_2$ containing $\Zout_j$ and $\Zout_k$.
Consider the map
$F=(F_1,F_2,F_3) \colon \BigModMatched\to \R^3$ whose three components are
the are evaluation maps
\begin{align*}
F_1&=t\circ \ev_{\psi(q_1)}-t\circ \ev_{q_1} \\
F_2&=t\circ \ev_{\psi(q_1)}-t\circ \ev_{\psi(q_2)}\\
F_3&=t\circ \ev_{q_1}-t\circ \ev_{q_2}.
\end{align*}
Clearly,
\[ \ModMatchedChanged^{B_1,B_2+{\mathcal D}}(\mathbf x,\mathbf y,\Source_1,\Source_2')=F^{-1}(\Delta\times \{0\}), \]
where $\Delta\subset \R^2$ is the diagonal.
For sufficiently small open subsets ${\widetilde U}\subset
{\widetilde \ModMatchedChanged}$, gluing gives a map
\[ \gamma\colon (-\epsilon,\epsilon)\times {\widetilde \ModMatched}\times_{\Tb} \ModDeg({\mathcal D})\times
[0,\epsilon) \to \BigModMatched, \]
where the $(-\epsilon,\epsilon)$ factor
specifies the $t$-coordinate where the
gluing is performed, and the $[0,\epsilon)$ represents the gluing scale.
The fibered product over $\Tb$ is taken with respect to an evaluation map
$\ev_t \colon {\ModFlow}\to \Tb$, defined by
\[ \ev_t(u_1)={u_1}^{-1}(0,t)\in \Tb; \] and the
(degree one) evaluation map $\ev^\beta\colon \ModWest\to \Tb$.
Assume that $\ev^{\beta}(w_1)$ is a regular value, so that there
is an open neighborhood $W$ of $w_2$ so that $\ev\colon W \to \Tb$
is a local diffeomorphism. Restricting to some sufficiently small
neighborhood ${\widetilde U}\subset {\widetilde \ModFlow}$, we
can guarantee that for $t\in (-\epsilon,\epsilon)$,
$\ev_t({\widetilde U}) \subset \ev(W)$, so
${\widetilde U}\times_{\Tb} W \cong {\widetilde U}$.
Further shrinking ${\widetilde U}$ if needed, we can assume that
\[ t\circ \ev_{q_1}-t\circ \ev_{q_2} \colon {\widetilde U} \to
(-\epsilon,\epsilon) \] is a homeomorphism, so that the preimage of
$0$ is the nearly self-matched curve pair $(u_1,u_2)$. We
abbreviate the gluing map (suppressing the choice of $W$), writing
instead
\[ \gamma\colon {\widetilde U}\times (-\epsilon,\epsilon)\times
(0,\epsilon)\to
{\BigModMatched}.\]
This map has a natural extension to
${\widetilde U}\times (-\epsilon,\epsilon)\times
[0,\epsilon)$ to the Gromov compactification of ${\BigModMatched}$.
For any fixed ${\widetilde u}\in {\widetilde U}$, $r\in (0,1)$,
\[ F_1\colon \gamma(\cdot,{\widetilde u},r)
\colon (-\epsilon,\epsilon)\to (-\epsilon,\epsilon) \]
has degree one, since the same statement holds when $r=0$.
Also, for any fixed $t\in (-\epsilon,\epsilon)$,
\[ F_2\circ \gamma(t,\cdot,r)
\colon {\widetilde U}\to (-\epsilon,\epsilon)\]
has degree one, since the same statement holds setting $r=0$
Finally,
\[
F_3(t,{\widetilde u},0)= 0\qquad{\text{and}}\qquad
F_3(t,{\widetilde u},r)> 0.
\]
It follows that
\[ F\circ \gamma \colon (-\epsilon,\epsilon)\times
{\widetilde\ModMatched}\times [0,\epsilon) \to
\R\times \R \times \R^{\geq 0}\] has degree $1$ near the origin; and the smooth
manifold $F^{-1}(\Delta\times \{0\})$ has one end over the origin.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:dChangedSqZero}]
As usual, we consider index two moduli spaces. Consider their ends,
as in Proposition~\ref{prop:SMCP-Ends}.
Combining Lemma~\ref{lem:JJ} and \ref{lem:XO}, it follows that
the count of ends of Type~\ref{SMCP:JJ} cancels with
the ends of Type~\ref{SMCP:XO}. Combining Lemma~\ref{lem:WX} and
\ref{lem:OXodd}, it follows that ends of of Type~\ref{SMCP:WX}
cancel with ends of Type~\ref{SMCP:OX}, where the orbit curve is
odd, as illustrated in Figure~\ref{fig:OXoddWX}. Combining
Lemma~\ref{lem:OXeven} and~\ref{lem:XW}, it follows that ends of
Type~\ref{SMCP:XW} cancel with ends of Type~\ref{SMCP:OX} where the
orbit curve is even, as illustrated in Figure~\ref{fig:OXevenXW}.
The ends that are not accounted for are of Type~\ref{SMCP:2Story};
and these ends count the $\mathbf y$ coefficient of
$\dChanged\circ\dChanged$.
\end{proof}
\subsection{Intermediate complexes}
\label{subsec:Intermediates}
Note that the self-matched compatible pairs from
Definition~\ref{def:SelfMatchedCurvePair} use the orbits curves
quite differently from Definition~\ref{def:MatchedPair}; and so the
chain complex defined using the two objects
$(C,\partial^{(0)})$ and $(C,\partial_{\natural})$ seem quite different.
To construct a homopy equivalence between these complexes, we will use a
sequence of intermediate complexes, defined here.
Fix an integer $\ell$ and a marked source $\Source$. Let
$\IntPunct^{\leq \ell}(\Source)\subset \IntPunct(\Source)$ denote the
subset of those punctures $q\in \Source$ that are marked by orbits
$\orb_j$ with $f(j)\leq \ell$. Define $\IntPunct^{\ell}(\Source)$ and
$\IntPunct^{>\ell}(\Source)$ analogously
\begin{defn}
\label{def:IntermediateComplexes}
Let $\ell$ be an integer between $0,\dots,2n$.
An $\ell$-self-matched curve pair is the following data.
\begin{itemize}
\item a holomorphic curve $u_1$ in $\Hdown$, with source $\Source_1$
\item a holomorphic curve $u_2$ in $\Hup$, with source $\Source_2$
\item an injection
$\phi\colon \IntPunctEv^{\leq \ell}(\Source_1) \to \East(\Source_1)$
\item an injection $\psi\colon \East(\Source_2)\to\East(\Source_1)$
\end{itemize}
with the following properties:
\begin{itemize}
\item $\East(\Source_1)$ is a union of three disjoint sets,
$\phi(\IntPunctEv^{\leq \ell}(\Source_1))$, $\psi(\East(\Source_2))$, and
$\IntPunct^{\leq \ell}(\Source_1)$
\item If $p\in \IntPunctEv^{\leq \ell}(\Source_1)$ is labelled by some orbit $\orb_j$,
then $\phi(p)$ is marked by a length one Reeb chord that covers
the boundary component of $\Zin_k$, where $\{j,k\}\in\Mup$, and
\[ t \circ u_1(\phi(p))=t\circ u_2(p). \]
\item If $q\in \East^{\leq \ell}(\Source_2)$ , then $\psi(q)$ is labelled by a
length one Reeb chord that covers the boundary component of
$\Zin_j$, and
\[ t\circ u_1(\psi(q))=t\circ u_2(q). \]
\item If $q\in \East^{> \ell}(\Source_2)\cup\East(\Source_2)$,
then the marking on the Reeb chord or orbit
of $q$ is the same as the marking on the Reeb chord or orbit of
$\psi(q)$, and
\[ (s\circ u_1(\psi(q)),t \circ u_1(\psi(q)))=
(s \circ u_2(q),t \circ u_2(q)).\]
\end{itemize}
Let $\ModInt{\ell}^{B_1,B_2}(\mathbf x,\mathbf y,\Source_1,\Source_2,\phi,\psi)$ denote the
moduli space of $\ell$-morphism matched curves.
\begin{align*} \ModMor{\ell}^B&(\mathbf x,\mathbf y)\\
&=
\bigcup_{\left\{\begin{tiny}\begin{array}{r}
B_1\in \pi_2(\mathbf x_1,\mathbf y_1) \\B_2\in\pi_2(\mathbf x_2,\mathbf y_2)
\end{array}\end{tiny}\Big|~B=B_1\natural B_2\right\}}
\bigcup_{\{(\Source_1,\Source_2,\phi,\psi)\big| \ind^{\natural}(B_1,\Source_1;B_2,\Source_2)=\ind(B)\}}
\ModMor{\ell}^{B_1,B_2}(\mathbf x,\mathbf y,\phi,\psi).
\end{align*}
\end{defn}
\begin{defn}
Fix $\ell\in \{0,\dots,2n\}$.
Consider $C$ equipped with the endomorphism
determined by
\[ \dInt{\ell}(\mathbf x)=\sum_{\mathbf y} \sum_{\{B\in\doms(\mathbf x,\mathbf y)\big| \ind(B)=1\}}
\# \left(\frac{\ModInt{\ell}^B(\mathbf x,\mathbf y)}{\R}\right)\cdot \mathbf y \]
\end{defn}
\begin{remark}
Note that $\dInt{0}$ is the operator from
Equation~\eqref{eq:DefD0}; and $\dInt{2n}$ is the operator
$\dChanged$ is the operator
from Equation~\eqref{eq:dChanged}
\end{remark}
Unlike the earlier cases, a sequence of $\ell$-self-matched curve
pairs can have a Gromov limit to a pair of curves
$((w_1,u_1),(w_2,u_2))$ where both $w_1$ and $w_2$ are simple boundary
degenerations, so that each puncture in $w_2$ has a corresponding
puncture in $w_1$, which is marked by the same orbit. This can happen
in the special case where the odd orbit $\orb_j$ in $w_2$ has $f(j)=\ell$.
We will formulate the end counts in terms of the following types of curves
(which naturally arise in Gromov limits of $\ell$-self-matched curve pairs):
\begin{defn}
\label{def:IntermediateComplexesx}
Let $\rho$ be a Reeb chord in $\Sigma_1$, and
let $\ell$ be an integer between $0,\dots,2n$.
An {\em $\ell$-self-matched curve pair with remaining $\{\rho\}$} is the following data.
\begin{itemize}
\item a holomorphic curve $u_1$ in $\Hdown$, with source $\Source_1$
\item a holomorphic curve $u_2$ in $\Hup$, with source $\Source_2$
\item an injection
$\phi\colon \IntPunctEv^{\leq \ell}(\Source_1)\to \East(\Source_1)$
\item an injection $\psi\colon \East(\Source_2)\to\East(\Source_1)$
\end{itemize}
with the following properties:
\begin{itemize}
\item $\East(\Source_1)$ is a union of four disjoint sets,
\[ \phi(\East^{\leq \ell}_+(\Source_1)), \qquad
\psi(\East(\Source_2)\cup\East(\Source_2)), \qquad
\East^{<\ell}_-(\Source_1), \qquad \{q_0\}, \]
where $q_0$ is a puncture labelled by the Reeb chord $\rho$.
\item If $p\in \IntPunctEv^{\leq \ell}(\Source_1)$ is labelled by some orbit $\orb_j$,
then $\phi(p)$ is marked by a length one Reeb chord that covers
the boundary component of $\Zin_k$, where $\{j,k\}\in\Mup$, and
\[ t \circ u_1(\phi(p))=t\circ u_2(p). \]
\item If $q\in \East^{\leq \ell}(\Source_2)\cup \East(\Source_2)$,
then $\psi(q)$ is labelled by a
length one Reeb chord that covers the boundary component of
$\Zin_j$, and
\[ t\circ u_1(\psi(q))=t\circ u_2(q). \]
\item If $q\in \East^{> \ell}(\Source_2)$,
then the marking on the Reeb chord or orbit
of $q$ is the same as the marking on the Reeb chord or orbit of
$\psi(q)$, and
\[ (s\circ u_1(\psi(q)),t \circ u_1(\psi(q)))=
(s \circ u_2(q),t \circ u_2(q)).\]
\end{itemize}
We denote the moduli space of these data by
$\NearModMatched{\ell,\{\rho\}}{B}(\mathbf x,\mathbf y,\Source_1,\Source_2)$.
\end{defn}
\begin{figure}[h]
\centering
\input{OXWW.pstex_t}
\caption{Ends of Type~\ref{SMCP:OX}
where the orbit is labelled by $\orb_j$ with
$f(j)=\ell$ is odd,
cancel against curves with boundary degenerations on both sides.}
\label{fig:OXWW}
\end{figure}
\begin{lemma}
\label{lem:WW}
Let $j$ be so that $f(j)=\ell$ is odd.
The number of
curves in
$\NearModMatched{\ell;\{\longchord_j\}}{B}(\mathbf x,\mathbf y;\Source_1,\Source_2)$
has the same parity as the number of ends of
$\ModInt{\ell}^{B'}(\mathbf x,\mathbf y;\Source_1',\Source_2')$ of
the form $((u_1,w_1),(u_2,w_2))$, where
$\Source_1'=\WestSource_1\natural \Source_1$,
$\Source_2'=\WestSource_2\natural\Source_2$, $w_1$ and $w_2$ are
simple boundary degenerations with sources $\WestSource_1$ and
$\WestSource_2$ both of which contain a puncture marked with
$\orb_j$, and
$B'_i=B_i+{\mathcal D}_i$, where ${\mathcal D}_i$ is the shadow of
$w_i$ for $i=1,2$.
\end{lemma}
\begin{proof}
Let $q_1$ and $q_2$ be the two punctures on $\Source_2'$ coming from
$w_2$, labelled so that $q_1$ is labelled by the odd Reeb orbit and
$q_2$ by the even one. In particular, $\psi(q_1)$ is a puncture on
$\Source_1'$ labelled by $\longchord_j$.
There is a
moduli space ${\BigModMatched}$ which is like
$\NearModMatched{\ell;\{\longchord_j\}}{B}(\mathbf x,\mathbf y,\Source_1',\Source_2')$, except we now drop the
conditions that
\[ t\circ u_1(\psi(q_1))=t\circ u_2(q_1)\qquad{\text{and}}\qquad
t\circ u_1(\psi(q_2))=t\circ u_2(q_2). \]
Thus, we have a map
\[ F=(F_1,F_2,F_3,F_4)\colon \BigModMatched\to \R\times \R\times (0,1)\times (0,1) \]
with components
\begin{align*}
F_1&=t\circ u_1(\psi(q_1))-t\circ u_2(q_1),\\
F_2&=t\circ u_1(\psi(q_2))-t\circ u_2(q_2),\\
F_3&=s\circ u_1(\psi(q_2)), \\
F_4&=s\circ u_2(q_2)
\end{align*}
so that $F^{-1}(0\times \Delta)=\NearModMatched{\ell;\{\longchord_j\}}{B}(\mathbf x,\mathbf y,\Source_1',\Source_2')$.
Fix $(u_1,u_2)\in\NearModMatched{\ell;\{\longchord_j\}}{B}(\mathbf x,\mathbf y;\Source_1,\Source_2)$.
Gluing $w_1$ and $w_2$ to $u_1$ and $u_2$ gives a map
\[ \gamma\colon \times (-\epsilon,\epsilon)\times
(0,\epsilon)\times (-\epsilon,\epsilon)\times (0,\epsilon)\to
\BigModMatched.\] (Note we are now gluing to both sides, giving two
time parameters and two scale parameters.)
The gluing map extends
to a map from $(-\epsilon,\epsilon)\times
[0,\epsilon)\times (-\epsilon,\epsilon)\times [0,\epsilon)$
to the Gromov compactification of $\BigModMatched$.
This extension satisfies the following properties,
for all $t_1,t_2\in(-\epsilon,\epsilon)$ and $r_1,r_2\in(0,\epsilon)$:
\begin{itemize}
\item
$F_1\circ \gamma(\cdot, r_1, t_1
r_2)\colon (-\epsilon,\epsilon)\to (-\epsilon,\epsilon)$
has degree one, since the same holds for $r_1=0$.
\item
$F_2\circ \gamma(t_1, r_1, \cdot,
r_2)\colon (-\epsilon,\epsilon)\to (-\epsilon,\epsilon)$
has degree one, since the same holds for $r_2=0$.
\item
$F_3(t_1,\cdot,t_2,r_2)\colon [0,\epsilon)\to (0,1]$
has degree one near $1$.
\item
$F_4(t_1,r_1,t_2,\cdot)\colon [0,\epsilon)\to (0,1]$
has degree one near $1$.
\end{itemize}
It follows at once that
$(F\circ \gamma)^{-1}(\{0\}\times \{0\}\times \Delta)$ is a one-manifold with a single
end over the origin.
\end{proof}
\begin{prop}
The endomorphism $\dInt{\ell}$ satisfies
$\dInt{\ell}\circ\dInt{\ell}=0$.
\end{prop}
\begin{proof}
This is a straightforward synthesis of Propositions~\ref{prop:dMatchedZqZero}
and~\ref{prop:dChangedSqZero}. Again, we look at two-dimensional
moduli spaces. We find that their ends are of the following types:
\begin{itemize}
\item Two-story $\ell$-self-matched curve pairs.
\item Type~\ref{def:JJ}.
\item Type~\ref{def:OO}, if the orbit curves are marked by $\orb_j$ with $j\not\in\Omega_k$
\item Type~\ref{SMCP:XO}, if the odd orbit $\orb_j$ has $f(j)\leq \ell$.
\item Type~\ref{SMCP:OX}, where the orbit curve on the left is marked by an odd orbit $\orb_j$,
with $f(j)\leq \ell$.
\item Type~\ref{SMCP:WX}.
\item Type~\ref{SMCP:XW}.
\item $\Omega$-matched story pair ends, of the form $(w_1,u_1),(w_2,u_2)$,
which occurs when $\ell$ is odd and
both $w_1$ and $w_2$ contain the puncture with $f$-value equal to $\ell+1$.
\end{itemize}
Ends of Type~\ref{def:OO} and~\ref{SMCP:XO} cancel with ends of
Type~\ref{def:JJ} by Lemma~\ref{lem:Join}
when the orbit is has $f$-value greater than $\ell$; and a combination
of Lemmas~\ref{lem:JJ} combined with Lemma~\ref{lem:XO} otherwise.
Similarly, ends of type~\ref{SMCP:OX} where the orbit is an odd orbit in $\Omega_k$
cancel with those of Type~\ref{SMCP:WX}, as in Lemmas~\ref{lem:OXodd} and~\ref{lem:WX}.
Ends of Type~\ref{SMCP:OX} where the orbit is even and and has $f$-value less than or equal to $\ell$
cancel with those of Type~\ref{SMCP:XW} where the orbit has $f$-value less than ore equal to $\ell$.
By Lemma~\ref{lem:WW} and~\ref{lem:OXodd},
the remaining ends of the form $(w_1,u_1),(w_2,u_2)$
cancel with
ends of Type~\ref{SMCP:OX} where the orbit is odd, has and has
$f$-value equal to $\ell$; see Figure~\ref{fig:OXWW}.
\end{proof}
Consider $C$ equipped with the endomorphism
\begin{align*}
\partial^{(k)} &(\mathbf x)
&=\sum_{\mathbf y}
\sum_{\{B\in\pi_2(\mathbf x,\mathbf y)|\ind(B)=1\}}
\#\left(\frac{\ModInt{\ell}^B(\mathbf x,\mathbf y)}{\R}\right)\cdot U^{n_\wpt(B)}V^{n_\zpt(B)}
\cdot \mathbf y.
\end{align*}
\subsection{Interpolating between the intermediate complexes}
\label{subsec:Interpolate}
In this section we prove the following:
\begin{prop}
\label{prop:Intermediates}
When $n>1$,
there is an isomorphism of chain complexes over
$\Ring$
\[ \Phi_{\ell}\colon (C,\partial^{(\ell)})\to (C,\partial^{(\ell+1)})\]
\end{prop}
The map $\Phi_\ell$ will be constructed by counting curves which generalize
the $\Omega$-matched curves, as follows.
\begin{defn}
\label{def:MorMatch}
Let $\ell$ be an integer between $1,\dots,2n$.
An $\ell$-morphism matched curve, is the following data.
\begin{itemize}
\item a holomorphic curve $u_1$ in $\Hdown$, with source $\Source_1$
\item a holomorphic curve $u_2$ in $\Hup$, with source $\Source_2$
\item a subset $X\subset \East^{\ell}_+(\Source_1)$,
(i.e. which is empty if $\ell$ is odd)
\item a subset $Y\subset \East^{\ell}_-(\Source_1)$
\item an injection
$\phi\colon \IntPunctEv^{<\ell}(\Source_1)\cup X \to \East(\Source_1)$
\item an injection $\psi\colon \East(\Source_2)\to\East(\Source_1)$
\item a real number $t_0\in \R$
\end{itemize}
with the following properties:
\begin{itemize}
\item $\East(\Source_1)$ is a union of four disjoint sets,
$\phi(\East^{<\ell}_+(\Source_1)\cup X)$, $\psi(\East(\Source_2))$,
$\East^{<\ell}_-(\Source_1)$, and $Y$.
\item $X$
contains all punctures $q\in \East^{\ell}_+(\Source_1)$
with $t\circ u_1(q)>t_0$
\item $Y$ contains all punctures $q\in \East^{\ell}_-(\Source_1)$
with $t\circ u_1(q)<t_0$
\item If $p\in \IntPunctEv^{<\ell}(\Source_1)\cup X$ is labelled by some orbit $\orb_j$,
then $\phi(p)$ is marked by a length one Reeb chord that covers
the boundary component of $\Zin_k$, where $\{j,k\}\in\Mup$, and
\[ t \circ u_1(\phi(p))=t\circ u_2(p). \]
\item If $q\in \East^{<\ell}(\Source_2)$ or $q\in\East^{\ell}(\Source_2)$
and $t \circ u_2(q)<t_0$, then $\psi(q)$ is labelled by a
length one Reeb chord that covers the boundary component of
$\Zin_j$, and
\[ t\circ u_1(\psi(q))=t\circ u_2(q) \]
\item If $q\in \East^{> \ell}(\Source_2)$ or
$q\in \East^{\ell}(\Source_2)$ and $t \circ u_2(q)>t_0$,
then the marking on the Reeb chord or orbit
of $q$ is the same as the marking on the Reeb chord or orbit of
$\psi(q)$, and
\[ (s\circ u_1(\psi(q)),t \circ u_1(\psi(q)))=
(s \circ u_2(q),t \circ u_2(q)).\]
\item
If $q\in \East^{\ell}(\Source_2)$ and $t\circ u_2(q)=t_0$,
then $\psi(q)$ is also labelled by the same Reeb orbit, and
\[ t\circ u_1(\psi(q))=t\circ u_2(q)=t_0.\]
Moreover, the following inequalities hold on the $s$ projection.
Order the punctures $q\in \East^{\ell}(\Source_2)$ with $t\circ u_1(q)=t_0$
$\{q_i\}_{i=1}^m$ so that the sequence $\{s\circ u_2(q_i)\}_{i=1}^m$ is
increasing; then
\[ s\circ u_2(q_i)< s\circ u_1(\psi(q_i)) \]
and, if $i<m$,
\[ s\circ u_1(\psi(q_i))<s\circ u_2(q_{i+1}).\]
\end{itemize}
Each $\ell$-morphism matched curve has three associated integers,
$m_-$, $m$, and $m_+$, where $m_-$ resp. $m_+$ denotes the number of punctures $q$
in $\East^{\ell}(\Source_2)$ with $t(q)<t_0$ resp. $t(q)>t_0$;
and $m$ (as above) is the number of punctures
in $\East^{\ell}(\Source_2)$ with $t(q)=t_0$.
The triple $(m_-,m,m_+)$ is called the {\em profile} of the $\ell$-morphism
matched curve.
Let $\ModMor{\ell}(\mathbf x,\Source_1,\Source_2,\phi,\psi)$ denote the
moduli space of $\ell$-morphism matched curves.
\begin{align*} \ModMor{\ell}^B&(\mathbf x,\mathbf y)\\
&=
\bigcup_{\left\{\begin{tiny}\begin{array}{r}
B_1\in \pi_2(\mathbf x_1,\mathbf y_1) \\B_2\in\pi_2(\mathbf x_2,\mathbf y_2)
\end{array}\end{tiny}\Big|~B=B_1\natural B_2\right\}}
\bigcup_{\{(\Source_1,\Source_2,\phi,\psi)\big| \ind^{\natural}(B_1,\Source_1;B_2,\Source_2)=\ind(B)\}}
\ModMor{\ell}^{B_1,B_2}(\mathbf x,\mathbf y,\phi,\psi).
\end{align*}
\end{defn}
\begin{defn}
Define a map $h_\ell\colon (C,\partial^{(\ell)})\to (C,\partial^{(\ell+1)})$
by the formula
\begin{align*}
h_\ell &(\mathbf x)
&=\sum_{\mathbf y}
\sum_{\{B\in\pi_2(\mathbf x,\mathbf y)|\ind(B)=1\}}
\#\ModMor{\ell}(\mathbf x,\mathbf y)\cdot U^{n_\wpt(B)}V^{n_\zpt(B)}
\cdot \mathbf y.
\end{align*}
\end{defn}
\begin{lemma}
\label{lem:ChainMap}
\[ \partial^{(\ell+1)}\circ h_\ell + h_\ell\circ \partial^{(\ell)}
= \partial^{(\ell+1)}+\partial^{(\ell)}. \]
\end{lemma}
\begin{proof}
Consider ends of one-dimensional moduli spaces
$\ModMor{\ell}(\mathbf x,\mathbf y)$.
There are two cases, according to the parity of $\ell$. Suppose that
$\ell$ is odd.
There are ends as involving punctures other than the ones whose
$t$-projection is $t_0$. These ends are as in the intermediate
complexes in Proposition~\ref{prop:Intermediates}. Many of these ends cancel in pairs
in the proof of that proposition, leaving the two-story buildings,
and the ends that involve the special $t_0$-level.
Those ends in turn can be classified, as follows. Let
$\{q_1,\dots,q_m\}\in \East^{\ell}(\Source_2)$ be the punctures with
$t\circ u_1(q_i)=t_0$, labelled as in Definition~\ref{def:MorMatch}.
\begin{enumerate}[label=($\flat$-\arabic*),ref=($\flat$-\arabic*)]
\item \label{end:OffRight} The end corresponds to $s(u_1(q_m))\goesto 1$;
in this case, there is a Gromov limit
to $((u_1,v_1),u_2)$,
where $v_1$ is an orbit curve (for an orbit whose $f$-value is $\ell$),
attached at the level $t_0$.
These ends are labelled by integers $(m_-,m-1,m_+)$,
where $m_-$ and $m_+$ are defined as in Definition~\ref{def:MorMatch}.
\item \label{end:OffLeft}
The end corresponds to $s(u_2(\psi(q_1)))\goesto 0$;
in this case, there is a Gromov limit
to $((w_1,u_1),(w_2,v_2))$, where
$w_1$ and $w_2$ are simple boundary degenerations,
both of which contain an orbit $\orb_j$
with $f(j)=\ell$.
These ends are labelled by integers $(m_-,m-1,m_+)$.
\item
\label{end:HorizCollision}
Pairs $(u_1,u_2)$ with $s(u_1(q_i))=s(u_2(\psi(q_i)))$
or $s(u_2(\psi(q_i)))= s(u_1(q_{i+1}))$.
These ends are labelled $(m_-,m,m_+,j)$ where
with the convention $j=2i-1$, if $s(u_1(q_i))=s(u_2(\psi(q_i)))$;
and $j=2i$ if $s(u_2(\psi(q_i)))= s(u_1(q_{i+1}))$.
\item
\label{end:OrbFromBelow}
there is some $q\in \East^{\ell}(\Source_2)$
with $t(u_2(q))=t_0$, but $q$ arises as a limit point
of punctures with $t(u_2(q))<t_0$.
Let $s_0=s(u_1(\psi(q)))=s(u_2(q))$.
These ends are labelled $(m_-,m,m_+,j)$
where
\begin{equation}
j =\left\{\begin{array}{ll}
1 & {\text{if $s_0< s\circ u_1(q_1)$}} \\
2i-1 & {\text{if $s\circ u_2(\psi(q_{i-1}))<s_0 <s\circ u_2(q_i)$}} \\
2i &{\text{if $s\circ u_2(q_i)< s_0< s\circ u_1(\psi(q_{i+1}))$}} \\
2m &{\text{if $s\circ u_1(q_{m})< s_0$}}
\end{array}\right.
\label{eq:DefOfj}
\end{equation}
Let $m_+$ here be one greater than the number of
$q\in\East^{\ell}(\Source_2)$ with $t(u_2(q))>t_0$;
this is $m_+$ for the curves before taking the Gromov limit.
\item \label{end:OrbFromAbove} there is some $q\in \East^{\ell}(\Source_2)$
with $t(u_2(q))=t_0$, but $q$ arises as a limit point
of punctures with $t(u_2(q))> t_0$.
These ends are labelled $(m_-,m,m_+,j)$,
where $j$ is defined as in Equation~\eqref{eq:DefOfj};
where now $m_-$ is computed before taking the Gromov limit.
\item \label{end:OrbFromAboveA} there is some $q\in \East^{\ell}(\Source_1)$
with $t(u_1(q))=t_0$, arising as a limit point
of punctures with $t(u_1(q))> t_0$.
These ends are labelled $(m_-,m,m_+,j)$,
where $j$ is defined as in Equation~\eqref{eq:DefOfj};
and $m_-$ is computed before taking the Gromov limit.
\end{enumerate}
\begin{figure}[h]
\centering
\input{MorMod1.pstex_t}
\caption{{\bf Ends of Types~\ref{end:OrbFromBelow} and~\ref{end:HorizCollision}
cancel}
We have drawn here the strip: the light dots represent the images under the
projection to $[0,1]\times \R$ of the punctures on $\Source_1$;
the dark ones represent the punctures on $\Source_2$.
\label{fig:MorMod1}}
\end{figure}
End of Type~\ref{end:HorizCollision} $(m_-,m,m_+,j)$
cancel with end of Type~\ref{end:OrbFromBelow} $(m_-,m-1,m_++1,j)$
except when $m=1$; see Figure~\ref{fig:MorMod1}.
Ends of Type~\ref{end:OrbFromAbove} $(m_-,m,m_+,j)$ cancel in pairs
except in the special case where $j=m$.
\begin{figure}[h]
\centering
\input{MorMod2.pstex_t}
\caption{{\bf Ends of Type~~\ref{end:OffRight}
cancel certain ends of Type\ref{end:OrbFromAbove},}
when the ennd of Type~\ref{end:OrbFromAbove} has the form
$(m_-,m,m_+,m)$.
\label{fig:MorMod2}}
\end{figure}
\begin{figure}[h]
\centering
\input{MorMod3.pstex_t}
\caption{{\bf Ends of Types~\ref{end:OrbFromAboveA} and~\ref{end:OffLeft}
cancel}
\label{fig:MorMod3}}
\end{figure}
Ends of Type~\ref{end:OffRight} $(m_-,m,m_+)$
cancel with ends of Type~\ref{end:OrbFromAbove}
$(m_-+1,m-1,m_+,m-1)$ except when $m=1$.
Ends of Type~\ref{end:OffLeft} $(m_-,m,m_+)$ cancel with ends of
Type~\ref{end:OrbFromAboveA} $(m_-+1,m-1,m_+)$
except when $m=1$.
The remaining ends are: Type~\ref{end:HorizCollision} $(m_-,1,m_+,1)$,
Type~\ref{end:OffRight} $(m_-,1,m_+)$, and Type~\ref{end:OffLeft}
$(m_-,1,m_+)$. Now, provided that $m_->0$,
ends of Type~\ref{end:HorizCollision} $(m_-,1,m_+,1)$
correspond to ends of Type~\ref{end:OffRight} and~\ref{end:OffLeft}
$(m_--1,1,m_++1)$.
After these further cancellations, the remaining ends are
of Type~\ref{end:HorizCollision} $(0,1,m_+,1)$ -- which
correspond to the terms in $\partial^{(\ell)}$ -- and ends of type
Type~\ref{end:OffRight} and~\ref{end:OffLeft}
$(m_-,1,0)$ -- which correspond to the terms in $\partial^{(\ell+1)}$.
The remaining two-story buildings count terms
in $\partial^{(\ell+1)}\circ h_{\ell}+h_{\ell}\circ \partial^{(\ell)}$,
verifying that
$\partial^{(\ell+1)}\circ h_{\ell}+h_{\ell}\circ \partial^{(\ell)}=0$ when
$\ell$ is odd.
This discussion requires slight modifications in case $\ell$ is even.
Let
$\{q_1,\dots,q_m\}\in \East^{\ell}(\Source_2)$ be the punctures with
$t\circ u_1(q_i)=t_0$, labelled as in Definition~\ref{def:MorMatch}.
\begin{enumerate}[label=($\flat'$-\arabic*),ref=($\flat'$-\arabic*)]
\item \label{end:eOffRight} The end corresponds to $s(u_1(q_m))\goesto 1$;
in this case, there is a Gromov limit
to $((u_1,v_1),u_2)$,
where $v_1$ is an orbit curve (for an orbit whose $f$-value is $\ell$),
attached at the level $t_0$.
\item \label{end:eOffLeft}
The end corresponds to $s(u_2(\psi(q_1)))\goesto 0$;
in this case, there is a Gromov limit
to $((w_1,u_1),u_2)$, where
$w_1$ is a simple boundary degeneration containing
an orbit $\orb_j$
with $f(j)=\ell$,
and $u_2$ contains an extra unmatched chord $\longchord_k$,
which covers the boundary component $\Zdown_k$ so that $f(k)=\ell-2$.
\item
\label{end:eHorizCollision}
Pairs $(u_1,u_2)$ with $s(u_1(q_i))=s(u_2(\psi(q_i)))$
or $s(u_2(\psi(q_i)))= s(u_1(q_{i+1}))$.
\item
\label{end:eOrbFromBelow}
There is some $q\in \East^{\ell}(\Source_2)$
with $t(u_2(q))=t_0$, but $q$ arises as a limit point
of punctures with $t(u_2(q))<t_0$.
\item \label{end:eOrbFromAboveA} There is some $q\in \East^{\ell}(\Source_1)$
with $t(u_1(q))=t_0$, arising as a limit point
of punctures with $t(u_1(q))> t_0$.
Note that in this case there as an extra puncture $q'$ (the limit of $\phi(q)$)
with $t(q')=t_0$ on $u_1$ labelled by $\longchord_k$, where
$f(k)=\ell-1$.
\end{enumerate}
With the above remarks in place, verification of the stated relation
when $\ell$ is even proceeds much as before. The most significant
difference is that the cancellation of ends of Type~\ref{end:eOrbFromAboveA}
with those of Type~\ref{end:eOffLeft} is slightly simpler than the corresponding
cancellation in the odd case; see Figure~\ref{fig:eMorMod}.
\begin{figure}[h]
\centering
\input{eMorMod.pstex_t}
\caption{{\bf Ends of Types~\ref{end:eOrbFromAboveA} and~\ref{end:eOffLeft}
cancel}
\label{fig:eMorMod}}
\end{figure}
\end{proof}
\subsection{Time dilation}
\label{subsec:TimeDilation}
Consider $(C,\dChanged)$. The differential $\dChanged$ counts self-matched curve pairs.
As in~\cite{InvPair}, we deform the matching appearing. In our case,
we deform the constraints appearing in the definition
of a self-matched curve pair (Equation~\eqref{eq:SelfMatchedCurvePair})
by conditions indexed by a real parameter $T$, as follows:
\[ T\cdot t\circ u_1(\psi(q))=t\circ u_2(q). \]
The corresponding moduli spaces are denoted
$\ModMatchedChanged(T;\mathbf x,\mathbf y)$, which we call the {\em moduli space of $T$-modified
paritally self-matched curve pairs}.
Taking the limit as $T\goesto\infty$, the curves converge to combs,
whose algebraic information is contained in their main components.
These limiting objects are natural analogues of the ``trimmed simple ideal matched curves'' from~\cite[Definition~9.31]{InvPair}:
\begin{defn}
\label{def:tsic}
A {\em trimmed simple ideal partially self-matched curve} is a pair of
holomorphic combs $(u_1,u_2)$ connecting two Heegaard states
generators $\mathbf x=\mathbf x_1\#\mathbf x_2$ and $\mathbf y=\mathbf y_1\#\mathbf y_2$, where $u_1$ is a
self-matched curve (in the sense of
Definition~\ref{def:SelfMatched}), equipped with a a one-to-one
correspondence $\varphi\colon \East(\Source_1)\setminus
\phi(\IntPunctEv(\Source_1))\to\East(\Source_2)$ such that
either
one of $u_1$ or $u_2$ is trivial, and the other has index $1$,
and
$\East(\Source_1)\setminus\phi(\IntPunctEv(\Source_1))$ and $\East(\Source_2)$ are empty; or all of
the following conditions hold:
\begin{enumerate}[label=(TSIC-\arabic*),ref=(TSIC-\arabic*)]
\item
\label{TSIC:LeftIsCurve} The comb $u_1$ is a holomorphic curve for $\Hdown$ asymptotic to a sequence of non-empty sets of
Reeb chords $\vec{\rhos}=(\rhos_1,\dots,\rhos_m)$
\item \label{TSIC:Ind1} $u_1$ has index $1$ with respect to $\vec{\rhos}$.
\item \label{TSIC:Ind2} $u_2$ is a height $m$ holomorphic building for $\Hup$ with no components at east infinity
\item \label{TSIC:Ind3} each story of $u_2$ has index one.
\item \label{TSIC:SBM}
$u_1$ and $u_2$ are strongly boundary monotone
\item
\label{TSIC:Composable}
for each $i=1,\dots,m$, the east punctures of the $i^{th}$ story of $u_2$ are labelled, in order,
by a non-empty sequence of Reeb chords $(-\rho^i_1,\dots,-\rho^i_{\ell_i})$ with the property that
the sequence of singleton sets of chords $\vec{\rho}^i=(\{\rho^i_1\},\dots,\{\rho^i_{\ell_i}\})$ are composable.
\item
\label{TSIC:Matching}
The composition of the sequence of singleton sets of Reeb chords $\rho^i$ on the $i^{th}$ story of $u_2$
coincides with the $i^{th}$ set of reeb chords $\rhos_i$ in the partition for $w$.
\end{enumerate}
Let $\ModMatchedTSIC^{B_1\natural B_2}(\mathbf x,\mathbf y)$ denote the moduli space of trimmed simple ideal partially
self-matched curves.
\end{defn}
\begin{prop}
\label{prop:TGoestoInf}
Fix $\mathbf x=\mathbf x_1\#\mathbf x_2,\mathbf y=\mathbf y_1\#\mathbf y_2\in\States(\HD=\Hup\#\Hdown)$,
$B_1\in\doms(\mathbf x_1,\mathbf y_1)$,
and
$B_2\in\doms(\mathbf x_2,\mathbf y_2)$.
For each generic $T$ sufficiently large,
\[ \#\ModMatchedTSIC^{B_1\natural B_2}(\mathbf x,\mathbf y)=
\#\ModMatchedChanged^{B_1\natural B_2}(T;\mathbf x,\mathbf y).\]
\end{prop}
\begin{proof}
The proof is as in~\cite[Proposition~9.40]{InvPair}; the key
difference being that in~\cite{InvPair}, there is no self-matching
(in particular $u_1$ a holomorphic curve rather than a self-matched
curve), but this does not affect the argument.
In a little more detail, Gromov compactness shows that as
$T\goesto\infty$, the $T$-selfmatched curves converge to a pair of
combs $U_1$ and $U_2$ (in $\Hdown$ and $\Hup$ respectively)
satisfying a matching condition. Throwing out the East infinity
curves, we arrive at a pair of combs $(u_1,u_2)$. The matching
conditions (Condition~\ref{TSIC:Matching}) is clear; the fact that
$u_1$ is a (one-story) holomorphic curve (Condition~\ref{TSIC:LeftIsCurve})
of index one (Property~\ref{TSIC:Ind1}) follows
from the index formula.
Boundary monotonicity of $u_1$ follows from
Lemma~\ref{lem:StrongMonotoneClosed}. It follows then from
Lemma~\ref{lem:NonZeroAlgElts} that the algebra elements from each
packet are non-zero. It now follows that $u_2$ has no
$\alpha$-boundary degenerations; for such a boundary degeneration
would give rise to a vanishing algebra element. Having eliminated
$\alpha$-boundary degenerations from $u_2$,
Condition~\ref{TSIC:Ind2} and~\ref{TSIC:Ind3} follows from the index
formula.
Boundary monotonicity of $u_2$ follows now from Lemma~\ref{lem:NonZeroAlgElts}
combined with Proposition~\ref{prop:SBD}.
The matching conditions~\ref{TSIC:Matching} is straightforward.
Conversely, the existence of $T$-self-matched curves for sufficiently
large $T$ follows from a gluing argument as in the proof
of~\cite[Proposition~9.40]{InvPair}.
\end{proof}
\subsection{Putting together the pieces}
We can now assemble the steps to provide the main theorem:
\begin{proof}[Proof of Theorem~\ref{thm:PairAwithD}]
Start from the complex $C_\Ring(\HD)=(C,\partial)$ for the
doubly-pointed Heegaard diagram, with differential as in
Equation~\eqref{eq:OriginalComplex}. When $n>1$, Theorem~\ref{thm:NeckStretch}
(neck stretching) identifies $C_\Ring(\HD)\simeq
(C,\partial^{(0)})$, where the latter differential counts matched
holomorphic curves. Proposition~\ref{prop:Intermediates}
gives the sequence of isomorphisms
\[ (C,\partial^{(0)})\cong\cdots \cong (C,\partial^{(2n)}). \]
Note that $(C,\partial^{(2n)})=(C,\dChanged)$.
When $n=1$,
Next, we replace the
differential $\dChanged$ by a new differential $\dChangedT$ which
counts $T$-modified partially self-matched pairs; i.e. points in
$\ModMatchedChanged(T;\mathbf x,\mathbf y)$. When $T=1$, clearly
$\dChanged=\dChangedT$. The chain homotopy type of
$(C,\dChangedT)$ is independent of the choice of $T$: i.e. varying
$T$ gives chain homotopy equivalences between the various choices of
complex. (This is the anlogoue of~\cite[Proposition~9.22]{InvPair},
with the understanding that now, in one-dimensional families, we
have orbit curve end cancellation ends in addition to the
cancellation of join curve ends as in~\cite{InvPair};
cf. Lemma~\ref{lem:Join} above.) Taking $T$ sufficiently large
as in Proposition~\ref{prop:TGoestoInf}, and composing homotopy
equivalences, we find that $C_{\Ring}(\HD)$ is chain homotopic to
$(C,\partial')$, where now $\partial'$ counts trimmed simple ideal
partially self-matched curves (Definition~\ref{def:tsic}).
Since $u_1$ is strongly boundary monotone (which can be phrased in terms
of chord packets, thanks to Lemma~\ref{lem:SBA}),
Lemma~\ref{lem:NonZeroAlgElts} guarantees that the
objects counted in $\partial'$ correspond to the algebraic
counts appearing in the differential on $\Amod(\Hup)\DT\Dmod(\Hdown)$.
\end{proof}
\subsection{The case where $n=1$}
\label{subsec:Nequals1}
The case where $n=1$ works technically a little differently from the
case where $n>1$. The key distinguishing feature is that in the case
where $n=1$, closed components do exist in the Gromov
compactification. (It is also, of course a bit simpler, since we have to
deal with deforming only one pair of matched orbits.)
As we shall see in our proof of Theorem~\ref{thm:MainTheorem}
(Section~\ref{sec:Comparison}), we will need the case $n=1$ only in a
very specific special case: gluing on the standard lower diagram,
which has the property that any homotopy class that covers both $Z_1$
and $Z_2$ also covers the two basepoints $\wpt$ and $\zpt$. This property
would allow us to simplify the arguments considerably; but in the interest
of giving a clean statement of Theorem~\ref{thm:PairAwithD}, we give a
proof when $n=1$ without these restrictions hypotheses.
\subsubsection{Matched curves}
Consider the notion of matched curves (as in
Definition~\ref{def:MatchedPair}), except where the objects
${\overline u}_1$ and ${\overline u}_2$ are stories, rather than
simply curves. When $n>1$, Lemma~\ref{lem:NoClosedCurves} shows that
in sufficiently small index (and in homology classes not covering both
$\wpt$ and $\zpt$), the combs contain no closed components;
Lemma~\ref{lem:NoBoundaryDegenerations} shows that they can contain no
boundary degenerations. Thus, with these hypotheses, the matched
stories are automatically matched curves.
This is no longer the case where $n=1$. Specifically,
Lemma~\ref{lem:NoClosedCurves} fails in this case: moduli spaces of
self-matched stories are expected to contain closed components (on the
$\Hup$ side); and indeed, after removing those components, we obtain a (suitably) generalized matched curve in a moduli space of the same expected dimension.
We formalize these curves as follows:
\begin{defn}
A {\em special matched pair}
consists of
\begin{itemize}
\item a holomorphic curve $u_1$ in $\Hdown$ with source $\Source_1$ representing homology class $B_1\in\doms(\mathbf x_1,\mathbf y_1)$
\item a holomorphic curve $u_2$ in $\Hup$ with source $\Source_2$ representing homology class $B_2\in\doms(\mathbf x_2,\mathbf y_2)$
\item a subset $X\subset \East(\Source_1)$ of punctures marked by the orbit $\orb_1$,
\item a subset $Y\subset \East(\Source_1)$ of punctures marked by the orbit $\orb_2$,
\item a one-to-one correspondence $\phi\colon X\to Y$
\item an injection $\psi\colon \East(\Source_2)\to \East(\Source_1)$
\end{itemize}
with the following properties:
\begin{itemize}
\item $\East(\Source_1)$ is a disjoint union of
$\psi(\East(\Source_2))$, $X$, and $Y$.
\item For each $q\in \East(\Source_2)$ is marked with a Reeb orbit
or chord in $\Hup$, the corresponding puncture $\psi(q)\in
\East(\Source_1)$ is marked with the Reeb orbit or chord in
$\Hdown$ with the same name.
\item For each $q\in\East(\Source_2)$,
\[ (s\circ u_1(\psi(q)),t\circ u_1(\psi(q)))=(s\circ u_2(q),t\circ u_2(q)).\]
\item For each $p\in X$
\[ (s\circ u_1(\phi(p)),t\circ u_1(\phi(p)))=(s\circ u_2(p),t\circ u_2(p)).\]
\end{itemize}
If $B_1$ and $B_2$ induce $B\in \doms(\mathbf x,\mathbf y)$, let $\SModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2;\Source_1,\Source_2;\psi,\phi)$
denote the modul space of matched pairs.
\end{defn}
We have the following analogue of Lemmas~\ref{lem:NoClosedCurves} for
special matched curves.
\begin{lemma}
\label{lem:NoBoundaryDegenerationsMatchedNone}
Suppose $n=1$.
Fix $B_1\in\doms(\mathbf x_1,\mathbf y_1)$ and $B_2\in\doms(\mathbf x_2,\mathbf y_2)$ so that
$\weight_i(B_1)=\weight_i(B_2)$ for $i=1,\dots,2n$, and at least one of
$n_\wpt(B_1)$ or $n_\zpt(B_1)$ vanishes, and so that $\ind(B_1\natural
B_2)\leq 2$. Then, curves in the Gromov compactification of
$\SModMatched^B(\mathbf x_1,\mathbf y_1;\mathbf x_2,\mathbf y_2)$ contain no
closed components or boundary degenerations.
\end{lemma}
\begin{proof}
For a closed component to form, at least two pairs of matched orbits
must come together (i.e. it could either be that two punctures in $X$ along with
their two corresponding punctures in $Y$; or punctures $q_1$ and $q_2$ in $\Source_2$
along with their matching punctures $p_1$ and $p_2$ in $\Source_1$).
In any case, this occurs in codimension $2$, and is therefore
excluded from the index computation.
Let $({\overline u}_1,{\overline u}_2)$ denote a Gromov limit.
Suppose that ${\overline u}_2$ contains a boundary degeneration. It
follows that there are two punctures in ${\overline \Source}_2$
marked by Reeb orbits that cover both boundary components, and which
project to $(1,\tau)$. There must be two matching punctures in
${\overline \Source}_1$,
which project to the same point $(1,\tau)$.
It follows that $\Source_1$ contains either a closed component projecting to $(1,\tau)$,
or two boundary degenerations (covering $\Zin_1$ and $\Zin_2$). In either case,
it follows that $n_\wpt(B_1)>0$ and $n_\zpt(B_1)>0$, violating our hypothesis.
Similarly, if ${\overline u}_1$ contains a boundary degeneration,
then it follows that ${\overline u}_2$ must also contain a closed component
or a boundary degeneration, contradicting the above.
\end{proof}
We can form an endomorphism of $C$, obtained by counting index one
special matched curves. We denote this by $(C,\partial^{(0)})$.
In view of the above compactness result, $\partial^{(0)}$ is a differential.
(Compare Proposition~\ref{prop:dMatchedZqZero}.)
\begin{thm}
\label{thm:NeckStretchNOne}
When $n=1$,
suitable choices of almost-complex structures $J$ used to define
$\CFKsimp(K)$, there is an isomorphism (of chain complexes)
$\CFKsimp(K)\cong (C,\partial^{(0)})$.
\end{thm}
\begin{proof}
Using the neck stretching from Theorem~\ref{thm:NeckStretch}, we
finding limiting objects which are now matched stories. Those
stories contain closed components on the $\Hup$-side. Removing each
such closed component, we arrive at a corresponding special matched
curve. Conversely, given a special matched curve, at each puncture
$p\in X$, and matching puncture $\phi(p)\in Y$, we attach a sphere
that covers $\Hup$. Gluing that sphere gives
the stated identification.
To justify this, we need to observe that the count of curves
after gluing spheres at each
$\{p,\phi(p)\}$ puncture (with $p\in X$) agrees with the original count of
special marked curves. It suffices to verify this in a special case;
see Figure~\ref{fig:StabN1}.
\begin{figure}[h]
\centering
\input{StabN1.pstex_t}
\caption{{\bf Local contributions of spheres}.
\label{fig:StabN1}}
\end{figure}
We have exhibited a region in a Heegaard diagram. It is known that
for any almost-complex structure, the displayed shadow from
$\{x_1,x_2\}$ to $\{y_1,y_2\}$ has an odd number of
representatives. View the annular region as an upper diagram. The
moduli space from the lower diagram now consists of a bigon from
$x_1$ to $y_1$ superimposed on a bigon from $x_2$ to itself. The
first moduli space is one-dimensional, with a free $\R$ action,
and an orbit $\orb_1$ in the interior. The second moduli space is
two-dimensional, parameterized by a cut and an $\R$ action, with
an orbit $\orb_2$ in its interior. We can scale the $\R$ action
and the cut perameter so that $\orb_1$ and $\orb_2$ occur at the
same $(s,t)$ coordinate; gluing in the sphere gives rise to the
needed differential.
\end{proof}
\begin{rem}
The gluing of closed components is very similar to the stabilization
invariance proof in Heegaard Floer homology~\cite[Section~10]{HolDisk}.
The key difference is that here we are gluing spheres with two punctures,
rather than the one puncture considered there.
\end{rem}
\subsubsection{Comparison with self-matched curves}
The intermediate complexes considered when $n=1$ have a slightly
different form. For $(C,\partial^{(1)})$, we count points in moduli
spaces satisfying the matching conditions for special matched curves,
except now punctures on the $\Hup$-side marked by $\orb_1$ are matched
with punctures on the $\Hdown$-side, marked by the corresponding length one
Reeb chord. Similarly, in $(C,\partial^{(2)}$, all orbits on the $\Hup$-side
are matched with the corresponding length one Reeb chord; while orbits on the $\Hup$ side are matched with long chords on the $\Hup$ side.
We have the following analogue of Proposition~\ref{prop:Intermediates}:
\begin{prop}
\label{prop:Intermediatesx}
There is an isomorphism of chain complexes over $\Ring$
\[ \Phi\colon (C,\partial^{(0)})\to (C,\partial^{(2)})\]
\end{prop}
\begin{proof}
We construct first an isomorphism
$\Phi_0\colon (C,\partial^{(0)})\to (C,\partial^{(1)})$
as in the proof of Proposition~\ref{prop:Intermediates}.
Specifically, we modify the definition of $0$-morphism matched curves
as in Definition~\ref{def:MorMatch}, which comes equipped with a special
time $t_0$ with the following properties:
\begin{itemize}
\item Each puncture in $\Source_2$
marked by $\orb_2$ is matched with a length one chord in $\Source_1$
that covers the corresponding boundary component, with the same $t$-projection.
\item Each puncture in $\Source_2$ marked with the orbit $\orb_1$
and which projects to $t>t_0$ is matched with an orbit on $\Source_1$
with the same $(s,t)$ projection.
\item Each puncture in $\Source_2$ marked with the orbit $\orb_1$
and which projects to $t<t_0$ is matched with a chord on $\Source_1$
marked with the corresponding length one chord, with the same $t$-projection.
\item The punctures $\{q_i\}_{i=1}^m$ in $\Source_2$
that project to $t=t_0$ are all marked by $\orb_1$
have corresponding punctures $\{\psi(q_i)\}_{i=1}^m$
in $\Source_1$ that project to $t=t_0$.
For all $i=1,\dots,m$
$s\circ u_2(q_i)<s\circ u_1(\psi(q_i))$; and moreover
for $i=1,\dots,m-1$,
and $s\circ u_1(\psi(q_i))<s\circ u_2(q_{i+1})$.
\item The remaining punctures on $\Source_1$ marked with $\orb_1$ are paired
off with punctures on $\Source_1$ marked with $\orb_2$, with the same $(s,t)$ projection.
\end{itemize}
Define a map $h_0\colon C\to C$ counting rigid such objects, as
in the proof of Lemma~\ref{lem:ChainMap}.
We adapt the proof of Lemma~\ref{lem:ChainMap}, to show that
\[ \partial^{(1)}\circ h_0 + h_0\circ \partial^{(0)}=\partial^{(0)}+\partial^{(1)}.\]
In this adaptation, we consider once again ends of one-dimensional
moduli spaces. Some of the ends considered in that proof cannot
occur: ends of Type~\ref{end:OffRight} can occur.
Ends of
Type~\ref{end:OffLeft} are excluded: the argument of Lemma~\ref{lem:NoBoundaryDegenerationsMatchedNone} would show that if a boundary degeneration occurs in $\Hup$, then in fact the homology class on $\Hdown$ has $n_\wpt>0$ and $n_\zpt>0$.
Ends of
Types~\ref{end:HorizCollision},~\ref{end:OrbFromBelow}, and~\ref{end:OrbFromAbove} can occur; but ends of
Type~\ref{end:OrbFromAboveA} are replaced by ends where there are
two matching punctures on $\Source_1$, $q$ and $\phi(q)$, labelled
by $\orb_1$ and $\orb_2$, which project to the same $(s,t)$
coordinate. These latter ends cancel in pairs (where the double
puncture comes from $t>t_0$ or $t<t_0$. The remaining ends cancel in
pairs as in the proof of Lemma~\ref{lem:ChainMap}. (Note
also that there are no join curve ends to the moduli spaces, since
the boundary Reeb chords appearing in join curves do not exist for
$n=1$ diagrams.)
The isomorphism $\Phi_0$ is now given by $\Id+h_0$.
A similar isomorphism is constructed
$\Phi_1\colon (C,\partial^{(1)})\to (C,\partial^{(2)})$ is constructed analogously.
\end{proof}
We wish to compare $(C,\partial^{(2)})$ with the chain complex
$(C,\dChanged)$ defined by counting self-matched curves
(Definition~\ref{def:SelfMatched}). (Note that for $\partial^{(2)}$,
we allow punctures in $\Source_1$ marked by $\orb_2$ to project to the
same $(s,t)$-coordinate as some puncture marked by $\orb_1$; while
punctures in $\Source_2$ marked by $\orb_i$ project to the same
$t$-coordinate as some boundary puncture on $\Source_1$ marked by a
corresponding length $1$ Reeb chord.) An isomorphism between the two
complexes is constructed as follows:
\begin{prop}
\label{prop:nEqualsOne}
When $n=1$, there is an isomorphism of complexes over $\Ring$
\[ (C,\partial^{(0)})\cong (C,\partial^{(2)}). \]
\end{prop}
\begin{proof}
As in the proof of Proposition~\ref{prop:Intermediates}, we
construct an isomorphism
\[ \Phi\colon
(C,\partial^{(2)})\to(C,\dChanged),\] by counting certain curves.
The curves we count in this morphism are equipped with a
distinguished $t$-value $t_0$, with the following properties:
\begin{itemize}
\item All punctures on $\Source_2$ labelled with some orbit
are paired off with punctures in $\Source_1$ labelled with
the matching length one Reeb chord.
\item The punctures $\{q_i\}_{i=0}^{2m-1}$ on $\Source_1$
with $t(q_i)=t_0$ have the following properties:
\begin{itemize}
\item
$q_i$ is marked by the orbit $\orb_j$ for $j=1,2$
where $i\equiv j\pmod{2}$
\item
$s(u_1(q_i))$ is a monotone increasing function of $i=0,\dots,2m-1$.
\end{itemize}
\item The remaining punctures $q$ on $\Source_1$ labelled with the orbit
$\orb_2$ are paired off with punctures $\phi(q)$ in $\Source_1$:
\item
if $t(q)>t_0$, then $\phi(q)$ is labelled with $\orb_1$,
and $q$ and $\phi(q)$ have the same $(s,t)$ projection.
\item
if $t(q)<t_0$, then $\phi(q)$ is labelled with the length one chord
covering $\Zin_1$; and $t(u(\phi(q)))=t(u(q))$.
\end{itemize}
Counting such curves induces a map
$h\colon (C,\partial^{(2)})\to (C,\dChanged)$.
We claim that
\[ \dChanged \circ h + h \circ \partial^{(2)}=\partial^{(2)}+\dChanged.\]
This is obtained by looking at ends of one-dimensional moduli spaces of the above kind.
The following kinds of ends can occur
\begin{enumerate}[label=($\flat$'-\arabic*),ref=($\flat$'-\arabic*)]
\item \label{end:OffRightx} $s(u_1(q_{2m-1}))\goesto 1$,
so that in the Gromov limit, we get $((u_1,v_1),u_2)$,
where $v_1$ is an orbit curve (with orbit $\orb_1$)
attached at the level $t_0$.
\item \label{end:OffLeftx}
Ends where $s(u_2(\psi(q_0)))\goesto 0$;
in this case, there is a Gromov limit
to $((w_1,u_1),u_2)$, where
$w_1$ is a simple boundary degeneration containing $\orb_2$ and
$\wpt$.
\item
\label{end:HorizCollisionx}
Pairs $(u_1,u_2)$ with $s(u_1(q_i))=s(u_2(\psi(q_i)))$
or $s(u_2(\psi(q_i)))= s(u_1(q_{i+1}))$.
\item
\label{end:OrbFromBelowx}
there is some $q\in \East(\Source_2)$
labelled by an orbit
with $t(u_2(q))=t_0$, but $q$ arises as a limit point
of punctures with $t(u_2(q))<t_0$.
\item \label{end:OrbFromAbovex} there is some $q\in \East(\Source_2)$
labelled by an orbit
with $t(u_2(q))=t_0$, but $q$ arises as a limit point
of punctures with $t(u_2(q))> t_0$.
\item \label{end:OffLeftxGen}
Ends where $s(u_2(\psi(q)))\goesto 0$
and $t(u_1(q))>t_0$;
in this case, there is a Gromov limit
to $((u_1,v_1),u_2)$, where
$v_1$ is an orbit curve.
\item \label{end:OffRightxGen}
Ends where $s(u_2(\psi(q)))\goesto 1$
and $t(u_1(q))>t_0$;
in this case, there is a Gromov limit
to $((w_1,u_1),v_2)$, where
$w_1$ is a simple boundary degeneration.
\end{enumerate}
Consider ends of Type~\ref{end:OffLeftxGen}. When $w_1$ contains
$\orb_1$, it also contains $\zpt$. These cases do not count
algebraically, since we have specialized to $UV=0$. Thus, ends of
Type~\ref{end:OffLeftxGen} count when $w_1$ contains $\orb_2$ and
$\wpt$; and these cancel against ends of
Type~\ref{end:OffRightxGen}.
Ends of type~\ref{end:OffRightx} drop out in pairs, or cancel with ends
of Type~\ref{end:OrbFromAbovex},
when the latter orbit is labelled $\orb_2$.
Ends of Type~\ref{end:OffLeftx} drop out with orbits of
Type~\ref{end:OrbFromAbovex},
when the latter orbit is labelled $\orb_1$.
Ends of Type~\ref{end:HorizCollisionx}
cancel with ends of Type~\ref{end:OrbFromBelowx} (noting that the latter
punctures come in pairs). See Figure~\ref{fig:MorNOne} for an illustration.
\begin{figure}[h]
\centering
\input{MorNOne.pstex_t}
\caption{{\bf Cancellation of ends in the moduli spaces in the
construction of $h\colon (C,\partial^{(2)})\to
(C,\dChanged)$.} The drawings are shorthand: we have
illustrated the projections of punctures in $\Source_1$ to the
strip, coloring the ones labelled by $\orb_1$ white and those
labelled by $\orb_2$ black. There are four moduli spaces,
with arrows coming out of them; and their ends are at the ends
of the arrows. Ends with two incoming arrows cancel, and ends
with only one incoming arrow
are the terms in $\partial_\sharp$ and $\partial^{(2)}$.}
\label{fig:MorNOne}
\end{figure}
\end{proof}
\section{Type $A$ modules}
\label{sec:TypeA}
Let $\Hdown$ be a lower diagram, and $\Matching$ a matching on
$\{1,\dots,2n\}$. Let $\Mdown$ be the equivalence relation
on $\{1,\dots,2n\}$ induced
by $\Hdown$. Together, $\Matching$ and $\Mdown$ generate
an equivalence relation on
$\{1,\dots,2n\}$.
The matching $M$ is called {\em compatible} with $\Hdown$ if the
equivalence relation has one equivalence class in it.
Our aim here is to prove the following:
\begin{thm}
\label{thm:DefTypeA}
Fix a lower diagram $\Hdown$ and a matching $M$ on $\{1,\dots,2n\}$
compatible with $\Mdown$. Fix also a generic almost-complex
structure for $\Hdown$. Let $\Amod(\Hdown,\Matching)$ be the free
$\Field[U,V]/UV$-module generated by lower states. This can be endowed
with the following further structures:
\begin{itemize}
\item A rational-valued Alexander grading $\Agr$ (Equation~\eqref{eq:DefAgr} below)
\item An integer-valued relative grading $\Mgr$ (Equation~\eqref{eq:MgrA})
\item A collection of maps
\[ m_{1+\ell}\colon \Amod\otimes\overbrace{\Blg\otimes\dots\otimes\Blg}^\ell\to \Amod \]
for $\ell\in\Z^{\geq 0}$,
defined by counting pseudo-holomorphic
flows (Equation~\eqref{eq:DefAction}).
\end{itemize}
The result is a curved $\Ainfty$ module over $\Blg$,
with curvature $\sum_{\{i,j\}\in\Matching} U_i U_j$,
which has the following grading properties:
\begin{itemize}
\item $U$ drops $\Agr$ by $1$; $V$ raises $\Agr$ by $1$; and
the operations $m_{1+\ell}$ preserve $\Agr$
\item $U$ drops $\Mgr$ by $1$; $V$ drops $\Mgr$ by $1$, and
the operations $m_{1+\ell}$ respect $\Mgr$,
in the sense that
if $\mathbf x\in\Amod$ is a homogeneous module element,
and $a_1,\dots,a_\ell$ is a sequence of homogeneous algebra elements, then
$m_{1+\ell}(\mathbf x,a_1,\dots,a_\ell)$ is also homogeneous, with grading given by
\begin{equation}
\label{eq:MgrTypeA}
\gr(m_{1+\ell}(\mathbf x,a_1,\dots,a_\ell))=\gr(\mathbf x)+\ell -1+ \sum_{i=1}^{\ell} \Mgr(a_i).
\end{equation}
\end{itemize}
\end{thm}
Constraint packets are related to algebra elements in $\Clg$ in
Section~\ref{subsec:AlgebraicConstraints}. Once that is complete, the
proof will be given in the end of
Section~\ref{subsec:ConstructA}. Invariance properties will be dealt
with in Subsection~\ref{subsec:VaryCx}, and examples will be given in
Subsection~\ref{subsec:ExA}
\subsection{Algebra elements and constraints}
\label{subsec:AlgebraicConstraints}
An ingredient to constructing type $A$ modules to lower diagrams is
the relationship between constraint packets
(c.f. Definition~\ref{def:ConstraintPacket}) and the algebra $\Clg$, which we formulate presently.
If $\rho$ is a Reeb chord for a lower diagram, then it has a
corresponding algebra element $\bIn(\rho)$ obtained by multiplying
together the letters that represent the chord, as in
Figure~\ref{fig:ChordNamesA}. For example, the chord $\rho$ that
covers $Z_i$ with multiplicity one and which starts and ends at
$\alpha_i$ has
\[a(\rho)=L_i \cdot R_i = \left(\sum_{\{{\mathbf{s}}\big| i\in {\mathbf{s}}, i-1\not\in
{\mathbf{s}}\}}\Idemp{\mathbf{s}}\right)\cdot U_i \]
We generalize this construction to certain kinds of chord packets, as follows.
If $\rho$ is a Reeb chord, let $\rho^-$ denote its initial $\alpha$-arc, and $\rho^+$ denote its
terminal $\alpha$-arc.
\begin{defn}
\label{def:AlgebraicPacket}
A set of Reeb chords
$\{\rho_1,\dots,\rho_j\}$ is called {\em algebraic} if
for any pair of distinct chords $\rho_a$ and $\rho_b$,
\begin{itemize}
\item the chords $\rho_a$ and $\rho_b$ are on different boundary components $Z_i$ and $Z_j$,
\item the initial points $\rho_a^-$ and $\rho_b^-$ are on different $\alpha$-curves; and
\item the terminal points $\rho_a^+$ and $\rho_b^+$ are on different $\alpha$-curves.
\end{itemize}
\end{defn}
Let $\rhos$ be a set of Reeb chords that is algebraic, in the above sense,
we can associate an algebra element to $\rhos$, defined as follows.
Let $\alpha(\rho^+)$ be the curve
$\alpha_i$ with $\rho^+\in \alpha_i$; define $\alpha(\rho^-)$ similarly.
Let
\[ I^-(\rhos)=\sum_{\{{\mathbf{s}}\big| \{\alpha(\rho_1^-),\dots,\alpha(\rho_j^-)\}\subset
\{\alpha_i\}_{i\in{\mathbf{s}}}\}} I_{\mathbf{s}}
\qquad\text{and}\qquad
I^+(\rhos)=\sum_{\{{\mathbf{s}}\big| \{\alpha(\rho_1^+),\dots,\alpha(\rho_j^+)\}\subset
\{\alpha_i\}_{i\in{\mathbf{s}}}\}} I_{\mathbf{s}}
\]
Then, $\bIn_0(\rhos)$ be the algebra element $a_0\in\BlgZ(2n,n)$ with
\[ a=I^- \cdot a\cdot I^+\] and whose weight $w_i(a)$ is the average local
multiplicity at $Z_i$. Let $\bIn(\rhos)$ be the image of $\bIn_0(\rhos)$
in $\Clg(n)$.
\subsection{Constructing the $\Ainfty$ module}
\label{subsec:ConstructA}
We will fix throughout a lower diagram $\Hdown$ and a compatible $\Matching$ on $\{1,\dots,2n\}$.
Sometimes it is useful to enlarge the equivalence relation to include
the points $\wpt$ and $\zpt$, as follows. We extend $\Mdown$ to a
matching $\Mdown_*$ on the set $\{\wpt,\zpt,1,\dots,2n\}$, so that $i$
is matched with $\wpt$ resp. $\zpt$ if $\Zdown_i$ can be connected to
$\wpt$ resp. $\zpt$ without crossing any $\alpha$-circles. Then, $M$
and $\Mdown_*$ extended in this manner define an equivalence relation
on $\{\wpt,\zpt,1,\dots,2n\}$. If $M$ and $\Mdown$ are compatible, the
associated one-manifold $W(M)\cup W(\Mup_*)$ is homeomorphic to an
interval from $\wpt$ and $\zpt$.
Traversing the arc $W(M)\cup W(\Mup_*)$ (starting at $\wpt$ and ending at
$\zpt$), we encounter placemarkers for the orbits $\{1,\dots,2n\}$ in
some order. Let $f_0\colon \{1,\dots,2n\}$ denote the order in which
the placemarkers are encountered along this arc. We will define a function
\[ f\colon \{1,\dots,2n\}\to \{1,\dots,2n\},\]
obtained by post-composing $f_0$ with the involution on $\{1,\dots,2n\}$
that switches $2i-1$ and $2i$ for $i=1,\dots,n$. Thus,
the resulting function has $f(i)=1$ if the orbit $\orb_i$ is the second orbit we encounter
on the arc; $f(i)=2$ if $\orb_i$ is the first;
$f(i)=3$ if $\orb_i$ is the fourth, and $f(i)=4$ if $\orb_i$ is the third, etc.
See Figure~\ref{fig:LabelOrbits}.
\begin{defn}
\label{def:InducedOrdering} The function $f\colon \{1,\dots,2n\}$ is
called the {\em induced ordering} on the boundary components,
induced by $\Mdown$ and $\Matching$.
\end{defn}
Observe that if
$i$ is chosen so that $Z_i$ and $\wpt$ are contained in the same
component of $\Sigma\setminus\betas$, then $f(i)=2n-1$. We call the
Reeb orbit $\orb_i$ {\em even} resp. {\em odd} if $f(i)$ is even
resp. odd; i.e. $f$ induces an orbit marking, in the sense of
Definition~\ref{def:OrbitMarking}.
\begin{remark}
The function $f$ can be thought of as a relabeling of the Reeb orbits,
which might seem somewhat artificial at the moment. It is, however,
a reordering which very convenient for the purpose of the pairing theorem;
cf. Section~\ref{sec:Pairing}.
\end{remark}
\begin{figure}[h]
\centering
\input{LabelOrbits.pstex_t}
\caption{{\bf Labeling the orbits.}
The matchings $\Mdown$, $\Matching$, and the basepoint $\zpt$ order the
Reeb orbits as indicated.}
\label{fig:LabelOrbits}
\end{figure}
We will give $\Amod(\Hdown,\Matching)$ the structure of a bimodule,
which is a right module over the idempotent ring
$\RestrictIdempRing(n)\subset \Clg(n)$ and a left module over
$\Ring=\Field[U,V]/U V=0$. As a left $\Ring$-module it is freely
generated by all lower states. The idempotent in $\Clg(n)$
associated to a lower state is defined be
\[ \Idown(\mathbf x)=\Idemp{\alpha(\mathbf x)},\]
where $\alpha(\mathbf x)$ is as in Definition~\ref{def:UpperState}. (Note
that this is different from the idempotent for upper states, as in
Equation~\eqref{eq:IdempOfUpper}.) The right action of the idempotent
subalgebra of $\Clg(n)$ is specified by the condition that $\mathbf x\cdot
\Idown(\mathbf x)=\mathbf x$.
\begin{lemma}
\label{lem:AgrDom}
Given $\mathbf x,\mathbf y\in\States$, the quantity
\[ \Agr(B)=n_\zpt(B)-n_\wpt(B)+
\sum_{i=1}^{2n} (-1)^{f(i)} \weight_i(B)\]
is independent of the choice of $B\in\doms(\mathbf x,\mathbf y)$.
\end{lemma}
\begin{proof}
This follows from the fact that $\Agr(B)$ vanishes on
each component $B$
of $\Sigma\setminus\betas$. (Compare the proof of Lemma~\ref{lem:GradingsWellDefined}.)
\end{proof}
Since $\Agr(B_1*B_2)=\Agr(B_1)+\Agr(B_2)$, there is a
function $\Agr\colon \States\to \mathbb Q} \newcommand{\R}{\mathbb R$, uniquely characterized up to
an overall additive indeterminacy, so that
\begin{equation}
\label{eq:DefAgrGen}
\Agr(\mathbf x)-\Agr(\mathbf y)=\Agr(B)
\end{equation}
for $B\in\doms(\mathbf x,\mathbf y)$.
This induces a $\mathbb Q} \newcommand{\R}{\mathbb R$-valued grading on $\Amod(\Hdown,\Matching)=\bigoplus_{s\in\mathbb Q} \newcommand{\R}{\mathbb R} \Amod(\Hdown,\Matching,s)$,
with the convention that
\begin{align*}
U\colon &\Amod(\Hdown,\Matching,s)\to \Amod(\Hdown,\Matching,s-1) \\
V\colon &\Amod(\Hdown,\Matching,s)\to \Amod(\Hdown,\Matching,s+1).
\end{align*}
Chose a generic admissible almost-complex structure for $\Hdown$.
We use this to endow $\Amod(\Hdown,\Matching)$
with the structure of a right $\Ainfty$ module over
$\Clg(n)$, as follows.
\begin{defn}
\label{def:CompatiblePacket}
Fix a Heegaard state $\mathbf x$ and a sequence $\vec{a}=(a_1,\dots,a_\ell)$ of pure
algebra elements in $\BlgZ(2n,n)$.
A sequence of constraint packets $\rhos_1,\dots,\rhos_k$ is called
\em{$(\mathbf x,\vec{a})$-compatible} if there is a sequence
$1\leq k_1<\dots<k_\ell\leq k$ so that the following conditions hold:
\begin{itemize}
\item the constraint packets $\rhos_{k_i}$ are algebraic, in the sense
of Definition~\ref{def:AlgebraicPacket}
\item
$\Idown(\mathbf x)\cdot \bIn_0(\rhos_{k_1})\otimes\dots\otimes \bIn_0(\rhos_{k_\ell})=
\Idown(\mathbf x)\cdot a_1\otimes\dots\otimes a_{\ell}$,
as elements of $\Amod(\Hdown,\Matching)\otimes \ClgZ(n)^{\otimes \ell}$
\item
for each $t\not\in \{k_1,\dots,k_\ell\}$,
the constraint packet $\rhos_t$ is either of the form $\{\orb_i\}$
where $f(i)$ is odd; or it is of the form
$\{\orb_i,\longchord_j\}$, where
\begin{itemize}
\item $f(i)$ is even
\item $\{i,j\}\in\Matching$
\item $\longchord_j$ is one of the two Reeb chords
that covers $Z_j$ with multiplicity one.
\end{itemize}
\end{itemize}
Constraint packets $\rhos_j$ with $j\in\{k_1,\dots,k_\ell\}$ are called
{\em orbitless}.
Let $\llbracket \mathbf x,a_1,\dots,a_\ell\rrbracket$
be the set of all sequences of constraint packets $\rhos_1,\dots,\rhos_h$
that are $(\mathbf x,\vec{a})$-compatible.
\end{defn}
When considering boundary-monotone sequences, we can use
$\Clg(n)$ instead of $\ClgZ(n)$, according to the following:
\begin{lemma}
\label{lem:NonZeroAlgElts}
If $(\mathbf x,\rhos_1,\dots,\rhos_k)$ is a strongly boundary monotone sequence
which is $\vec{a}=(a_1,\dots,a_{\ell})$-compatible for some sequence
of algebra elements in $\BlgZ$;
then the projection of
$\mathbf x\otimes a_1\otimes\dots\otimes a_\ell$ in $\Amod\otimes\Clg(n)^{\otimes \ell}$
is non-zero.
\end{lemma}
\begin{proof}
We use Proposition~\ref{prop:Ideal}. Consider the packet at time
$t$, $\rhos_t$, and let $\Idemp{\mathbf x_-}$ and $\Idemp{\mathbf x_+}$ be the
idempotents immeddiately before and after.
We claim that $\mathbf x_-$ and $\mathbf x^+$ cannot be too far. This follows
from the fact that $\rhos$ contains two chords $\rho_1$ and $\rho_2$
so that the terminal point of $\rho_1$ is the initial point of
$\rho_2$, then the initial point of $\rho_2$ also appears in
$I_{\mathbf x_-}$. This is immediate.
It remains to exclude the other possibility from Proposition~\ref{prop:Ideal}.
That can be excluded by the following reasoning. Suppose that
$\rhos$ contains a sequence of chords $\rho_i,\dots,\rho_j$
with the following properties:
\begin{itemize}
\item $\rho_t$ is supported on $\Zin_t$ for $t=i,\dots,j$
\item either $\Idemp{\mathbf x_-}$ or $\Idemp{\mathbf x_+}$ does not contain $\alpha_i$.
\item $\weight(\rho_t)\geq 1$ for $t=i+1,\dots,j-1$
\end{itemize}
A straightforward proof by induction, using boundary monotonicity,
shows that endpoints of $\rho_j$
are on $\alpha_j$.
From Proposition~\ref{prop:Ideal},
it follows that the pure algebra element
$\Idemp{\mathbf x_-}\cdot b_0(\rhos)\Idemp{\mathbf x_+}$ is not in ${\mathcal J}$;
i.e. its projection to $\Blg$ is non-zero.
\end{proof}
\begin{defn}
\label{def:UweightVweight}
Given $(B,\rhos_1,\dots,\rhos_h)$, the {\em{$U$-weight}}
is the quantity $\gamma$ which is be the multiplicity
of $B$ at $\wpt$ plus the number of constraint packets among
the $\rhos_1,\dots,\rhos_h$ consisting of a single Reeb orbit
(or, equivalently, the total count of odd Reeb orbits appearing in the
$\rhos_1,\dots,\rhos_h$); and the {\em{$V$-weight}}
$\delta=\delta(B,\rhos_1,\dots,\rhos_h)$ is the
local multiplicity of the domain $B$ at the basepoint $\zpt$.
\end{defn}
\begin{lemma}
\label{lem:MgrDefined}
If $\doms(\mathbf x,\mathbf y)$ is non-empty, then for $B\in\doms(\mathbf x,\mathbf y)$,
the integer
\begin{equation}
\label{eq:MgrA}
\Mgr(B)=e(B)+P(B)-\weight_\partial(B)-n_\wpt(B)-n_\zpt(B)
\end{equation}
is independent of $B$.
\end{lemma}
\begin{proof}
This follows as in Lemma~\ref{lem:GradingsWellDefined},
with the observation that now
$e(\cald)+P(\cald)-\weight_\partial(\cald)-n_\wpt(\cald)-n_\zpt(\cald)$ vanishes
components $\cald$ of $\Sigma\setminus\betas$.
\end{proof}
Correspondingly, we have a function $\Mgr\colon \States\to \Z$
uniquely characterized up to an overall additive constant by the property that
\begin{equation}
\label{eq:DefMgrGen}
\gr(\mathbf x)-\gr(\mathbf y)=\Mgr(B)
\end{equation}
for any $B\in\doms(\mathbf x,\mathbf y)$.
\begin{lemma}
\label{lem:CompatWithMgr}
Fix $\mathbf x,\mathbf y\in\States$, a sequence of pure algebra elements
$\vec{a}=(a_1,\dots,a_\ell)$, an $(\mathbf x,\vec{a})$-compatible sequence
of constraint packets $\rhos_1,\dots,\rhos_h$, and
$B\in\doms(\mathbf x,\mathbf y)$. If there is a pre-flowline $u$ whose shadow is
$B$ and whose packet sequence is $(\rhos_1,\dots,\rhos_\ell)$,
then
\[ \gr(\mathbf x)+\ell-\sum_i\weight_{i=1}^{\ell}(a_i)=\gr(\mathbf y)-\Uweight(B)-\Vweight(B)+
\ind(B,\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_h).\]
\end{lemma}
\begin{proof}
By the definition of $\gr$ and $\Mgr$,
\[ \gr(\mathbf x)-\gr(\mathbf y)=e(B)+n_\mathbf x(B)+n_\mathbf y(B)-\weight_\partial(B)-n_\wpt(B)-n_\zpt(B).\]
Since all the orbits in appearing in $\rhos_i$ have weight $1$, the index formula gives
\[ \ind(B,\vec{\rhos_i})=e(B)+n_\mathbf x(B)+n_\mathbf y(B)+h-\weight_\partial(B)+\sum \iota(\chords(\rhos_i)).\]
Indeed, $\iota(\chords(\rhos_i))=-\weight(\chords(\rhos_i))$, so
taking the difference of these two equations, we find that
\begin{align*}
\gr(\mathbf x)-\gr(\mathbf y)-\ind(B,\vec{\rhos})&=-h-n_\wpt(B)-n_\zpt(B)+\sum_{i=1}^h \weight(\chords(\rhos_i)) \\
&= -\ell-n_\wpt(B)-n_\zpt(B)-\#(\text{odd orbits})+\sum_{i=1}^{\ell}\weight(a_i) \\
&=-\ell-\gamma(B)-\delta(B)+\sum_{i=1}^{\ell}\weight(a_i).
\end{align*}
Going from the second to the third line uses the fact that the packets containing
an even orbit also contain a weight one chord.
\end{proof}
Fix a lower Heegaard state $\mathbf x$ and a sequence of pure
algebra elements $a_1,\dots,a_\ell$ so that
$\Idown(\mathbf x)\cdot a_1\otimes \dots\otimes
a_\ell\neq 0$.
Define
\begin{equation}
\label{eq:DefAction}
m_{1+\ell}(\mathbf x,a_1,\dots,a_\ell)=
\sum_{\{\mathbf y\in\States,
(\rhos_1,\dots,\rhos_h)\in \llbracket \mathbf x, a_1,\dots,a_\ell\rrbracket\}}
U^{\gamma}V^{\delta}\#\ModFlow(\mathbf x,\mathbf y,\rhos_1,\dots\rhos_h)\cdot \mathbf y,
\end{equation}
where $\gamma=\gamma(\rhos_1,\dots,\rhos_h)$ is the multiplicity at
$\wpt$ of the domain $B$ determined by the sequence
$\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_h$ plus the number of constraint packets
among the $(\rhos_1,\dots,\rhos_h)$ consisting of a single Reeb orbit
When
$\Idown(\mathbf x)\otimes a_1\otimes\dots\otimes a_\ell=0$, we define the
action $m_{1+\ell}(\mathbf x,a_1,\dots,a_\ell)=0$.
For example, the $\mathbf y$ coefficient of $m_1(\mathbf x)$ is computed by counting
points in
\[ \ModFlow(\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_h),\] where each
$\rhos_t$ is either an odd Reeb orbit, or it is the constraint packet
consisting of some even Reeb orbit covering some boundary component once
together with a Reeb chord that covers its matching (using $M$) boundary component
exactly once.
As another example, the $\mathbf y$ coefficient of $m_2(\mathbf x,U_2)$ is computed
by counting points in $\ModFlow(\mathbf x,\mathbf y;\rhos_1,\dots,\rhos_h)$, where
exactly one of $\rhos_t$ is Reeb chord $R_2 L_2$ or the Reeb chord
$L_2 R_2$, and all other constraints packets have an orbit or
an orbit and a chord as above.
We claim that the sum appearing in Equation~\eqref{eq:DefAction} is
finite, according to the following:
\begin{lemma}
\label{lem:FiniteSum}
Given $(\mathbf x,a_1,\dots,a_\ell)$ and $\mathbf y$, there are only finitely many
homology classes $B$ of holomorphic disks that can represent
$\ModFlow(\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_h)$, where
$\rhos_1,\dots,\rhos_h$ is $(\mathbf x,\vec{a})$-compatible,
and for which one of $\Uweight=0$ or $\Vweight=0$.
\end{lemma}
\begin{proof}
Fix $\mathbf x,\mathbf y$, and let $B\in\doms(\mathbf x,\mathbf y)$ representing some
$(\mathbf x,\vec{a})$ compatible sequence $(\rhos_1,\dots,\rhos_h)$.
The following bounds are immediate:
\begin{enumerate}[label=(b-\arabic*),ref=(b-\arabic*)]
\item
\label{b:EvenOdd}
For
$\{i,j\}\in\Mdown$, the quantity $\weight_i(B)-\weight_j(B)$ is
independent of the choice of $B$ (depending only on $\mathbf x$ and $\mathbf y$).
\item
\label{b:Two}
$\weight_{f^{-1}(2)}(B)-n_\wpt(B)$ is independent of $B$.
\item
$\weight_{f^{-1}(2n-1)}(B)-n_\zpt(B)$ is independent of $B$.
\item while for
$\{i,j\}\in M$ with $f(i)$ odd,
\[\weight_i(B)-\weight_j(B)-\#(\orb_i\in \rhos_1,\dots,\rhos_h)\]
depends only on $\mathbf x$, $\mathbf y$, and $\vec{a}$.
\end{enumerate}
Thus, for fixed $(\mathbf x,\vec{a})$ and $\mathbf y$,
there are constants $c_i$
so that
\begin{equation}
c_{i} = \weight_{f^{-1}(2i-1)}(B)-n_\wpt(B)-
\# (\orb_{k}\in\rhos_1,\dots,\rhos_h
~\text{with $f(k)$ odd and $f(k)\leq 2i-1$}). \label{eq:WeightBounds}
\end{equation}
The case $i=n$, together with the fact that $\weight_{f^{-1}(2n-1)}(B)-n_\zpt(B)$,
is independent of $B$ shows that
\[ n_\zpt(B)-n_\wpt(B)-\# (\orb_{k}\in\rhos_1,\dots,\rhos_h
~\text{with $f(k)$ odd})=\delta-\gamma\] is
independent of the choice of $B$.
Since $U V=0$, we obtain an upper bound on the $U$-power $\gamma$ appearing on any term in
$m_{1+\ell}(\mathbf x, a_1,\dots, a_\ell)$.
Equation~\eqref{eq:WeightBounds} implies that for any odd $i$,
$\weight_{f^{-1}(i)}(B)\leq c_i + \gamma$. By~\ref{b:Two}
and~\ref{b:EvenOdd}, we can conclude a similar bound for even $i$,
as well. Since $\gamma$ is bounded above as above, we obtain a universal
bound on $\weight_i(B)$ for any $B$ that contributes to
$m_{\ell}(\mathbf x,a_1,\dots,a_\ell)$. It is elementary to see that there
are only finitely many non-negative homology classes of $B\in
\doms(\mathbf x,\mathbf y)$ with a universal bound on $\weight_i(B)$ bounded for
all $i$.
\end{proof}
\begin{prop}
\label{prop:CurvedTypeA}
The operations defined above endow $\Amod(\Hdown,\Matching)$ with
the structure of right $\Ainfty$ module over $\Clg(n)$ (which is
also a free left module over $\Ring$).
\end{prop}
\begin{proof}
Let $\mathbf x$ be an lower Heegaard state, and $a_1,\dots,a_\ell$ be a
sequence of algebra elements so that $\mathbf x\otimes
a_1\otimes\dots\otimes a_\ell\neq 0$. As usual, the $\Ainfty$
relations are proved by looking at ends of one-dimensional moduli
spaces. Specifically, we consider $\UnparModFlow(\mathbf x,\mathbf z,\rhos_1,\dots\rhos_h)$, where we take
the union over all sequences of algebraic constraint packets
$(\rhos_1,\dots,\rhos_h)\in\llbracket \mathbf x,a_1,\dots,a_\ell\rrbracket$.
The condition that $\Idown(\mathbf x)\cdot a_1\otimes \dots\otimes
a_\ell\neq 0$ ensures that the corresponding compatible constraint
packets $(\mathbf x,\rhos_1,\dots,\rhos_h)$ are strongly boundary monotone. (See
Lemma~\ref{lem:SBA}.) The homology classes that have non-zero
contribution cannot have positive multiplicity at both $\wpt$ and
$\zpt$, since we have the relation in our algebra that $UV=0$.
It is easy to see that each constraint packet is
allowed, in the sense of Definition~\ref{def:Allowed}.
Contributions of the two-story ends (Type~\ref{endA:2Story})
correspond to the terms
\[m_{\ell-i+1}(m_{i+1}(\mathbf x,a_1,\dots,a_{i}),a_{i+1},\dots,a_\ell)\]
appearing in the $\Ainfty$ relation.
Orbit curve ends (Type~\ref{endA:Orbit}) occur in two types. When
the constraint packet is of the form $\{\orb_j,\longchord_k\}$ (so that
$f(j)$ is even), its corresponding orbit curve ends corresponds to a
term in
\begin{equation}
\label{eq:DiffAinf}
m_{\ell+2}(\mathbf x,a_1,\dots,a_{i},\mu_0,a_{i+1},\dots,a_\ell).
\end{equation}
If the constraint packet is of the form $\{\orb_j\}$ (so that $f(j)$ is odd),
we call the corresponding orbit curve end an {\em odd orbit curve end}.
In the course of this proof, we will find other ends that cancel these odd orbit curve ends.
We turn now to visible collision ends (Type~\ref{endA:ContainedCollisions}),
first considering visible collision ends where the two constraint packets
$\rhos_i$ and $\rhos_{i+1}$ are orbitless. These contribution
correspond to the terms in the $\Ainfty$ relation of the form
\[ m_{\ell}(\mathbf x,a_1,\dots,a_{i-1},a_{i}\cdot a_{i+1},a_{i+2},\dots,a_\ell).\]
Consider next collision ends where exactly one of $\rhos_i$ or
$\rhos_{i+1}$ is orbitless. In this case, we can find an
alternative choice of $i^{th}$ and $(i+1)^{st}$ constraint packet
$\rhos_i'$ and $\rhos_{i+1}'$, so that the corresponding ends of
$\ModFlow(\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_{i-1},\rhos_i,\rhos_{i+1},\rhos_{i+2},\dots,\rhos_\ell)$
and
$\ModFlow(\mathbf x,\mathbf y,\rhos_1,\dots,\rhos_{i-1},\rhos_i',\rhos_{i+1}',\rhos_{i+2},\dots,\rhos_\ell)$
cancel. We choose $\rhos_i'=\rhos_{i+1}$ and $\rhos_{i+1}'=\rhos_i$
except in the special case where one of $\rhos_i$ or $\rhos_{i+1}$
(which we can assume without loss of generality is $\rhos_i$)
consists of an even orbit $\orb_j$ and matching chord $\longchord$
that covers $Z_k$ with multiplicity $1$, and $\rhos_{i+1}$ also
contains some chord $\rho$ supported in $Z_k$. In that case, there
is a unique, possibly different long chord $\longchord'$ covering
$Z_k$ with multiplicity one, so that $\longchord\uplus
\rho=\rho\uplus \longchord'$. Then,
$\rhos_{i+1}'=\{\orb_j,\longchord'\}$, and $\rhos_{i}'=\rhos_{i+1}$;
see Figure~\ref{fig:CollisionCancels}. (We are cancelling here
contributions corresponding to ends of different moduli spaces; but
the domains $B$ each pair of moduli spaces is the same, as are the
total number of odd orbits in the corresponding Reeb sequences; so
the $U$ and $V$ exponents for the contributions are the same, and
the cancellation occurs.)
\begin{figure}[h]
\centering
\input{CollisionCancels.pstex_t}
\caption{{\bf Cancellations of collision ends.}
For this collision cancellation, we use the two decompositions
of $L_k R_k L_k$ as
$L_k\uplus R_k L_k = L_k R_k \uplus L_k$.
(In the picture, we assume that
$f(k)$ and $f(j)$ are consecutive integers,
and $f(k)$ is odd.)}
\label{fig:CollisionCancels}
\end{figure}
Similar
cancellations occur when both $\rhos_i$ and $\rhos_{i+1}$ contain
orbits, but the orbit in $\rhos_i$ is not matched (via $\Mdown$) with the one in
$\rhos_{i+1}$.
When the orbit in $\rhos_i$ is matched with the orbit in
$\rhos_{i+1}$ there are two kinds of collision ends: those that are
contained (Type~\ref{endA:ContainedCollisions}), and those that are
not (Type~\ref{endA:BoundaryDegeneration}); see
Figure~\ref{fig:OrbitCancels}. Ends where the collision is contained
once again cancel corresponding ends of moduli spaces where the
order of the two packets is permuted.
The total number of remaining (i.e. uncontained) collision ends of this
moduli space and the one obtained by permuting $\rhos_i$ and
$\rhos_{i+1}$ counts points in
\[\ModFlow(\mathbf x,\mathbf z,\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+2},\dots,\rhos_\ell),\]
where $\sigmas=\{\longchord_k\}$ is one of the two Reeb chords
that covers the component $Z_k$ with $f(k)$
odd. These cancel against the odd orbit curve
ends described above.
See Figure~\ref{fig:OrbitCancels}.
Note that in this case, the homology classes $B$ and $B'$ of the curves representing the two
cancelling ends do not coincide: an uncontained collision end removes a boundary degeneration.
Nonethless, the boundary degenerations considered here have $n_\wpt(v)=n_\zpt(v)=0$;
and the total number of odd orbits remains unchanged, so the $U$ and $V$ exponents
of the two ends coincide.
The same cancellation occurs for ends of
Type~\ref{endA:SpecialBoundaryDegeneration} (the ``special boundary
degenerations''), where the even unmatched Reeb orbit (i.e. $\orb_j$
with $f(j)=2$) is removed: it cancels with a corresponding odd
orbit end. (See Figure~\ref{fig:SpecialCancellation}.) Cancellation
occurs because the moduli space with the boundary degeneration end
contributes the same $U$ and $V$ powers as the moduli space with the
odd orbit end. To see this, let $B$ denotes the homology class with
the special boundary degeneration; let
$(\rhos_1,\dots,\rhos_{\ell})$ denote the Reeb sequence for the
moduli space containing the special boundary degeneration end; let
$(\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_{\ell})$
be the sequence associated with the boundary degeneration end; and
let $B'$ be the homology class where the boundary degeneration is
removed (and the one where the corresponding odd orbit end
occurs). Clearly, $n_\wpt(B')=n_\wpt(B)-1$, while the odd orbit end
is associated with the sequence
$(\rhos_1,\dots,\rhos_{i-1},\{\orb_j\},\rhos_{i+1},\dots,\rhos_\ell)$,
which has one odd orbit more than the original sequence moduli space
that has one additional odd orbit in its interior than
$\rhos_1,\dots,\rhos_{\ell}$. Thus, the $U$ exponents of both moduli
spaces coincide. Moreover, since $n_\zpt(B)=n_\zpt(B')$, the $V$
exponents coincide, as well.
\begin{figure}[h]
\centering
\input{SpecialCancellation.pstex_t}
\caption{{\bf Special boundary degeneration cancelling
an odd orbit end.} Orbits here are labelled by their $f$-values.}
\label{fig:SpecialCancellation}
\end{figure}
By contrast, ends of Type~\ref{endA:SpecialBoundaryDegeneration},
involving the other unmatched Reeb orbit $\orb_j$ (with $f(j)=1$) do not contribute.
This is true because
${\mathcal B}_{\{j\}}$ also crosses the $\zpt$ basepoint, so the homology class
contributes a multiple of $V$.
By Equation~\eqref{eq:DefAction}, moduli spaces containing this orbit
are counted with a multiple of $U$.
Since we have specialized to $UV=0$,
this end does not contribute.
\begin{figure}[h]
\centering
\input{OrbitCancels.pstex_t}
\caption{{\bf Cancellations when two packets contain orbits.} The top row indicates
cancellations of two contained collision ends. The bottom row
indicates the cancelation of an uncontained collision end with an
odd orbit end. In this picture, we subscript each orbit by its $f$-value.}
\label{fig:OrbitCancels}
\end{figure}
Consider next the join ends (Type~\ref{endA:Join}). We argue
ultimately that these all count with even multiplicity, as follows.
Suppose that $\rhos_i$ contains some chord $\rho$ supported in
$Z_i$, which we can split as
$\rho=\rho_1\uplus \rho_2$. We investigate the
corresponding join ends
$\rhos_1,\dots,\rhos_{i-1},\sigmas,\rhos_{i+1},\dots,\rhos_\ell$,
where
\[\chords(\sigmas)=(\chords(\rhos_i)\setminus \{\rho\})\cup \{\rho_1,\rho_2\}.\]
If the two endpoints of $\rho$ are distinct, then this corresponding
join curve end cannot appear in a moduli space of boundary monotone
curves. That is, suppose that $\rho$ has two distinct endpoints and
it has a decomposition as $\rho=\rho_1\uplus \rho_2$. Then it is
elementary to see that either $\rho_1^+=\rho_2^+$ or
$\rho_1^-=\rho_2^-$; i.e. the chords $\rho_1$ and $\rho_2$ cannot
appear in the same constraint packet, for a boundary monotone sequence.
Consider next the case where the two endpoints of $\rho$ are the
same, and consider there is a join curve end corresponding to a
splitting $\rho=\rho_1\uplus \rho_2$. Such ends appear boundary
monotone moduli spaces only when the two endpoints of $\rho_1$ (and
hence also of $\rho_2$) are distinct. For example, we know
$\rho_1^-=\rho_2^+$, so if $\rho_1^-=\rho_1^+$, then
$\rho_2^+=\rho_1^+$, so $\rho_1$ and $\rho_2$ cannot appear in the
same constraint packet.
We are left with the case where $\rho=\rho_1\uplus\rho_2$, and the
two endpoints of $\rho_1$ are distinct, as are the two endpoints of
$\rho_2$. We can then form $\rho'=\rho_2\uplus\rho_1$.
When $\rho_1$ or $\rho_2$ covers only half of the
corresponding boundary component $Z_i$, this
splitting of $\rho'$ gives rise to a join curve end in moduli space
\[ \UnparModFlow(\rhos_1,\dots,\rhos_{i-1},\rhos_i',\rhos_{i+1},\dots,\rhos_h),\]
where $\orbits(\rhos_i')=\orbits(\rhos_i)$ and
\[ \chords(\rhos_i')=(\chords(\rhos_i)\setminus \{\rho_1\uplus
\rho_2\})\cup \{\rho_2\uplus \rho_1\}.\] Clearly,
both $(\rhos_1,\dots,\rhos_{i-1},\rhos_i',\rhos_{i+1},\dots\rhos_h)$
is also consistent with $(\mathbf x,a_1,\dots,a_h)$.
Moreover, this join curve end occurs with the same multiplicity as
the corresponding join curve of
\[
\UnparModFlow(\rhos_1,\dots,\rhos_{i-1},\rhos_i,\rhos_{i+1},\dots,\rhos_h),\]
corresponding to the splitting $\rho=\rho_1\uplus \rho_2$.
We have thus set up a one-to-one correspondence between pairs of
join curve ends of different moduli spaces that are consistent with
the same algebra actions, so the join curve ends of the moduli
spaces counted in the $\Ainfty$ relations cancel in pairs.
See
Figure~\ref{fig:JoinsCancel} for some examples.
\begin{figure}[h]
\centering
\input{JoinsCancel.pstex_t}
\caption{{\bf Cancellations of join ends.} The two one-dimensional
moduli spaces shown at the left (containing Reeb chords $L_k R_k$
and $R_k L_k$) have the same ends. The two one-dimensional moduli
spaces on the right (containing length two Reeb chords $L_k R_k L_k
R_k$ and $R_k L_k R_k L_k$ and) each now have two different kinds
of join curve ends; but again these two ends of the two moduli spaces
are in one-to-one correspondence with each other.}
\label{fig:JoinsCancel}
\end{figure}
(Observe that join curve ends can appear for splittings of the chord
$(L_k R_k)^m$ or $(L_k R_k)^m$ for arbitrarily large $m$. In this
case, the join curve at East infinity covers the cylinder with
multiplicity $m$. Moreover, according to Theorem~\ref{thm:AEnds}
there are only two codimension one join ends corresponding to
splittings of $(L_k R_k)^m$; and they are
$\{L_k,(R_k L_k)^{m-1} R_k\}$ and $\{R_k,(L_k R_k)^{m-1} L_k\}$.
The same two ends appear when the
chord is replaced by $(R_k L_k)^m$.
Figure~\ref{fig:JoinsCancel} illustrates $m=1$ and $2$.)
This cancellation completes the verification of the $\Ainfty$ relation.
\end{proof}
We can synthesize these parts to give the following:
\begin{proof}[Proof of Theorem~\ref{thm:DefTypeA}]
The Alexander grading was defined in Equations~\eqref{eq:DefAgrGen}
(which is valid thanks to Lemma~\ref{lem:AgrDom}. Actions were
defined in Equation~\eqref{eq:DefAction}, which was shown to be a
valid definition in Lemma~\ref{lem:FiniteSum}. The grading $\Mgr$
was defined in Equation~\eqref{eq:DefMgrGen} (which is valid thanks to
Lemma~\ref{lem:MgrDefined}). The $\Ainfty$ relations were verified in
Proposition~\ref{prop:CurvedTypeA}. The fact that the operations
preserve $\Agr$ is obvious from Lemma~\ref{lem:AgrDom}.
The fact that they respect $\Mgr$ (Equation~\eqref{eq:MgrTypeA}) follows
readily from Lemma~\ref{lem:CompatWithMgr}.
\end{proof}
\subsection{Invariance properties}
\label{subsec:VaryCx}
The following is an adaptation of~\cite[Proposition~7.19]{InvPair}:
\begin{prop}
If $J_0$ and $J_1$ are any two generic paths of almost-complex structures,
there is an $\Ainfty$ homotopy equivalence
\[ \Amod(\Hdown,\Matching,J_0)\simeq \Amod(\Hdown,\Matching,J_1).\]
\end{prop}
\begin{proof}
This is mostly standard; so we sketch the proof.
A morphism
\[ f\colon \Amod(\Hdown,\Matching,J_0)\to \Amod(\Hdown,\Matching,J_1) \]
is constructed by counting points in moduli spaces
$\ModFlow^B(\mathbf x,\mathbf y;\rhos_1,\dots,\rhos_h)$, where now the moduli
spaces are taken with respect to a path of paths of almost-complex
structures that interpolate from $J_0$ to $J_1$. The proof that this
morphism satisfies the structure equation of an $\Ainfty$
homomorphism is similar to the proof of
Proposition~\ref{prop:CurvedTypeA}.
An analogue
of Theorem~\ref{thm:AEnds} identifies ends of these moduli spaces,
and cancellations are analogous to the ones in
Proposition~\ref{prop:CurvedTypeA}: two-story ends correspond to
terms of the form
\[ \sum_{i=1}^{1+\ell} f_{\ell-i+1}(m_i(\mathbf x,a_1,\dots,a_{i-1}),a_i,\dots,a_\ell) +
m_{\ell-i+1}(f_{i}(\mathbf x,a_1,\dots,a_{i-1}),a_i,\dots,a_\ell); \]
even orbit ends contribute terms of the form
\[ \sum_{i=1}^\ell
f_{\ell+2}(\mathbf x,a_1,\dots,a_{i-1},\mu_0,a_i,\dots,a_\ell); \] while
odd orbit ends cancel with boundary degeneration collisions. Join
curve ends cancel in pairs, and contained collisions correspond to
terms of the form
\[ \sum_{i=1}^{\ell} f_{\ell}(\mathbf x,\dots,a_{i-1},\mu_2(a_i,a_{i+1}),a_{i+2},\dots,a_\ell).\]
Adding up these contributions give the $\Ainfty$ homomorphism relation.
The homotopy inverse $g$ is constructed by turning the one-parameter
family of paths around, and the homotopy relation
\[ f\circ g \simeq \Id \] is verified by considering a variation of one-parameter families.
\end{proof}
More topological invariance properties of this module can be established by adapting methods from~\cite[Section~7.3]{InvPair};
the above result is the only one necessary for the purposes of this paper.
\subsection{Examples}
\label{subsec:ExA}
Consider the algebra $\cClg(1)$, using the only matching on the single
pair of strands.
There is an isomorphism
\begin{equation}
\label{eq:IsoClg1}
\Psi\colon \cClg(1)\to \Ring=\Field[U,V]/UV,
\end{equation}
with $\Psi(U_1)=U$ and $\Psi(U_2)=V$. Let $\lsup{\Ring}[\Psi]_{\Clg(1)}$ be the associated bimodule.
Note that
$\lsup{\Ring}[\Psi]_{\Clg(1)}=\lsup{\Ring}[\Psi]_{\cClg(1)}$, as the curvature in $\cClg(1)$ vanishes.
\begin{figure}[h]
\centering
\input{SmallLower.pstex_t}
\caption{{\bf Standard lower diagram with $n=1$.}}
\label{fig:SmallLower}
\end{figure}
\begin{lemma}
\label{lem:GlobalMinimum}
For the standard lower diagram $\Hdown$ with two strands,
there is an identification
$\Amod(\Hdown)=\lsup{\Ring}[\Psi]_{\cClg(1)}$.
\end{lemma}
\begin{proof}
Consider the diagram pictured in Figure~\ref{fig:SmallLower}. It
has a unique generator $x$. Homology classes of disks are multiples
of the two components $\dom_1$ and $\dom_2$ of $\Sigma\setminus
\beta_1$, labelled so that $\dom_i$ contains the boundary component
$Z_i$. A curve representing the homology class
$k \cdot \dom_1$ has index one if and only if contains a single Reeb
chord on its boundary with length $k$ (and no internal
punctures). In that case, there is a unique holomorphic
representative, giving rise to the action $m_2(x,U_1^k)=U^k\otimes
x$. Arguing similarly on the other side, we find that
$m_2(x,U_2^k)=V^k\otimes x$. This completes the verification.
\end{proof}
\begin{figure}[h]
\centering
\input{Lower2.pstex_t}
\caption{{\bf Standard lower diagram with $n=2$.}}
\label{fig:Lower2}
\end{figure}
We consider a less trivial example; the standard lower diagram with
$n=2$, and (compatible) matching
$\Matching=\{\{1,4\},\{2,3\}\}$. (See Figure~\ref{fig:Lower2}.)
For
simplicity, we consider the $V=0$ specialization. In this case, the
module has a simple description, described by the following graph:
\begin{equation}
\label{eq:MinGraph}
\begin{tikzpicture}
\node at (3,0) (YR2) {${\mathbf x}$} ;
\node at (0,2) (Y) {$*$} ;
\node at (0,-2) (X) {$*$} ;
\draw[->] (Y) [bend left=30] to node[above,sloped] {\tiny{$U^{a+b} \otimes U_1^a U_3^{b+1}$}} (X) ;
\draw[->] (YR2) [bend left=15] to node[below,sloped] {\tiny{$U^{a+b}\otimes
U_1^a U_3^b R_3$}} (X) ;
\draw[->] (X) [bend left=30] to node[above,sloped] {\tiny{$U^a\otimes U_1^a U_4$}} (Y) ;
\draw[->] (Y) [bend left=15] to node[above,sloped] {\tiny{$U^{a+b}\otimes U_1^a U_3^{b} L_3$}} (YR2) ;
\draw[->] (YR2) [loop right] to node[above,sloped] {\tiny{$U^{a+b}\otimes U_1^a U_3^{b}$}} (YR2);
\end{tikzpicture}
\end{equation}
Each path from $x$ to itself (and choices of $a$ and $b$ for each edge)
gives an ${\mathcal A}_\infty$ action. The inputs are obtained by concatenating
the second tensor factor on each edge, and the output is obtained by
multiplying together the first tensor factor in each edge. So, for example, we have
actions
\begin{align*}
m_2(x,U_1)&= U\otimes x \\
m_4(x,R_3,U_4,L_3)&=1\otimes x.
\end{align*}
The above two actions can be seen by looking at embedded disks in the Heegaard diagram.
Indeed, all of the other above actions are determined by the existence of the
above two actions, and the (curved) ${\mathcal A}_\infty$ structure relations.
For example, applying the above two actions, and the ${\mathcal A}_{\infty}$
relation with inputs $R_3, U_4, L_3, U_1$, we can conclude that
\[ m_4(x,R_3, U_4, L_3 U_1)= U \otimes x.\]
Applying the ${\mathcal A}_{\infty}$ relations with input sequence
$R_3, U_4, U_1, L_3$ gives
\[ m_4(x,R_3, U_1 U_4, L_3)= U \otimes x.\]
Finally, applying the input sequence
$R_3, L_3$, we can conclude that
\[ m_2(x, U_3)= U\otimes x.\]
It is straightforward, if a little cumbersome, to generalize to the
$V$-unspecialized case; compare~\cite[Section~\ref{BK2:sec:Min}]{Bordered2}.
\section{Type $D$ modules}
\label{sec:TypeD}
Let $\Hup$ be an upper diagram, and let $\Matching$ be its
corresponding matching. Let $W$ be the one-manifold associated to
$\Matching$ (as in Definition~\ref{def:AssociatedW}), and fix an
orientation on $W$; i.e. a choice of preferred $i$ for each
$\{i,j\}\in\Matching$. (This latter data is needed to define the
$\mathbb Q} \newcommand{\R}{\mathbb R^n$-valued Alexander grading.)
Choose an admissible almost-complex structure $J$ over
$\Hup$ (as in Definition~\ref{def:AdmissibleAlmostCx}).
Let $\Dmod(\Hup)$ be the vector
space generated over $\Field$ by the upper states.
For $\alpha(\mathbf x)$ as in Definition~\ref{def:UpperState}, let
\begin{equation}
\label{eq:IdempOfUpper}
\Iup(\mathbf x)=\Idemp{\{1,\dots,2n-1\}\setminus \alpha(\mathbf x)}
\end{equation}
thought of as an idempotent in the algebra $\Clg(n)$. The left action of
the idempotent subalgebra of $\Clg(n)$ is specified by the condition that
$\Iup(\mathbf x)\cdot \mathbf x = \mathbf x$.
Consider the functions $\Mgr\colon \States(\Hup)\to \Z$ and
$\Agr\colon \States(\Hup)\to \OneHalf \Z^n$ defined in
Equations~\eqref{eq:DefMgr} and~\ref{eq:DefAgr} above. These endow
$\Dmod(\Hup)$ with a $\Z$-valued $\Delta$-grading and a $\OneHalf
\Z^n$-valued Alexander grading, denoted $\Agr$.
Let $\mathbf x,\mathbf y$ be upper Heegaard states, and $B\in\doms(\mathbf x,\mathbf y)$. Let
$\ModFlow^B(\mathbf x,\mathbf y)$ be the union of $\ModFlow^B(\mathbf x,\mathbf y;\vec\rho)$ (as
in Equation~\eqref{eq:EmbedMod}), taken over all typical sequences
$\vec\rho$ of Reeb chords and orbits that are compatible with $B$.
Recall that by Theorem~\ref{thm:GeneralPosition}, $\ModFlow^B(\mathbf x,\mathbf y)$
has expected dimension $\Mgr(B)$ (independent of the typical sequence
$\vec\rho$).
Define the operation
\[ \delta^1\colon \Dmod(\Hup)\to \Clg(n)\otimes \Dmod(\Hup) \]
by
\begin{equation}
\label{eq:TypeDOperation}
\delta^1(\mathbf x) = \sum_{\{\mathbf y\in \States, B\in\doms(\mathbf x,\mathbf y)\big|
\Mgr(B)=1\}} \#\UnparModFlow^B(\mathbf x,\mathbf y) \cdot \bOut(B)\otimes \mathbf y,
\end{equation}
where
$\bOut(B)\in\Clg(n)\subset\Blg(2n,n)$ is as in
Section~\ref{sec:Shadows}.
\begin{prop}
The sum appearing on the right of Equation~\eqref{eq:TypeDOperation}
is finite.
\end{prop}
\begin{proof}
The non-zero terms arise from $B\in\doms(\mathbf x,\mathbf y)$ with $\Mgr(B)=1$,
which have pseudo-holomoprhic representatitves. We must show that there are only finitely many
such $B$. To this end,
fix some $B_0\in\doms(\mathbf x,\mathbf y)$.
Our hypothesis on upper diagrams (Property~\ref{UD:NoPerDom})
ensures that for any other $B\in\doms(\mathbf x,\mathbf y)$,
we can find integers $n_{\{r,s\}}$ so that
\[ B=B_0 + \sum_{\{r,s\}\in\Matching} n_{\{r,s\}} \cdot \Brs.\] If $B$ has a holomorphic representative, then
all of its local multiplicities must be non-negative, giving a lower bound on each
$n_{\{r,s\}}$. Since
\[ \Mgr(B)=\Mgr(B_0)+ 2 \sum_{\{r,s\}\in\Matching} n_{\{r,s\}},\]
condition that $\Mgr(B)=1$ also places an upper bound on all the $n_{\{r,s\}}$, proving the desired
finiteness statement.
\end{proof}
\begin{prop}
The map $\delta^1$ respects (relative) gradings in the following sense:
if $b\otimes \mathbf y$ appears with non-zero multiplicity in $\delta^1(\mathbf x)$,
then
\begin{align*}
\Mgr(\mathbf x)-1=\Delta(b)+\Mgr(\mathbf y) \\
\Agr(\mathbf x)=\Agr(b)+\Agr(\mathbf y).
\end{align*}
\end{prop}
\begin{proof}
The above equations follow at once from
Equations~\eqref{eq:DefMgr},~\eqref{eq:DefAgr}, and the definitions
of the gradings on the algebra, Equations~\eqref{eq:DefDeltaAlg}
and~\eqref{eq:DefAgrAlg}.
\end{proof}
\begin{prop}
\label{prop:CurvedTypeD}
The map $\delta^1$ defined above
satisfies a curved type $D$ structure relation,
with curvature $\mu_0=\sum_{\{r,s\}\in\Matching} U_r\cdot U_s$.
\end{prop}
\begin{proof}
Choose $\mathbf x$ and $\mathbf z$ so that there is some
$B\in\doms(\mathbf x,\mathbf y)$ with $\Mgr(B)=2$.
Consider the ends of the moduli spaces
$\UnparModFlow^B(\mathbf x,\mathbf z,\Source;\vec{P})$, where we take
the union over all choices of typical Reeb sequences
$[\vec{P}]=(\rho_1,\dots,\rho_\ell)$ and all choices of source
$\Source$. We can assume without loss of generality that the homology
class $B$ of the curve does not cover all of $\Sigma$; for otherwise,
the corresponding term in
$(\mu_2\otimes \Id_{\Dmod(\Hup)})\circ (\Id_{\Clg}\otimes \delta^1)\circ\delta^1$ vanishes;
there are no non-zero algebra elements with positive weight everywhere.
These moduli spaces are one-dimensional according to
Theorem~\ref{thm:GeneralPosition}. Next we appeal to
Theorem~\ref{thm:DEnds}, observing cancellations of the counts of
various ends cancel. The various collision ends where at least one
of $\rho_i$ or $\rho_{i+1}$ is a Reeb orbit cancel with one
another. Specifically, consider a typical Reeb sequence
$(\rho_1,\dots,\rho_i,\rho_{i+1},\dots,\rho_\ell)$ with a visible
collision end for $\rho_i$ and $\rho_{i+1}$, where at least one of
the two is a Reeb orbit (i.e. in the terminology of
Theorem~\ref{thm:DEnds}, this is an end of
type~\ref{typeDE:CollisionWithOrbit}). These ends correspond to the
corresponding collision end of the moduli space where the order of
$\rho_i$ and $\rho_{i+1}$ are permuted. (This is a different moduli
space, since the collision is visible.) Similarly, if $\rho_i$ and
$\rho_{i+1}$ are two Reeb chords that are not weakly composable (a
subcase of~\ref{typeDE:ChordChord}), we can permute them to get
another moduli space with a corresponding end. When $\rho_i$ and
$\rho_{i+1}$ are strongly composable chords, the corresponding ends
cancel against orbit ends (Type~\ref{typeDE:OrbitEnd}).
The two types of ends left unaccounted for are the two-story ends
(Type~\ref{typeDE:2StoryEnd}) and the boundary degeneration ends
(Type~\ref{typeDE:BoundaryDegeneration}). The fact there is an even
number of remaining ends gives the type $D$ structure relation.
\end{proof}
The above three propositions can be summarized as follows: the vector
space $\Dmod(\Hup)$, with differential as in
Equation~\eqref{eq:TypeDOperation} is a curved type $D$ structure,
with a homological grading induced by $\Mgr$ and Alexander gradings
$\Agr$.
The following invariance property of this curved type $D$ structure
will be important for us:
\begin{prop}
If $J_0$ and $J_1$ are any two generic almost-complex structures,
there is a type $D$ structure quasi-isomorphism of graded type $D$ structures
\[\Dmod(\Hup,J_0)\simeq \Dmod(\Hup,J_1).\]
\end{prop}
\begin{proof}
As usual, one must show that a path $\{J_t\}_{t\in[0,1]}$ induces a
type $D$ morphism. This is done via the straightforward
modification of Theorem~\ref{thm:DEnds} to varying $\{J_t\}$, and
considering moduli spaces between generators $\mathbf x$ and $\mathbf y$ where the
with index $1$.
Specifically, fix a generic one-parameter family
$\{J_t\}_{t\in[0,1]}$ of almost-complex structures, and consider the
moduli space $\ModFlow^B(\mathbf x,\mathbf y;\{J_t\})$ of $J_t$-holomorphic curves,
where $t$ is the second parameter in the projection to $[0,1]\times
\R$ (parameterized by pairs $(s,t)$). For such moduli spaces, the
ends of Type~\ref{typeDE:BoundaryDegeneration} do not exist (such
moduli spaces connect a generator to itself; and indeed they count
curves with index $2$). With this remark in place, the above proof
of Proposition~\ref{prop:CurvedTypeD} shows that there is an even
number of two-story ends. This immediately shows that the map
\[ h^1(\mathbf x)=\sum_{\mathbf y\in\States,B\in\doms(\mathbf x,\mathbf y)}
\bOut(B)\otimes\#\ModFlow^B(\mathbf x,\mathbf y;\{J_t\})\cdot \mathbf y.\] gives a type
$D$ morphism
\[ h^1\colon \Dmod(\Hup,J_0)\to \Clg(n)\otimes \Dmod(\Hup,J_1). \]
Homotopies of paths of complex structures induce homotopies of type
$D$ morphisms and the identity path induces the identity map as
usual; so it follows that $h^1$ is a type $D$ quasi-isomorphism.
Verifying that the maps are graded is straightforward.
\end{proof}
\begin{remark}
\label{rem:NoInvariance}
The above proposition shows that the quasi-isomorphism type of the
type $D$ structure of an upper Heegaard diagram is independent of
the analytical choices (of almost-complex structures) made. One
could aim for more invariance. One could think of an upper Heegaard
diagram as in fact representing an upper knot diagram, and then try
to prove dependence of the type $D$ structure only on the upper knot
diagram; and indeed one could try to show that it is an invariant of
the tangle represented by the diagram. We do not pursue this route,
in the interest of minimizing the road to
Theorem~\ref{thm:MainTheorem}.
\end{remark}
\subsection{Examples}
\label{subsec:ExampleTypeDs}
\begin{figure}[h]
\centering \input{UpperHeegTref.pstex_t}
\caption{{\bf Upper Heegaard
diagram.} This diagram has five Heegaard states, corresponding
to the five intersection points of $\beta$ with
$\alpha_1\cup\alpha_2\cup\alpha_3$. One of these ($x_2$ in the text)
is indicated in
black.} \label{fig:UpperHeegTref} \end{figure}
We start with a simple example. Consider the upper Heegaard diagram
from Figure~\ref{fig:UpperHeegTref}. This has five Heegaard states, which
we label from left to right, $x_1$, $x_2$, $t$, $y_1$, $y_2$. Since we
have $d=1$, the curve counting is straightforward, and we find that
the type $D$ structure is as indicated in the following diagram:
\begin{equation}
\label{eq:Dmod}
\begin{tikzpicture}[scale=1.8]
\node at (0,0) (x1) {$x_1$} ;
\node at (1,0) (x2) {$x_2$} ;
\node at (2,0) (t) {$t$} ;
\node at (3,0) (y1) {$y_1$} ;
\node at (4,0) (y2){$y_2$} ;
\draw[->] (x1) [bend left=15] to node[above,sloped] {\tiny{$U_4$}} (x2) ;
\draw[->] (x2) [bend left=15] to node[below,sloped] {\tiny{$U_3$}} (x1) ;
\draw[->] (x2) [bend left=15] to node[above,sloped] {\tiny{$L_2 U_1$}} (t) ;
\draw[->] (t) [bend left=15] to node[below,sloped] {\tiny{$R_2$}} (x2) ;
\draw[->] (t) [bend left=15] to node[above,sloped] {\tiny{$L_3$}} (y1) ;
\draw[->] (y1) [bend left=15] to node[below,sloped] {\tiny{$R_3 U_4$}} (t) ;
\draw[->] (y1) [bend left=15] to node[above,sloped] {\tiny{$U_2$}} (y2) ;
\draw[->] (y2) [bend left=15] to node[below,sloped] {\tiny{$U_1$}} (y1) ;
\draw[->] (x2) [bend left=40] to node[above,sloped] {\tiny{$L_2 L_3$}} (y2) ;
\draw[->] (y1) [bend left=40] to node[below,sloped] {\tiny{$R_3 R_2$}} (x1) ;
\end{tikzpicture}
\end{equation}
We give a (very simple) family of examples which will play a
fundamental role in our future computations.
For $n>1$, consider the
matching $M=\{\{1,2\},\{3,4\},\dots\{2i-1,2i\},\dots,\{2n-1,2n\}\}$ on
$\{1,\dots,2n\}$ where $2i-1$ is matched with $2i$ for
$i=1,\dots,n$. Let ${\mathbf s}=\{2i-1\}_{i=1}^{n}$, and
$\mu_0=\sum_{i=1}^n U_{2i-1} U_{2i}$. The type $d$ structure
with a single generator $\mathbf x$ satisfying
$\Idemp{\mathbf s}\cdot \mathbf x = \mathbf x$ and $\delta^1(\mathbf x)=0$ can be viewed
as a curved module over $\Clg(n)$ since $\Idemp{\mathbf s}\cdot
\mu_0=0$. We write this type $D$ structure $\lsup{\cClg}k$,
\begin{lemma}
\label{lem:StandardTypeD}
For any $n>1$, let $\Hup$ denote the standard upper diagram
with $2n$ local maxima (pictured in
Figure~\ref{fig:StandardUpperDiagram}; after deleting
$\beta_4$ and the basepoints $\wpt$ and $\zpt$).
There is an identification of type $D$ structures
\[ \Dmod(\Hup)\cong~\lsup{\cClg}k. \]
\end{lemma}
\begin{proof}
The diagram has exactly one Heegaard state $\mathbf x$; it has
$\Idemp{{\mathbf s}}\cdot \mathbf x = \mathbf x$; and there are no holomorphic
curves that can induce $\delta^1$-actions.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 330 |
Q: diamond character(�) appeared after fetch url in google script newbb got a question about character problem between google and webpage table.
I'm using google sheet and script to get a table from a opened website to google spreadsheet.
the page url is below (um.. I have reputation limit)
http://www.mogef.go.kr/system/korea/board/photo/photo.jsp?bid=341
At first time I used google spreadsheet function 'importhtml'.
The tables works fine but the language is changed weird.
I've changed google language option but didn't work.
picture of character problem at google sheet
so I trying to get HTML of table in google script by following api fetch example and
I thought this problem comes from charset.. encoding.. or something
so added an option about charset ('euc-kr' from the page HTML source)
//in google apps script
function myFunction() {
var option =
{
"contentType" : "charset=euc-kr"
};
var response = UrlFetchApp.fetch("~~~url~~~", option);
Logger.log(response.getContentText());
}
And It success to take HTML as text
but the language(same above language) looks like this..
diamond characters
How can I solve this problem..? I need normal text table or HTML text in google sheet or script. without MS excel.
A: According to the docs you can pass a charset to getContentText:
response.getContentText('EUC-KR')
This should return the characters correctly.
https://developers.google.com/apps-script/reference/url-fetch/http-response#getcontenttextcharset
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,568 |
{"url":"http:\/\/materialy.rubesz.cz\/rovnice\/kvarce\/subtopic12-example\/","text":"1. \u0158e\u0161te pro $x\\in\\mathbb R$. $(a)\\ x^2-5x+6=0$ $(b)\\ 6x^2+x-1=0$ $(c)\\ x^2+4x+13=0$ $(d)\\ 3x^2=12x+12$ $(e)\\ (x^2-4x-5)(x-3)=0$ $(f)\\ 5x^6-20x^4=0$ $(g)\\ x^3-4x^2+4x=0$ $(h)\\ x^4+3x^2-4=0$ $(i)\\ \\dfrac{x}{x^2-4}=1$ $(j)\\ \\dfrac{x}{x-1}-\\dfrac{x-1}x=1$ $(k)\\ \\dfrac{x^2+2x}{2x^2+2x-4}=1$","date":"2022-05-25 21:42:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 23, \"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\": 23, \"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\": 0.5921856164932251, \"perplexity\": 78.31188769101627}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662594414.79\/warc\/CC-MAIN-20220525213545-20220526003545-00398.warc.gz\"}"} | null | null |
\section{Introduction}
The {\em length} of a path in an {\em edge-weighted} graph is the sum
of the weights of edges on the path. A {\em shortest} path between two
vertices in a graph has minimal length among all paths between these
vertices. The {\em distance} between vertices $v$ and $u$
is the length of a shortest path from $v$ to $u.$
If the graph is directed, the paths are supposed to be also
directed.
Shortest path problems, in particular the single-source shortest paths
problem (SSSP) and the all-pairs shortest paths problem (APSP), belong
to the most basic and important problems in graph algorithms
\cite{CLR,Z01}. There are several variants of SSSP and APSP depending
among other things on the restrictions on edge weights and the input
graphs. The input to these problems is a directed or an undirected
edge-weighted graph. The output is a representation of shortest
paths between the source and all other vertices or between all pairs of
vertices in the graph, respectively.
In the general case of directed graphs (without negative cycles), when both positive and
negative real edge weights are allowed, the difference between the best
known asymptotic upper time-bounds for SSSP and APSP respectively is
surprisingly small. Namely, if the input directed graph has $n$
vertices and $m$ edges with real weights, then the best known SSSP
algorithm due to Bellman \cite{Bel58}, Ford \cite{For56}, and Moore
\cite{Mo59} runs in $O(nm)$ while the APSP can be solved already in
$O(nm+n^2\log n)$ time \cite{sur17,Z01}. The APSP solution uses
Johnson's $O(nm)$-time reduction of the general edge weight case to
the non-negative edge case and then it runs Dijkstra's algorithm
\cite{Dij59} $n$ times \cite{sur17,Z01}. The latter upper time-bound
for APSP with arbitrary real edge-weights has been more recently improved to
$O(nm+n^2\log \log n)$ by Pettie in \cite{Pet04}.
Note that the aforementioned
best asymptotic upper time bounds for SSSP and APSP are different only for
sparse graphs with $o(n\log \log n)$ edges. Interestingly,
when edge weights are integers,
the best known upper time-bound for APSP
just in terms of $n$ is $n^3/2^{\Omega(\sqrt {\log n})}$
\cite{CW16}.
The situation alters dramatically when the input directed graph is
acyclic, i.e., when it does not contain directed cycles. Then, a
simple dynamic programming algorithm processing vertices in a
topologically sorted order solves the SSSP problem in $O(n+m)$ time
\cite{CLR}, an $O(n(n+m))$-time solution to the APSP problem in this case
follows.
In fact, Yen could use the aforementioned method for SSSP in DAGs
iteratively in order to improve the time complexity of Bellman-Ford
algorithm for directed graphs by a constant factor \cite{Yen70}.
Bellman-Ford algorithm runs in $n-1$ iterations. In each iteration, for
each edge $e$, the current distance (from the source) at the head
of $e$ is compared to the sum of the current distance at the tail of
$e$ and the weight if $e.$ If the sum is smaller the distance at the
head of $e$ is updated. To achieve the improvement, Yen imposes a
linear order on the vertices of the input directed graph which yields
a decomposition of the graph into two DAGs. Next, the SSSP method for
DAGs is run on each of the two DAGs instead of an iteration of
Bellman-Ford algorithm \cite{Yen70}. Bannister and Eppstein
obtained a further improvement of the time complexity of Bellman-Ford
algorithm by a constant factor using a random linear order \cite{BE11}.
A pair of vertices in an edge weighted undirected or directed graph
can be connected by several paths, in particular several shortest
paths. Beside the length of a path, the number of edges forming it can
be an important characteristic. For example, Zwick provided several
exact and approximation algorithms for all pairs {\em lightest} (i.e.,
having minimal number of edges) shortest paths in directed graphs with
restricted edge weights in \cite{Z99}.
\junk{Recall also the Bellman-Ford
algorithm for SSSP in directed graphs with positive and negative edge
weights. It runs in $n-1$ iterations, in each iteration for each edge
$e$ the current distance (from the source) at the head of $e$ is
compared to the sum of the current distance at the tail of $e$ and the
weight if $e.$ If the sum is smaller the distance at the head of $e$
is updated.
Note that after $k$ iterations the current distances are
not greater than the lengths of shortest corresponding paths using at
most $k$ edges.}
In this paper, first we consider {\em $t+$light paths}, i.e., directed
paths that have at most $t$ more edges than the paths with the same
endpoints having the minimal number of edges. In part following
\cite{Yen70}, we iterate $O(t)$ times the SSSP method for DAGs on
two implicit
DAGs yielded by an extension of the BFS partial order to a
linear order. The iterations alternatively process the vertices in a
breadth-first sorted order and the reverse order. In result, we obtain
path distances from the source to all other vertices that are not
greater than the corresponding shortest-path distances for $t+$light
paths. It takes $O(tm)$ time totally. For $t=n-2$, our method matches
that of Bellman-Ford for SSSP in directed graphs with real edge
weights.
A vertex $v$ is an {\em ancestor} ({\em direct ancestor},
respectively) of a vertex $u$ in a DAG if there is a directed path
(edge, respectively) from $v$ to $u$ in the DAG.
Our main result is an output-sensitive algorithm for the
APSP problem in DAGs.
It runs in time
$O(\min \{n^{\omega}, nm+n^2\log n\}+\sum_{v\in V}\text{indeg}(v)|\text{leaf}(T_v)|),$ where
$n$ is the number of vertices, $m$ is the number of edges,
$\omega$ is the exponent
of fast $n\times n$ matrix multiplication
\footnote{$\omega$ is not greater than $2.3729$
\cite{AV21}.},
$\text{indeg}(v)$ stands for the indegree
of $v,$ $T_v$ is a tree of lexicographically-first shortest
directed paths from all ancestors of $v$ to $v$, $\text{leaf}(T_v)$ is
the set of leaves in $T_v,$ and for a set $X$, $|X|$ stands for its size.
Note that if $T_v$ is a path
the term $O(\text{indeg}(v)|\text{leaf}(T_v)|)$ equals
$O(\text{indeg}(v))$ while when $T_v$ is a star with $v$ as a
sink the term becomes $O(\text{indeg}(v)|T_v|).$
Thus, the running time of the APSP algorithm
can be so low as $O(n^{\omega})$ and so high as $O(n^{\omega}+nm).$
It follows also that if $\alpha$ is defined by
$\max_{v\in V} |\text{leaf}(T_v)|=O(n^{\alpha})$
then the algorithm
runs in $O(n^{\omega}+mn^{\alpha})$ time.
Similarly, if $\beta $ is defined by $\frac {\sum_{v\in V} |\text{leaf}(T_v)|} n=O(n^{\beta})$
then the algorithm runs in $O(n^{\omega}+n^{2+\beta})$ time.
Next, we provide an extension of hypothetical, improved upper
time-bounds for APSP in DAGs with non-negative edge weights to include
directed graphs with a polynomial number of large directed cycles.
Finally, we present experimental comparisons of our SSSP
algorithm with the Bellman-Ford one and our output-sensitive APSP algorithm
for edge-weighted DAGs with the standard APSP algorithm for edge-weighted DAGS.
In particular, they show that our SSSP algorithm
converges to the true shortest-path distances
on dense edge-weighted pseudorandom graphs faster than
the Bellman-Ford algorithm does. On the other hand, they
exhibit only a slight time-performance advantage of our APSP algorithm
over the standard APSP algorithm on dense edge-weighted pseudorandom DAGs.
Presumably, the shortest-path trees in the aforementioned DAGs
have large number of leaves.
\subsection{Paper organization}
In the next section, we provide our solution to the SSSP problem in
directed graphs with real edge weights based on the SSSP method for
DAGs and the BFS partial order in terms of $t+$light paths.
Section 3 is devoted
to our output-sensitive algorithm for the APSP problem in DAGs with
real edge weights and its analysis. In Section 4, we discuss the
extension of hypothetical, improved bounds for APSP in DAGs with
non-negatively weighted edges to directed graphs with a
polynomial number of large directed cycles.
Section 5 presents our experimental results.
We conclude with final
remarks.
\section{An application of the SSSP method for DAGs}
The SSSP problem for directed acyclic graphs can be solved by
topologically sorting the DAG vertices and applying straightforward dynamic
programming. For consecutive vertices $v$ in the sorted order, the distance
$dist(v)$ of $v$ from the source
is set to the minimum of $dist(u)+weight(u,v)$ over all
direct ancestors $u$ of $v$, where $weight(u,v)$
stands for the weight of the edge $(u,v)$. It takes linear (in the size of the DAG)
time. Yen used the dynamic programming method iteratively to improve
the time complexity of Bellman-Ford algorithm for directed graphs by a
constant factor in \cite{Yen70}. Interestingly, we can similarly
apply this method iteratively to determine shortest-path distances
among paths using almost the minimal number of edges. To formulate our
algorithm (Algorithm 1), we need the following definition and two
procedures.
\begin{definition}
A directed path from a vertex $u$ to a vertex $v$
in a directed graph is {\em lightest} if it consists
of the smallest possible number of edges.
A path from $u$ to $v$ is $t+$light if
it includes at most $t$ more edges than
a lightest path from $u$ to $v.$
\end{definition}
\par
\noindent
{\bf procedure} $SSSPDAG(G,D)$
\par
\noindent
{\em Input:} A directed graph $(V,E)$
with real edge weights, linearly ordered
vertices $v_1,....,v_n,$
and
a $1$-dimensional table $D$ of size $n$
with upper bounds on the distances
from $v_1$ to all vertices in $V.$
\par
\noindent
{\em Output:} Improved upper bounds on the shortest-path distances from
$v_1$ to all vertices in $V$ in the table $D.$
\par
\vskip 2pt
\noindent
{\bf for} $j=2,...,n$ {\bf do}
\par
\noindent
For each edge $(v_i,v_j)$ where $i<j$\\
$D(v_j) \leftarrow \min\{ D(v_j),D(v_i)+weight(v_i,v_j)\}$
\par
\vskip 5pt
\noindent
{\bf procedure} $reverseSSSPDAG(G,D)$
\par
\vskip 3pt
\noindent
{\em Input and output:} the same as in $SSSPDAG(G,D)$
\par
\vskip 2pt
\noindent
{\bf for} $j=n-1,...,1$ {\bf do}
\par
\noindent
For each edge $(v_i,v_j)$ where $i>j$\\
$D(v_j) \leftarrow \min\{ D(v_j),D(v_i)+weight(v_i,v_j)\}$
\par
\vskip 6pt
\noindent
{\bf Algorithm 1}
\par
\noindent
{\em Input:} A directed graph $(V,E)$
with $n$ vertices, real edge weights and a distinguished
source vertex $s$, and a positive integer $t.$
\par
\noindent
{\em Output:} Upper bounds on the shortest-path distances from
$s$ to all other vertices in $V$ not exceeding the
corresponding shortest-path
distances constrained to $t+$light paths.
\begin{enumerate}
\item Run BFS from the source $s$.
\item Order the vertices of $G$
extending the BFS partial order according to the levels
of the tree, i.e., $s$ comes first,
then the vertices reachable by direct edges from $s$,
then the vertices reachable by paths composed of two
edges and so on. We may assume w.l.o.g. that all vertices
are reachable from $s$ or alternatively extend the aforementioned
order with the non-reachable vertices arbitrarily.
\item Initialize a $1$-dimensional
table $D$ of size $n,$
setting $D(v_1)\leftarrow 0$ and $D(v_j)\leftarrow\infty$ for $1<j\le n$
\item $SSSPDAG(G,D)$
\item {\bf for} $k=1,...,t $ {\bf do}
\begin{enumerate}
\item $reverseSSSPDAG(G,D)$
\item $SSSPDAG(G,D)$
\end{enumerate}
\end{enumerate}
\junk{
\begin{figure}
\label{fig: dag1}
\begin{center}
\includegraphics[scale=0.5]{dag1}
\end{center}
\caption{An example of a graph with a BFS vertex numbering and the
two DAGs implied by forward and backward edges, respectively.}
\end{figure}}
\begin{theorem}\label{theo: light}
Let $G$ be a directed graph with $n$ vertices,
$m$ real-weighted edges, and a distinguished source vertex $s.$
For all vertices $v$ of $G$ different from $s$,
an upper bound on their distance
from the source vertex $s$, not exceeding the length of a shortest path
among $t+$light paths from $s$ to $v,$
can be computed in $O((t+1)(m+n))$ total time.
\end{theorem}
\begin{proof}
Consider Algorithm 1 and in particular
the ordering of the vertices specified in its second step.
We shall refer to an edge $(v_i,v_j)$ as forward if
$i<j$ otherwise we shall call it backward.
Note that the vertices at the same level of the BFS tree
can be connected both by forward as well as backward edges.
See also Fig. 1.
Let $\ell$ be the number of (forward) edges
in a lightest path from $s$
to a given vertex $v.$
It follows that any path from $s$ to $v$,
in particular a shortest $t+$light one,
has to have at least $\ell$ forward edges.
Consider the BFS tree from the source $s.$
Define the level of a vertex in the tree as
the number of edges on the path from $s$
to the vertex in the tree. Thus, in particular,
$level(s)=0$ while $level(v)=\ell.$
Recall that the linear order
extending the partial BFS order used in
Algorithm 1 is non-decreasing with respect of
the levels of vertices. Also, if $(u,w)$
is a forward edge then $level(u)\le level(w)\le level(u)+1$
and if $(u,w)$ is a backward edge then $level(u)\ge level(w).$
Hence, any path from $s$ to $v$ has to have at least
$\ell$ forward edges, each increasing the level by one.
Consequently, a shortest $t+$light path from $s$ to $v$
can have at most $t$ backward edges.
Thus, it can be decomposed into at most $2t+1$ maximal
fragments of consecutive edges of the same type (i.e.,
forward or backward, respectively), where the even
numbered fragments consist of backward edges.
Thus, the at most $2t+1$
calls of the procedures
$SSSPDAG(G,s,D)$,\\
$reverseSSSPDAG(G,s,D)$ in the algorithm
are sufficient to detect a distance from $s$ to $v$
not exceeding the length of a shortest path
among $t+$light paths from $s$ to $v.$
The asymptotic running time of the algorithm is dominated
by the aforementioned procedure calls.
Hence, it is $O((t+1)(m+n))$.
\end{proof}
\begin{figure}
\label{fig: dag1}
\begin{center}
\includegraphics[scale=0.5]{dag1}
\end{center}
\caption{An example of a graph with a BFS vertex numbering and the
two DAGs implied by forward and backward edges, respectively.}
\end{figure}
We can obtain a representation of directed paths achieving the upper
bounds on the distances from the source provided in Theorem \ref{theo:
light} in a form of a tree of paths emanating from the source by
backtracking. By setting $t=n-2$ in this theorem, we can match the
best known SSSP algorithm for directed graphs with positive and
negative real edge weights, i.e., the Bellman-Ford algorithm and
its constant factor improvements \cite{sur17,Z01}, running in $O(nm)$
time. Similarly as in the case of Bellman-Ford algorithm, by calling
additionally $reverseSSSPDAG(G,D)$ and $SSSPDAG(G,D)$ after the
last iteration in Algorithm 1, we can detect the existence of negative
cycles.
Comparing our algorithm with the Bellman-Ford one, note that if the
lightest path from the source to a vertex $v$ has $\ell$ edges then
$\ell +t$ iterations in the Bellman-Ford algorithm may be needed to
obtain an upper bound on the distance of $v$ from the source comparable to
that obtained after $O(t)$ iterations in Algorithm 1.
\junk{
\begin{theorem}
The SSSP problem for directed graphs with $n$ vertices,
$m$ edges and non-negative edge weights
of $O(1)$ value can be solved in $O(nm)$ time.
\end{theorem}
\begin{proof}
Use Dijkstra's' algorithm with a $1$-dimensional table of size $O(n)$
instead of a priority queue.
\end{proof}}
\section{An output-sensitive APSP algorithm for DAGs}
The APSP problem in DAGs with both positive and negative real edge
weights can be solved in $O(n(n+m))$ time by running $n$ times the SSSP
algorithm for DAGs. It is an intriguing open problem if there exist
substantially more efficient algorithms for APSP in edge-weighted
DAGs. In this section, we make a progress on this question by
providing an output-sensitive algorithm for this problem.
Its running time depends on the structure of shortest path
trees. Although in the worst-case it does not break the $O(nm)$
barrier it seems to be substantially more efficient in the majority of
cases.
The standard algorithm for APSP for DAGs just runs the SSSP
algorithm for DAGs for
each vertex of the DAG as a source separately. Our APSP algorithm does
everything in one sweep along the topologically sorted order. Its main
idea is for each vertex $v$ to compute the tree of lexicographically-first
shortest paths from the ancestors $u$ of the currently processed
vertex $v$ to $v$, in the topologically sorted order. In case the tree
of lexicographically-first shortest paths from the already considered
ancestors of $v$ includes $u$ (as some intermediate vertex) then we
are done as for $u.$ Otherwise, we have to find the direct ancestor of
$v$ on the lexicographically-first shortest path $P$ from $u$ to $v$
and add an initial fragment of $P$ to the tree. By the topologically
sorted order in which the ancestors $u$ of $v$ are considered, this
can happen only when $u$ is a leaf of the (final) tree of
lexicographically-first shortest paths from the ancestors of $v$ to $v.$ The
direct ancestor of $v$ on $P$ can be found by comparing the lengths of shortest
paths from $u$ to $v$ with different direct ancestors of $v$ as the
next to the last vertex on the paths in time proportional to the
indegree of $v.$ In turn, the initial fragment of $P$
to add can be found by using the
link to the lexicographically-first shortest path from $u$ to the direct
ancestor of $v$ that is on $P.$ The correctness of the algorithm is
immediate. The issues are an implementation of these steps and an
estimation of the running time.
To specify our
output-sensitive algorithm (Algorithm 2) more
exactly, we need the following definition.
\begin{definition}
Assume a numbering of vertices in an edge-weighted DAG
extending the topological partial order.
A shortest (directed) path $P$ from $v_k$ to $v_i$ in the DAG
is {\em first in a lexicographic order} if the direct
ancestor $v_j$ of $v_i$ on $P$ has the lowest number $j$
among all direct ancestors of $v_i$ on shortest
paths from $v_k$ to $v_i$ and the subpath of $P$
from $v_k$ to $v_j$ is the lexicographically-first
shortest path from $v_k$ to $v_j.$ For a vertex $v_i$ in the DAG,
the tree $T_{v_i}$ of (lexicographically-first) shortest paths
is the union of lexicographically-first paths from
all ancestors of $v_i$ to $v_i.$ Note that the vertex $v_i$ is a sink
of $T_{v_i}.$ It is assumed to be the root of $T_{v_i}$ and
$\text{leaf}(T_{v_i})$ stands for the set of leaves of $T_{v_i}.$
\end{definition}
\par
\vskip 4pt
\noindent
{\bf Algorithm 2}
\par
\noindent
{\em Input:} A DAG $(V,E)$
with real edge weights.
\par
\noindent
{\em Output:} For each vertex $v\in V,$
the tree $T_v$ of lexicographically-first shortest paths from all
ancestors of $v$ to $v$ given by the table $NEXT_v$, where for
each ancestor $u$ of $v$, $NEXT_v(u)$ is the direct
successor of $u$ in the tree $T_v,$ (i.e., the head of the unique
directed edge having $u$ as the tail in the tree).
\begin{enumerate}
\item
Determine the source vertices,
topologically sort the remaining vertices in $V$,
and number the vertices in $V$ accordingly,
assigning to the sources the lowest numbers.
\item Set $n$ to $|V|$ and $r$ to the number
of sources in $G.$
\item Initialize an $n\times n$ table
$dist$ by setting $dist(u,u)=0$
and $dist(u,v)=\infty$ for $u,\ v \in V, \ u\neq v.$
\item {\bf for} $i=r+1,...,n$ {\bf do}
\begin{enumerate}
\item Compute
the set $A(v_i)$ of ancestors of $v_i.$
\item Initialize a $1$-dimensional table $NEXT_{v_i}$ of size $|A(v_i)|$,\\
setting $NEXT{v_i}(v_j)$ to $0$ for $v_j\in A(v_i)$.
\item {\bf for} $v_k\in A(v_i)$ in increasing order of the index $k$ {\bf do}
\begin{enumerate}
\item {\bf if} $NEXT_{v_i}(v_k)\neq 0$ {\bf then}
proceed to the next iteration of the interior for block.
\item Determine a direct ancestor $v_j$ of $v_i$ that
minimizes the value of $dist(v_k, v_j)+ weight(v_j,v_i)$.
In case of ties the vertex $v_j$ with the smallest index $j$ is
chosen among those yielding the minimum.
\item
$v_{current}\leftarrow v_k$
\item
{\bf while} $v_{current}\neq v_j$ $\&$ $NEXT(v_{current}, v_i)=0$ {\bf do}\\
$dist(v_{current}, v_i)\leftarrow dist(v_{current}, v_j) + weight(v_j, v_i$)\\
$NEXT_{v_i}(v_{current})\leftarrow NEXT_{v_j}(v_{current})$\\
$v_{current}\leftarrow NEXT_{v_i}(v_{current})$
\item
{\bf if} $NEXT_{v_i}(v_j)=0$ {\bf then} $dist(v_j, v_i)\leftarrow weigh(v_j, v_i)$ $\&$
$NEXT_{v_i}(v_j)\leftarrow v_i$
\end{enumerate}
\end{enumerate}
\end{enumerate}
\begin{lemma}\label{lem: path}
Steps 4.c.iii-v add the missing fragments of a lexicographically
shortest path from $v_k$ to $v_i$ and set
the distances from vertices in the fragments to $v_i$ in time proportional
to the number of vertices added to $T_{v_i}$.
\end{lemma}
\begin{proof}
Follow the path from $v_k$ to $v_j$ in $T_{v_j}$
extended by $(v_j,v_i)$ until
a vertex $v_q\in T_{v_i}$ is encountered.
This is done in Steps 4.c.iii-v.
The membership of $v_{current}$ in $T_{v_i}$
is verified by checking whether or not
$NEXT_{v_i}(v_{current})=0.$
Also, if $v_{current}$ is not yet in $T_{v_i}$
then its distance to $v_i$ is set
by $dist(v_{current}, v_i)\leftarrow dist(v_{current}, v_j) + weight(v_j, v_i)$
and it is added to $T_{v_i}$ by
$NEXT_{v_i}(v_{current})\leftarrow NEXT_{v_j}(v_{current})$
in Step 4.c.iv.
By the inclusion of $v_q$ in $T_{v_i}$,
a whole shortest path $Q$ from $v_q$ to $v_i$
is already included in $T_{v_i}$ by induction
on the number of steps performed by the algorithm.
We claim that $Q$ exactly overlaps with
the final fragment of the extended path starting from $v_q$.
To see this encode $Q$ and the aforementioned
fragment of the extended path
by the indices of their vertices in the reverse
order. By our rule of resolving ties in Step 4.c.ii
both encodings should be first in the
lexicographic order so we have an exact overlap.
For this reason, it is sufficient to add
the initial fragment of the extended path
ending at $v_q$ to $T_{v_i}$ and if necessary
also the edge $(v_j,v_i)$ to $T_{v_i},$ and to update
the distances from vertices in the added fragment to $v_i,$
i.e., to perform Steps 4.c.iii-v.
\end{proof}
\begin{theorem} \label{theo: main}
The APSP algorithm for a
DAG $(V,E)$ with $n$ vertices, $m$ edges and real edge weights
(Algorithm 2) runs in time
$O(\min \{n^{\omega}, nm+n^2\log n\}+\sum_{v\in V}\text{indeg}(v)|\text{leaf}(T_v)|).$
\end{theorem}
\begin{proof}
The sets of ancestors can be determined in Step 4.a
by computing the transitive closure of the input
DAG in $O(\min \{n^{\omega}, nm\})$ time by using fast matrix multiplication \cite{Mu71}
or BFS \cite{CLR}, first.
In fact, to implement the loop in Step4.c,
we need the sets of ancestors to be ordered according to the numbering
of vertices provided in Step 1. If the transitive closure matrix
is computed such an ordered set of ancestors can be easily
retrieved in $O(n)$ time. Otherwise, additional
preprocessing sorting the unordered sets of ancestors is needed.
The total cost of the additional preprocessing is $O(n^2\log n).$
All the remaining steps, excluding Steps 4.c.ii-v for vertices
$v_k$ not yet in $T_{v_i}$,
can be done
in total (i.e., over all iterations) time
$O(\sum_{v\in V}(1+|A(v)|))=O(n^2),$ where $A(v)$
stands for the set of ancestors of $v$ in the DAG.
The time taken by Step 4.c.ii,
when $v_k$ is not yet in the current $T_{v_i}$, is $O(\text{indeg}(v_i)).$
Suppose that $v_k$ is not a leaf of the final tree $T_{v_i}$.
Then, there must exist some leaf $v_p$ of the final tree
such that there is
path from $v_p$ via $v_k$ to $v_i$ in this tree.
By the numbering of vertices extending the partial topological
order, we have $p< k.$ We infer that the aforementioned
path is already present in the current $T_{v_i}$.
Thus, in particular the vertex $v_k$ is in the current tree.
Hence, the total time taken
by Step 4.c.ii is $O(\sum_{v\in V}\text{indeg}(v)|\text{leaf}(T_{v})|).$
Finally,
the total time taken by Steps 4.c.iii-v is $O(\sum_{v\in V}(1+|A(v)|))$
by Lemma \ref{lem: path}.
\end{proof}
Note that
the following inequalities hold:
$$\sum_{v\in V}\text{indeg}(v)|\text{leaf}(T_v)|\le m\max_{v\in V} |\text{leaf}(T_v)|,$$
$$\sum_{v\in V}\text{indeg}(v)|\text{leaf}(T_v)|\le n^2\frac {\sum_{v\in V} |\text{leaf}(T_v)|} n.$$
They immediately yield the following corollary from Theorem \ref{theo: main}.
\begin{corollary}
Let $G=(V,E)$ be an $n$-vertex
DAG with $n$ vertices and $m$ edges
with real edge weights.
Suppose $\max_{v\in V} |\text{leaf}(T_v)|=O(n^{\alpha})$
and\\
$\frac {\sum_{v\in V} |\text{leaf}(T_v)|} n=O(n^{\beta})$.
The APSP problem for $G$ is solved by
Algorithm 2 in time
$O(\min \{n^{\omega}, nm+n^2\log n\}+\min \{ mn^{\alpha},n^{2+\beta}\}).$
\end{corollary}
Observe that $ |\text{leaf}(T_v)|$ is equal to the minimum
number of directed paths covering the tree $T_v.$
Hence, $\alpha < 1$ if the maximum of the minimum number
of paths covering $T_v$ over $v$ is substantially sublinear.
Similarly, $\beta < 1$ if the average of the minimum number
of paths covering $T_v$ over $v$ is substantially sublinear.
To illustrate the superiority of Algorithm 2 over
the standard $O(n(n+m))$-time method for APSP in DAGs, consider
the following simple, extreme example.
\par
\noindent
Suppose $M$ is a positive integer.
Let $D$ be a DAG with vertices $v_1, v_2$,...,$v_n,$
and edges $(v_i,v_j),$ where $i<j,$ such that
the weight of $(v_i,v_j)$ is $-1$ if $j=i+1$ and $M$ otherwise.
\par
\noindent
It is easy to see the tree $T_{v_i}$ is just the path
$v_1,\ v_2,...,v_i$ and hence $|\text{leaf}(T_{v_i})|=1.$
Consequently, Algorithm 2 on the DAG $D$ runs in $O(n^{\omega})$
time while the standard method requires $O(n^3)$ time.
If $M=1,$
one could also run Zwick's APSP algorithm for directed
graphs with edge weights in $\{-1,0,1\}$ on this example
in $O(n^{2.575})$ time \cite{Z02}.
To refine Theorem \ref{theo: main}, we need the following
definition.
\begin{definition}
For an edge weighted DAG $G$, let
$\tilde{G}$ be the edge weighted DAG resulting from
reversing the direction of edges in $G.$
For a vertex $v_i$ in the DAG $G$,
the tree $U_{v_i}$ of (lexicographically-first in reversed order) shortest paths from $v_i$
to all descendants of $v_i$ in $G$ is just the tree
resulting from the tree $T_{v_i}$ in $\tilde{G}$
by reversing the edge directions.
Note that the vertex $v_i$ is a source
of $U_{v_i}.$ It is assumed to be the root of $U_{v_i}$ and
$\text{leaf}(U_{v_i})$ stands for the set of leaves of $U_{v_i}.$
\end{definition}
The APSP for edge weighted DAGs can be solved
by providing the trees $U_v$ instead of the trees $T_v.$
Hence, we obtain immediately the following strengthening
of Theorem \ref{theo: main} by symmetry.
\begin{theorem} \label{theo: gmain}
The APSP problem for a
DAG $(V,E)$ with $n$ vertices, $m$ edges and real edge weights
can be solved in time
$$O(\min \{n^{\omega}, nm+n^2\log n\}+
\min \{ \sum_{v\in V}\text{indeg}(v)|\text{leaf}(T_v),
\sum_{v\in V}\text{outdeg}(v)|\text{leaf}(U_v)\}
).$$
\end{theorem}
\begin{proof}
Alternate the steps of Algorithm 2 run on the input DAG $G$
with those of Algorithm 2 run on the DAG $\tilde{G}$.
When any of the two runs finishes we are basically
done. In case the run of Algorithm 2 on the DAG $\tilde{G}$
finishes first, we obtain the trees $U_v$ in $G$
from the trees $T_v$ in $\tilde{G}$ by
reversing the direction of edges.
The upper time bound follows from Theorem \ref{theo: main}
and the fact that the indegree of a vertex in
$\tilde{G}$ is equal to its outdegree in $G.$
\end{proof}
\junk{
\par
\vskip 4pt
\noindent
{\bf Algorithm 3}
\par
\noindent
{\em Input:} A DAG $(V,E)$
with real edge weights.
\par
\noindent
{\em Output:} For each vertex $v\in V,$
the tree $T_v$ of lexicographically first shortest paths from all
ancestors of $v$ to $v.$
\noindent
\begin{enumerate}
\item Topologically sort the vertices in $V$
and number the vertices in $V$ accordingly.
\item Set $n$ to $|V|$ and $r$ to the number
of sources in $G.$
\item Initialize an $n\times n$ table
$dist$ by setting $dist(u,u)=0$
and $dist(u,v)=\infty$ for $u,\ v \in V, \ u\neq v.$
\item Set $L$ to an empty set.
\item {\bf for} $i=r+1,...,n$ {\bf do}
\begin{enumerate}
\item Compute
the set $A(v_i)$ of ancestors of $v_i.$
\item Initialize a $1$-dimensional table $L_{v_i}$ of size $|A(v_i)|$,
setting $L_{v_i}(v_j)$ to an empty link for $v_j\in A(v_i)$.
\item Set $T_{v_i}$ to the singleton tree $v_i$.
\item {\bf for} $v_k\in A(v_i)$ in increasing order of the index $k$ {\bf do}
\begin{enumerate}
\item {\bf if} $v_k$ is in $T_{v_i}$ {\bf then}
proceed to the next iteration of the interior for block.
\itrm {\bf if} $v_k$ is not in $L$ {\bf then}
solve the SSSP problem with the source $v_k$
on the sub-DAG induced by vertices $v_k$ through $v_n$,
in particular computing $T^{v_k}$ and updating
the entries $dist(V_k,*).$
\item Set $v_j$ to the direct ancestors of $v_j$
in $T^{v_k}$.
\item Using the link $L_{V_j}(v_k)$ find and
add the not yet added fragments
of the path from $V_k$ to $v_j$
and the edge $(v_j,v_i)$ to $T_{v_j}$.
\item Set $L_{v_i}(v_k)$ to the link to the beginning of the added
path in $T_{v_j}.$ Analogously, for all newly added to $T_{v_i}$ vertices $v_q,$
set $L_{v_i}(v_q)$ to the link to the position
of $v_q$ in $T_{v_j}.$
\end{enumerate}
\end{enumerate}
\end{enumerate}
\begin{theorem}
Algorithm 3 solves the
APSP problem on a DAG with $$ vertices and $m$ real-weighted
edges in time
$O(n^{\omega}+m |\bigcup_{v\in V}\text{leaf}(T_v)|).$
\end{theorem}}
\section{A potential extension to digraphs with large cycles}
As we have already
noted the APSP problem in DAGs with both positive and negative real edge
weights can be solved in $O(n(n+m))$ time.
It is also an interesting open problem
if one can derive substantially more efficient algorithms for APSP in DAGs
than the $O(n(n+m))$-time method in case of
restricted edge weights, e.g., non-negative edge weights etc.
In this section, under the assumption of the existence
of such substantially more efficient algorithms for DAGs
with non-negative edge weights, we show that
they could be
extended to include directed graphs having
a polynomial number of large cycles.
The idea of the extension is fairly simple, see Fig. 2. We pick
uniformly at random a sample of vertices of the input directed graph
that hits all the directed cycles with high probability
(cf. \cite{Z02}). Here, we use the assumption on the minimum size of
the cycles and on the polynomially bounded number of the
cycles. Next, we remove the vertices belonging to the sample and run
the hypothetical fast algorithm for APSP in DAGs on the resulting
subgraph of the input graph which is acyclic with high probability.
In order to take into account shortest path connections using the
removed vertices, we run the Dijkstra's SSSP algorithm from each
vertex in the sample on the original input graph two times. In the
second run we reverse the directions of the edges in the input
graph. Finally, we update the shortest path distances
appropriately. See Algorithm 3 for a more detailed description.
\par
\vskip 4pt
\noindent
{\bf Algorithm 3}
\par
\noindent
{\em Input:} A directed graph $(V,E)$ with $n$ vertices,
$m$ non-negatively weighted edges
and a polynomial
number of directed cycles,
each with at least $d$ vertices.
\par
\noindent
{\em Output:} The shortest-path distances for all
ordered pairs of vertices in $V.$
\noindent
\begin{enumerate}
\item
Initialize an $n\times n$ array $D$ by setting
all its entries outside the main diagonal
to $+\infty$ and those on the diagonal
to zero.
\item Uniformly at random pick a sample $S$
of $O(n\ln n/d)$ vertices from $V.$
\item Run the hypothetical APSP algorithm for DAGs on the graph\\
$(V\setminus S,\ E\cap \{(u,v)|u,v \in V\setminus S\})$
and for each pair $u,\ v\in V\setminus S,$
set $D(u,v)$ to the distance determined
by the algorithm.
\item
For each $s\in S$, run the Dijkstra's SSSP algorithm
with $s$ as the source in $(V,E)$
and for all $v\in V\setminus \{s\}$
update the $D(s,v)$ entries respectively.
\item
For each $s\in S$, run the Dijkstra's SSSP algorithm
with $s$ as the source on the directed graph resulting
from reversing the directions of the edges in $(V,E),$
and for all $ v\in V\setminus \{s\}$
update the $D(v,s)$ entries respectively.
\item
For all pairs $u,\ v$ of distinct vertices in $V\setminus S,$
and for all vertices $s\in S,$
set $D(u,v)=\min \{D(u,v), D(u,s)+D(s,v)\}$.
\end{enumerate}
\begin{figure}
\label{fig: dag2}
\begin{center}
\includegraphics[scale=0.5]{dag2}
\end{center}
\caption{An example of a directed cycle that can be broken
by removing the encircled vertex belonging to the sample.
To find shortest-path connections passing through this vertex
two SSSP from it are performed, in the original and the reversed
edge directions, respectively.}
\end{figure}
\begin{theorem}\label{theo: 3}
Let $t(n,m)$ be the time required by APSP in DAGs with $n$
vertices and $m$ non-negatively weighted
edges. Algorithm 3 solves the APSP problem for a
directed graph with $n$ vertices, $m$ non-negatively weighted edges
and a polynomial number of directed cycles, each with at least $d$
vertices, in $O(t(n,m)
+n^3\ln n/d)$ time with high probability.
\end{theorem}
\begin{proof}
Suppose that the number of directed cycles in
the input graph $(V,E)$ is
$O(n^c).$ By picking enough large constant
for the expression $n\ln n /d$ specifying the size of the sample $S,$
the probability that a given directed cycle in $G$
is not hit by $S$ can be made smaller than $n^{-c-1}.$
Hence, the probability that the graph resulting from
removing the vertices in $S$ is not
acyclic becomes smaller than $n^{-1}.$
It follows that Algorithm 3 is correct with high probability.
It remains to estimate its running time.
Steps 1, 2 can be easily implemented in $O(n^2)$ time.
Step 3 takes $t(n,m)$ time.
Steps 4, 5 can be implemented
in $O((n\ln n/d) \times m +n^2 \ln^2 n /d)$ time \cite{CLR}.
Finally, Step 6 takes $O(n^3\ln n/d)$ time.
\end{proof}
Note that because of the term $n^3\ln n/d$ in the upper time-bound
given by Theorem \ref{theo: 3}, the upper bound can be substantially
subcubic only when $d=\Omega (n^{\delta})$ for some $\delta > 0.$
\section{Experimental results}
We have implemented Algorithm 1 and the Bellman-Ford algorithm in order to
compare the quality of their estimation of the shortest-path distances
after corresponding iterations. We have also implemented Algorithm 2
and the standard APSP algorithm for DAGS ($n-1$ runs of of the SSSP
dynamic programming algorithm for DAGs) in order to compare their
running times.
For the comparison sake, we used Erdős–Rényi $G(n, p)$ random graph
model, and generated $100$ pseudorandom graphs for $n \in \{10, 100,
1000\}$ and $p \in \{0.2, 0.4, 0.6, 0.8\}$.
We used
\texttt{mt19937} implementation of Mersenne Twister pseudorandom
number generator from GNU C++ Standard Library version 10.2.
Pseudorandom integer
weights from the interval $[-1000, 1000]$ were assigned to the edges.
In case of the APSP algorithms for DAGS, the generated pseudorandom
graphs were converted into DAGs simply by directing each edge
$\{v_i,v_j\}$, where $i<j,$ from $v_i$ to $v_j.$
All four algorithms
were implemented in C++ and Google Benchmark library
was used to measure the CPU time. High-resolution clock with
nanosecond precision was
used for time measurement. The code
was
compiled with \texttt{-O2} optimization flag using GNU C++ Compiler
version 10.2, and was executed on a PC with Intel Core i5-2557M 2.7
GHz CPU and 4 GB RAM running Linux kernel version 5.11.15.
\subsection{Algorithm 1}
We have compared the quality of estimations of shortest-path distances
in initial iterations of Algorithm 1 and the Bellman-Ford
algorithm. We count Step 4 as the first iteration, and then each
performance of Steps 5.a and 5.b as consecutive iterations of
Algorithm 1. In an iteration of the Bellman-Ford algorithm, for each
edge $e$, the current distance (from the source) at the head of $e$ is
compared to the sum of the current distance at the tail of $e$ and the
weight if $e.$ If the sum is smaller the distance at the head of $e$
is updated.
Figures 4 and 5 (see Appendix) show the proportions between the numbers
of vertices for which Algorithm 1 or the Bellman-Ford algorithm
respectively provides a sharper estimation of the shortest-path
distance from the source in corresponding iterations for pseudorandom
graphs on $10$ and $100$ vertices. The figures support the claim
that Algorithm 1 provides reasonable estimation substantially
faster than the Bellman-Ford algorithm does.
\subsection{Algorithm 2}
In one of the initial steps of Algorithm 2, the transitive closure of
the input DAG is computed. For dense DAGs, the computation of the
transitive closure involves fast matrix multiplication algorithm
known to have huge overhead. Since we run Algorithm 2 on relatively small DAGs
where the aforementioned overhead could shadow the time performance of
the core of the algorithm, we do not account the time taken by the
transitive closure step in our results. See Figure 3.
\junk{
The mean and standard deviation values of the measured CPU time are
reported in the table in Fig. 5.
\begin{figure}[h]
\begin{tabular}{llllll}
\toprule
& & $n=10$ & $n=100$ & $n=1000$ & $n=10000$ \\
\midrule
\multirow{2}{*}{$p=0.2$}
& Baseline & $0.000371\pm0.000035$ & $0.285\pm0.012$ & $236\pm2$ & $238347\pm376$ \\
& Algorithm 2 & $0.000221\pm0.000040$ & $0.162\pm0.009$ & $94\pm2$ & $120679\pm929$ \\
\midrule
\multirow{2}{*}{$p=0.4$}
& Baseline & $0.000504\pm0.000047$ & $0.433\pm0.016$ & $461\pm4$ & $458933\pm1631$ \\
& Algorithm 2 & $0.000353\pm0.000063$ & $0.230\pm0.015$ & $150\pm4$ & $175463\pm947$ \\
\midrule
\multirow{2}{*}{$p=0.6$}
& Baseline & $0.000646\pm0.000053$ & $0.557\pm0.018$ & $697\pm7$ & $683306\pm917$ \\
& Algorithm 2 & $0.000496\pm0.000076$ & $0.277\pm0.0230$ & $196\pm7$ & $221904\pm2371$ \\
\midrule
\multirow{2}{*}{$p=0.8$}
& Baseline & $0.000789\pm0.000052$ & $0.705\pm0.018$ & $945\pm7$ & $913757\pm1281$ \\
& Algorithm 2 & $0.000607\pm0.000089$ & $0.309\pm0.029$ & $233\pm7$ & $259436\pm2522$ \\
\bottomrule
\end{tabular}
\caption{Mean and standard deviation for CPU time in milliseconds}
\end{figure}
}
\begin{figure}[h]
\begin{tabular}{lllll}
\toprule
& & $n=10$ & $n=100$ & $n=1000$ \\
\midrule
\multirow{2}{*}{$p=0.2$}
& Baseline & $0.000371\pm0.000035$ & $0.285\pm0.012$ & $236\pm2$ \\
& Algorithm 2 & $0.000221\pm0.000040$ & $0.162\pm0.009$ & $94\pm2$ \\
\midrule
\multirow{2}{*}{$p=0.4$}
& Baseline & $0.000504\pm0.000047$ & $0.433\pm0.016$ & $461\pm4$ \\
& Algorithm 2 & $0.000353\pm0.000063$ & $0.230\pm0.015$ & $150\pm4$ \\
\midrule
\multirow{2}{*}{$p=0.6$}
& Baseline & $0.000646\pm0.000053$ & $0.557\pm0.018$ & $697\pm7$ \\
& Algorithm 2 & $0.000496\pm0.000076$ & $0.277\pm0.0230$ & $196\pm7$ \\
\midrule
\multirow{2}{*}{$p=0.8$}
& Baseline & $0.000789\pm0.000052$ & $0.705\pm0.018$ & $945\pm7$ \\
& Algorithm 2 & $0.000607\pm0.000089$ & $0.309\pm0.029$ & $233\pm7$\\
\bottomrule
\end{tabular}
\caption{Mean and standard deviation for CPU time in milliseconds}
\end{figure}
\junk{
For further illustration the benchmark results see Fig. 6, 7
in Appendix. Fig. 6 presents a plot showing the mean
and the standard deviation for CPU time versus $n$
in the logarithmic scale. Fig. 7 presents
scatter plots showing CPU time versus
$m$ (number of edges) for different values of $n.$}
\section{Final remarks}
In the absence of substantial asymptotic improvements to the time
complexity of basic shortest-path algorithms, often formulated at
the end of 50s, like the Bellman-Ford algorithm and Dijkstra's
algorithm, the results presented in this paper should be of
interest. Our output-sensitive algorithm for the general APSP
problem in DAGs possibly could lead to an improvement of the
asymptotic time complexity of this problem in the average case.
A probabilistic analysis of the number of leaves in the
lexicographically-first shortest-path trees is an interesting open
problem.
In the vast literature on shortest path problems, there are several
examples of output-sensitive algorithms. For instance, Karger et
al. \cite{KKP} and McGeoch \cite{Mc} could orchestrate the $n$ runs of
Dijkstra's algorithm in order to solve the APSP problem for
directed graphs with non-negative edge weights in $O(m^*n+n\log n)$
time, where $m^*$ is the number of (essential) edges that
participate in shortest paths.
Finally, note that DAGs
have several important scientific and computational applications in
among other things scheduling, data processing networks,
biology (phylogenetic networks, epidemiology),
sociology (citation networks), and data compression.
For these reasons, efficient algorithms for shortest paths in
DAGs are of not only theoretical interest.
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_the lives of_
ROBERT RYAN
_the lives of_
ROBERT
RYAN
J.R. JONES
_Wesleyan University Press_ | _Middletown, Connecticut_
Wesleyan University Press
Middletown, CT 06459
www.wesleyan.edu/wespress
© 2015 J.R. Jones
All rights reserved
Manufactured in the United States of America
Designed by Mindy Basinger Hill
Typeset in Garamond Premier Pro
Wesleyan University Press is a member of the Green Press Initiative.
The paper used in this book meets their minimum requirement for recycled paper.
Portions of this book appeared previously in "The Dark Shadings of Robert Ryan: A Brief Biography," _Noir City_ , Vol. 6, No. 2, Summer 2011, and in "The Actor's Letter" and "The Essential Robert Ryan," _Chicago Reader_ , October 29, 2009.
Library of Congress Cataloging-in-Publication Data
Jones, J.R., 1963–
The lives of Robert Ryan / J.R. Jones.
pages cm
Includes bibliographical references and index.
Includes filmography.
ISBN 978-0-8195-7372-8 (cloth: alk. paper)—
ISBN 978-0-8195-7373-5 (ebook)
1. Ryan, Robert, 1909–1973. 2. Actors—United States—Biography. I. Title.
PN2287.R88J66 2015
791.4302'8092—dc23
[B] 2014033019
5 4 3 2 1
Cover illustration: Ryan relaxes on the RKO lot (1946) during production of The Woman on the Beach. Courtesy of the Wisconsin Center for Film and Theater Research.
_For Margaret_
Contents
Introduction | ix
---|---
_one_ Inferno |
_two_ The Mysterious Spirit |
_three_ Bombs Away |
_four_ You Know the Kind |
_five_ We Will Succeed, You Will Not |
_six_ Caught |
_seven_ Learning by Doing |
_eight_ The Whiz Kids |
_nine_ Rum, Rebellion, and Ryan |
_ten_ The Gates of War |
_eleven_ Beautiful Creatures |
_twelve_ The Longest Day |
_thirteen_ One of the Boys |
_fourteen_ My Good Bad Luck |
_fifteen_ The Loneliest Place in Town |
Acknowledgments |
Appendix |
Notes |
Selected Bibliography |
Index |
Introduction
He was well liked in Hollywood but hardly well known. Tall and trim, with a winning Irish grin and a politician's firm handshake, he listened more than he spoke, his small brown eyes taking everything in. By the mid-1950s he had worked with some of the best directors in the business — Jean Renoir, Pare Lorentz, Jacques Tourneur, Joseph Losey, Fred Zinnemann, Max Ophuls, Robert Wise, Nicholas Ray, Fritz Lang, Budd Boetticher, Anthony Mann, Samuel Fuller — and none of them had an ill word for Bob Ryan. He dug into his part, he showed up on time, he delivered on the first take. He was generous with other actors, patient with young performers who might be having trouble. He got to know the crew and looked out for their interests; in stressful situations he was always good for a wisecrack to break the tension. But at 6 PM every night he disappeared, home to his wife and three children in the San Fernando Valley. Even his close friends found him a puzzle; director Harold Kennedy echoed many when he called Ryan "the most private person I have ever known."
Forty years after Ryan's death, his artistic reputation has only grown. Martin Scorsese called him "one of the greatest actors in the history of American film," and when Film Forum in New York mounted a twenty-three-film Ryan retrospective in August 2011, critics recognized it as a powerful body of work with its own thematic coherence. Schooled by the great Austrian theater director Max Reinhardt, Ryan was hired by RKO (Radio-Keith-Orpheum) Radio Pictures in 1942 and groomed as a handsome male lead, but all that changed with his unnerving performance as a bigoted army sergeant concealing his murder of a Jewish civilian in the film noir classic _Crossfire_ (1947). His career ignited just as noir was developing into a shadowy interrogation of American values; with his strength, intelligence, and willingness to explore the soul's darker corners, he invested the genre with a string of neurotic and troubling portrayals that still reverberate through the popular culture.
Ryan liked to upset the easy morality of genre pictures, and he was drawn to men with complicated motives: the insecure millionaire who validates himself by controlling his wife's every move in _Caught_ (1949), the closeted crime lord coveting the cop who's out to get him in _House of Bamboo_ (1955), and the ruthless California rancher who avenges the attack on Pearl Harbor by killing a Japanese farmer in _Bad Day at Black Rock_ (1955). Long after Ryan had grown frustrated with his sinister screen persona, he continued to play men twisted by hatred or bigotry if they promised great drama that would change minds. By all accounts he was a good man, but often he expressed his goodness by playing evil men — with an alarming relish and conviction. That curiosity and daring set him apart from his '40s and '50s peers; his coiled performances widened the parameters of what moviegoers might expect from a leading man and helped pave the way for such volatile personalities as Robert De Niro, Harvey Keitel, and Tommy Lee Jones.
His reputation as a heavy obscures his great versatility: by the time Ryan died in 1973, he had played everything from Jay Gatsby to John the Baptist. Against his agent's advice, he grabbed the role of Ty Ty Walden, the elderly patriarch of _God's Little Acre_ (1958), and turned in a tender and funny performance as the grizzled old coot. In search of acting challenges, he struck out into legitimate theater, playing political satire (Jean Giraudoux's _Tiger at the Gates_ ), theological drama (T. S. Eliot's _Murder in the Cathedral_ ), and his beloved Shakespeare ( _Antony and Cleopatra_ with Katharine Hepburn, _Coriolanus_ with director John Houseman). Near the end of his career he was hailed for two performances on the New York stage, as the scheming newspaper editor in Ben Hecht and Charles MacArthur's comedy _The Front Page_ and the angry, self-pitying father in Eugene O'Neill's tragic _Long Day's Journey into Night_.
Such was his life on the stage and screen. In public Ryan was the antithesis of the right-wing characters he often portrayed; raised in the Chicago Democratic machine, married to a Quaker woman of strong pacifist ideals, he campaigned tirelessly for liberal causes throughout his career, tracing a careful route through the political booby traps of the blacklist era and into the tumultuous '60s. His experience as a Marine Corps drill instructor during World War II turned him against the war machine forever; he championed "world peace through world law" as a member of the United World Federalists, and in the late '50s he cofounded the Hollywood chapter of the National Committee for a Sane Nuclear Policy. Playing a violent, racist hick in the late-period noir _Odds against Tomorrow_ (1959), he grew close to Harry Belafonte and got involved in the civil rights struggle. In the mid-1960s he spoke out against the Vietnam War, stumping for Eugene McCarthy in the New Hampshire primary that drove Lyndon Johnson from the White House — even as, onscreen, he played hardened military men in _Battle of the Bulge, Anzio_ , and _The Dirty Dozen_.
His partner in all this was Jessica Cadwalader, a freethinker from Berkeley, California, who, having married Ryan, abandoned an acting career and became a writer, publishing five novels. He trusted and admired her; to some extent she became his social conscience, inspiring him in his political work. Their civic ambition manifested itself most impressively with the Oakwood School, a progressive grade school they launched in North Hollywood in 1951 with a handful of other parents. Combining his star power and her determination, the couple managed to guide the school through its rocky first years, when political conflicts among the parents of the enrolled students threatened to tear it apart. They invested heavily in the project, donating thousands of dollars to keep it afloat and immersing themselves in the scholarship of education. Their own children graduated from Oakwood, now considered one of the better private schools in Los Angeles. Ryan often told people the school was the most important thing he had ever done.
His private life Ryan reserved for himself and his family, avoiding the Hollywood social scene to concentrate on raising his children. Movie magazines invariably portrayed him as a contented spouse and dad, and there was some element of truth in this; interviewed at home in the mid-1960s, he remarked with touching sincerity, "All my best friends live in this house." But there was a dark side to Ryan as well. He could be silent and withdrawn; he drank too much and suffered from debilitating depressions — "Black Irish moods," he called them. Jessica grappled with similar problems, and through the '50s a good deal of the hands-on parenting in the Ryans' home was administered by Solomon and Williana Smith, a childless black couple who lived with the family. Millard Lampell, one of Ryan's few close friends in the '60s, shrewdly observed, "I think Robert would [have liked] to be remembered as a loving husband and a good father, neither of which he always was."
Ryan's own father, who died in 1936, taught him by example that a man keeps his problems to himself, and as Ryan matured and became a celebrity, he grew increasingly adept at compartmentalizing his life. This permitted and, to some extent, encouraged the sharp contradictions in his character. He loved acting more than anything else, but his tireless political activities sprang from a gnawing sense that his chosen profession really was shallow and narcissistic. He recoiled from the hobnobbing and false friendships of the movie business, then fumed when the good roles went to more enterprising actors. He kept his frustrations buttoned up, and when they had a chance to burst out in some of his more unhinged characters, they hinted at a man with more issues than he would ever let on. "Every actor has at least two selves," he said. "There's the _outside_ self that takes part in family life and society and the _inside_ self who is someone else."
I gained an unexpected insight into Ryan's inner life in 2009, when I got the chance to read an undated, twenty-page manuscript he had written for his children and then filed away and forgotten. Uncovered by his youngest child, Lisa, and passed along to Michael Miner, my colleague at the _Chicago Reader_ , it was a brief history of Ryan's years growing up in the city, warmly nostalgic in its recollections of the North Side and his extended Irish family. But it also contained references that, as I began to investigate, led me to a scandal undocumented in any account of Ryan's life. His father, Timothy, and three uncles operated the politically well-connected Ryan Company, a firm that specialized in rail and sewer tunnel construction. In April 1931 Timothy Ryan was personally responsible for a South Side project where a disastrous subterranean fire lasting some twenty hours claimed the lives of twelve men and injured another fifty.
One should always take care when connecting an actor's life to his roles. But if Ryan was indeed the puzzle that so many claimed, this tragic story supplies at least one piece of it, helping us understand the power and insight he brought to so many of his tortured characters. Conscience runs like a gold thread through many of his key performances. Nicholas Ray's _On Dangerous Ground_ (1952) presents Ryan as a brutal policeman forced to reckon with his rage when he meets a blind woman, played by Ida Lupino, who challenges him to find his better self. In _The Professionals_ (1967) he's a horse wrangler who hires on to help rescue a kidnapped woman but antagonizes his partners by peeling away the heroic façade of their mission; in _The Wild Bunch_ (1969) he's an outlaw who can barely live with himself after cutting a deal with the law to track down his old friend. More broadly, Ryan's political and social conscience sharply influenced his choice of roles, especially after he was freed from his RKO contract in the early 1950s and could exercise somewhat more control over the films he was making.
Even more revealing than Ryan's manuscript are the several unpublished memoirs Jessica Ryan left behind at the time of her death in 1972. Witty and acutely observed, these pieces illuminate her husband's character and her own, particularly their aversion to Hollywood social life. They provide the clearest picture of Ryan's political skills, honed from years of exposure to the inner workings of machine politics. They also offer a rare female perspective on a Hollywood dominated by men and, in Ryan's case, populated by such macho characters as Mann, Fuller, Lee Marvin, Robert Mitchum, Richard Brooks, André de Toth, Sam Peckinpah, and John Wayne. Ryan may have been famous for his tough-guy roles in westerns and crime pictures, but when his wife passed away, his sense of self began to crumble.
The more I explored the Ryans' lives, the more I realized that here was not just the story of a movie star but a pocket history of American liberalism, stretching from a war against fascism in Europe that united the country to a war against communism in Southeast Asia that bitterly divided it. This struggle played out in Ryan's screen life, which he began as an eager army flyboy in _Bombardier_ (1943) and ended as a right-wing millionaire conspiring to kill President Kennedy in _Executive Action_ (1973). It defined his public life, where he fought the good fight in the coldest years of the Cold War, his compromises as revealing as his victories. It also animated a good deal of his inner life, a place where men guard their secrets and, sometimes, take them to their final rest.
_the lives of_
ROBERT RYAN
_one_
Inferno
The day Robert Ryan turned nine, the entire nation celebrated. All weekend long had come word that the Armistice was about to be signed, bringing home a million American soldiers from the trenches of France. In Chicago, where the boy lived, whistles began to sound and guns to go off in the predawn darkness of Monday, November 11, 1918. Women ran from their homes with overcoats tossed over their nightgowns, beating on pots and pans. The elevated trains coming from the Loop tied down their whistles and went screaming through the neighborhoods, confirming that the nation was at peace. People who ventured downtown for work were sent home by their employers, and by noon the neighborhood parties were rolling. In Uptown, on the city's North Side, young Bob ran around telling people this was his birthday and returned home with a few dollars in change. His parents, Tim and Mabel, made him give back most of the money, but even so this was a great day. Everyone had called this "the war to end war" — if that were true, then he would never have to die in a trench.
The Ryans had no need for their neighbors' charity; they were respectable, middle-class people who had worked their way up. Bob's great-grandparents, Lawrence and Ellen Fitzpatrick Ryan, had immigrated from County Tipperary, Ireland, in 1852 during the Great Famine and settled in Pittsburgh, where times were tough (their son John would later tell Bob about the "No Irish Need Apply" signs that greeted them on their arrival). The family moved to Chicago four years later and eventually retreated about thirty miles south to the heavily Irish Catholic river town of Lockport, Illinois, along the Illinois and Michigan Canal.
John and his older brother, Timothy E. Ryan, worked together as boat builders in the 1860s, then went their separate ways as John established his own business in town and Timothy (known as "T. E.") returned to Chicago to try his hand at real estate speculation. John served as superintendent of the canal at one point and, with his wife, Johanna, raised a family of eight children. He liked his glass. "Although my grandfather drank a quart of whiskey a day for sixty-five years, he was never drunk or out of control," Bob later recorded in a memoir for his children.
Up in Chicago, T. E. Ryan prospered, cofounding the real estate firm of Ryan and Walsh and building his family a mansion on Macalister Place on the Near West Side. He also established himself as a political brawler in the city's well-oiled Democratic machine. Through the 1890s he won five terms as West Side assessor, and from 1902 to 1906 he served as Democratic committeeman for the Nineteenth Ward. T. E. was widely regarded as boss of the West Side, so popular and influential that, during the World's Columbian Exposition in 1893, he was named grand marshal of the Irish Day parade. A portrait reproduced in an 1899 guidebook to state politics shows a handsome man with swept-back hair, a handlebar mustache, and a hungry glint in his eye. "One of the most popular men on the West Side," the guidebook reported, "and a politician whose power is as strong as ever." His success exerted an irresistible pull on John's sons, and one by one they all drifted to Chicago.
Timothy Aloysius Ryan was the second of John's children, born in 1875, and in the 1890s he headed north to board with his illustrious uncle and get into business in the city. Tim proved to be an eager political protégé: in 1899 he was appointed chief clerk in the city attorney's office, and five years later he ran for the state board of equalization in the Eighth Congressional District, billing himself as "T. A." Ryan. His uncle bankrolled all this, apparently seeing in his tall and handsome young nephew a rising political star. Tim got himself started in the construction business and ran an unsuccessful campaign for West Town assessor, his uncle's former position. "Father's duties have always been somewhat vague in everyone's mind," Bob wrote. "In his twenties he seems to have been occupied principally with fancy vests, horse racing, attending prizefights, and a great deal of social drinking. In short, a rather well-known and well-liked man about town."
By 1907, Chicago was home to five of John and Johanna's sons. They were big men — one of Bob's uncles stood six feet eight inches tall — with ambitions to match. Larry, Tim's younger brother by eight years, had come north to clerk for T. E.'s real estate firm, and Tom, Joe, and John Jr. wanted to start their own construction firm so they might capitalize on their uncle's political influence. But the brothers' relationship with their uncle ruptured. According to Bob, Larry's job "involved handling some funds and he was ultimately accused by his uncle of a minor embezzlement. Larry was about as liable to have done this as to burn down the Holy Name Cathedral. Father sided with his brother and left his uncle's bed, board, and generous patronage for good." From T. E.'s power base in the west, Tim and Larry relocated to the relatively unpopulated North Side, where they banded together with their siblings to turn the newly christened Ryan Company into a going concern.
Timothy Aloysius Ryan, the actor's father. Informed once that a gubernatorial candidate had been accused of embezzling fifty thousand dollars, he remarked, "Any man who could only steal fifty thousand dollars in that job isn't smart enough to be governor." _Robert Ryan Family_
Tim was thirty-two the night Larry introduced him to Mabel Bushnell, a lovely twenty-four-year-old secretary at the _Chicago Tribune_. Raised in Escanaba, a port town on Michigan's Upper Peninsula, Mabel was descended from some of the first English families of New York, though her father was a cruel and alcoholic newspaper editor from Gladstone, Michigan, whose career ultimately had given way to a tougher life as a tramp printer operating out of Rhinelander, Wisconsin. Tim took Mabel out on the town, squiring her to restaurants and theaters, springing for hansom cabs. He wanted her badly, but she took a dim view of his boozing, not to mention his political ambitions. Tim agreed to swear off liquor and politics, and in 1908 they were married, in a ceremony conducted by both a priest and a protestant minister. They moved into the apartment on Kenmore, and Robert Bushnell Ryan arrived late the next year — November 11, 1909.
Robert Bushnell Ryan (circa 1912). "I was a completely nonaggressive youngster," he later recalled. _Robert Ryan Family_
Two years later Mabel gave birth to a second child, John Bushnell, and the two boys slept in the same bed. "Very early in my life I remember the lamplighter," Bob wrote, "a solitary youth who went around lighting the street lamps." He and Jack enjoyed an idyllic life in Uptown, frolicking every summer on Foster Avenue Beach and running up and down the alley behind their house, an avenue for commercial activity. "Almost all heavy hauling was done by horse and wagon," Bob remembered, and the alley "was full of various dobbins hauling ice, garbage, groceries, etc. In the hot summers the horses wore straw hats. The horses got to know the various stops and often would break in a new driver by showing him where to go."
The brothers' friendship ended in June 1917 when Jack — "a rather solemn, gentle little fellow," Bob wrote — died of lobar pneumonia, probably brought on by flu. He was not quite six years old. "I remember the terrible day that he died,"* Bob would write, "and the feeling of my mother and father that he might have been saved." Devastated by the boy's death, Tim and Mabel vacated their little apartment at 4822 Kenmore, blocks from Lake Michigan, and moved slightly northwest to a one-bedroom on Winona Street. "The neighborhood was somewhat less desirable," Bob wrote. "But nothing mattered. We had to move and we did." His parents, craving a portrait of little Jack, took a photograph they had of their sons on a dock and had Bob airbrushed away.
Now Bob slept alone, in a Murphy bed that folded out from the wall, like the one Charlie Chaplin had wrestled with in his two-reeler _One a.m_. He went to school alone, having transferred from Goudy Public School, which he remembered as mostly Jewish, to Swift Public School nearer his home. His parents were Victorian people, reserved even with their own child; and as the years passed, Bob learned to keep his own company, reading endlessly and roaming around the new neighborhood.
One unique attraction was the Essanay Film Manufacturing Company on Argyle, founded a decade earlier and now the city's premiere movie studio. Chaplin had made films for Essanay in 1915, and Gloria Swanson and Wallace Beery had gotten their start there; Bob would remember seeing them all on the streets of Uptown. He and his school friends even spent their Saturday afternoons appearing as extras in the two-reel comedies of child star Mary McAllister, each earning the princely sum of $2.50 a day.
He was naturally quiet, even withdrawn, and his parents worried over his introverted nature. Mabel gave him a violin that once had belonged to her brother and every Friday marched Bob to the elevated train and downtown to Kimball Hall for a lesson. His teacher, a Scandinavian player for the Chicago Symphony, couldn't do anything with him. Tim, knowing full well that a boy carrying a violin down the streets of Chicago would be a magnet for bullies, signed Bob up for boxing lessons at the Illinois Athletic Club, where a coach by the name of Johnny Behr taught him how to fight. Bob loved boxing: he was smart and quick in the ring, and he realized that if you didn't worry about the punch it didn't hurt as much. "Athletic prowess did a lot for my ego and my acceptance in school," he later told an interviewer. "The ability to defend yourself lessens the chance you'll ever have to use it."
Chicago could be an ugly place. Eight months after the Armistice was signed, Bob saw the city erupt again, this time in violence. Temperatures in the nineties had irritated tensions on the Near South Side between blacks confined to the Twenty-Fifth Street Beach and their white neighbors on the Twenty-Ninth Street Beach. On July 27, 1919, a black boy rafting near the shore at Twenty-Ninth Street was killed by a white man hurling rocks, and the incident touched off five days of murderous rioting. "As rumors of atrocities circulated throughout the city, members of both races craved vengeance," wrote historian William M. Tuttle Jr. "White gunmen in automobiles sped through the black belt shooting indiscriminately as they passed, and black snipers fired back. Roaming mobs shot, beat, and stabbed to death their victims."
Thirty-eight people died, and more than five hundred were injured. An official report would blame much of the initial violence on Irish athletic clubs such as Ragen's Colts and the Hamburg Club, but the rage had spread like an infection, creeping into the West and North Sides. (Just south of Uptown lay one of the North Side's isolated pockets of blacks.) For a boy not yet ten, the riot must have been a frightening experience. Not only could war go on forever, it could happen right in your own backyard.
THE RYAN FAMILY'S FORTUNES began to turn in 1920 when Tim's friend Ed Kelly was appointed chief engineer of the Chicago Sanitary District. Son of a policeman, Kelly had started out with the district at age eighteen, and though he had studied engineering at night school, he displayed more talent as a South Side politician, having founded and been elected president of the two-hundred-member Brighton Park Athletic Club. The Irish athletic clubs were mainly social, organizing team sports, but they were also politically oriented, and Kelly soon made a name for himself in the Cook County Democratic Party. By the time he became chief engineer, he had put in more than thirty years with the district. His spotty formal training was much noted in the press (one muckraking journalist accused him of farming out his technical work to consultants). Yet Kelly understood and had mastered the operating principle of Chicago politics: take care of your friends and they'll take care of you.
Under Kelly, the Ryan Company won lucrative city jobs paving streets and building sewer tunnels. Tim, who supervised sewer construction, worked from 5:30 AM until 8 or 9 PM at night; he and his son barely saw each other except for weekends. With his winning manner and many connections, Tim was critical to the operation, though according to Bob, the man who really ran the company was his Uncle Tom, "a rather cold and shrewd businessman." Flush with the company's profits, Tim and Mabel decided to move again, this time to a bigger apartment, in the northerly Edgewater neighborhood, that was only a block from the lake. They bought their own automobile and furnished their new home well. During the summers Bob went to Camp Kentuck in Wisconsin, while his parents enjoyed golfing weekends in Crystal Lake, northwest of the city. Mabel might have succeeded in keeping Tim away from drink, but politics was another matter, and Kelly could always rely on T. A. Ryan as a Democratic Party committeeman for the Twenty-Fifth Ward.
Haunted by the memory of little Jack, Tim and Mabel would never have another child, choosing instead to spoil and smother Bob. "You cannot know the difficulties that attend an only child," he would write years later, in a letter to his own children. "Two big grown-ups are beaming in on him all the time — even when he isn't there. It is a feeling of being watched that lingers throughout life." He hid in the darkness of the movies, spending countless afternoons at the Riveria Theater on Broadway or the smaller Bryn Mawr near the "L" stop. The charm and dash of Douglas Fairbanks were his greatest tonic, and he never missed a picture: _The Mark of Zorro, The Three Musketeers, Robin Hood_. Bob had seen how motion pictures were made and was fascinated by the results. Yet he could barely conceive of the movies as an occupation; his father and uncles considered the Ryan Company a legacy for their children.
After Bob graduated from Swift in 1923, his father pulled some strings to get him a summer job as a fireman on a freight locomotive, which satisfied the thirteen-year-old boy's appetite for freedom and Tim's desire that he learn the value of a dollar. Rumors of petting parties at the local public high school had persuaded Mabel that Bob needed a private education, and that fall his parents enrolled him at Loyola Academy, a Jesuit college prep school for young men that was located near the Loyola University campus to their north. The experience would shape him not only as a person but also as an artist.
Loyola was heavily Irish Catholic, the sons of an aspiring middle class, and the class of 1927 would produce an unusual number of Jesuit priests. Tim must have been pleased that his son would be schooled in the Catholic faith, though Mabel valued Loyola more for its academic reputation. The priests were known as stern taskmasters, and the curriculum was tough — along with the arts and sciences, the boys learned Latin, Greek, and Christian doctrine. Later in life, when Bob Ryan's interests had turned to education, he would take a more skeptical view of Jesuit schooling. "The fathers were well-seasoned men who had a good deal of authority that they seldom used," he remembered. "Huge areas of a fruitful life were almost ignored. Jesuit education was books and drill and writing and _some_ discussion."
At the new school Bob began to distinguish himself in athletics, especially after a growth spurt propelled him to a height of six-foot-three, only an inch shorter than his father. He played football all four years and competed in track and field. Formidably big and agile on the gridiron, he was an All-City tackle his senior year. In school he struggled with Latin and especially chemistry but excelled in English, joining the literary society and working on the school magazine, _The Prep_. He read voraciously. "Truly, I may say that a man's best friends are his books," he wrote in the magazine his junior year. "Your companions may desert you, but your books will remain with you always and will never cease to be that source of enjoyment that they were when you first received them."
Ryan with his parents, Mabel and Tim. "You cannot know the difficulties that attend an only child," he later wrote. "It is a feeling of being watched that lingers throughout life." _Robert Ryan Family_
The book that changed his life was _Hamlet_ , which he spent an entire semester studying under the instruction of his beloved English teacher, Father Joseph P. Conroy. The priest led the boys through the Elizabethan verse into the dark heart of the play, the young prince charged by the ghost of his dead father to avenge the treachery of his uncle, Claudius, and the unfaithfulness of his mother, Gertrude. _Hamlet_ was full of moral conundrums, the hero torn between his conscience and his thirst for revenge. Bob was captivated: such rich language, such profound thoughts, such high drama. By the end of the semester he could recite practically the entire text. He fell in love with theater, reading Shakespeare, Chekhov, Shaw, and O'Neill, a writer who spoke to his own Irish melancholy. Their work awakened in him a hunger for self-expression, and he wondered if, instead of following his father into construction, he might become a playwright himself.
The money kept rolling in at the Ryan Company, and before long the family bought a Cadillac, then a Pierce-Arrow with a chauffeur to drive Tim to work. Bob got his own Ford and tooled around in bell-bottom suits and a fur coat. Tim became a patron of the Chicago Opera Association; he took Mabel to New York City to see all the shows. (Bob shared their love of musical theater; among his favorite performers were Fanny Brice and the great Irish-American showman George M. Cohan.) Tim Ryan, Bob wrote in a letter to his own children, "was always generous and kind to me — in a day when father-son relationships were not thought of as they are now." His father was "a big man (6′ 4″ — 250 lbs.) with a radiant personality and strong sense of humor, and was idolized by many people. His other side was only displayed at home and was very hard to take."
Bob wouldn't elaborate on this statement, but he would note his father's ambivalence toward the construction business, which hardly inspired one to join him. "Dad, I think, would have been content to have enough money to live well, eat well, play bridge, and tell stories to his rather small circle of friends." Friction between father and son began to build as Bob's graduation from Loyola drew near. Tim had mapped out his son's future: he would stay at home, earn a professional degree at Loyola or DePaul or the University of Chicago, and find a good living for himself as the next generation of the Ryan Company. Bob insisted on going east to school and won admission to Dartmouth College in Hanover, New Hampshire.
That summer he accepted an invitation from his former camp counselor, a wealthy Yale graduate named Frank Scully, to work at a dude ranch Scully was trying to start on some land his family owned in Missoula, Montana. Bob took the train out West, spent the summer sharpening his horseman skills, and even found time for a first romance with a girl named Thora Maloney. He would remember his awe at seeing "plains that never ended — where one seemed to be becalmed in a purple ocean. As we got into the foothills of the Rockies and finally saw some of the high peaks I was aware of a lift of spirit that I shall never forget. It was strange to be so far from home and yet to feel as if I was coming home."
Back in Chicago he gathered his belongings for school and at long last left his parents behind. His father was pained to see him leave. "He didn't get the point — packing off 1,300 miles to the state of New Hampshire when there were five colleges to be had within an hour's drive," Bob would write. "Mother must have sensed that I _should_ go — though I hope she didn't know how much I _wanted_ to go."
At Dartmouth he pledged Psi Upsilon (one of his fraternity brothers was Nelson Rockefeller) and went out for track and football. But his real claim to fame was boxing: in his freshman year he won the college its first heavyweight title. His grades were unspectacular; he maintained a C average, studying Greek, French, English, physics, evolution, philosophy, and citizenship. The following summer he returned to Scully's ranch, pursuing romance with another girl, Thula Clifton, and in the fall he played football again, though his career ended ignominiously after he broke his knee in a game against Columbia University. The injury threw his schoolwork into disarray, and in December 1928 he withdrew from all his classes without receiving any grades, standard procedure for someone flunking out.
For the next eight months Bob returned home to his parents, who had moved to a new apartment on Lake Shore Drive. Tim insisted that Bob work, so he got a job as a salesman, first for a steel company and then for a cemetery. "I'm offering a permanent product," he would tell his customers. That fall he reenrolled at Dartmouth, starting over as a sophomore, and though he would continue to box, he had resolved to get serious about his studies.
A month after he returned, the stock market crashed. October 23 brought the first wave of sell-offs, then on October 29 — "Black Thursday" — the bottom dropped out. Crowds gathered outside the Chicago Stock Exchange, where a record one million shares changed hands in a single day. The Ryan Company was privately held and, at that point, worth at least $4 million. But each of the brothers was personally invested in the market, and they were all wiped out. All they had left was the promise of more construction work.
Even that seemed precarious: earlier that year Assistant State's Attorney John E. Northrup had returned indictments against Ed Kelly and a dozen other men at the sanitary district, charging that they had defrauded taxpayers of $5 million over the past eight years and done a healthy business in bribes and kickbacks from contractors. "A well-greased palm was essential to doing business with the department," wrote Kelly's biographer, Roger Biles. "Some trustees received gifts of twenty-five cases of liquor a month from favored contractors." Others "admitted financing lengthy European vacations with illegally solicited contributions." Kelly would later concede to the IRS that from 1919 to 1929 his income was $724,368, though his salary for that period totaled only $151,000.
More than seven hundred people were called to testify, many of them against their will. Witnesses exposed gaping discrepancies between the district's stated expenditures and what contractors were actually paid: the payroll was said to be padded by as much as 75 percent. The trial revealed that bids were submitted in plain envelopes that were later opened and altered so that favored firms could be awarded lucrative contracts. Elmer Lynn Williams, publisher of the muckraking newsletter _Lightnin_ ', alleged that the district's central auto service had provided high-ranking officials with "young women procured for these tired business men by an older woman who was on the pay roll. The taxpayers were charged for vanity cases, whiskey and the time of the 'entertainers.'"
None of the Ryan brothers was ever implicated, but the scandal soiled the reputations of everyone doing business with the district. Kelly escaped conviction only when the judge in the case, who was pals with a local Democratic boss, quashed the indictments and Northrup, forced to reassemble his case before the statute of limitations ran out, dropped the chief engineer as a defendant. Years of hardball Chicago politics had turned Tim Ryan into a cynic when it came to graft; informed once that a gubernatorial candidate had been accused of embezzling fifty thousand dollars, he remarked, "Any man who could only steal fifty thousand dollars in that job isn't smart enough to be governor."
EIGHTEEN MONTHS AFTER THE CRASH, in April 1931, Tim suffered another devastating blow. One of his sewer projects for the city, southwest of the Loop in the Pilsen neighborhood, was engulfed in a horrific fire that burned for nearly twenty-four hours and claimed at least a dozen lives. Bob would come to view the disaster as a key factor in his father's death.
The Ryan Company had contracted to build the Twenty-Second Street section of a huge, $2.1 million concrete intercepting sewer that would travel southwest to the sanitary drainage and ship canal. During construction each block-long section of the ovoid, seventeen-foot tunnel was sealed off to maintain air pressure and prevent collapse; the only exit was a short, perpendicular work tunnel that led to an elevator shaft. The cause of the fire was never officially determined, but according to several newspaper reports — including one that cited Tim Ryan as its source — a cement worker had dropped a candle (used to detect air leaks) into a pile of sawdust. Timber and sawdust were major components in tunnel construction: wooden forms used to mold the concrete were braced against the earthen walls and anchored in place with sawdust packs. The fire began to spread underneath the concrete, pumping black smoke into the tunnel.
At street level a foreman noticed a ribbon of smoke drifting up from the elevator shaft and, fearing an electrical fire, sent three electricians down to check the wiring; they found nothing wrong. Tim learned of the fire around 6 PM, and the first workmen to flee the tunnel reported a smell of burning insulation, which led him and his crew to believe the cause was indeed electrical. Morris Cahill, the construction superintendent, warned them that if the fire reached the east end of the tunnel and destroyed the hoses maintaining the air pressure belowground, the entire tunnel section would collapse.
According to the _Daily News_ , loyal employees begged Ryan to let them extinguish the fire: "We'll be okay, boss. Let us go, please. It'll mean your contract if we don't." Without waiting for Ryan's permission, an assistant foreman led a party of men down into the tunnel; Cahill made three trips down but each time was overcome by smoke. With no word from the men below, Ryan summoned the fire department around 7 PM.
"My men are in there!" Tim exclaimed as the first engine company arrived on the scene. "What are we going to do?" Confusion over the fire's cause and ignorance of its severity may have been as deadly as the blaze itself: the first two rescue parties descended into the tunnel without the benefit of gas masks. The operation went on for hours, slowed by the thick smoke and the difficulty of getting at the burning material. When the fire broke out, panicked workmen had retreated into the metal chambers at either end of the tunnel section, which were sealed by an air lock and offered fresh air pumped in from street level thirty-five feet above. As the fire raged out of control, it pushed firefighters back into the chambers as well, and the trapped men waited through the night, praying and trying to lie still.
By midnight the construction site looked like the scene of a mining disaster. A light wagon trained its searchlight on the mouth of the elevator shaft, and thousands of spectators, some of them distraught family members of Ryan employees, were being held back by a police cordon. Hospital squads had arrived on the scene and set up shop in a neighboring lumberyard. More than two dozen firefighters had already been taken to Saint Anthony Hospital, and the fire department had by now dispatched a full quarter of its forces to the site.
Firefighters attacked the superstructure over the elevator shaft and eventually managed to tear the roof off in an effort to provide more ventilation. Mining equipment arrived, and mine workers from around the city converged on the site to volunteer their services. After the utility companies shut off the electricity and the Twenty-Second Street gas main (located a perilous ten feet from the tunnel), crews of men with picks, shovels, and pneumatic drills started three new ventilation holes in the concrete — one above each air chamber and another at the center of the tunnel.
No plan was too far-fetched: a professional diver who lived on the North Side was recruited to venture into the tunnel in his wet suit, but after only a few minutes he signaled for help and was brought back up — the rubber was melting. A description in the _Chicago Evening Post_ sounds like a scene from Dante: "Terrific heat developed in the cramped quarters underground. Blazing timbers fell.... Water, poured above the tunnel in a vain effort to cool it and dissipate some of the fumes, eddied, four feet deep in spots, and made it impossible to see even inches ahead in the thick white mist." Sometime during the night, the air supply inside the east air chamber failed, and the laborers and firefighters trapped inside decided to make a break for it, but most them died of smoke inhalation before they could reach the elevator shaft.
Outside, the rescue effort was beginning to reach across state lines. Henry Sonnenschein, secretary to Mayor Anton Cermak, brought word from his boss, who was vacationing in Miami Beach, that the city would spare no ex pense in addressing the crisis, which threatened to become a citywide calamity if the fire breached the east and west walls of the tunnel into the remainder of the sewer line. By 3 AM a rescue squad from the federal mining bureau had roared out of Vincennes, Indiana, for Chicago, escorted by state police. A squad from the state mining bureau in Springfield boarded a special train with right-of-way cleared to the site of the disaster. But the critical arrival, just after dawn on Tuesday, was an experimental smoke-ejector truck designed by an inventor in Kenosha, Wisconsin. A modified fire truck, the smoke ejector was essentially a gigantic vacuum cleaner on wheels, and its long, flexible fourteen-inch tubes were extended down the mine shaft to suck the smoke out of the tunnel.
The crowd roared later that morning when sixteen men trapped in the metal chamber and already given up for dead began emerging from the elevator shaft. Early that afternoon rescuers recovered the last dead man from the tunnel: Captain James O'Neill, one of the first firefighters on the scene, who had been trapped in the east chamber and was trampled near the air lock by the stampeding workmen as they tried to escape. The final death toll was four firemen and seven laborers, plus a policeman who had been run over by an ambulance. Nearly fifty other people had been injured, some seriously. Later that afternoon, the young widow of Edward Pratt, a firefighter whose body had been recovered overnight, broke past the police cordon and tried to hurl herself down the elevator shaft.
By that time Tim had been summoned to the county morgue, where Dr. Herman N. Bundesen, the Cook County coroner, was convening an inquest to determine how the fire had started and how the eleven men had died. Bundesen had a long history with Ed Kelly, having worked for the sanitary district during the Whoopee Era; according to journalist Elmer Lynn Williams, he had proved himself "one of the pliable tools of the machine." Kelly, still holding firm in his capacity as chief sanitary engineer, served as technical advisor to the inquest.
Called to testify, Tim Ryan wept as he recalled the first crews of firefighters going after his trapped workmen: "I saw men going down into that reeking tunnel without gas masks — without masks. I never saw such courage displayed in my life." Neither he nor his construction superintendent could state with certainty what caused the fire, and the news accounts of a workman igniting a pile of sawdust never were introduced.
When the inquest reconvened a week later in a courtroom at City Hall, the panel ruled that all eleven men had died of smoke inhalation but declared the cause of the fire unknown. "Unofficially," reported the _Chicago American_ , "the jury members expressed the view that no human agency was at fault in the fire and tragedy that followed; that all precautionary measures were maintained by the contractors to safeguard life." The city was indemnified against liability for the workmen's deaths; the Ryan Company would pay any settlements to the families through its compensation insurance. A pall hung over the firm, exacerbated by the Kelly corruption charges still crawling through the court system.
IN THE QUIET SECLUSION OF HANOVER, Bob must have been even more determined not to join the Ryan Company. His grades had improved substantially; he was earning mostly B's now and the occasional A in English or comparative literature. He had reclaimed his record as an intercollegiate boxing champion and — encouraged by his coach, Eddie Shevlin — even entertained thoughts of becoming a professional fighter. But his father talked him out of it: most boxers, he pointed out to Bob, were washed up at thirty. Bob was tired of athletics anyway. Having defended the heavyweight title in his sophomore and junior years, he retired from the ring to devote himself to his degree in dramatic literature.*
His best friends were still his books. The 1920s had brought a great revival of interest in Herman Melville, and Bob was floored by _Moby-Dick_. Something in Ahab's lonely obsession spoke to him; his daughter, Lisa, would remember him ritually reading the book every year. He adored Joyce, especially _Ulysses_ , but his tastes also ran to more popular fare; at Dartmouth he sold a professor and several of his classmates on Joseph Moncure March's 1928 narrative poem _The Set-Up_ , about a black boxer who runs afoul of gangsters.
As an admired upperclassman, Bob drove around campus in a Buick roadster, took up smoking a pipe, and made bathtub gin. Prohibition had been in effect since 1919, and overturning it had become a touchstone for Democrats. In a nod to his father's electoral ambitions, he ran for class marshal on the slogan "Rum, Rebellion, and Ryan." His flyers declared him in favor of "free beer, free love, and free wheeling." But that summer would bring him closer to genuine lawlessness than he could stomach. "I answered an ad," he later recalled. "An oil man wanted a chauffeur. He took one look at me and said I was it. I ferried him around for two weeks before I discovered he was a bootlegger and that he was taking me along as a bodyguard." Bob soon quit the job.
As an undergraduate at Dartmouth College, Ryan became an intercollegiate boxing champ and ran for class marshal on the slogan "Rum, Rebellion, and Ryan." _Robert Ryan Family_
Without the athletics, his academic performance improved; he made Phi Beta Kappa in his junior and senior years, wrote an essay on Shakespeare that was anthologized in a collection of undergraduate writing, and won a hundred-dollar prize for his experimental one-act play _The Visitor_ , whose title character was the grim reaper and whose one and only performance took place in the college's Robinson Hall. Now twenty-two, Bob had hung onto his blissful ignorance for as long as possible, but he began to understand that he would be graduating into harder times than any he had ever known. The Ryans' life of luxury had evaporated as the country spiraled into depression. Tim wanted Bob to come home and help with the business, but Bob resisted. He would do anything but seal himself up inside an office.
After graduating in June 1932, Bob took what little money he had and moved to Greenwich Village with two fraternity brothers, intending to find a job as a newspaper reporter and work on his playwriting. A third of the country was out of work, and along the streets of New York people queued at breadlines and soup kitchens. Bob couldn't figure out what he wanted to do with his life; he only knew he couldn't go into business. He fought a professional bout under an assumed name to raise some cash, but otherwise the boxing went nowhere. A girlfriend got him gigs modeling for true-confession magazines and department store ads — he later claimed to be the first man in America to model French jockey shorts — but his pals gave him so much grief over this that he quit. For a while he worked as a sandhog, pushing rock barges through tunnels under the Hudson River.
In this economic climate the pampered young man oscillated between realism and sheer fantasy. Some pals from Psi Upsilon persuaded him to come in with them on a gold mine in Libby, Montana, and Bob moved out West to prospect with a friend. The living was rough; they had to break ice on a stream for bathing water. After four months they had managed to extract about eight dollars' worth of gold. When Bob heard about a cowpuncher job in Missoula paying that much every week, he gave up on the mine, and eventually he returned to New York City, wearing a long beard and hitting up his classmates for money to get back on his feet.
Magazine profiles would offer differing accounts of how Bob managed to wind up a sailor aboard _The City of New York_ , a diesel freighter making runs to South Africa, in 1933. According to one, he was strolling along the Brooklyn waterfront one day, visiting a friend, and when he saw the ship loading on the wharf, he impulsively asked for a job. According to another, he "accepted drinks one night from a jovial tramp steamer captain" and "woke next morning bound for Lourenzo Marques, Portuguese East Africa." In any event, Bob shipped out as an engine room wiper, cleaning up oil that leaked from the cylinders and various pumps, oiling the pumps, and fetching coffee. Owned by the private Farrell Lines, _The City of New York_ headed down the East Coast to New Orleans and then across the Atlantic, carrying manufactured goods. It probably docked in Cape Town, East London, and Durban, and it returned to New York two or three months later with shipments of raw asbestos or chrome.
Bob might have been surrendering to his love of Melville and Eugene O'Neill, who had written of the seafaring life in _Anna Christie_ and _The Hairy Ape_. He spent more than two years at sea, collecting stories of hardship and adventure. The equatorial heat was unbearable; once he had to intervene when a delirious female passenger tried to push her baby through a porthole. Another time, after the ship's store of food spoiled, he subsisted for days on lime juice.
Whenever Bob heard from his parents, the news was grim. In December 1934 his Uncle Tom died, leaving the presidency of the Ryan Company to Tim. Soon after that both Joe Ryan and John Ryan died. The pressure of the construction industry was crushing them out like the cigarettes Bob now smoked daily. In January 1936, not long after returning home from a run, he received a phone call from his mother: his father had been hit by a car, and Bob was to return to Chicago at once to look after him and help out with a subway tunnel project. Bob made an inglorious return to Chicago as a common sandhog, pushing rock barges beneath the streets of the city by day and struggling to understand the business by night.
Tim's accident had exacerbated a heart condition, and on April 27, 1936, he died of a coronary occlusion at Passavant Memorial Hospital. He was sixty. Writing to his children, Bob would quickly recount the stock market crash and the tunnel disaster, adding, "I am sure that both of these events caused my father's early death." Tim was laid to rest in Calvary Cemetery in the north suburb of Evanston, beside his little son Jack.
Bob knew he had to look after his mother and made a game effort to help his uncle Larry, now president of the Ryan Company and the last surviving brother at the firm. But he wanted out of the tunnels: one time, as he was breaking up rocks with a sledgehammer, he turned over a rock to find an abandoned dynamite charge. Another time he worked forty-eight hours straight when a power plant failure imperiled the air pressure in a tunnel. Eventually, he quit the company, drifting from one job to the next. One oft-repeated story had him working as a collector for a loan shark on the blighted West Side and, moved by the poverty he saw, coming back to return one family's money. He was working as a gang boss on a WPA road paving crew for thirty dollars a week when his uncle Larry Ryan died in December 1937, only fifty-five years old. The Ryan Company would endure into the 1940s, but there was nothing left of the brothers now except their name.
Frustrated with her son's career drift, Mabel finally called on Tim's old friend Ed Kelly, who had not only survived the sanitary district scandal but ascended through a city council vote to become mayor of Chicago. After Anton Cermak was fatally wounded during an assassination attempt on President-elect Roosevelt in February 1933, Kelly had been pushed through the council by his old friend Patrick Nash, the Twenty-Eighth Ward alderman, and they controlled a formidable vote-getting operation that gave them enormous power over the city. Bob would remember Big Ed Kelly as an avatar of ward politics and no dreamer. One night in 1928, when Bob was home from college, he had been sitting in his parents' living room when Kelly came calling for Tim, having just met Al Smith, the progressive New York governor and Democratic nominee for president. "He's talking about things like welfare and human rights and all that shit," Kelly complained.
As mayor, Kelly had relaxed enforcement of gambling laws; according to the Chicago Crime Commission, the administration pocketed $20 million from organized crime one year to ignore illegal operations. At the same time Kelly had forged an alliance with Roosevelt and brought much-needed New Deal funding to the city. He went out on a limb politically with his vocal advocacy of open housing and school integration. To some extent this was pure politics: his success at drawing blacks away from the Republican Party contributed to his success at the polls. But Kelly acted too, appointing blacks to more influential posts, working to integrate the police department, and, at one point, shutting down a local screening of D. W. Griffith's _The Birth of a Nation_. "The time is not far away," he told one audience, "when we shall forget the color of a man's skin and see him only in the light of intelligence in his mind and soul."
Now Big Ed would come through for the Ryans one more time, with a white-collar patronage job for Tim's aimless son. Bob joined the Department of Education as an assistant vice superintendent, though his job consisted of little more than filling requisitions for school supplies. Under the leadership of James B. McCahey, a coal company executive and crony of the mayor's, the board had developed a reputation to rival the sanitary district's; local muck-raker Elmer Lynn Williams called it "the most corrupt Board of Education that ever cursed the Chicago schools." Down in his little basement office, Ryan recalled, he "had little to do except combat hangovers," so he spent a good deal of time writing, an infraction ignored by the other patronage hires. The boredom drove him mad — this was everything he'd struggled to avoid in his vagabonding days. He was pushing thirty, his father was dead, and he still hadn't decided what to do with his life.
The answer came to him one afternoon when he ran into a friend and she talked him into taking a role in a local theater production. Despite his passion for theater, Bob could be painfully inhibited; he still winced at the memory of delivering a speech in grade school and hearing laughter ripple through the audience when his voice cracked. But he took the role, and something happened. "I never even thought of acting until I was twenty-eight," he later recalled. "The first minute I got on the stage, I thought, 'Bing! This is it.'"
Electrified by the experience, Bob signed up for acting classes with Edward Boyle, a stock company actor who charged five dollars a week. "What an audience most likes to feel in an actor is decision," Boyle would tell him. "Always keep saying to yourself, 'Decision, decision, decision.'" After Bob's mother informed him that the Stickney School, whose upper classes were college preparatory instruction for girls, would have to cancel its senior class play because the drama coach had taken ill, Bob managed to convince the principal that he was an experienced stage director and took over the production. The play was J. M. Barrie's comic fantasy _Dear Brutus_ , and the performance, on May 6, 1938, went off reasonably well. "With kindest regards for the first person who ever wanted my autograph," Bob would write on a program for a friend.
Bob silently hatched a plan that would get him out of Chicago for good: over the next couple of years, he would save a few thousand dollars, move to Los Angeles, and enroll in the acting school at the Pasadena Playhouse. In another era he might have set his sights on New York, but Bob was still smitten with the movies. "The very mention of them excites the imagination and stirs the blood," he'd written in a high school essay. "We may walk out of our own world into another." By now his focus had shifted from Douglas Fairbanks to the new generation of talking actors: Clark Gable, Spencer Tracy, and James Cagney, the latter of whom had become a star playing a Chicago gangster in _The Public Enemy_.
His ticket out of town arrived in summer 1938. Years later a couple of news stories about Ryan would refer to an inheritance, but the story most frequently told had him unexpectedly striking it rich on a friend's oil well near Niles, Michigan, his three hundred dollars' worth of stock paying a sudden dividend of two thousand dollars. His mother was dumbfounded when he informed her of his plan. "You can't earn a living that way," she insisted. "This little acting group you play with is nice, as a hobby. But I know you. You can't act." Act he would, and before long he had kissed his mother good-bye and boarded a westbound train from Union Station. Surely his father would have disapproved. "How sharper than a serpent's tooth it is / To have a thankless child!" But then, if his father hadn't struck out on his own as a young man, he would have spent his life caulking boats in Lockport, Illinois. Whatever sort of life Bob found for himself in Los Angeles, at the very least it would be his own.
*His death predated by only a few months the first recorded cases of the Spanish influenza, which would kill at least half a million people in the United States alone.
*Countless news stories would misreport that Ryan retired from collegiate boxing undefeated; in fact, Dartmouth yearbooks indicate he lost to his opponent at Western Maryland College on a close decision in the 1930 season and fought his opponent at University of New Hampshire to a draw in the 1931 season.
_two_
The Mysterious Spirit
She was gorgeous. Five-foot-seven at least, with dark red hair and cutting, observant brown eyes. Ryan first spotted her in the hallway of the Max Reinhardt School of the Theater on Sunset Boulevard. He had arrived in Los Angeles to discover that the theater school at the Pasadena Playhouse was full, but a fellow named Jack Smart, whom he had met through a girlfriend in Chicago, recommended the Reinhardt School, which had opened just that summer. As Ryan liked to tell it, he decided to enroll the moment he saw the girl in the hallway. Through a school administrator he managed to arrange an introduction; her name was Jessica Cadwalader, she was studying acting as well, and they would begin classes together the next day with Professor Reinhardt. Feeling impetuous, Ryan asked her to dinner, and she accepted.
Jessica Cadwalader was twenty-three, born in Los Angeles to Quaker parents and, after they divorced, raised in Berkeley by her mother. She had graduated from the private Anna Head School, where she had been a tennis champ, and shortly thereafter she boarded a bus for New York City, where she found an apartment in the Murray Hill neighborhood of Midtown Manhattan, took modeling jobs through a Park Avenue agency, and tried to establish herself as an actress under the name Jessica Cheyney. For some time she had performed with the Wayfarers, a theater group in San Francisco. Now she was back in Los Angeles looking for movie work; she had been an extra in the W. C. Fields comedy _Poppy_ (1936) and gotten a line, only to see it cut, in the Gary Cooper drama _The Adventures of Marco Polo_ (1938). She was formidably well read despite the fact that she had never attended college, and she looked a little startled at dinner when Ryan informed her that the piece he had been rehearsing for the first class was no less than Hamlet's second soliloquy. He wanted to get Reinhardt's attention.
As Jessica already knew, Reinhardt's attention was a force to be reckoned with. Quiet and stout, with hypnotic blue eyes, the aging Austrian studied you so intensely, and listened with such force, that he seemed to be penetrating your very soul. Reinhardt had made his name in Europe and the United States with spectacular, expressionist stagings of _Everyman_ (for the Salzburg Festival, which he cofounded in 1920 with Richard Strauss), _The Miracle_ , and _A Midsummer Night's Dream_. His 1934 production of the latter at the Hollywood Bowl became the talk of the town, and the following year Warner Bros. hired him to direct a lavish screen version with James Cagney, Olivia de Havilland, and Mickey Rooney. Born to Jewish parents in Austria-Hungary, Reinhardt had fled the Third Reich in 1938 and settled in Los Angeles. Though he never managed to land another movie assignment, he continued to direct stage productions on both coasts; in fact, the new school would serve as a workshop for plays he wanted to mount commercially.
Jessica braced herself the next day as her new friend from Chicago came forward to butcher Hamlet's second soliloquy: "O villain, villain, smiling, damned villain! My tables — meet it is I set it down / That one may smile, and smile, and be a villain!" Reinhardt's only response was, "With training..." Ryan took this as a great triumph when Jessica spoke to him afterward. "There is a young man who has just enrolled that I like very much," she reportedly wrote to her mother, "but he's the worst actor I've ever seen in my life."
Meeting with Ryan later, Reinhardt told the young man he had a quality that reached out over the footlights, and with enormous work and commitment he might one day become a great performer. These were the right words coming from the right man at the right time, and from that moment onward Ryan entrusted himself to Reinhardt. "Max Reinhardt was not only my first teacher," he would write near the end of his life (forgetting Ed Boyle in Chicago), "but remains to this day, thirty-two years later, the most tremendous and important person who has ever influenced my career and my work." Though Reinhardt was best known for his elaborate productions, incorporating music, choreography, and lighting effects, Ryan saw that the old man was also deeply and personally invested in his smaller projects. "His own obsession was the inner life of man," Ryan wrote, "the mysterious spirit that both flickers and flames in all of us."
Reinhardt felt that human emotion was stifled by bourgeois life. "Unconsciously we feel how a hearty laugh liberates us," he wrote in an essay on acting, "how a good cry or an outbreak of anger relieves us. We have an absolute need of emotion and its expression. Against this our upbringing constantly works. Its first commandment is — Hide what goes on within you. Never let it be seen that you are stirred up, that you are hungry or thirsty; every grief, every joy, every rage, all that is fundamental and craves utterance, must be repressed."
How profoundly this idea must have struck his new student from Chicago, this powerfully built but painfully shy man whose parents had shown him the good life but always taught him to keep his feelings to himself. "Only the actor who cannot lie, who is himself undisguised, and who profoundly unlocks his heart deserves the laurel," Reinhardt wrote. Not until years later, after working with numerous pedestrian directors, would Ryan recognize what an enormous gift Reinhardt had given him so early in his development. Yet implicit in that gift lay a great moral and emotional challenge.
Reinhardt cut an imposing figure, yet he tended to put people at ease because he listened so closely. "He never listened passively," recalled the composer Bronislaw Kaper, "he listened actively, with the greatest interest reflected in his eyes and his half open lips." In fact, Reinhardt's ability to listen defined his whole approach to acting. "The best piece of advice I've ever received as an actor was given me by Max Reinhardt," Ryan told a reporter years later. "He put it in one word — 'Listen.' If you really hear what other actors say to you, your own reaction and the proper reading of your lines will be easy."
Actors who worked with Reinhardt, among them Stella Adler and Otto Preminger, testified to his talent for bringing an actor out of himself, quite literally — for locating personal traits that one might heighten and project onstage. If you engaged Reinhardt imaginatively, he invested himself in your performance, and you immediately felt the thrill of shared discovery. "He was most effective when he liked an actor, and perhaps only when he liked him," remembered Preminger. "If he felt that the actor really wanted to be directed by him, then his imagination, the variety of advice, the way he worked the actor in the scene and _for_ the scene, was just fantastic. I don't think any director ever had that gift. Maybe it was because he was an actor originally."
The Reinhardt School offered a well-rounded education, and Ryan threw himself into his studies, learning about lighting, set design, and direction. But acting was his great love now. His workshop teacher, Vladimir Sokoloff, had performed with the Moscow Art Theatre under the great director Constantin Stanislavski, and from him learned the principle that movement expressed a character's motivation better than anything else. Yet Sokoloff's classes were more traditional than the Stanislavski-inspired "method acting" then gaining traction at the Group Theatre in New York, in which the performer used powerful personal memories to trigger onstage responses. " 'The Method' would have driven Sokoloff out of his skull," Ryan later mused. "He taught action, not 'memory of emotion.'"
Under Sokoloff's instruction the young man improved rapidly, and during the fall 1938 semester Reinhardt cast him as Silvio and Jessica as Beatrice in a workshop production of Carlo Goldoni's _At Your Service_. Ryan played Bottom in _A Midsummer Night's Dream_ and the father in Pirandello's _Six Characters in Search of an Author_ , "at one of whose unforgettable rehearsals," wrote Gottfried Reinhardt, "my father showed Bob Ryan how literally to collapse after the discovery of his daughter in a brothel, how to fold up like a jackknife and to exit, his torso bent horizontal, a destroyed human being." Clearly Reinhardt appreciated the physicality of this boxer who had graduated to the stage, and Ryan would embrace the idea of movement as character.
ALL THROUGH this great artistic awakening, Ryan was falling in love with Jessica Cadwalader. Their courtship took a rocky turn when he invited her to dinner at the Brown Derby and a miscommunication resulted in each of them sitting alone, waiting for the other to materialize, on successive days. When he called her to complain about being stood up, she hung up on him and went to San Francisco with a girlfriend. But before long the two thespians had become inseparable, going out for drinks when they could afford it or talking all night about books and movies and politics and, of course, acting. Ryan had never met anyone like her; she was introverted, but smart as a whip and passionately idealistic. The more time he spent with her, the more he wanted her in his life. For some reason she always called him Robert; friends and family had called him Bob for years, but to Jessica he would always be Robert Ryan.
Ryan might have thought he had experienced the West in his Montana adventures, but Jessica's people were real westerners. Her maternal grandmother, Anno, told Jessica all about the old days. Born Annie Neal in 1859 to an undertaker in Atchison, Kansas, she had been worshipping at the town's Episcopal church one Sunday morning when she met George Washington Cheyney, a young Philadelphian five years her senior whose wealthy family, alarmed by his indolence, had set him up as manager of a silver mine that some of his father's colleagues owned in Tombstone, Arizona. On his travels back and forth, George Cheyney changed trains in Atchison, and before long he and Annie had married and moved to Tombstone, to a large house on the hill overlooking the town.
Jessica Cadwalader (late 1930s). Ryan met her in the lobby of the Max Reinhardt School of the Theater on Sunset Boulevard; they spent the next thirty-three years together. _Robert Ryan Family_
By then Tombstone was the fastest-growing boomtown in the Southwest, with a fair amount of culture alongside the roughnecks who poured in hoping to strike it rich. There were decent restaurants, an ice cream parlor, and opera performances at Schieffelin Hall, named for the prospecting family that had founded the town. Jessica pressed her grandmother for details about the famous shootout at the OK Corral in 1881. "I never knew anything about all that riff-raff," Anno replied. Her husband "did not think such goings-on should be talked about in front of ladies.... I have a feeling George said it was good riddance to bad rubbish." Later Jessica dug up a history of Tombstone that described one George Cheyney ducking behind a counter during the armed robbery of an assayer's office.
As superintendent of the Tombstone Mill and Mining Company, George Cheyney branched out from Tombstone and developed a new mine in the Oro Blanco Mining District, but in the late 1880s Tombstone's mining industry collapsed after the miners began to hit water and the town's pumping plant was destroyed in a fire. George ran for Congress as a Republican in 1890 and served as school superintendent for the territory, then moved his family to Tucson, where he was appointed postmaster in 1898 and four years later ran a successful campaign for probate judge. Shortly after his election George traveled to San Francisco, seeking treatment for a liver ailment from a Tucson physician who had moved there, and died at age forty-nine from cirrhosis.
Three years later his second daughter, Frances — Jessica's mother — married Richard Bacon Cadwalader, a young Quaker in his early twenties who had come West from Cincinnati with his mother, Ella Bacon Cadwalader, after suffering a nervous breakdown in his first semester at Harvard. Ella Cadwalader fought against the union between Richard and Frances, but when Anno traveled from Tucson to Los Angeles to visit her sister, she took the young lovers along and had them married by an Episcopal clergyman. This would have been the ultimate horror for Ella and her husband, Pierce Jonah Cadwalader, whose family had followed the Society of Friends since the seventeenth century and been part of the influential Philadelphia Quakers Meeting.
In 1907, Frances gave birth to a son, Richard Jr., and seven years later, on October 26, 1914, Jessica Dorothy Cadwalader arrived. The family was living in Tucson when Richard Jr., only ten years old, died of influenza in September 1917 (just three months earlier, little Jack Ryan had succumbed in Chicago). Jessica grew up an only child, an introvert, and a voracious reader, closely instructed in her religious beliefs by her great aunt Dora, whom she remembered as "a great and determined Quaker lady." From childhood Jessica learned to value peace over war, mercy over revenge; she learned that God's spirit, dwelling within her, not only permitted but obliged her to work for peace. Dora liked to recite the "Quality of Mercy" speech from _The Merchant of Venice_ , in which Portia describes mercy as "twice blest: / It blesseth him that gives and him that takes."
Since the beginning the Society of Friends had preached the equality of men and women, allowing women into ecclesiastical positions, and in America the Quakers had provided not only the idealism but also some of the early leaders of the women's movement: the Philadelphia abolitionist Lucretia Mott, who had helped organize the 1848 Seneca Falls Convention on women's rights; the great speaker and activist Susan B. Anthony, who spent a lifetime trying to win women the vote; and Alice Paul, who helped pass the Nineteenth Amendment in 1920 and wrote the Equal Rights Amendment proposed to Congress three years later.* Dora wanted Jessica to get a good education and become a lawyer like Dora's brother, Jonah; there was no reason she should have to spend her life in her husband's shadow.
ON SATURDAY, MARCH 11, 1939, Bob and Jessica exchanged vows at St. Thomas Episcopal Church in West Hollywood, with their mothers, the Reinhardts, the Sokoloffs, and about fifty of their fellow students attending (including Nanette Fabray, the other big star who would emerge from their graduating class). Anno must have been there as well, a reminder to Ryan of the iron female will surging through his bride's family. A respectable matron in Tombstone and an example to her children in late middle age, Anno had decided upon her seventieth birthday to please no one but herself. "That evening she drank her first highball and smoked her first cigarette," her granddaughter wrote. "She went on doing both to the end, chain smoking without inhaling, puffing out great clouds of smoke to wreathe her white head, looking like something between a Chinese ancient and an old madame, while the cigarette ashes spilled down the front of her massive bosom."
Two more productions — Goldoni's _Servant of Two Masters_ , which had been one of Reinhardt's early triumphs, and _Holiday_ , a romantic comedy by Phillip Barry that had become a screen hit for Katharine Hepburn and Cary Grant — followed before the end of the year's study, at which point the two newlyweds began to reckon with the question of money. As the story goes, word came shortly after their wedding that Ryan's oil well in Michigan had run dry, which meant an end to their steady dividend.
They supported themselves as best they could: Ryan worked as an assistant director to Reinhardt and taught boxing lessons for a dollar a pop, but Jessica was the real breadwinner, modeling for a photographer and then hiring on with vaudeville producers Franchon and Marco as a chorus girl at the Paramount Theater. "It was a rugged job, and she hated it," Ryan would write, "but it made it possible for me to work and study and pound on doors and try a little longer to make somebody believe I could really act." The first agent Ryan approached told him to go out the door and come back in again. "Make an entrance. Get it?" When Ryan did, the agent said, "Go back to Chicago."
From the house they had rented after their marriage, they moved to a small cottage and then to an apartment above someone's garage. Their situation was precarious, but Ryan was relatively sanguine. "I thought of what had happened to my father and knew that it was worse than useless to worry," he recalled. "The moment I stopped worrying, things began to come right for us."
In late December 1939, Reinhardt cast Ryan in a commercial production of Somerset Maugham's drawing room farce _Too Many Husbands_ , to open the following month at the Belasco Theater in Los Angeles. Promoted as "a saucy comedy with music," the play centered on a woman who believes her first husband has been killed in action during the Great War and takes a second, only to have the first return home; by then she has a child by each man. Marsha Hunt, a young actress who had recently signed to MGM, went to see a friend in the play and was struck by Ryan and the other male lead, former Olympic shot putter Bruce Bennett. "They were remarkable, both of them," Hunt recalled. "Tall, wonderfully good-looking but, most of all, graceful in their movements onstage." The engagement brought Ryan his first serious attention around town, and by the end of its run a casting director for Paramount Pictures had recommended him to director Edward Dmytryk for the lead in _Golden Gloves_ , an upcoming picture about amateur boxing.
_Golden Gloves_ told the story of a young fighter, mixed up with a crooked promoter, who sees the error of his ways and throws in with a crusading journalist to clean up the sport.† Dmytryk shot a screen test with Ryan as the fighter, then decided to give the role to Richard Denning; as a consolation prize he cast Ryan as Denning's opponent in the climactic bout. They began shooting the fight in mid-December and finished in seven days; given nothing but a soundstage and three hundred extras, Dmytryk managed to evoke an entire stadium by dimming the lights on the audience, as at a real fight, and using bee smokers to create a cigarette haze over the crowd. Dmytryk was impressed with Ryan in the ring: "He was 6′ 4″, weighed 198 pounds, boxed beautifully, and hit like a mule. He tapped Denning in the ribs during their fight, and Dick made three trips to the hospital for X-rays. To this day he insists his ribs were broken, though the pictures showed nary a crack."
With the role came a contract as a stock player at Paramount for $125 a week, and the chance to experience a moviemaking operation from the inside. As Ryan sat with photographers and makeup artists and casting people, his physical attributes were evaluated with cold precision. At thirty years old, he was a seriously handsome Black Irishman, lean and muscular, with a strong jaw and a warm, brilliant smile. Yet his forehead was already lined from years of hard labor, and his brown, crinkly eyes were rather small in his face; if he narrowed them even slightly, they took on a beady, menacing quality. His height was impressive but hardly ideal for someone trying to get a leg up in supporting roles. "The men stars wouldn't have me in a picture with them," he recalled. "I towered over so many of them."
Paramount threw him bit roles: one morning in January 1940 he shot a scene for _Queen of the Mob_ , based on the story of Kate "Ma" Barker and the Barker-Karpis gang, and a month later he put in two days playing an ambulance driver, barely glimpsed on-screen, in the Bob Hope comedy _The Ghost Breakers_. From mid-March to early May he was a Canadian mountie in Cecil B. DeMille's _North West Mounted Police_ , starring Gary Cooper and Madeleine Carroll, and that same month he played a train passenger in the nondescript western _The Texas Rangers Ride Again_. Ryan was disappointed but not exactly surprised when Paramount cut him loose after six months. Rather than hanging around Hollywood, waiting for something to happen, he and Jessica resolved to look for stage work in New York.
Back in Manhattan, the couple scraped by on Jessica's modeling gigs and whatever Ryan could find. A year after Hitler's invasion of Poland had ignited the war in Europe, President Roosevelt succeeded in passing the Selective Service Act, which established the country's first peacetime draft and required the registration of all men from twenty-one to thirty-five years old. As a married man, Ryan was unlikely to be drafted soon or at all, but Jessica was horrified by the idea of him going to war. Ryan "believed that people should fight their own fights," their son, Cheyney, later wrote. "Hence, if you believed in a war, you should be ready to fight it yourself." Yet Jessica had been raised to believe that all war was immoral. "For her, war was not a story of people fighting their own fights. It was one of the privileged sending others to pay the costs while they reaped the benefits and attacked the patriotism of others along the way."
By June 1941 they had hired on at the Millpond Playhouse, a summer stock theater in Roslyn, Long Island. The productions tended toward mystery and comedy; the company, Ryan recalled, was "appalling, being mostly bad amateurs." In _The Barker_ he played a carnival barker and Jessica a hootchie cootchie dancer; two weeks later they costarred again in something called _Petticoat Fever_. Millpond staged a mystery play Jessica had written, _The Dark Corner_ , and in the comedy _Angel Child_ , Ryan costarred with twenty-two-year-old Cameron Mitchell. The highlight of the season was William Saroyan's philosophical barroom comedy _The Time of Your Life_ , starring Ryan as the rich drunk, Joe, who encourages the other barflies to live life to the fullest.
The Ryans bailed out soon afterward, landing first at the Robin Hood Theater in Arden, Delaware, and then at the Cape Playhouse in Dennis, Massachusetts, where Ryan won a romantic role opposite the celebrated Luise Rainer in J. M. Barrie's comic fantasy _A Kiss for Cinderella_. Set in London during World War I, the class-conscious fantasy told the story of a poor cleaning woman, played by Rainer, who dreams that she is Cinderella and the neighborhood constable, to be played by Ryan, is Prince Charming. _This guy is going to be a big star_ , thought Robert Wallsten, a fellow cast member, as he watched Ryan rehearse. "I had no idea about his dramatic ability, and playing this Irish bobby was not a very serious role. But he had a corner on that Irish charm, and there was that magic grin.... It was the smile that was so warm and engulfing, and so endearing." Wallsten would become one of the Ryans' oldest friends.
From Dennis the production moved to the Maplewood Theatre in Maplewood, New Jersey, where Ryan caught an extraordinary break. Rainer had been married to Clifford Odets, a founder of the Group Theatre and one of the most daring American playwrights of the day ( _Waiting for Lefty, Awake and Sing!_ ); with the recent demise of the Group, Odets had sold his play _Clash by Night_ to showbiz impresario Billy Rose, who was mounting a Broadway production with Lee Strasberg, another Group founder, as director. The play dealt with an unhappy working-class couple on Staten Island, but in a larger sense it considered the restive political mood in America as the war in Europe raged on. Tallulah Bankhead, hailed for her recent performance in Lillian Hellman's _The Little Foxes_ , had signed to play the bored and frustrated wife; Lee J. Cobb, among the Group's most gifted actors, was cast as her dense but devoted husband; and Joseph Schildkraut, a longtime stage and screen veteran who had won an Oscar playing Alfred Dreyfus in _The Life of Émile Zola_ , was the husband's cynical friend, who moves in on the wife. For the minor role of Joe Doyle, a young neighbor with romantic problems of his own, Rainer urged Odets to consider her handsome young lead in _A Kiss for Cinderella_.
Rose took Bankhead out to Maplewood to see the show, and she liked Ryan. Soon after _A Kiss for Cinderella_ closed on September 23, 1941, he was rehearsing _Clash by Night_ in New York City. One can only imagine his excitement: four months earlier he had been slugging it out at the Millpond Playhouse, and now he would be making his Broadway debut in a cutting-edge social drama, alongside some of the most respected talents in the American theater. He had seen Bankhead in _The Little Foxes_ and thought her an extraordinary actress. A world-class diva, she could be witheringly cruel to colleagues, but she took a shine to him during rehearsals. When he introduced her to Jessica, who had been modeling to help meet the rent, Bankhead quipped, "If I was fifteen years younger I'd take him away from you." The Ryans laughed, though Jessica couldn't have been too pleased. She would spend the next thirty years meeting women who were less frank but similarly inclined.
"Tallulah was a stereotype of what the public thinks star actresses are like: they really aren't except in her case," Ryan would remember. "She liked some kind of excitement going on and didn't much care where it came from." At the same time Bankhead was a consummate professional, the first to arrive and the last to leave, and always with her part down cold. She might challenge Strasberg or Odets in rehearsal, yet in performance she could be remarkably generous toward other players. "She was a great experience," Ryan would conclude, "and she came along at a most important time in my life."
Unfortunately, the production quickly degenerated into a snake pit of professional rivalries and personal grudges, from which Ryan was lucky enough to be excepted. Bankhead despised Billy Rose, a diminutive casting-couch type whose theatrical résumé consisted mainly of brassy revues. "He approached the Odets play as if he were putting on a rodeo," she later wrote. An elegant presence onstage, Bankhead had taken the role of the drab housewife as a dramatic stretch, but when the play began its out-of-town tryouts in Detroit, critics decided she had been miscast, favoring Lee Cobb's performance as the husband. "That was when the shit hit the fan," Ryan remembered. Bankhead and Katherine Locke, who played Ryan's girlfriend, soon fell out, united by nothing except their dislike of Schildkraut, whom Locke later accused of putting the moves on her.
Though some of these conflicts sprang from ego or personal enmity, the production was built on an artistic fault line that would become more apparent in years to come: on one side were the more traditionally trained actors such as Ryan, Bankhead, and Schildkraut, and on the other were proponents of the Method such as Cobb and Strasberg, the latter of whom would institutionalize the techniques of tapping into one's own emotional experience when he founded the Actors Studio six years later. Method acting could be fresh, genuine, even explosive, but it could also be unpredictable and inconsistent from night to night. Cobb, the most ardent Method actor among the cast, often seemed to be working through his role onstage, and for someone such as Bankhead, playing against him was one curveball after another. Ryan sympathized with her, and later in his career, colleagues would note his annoyance and even anger over onstage surprises.
From Detroit, _Clash by Night_ moved on to Baltimore, Pittsburgh, and Philadelphia, where Bankhead came down with pneumonia and the show was shut down (as the star, she had no understudy). While the cast and crew cooled their heels in New York, waiting for her to recover, Ryan scored an interview for the lead role in a Hollywood prestige picture to begin shooting the next year. Pare Lorentz — whose acclaimed documentary shorts _The Plow That Broke the Plains_ (1936) and _The River_ (1938) had won him a brief but controversial tenure as director of the US Film Office — had signed with RKO Radio Pictures to direct a dramatic feature about a war veteran trying to make ends meet during the Depression, to be titled _Name, Age and Occupation_. For six months he had been crisscrossing the country in search of an actor skillful enough to play the role but credible enough to function in the semidocumentary format Lorentz envisioned. Finally, he turned to his friend John Houseman, an erudite British producer who had collaborated famously with Orson Welles.
Working for the Federal Theatre Project, Houseman and Welles had staged the "voodoo" _Macbeth_ (1935), which transplanted the Shakespeare play to a Caribbean island, and the proletarian musical _The Cradle Will Rock_ (1937), which proved too hot for the government and inspired them to launch the independent Mercury Theatre. Houseman and Welles had gone on to create the CBS radio broadcast _The War of the Worlds_ , which had terrified the nation with its too-convincing account of a martian invasion, and the RKO drama _Citizen Kane_ (1941), whose critical acclaim had now emboldened the studio to bankroll Lorentz's ambitious project. Houseman arranged for Lorentz and himself to spend a week interviewing actors in a Manhattan hotel suite. When Ryan arrived to read for the part, his acting must have impressed them, but what really won over Lorentz was Ryan's endless litany of soul-crushing jobs in the depths of the Depression. Here was a man who not only could play the part but already had lived it.
In the 1972 memoir _Run-Through_ , Houseman would remember traveling by train with Lorentz through western Kansas and hearing on the radio in the club car that the Imperial Japanese Navy had attacked the US air base at Pearl Harbor, Hawaii, killing and wounding thousands of Americans. The next day the United States and United Kingdom declared war on Japan. Three weeks later, _Clash by Night_ opened on Broadway at the Belasco Theatre, its take on the national mood decisively outpaced by world events. Reviews were scathing, though the players got good notices for their work, Ryan included; most critics went to town on Odets, citing a lack of passion and fresh characterization. The play closed on February 7, 1942, after only forty-nine performances, but not before Ryan was seen by such luminaries as Greta Garbo, Judith Anderson, Ruth Gordon, and Thornton Wilder.
"Ryan's was a small part," remembered Tony Randall, then a young actor starting out in New York, "but he was very, very good." According to one news story, Ryan was "showered" with offers from New York producers, including one from the Theater Guild to appear in a new play with Katharine Hepburn. The attention went to his head. He would remember "swaggering" into Bankhead's dressing room one night and "demanding to know how long it was going to take before I was a really great actor. I expected her to say a year or so. But instead she said very quietly, 'In 15 or 20 years you may be a good actor, Bob — if you're lucky.'"
*For more on this fascinating topic, see Martha Hope Bacon, _Mothers of Feminism: The Story of Quaker Women in America_ (San Francisco: Harper and Row, 1986).
†The picture was loosely based on the story of Arch Ward, a _Chicago Tribune_ sports editor who had founded the tournament in the 1920s.
_three_
Bombs Away
Soon after the Odets play breathed its last, Ryan found himself in Tennessee shooting locations for _Name, Age and Occupation_ with director Pare Lorentz and actress Frances Dee. The movie's story dated back to a novel Lorentz had begun in 1931: an eighteen-year-old boy from North Carolina fights overseas in the Great War but finds nothing waiting for him back home except a series of dehumanizing farm and factory jobs. As Houseman explained, the movie would explore "the condition of the US industrial worker with special emphasis on the economic and emotional effects of the production line."
Lorentz already had tried mixing actors with real people, to less than stellar effect, in his documentary _The Fight for Life_ (1940), about the Chicago Maternity Center and infant mortality in the slums. But George Schaefer, president of Radio-Keith-Orpheum, was prepared to take a gamble on the director; before Ryan even reported for work, Lorentz and cinematographer Floyd Crosby had spent twenty days shooting industrial operations at Ford's River Rouge Plant and a US Army facility. Location shooting continued through the spring, and in June the company arrived in Los Angeles to spend four weeks shooting interiors on the Pathé lot in Culver City.
That same month, fed up with Schaefer's artistic pretensions and dismal bottom line, the RKO board replaced him with N. Peter Rathvon and installed Ned Depinet as president of the movie division, RKO Radio Pictures. Charles Koerner, the new, commercial-minded head of production, immediately targeted two runaway films: _It's All True_ , which Orson Welles had been shooting in Brazil since early that year, and _Name, Age and Occupation_. Lorentz, observed director Edward Dmytryk, was "a fine critic, a top maker of documentaries, but completely lost in straight drama. After 90 days of shooting, he was 87 days behind schedule."
Ryan on location with director Pare Lorentz for the ill-fated _Name, Age and Occupation_. Their RKO colleague Edward Dmytryk called Lorentz "a fine critic, a top maker of documentaries, but completely lost in straight drama. After 90 days of shooting, he was 87 days behind schedule." _Robert Ryan Family_
In late June, RKO halted production of _Name, Age and Occupation_ and asked Lorentz for a financial accounting. He must have seen the writing on the wall when Koerner announced his production plans for 1942–43: $12 million was budgeted for only forty-five features, and in contrast to the literary projects favored by his predecessor, RKO would be aiming for good, solid box office by making patriotic movies for the home front. _Name, Age and Occupation_ , with its Depression setting and heavy themes, hardly filled that bill, and after screening rushes, RKO executives killed the project.
They liked Ryan, however, and signed him to a $600-a-week contract; under Schaefer the movie division had developed a shortage of leading men, exacerbated now by the many actors enlisting in the armed forces. "Without the talent shortage I would very likely have still been grubbing around New York for 40 a week jobs where I more or less belonged at my stage of the game," Ryan confessed in a letter to a friend. He and Jessica moved their belongings back to Los Angeles and rented a house in Silverlake, which they shared with Ryan's fifty-nine-year-old mother, Mabel. By October he had his first assignment from RKO, a picture about the US Army Air Forces called _Bombardier_. With this new job, the couple decided the time had come for children. Before long Jessica was pregnant, but she miscarried early the next year, another sad bond for two partners who had each lost a sibling in childhood.
With _Clash by Night_ and now _Name, Age and Occupation_ , Ryan had been involved with two prestigious dramas that crashed and burned yet elevated him professionally. Two years after his pink slip from Paramount, he was back in Hollywood earning nearly five times as much from RKO. Boom times had returned to Hollywood with the US military mobilization against Germany and Japan: as defense plants hummed in the nation's industrial centers, workers with good wages packed the movie palaces in search of solace, inspiration, or just relief from their worries. The studios cooperated eagerly with the Office of War Information to rally moviegoers to the war effort, and of the seven features Ryan would make over the next sixteen months, every one addressed the war in some way.
Even as he cranked out these patriotic pictures, Ryan waited for his own draft notice to arrive. After Pearl Harbor the draft age had been widened to include all men from twenty to forty-four years old; he was thirty-two, but Congress and public opinion favored drafting single men over husbands and fathers. In February 1942 the director of selective service, Lewis B. Hershey, had ruled that movies were "an activity essential in certain instances to the national health, and in other instances to war production" and had granted deferments for essential "actors, directors, writers, producers, camera men, sound engineers and other technicians." The outcry in Congress and around the country was immediate, and within forty-eight hours the board of the Screen Actors Guild had voted to oppose the order, arguing that "actors and everyone else in the motion picture industry should be subject to the same rules for the draft as the rest of the country." Hershey soon reversed the policy, but California draft boards were generally cooperative toward the big studios.
_Bombardier_ had been in development for two years already and took as its inspiration not a play or novel but a piece of military hardware, the top secret Norden bombsight, which used a mechanical computer to calculate precision bombing at high altitudes. Pat O'Brien starred as an air force major preaching the virtues of the new contraption, and Randolph Scott was his friendly antagonist, a captain who favors traditional dive-bombing attacks. Ryan was cast as a jaunty young cadet at the new aerial bombardment training school (one scene has him reciting for O'Brien a pledge similar to that taken by real-life bombardiers, that he will "protect the secrecy of the American bombsight, if need be with my life itself," as strings swell and O'Brien looks on in misty-eyed sanctification).
Most of the movie was shot on Kirtland Army Air Base in New Mexico, whose bombardier training program was the model for the one in the movie. RKO vetted the script with the Office of War Information in exchange for access to planes and other resources; one fascinating scene in _Bombardier_ shows Ryan's cadet practicing atop a bomb trainer, a twelve-foot frame on wheels that simulates bombing trajectories as it rolls toward a small, motorized metal box representing the target. In return the army got a wholesome, rousing picture that reasoned away any qualms one might have had about raining death from above. One cadet is torn by letters from his mother, who belongs to a peace organization and fears for his soul. "Peace isn't as cheap a bargain, Paul, as the price those people put on it," his commander explains. "Those people lock themselves up in a dream world. You see, there are millions of other mothers that are looking to you."
Ryan put in five weeks on the shoot, though the cast was large and he didn't get much time in the foreground. At one point he bursts into a funeral service for a young trainee to announce in close-up, "The Japs have bombed Pearl Harbor!" During the climax, as Scott and Ryan fly a nighttime scouting mission over Japan in advance of the squadron, their plane is hit; instead of bailing out with the others, Ryan stays behind to dismantle the bombsight with a pistol (actual military protocol) and dies in a giant fireball when the plane crashes. The absurd ending has Scott, captured behind enemy lines, escaping from the Japanese to drive a flaming truck around the munitions plant for the benefit of O'Brien's squadron above. With its fiery payoff, _Bombardier_ validated Charles Koerner's new production strategy when it opened the following spring: budgeted at $907,000, it grossed $2.1 million.
More so than the bit parts at Paramount, Ryan's roles at RKO gave him a chance to learn the craft of screen acting, which favored subtlety of expression and demanded incredible mental focus. "On the stage you can coast along," he explained to a journalist years later.
You don't have to concentrate so intensely on small details as you do in a movie.... Let's say in this scene, you're talking to me and I'm supposed to be taking a sip from this cup while I listen to you.... on the stage, it doesn't make any difference when the cup goes back into the saucer because nobody can hear it. But in a movie scene, while I'm listening to your lines and thinking of the line I have to say next, I must also remember to time the return of the cup to the saucer so that it won't get there until after you finish the last word from your speech, and not a split-second before you finish. If the cup hits the saucer while you're still talking, the clack it makes on the soundtrack will clash with your last words and ruin the scene. A half-hour later we have to do the same scene over again for a close-up or from a different camera angle and it has to be done exactly the same as we did it before.
But even more than acting experience, Ryan took away from _Bombardier_ a long, warm friendship with Pat O'Brien, who liked the young man's professionalism and soon began lobbying to have him in his pictures. They shared some striking similarities, including a birthday (O'Brien was exactly ten years older), a Catholic upbringing in the Midwest (he had attended Marquette Academy and Marquette University in Milwaukee), and a love of Chicago (he had met his wife, Eloise, while appearing in a show at the Selwyn Theater in 1927). O'Brien had been summoned from the New York stage to Hollywood by Howard Hughes, who cast him as Hildy Johnson in the movie version of Ben Hecht and Charles MacArthur's _The Front Page_ (1931). O'Brien soon moved to Warner Bros. and became a professional Irishman, costarring no fewer than eight times with his pal James Cagney, before memorably embodying the title Norwegian in _Knute Rockne, All American_ (1940).
That hit allowed him to end a long, frustrating relationship with Warners and eventually sign with RKO. "I loved that RKO lot, as did most who worked there," O'Brien later wrote. "It exuded more friendliness and warm camaraderie than any studio in which I ever worked." In addition to Cagney, O'Brien was tight with Frank McHugh, a pudgy comic actor who had been with them at Warners, and Spencer Tracy, an old classmate at Marquette Academy. Ryan got to meet them all, though he was never social enough to be considered part of this "Irish mafia." When Ryan asked O'Brien if his natural reticence would hurt him in the movie business, O'Brien pointed to Cagney, who was equally private but remained one of Hollywood's biggest stars. "That was all I needed to know," Ryan recalled. "I became a Cagney."
Ryan's next assignment brought him into close quarters with another big talent. _The Sky's the Limit_ , which began shooting in February 1943, starred Fred Astaire as a heroic Flying Tiger who goes AWOL during a publicity tour of the United States and falls for news photographer Joan Leslie; chasing after him are his two pilot buddies, played by Ryan and Richard Davies. Ryan's character, Reggie Felton, is a snide comedian: riding in a parade, he prepares to poke Davies in the eyes Three Stooges–style but then remembers where he is and flashes the crowd a "V for Victory" sign. He spends most of his screen time needling Astaire, and in one memorable scene, set in an army canteen, blackmails him into doing a "swami dance" atop their table. Choreographed by Astaire, the dance took several days to film, during which Ryan sat in a chair looking up at the great performer. Ryan even scored some waltz lessons from Astaire when the scene called for him to share a dance with Leslie.
_Behind the Rising Sun_ , which began shooting in late April, was the darkest and most interesting of Ryan's wartime releases, an anti-Japanese propaganda picture of some journalistic substance but even more racial hysteria. Its source material was a 1941 book by James R. Young, an American journalist who had spent thirteen years working for the influential _Japan Advertiser_ before his reporting from occupied China, published in a variety of Japanese papers, got him arrested in Tokyo and held by police for sixty-one days. _Behind the Rising Sun_ offered a variety of insights into Japanese culture and a ringing indictment of the Imperial Army's misadventure on the continent. Young's sympathy and affection for the people of Japan was evident throughout, yet Doubleday Doran had packaged the book with a cover drawing of a slit-eyed, hideously grinning man, a fan in one hand and a revolver in the other.
After directing Ryan in _Golden Gloves_ over at Paramount, Eddie Dmytryk had landed at RKO and, with screenwriter Emmet Lavery, had assembled an anti-German propaganda piece called _Hitler's Children_ , about the Hitler Youth movement. For _Behind the Rising Sun_ the two men banged out a script that bore little resemblance to the book, but incorporated various incendiary news stories from prewar Japan. "On a not very original plot, we strung ten or twelve incidents calculated to increase the flow of patriotic juices," Dmytryk recalled. One of them involved a fighting match between an American boxer and a Japanese sumo wrestler, and Dmytryk had just the guy to play the boxer.
The not-very-original plot involved a Tokyo businessman (J. Carrol Naish), whose son Taro (Tom Neal) returns from the United States with an engineering degree from Cornell, falls for a pretty secretary (Margo), and clashes with his father over her. Nearly all the Japanese characters were played by American actors in eye makeup; Neal is particularly unconvincing, bounding down a ship's gangplank to announce in pure Americanese, "Gee, Dad, it's good to see you!" Later Taro serves with the Imperial Army in an occupied province of North China, where he hardens himself against atrocity. Confronted by an American reporter (Gloria Holden) in his office, he watches from a window as soldiers throw a child into the air and — Dmytryk implies with a jump cut — catch it on a bayonet. "They're not my men," Taro replies. "It's not my responsibility."
Billed fourth in the credits, Ryan played Lefty O'Doyle, a bushy-headed American baseball coach in Tokyo, and of the few incidents or observations from Young's book that found their way into the movie, many involved him. When Taro shows up at a game with his sweetheart, O'Doyle points out the flag display interrupting the game on the field. "Can you beat it?" he asks them. "Telling them that baseball isn't just baseball anymore? They mustn't come here to enjoy it just as a sport. They must come here to enjoy it as a military exercise." Another scene, lifted directly from the book, takes place during a late-night poker game at a geisha house where O'Doyle, who's had a few drinks too many, loses his temper over a mewling cat and fires his pistol into the darkness, scaring it away. Almost immediately a trio of police appear at the door to grill and browbeat him and his companions about the fired gun and the "arreged cat."
The big fight scene, which began shooting one Saturday afternoon in mid-May and continued the following Monday, would wind up the movie's oddest and best-remembered moment. O'Doyle is called upon to defend the honor of an American engineering executive who has clashed with Taro; defending Taro's honor is a towering sumo wrestler, played — in eye makeup — by Austrian-American wrestling champ Mike Mazurki.* At thirty-six, Mazurki stood an inch taller than Ryan and was built like a brick wall. The ensuing fight is less a match than a melee, O'Doyle throwing roundhouse punches as the wrestler kicks, chops, and grapples with him. In the end the boxer triumphs, yet even this wacky contest has a sharp edge: for dishonoring Japan, the wrestler is later executed.
Ryan triumphs as the American boxer pitted against Japanese sumo wrestler Mike Mazurki in RKO's propaganda item _Behind the Rising Sun_ (1943). When Ryan joined the Marine Corps, his reputation from the picture preceded him. _Wisconsin Center for Film and Theater Research_
THREE DAYS BEFORE MIXING IT UP with Mazurki, Ryan was inducted into the US Army. His draft notice finally had arrived, and RKO had arranged for a deferment so he could finish _Behind the Rising Sun_ and _The Iron Major_ , which were shooting simultaneously. The latter film was RKO's attempt to score with Pat O'Brien playing another legendary college football coach — in this case Frank Cavanaugh, whose long career was interrupted only by his meritorious service in France. O'Brien had lobbied for Ryan to play Timothy Donovan, a football hero under Cavanaugh who later became a priest and served alongside him as an army chaplain. Ryan ages unpersuasively from his early twenties to his sixties, with the usual graying temples, and brings the story to a close with a mawkish prayer promising his late friend Cavanaugh that the fight for freedom continues: "We thank you, Cav, and we salute you. God rest your gallant soul."
Another deferment was granted so Ryan could appear in the low-budget _Gangway for Tomorrow_ , an inspirational tale for the home front about five random folks riding in a carpool to their jobs at a defense plant. As they travel, flashbacks reveal stories from their past; Margo had a pretty good one, playing a French cabaret singer and resistance member who escapes from the Nazis, but Ryan's was a hokey number about an auto racer who wipes out in the Indianapolis 500 and has to stay home while his two buddies join the Army Air Corps. Originally titled "An American Story," the picture wore its Office of War Information credentials on its sleeve: in the final moments the workers arrive at the plant, lock arms, and head through the gates to the strains of "The Battle Hymn of the Republic."
By summer _Bombardier_ had opened to tremendous box office, and preview screenings of _The Sky's the Limit_ and _Behind the Rising Sun_ had brought Ryan overwhelmingly positive response cards from patrons. David Hempstead, producer of _The Sky's the Limit_ , was about to start an A picture with Ginger Rogers called _Tender Comrade_ , and he urged her to consider Ryan for the male lead. Written by the talented Dalton Trumbo, _Tender Comrade_ told the story of four women working in a Los Angeles airplane factory who decide to rent a house together; interspersed with this were flashbacks focusing on Rogers and her man, who's preparing to go to war. For an unknown actor this would be quite an assignment — seventeen solitary love scenes with one of Hollywood's biggest stars.
Rogers had caught a preview of _The Sky's the Limit_ at the invitation of her old dance partner Fred Astaire; she thought Ryan was too tall and too mean looking, but she agreed to let him read for the part. When Ryan showed up to audition, he found about a hundred other actors waiting. Yet as he and Rogers talked and played scenes together, she slipped Hempstead a note: _I think this is the guy_. Later, when Hempstead offered Ryan the job, he gave him the slip of paper, which Ryan kept to the end of his life.
_Tender Comrade_ would be Eddie Dmytryk's first A picture, and he spent a full month, from mid-August to mid-September, directing Ryan's love scenes with Rogers. The first of them neatly demonstrated Ryan's skill at expressing character through action: Jo (Rogers), alone one night in her studio apartment, hears a knock at her door and is overjoyed to discover it's her husband, Chris (Ryan), home on furlough; as he steps into the room, he swings his overnight bag into the air and sends it sailing across the room onto her bed. Ryan plays straight man to Rogers in the forced comic scenes detailing their early relationship, but he comes into his own as the couple begin wrestling with the fact that he wants to enlist. "I've never felt so at-home in a role in my life," Ryan told _Photoplay_. "Y'know, a lot of these scenes are retakes of things that have happened between Jessica and myself."
Ryan and Pat O'Brien in _Marine Raiders_. O'Brien mentored the younger actor at RKO, but they wound up on opposite sides when the House Un-American Activities Committee came to Hollywood. _Franklin Jarlett Collection_
Trumbo was one of Hollywood's more politically outspoken writers — he had participated in the founding of the Screen Writers Guild, the movie industry's bitterest labor battle of the mid-1930s — and with _Tender Comrade_ he added a provocative subtext to the standard women's picture. Jo (Rogers) can barely contain her heartache after Chris ships out; but while he's gone, she comes up with the idea of pooling her rent money with three fellow riveters. They agree to a majority vote on all matters, and Jo proposes that they share their resources further: "Now the four of us here have two cars, two sets of tires wearing out. We could sell one car and use the other on a share-and-share alike basis." Rogers, a determined anticommunist, had balked at Trumbo's original line: "Share and share alike — that's American."
Ryan was scheduled to report for duty on October 20, but as the date approached, RKO offered him yet another script. Pat O'Brien wanted Ryan to costar with him in a war picture called _Marine Raiders_ , about the Marines' new amphibious commando units. The script was lousy, with tedious love scenes and chest-thumping heroics. In one jungle scene Ryan's impetuous captain finds a fellow marine who has been killed and desecrated by the Japanese; enraged, he goes charging into the enemy's position spraying machine gun fire.
To play something like this at Camp Pendleton in San Diego County, where _Marine Raiders_ would be shot as less fortunate men actually shipped out for the Pacific, must have filled Ryan with the sort of manly shame he had felt as a male model. Stars the stature of James Stewart and Robert Montgomery had enlisted and taken combat assignments; even Pat O'Brien put himself in harm's way entertaining troops in northern Africa and Southeast Asia. As part of the deal struck by RKO, Ryan asked to be discharged from the army so that he could enlist in the Marines, which would mean a greater chance of seeing combat. The Marines, in turn, would give him a deferment until January 1, 1944, so that he could make the picture.
_Marine Raiders_ didn't wrap until late January, however, and by that time the Marines had granted Ryan a second deferment through February 15. Shortly before the picture was completed, the commanding general at Camp Elliott in San Diego wrote to Marine Corps Commandant Alexander Vandergrift to request a third deferment through April 15, so that Ryan could appear opposite Rosalind Russell in _Sister Kenny_ , a biopic of the Australian nurse who had developed a radical new treatment for polio. Vandergrift would have none of this, and Ryan was ordered to report for duty on the fifteenth as previously agreed. RKO was offering him a "duration contract," which meant that he would be welcomed back to the studio upon his discharge from the service. _Behind the Rising Sun_ had been released in August and, partly on the strength of Ryan's much-talked-about fight scene, turned a jaw-dropping $1.5 million profit.
In a movie magazine piece that appeared under her byline, Jessica recalled "the dreary building in downtown Los Angeles" where she dropped Robert off for his Marine Corps induction. "It was that ungodly hour of the morning, at which time all good men seem to have to go to the aid of their country." They said their good-byes, she drove away weeping, and Ryan finally joined the war.
*Though still working his way up in bit roles, Mazurki would play heavies in movies and TV for another fifty years, most memorably in _Murder, My Sweet_ (1944).
_four_
You Know the Kind
If Ryan had any hope of remaining unnoticed in the ranks, they were diminished when he learned from a fellow recruit at the LA induction center that a letter from the Marine Corps — which Ryan had never gotten — listed toiletries and other items they should bring with them. A private on duty offered to pick up some things for him, and Ryan got off the bus in San Diego carrying his belongings in a brown paper bag. A Marine Corps photographer was there to meet him, snapping pictures as he turned in his travel orders, got fitted for fatigues, sat for a regulation army haircut, went through a classification interview, and picked up his gear from the quartermaster's depot. After that he was on his own and wondering how he would be received. When he had been at the base earlier, shooting _Marine Raiders_ , an officer had told him that movie boys were liable to get roughed up in the Corps, but Ryan didn't have any trouble. He mentioned this to a bunkmate; the man replied, "Most of these guys saw you bat that Jap around in _Behind the Rising Sun_."
Basic training commenced at Camp Pendleton, about an hour north of the San Diego base. Established as a Spanish mission in 1769 and built up through land grants into the vast Rancho Santa Margarita, the property had been purchased in 1942 by the US government, which was converting it into the nation's largest Marine Corps base. It was enormous — about two hundred square miles, with eighteen miles of shoreline for amphibious training. According to Pendleton historians Robert Witty and Neil Morgan, the terrain "stretches eastward across twelve miles of rolling hills, broad valleys, swampy stream beds, and steep-sided canyons, rising on its northeast perimeter to a height of 2,500 feet." By the time Ryan arrived, Pendleton had sent two divisions into the war and was home to more than 86,000 people.
He and Jessica had resolved to keep a stiff upper lip in each other's absence, but by the fourth week Ryan had written asking her to visit him that Sunday, and she endured the four-hour bus ride to meet with him at the reception center. They went outside, and he spread his poncho over the grass so they could sit and talk. He had learned how to use an automatic rifle, Thompson submachine gun, mortar, bayonet, and hand grenade. The infiltration course, in Wire Mountain Canyon, forced recruits to crawl through three trenches and penetrate a single and then a double apron of barbed wire while dynamite charges went off all around them and live rounds were fired over their heads. The obstacle course, built over a cactus patch, included a 125-foot wooden tunnel, a house whose only exit was through the roof, and a 100-foot cable bridge. He was mastering more mundane skills as well — how to mend his clothes, for instance — and drilling with his platoon. As the tallest marine, he was named honor man and placed in the front rank to set the pace; he took direction well, of course, and had to admit that the theatricality of it appealed to him.
Following a ten-day furlough in April, Ryan got his first assignment: effective immediately, he would be a "recreation assistant" at Pendleton. This was good news for Jessica, who wanted him out of harm's way and far from the trigger of a gun, but not for him. The whole idea of enlisting in the Marines was to erase the stigma of all those deferments; now he would be running a sixteen-millimeter projector and directing amateur plays. After fifteen weeks of this, during which time the D-Day landing commenced, he was transferred to the San Diego base, where he continued to thread a projector and also performed on _Halls of Montezuma_ , a weekly radio show broadcast coast to coast. Once Jessica realized he was unlikely to see action, she decided to leave Mabel on her own in Silverlake and moved to San Diego, where she occupied a "tiny box of a house on the pier at Pacific Beach."
By this time Jessica had stopped acting entirely. Back when they were with Reinhardt, she had been considered the better actor, but over the years she had watched Robert work and grow, and she was proud of his success. She had been at it for ten years now, and once Robert had started pulling down $600 a week at RKO, she decided she had had enough. She hated the stage fright and the tedium and the itinerant lifestyle. Instead she would turn to her second love, writing. In addition to the first-person piece about Robert's induction, which would appear in the October 1944 issue _of Movieland_ , she began placing stories in fan magazines such as _Photoplay_ and _Motion Picture_ and women's magazines such as _Coronet_ and _Mademoiselle_. Her immediate success would bring a weird parity to the marriage, since Robert had started out writing and, frustrated, turned to acting.
Serving on the sidelines must have gotten to Ryan, because on August 25 — the day Paris was liberated — he applied for a commission as a second lieutenant to serve on an aviation ground crew. "I feel that my background would qualify me for any branch of ground duty not requiring technical knowledge or expertise," he wrote on his application. A complete physical found him fit for overseas duty, and his commanding officer wrote him no fewer than three letters of recommendation. He waited the rest of the year for an answer. The Marine Corps was hardly generous with promotions, and he had no way of knowing whether the scuttlebutt about his deferments would hurt his chances, or what RKO might be doing behind the scenes to protect its investment.
In January 1945, as the Battle of the Bulge raged, Ryan was assigned to the Fortieth replacement draft; he would be shipping out as an infantryman, a development he would later describe to a friend as "swell." His application for a commission was turned down a week later. Ryan would be leaving for the Pacific in late February, which was alarming news to his wife and mother. But things didn't work out that way: as he would tell his Dartmouth class newsletter, "I was yanked out of the infantry with the proverbial foot on the gangplank and put to instructing troops in bayonet, judo, boxing and such at Camp Pendleton." Now classified as a combat conditioner, Ryan would spend his days training men for battle; in May he was promoted to private first class, and in August he was reclassified again, as a combat swimming instructor.
Thousands of Americans spent the war this way — not quite at home, not quite at battle. "Theirs is the task of the damned," wrote Richard Brooks in a prefatory note to his novel _The Brick Foxhole_. "These men see others trained and shipped off to ports of embarkation, but they themselves are always left behind. They brood over it, and in the end they become disappointed, introverted, and embittered." Ryan read the book with great interest when it was published in May 1945; the author was actually stationed at Pendleton, and according to scuttlebutt, the publication had brought him a court-martial. At the center of _The Brick Foxhole_ was an ugly sequence in which soldiers at liberty in Washington, DC, go home with a homosexual man for some late-night drinking and wind up beating him to death. Ryan was struck immediately by the book's frank depiction of the bilious prejudice on open display at Pendleton.
Robert and Jessica (circa 1944). While he drilled recruits at Camp Pendleton, she struck out on her own as a magazine writer and novelist. _Robert Ryan Family_
Of particular fascination was the character of Montgomery Crawford, the bullying sergeant who drags the other men into this murderous episode. Monty has served as a beat cop in Chicago and likes people to know he's killed a Jew and two black men in the line of duty. He "always shot niggers in the belly because then they didn't die right away and they squirmed like hell." Ryan had known guys like this before — cruel, jingoistic, worshipful of authority. Monty "shook hands with too firm a grip and he would openly cry when the post band played 'God Bless America.'"
Ryan tracked Brooks down to tender his compliments, and the two met in the library at Camp Pendleton. Brooks was tall and athletic — for a while he had considered a career as a pro baseball player — and favored a pipe that belied his short temper. Born to Russian Jewish parents in Philadelphia, he had grown up in poverty and gotten his start as a writer during the Depression by riding the rails and reporting on his experiences for local newspapers. From there he had moved into radio drama in New York and, in a weird coincidence, cofounded and then quit the dreaded Millpond Playhouse the summer before the Ryans performed there. Out on the West Coast Brooks found work writing for NBC Radio in Los Angeles, and as a screenwriter at Universal he knocked out a couple of jungle pictures for Maria Montez before enlisting in the Marines. Rumors of a court-martial over _The Brick Foxhole_ were true, though the Marines, realizing that more publicity would only enlarge the book's audience, had ultimately dropped the case.
Brooks undoubtedly knew Ryan from his good-guy roles at RKO, and to his surprise the actor told him that, if _The Brick Foxhole_ were ever filmed, he wanted to play Monty. Anyone could see the physical resemblance — Brooks had described Monty as "more than six feet tall" with "a pair of small, bright eyes" — but why would a lead player want a role like this? "I know that son of a bitch," Ryan explained. "No one knows him better than I do."
In fact, Ryan's experience as a combat conditioner — teaching men to kill and wondering if they would ever come back alive — was turning him against the military. When he wrote to his old Dartmouth pal Al Dickerson in early summer 1945, he took a dim view of his own contribution to the war. "I certainly haven't made any 'sacrifice,'" he admitted, "especially when you add the fact that I have sat on my ass stateside for 16 months while a lot of my buddies went on to Saipan and Iwo.... I will not bore you with the too well known complaints against the military. War is a stupid institution when it isn't being sinful and tragic and catastrophic." By the time he got out of the Marines he would come to share much of Jessica's pacifist philosophy.
Meanwhile Jessica had come into her own as a writer. To Ryan's surprise she announced one day that she had written a mystery novel and wanted him to read it. _The Man Who Asked Why_ was a "literary mystery," mindful of formula but written with intelligence and wit. Its eccentric sleuth, Gregory Sergievitch Pavlov, "looked like a retired clown" but was in fact an eccentric professor of languages, transparently based on the Ryans' dear friend and acting teacher Vladimir Sokoloff. Ryan passed the manuscript along to someone he knew at Doubleday Doran (publisher of _Behind the Rising Sun_ ), and to his and Jessica's delight the book was accepted for publication as part of Doubleday's Crime Club imprint, scheduled to appear in November.
In August the war, and the very concept of war, attained new levels of sin, tragedy, and catastrophe when President Truman ordered the atomic bombing of Hiroshima and Nagasaki. The blast in Hiroshima on August 6 flattened nearly five square miles and killed seventy thousand people, with tens of thousands more to die from burns and radiation by the end of the year. Few in America could grasp the devastation, but one thing everyone understood was that now Japan would surrender. RKO wasted no time in reasserting its claim to Ryan's services; the day after Nagasaki was bombed, the Marine Corps director of public information dispatched a letter to the commandant asking that Ryan be allowed to make a "marine rehabilitation picture" called _They Dream of Home_.* This request was denied, and ten days later Ryan was assigned to the Seventy-Ninth Replacement Draft. His feelings about shipping out may have been different this time, though, because Jessica had discovered that she was pregnant.
Two weeks after receiving his new assignment, Ryan reported to the Bureau of Medicine and Surgery complaining of neck and back pain. According to the doctor's summary, Ryan said that he had wrenched his back in May 1939, while lifting a car to change a tire, and subsequently suffered periodic attacks that had sidelined him for two to four days, but that "he did not mention this condition on induction because he considered it of minor importance and he wished to get into the service." Diagnosed with epiphysitis of the spine, he was pronounced unfit for service and recommended for discharge. By this time the Fourth Division had begun arriving home, and Pendleton was discharging 175 men daily. On October 30, 1945, Ryan won an honorable discharge and returned to civilian life, the prospect of fatherhood, and a steady job at RKO. The studio had already slotted him for a melodrama called _Desirable Woman_ that would give him a chance to work with Joan Bennett and the great French director Jean Renoir.
Years later, press accounts would note simply that Ryan had enlisted in the Marines, which was true but hardly the whole story. His wife's pacifism, his employer's opportunism, and his own professional ambition had kept him out of uniform for nine months, but then he had served and sought a combat assignment. He would note with disdain how his friend John Wayne had avoided doing his duty, and his own stint in the Marines would become a much-valued credential as he became more vocal in his commitment to peace. When his teenage son, Cheyney, asked him why he had served, Ryan's only response was, "What else was I gonna do?"
SON OF THE GREAT PAINTER, Jean Renoir had directed some of the best French films of the 1930s — _The Crime of Monsieur Lange, Grand Illusion, The Rules of the Game_ — before fleeing the Nazi invasion in 1940. Since then he had bounced around Hollywood, making one picture at Twentieth Century Fox, another for RKO, and two more as independent productions released through United Artists. His first UA project, _The Southerner_ , was a moving tale of struggling cotton farmers in Texas, but in general Renoir found the Hollywood of the war years to be rocky soil for his kind of left-wing humanism. (His limited fluency in English didn't help.) After finishing _Diary of a Chambermaid_ for UA, he returned to RKO at the invitation of Joan Bennett to direct _Desirable Woman_ , a romantic melodrama that had been in development for some time. Renoir had enjoyed working at RKO, and he looked forward to collaborating with producer Val Lewton, who had delivered for the studio with a series of artful, low-budget horror films ( _Cat People, I Walked with a Zombie, The Seventh Victim_ ).
Ryan's excitement about the picture only grew as he got to know the director. "One of the most remarkable men I've ever met," he would say of Renoir. "Working with him opened my eyes to aspects of character that were subtler than those I was accustomed to." His character was notably darker than anything he had played on-screen, a shell-shocked Coast Guard lieutenant now relegated to patrolling the misty Pacific coast on horseback. In one scene a friend at the base hesitantly informs him that the ship he was serving on has gone down, and the lieutenant is crushed. Not long afterward, on one of his lonely rides, he passes a wrecked ship, where he encounters a beautiful woman gathering firewood (Bennett). She brings him home to meet her husband (Charles Bickford), a famous painter now blind and embittered, and the lieutenant, consumed by lust for the woman, becomes an uneasy companion to the fractious couple.
In fact, Val Lewton had never really been interested in the project, and by the time principal photography commenced in late January 1946, he had been replaced by producer Jack Gross, who let Renoir do pretty much as he pleased. A month into the shoot, Charles Koerner — the head of production who had axed Pare Lorentz and Orson Welles — died suddenly of leukemia, which left Renoir even more unsupervised. He had never made a picture with so much improvisation on the set. "I wanted to try to tell a love story based purely on physical attraction, a story in which emotions played no part," Renoir said. The open adultery of the source novel, _None So Blind_ , already had been scrubbed away by the Production Code Administration, but there was something haunting about the lovers' wordless attraction playing out right under the blind man's nose.
Renoir also was intrigued by the story's sense of solitude, something he felt was increasingly prized amid the chaos of modern life. "Solitude is the richer for the fact that it does not exist," Renoir wrote. "The void is peopled with ghosts, and they are ghosts from our past. They are very strong, strong enough to shape the present in their image." One scene showed the lieutenant, Scott, in his bed at the base, consumed by a nightmare. In a feverish montage, an Allied ship hits a mine and goes down, the image of a whirlpool pulls the eye in, bodies and ropes drop through the water, and Scott strides across the ocean floor in slow motion, stepping over the skeletons of his dead crewmates, on his way toward a lovely woman in a flowing gown. Before they can kiss, there's an eruption of flame, an inferno that jolts Scott out of his dream.
Ryan — whose brother, father, and uncles all had preceded him to the grave — knew all about ghosts, and his strong streak of willful self-isolation made him an ideal collaborator for this kind of story. He would marvel at Renoir's ability to "discover the true personality of the actor" and integrate it into a performance, a skill he would recognize in no other director but Max Reinhardt. Renoir found a neurotic quality in Ryan that had never been captured on-screen and would become the key element in his screen persona. Lying in bed, Scott confesses to his commanding officer that the nightmares have become chronic since he was released from the hospital. Ryan's gaze shifts back and forth between two fixed points — the officer's face and something awful a million miles away — as the tension gathers in his voice. The doctors have pronounced Scott healed. "But my head!" he exclaims in anguish, gesturing at it as if it were strange to his body.
The picture wrapped in late March, leaving Ryan free to tend to his expect ant wife. On Saturday, April 13, Jessica gave birth to a healthy baby boy, whom they named after his grandfather, Timothy, in the Irish tradition. Ryan's next picture, a mediocre western called _Trail Street_ , didn't start shooting for three months, so the couple had plenty of time to care for and enjoy their new child and each other. Now that Ryan was back from the military, he worked either six days a week or not at all, loafing around in his mismatched pajamas until noon and working out later in the day. Jessica usually started writing in the morning — her second mystery novel, _Exit Harlequin_ , was scheduled for publication in January 1947 — and worked until mid-afternoon. Determined homebodies, she and Robert relaxed in each other's company, smoking, drinking, reading, and talking into the night.
Joan Bennett, Jean Renoir, and Ryan rehearsing _The Woman on the Beach_ (1947). "One of the most remarkable men I've ever met," Ryan called Renoir. "Working with him opened my eyes to aspects of character that were subtler than those I was accustomed to." _Franklin Jarlett Collection_
_Trail Street_ starred Randolph Scott as western hero Bat Masterson; Ryan got second billing as the villain, and whiskered "Gabby" Hayes dispensed cornpone comedy as Scott's sidekick. (Four-month-old Tim Ryan made his screen debut as a baby being held by a woman in the frontier town.) Director Ray Enright had spent twenty years in the business without making a notable picture; he was quite a comedown after Jean Renoir. At the same time, Ryan's collaboration with Renoir was in trouble. _Desirable Woman_ was test screened on August 2 in Santa Barbara, where it was laughed at and jeered by an audience full of students. Renoir would confess later that he was the first to get cold feet, and he offered to reedit the film. Five or six weeks later, he emerged with a version that was shown to two disinterested parties: screenwriter John Huston, who argued that the lieutenant's war trauma should be eliminated entirely, and director Mark Robson, who argued that it was central to the story. Renoir listened to Robson and moved the lieutenant's fiery nightmare to the beginning of the picture.*
By late November, when Ryan and Bennett were called in for reshoots, Renoir had lost confidence in his original conception, and the love relationship became more conventional. Several dialogue scenes that explained the characters' motivations were excised, which gave the action a detached quality. This garbled, seventy-minute cut of the picture, retitled _The Woman on the Beach_ , would flop at the box office eight months later and end Renoir's association with RKO. By then Renoir had grown close to the Ryans — Jessica adored him and his wife, Dido — and the two couples would keep in touch long after the Renoirs returned to France. "Bob Ryan is a marvelous person," Renoir would later attest. "Professionally he's absolutely honest in everything he does." Almost everything — Ryan admired Renoir too deeply ever to tell him he thought _The Woman on the Beach_ was a failure.
THE MOVIE BUSINESS boomed in 1946 as servicemen rejoined their families, which may explain why Peter Rathvon, the new president of Radio-Keith-Orpheum, allowed most of the year to pass before finally choosing forty-one-year-old Dore Schary to replace the late Charles Koerner as head of production. Schary was a comer: born to Russian Jewish immigrants in Newark, New Jersey, he had written plays in New York before arriving in Los Angeles to write for the screen and winning an Oscar for the MGM classic _Boys Town_ (1938). Since then the tall, bespectacled young man had supervised B-movie production for Louis B. Mayer at MGM, and independent producer David O. Selznick had tapped Schary to head his new company Vanguard Pictures. Schary had great story sense, and he knew how to get the most out of a dollar. Selznick was generous enough to let RKO buy out Schary's contract, and on January 1, 1947, Schary took charge of the studio's production slate.
Two days later the United States experienced a dramatic political shift when the Eightieth Congress convened, its opening session carried for the first time on broadcast television. President Truman, battered by union struggles as he served out Franklin Roosevelt's fourth presidential term, had been rebuked at the polls in November, when Republicans picked up fifty-five congressional seats and took control of the House of Representatives. Once the new Congress was sworn in, Republicans wasted no time in mounting a frontal assault on the Roosevelt legacy, and the House Un-American Activities Committee (HUAC), created in 1938 to investigate subversion against the US government, announced that a top priority would be uncovering communist influence in the motion picture industry.
A liberal Democrat, Schary took little notice of this as he moved into position at RKO. His formula for success involved socially conscious films that could be made on relatively small budgets, and the first script he sent into production was a murder mystery adapted from _The Brick Foxhole_ , the novel that had so intrigued Ryan when he read it in the Marines. Producer Adrian Scott, who had scored at RKO with the Dick Powell mysteries _Murder, My Sweet_ (1944) and _Cornered_ (1945), had read _The Brick Foxhole_ and was struck by the sequence in which soldiers beat a homosexual man to death. This would never get past the Production Code Administration, but what if the victim were a Jew instead? Scott hired screenwriter John Paxton to take a crack at the novel; their project, _Cradle of Fear_ , would be the first Hollywood picture to deal openly with anti-Semitism in the United States.
The script had gone nowhere with Charles Koerner in charge, and market research indicated that only 8 percent of moviegoers would go for such a picture (compared to 70 percent for _Sister Kenny_ , the Rosalind Russell drama RKO was still trying to get made three years after the Marines had refused to let Ryan appear in it). Schary was a different story — he read _Cradle of Fear_ one night and pulled the trigger on it the next day, naming Scott as producer and Eddie Dmytryk as director. The budget was around $500,000, but half of that would go for the stars Schary felt would be needed to sell such a controversial picture to the public. Scott and Dmytryk would have to get _Cradle of Fear_ in the can with what remained, shooting for about twenty days on existing sets. Paxton would remember his excitement after Schary gave them the go-ahead, as "a little parade went off around the lot (the writer just tagged along) looking for sets that could be borrowed or adapted, or stolen. An unusual procedure with front office blessing."
"What's-a-matter, Jewboy? You 'fraid we'll drink up all your stinkin' wonderful liquor?" Montgomery (Ryan) and Floyd (Steve Brodie) close in on their victim, Samuels (Sam Levene), in _Crossfire. Franklin Jarlett Collection_
Ryan got along well with Schary, and when he learned the picture was in preproduction, he begged the new chief to let him play Montgomery. Schary must have been surprised: this wasn't the sort of role that would lead to more love scenes with Ginger Rogers. Monty was repellent — ingratiating one moment, bullying the next, especially when he and his drunken pals are boozing it up with Samuels, who has met them at a bar and invited them back to his place. "Sammy, let me tell you something," Monty slurs. "Not many civilians will take a soldier into their house like this for a quiet talk. Well, let me tell you something. A guy that's afraid to take a soldier into his house, he stinks. And I mean, he _stinks!_ " Things only get worse from there: when Samuels tries to get rid of them, Monty snaps, "What's-a-matter, Jewboy? You 'fraid we'll drink up all your stinkin' wonderful liquor?" The word had never been uttered in a Hollywood picture.
The role might well blow up in Ryan's face. But he loved the script, valued the idea behind the picture, and knew he was the man to play Monty. "I thought such a part would make an actor — not break him," he later wrote. He lobbied Dmytryk — who, by this time, had directed him in his first picture ( _Golden Gloves_ ), his biggest hit ( _Behind the Rising Sun_ ), and his first romance ( _Tender Comrade_ ). Schary and Dmytryk acceded, billing Ryan third behind Robert Young as Finlay, the pipe-smoking police detective who investigates the crime, and Robert Mitchum as Keeley, a jaded sergeant who tries to save the confused young Private Mitchell from being framed by Monty. Schary also brought in some first-rate supporting players: Sam Levene as Samuels; sultry, blond Gloria Grahame ( _It's a Wonderful Life_ ) as a hooker who briefly adopts the private during his nocturnal wanderings; and, in the picture's second-creepiest role, craggy Paul Kelly as a man who hangs around her apartment and keeps changing his story about their relationship.
When the picture came out, Ryan would publish two stories under his own byline, in publications no less divergent than _Movieland_ and the _Daily Worker_ , that explained his rationale for taking the role. "Convictions are nice things to have," he wrote in the _Worker_ , "but when close friends tell you that you're jeopardizing your career by taking a role you believe in — well, it makes you stop and think." The picture was unlikely to convert any hardened anti-Semites, he conceded. "No one picture, no one book, no one speech could accomplish that. It's the cumulative effect that counts." In _Movieland_ he argued that the picture's subject was broader than anti-Semitism: "We all stand to lose if fascism comes. Not just the Jews. The Irish, the Catholics — and I'm both of those — the Negroes, labor, the foreign born, everyone is done for whose color, or religion, occupation or political belief is distasteful to some new paperhanger-turned-Strong Man."
Once he had been cast, Ryan dove into the part. He studied back issues of _Social Justice_ , a frequently anti-Semitic magazine edited by the Roman Catholic priest and populist demagogue Father Charles Coughlin. Launched in 1936, it had serialized the fraudulent _Protocols of the Elders of Zion_ , which purported to be a Jewish plan for global conquest, and published one article by Coughlin that borrowed passages from a speech by Joseph Goebbels, Hitler's propaganda minister, about the threat posed by communism, atheism, and the Jewish people. Ryan also paid a visit to Jean Renoir, who was still wrestling with _The Woman on the Beach_ , and asked him about the fascist sympathizers he had known in France. Renoir spent the afternoon telling him stories, and Ryan came away convinced that the key to Montgomery was a deep-seated sense of inferiority.
If Schary wanted to test the limits of his authority at RKO, he succeeded; in early February, Rathvon sent him a memo expressing his doubts that _Cradle of Fear_ would do anything to reduce racial intolerance. "Prejudiced Gentiles are not going to identify themselves with Monty and so feel ashamed of their prejudices," wrote Rathvon, a smart and cultured man whom Schary respected. "Rather they may be resentful because they feel we have distorted the problem by using such an extreme example of race hatred."
On another front, Darryl F. Zanuck, president of Twentieth Century Fox, informed Schary that Fox had a picture about anti-Semitism on the boards, _Gentleman's Agreement_ , and suggested he cease and desist. "We exchanged a few notes," Schary recalled, "then a phone call during which I was compelled to tell him he had not discovered anti-Semitism and that it would take far more than two pictures to eradicate it." Determined to beat _Gentleman's Agreement_ out of the box, Schary stepped up production on _Cradle of Fear_ ; principal photography would begin Monday, March 3.
Scott and Dmytryk went over the script carefully, working out every shot in advance to save time on the set. Once the cameras began rolling, Dmytryk fell into a pattern of shooting for about six-and-a-half hours each day, then using the last couple of hours to rehearse the next day's scenes; this gave the crew time to set up the first shot and enabled the players to come in the next morning ready to go. The sets looked cheap, so Dmytryk placed his key lighting low in the frame to throw lots of shadows; for a scene in which Monty bullies his accomplice, Floyd, the only light source was a table lamp, revealing some of the uglier lines in Ryan's face. Dmytryk also chose his lenses to make Monty look increasingly crazed: at first his close-ups were shot with a fifty-millimeter lens, but this was reduced to forty, thirty-five, and ultimately twenty-five-millimeter. "When the 25mm lens was used, Ryan's face was also greased with cocoa butter," Dmytryk recalled. "The shiny skin, with every pore delineated, gave him a truly menacing appearance."
The real menace, though, lay in Ryan's deft underplaying. Critics would stress the intelligence he brought to his heavy roles, but in the case of Monty, an ignorant blowhard, the defining characteristic was an animal cunning. In his first two speaking scenes, Monty is interrogated by Detective Finlay, and in both instances he hastens to defend his pal Mitchell, whose wallet has been found at the crime scene, even as he directs suspicion toward him and away from himself. In the second interrogation, with Sergeant Keeley looking on, Monty grows angry at Finlay's questioning and barks at him, promising, "You won't pin anything on Mitch, not in a hundred years!" Catching himself, he drops his gaze, glances back and forth at the two men, and apologizes, pleading, "It's just that I'm worried sick about Mitch."
This was Ryan's first picture with Mitchum, whose roughneck adventures during the Depression (boxing, riding the rails, doing time on a chain gang) were even more dramatic than his. The men liked and respected each other, but their upbringings set them apart; Mitchum had grown up poor and dropped out of high school, and his politics were more conservative. Ryan might have held forth on the dangers of fascism, but according to Dmytryk, when a reporter on the set asked Mitchum why he was making the picture, the actor replied, "Because I hate cops." In fact, he was annoyed at having been lured back from a Florida vacation by Scott with the promise of a great part, only to learn it was no such thing (Scott confessed that they needed him for his box office clout). Mitchum must have realized at some point that Ryan was walking away with the picture.
The only real complication to emerge during production was how to get rid of Monty after the detective has tricked him into exposing himself. Screenwriter John Paxton wanted to add a scene in which Monty goes to trial, but Schary scotched this idea. Schary later claimed that the picture's original ending had MPs cornering Monty and shooting him down "like a rat," which might have increased the audience's sympathy for him. Instead Monty breaks away from the cops and runs out into the street; from a second-floor window, the detective orders him to halt and then calmly dispatches him with a single bullet in the back. Paxton was appalled when he saw this, but he had no say in the matter. Schary also changed the release title to _Crossfire_ , which had no relevance to the story but sounded great.
Principal photography wrapped on Saturday, March 29, after only twenty-four shooting days. The project had come together so quickly that there was no time for the sort of front-office meddling that might have watered down the story. A few weeks later Ryan attended a rough-cut screening with Scott, Dmytryk, and a handful of RKO executives. None of the other cast members was there, but he was eager to get a look at his performance. Watching the story unfold, Ryan knew he had nailed the character. About fifteen minutes into the picture, the detective interrogates Monty, asking him about the victim. "I've seen a lot of guys like him," Monty explains, conspiratorially. "Guys that played it safe during the war? Scrounged around keepin' themselves in civvies? Got swell apartments, swell dames? You know the kind.... Some of them are named Samuels, some of them got funnier names." Later, as Monty smacks Floyd around, his rage boils over: "I don't like Jews! And I don't like nobody who likes Jews!"
After the screening was over and the lights came up, the room was silent. Finally, one of the RKO suits spoke up: "It's a brave thing you've done, Ryan. You're gambling with your career, of course." Another piped up: "Really courageous." Taken aback, Ryan walked out of the screening room, crossed the lot, picked up his car, and headed home. Given what he had seen in the Marines, talk of bravery embarrassed him. But the executives' remarks were the first reaction he had received outside of the cast and crew, and their subtext was obvious: if the public turned against Ryan, RKO would simply cut him loose.
*The film would ultimately be released as _Till the End of Time_ (1946), starring Robert Mitchum and directed by Edward Dmytryk.
*A definitive genesis of the movie can be found in Janet Bergstrom's "Oneiric Cinema: _The Woman on the Beach," Film History 11_ (1)(1999): 114–125.
_five_
We Will Succeed, You Will Not
Jessica Ryan hated guns: she had no intention of letting her lovely Tim play with toy guns, learning to fantasize about combat and killing. Robert, a capable marksman in the Marines, didn't feel that strongly, but he had no fondness for firearms either. "He went hunting once with his father and shot something," his son Cheyney remembered. "He said he'd never do it again."
Regardless, the RKO publicists liked nothing better than to send Ryan on a hunting expedition. The previous November he had driven up to Oregon with actor Lex Barker to be photographed hunting geese, and that spring Jessica swallowed her pride and accompanied him on a jaunt out to the desert with a photographer for _Photoplay_ and actress Jane Greer, who had just starred in _Out of the Past_ for RKO. Jessica posed at the wheel of a jeep and stood by as Robert held up a dead jackrabbit for Greer's inspection; the resulting story claimed that she and Greer each had bagged a rabbit as well. This was followed by another trip to a ranch in the San Fernando Valley to hunt pheasants for a four-page pictorial in _Screen Guide_ ; one photo showed the couple heading out from their car, Jessica scowling as she carries a rifle at her waist, the barrel pointed to her side.
After _Crossfire_ , Ryan strapped on his six-guns again for _Return of the Bad Men_ , another B western with Randolph Scott and "Gabby" Hayes. He couldn't wait to finish with this tired oater and move on to _Berlin Express_ , an espionage thriller scheduled to begin shooting overseas in July. Dore Schary had been mightily impressed by the documentary authenticity of Roberto Rossellini's Italian postwar drama _Open City_ (1945), and he wanted _Berlin Express_ to be the first drama filmed inside Germany since the fall of the Third Reich. (Director Billy Wilder would be arriving at the same time to shoot _A Foreign Affair_ for Paramount.) _Berlin Express_ centered on an international group of passengers riding a US military train from Paris to Berlin, and like _Crossfire_ , it would mix genre entertainment with liberal politics, stressing the imperative of world peace.
Ryan would be gone for more than two months, flying from New York to London and then traveling with cast and crew to Paris, Frankfurt, and Berlin. He was excited about the picture and eager to get a firsthand look at the ravages of war. General George Marshall had just delivered a commencement address at Harvard in which he stressed the danger of allowing the European economy to deteriorate any further; he called for a massive economic aid plan to rehabilitate the victors and the vanquished alike. _Berlin Express_ would carry Ryan right into the heart of this debate. He finished _Return of the Bad Men_ in mid-July 1947, and yet another photographer arrived, this time at the house in Silverlake, to shoot him packing his bags and bidding Jessica and Tim farewell on his way to the LA airport. Jessica was afraid of airplanes and begged him to take a train east, but Ryan never passed up a chance to fly.
The producer, thirty-seven-year-old Bert Granet, was shouldering quite an operation. Military permits were required at each point of travel through occupied Germany, which meant that twenty-seven cast and crew members had to be screened by the FBI* and then cleared by the army and State Department. Cinematic resources in Europe were so scarce that nearly all camera and lighting equipment had to be brought over, along with one hundred thousand feet of film that required cool, safe storage in nations where any kind of storage space was prized. Because local film laboratories were so dodgy, all footage had to be shipped back to the United States for processing, so none of the completed scenes could be viewed until the expedition returned to Hollywood. Lodging and automobiles were in short supply (there was only one camera truck available in all of Paris); even simple items such as nails, lumber, and rope had to be bought on the black market.
A native Parisian, director Jacques Tourneur had come to Hollywood in the 1930s and distinguished himself at RKO with subtle, low-budget chillers such as _Cat People_ (1942) and _I Walked with a Zombie_ (1943). He had just completed his masterpiece, the wistful film noir _Out of the Past_. Unfortunately, _Berlin Express_ didn't have much of a script; inspired by a _Life_ magazine story, it would be a rather awkward marriage of journalism and Hitchcock-style suspense, its harsh scenes of a ravaged Germany punctuating an increasingly far-fetched tale in which a German diplomat critical to the reunification effort is kidnapped by right-wing terrorists. The four heroes pulling together to foil this plot were obvious stand-ins for the occupying powers: Ryan is an American agricultural expert, Roman Toporow a Russian military officer, Robert Coote a British veteran of Dunkirk, and actress Merle Oberon the French secretary of the kidnapped politician.
Decades later Granet left behind in his papers an account of the personal chaos inspired by Oberon during the trip. The delicate British beauty had outmaneuvered him already by getting Schary to name her husband, Lucien Ballard, as cinematographer; the couple would be rooming together. The company was lodged at the Hôtel George V, just off the Champs-Élysées, but Oberon, in a standard movie-star power play, insisted that she and Ballard stay at the palatial Hôtel Ritz on the Place Vendôme. Ballard, an Oklahoman of part-Cherokee descent, was tall, athletic, cultured, and handsome; he had won Oberon's heart by inventing a small spotlight that attached to the camera and helped minimize her facial scars from a 1937 car accident.*
"By the time we settled in Paris, Merle had developed a deep passion for Robert Ryan," wrote Granet. "He was tough looking but at heart he was a happily married pussycat. He was not even fair game for someone of Merle's sexual talents. She would tease him then cool it." Born in Bombay to a Welsh father and an Indian mother, Oberon had spent her adult life concealing the mixed parentage that would have ended her career as an actress in Britain and the United States. For six years she had been married to the great British producer Alexander Korda, who cast her opposite Laurence Olivier in _Wuthering Heights_ (1939), but in 1945 she had left Korda for Ballard. She obsessed over her beauty and exulted in her status, spoiling herself with clothing and gems. Oberon was high-strung and wildly romantic — among her previous lovers were Leslie Howard, David Niven, George Brent, and the heroic RAF pilot Richard Hillary.
During the company's stay in Paris, wrote Granet, Oberon urged him and his wife, Charlotte, to throw a dinner party in their suite and invite Ryan. That evening she arrived hours late, dressed to the nines in a black evening gown and accompanied by a dapper Englishman; later she confessed to Charlotte that she was trying to make Ryan jealous. "By the time we were shooting in Frankfurt, she had successfully bedded Ryan," Granet reported. "Since Lucien... was constantly on location, all he could do was develop suspicions. Merle successfully made him believe that it was Charles Korvin who was making a pass at her."
Ryan with Merle Oberon in _Berlin Express_ (1948). Their affair unfolded amid the chaos and deprivation of postwar France and Germany. _Film Noir Foundation_
Korvin, a Hungarian actor playing one of the villains, had already shot two pictures with Oberon, and the two despised each other. More than thirty years later, after her death, Korvin told celebrity biographers Charles Higham and Roy Moseley that Oberon deserted Ballard on more than one occasion to spend the night with Ryan, first on the cross-country train from Paris and then in Frankfurt (where the crew lodged at hotels in the center of town and the cast was billeted at a castle in Bad Nauheim, thirty-five kilometers north of the city). "I know that she slept with Ryan both in Hollywood and in Europe and I thought it unfair and cruel of her," Korvin remembered. "I objected to the affair and so did everyone else on the picture."
Political argument only added to the tension. When Ryan asked his fellow cast members how they felt about General Marshall's vision for postwar Europe, the idea of economic aid for Germany got a cool reception. Coote and Oberon had endured the London blitz. Korvin and Paul Lukas, both Hungarian, had been personally touched by the Holocaust, and Toporow, who was Polish, loathed the Germans and the Russians alike. "How can you let 80 million people starve?" Ryan would ask. Invariably they dismissed him as naïve or softhearted; mass starvation, said one, would be no less than the German people deserved.
Their resolve began to melt away as they got a look at Frankfurt: entire neighborhoods reduced to rubble, middle-class people reduced to beggars. More than fifty-five hundred had been killed in the bombardment, and the medieval city center, the Römer, had been completely destroyed. In _Berlin Express_ , Ryan and Oberon venture into the neighborhood and discover a maze of shoulder-high rubble, like a bizarre sculpture garden. Another scene shows Ryan staring grimly out a bus window as people walk the streets with suitcases full of belongings for sale; in the train station he tosses away a cigarette and two shabby men race like pigeons to scoop it up. The children they encountered on location were "emaciated, shocked and sick," Ryan later wrote, with "old faces and rickety bodies." By the end of the first week, he remembered, no one talked anymore about the justice of letting people starve.
From Frankfurt the company flew to Berlin, where principal photography began on Saturday, August 2. This time the company stayed in Zehlendorf, about fifteen miles from downtown, near the US occupation forces headquarters, and cast members were chauffeured about in a car that had belonged to Hitler's foreign minister, Joachim von Ribbentrop. More than three years after the Allied bombing, Berlin was still a boneyard of gray, jagged, hollowed-out buildings, block after block, mile after mile. Unter den Linden, once the capital city's most majestic boulevard, was bare now, its namesake linden trees destroyed or chopped down for firewood. The great Reichstag was an empty shell, the lush Hotel Adlon bombed out and boarded up. Out for a stroll one night, Ryan fell into a bomb crater.
The poverty on the streets was overwhelming: Germans clustered around the locations, pleading for work as grips or extras. One well-known theater actor offered to work for a pair of pants, then came back the next day and said he wanted food for his family instead. Granet gave him both, and a check. "It is hard to visualize a world where the standard of currency is simply the cigarette," he wrote. Chocolate bars were equally prized, and Ryan used them to pay the woman who ironed his shirts. "There seemed to be very little bitterness on the part of the Germans who worked with us," Ryan wrote. "A grip, hoisting a heavy prop one day, laughed and said, 'There goes my 1,500 calories.'"
Apart from soldiers and government staffers, most of the other Americans in Berlin were journalists, who congregated at the press club and treated the movie people with smirking condescension. "As our visit wore on, the frost melted," Ryan would write. "Fortunately there were no jokers in our company. Nobody tried to dress up like Hitler and make a speech from the famous balcony. Nobody got drunk or was carted off to jail." The Russians were suspicious when cast and crew arrived to shoot in the Soviet sector, though Ryan saw no evidence of the military might he had expected, "no streets bristling with machine guns, no bayonets — as a matter of fact, almost no Russians." Their chilly reception contrasted sharply with the picture's final scene, in which the American and the Russian mend their ongoing political quarrel with a brotherly wave outside the Brandenberg Gate.
The last week in Berlin the company enjoyed a picnic on the Rhine River, courtesy of the US Army, and a party at the press club, attended by reporters and military people. There followed another nine days of photography in Frankfurt and four more in Paris. According to Granet, Ballard and Korvin came to blows during one train trip. When the company arrived in London, Oberon refused to fly back to New York and persuaded Granet to send her, Ballard, and Ryan on the _Queen Mary_ out of Southampton. News photographers snapped photos of Ryan, grinning angrily and fiddling with his hat, as he escorted Oberon through Waterloo Station.
Decades later, when the affair had become a distant memory, Ryan would share with Harold Kennedy, his theatrical colleague and drinking buddy, a curious anecdote about his Atlantic crossing with a beautiful costar and her cameraman husband. As Ryan framed it, the woman had been making passes at him throughout the journey and cornered him late one night as he was taking the air on deck; getting no response to her come-ons, she pounced, knocked him down, and refused to let him up. Suddenly her husband "materialized on the deck, lifted her up, reached down and took hold of Bob's shoulder, assisted him to his feet, and then, after apologizing to him profusely, blackened both of the lady's eyes."
Ryan had an Irishman's way with a story — he wasn't the sort of man to stand by while someone was hitting a woman — but then Granet also reported rumors of noisy fights between Ballard and Oberon on the voyage home, and said that she returned to Hollywood with broken teeth. Ryan was staying on in New York for a few days to do promotional work for _Crossfire_. When the _Queen Mary_ docked on Tuesday, September 9, the first thing he did was to call Jessica in Los Angeles. She put Tim on the phone to hear his father's voice, and she must have informed Ryan, if she hadn't already, that she was expecting another child.
WHILE RYAN WAS IN EUROPE, _Crossfire_ had exploded. The picture opened in late July at the Rivoli, a 2,092-seat movie palace on Broadway, and broke box office records in its first week. "One of the most startling pictures ever to come out of Hollywood," wrote the _New York Morning Telegraph_. "A film to be praised, praised again, and seen by all," wrote the _New York Post_. "An important, stirring film," declared the _Daily Mirror_. "Robert Ryan gives one of the performances of the year." The picture had transformed his reputation overnight: critics and colleagues who had regarded him as a confident but unspectacular leading man now recognized him as an exciting, first-class character actor. There was talk of an Oscar nomination. "I came back to the sort of reception reserved by the New York press for people who had done something," he later recalled. "Everybody wanted an interview; photographers were everywhere."
Ryan made a rare personal appearance at the Rivoli, where _Crossfire_ was in its eighth week. "It was the first time I had seen the picture with an audience, and I was elated at the reaction," he said. Afterward, when he was introduced to the crowd and walked onstage, the room suddenly quieted. Monty was the black, unfathomable heart of the picture; moviegoers who reacted to the anti-Semitism in _Crossfire_ inevitably zeroed in on Ryan as the embodiment of all that fear and hatred. "I'm really not that kind of a guy," he said, bringing down the house.
By fall _Crossfire_ had gone into general release and was performing well across the country, racking up impressive numbers not only in the more liberal metropolitan centers but in many small towns and in such conservative communities as Memphis, Omaha, and Oklahoma City. The RKO sales force played down the picture's anti-Semitic angle, stressing the mystery element and Ryan's costar Robert Mitchum, whose popularity was on the rise. There were pockets of resistance — "We never have had any racial troubles in this town and I don't want to put anything before the people that might put ideas into their heads," declared one theater owner — but RKO pursued a sharply effective strategy of establishing the picture in one cluster of towns and then expanding it to the next.
Given the taboo-smashing nature of the story, some conflict was inevitable. The US Navy found the idea of an American soldier murdering a Jewish civilian so inflammatory that it banned _Crossfire_ , refusing to screen it for troops at bases foreign or domestic. The army allowed it to be shown to soldiers at home but nixed any screenings overseas, arguing it might be seen by foreigners on the bases and reflect poorly on the United States. The Motion Picture Export Association, which cleared movies for the international market, turned down _Crossfire_ , citing the same concerns. And though most leaders of the American Jewish community hailed the picture — in Chicago, the Anti-Defamation League had launched a vigorous campaign encouraging local lodges to sponsor private screenings for civic leaders — some argued that _Crossfire_ might harden anti-Semitic feelings and even provoke bigots to violence.
This opinion emerged most strongly in the American Jewish Committee's monthly magazine _Commentary_. Editor Eliot Cohen noted the malign magnetism of Ryan's hypnotic performance: "You're drawn to him. He's big, he catches your eye. His personality overshadows the others. A plain, husky fellow, not much education, visibly troubled, up against a world too smart for him, fighting shrewdly, stupidly, blindly against the 'others' who hem him in — before his crime, after his crime. (For the millions near enough like him to identify with him, will Montgomery be the simple bully and villain the producer intended, assuming that was his intention? The chances are just as good that he will be taken as a kind of hero-victim — the movie equivalent of the Hemingway-Faulkner-Farrell male, hounded and struck down by a world he never made.)"
Ryan was less concerned with anti-Semites or the Jews they hated than with the much larger middle ground of Gentiles who were innocent but ignorant. "What I hope for is that the mass of Americans — those who have never come directly, first-hand, against intolerance — will think about those who daily are exposed to it, and will reflect on their actions to those groups in a new light," he wrote in the _Daily Worker_. "Most Americans aren't intolerant, but neither are they concerned with those who are. Pictures like this will help show how senseless, how ignorant, how detrimental to fundamental American principles... any kind of bigotry is. When people fully realize that, they will stop the careless thinking and the even more careless talk."
One thing Ryan had understood better than the friends who had discouraged him from playing Monty: a controversial role can help an actor's career. Ray Milland had won an Oscar playing a raging alcoholic in _The Lost Weekend_ (1945). Yet, as Ryan pointed out in yet another first-person piece about _Crossfire_ , Stephen Crane's _The Red Badge of Courage_ had never been filmed because no movie star wanted to play a coward.* "The controversial role, like no other, can meet the needs of the actor who feels the void of not achieving professional stature," he wrote, inadvertently revealing the career frustration that had driven his choice. "It gives one the feeling of accomplishment, of acting with a purpose." When he reflected on the risk taken by Dore Schary, Adrian Scott, Eddie Dmytryk, and John Paxton, who had dreamed up the picture, his own gamble paled in comparison.
Ryan looked forward to more such projects, but the political winds were shifting. That fall Schary was approached by two investigators from the House Un-American Activities Committee — "rather gray-looking gentlemen," he wrote. The committee was moving forward with its hearings into communist infiltration of the movie industry, and they wanted to know if he might have any relevant information about Ryan. Schary pointed out that Ryan was a former marine, a credential he felt spoke for itself. They asked him about Scott and Dmytryk, the producer and director of _Crossfire_. They requested screenings of _Crossfire_ and _The Farmer's Daughter_ , a Loretta Young comedy that RKO had released in March, and afterward they declared both pictures to be "pro-Communist."
Schary later wrote that he gave the investigators nothing and expressed his lack of regard for the committee. On September 22, Schary, Scott, and Dmytryk all received subpoenas to testify in Washington. Forty other Hollywood professionals were summoned as well, ranging from such right-wingers as Adolphe Menjou, Ayn Rand, Leo McCarey, and Walt Disney to such left-wingers as Charles Chaplin, Clifford Odets, Robert Rossen, and Bertolt Brecht. The Red-baiting _Hollywood Reporter_ labeled nineteen of the forty-three — including Scott and Dmytryk — as "unfriendly" witnesses on the basis of their previous public statements about the committee. The hearings would convene a month later.
Ryan always would attribute his narrow escape from the blacklist to his war record and his Irish-Catholic heritage (the committee's equation of communists and Jews was well known). He had just been investigated by the FBI and cleared for travel in the Soviet sector of Berlin. The fact was that Scott and Dmytryk _had_ been Communist Party members, whereas Ryan (for all his willingness to publish in the _Worker_ ) was a solid Democrat who could always be counted on to inject a note of ward-heeling realism into the unmoored radicalism of friends and colleagues. During this period, Jessica would write, he had "his first brush with the doubletalk, the rigid doctrinaire attitudes, the attitude of take over or destroy, of some people involved who were or had been truly Communist-minded. At the same time he would not nudge one inch from the position of defending their right to believe as they did." Ryan quickly threw in with the Committee for the First Amendment (CFA), an organization formed by his screenwriter pal Philip Dunne, as well as John Huston and director William Wyler, to protest the hearings.
Wyler hosted an overflow meeting of the new group at his Beverly Hills home in early October. Outside, FBI agents took down license plate numbers, yet the CFA was a safely liberal group: it defended civil liberties in general, not the "Hollywood Nineteen" in particular, and the founders actively discouraged communists and fellow travelers from joining. The group resolved to protest the congressional probe in full-page newspaper ads and organized a large delegation of celebrities to fly east for the hearings. Ryan was stuck in town shooting interiors for _Berlin Express_ , but he agreed to take part in _Hollywood Fights Back_ , a pair of radio programs to be broadcast nationwide on October 26 and November 2.
Even before that, on Wednesday, October 15, Ryan appeared at the giant "Keep America Free!" rally at the Shrine Auditorium, which benefited a defense fund for the Nineteen. Presented by the Progressive Citizens of America (PCA) — a more radical group that was the Communist Party's last real lobbying presence in Hollywood — the rally drew some seven thousand people. "We protest the threat to personal liberty and the dignity of American citizenship represented by this police committee of Dies, Wood, Rankin, and Thomas," Ryan declared, naming the congressmen on the committee as he read a proclamation from the PCA. "We demand, in the name of all Americans, that the House Committee on Un-American Activities be abolished, while there still remains the freedom to abolish it."
The following Monday the hearings commenced in the Caucus Room of the Capitol Building, with every seat filled and the proceedings recorded by newsreel cameras, nationwide radio, and a battery of reporters and press photographers. J. Parnell Thomas, the New Jersey Republican who had assumed chairmanship of the committee with the Eightieth Congress, presided over the hearings, which got off to a bang when studio head Jack Warner volunteered the names of twelve people who had been identified as communists and fired from Warner Bros. His action stunned the Hollywood community, especially his colleagues at the Motion Pictures Producers' Association (MPPA), which had agreed to close ranks against the committee. As the week progressed, the committee called a succession of friendly witnesses, who named some three dozen people as communists.
The week's events failed to dent the enthusiasm of the Committee for the First Amendment, whose members took heart from editorials condemning the hearings in the _New York Times_ , the _Washington Post_ , and other dailies. On Sunday morning the CFA's star contingent — including Humphrey Bogart, Lauren Bacall, Myrna Loy, Gene Kelly, Judy Garland, and Danny Kaye — took off for New York and then Washington, having already recorded their contributions for the _Hollywood Fights Back_ broadcast. Ryan delivered his thirty-second bit live in the studio: "President Roosevelt called the Un-American Committee a sordid procedure, and that describes it pretty accurately," he declared. "Decent people dragged through the mud of insinuation and slander. The testimony of crackpots and subversives accepted and given out to the press as if it were the gospel truth. Reputations ruined and people hounded out of their jobs."
The tide of public opinion began to turn against the Nineteen on Monday morning, when writer John Howard Lawson accused the committee of Nazi tactics, was charged with contempt of Congress, and had to be forcibly removed from the chamber. As all this was going on, Dmytryk turned to Schary and asked, "What are my chances at the studio now?"
"You have an ironclad contract," Schary replied.
Adrian Scott brought a four-page statement defending _Crossfire_ and noting the anti-Semitism of Mississippi Democrat John E. Rankin, a committee member, which Thomas refused to let him read. Both Scott and Dmytryk were asked repeatedly if they were communists; they declined to answer, citing their Fifth Amendment rights, and were charged with contempt. Schary, asked if he would knowingly employ communists at RKO, replied that he would, "up until the time it is proved that a communist is a man dedicated to the overthrow of the government by force or violence, or by any illegal methods."
Seven more unfriendly witnesses defied the committee and were cited for contempt, among them screenwriter Dalton Trumbo — whose wartime romance _Tender Comrade_ (1944), directed by Dmytryk, had given Ryan his first big break. The committee had absurdly labeled the movie communist propaganda for its story of four women sharing a house while their men fight in World War II. When Thomas suddenly suspended the hearings on October 30, with Brecht having broken rank and eight witnesses still to be heard, _Variety_ reported that one factor was the reluctance of several committee members to release a long-promised list of subversive pictures. Once these innocuous and well-known titles were made public, the members argued, the committee would become "a laughing stock."
If Ryan was afraid of the committee, he didn't show it: while the hearings were in progress, he and his _Crossfire_ costar Gloria Grahame spoke at the annual convention of the American Jewish Labor Council, which would turn up on the US attorney general's list of communist (but not subversive) organizations. The studio moguls, however, were badly spooked by the hearings. On November 24 — the same day the House of Representatives voted 346 to 17 to uphold the contempt citations — the Motion Picture Producers' Association met at the Waldorf-Astoria Hotel in New York to hammer out a strategy. President Eric Johnston insisted that the studios purge their ranks; Schary led the charge against him, backed by independent producers Samuel Goldwyn and Walter Wanger, but Johnston carried the day. The MPPA announced that members would no longer employ known communists and would fire the unfriendly witnesses (now labeled the Hollywood Ten), whose actions "have been a disservice to their employers and have impaired their usefulness to the industry."
RKO was the first studio to act, firing Scott and Dmytryk. Schary refused to drop the ax, so Floyd Odlum, chairman of the board, handed the job to Schary's boss, RKO president Peter Rathvon. Every studio contract included a vaguely worded morals clause allowing the studio to terminate any employee deemed to have disgraced the company. Barred from the lot, their current projects canceled or reassigned to other producers, Scott and Dmytryk turned their attention to the pressing matter of defending themselves against the contempt citations, which could land them in federal prison.
Support for the unfriendly witnesses wilted. Humphrey Bogart, whose iconic tough-guy persona had been a potent weapon for the CFA, issued a statement describing the PR tour to New York and Washington as "ill-advised and even foolish." He had never been a communist or communist sympathizer, he declared, and he detested communism. The statement caused a collective shudder in Hollywood — if a star of Bogart's magnitude felt the need to distance himself from the Ten in such strident terms, could anyone be safe? Donations to the Committee for the First Amendment dried up immediately, and members reported pressure to resign. Within three months the organization would fold.
Amid all this, _Berlin Express_ was still shooting on the RKO lot. The picture's final scene, with the American and the Russian expressing their fellowship outside the Brandenberg Gate, must have seemed like fantasy now. Closer to the mark was the little speech delivered to the kidnapped peacemaker by the malevolent leader of the right-wing underground: "I too believe in unity. But unlike you I know that people will only unite when they are faced with a crisis, like war. Well, we are still at war; you are not. So we are united; you are not. So we will succeed; you will not."
RYAN LIKED TO TELL INTERVIEWERS he wasn't "a chaser" (which was true — the way women responded to him, he never had to chase anyone). For a man so proud of his family, the affair with Merle Oberon was a strange anomaly, an ongoing adulterous relationship that became an open secret among the cast and crew. Charles Korvin contended that the affair continued on the RKO lot, though production records suggest some turbulence as the picture was drawing to a close. On Wednesday, November 5, Oberon went home sick at noon, forcing Tourneur to scrap the rest of the day's scenes. The following Monday she didn't show up for work, and that Friday she left in the middle of the afternoon. According to biographers Higgins and Moseley, she and Ryan never saw each other again after _Berlin Express_ , though Ryan and Lucien Ballard would make four more pictures together.*
Somehow RKO managed to keep the whole mess out of the scandal sheets; however, the much-feared gossip columnist Louella Parsons twitted Ryan about it in a February 1948 profile. ("There had been a lot of talk about feuding in the 'Berlin Express' troupe, and I asked Bob if that were true," wrote Parsons. "I had heard that he and Merle Oberon had been particularly bitter in their quarrel.") From that point on, Ryan's movie-magazine pictorials stressed fatherhood, with Tim becoming a frequent participant. How Bob and Jessica dealt with the affair would remain private, but soon after he returned from Europe, they decided to buy a house in the San Fernando Valley, far from the Hollywood social scene.
A certain amount of hobnobbing was required to keep one's career going, but Jessica didn't like actors or the parties they threw. "As a wife, you met the same people over and over again," she wrote in a later memoir, "because they didn't recognize you unless you were standing right beside your husband, and even then they weren't always sure you were the wife. It was spooky." By now she had published her second mystery for Doubleday and was working on a third, but no one was interested in that. She would start conversations with people and then see their eyes darting about in search of someone more important. "If you were a wife you got very tactful about releasing any poor sap quickly to go do business... and then ended up sitting tensely with other tense wives trying their best to look as if they were having a good time."
She reached her limit one night when she and Robert attended a swank party and she was immediately shunted off to the side with her friends Amanda Dunne and Joan Houseman. Robert, Philip Dunne, and John Houseman were off somewhere having lively conversations. "That night Joan Houseman's solution to the condition of non-being was to retreat to a corner of the vast living room of the vast house and get quietly smashed," Jessica wrote, "while she stared at the crowd with an expression of splendid French contempt." Amanda and Jessica began tossing back drinks as well, until Amanda stood up suddenly, looking as if she might be ill, and went off in search of a bathroom.
Left alone, Jessica strolled into the host's library to find some reading material, and before long Amanda burst into the room, looking rather crazed. "There's a room full of dead animals out there!" she exclaimed. Jessica followed her back into a coatroom where all the women's furs were hung. This was too much for Jessica, and she told Robert she was going out for some air. "Once outside in the car, I went quietly into hysterics," she wrote. "The condition of non-being produces intense anxiety."
On Kling Street, just east of Cahuenga Boulevard in North Hollywood, the Ryans found an A-frame ranch house with a paved terrace and a bare, spacious yard. "It was the biggest house we could get with the most ground for the least money at a time when we still did not trust — _I_ didn't trust — that the money would keep coming in," Jessica wrote. "Robert never doubted it, but he had never been as poor as I had been." The couple landscaped the place themselves (planting ivy that eventually ran riot over the house) and began adding wings. The shed in the backyard was converted into Ryan's private office and workout room. This was the first time Ryan had actually owned a home — his parents had rented all their lives — and the suburban locale suited his reclusive nature.
The place was modest but comfortable, with plenty of room for the kids to run around; he and Jessica installed a sandbox, a swing set, and a wading pool. "Facing the garden is a wide, airy living room with almost one whole wall of glass, opening onto the terrace," noted a visiting journalist. "A beautiful antique chest dominates one end. The chairs and divans are tailored and comfortable; the tables low and wide... The muted greens and grays and blues of walls, carpets, and upholstery are brightened by huge bouquets of fresh garden flowers."
Ryan made sure the reporter understood that social gatherings at their home were limited to their close circle of friends, not the movers and shakers of the picture business. He and Jessica were perfectly happy with each other's company. Philip Dunne would marvel at Ryan's "tremendous devotion to his family. He was the most family-oriented man I ever knew."
Ryan tending to chores at the new house on Kling Street in North Hollywood. His years there with Jessica and their young children were among his happiest. _Robert Ryan Family_
In December 1947, Ryan made a quick trip to Chicago to address the national Conference of Christians and Jews, pinch-hitting for Dore Schary. "He began to be asked to speak before Jewish groups to discuss anti-Semitism," Jessica recalled. "In the beginning, the doing of it appeared to be for publicity for the movie... but when that phase was over, they wouldn't let him go. For a long time there he was playing what he called the Synagogue Circuit."
From there Ryan flew to New York to see some plays. Since _Crossfire_ had hit, he had been fielding offers from Broadway, but his calendar for 1948 already was filling up with pictures. RKO announced that he would costar with Cary Grant, Frank Sinatra, and Robert Mitchum in _Honored Glory_ , an episodic drama about nine unidentified men, killed in action during World War II, whose stories make them candidates for the Tomb of the Unknown Soldier (the film would never be made). MGM wanted to borrow Ryan for the revenge drama _Act of Violence_. And Schary, who had been trumpeting _Crossfire_ as proof that A pictures could be made on B budgets, was ready to move forward with his next such experiment: _The Set-Up_ , a boxing drama about a washed-up fighter staring down his bleak future. The source material was Joseph Moncure March's narrative poem of the same title, a favorite of Ryan's at Dartmouth.
His first assignment that year was RKO's antiwar parable _The Boy with Green Hair_ , adapted from a short story by Betsy Beaton. Filmed in Technicolor, it starred eleven-year-old Dean Stockwell as a schoolboy who has been passed from relative to relative while his parents are overseas.* Eventually he lands in a bucolic small town with a kindly old Irish-American gent (Pat O'Brien) and begins to make a life for himself, but then he learns the truth about his parents — they were killed in London during the blitz — and the trauma turns his hair green overnight.
The project had originated with Adrian Scott, himself the adoptive father of a traumatized British war orphan; but after Scott was fired by RKO, Schary handed _The Boy with Green Hair_ over to producer Stephen Ames and firsttime movie director Joseph Losey. A senior at Dartmouth when Ryan was a freshman, Losey had studied with Bertolt Brecht in Germany and in 1935 had traveled to the Soviet Union, where he staged Clifford Odets's _Waiting for Lefty_ in Moscow. His latest theatrical project had been an acclaimed Broadway production of Brecht's _Galileo_ , performed in English for the first time and starring Charles Laughton.
Fresh from the rubble and hungry children of Frankfurt and Berlin, Ryan couldn't have been more sympathetic to _The Boy with Green Hair_. His second-billed part consisted of only one extended scene with Stockwell, which took two days to shoot; even so, it would remain one of the picture's best-liked sequences. At a police station one night, cops fire questions at Peter, the brooding and now bald-headed boy. Ryan plays Dr. Evans, a laid-back child psychologist who arrives with a brown-bag dinner and asks the cops to leave them alone. Children who grew up around the actor would remember his uncondescending manner toward them, and he incorporates it here to fine effect. Evans wordlessly changes the lighting in the room, taking an overhead spot off them, and asks Peter to move to a chair so he can have the bench for his dinner. "Chocolate malted milk," he notes, frowning into the cup. "I'm sure I asked for strawberry." They both know it's a game, but Peter is starving; he takes the malted and digs into a hamburger, and his responses to the doctor's questions trigger a series of flashbacks.
_The Boy with Green Hair_ can be cloying and moralistic, but there are genuine moments of fear and anger as well. Peter, having learned of his parents' death, is stocking shelves at a local grocery and overhears three women debating the Cold War. Losey follows his face, shooting him through cabinets and shelves as the women's voices hover off-screen. "People say another war means the end of the world," says one. "War will come, want it or not," her friend replies. "The only question is when." A third adds: "Just in time to get more youngsters like Peter." This so frightens the boy that he drops a bottle of milk, which smashes on the floor. A low-angle shot shows the three ladies gathered above, grinning in amusement.
The central scene is a powerfully weird and stylized dream sequence in which Peter awakes in a forest clearing and encounters the very war orphans he and his classmates have been studying on posters in school. One girl has lost a leg; another holds an Asian infant. The oldest orphan explains to Peter that his green hair marks him as a messenger: "You must tell all the people — the Russians, Americans, Chinese, British, French, all the people all over the world — that there must not ever be another war."
_The Boy with Green Hair_ crystallized a public sentiment for world government that had been growing in the United States since the end of the war. Norman Cousins, editor of the _Saturday Review of Literature_ and a founder of the United World Federalists, had framed the issue before any other journalist. His celebrated editorial "Modern Man Is Obsolete," written the night after Hiroshima was destroyed, argued that the event marked "the violent death of one stage in mankind's history and the beginning of another." Now that man had the power to incinerate whole cities, he would have to evolve past the need for war, which would mean eradicating global inequality and establishing world government. To this end, Cousins wrote, modern man "will have to recognize the flat truth that the greatest obsolescence of all in the Atomic Age is national sovereignty." By 1946 a Gallup poll found that 52 percent of Americans favored the liquidation of the US military in favor of an international peacekeeping force. Ryan was one of them, and he would get to know Cousins later that year when he joined the Federalists, a rapidly growing organization that advocated "world peace through world law."
The week before Ryan shot his scene for _The Boy with Green Hair_ , the Academy of Motion Picture Arts and Sciences announced the Oscar nominations for 1947. _Crossfire_ was honored in five categories: best picture, best director, best screenplay, best supporting actress (Grahame), and best supporting actor (Ryan). But as more than one industry observer noted, this good fortune put RKO in a ticklish position, given that it had fired the picture's producer and director. There was another twist as well: in every category except Ryan's, _Crossfire_ was competing with _Gentleman's Agreement_ , the Fox production Schary had beaten to the box office by four-and-a-half months. Released in December and carefully marketed with _Crossfire_ as its model, _Gentleman's Agreement_ was still doing big business across the country and had topped the nominations race with a total of eight.
Jessica and Robert attend the 1948 Academy Awards ceremony. "We don't ask actors home," she would later write. "We haven't, Robert or I, much to say to them privately." _Franklin Jarlett Collection_
Though RKO had beaten Fox to the punch, _Gentleman's Agreement_ had effectively stolen _Crossfire's_ thunder as an exposé of anti-Semitism, to Ryan's great irritation. Adapted from a novel by Laura Z. Hobson, it starred Gregory Peck as a journalist who poses as a Jew in order to write a magazine story. In some respects _Gentleman's Agreement_ was bolder than _Crossfire_ ; it confronted prejudice head-on instead of sneaking it into a murder mystery, and in contrast to the other film's psychopathology, it revealed more casual and insidious forms of bigotry. It was also the kind of picture Academy voters could feel good about honoring: this was no crummy little crime story shot on borrowed sets, but a big, long prestige drama set in the penthouses and boardrooms of Manhattan, produced by the great Darryl F. Zanuck.
The other nominees for best supporting actor were Charles Bickford as the starchy butler in RKO's _The Farmer's Daughter_ , Thomas Gomez as the warmhearted carny in Universal's _Ride the Pink Horse_ , Richard Widmark as the giggling killer in Fox's _Kiss of Death_ , and Edmund Gwenn as Kris Kringle in Fox's _Miracle on 34th Street_. Ryan relished the attention, though his chances of winning seemed fairly slim: he would be dividing the psycho vote with Widmark, and really, who was going to choose a Jew-hating murderer over Santa Claus?
Cheyney Ryan arrived on March 10, at Cedars of Lebanon Hospital in Los Angeles, and ten days later Jessica had recovered sufficiently to accompany her husband to the Shrine Auditorium. As most had predicted, Gwenn won best supporting actor. _Crossfire_ was shut out by _Gentleman's Agreement_ , which took best picture, best director (Elia Kazan), and best supporting actress (Celeste Holm). According to Dmytryk, the right-wing Motion Picture Alliance for the Preservation of American Ideals had conducted a vigorous campaign against _Crossfire_. The picture had made the year-end lists of all the major critics and collected honors ranging from an Edgar Allan Poe Award (for best mystery film) to a Cannes Film Festival award (for best social film). But in Hollywood, _Crossfire_ was still a double-edged sword. A few months earlier, when MPPA president Eric Johnston had praised the picture in a speech, the legal counsel for the Hollywood Ten had puckishly invited him to serve as a character witness for Scott and Dmytryk. Johnston declined.
*In response to my Freedom of Information Act request, the FBI reported that its central records system contained no file for Ryan.
*Cinematographers still refer to this device as "the obie."
*When John Huston finally brought the novel to the screen in 1951, his star was the legendary war hero Audie Murphy.
* _Inferno_ (1953), _The Proud Ones_ (1956), _Hour of the Gun_ (1967), and _The Wild Bunch_ (1969).
*Stockwell had just won a special Golden Globe Award for his performance as Gregory Peck's son in _Gentleman's Agreement_.
_six_
Caught
The Shrine Auditorium may have been a temple of self-congratulation on Oscar night, but outside its walls the movie business was in serious trouble. Ticket sales had boomed after the war when soldiers were streaming home, but in 1947 domestic box office revenue plummeted as people like the Ryans started families and moved into the suburbs. Britain and other countries, hoping to revive their own war-ravaged film industries, levied tariffs on US imports, diminishing the once-lucrative European market. And the federal government renewed its antitrust campaign against the major film studios, pressuring them to sell their theater chains. If that happened, the entire business model for the studio system would collapse.
Dore Schary had come to RKO promising to cut costs, and the board of directors reaffirmed its confidence in him after the HUAC hearings. But in his first year as production chief, the studio's annual profit had plunged from $12 million to $5 million. In February 1948 the trade papers reported that Floyd Odlum, RKO's chairman and majority stockholder since 1936, would sell his controlling interest in the studio to Howard Hughes, the aviation giant and mercurial moviemaker who had produced such landmark pictures as _Hell's Angels_ (1930) and _Scarface_ (1932).
In early May, Ryan made a quick trip to New York for the _Berlin Express_ premiere, and by the time he returned to the West Coast, Hughes had struck a deal with Odlum, purchasing 24 percent of RKO for the grand sum of $8.8 million — then the largest cash transaction in the history of the movie business. The announcement sent shock waves through the studio: Hughes had a reputation as a controlling and capricious moviemaker. For years his pet project had been _The Outlaw_ , a sexually suggestive western starring Jane Russell and her thirty-eight-inch bust; shot over a maddening nine months in 1940 and 1941, the picture had been held up first by the Production Code Administration and then by various state censorship boards, finally winning wide release in 1946. Hughes's latest infatuation was starlet Faith Domergue, and many thought he was buying RKO in order to distribute _Vendetta_ , the film he had been constructing around her to the tune of $2 million.
Hughes moved quickly to tamp down the paranoia raging around the Gower Street lot: Peter Rathvon, president of Radio-Keith-Orpheum, issued a memo to all employees assuring them that the new owner had "no hungry relatives looking for your jobs, nor substitutes waiting to step into the RKO management." Schary, whose contract permitted him to leave the studio in the event of an ownership change, was particularly apprehensive about the prospect of Hughes, an anticommunist and anti-Semite, meddling in his production slate. But when Rathvon arranged a meeting between the two men at his home, Hughes promised Schary that nothing would change under his leadership. In early June, Schary issued a reassuring memo to studio personnel: "I have had a number of talks with Mr. Howard Hughes, and we are in complete agreement on present policy and on the projected program for RKO."
While all this transpired, Ryan was on loan to MGM, making _Act of Violence_ with director Fred Zinnemann. Like _Crossfire_ it was a hard-bitten suspense film that touched on the harsh realities of the war, with Ryan as a fearsome heavy, though in this case the theme was not bigotry but betrayal. Ryan and bug-eyed Van Heflin play survivors of a Nazi prison camp, once comrades but now deadly enemies. Just as _Crossfire_ began with a shadowy silhouette of Ryan beating someone to death, the new movie opened with a predawn sequence of a mysterious figure in trench coat and fedora limping across a New York street, climbing the stairs of a seedy tenement building, and extracting a pistol from a bureau drawer. Ryan's face is revealed finally in close-up as he holds the gun before him, and thick white letters scream out the title: ACT OF VIOLENCE.
Pictures about returning soldiers had become commonplace (a year earlier, _The Best Years of Our Lives_ had won seven Oscars), but seldom were they so bitter. Ryan's character, Joe Parkson, catches a Greyhound bus to the West Coast and arrives in the small town of Santa Lisa, where he's immediately halted by a cop as he tries to cross the street: it's Memorial Day, and a parade is passing. "I liked the idea of this man, who was the veteran of an unhuman [ _sic_ ] experience in the war, having to step back because a few old guys were walking past him, carrying the American flag, as though they owned it," Zinnemann recalled. Parkson has come to town to murder Frank Enley (Heflin), a local contractor, family man, and pillar of the community. Enley is a war hero in Santa Lisa, but Parkson knows the truth about him: when starvation drove Parkson and other POWS to attempt a tunnel escape, Enley ratted them out to the commandant.
As Enley, Heflin had nabbed the better role, yet Parkson was iconic — an avenging angel, slouching toward Southern California to be born. Early in the picture he shows up at the contractor's home to find only his flinty wife, Edith (twenty-one-year-old Janet Leigh). Sweating and absurdly out of place in his noirish getup, Parkson forces his way in. "He's got it nice here, hasn't he?" Parkson observes as he peers around their pleasant home. "Real nice." Edith tells him to get lost, and in a rage Parkson spills the whole story, his eyes drilling through her. He remembers how he and ten other prisoners were ambushed by guards, tortured with bayonets, and left on the ground to die. Only he survived, listening to the others all night: "One of them lasted till morning. By then you couldn't tell his voice belonged to a man. He sounded like a dog that got hit by a truck and left in the street." This last detail reduces Edith to tears, and Parkson leaves her to weep in her sparkling kitchen.
Written by Collier Young, the story had bounced around Hollywood (it was briefly a vehicle for Gregory Peck and Humphrey Bogart at Warner Bros.) before landing at MGM with Zinnemann directing and Robert Surtees as cinematographer. The two men had worked together in the early '30s as camera operators at Berlin's EFA Studio, where Zinnemann was mentored and powerfully influenced by the pioneering documentary maker Robert Flaherty ( _Nanook of the North_ ). Since then, Zinnemann, a Viennese Jew, had fled the Nazis and found an unlikely home for his grim style at the MGM fantasy factory: _The Seventh Cross_ (1943) starred Spencer Tracy, Hume Cronyn, and Jessica Tandy as Germans who have escaped from a concentration camp; and _The Search_ (1945), with Montgomery Clift, was about a boy who has survived Auschwitz and goes looking for his mother across the German countryside. Only after the war would Zinnemann discover that his own parents had died in a concentration camp.
From Flaherty, Zinnemann had learned the value of authentic locations, and this new picture would give him and Surtees a chance to photograph the real LA, where Enley flees and descends into the criminal underworld. Zinnemann would recall "the many sleepless nights we spent shooting exteriors in the eerie slums of downtown Los Angeles," most notably the Bunker Hill district, with its hilly terrain, its slanting, funicular railway cars, and its long flights of cement steps hugging the run-down buildings. Other scenes were shot at the Hill Street railroad tunnel, the Santa Fe freight yards, and as far afield as Big Bear Lake in San Bernardino County. This sense of realism extended to the actors as well. "No make-up of any kind was used on any member of the cast," wrote Surtees. "We tried to maintain on the screen a high standard of skin texture." The technique heightened the hard set of Ryan's face, with its lined brow and sneering mouth.
Ryan and Janet Leigh in _Act of Violence_ (1948). The flinty young actress gave as good as she got when she and Ryan faced off" in this film and _The Naked Spur_ (1953). _Franklin Jarlett Collection_
As a Catholic, Ryan had no trouble grasping this study in guilt and damnation. Enley's confession to Edith, detailing the hunger and desperation inside the camp, was the sort of thing most Americans wanted to forget about. "The Nazis even paid me a price," Enley exclaims. "They gave me food, and I ate it.... There were six widows, there were ten men dead, and I couldn't even stop eating." Parkson turns out to be similarly cursed by his thirst for vengeance. In his cheap hotel room he is visited by Ann (Phyllis Thaxter), his girl in New York, who has followed him to LA in the hope of derailing his homicidal plan. She appeals to his conscience, begging him to let go of his hatred. Ryan plays the scene in near silence, considering her words, and looks up in shock when she refers to him being "as crippled in your mind as you are in your — " Then the telephone rings, bringing news of a rendezvous with Enley, and Parkson's face hints at a smile as he's pulled back into his wicked dream.
BY THE TIME _Act of Violence_ wrapped in mid-July, RKO was in chaos. Dore Schary had resigned on July 7, less than a month after his calming memo to employees, when Hughes ordered him to suspend production of one picture and fire contract player Barbara Bel Geddes from another (for which she and Ryan had already done some test scenes). Following a weekend meeting with the board, Hughes announced that all production would shut down from August to October as he reorganized the studio. Three hundred RKO employees were immediately terminated, and _Variety_ reported that another seven hundred would be fired in the near future; by the time Hughes was finished, the studio workforce of twenty-five hundred would be reduced to six hundred. Numerous contract players were let go; by the following winter, only six remained, among them Ryan, Gloria Grahame, western star Tim Holt, and Robert Mitchum (whose contract RKO shared with producer David O. Selznick).
Hughes could be a ruthless enemy but also a staunch ally, and he was vocal in his admiration for Ryan and Mitchum; he backed Mitchum after the actor was busted for marijuana possession in September 1948. With his connections to the Defense Department, Hughes was a good man to have in your corner during the Red Scare. But he was a strange individual: one time Ryan, summoned to the millionaire's house, found him shuffling around with empty tissue boxes on his feet; when Ryan asked about them, Hughes told him they were disposable and thus more sanitary than socks. On another occasion Hughes invited the Ryans over for a dinner party; they arrived to a house full of guests but no Hughes. After an hour and a half of cocktails, the guests were called in to dinner. Finally, the host arrived and greeted his guests. "I'm so glad that you're here tonight, Mrs. Ryan," Jessica remembered him saying. Then he left the room, and no one saw him again.
Ryan returned to RKO expecting to begin principal photography for his coveted boxing drama _The Set-Up_ , but like everything else it had been shelved when Hughes shut down the studio. Instead Ryan found himself swept into a bizarre professional intrigue that not only touched on intimate details of Hughes's private life but also provided Ryan with one of the more neurotic roles of his career: Smith Ohlrig, the controlling multimillionaire in Max Ophuls's caustic melodrama _Caught_.
Ophuls, another German Jewish exile, was one of many directors Hughes had walked on and discarded over the years. Despite an impressive track record in Germany and later France, Ophuls's first assignment in Hollywood didn't materialize until 1946, when his fellow émigré Preston Sturges hired him at California Pictures Corporation, the small indie Sturges had formed with Hughes, to develop and direct _Vendetta_ , Hughes's project for his nineteen-year-old lover, Faith Domergue. When production began, Ophuls — a master stylist who favored long, sinuous tracking shots and dramatic use of perspective — immediately fell behind schedule and soon was relieved by Sturges, who had wanted to direct the picture himself anyway. Hughes took to ridiculing Ophuls,* referring to him as "the Oaf."
Since then Ophuls had gained a foothold in Hollywood, directing the hit swashbuckler _The Exile_ and the sublime romance _Letter from an Unknown Woman_ for Universal. Charlie Einfeld, founder of the small Enterprise Studio, engaged Ophuls to develop a screen adaptation of _Wild Calendar_ , a popular novel by Libbie Block in which a middle-class woman marries her wealthy playboy of a cousin and, following his heart attack, takes up with another man. Einfeld had bought the rights at the suggestion of Ginger Rogers, who wanted to star, and commissioned a script by Abraham Polonsky; when Rogers dropped out, Einfeld gave the property to Ophuls, who threw away Polonsky's script and started over with Arthur Laurents. "I'm not going to make a picture from that lousy book," Ophuls told Laurents over lunch. "I'm going to make a picture about Howard Hughes."
Enterprise approached RKO for a loan of Ryan and Barbara Bel Geddes, but Peter Rathvon, who had taken over as production chief temporarily, didn't like the script and nixed the deal. Ryan must have been eager to make the picture; its producer, Wolfgang Reinhardt, was the son of his beloved acting teacher Max Reinhardt (who had died in 1943). After Einfeld invited Ryan over to the studio to discuss the situation, they placed a conference call to pitch the project to Hughes, who was notoriously hard to see in person. Hughes read the script and insisted that certain of his own idiosyncrasies (the sneakers, the rumpled clothing, the refusal to drink anything but milk) be deleted, but in the end he approved the loan. "Max could have Ryan and Bel Geddes," Laurents recalled, "but the dailies had to be delivered to Mr. Hughes in person at his house at midnight by the editor." When Rathvon learned that Hughes had overruled him, he resigned.
By the time Ryan reported for work on July 30, _Caught_ was already twelve days behind schedule; John Berry had taken over as director after Ophuls came down with shingles, but then Ophuls recovered, took over the picture again, and junked all of Berry's footage. Enterprise was sliding toward bankruptcy after the recent flop of its Ingrid Bergman vehicle _Arch of Triumph_ , and Ophuls was under serious pressure to get his picture back on track; the twelve days abandoned would have to be made up in four. Luckily for Ophuls, he had a trio of highly professional stars in Ryan, Bel Geddes, and James Mason, making his US screen debut as a compassionate ghetto doctor who completes the story's love triangle. Mason was a superb actor, though according to Laurents he wore a false chest to make himself look more manly. Ryan, who had no need for a false chest, once confided to his friend Robert Wallsten that Bel Geddes came on to him before the shoot, explaining that she liked to sleep with her leading men. "I'm sorry to tell you," Ryan replied, "but I have a wife and children."
Ophuls was himself a disciple of Max Reinhardt, and Ryan's performance delighted him. "Max was very fond of Robert Ryan," recalled assistant director Albert van Schmus. "He thought that he was a natural human being in front of the camera. And that's what Max wanted, in that particular part at least." Smith Ohlrig, the millionaire, was a complicated role; for the story to work, viewers would have to believe that Leonora had married him for love as well as money. When Leonora and Ohlrig first meet, he takes her on an aggressively fast midnight drive and interrogates her mercilessly, mocking her charm school training; Ryan captures the strange mix of charm, confidence, and vulnerability that made Hughes irresistible to so many women.
A later scene in a psychiatrist's office tells a different story: Ohlrig is a serious head case, warped by his wealth and icy in his calculation of other people's motives. He's come to the shrink hoping to alleviate his recent heart trouble, and when the doctor suggests his attacks are an attention-getting device, Ohlrig leaps off the couch in a rage. "What are some of your other little gems?" he exclaims. "I must destroy everyone I can't own? I'm afraid all anyone wants is my money? I'll never marry because I'd only be married for my money?" To prove the doctor wrong, Ohlrig picks up the phone and directs an assistant to offer Leonora a wedding proposal.
Leonora soon finds herself confined to Ohlrig's mansion on Long Island as he goes about his business and mercilessly antagonized when he deigns to come home — in one case, late at night, barking orders, with an entourage of business associates. His ensuing confrontation with Leonora plays out in the game room, where Ryan, in a wonderful bit of business, banks a billiard ball around the edges of a pool table as Ohlrig calculates Leonora's motives for marrying him and her options for breaking free. "Every one of my corporations, every single one, has a different staff, a different lawyer, a different accountant," he explains. "Not one of them knows anything about each other. I run it all. Each one has his place and he stays there.... And that's what you've got to learn, Leonora. You're better paid than any of them."
BACK AT RKO, Ohlrig's real-life counterpart was remaking the studio in his own image. Hughes had installed two of his men, C. J. Tevlin and Bicknell Lockhart, on an executive committee overseeing the studio, and through them he declared that the era of Dore Schary's "message pictures" was over. Studio president Ned Depinet moved to New York to replace Peter Rathvon as corporate president, and longtime veteran Sid Rogell became head of production at RKO Radio Pictures. Yet the new owner's management style caused no end of frustration. With his phobia of germs, Hughes kept counsel from his house or his office at the nearby Samuel Goldwyn Studio; RKO executives couldn't reach him, which caused endless delays on critical decisions, but when he decided he was ready to talk, he would phone them at home in the middle of the night.
Cameras finally rolled again that fall as Hughes launched a slate of six modestly budgeted A pictures — the second of which, _The Set-Up_ , began shooting in mid-October. Ryan had been coveting this assignment ever since he learned that RKO owned the rights to Joseph Moncure March's pungent narrative poem. When it made the _New York Times_ best-seller list in 1928, there hadn't been an African-American heavyweight champion since Jack Johnson thirteen years earlier, and as March later observed, "The fight racket was still tainted by a strong residue of race prejudice." Pansy Jones, the tragic hero of _The Set-Up_ , is a black middleweight nearing the end of an undistinguished career,
A dark-skinned jinx
With eyes like a lynx,
A heart like a lion,
And a face like the Sphinx:
Battered, flat, massive:
Grim,
Always impassive.
Pansy has lost so many fights lately that his manager doesn't even bother to tell him when he cuts a deal with Tony Diamond, a local racketeer, for Pansy to throw his next bout. To everyone's astonishment, however, Pansy comes on strong and knocks out his opponent. Walking home from the fight, Pansy is stalked by Diamond and one of his boys; they chase him down to the subway, where Pansy falls onto the tracks and dies under the wheels of a train.
Soon after publishing _The Set-Up_ , March had come to Hollywood as a screenwriter and contributed story and dialogue to Hughes's early triumph _Hell's Angels_ ; since then, however, he had fallen into an unbroken run of forgettable pictures. When he heard that _The Set-Up_ was being produced, he offered his services to RKO, but instead the job went to first-timer Art Cohn, a former sportswriter, and Pansy Jones became Bill "Stoker" Thompson, a two-bit fighter who has spent twenty years getting pummeled in bottom-of-the-card bouts. Julie, his disillusioned wife, begs him to retire before he gets killed, but Stoker still dreams of getting a title shot and thinks he can beat Tiger Nelson, the young up-and-comer he faces that night in the heartless tank town of Paradise City. "I thought the story was wonderful," Ryan later told an interviewer, "because it had none of the usual mawkish glamour that is falsely attached to prize fight stories. It's not a glamorous business."
RKO offered the script to Fred Zinnemann, but after _Act of Violence_ the director was tired of brutality. Instead the job went to Robert Wise, an RKO contract director who had gotten his start as an editor ( _The Hunchback of Notre Dame, Citizen Kane, The Magnificent Ambersons_ ). In keeping with the poem, Wise wanted the hero to be a black man, and he had his eye on Canada Lee, a former boxer who had played John Garfield's sparring partner in _Body and Soul_. But at that point no black actor had ever starred in a big studio release, and as Wise recalled, Ryan was "dying to do it." Even without the racial element, _The Set-Up_ could be a relevant picture, exposing the seedy world of small-time boxing: the corruption, the bloodthirsty crowds, the athletes chewed up and spit out.
Wise and Cohn had plenty of time to research the picture while RKO was shut down. They saw fights at Hollywood Legion Stadium, and Wise toured the small arenas in Long Beach, observing the crowds and hanging out in the dressing rooms with the fighters. _The Set-Up_ would be heavily populated with vivid minor characters: Shanley (Darryl Hickman), a nervous teenager facing his first professional bout; Luther Hawkins (James Edwards), a black boxer in his prime, his route to the top all mapped out; "Gunboat" Johnson (David Clarke), an old-timer with a face full of scar tissue who leaves for the ring promising he will be champ someday and returns on a stretcher, unable to remember his name.
At the center of this ensemble, however, was Stoker Thompson, a dull-witted, good-hearted man and the most sympathetic figure Ryan had ever played onscreen. "I liked the character of Stoker," he later wrote. "I liked his decency in a pretty grim business." Ryan's moving performance was even more impressive coming on the heels of _Caught_ : Smith Ohlrig is rich, ruthless, and articulate, whereas Stoker is poor, empathetic, and plainspoken. Except for his opening argument with Julie (Audrey Totter) in their cheap room at the Hotel Cozy, his dialogue is sparse and mostly functional, which forces Ryan to communicate almost everything about the character physically. In the lengthy dressing room sequence he hugs the periphery, watching the other fighters and silently weighing the price he's paid for a life in the ring.
After the fashion of Alfred Hitchcock's recent mystery _Rope, The Set-Up_ would transpire in real time, from 9:05 to 10:16 PM, with the four-round bout between Stoker and Tiger Nelson commencing at the midpoint of the picture. Ryan had boxed on-screen already in _Golden Gloves_ and _Behind the Rising Sun_ , but this would be his most demanding match. "I had to learn to fight like a professional instead of an amateur, and that took months of training," he wrote. He and Hal Baylor, playing Nelson, rehearsed carefully with fight choreographer Johnny Indrisano, a former welterweight boxer who had found a second career in Hollywood. Wise spent about a week on the scene, using three cameras — one for a long shot, another for a two-shot, and a handheld camera to crowd the fighters — and edited the sequence himself. The result was thrilling, more intense and chaotic than any boxing match ever filmed. One former prizefighter on the set told Ryan, "This is so true it makes me sick."
The photojournalist Weegee (aka Arthur Fellig) was cast in a nonspeaking role as the timekeeper, and in a story for the _Los Angeles Mirror_ he described the shoot as "a social register of the fighting racket," observed by no less than eight former professionals. "HOW THAT GUY CAN FIGHT," Weejee said of Ryan. "Usually on a fight story the make-up men (they prefer to be called ARTISTS) are busy painting on BLACK EYES... but here it was different... they were busy painting OUT real black eyes, as the fighters forgot about the camera and were really slugging it out." Stoker is supposed to go down in the third round, but he hasn't gotten the message and gives Nelson a run for his money. Back in Stoker's corner, his manager (George Tobias) and corner man (Percy Shelton) spill the beans about the fix and beg him to lie down, but by now the struggle has taken on a life of its own.
Stoker wins by a knockout in the fourth, cheating the stony-faced racketeer Little Boy (Alan Baxter) out of the dive he was promised. As soon as the fight is over and the arena empties out, the power dynamic is suddenly reversed; when Stoker learns that his manager and corner man have vanished, the victory drains from his face, leaving only fear. Little Boy and his goons pay Stoker a visit in the dressing room, now silent and nearly deserted, and the racketeer promises they will be waiting for him outside. Once they're gone, Stoker races through the arena in a panic* and tries to escape into the alley, but they're waiting for him, along with Tiger Nelson. They close in, backing Stoker up against the corrugated metal shutter of a loading dock, his face slack with terror. For an actor such as Ryan, whose strength was key to his screen persona, it was a shockingly vulnerable moment. He would never have another quite like it.
"He takes more pride in that movie than any other he ever made," Jessica Ryan wrote. "It was an original." Howard Hughes certainly thought so: the following March, as RKO was readying the picture for release, he sued United Artists for copyright infringement, arguing that its forthcoming drama _Champion_ (directed by Wise's friend Mark Robson) duplicated key scenes from _The Set-Up_. A federal judge ruled for Hughes, ordering UA to delete from _Champion_ a sequence in which the hero (Kirk Douglas), who has refused to throw a fight, is stalked and beaten by hoods. _The Set-Up_ opened to glowing reviews, and that fall it won Wise the FIPRESCI critics' prize at the Cannes Film Festival. But it would be snubbed at Oscar time, possibly because Hughes had rankled so many industry people by going after UA. _Champion_ , however, was nominated for five Oscars, including Douglas as best actor.
"This is so true it makes me sick," one former prizefighter told Ryan during production of _The Set-Up_ (1949). The boxing classic would become a primary inspiration for Martin Scorsese's _Raging Bull_ (1980). _Franklin Jarlett Collection_
Stoker Thompson, cornered by Little Boy and his thugs, at the climax of _The Set-Up_ (1949). "It had none of the usual mawkish glamour that is falsely attached to prizefight stories," Ryan observed. "It's not a glamorous business." _Film Noir Foundation_
There were compensations. One night, after a preview screening at the studio, Ryan was approached on the street by Cary Grant. "You're Robert Ryan," Grant said, offering his hand. "My name's Cary Grant." The self-introduction was almost comical: Grant was one of the most famous movie stars in the world, and Ryan had admired him for years. "I want you to know that I just saw _The Set-Up_ ," Grant went on, "and I thought your performance was one of the best I've ever seen." Ryan never forgot the experience.
WHAT SPARE TIME RYAN HAD during _The Set-Up_ went into the 1948 presidential campaign. President Truman was running against Thomas Dewey, the Republican governor of New York, but also was being challenged on the right by Strom Thurmond, who had led a walkout of Southern Democrats from the convention over the party's new civil rights plank, and on the left by Progressive Party candidate Henry Wallace, whom Truman had fired from his cabinet two years earlier. Wallace wanted to give equal rights to women and racial minorities, abolish the Un-American Activities Committee, and dismantle America's nuclear arsenal, all attractive positions to Ryan. Yet he was going with Truman. Years later, in notes for a magazine article, Jessica Ryan would remember her husband's insistence that votes for Wallace would only throw the presidency to the Republicans. "Through those years, he repeated again and again the dogma he had been raised on, Vote the Party, not the Man."
Wallace's man in Hollywood was John Huston, director of _The Maltese Falcon, The Treasure of the Sierra Madre_ , and _Key Largo_. Philip Dunne, who had formed the Committee for the First Amendment with Huston, recalled that "when John took on the job, the only people who were really supporting Wallace out here were the Hollywood Ten. They were all going around with big 'Wallace in '48' buttons on. But nobody was supporting Truman very much, either. Bob Ryan and Dore Schary and [screenwriters] Lenny Spiegelgass, Allen Rivkin, and I were about the only liberals who were doing anything for Truman." Ultimately, Wallace's candidacy generated more excitement in Hollywood; Jessica would remember a banquet for Truman that failed to attract any more star power than her husband and Alan Ladd, "while Henry Wallace was wreathed in beauty glamor and fame." But in the end Truman won, carrying Los Angeles and the State of California and dominating the electoral map against Dewey in a stunning upset.
The campaign impressed upon Jessica how hardheaded her husband's politics were. When she had met him in the late '30s, he had been full of colorful stories about Chicago ward heelers, though at that point he seemed disengaged from it all, more focused on acting and the theater. "But when he came to a broader political consciousness during the war and afterward, he came to it with an infinitely greater sophistication and sense of reality than the intellectual and artistic liberals we met in Hollywood had.... He approached the issues, always, with an insistence on what was possible, what would work... with a degree of frustration and often contempt for the vagaries of the liberals who spent immense amounts of energy talking endlessly about things that ought to be done, but obviously could not be done in the framework of _how_ things _got_ done. I think that neither one of us were ever liberals in that sense of the word."
After _The Set-Up_ wrapped in mid-November, Ryan found himself without a pending assignment for the first time in nearly two years. Production at RKO was proceeding at a snail's pace under Hughes, who managed to appease the studio's distributors by doling out the fifteen completed pictures Schary had left behind. One of these was _The Boy with Green Hair_ , which had finished shooting eight months earlier but sat in limbo as Hughes tried to turn its pacifist philosophy inside out. Director Joseph Losey remembered an endless succession of notes from Hughes, written in pencil on yellow scrap paper, with orders for recutting the picture, but Losey had shot so little excess footage that not much could be done without reshoots. When the strange boy in the forest admonishes Dean Stockwell that war is harmful to children, Hughes wanted Stockwell to reply, "And that's why we must have the greatest army, the greatest navy and the greatest air force in the world." The twelve-year-old actor refused.
Three months and $150,000 later, RKO executives and board members screened the new version, which was so bad they persuaded Hughes to bite the bullet and release _The Boy with Green Hair_ in something close to its original form. Losey stated later that only a few lines of offscreen dialogue were struck, though as he recalled, all the publicity "sort of militated against it because it made it appear to be a more important film than it was." Reviewing the picture in the _New York Times_ , Bosley Crowther called it "a novel and noble endeavor" but also "banal" and "weakly motivated." Still, it was warmly received in liberal quarters, not least the Ryans' dinner table. "For some reason _The Boy with Green Hair_ was a movie that had a big presence when I was a kid," Cheyney Ryan remembers.
By the time _The Boy with Green Hair_ finally opened nationwide in January 1949, the man who had initiated the project, Adrian Scott, was persona non grata in Hollywood. The Ten were still fighting their contempt citations in the court system, an ongoing financial challenge, and while the screenwriters among the group began to find work under assumed names, Scott and Dmytryk didn't have that option. Shortly after RKO fired them, they had launched their own independent production company and tried to secure funding for _Albert Sears_ , a drama about a black family moving into a white neighborhood, but they got nowhere with the project. Dmytryk found work directing two pictures in England and was soon followed there by Scott and his family.
_Act of Violence_ had opened in December to strong reviews; _Caught_ followed in February, was savaged by critics, and flopped miserably, though by that time Enterprise Studio had already folded. Ryan had high hopes for himself professionally once _The Set-Up_ opened in March. But one assignment weighed heavily on him as he puttered around his new home in the Valley, sleeping late, working out in the shed behind the house, and playing with his little boys. Of all the pictures on RKO's new production slate, none was dearer to Howard Hughes than _I Married a Communist_ , a Red-Scare melodrama that he hoped would establish the studio's patriotic credentials under his leadership. As one of the few male stars still employed by RKO, Ryan was a prime candidate for the lead, but to star in something like this would be a slap in the face to his colleagues who were losing their careers.
_I Married a Communist_ was a title in search of a script, and it quickly became the picture nobody at RKO wanted to make. It gained a reputation as a political litmus test, a way for Hughes to weed out suspected communists at the studio. Joe Losey, the first director to pass on it, recalled Hughes threatening to keep him idle for the full duration of his seven-year contract, though eventually he was fired. "Howard Hughes dropped my option when I refused to work on _I Married a Communist_ ," remembered Daniel Mainwaring, who had written _Out of the Past_ for RKO. "He used that project to get rid of a lot of writers, directors, and actors. If you turned it down, out you went."
John Cromwell, the left-wing director first announced for the project, confirmed this story but said he took the job figuring that the god-awful script, something about a San Francisco shipping executive being blackmailed by the party for his past membership, could never be salvaged. Hughes dropped Cromwell anyway (after a fifteen-year career at RKO) to avoid paying him a scheduled salary increase. The picture's dire reputation couldn't have been much help in recruiting stars. RKO's shrinking list of contract players included only two suitable actors, Ryan and Mitchum — and Mitchum was currently serving a fifty-day jail term for drug possession, not exactly the credential Hughes needed for his patriotic picture. Hughes had indulged Ryan with _Caught_ and _The Set-Up_ and must have felt he was owed a favor. Ryan talked long and hard about it with Jessica, and in the end he capitulated; _Variety_ announced on February 23 that he would star in _I Married a Communist_.
By that time Hughes had finally collared a director: Robert Stevenson, an Englishman whose most respected picture was a Fox adapation of _Jane Eyre_ starring Joan Fontaine and Orson Welles. Script and casting problems delayed principal photography _of I Married a Communist_ until April, when Ryan was joined on the set by Laraine Day, John Agar, and Thomas Gomez, the portly and commanding actor who had competed with Ryan for the Oscar a year earlier and was haunted professionally by his attendance at two Communist Party meetings back in the '30s. Now he was doing penance as the sinister party boss, who summons Ryan's compromised hero to the docks one night and lets him watch as two thugs tie up a recalcitrant member and throw him into the ocean to drown.
Once the picture was shot, Hughes began his usual tinkering. He tended to fixate on odd details; during production of _The Outlaw_ he had dispatched detailed memos on the proper presentation of Jane Russell's breasts, and on _I Married a Communist_ he decided that Ryan and supporting actor William Talman each needed to be taught how to handle a gun. Before retakes commenced in June, RKO executive Jack Gross sent a memo to producer Sid Rogell informing him that Hughes wanted Ryan and Talman "taken out to a target range and taught to shoot, particularly how to draw, shoot, and not flinch when shooting." Hughes further insisted that the instruction be followed by a screen test of their progress, to be delivered to him personally.
In later years Ryan could barely bring himself to mention the picture. When a teenaged Cheyney remonstrated with his mother over his father's decision, arguing that his father should have stood his ground, Jessica replied, "Let's have this conversation when you have a career yourself, and see how you feel about it."
*Hughes fell out with Sturges too, and their partnership collapsed before the picture could be completed; when Hughes finally released _Vendetta_ through RKO in 1950, it was directed by Mel Ferrer.
*The scene was shot at Ocean Park Arena in Santa Monica.
_seven_
Learning by Doing
In March 1949 the Ryans decided to celebrate their tenth wedding anniversary with a big blowout at their new home. Everyone they knew in Hollywood was invited, including their classmates from the Reinhardt School who had attended the ceremony a decade earlier. For the most part, though, they preferred small dinner parties with their more educated, writerly, serious-minded friends. "We don't ask actors home," Jessica later wrote. "We haven't, Robert or I, much to say to them privately. Nor do they have much to say to us. We aren't interested in the same things."
Her own creative life was in flux. After publication of her second mystery, _Exit Harlequin_ , she decided to try her hand at a romance novel, but it was tough going. She wrote in the morning, as Tim, now three years old and the image of his father, roamed around the house and Cheyney entertained himself in his playpen. (Writing to Dido and Jean Renoir, she predicted needing "an eight-foot steel fence to contain these two men.") No one expected her, the wife of a movie star, to do anything but care for him and his children, but being a mother wasn't enough. She devoted her spare time to the American Friends Service Committee, a Quaker-affiliated organization that ministered to Japanese-American interns during the war and now mobilized support for refugees in Europe and Asia. A native of freethinking Berkeley, she studied yoga with Indra Devi, a Russian-Swedish immigrant who had arrived in LA two years earlier, and experimented with homeopathic remedies prescribed to her by Devi's friend (and eventual husband) Siegfried Knauer, a physician on Sunset Strip whom Jessica referred to as "the Witch Doctor."
She had given up on tennis, given up on acting, and now might fail as a writer. She was prone to anxiety, and nothing stoked it more than the women who threw themselves at Robert. "There was a time when a lot of silly girls wanted my autograph," Ryan would recall a decade later. "Today my wife remembers that period as the most unbearable part of our... years together." Especially after _The Set-Up_ , Ryan had become a beefcake idol to bobby-soxers; fans often requested photos of him bare chested, and photographers asked him to strip for the camera. "Heck, I'm no Tarzan," Ryan told the _Hollywood Citizen-News_ that summer. "But if that's what they want, I'll give it to them."
Despite his exhibitionism, Ryan struck most people as a good husband. Laraine Day, who costarred with Ryan in _I Married a Communist_ , would describe him years later as "such a gentleman. It was a pleasure to work with him. And especially because he was so devoted to his wife. It was wonderful to listen to him talk about her and their life together, because you felt there was real devotion."
After moving to North Hollywood, the Ryans made another positive change to their lives by hiring Solomon and Williana Smith, a black couple in their early fifties, as household help. Born in Waterproof, Louisiana, a delta town on the Mississippi state line, Solomon had run a radio repair shop in New Orleans, where he married Williana in 1925. During the war, as part of the great migration, they had moved to Vallejo, California, where Solomon worked at the Mare Island Naval Shipyard, and since then they had come to Los Angeles, buying a house in the Mid-City neighborhood. They had no children. "Smith," as everyone called him, tended to the Ryans' house and garden, chauffeured Bob back and forth to the studio, and took Jessica on errands (she hated driving). Willie cooked, cleaned, and minded the children. "Smith was one of these guys that knew how to do everything," Cheyney Ryan recalls. "They were people who had an enormous amount of practical intelligence." During the week the Smiths stayed in a guest room the Ryans had added to the house, and before long they became part of the family.
A month after completing _I Married a Communist_ , Ryan began shooting a romance with Joan Fontaine called _Bed of Roses_. This would be his first women's picture since _Tender Comrade_ six years earlier; Jessica had urged him to take it, fearing he was in danger of becoming typecast as a heavy. "I spend a lot more time now posing for romantic stills," he told fan-magazine writers Reba and Bonnie Churchill. "All the photos the studio had on file were shots of me snarling at the camera."
Williana and Solomon Smith worked for the Ryan family from 1948 through the early 1960s, stabilizing a household that was buffeted by the demands of a movie star's career. _Robert Ryan Family_
No less than seven screenwriters and five directors had labored over this adaptation of a 1928 romance novel called _All Kneeling_ , to which Fontaine owned the rights. When Hughes assigned Nicholas Ray to direct, Ray and writer Arthur Schnee took yet another crack at the script, working more feverishly perhaps than the material required. Producer Robert Sparks advised Ray not to get so wrapped up in this star vehicle, but Ray was adamant: "This picture shows the turmoil inside a woman's heart." Sparks replied, "The only turmoil inside Joan Fontaine's heart is whether her dressing room is heated in the morning."
Born in 1911, Ray had grown up in LaCrosse, Wisconsin (also the hometown of Joseph Losey), and in the early '30s studied under playwright Thornton Wilder at the University of Chicago. Through Wilder he won a fellowship to the experimental arts community created by Frank Lloyd Wright at his Taliesin estate near Spring Green, Wisconsin. After moving to New York City in 1934, the young man acted in and directed left-wing political theater for the ragtag Theatre of Action company, during which time he joined the Communist Party. Eventually the Theatre of Action was absorbed into the Federal Theatre Project, and before long Ray was staging community-based theater in the rural Southeast. During the war years, John Houseman had hired Ray to direct radio programs for the Voice of America, and eventually launched him as a Hollywood director.
Ryan was struck by Ray, who summoned the five lead actors for an initial table reading of the script before shooting commenced, something that rarely happened at RKO. Ray had fought to get Ryan on the picture, having admired his low-key charm in _The Boy with Green Hair_ , and the two men connected, though Ryan was a little frustrated to be working with Ray on such a weak story. Christabel Scott (Fontaine), the beautiful schemer at the center of _Bed of Roses_ , climbs the ladder of high society by marrying a millionaire (oily, mustachioed Zachary Scott) but then cheats on him with a mischievous, razor-sharp novelist (Ryan). Ray's dialogue was witty, and at the very least he concocted a meet-cute that ranked as one of Ryan's few genuinely comic moments on-screen. Staying with friends, Christabel thinks she's alone in the house, but her phone call is interrupted by a man's voice ordering her to get off the line. She races back into the kitchen, where the other phone handset is located, and Ryan's grinning face pops up from behind an open refrigerator door.
Ray and Ryan would make two more pictures together in fairly quick succession before the director left RKO; one of the enduring mysteries of the blacklist period is why Hughes protected Ray, a former party member. Earlier in Ray's career, Hughes had tried to force _I Married a Communist_ on him; after Ray asked his agents to free him from the project, he received a Christmas Eve summons to meet with Hughes at the Goldwyn studios (where he found the boss watching _Caught_ ). Somehow Ray managed to wriggle out of the assignment without losing his job.
All spring and summer there had been worrying headlines from Berlin and mainland China. Following the massive Berlin airlift, which had thwarted Joseph Stalin's blockade of the city, the United States had established the German Federal Republic and the Soviets began constituting the German Democratic Republic. In the East, Mao Zedong's Communist forces had captured Nanjing, capital of the Republic of China, and driven Chiang Kaichek's nationalist government south to Canton, hastening the end of the decades-long civil war. Then, on September 23, President Truman grimly announced "evidence that in recent weeks an atomic explosion occurred in the USSR." The atomic monopoly of the United States was over.
Domestically the news itself had the force of a blast, and the confluence of international developments only fed the flames of anticommunism in Washington. A week after Truman's fateful announcement, Mao Zedong declared the People's Republic of China; a week after that, _I Married a Communist_ test screened in Los Angeles and San Francisco. Audience response was dismal. After the blacklist hit, several studios had defensively launched anticommunist dramas, but the ones that had opened — MGM'S _Conspirator_ , with Elizabeth Taylor; Republic's _The Red Menace_ — had bombed. Now people were confronted with the possibility of nuclear annihilation on American soil; in the era of the bomb shelter and the unspeakable end, who wanted to relax at the movies with a picture about scheming communists?
RKO executives pleaded with Hughes to tone down the politics and change the title; he was incredulous, considering the title "one of the most valuable parts of the picture," but reluctantly agreed. _Variety_ reported that _I Married a Communist_ would open in January 1950 as _Where Danger Lives_ , but when it finally arrived in theaters six months later, collecting ho-hum reviews and expiring at the box office, it was titled _The Woman on Pier 13_.
_The Secret Fury_ , which began shooting in October and continued through early December, looked to be even worse, a tepid mystery starring forty-six-year-old Claudette Colbert ( _It Happened One Night_ ) as a classical pianist who may be suffering from amnesia and Ryan, who turned forty during the shoot, as her concerned fiancé. After the holidays Ryan reported back to RKO for a series of retakes on _Bed of Roses_ ; Hughes wanted a new ending, but even this failed to mollify him, and four months later he was still dispatching instructions for reediting the picture. Like _I Married a Communist_ , the caustic romance would sit on the shelf for the better part of a year, come out under a different title (the saucier _Born to Be Bad_ ), and do lackluster business.
The glacial production pace at RKO had become par for the course under Hughes; all through 1949 the studio had been promising to ramp up the release schedule, but of thirty pictures announced, only a dozen were produced. One executive recalled, "Working for Hughes was like taking the ball in a football game and running four feet, only to find the coach was tackling you from behind." RKO Radio Pictures had lost $5.2 million in 1948, the year Hughes took over, and hemorrhaged another $3.7 million in 1949. "I think he bought RKO as a tax liability," said director Joe Losey. "He wanted to run it into the ground so he could take a huge loss."
Jean Renoir and Eddie Dmytryk were long gone. Losey, Bob Wise, and Jacques Tourneur had escaped to other studios. Given the dearth of directing talent at RKO, Ryan looked forward to his next project with Nicholas Ray. Ryan's friend John Houseman had approached him to star in a crime thriller he and Ray were trying to get off the ground at RKO, to be adapted from a 1946 British novel by Gerald Butler called _Mad with Much Heart_. Ray was fascinated by the book, in which a London police detective tracks a mentally disabled child murderer through the snowbound English countryside, but RKO had passed, and when Houseman sent the book to his friend Raymond Chandler, the writer replied with a withering critique. Ryan agreed to make the picture if he could approve the script, and Sid Rogell, head of production at RKO, relented. The film that finally emerged two years later, retitled _On Dangerous Ground_ , would include one of Ryan's most indelible performances and become a key film in his screen persona. But first it had to get past Hughes.
_MAD WITH MUCH HEART_ was an odd novel, a suspense story that evolved into a melancholy romance. In a quiet farming community, two little girls have been strangled, and one dies; the father of the other, Walter Bond, seethes with anger. "Why are such things allowed to happen?" he asks his wife. "Has the eye of the Lord left our village?"
Shotgun in hand, Bond hopes to kill the culprit in the ensuing manhunt; holding him in check is James Wilson, a plainclothesman who has been dispatched from London to command the investigation. The two men follow the murderer by car in a blinding snowstorm and eventually spin out into a ditch; after Bond finds the suspect's deserted vehicle, he and Wilson trudge to the nearest house. Its occupant is a beautiful blind woman, Mary Maldon, who invites them in but acts suspiciously. Wilson guesses correctly that she's hiding something. The killer is her younger brother, whom she cares for; she says he's been missing for days, but Bond doesn't believe her and Wilson isn't sure if he should.
Packed with action, Butler's book was tailor-made for the screen, but what really fascinated Ray was a short passage at the midpoint, after the snowstorm forces Bond and Wilson to suspend the search until morning and the blind woman offers them shelter. Tucked into a chair before the fire, Wilson dwells on his unhappy life in the police force: "There was never anything clear and clean, never any gift without a hook in it, never a meeting without some undercover deceit.... You thought it was going to be a romantic life.... You didn't know you were simply putting your head into a world that stinks from top to bottom. You didn't know you were choosing the life of a garbage man, digging and prodding and letting the smell out from human dregs."
Ray wanted to relocate the story to the United States and envisioned a prefatory sequence showing Wilson as a cop driven to savagery by the ugliness and venality all around him. To adapt _Mad with Much Heart_ he recruited screenwriter A. I. Bezzerides — "Buzz" to his friends — a Greek-Armenian immigrant who had grown up in Fresno, California, and published a hard-boiled novel about truckers, _They Drive by Night_ ,* that was adapted to the screen for George Raft and Humphrey Bogart. Since then Buzz had become a screenwriter himself and adapted his novel _Thieves' Highway_ for director Jules Dassin. Ray sold him on the idea of riding with big-city cops to learn what their daily lives were like; Bezzerides spent a few nights with the LAPD, and Ray, visiting the East Coast, rode with Irish cops in Boston. In New York, Ray saw Sidney Kingsley's new play _Detective Story_ , which also dealt with a brutal cop in a big-city precinct.
"We start with the cop in the city being called up for his violence," Bezzerides remembered. "He's a vicious cop, vicious to criminals because he can rationalize it. Criminals are criminals to him, they're not people. So he's sent out of the city for his behavior, into the mountains." The urban section, shot in Los Angeles, would take up the first half hour of an eighty-minute picture, and its tone was uniformly harsh, its city a nocturnal cesspool of drunks, hookers, and hustlers. The ostensible story line, a police procedural in which Jim Wilson and his two partners search for a cop killer, was simply dropped when the action moved out to the country; the real narrative was psychological, a series of encounters with urban lowlifes that exposed Wilson's disgust with humanity.
The bifurcated story made Houseman uneasy, but for Ray it was the key to the picture: he even wanted to heighten the contrast by shooting the city scenes in black and white and the natural scenes in color (an idea the studio quickly nixed). Ryan signed off on the script, and on Monday, March 27, he set off from Union Station in Los Angeles to Denver, Colorado, and from there to Granby, Colorado, in the Rocky Mountains. According to Ray biographer Bernard Eisenschitz, "Ryan was in a scene as soon as he arrived, at 4:30 in the afternoon." Ida Lupino, a luminous actress ( _High Sierra_ ) who had become one of the few women directors in Hollywood, was cast as Mary, and Ward Bond, a hard-line anticommunist active in the right-wing Motion Picture Alliance, would play Bond, renamed Walter Brent. For Danny, the frightened man-child on the run, Ray managed to sneak in his nephew, Sumner Williams (whose anguished, inarticulate performance hinted at the work Ray would coax from James Dean in _Rebel without a Cause_ ).
Staying at the El Monte Inn, the cast and crew got a warm reception from the townspeople of Granby and nearby Tabernash, some of whom appeared in the film. Ray would remember the Granby shoot as a wonderful experience, and Ryan was sufficiently at ease there to show up at the local high school, speak to the boxing club, and sign autographs. Heavy snowfall impeded the two-week shoot, and at one point the generators broke down and the camera tripods began to freeze up. But the fresh snows created a blinding whiteness for the chase scenes, and the mountain terrain was stirring, a stark backdrop for the contest between Wilson and Brent over whether Danny will be apprehended or executed.
Back in Los Angeles, the focus shifted to the interiors and later the city locations, where Ray and Ryan really began digging into their embittered hero. The picture was ahead of its time in noting how cops are isolated and worn down not only by the repellent characters they deal with every day, but also by the fear and contempt of law-abiding citizens. Prowling the streets, Wilson and his two partners think they've spotted their man, and Wilson runs him down; a crowd gathers, but the guy is clean. "Dumb cops!" he blurts out angrily. "I was only running." Wilson has to be held back from clocking the guy, and the policemen retreat to more snide comments from the crowd. Shortly after this altercation the trio stop at a drugstore, where Wilson sits at the soda fountain and flirts with the counter girl. When someone teases her about her boyfriend, she replies, "That's all he'd need to know — me going out with a _cop_." Wounded, Wilson spins around on his stool to hide his face, and his mouth tightens in resignation.
The night is full of users. At a dive bar Wilson is trailed by Lucky (Gus Schilling), an alcoholic trembling for his next drink, and gets hit on by an underage B-girl (Nita Talbot). At a side table a balding, bespectacled, obscenely grinning man (played by Bezzerides) tries to force some money on Wilson. Searching for a suspect named Bernie Tucker, Wilson pays a visit to Myrna (Cleo Moore), a slatternly platinum blond who shows him the bruises Tucker left on her and insinuates that he can leave a couple of his own if he likes. "The dissolve at the end of [the scene] will be played in such a way as to avoid the direct impression that Jim is about to indulge in a sex affair with Myrna," Joseph Breen, head of the Production Code Administration, had instructed Harold Melniker of RKO. Ray cross-fades to a shot of Wilson coming down the stairs of the apartment building alone, but the sexual implication hangs in the air.
Breen was even more concerned about the graphic scene in Bernie Tucker's apartment, where Wilson corners the suspect. Bernie, a grinning slimeball, dares Wilson to hit him, but his smile fades when he realizes he's taunted the wrong man. "Why do you make me do it?" Wilson sputters, his voice rising in desperation. "Why do you make me do it? _You_ know you're gonna talk. I'm gonna _make_ you talk! I _always_ make you punks talk! Why do you do it? _Why? Why?_ " In an era when studio releases were subjected to additional censorship from local boards, Breen pointed out that the ensuing beat-down "would unquestionably subject the picture to extensive cutting and possibly even rejection, especially in the many municipalities where censor control is exercised by the police department." In the release version Ray fades out as Wilson comes down on Bernie, but no on-screen punch could be as unnerving as his twisted reasoning: _Why do you make me do it?_
The picture supplied Ryan with not only his most demented on-screen moment but also, ironically, the most moving love scenes he had ever shot. Ryan liked Lupino; as the blind Mary, she invested what might have been a mawkish character with an arresting combination of strength and empathy. When Mary asks Wilson what it's like being a cop, he confesses, "You get so you don't trust anybody." Mary replies, "You're lucky. You don't have to trust anyone. I do. I have to trust everybody." Lupino disliked the downbeat ending, which showed Mary alone and weeping after Wilson returns to the city, and Ray was sufficiently swayed to let her and Ryan improvise a new one in which Wilson turns around and comes back, and the characters are reunited.
Ray had elicited mesmerizing performances from both his leads. To hear Ryan tell it, Ray never gave him explicit instructions, which was fine with him. "I hate film-makers who want long discussions with actors over a scene," he explained. "An actor who doesn't know what a scene he's going to play is all about is in the wrong profession. Nick had, I think, great respect for me. Right from the start of our collaboration, he only offered me a few suggestions." Mostly Ray would tell Ryan about the Irish Catholic cops he had ridden with in Boston and how they had behaved in certain situations, and then let him take it from there.
Ida Lupino and Ryan in _On Dangerous Ground_ (1952). His role as an angry, despairing cop supplied him with not only his most demented screen moment but also the most moving love scenes he had ever shot. _Film Noir Foundation_
In Wilson's apartment, the dresser is decorated with high school athletic trophies and a crucifix, evidence of the personal route Ryan was taking into his damned character. He had long since left the church, but the church had never left him; raised by Jesuits, he had come to manhood believing in the horrible stain of original sin, the sin of Adam, which had corrupted mankind forever. Lamont Johnson would remember commiserating with Ryan over "the hangovers that we both shared as ex-Catholics." These involved "a hell of a lot of residual anger that I have, and I could sense that that was part of what there was with Bob.... I mean, we all had other things too, you can't blame it all on the Catholic Church, but it was certainly a considerable portion of it, and you would see Bob just retreating into a cynical, cold, and conceivably dangerous guy in some moods. [He] was by no means the great, good-hearted Herbert that a lot of people think." Whatever repressed rage Ryan may have been carrying around came bursting out in _Mad with Much Heart_. Then, like any other job, it was over, and he went home to his wife and kids.
FOUNDED IN 1947 on the University of California campus in San Diego, the La Jolla Playhouse had become a magnet for movie actors looking to get back onstage. Ryan had appeared there in the romantic farce _Petticoat Fever_ in summer 1949, and on July 4, 1950, he returned to play the low-rent tycoon Harry Brock in a six-day run of Garson Kanin's comedy _Born Yesterday_.
At one performance Jessica met Irene Selznick, the wife of producer David O. Selznick and daughter of MGM lion Louis B. Mayer, and over a drink in the Valencia Hotel, the two women started talking about their children and the chronic school overcrowding in California. Since the war began, the state population had swelled by 3.5 million, and there was an ongoing shortage of teachers and classrooms. Schools in Los Angeles County were operating on double shifts, with students receiving only a half day of instruction; one school had four kindergarten sessions stacked up from morning to late afternoon. The US birth rate had spiked in 1946, increasing nearly 6 percent, and a year from now those children — including Tim Ryan — would all be old enough for kindergarten.
Jessica mentioned that her pediatrician, Siegfried Knauer, had suggested she start her own school, and to her surprise, Selznick explained that during the war she had done just that: "You get together some children and find a place. Then you get a teacher to run it."
Years later, in a memoir about the founding of the Oakwood School, Jessica would trace her interest in starting a school to her own insecurity. "My hang-up was a simple one: I felt I had not had enough education. Meaning college. With the passing of time this want had become, family and friends tell me, something of an obsession." Robert had to drive out to Kenab, Utah, in August to shoot locations for an RKO western called _Best of the Badmen_ , but not long after his return, Jessica talked him into meeting with Tim's nursery school teacher to discuss the idea. The teacher referred them to Sidney and Elizabeth Harmon, similar-minded parents who lived in nearby Studio City and had four children, and the two couples got together for cocktails.
Cheyney, Jessica, and Timothy Ryan (circa 1951). _Robert Ryan Family_
"Lizzie was a small, pretty woman with a breathless manner and childlike eyes that gazed with some bewilderment on the world," wrote Jessica. Sid Harmon "wore horn-rimmed glasses on myopic brown eyes that looked warmly upon all the people he liked which, together with a fondness for talking, often made him resemble a benign rabbi." In fact, Harmon was a producer — in the early '30s, he had mounted a Pulitzer Prize-winning production of Sidney Kingsley's _Men in White_ with the Group Theatre, and a few years later he had come out to California to break into the picture business.* Lizzie had attended the private Ethical Culture Fieldston School in New York City, whereas Sid, like the Ryans, had gotten his primary education at public schools. The two couples agreed to host an open meeting for interested parents at the nursery school where Tim was enrolled with Andy Harmon.
Through director Joe Losey, the Ryans met producer Frank Taylor and his wife, who had started the Westland School near Beverly Hills; they sent the Ryans to meet its director, Lori Titelman. "Her advice was refreshingly uncluttered and to the point," wrote Jessica. "In starting a school we must make up our minds to call a spade a spade — meaning, calling progressive _progressive_ , even though the word had lately become suspect in both its educational and political context."
_Progressive_ was a code word for _communist_ , yet progressive education was actually rooted in the philosophy of John Dewey, embracing the notion that children learn better when engaged with the world around them. When the parents' meeting at the nursery school took place, drawing in about fifty people, someone asked Ryan if he was proposing a progressive school. Winging it, he replied, "Modified progressive." Harmon expressed the idea that the school would be open to all races and religions, to which someone commented, "Sounds pinko to me." That first meeting did flush out one more interested couple: Ross Cabeen, a ruddy-faced petroleum engineer and rock-ribbed Republican, and his wife, Wendy. Cabeen "was out to make a great deal of money," Jessica recalled. "But he was bugged by a conscience (partly his wife) telling him that he should do more."
A second parents' meeting was called, with Lori Titelman as guest speaker, but her left-leaning philosophy and the idea of opening a racially integrated school seemed to scare off many of those attending. By this time loyalty oaths had become part of civic life in California, required of municipal and county employees as well as faculty and staff at state universities; that fall the Regents of the University of California had fired twenty-six tenured professors who refused to sign. The Levering Act, which passed the California legislature weeks later, barred state employees from collecting their checks unless they signed a statement denying membership in any organization deemed subversive by the US attorney general.
Communism had become a hot issue in the 1950 midterm elections, especially after North Korean forces crossed the thirty-eighth parallel into South Korea in late June. In California, Democratic Congresswoman Helen Gahagan Douglas — a former actress married to movie star Melvyn Douglas — was running for an open Senate seat against Republican Congressman Richard M. Nixon. As a member of the House Un-American Activities Committee, Nixon had come to prominence investigating whether former US State Department official Alger Hiss had passed secrets to the Soviet Union, and his work had led to Hiss being convicted of perjury. Nixon decided to make his anticommunist credentials central to his campaign, one so cunning in its attempts to smear Douglas that it would forever saddle him with the nickname Tricky Dick. The masterstroke was a flyer, printed on pink paper, that compared her voting record with that of New York Congressman Vito Marcantonio, widely thought to be a communist. A half million copies of this "Pink Sheet" were distributed across Southern California.
Ryan stumped for Douglas and contributed two hundred dollars to her campaign, but for the most part the Hollywood community shied away from her. Nixon, grasping the power of television, finished out the election cycle with a flood of commercials in which he accused Douglas of being soft on communism. A whispering campaign against her husband insinuated that he had changed his name from Melvyn Hesselberg to conceal his father's Jewish roots in Russia. In the end Nixon clobbered Douglas, winning 59 percent of the popular vote. "There, in that murderous character assassination campaign," wrote Jessica, "we saw that the horror of what had been going on in Hollywood with the rise of the blacklist was not a particular attack on the movie business by HUAC but had entered the state and national scene... _and_ was winning." The Republicans picked up twenty-eight seats in the House and five in the Senate, and like the Eightieth Congress, elected four years earlier, the Eighty-Second would bring a blast of red-baiting.
SYMPATHY FOR DOUGLAS was in short supply on the set of Ryan's latest picture, a big-budget war movie teaming him with John Wayne. _Flying Leathernecks_ chronicled the exploits of a Marine aviation unit in the Battle of Guadalcanal, allowing Howard Hughes to indulge two of his great passions — aerial heroics and knee-jerk patriotism. Wayne was currently president of the Motion Picture Alliance for the Preservation of American Ideals. Producer Edmund Grainger and screenwriter James Edward Grant, who had created Wayne's giant hit _Sands of Iwo Jima_ (1949), were active in the Alliance as well. Crusty character actor Jay C. Flippen and on-set screenwriter Rodney Amateau were staunch conservatives. Cornered on the left were Ryan and director Nick Ray. "We often asked ourselves what we were doing on a film like this," Ryan would recall. "I hate war films."
By this time Wayne had been in pictures for twenty years, but in the last few he had really caught fire, giving iconic performances in _Fort Apache_ (1948), _Red River_ (1948), _She Wore a Yellow Ribbon_ (1949), and _Rio Grande_ (1950). In _Flying Leathernecks_ he played Major Dan Kirby, who arrives in Oahu, Hawaii, to take command of the VMF 247 Wildcats unit but prefers not to get too close to men he may have to sacrifice. Ryan was his philosophical antagonist, Captain Carl Griffin, beloved by the men but passed over for promotion. "I cast [Ryan] opposite Wayne because I knew that Ryan was the only actor in Hollywood who could kick the shit out of Wayne," Ray wrote in a memoir. "That conflict was going to be real, so I'd have two naturals."
Ryan wasn't inclined to kick the shit out of anyone, though his skill with his fists always guaranteed him a wide margin of respect from Wayne, who admired not only his strength but his education and intellect. In any case Wayne was in no position to question Ryan's patriotism, given his own lack of World War II service; unwilling to let his career languish, Wayne had passed up numerous opportunities to serve with his friend and director John Ford in the US Navy's photographic unit, which earned him Ford's eternal scorn. The day after Thanksgiving, principal photography for _Flying Leathernecks_ commenced at Camp Pendleton, where Wayne had shot _Sands of Iwo Jima_ but Ryan actually had drilled recruits during the war.
_Flying Leathernecks_ had been blessed by the military, and the Marines came across with men and materiél; production files show Grainger requesting more than three dozen fighter planes (F6F Hellcats and F4U Corsairs) for aerial photography, another twelve planes for set decoration at Henderson Field, a long list of ground equipment, and the services of one hundred marines. As usual with Hughes's projects, the story was a mess. Ray actually was working from three different scripts, shooting elements he liked from each, and he hired Amateau to collate them into a single narrative. The result was so disjointed and generic that, by default, the conflict between Kirby and Griffin, over how much empathy to show the men, became the picture's most interesting story element. Ray was right about the two stars: both men were 6′ 4″ and they filled the frame, Ryan often hovering silently behind Wayne and stealing scenes with his sidelong glances. "It's all in the eyes," he would tell his son Tim about the art of acting. "That's where you do most of your work."
Ray purposely staged their big showdown inside a tent, "using the space for tension, so you could expect that the moment Duke dropped his right, Ryan would stiffen, and pretty soon they'd bring the tent down around them." Unfortunately, the dialogue didn't live up to the staging. "You just can't bring yourself to point your finger at a guy and say, 'Go get killed!'" Kirby says. "You've gotta tear your guts out worrying about his flight record, or because some dame back in the States is giving him the brush-off!" Ryan does his best with Griffin's overripe response: "Four hundred years ago a poet put it better than I ever could: 'No man is an island.' When the funeral bell rings, it isn't just for the dead guy. It's a little bit for all of us." He was glad to get the whole thing behind him.
With the new year, the Ryans, Harmons, and Cabeens pushed forward with their school. Harmon suggested that they relocate their meetings to the Country Schools, a nursery school and summer day camp in the area, and the group circulated a letter, signed "Robert Ryan, Chairman, The Oakwood School," inviting people to get involved:
For the past eight months a group of parents in the San Fernando Valley has been working to lay the groundwork for an Independent Elementary School... which would be dedicated to the principles of the best in modern education.
We plan a non-sectarian, non-profit school with a serious program of parent participation.
We plan a school
where — The children are encouraged to work, play and learn together as responsible parts of a group and a community.
where — The teacher guides the child to achieve learning not by rote but through his curiosity and activity.
where — The classes are smaller, and closer individual attention is possible than in the overcrowded public schools.
where — The school belongs to the child and the parents as well as to the Professional Teaching Staff.
Parents at the Country Schools proved more receptive to these ideals. "The word _progressive_ could be used without fear of it reflecting upon one's loyalty to one's country," Jessica wrote in her memoir about the school. "Certain common goals became clear: the school certainly should be open to all chil dren regardless of race, color or creed. In fact, we should make every effort to assure a broad democratic base.... Scholarships should be made available to children of working-class parents."
The Ryans had begun immersing themselves in the philosophy of education, especially the writings of John Dewey. "I believe that the only true education comes through the stimulation of the child's powers by the demands of the social situations in which he finds himself," Dewey wrote in 1897. "Through these demands he is stimulated to act as a member of a unity, to emerge from his original narrowness of action and feeling and to conceive of himself from the standpoint of the welfare of the group to which he belongs." _Sounds pinko to me_ , parents at their early meetings might have commented. But for Ryan, who had championed motion pictures as a tool for enlightenment four years earlier, the idea of reaching people at a more impressionable age must have been enticing — in any case, more enticing than _The Secret Fury, Best of the Badmen_ , or anything else RKO might have in store for him.
THE NEW CONGRESS was sworn in on January 3, 1951, and two months later the House Un-American Activities Committee returned to Hollywood with a vengeance. By this time the original Hollywood Ten had all gone to prison for contempt of Congress, and those who had been paroled found themselves unemployable under their own names. The sole exception was Edward Dmytryk, the RKO director whose career had been so intertwined with Ryan's; in September 1950, from inside a federal prison in Mill Point, West Virginia, Dmytryk released a statement affirming his loyalty to the United States and swearing he was no longer a Communist Party member or sympathizer. He was paroled two months later, and in mid-April 1951, as a second wave of congressional subpoenas rolled through the studios, Dmytryk testified before the committee as a friendly witness and named names. This time there wasn't a whisper of protest from the Hollywood Left. Of the 110 people called to testify, more than half would recant their radical beliefs and inform on their past associates.
Interviewed more than thirty years later, Dmytryk would reveal that he quit the Communist Party of America after cell leaders began pressuring him and Adrian Scott to alter the story of _Crossfire_ (he gave no details). To hear him tell it, he stuck with the Ten in 1947 because their Fifth Amendment strategy required solidarity, but he couldn't tolerate being linked with some of his fellow defendants. "The first day the unfriendly witnesses hit the stand, I knew I was not with them," said Dmytryk. "The question from then on and for the next two and a half years or more, 1947–1950, was how do I do it? When do I do it as gracefully as possible?" Dmytryk would be reviled by some, but he returned to Hollywood and within a year scored a four-picture deal with liberal producer Stanley Kramer.
Dmytryk had plenty of company: writers Clifford Odets and Budd Schulberg, directors Elia Kazan and Robert Rossen, and actors Edward G. Robinson, Lee J. Cobb, Sterling Hayden, and Lucille Ball all prostrated themselves before the committee and were spared. Other industry people, such as actor Lloyd Bridges, were allowed to testify secretly.* As the panic took hold, a whole cottage industry of red-baiting organizations sprang up around the entertainment industry, companies such as American Business Consultants, which published the infamous _Red Channels_ index of alleged communists and contracted with the studios to help "clear" employees with suspect activities or associations. Compared to the blacklist, which banned specific people from working at the studios, the "graylist" of independent pamphlets and newsletters was even more insidious, a creeping mist of rumor and innuendo.
While this inquisition played out, much of the nation was also mesmerized by the hearings of the Kefauver Crime Committee, established by the US Senate in May 1950 to investigate organized crime in America. Chairman Estes Kefauver was a liberal Tennessee Democrat determined to eradicate crime syndicates in America, and over the next fifteen months he and his four-member committee traveled across the country to hear testimony from local mobsters. Sessions in New Orleans, Detroit, and New York City were broadcast on live television and drew rapt audiences; for many Americans, this was their first exposure to the Mafia. Mobster Frank Costello refused to show his face on camera, and a close-up showed his twitching hands as he spoke.
Watching all this unfold, Howard Hughes decided he had the perfect topical hook for a remake of his silent 1928 gangster drama _The Racket_. Adapted from a play by Bartlett Cormack, it told the story of an NYPD captain, Tom McQuigg, trying to get the goods on a fearsome crime boss, Nick Scanlon (the role had made Edward G. Robinson a Broadway star in 1927). Reporter-turned-screenwriter Samuel Fuller was tapped to update the story, but Hughes rejected Fuller's florid script in favor of a more prosaic draft hammered out by William Wister Haynes and hard-boiled novelist W. R. Burnett (author _of Little Caesar, High Sierra_ , and _The Asphalt Jungle_ ). Their main innovation in the wake of the Kefauver hearings was to saddle the old-school gangster Scanlon with a new, nationwide syndicate of smooth corporate operators who considered his bare-knuckled style embarrassingly passé.
_The Racket_ was Ryan's first picture with Robert Mitchum since _Crossfire_ four years earlier (Hughes preferred to parcel out the few stars still on his payroll), though the two actors and their pal Jane Russell sometimes drank together in Mitchum's trailer on the RKO lot. As in _Crossfire_ , Mitchum sauntered through the picture as the police captain, letting Ryan tear it up as the ruthless thug. Whenever the two of them squared off, the picture crackled: newly assigned to the district, McQuigg barges into Scanlon's office to put him on notice, and the gangster emerges from a side room eating an apple. For Ryan the apple was a typical bit of business that helped him define the character: Scanlon talks with his mouth full, stops short of taking a bite in reaction to something McQuigg has told him, and, when McQuigg pushes him too far, angrily flings the half-eaten fruit aside.
Fuller, who would later direct Ryan in _House of Bamboo_ , thought he had a "charismatic gift for making you like the bastard he played, because he understood what made that bastard tick — and he made the audience understand it." What makes Scanlon tick is Joe, the younger brother he's been grooming for a more respectable life than his own. To Scanlon's dismay, he learns that Joe has proposed to a two-bit cabaret singer (Lizabeth Scott), and for a moment McQuigg becomes his unlikely confidante. "You'd never guess what I've done for that kid," Scanlon fumes, pacing around his office with his hands jammed in his trouser pockets. "Made a gentleman out of him. Sent him to four colleges. _Four!_ And the last one, I had to buy a _chair_." Scanlon kicks a nearby armchair. "Not like that! An endowment, they call it."
Amid the interest generated by the Country Schools meetings, the parents in North Hollywood decided to move forward and find a school supervisor. David Walden, finance secretary for the American Friends Service Committee, referred them to a Quaker educator named Lloyd Nixon, who was skeptical of actors but soon recruited a promising candidate for the job. Bryson Gerard, a teacher at Pacific Ackworth Friends School in far-off Temple City, California, impressed them with his notion of a school rich in the humanities, responsive to the parents, and inspired by the democratic principles of the Quaker meeting. Jessica, who by that time was quite pregnant with her third child, talked Robert and the others into funding the school. Classes would begin in September, around the time she was due to give birth.
Right after Independence Day, Ryan began rehearsing a low-budget thriller with Ida Lupino called _Day without End_ , about a woman held captive in her home by a violent schizophrenic. Mel Dinelli's story had been through several incarnations already: it debuted in January 1945 as an episode of the CBS radio anthology _Suspense_ , starring Agnes Moorehead and Frank Sinatra, and since then Dinelli had turned it into a short story and a successful play. Lupino had bought the screen rights for her production company, The Filmmakers; Hughes agreed to loan Ryan out for the picture, which would be shot in three weeks, mostly on one set, and to distribute the result through RKO. Lupino, despite her own résumé as a director, handed this project over to Harry Horner, an Oscar-winning production designer.
Like _On Dangerous Ground_ , the new picture relied heavily on the strange chemistry between Lupino, playing a similarly benign and ethereal character, and Ryan, cast again as a dangerously volatile, physically intimidating man. "I know what it is to be lonely too," her character tells his, in a distinct echo of the blind woman and the cop from the earlier picture. This time, however, beauty fails to tame the beast. The opening scene finds Ryan's character, Howard Wilton, tending to his handyman chores in a woman's home, calling out for his employer, and then swinging open a door to discover her dead body sprawled on the floor. (In a close-up of Ryan's hand, his fingers jump with fright.) Howard doesn't remember having killed her, but he flees anyway, hopping a freight train and landing in a new town where he hires on with Helen Gordon (Lupino), a recent war widow. (For some reason the action is set in 1918.)
Howard seems like a model employee at first, gentle and courteous, but Helen is soon alarmed by his moodiness and paranoia. Once again Ryan shows his talent for startling physical movement; after Howard has fallen to the floor of the parlor in anguish, he grabs his forehead, pulls himself to his feet, and lurches unexpectedly into the foreground of the shot, where he collides with a little Christmas tree. During another such spell Howard lightly bumps a table lamp, setting the crystal decorations that hang from the shade tinkling like his own disordered mind. "Scenes, like life itself, are mostly a matter of feeling and action," he once said. "Hardly anyone ever thinks of what he wants to say — almost everyone thinks about what he wants to do." The role gave Ryan a chance to essay a variety of moods: at one moment Howard is quiet and humble, at another he accuses Helen of conspiring against him, his voice shaking with rage. It was a real tour de force, though the premise, so effective in a half-hour radio play, was stretched too thin for a feature-length film.
Less than a week after they finished the picture (whose title would be changed to _Beware, My Lovely_ ), Ryan flew into Chicago on TWA — the boss's airline — with producer Edmund Grainger and actress Janis Carter to attend the gala premiere of _Flying Leathernecks_. "A spectacular aerial exhibition by a squadron of Marine Corsairs, piloted by Korean War aces, was staged over Lake Michigan [Sunday] night," press materials reported. "The Marine air-devils, as climax to their stunts, released flares spelling in the skies at high altitude 'FLYING LEATHERNECKS,' which was visible as far west as Aurora." The premiere on Monday, August 14, brought even more pomp, including a three-mile parade down Michigan Avenue with Marine bands, color guards, and fife-and-drum corps. Ryan rode in a car with Grainger, Carter, and Radio-Keith-Orpheum president Ned Depinet (who was nearing the end of his rope with Hughes and would soon resign). In Washington, Richard Nixon caught an advance screening of _Flying Leathernecks_ and commended Grainger on the floor of the Senate.
All that summer the parents had scouted North Hollywood and Studio City for a suitable classroom facility, as Jessica struggled with the minutiae of health and fire-code regulations. "The job of starting a school," she remarked, "often seemed entirely to do with fires and toilets.... Meanwhile what you taught in the school, who taught it, and how it was taught, was of no interest to anybody." Originally, they had hoped to find a vacant house, but such properties were in short supply. Following the example of Westland School, they began inquiring at places of worship, which often owned multiple buildings, and settled on one belonging to Temple Beth Hillel. It was located near Magnolia Boulevard at the end of a dirt road, near a giant flood control project; as Jessica recalled, its concrete channels "were of such unreasonable width and depth they got a rumor going that they were really secret military highways to be used in case of atomic attack."
Ryan's business manager, Henry Bamberger, had advised him and Jessica to form a nonprofit corporation; according to Jessica's memoir, she and Robert hosted the other two couples one evening to come up with a name for the institution. "As the evening progressed (and the liquor flowed)," wrote Jessica, "we went from ideas like San Fernando Elementary School and North Hollywood Elementary School to things like Ryan-Harmon-Cabeen's Folly, Disturbed Children's Lyceum, Blacklisted Writers' Refuge, and The Little Red School House by the Flood Control Project. And laughed and laughed." Despite her recollection, the project had been announced months earlier as the Oakwood School, after a Quaker school by that name that Sid Harmon recalled in his native Poughkeepsie, New York. The name stuck.
Ryan was named president, Harmon vice president, and Ross Cabeen secretary-treasurer. Tuition was set at fifty dollars, with some thirty-two parents enrolling thirty-five students. In the weeks before the school opened, the parents all pitched in to clean up the building, scrubbing the floors and the desks. One conspicuous absentee was Jessica, who had gone into labor and given birth to a baby girl on September 10. The Ryans named their new child Lisa — after Lisa Sokoloff, wife of their old acting coach, Vladimir Sokoloff. Ryan was proud of the child, proud of his wife, and proud of the classes starting up across town, which had been all her idea. The harmony of Jessica giving birth as Oakwood opened its doors was lost on no one, though of the two offspring, Lisa Ryan would prove much less troublesome.
*Not to be confused with Ray's first feature, _They Live by Night_ (1948).
*Harmon was nominated for an Oscar for the original story of George Stevens's _The Talk of the Town_ (1942), starring Cary Grant and Jean Arthur.
*Although Howard Hughes had the connections to make this happen for Ryan, no evidence has ever emerged that he did.
_eight_
The Whiz Kids
Nearly a decade had passed since Ryan signed with RKO. In that time he had made more than twenty pictures for the studio, but gems such as _The Set-Up_ and _On Dangerous Ground_ were few and far between. Max Reinhardt would have been disappointed to see him squandering his talent on potboilers like _Flying Leathernecks_ and _Beware, My Lovely_. "He could have written his own ticket after the war, on the strength of _Crossfire_ ," observed columnist Louis Berg earlier that year in the _Los Angeles Times_. "He let the golden opportunity dangle. 'I'm doing fine,' is his invariable reply. 'I've got time.'" Ryan must have been stung by the observation that his career had crested, but Berg was right. Beholden to Hughes for his political protection, Ryan took what he could get at the studio. Too many people were depending on his paycheck — not just a wife and three children, but a private school now as well.
Luckily for Ryan, the fall of 1951 brought an intriguing new assignment. Earlier that year the talented Warner Bros. producer Jerry Wald ( _Mildred Pierce, Johnny Belinda, Key Largo_ ) and comedy screenwriter Norman Krasna had formed the independent Wald-Krasna Productions (whose acronym earned them the industry tag the Whiz Kids) and struck a deal with Hughes to release sixty features over six years. Now Wald wanted to film _Clash by Night_ , the Clifford Odets flop that Ryan had performed ten years earlier on Broadway with Tallulah Bankhead and Lee J. Cobb. The screen rights had been parked at RKO for years. Wald sold Hughes on the idea of relocating it from Staten Island to Monterey, California, and stripping away its outdated social commentary to focus on the adulterous love triangle.
Barbara Stanwyck signed to play Mae, the smothered wife, and Paul Doug las was cast as her simple, devoted husband, Jerry. Ryan took third billing as Jerry's malignant friend Earl, the part Joseph Schildkraut had performed back in 1941, and Keith Andes inherited Ryan's original role as Jerry's younger brother, Joe. To play Joe's fiancée, Peggy, Wald cut a deal with Twentieth Century Fox to borrow twenty-five-year-old starlet Marilyn Monroe. At Stanwyck's urging, Fritz Lang was hired to direct; his brilliant career in the German cinema ( _Metropolis, M_ ) was now two decades behind him, but he had eked out a second act in Hollywood that included such haunting dramas as _Fury_ (1936), _The Woman in the Window_ (1944), and _Scarlet Street_ (1945). Wald indulged Lang by sending him and cinematographer Nick Musuraca up to Monterey to shoot extensive footage of fishermen and canners at work, with Monroe and Douglas in tow; after three days they came back with ten thousand feet of footage that Wald had edited into a documentary-type preface for the beginning of the picture.
"It was the first time I could convince any producer that we should have rehearsals, as is done for the stage," recalled Lang. "Because it dealt mainly with three people, you could, in a certain way, rehearse the main scenes.... We marked the exact positions of the camera, its movements and so on. It was wonderful to work with all three: Barbara Stanwyck, Bob Ryan and Paul Douglas." Ryan ranked the director alongside Jean Renoir and Max Ophuls in his ability to recognize and heighten an actor's best qualities, though in contrast to Renoir, who made everything feel spontaneous, Lang wanted complete control over every aspect of a scene. "He leaves nothing to chance," Ryan explained. "He plans everything in advance."
The one element Lang couldn't control was Monroe, whose chronic inability to remember her lines began to slow down production almost as soon as shooting commenced in October 1951. She was terrified of Lang, who tried to banish her trusted acting coach, Natasha Lytess, from the set; sometimes the pressure brought out red splotches on Monroe's skin. Lang watched from behind the camera, fuming, as Stanwyck tried to pull off a complicated stretch of dialogue while hanging clothes on a line and Monroe wrecked the scene again and again. Stanwyck never complained, but Lang took to berating Monroe. At some point Ryan intervened, taking the director aside and urging him to lay off. Ryan took a dim view of colleagues who were unprofessional, but clearly the girl was trying, and haranguing her would only exacerbate the situation. Besides: _look_ at her!
Earl Pfeiffer (Ryan) gets tanked up with Peggy (Marilyn Monroe) in _Clash by Night_ (1952). Ryan took Monroe's side against director Fritz Lang, and she never forgot it. _Franklin Jarlett Collection_
Something about _Clash by Night_ brought out the worst in people, and Ryan watched in dismay as the cast, like that of the Broadway production, was riven by professional jealousy. A rumor began circulating around town that Monroe was the young woman posing nude in a new girlie calendar (according to one account, the information was leaked by RKO's publicity man in the hope of drumming up interest in the new picture). Reporters flocked to the set, ignoring the forty-four-year-old Stanwyck as well as Douglas and Ryan. Monroe had sat for the photo back in 1949, when she was still unknown and needed the fifty dollars for a car payment. "The calendar business was no secret in Hollywood, but the public didn't know about it," recalled Ryan. "One of the reporters asked me, 'Where's the babe with the big tits?' He didn't even know her name." At one point Wald received an anonymous call from someone demanding $15,000 to keep quiet about Monroe's deep, dark secret; instead the producer wanted to put her name above the title with those of the other three stars.
Earl (Ryan) moves in on Mae Doyle (Barbara Stanwyck), his best friend's wife, in _Clash by Night_ (1952). _Franklin Jarlett Collection_
Lang remembered their reaction when they heard the news. "Douglas said, 'I will never give my permission, never! Who is she? A newcomer! She will never make the top grade.' Ryan didn't say anything, but Barbara said, 'What do you want — she's an upcoming star.'" Monroe got her above-the-title billing, fourth after Ryan. Douglas was furious; on the set, after Monroe referred to him in passing as "Paul," he ordered her to address him as "Mr. Paul Douglas." According to Keith Andes, Stanwyck eventually grew frustrated with Monroe, who would "always come in late and all f_____ up. Stanwyck finally said, 'Look, unless she's working, keep her off the set. I don't want her around.' After all, Barbara was a good-humored woman but she was also a professional." After Monroe's death, Stanwyck would remember that she "drove Bob Ryan, Paul Douglas and myself out of our minds... but she didn't do it viciously, and there was a sort of magic about her which we all recognized at once."
Ryan tried to stay above the fray, just as he had with Lee Cobb and Tallulah Bankhead. He had a much better part this time, though Earl Pfeiffer would be his third heavy in a row after _The Racket_ and _Beware, My Lovely_. A projectionist at the local movie house, Earl despises his wife, a burlesque performer who is constantly on the road. "Someday I'm going to stick her full of pins, just to see if blood runs out," he tells Mae. Earl can be an embarrassment — at a restaurant, as a joke, he pulls his eyes into slits and jabbers obnoxiously in mock Chinese — but Mae is attracted to his hard body and cynical talk, and tired of the dully unimaginative Jerry. When Mae and Earl go into a clinch, Mae hungrily reaches up under the back of his wife-beater, a sharply sexual moment probably inspired by the recent release _A Streetcar Named Desire_. Lisa Ryan would remember her mother always getting angry when Stanwyck's name came up.
To produce _Clash by Night_ , Wald and Krasna had chosen Harriet Parsons, one of very few women in the business and the daughter of gossip columnist Louella Parsons. Harriet persuaded her mother to write a column on Ryan, her first since the 1948 profile in which she had hinted at his affair with Merle Oberon. This time around Louella was squarely in his corner: "With all this talk of divorce and scandal in Hollywood, Robert Ryan is almost too good to be true. None of these evils has ever touched him, and I'm going to put my neck out a mile and say I'm sure none ever will." Louella visited the set to watch Ryan play a scene as Earl, and according to the column, Ryan asked her afterward, "How did you like me as a home wrecker?" The complex social transaction was completed when Ryan credited Harriet for having persuaded him to take the role. "When Howard Hughes first sent word that I was to play this dubious gentleman for Wald and Krasna, I had plenty of reservations. I had never been a no-good character who steals another man's wife."
Production wrapped in early December, leaving Ryan with a two-month interim before his next job. When the story about Monroe's nude photos finally broke the next year, the young actress made a frank statement about it that disarmed critics, and as Wald and Krasna had hoped, the publicity drove ticket sales for _Clash by Night_ , which connected at the box office and drew good reviews as well. Seven years later, when Ryan was shooting _Lonelyhearts_ at Samuel Goldwyn Studio, he would take his son Cheyney over to meet Monroe as she filmed _Some Like It Hot_ on a neighboring soundstage, and the boy would be surprised by how warmly she received his dad. Word had gotten back to Monroe that Ryan took her side with Lang, and she never forgot it. "Poor kid, she was so bewildered," Ryan said after her death. "Right after the picture was finished she sent me a big box of candy with a very touching note."
Once 1952 arrived, the Ryans grew busy again with the Oakwood School. After searching fruitlessly for a better facility than the temple building, Ryan and Ross Cabeen decided to buy land and put up a building themselves. The parcel they chose was on Moorpark Street in North Hollywood, a few miles southwest of the Ryans' home. "Circling the property on two sides was a dry wash as yet unreclaimed by the Flood Control System; it still presented a sandy bottom lined with scrub willows," wrote Jessica in her memoir. "The property itself consisted of close to three acres of ground with a magnificent stand of eucalyptus trees down one side bordering the wash."
The two men bought the property for $6,500 and went before the parents' group proposing that everyone contribute toward a building fund. According to Jessica, this idea met with controversy because the parents, now numbering about thirty-five, would neither own the property nor have any legal power over the buildings' disposition — only Ryan, Cabeen, and Sid Harmon had incorporated the school. With or without the parents' participation, Ryan and Cabeen resolved to go ahead and construct two classroom buildings on the lot.
Their first hurdle was to win a zoning variance from the city, which meant collecting signatures from all the neighbors. Whenever Ryan had a day off from work, he and Cabeen spent the afternoon making the rounds with their petition. One tough customer, wrote Jessica, told them he didn't like actors, children, or Jews, but after several visits they caught him after a few drinks, listened politely as he recalled his days as a Klansman back in Illinois, and finally won his signature by promising to "keep _them_ out." A building fund of $14,000 was established, heavily endowed by the Ryans, Harmons, and Cabeens, and construction began in April. Instead of laying a cornerstone, the parents staged a little ceremony in which each child at the school laid a concrete block on one row of a wall.
With production at RKO slowing to a crawl again, Hughes loaned Ryan out to Universal-International for a two-picture deal: the first, _Horizons West_ , began shooting in February 1952, and the second, _City Beneath the Sea_ , followed soon afterward, keeping him busy through early May. The deal must have seemed like a good move: he would get top billing in both films, which would be shot in Technicolor and directed by the capable Budd Boetticher. In _Horizons West_ he plays a former Confederate soldier who returns to his native Austin, Texas, and, frustrated in his plans to establish himself in business, turns to horse rustling and amasses a small fortune; Rock Hudson is his brother, whose new job as sheriff of Austin puts them on a collision course. ("He's not getting married again soon," Ryan would remark whenever Hudson's name came up.) _City Beneath the Sea_ teamed him with Anthony Quinn in a tale about deep-sea divers. Universal was a lesser major studio, carried by the tireless Abbott and Costello, yet it was in better shape than RKO.
Jessica plants a tree at the Oakwood School in North Hollywood as Tim and Bob look on. "More than anyone else, she was responsible for Oakwood's survival," wrote the school's director, Marie Spottswood. _Robert Ryan Family_
The first time RKO had loaned Ryan out — to MGM for _Act of Violence_ — all hell had broken loose in his absence, and the same thing happened again as he was shooting at Universal. _Variety_ reported in February that both Edmund Grainger Productions and Wald-Krasna Productions were at wits' end, waiting endlessly for Hughes to approve their scripts. Then, two months later, as the American Legion and other right-wing groups massed for another anticommunist assault on the entertainment industry — not just movies this time but also TV, radio, and theater — Hughes shut down production on the Gower Street lot for a second time, announcing that he would conduct a systematic purge of communists and their sympathizers from the studio ranks. Industry observers wondered if this were just a prelude to Hughes selling his interest in the studio; the president of the Screen Writers Guild, which had been feuding with Hughes over giving screen credit to blacklisted writer Paul Jarrico, argued that Hughes had "thrown a mantle of Americanism over his own ragged production record."
As part of this crusade, Hughes established a new security office at RKO to screen all employees for suspect activities or associations. Many were suspended, and at some point Hughes must have made up his mind that Robert Ryan would have to go. Publishing in _The Worker_ , launching this bohemian school out in the Valley, stumping for the Progressive Citizens of America, American Civil Liberties Union, and United World Federalists — now that Hughes had to put up or shut up, he may have decided Ryan was too far over the line to be defended. Before long the actor had a new contract with RKO stipulating only that he make one picture a year. For the first time since signing with the studio in 1942, Ryan was a free agent.
THE BREAK WITH RKO opened up a world of possibilities for Ryan — he could return to Broadway, even play Shakespeare — but more immediately he needed to land a good picture, just to prove he was still bankable. Once he found himself on the open market, he gravitated immediately toward Dore Schary, the liberal producer who had cast him in _Crossfire_ five years earlier. Since fleeing the Hughes regime at RKO, Schary had returned to his previous employer, MGM, where he kept up the good fight with such stark, serious-minded dramas as _Battleground_ (1949) and _The Red Badge of Courage_ (1951). In a studio shake-up he had recently replaced the aging Louis B. Mayer as president, and now he came through for Ryan with a plum assignment, supporting James Stewart and Janet Leigh in Anthony Mann's harsh psychological western _The Naked Spur_.
Ryan had never cared much for westerns, and they were hell to make, with remote locations and strenuous action shots. But the _Naked Spur_ script, by first-timers Sam Rolfe and Harold Jack Bloom, read like a chamber drama; aside from a brief shootout with some Blackfoot Indians, the action was re stricted to only five characters, all locked together in a treacherous mountain journey. Bounty hunter Howard Kemp (Stewart) has arrived in the Rocky Mountains searching for Ben Vandergroat (Ryan), who's accused of murdering a marshal in Abilene, Kansas. Aided by an elderly prospector, Jesse Tate (Millard Mitchell), and a disgraced cavalryman, Roy Anderson (Ralph Meeker), Kemp manages to capture the outlaw and his unwitting young accomplice, Lina (Leigh), but as they cross the mountains on horseback to collect the reward, Vandergroat begins sowing discord among the three men.
The role had originally been earmarked for Richard Widmark, and Ryan was happy to have aced him out: here was a smart, layered character who threatened to turn the pat morality of most westerns on its head. Ryan played him as a cracker-barrel philosopher, his homespun wisdom dispensed in a mirthless chuckle. When a frustrated Kemp offers to shoot Vandergroat on the spot, the outlaw peers up at him, his brow lined with the years, and replies, "Choosin' a way to die, what's the difference? Choosin' a way to _live_ — that's the hard part." Ryan tosses off this Will Rogers dialogue with ease, yet it masks an ugly soul. Near the end of _The Naked Spur_ , Vandergroat succeeds in peeling away the old prospector, Tate, with the promise of a gold mine, then wrestles away his rifle and coldly murders him. "Look at him, lyin' there peaceful in the sun," he tells the horrified Lina, slipping back into his gentle drawl. "Ain't never gonna be hungry again, want anything he can't have."
Four years earlier Ryan and Leigh had shared an electric scene together in _Act of Violence; The Naked Spur_ gave them much more screen time, and they worked well together. The ambiguous relationship between the outlaw and his young charge turns out to be the most troubling aspect of _The Naked Spur_. Vandergroat presents himself as the only family Lina has left, and they're tenderly affectionate with each other. Bound at the wrists, he drapes his arms over Lina's head, even brushing her forehead with a kiss; plagued by a sore back, he periodically calls on her to "do me," and she faithfully massages his shoulders. At the same time he's not above exploiting her sexually, especially after he realizes that both Kemp and the low-rent Anderson are taken with her. "The more they look at you, the less they'll be lookin' at me!" he whispers to her in a quiet moment.
The entire film was shot on location in the San Juan Mountains, near Durango, Colorado, from late May to early July. The cast was lodged about fifteen miles from town in a group of cabins at El Rancho Encantado. "It was a congenial, pleasant, cheerful group, and I believe everyone thoroughly enjoyed themselves," Leigh recalled. Jimmy Stewart and his wife, Gloria, threw an anniversary party for Leigh and Tony Curtis, during which Curtis buttonholed Ryan for a program in which they would visit with underprivileged kids in East Los Angeles. Ryan was joined by Jessica and the children; the boys got to watch stunt work being filmed, and one evening the family drove into town and caught an early show of _Best of the Badmen_ at a drive-in theater. It was the first time the children ever saw their father on-screen.
Ryan loved working with Stewart — a class act who, on one occasion, finished his own shots early but stuck around all afternoon feeding lines to Ryan and Leigh from behind the camera. But the best professional relationship Ryan would carry away from _The Naked Spur_ was with director Tony Mann. " What made Mann so brilliant," remembered Ralph Meeker, "was his ability to pick a backdrop that might be green and lush or barren and stark, or a rushing river, and he could put his actors against these backdrops, and it all became _one_." Mann's camera sense was superb; he knew not only how to tell a story in pictures but how to tell it in landscapes. He liked vertiginous overhead shots and put his actors through quite a gauntlet as they rode and climbed, using doubles only when absolutely necessary. Leigh would recall the tall rock above a roaring river where she, Ryan, and Stewart filmed the climax: "The ultimate panic was being on top and peering down down down at the angry water and rocks below. With no guard rails, with no nothin'!... The fear we registered was genuine."
Tall and athletic like Ryan, Mann had grown up in San Diego and bounced around RKO and the low-rent Republic Studio before distinguishing himself with some dynamic crime pictures ( _Raw Deal, Border Incident, Side Street_ ) and Stewart's hit western _Winchester '73_. No one could ever get a word out of the guy, though Ryan liked this. "I understand that for certain people it became very difficult to work with him," he later said. "For young people especially, since they love to talk about their character for hours.... There are certain actors who, to light a cigarette, need to create a back story. Me, I prefer not to talk." He and Mann would make three pictures together, all of them excellent, and Ryan would rank his performance in _The Naked Spur_ as one of his best. It gave him just what he needed at this career juncture: a chance to shine as an actor, working alongside the best in the business.
By August the two school buildings on Moorpark Street had been com pleted, but they were only unfinished concrete shells: the parents would have to finish the buildings and construct a school grounds around them. Chuck Haas — who, with his wife Emilie, became such a devoted backer of the school that Oakwood would later credit them equally with the three founding families — supervised the carpentry work. Bryson Gerard took charge of laying a concrete walk, and Ryan paired off with Sid Harmon to erect a rickety fence around a small yard for the kindergarten. Mothers painted; fathers built furniture. Even then Oakwood was strapped for space; word of the school had spread and enrollment had risen. The school acquired a substantial loan for more building construction, but in the meantime, Gerard suggested that the older children, second through fifth grade, be quietly relocated to parents' homes.
For several months the fourth- and fifth-grade teacher took up residence in Ryan's Refuge out in the backyard, while Ryan found refuge elsewhere. Children would come into the house to use the bathroom, Jessica remembered, and afterward might talk to Smith as he gardened or play in the yard with little Lisa trailing after. "Often, having been to the toilet, they would stay inside the house playing with our boys' things or just wandering around," she wrote. "It was disconcerting sometimes if one was trying to take a bath or get dressed when one or another unfrustrated child would walk blithely in." This surreptitious home schooling continued until a third building was erected on the Moorpark lot.
THAT SUMMER the Democratic Party, convening at the International Amphitheatre in Chicago, nominated the liberal, well-spoken Governor Adlai Stevenson of Illinois for president. The Ryans watched on the TV set in their den as Stevenson was chosen on the third draft, beating out Estes Kefauver and Senator Richard Russell of Georgia. "Robert let out a whoop of joy — a whoop repeated that day, I have no doubt, in thousands, no, millions of Democrat homes, ranch-style or otherwise," wrote Jessica in a later magazine piece that would never see publication.
In accepting the nomination, Stevenson promised to "talk sense to the American people," and he made good on that promise in late August when he attacked red-baiting in a speech to the American Legion convention in New York. "True patriotism, it seems to me, is based on tolerance and a large measure of humility," Stevenson declared, denouncing those who would ex ploit patriotic feeling to oppress minority groups or silence minority opinions. "The tragedy of our day is the climate of fear in which we live, and fear breeds repression. Too often sinister threats to the Bill of Rights, to freedom of the mind, are concealed under the patriotic cloak of anticommunism."
Ryan was itching to get involved in the campaign, though as Jessica would observe, many actors still were skittish about supporting liberal candidates; to the self-appointed cops who published red-baiting pamphlets, the merest hint of involvement in the Progressive Party or Henry Wallace's 1948 presidential bid was considered subversive. A Hollywood for Stevenson committee was formed, and Dore Schary hosted a glamorous launch party at his home in Brentwood; though Jessica noticed a heavy star quotient, she attributed that to Schary more than Stevenson: "When the head of MGM called, you went!" She and Robert shook hands with Stevenson and had their picture taken with him and Schary. Then in mid-October, Ryan got a call from screenwriter Allen Rivkin, head of Hollywood for Stevenson. Rivkin was having trouble finding stars to take part in a rally on Wednesday, October 15, at the Cow Palace in San Francisco, so the Ryans made the trip, joining Mercedes McCambridge, Lauren Bacall, and Humphrey Bogart.
Despite his gangster image, Bogart was a cultured man who had grown up in New York City as a child of privilege just like Ryan. The two men got along well; Ryan especially appreciated the fact that Bogart always turned up on time. "Bogey is the only other man in town with a punctuality complex," he once joked to a fan magazine. "He's a great comfort to me and the hours we've spent waiting for parties to start have allowed us to become intimately acquainted." Appearing at the Cow Palace without his toupee, Bogart did a little comedy routine in his familiar tough-guy persona, ordering the audience to vote for Stevenson or else; Ryan spoke extemporaneously, giving what Jessica considered "a short, articulate, knowledgeable and sometimes funny speech." Ryan claimed he had spoken off the top of his head, and though Jessica knew he had prepared carefully, she still was floored by his poise.
Soon after this the Ryans traveled out to the East Coast to campaign for Stevenson in the final week before Election Day, and during the trip they traveled with Bogart and Bacall. The couples had adjoining rooms at the Statler Hotel in Boston, where the three stars appeared at a rally at Mechanics Hall. "Robert did his extempore thing, but it was strikingly changed from the Cow Palace," wrote Jessica. "He had familiars before him, Irish city machine politicians, and the blood of Old Tim rose; he almost talked with a brogue, not a new experience for me."
Back at the hotel Ryan and Bogart invited their driver up for a drink, though Ryan made "a slightly slurring remark" to Jessica that the little Irishman was "a typical ward heeler." The man had no interest in the Stevenson campaign, only the local races that affected patronage hiring. "Bogart and Betty listened with distress and horror, Bogart particularly," Jessica observed. Ryan enjoyed the experience, always amused to see Hollywood liberals confronted by the sort of down-and-dirty machine politics he had known as a child in Chicago.
The next day began with a breakfast for Governor Stevenson that was being hosted by Governor Paul Dever of Massachusetts at a Cambridge hotel. Out in the hall, Ryan was approached for directions to the breakfast by the young Congressman John Kennedy, who was running for the US Senate against the moderate Republican Henry Cabot Lodge Jr. — and whose father, Joseph, had formed RKO back in 1928. When Ryan introduced himself, Kennedy replied, "I know who _you_ are."
After breakfast the Ryans and Bogarts joined a daylong motorcade through the southeast suburbs of Boston. As they neared each stop and the caravan slowed to a crawl, gawkers would move down the line of cars looking for celebrities and waving and yelling at Ryan and the Bogarts. When they peered in at Jessica, though, their faces darkened. "It wasn't the first time I'd experienced this strange hostility from people who had come to see movie stars and suddenly were cheated by people who weren't anybody being there too," she wrote in her notes. Bogart won her heart by yelling at fans through the window glass that she was Rosalind Russell, which some of them seemed to buy. Jessica had always swooned for Bogart in the movies, and in person she found him "a bright, gentle, warm, human being.... Sure, there had always been crazy stories about him in Hollywood. He drank a lot and fought with various wives... that's what they said. It was nothing to me. Bogart was a gentleman." Also a comedian: at one point, when he and Ryan had been signing autographs, Bogart observed, "I notice that your fans are younger than mine, and I don't like that."
There followed a whistle-stop tour from Boston to New York City, where Stevenson addressed a late-night rally in Harlem. Walking down 125th Street toward the Hotel Theresa, Jessica got separated from the group, and Ryan was mobbed in the lobby of the hotel until none other than boxer Joe Louis pulled him out of the crowd to safety. "Respect from the champ," Jessica wrote. "R has always had respect from them. They recognized in the fight pictures he made that he really knew how to box. They all value him for making _The Set-Up_." Louis took Ryan out a back exit and enlisted a friend to drive the actor back to his own hotel.
New York City was the final stop on the campaign swing, with a giant rally at Madison Square Garden on Wednesday, October 29. Broadcast live on TV, the program opened with a chorus singing "Stevenson for President" (updated from the old Gershwin tune "Wintergreen for President"). Ryan was supposed to speak first, but through some miscommunication he missed his cue. When he heard himself announced over the loudspeakers, he had to race for the stage. On-screen there was an awkward moment of dead air before a singer launched back into the song and the other performers shifted around nervously; more than a minute passed before the star appeared to recite his piece. Ryan was angry and embarrassed about the incident, though it seemed to be par for the course in the chronically unorganized campaign.
Election night brought heartache when General Dwight D. Eisenhower handily defeated Stevenson with 55 percent of the popular vote. Ryan was devastated; he generally took a live-and-let-live attitude toward Republicans, but he despised Eisenhower. "My father's hatred of Eisenhower was over the top," said his son Cheyney. "And it would always come out as, 'You can't let the military do anything. You can't let a general do anything. Generals and officers just screw it up.'" Not only did the Eisenhower victory usher in four years of Republican supremacy, with functional majorities in the House and Senate, it elevated Richard Nixon to the vice presidency, only two years after he had destroyed Helen Gahagan Douglas. At one point during the campaign swing, Jessica was amused when some idiot thought Ryan was Dick Nixon. "It was many long years before I ever told Ryan that someone had taken him for Nixon," she wrote. "But even that many years after the fact he still got furious."
BY THE END OF 1952 about seventeen million homes in the United States had TV sets; the figure had doubled in just two years. The Hollywood studios, scrambling to meet this new assault on their viewership, embraced a strategy of playing up technological innovations that couldn't be reproduced by a cathode ray tube. At Twentieth Century Fox, Darryl F. Zanuck had placed a large bet on CinemaScope, mounting the widescreen biblical epic _The Robe_ , but while that was in preproduction, the independent Arch Oboler Productions made a small fortune with the first 3-D feature, a low-rent jungle adventure called _Bwana Devil_. Eager to jump on this bandwagon, Zanuck decided that _The Waterhole_ , a desert survival drama to be directed by Englishman Roy Ward Baker, would be shot in 3-D.
Baker had made two pictures for Fox already (the second a creepy thriller with Richard Widmark and Marilyn Monroe called _Don't Bother to Knock_ ), and he was fascinated by the long stretches of action in Francis Cockrell's short story "The Waterhole," about a spoiled millionaire stranded in the Mojave Desert by his scheming wife and her lover. "I had always had an ambition to make a picture in which the leading character spends long periods alone on the screen, where the interest would be in what he does, rather than what he says," Baker wrote. To handle the complicated process of shooting in 3-D, Baker recruited cinematographer Lucien Ballard, who had worked with him on _Don't Bother to Knock_.
Ryan couldn't have been pleased to hear that Ballard was on the picture, having slept with his wife throughout production of _Berlin Express_ , but apparently the shoot passed without incident. Merle Oberon had left Ballard in 1948, shortly after her affair with Ryan, taking up with an Italian shipping magnate. Ironically, an adulterous affair lay at the center of the new picture (later retitled _Inferno_ ), yet that element of the story was confined almost entirely to William Lundigan and redheaded Rhonda Fleming as the conniving lovers. Ryan's scenes as the beleaguered millionaire, shot in the Mojave near Apple Valley, California, were almost entirely wordless (though a superfluous internal monologue would be dubbed in later). Ryan always rose to a creative challenge, and he loved Baker's idea of shooting a modern silent picture; in fact, the idea of a man trapped in a hellish desert terrain recalled the climax of one of the great silent features, Eric von Stroheim's _Greed_ (1925).
Stereoscopic images had been around since the early nineteenth century, and the anaglyph process, which split the image into red and green and recombined them with polaroid glasses, dated back to the 1939 New York World's Fair. Over the years people had tried to release stereoscopic films, but the big studios had never gotten involved until now: _Inferno_ would join a surge of 3-D releases that year, including MGM's _Kiss Me Kate_ , Warner Bros.' _House of Wax_ , and Universal's _It Came from Outer Space_. Baker described the contraption he and Ballard were using: "Two cameras were bolted onto a large plate at right angles to each other and mounted on the usual dolly. A polar screen was placed in front of the lens of each camera and the two screens were set in opposition to each other. A two-way mirror was set in front of both cameras at 45 degrees: the right-hand camera shot straight through the mirror and the left-hand camera received the mirror image, which was then [flipped] in the processing so as to present it the right way round.... The cameras were interlocked and run in synch. Thus we had two matching films, left eye and right eye."
This setup was so unobtrusive that Fleming didn't even realize until later that the picture was being shot in 3-D. Baker mainly avoided the gimmickry of objects flying toward the camera; he was more interested in the way 3-D allowed him to place his actors in front of mountain ranges and capture the skyline in all its wondrous depth.
Four years earlier Ryan had played a fictionalized version of Howard Hughes in _Caught_ , and now he had been cast as another capricious millionaire wrestling with the limits of his wealth. Like Hughes, Donald W. Carsons II disappears on people; his secretary refers to his desk as "the bottleneck," and his second-in-command at the mining firm is paralyzed waiting for the boss's signature. "I think he's always had the fear that without his money he'd be nothing, helpless," the executive remarks at one point. Meanwhile in the desert, Carsons is discovering the exact opposite: stranded on a mountainside with a broken leg, he manages to straighten the leg, bind himself with splints, lower himself down the rocks to the ground, find a water hole under the surface of the sand, and kill a deer for food. The rock-climbing sequences rival those in _The Naked Spur_ , particularly the tense moment when Carsons lowers himself past a rattler coiling on a rock ledge.
The picture ended with Carsons being rescued, his wife returning guiltily to his side, and her lover, Duncan, fleeing for Mexico. But when Zanuck saw this cut, he decided the picture needed a slam-bang ending that would deliver on the promise of 3-D action. Assisted by fight choreographer Dick Talmadge, Baker staged a scene in which Duncan trails Carsons to a little shack where he's being sheltered by a local desert rat (Henry Hull); the antagonists go after each other, a flung oil lamp ignites the shack, and Carsons is pulled to safety while the flaming roof caves in on Duncan (in a hair-raising point-of-view shot, to make Zanuck happy).
Retitled _Inferno_ , the picture opened in London, New York, Chicago, and Los Angeles, but so few theaters in the United States were equipped for 3-D projection that it never got a wide release and soon disappeared, despite strong reviews that singled out Ryan's work. The picture's failure was disappointing to him; interviewed in the early '70s, Ryan would list his best screen work as _Crossfire, The Set-Up, God's Little Acre_ , and "a picture nobody ever heard of called _Inferno_ which I made for Fox."
THAT SUMMER the Ryans rented a beach house in Malibu. The children — seven-year-old Tim, five-year-old Cheyney, and little Lisa, coming up on her second birthday — frolicked in the waves as their parents read or had drinks with Joan and John Houseman, who had a house nearby. Ryan was shooting _Alaska Seas_ , a romantic drama with Jan Sterling and Brian Keith, as part of a two-picture deal with Paramount. The studio that had given him bit parts and then dropped him in 1940 was now paying him $125,000 per picture. The Ryans lived modestly out in the Valley, and the summer rental was something they could easily afford. Besides movies, Ryan moonlighted on radio and TV, and a few years earlier he had signed a lucrative deal as a pitchman for Chesterfield cigarettes. His business manager, Henry Bamberger, advised him to invest in real estate, and Ryan had bought several apartment buildings, as well as a shopping mall in Beverly Hills. Then there was the school, which would cost the Ryans some $40,000 by the end of the decade.
Houseman, who understood that Ryan was "trying to shed the stigma of playing only brutal and violent parts," recruited him for _Her Twelve Men_ , a women's picture he was producing at MGM with Greer Garson. The British actress had been at MGM since _Goodbye, Mr. Chips_ (1939), but at forty-eight she was nearing the end of her tenure at the studio; shot in color, this last picture evoked _Mr. Chips_ with its story of a beloved teacher at a tony boarding school for boys, coincidentally named the Oaks. Ryan played another teacher at the school and, eventually, Garson's love interest, but the script was weak. Shortly after the cameras began rolling in August, Houseman disappeared, busy with the editing of _Julius Caesar_ and the preproduction for _Executive Suite_ , and Ryan fended for himself with director Robert Z. Leonard, a longtime MGM hack who had been making pictures since Ryan was in short pants.
The tweedy, upper-class world of the Oaks stood in ironic contrast to the Oakwood School, which seemed to be spinning out of control. The founding parents had launched the school with no clear vision for how it should operate; that had come mainly from Bryson Gerard, who modeled the parent-teacher partnership after Quaker meetings with their vigorous debate and eventual consensus. This proved unworkable: a like-minded congregation might arrive at this "sense of the meeting," but the Oakwood parents more often fell prey to bickering, backbiting, and political point scoring. The group had split into two factions: a radical wing that kept trying to use the school as a political arena, and the more responsible parents who had sunk their own money into the project. According to Jessica's memoir, one father, a blacklisted writer, refused to admit a student whose father had informed on him to HUAC; this angered oilman Ross Cabeen, who thought they should keep their politics out of the school.
Gerard had arrived with plenty of ideas about hands-on learning, his curriculum "built around hand-crafts, bug, tree, rock, and bird studies, cook-outs, animal husbandry, and camping trips," as Jessica reported. The children sang, danced, and acted in plays. But two years in, parents were beginning to complain that their children couldn't read. When Jessica conferred with Tim's teacher about his slow progress in reading and writing, the teacher blamed Gerard for failing to provide the proper teaching materials. Mothers were getting back to her with reports that the staff couldn't get a curriculum or any supervision from him. Rumors were circulating that he had been avoiding parents, and Wendy Cabeen wanted him out.
Jessica called a meeting to clear the air, and the parents unloaded on Gerard, their grievances seeming to come from all directions. As she recalled it, the radical wing argued that Oakwood should be "making a contribution to the community" by "having a series of lectures and discussion groups — child psychology, adult psychology, ceramics, noted psychoanalysts, political experts." It should "take political action in terms of the reactionary forces in the public school system, fighting for higher salaries, smaller classes, and integration." This provoked Ryan, who had clocked so many hours with his conservative friend Ross Cabeen trying to build up the school. Chuck Haas was absent from the meeting, but he received a report from Sid Harmon: "Bob stood up and said, 'Look, I agree with your position, but this is not the place for it. If that's how you feel, goodbye.'"
Whenever the going got tough, the founders would retreat to the core of families that seemed to share their commitment and good sense. "It was a small group of us who knew exactly what we wanted and that's what we got," Haas remembered. The Quaker-meeting model, they decided, was too unwieldy, and what the school needed was a board of directors. According to Jessica's account, a long discussion over Gerard's future was capped off by her husband, who voiced what no one else would: "It looks as though maybe he will have to go." Gerard was asked to leave at the end of the school year; and according to Haas, the three original founders — Ryan, Sid Harmon, and Ross Cabeen — went to their attorney and drafted by-laws "setting them up as general members perpetually, with the ability to take over the school if necessary, and setting up a board of directors which... would run the school instead of the town meeting."
RYAN'S CONCERN about being typecast as a thug was well-founded: in the past two years alone, as Hughes finally coughed up some long-delayed projects at RKO, moviegoers had seen Ryan play nasty, sometimes violent characters in _The Racket; On Dangerous Ground; Clash by Night; Beware, My Lovely_ ; and _The Naked Spur_. He hoped the just-completed _Alaska Seas_ and _Her Twelve Men_ would help reverse this trend, and as the second picture of his deal with Paramount, he would play a quiet, bookish gentleman closer to his actual personality than any other character he had taken on. Adapted from a novel by Vina Delmar, _About Mrs. Leslie_ starred Shirley Booth as a lonely New Yorker and Ryan as a wealthy married man who falls in love with her and funds their annual rendezvous at a little cottage along the Pacific coast. A respected stage actress, Booth had just won an Oscar for her feature film debut in _Come Back, Little Sheba_ , costarring Burt Lancaster; now that picture's producer, Hal Wallis, and director, Daniel Mann, had sold her on _About Mrs. Leslie_ as a suitable follow-up.
Bringing Delmar's book to the screen required a fair amount of narrative convolution; the Production Code Administration, undaunted after all these years, wouldn't permit a story in which adultery was glamorized, so the screenwriters contrived to make the love relationship between George Leslie and Vivien Keeler completely chaste. When Vivien meets George on a Los Angeles airfield and drives with him to his beach home, she doesn't know he's married, and there's a conspicuous scene in which he shows her to a separate bedroom. They meet again the next year before Vivien, back in New York, learns from a newsreel that he is George Leslie Hendersall, an aviation giant vital to the Allied war effort, with a wife and two grown boys.
Shirley Booth and Ryan in _About Mrs. Leslie_ (1954). His role as a quiet, bookish gentleman was closer to his actual personality than any other character he played onscreen. _Franklin Jarlett Collection_
Ryan would remember Shirley Booth as "uncomfortable working in pictures." Off-screen she was even more meek than her character. "I picked her up in my car about a quarter of a mile from the studio on three consecutive days, and on the third day I finally asked her why she walked. She said she parked her car where she did because it was the only parking lot she could find — and she paid $3 a day to do it. So I informed her that, as the star of the picture, she had the right to park on Paramount's lot." Daniel Mann brought a warm, nicely melancholy tone to the picture, but _About Mrs. Leslie_ flopped at the box office and, like _Inferno_ , disappeared completely from circulation, an unjust fate for such a gentle, heartfelt drama.
That fall John Houseman, a little guilty over having abandoned Ryan during _Her Twelve Men_ , approached him with a head-turning offer: he wanted Ryan to star in an off-Broadway production of Shakespeare's _Coriolanus_. Ryan liked to tell people he'd perform in the men's room at Grand Central Station if he could do Shakespeare, and this would be considerably better: the 1,100-seat Phoenix Theatre on Second Avenue and Twelfth Street, whose new owners wanted to present challenging plays at reasonable ticket prices and had solicited Houseman to direct their second production.
Ryan and Houseman got together to discuss the project, and Houseman was frank with his old friend: this would be a real challenge for an actor who had never performed blank verse and lacked the vocal training of a Shakespearean actor. Ryan's voice, Houseman later wrote, "was pitched rather higher than it should have been in a man of his size, and his speech — though that of an educated man — had the ineradicable nasality of his Chicago origin." The New York critics might murder him. Ryan understood, but there was no way he could turn this down. Curled up inside him was the solitary teenager who had memorized _Hamlet_.
_nine_
Rum, Rebellion, and Ryan
Among Shakespeare's plays, _Coriolanus_ had the distinction of being both highly regarded and rarely staged. T. S. Eliot called it the Bard's greatest tragedy, yet the proud protagonist, Caius Martius Coriolanus, is a hard man for audiences to like. A military hero in Rome, he runs for consul of the senate, but his aristocratic contempt for the popular will gets him banished from the city, and he takes up arms against his own people. Houseman understood that the play tended to refract the politics of the day, and three years after President Truman had relieved General Douglas MacArthur of his command in Korea, civilian control of the military was still a provocative issue. "De Gaulle and Churchill (and to a lesser degree Eisenhower) had raised the question of the wartime hero as political leader," wrote Houseman. "Inevitably, as I edited and prepared the play for production, I found myself emphasizing its political aspects." Ryan, with his long-standing antipathy for generals, couldn't have been more sympathetic to Houseman's conception of the play.
_Coriolanus_ takes place in the fifth century BC, when Rome is a republic but not yet a democracy, and Shakespeare is notably ambivalent about the wisdom of popular rule. Coriolanus can barely restrain his frustration with the people: "He that trusts to you, / Where he should find you lions, finds you hares; / Where foxes, geese.... With every minute you do change a mind / And call him noble that was now your hate, / Him vile that was your garland." His controlling mother, Volumnia, persuades him to take up politics, and the senate welcomes him, but street protests organized by his political enemies send him into a rage. Coriolanus reminds the senators that commoners were the downfall of ancient Greece and argues that catering to them will only "break ope the locks o' the senate and bring in / The crows to peck the eagles." As Houseman noted, _Coriolanus_ had gained a new currency in the 1930s as Hitler and Mussolini took power. Yet its tragic hero was a timeless figure: the warrior with no place in a civil society.
Ryan had links to several of the cast members. Aufidius, the enemy commander, was played by John Emery, who had been married to Tallulah Bankhead when Ryan was appearing with her in _Clash by Night_. ("I never understood how an essentially gentle man like him could get mixed up with a dreadnought like Tallulah," Ryan later remarked.) Lauri March, cast as Coriolanus's wife, Virgilia, was the daughter of poet Joseph Moncure March, author of _The Set-Up_. (Her father disliked the movie version because it had turned his black fighter into a white one.) The players were first-rate: Mildred Natwick as Volumnia, Alan Napier as the wise patrician Menenius, and, providing comic relief as a trio of servants, the young actors Jack Klugman, Jerry Stiller, and Gene Saks (the first two would become TV stars, the third a successful director). Houseman also made the nervy decision to cast Will Geer as Sicinius, one of Coriolanus's rabble-rousing antagonists. Blacklisted in Hollywood, Geer had been reduced to working as a gardener in LA (for Sidney Harmon, among others), and Houseman would catch hell at MGM for having hired him.
With this largely American cast, Houseman decided the best vocal strategy for Ryan was to stick with his plain midwestern accent rather than strive for a classical delivery. "Shakespeare can be enjoyed — and understood — just as well if actors perform it in modern theatrical style," Ryan told a columnist. "Many performers have a tendency to over-act when they get their teeth into a Shakespearean passage — not only with their bodies, but with their voices. The result can quite often be unintelligible." Whatever his vocal limitations, Ryan was a commanding presence onstage. Houseman brought in a judo expert to choreograph Ryan and Emery in hand-to-hand combat, though these rough-and-tumble sequences proved too bruising on the stage's hardwood floor. Most important, the director understood Ryan's dark appeal as a performer, the "disturbing mixture of anger and tenderness," as Houseman phrased it, that had powered _On Dangerous Ground_.
The Ryans decided to move their children to New York for the six-week run, and the boys got to see two rehearsals and three performances of _Coriolanus_. They watched their father applying his makeup, and Cheyney took home one of his putty noses. "This was wonderful for me," Ryan said. "Movie making is hard to explain to children.... But the theater was different. They could see and feel and understand what I was doing." While he was working, Jessica filled the children's days with museums and other cultural activities. Having grown up in the San Fernando Valley, the children were a little overwhelmed by the city, where their father's fame was more of an issue. On one occasion Jessica took them ice skating in Central Park and their father chanced to join them; before long a crowd gathered, and the children were frightened by the crush of bodies. Eventually police had to wade into the crowd, form a protective cordon around them, and remove them to safety.
_Coriolanus_ opened on January 19, 1954, to positive reviews, with much credit going to Houseman for his incisive framing of the play's politics. Ryan's delivery was faulted by George Shea of the _Wall Street Journal_ ("His voice was not always under control, and it quickly became obvious that he is not yet sufficiently accustomed in the verse form") but excused by Brooks Atkinson of the _New York Times_ , who liked Ryan's conception of the Roman general as "an attractive, well-bred son of the upper classes who despises the people more out of intellectual sluggishness than malice.... This is a refreshing interpretation of the massive personality of Coriolanus. Mr. Ryan plays it with warmth, candor, grace, and a kind of artless sincerity." The play enjoyed a successful run of forty-eight performances before closing on February 28 to accommodate the next production in the series. Ryan loved the experience, and he and Houseman made vague plans to collaborate on another Shakespeare play. This would never happen, but as the family returned to Los Angeles, Ryan could take heart in knowing he had grown substantially as an actor.
Back in North Hollywood, the Oakwood School was unraveling again. After Bryson Gerard was let go, the parents launched a national search committee to find a new school director, but it was badly organized and failed to produce a suitable candidate. In desperation the founders hired Mary Bernick, who had studied school administration at City College and been recommended by one of the teachers at Oakwood. According to Jessica's account, Bernick became a polarizing figure, finding common cause with the more radical parents and communicating to the staff that "the reason the children were not learning any more than they had before was because it was impossible to teach the spoiled children of the rich."
Bernick was let go at the end of the school year after the volunteer book keeper, Marvin Brown, announced a $12,000 deficit. Frustrated and confused, the founders debated throwing in the towel and giving the school away. Ryan and Ross Cabeen appeared before the Country Schools' board of directors to offer them the school — land, buildings, and all — but the board wasn't interested. The couples discussed simply liquidating the corporation's assets and giving the money to the United Jewish Appeal. But Sid Harmon insisted on making one last stand and suggested that his wife, Liz, search for a new director while she was visiting her mother in New York City. Liz was still there in late June when Ryan flew out to promote _About Mrs. Leslie_ , and she urged him to meet with a woman who might be the ideal candidate: fifty-seven-year-old Marie Spottswood.
Unlike Bryson Gerard or Mary Bernick, Spottswood was a seasoned educator with a sophisticated understanding of progressive thought. Born in Mobile, Alabama, she had graduated from Randolph-Macon College in Virginia and studied at Tulane University and the University of Chicago before earning a graduate degree from Columbia University. Since 1929 she had been at Ethical Culture Fieldston School in New York, Liz's alma mater, though she had recently resigned as principal of the Lower School and was considering job prospects in Chicago, Vermont, and Massachusetts. According to Jessica's account, Ryan came to her office wearing gray gloves and a hat, the sort of patrician clothing George Leslie Hendersall might have worn. Spottswood, a white-haired woman with a fondness for cats, didn't know him from the movies, but Ryan could always present himself as an urbane, intelligent man, and did. Attracted by the idea of a warmer climate, she accepted an invitation to come visit the Ryans in North Hollywood.
Jessica liked her immediately. Spottswood had "prematurely white hair brushed back from a strong-featured face — a handsome woman and sometimes beautiful, even awe-inspiring, when the visionary gleam appeared in her eyes. One sensed the presence of a passion in her, put to the service of education and cats."
Ryan had promised Spottswood complete control over the educational policy, and she began outlining her plans for a curriculum based on social studies, with a particular focus on California's early Spanish and Indian cultures. In addition to this, heavy emphasis on the arts would engage the students creatively. But Spottswood also understood that reading instruction at Oakwood was seriously deficient, and she wanted to recruit her friend Mary Davidson, a phonetic reading specialist at Fieldston Lower, to join the staff. The Ryans held a buffet dinner for Spottswood and the parents, at which she was offered, and accepted, the job. That fall, when she took over at Oakwood, the parents began to realize that they had finally turned the corner.
DORE SCHARY LOVED BEING FIRST: no one had ever made a picture about anti-Semitic violence in America before _Crossfire_ , and now MGM would make the first picture about the attacks on Japanese-Americans during World War II. Howard Breslin's short story "Bad Time at Honda," about a southwestern town covering up the lynching of a Nisei (second-generation) farmer, had appeared in the January 1947 issue of _American_ magazine, but seven years later it was still controversial for the movies. Having bought the rights, Schary requisitioned an in-house report on violence against Japanese-Americans and read of numerous incidents (mainly arson and dynamite attacks, but also shootings) cited in _Life_ , the _Saturday Evening Post_ , and other publications. He assigned Millard Kaufman to adapt the Breslin story, and for the lead role he set his sights on Spencer Tracy.
Despite the desert setting, "Bad Time at Honda" was more of a mystery than a western. Honda, located between a bluff and a rail line, is so isolated that the Silverliner train screaming westward every morning has become "an event to Honda, a glimpse of the sleekness and wealth, the silver-chromium speed, that belong to other places." One morning, to the townspeople's shock, the train slows to a halt and disgorges Mr. George Macreedy of Chicago, who checks in to a local hotel but deflects questions about his business. He hires a young woman, Liz Brooks, to drive him out to a place in the desert called Adobe Wells, where he finds a small complex of burned-out buildings. Someone fires a warning shot at them, and when they return to town, everyone is watching Macreedy. In the hotel he's confronted by a trio of local ranchers, the leader of whom, Coogan Trimble, alludes to the lynching but warns Macreedy that he'll never prove anything. "Other places... settled it other ways," Trimble remarks. "Camps. Things like that. We only had the one. We ran him out. Burned him out. That's all."
Macreedy knows he's licked, and as he waits for his train out of town, everyone in Honda celebrates at the local watering hole. But before he leaves, Macreedy barges into the tavern. Someone silences the jukebox, and they all stare. " 'Now, listen,' said Macreedy. 'All of you know I came here to find Old Man Kamotka. You know what happened to him. So do I — now.' He could hear the breathing in the room, and he went on: 'This is why I came. There was a kid named Jimmy Kamotka. He left here years ago. He never wrote his father. The old man couldn't read. I met Jimmy in the Army. In Italy. He asked me to look in here.... Jimmy Kamotka was killed in Italy. I think maybe this town should know that. And remember it.'"
Kaufman made a number of changes to the story, the most significant of which was turning the mob violence into a smaller, more focused attack. In "Bad Time at Honda" Kamotka dies of a heart attack while fleeing the crowd; in the movie, retitled _Bad Day at Black Rock_ , Komoko has been shot to death by the town's fearsome boss man, Reno Smith (which sounded more menacing than "Coogan Trimble"). The conflict between Smith and Macreedy generated more of a showdown, and Kaufman's elongated story line borrowed significantly from _High Noon_ , with Macreedy boxed in by the uncooperative townspeople. When Tracy read the script, he complained that Macreedy wasn't dimensional enough, so Schary, salvaging a story idea from an abandoned project, suggested that Macreedy have a paralyzed arm, which satisfied Tracy. He would play the part in a baggy suit with his left sleeve tucked into its pocket.
_Bad Day at Black Rock_ quickly became a sticking point between Schary and Nicholas Schenck, the seventy-two-year-old president of Loews, Inc. (MGM's parent company). Early in the development, when Schary was at home recovering from a kidney stone, Schenck dropped in on him to insist he cancel the picture. A shouting match ensued, Schary prevailed, and when none of his producers expressed interest in the script, he took on the project himself. A month before production commenced, an interoffice memo notified him that a State Department official had informally complained to the studio about the damage the picture might do to the United States overseas. "Historically there never was a lynching of this kind," MGM's Kenneth MacKenna wrote to Schary, "and the whole issue of Asiatic-American relationships is such a touchy matter that such an un-historical incident might be used by our enemies to stir up further misunderstandings." Schary had been through all this before with _Crossfire_ , which turned out to be a huge success, so he simply plowed ahead.
Given the similarity between the two pictures, Ryan was a natural choice to play the bigoted Reno Smith, and whatever reservations he may have har bored about the role, which was rather thinly written, they paled before the opportunity to make an A picture with Tracy. Ryan trusted Schary, who demonstrated once again his skill at assembling a superb cast. Ernest Borgnine, the brutal sergeant in _From Here to Eternity_ , would play Reno Smith's hot-tempered strong man, Coley, and lean, mean Lee Marvin, who had given bare-knuckle performances in _The Big Heat_ and _The Wild One_ , was cast as the coolly bullying Hector. Walter Brennan, a three-time Oscar winner, signed on as Doc Velie, the local undertaker who becomes Macreedy's sole ally and the town's conscience. The story was a welcome corrective to _Behind the Rising Sun_ , the anti-Japanese propaganda picture that had become Ryan's calling card in the Marines.
Production began on Monday, July 19, in Lone Pine, California, about two hundred miles north of LA in the Owens Valley. "The temperature was about a hundred degrees," recalled Anne Francis, who had been cast as Liz. "And in those days, they used klieg lights to offset the sun. So, _with_ those lights, we were working in 115, 120 degrees. We all lost a tremendous amount of weight; I mean, at the end of the day, who was hungry? Spencer Tracy had a _very_ hard time. They had to coax him more than once to please, _please_ see it through, because it was terribly draining for him." In fact, Tracy had tried to drop out of the picture the week before shooting began, but Schary reminded him that his pay-or-play clause would obligate him to reimburse MGM for nearly half a million dollars. Tracy caved, and to mollify the star, Schary promised to visit the set and suffer in the heat with the rest of them, which he did on one occasion.
"When I was starting out in Hollywood, I would spend any day off I had or any free time on the set watching Spencer Tracy, who was one of the great masters," Ryan recalled. In getting to know the actor, he was gratified to learn that Tracy had been the same way. "When he was a young man in New York, he would wait outside a certain theater at a certain time just to see Lionel Barrymore leave. He couldn't afford to see him act on the stage but at least he could watch and see him walk out of the theater. I think this is terribly important." Tracy, a political conservative, seemed unsure of what to make of Ryan. Anne Francis recalled that after she and Ryan borrowed Tracy's car to get dinner, "I got the silent treatment because he felt Bob and I were having an affair, which we weren't." After shooting one of Macreedy's confrontations with Reno Smith, Tracy asked Millard Kaufman, "Does Ryan scare you?" Kaufman replied, "No, I've known Bob Ryan for years. He's a fine man." Tracy replied, "Well, he scares the hell out of me.'"
Ryan got a chance to hold his own against Tracy at the picture's midpoint, when Macreedy, trapped in Black Rock until the next day's train, sits outside a gas station and Smith fills up his wagon. Shooting in CinemaScope, director John Sturges placed Tracy at the left of the frame in his dark suit and hat and Ryan at right, flanked by a pair of ten-foot gas pumps painted bright red, the vast blue sky behind him. "Somebody's always looking for something in this part of the west," Smith tells Macreedy. "For the historian it's the Old West, to the book writer it's the Wild West, to the businessman it's the undeveloped west.... But to us, this place is _our_ west, and I wish they'd leave us alone!" The scene ran more than five minutes and was filled with dialogue; as Tracy sat immobile, Ryan kneeled down to face him and paced back and forth. Watching them work, Borgnine thought at first that Ryan was stealing the scene but then realized his gaze had been fixed on Tracy the whole time. When they finished the scene and Sturges asked for a second take, Tracy refused.
Movie publicists routinely fabricated stories about actors doing their own stunts, but in fact Ryan liked the physical part of his job and would take a crack at something if it didn't look too dangerous. On _Bad Day at Black Rock_ , though, he gambled and lost. The picture climaxes with a night scene in which Macreedy, pinned down by Smith's rifle fire, improvises a Molotov cocktail and sets the rancher ablaze. A long shot of a stunt man wearing an asbestos suit resulted in the man scorching his lungs and being taken to the hospital. Ryan completed the sequence with a less risky shot in which Smith, now partially extinguished, flops to the ground with flames still clinging to his left side. "There was a hole in his asbestos suit, so he actually got burned doing that," said Cheyney Ryan. "That was one of the few times that anything like that had ever happened."
Years later twelve-year-old Lisa Ryan phoned her father during a sleepover at a friend's house to tell him that _Bad Day at Black Rock_ was playing on the late show. Ryan always disliked the idea of his children seeing him injured or killed on-screen, and he forbade Lisa to watch the movie. "So of course I had to watch it," Lisa recalled. "And he gets set on fire at the end.... I was so traumatized by that scene that I called him, and I said, 'Are you all right?' He said, 'I _told_ you not to watch that movie!'"
TALKING TO COLUMNIST ERSKINE JOHNSON of the _Los Angeles Daily News_ , Ryan disparaged fan-magazine stories about life with the Ryans: "They come rushing over for home layouts and photograph me with my children. And they've got their stories already written before they even see me." But an "as-told-to" piece he published in _Parents_ magazine in September 1954 turned out to be unusually reflective and revealing. With all his reading on early childhood development, Ryan had a lot on his mind, not least his own three children: "Tim, the older, is extremely sensitive. I was so much like Tim as a boy that — and I say it in all humility — I truly believe that I have a deep understanding of him through my own past. I too was a completely nonaggressive youngster.... Cheyney gives evidence of being of a totally different mold. He has, we think, just the right amount of aggressiveness. Not so much that he needs his ears pinned back by either parents or playmates, but enough to stand solidly on his rights.... As for Lisa — well, she's not three yet and we haven't taken a shot at cataloguing her.... She fits into family life in her own way and altogether is very much the fair-haired girl any father of two sons dreams about."
Ryan took advantage of the piece to plug the Oakwood School, crediting Jessica for her initiative and reporting that the present enrollment had hit sixty-five. The month the story appeared, Marie Spottswood began her first semester as director of the school, and the change was dramatic. To address the reading crisis, she divided all the students into "readiness groups" regardless of grade, and Mary Davidson, the reading specialist newly arrived from Fieldston Lower, began working with the groups individually, using the phonics-based approach advocated by educator and psychologist Anna Gillingham in her book _Remedial Training for Children with Specific Disability in Reading, Spelling and Penmanship_. By the end of the school year, wrote Jessica, every student was reading at his or her grade level.
A new board was voted in, with Jessica as president and Marvin Brown, who had uncovered the gaping budget deficit, as treasurer. He set about collecting tuition from deadbeat parents, and the board announced that, for the survival of the school, tuition would have to be hiked from fifty dollars a month to seven hundred annually. Even after that the school would continue to run an annual deficit of about five thousand dollars, which the Ryans and others would cover. That included a scholarship program for minority students, something the founding parents considered crucial. With Spottswood in place, the Oakwood School seemed like a much worthier cause, and the Ryans took pride in what they had managed to accomplish. Ultimately, they wanted to expand the school to include the seventh and eighth grades.
That fall Ryan reteamed with Barbara Stanwyck for _Escape to Burma_ , a ridiculous jungle adventure that RKO was distributing for the independent Filmcrest Productions. His character, an American businessman, goes on the run after being framed for the murder of his partner, the Burmese prince, and hides out with Stanwyck's character, who owns a teak plantation. At one point they welcome some locals into her courtyard with a trained baby elephant that does tricks and performs a little dance routine, kicking its back legs in the air. The picture fulfilled Ryan's one-a-year obligation to Howard Hughes, who had recently cemented his control over the studio by purchasing all the outstanding stock of Radio-Keith-Orpheum for $23 million. But after an experience like _Bad Day at Black Rock_ , the new picture was a pitiful reminder of how erratic Ryan's career had grown. It was enough to drive a man to drink.
And it did. Alcohol had been an integral part of Ryan's life since his Dartmouth days, when he ran for class marshal on the slogan "Rum, Rebellion, and Ryan." He and Jessica bonded over drinks, and cocktails were central to their little social scene. In her memoir about Oakwood, Jessica remembers going out with Ryan and Sid Harmon to discuss some vexing problem and winding up at a family place called Fatso's Mile High Ice Cream Cones, where Ryan tried to order scotch and soda, and then beer, to no avail. Harmon asked for an ice cream soda; her husband "ordered two coffees and sat back in a state of shock."
He was a big man who could hold his liquor, and in the booze-soaked '50s that was good enough. "He told me that when they were doing movies out in the desert, he was drinking about two fifths of whiskey a day, just to get through the day," said actor James Naughton, who worked with Ryan years later. "That didn't count wine with dinner, or beer and so on. Because making movies out there in the desert was about the most boring thing anybody could do."
A drink made him more genial and charming, but he could also slip into angry silences that pulled him away from the family and into his little refuge out back (now that he had reclaimed it from the fourth and fifth grades). Everyone knew not to bother Daddy when he was in one of his Black Irish moods. "He had difficulty speaking about personal things," his friend Millard Lampell explained. "It just wasn't in his past, it wasn't in his tradition." Later in life Ryan would confess that he had always been haunted by the death of his brother, Jack, so many years earlier. There was unfinished business with his father, another man of dramatic mood swings. There was the nagging sense that the success he had chased all his life was hollow. And there was the world going to hell all around him.
"When my father got depressed, it was usually about Richard Nixon, it wasn't about his life," observed Cheyney Ryan. "It would be more about what was happening in the world." Ryan was increasingly distressed by the threat of nuclear weapons. Earlier that year the US military had detonated a hydrogen bomb on the Bikini Atoll in the Marshall Islands; the fifteen-megaton blast, the most powerful in human history, was supposed to be a secret, but news of it had spread around the world after the wind carried radioactive fallout to neighboring islands, which had to be evacuated, and infected the crew of a Japanese fishing boat, killing one man. The fallout had spread as far as Southern California. If that didn't kill Ryan, he'd damn well do it himself. "Cancer by the Carton," a story in the December 1952 _Reader's Digest_ , had revealed that cigarettes almost certainly caused lung cancer: "Above the age of 45 the risk of developing the disease increases in simple proportion with the amount smoked." Ryan smoked two packs a day.
_Bad Day at Black Rock_ opened the first week of January 1955 to stellar reviews. "This is one of the finest pictures ever made," raved John O'Hara in _Collier's;_ 28 a more measured assessment from Robert Hatch in the _Nation_ lauded it as "a tight, economical work, directed and acted with conviction" that "enlarges the stature of everyone in connection with it." The State Department's concerns notwithstanding, _Bad Day at Black Rock_ was released overseas in March and collected even more superlatives from critics. That year it represented the United States in the Cannes Film Festival (along with Elia Kazan's _East of Eden_ ), and the following year brought Oscar nominations for Spencer Tracy, John Sturges, and Millard Kaufman. Ticket sales were less impressive: after a year and a half in release the picture had posted a profit of only $600,000.
Shortly after Ryan hit movie screens as the murderer of a Nisei farmer, he flew to Tokyo to shoot a big-budget crime thriller for Twentieth Century Fox. _House of Bamboo_ was an A remake of an old B movie, _The Street with No Name_ (1948), in which an FBI agent tracks an organized crime operation; Darryl Zanuck, eager to make the first Hollywood feature shot in Japan, had screenwriter Harry Klein write a new version set in Tokyo, where an army investigator going under the name Eddie Spanier (Robert Stack) infiltrates a gang of American ex-G.I.s led by the dapper, calculating Sandy Dawson (Ryan).
The cigar-chomping director, Samuel Fuller, had worked as a crime reporter and served as an infantryman in the "Big Red One," experiences that informed his gripping Korean war drama, _The Steel Helmet_ (1951), and his nineteenth-century period picture about newspapermen, _Park Row_ (1952). Having scored at Fox with the itchy crime picture _Pickup on South Street_ (1953), he was looking to move up, and _House of Bamboo_ was a classy project in color and CinemaScope, with exotic locations and a big star. "Robert became a true friend," Fuller would write. "He was well read and balanced, a kindhearted man with grand democratic ideals."
Fuller had a few twists of his own to add to _House of Bamboo_. "I moved the entire shebang to Tokyo," he would later write, ignoring Klein's contribution, "added stuff about Japanese contemporary life, threw in some sexual exploitation and interracial romance, and then, for some unexpected pizzazz, wrote a violent love scene between two hardened criminals.... Zanuck loved it, even the homoerotic scene with the two gangsters." As conceived by Fuller, _House of Bamboo_ included a male love triangle: Sandy Dawson is intimate with his second-in-command, Griff (Cameron Mitchell), but then falls for the new man, Spanier, and elevates him to Griff's role. To slip this romantic subtext past the Production Code Administration would require skillful underplaying, and Ryan embraced the challenge. "I like Sam, he's crazy!" he told an interviewer in the early '70s. "We were very much in agreement during the filming of _House of Bamboo_."
Sometimes Fuller seemed like a character in one of his own movies. "He liked to keep a gun strapped to his hip while we were shooting in Japan — he looked like General Patton," wrote Stack. "And instead of saying 'Action!' to start a scene, he would take a .45 out of the holster and shoot it in the air — _Boom!_ And people all around would run and scurry and hide." As Stack remembered it, Fuller didn't seem to comprehend how the conquered Japanese would feel about an American firing a pistol for emphasis. Still, people stayed out of his way.
Soon after they arrived in Tokyo, Fuller needed to shoot a simple scene in which Ryan would drive up to a curb and get out, but when the company arrived at the location, they were greeted by anti-American protesters. The director told his cinematographer, Joe MacDonald, to film the people who had gathered, and as soon as the camera was turned on them, the crowd dispersed.
Ryan and director Samuel Fuller agreed privately that Sandy Dawson (Ryan), the menacing crime lord in _House of Bamboo_ (1955), would be a closeted man lusting for undercover cop Eddie Kenner (Robert Stack). _Franklin Jarlett Collection_
Fuller already had shot one picture in CinemaScope (the submarine adventure _Hell and High Water_ ), and he took advantage of its extremely long aspect ratio — 1 to 2.55 — to suggest on-screen what he couldn't get past the censors in typescript. For the interior scenes with Dawson's gang, he created a series of elegant tableaux showing male bodies in repose, with Ryan bisecting the group and often elevated in the frame to stress his tall, lanky physique. Dawson doesn't appear until nearly twenty minutes into the picture, but when he does, Fuller employs a startling shift in perspective. Griff finds Spanier trying to shake down a club owner and punches him out; Spanier crashes through a paper screen into a private room, and the tear left by his body frames Dawson, cool and lean as he sits atop a table, surrounded by his boys. From _Caught_ , Ryan recycled the business in which his character banks a cue ball around a billiard table as he holds forth, and as in the earlier picture, the ball's movement traces a triangle.
What Fuller described as a "violent love scene between two hardened criminals" was not really a love scene but an execution: Dawson decides that Griff has ratted him out to the cops and arrives at Griff's home to find him bathing in a wooden tub. The gang leader steps into the room and pumps eight shots into his underling, the bathwater squirting out of the tub through the entrance holes. "You didn't know what you were doing," Dawson tells his dead lover, cradling his head tenderly. "I could see you had no control of yourself, absolutely none. And I knew, Griff. I knew when you started blowing your buttons for no reason whatsoever. Griff, I wish I hadn't been right. But I was, Griff. Like always." Fuller loved this psychotic moment: "Sandy is gentle for the first time, almost sensual," Fuller later wrote. "Except the object of his affection is his dead victim, showing just how insane the sonofabitch has really become!"
While shooting in the streets of Tokyo, Fuller found the perfect location for Dawson's violent end: a twenty-story building anchored by a department store that operated a children's nursery on the roof. Carnival rides had been erected on one side, including a little carousel in the shape of Saturn; a gold half-sphere spun clockwise as a ring of seats turned counterclockwise. In the climactic sequence the bad guy is cornered high above ground, like Bogart in _High Sierra_ (1942) or Cagney in _White Heat_ (1949). Stack would recall his unease as he and Ryan filmed their showdown on the contraption, high above the street; the carousel was rusty and the gigantic CinemaScope camera unbalanced it. But the results were spectacular, a yawning vista of Tokyo behind them as Sandy Dawson is shot to death. A subsequent reverse angle shows Spanier stepping off the carousel and Dawson's body draped over the edge of the ring as it continues its slow rotation.
RYAN'S STEADY WORK SCHEDULE always prevented him and Jessica from vacationing at length; they might pack up the kids for a quick trip to La Jolla, Santa Barbara, or the Ojai Valley, and the previous year they had all gone to New York while Ryan was performing _Coriolanus_. But in April, Jessica wrote to Dido and Jean Renoir to announce, "We are coming to France! It seems quite astonishing that we actually have definite plans to make such a trip but it is true — and all for vacation and no work." She and Robert, along with the children and Williana Smith, would set off from New York in early July on the steamship _Independence_ , arrive in Cannes eight days later, and spend the next six weeks "vagabonding" around France and Italy by auto before departing from Naples on September 4. Robert was about to leave for Durango, Mexico, to costar with Clark Gable in a big-budget western for Fox called _The Tall Men_ , but after that he was free and clear.
Raoul Walsh, director of _The Tall Men_ , was another grand old man of the movie business, his career stretching back to the Silent Era and encompassing such hard-charging talkies as _The Roaring Twenties, High Sierra, They Died with Their Boots On_ , and _White Heat_. He had made more than a hundred features, and _The Tall Men_ was just another of them, with a weak script about antagonistic ranchers that was mainly an excuse to stage an epic cattle drive in CinemaScope. Jane Russell is Gable's love interest, and their dialogue scenes are excruciating. Walsh was still working on the script as the cameras began rolling in mid-April, the first order of business being the big, expensive cattle-drive sequences that had brought them to Mexico (no US location offered a sufficient number of cattle).
First Tokyo, now dusty Durango. Hoping to pull people away from their TV sets, the big studios turned increasingly to dramatic location photography, which meant Ryan was spending more and more time away from the family. In Durango he shared a bungalow with costar Cameron Mitchell, whom he had known since the days of the Millpond Playhouse on Long Island; Russell was lodged in a nearby hacienda, and Gable and Walsh shared another. "I went through the location in a haze," Ryan later said. "I've hardly any recollection of Durango — Gable and Jane and I would sit around getting swacked! I can't even remember where I lived!"
A generation after _It Happened One Night_ (1934), Gable was a craggy fifty-four and humbly resigned to playing himself in picture after picture. Ryan found him to be an uncommonly gracious and decent guy; one time, when Ryan ran out of his brand of cigarettes, Gable got up early and drove into town to fetch him a few cartons. Walsh was great company too, and Ryan pumped him for stories about his hero Douglas Fairbanks, whom Walsh had directed in _The Thief of Bagdad_ (1924).
These pleasant moments notwithstanding, _The Tall Men_ was a tough shoot. According to Walsh, four hundred people in the cast and crew contracted dysentery. Then, after years of heavy drinking, Ryan came down with alcoholic hepatitis. Shipped back to Los Angeles for treatment, he learned he was also suffering from cirrhosis of the liver. The trip to France, which Jessica had so looked forward to, was canceled. "He didn't go to the hospital, but he was in bed for quite a long time because of that," Cheyney Ryan recalled. "I think [he] cut back on alcohol because he realized he was going to die." Ryan's doctors told him he couldn't touch any booze for a year. "I damn near died when I heard that," he told a reporter years later. "But I got through the first two weeks, and I never had the same urge again." Eventually he would fall off the wagon, holding himself to a couple Löwenbräus every night. But in the immediate aftermath of his illness, at least, Ryan got a chance to look at the world sober for the first time in decades.
_ten_
The Gates of War
Recovering from his illness, Ryan had plenty of time to take stock of his career. Since parting ways with RKO, he had landed some nice deals, first with Paramount and then with Fox, that had given him starring roles in A pictures. Dore Schary came to him with good supporting roles at MGM. He had shared the screen with James Stewart, Spencer Tracy, and Clark Gable. Around town he was known as a consummate professional, quietly cooperative and reliably inspired. Yet somehow he always got shut out of the big parts. Early in 1954 he had eaten his guts out as Gregory Peck landed the role of Ahab in John Huston's lavish adaptation of _Moby Dick_ ; Ryan still read the novel once a year, and watching the part slip away was like seeing the great white whale disappear into the blue waters.
His reclusiveness didn't help. "I don't want any identification between my personal life and my acting life," he once said. "An actor's private life should be very private. The public should see nothing but what they see on the screen." Jessica had grown to loathe photo sessions. "Asking her to pick up a coffee pot and have her pour coffee while I grin over her shoulder for a photographer is murder." Their home out in the Valley had pulled him away from the movie business, and getting so involved in the Oakwood School had sapped all his free time. All around him stars were making their way outside the studio system, cutting profit-participation deals or even forming their own production companies. Ryan didn't want to produce or direct — that would mean even more time away from home — but there had to be a way of finding projects that were more engaging and ambitious.
An intriguing possibility emerged that fall when his friend Sidney Harmon landed a three-picture financing and distribution deal with United Artists for his independent production company, Security Pictures, Inc. The first picture, _Variety_ reported, would be _Men in War_ , set during the Korean conflict and based on Van Van Praag's 1949 novel _Combat_. Soon after came news that Harmon and his partner, screenwriter Philip Yordan, would produce a screen adaptation of Erskine Caldwell's best-selling, notoriously salacious novel _God's Little Acre_ , with Anthony Mann directing. Ryan trusted Harmon from their many years in the trenches at Oakwood, and through him had gotten to know Yordan, a Chicago native and a voracious reader. "We used to socialize quite a bit," remembered Yordan, "and by socialize I mean just come over [to] the house and sit down, and smoke and drink beer." Quite naturally the three men had kicked around the idea of working together, and now they had financing.
Yordan would earn a reputation as a classic Hollywood hustler.* Originally an attorney and entrepreneur, he had come to Hollywood in the late '30s and written such hard-boiled pictures as _Dillinger, House of Strangers_ , and _Detective Story_. But he also served as a front for blacklisted writers, passing their work off as his own and giving them a percentage. Harmon had sent several people his way. "Sidney, in a sense, was my conscience," recalled Yordan, who was so apolitical he claimed he never read a newspaper until he was fifty. "He says, 'Phil, you've got so much money, you've got a play on Broadway, you're making $5,000 a week, and these guys are starving to death. Give them some work, for chrissakes.'" Ben Maddow, who hadn't written under his own name in Hollywood since 1952, remembered Yordan fronting for him on _Men in War_ and several other scripts; Yordan offered him a 50 percent cut, though as Maddow noted, "I was never sure of what percentage it actually was."
Maddow watched in amazement one afternoon as Yordan tried to parlay a western script Maddow had written into a novel and a picture deal. As Maddow related to writer Patrick McGilligan, Yordan phoned Simon and Schuster and asked if they were interested in publishing a novel that Warner Bros. was adapting to the screen. Then he called Warners and asked if they were interested in adapting a novel that Simon and Schuster was publishing. To clinch the deal, Yordan contacted a "minor executive" at Warners and offered to retire the man's $14,000 gambling debt if he would pimp the script to studio head Jack Warner, claiming he had picked it up by accident and been bowled over when he read it. Maddow was then dispatched to write the nonexistent novel, though the picture was never made; when he later saw the book in print in Europe, he said, Yordan had attached his own byline.
In December 1955, Ryan returned to work, playing Abraham Lincoln in a half-hour TV play called _Lincoln's Doctor's Dog_.* Lincoln spends most of this little fable laid up in bed, the human cost of the war weighing heavily on his mind. His friend and physician, Robert K. Stone (Charles Bickford of _The Woman on the Beach_ ) examines him, taking his pulse from his ankle as his feet stick out from the covers, and insists that he get at least twenty-four hours of rest. The extended scene between them is beautifully written and gently played; at one point Lincoln pages through _Henry IV_ , part 2, and takes a solemn pass at one of Prince Hal's soliloquies: "O polish'd perturbation! golden care! / That keep'st the ports of slumber open wide / To many a watchful night!" Unfortunately, this gives way to a sappy resolution in which the doctor cheers Abe by getting him a puppy.
The new year brought two more mediocre pictures whose sole consolation was top billing. _The Proud Ones_ , another CinemaScope western for Fox, costarred Virginia Mayo and Jeffrey Hunter, with location work in Nogales, Arizona. _Back from Eternity_ , a jungle survival drama, was being shot — thank God — at RKO Pathe Studios, with a $300,000 tropical set and sixty exotic birds. Ryan plays an alcoholic airline pilot who tries to save his passengers after their plane goes down somewhere in South America; among them are Swedish bombshell Anita Ekberg and the tightly wound Rod Steiger. By this time Howard Hughes had severed his ties with RKO Radio Pictures, selling his share to General Tire and Rubber Company for a $6.5 million profit. His bizarre stewardship of RKO would be remembered as the most baroque episode in the era of the great movie moguls, an era that was now, clearly and irrevocably, coming to a close. When Harmon and Yordan approached Ryan with a profit-participation deal to star in their first picture, _Men in War_ , he grabbed it.
RYAN HADN'T MADE A PICTURE about ground combat since he and Pat O'Brien starred in the trite _Marine Raiders_ back in 1943. _Men in War_ took a much darker view of its subject: the story of an army patrol stranded behind enemy lines in the early days of the Korean conflict, it had no romantic subplot, no flag-waving speeches, no moral compass. Since the end of World War II, Hollywood war movies had grown more adult, ambivalent, and even philosophical, but _Men in War_ was downright existential, focused like a telescope on the wild terrain and the men's desperate, improvised tactics against an encroaching but unseen enemy. "The battalion doesn't exist," the despairing Lieutenant Benson (Ryan) tells his loyal radioman, Sergeant Riordan (Phillip Pine). "The regiment doesn't exist. Command headquarters doesn't exist, the USA doesn't exist. They don't exist, Riordan. We'll never see 'em again." The picture ended with a bitter scene in which Benson, Riordan, and Sergeant Montana (Aldo Ray) toss a dead officer's silver stars into the brush as they read off the names of all the men they've lost.
Sergeant Riordan (Phillip Pine) and Lieutenant Benson (Ryan) in _Men in War_ (1957). "The battalion doesn't exist," Benson tells Riordan. "The regiment doesn't exist.
Command headquarters doesn't exist, the USA doesn't exist.... We'll never see 'em again." _Wisconsin Center for Film and Theater Research_
The entire story transpired out in the field, which made Tony Mann an inspired choice to direct: all he needed to create drama was good actors and a good location. Yordan had known Mann for years, and Ryan was eager to work with him again after _The Naked Spur_. According to Yordan, Mann read the 150-page script and cut it down to 82 pages, stripping out nearly all the dialogue. "Of course, I put all of the dialogue back in to get Aldo Ray and Robert Ryan to play it," remembered Yordan. "I said to [Mann], 'What am I going to do if you send them this script? They won't show up!'"
Fortunately, they did show up, to a hilly, overgrown area of Thousand Oaks, California, thirty-five miles northwest of Los Angeles in the Conejo Valley, where shooting began in July 1956. Harmon had rounded up a sharp supporting cast, including Vic Morrow, L. Q. Jones, Nehemiah Persoff, and James Edwards (who had played the black fighter Luther Hawkins in _The Set-Up_ ). Aside from the players, Mann had little at his disposal but a few old jeeps and some explosive charges, but that didn't matter — his muses were the open sky, the billowing smoke, and the tall, bleached grass that seemed to stretch in every direction, enveloping the men and concealing the enemy.
This bare-bones approach lay at the opposite end of the spectrum from something such as _Flying Leathernecks_ , with its fleet of fighter planes supplied by the Marine Corps, but shooting on the cheap bought the filmmakers some artistic freedom. _Flying Leathernecks_ features a corny scene in which John Wayne struggles to write a condolence letter to the parents of a fallen marine under his command; its corollary in _Men in War_ is a grim, wordless shot in which Lieutenant Benson cuts a dog tag off a dead G.I. and adds it to a thick stack he keeps on a key ring. The entire division has retreated under a tank assault from the North Koreans, who are advancing to Pusan, and Benson is responsible for leading his men to safety. He's a good commander, cool-headed and resolute, but his mission is complicated when the patrol collides with Sergeant Montana (Ray), the chauffeur and self-appointed caretaker of a mute, shell-shocked colonel (Robert Keith). From Ryan's perspective, nothing could have symbolized the military command structure better than a catatonic officer strapped into a jeep and staring blankly into space.
"Tell me the story of the foot soldier and I will tell you the story of all wars," declares an opening title, signaling not only the picture's antiwar leanings but also Mann's minimalist approach. The men here are vividly human — scared, confused, fiercely devoted to one another — but there are no reminiscences about girlfriends back home or dreamy soliloquies outlining plans for the future. Forced to fill in the blanks, Ryan drew on his own life, adding a rare personal touch that would become the picture's sole reminder of an outside world: when the lieutenant takes off his helmet, nestled inside is a snapshot of Jessica, Tim, Cheyney, and Lisa. Ryan didn't often lose track of himself in a character, but during _Men in War_ he began to feel he _was_ Benson. "With each day I felt dirtier, grimier, lousier and more forsaken," he told one reporter. "It became more and more difficult to sleep in my clean bed at night. I found it finally, impossible. I had to sleep on the floor."
Shooting close to home was a blessing; Ryan could commute to the location, and there were frequent visits from Jessica, the kids, and the Harmon family. Tim and Andy Harmon, good friends at Oakwood, latched onto a copy of the screenplay and wrote their own digested version of _Men in War_ , which they rehearsed with Cheyney and some of their classmates and performed for their fathers. Back then the empty lot just west of Oakwood on Moorpark Street was a sandy wash with overgrown bushes and dried-out drainage canals that were perfect for playing war. "Tim and Andy were always writing a movie," recalled Toya Harrison, daughter of Chuck Haas. "Tim was always the star, and the girls in the class, me included, we were always the nurses.... Tim and Andy were fantastic at persuading us all that these movies were gonna be made and we had to rehearse them.... I remember Tim as being a very gentle soul, really quiet-spoken. A lot like Bob, really."
That summer the Democrats nominated Adlai Stevenson to run again for president. He faced an even tougher race than he had in 1952: the Korean conflict had ended, the economy was booming, and polls showed widespread support for President Eisenhower. Stevenson staked out courageous positions supporting a nuclear test ban, an issue dear to Ryan's heart, and proposing an end to the military draft. In the final weeks of the race Ryan flew out to Washington to appear at a $100-a-plate Stevenson dinner, which was being carried by closed-circuit TV to thirty-three other locations around the country. Henry Fonda, Frank Sinatra, Bette Davis, Marlon Brando, Harry Belafonte, Geraldine Fitzgerald, and Lauren Bacall were among his costars; sorely missed was Humphrey Bogart, who was battling cancer and had only months to live. Ryan got to introduce President Truman, who lambasted Eisenhower and Nixon and promised, "The day of liberation for the city of Washington is close at hand." Stevenson carried only seven states on election day, but he would always be a hero to the Ryans. "Till the day they died they had a picture of Adlai Stevenson over the mantelpiece," Cheyney recalled.
Robert and Tim Ryan during location shooting for _Men in War_ (1957). That summer Tim and his brother, Cheyney, got their hands on a script and staged scenes with their playmates on the grounds of the Oakwood School. _Robert Ryan Family_
After _Men in War_ wrapped, Ryan didn't work for the rest of the year. By this time Universal, Paramount, and Fox all had tried him out as a leading man, with decent but unspectacular financial returns, and Nicholas Schenck had finally canned Dore Schary as president of MGM, leaving Ryan without a connection at that studio. He was forty-six and looked it, in a business whose hottest new star was twenty-one-year-old Elvis Presley. Hollywood legends such as Clark Gable, Gary Cooper, and Cary Grant were still playing romantic leads well into their 50s, but Ryan was hardly in that league; the more lined his face, the more menacing he looked. The Dartmouth alumni magazine reported in December that he was trying for the lead in a Jack Dempsey biopic, though nothing came of this. He could have gotten more television gigs, but that was a step down from pictures.
Near the end of the year Ryan got an irresistible offer when theatrical producer-director Harold J. Kennedy asked him to appear in a local production of Jean Giraudoux's antiwar satire _Tiger at the Gates_. First staged in Paris in 1935, as the Nazi threat gathered in Europe, _Tiger at the Gates_ was a modern-language take on the _Iliad_ , though in contrast to Homer's epic poem, set near the end of the ten-year Trojan War, the French play takes place on the eve of that conflict. Hector, the Trojan commander whom Kennedy wanted Ryan to play, has just returned from battle and craves peace; reunited with his loving wife, Andromache, he plans to march into the city's courtyard and close the Gates of War forever. But in his absence another conflict has been brewing: his brother Paris has kidnapped Helen, the beautiful queen of Sparta. "Those poor gates," observes Andromache's caustic friend Cassandra. "They need more oil to shut them than to open them."
Ryan loved the play, particularly the elegant translation by Christopher Fry. Hector was like a well-spoken version of the weary Lieutenant Benson: confiding in Andromache, the Trojan warrior remembers the precise moment when his sense of glory deserted him, just as he was preparing to slay an opponent. "Up to that time, a man I was going to kill had always seemed my direct opposite," he says. "This time I was kneeling on a mirror, the death I was going to give was a kind of suicide." He and Andromache plead with the people of Troy to surrender Helen, but their fellow citizens, unacquainted with battle, all have their own agendas. In protest Hector refuses to make the expected oration for the men slain in the last conflict. "An Oration for the Dead of a war is a hypocritical speech in defense of the living, a plea for acquittal," he declares. "I am not so sure of my innocence."
Witty and flamboyant, Kennedy had won a Tony in the mid-1940s performing his own play _A Goose for the Gander_ on Broadway before he branched out as a director on the West Coast. Ryan first met him in 1951, when Kennedy invited him to Lucey's Restaurant, across the street from the RKO lot, and asked him to play the brutal cop in an LA production of Sidney Kingsley's _Detective Story_. Having just played a similar character in _On Dangerous Ground_ , Ryan politely declined, telling Kennedy, "If I'm going to work in the theater, and for no money, I have to have a chance to do something I would never be allowed to do in films, and that probably would never be done in films." _Tiger at the Gates_ certainly fit that description, and Ryan agreed to play Hector for Actors Equity scale, which was fifty dollars a week. A one-week tryout was scheduled in mid-January 1957 at the Sombrero Playhouse in Phoenix, followed by a two-week run at the four-hundred-seat Ivar Theatre in Los Angeles.
A Republican stronghold, Phoenix was hardly the ideal town for a pacifist play, only two months after the Soviets had crushed the Hungarian Revolution. According to Kennedy, opening night was a fiasco, as patrons "stormed out in droves during the first act and fled to the bar, which set a new liquor record for the night." The next morning Ryan came to his hotel room, perspiring heavily — "flop sweat," he explained. (This was a common physical reaction on his part; Jessica recalled him dripping with sweat at the giant Stevenson rally in San Francisco in 1952.) Kennedy assured Ryan that the play would be more warmly received in Los Angeles, and he was right. Buoyed by positive reviews, _Tiger at the Gates_ sold out in its final week and set a new box office record for small venues in LA. Unable to secure seats, Howard Hughes sent his chauffeur to buy four tickets from waiting patrons at fifty dollars each, and experienced Ryan in the full flower of his liberal pacifism.
A certain amount of offstage drama attended the run. During the final week, Ryan's seventy-three-year-old mother, Mabel, was hit by a car on her way to the theater, sustaining cuts, bruises, and a five-inch laceration on her head. She asked police not to notify anyone until the performance was over, and given the fact that Old Tim had died of complications from being struck by an auto, the news must have upset Ryan when he got it. According to Kennedy, Ryan also stepped out on his wife during a drunken cast party, disappearing with a young actress in the cast who was enthralled by him. Early the next morning, Kennedy wrote, Ryan showed up at his house, sat down in the living room, and told him, "I've been on this couch all night." After Kennedy agreed to stick with this fiction, Ryan remarked, "I wish I had been." Kennedy waited for a phone call from Jessica Ryan, but it never came.
_TIGER AT THE GATES_ might have played well past its scheduled two-week run, but Ryan had committed to joining Aldo Ray on a two-week promotional tour for _Men in War_ , and Kennedy considered Ryan so essential that he decided to close the show rather than replace him. With a financial stake in the new picture, Ryan had promised Harmon and Yordan an intensive publicity push; according to another item in his alumni magazine, he lost thirteen pounds on the road. The effort must have paid off, because _Men in War_ turned out to be a critics' favorite. " _Men in War_ ranks with the great war pictures," raved the _Washington Post_ , praising its "commanding use of silence, detail and accent on the crude monotony of warfare." Writing in the _Los Angeles Times_ , Philip Scheuer noted the picture's documentary feel: "It has something of the on-the-spot reportage of, say, John Huston's memorable _San Pietro_ , made on the Italian front in World War II." For relatively little money, Security Pictures had made a daringly modern war movie, though its box office performance left something to be desired.
That May the Oakwood School held a groundbreaking ceremony for its new classroom expansion, which would add two classrooms, an instructors' room, and a conference room for the board, and allow Oakwood to expand its enrollment from sixty-five to ninety-seven students. The school continued to operate at a deficit every year, but where Oakwood was involved, Ryan never seemed to reach the bottom of his pocket. Nor was he averse to using his Hollywood connections to help out the school; Lamont Johnson, an Oakwood parent from the beginning, remembered stage-managing one benefit that included Frank Sinatra, Sammy Davis Jr., Peggy Lee, Peter Ustinov, Dean Martin, and Jerry Lewis.
"Marie Spottswood, who was absolutely brilliant, and Bob and Jessica were the soul and the spine of the whole thing," said Johnson. Jessica was still deeply involved in Oakwood, but on a less conspicuous level than her husband, working with Spottswood to create teaching materials for the social studies curriculum. When Spottswood lamented the lack of any suitable children's books about the Spanish in early California, Jessica wrote one, called _Mañana_ , and followed it with a book about the Aztecs, _Teca and the Plumed Serpent_. "We needed all sorts of unavailable illustrative material," Spottswood later wrote, "such as charts of Norse runes, Egyptian hieroglyphics, Chinese calligraphy. No problem: she would do the job. Soon the children were using her beautiful handiwork." The two women grew close. Jessica would portray Spottswood as devoted only to her students and cats, but according to the Ryans' old friend Robert Wallsten, the older woman "went into an absolute flutter any time Ryan came her way. It wasn't all education she had in her mind or at least it was another kind of education. And I'm not sure how conscious she was of it. I'm sure Jessica was aware of it."
With _Men in War_ in release, Harmon and Yordan turned to _God's Little Acre_ , an even more ambitious project with serious literary credentials. Published in 1933, this story of a poor, hard-lusting family in rural Georgia had become the center of a landmark First Amendment case when the New York Society for the Suppression of Vice sued the publisher, Viking Press, for violating a state statute against disseminating pornography. The judge in the case ruled for Viking, establishing two important legal precedents: first, the work was judged in its entirety, not on the basis of selected passages, and second, the court considered the opinions of other writers in evaluating the work's merit (a committee organized to defend Caldwell had included Sinclair Lewis, H. L. Mencken, Sherwood Anderson, Dorothy Parker, and Alexander Woollcott). The publicity from the trial, and from censorship efforts in other localities, stoked the sales of _God's Little Acre_ ; by the time Security Pictures got hold of the book in December 1955, it had sold eight million copies.
Ryan, Marvin Brown, and Marie Spottswood break ground for a classroom expansion at Oakwood School. It continued to operate at a deficit every year, but where Oakwood was involved, Ryan never seemed to reach the bottom of his pockets. _Robert Ryan Family_
Ben Maddow would later claim authorship of the screenplay, which Yordan flatly disputed; one can understand why either man would want credit for the elegant adaptation, which highlighted the book's humor and humanity while deleting or toning down its randier episodes. Even after a quarter decade, _God's Little Acre_ was too hot for the screen, with open infidelity and crudely sexual talk. Ty Ty Walden, the cracked patriarch at the center of the story, is widowed and celibate, but he cheerfully praises his daughter-in-law's "rising beauties" in front of his grown children: "They're that pretty it makes me feel sometime like getting right down on my hands and knees like these old hound dogs you see chasing after a flowing bitch. You just ache to get down and lick something. That's the way, and it's God's own truth as He would tell it Himself if He could talk like the rest of us." The picture was a little more decorous: "If the good Lord seen fit to put a beauty like you in our house, I'm gonna take my fill of lookin' while I can." Harmon and Yordan told _Variety_ they were going to bypass the Production Code Administration and seek independent distribution, but ultimately they submitted to the usual negotiations with the Breen office.
Ryan's representative at the William Morris Agency begged him not to accept the role of Ty Ty; the character was seventy years old, and if Ryan pulled it off, he might find himself typecast as an old man. But Ty Ty was too juicy to pass up, a grand, earthy, philosophical fool. The Waldens live in cotton country, but for the past fifteen years Ty Ty has been gripped with gold fever, digging gigantic holes on his land in search of a treasure chest described to him by his dying grandpappy. A good Christian man, Ty Ty reserves one acre of his farm for the Lord, giving everything grown on it to the church; he keeps moving its location around, though, because he needs new places to dig and can't stand the thought of his preacher getting all the gold. As scripted, the role ranged from raucous physical comedy to quiet drama, with the sort of poignant moments that seldom came Ryan's way. This was the sort of big, rich part the studios never offered him. According to Yordan, Ryan "just went over his agent's head, and he did it."
Harmon, Yordan, and Mann did their best to reunite the company that had collaborated so well on _Men in War_ : cinematographer Ernest Haller, composer Elmer Bernstein, actors Vic Morrow and Aldo Ray. Tall, curvaceous Tina Louise, a Broadway actress who had appeared in the musical _Li'l Abner_ , would make her screen debut as Ty Ty's daughter-in-law, Griselda, and rotund comedian Buddy Hackett signed on to play Pluto Swint, the hapless political candidate who's too busy sniffing around Ty Ty's youngest, Darling Jill, to canvas for votes. As with _Men in War_ , Aldo Ray shared top billing with Ryan, holding down another story line as Will, the striking millworker married to Ty Ty's elder daughter, Rosamond. Locked out of work for eighteen months, Will has begun scheming with his fellow strikers to seize the mill and start it up again; this situation added yet another wrinkle to the picture's release, since it might be attacked as subversive.
The producers had scouted locations in Georgia, but the book's reputation preceded it, and pressure from business and civic leaders in Atlanta prompted them to look closer to home; in early September, principal photography began in Stockton, California, five hours north of Los Angeles in the San Joaquin Valley. Tina Louise remembered Ryan as distant; she gravitated toward Aldo Ray and Jack Lord, who played her husband, Buck. Ryan preferred the company of Hackett, who could make him laugh until his sides hurt. "He adored Buddy Hackett," remembered Lisa Ryan. "I think my dad felt more comfortable being around people like Buddy Hackett than people who were more like my dad, who made him nervous." Ryan and Hackett shared some memorably funny scenes together; after learning of Ty Ty's quest, Pluto persuades him that what he needs is an albino, because these white-skinned people have a special power for divining gold. Ryan's expression widens in wonder as Ty Ty considers this, and before long the old coot has kidnapped a swamp-dwelling albino named Dave (played by young Michael Landon) and brought him back to the farm.
The character's saving grace was his humility, which allowed his ludicrously comic moments to coexist with his sincere religiosity. When Dave the albino follows his quivering dowser wand right to the makeshift cross that marks God's little acre, Ty Ty uproots it once again and ultimately moves it out to the river's edge. "Now God, I don't aim to cheat you none, I swear I don't," prays Ty Ty, as Bernstein's score swells. "But what with this unseasonable weather and all, I believe you'd admire to have your acre in a cooler spot. If you don't like this, if you don't approve of what I'm doing, Lord, then strike me down dead right here where I stand." Ty Ty shuts his eyes to wait for the lightning, then opens one eye and grins with satisfaction. "Thank you, Lord, Glory be. Amen."
Ryan must have known they had something good here, though in a quick note to the president of the Dartmouth boxing club, he was typically self-deprecating. "[Erskine] Caldwell came up to watch us shoot," he reported, "and is either the quietest man ever born or was stunned at the awfulness of what we were doing to his little epic. It has sold second only to the Bible but as one of our actors said, they do not come as a set."
WHILE RYAN WAS SHOOTING _God's Little Acre_ , an organization that would become his political focus for the next few years was taking shape in New York City. Earlier that year the revered theologian Albert Schweitzer had issued a "Declaration of Conscience" that explained how radioactive fallout from nuclear test explosions was infiltrating the water and food supply. "That radioactive elements created by us are found in nature is an astounding event in the history of the earth and of the human race," Schweitzer wrote. "To fail to consider its importance and its consequences would be a folly for which humanity would have to pay a terrible price." Published in the _Saturday Review_ , this statement ignited a fierce debate over nuclear testing in the United States, where stories about milk tainted with the radioactive isotope Strontium-90 had been cropping up in the news for the past year. That summer the crusading editor of the _Review_ , Norman Cousins, and two prominent figures in the American Friends Service Committee, Lawrence Scott and Clarence Pickett, summoned various civic leaders to discuss the issue, and from this gathering emerged the National Committee for a Sane Nuclear Policy.
Ty Ty Walden (Ryan) succumbs to gold fever in _God's Little Acre_ (1958). Erskine Caldwell's salacious novel, Ryan noted, "has sold second only to the Bible, but as one of our actors said, they do not come as a set." _Wisconsin Center for Film and Theater Research_
The new organization, nicknamed SANE, met again that fall and published a full-page ad in the _New York Times_ , asking readers to press President Eisen-hower for an international test ban, to be enforced by the United Nations, and new powers for the U.N. to monitor all missiles and satellites worldwide and to pool all world science for the purpose of space exploration. The public response was overwhelming, and what began as a small group of intellectual elites broadened into a mass movement with 25,000 members in about 130 chapters nationwide. Ryan was already affiliated with the United World Federalists (UWF) and the American Friends Service Committee, and he threw in with the new group as well, participating in rallies at Madison Square Garden and the Manhattan Center on Thirty-fourth Street, and in smaller public meetings convened to raise awareness of the issue.
Cousins liked Ryan a great deal, finding him not only passionate but also smart, informed, and pragmatic. "He had obviously done the proper homework on these issues," Cousins recalled, though their discussions focused less on the science and more on political strategy — how to combat the Eisenhower administration's arguments for nuclear readiness, how to address the Atomic Energy Commission's promotion of nuclear power, how to counter the pronuclear lectures being delivered across the country by physicist Edward Teller, father of the H-bomb. Ryan, Cousins discovered, had a shrewd sense of how to put ideas across to the public. "What should a program be, how do we stage them to appeal to people, these were the operational questions. Here, he had a very searching knowledge of the people involved, and could tell you who could do what, and not much time was wasted." Actors involved in political issues could be egotistical and uninformed, Cousins learned, but Ryan was just the opposite, more interested in results than in scoring points.
Though the public support for SANE was encouraging, both Ryan and Cousins knew from bitter experience just how easily pacifist movements could be derailed by assaults from the right and bad news from overseas. As members of the United World Federalists, they had seen the organization swell to a membership of fifty thousand in 1949 and then shrivel after the Korean conflict erupted in June 1950. Protecting its right flank, the UWF supported President Truman's decision to send US troops to the Korean peninsula, which put the group in the untenable position of seeking world peace even as it endorsed armed conflict. Still the attacks came: in 1951 the Senate Appropriations Committee voted to prohibit funding of groups that promoted world government, and by 1953 government employees were being asked if they had ever belonged to the UWF, regarded now as a communist front organization. By the end of the decade Ryan was serving as president of the Southern California chapter, though the fact that the nation's third largest metropolitan center sustained just one chapter only highlighted the UWF's diminishing relevance.*
Ryan had plenty of time for activism, because his picture offers were drying up. After _God's Little Acre_ he decided to take some television work, appearing simultaneously in three different anthology series during the 1957–58 season. Both _Goodyear Theatre_ and _Alcoa Theatre_ were broadcast Monday nights on NBC and aspired to quality drama, though the half-hour playlets Ryan appeared in sometimes fell short of the mark. He also appeared on an episode of the western anthology _Zane Grey Theater_ , with two more to follow in the 1958–59 season. But no movie star had ever forged a TV career appearing in drama anthologies. Many of Ryan's young costars in _God's Little Acre_ would become household names on TV over the next decade, but all in continuing series that identified them with single characters: Vic Morrow as a foot soldier on _Combat!_ , Tina Louise as the starlet on _Gilligan's Island_ , Michael Landon as the junior brother on _Bonanza_ , Jack Lord as a detective on _Hawaii Five-O_.
A more enticing TV project came up in summer 1958 when Ryan landed the title role in a _Playhouse 90_ adaptation of _The Great Gatsby_. Ryan had always cherished the book. "I was a child of the 1920s — a dream world," he recalled. "Life was a ball: two Cadillacs, the trip to Hawaii, lots of fun, no problems — and everyone was going to make a million. That's why Scott Fitzgerald is the best; he got it down the way we lived it." His long history of playing strong, confident men with secret vulnerabilities made him a natural for Jay Gatsby, the wealthy bootlegger pining for his lost love.
Unfortunately, the broadcast would fail to capture the novel's mystery and only emphasized Ryan's advancing age. Fitzgerald describes Gatsby as being about thirty-two years old; Ryan was visibly forty-eight. Rod Taylor, as the story's narrator, Nick Carraway, and Jeanne Crain, as Gatsby's coveted Daisy Buchanan, were much closer to their characters' ages. Ryan shone in the scenes where Gatsby, having contrived a meeting with Daisy at his palatial home in West Egg, frets and fusses prior to her arrival, then relaxes into himself when they finally meet. As an adaptation of Fitzgerald, the script was functional but uninspired, and the broadcast, recorded and edited on the new technology of magnetic videotape, suffered from Franklin Schaffner's clunky direction.
The challenge of adapting classic literature to the movies hit home even harder in the weeks after the broadcast, when Ryan reported for work on a screen adaptation of Nathanael West's corrosive 1933 novel _Miss Lonelyhearts_. Fired from MGM, Dore Schary had gone back East to produce a play he had written about Franklin Roosevelt, _Sunrise at Campobello_ , which became a Broadway hit; while in New York he had seen a stage version of _Miss Lonely-hearts_ and bought the rights, then tossed out Howard Teichmann's script and wrote his own version of West's story. Financed by United Artists for a modest $750,000 (about the same as _God's Little Acre_ ), the project heralded Schary's triumphant return to Hollywood as an independent producer, and he filled the cast with old friends: Montgomery Clift as the tormented advice columnist, Ryan as his cynical editor, Myrna Loy as the editor's unloved and unhappy wife. Vincent Donehue, who had directed _Sunrise at Campobello_ , was brought out West to make his screen directing debut and signaled his stage inclinations immediately by calling for two weeks of rehearsal prior to shooting.
Unfortunately, no amount of rehearsal could compensate for Schary's script, which suffered from a playwright's verbosity and a Jewish writer's indifference to a deeply Catholic work. West's protagonist, a newspaper writer assigned to his paper's lovelorn column, has become a connoisseur of human misery: a severely deformed girl who pines for a boyfriend, a woman who fears her eighth pregnancy will kill her, and a deaf girl who has been sexually assaulted are only a few of the anguished souls introducing themselves in the letters that cross his desk. Shrike, his jaundiced boss, has written a little prayer mocking his sense of having taken on the world's troubles: "Soul of Miss L, glorify me / Body of Miss L, nourish me / Blood of Miss L, intoxicate me / Tears of Miss L, wash me." Schary updated the story to the '50s and got rid of all the Jesus talk; in place of the novel's dark ending, when Miss Lonelyhearts finally wins the crucifixion he's been seeking, he wrote a sunny-side-up conclusion in which the hero — renamed Adam in Old Testament fashion — lives happily ever after with his sweetheart, and even the rancid old Shrike learns to be a little nicer to his woman.
"The picture was a misfire — a compromise," Ryan would later say. "It would have been much more interesting, and equally commercial, if they had made it really like the book." He admired Nathanael West and must have related to the novel's eerie mingling of Catholic mysticism and grotesque black humor; as a lapsed Catholic, Ryan was drawn more to the Passion than the Resurrection. "He only went to church once a year; he'd go on Good Friday," remembered Lisa Ryan. "I went with him a couple of times. Talk about the most depressing service you could go to!"
Montgomery Clift, doing the picture for a mere $25,000 as a favor to Schary, thought the script was horrendous. It was full of overripe dialogue, and Ryan had more than his share, his arch delivery only emphasizing its stiffness: "Love and kindness. Man is good. Well listen, Little Boy Blue. You'd better take a bath and wash off the eau de cologne — it stinks." The hero's Christ complex was watered down to something more revealing of Dore Schary, that tireless supporter of causes and charities: "Is it a sin to feel?" asks Adam. "Is do-gooder a dirty name? Why should it be?" Clift thought the whole thing read like an Andy Hardy movie.
Perpetually drunk and whacked out on pills, Clift struggled to get through the picture. "He would start to cry in the middle of scenes and they'd have to stop the film," recalled Cheyney Ryan, who visited the set. "One time he literally lay down and got into a fetal position in the middle of a scene and started crying and rolling around, and everyone's sitting around waiting for this to end." Some of the crew and minor players began to get fed up with Clift's behavior, and the situation came to a head during a barroom scene in which Adam punches a colleague at the paper. The punch was choreographed as a right cross, but in the take Clift twisted and around and clocked actor Mike Kellin with his left. Kellin cursed him out and Clift was taken away, after which Robert Ryan donned Clift's shirt and jacket and stood in for a close-up of the punch that was inserted to rescue the botched take. "I've always wanted to work with you," Ryan told Loy during the production, "and now that I am, I hardly see you. You're too busy taking care of Monty."
_Lonelyhearts_ , as Schary had titled the picture, was Ryan's first screen work in nine months. He had quit smoking and put on weight. A vague sheen of perspiration — the old flop sweat — clung to him as he tried to put across the turgid dialogue. When the picture was released at the end of the year, critics were merciless. "The play had at least tried to capture some of the book's agony, but in the film everything is munched down into pablum," wrote Stanley Kauffmann in the _New Republic_. In _Esquire_ , Dwight Macdonald published a feature story on Schary's return to Hollywood, labeling _Lonelyhearts_ "the apotheosis of the adult soap opera" and observing, "Robert Ryan, a self-conscious and wooden actor who depends for his effectiveness on a sinister cast of countenance, plays Shrike like a Western bad man coached by Noel Coward, nor is he more successful than Clift in making contact with his opposite number, Myrna Loy, who plays Mrs. Shrike in a world-weary manner more suitable to the Oriental vamps she used to do in the Twenties." Macdonald's piece ended with Schary admitting that he and his wife had felt like outsiders in Hollywood; he would produce only two more pictures before his movie career petered out five years later.
Ryan wound up as the highest paid performer in _Lonelyhearts_ , taking home $75,000. But the job must have put him in an awkward position with his old friend Pat O'Brien, who had played Shrike in the Broadway production but been passed over by Schary for the screen role. Unloaded from RKO in 1949, O'Brien had fallen on hard times in LA, scratching around for B movies and TV guest shots and wondering why every door in town seemed to be shutting on him. In his 1964 autobiography he would write of being blackballed by liberal studio executives who considered him part of the anticommunist right. Schary had a reputation for getting even with red-baiters — John Wayne liked to say that the only blacklist in Hollywood was the one Schary ran at MGM — and O'Brien would recall Spencer Tracy threatening to quit MGM's _The People vs. O'Hara_ unless Schary gave O'Brien a role. "I was against communists, I was against the methods and the procedures by which they and fellow travelers had, it was reported, infiltrated the studios," wrote O'Brien. "But I hadn't made a full-time crusade of it." Ironically, Joseph Losey would remember O'Brien, whom he directed in _The Boy with Green Hair_ , as one of the few people who came to his aid when he was blacklisted.
"SOME ACTORS think they're businessmen," Ryan told _Variety_ earlier that year, "but few of them are. Most of them are going through the motions of playing executives. I'd rather let somebody else take care of the production details. If you are your own producer, you might start making concessions. You can't do a good acting job and be a producer at the same time." Most actors didn't know how to judge a script, he observed, and as producers, "they all wind up making westerns."
He preferred the kind of profit-participation deals he had cut with Security Pictures. In his words they constituted "income roulette," and the promotional tours were grueling: two or three weeks, city after city, each day beginning with press interviews at breakfast and continuing on to nightfall or even midnight screenings. Yet as Ryan explained to one reporter, "The odds are generally in your favor if you get enough of these deals and one comes through to tip the scales." _God's Little Acre_ had been that picture—according to Phil Yordan, Ryan pocketed a quarter million dollars for it. When Yordan and Sid Harmon offered Ryan another participation deal for their last United Artists release, he readily accepted.
Of course, it was a western, adapted from a novel by Lee Wells called _Day of the Outlaw_. Director André De Toth had directed six westerns with Randolph Scott, making the occasional detour into crime ( _Pitfall_ ) or horror (the 3-D release _House of Wax_ ). Harmon and Yordan brought back Tina Louise to play Ryan's love interest and recruited another cast of talented character actors: William Schallert, Nememiah Persoff, and Elisha Cook Jr. Yet _Men in War_ and _God's Little Acre_ had been happy experiences; _Day of the Outlaw_ turned into a nightmare.
Yordan's script was intriguing, a chamber drama set in a snowy western outpost, with the balance of power shifting radically at different points. Rancher Blaise Starrett (Ryan) rides into Bitter, a frontier town in the Wyoming Territory, itching to kill a local farmer who has married Starrett's true love (Louise). But just as this conflict comes to a head, the town is invaded by a renegade army officer (the immense, gravel-voiced folksinger Burl Ives) and his band of violent derelicts. Hunted by a cavalry outfit, they take over the town and threaten to rape the four women. Starrett, once a gunslinger, is forced back into his old role as the town's protector, though this time he succeeds through cunning: with the cavalry nearing the town, he persuades the outlaws to follow him to safety on a secret path through the snowy mountains, and leads them into the face of a deadly blizzard.
De Toth loved the idea of the fearsome gang "terrorizing a small Western village, and then, by a quirk of nature, becoming equally the prisoners of the white silence in the middle of nowhere." Harmon and Yordan, he would later argue, "didn't understand where I was heading — a sphere I had been exploring for some time: is it worse being the jailer, instead of the prisoner?" The producers wanted him to shoot in color, on a soundstage; De Toth insisted on black and white, to make the blizzard sequences as stark as possible, and persuaded them to erect an exterior set of half a dozen building fronts on Dutchman Flat, located southwest of Bend, Oregon, in full view of the Cascade Range. He wanted the buildings up by the fall, so that they would be properly weathered and snow-encrusted when shooting commenced in mid-November. Ryan backed De Toth on this point, and the set construction began. "He was a gentleman, a sincere human being — and what an actor," remembered De Toth. "He was with me all the way. Without him, I would've been laid out in the snow and counted out quickly."
The relationship between Harmon, Yordan, and De Toth deteriorated further when the director discovered that the art director had ignored his compass coordinates for the set and built it facing in the wrong direction, which ruined his plans for taking advantage of available light and keeping the snow in the background pristine and untouched. The producers, who had spent most of their UA money on the last two pictures and budgeted this one for a measly $400,000, considered simply firing De Toth, but Ryan stuck by him and the production went forward. "[Ryan] was a loyal man, which is very uncommon in Hollywood," Yordan said, "in fact it's very rare in the business world." The set was rebuilt just before the cast and crew arrived, and De Toth shot a few scenes there before taking off for Mount Bachelor, twenty-two miles west, to shoot the blizzard sequences.
The work was punishing, as Ryan, Ives, and the actors playing his henchmen rode on horseback through three-foot snowdrifts. After a few days Ryan was diagnosed with pneumonia; a week was lost before he was well enough to return to work. Snowstorms pushed the production even further behind schedule, and according to Yordan, the location shoot was finally shut down and the cast and crew brought back to Hollywood for interior scenes that would be needed to patch up the story.
Yordan was proud of his script and bitterly disappointed with how the picture turned out: "It could have been a real winner," he argued. The picture's reputation would grow steadily in the decades ahead, as people began to appreciate its novel plot structure and brutal naturalism. But when _Day of the Outlaw_ opened in July 1959, it bored critics and tanked at the box office, bringing Ryan's relationship with Security Pictures to an ignominious close. The next time he worked with Phil Yordan and Sid Harmon, he would need them more than they needed him.
*For a definitive account of Yordan's storied career, see Alan K. Rode's "'First Is First and Second Is Nobody': The Philip Yordan Story," _Noir City Sentinel_ , November/December 2009.
*This homespun tale was part of the NBC anthology series _Screen Directors Playhouse_ , which showcased the small-screen work of veteran Hollywood filmmakers (in this case, H. C. Potter).
*For a biting history of the UWF, see John A. Yoder's "The United World Federalists: Liberals for Law and Order" in _Peace Movements in America_ (New York: Schocken Books, 1973), 95–115.
_eleven_
Beautiful Creatures
Andy Harmon often came over to the Ryans' house on Kling Street to play with Tim. He liked the atmosphere there, so cool and quiet compared to his own home. "I thought the Ryans were the height of elegance," he recalled. "It certainly seemed to me like they never argued, whereas we always argued." The Ryans had dinner every night at 6:30; Jessica said a Quaker grace, and they would talk about politics or current events at the table. But Jessica could be distant, and at times the children were admonished to be quiet around her. "There would be this mysterious thing where one of [the parents] would be in bed for a long time in the mornings, which I didn't understand, 'cause my dad was literally out of bed and in his tie and jacket at 7:30 in the morning." Like any child, Andy could only puzzle out what was going on in the grown-up world, but Bob and Jessica Ryan seemed even more private than most parents.
As the decade wore on, Jessica grew increasingly troubled, until she suffered a breakdown sometime in 1958. "I don't think she was hospitalized," said Cheyney Ryan. "But you certainly had the sense that there was something wrong with her. I remember this in part because we were all supposed to go to Palm Springs and we didn't do that, and then there was some plan to take a cruise somewhere, we didn't do that. So things were being canceled because of this, and she went into psychoanalysis." Jessica had long been an avid student of Jungian psychology, and she became a patient of therapist Max Zeller, a founder of the C. G. Jung Institute of Los Angeles. "What she said about it afterwards was that... she didn't want to be just a regular housewife, and people were telling her that she should be."
Years later, in the introduction to a scholarly manuscript, Jessica would identify herself as "a middle-aged American woman, wife, and mother of three children, wondering if I can find a better answer than I have yet had, to why I am the way I am, to why I have had an uneasy life with men and them with me." All around her she saw women who "feel that they were promised something, even if they have no notion of what, or by whom, or when or why; the promise unfulfilled, they continue to resent and rage, while they play the roles they feel have been assigned to them. But underneath, it is as if they suffer a feeling of some great disappointment."
And so she withdrew — from the Oakwood School, from her friends, and, to some extent, from her children. She stopped attending school functions and no longer accompanied her husband to political or charity events, not to mention movie promotions or Hollywood parties. She shut herself up in the house and buried herself in books, searching for an intellectual answer to a problem that, as some of her children would later suppose, might have been addressed more effectively with less booze, more exercise, and a mild antidepressant. She wrote incessantly, filling up page after page in longhand. (Despite her feelings of having been squashed professionally, her career wasn't going badly; her novel _City of Angels_ had been published in France, and she had recently placed a children's mystery, _The Malibu Monster_ , with Bobbs-Merrill.) Jessica always had been the center of the family, and now her husband and children rallied around her. "There was this kind of thing happened in the family, that Mom was fragile," said Cheyney. "When you have someone like that, the whole family thinks it's their role to prop her up."
The Ryans drew a cloak of privacy over Jessica's condition. Cheyney would wonder if his parents were ever physically intimate again after his mother's breakdown, but there was no doubt of their mutual love and respect. "I never remember their being sarcastic with each other," he marveled. "I never remember their raising their voices with each other. I never remember them ever acting in any way that suggested that they weren't taking each other's views about whatever the issue was with complete seriousness." Ryan may not have understood how women felt in a world dominated by men, or what Jessica experienced as the wife of a movie star, or how his occasional, guilty infidelities had damaged their relationship. But he valued his family, and now his family was in trouble. In any case, he was so reclusive himself that Jessica's growing agoraphobia seldom presented much of a practical issue.
Regardless of what transpired at home, Ryan was still the breadwinner, so in February 1959 he flew off to New York City to begin shooting a crime thriller with Harry Belafonte called _Odds against Tomorrow_. At that point Belafonte was one of the biggest recording stars in the world: in the new age of the long-playing record, he had shot to the top of the _Billboard_ album chart with a self-titled LP in early 1956 and four months later followed it with _Calypso_ , which spent a staggering thirty-one weeks at number one and another year on the chart after that. He had started out in show business as an actor in New York, but he was unhappy with the screen roles that had come his way and decided to start his own production company, HarBel, to release through United Artists. Adapted from a novel by William P. McGivern, _Odds against Tomorrow_ combined a heist plot with tense racial drama, as Johnny Ingram, a Harlem musician in debt to gamblers, agrees to stage a daring bank stickup with Earl Slater, a bigoted white Oklahoman. Belafonte hired Robert Wise to direct, and they agreed that the man to play Slater was Robert Ryan.
Ryan didn't agree at first — when they had sent him the script the previous fall he passed, explaining that he didn't want to go down that road again. "A great many people realize that the characters they see on the screen are fictional or created but there is a substantial group that does not make that distinction," he later wrote in a self-exculpatory piece for _Ebony_ , reminiscent of the things he had published when _Crossfire_ was released twelve years earlier. "I changed my mind about playing Slater after re-reading the script and appreciating its excellent qualities.... _Odds against Tomorrow_ says something... of significance and says it well, dramatically, without preaching. The drama strongly suggests that bigotry is based on fear and envy and that the most important thing that keeps a bigot operating is the feeling that he is better than another man."
He also liked the script's evenhandedness: unlike the quietly suffering role models Sidney Poitier often played, Belafonte's character was embittered by racism and ultimately pulled down to Slater's level. And here was a chance to work again with Bob Wise, who had done a spectacular job directing _The Set-Up_ and since moved on to such highly regarded dramas as _The Day the Earth Stood Still_ (1951), _Executive Suite_ (1954), and _I Want to Live!_ (1958).
Belafonte had admired Ryan's work for years, and their paths had crossed during the Stevenson campaign in 1956, but they had never met properly before shooting the picture. "What really surprised me was that, in many of the films that he did, he had always played a heavy villain," Belafonte recalled, "and to meet the man in person, to find out that his whole persona, and his way of life, and his thinking and his philosophy, was so the opposite.... The contrast was so glaringly evident." Earl Slater would complete Ryan's big trinity of intolerance, along with Monty in _Crossfire_ and Reno Smith in _Bad Day at Black Rock_ — "I'm either killing a Jew, a Jap, or a Negro," he would lament to one reporter."
Yet Belafonte came to realize that Ryan's acceptance of these roles, and the penetrating intelligence and empathy he brought to them, was a kind of activism in itself. "I think he took them because he really believed that he was making a contribution to people's overall sense of what it was to be a minority or to be discriminated against." Ryan, in turn, was impressed by Belafonte's humanity and political commitment; he would maintain an affectionate friendship with the younger man for the rest of his life.
Ten years had passed since Ryan and Wise had collaborated on _The Set-Up_ ; in the interim, that postwar cycle of shadowy, morally conflicted crime dramas had petered out at the box office, even as French critics dubbed it _film noir_ and, breaking into the movie business themselves, began to draw on its realism and immediacy. Ryan had appeared in numerous pictures now regarded as part of the noir canon — not only _Crossfire_ and _The Set-Up_ but also _Act of Violence, The Woman on Pier 13, On Dangerous Ground, The Racket_ , and _Beware, My Lovely_. Wise would consciously revisit the genre with _Odds against Tomorrow_ ; along with Ryan, the cast included such noir veterans as slimy Ed Begley (On _Dangerous Ground_ ) and saucy Gloria Grahame ( _Crossfire_ ). For the climactic bank robbery sequence, shot on location in the small town of Hudson, New York, Wise would recycle a prominent visual motif from _The Set-Up_ , a street scene with a public clock marking the time.
Belafonte first gave the book to John Oliver Killens, a talented black novelist and a cofounder of the Harlem Writers Guild. But Killens had never written a screenplay; when his effort proved unsatisfactory, Belafonte turned to veteran screenwriter Abraham Polonsky ( _Body and Soul, Force of Evil_ ), who had been blacklisted since the second HUAC investigation in 1951. Killens gladly agreed to front for Polonsky, and Belafonte approached both Wise and Ryan to explain what was going on. Ryan "threw his lot in with us and said, absolutely, he'd take a stand, and he thanked me for informing him that it was Abe Polonsky. He and Abe became very good friends after that." The script was expertly structured, and Polonsky was an old hand at the cynical poetry of film noir. When Ingram learns about the bank job from seedy Dave Burke (Begley), the disgraced cop masterminding the operation, his first response is, "I did all my dreaming on my mama's knee."
Ryan with Harry Belafonte on the set of _Odds against Tomorrow_ (1959). The younger man awakened his interest in the civil rights movement and would introduce him to Martin Luther King Jr. _Frankin Jarlett Collection_
Ryan initially turned down the role of Earl Slater, the bigoted criminal in _Odds against Tomorrow_ (1959). "A great many people realize that the characters they see on the screen are fictional or created," he wrote in _Ebony_ , "but there is a substantial group that does not make that distinction." _Franklin Jarlett Collection_
"You didn't say nothin' about the third man bein' a nigger!" Slater complains to Burke as the two men case the bank from a hotel room across the street, and Wise accents this declaration of theme with one of the oldest noir tricks in the book, using venetian blinds to cast horizontal stripes across Ryan's face. Ryan always felt for the soft spot in his most despicable characters, and he would note how Slater was "peculiarly provoked by the kind of Negro he finds himself involved with. It just so happens that this young Negro is better looking, better dressed, more intelligent; is, in fact, everything that Slater would like to be but isn't.... Slater, therefore, is a many-sided character and this made the problem of projection a lot more difficult." Initially, it also had discouraged him from taking the role; as with Monty and Reno Smith, there was the danger of making this heel a little too sympathetic.
McGivern's novel ended with Slater and Ingram learning to trust one another, but that denouement was too reminiscent of Stanley Kramer's recent hit _The Defiant Ones_ , another tale of a black man and a white man yoked together, so Belafonte and Polonsky agreed on something darker. The carefully planned robbery falls apart after Slater, unwilling to leave Ingram with the keys to the getaway car, gives them to Burke, who is shot down by police outside the bank. Slater and Ingram fire angrily on each other even as they flee the cops, and Ingram chases Slater across the giant tanks of a nearby oil refinery. An errant bullet ignites one of the tanks, and — shades of James Cagney in _White Heat_ — the antagonists are blown to kingdom come; when their bodies are carted off by the police, the black man and the white man are indistinguishable. The finishing touch would be an edgy symphonic score by John Lewis of the Modern Jazz Quartet.
The shoot was mostly uneventful, though actress Shelley Winters, who played Slater's blowsy girlfriend, was coming unglued over her rocky marriage to actor Tony Franciosa. "My God, we had to nurse her through that sometimes, and Bob was a big help on that," said Wise. "Bob was very, very patient with her and worked with her and helped her and didn't get out of sorts, which some actors might have." For his trouble, Ryan would turn up in one of the kiss-and-tell memoirs Winters published in the 1980s, which detailed her affairs with Errol Flynn, William Holden, Sterling Hayden, Burt Lancaster, and Sean Connery. According to Winters, she and Ryan had known each other "not-so-casually" after being introduced by Marilyn Monroe on the set of _Clash by Night_ back in 1951. Reunited for _Odds against Tomorrow_ , she wrote, they spent a good deal of time together between takes, talking about "the theater, organic farming, where Hollywood was going in this age of television, anything but what we were really thinking about."
Ryan had more on his mind than Winters might have imagined: he and Jessica had decided to sell the ranch house in North Hollywood and move out of the San Fernando Valley, back to the movie colony in Los Angeles. Numerous factors influenced their decision: Only a block south of Kling Street, construction had begun for the Ventura Freeway, which the Ryans were certain would spoil the quiet little neighborhood. Jessica wanted some physical distance from the Oakwood School, and Ryan hoped a big house might raise his professional profile. After he returned from New York, they settled on a beautiful Georgian Colonial residence in Holmby Hills, just east of the UCLA campus.
The family moved that summer, and the children all started at public schools in the fall, Tim and Cheyney at Emerson in West Los Angeles and Lisa at Warner Avenue Elementary in Holmby Hills. They couldn't understand why they were being uprooted and moved into this fancy new home, though Cheyney would later attribute the decision to his mother's unhappiness: "The kind of thing my parents would do is move when there was a problem." His remark echoed Ryan's recollection of his own parents pulling up stakes after his brother, Jack, died back in 1917: "We had to move and we did."
"WHEN WE MOVED TO HOLMBY HILLS, it was like living in a fortress," said Lisa Ryan. The old neighborhood in North Hollywood was built on a street grid, but Holmby Hills discouraged outsiders with its maze of hilly, winding streets, and all the homes were protected by high walls. The one at 301 N. Carolwood Drive had been built in 1937 for Joan Bennett, Ryan's costar in _The Woman on the Beach_ , by noted architect Wallace Neff, an originator of the California style. Walt Disney lived nearby. The house was luxurious by the Ryans' standards, backed up by a large patio that gave way to a lawn and then a swimming pool with beach houses on either side. For the first time each of the boys had his own room, but the streets were too steep for bicycle riding and the grounds too landscaped for playing catch. "If you missed the ball, you'd have to spend twenty minutes trying to find it in some thicket or something," Cheyney Ryan recalled. In North Hollywood they could walk a few blocks to the drugstore, but here they had to get in the car and drive out the security gate that cordoned them off from the rest of the world.
The move baffled many of the Ryans' friends, who had never considered them materialistic despite their comfortable lifestyle. Lamont Johnson thought he understood: "There was a part of [Ryan] that was sort of lace-curtain Irish, that loved not only keeping up with the Joneses but beating the shit out of them." Interviewed at length by columnist Sidney Skolsky, Ryan presented the move as a simple matter of needing more space. "When the children were babies, it was fine; we were all close together, and we had a nice yard and a wonderful life there; but my children are getting quite large now, and we found that we were all bumping into each other. Also, there didn't seem to be room enough for anything, and I had about exhausted the expansion possibilities of the house we had, having built on it six times already." The new house, he pointed out, had "a huge playroom downstairs where the kids can go down and turn on the TV full blast, which is the only way they seem to enjoy it. None of us can hear it upstairs. That I think is the outstanding feature."
Saddled now with a hefty new mortgage, Ryan took off in August for Petersburg, Alaska, to begin location shooting for _Ice Palace_ , his very first picture for Warner Bros. and the first in three years that he had made for his straight salary of $125,000. A big-budget melodrama spanning almost fifty years, _Ice Palace_ was adapted from a fat best seller by Edna Ferber, author of _Giant_ ; when Ryan got a look at the Mendenhall Glacier, which figured in the story, he joked that the picture should be retitled "Giant-on-the-Rocks." He and Richard Burton starred as young entrepreneurs who build a fishing and canning empire together but then grow apart, Burton's character lusting for wealth and Ryan's becoming a political crusader for Alaskan statehood and environmental protection; of course, they both love the same woman.
The picture was junk, but at least it presented Ryan as a sympathetic figure. "I think it's good for business to be liked by the audience every few years," he told Skolsky. "This particular part involved portraying what you might call a practical saint, and this is a very unusual experience for me." The role also required him to age forty-seven years over the course of the story. "It took me two hours to make up for the twenty-eight-year-old," he would crack, "and about fifteen minutes for the man of seventy-five." Three weeks later Ryan was back in Los Angeles shooting interiors for the picture, and work continued through December (the final cut would clock in at nearly two-and-a-half hours).
His free time during this period was largely consumed by politics; he and late-night comedian Steve Allen were trying to launch a new chapter of the National Committee for a Sane Nuclear Policy, to be called Hollywood for SANE. The inaugural event, a buffet dinner at the Beverly Hills Hotel on September 26, drew about two hundred industry professionals, a healthy turnout by Hollywood standards, and the cochairs announced a drive to raise $50,000 from the entertainment community by the end of the year. Three weeks later, on Saturday, October 17, the Ryans hosted a second meeting at their new home on Carolwood Drive, with Norman Cousins as the guest of honor. "An executive present at both meetings said that film writers were the most prominent industry figures at the initial dinner," reported the _New York Times_. "At Mr. Ryan's house, many more actors and directors attended."
This was significant: writers could hide behind fronts, but the large contingent of actors — Rod Steiger, David Niven, James Whitmore, Mercedes McCambridge, Keenan Wynn — suggested that for Hollywood liberals the frost was beginning to thaw. Three months later, director Otto Preminger would announce that Dalton Trumbo, one of the Hollywood Ten, was the screenwriter of his forthcoming United Artists release _Exodus_ , and even before that picture came out, Kirk Douglas credited Trumbo on-screen as the author _of Spartacus_. After twelve years the blacklist was unraveling. "People in Hollywood are finally ready to speak out for something besides mother love," Ryan declared after the second Hollywood for SANE gathering. Cousins seemed to share his optimism: "The long silence is over," he observed. "Hollywood is again putting itself on the line for public issues."
Ryan's continuing commitment to _Ice Palace_ prevented him from doing a publicity tour for _Odds against Tomorrow_ , which opened in November, but whenever he got time away from the set, he sat down with press people to talk about the picture and the problem of race prejudice in America. Doing the picture, he wrote in _Ebony_ , had forced him to reexamine his own attitudes and increase his own commitment to ending bigotry. Unfortunately, even as the magazine was hitting newsstands, Ryan committed a cringe-worthy racial gaffe on network TV. After completing his day's work on _Ice Palace_ , he rushed over to the NBC studios, the fake gray still in his hair, to appear on a prime-time game show called _It Could Be You_ , a hidden camera catching him out in the hall as he got his shoes shined. As part of the show, Ryan surprised the owner of the shoeshine stand, O. C. Jones, by bringing him in front of the studio audience to accept a $200 tip. Ryan towered over Jones, and as he was exiting, he impulsively rubbed the man's head as if he were a dog.
One benefit of the Ryans' new house in Holmby Hills was its proximity to UCLA, where Ryan had helped found a fledgling experimental theater company earlier that year as part of the extension program. Along with Sidney Harmon and theatrical producer Eddie Cook, Ryan proposed the idea to Abbott Kaplan, head of the UCLA Extension, who allowed them to use the university's conference center at Lake Arrowhead for a brainstorming session. Lee Strasberg addressed this gathering of about eighty people, and a subsequent meeting on campus drew such big names as Paul Newman, Joanne Woodward, Anthony Quinn, and Eva Marie Saint. That summer the newly christened Professional Theatre Group of the UCLA Extension presented three staged readings — Dylan Thomas's _Under Milkwood_ , Bertolt Brecht's _Mother Courage_ , and Nikos Kazantzakis's _Sodom and Gomorrah_ — that nearly sold out their six-day runs. Emboldened by this response, Kaplan hired John Houseman as artistic director, and for the first full production, in January 1960, Houseman asked Ryan to play Thomas Beckett, the martyred Archbishop of Canterbury, in T. S. Eliot's _Murder in the Cathedral_.
Remembering the production in one of his memoirs, Houseman was typically rigorous in his assessment of Ryan. "Though he had done some work on his voice since _Coriolanus_ , it remained flat and unresonant," Houseman wrote, "but, as in _Coriolanus_ , his vocal weakness was offset by his physical presence, his intelligence and his personal experience... of the emotional problems of this profoundly Catholic play."
As it opens, Beckett is returning to England in 1170 after seven years in exile, having offended King Henry II by proclaiming his primary allegiance to the Pope; the faithful celebrate the archbishop's return, but the king dispatches four knights to assassinate him. Beckett is visited by a succession of tempters, each of whom offers him some compromise that will save his life, but he rejects them all and waits in the peace of God for the end to come. After the knights arrive and hack him to death, each offers his justification to the crowd, and the last of them argues with some force that Beckett was no martyr at all but "a monster of egotism." Eliot's aim, then, was not simply to celebrate Beckett's spiritual devotion but to note how even the most perfect devotion can be corrupted by the sin of pride.
_Murder in the Cathedral_ opened on January 18 at UCLA's Schoenberg Hall, a 526-seat classical music venue whose hydraulic orchestra pit enabled Houseman to create various levels in the action as Beckett was addressed by the tempters and a chorus of the faithful. Writing in the _Los Angeles Times_ , Philip Scheuer praised the production but faulted the casting of Ryan, who "appears to have bent over backward to emphasize the turn-the-other-cheek side of this dedicated priest." The review pinpointed what a stretch this passive, contemplative character was for Ryan, who had played so many angry, dynamic men. In the Christmas sermon that bisects the play, Beckett explains: "A martyrdom is always the design of God, for His love of men, to warn them and to lead them, to bring them back to his ways. It is never the design of man; for the true martyr is he who has become the instrument of God, who has lost his will in the will of God, and who no longer desires anything for himself, not even the glory of being a martyr."
The two-week run was well attended, and the Theatre Group would continue on the campus for eight seasons, staging forty-one productions before it moved to the Mark Taper Forum in 1966. Houseman's insistence on casting big names from movies and television led to some friction between the company and the university's theater department, but the Theatre Group became a source of great cultural excitement in Los Angeles. "Going to the plays had the feeling of attending a private party," wrote critic Cecil Smith in 1981. "You ran into friends there, people of like interests. Intermissions were alive with spirited discussions." Ryan's founding role in all this would go largely overlooked, and he would never get another chance to perform with the group.
After the play closed, Ryan hosted _A Call from_...,* a star-studded, hour-long TV special, produced by his friend Marsha Hunt, about the United Nations' yearlong campaign to address the humanitarian crisis of some fifteen million refugees worldwide. Over the past decade Ryan had developed a reputation in Hollywood as someone who never said no to a good cause — the ACLU, the NAACP, the United World Federalists, the American Friends Service Committee, and so on. "Everyone would like to do things for others," he told one reporter. "Let's just say that between acting assignments, I have time to do them."
He had an especially hard time turning down Hunt, a probing and articulate woman who had been blacklisted after her name appeared in _Red Channels_. Her professional hardship hardly had dented her social commitment; she may have been the only liberal in Hollywood capable of exhausting Ryan. At one point she called asking him to appear on a program and heard a sigh on the other end of the line. "Marsha, you know I'm with you, and all of the things that we're working toward," Ryan replied. "I think maybe it is someone else's turn for a while." Hunt would chuckle at the memory: "He put it so kindly and so gently, but he was perfectly right."
Always on the lookout for literate scripts, Ryan agreed to fly to New York in mid-March to appear in a TV adaptation of Ernest Hemingway's "The Snows of Kilimanjaro" for CBS. Sponsored by General Motors, _Buick-Electra Playhouse_ was a series of four 90-minute Hemingway specials scheduled throughout the 1959–60 season, the first two of which — an adaptation of "The Killers" starring boxer Ingemar Johansson and a production of _The Fifth Column_ , Hemingway's only play, with Richard Burton and Maximilian Schell — had aired the previous fall. _The Snows of Kilimanjaro_ , scheduled for broadcast on the evening of Friday, March 25, was to be shot and edited on videotape by thirty-year-old John Frankenheimer, one of the most talented and innovative of the young directors then coming up through TV. The strong supporting cast included Mary Astor, James Gregory, Brock Peters, and Liliane Montevecchi. But Ryan would be the lynchpin: as Harry Walters, the washed-up writer dying of gangrene on safari in Africa, he appeared in almost every scene, with flashback sequences that followed him to Paris and New York.
Hemingway famously had disliked the 1952 movie version starring Gregory Peck, which replaced the protagonist's death and spiritual deliverance with a happy ending in which a rescue plane arrives with penicillin just in the nick of time. The TV version promised to be much better: writer A. E. Hotchner had done several Hemingway adaptations, and he made liberal use of Harry's haunting interior narration, turning it into dialogue or voice-over. He had certainly nailed the character, whose regrets eat away at him even worse than the gangrene. Once a promising artist, Harry has squandered his talent doing mediocre work he thinks will sell, something Ryan might have identified with after _Ice Palace_. Given the actor's envy of Peck, he must have relished the opportunity to give a better performance in the same role.
But when Ryan arrived in New York a week before the broadcast, Frankenheimer came to him with bad news: someone at CBS had learned that taping the program would constitute a copyright infringement on the Fox release, so the teleplay would have to be performed live. Ryan, who had never done live TV, asked Frankenheimer what this would mean in terms of staging. "What it means is that you're gonna be in for the ride of your life," Frankenheimer replied. He was still trying to figure out how on earth they would make the transitions from the jungle set, where Harry is tended by his wife and their African guides, to the flashbacks, which showed his life leading up to the fateful safari.
Brainstorming with art director Burr Schmidt, Frankenheimer came up with a novel solution: their elaborate jungle set would remain, but the bed on which Harry lay dying would be placed on a large turntable unseen by the viewer, and a camera would be mounted on the turntable as well, at the foot of the bed. Whenever Harry slipped into one of his reveries, the camera would zoom in tight on his face, grips would push the turntable counterclockwise into the flashback setting, the camera would zoom out again, and Ryan would simply climb out of bed and walk into the next scene. To make this elaborate floor plan more manageable, Schmidt painted some of the flashback settings on giant sheets of paper that could be drawn behind the action like a curtain, and his impressionistic imagery would heighten the sense of a fevered memory play, turning a problem into a creative advantage. Frankenheimer thought their scheme would work, but the fact that his star had never done live TV was worrisome, to say the least.
To execute this high-wire act, CBS moved the production to Los Angeles, where Frankenheimer could take advantage of the network's giant Television City complex. The day he began blocking the program, CBS executives brought in the legendary Warner Bros. director Michael Curtiz ( _Captain Blood, The Adventures of Robin Hood, Yankee Doodle Dandy, Casablanca, Mildred Pierce_ ) to watch him work. Curtiz, whose career had declined in the '50s, was considering a move into TV, but this was hardly the best introduction.
"The madness of trying to block this thing, I mean, I cannot describe it to you," Frankenheimer told a seminar audience years later. "Cameras would come crashing through and just missing people... and booms coming and going through the paper, and we'd knock [the paper sets] down and put 'em up again.... We did kind of a stagger-through of this thing — you couldn't call it a run-through, because it was just insane." At one point Frankenheimer glanced over to the control room and saw the elderly, well-dressed Curtiz with his collar open and sweat running down his face. The old man watched for an hour and then fled, telling Frankenheimer, "You are crazy! This whole business is crazy!" He would never work in television.
The pressure of doing a live drama for network television was incredible; many actors vomited from nerves. As Frankenheimer would recall, the hour between dress rehearsal and airtime was the scariest: "You were sitting there, most of the time, with your own thoughts. And it was a very private time, because going on the air with one of these things — it really wound your watch, let's put it that way." Ryan always had sought to challenge himself as an actor, but this time he had his work cut out for him: barely leaving the screen for ninety minutes, he would have to turn on a dime from the African scenes, where Harry is swept into delirium by his advancing illness, to flashbacks that took place years earlier, played in a variety of moods. Twenty years into his career, Ryan never had so much riding on a single performance, and at 5:30 PM,* when an assistant director counted down the seconds to air and then cued the action, the actor would draw on everything he had ever learned onstage or in front of a camera.
Not only did Ryan pull off the live broadcast, but he also delivered one of his best screen performances. The extreme close-ups of Harry in his sickbed, closing his eyes and slipping away into the past, showed Ryan's great subtlety of expression, every thought registering in his strong features as the bed slowly spun beneath him (a faint grinding could be heard at one point, but otherwise the turntable worked like a charm). The flashbacks, by contrast, took advantage of his physical agility: when Harry gets into a fistfight with a British soldier in Paris, Frankenheimer concocted an elaborate ground-level shot in which the triangle formed by a woman's legs frames a silhouette of the men beating each other, cast in shadow against a brick wall.
The shot would have been difficult even for a movie, and there were plenty more like it. Yet aside from a bit of fumbled dialogue in the first act, Ryan negotiated it all with ease, giving an assured, impressively rich interpretation of the Hemingway character. "For anybody to do something like this, it was fabulous," Frankenheimer said. "For him to do it, never having done live television, was unbelievable." Then, at 7 PM, the performance was over, and few ever saw it again.*
RYAN LIKED TO TELL PEOPLE that when his agent phoned to offer him a part in _King of Kings_ , a biblical epic being shot in Madrid, he turned to Jessica and said, "Here we go again — Judas." But, in fact, his old friends Phil Yordan and Nicholas Ray, who were making the picture for producer Samuel Bronston, wanted him to play John the Baptist. The salary was $50,000, well below his usual fee, but his scenes would take only a week and then he could go home. From their perspective, attaching Ryan's name to _King of Kings_ provided some much-needed credibility, given that the _New York Times_ had just published a story about the picture's shaky financing.
The diminutive Bronston embodied the new breed of international film hustler: born in Bessarabia, then part of Russia, he had made a few pictures in Hollywood during the war but resurfaced more recently as a producer of historical spectaculars in Franco's Spain, where dirt-cheap labor and favorable banking policies allowed a producer to put a lot on-screen for very little money.** By this time Ray had peaked commercially with _Rebel without a Cause_ (1955), which crystallized the emerging youth culture in America; his masterful _Bigger Than Life_ (1956) and _Bitter Victory_ (1957) both had under-performed at the box office, and a pair of out-and-out flops ( _Wind across the Everglades, Party Girl_ ) had ended his career in Hollywood. He looked on this Spanish adventure as a chance to get his career back on track.
Attacking the project with his usual vigor, Ray immersed himself in the historical Jesus, fascinated by the political conflicts swirling around this so-called king of Judea. He hired Yordan to crank out a new screenplay presenting Jesus as a radical humanist, and Yordan hit on the plot gimmick of playing up Jesus' relationship to the criminal Barabbas, portrayed in his script as a violent revolutionary. Script in hand, Ray, Yordan, and Bronston managed to secure $5.5 million in financing from MGM, which hired Yordan to supervise production.
Ryan, having just played a "practical saint" in _Ice Palace_ and a genuine saint in _Murder in the Cathedral_ , now took a crack at John the Prophet, portrayed as a muscular evangelist in animal skins, a bushy wig, and a long beard. After Ryan arrived in Madrid, the company drove out to the countryside to shoot John's baptism of Jesus in the River Jordan. (Ryan would recall being chauffeured to the location in costume with costar Jeffrey Hunter, who played Jesus, and the two of them startling onlookers when they got out of their stalled car to give it a push.) There were a few more scenes, shot on a local soundstage, where John is imprisoned by Herod in a lonely dungeon. When Jesus appears at the window of John's cell, reaching a hand through the bars to comfort him, John scrambles up the stone incline to the window and manages to clasp his hand. "For the first time in my life," reported film editor Renee Lichtig, "I saw technicians weeping when silent rushes were screened."
Ryan was intrigued by the Spanish movie industry. Cinema was still the national entertainment there, and movie production was a more freewheeling affair than in Hollywood. Bronston burned through MGM's money at an astonishing rate; a story in _Family Weekly_ described a Sermon on the Mount featuring seven thousand extras "costumed in hand-woven desert fabrics scoured from every corner of the Mediterranean world.... Camels from the Canary Islands mingled with uncounted horses and burros. The sequence is so vast it had to be planned like a military operation." Originally budgeted at $5.5 million, the movie would top out at $8 million; talking to reporters upon its release, Ryan would hasten to point out that his $50,000 salary was for only a week's labor.
Back home from Madrid, Ryan looked forward to a blissful summer: Katharine Hepburn had asked him to costar with her in _Antony and Cleopatra_ at the American Shakespeare Festival Theatre in Stratford, Connecticut, and the family was coming along for a three-month stay. The 1,500-seat Festival Theatre was located on a lovely stretch of land that ended at the Housatonic River, and the Ryans had a house overlooking the water, only a hundred yards from the theater. Hepburn had been heavily involved with the festival from its inception, performing in three productions there since the theater opened in 1955. She adored Ryan personally and thought him a marvelous talent, and once they arrived she immediately befriended his family, bonding especially with his two sons and taking them on sailing expeditions. Eight-year-old Lisa, who often peeled off from the boys, spent many long afternoons watching in fascination as the company rehearsed. "When I first showed up, Katharine Hepburn was angry that there was a child sitting in the audience, and demanding to know why I was there. And my father said, 'Well, that's my daughter. She'll be quiet.' I'd just sit there all day."
Cheyney, Jessica, Tim, Robert, and Lisa Ryan in Stratford, Connecticut. "The family struck me as a rather private group," remembered Mike Metzger, who worked for the Ryans while studying at UCLA. "They were quiet and contemplative." _Robert Ryan Family_
Bob and Jessica in Stratford, Connecticut. "He was dependent on her for her critical attitudes," director Arvin Brown observed. "He admired a great deal what she had to say about him in performances, and he took her very seriously." _Robert Ryan Family_
Performing Shakespeare and relaxing with his wife and kids — for Ryan, show business didn't get any better than this, and as he sat in the nearby Fagan's-in-Stratford pub, having steak and a beer with a reporter from _Cue_ magazine, he was unusually frank about his career. "The junk I've played!" he exclaimed. "An actor spends years on junk. And then comes a chance to do something better — by the greatest dramatist that ever lived.... Just the effort to master the master increases the stature of any actor. I think that's why so many Shakespearean actors get better as they grow older. They grow greater with greater understanding of him." He admitted that his 1954 run as Coriolanus left something to be desired and that he would play the role again, given the chance.
Cleopatra was one of Shakespeare's strongest female characters, so the play naturally favored Hepburn, but in Ryan she had selected as her stage lover an actor whose power and virility projected to the back row. Writing about Ryan for the _Dartmouth Varsity_ , fellow alumnus Raymond Buck noted that teenage girls lined the hall outside his dressing room and swooned when he opened the door to admit Hepburn, who had come to introduce her father.
The critics were considerably less gaga: "Mr. Ryan's Roman clumps about in what seems to be a perpetual hangover, more stumblebum than fallen hero," wrote Judith Crist in the _New York Herald Tribune_. "We see no flashes of past greatness in his meeting with Lepidus and Caesar; he gives no impression of strength beyond the physical. True, he is besotted, but his monotone and single-keyed performance fails to evoke a past image or a present sympathy." Ryan always made a show of laughing off bad reviews, but when it came to Shakespeare, they cut deeper.
The family returned home after Labor Day, and the kids went back to school, though their parents, who knew a thing or two about educating children, had decided they didn't like the public schools in West Los Angeles. Lisa, who had been miserable at Warner Avenue Elementary, returned to Oakwood, while Tim and Cheyney were enrolled at the Harvard School for Boys, a military academy on Coldwater Canyon Drive in Studio City. Beginning that fall, Solomon Smith began driving them out to the Valley in the morning and picking them up in the afternoon. Lisa was overjoyed to be back at Oakwood, but the boys despised the Harvard School. Their parents were impressed by the school's academic reputation, though given their politics, the decision to send the boys to a military academy was surprising. "The problem with the school was not just the military side," Cheyney explained. "It was drawing on a kind of a conservative LA business element that was very racist, anti-Semitic, believe it or not. I don't think they understood that that was what they were getting into with the school."
With the kids squared away, Ryan took off for Cypress Hills, Saskatchewan, to shoot _The Canadians_ , a British adventure about the Royal Mounted Police that would be released in the United States by Fox. It had an interesting historical angle — the Mounties must deal with Sioux populations that have been driven north into Canada following the Battle of the Little Bighorn in 1876 — but the picture would die at the box office. ("Ryan, expressionless as his horse, gives a stolid performance," noted _Variety_.) The assignment pre cluded him from doing anything in the 1960 presidential campaign, though he contributed enough to Democratic nominee John F. Kennedy to be invited to the inauguration the following year (he did not attend). Jessica disliked Kennedy for his philandering, which was an open secret around Hollywood, and decided to vote for Eric Hass, the Socialist Labor candidate.
As a prominent member of SANE, Ryan was concerned chiefly with the nuclear test ban treaty being negotiated between the United States and the USSR in Geneva. The Soviets had unilaterally suspended atmospheric testing in March 1958, to be followed by the United States in August, but since then the talks had stalled. In January 1960, Ryan and Philip Dunne had written an open letter from Hollywood for SANE to the likely presidential candidates, asking them to express their support for the test ban, and the following month Ryan signed and helped pay for a full-page SANE advertisement in the _Washington Post_ and other papers nationwide. "Agenda for Geneva" suggested incremental steps toward disarmament that included dismantling all missile bases and suspending production of all nuclear, chemical, and biological weapons.
The letter to candidates notwithstanding, Hollywood for SANE concerned itself mainly with raising funds and corralling celebrities to take part in public-awareness campaigns. A series of radio spots had been recorded and distributed to some 650 disc jockeys and SANE chapters, featuring not only Ryan, Steve Allen, and Allen's wife, Jayne Meadows, but also such stars as Janet Leigh, Tony Curtis, Jack Lemmon, Anthony Quinn, Keenan Wynn, and Mercedes McCambridge. The two chapter presidents, Ryan and Allen, spoke at civic organizations around the Los Angeles area, along with James Whitmore and Ryan's friend Lee Marvin. Hollywood for SANE answered letters addressed to the many celebrities on its letterhead and distributed informational audiotapes to schools, universities, and discussion groups (one featured a panel with noted atomic scientists, recorded after the premiere of Stanley Kramer's postapocalyptic drama _On the Beach_ ). SANE chairman Norman Cousins would remember Ryan as a tireless advocate: "I can't think of an affair for the Federalists or for SANE that he didn't accept."
Earlier that year SANE had suffered the sort of internal convulsion that Cousins always dreaded. A giant rally was planned for Madison Square Garden in May 1960, to be followed by a Harry Belafonte concert at the Shrine Auditorium in Los Angeles. But in the days leading up to the rally, Democratic Senator Thomas J. Dodd of Connecticut — a vocal critic of the test ban — charged that the rally's chief organizer, Henry Abrams, was a member of the Communist Party of America. Abrams, a cochairman of SANE's West Side New York chapter, was summoned to appear before Dodd's Senate subcommittee on internal security, and when asked about his political association, he invoked his Fifth Amendment rights. Cousins asked Abrams to come clean with the SANE board, but Abrams refused. Faced with the prospect of SANE going down in flames, Cousins fired Abrams, and the national board instituted a new rule requiring chapters to screen out members of the CPA. A wrenching internal debate followed, and a quarter of the organization's local chapters, all located in the New York area, were ejected for noncompliance.
Cousins would be pilloried for his decision, yet he knew SANE could never achieve its goal of marshaling public opinion if it were tarred as a front for communists. The skittishness of the Hollywood chapter played no small part in this calculation; as Steve Allen would explain, SANE owed its unusually large star contingent to the fact that it was a squarely liberal organization, "a center to coalesce around that was not extremist, not considered hopelessly idealistic, not denominational, not unrealistically radical... an organization to whose center flocked respectable people of all sorts." Allen served as master of ceremonies that October when SANE met for its annual conference in Chicago; the attendees voted to endorse the new membership standards, but the organization was severely wounded by the whole episode.
After the new year nothing of interest materialized for Ryan. He slept late and puttered around his vast new home, enjoying the quiet while the kids were in school. Since the family's move to Holmby Hills, Jessica had begun work on a new novel and hired young Priscilla Ulene, a student at UCLA, to type up her longhand drafts, help with the shopping, and pick up the kids from school. She was followed in early 1961 by Mike Metzger, another UCLA student, who took over the same duties. Metzger came from a show business family — his grandfather was the great entertainer Eddie Cantor — and he grew chummy with the kids, introducing Tim to the finger-style guitar playing that would become his lifelong passion. Hammering away at the typewriter in Ryan's office, Metzger got a firsthand look at the household. "The family struck me as a rather private group," he said. "They weren't real social. There weren't a lot of people hanging around. They kind of went off and did their thing, whatever that was. They were quiet and contemplative."
Metzger knew Ryan from the movies, but in real life he was a different guy, always deep in thought. One afternoon Metzger was typing away in the office, a TV playing silently in the corner. Passing by the open door, Ryan glanced at the screen and saw one of his old black-and-white RKO pictures playing. "Oh my God, that thing," he remarked to Metzger. They watched it for a few seconds, then Ryan grunted and walked off. "Two minutes later he doubled back, came back, sat transfixed, turned the sound up, and watched the whole film, with great interest," Metzger recalled. "And he would make comments like, 'Ah, Christ!' as though he were reliving the making of that film. I probably said, 'Was that a particularly memorable film?' 'Nah!' You know, that kind of thing. But he was definitely interested."
Ryan always struck Metzger as "a giant animal in a small cage.... He seemed to me like a person who was really born to be out in the woods." Often Ryan would disappear to go hiking, either in the ravine behind the house or, as Cheyney recalled, up in the Hollywood Hills, around the reservoir where Hollis Mulwray's body would be recovered in the movie _Chinatown_. After one of his hikes out back, Ryan came into the office carrying something in a rag and invited Metzger to take a look. "Here was this dead hawk, and it was in perfect condition. He was amazed by it. He said, 'Look at this thing! It's beautiful.' He spread the wings and he said, 'There isn't a mark on this thing. Must have just died naturally. This is a beautiful creature.'" He was going to have it stuffed. Ryan walked off with his prize, and Metzger forgot about it.
A week later, Ryan was sitting in his armchair reading as Metzger worked, and from the basement they heard a shriek: Williana Smith had gone into the freezer looking for a package and discovered the dead hawk stowed there. "He jumped out of his chair. I mean, this guy is six-foot-six or something. And she's four-foot-eleven.... He did one of those, it was almost like a silent-movie take, where he went, 'Oh, _shit!_ ' And he made it through the French doors out into the yard.... She comes storming into the den, screaming, with this hawk in her hands.... 'Where is he? Where is he? I'm gonna skin him alive!' If you can picture seeing this huge man sneaking past the glass windows back and forth, trying to dodge her."
WHEN RYAN HEARD that British actor and director Peter Ustinov was developing a screen adaptation of Herman Melville's seafaring story _Billy Budd_ , he leapt into action, calling Ustinov personally to lobby for a role in the picture. Melville had left this Christian allegory unfinished when he died; published in 1924, it fueled the giant resurgence of interest in his work that had engulfed Ryan in his college years. The story opens in 1797, when the Revolutionary French Republic is flexing its muscles on the high seas and the Royal Navy is struggling to maintain order after major mutinies aboard two of its warships. Captain Edward Fairfax Vere, commander of the HMS _Bellipotent_ , fears that the rebellion may spread to his own vessel, and John Claggart, his black-hearted master-of-arms, carefully monitors the crew for any hint of conspiracy. Ryan must have known he would be offered the role of Claggart, but his love of Melville ran so deep that in this instance playing the villain didn't bother him.
As it turned out, he was a perfect fit for the project: Ustinov needed a Hollywood star to bolster the picture's commercial prospects in the United States and, as he later wrote, thought Ryan "a massive and wicked presence on the screen." Making Claggart an American actually was consistent with the story, in which he is rumored to be a foreigner — possibly a criminal — serving in a lowly rank with the British Royal Navy. Ustinov gave Ryan top billing but reserved for himself the more sympathetic role of Captain Vere; the title character, an angelic teenage sailor whom Claggart sets out to crush, would be played by a twenty-two-year-old theater actor from London named Terence Stamp, making his screen debut.
Ryan had played some sinister men in his day, but none so cold as Claggart, whose evil, Melville notes, is "not engendered by vicious training or corrupting books or licentious living, but born with him and innate, in short 'a depravity according to nature.'" Perceptive and intelligent, Claggart recognizes in Billy a sort of divinity: "If askance he eyed the good looks, cheery health, and frank enjoyment of young life in Billy Budd, it was because these went along with a nature that, as Claggart magnetically felt, had in its simplicity never willed malice or experienced the reactionary bite of that serpent.... One person excepted, the master-at-arms was perhaps the only man in the ship intellectually capable of adequately appreciating the moral phenomenon presented in Billy Budd."
Melville's story was hardly cluttered with incident — barely anything happens in its first half — but Ustinov and coscreenwriter DeWitt Bodeen had at their disposal an excellent stage version by Louis O. Coxe and Robert Chapman, who had first presented it at New York's Experimental Theatre in 1949 under the title _Uniform of Flesh_. From their version came the scene in which Billy approaches Claggart on deck one moonlit night and tries to reach him emotionally. When Billy notes the sea's calm, Claggart replies, "The sea's deceitful, boy: calm above, and underneath, a world of gliding monsters preying on their fellows. Murderers, all of them. Only the sharpest teeth survive." When Billy offers to keep Claggart company during his watches, the older man softens but then recoils: "No! Charm me too, would you? Get away!" Ustinov drew heavily on the play's structure and scenic development, steadfastly ignoring the homoerotic subtext of Melville's story; he would have enough trouble selling this as a family picture with its multiple lashings and its climactic scene of the hero being hanged.
Ryan and Peter Ustinov shooting _Billy Budd_ (1962) off the coast of Alicante, Spain. For Ryan, starring in a screen adaptation of Herman Melville was a dream come true. _Franklin Jarlett Collection_
_Billy Budd_ began shooting June 1 off the coast of Alicante, Spain, where Ustinov had expended much of his $1.4 million budget hiring two ships, one to play the _Bellipotent_ (here renamed the _Avenger_ ) and another to double as both the _Rights-of-Man_ , from which Billy is impressed, and a French warship that later attacks the British. A solid lineup of English character actors — John Neville, Cyril Luckham, John Meillon, Robert Brown — was embellished by handsome David McCallum (soon to become an American TV star on _The Man from U.N.C.L.E_.) and Melvyn Douglas, whose wife Helen had been pulverized by Nixon in the 1950 Senate race. Terence Stamp, who was tying himself in knots worrying about his performance, found Ryan distant on the set: "He never said two words to me. And it wasn't really until after the movie that I realized what a great favor he'd done for me, because the big scene, the pivotal scene between Claggart and Billy, was really difficult and really subtle.... He was a wonderful actor, and I think he sort of anticipated that. And he kept me at arm's length."
The company were met with a terrible storm the first day of shooting but continued undeterred for six weeks, shooting six days a week from dawn. Interviewed by _Variety_ , Ryan said his experiences aboard the _City of New York_ in the early '30s had steeled him against seasickness. Many of the cast and crew were dosed with Dramamine, and the cameras were equipped with a stabilizer to balance out the swelling and ebbing seas. When the exteriors were completed, the company took a short break, and Ryan traveled to County Tipperary in Ireland to search for his family's roots. Production resumed at British Elstree Studios in Hertfordshire, England, and continued through August; by the time he returned home in September, another school year had started.
"John the Baptist and Claggart mark a new epoch," Ryan observed. "I used to feel like a plumber in most of my past movies." _Billy Budd_ collected glowing reviews in the United Kingdom and United States when it opened more than a year later, bolstering the independent Allied Artists. Ryan was delighted with the picture and proud of his performance, a remarkably detailed piece of work reaching deeper into a malignant soul than any other he had given. He dominated almost every scene, finding his match only in Ustinov. His most incisive moment may have been the one in which Claggart, counting off the lashes of a man's corporal punishment, reaches his proscribed limit, and the grim pleasure in his face gives way to weakness and even need as he is denied any more. In one of the subtle movements that were Ryan's stock in trade, Claggart turns away, unconsciously swatting his leg with a swagger stick to deliver the additional strokes. No matter how many saints Ryan might play, he would always be more intimately acquainted with the serpents.
*The special was restored in 2009 as _A Call from the Stars_.
*The program was performed live for the Eastern and Central time zones, and videotaped for broadcast later that evening in the Mountain and Pacific zones.
* _The Snows of Kilimanjaro_ has never been released to home video, though it can be viewed by request at the New York or Los Angeles facilities of the Paley Center for Media.
**His first such effort, the flag-waving flop _John Paul Jones_ (1959), was bankrolled by Pierre du Pont III, an heir to his family's fortune.
_twelve_
The Longest Day
While Ryan was in England finishing _Billy Budd_ , he saw a newspaper story about himself whose headline noted he had been married for twenty-two years. "They seemed to be more impressed with this fact than what I did as an actor," he later joked. Yet filming this last picture had kept him away from the family for three long months, during which time he could only monitor by phone and letter the condition of Solomon Smith, their longtime handyman, who was dying of lung cancer.
Smith was more than a servant; after more than twelve years around the house, he had become a surrogate father to Ryan's children — especially Lisa, who liked to eat dinner in the kitchen with the Smiths while her parents and brothers were yacking about politics in the dining room. "If I was afraid in the middle of the night, I would go and climb into bed with Willie and Smith," she explained. "They were the people that I would go to before I would go to my parents. I don't know why exactly, but they were much more accessible. And they were around a lot more than my dad was." Ryan went to visit Smith on his deathbed, where the big man had shriveled to about a hundred pounds, and came home badly shaken. Smith died on November 26, 1961, at age sixty-three.
His passing left a hole in the family that Christmas season, and only a few weeks later Ryan departed again, heading this time to France to shoot a few scenes for Darryl Zanuck's star-studded war epic _The Longest Day_. Based on a gripping nonfiction book by Cornelius Ryan, the picture chronicled the Allied invasion of Normandy on June 6, 1944, and featured a giant international cast, though few of the actors had much more screen time than Ryan.
As General "Jumpin' Jim" Gavin, who commanded the parachute assaults of the Eighty-Second Airborne Division (and who now served as a technical advisor on the picture), Ryan shared his only major scene with John Wayne, playing paratrooper Lieutenant Colonel Benjamin Vandervoort. They enjoyed each other's company during the shoot, though Ryan never had considered Wayne to be the sharpest knife in the drawer. Wayne was appalled by SANE and all it represented. "He probably wonders why I think the way I do," said Ryan. "He figures I ought to wear horn-rimmed glasses and be five-foot-four. He's fairly conservative and I'm fairly liberal — whatever that means."
In practical terms it meant a good deal. On Monday, February 5, 1962, Ryan received a long-distance call from Jessica: the previous evening, someone had phoned the house on Carolwood Drive and promised Williana Smith that a bomb attack would follow if Ryan took part in a scheduled radio program about the archconservative John Birch Society. Listener-supported KPFK-FM in Los Angeles had produced a weeklong series, to begin Monday night, that combined panel discussion with recordings of Hollywood stars reading from the society's "Blue Book."
The previous week, bombers had struck the homes of two local ministers, John G. Simmons of North Hollywood and Brooks Walker of Canoga Park, as they took part in a public discussion about the radical right. Ryan's friend Marsha Hunt participated in that same event and remembered leaving the synagogue where it was held to find every car in the parking lot with a leaflet pinned to its windshield: "It had on it the communist hammer and sickle, the Jewish Star of David, and the United Nations wreath of peace, all concentric... and three words across the bottom: _Know Your Enemy_."
KPFK had received three bomb threats as well; one caller warned, "You commies are next." Jessica instructed the station to broadcast her husband's taped reading as planned, and though Ryan wanted to drop everything and come home, she persuaded him to stay put. He arranged for a security company to guard the house and escort the children to school, which excited Lisa and embarrassed the boys, but there wasn't much else he could do from across the Atlantic. "I have talked to Mr. Ryan in France and he was quite disturbed," Leonard Kaufman, a family friend, told the _Hollywood Citizen-News_.
Wayne was furious when he heard the news and volunteered to fly back with Ryan and be photographed guarding the family's home personally, an offer that touched Ryan and later morphed into one of his Irish stories. As Philip Dunne repeated it in his autobiography, Bob and Jessica arrived home one night and "spotted a man armed with a rifle standing at their front door. Bob slammed on the brakes, and was starting to back out, when the intruder waved and stepped forward into the glare of the headlights. It was dedicated right-winger Wayne, on sentry duty to protect his friend and colleague, liberal or not." According to the Ryan children, this episode was pure fantasy.
Despite Jessica's firm response to the incident, the couple was spooked; and after Ryan returned home from France, they decided to vacate the house in Holmby Hills temporarily and relocate to Ojai, California, an hour and a half northwest of Los Angeles. At first Tim and Cheyney were told they would be boarding at their school, which set Cheyney to wondering, "What can I ingest that will make me so sick that I can't board at Harvard Military School?" But a week later his parents changed their minds and announced that all three children would be accompanying them and transferring to new schools; Willie, still grieving for her late husband, would remain in Los Angeles. At first they rented a cabin at the Ojai Valley Inn and Country Club, a posh resort where Spencer Tracy and Katharine Hepburn had shot their beloved comedy _Pat and Mike_ (1952), then they sublet a home in the area. Tim and Cheyney, whose school routine had included drilling in uniform, now enrolled in a school run by the Krotona Institute of Theosophy, which was even freakier than the Oakwood School with its curriculum steeped in Eastern mysticism.
The kids had been told they would return to LA after things cooled down, but after three or four months their parents dropped another bombshell on them: now the family was moving, at least temporarily, to New York, where their father hoped to jump-start his ailing career by starring in a new Broadway musical called _Mr. President_. "You don't mean to tell me that people are going to pay money to hear you sing?" asked fourteen-year-old Cheyney when he heard the news. But Ryan was much taken with the idea: Rex Harrison didn't have much of a voice either, yet he had transformed his career in 1956 with _My Fair Lady_ , and Robert Preston had pulled off the same trick the following year with _The Music Man_.* Jessica considered the Broadway show a wonderful idea and couldn't wait to get out of California; after Ryan headed East in July to start working on the show, she began arranging the move to New York.
_Mr. President_ looked like a good bet: producer Leland Hayward, director Joshua Logan, and the writing team of Howard Lindsay and Russel Crouse had already collaborated, in various combinations, on such classics as _State of the Union, Mister Roberts, South Pacific_ , and _The Sound of Music_. Most impressive of all, the songs for _Mr. President_ would be penned by the great Irving Berlin, returning to the theater at age seventy-four after a decade in retirement. Katharine Hepburn, knowing of Ryan's secret ambition to try musical comedy, had recommended him to her friend Howard Lindsay for the title role, and Ryan had flown out to audition in June, singing Kurt Weill's "September Song" and Berlin's "Always" for the composer and Leland Hayward. According to Ryan, Berlin told him, "That last note was great," and a half hour later he was signing a contract. Nanette Fabray, his old friend from the Reinhardt School, would costar as the long-suffering First Lady. After seven weeks on the road in Boston and Washington, DC, _Mr. President_ would open in New York on October 20, at the seventeen-hundred-seat St. James Theatre on West Forty-Fourth Street.
As promising as all this seemed from a distance, Ryan realized once rehearsals began that the show was in deep trouble. Howard Lindsay was suffering from the early effects of leukemia, and Russel Crouse had just undergone surgery to remove a blood clot; in their weakened conditions they couldn't assemble a decent script by the time rehearsals commenced, and as Logan remembered, they resisted any revisions. Ryan later described the show's genesis as "a dogfight, and one unhappy experience after another. In the first two weeks, the dance director wasn't speaking to the composer, and the feuds just grew after that." Logan and Crouse clashed over the length of the show and Logan's desire to underscore some of the dialogue with music. Berlin's songs were good, and his enthusiasm drove the production along, but Ryan braced himself for the worst as the show lurched toward its Boston opening. Once the curtain rose, he would be at the center of this whole thing, with his homely baritone and negligible dance skills, and at this rate it might crumble around him.
The hype surrounding _Mr. President_ only increased the pressure on everyone involved: news of Berlin's comeback, combined with the sterling reputations of the producer, director, and writers, had driven advance ticket sales to an astounding $2.5 million. At the same time, rumors swirled that _Mr. President_ would satirize the Kennedy family, when in fact, Lindsay and Crouse had written a rather melancholy story about a lame-duck president, Stephen Decatur Henderson, preparing to leave office, move back to the Midwest, and resume life as a private citizen (only Berlin's participation had turned the play into a musical). Talking to the press, Ryan did his best to deflate the Kennedy rumor, which might prove fatal once people discovered how apolitical and distinctly old-fashioned the show was. He must have kicked himself for committing to the project without having read the script, but almost everyone else had made the same mistake.
As usual, Ryan kept his head down and worked hard, worrying most about his singing. Berlin had given him some good numbers: "In Our Hide-Away" was a loping, eminently hummable tune in which he held down the melody while the more skilled Fabray snaked around him providing harmony, and the husky ballad "It Gets Lonely in the White House" considered the solitary burden of the presidency, concluding, "The White House is the loneliest place in town." But the book refused to come to life; in the age of the Cuban Revolution and the Berlin Wall, it seemed irrelevant even for musical theater. _Mr. President_ was savaged by the Boston critics, as well as _Time_ and _Newsweek_ ; in late September the cast and crew moved on to the National Theatre in Washington, where the opening performance, a benefit for the Kennedy Foundation, was attended by Jacqueline Kennedy. "There was literally no laughter at all," Logan wrote. "Every time a joke was launched from the stage, the audience, like an audience at a tennis match, looked to the First Lady to see if she was laughing, and then turned back to the play, stony-faced."
The president, preoccupied with top-secret reports of Soviet missile silos in Cuba, missed most of the performance, though he came backstage afterward and, to Ryan's surprise, brought up their momentary encounter ten years earlier in Boston, during the first Stevenson campaign. A gala party followed at the British embassy, heavily attended by the Kennedy family, administration figures, and the diplomatic corps. _Variety_ reported that the show was being revised throughout its Washington run, as Logan pumped up the production numbers to compensate for the lifeless script. The flag-waving finale, wrote Les Carpenter, could have been topped only by having Ryan "rip off his shirt to display the US Constitution tattooed on his chest."
By the time _Mr. President_ arrived on Broadway it was already damaged goods. "It is always painful when a man you admire introduces you to his awkward, charmless fiancée," wrote Walter Kerr in the _New York Herald Tribune_ , encapsulating the sense of disappointment that surrounded the show. Two days after the New York opening, President Kennedy addressed the nation to reveal the presence of Soviet missiles in Cuba, and Americans held their breath, wondering if nuclear annihilation was at hand, until Premier Khrushchev announced on October 28 that the missiles would be removed. Jessica would remember her eldest, Tim, observing that the crisis must be worse for her generation than for his: "We have always sort of taken it for granted that sooner or later they'd blow up the world. But you can remember a time when it couldn't be done. So it must be harder for you to get used to the idea that it may happen tonight. Or tomorrow."
Ever the good soldier, Ryan always talked up his projects in the press — when _King of Kings_ had opened to scathing reviews a year earlier, he assured one reporter that it would clean up at the Academy Awards (it failed to receive a single nomination). Now he would go to bat for _Mr. President_ too. "Perhaps our mistake was in opening in New York after the critics had been torn to pieces by 'Who's Afraid of Virginia Woolf?' which is not at all like 'Mr. President,'" he said. "But the old people like it and I think we should make the old people happy." Privately he conceded the show was a dud and, according to Harry Belafonte, discouraged his friends from seeing it.
Unfortunately for Ryan, the advance sales guaranteed _Mr. President_ a long run despite its dim critical reception, and it held out for 265 performances, closing in June 1963. For days, weeks, and then months, Ryan tried to give his best to a show that refused to die despite the fact that nobody seemed to like it much. The loneliest place in town, it turned out, was center stage at the St. James Theatre.
INSTEAD OF MOVING TO MANHATTAN, which seemed too wild for the kids, the Ryans decided to find a place in Westchester County, close enough for Bob to commute into the city every day. Sunday was his only day off from rehearsals, and with Jessica and the children arriving soon, he quickly rented a house in the wealthy village of Bronxville. "The furnishings aren't very fancy," he explained, "but it was the _only_ house for rent in Bronxville, and we wanted to live in Bronxville because the public schools here are very good." The children thought they would be returning to Los Angeles after the show closed — whenever that was — but privately their parents had more or less decided to stay on the East Coast for good. For financial reasons, more and more American movies were being shot in Europe now; living near New York would lessen Bob's travel time and even enable him to come home occasionally during long shoots. The house on Carolwood Drive was put on the market, where it would stay for some time, owing to a recent dip in demand for luxury properties, before finally selling to writer George Axelrod.
The move was rough on Tim, Cheyney, and Lisa — by this time they had changed schools four times in three years — and though Bronxville might have struck Ryan as the sort of quiet bedroom community the family had enjoyed in North Hollywood, they soon discovered that their neighbors were nothing like the ones they had known in the San Fernando Valley. Bronxville was a lily-white suburb, hostile to blacks and Jews and not terribly fond of Catholics either, especially if they were in show business. "I was so shocked, because I'd grown up in this completely Jewish environment," remembered Lisa Ryan. "I couldn't understand it. And racist — I mean, it was just awful."
Williana Smith had come along with the family but found the experience so dismal that she returned to Los Angeles after a few weeks, ending her employment with the Ryans after fourteen years (the children would keep in touch with her until her death in 1988). Clearly Bronxville was a mistake, but they had signed a lease, and doing _Mr. President_ six days a week was so grueling that for the time being they would just have to stick it out. The bitterly cold winter took the children by surprise, and everyone in the family came down with chicken pox.
Though _Mr. President_ kept Ryan at home, his schedule barely overlapped with the children's. He slept late and sat around in his pajamas and robe until late afternoon, then showered, dressed, and drove down to the theater off Times Square, returning home around midnight and going to bed a couple of hours later. During the run he granted a long and fascinating interview to _Holiday_ magazine writer Joe McCarthy, who paid him a visit in Bronxville and noted that, in contrast to the Cadillacs and Jaguars parked at other houses in the neighborhood, the Ryan fleet consisted of a Falcon station wagon and a black Buick sedan.
During the interview (which would never run), Ryan reflected on the new experience of doing musical comedy onstage. "There are certain things about acting in a musical comedy that I'll never learn," he admitted. "Did you ever notice that a good musical comedy performer, like Bert Lahr or Ethel Merman, or Nanette Fabray in our show, doesn't speak the lines to the other actors in the scene? They turn and speak to the audience, like the comedians in the old-time burlesque shows. I can't do that."
Bronxville aside, Ryan quite liked being back in New York, where he had lived first as a young man out of college and later as a newlywed husband hitting the straw-hat circuit with his fine young wife. The Ryans had good friends here: Robert Wallsten, who had acted alongside them at the Millpond Playhouse, and his wife Cynthia; married screenwriters Frances Goodrich and Albert Hackett, whose track record included _The Thin Man_ (1934), _It's a Wonderful Life_ (1946), _Easter Parade_ (1948), and _Father of the Bride_ (1950); and Millard Lampell, who had met Ryan through director Dick Brooks back in the late '40s, before Lampell was listed in _Red Channels_ and his screenwriting career ended. Starring in a Broadway show gave Ryan a good opportunity to renew old acquaintances; he looked forward to the time when he and Jessica could move the family into the city, as they ought to have done all along.
Unwilling to relocate to the East Coast, Mabel Bushnell Ryan had remained behind in Los Angeles, and in March 1963 her son received word that she had died of a heart attack at her home in North Hollywood, at age seventy-nine. "I never saw him so distraught," said Wallsten, who came over to the house in Bronxville and found Ryan in his bathrobe. "When he burst into tears I was very surprised that he was that moved, although these things are always more important to people than they let on. I remember him putting his arm around my shoulder and squeezing so hard that it hurt as he wept." The family traveled by train to Chicago, and Mabel was buried in Calvary Cemetery beside her son Jack, who would have been fifty-one now, and Old Tim, dead for twenty-seven years. Ryan was the last survivor of the little family in Uptown, yet he couldn't conceive of being laid to rest anywhere but beside Jessica.
Once the lease in Bronxville expired, the Ryans wasted no time in getting out. That summer they stayed at a place in Westport, Connecticut, and in the fall they moved into a spacious apartment on the top floor of the Brentmore, located at 88 Central Park West and facing the park at the corner of Sixty-ninth Street. "Frances and Albert Hackett lived there," said Cheyney, "and I remember that was important to Dad because he felt that people he knew lived there."
Ryan was entering a scary new period in his career: apparently the stink of _Mr. President_ clung to him, because he couldn't get a movie to save his life. _The Canadians_ had sunk without a trace in the United States, and _King of Kings_ had been ridiculed in the press (with Jeffrey Hunter tagged the "teen age Jesus"). Ryan's performance in _Billy Budd_ had won glowing reviews, but the picture was poorly marketed in the United States and failed to have the impact he had hoped for; Terence Stamp won an Oscar nomination for best supporting actor, but the picture had been eclipsed by another maritime drama, MGM's remake of _Mutiny on the Bounty_ with Marlon Brando. Ryan had abandoned Hollywood to become a Broadway sensation, and now Hollywood had abandoned him.
He kept busy as best he could. Millard Lampell had written a documentary feature called _The Inheritance_ , about immigrant labor and the union movement, and Ryan offered to supply the narration for a token fee of one thousand dollars. Before getting into pictures, Lampell had performed in the Almanac Singers with Woody Guthrie, Pete Seeger, and Lee Hays, and the documentary reflected his folk roots; over one section Ryan recites:
Layin' down track for the west-bound train
Stackin' up timber in the State of Maine
Diggin' out coal in the West Virginia hills
Hammerin' steel in the Pittsburgh mills....
Six-day week and a 12-hour day
And it's welcome boys to the USA
That November Ryan also managed to pick up a starring role in an hour-long TV drama for NBC's _Kraft Suspense Theatre_. "Are There Any More Out There Like You?" was an interesting little piece about a wealthy man whose collegiate daughter, played by young Katharine Ross, is arrested after she and her friends drunkenly run over a pedestrian. The rest of the time Ryan loafed around the house playing pool on his new, regulation-size table in the apartment, or at some of the old billiard parlors on Broadway. By now he was smoking again, and his daily diet of a couple Löwenbräus had given way to a couple glasses of J&B Scotch.
Ryan especially liked slipping out for a game of pool late at night, at some of the old halls where legendary players might turn up. But going out in public could turn sour at a moment's notice. One evening Ryan took Cheyney with him to shoot some pool, and a crowd gathered around them as they played. "What a nightmare that was," Cheyney recalled. "These guys were big admirers of the war movies and stuff. _Odds against Tomorrow_ apparently had been on television within the last week, and there's a scene where Dad is supposed to throw some keys to Harry Belafonte and he doesn't do it 'cause he's a racist. We're trying to play pool and this guy keeps yelling at us, 'Hey Bob, why didn't you throw Harry the keys?' Over and over again, and then laughing. Even if he'd been inclined to do dad things with his son, you'd go out and weird shit like that happens."
Now that Ryan lived in New York, he spent more time with Belafonte, getting together for the occasional dinner or game of pool but also becoming more involved in the civil rights movement, which hadn't reached the same level of intensity on the West Coast as it had in the South and the northern industrial cities. Belafonte had a sixteen-millimeter projector in his home and sometimes hosted movie nights for his friend Martin Luther King Jr., who loved movies but seldom went out to theaters. On one occasion, Belafonte invited Ryan, Anthony Quinn, and other friends to meet the minister, and they watched _Come Back, Africa_ (1959), Lionel Rogosin's striking drama about apartheid in South Africa, which had introduced the United States to singer Miriam Makeba. Like Ryan, King was a longtime SANE sponsor, and the two men seemed to connect. Ryan "was quite enamored of Dr. King and quite humbled by the experience," Belafonte remembered.
Ryan signed and bankrolled advertisements supporting the Southern Christian Leadership Conference, and at Belafonte's behest he took part in several civil rights forums around town. In August 1963, he and Belafonte flew to DC for the March on Washington, joining a large contingent of Hollywood celebrities that included Sidney Poitier, Marlon Brando, Burt Lancaster, Tony Curtis, Sammy Davis Jr., Tony Bennett, Lena Horne, and Charlton Heston. At the Washington Monument, Lancaster delivered a scroll brought from Paris with the signatures of some fifteen hundred French artists expressing their support, and the entertainers joined the march to the Lincoln Memorial, where King captured the nation's imagination with his historic speech. Tim Ryan attended the march on his own and was startled to see his father onstage. A few months later Ryan would help bankroll an ambitious production slate by the Free Southern Theatre in Jackson, Mississippi, an independent black company that mounted plays by Langston Hughes, James Baldwin, Ossie Davis, and John O. Killens.
Upon returning to New York, Ryan had also reengaged with SANE, serving as emcee for a program at New York Town Hall in spring 1963, and on November 15 the organization celebrated its fifth anniversary with a dinner honoring Steve Allen at the Biltmore Hotel. SANE had reason to celebrate. After the United States and the USSR had resumed atmospheric testing in March 1962, the organization mounted a vigorous campaign to restart the Geneva talks, including a highly influential advertisement that featured the trusted pediatrician and author Benjamin Spock. In the wake of the Cuban missile crisis, Norman Cousins had served as a secret envoy from President Kennedy to Premier Khrushchev, and subsequently the two superpowers managed to agree on a partial ban that would permit tests underground but prohibit them in the atmosphere. The Senate's ratification of the ban in September 1963 gave new hope for détente and disarmament.
A week after the dinner, Kennedy was assassinated in Dallas. "I remember walking to the window, and cars were going through red lights and driving rather aimlessly," Ryan recalled. "The whole city seemed to be in shock at that moment, and it stayed that way, all that day and the next." Schools let out early, and Ryan drove across town to pick up Lisa. "I could have just gotten home by myself, but he made a point of coming and getting me," she said. "He was crying." Like the rest of the country, the Ryans spent the weekend clustered in front of the TV, trying to make sense of what had happened and recoiling in shock as the suspect, Lee Harvey Oswald, was murdered on camera by another mysterious assailant. "Dad was beside himself," said Cheyney. "I think he felt the whole country was just falling apart, and God knows what was gonna happen next. He didn't put on his clothes for three days.... I remember him ranting about Texas and how stupid they were down there, and these idiots with their ten-gallon hats."
In early 1964, Ryan turned to TV, doing guest spots on dramatic series and hosting _The Bell Telephone Hour_ ; the latter gave him a chance to perform the recitative for an orchestral performance of Aaron Copland's _Lincoln Portrait_ as Matthew Brady photos of Lincoln filled the screen. It was pretty square stuff compared to _The Ed Sullivan Show_ two nights previous, which had introduced America to the Beatles, but in fact Ryan often characterized himself as a square. Millard Lampell's documentary, _The Inheritance_ , opened in May at the Carnegie Hall Cinema, and Ryan took part in the premiere, which included folk music by Judy Collins and Pete Seeger. His intelligent handling of the picture's narration was much noted and led to more voice-over and spoken-word gigs, the honorable piecework of the fading movie star. Later that year CBS hired him to perform the words of Abraham Lincoln for a one hour special, _The Presidency: A Splendid Misery_ , and to narrate an ambitious eleven-hour series, _World War I: The Complete Story_.
That summer Ryan finally managed to score a picture, dodgy as it may have seemed, when Argo Film Production, a British independent, cast him alongside English actor Stewart Granger in _The Crooked Road_ , an international intrigue to be shot in Zagreb, Yugoslavia. "When they first discussed this project with me they inquired what my demands might be," Ryan told the _Los Angeles Times_. "Only two, I told them — a good script and my salary in advance. I got both." The script wasn't bad at that; adapted from a novel by Morris L. West, it critiqued the United States' support of corrupt governments abroad, though its more serious intentions were undercut by a far-fetched love triangle. Richard Ashley (Ryan), an American journalist in a fictional Balkan nation, comes into possession of some letters proving that its silver-haired leader, the Duke of Orgagnia (Granger), has been lining his own and his backers' pockets with economic aid from the United States. Ashley's pursuit of the story throws him back into the arms of a former lover, Cosima (Romanian actress Nadia Gray), who is now married to the duke but disgusted by the venality of his rule.
Ryan must have been drawn to the script's realpolitik angle. "You, my dear Ashley, are a man of a new world, America," the duke points out. "A great country, a great people, but sometimes so naïve.... Your country invested millions and millions of dollars here for one reason, to prevent a communist government. Is there a communist government? No! Well then, why are you so outraged at some of the methods we adopt to reach that end?"
Of course, Yugoslavia had a communist government, one whose leader, Josip Broz Tito, loved Hollywood movies (his favorite stars were John Wayne and Kirk Douglas). Shortly after the Federal People's Republic of Yugoslavia was declared in 1945, a government order established the Central Film Studio in Belgrade; originally, it operated in partnership with the venerable Soviet studio Mosfilm, but after Tito broke with Stalin in 1948, the Yugoslavian outfit, better known as Avala Film, began courting international co-productions. Studio boss Ratso Drazevic, formerly with the state security apparatus, knew how to move money around, and production funds from Western partners had a way of migrating to other state projects.*
"For the filmmaker, the conditions are close to ideal," Ryan told the _Los Angeles Times_. "They have Westrex sound, all brand-new equipment. The technicians are a little slow, but excellent. The producer has his choice of the finest classical actors in Europe. They speak a myriad of languages and sometimes the set sounds like Babel, but it doesn't matter." The dialogue, he explained, was usually postdubbed, though in the case of _The Crooked Road_ , direct sound was recorded and "everyone spoke English, sort of. I imagine there will be considerable rerecording prior to release." Visiting movie stars were treated like royalty in Belgrade, lodging at the lavish Hotel Metropol, and the countryside was stunning; the picture's big action sequence, with Ryan and Gray tooling down the highway in a Mercedes convertible, was shot along the Dalmatian coast.
The job forced him to miss a big "Stars for SANE" event, which was a bit embarrassing given that he still cochaired Hollywood for SANE. "I am now in Split, Yugoslavia, trying to start a SANE branch to be known as Split for SANE," he wrote in a letter to the attendees. "Of secondary importance is the fact that I am also here trying to make a movie and a living." Ryan took advantage of the opportunity to pay tribute to JFK: "The Test Ban Treaty, for which our late President will indeed be remembered, is the first step of a long trip. We still have the atom and we always will; we still have the bomb and the ever dwindling time before it falls into possibly maniacal and homicidal hands. We must, therefore, work unceasingly to extend and enrich the period of grace in which we now live."
Back in New York, Ryan tried to get used to the idea that his children were growing up. Tim had graduated the previous spring from the Collegiate School on the Upper West Side and spent the ensuing year with the American Friends Service Committee, caring for disabled children at a rehabilitation center in New Hampshire and helping an Episcopal priest dispense social services to the poor in Jersey City. That summer he had returned to Los Angeles, working for an American Friends day camp in Watts prior to his enrollment that fall at Pomona College in Claremont, California. Cheyney, now sixteen, had pulled himself out of an academic rough patch and become a star student at the Collegiate School; he was spending the summer in rural Kentucky, teaching literacy for the American Friends. The tumult of moving to New York, combined with the usual stresses of adolescence, had put some distance between the boys and their father; once they grew old enough to converse on his level, they began to understand what a sealed envelope he was. Friends would recall Ryan's enormous pride when he spoke about his children, but around them he kept his feelings to himself.
That fall the family moved again, but only three blocks, to a grand twelve-room apartment at the Dakota on Central Park West and Seventy-Second Street. Built in the 1880s, the seven-story, Gothic-style apartment building featured an interior courtyard with an entrance on Seventy-Second; the Ryans' unit had a large living room facing east and overlooking the park below, while the dining room and kitchen were at the west end of the unit, overlooking the courtyard. "It was an elegant apartment, but he and Jessica didn't spend any money fixing it up, or making it lavish," said Millard Lampell. "The old, worn furniture that they had from the California days did them perfectly fine; they weren't interested in impressing anybody, and although it was a big apartment, it was somehow probably the only apartment in the Dakota that I was ever in that felt kind of homey and comfortable." Lauren Bacall lived in the Dakota with her second husband, Jason Robards; among the other tenants were designer Ward Bennett, author Betty Friedan, and actress Judy Holliday (whose kitchen window was across the courtyard from the Ryans').
Ryan took on more TV work, shooting a pilot episode for _Indictment_ , a legal drama to debut the next year on NBC. Produced by Universal-TV and shot in color, it featured Ryan as a crusading assistant DA, with Robert Duvall of _To Kill a Mockingbird_ and Richard Beymer of _West Side Story_ as attorneys on his staff (the series never got picked up). Ryan also guest-starred on episodes _of Wagon Train_ and a new CBS series called _The Reporter_ , starring Harry Guardino.
Years later, Cheyney met Jerome Weidman, who had created the series and invited Ryan to appear, and Weidman recounted Ryan's on-set clash with Guardino. Production of TV series was tightly scheduled, and on the first day of shooting Guardino didn't show up until the afternoon, letting the cast and crew cool their heels. When this happened again the second day, said Weidman, Ryan strode up to Guardino and told him off in front of everybody: "If you pull this again, I'm gonna punch your lights out. Because we're professionals here, and this is unacceptable."After that, Guardino showed up on time.
"I'm available," Ryan told the _Los Angeles Times_. "All I want is a good script. Here at home I won't require my salary in advance." Even this abject plea failed to elicit a good offer from the studios, so after the holidays Ryan flew off to West Germany to appear in another international co-production, a spy thriller called _The Secret Agents_. The picture would be shot by four different directors in Berlin, Paris, and Rome, and Ryan would be the character linking these episodes, a US intelligence officer who negotiates with the Soviets for the exchange of captured spies. Henry Fonda had been signed for the Berlin segment, playing an American operative who returns from the German Democratic Republic; the other principals were Italian actor Vittorio Gassman (Big _Deal on Madonna Street_ ) and French actress Annie Girardot ( _Rocco and His Brothers_ ). All three were billed above Ryan, but he liked his world-weary character, and the picture had been conceived as a TV pilot, which might mean a full-time job later. "What I'd really like," he told one reporter, "is to get a TV series in New York so I could stay home all the time."
Berlin was much changed from the ruined city he had visited in August 1947, when he and Merle Oberon were sneaking around behind her husband's back during location shooting for _Berlin Express_. Oddly, both pictures ended the same way, with a friendly encounter between an American and a Russian at the border between East and West. In _The Secret Agents_ two cars meet on a suspension bridge, and Ryan's general makes an exchange of prisoners with his Soviet counterpart, played by Wolfgang Lukschy. Their business concluded, the men share a smoke; the American promises to have a carton of the cigarettes delivered to the middle of the bridge, and the Russian tells him there will be some caviar waiting. As they're returning to their cars, Ryan concludes in voice-over, "Hell of a way to make a living, isn't it?" Unfortunately, _The Secret Agents_ was distributed in the United States by American International Pictures, the cheapo outfit that had given the world _Beach Party_ and _I Was a Teenage Frankenstein_ ; chopped down from two hours to ninety minutes, furnished with a cheesy organ score, and retitled _The Dirty Game_ , it quickly disappeared from theaters.
Back in the States, Ryan played Abraham Lincoln for the fourth and last time when Dore Schary asked him to come to Washington and perform in costume as the sixteenth president on the east steps of the Capitol. Schary was re-creating Lincoln's second inaugural exactly one hundred years later, on March 4, 1965, and for this kind of oratory, at least, Ryan's flat, midwestern voice worked to his advantage. Captured on film for later broadcast on TV, the event drew some thirty thousand people, the largest live audience of Ryan's career.
"Robert waited for work to come to him, he didn't go out and seek it," Millard Lampell observed. But when Ryan heard that Sid Harmon and Phil Yordan had cut a deal with Warner Bros. to shoot a Cinerama epic about the Battle of the Bulge, he phoned them and asked if they could find him a part. They did more than that: according to Yordan, when he went to Jack Warner to propose Ryan for the cameo role of General Grey — whose subordinates would be played by Henry Fonda, Robert Shaw, and Dana Andrews — he persuaded the studio chief to pay Ryan his top salary of $125,000. "He would have worked for short money to come back," Yordan said, "but then it would have been a hard road coming back because everybody watches the salaries. When he came back he worked for his full salary, and from then on he made more money than he ever made in his life." Ryan departed for Spain in March and soon completed his scenes, the best of which paired him with Fonda. The two men became friends and even talked about doing theater work together; Fonda had long wanted to start a company on the East Coast and develop work for the New York stage.
Cheyney returned to Kentucky that summer to continue his literacy work, while the others — including Tim, who had finished his first year at Pomona — stayed in a rented home on Martha's Vineyard. Ryan had heard about the place for years, and he looked forward to seeing the Hacketts and Jim Cagney, a quiet and reclusive man who had retired from the screen four years earlier and now spent all his time painting. Millard Lampell and his wife, Elizabeth, visited the Ryans one weekend, though Millard recently had fallen in love with another woman. He and Ryan opened up to one another as they strolled along the beach together, or so Lampell thought. "That was the time I really found out that Robert had never been unfaithful to Jessica," Millard later said. "He told me about a couple of passes that had been made at him, one by Joan Crawford and one by Rita Hayworth, and some other actresses that I don't remember... he seemed about women a mixture of naivete and boyishness."
_Battle of the Bulge_ wouldn't be released until December, but already the news of Ryan's well-paid cameo brought dividends. That summer his old friend Dick Brooks, author of _The Brick Foxhole_ , offered him a starring role in a new western he was writing and directing for Columbia Pictures called _The_ _Professionals_. Based on Frank O'Rourke's potboiler _A Mule for the Marquesa_ , the picture would star Burt Lancaster, Lee Marvin, Woody Strode, and Ryan as soldiers of fortune hired by a millionaire to rescue his kidnapped wife from a Mexican revolutionary. Ryan's part wasn't as flashy as Marvin's or Lancaster's, but at this point both Marvin and Lancaster were Hollywood A-listers, and just working alongside them would restore some of Ryan's luster. Brooks knew how to write a movie — he had won an Oscar in 1961 for adapting Sinclair Lewis's _Elmer Gantry_ to the screen — and his script for _The Professionals_ was both punchy and philosophical. After two years of forgettable TV shows and half-assed international thrillers, Ryan set out for the West Coast in October to shoot the picture that would become his biggest hit since _Crossfire_. All he had to do was survive six weeks in Las Vegas with Lee Marvin.
*Preston costarred with Ryan in the RKO western _Best of the Badmen_ (1951).
*For an eye-opening, if somewhat sentimental, history of Avala Film, see Mila Turajlic's 2012 Serbian documentary _Cinema Komunisto_.
_thirteen_
One of the Boys
_The Professionals_ takes place mostly in Mexico, but shooting there would have required Richard Brooks to submit his script for government approval and transport cast and crew across great distances to get the locations he needed. Instead he stayed in the United States, filming along a rail line in Indio, California, and in Death Valley before moving the cast and crew to Las Vegas for more location work at nearby Lake Mead and Valley of Fire State Park. Woody Strode would remember the trip from Death Valley to Vegas: "Robert Ryan, Lee Marvin, and I made the ride in a limousine. I was riding in the jump seat, and I mixed martinis for 250 miles. By the time we arrived in Las Vegas, we were falling down drunk." (They must have been, because the drive from Death Valley to Las Vegas is less than 120 miles.)
The party rolled on at the Mint Hotel, where the cast lodged for six weeks; Marvin remembered it having "seven bars, twenty-seven hours a day gambling, anything you wanted, twenty-one topless Watusi girls in the basement." Their second night at the hotel, Strode — whose character in the picture was an expert archer — crawled out the window of his room onto a ledge and shot an arrow at Vegas Vic, the giant, waving cowboy that decorated the Fremont Hotel across the street. When the arrow connected, the whole statue shorted out. Strode and a drinking buddy raced upstairs to Marvin's room and asked him to hide the bow before the police arrived. "Well, that crazy son of a bitch got so excited he fired a shotgun out of his window," Strode wrote. "The cops came and found the bow in his room. Lee was so proud; it got to be the biggest joke in town." Ryan wasn't inclined to this sort of mischief — for him the highlight of the stay was a visit from Tim, now a sophomore at Pomona — but when you hung around with Marvin, you were bound to hoist a few.
Theirs was a complicated friendship. Both men had served in the Marines during the war, though Marvin, fifteen years younger, had seen action, narrowly surviving the bloody Battle of Saipan in the summer of 1944. As a young supporting player on _Bad Day at Black Rock_ , he had looked up to Ryan, and his combat experience made him a highly credible spokesman for SANE. But the last few years had reversed their status in the Hollywood pecking order: Ryan's professional decline had been noted around town, whereas Marvin had just turned in a star-making performance in the western comedy _Cat Ballou_ , with Jane Fonda. They both could put away the whiskey, but as Phil Yordan once explained, Ryan was "a sober drinker. He drank by himself; he never gave anybody any trouble." Marvin was trouble personified, a loud, sometimes belligerent drunk who infuriated Burt Lancaster by showing up blasted one Friday for a scene atop a twenty-five-foot rock and had to be straightened out by Brooks over the weekend. Lancaster, a serious and highly professional man, much preferred the dependable Ryan (and would recruit him for two more pictures down the road).
In keeping with the popular heist movies of the '50s, Brooks gave each of his four heroes a specialty: Fardan (Marvin) is an automatic weapons instructor, Dolworth (Lancaster) a dynamite expert, Sharp (Strode) a peerless scout and tracker, and Hans Ehrengard (Ryan) an experienced horseman. Described as an "ex-cavalry man, cattle boss, wrangler, bullwhacker, pack master," Ehrengard is the group's senior member and sole bleeding heart: Brooks introduces him on his ranch, where he catches a hired man brutalizing a wild horse and punches the guy out.
Once these four have set off on their mission to rescue the kidnapped Maria (Italian sex symbol Claudia Cardinale) from the formidable Jesus Raza (Jack Palance), Ehrengard emerges as the weak link in the unit, too old to handle the brutal heat and too merciful to be trusted in such a cutthroat operation. After the professionals fend off an attack by Raza's men, killing all ten, Ehrengard persuades Fardan to set loose the men's horses rather than shoot them, which tips off their enemies and results in another attack. As the mission winds on, Dolworth and Ehrengard become moral antagonists, though Brooks comes down time and again with the strong and pitiless. At least two of Ryan's professional pals, screenwriter Philip Dunne and director Michael Winner, thought his character sadly underwritten.
Protesting, Ryan would joke with Winner that he had the most important line in the picture — "If it isn't hot, it's cold. If it isn't cold, it's raining" — because it explained away the seesawing weather conditions as the company tried to complete an ambitiously long action story shot almost entirely outdoors. Brooks recalled temperatures as high as 115 degrees Fahrenheit during the Death Valley sequences (the heat so overwhelms Ehrengard that he collapses, telling Dolworth, "I hate the desert. It's got no... pity"). After the company moved on to Valley of Fire State Park, they were pelted with rain, sleet, and even snow. As Lancaster biographer Gary Fishgall reported, "On December 13 a flash flood swept through the area, trapping cast and crew in a box canyon until workers with the requisite road grader and shovel loader could rescue them."
Ryan must have been relieved to wrap the picture and get back to New York, but his time on _The Professionals_ was well spent. Released the following November, it would collect strong reviews from _Life, Newsweek_ , and the _New York Times_ and go on to become Columbia's highest grossing picture of 1966. Almost as important for Ryan, its modern take on the western would prove influential over the next few years, and a younger generation of filmmakers, such as Winner and Sam Peckinpah, would make similar use of Ryan as they tried to reinvent the genre themselves. In their pictures he would also figure as the odd man out, an outsider among outsiders. He had always been drawn to the offbeat roles anyway; the conventional machismo of westerns and war movies never had appealed to him. In a sense, though, his casting in this new breed of action pictures also reflected how the industry had begun to perceive him; having gone his own way for so long, Ryan now was considered rather an eccentric character himself.
"HE WAS A WONDERFUL MAN, and it was a privilege knowing him," Ryan had told a reporter after Adlai Stevenson died of a heart attack in July 1965. "He always wanted to be called governor, rather than ambassador. He considered that his greatest honor." Now, with Kennedy and Stevenson gone, the face of the Democratic Party was President Johnson. "I remember getting into some rant about Lyndon Johnson," recalled Lisa Ryan, "and my father defending Johnson and saying, 'Listen, you don't know what you're talking about. He's done a lot of good.' He was always trying to point out the humanity in everybody."
Ryan admired LBJ for pushing through the Civil Rights Act, the Voting Rights Act, and new educational initiatives such as the Head Start Program, but Vietnam was going to pull him down. SANE had come out against the war in early 1965 as the troop numbers escalated and Johnson began bombing North Vietnam. That November SANE staged a march on Washington, with Ryan as one of its sponsors, and drew an orderly, responsible crowd of thirty-five thousand people, countering the media myth of a peace movement dominated by angry radicals.
Though born and raised in the Democratic Party, Ryan was beginning to waver. When liberal Republican Congressman John Lindsay — a key vote on Johnson's civil rights legislation — ran for mayor of New York in fall 1965, in a three-way race against conservative writer William F. Buckley Jr. and Democratic machine politician Abraham Beame, Ryan finally broke ranks and voted for Lindsay. "By then the dyke of any rational political system — two-party system — had cracked and the waters of chaos were beginning to rush in," Jessica Ryan observed.
They lifted the boat of Hollywood song-and-dance man George Murphy, who mounted a successful campaign against JFK confidante Pierre Salinger that fall for California's open US Senate seat. Ryan would tell one reporter that he himself was approached to run against Murphy: "So help me God, I was. A very powerful Democrat urged me to consider running for the nomination. I didn't consider it for a second. I want to make a contribution. I would not like to make an ass of myself. Nor would I want to be a front man for the politicians backstage."
Television was turning politics into show business, which gave any skilled performer the edge. Ryan had become an occasional guest on TV talk shows, especially in the local New York media, where his facility with political issues tended to blur the line between him and the real thing. "Not long ago, Robert was on a TV panel show with a well-known and distinguished United States senator," Jessica wrote around this time. "After the show, R. got a number of letters from [viewers] that said, not only did he seem to know all about politics, but he looked more like a senator ought. Therefore, they thought he should be the senator."
Director Robert Aldrich reinforced the notion of Ryan as a marginalized figure when he and producer Kenneth Hyman signed him for _The Dirty Dozen_ , a big-budget World War II epic they were shooting for MGM in England in April 1966. Lee Marvin, fresh from a triumphant Oscar win for _Cat Ballou_ , starred as John Reisman, a pugnacious army major handed the assignment of staging a suicide mission against the Germans with a motley crew of military convicts (among them Charles Bronson, Jim Brown, John Cassavetes, Telly Savalas, and Donald Sutherland). To Ryan fell the thankless task of playing Marvin's nemesis in the first of the picture's three acts, a prim, vainglorious West Point man with the pedigreed name of Colonel Everett Dasher Breed. Commander of the parachute school for the 101st Airborne Division, Breed bristles when Reisman shows up with his men to train for their top-secret mission, and eventually Reisman's crew are forced to prove themselves to the shrewd General Worden (Ernest Borgnine) by going up against Breed's company in a war games operation.
By 1966 there were 400,000 American troops in South Vietnam, and _The Dirty Dozen_ , with its jaundiced take on the military, would connect with a generation of kids questioning the war and the draft. Yet the picture, adapted by veteran screenwriter Nunnally Johnson ( _The Grapes of Wrath_ ) from a best-selling novel by E. M. Nathanson, was antiauthoritarian without really being antiwar; opening someone's throat was okay, it suggested, as long as you were doing it for your own purposes and not getting suckered by the brass. In this context Breed was a priggish heavy, ramrod straight and devoted to the chain of command. By now Ryan had played quite a few career soldiers — in _The Longest Day, The Secret Agents_ , and _Battle of the Bulge_ — but as Colonel Breed he's the butt of every joke, scowling impressively as he absorbs one insult to his dignity after another. Ryan shouldered the challenge with his usual skill and resolve, turning in a spirited comic performance amid a rogue's gallery of posturing tough guys.
Ryan's standout scene comes when Reisman first arrives with his truck full of reprobates, and Breed, instructed by the higher-ups that a general will command the hush-hush operation, welcomes the truck with a military band and troop inspection. Trim and handsome in his dress uniform, sunglasses, and leather gloves, Breed glows with excitement as the party rolls in. Reisman hasn't brought any general, however, so he drags the grubby, unshaven goof-ball Vernon Pinkley (Sutherland) from the back of the truck to impersonate one. This charade reduces his fellow convicts to hysterics, and Breed gives Reisman a tongue-lashing. In the sort of comeuppance that was Ryan's lot for the picture, Reisman silences him with the barbed implication that he's a bit of a swish: "I owe you an apology, colonel. I always thought that you were a cold, unimaginative, tight-lipped officer. But you're really... quite _emotional_ , aren't you?"
Colonel Everett Dasher Breed, the vain martinet of _The Dirty Dozen_ (1967). In the westerns and war movies of the late 1960s, Ryan increasingly would figure as the odd man out, an outsider among outsiders. _Franklin Jarlett Collection_
Despite the large male ensemble, _The Dirty Dozen_ was Marvin's picture, just as _Bad Day at Black Rock_ had been Spencer Tracy's. But Ryan was used to that. The job paid well, and the picture would be an even bigger hit than _The Professionals_ , solidifying his comeback as a graying character actor. He always enjoyed England. Lisa, now fifteen, flew in from New York with a girlfriend to visit him and was hanging around the set one day when a soused Lee Marvin reeled into her orbit and began hitting on her. Starstruck, she talked to him for a while until her father spotted them. "The next thing I remember is my dad came marching over and said, 'Lee! That's my daughter!'... [Marvin] literally jumped backwards. I mean it really was like he got zapped with a cattle prod or something."
AFTER RETURNING HOME FROM THE UK, Ryan flew out to Hollywood to costar with TV comedian Sid Caesar in a mystery spoof adapted from Donald Westlake's novel _The Busy Body_. Ryan may have treasured the great works of literature — he gave Lisa a copy of _Ulysses_ when she was only fourteen — but he also loved farce; among the few TV shows he could be bothered with were _Get Smart_ , starring Don Adams as a bumbling secret agent, and the campy _Batman_ , whose inane, monosyllabic theme song his daughter heard him singing around the house.
_The Busy Body_ had a funny plot: Ryan is a Chicago mob boss and Caesar his wacky lieutenant, who makes the funeral arrangements for one of their men but accidentally buries him in a jacket lined with a million dollars. Unfortunately, the two stars were severely mismatched, Caesar mugging as Ryan turned in his usual meticulously detailed performance. Producer-director William Castle, best known for gimmicky horror movies such as _The Tingler_ and _House on Haunted Hill_ , proved the old showbiz adage that death is easy but comedy is hard, squandering a lively young cast that included Dom DeLuise, Godfrey Cambridge, Bill Dana, and Richard Pryor (he and Ryan shared no scenes).
From Los Angeles, Ryan then flew to Torreón, Mexico, to costar with his Dakota neighbor Jason Robards and James Garner in an MGM western about the gunfight at the O.K. Corral. Director John Sturges had worked with Ryan on _Bad Day at Black Rock_ and since directed two action classics, _The Magnificent Seven_ (1960) and _The Great Escape_ (1963). He had already made one picture about the famous shootout — _Gunfight at the O.K. Corral_ (1957), with Burt Lancaster and Kirk Douglas — but that one climaxed with the gun battle, whereas this one would open with it. Carefully researched by screenwriter Edward Anhalt, _The Law and Tombstone_ followed the repercussions of the gunfight as Wyatt Earp and the Clanton gang fight for political control of the town. Garner played the legendary marshal, Robards the hard-drinking Doc Holliday, and Ryan the ruthless cattle rustler Ike Clanton, who skips out on the gunfight but then mounts a legal and public-opinion crusade against the two lawmen.
"It's a very good part, a very interesting part," Ryan told a German radio reporter on the set, "because [Clanton] pretends to be a very substantial citizen, a very fine man, but actually he's using all these killers to do his work." Anhalt had written a sober, thoughtful script, pondering the issues of civil authority that were implicit in the famous tale. After Earp and Holliday kill Clanton's nineteen-year-old brother, Billy, and two other outlaws, Ike has the corpses displayed for the people of Tombstone in a storefront window and leads a memorial procession through town to protest the killings, glowering at the two lawmen as he passes. Ryan's interest in the picture must have been stoked by the fact that his wife's grandfather, George Washington Cheyney, had held a prominent position as superintendent of the Tombstone Mill and Mining Company when the gunfight took place in October 1881.
Torreón was another of those soul-killing Mexican towns that hosted one Hollywood western after another, and Ryan took advantage of the shoot to get to know Robards, another heavy drinker with a love for Eugene O'Neill. Born in Chicago but raised in LA, the raspy-voiced actor had delivered some extraordinary performances in O'Neill plays: he had made his Broadway debut in 1956 as James Tyrone Jr., the dissolute elder brother, in the original production of _Long Day's Journey into Night_ , and given what many considered a definitive portrayal of Hickey, the dream-weaving salesman, when Sidney Lumet directed a two-part TV version of _The Iceman Cometh_ for public television in 1960. Two years later Robards had reprised his role as Jamie Tyrone when Lumet directed the heralded screen version of _Long Day's Journey_ , starring Ralph Richardson, Katharine Hepburn, and twenty-five-year-old Dean Stockwell ( _The Boy with Green Hair_ ). Ryan's conversations with Robards must have set him to thinking, because four months later he would star in his own version of the play, tackling the role of the bitter patriarch, James Tyrone Sr.
That November, to Ryan's dismay, Ronald Reagan completed his metamorphosis from movie star to president of the Screen Actors Guild (where he helped enforce the Hollywood blacklist) to TV pitch man for General Electric (one of the nation's top defense contractors) to governor of California. "Ronnie's a pleasant guy, but I don't believe the man has any degree of knowledge of what he's talking about, and I don't believe he has any strong convictions," said Ryan on the eve of Reagan's victory. Most actors, he pointed out, were introverts, but Reagan was "the most gregarious man you ever met.... Ronnie loves campaigning and meeting people and giving speeches." Talking to the Dartmouth College paper a few months later, Ryan found the perfect line for Reagan: "There aren't any series around, so I might as well be governor."
A clause in Ryan's contract guaranteed him time off for Christmas, and back in New York he attended to Jessica, who was grieving over her mother's death earlier that month at age eighty-one. ("While it was a blessed release from what had become a meaningless existence, it was difficult," she wrote to Jean Renoir. "As I guess these things are, regardless of how well-prepared you think you are.") Jessica was always encouraging Ryan to return to the theater, and now that he had four successive Hollywood pictures under his belt, he decided to take her advice and also cheer her up with a trip to Europe. Shooting _The Dirty Dozen_ in the UK, he had crossed paths with actor Paul Rogers, one of the sailors in _Billy Budd_ , and through Rogers got back in touch with their old cast mate John Neville, now artistic director of the Nottingham Repertory Theatre in the East Midlands. In March 1967, Ryan flew to England and met with Neville to discuss a residency at the theater that coming fall; announced in the trades a month later, the deal called for Ryan to play the title role in _Othello_ and the father in _Long Day's Journey into Night_. His pay would be $150 a week.
With this engagement in place, Ryan also picked up cameo roles in three pictures that were shooting in Europe that summer; he set off for Italy in late May, and Jessica and Lisa followed in early June, spending the entire summer with him. None of the pictures threatened to set the world on fire: _The Prodigal Gun_ , released in the United States as _A Minute to Pray, A Second to Die_ , was a cheapo spaghetti western with Alex Cord as a storied gunslinger and Ryan as governor of the New Mexico Territory, who has created a volatile situation by declaring an amnesty for local outlaws. _Anzio_ was another thundering World War II epic, this one from Columbia Pictures, that reunited Ryan with his old RKO collaborators Bob Mitchum and Eddie Dmytryk. The picture chronicled the botched amphibious landings at the title beach, with Ryan as an incompetent general based on real-life Allied Commander Mark W. Clark. Before Ryan even arrived, the conservative Mitchum, having just visited American troops in Vietnam, prevailed upon Dmytryk to lighten up on Clark in the script.
The Ryans stayed in Florence for six weeks, making a quick side trip to Madrid so that Robert could appear in _Custer of the West_ for his friend Phil Yordan. Like _Battle of the Bulge_ , it was being shot in Cinerama, an elaborate widescreen process that involved three synchronized cameras, and its intimate character study of the doomed general, played by Robert Shaw, alternated with screaming long shots that showcased the stunning countryside and hundreds of horses and extras. The role gave Ryan a chance to repay the great favor Yordan had done for him two years earlier, when he and Sid Harmon (with whom Yordan had since parted company) hired Ryan at his top salary for _Battle of the Bulge_ and launched him on the comeback trail. "He told his agent that it was between he and I, and that he was going to do the picture for me for nothing," Yordan recalled. "That's when he was drawing about $150,000 a picture. It was a magnificent gesture. Even though it was an expensive picture, there was no money at that time, and I would have had to hire some character actor."
As Sergeant Patrick Mulligan, a jolly deserter whom Custer sends to the firing squad, Ryan had only three scenes, his little episode taking up ten minutes of a 143-minute release, but he injected more juice into them than Shaw could manage for the balance of the picture. Jessica and Lisa accompanied him to Madrid, and Lisa rarely had seen her father in higher spirits. "He seemed to be incredibly happy when he was making movies," she recalled. "No matter what the movie was." The man striding around on set couldn't have been more different from the one she saw at home, lapsing into silent dejection for hours on end.
John Wayne happened to be in Madrid at the same time, and he, Ryan, and Yordan got together for dinner one night. Yordan saw a great many similarities between the two men, politics aside: both were quiet, modest, loyal to their friends, and highly cooperative on the set. "Once they signed to do a picture, you never heard a peep," Yordan recalled. As the evening progressed and the two actors threw back the drinks, Wayne couldn't help needling Ryan about some political issue. "Wayne had a real devilish sense of humor.... It got to the point where Ryan said, 'Okay, Duke, let's go outside and settle it.' Now, Duke's a big man, you know, but Ryan was bigger than Duke, and not only that, he was the heavyweight champion in college. Well, Duke had no intentions of going outside with Bob. He realized he had gone too far, and we both calmed him down."
By early August the Ryans had fled the summer heat of Florence and settled in a London flat, where Ryan studied his lines for Nottingham and wrote his old friend Dore Schary to turn down a role in a play back home. "I am very concerned about what I may do (if ever) on the New York stage after the 'President' fiasco," he admitted. When the Nottingham engagement ended, he wrote, his next order of business would be getting another picture to build up their bank account. "As Bernard Shaw said, art is expensive." The letter was illuminating: four years after _Mr. President_ closed, Ryan still couldn't shake the humiliation of having to front a bad production night after night on the Great White Way, which might explain why his first stage performances since then would be tucked away in the English countryside. At the same time, he expressed little affection for contemporary American theater, telling one reporter, "It stinks." Working with Max Reinhardt and John Houseman had inclined him toward adventurous work; the only recent play he really liked was Harold Pinter's _The Homecoming_.
Founded in 1948, the Nottingham Repertory Theatre had moved recently into a modern new building in Wellington Circus, designed by architect Peter Moro, whose circular auditorium accommodated 750 people and whose highly functional performing space could be easily converted from a proscenium to a thrust stage. Hired shortly thereafter, John Neville had revitalized the company, bringing in such fine young talents as Judi Dench and Ian McKellen, and added a youth group, poetry readings, and a weekly jazz night to the theater's offerings. Ryan's residency at the Playhouse was another innovation: "There had never been a Hollywood star in regional theatre before," said Neville. _Othello_ and _Long Day's Journey into Night_ opened in mid-September, shortly after Jessica took Lisa back to the United States to begin the school year, and both productions drew big houses that welcomed Ryan heartily. "The camera never gives you six curtain calls," he boasted to the _New York Times_.
The critics were less starstruck. "Mr. Ryan takes the stage with a brooding, dignified presence, stiff in gait but noble in bearing," observed John Peter of the _Times_ in his review of _Othello_. The American's interpretation, he wrote, lacked "the controlled animal impulsiveness that sets Othello apart from his Venetian masters. Thus portrayed, he is not much of a prey for Iago, and John Neville plays that part in splendid isolation." When _Long Day's Journey into Night_ premiered a week later, Peter was more impressed, calling Ryan's performance as the aging matinee idol James Tyrone Sr. "a finely muted portrait of glamour in decay." Ryan had grown philosophical about his limitations as a classical performer — "Shakespeare requires a dimension of acting I probably never will attain," he confessed to _Variety_ 28 — but he considered such work its own reward. Playing O'Neill was gratifying too, though he felt his English costars failed to grasp the Irish-American melancholy so central to the play. _Long Day's Journey_ , he thought, might be just the thing to revive his stage career in New York.
A few weeks before returning to the United States, Ryan corresponded with Lisa to inform her that he had met her "dreamboat," Albert Finney, who came backstage and stuck his head through the shower curtain in Ryan's dressing room to introduce himself. As the performance schedule wound down, Ryan had gotten a chance to see two American pictures at the local cinema: _Bonnie and Clyde_ , which he disliked, and _Cat Ballou_ , which he enjoyed only for "Lee M. who was marvelous." To his relief, he had just played the last of the school matinees offered by the Playhouse. "Shakespeare naturally is sacrosanct, so 6 yr. old moppets are allowed to watch a play about murder, sexual jealousy, and foul conniving," he reported. "The results are what you might expect: last week we played the big O to a constant storm of laughter which reached a crescendo in the bedroom scene. My killing of the lady was evidently funnier than the 3 Stooges. We did 6 of these horrendous things and I feel that I have paid for all my sins in this world."
WHILE THE RYANS EXPLORED EUROPE, American opposition to the war in Vietnam continued to build, broadening from the loyal ranks of students, professors, and clergymen into a genuine middle-class movement. The father of two draft-age sons, Ryan had a personal stake in the issue; he had never supported the war, and that summer he took advantage of an interview with the _New York Sunday News_ to make his opposition clear. "The thought of sending them off to a war we shouldn't be in is something that's awfully hard to live with," he said. Asked if he thought such statements would hurt his career, he replied, "During the McCarthy era, I had my say — and it didn't injure me at all. If I didn't keep my trap closed in those days, I'm certainly not going to remain quiet now." His proud recollection didn't really square with the careful balancing act he had maintained all through the blacklist years; in fact, a more private concern for him now was how to encourage his sons in the pacifist principles they had been taught but discourage them from marginalizing themselves for the rest of their lives.
Both his sons had made clear that they weren't going to Vietnam. Tim had applied for conscientious objector status several years earlier but was denied; when he was called for his physical that year, he turned in his draft card in protest but was never prosecuted. Cheyney had been radicalized by his summer social work; once a neighbor of Walt Disney, he had suddenly found himself in the desolate strip-mining community of Hazard, Kentucky. Back in New York he had joined the Catholic Worker Movement, whose founder, Dorothy Day, left a deep impression on him; when his draft card arrived, he informed his parents that he intended to burn it. "My parents correctly said, 'Well, you haven't even gone to college yet, and you'll get yourself thrown in federal prison for doing something like this. I don't see [Dorothy Day] having to face going to jail for the Vietnam War.'" When Cheyney won a student deferment and enrolled at Harvard (where he skipped a year and started as a sophomore) the issue subsided temporarily, but the draft wasn't going away. By the end of 1967 more than twenty thousand Americans had died in the conflict.
Once Ryan returned to the United States, he dove back into the roiling protest culture that had always been a key attraction of life in Manhattan. On a single weekend in January 1968 he participated in two high-profile benefit concerts that brought together some of the biggest stars in the country: on Saturday, January 20, he provided narration for matinee and evening performances of a show memorializing folk singer Woody Guthrie at Carnegie Hall, and the following night, at Philharmonic Hall, he performed in a "Broadway for Peace" concert to raise campaign funds for US congressmen who had taken stands against the war. With the presidential election less than eight months away, the hot topic that weekend was Senator Eugene McCarthy, a vocal opponent of the war who had declared himself a candidate for the Democratic nomination and said he intended to beat President Johnson in the New Hampshire primary that coming March.
"Bob was a great fan of folk music, and his kids were very involved with it," remembered Millard Lampell, "particularly Tim, who wanted nothing better than to be another Woody Guthrie." The revered songwriter had died the previous October after a fifteen-year battle with Huntington's disease, and his former manager, folk impresario Harold Leventhal, had organized the Carnegie Hall shows to raise money for Huntington's research, enlisting Lampell (who had performed with Guthrie in the Almanac Singers) and Will Geer (who had helped care for Guthrie in his last years). Lampell wrote a narration drawn from Guthrie's writings, to be performed by Geer and Ryan; the singers on the bill included Seeger, Odetta, Judy Collins, Arlo Guthrie, and the little-known Richie Havens, whom Lampell had discovered playing in the East Village. The star attraction, however, would be Bob Dylan, out of the public eye now for twenty months following a much-publicized motorcycle accident. "Carnegie Hall was entirely sold out two hours after the tickets went on sale," remembered Lampell. "There was a kind of electric excitement and anticipation."
Cheyney came home for the show, and Lampell recruited Lisa as a stage manager. Outside the concert hall, Seventh Avenue was mobbed with people trying to score tickets; inside, the performers gathered for a single rehearsal of the program. Dylan had brought his backup band, the Hawks, and when they launched into Guthrie's "The Grand Coulee Dam" that afternoon and evening, the house went wild. After the evening concert, the performers all were invited back to the Dakota for a party at the Ryans' apartment, which also drew Paul Simon, Art Garfunkel, and Allen Ginsberg. Lisa listened incredulously as her father — who loved to tease her with his Dylan impersonation — told the wild-haired singer how much he admired his music. "Bob [Ryan] sat on the floor with his kids all around him," Lampell said, "and he listened to the singing, which lasted until two or three in the morning, and then a few of us stayed on rehashing what had gone on, and I had never seen Bob grin so much in my life."
The concerts generated $7,500 for Huntington's research, a relatively meager amount compared to the $100,000 raised for antiwar candidates at Philharmonic Hall on Sunday. "Broadway for Peace" was more of a showbiz affair, with appearances by Ryan, Harry Belafonte, Paul Newman, Tony Randall, Barbra Streisand, Leonard Bernstein, Diahann Carroll, Joel Grey, Alan Arkin, and Carl Reiner. The proceeds all were earmarked for legislative candidates — Senator Wayne Morse of Oregon, Representative John Conyers of Michigan — who had voted against either the Gulf of Tonkin Resolution, which authorized President Johnson to prosecute the war, or the $700 million appropriations bill that followed in 1965.
Newman and Randall both liked Eugene McCarthy and planned to campaign for him as the March 12 primary approached. McCarthy was the sort of politician Ryan respected; he was Catholic and, like Stevenson, a man of learning and of principle. "When he was alone he acted bravely, and I was moved," Ryan later said. On January 27 the national board of SANE cast its lot in a presidential race for the first time, endorsing McCarthy by a vote of thirty-six to zero.
To some degree the McCarthy endorsement was SANE's attempt to heal a terrible internal schism between energized radicals who wanted to end the war by any means available and conventional liberals who wanted to work inside the system. The leadership maintained a careful distance from communist or radical elements that might besmirch the group's reputation, even as the grassroots membership pushed for a broad coalition with antiwar groups across the political spectrum. The issue reached a boil in spring 1967 when SANE sat out the giant Spring Mobilization Conference in Washington, and boiled over in the fall when the organization declined to endorse the March on the Pentagon planned for October. Both Norman Cousins, who favored the more isolated approach, and Dr. Benjamin Spock, whose membership in other peace organizations had added to the friction, resigned from SANE. In the wake of this crisis, McCarthy offered SANE something most of its members could latch onto: a credible, articulate candidate who wanted a negotiated settlement between the United States and North Vietnam.
Whatever residual loyalty Ryan may have felt for Lyndon Johnson probably evaporated on January 30, the day he took part in the historic reopening of Ford's Theatre in Washington, DC. Following the Lincoln assassination, the building had stood dormant for nearly 103 years, but now John Houseman was directing an opening-night tribute to the Great Emancipator, to be broadcast live on CBS. Ryan shared the narration with Henry Fonda and Fredric March, the three men positioned at lecterns across the very stage where John Wilkes Booth had cried, "Sic semper tyrannis!" Odetta, Harry Belafonte, Andy Williams, Helen Hayes, Nina Foch, and Richard Crenna rounded out the cast, and the show was to begin with a statement from President Johnson. But that afternoon came reports that the Vietcong and North Vietnamese Army had launched a giant surprise attack on key tactical zones in South Vietnam; the Tet Offensive, so called because it violated a cease-fire set for the Vietnamese new year, severely undercut the administration's rosy statements about the situation on the ground. Vice President Hubert Humphrey appeared on the program in place of LBJ, delivering a stone-faced introduction on the Lincoln legacy.
Public support for the war crumbled. Later that month, more than five hundred Americans were reported killed in a single week, and the Selective Service System issued a new call for forty-eight thousand soldiers. Ryan's next picture, a Warner Bros. western called _The Wild Bunch_ , didn't start shooting until just after the New Hampshire primary, so the month before the vote he made himself available to the McCarthy campaign for the sort of retail politics his father had always practiced. Cheyney had decided to back McCarthy as well and stayed with his father a few times.
Seymour Hersh, the campaign's thirty-year-old press secretary, had graduated from the University of Chicago and worked as a crime reporter for the City News Bureau. Later he would win a Pulitzer Prize for exposing the My Lai massacre, but at that point he was taking a break from journalism and, with some ambivalence, trying his hand at politics. He remembered eating dinner with Ryan at the Wayfarer Inn in Manchester, ordering a hamburger and slathering it with ketchup. "[Ryan] looked at me and he said, 'So, what part of Chicago are you from?' Very funny line." Ryan told him about his father's experiences as a Democratic committeeman, and they traded stories of civic corruption. Hersh was impressed by the depth of his affection for the city: "We talked about the fun of Chicago." With luck, they would be there in August for the convention, with McCarthy as the party's nominee.
"He was a bit mystified by his son," remembered Hersh, who lunched once with Ryan and Cheyney during the primary. "As I got to be a father, I could understand it. You know, they grow old, they separate.... There's this sort of mystery of why they're not eight anymore." The campaign staff were all young, and college kids flocked to support McCarthy, losing their beards, long hair, and psychedelic threads in response to the campaign's edict that they go Clean for Gene. Yet a large quotient of McCarthy's financial support came from older, straighter New York liberals like Ryan, who had congregated around Stevenson. Paul Newman and Tony Randall turned up in New Hampshire, along with the poet Robert Lowell. The last weekend of the campaign, Ryan and Randall were "carefully juggled at shopping centers," according to one memoir, while _Time_ reported that "Paul Newman's appearances had to be circumscribed for fear of a riot among Hampshire women."
Watching all this, Hersh would note how savvy Ryan was politically; Newman was always seeking advice on how to handle certain questions, but not Ryan. "He didn't have to be educated about what the best thing to say about the draft was. And there was never blowback on anything he said. Newman would sometimes be maladroit a little bit, but not really. He was smart enough to know what he didn't know. Ryan didn't have to be." The Saturday night before the primary, Ryan took to the podium in a ballroom of the Sheraton Carpenter Hotel to introduce McCarthy as the next president of the United States. That Tuesday, McCarthy won 42 percent of the Democratic vote, and Johnson only 49 percent. Exit polls suggested that McCarthy's strong showing may have been more of a generic no-vote against the president than a protest against the war, but that didn't matter: Johnson was wounded.
A week later Ryan was back in Torreón, Mexico, rehearsing _The Wild Bunch_ with director Sam Peckinpah. Like _The Professionals_ , it was a strenuous action picture with a tight shooting schedule of eighty days; for the first ten weeks they would be shooting in and around the sun-baked town of Parras, a hundred miles east of Torreón. "Sam takes you to the asshole of creation," explained crew member Gordon Dawson. "Everyone was worried about dying. You're rehearsing with full loads in the guns and horses that are skittish. When you're dealing with thirty or forty horses, a lot of things can go wrong.... Off the set, we spent our time drinking and trying to get good food." Ryan had seen a lot of miserable locations, but this one was like a ghost town. Peckinpah had scouted it out with cinematographer Lucien Ballard, who was working with Ryan for the fifth and last time.
Scripted by Peckinpah and Walon Green, _The Wild Bunch_ recalled _The Professionals_ with its story of aging outlaws chafing against the modern age. The year is 1913: Pike Bishop (William Holden), Dutch Engstrom (Ernest Borgnine), and their gang ride into a Texas town, disguised as soldiers, to steal a silver shipment from a railroad office, not realizing that Bishop's old partner, Deke Thornton (Ryan), waits on the roof of a building across the street with a gang of bounty hunters. Meanwhile, Peckinpah follows the progress of a local temperance meeting as a sermon gives way to a march through town with a brass band. These three narratives intersect when the outlaws emerge from the office with their booty, the parade crosses in front of them, and a rifle fight erupts between the outlaws and Thornton's crew. The next four minutes were complete chaos, with bullets tearing not only into people but out of them, and blood everywhere. The Motion Picture Association of America, representing the major studios, was in the process of scrapping the old production code in favor of a new ratings system, and _The Wild Bunch_ would put it to the test.
As in _The Professionals_ and _The Dirty Dozen_ , Ryan's character was isolated from the macho crew of the title. Thornton once rode with Pike Bishop, until he was captured and sent to prison; now he has been offered his freedom if he captures or kills his old pal. Harrigan (Albert Dekker), the money man behind the posse, tells him, "You're my Judas goat, Mr. Thornton." Thornton wrestles with this epithet, and a quick flashback shows him stripped to the waist and suffering under the lash.
Thornton and his band of grimy reprobates set off in pursuit of Bishop, though Thornton has more respect for his old partner than his new ones. "We're after _men_ ," he tells the noxious Coffer (Strother Martin), "and I wish to God I was with them." By this time Bishop, Engstrom, their surviving gang members (Warren Oates, Ben Johnson, Jaime Sánchez), and the cackling desert rat Freddie Sykes (film noir veteran Edmund O'Brien) have ridden into Mexico, where they enjoy themselves in the small town of Agua Verde and get caught up in a train robbery scheme on behalf of General Mapache (Emilio Fernández), its debauched ruler. The picture climaxes with an epic gun battle in the town square between the outlaws and the _federales_ , replete with explosives and machine-gun fire. Thornton watches from a distance through binoculars as this unfolds. By the end of the bravura five-minute sequence, 112 people are dead; the total body count for _The Wild Bunch_ would be 145.
Deke Thornton, the conflicted bounty hunter in Sam Peckinpah's western classic _The Wild Bunch_ (1969). The outcry over its graphic bloodshed put Ryan in a difficult position after his many years of peace activism. _Franklin Jarlett Collection_
With plenty of down time during the shoot, Ryan pored over American papers for political news. Four days after the New Hampshire upset, Senator Robert F. Kennedy of New York had declared his candidacy, angering the McCarthy faithful. By the end of March, President Johnson had seen the writing on the wall and announced that he would not seek reelection. Four days after that, on April 4, Martin Luther King Jr. was gunned down on a motel balcony in Memphis; a single bullet from a Remington pump-action rifle broke his jaw and neck and severed his jugular vein. Riots erupted that night in several American cities, including Washington, Baltimore, and Chicago. In Ryan's hometown — nearly a half century after the 1919 race riots of his youth — a three-mile stretch of Madison Street on the West Side was consumed by fires and looting. Ryan had brushed shoulders with King not only at civil rights events but through SANE; production records show Ryan missing from the set after King was killed, and _Variety_ mentioned him flying to New York. Two weeks later he was back in Parras, where actors were being outfitted with exploding squibs to mimic bullet entry and exit wounds.
The McCarthy campaign had been turned on its ear by Kennedy's late entry and President Johnson's stunning abdication; now the Minnesota senator faced not an embattled president but the young heir to Camelot. Vice President Hubert Humphrey was the party establishment's choice to succeed Johnson, and though he had entered the race too late to compete in the elective primaries, he had the edge in the ones that were still negotiated in smoke-filled rooms. McCarthy's uneasy relations with organized labor and the black community made him a problematic candidate to win in the fall, and the contest between him and Kennedy turned bitter as RFK took Indiana and Nebraska and McCarthy defeated him in Oregon. In late May the _Wild Bunch_ company returned to Torreón, where news arrived the morning of June 6 that Kennedy, a decisive victor in the California primary, had been shot by Sirhan Sirhan in the kitchen of the Ambassador Hotel in Los Angeles. His assailant emptied a .22 revolver, sending one bullet into Kennedy's head, a second into his neck, and a third tearing through his chest. Kennedy died twenty-six hours later.
These horrible events could only have exacerbated the ongoing tension on the set. Peckinpah was a terrible hothead, and he liked to goad and bully his actors. Holden and Borgnine both had run-ins with him, and according to Holden biographer Bob Thomas, Ryan lost his temper with Peckinpah as well. After the company moved to Torreón, wrote Thomas, Ryan asked Peckinpah for a few days off so he could do some campaigning,* but the director turned him down. "For ten days, Ryan reported to the set in makeup and costume. He never played a scene. Finally he grabbed Peckinpah by the shirt front and growled, 'I'll do anything you ask me to do in front of the camera, because I'm a professional. But you open your mouth to me off the set, and I'll knock your teeth in.'"
Peckinpah may have provoked Ryan, but he also coaxed a superb performance from him. Like Hans Ehrengard in _The Professionals_ , Deke Thornton is peripheral to the main action, yet he takes center stage in the denouement, after the wild bunch are wiped out and the bounty hunters arrive at the stricken town. Thornton finds his old friend Pike Bishop bloodied and dead, his arm hanging from the handle of a machine gun, and pockets Bishop's revolver as a memento. As his crew of scalawags loots the bodies, Thornton seems to buckle under the weight of it all; for hours on end he sits at the town gate, his horse standing by, as the bodies are carted out by the townspeople. Finally, the old codger Sykes rides up with a couple of Mexicans and an invitation to head out with them on some unspecified adventure. "It ain't like the old days," chuckles Sykes, "but it'll do." Thornton laughs, mounts his horse, and rides off with them, into the past.
Ryan flew home at the end of June, and that summer he and Jessica stayed in the little town of Holderness, in central New Hampshire on Squam Lake. Campaigning in New Hampshire that winter had reawakened Ryan's love for the state, and the couple had a friend in Holderness — Harold Taylor, who had been president of Sarah Lawrence College and a human-rights advisor to Adlai Stevenson. That summer the film board at Dartmouth College programmed a Jean Renoir festival, and Ryan drove out to Hanover for a screening of _The Woman on the Beach_. "Most of what I said was about you," Ryan wrote to Renoir the next day, "how important it has been in my life to have worked with you and to call myself your friend." By the end of the summer the Ryans had bought a piece of land in the Shepard Hill neighborhood of Holderness and were planning to build a second home on it.
Before returning to New York, Ryan traveled to Chicago to serve as a McCarthy delegate at the Democratic National Convention. McCarthy had no path to the nomination: by the time the elective primaries wrapped up in New York on June 16, Humphrey had a commanding lead in the delegate count, and McCarthy was too diffident a character to woo uncommitted delegates to his side. Senator George McGovern of South Dakota had leapt into the race as well, which would probably divide the anti-Humphrey vote at the convention. Ryan went anyway, largely to please a friend who was the head of the New York delegation. In Chicago he shared a hotel room with his son Cheyney and tended to his duties at the International Amphitheatre, while the younger man demonstrated against the war in Grant Park.
Cheyney asked his father if he could get him into the convention hall, and Ryan sounded out the head of the New York delegation for a second pass, with no luck. "The convention was being run by Mayor Daley, who was a big Humphrey supporter," Cheyney recalled. "So he wasn't gonna give anything to the McCarthy people." While Ryan was on the floor of the convention, however, a representative of the Pepsi-Cola Company approached bringing good wishes from Joan Crawford, whose late husband, Alfred Steele, had been president of the company, and who now served on the board of directors. "Ten minutes later, we have a pass to the convention!" said Cheyney. "I always thought, 'Well, that's an interesting anecdote about who runs the world.'"
Tension filled the convention hall: TV screens showed Chicago police clashing with protesters on Michigan Avenue outside the Congress Hotel, where Humphrey was staying. "When you were in the convention they had TV coverage everywhere," said Cheyney. "And most of the time they weren't showing what was going on in the convention, they were showing all these battles going on.... I was kind of bouncing back and forth. There was one night when I was there when all the police stuff was going on. And I remember another night I was actually in the convention. So it was quite an experience." Disgusted with the whole situation, his father decided to go home, leaving Cheyney with the hotel room for another few days. Ryan missed the climactic police riot on August 28, but it was all over TV, and as the protesters pointed out, the whole world was watching. Anyone with a grasp of electoral politics knew what all this meant: Richard Nixon was going to be president of the United States.
*Thomas incorrectly reports that Ryan wanted to campaign for Robert Kennedy, a candidate he never supported and who, on the basis of Thomas's chronology of events, already would have been dead. A more likely supposition is that Ryan wanted to campaign for McCarthy in the New York primary, which took place on June 16.
_fourteen_
My Good Bad Luck
Among Jessica Ryan's papers is a curious fifteen-page typed manuscript that may have been begun by her husband but was certainly finished by her. "My name is Robert Ryan," it opens. "The vital statistics? Born: Chicago, Illinois. Educated in a Jesuit academy and at Dartmouth College. Married — thirty years — to the same woman. Father of three children, two boys and a girl. (Well, in order not to offend women's lib — a girl and two boys.)" The next four pages meander, touching on the social upheaval of the '60s and the author's experience of being confused with the vicious characters he had played onscreen. By the fifth page, however, another writer clearly has taken over, and the piece becomes a scholarly consideration of movie westerns and the white, patriarchal society they champion. In the old silent westerns of William S. Hart, the author explains, "Women were dance hall girls (never openly identified as whores) or Innocent girls. When an innocent girl appeared, the good man's concern for her was to rescue her from the bad man. Not for himself, but to protect her virginity. In the end, of course, riding off into the sunset, alone."
The piece was never published; Jessica must have recognized that this wasn't the proper approach to such a rich topic, and alongside the ghostwritten piece are two longer manuscripts under her own name ( _Woman — The Mythless American_ and _America — Dream or Nightmare?_ ) in which the same ideas are worked out in much greater detail. She wrote constantly, which encouraged and to some extent excused her cloistered existence. Bobbs-Merrill had published her second children's book, _The Mystery of Arroyo Seco_ , in 1962, and for a long period afterward she had labored over another novel called _The SmokingMountain_, but it never saw print. By the end of the decade she had turned to nonfiction, cranking out not only the two scholarly manuscripts but _If School Keeps_ , a 150-page account of starting the Oakwood School; _Recollections of a Pioneer Grandmother_ , about the frontier women in her family line; and the memoir _Campaign–'52, or A Camera's-Eye View from Two Odd Birds_.
Writing under her husband's name, and then realizing that she had to take ownership of the ideas herself, must have been a telling experience. Jessica had read Betty Friedan's book _The Feminine Mystique_ when it shot up the bestseller charts in 1964, and she was taken with its clearheaded diagnosis of "the problem with no name" — that vague but gnawing sense among American women that there must be more to life than marriage, children, and creature comforts. "Sometimes a woman would tell me that the feeling gets so strong she runs out of the house and walks through the streets," wrote Friedan. "Or she stays inside her house and cries. Or her children tell her a joke, and she doesn't laugh because she doesn't hear it. I talked to women who had spent years on the analyst's couch, working out their 'adjustment to the feminine role,' their blocks to 'fulfillment as a wife and mother.'" Even a woman such as Jessica, who had published five books and launched one of the most respected private schools in Los Angeles, understood. To the world, she was still Mrs. Robert Ryan.
Since moving to New York, Jessica had become a patient of psychologist Rollo May, whose books had helped introduce the idea of existential anxiety into American psychotherapy. Interviewed by _Psychology Today_ , May defined anxiety as "the awareness of death.... This comes out in the loss of love, which is a partial death, it comes out when you write a book that turns out not to be publishable or to be a success, when something is not as good as you hoped. All of these things are partial deaths that precede our ultimate death." From his perspective, some anxiety was a good thing, "the struggle of being against non-being." May became a powerful influence on Jessica, who had wrestled with anxiety all her life, and some of his ideas and terminology would creep into her work.
When Jessica set out to write a feminist study of her own, she naturally gravitated toward her family history of strong, independent women, and to the Jungian scholarship that so fascinated her, with its emphasis on the psychological repercussions of cultural myths. Jessica opened _Woman — The Mythless American_ by explaining that she had once taken part in a study of violence in America, doing an analysis of how it was fueled by national mythology. "In the course of the work, two things began to be apparent to me," she wrote. "One, American mythology is entirely for men — and for men at a pre-adolescent, gang-age stage of human development; two, _there are no myths for women, nor are there women of any significance in the myths of men_." For her, this cultural mind-set posed a greater problem for women than the more practical matters of equal pay and career opportunities. "All the legislation in the world with respect to women can change nothing if the mythology, the cultural mores of the nation do not change."
The manuscript dead-ends after fifty pages, before the writer can follow her idea from the tall tales of the nineteenth century into the Hollywood dream world of the twentieth; one wonders what Jessica might have said about Ryan's endless schedule of westerns and war movies. _The Professionals, The Dirty Dozen_ , and _The Wild Bunch_ are all male romances, arguably "for men at a pre-adolescent, gang-age stage of human development," and their treatment of women ranges from chauvinism to violent misogyny. In _The Professionals_ , the Claudia Cardinale character is a big-breasted trophy, fought over by various macho factions. The only female characters intruding on the large-scale buddy romance of _The Dirty Dozen_ are a band of whores imported by Lee Marvin to service his boys. _The Wild Bunch_ is even worse: when one of the outlaws, Angel (Jaime Sánchez), catches his former lover in the arms of the sinister General Mapache, he pulls out his revolver and murders not the general but the woman.
Even in its incomplete form, Jessica's manuscript provides some offhand insights into her personal experience of "the woman problem." Her parents' generation, she wrote, came of age as the Victorian era faded away, leaving behind "an aggregation of thrashing about, dissatisfied, frustrated women, taking out their repressed rage, not only on their husbands, but on their children — over-demanding, over-exploitative, over-protective, over-rejecting, over-almost everything, out of a lack of any surety within themselves." Jessica knew Robert had given her more respect for her ideas than a woman of her mother's generation might have enjoyed, but he was old-fashioned in his attitudes. "Most women want to discover the truth of their existence within the reality of themselves as women," Jessica wrote. "They find it mysterious and infuriating when men react to their plea with disinterest, if not out-and-out anger; it is even more mysterious when the reaction comes from the kind of liberal, intellectual-type man who pays lip-service to the cause of women's rights, publicly supporting their desire for the freedom to assert their power, but who, when that assertion interferes with his own personal preoccupations, retreats in boredom from the engagement."
Jessica found a more sympathetic ear in young Ramona Lampell, the second wife of their friend Millard. Born to a coal-mining family in West Virginia, Ramona had boarded a Greyhound bus for New York City in 1947, on the eve of her seventeenth birthday; and after attending the Barbizon School of Modeling, she got a job in the fur department at Bergdorf Goodman. Millard had met her at a Bloody Mary brunch in Redding, Connecticut, in 1965, shortly before he and his first wife, Elizabeth, visited the Ryans in Martha's Vineyard; by 1968, when Millard helped stage the Woody Guthrie shows, he had divorced Elizabeth and married Ramona. Jessica loved Millard and took a real shine to Ramona, especially given their shared past as working women modeling clothes in Manhattan. The younger woman was touched and grateful for her friendship. "She was really very sweet to me," Ramona recalled. "When I first met Millard, he was married previously, and they knew him first. And it was difficult for me to come into the relationship as 'the other woman.'"
Ramona was captivated by Jessica, who introduced her to yoga and who stood on her head every day for fifteen minutes. Jessica struck her as enormously shrewd in sizing up interpersonal situations, and careful about whom she allowed into her orbit. "She was fun, she was funny," Ramona recalled. "She had a very intelligent sense of humor.... And she was beautiful to look at — she was tall and had beautiful skin, and she exercised and kept her weight down.... She was the first person that I met that made her own noodles. She bought a machine, and we had such fun going over there, and her cooking her specialties." Robert and Millard indulged all the talk of equal rights for women, but they were also strong-willed men. "Millard was very bright and very aware, and so was Robert. But walking the talk didn't happen. I think they tried, and they were somewhat sensitive to it, let's say. Intellectually."
The Lampells became part of the Ryans' inner circle in New York (along with Robert Wallsten and the writers George Bradshaw and Robert Thomsen), which put them in a privileged position when Ryan, bowing to the wishes of his publicist, threw one of his semiannual showbiz parties at the Dakota. "Everything else would have to be arranged, usually by public relations firms, and even the guest list was sort of drawn up, about half of it with people Robert didn't care about, or really know well," Millard recalled. "It was in order to get mentions in columns, but he did that really with a kind of amused contempt for the whole prospect, and the best part of it would be afterwards when all the guests went home, and Jessica and Robert and Ramona and I would sit around and do a sort of sardonic take-off on the people who had been at the party."
At least one of these parties threatened to turn unpleasant — when Millard ran into director Elia Kazan, who had named names before the House Un-American Activities Committee. Kazan had always portrayed his friendly testimony to HUAC as a matter of principle, but many in Hollywood never forgave him. "We went to a social thing in Connecticut once, and Kazan was there," Cheyney Ryan recalled. "All that my father talked about on the way there was about what a f___head he was, and how he turned in his friends. And of course, when we get there we're all very nice to each other." Millard had been blacklisted for years before resuscitating his career in the early '60s; when he won an Emmy in 1966, he took advantage of the opportunity to note from the podium that he was a victim of the blacklist, and followed this up with a piece about his experience in the _New York Times_. After he gave Kazan the cold shoulder at the Dakota, the director followed Ramona around, trying to explain his position, while Ryan took Millard aside and assured him that he understood how he felt.
Among the guests watching this play out were Robert Mitchum, John Houseman, and Henry Fonda, whose friendship with Ryan recently had blossomed into a theater collaboration. On location together for _Battle of the Bulge_ and _The Dirty Game_ , the two men had discussed starting a regional repertory company that would allow them to mount classic plays and even develop productions for New York. Now Fonda had drummed up some interest from actress Martha Scott, who had been hired by the Theatre Society of Long Island as artistic director for the local Mineola Theater. Renamed the Plumstead Playhouse, it would open in fall 1968 with stock performances of two chestnuts: Thornton Wilder's _Our Town_ , with Fonda as the Stage Manager and Ryan in a minor part as small-town newspaper editor George Webb, and, two weeks later, Ben Hecht and Charles MacArthur's _The Front Page_ , with Ryan as conniving big-city editor Walter Burns and Fonda in a small role as one of the reporters populating the press room at the Criminal Courts Building in Chicago.
_The Front Page_ had been Ryan's suggestion, inspired perhaps by his recent visit to Mayor Daley's Chicago and his chats with former City News Bureau reporter Sy Hersh during the McCarthy campaign. The play was forty years old at this point, but its breathtaking cynicism kept it evergreen. Hildy Johnson, a burned-out city desk reporter for the _Chicago Herald Examiner_ , has finally walked out on Burns, his roaring, exploitative boss; but after a condemned man escapes from jail, Burns manages to rope Hildy into covering one last story. Ryan's old friend Harold Kennedy, who had directed _Tiger at the Gates_ back in 1957, petitioned him for the part of the effeminate reporter Bensinger, which he had played in a previous production, and according to Kennedy's account, Ryan strong-armed Scott and director Leo Brady into casting him.
Kennedy would recall his growing unease as the director and the supporting players, many of them drawn from Brady's classes at Catholic University, squandered day after day of valuable rehearsal time, waiting for Ryan and Fonda to open _Our Town_ and then turn their attention to _The Front Page_. When Ryan arrived and apprehended the situation, he whispered to Kennedy, "What have these people been doing for ten days?"
Ryan's anger mounted as he and Fonda — who were already performing _Our Town_ at night — tried to get _The Front Page_ moving and some of the students proved unprofessional. Particularly irritating to Ryan was one student who kept missing his entrance because he was in the basement watching the World Series on TV. "What does he want to be, an actor or a ballplayer?" Ryan asked Kennedy. "He could learn more watching Hank Fonda in rehearsal for half an hour than he could learn at the Actors Studio in twenty years." When the student missed his entrance for the fifth time in a row, wrote Kennedy, Ryan walked up to the footlights and told Brady he would quit the production unless something changed. _The Front Page_ opened October 9, drawing considerably less attention than _Our Town_. Ryan was doubly frustrated by the experience because the professional actors in the cast — Kennedy, Estelle Parsons, John McGiver, Anne Jackson — were so good.
Privately, Ryan began hatching a scheme to restage the play in New York City. Five days after _The Front Page_ closed on Long Island, he invited Kennedy over to the Dakota and asked him to direct the new production, offering to do the play for nothing but insisting that they recruit a first-rate cast. "I want you and me to have complete artistic control," he said. An experienced producer, Kennedy crunched the numbers and determined that _The Front Page_ , with its twenty-four speaking parts, would be prohibitively expensive off Broadway and could turn a profit on Broadway only if they could persuade star players to work for the Actors Equity minimum of $167.50 a week. Ryan signed on immediately, his name serving as bait for other performers, and Kennedy found financial backing from Theatre 1969, a nonprofit founded by Edward Albee and Richard Barr, for a two-week tryout in Paramus, New Jersey, and a limited, four-week run at the Ethel Barrymore Theatre. Leaving Kennedy to assemble the production, Ryan shipped out for the UK in late November to play the title character in MGM's nautical fantasy _Captain Nemo and the Underwater City_.
Since Ryan's comeback three years earlier, his career had evolved into a schizoid cycle of low-paying theater work that gratified him creatively and high-paying pictures that he often preferred to forget. But his years in the desert had taught him a lesson: a movie actor could be so choosy about his scripts that he wound up backing out of the business. "It's important to continue working in films to keep your image warm," he told one reporter. "If you insist on turning down bad pictures, people are going to say: 'Whatever happened to Robert Ryan?'" The fact was that he'd done most of his best screen work when he was making pictures back to back; taking more jobs simply increased the odds that once in a while something would turn out well.
But sometimes that logic deserted him. Lisa Ryan, now a senior at the Nightingale-Bamford School, came home to the Dakota one night and found her father sitting in the kitchen alone. He hadn't heard her come in, and as she neared the door she overheard him muttering incredulously to himself: "God! Captain Nemo!"
BEN HECHT got his start in the newspaper business as a picture snatcher for the _Chicago Daily Journal_ , using all manner of skullduggery to score sensational photos of crime and accident victims, before he was promoted to reporter and eventually moved over to the _Chicago Daily News_ as a columnist. When he and Charles MacArthur, another veteran Chicago journalist, collaborated on _The Front Page_ in New York in 1928, they drew on a shared reservoir of colorful characters and outrageous anecdotes. Many of the caustic reporters who clustered together playing cards in the play's rat-a-tat opening scene were not only based on but named after real-life counterparts from Chicago, who basked in their newfound glory after the play opened. Walter Burns, the barking, scheming editor of the _Herald Examiner_ , was based on Walter Howey, MacArthur's boss first at the _Chicago Tribune_ and then at the _New York Mirror_ ; Hecht described him as "an invisible menace who sat in a Hearst tower, and with the aid of witches' brews, second sight, and other unethical trumperies, outwitted the town's honest news hounds."
Ryan had played this sort of comic heavy already in _The Busy Body_ , but that picture was stupid and _The Front Page_ was one of the great literary works of the American stage, perfecting the sort of staccato dialogue that Hollywood would embrace in its early talkies (the play was rapturously received in LA, and Hecht became a top screenwriter). For the first half of _The Front Page_ , Walter Burns never appears onstage; he's heard only through the earpiece of a telephone, ranting at his long-suffering reporter Hildy Johnson. Sick of the newspaper business, Hildy has turned in his resignation and plans to live happily ever after with his new wife, which invites a cascade of abuse from Burns: "You dirty double crossing Swede!... Walkin' out on me like a stinkin' yellow belly.... You two-faced bastard!... You goddamn tittering Swede moron — you lousy stewbum." Olivier might do a better Othello, but this was the gutter poetry of Ryan's youth, and no one could deliver it like he could.
As soon as Ryan returned from shooting _Captain Nemo_ in Britain, he got back to work with Kennedy on the new production. Kennedy's search for a suitable Hildy was impeded by the low wage they were offering as well as Ryan's insistence on top billing (he didn't often fuss about such things, but this was his own project and his Broadway comeback after _Mr. President_ ). Van Johnson, Peter Falk, George Segal, Jason Robards, and Richard Benjamin all had been approached but for one reason or another didn't work out; in the end Ryan and Kennedy went with young Bert Convy, who had appeared in the original productions of _Fiddler on the Roof_ (1964) and _Cabaret_ (1966). Kennedy reprised his role as Bensinger, and from the Plumstead cast they had snagged John McGiver as the mayor and Charles White as the Cook County sheriff. Peggy Cass played Mollie Malloy, the condemned man's girlfriend, and Katharine Houghton (who costarred with her aunt, Katharine Hepburn, in _Guess Who's Coming to Dinner_ ) was Hildy's love interest, Peggy Grant.
_The Front Page_ opened at the Ethel Barrymore on May 10, slaying the critics and selling out its scheduled run. A few doors down, the Biltmore Theatre was presenting James Rado and Gerome Ragni's "tribal love-rock musical" _Hair_ , and in the East Village, Kenneth Tynan was about to open his nude revue _Oh! Calcutta!_ Compared to these much-talked-about shows, _The Front Page_ might have seemed hopelessly retrograde, but as _New York Times_ critic Walter Kerr pointed out, "Plays that perfectly represent their own times never have to worry about what time it is." According to Ryan, his old _Born to Be Bad_ costar Joan Fontaine came backstage after a performance and told him, "Oh, Bob, it's so good to see a real play again!" There were plenty of New York theatergoers who felt the same way, and another four weeks were added to the run, extending _The Front Page_ into early July. After that, Albee's nonprofit group had to bow out, but plans were made to reopen the show under a new producing partnership in October.
Backstage at the Ethel Barrymore Theatre after performing _The Front Page_ ; at right are Lisa Ryan and Helen Hayes. The show's critical and commercial success marked a triumphant return to Broadway for Ryan after the humiliating _Mr. President_ six years earlier. _Robert Ryan Family_
That summer _The Wild Bunch_ was released, collecting both critical raves and harsh condemnation for its graphic killing; Ryan hadn't appeared in such a hotly debated picture since _Crossfire_. "Never have I seen such a bloody, violent, senseless, tasteless manifestation of what ails America as 'The Wild Bunch,'" wrote one _Times_ reader. "When such violence is depicted on movie screens without eliciting a picket line of protest, it can only mean that our citizenry has become completely immune to mass killing." According to one story, patrons at a Kansas City preview of the new western went out into the alley to puke. "I feel sorry for those who saw 'The Wild Bunch' and were so repelled they vomited," read another letter to the editor. "I wonder if they vomit from the real violence that we live with every day. The youths who won't return home or the ones who return maimed for life." The _Times_ even published a column satirizing the controversy, in which a cosseted Hollywood director is praised by a colleague for his recent feature _Mangled Entrails_. "That close-up of the girl's open carotid artery! And the way the sheriff's face exploded when the hand grenade hit him! You've got a beautiful grasp of the medium, baby."
Ryan was impressed by _The Wild Bunch_ , though the outcry over the violence gave him pause. After two decades with the American Friends Service Committee, United World Federalists, and SANE, how was he supposed to sell a picture that climaxed with an orgy of murder, a so-called "blood ballet" whose slow-motion shots, as critic Vincent Canby observed, rendered the violence " beautiful, almost abstract"? The contradiction between his politics and the pictures he made for a living had seldom seemed more acute. "I think it put Dad in a difficult situation, because he felt he needed to defend the violence in the movie, but I don't think he had a very good defense of it," said Cheyney Ryan. His father, he said, fell back on the canard that "Peckinpah wanted to make a movie that was so bloody that no one would ever want to make this kind of movie again." Of course, Peckinpah made another one just like it, and another after that.
The debate over _The Wild Bunch_ took place in the wider context of the racial rioting that had rocked American cities every summer since 1965 and the conservative backlash against it that had carried Richard Nixon into the White House. "It is time for some honest talk about order in the United States of America," Nixon had declared in a voice-over for one of his campaign commercials, as images of rioters and burned-out buildings filled the screen. "Dissent is a necessary ingredient of change. But in a system of government that provides for peaceful change there is no cause that justifies resort to violence."
This sort of rhetoric incensed Jessica Ryan, who, in her unpublished essay _America — Dream or Nightmare?_ , was quick to broaden the definition of violence: "Law and order, as the term is used by politicians and defenders of the status quo, in itself, represents an act of violence; it blunts awareness of who the victims of violence are: ten percent of the population denied full rights of citizenship; millions of Americans living in poverty and degradation; students demanding that the academies become more relevant to the times.... Law and order allows violence to be done to the dissenters while it frees the general public of the necessity to feel any personal or individual responsibility for the conditions that produced the dissent."
Interviewed two years after the initial controversy, Ryan was more philosophical about _The Wild Bunch_. "Whether or not the portrayal of violence is a good or bad thing no one will ever really know.... I, frankly, don't much care for the amount of violence shown in pictures. I thought _The Wild Bunch_ in some cases — although it had style and distinction — overstressed the bloodletting.... Violence, however, is an integral part of modern life. You can't blink at it. I just wish we could find another way." Talking to another reporter, Ryan hastened to put the issue in its historical perspective: "For a good many decades we were the country that led the world in lynching. We almost exterminated the Indians. We treated the blacks in the shameful way we still do.... When we don't like somebody, we shoot them!"
During his summer break from _The Front Page_ , Ryan took part in an independent short film that explored man's violent nature more wisely and subtly than any of the political invective flying around. Written by Arthur Miller as a one-act play, but turned into a thirteen-minute short by the young filmmaker Paul Leaf, _The Reason Why_ centers on a couple of old friends, Roger (Ryan) and Charles (Eli Wallach), relaxing one fine morning on Charles's farm. The only props of note are a pair of binoculars, which Roger has been admiring, and a rifle with a telescopic site. Through the binoculars Roger spies a woodchuck about 350 yards away, and Charles tells him how he used to pick off woodchucks with his rifle — dozens of them — because they kept destroying his vegetable garden. Eventually he gave up, though, because the bullets cost more than the vegetables were worth. "Seriously — it kept reminding me of a war," says Charles. "For what it costs to kill these days, we could put a tractor on every farm in the world and send all their kids to the University of Texas."
_The Reason Why_ must have struck a chord for Ryan, who had been sickened by his first hunting expedition with his father decades earlier. When Charles asks Roger if he's ever hunted, he replies, "Years ago. Birds. But I never really liked it. They're so beautiful, it breaks your heart." Yet Roger is a combat veteran who has killed two men in battle. When Charles brings up Vietnam, Roger observes, "These goddamn wars — they make everything seem so senseless." Eventually Charles goes into the house to retrieve his rifle and sights the woodchuck as Roger watches through the binoculars. Charles fires, the woodchuck goes down, and the two men walk out to the corpse and examine it. Charles tells Roger to leave the woodchuck for the circling hawks. When Roger asks him why he killed the animal, Charles replies: "I don't know. I probably won't anymore, though."
Ryan occupied an interesting position in popular culture at that moment: after twenty-five years as a movie tough guy, he had become a trusted face to a generation of middle Americans, someone who could articulate liberal values without scaring the hell out of people. Later that year Millard Lampell enlisted Ryan to help his friends Paul Simon and Art Garfunkel when their hour-long TV special, _Songs of America_ , ran into trouble prior to its scheduled broadcast on CBS. The original sponsor, AT&T, withdrew after it saw the boldly political program, directed by actor Charles Grodin, in which the duo's songs accompanied news footage of urban rioting, chaos in Vietnam, Cesar Chavez, and the Poor People's March. CBS backed the singers, Alberto VO5 stepped in as sponsor, and Ryan taped a thirty-second introduction to explain, "These two young men have attracted a tremendous following among the youth of America with their lyrical interpretation of the world we live in. We think you will find the next hour both entertaining and stimulating." (It was stimulating, all right: according to Grodin, one million viewers had turned it off by the first commercial break.)
Much of the cast of _The Front Page_ carried over when the show reopened at the Ethel Barrymore in October, though a few actors had moved on and one small role — Mrs. Grant, Hildy Johnson's prospective mother-in-law — went to Helen Hayes, not only the first lady of the American theater but the widow of Charles MacArthur. During this third incarnation, Martha Scott of the Plumstead Playhouse produced a fourth for TV broadcast, sponsored by Xerox and starring Ryan and George Grizzard as Hildy. Ryan was proud of the stage production, and like his idol Tallulah Bankhead in _Clash by Night_ , he had no understudy, playing every single performance himself. But as he liked to quote Shaw, art was expensive, and by February 1970, he had left _The Front Page_ and flown out to Durango, Mexico, for another western with Burt Lancaster, _Lawman_.
Canadian screenwriter Gerald Wilson had conceived of _Lawman_ as a commentary on the US political scene, particularly the recent drumbeat for civil order. After a crew of drunken cowpunchers (led by Robert Duvall) shoot up the town of Bannock, Marshal Jared Maddox (Lancaster) tracks them to the neighboring town of Sabbath, where the cowardly Marshal Ryan Cotten (Ryan) grovels to a local cattle baron. Once a respected lawman, Cotten has long since lost his nerve, tumbling from one bad assignment to the next; the people of Sabbath mock him as "Cotton Ryan." He covers his cowardice with cynicism: "If you're a lawman, you're a disease," he tells Maddox. "They want you but they hate you."
Ryan hadn't been in Durango since 1955, when he was stricken with alcoholic hepatitis during production of _The Tall Men_. ("The place hasn't changed much, though I don't remember it any too well," he told one reporter.) _Lawman_ turned out to be a positive experience: Lancaster ran a tight ship, and Michael Winner, the Englishman directing the picture for United Artists, knew his way around a camera. First-rate actors filled out the cast: Duvall, Albert Salmi, Sheree North, John Hillerman, Ralph Waite, John McGiver (as Sabbath's deaf mayor, following the action with a listening horn), and, as the powerful cattle baron, Lee J. Cobb, Ryan's cast mate in _Clash by Night_ on Broadway almost three decades earlier.
"Robert Ryan was an actor I'd admired nearly all my life," remembered Michael Winner. "He was the sweetest man in the world. He came to my house in Durango to say good-bye when he was leaving. He started to cry. He said, 'Michael, you'll never know what you've done for me. I can't thank you enough.' Tears were rolling down his cheeks." Winner was taken aback by this effusive display, but it began to make sense a few weeks later; after notifying Ryan's agent that he would need the star to rerecord some dialogue in New York, he received a nonsensical reply that Mr. Ryan had broken his leg and could not oblige, followed by a confidential call from Ryan himself explaining that he was getting radiation treatments for lymphoma. An actor had to keep something like this quiet if he ever wanted to work again, though Ryan assured Winner he could do the recording session. He didn't tell Winner what his doctors had told him: his chances of survival were less than fifty–fifty.
The cancer was inoperable, so for four months Ryan received cobalt radiation therapy at New York Hospital, suffering the usual side effect of crushing fatigue. The rest of the time he and Jessica retreated up north to their house in Holderness, which had been completed now and offered the utmost privacy. Ryan always had been proud of his looks and his physique, and he refused to let anyone but his family see him this way. Harry Belafonte met with Ryan after he got sick, but not often. "He really went into a social retreat during that period," said Belafonte. "I think he was just so discomfited by the disintegration, and I think his capacity to maintain social engagement and interest began to wane." Ryan's doctors had told him that, even if the cancer went into remission, they wouldn't know he was cured for another five years. Squirreled away in New Hampshire, the Ryans began to reckon with the fact that Robert might not be around much longer.
Jessica had seen many sides of her husband over the past thirty years. "I can almost guess, when we walk into a room full of strange people, which will be his persona for the evening," she wrote. "If the gathering is primarily one of WASPs or Jews, he immediately becomes Roman Catholic. And with certain non-Catholic, grand types — John Houseman, for instance — also the simple Irish boy, grandson of immigrants.... If we are at a party where there are uninformed R.C.s, he becomes the intellectual Catholic, giving accounts of the sophistication, the intellectualism and the historical conniving of the Church.... On the other hand, should intellectual Catholics be there, Ryan goes Protestant."
There were Ryans she didn't know — according to her friend Robert Wallsten, she once recalled opening a letter addressed to her husband from the Los Angeles Health Department and learning that he had been exposed to syphilis. But no one had more pieces of the puzzle than she did, and no one but she saw the Ryan who sat silently in the little A-frame house overlooking Squam Lake, contemplating the end.
During this period, another drama was playing out in Cambridge. The previous fall, Cheyney Ryan and other members of Students for a Democratic Society had occupied a dean's office at Harvard to demand better treatment of black service workers, and the school had expelled him. He was still in town that spring, working as a dishwasher at Massachusetts Institute of Technology, when revelations that President Nixon had widened the war to Laos and Cambodia sent the student antiwar movement off the rails. On Monday, May 4, the nation was shocked by images of students shot and killed by National Guardsmen on the campus of Kent State University. A week later, Cheyney, forbidden to set foot on the Harvard campus, showed up for a protest, and a few months after that he was arrested and charged with criminal trespass; convicted and sentenced to sixty days in jail, he would appeal and be given probation.
Tim had dropped out of school and was trying to launch a career as a folk singer, hitchhiking up and down the West Coast and often playing on the street. On May 15, he was part of the growing protests over the impending shutdown of People's Park, the free-speech area on the Berkley campus, when riot police waded into the crowd cracking heads and Governor Reagan called in the National Guard. Ryan told one interviewer that his eldest was "going through a great deal of self-searching and examination to find out what he wants to do." Around the same time, in a letter to the alumni office at Dartmouth, Ryan confessed his doubts about the value of a higher education: "My oldest son dropped out of Pomona in his senior year. He didn't feel what he was doing had any relevance to his life or the world around him. My youngest son was suspended in his senior year for nonviolent political activity. In neither case was the subsequent alternative a life of drugs or blissful inactivity. My youngest son was headed (easily) for a magna cum and is intellectually voracious. Whether or not he gets his Harvard BA is no longer of interest to him. Why?"
Amid all this youthful alienation, Ryan crossed paths again with Nicholas Ray, whose _Rebel without a Cause_ was only a distant memory now. After _King of Kings_ flopped, Ray collaborated with producer Samuel Bronston and screenwriter Phil Yordan on one more Spanish superproduction, _55 Days at Peking_ (1963) with Charlton Heston and Ava Gardner, but like the earlier one it spun out of control and the alcoholic director broke down (or was fired, depending on whose story one believed). Subsequent projects had crashed and burned, most recently a documentary about the Chicago conspiracy trial. "Mom, who is that guy sleeping on our couch?" Lisa asked her mother after she found Ray conked out in their living room at the Dakota. His shaggy hair had turned white, and a recent embolism had forced him to wear an eye patch. "There's a pirate here now," Lisa told Cheyney when he phoned. According to Cheyney, his father tried without success to get Ray a teaching job at Brandeis University, though the director would eventually land a two-year gig at Harpur College in upstate New York.
By the fall Ryan's doctors had pronounced his cancer in remission and said he might return to work, though he was greatly weakened by his ordeal and spending months in the desert on a western location was out of the question. Jessica, knowing how happy _The Front Page_ had made her husband, urged him to return to the stage (before his illness, he and Harold Kennedy had made plans to move their hit out to Los Angeles, and the Plumstead Playhouse had announced that Ryan would star in Archibald MacLeish's _J.B_.). But Ryan understood that his big problem now was proving to the movie industry that he could still be insured. His old friend Phil Yordan managed to land him a second-billed role as a TV executive in Columbia Pictures' _The Love Machine_ , adapted from a trashy best seller by Jacqueline Susann ( _Valley of the Dolls_ ). Before the deal was sealed, executive producer Irving Mansfield, who was married to Susann, paid Ryan a visit to inquire about his health. "Look, there is one thing about cancer: you don't die quickly," Ryan assured him. "I'll be able to make the picture."
The script was putrid, and Ryan disliked costar Dyan Cannon, who played his cheating wife. Before long the picture wrapped, though, and he was back in New York at the Dakota with Jessica and Lisa, now nineteen, who had quit school also and was driving a hansom cab in Central Park. Now that Ryan had overcome the professional stigma of cancer, Jessica had less trouble persuading him to consider a play, and a dream project materialized when Jay Fuchs, a producer on _The Front Page_ , put together a deal for Ryan to star in _Long Day's Journey into Night_ at the Promenade Theater at Broadway and Seventy-sixth. Ryan had been blown away by the first Broadway production in 1958, and his experience playing James Tyrone Sr. in Nottingham four years earlier had only whetted his appetite for a second staging, one that might better capture O'Neill's bleak vision of an Irish-American clan coming apart at the seams one day in 1912, at their shabby summer home. Ryan's doctors had advised him against doing the play, which ran nearly three hours, but he ignored them.
With Ryan on board, Fuchs and his partners took a chance on thirty-year-old director Arvin Brown, a Yale School of Drama graduate who had done good things at the Long Wharf Theatre in New Haven. Brown would recall his trepidation when he arrived at the Dakota to meet the star: "My impressions of him before I met him were from movies, and he scared the shit out of me." Ryan seemed aloof as he invited Brown in, which did nothing to settle the young man's nerves. But they connected quickly enough over the play, which Brown had directed at Long Wharf five years earlier (with Frank Langella as young Edmund). Ryan "began to get a little more comfortable, and I saw a man begin to emerge who was so the opposite of... his film persona, that I could hardly believe what I was seeing." Here was a man with a deep emotional, religious, and cultural connection to the play; he still remembered the racial discrimination faced by his grandfather and father, and his own Black Irish moods had made him a connoisseur of O'Neill's melancholy.
Casting was critical: aside from a maid who appears briefly, _Long Day's Journey_ has only four roles, all of them demanding. Tyrone, the rigid patriarch, is a mass of contradictions: he frets over the health of Mary, his wife of thirty-six years, and Edmund, the younger of his grown sons, but his own childhood of dire poverty has left him a skinflint who paces around the house extinguishing lightbulbs and skimps on the family's medical care. To play Jamie — the elder son, whose disillusionment has driven him to a life of whores and whisky — Brown chose twenty-nine-year-old Stacy Keach, another Yale alumnus and already an established Broadway actor. James Naughton, a handsome, twenty-five-year-old member of the Yale Repertory Theatre, leapt at the chance to play Edmund, who learns during the course of the play that he suffers from tuberculosis. For Mary Tyrone, who has slid back into morphine addiction after a brief period of recovery, Brown considered Kim Stanley but ultimately, at Ryan's urging, cast fifty-seven-year-old Geraldine Fitzgerald, who was Irish through and through.
O'Neill had drawn on memories of his family all through his writing career, but _Long Day's Journey into Night_ , his last completed work, was so nakedly autobiographical that he never allowed it to be produced in his lifetime. His father, James O'Neill, had shown enormous promise as a young actor but then made a fortune playing the title character in _The Count of Monte Cristo_ and stuck with the play until he was trapped in it for the rest of his career. In the fourth and final act, as Tyrone and Edmund sit alone in the parlor playing cards and waiting for Jamie to arrive home, the father opens up to his son, venting his disappointment in himself as he looks back over his career: "I loved Shakespeare. I would have acted in any of his plays for nothing, for the joy of being alive in his great poetry. And I acted well in him. I felt inspired by him. I could have been a great Shakespearean actor, if I'd kept on.... But a few years later my good bad luck made me find the big money-maker.... What the hell was it I wanted to buy, I wonder."
The words might have come from Ryan's own lips as he considered his straitened career and untapped potential. Friends who saw him perform _LongDay's Journey_ would be struck and in some cases disturbed by the parallels to his own life: the famous father, the fragile mother, the angry, idealistic sons.
That long, searching conversation between Tyrone and Edmund — over forty minutes of stage time — was the heart of the play, and Brown watched carefully as Ryan rehearsed with Jim Naughton, who was making his New York debut. Looking irritated, Ryan asked to speak with Brown afterward, and the director feared that he would demand another actor. Instead Ryan remarked, "The kid is really good, this was a good choice. But I don't wanna tell him that 'cause he'll get a swelled head. So I think you should say that to him." The incident revealed to Brown what a buttoned-down man Ryan was, though in fact the actor had taken this tack before with rising young talents, such as Terence Stamp in _Billy Budd_. From his perspective, egotism was the Achilles' heel of many a performer.
Ryan had more trouble acclimating himself to Fitzgerald, who liked to blurt out her character's thoughts in front of the other actors and whose interpretation of Mary Tyrone turned out to be radically different from the way previous actresses had played her. Researching the role, Fitzgerald had learned that, while morphine abuse reduces most people to dreamy indolence, it can drive others to shrieking fits, and in contrast to the shrinking violet essayed by Florence Eldridge and Katharine Hepburn, her Mary — in keeping with Tyrone's recollection that she once tried to throw herself off a dock — was forceful, even manic. "Ah, she's doing Medea tonight," Ryan cracked one evening as he and Naughton stood in the wings watching her carry on. He made the necessary adjustment, but he often thought Fitzgerald was overacting, a cardinal sin for him. Asked about Ryan's technique, Brown observed, "Above all things, it was economical. I think he loathed extravagance or wasted gesture." Fitzgerald's performance, however, added a strong feminine will to what always had been a male-oriented play, shifting the balance of power and creating a fresh dynamic.
_Long Day's Journey into Night_ opened Wednesday, April 21, to rave reviews from the _New York Times_ ("a towering achievement"), _New Yorker_ ("a triumph"), _New York Post_ ("a stunningly acted production"), Associated Press ("a production of searing splendor"), _Cue_ ("one of the decade's most memorable theatrical events"), _Variety_ ("best single legit offering off-Broadway this season"), _Newsday_ ("one of the major events of this season"), and on and on. Ryan liked to claim that, following Katharine Hepburn's advice, he never read reviews until a show was over, but his colleagues knew better, and after getting slammed so often for his serious stage work, he must have been elated by the universal acclaim. As _Times_ critic Walter Kerr noted, the key to Ryan's performance was his strength: "He silences [Edmund] by the steel in his eyes and the sores on his soul he is perfectly ready to expose. His very candor is kindness; he disembowels himself to show that he was made of good stuff.... In his cups, he has a grip on his psyche that no one can dislodge. He is character locked into itself, aware, obtuse, knowing and unalterable. The portrait, in its all-of-a-piece complexity, is beautifully composed, and Mr. Ryan explores the mea culpa in which no forgiveness is asked with admirable, leather-tough control."
That strength may have been illusory; as director of the show, Brown understood better than the other actors how cancer treatment had weakened this vibrant, athletic man. "Compared to what he was used to in himself, I think it was a hard adjustment," said Brown. "Jessica told that to me privately, too, in the early times of our relationship. It had just not occurred to him that he would be seriously ill.... He had been hit harder by it in certain ways than she ever expected." Brown had assembled and rehearsed a second company to perform the matinees, keeping Ryan down to six shows a week, though even this accommodation upset the actor. "We knew that the doctors had told him they didn't think he should be doing it now," remembered Naughton. "So his actions spoke very loudly to us, and set an example for me and for Stacy."
Keach and Naughton both adored Ryan, who was generous with them onstage and never pulled rank. "He loved acting, he loved the challenge of a great part," said Keach, who had admired Ryan on-screen for years. "[He had] an appetite, a wonderful appetite that was very inspiring to me as a young actor because I felt that it was something that I shared." Once the play had opened, Ryan initiated a little ritual with the young men. "At the end of the night," Naughton remembered, "when we were all changing to go out into the evening, he'd say, 'Okay, boys.' And Stacy and I would go into his dressing room, and he had a bottle of bourbon and three glasses. And he'd pour a shot glass for each of us. We'd have a drink together, and then off we'd go into the darkness, until the next night."
As Brown got to know the Ryans better, Bob began talking more openly about his and Jessica's alcoholism, which figured heavily in his regard for _Long Day's Journey_. "That seemed to be an area that, at least privately with me, he wanted to deal with, that he understood in the character and did not want to romanticize," said Brown. Talking to reporters, Ryan always claimed his hard-drinking days were over, and all during _The Front Page_ he had stuck to two beers a night as he dined with Harold Kennedy after every show. But the sixth member of the Ryan family had always been liquor. "I went to a Quaker weekend with my parents once, when I was in high school," said Lisa Ryan. "I remember very little of it except that they were unhappy because there was no alcohol there. [They] brought booze and they would go back to their room to drink so that they could then go back out and deal with these Quakers."
_Long Day's Journey into Night_ ran for three months, enough time for the quartet onstage to really take each other's measure. The polarity between Ryan and Fitzgerald persisted all through the run: she loved to improvise, throwing new notes into each performance, whereas Ryan hated surprises onstage. "He planned things, and thought it through," said Brown, "and needed to rehearse and rehearse and rehearse, and feel absolutely comfortable in what he was doing, so that his process was always this one of stripping away and making it more economical."
One night, when an onstage altercation between the brothers got out of hand, Edmund's shirt was badly torn and Naughton, deciding it was a distraction, shed the garment. No one thought to tell Ryan about this; when he made his next entrance and found Naughton bare-chested, he was flummoxed and more than a little angry. But Ryan might fool around backstage: one night Fitzgerald and Naughton, gazing through a window and ostensibly watching James Tyrone out in the yard, saw Ryan, hidden from the audience, peering back at them in a cat's-face mask.
Cancer hadn't impaired his sense of humor. Sometime during the run, Ryan wrote a little prose poem, "The Next Time You Want to Do a Play," in which he noted the various ego punishments of the New York stage. Among its warnings:
The bad notices will bother you more than the good notices will please you.
Kind friends will not fail to say things like, "I was furious at what John Simon said about your performance."
Every night at curtain call you will stand up and humbly solicit the approval of a gang of faceless ass-shifters who have spent most of the evening coughing and yawning.
Your days will be completely wrecked, spent in either doing the play or dreading it.
You will be besieged to do "talk" shows which pay nothing and operate solely to enhance the careers of the hosts. Your heart-warming reward will be a letter to your wife from an aging aunt in Round Rock, Texas, who will say, "I saw Robert on the Dick Cavett show." Her total comment.
Local tradesmen will say something like, "I saw your show last night. That Keach fellow is some actor, isn't he?"
Ryan wrote the piece at a time when he could afford to poke fun at himself: _Long Day's Journey_ was a genuine triumph and the pinnacle of his stage career. The following year, when the New York Critics' Poll was released, the play took best director, best actress, best supporting actor (Keach), and most promising new off-Broadway actor (Naughton). Even in this instance Ryan had to settle for being an also-ran, placing third behind Jack MacGowran and Harold Gould for best actor, but everyone knew he had left his mark on one of the great roles.
Keach left the show in early June to star in the boxing picture _Fat City_ , and in mid-July, Ryan, Naughton, and Fitzgerald handed the show over to the matinee cast. Jessica wanted Robert to keep working with Arvin Brown, and the young man became an occasional guest at their second home up in Holderness. "They had some of their happiest times up there," Brown recalled. "The family was very relaxed, and it was wonderful to see them all not kind of edgy with each other. It was really open." At such close quarters, Brown began to get a sense of the couple's history. "The marriage that I saw was a very devoted marriage, which you felt had had its problems and traumas — you certainly felt that, it was nothing easy about the marriage or the relationship — but it had become very strong.... He was dependent on her for her critical attitudes, he admired a great deal what she had to say about him in performances and whatnot, and he took her very seriously."
Ryan's next picture was a French crime drama with locations in Montreal and interiors at Boulogne Billancourt Studios in Paris. Director René Clément had made some distinguished films after World War II ( _The Battle of the Rails, The Walls of Malapaga_ ), but he had reached the tail end of his career, and the script, adapted from David Goodis's pulp novel _Black Friday_ , undercut its rich characters with an obscure and illogical plot. Charley (Ryan), a career criminal, is the patriarch of a little crime family that occupies a rural house on the water, including his common-law wife (Italian beauty Lea Massari), his thuggish enforcer Mattone (Aldo Ray), and a grown brother and sister whom Charley adopted when they were children. Jean-Louis Trintignant (And _God Created Woman_ ) took second billing as Tony, a New York crook being held captive by Charley and his clan because he knows the location of some cash Charley wants.
Ryan excelled in this claustrophobic stretch: there's a funny scene in which Tony manages to stack three cigarettes end to end, collecting ten dollars from Charley, and the older man, with a series of elegant hand flourishes and wrist flicks, tries to execute the stunt but can't get the third cigarette balanced without the cylinder crumpling. At the midpoint, though, the movie turns into a garbled heist adventure in which Tony bands together with Charley and his family to kidnap a government witness against an organized crime figure. In keeping with Clément's pretensions, the French release title was _La Course du lièvre à travers les champs_ (The race of a hare through the fields), though in the United States the picture would open (and quickly close) as _And Hope to Die_.
Jessica accompanied Ryan to Paris, where the growing fascination with film noir had begun to elevate him to cult-hero status. While they were there, a local cinema held a weeklong retrospective of his films, and according to Lisa, her mother and father ran into a group of film students on the street who began kneeling to Ryan and calling out the names of his old crime pictures. He thought they were nuts.
Back at the Dakota, Ryan took a breather and began looking into new projects. As United Artists produced a screen version of the Broadway musical _Man of La Mancha_ , a small outfit called International Producing Associates, with offices in Churubusco, Mexico, announced that Ryan would play the title character in a nonmusical adaptation of _Don Quixote_ , with Buddy Hackett as Sancho Panza. That spring the Ryans laid down their marker in the 1972 presidential race by holding a party at the Dakota for the dovish Senator George McGovern, then considered a long shot for the Democratic nomination. _New York Post_ columnist Pete Hamill attended, and remembered Ryan telling him, "Maybe this is the last chance we're going to have." Ryan had received an interesting script from MGM called _The Lolly Madonna War_ , about two feuding families in rural Tennessee, but the shooting schedule conflicted with a trip to Europe he and Jessica were planning, so he turned it down.
Jim Naughton had formed a close bond with Ryan, and as an avid pool player, he had an open invitation to drop in at the Dakota for a drink and a few games. One day in May 1972, he phoned Ryan to see if he was free, but Ryan begged off, explaining that Jessica was ill. Feeling unwell, she had gone into the hospital for some tests, and on Friday, May 12, her doctors came to Ryan with the grave news that she was dying, quickly and incurably, of liver cancer. Stunned, Ryan summoned the children and, at the doctors' urging, concealed the truth from Jessica. "I asked myself if I was a good enough actor to keep the news from her," Ryan recalled. "Somehow I did. But after a few days, she knew. 'You're all lying to me, aren't you?' she asked. I had to admit we were."
The news got out, and friends telephoned asking to see Jessica, but they were gently turned away. "It was very hard for me, because I loved her so much," said Ramona Lampell. "I'd never known anyone like her. And I'd never had anyone care about me the way Jessica did." But once the Ryan family closed ranks, that was it. According to Arvin Brown, Jessica spoke privately with her husband and each of her children before the end, and her dying request to Ryan was that he stick to the theater and stay true to himself as an artist. Ten days after her diagnosis, Jessica Cadwalader Ryan died at age fifty-seven.
Ryan was thunderstruck: after all these months worrying about his health, how could she be the one who had died? Nothing made sense anymore, but one grim certainty confronted him: like a western hero from the silent era, he would be riding off into the sunset alone.
_fifteen_
The Loneliest Place in Town
"Dido and I woke up this morning thinking of Jessica," Jean Renoir wrote his old friend Bob Ryan three days after her death. "It is only a moment of weakness: when we think of her, we know perfectly well that she is still with us. She won't abandon you: from now on, her spirit is around you." Jessica's private memorial service was held at the Greenwich Village meeting house of the American Friends Service Committee, with prayers from family and friends and songs from Tim and Cheyney; her ashes were given to Ramona and Millard Lampell, to be mulched into the ground of the New Jersey farm where they had recently moved, and where Bob and Jessica had been guests.
Once the service was over, Tim returned to California and Cheyney to Boston University, where he was teaching and earning a master's degree in philosophy. Lisa was getting an undergraduate degree at the School of Visual Arts in Manhattan and had long since moved out of the Dakota and into her own place. Ryan came home to a twelve-room apartment that now seemed shockingly empty. Eager to get out, he phoned his agent to see if MGM still wanted him for _The Lolly Madonna War_ , and a month after Jessica died he found himself in Knoxville, Tennessee, shooting out in the woods with director Richard Sarafian, costar Rod Steiger, and a fine ensemble of up-and-coming players that included Jeff Bridges, Scott Wilson, Gary Busey, Randy Quaid, Season Hubley, Ed Lauter, and Kiel Martin. " 'One-Take Ryan,' that's what we call him," Sarafian told the _Los Angeles Times_. "When he came in he was great with the young actors, and they have almost adopted him as a father."
Adapted from a novel by Sue Grafton (later a best-selling mystery writer), _The Lolly Madonna War_ harked back to the Hatfields and the McCoys but took place in the '70s: Pap Gutshall, the rustic farmer played by Ryan, tools around in a dilapidated station wagon. He and his neighbor, Leonard Feather (Steiger), have been enemies ever since a piece of Feather's land was put up for auction by the county to retire back taxes and Gutshall bought it. The men's grown sons have been pranking each other back and forth for months, but things get out of hand when a young stranger (Hubley) arrives in town, waiting for a connecting bus, and is kidnapped by Hawk Feather (Lauter) and his brother Thrush (Wilson), who have mistaken her for an accomplice in one of the Gutshalls' conspiracies. Bridges, who had already turned heads with his charismatic performance in _The Last Picture Show_ (1971), played Zack Feather, sobered by the death of his young wife, who strikes up a romance with the captured woman.
"I discovered that the only possibility for now was work," Ryan wrote to the Renoirs in mid-July, "so I am sweating it out in this steaming place doing a not-bad picture.... Jessica loved both of you so and we talked about you so many, many times — always with love, affection and admiration. She was a very rare lady and I feel a little ashamed to mourn when I should really rejoice that I knew her and that we had so many wonderful years together." The Renoirs were back in Hollywood at this point, and Ryan promised them a visit when he came west to shoot interiors for the picture. People knew him as a solitary man, but now he was desperate for company. With Jessica to confide in, solitude could be rich; without her, he was inconsolably lonely. Cheyney Ryan, who had been covering the Democratic National Convention in Miami for the magazine _World_ , stopped off in Knoxville on his way back to Boston, but then he was gone. "Something very big is missing," Ryan told a reporter on the set, "and I don't know what to put in its place."
There was always alcohol. Friends and family noticed immediately that Ryan had hit the bottle with a vengeance. Millard and Ramona Lampell were worried by his emotional state — what Millard described as a "black funk" — when Ryan came out to New Jersey to see the pear tree that was growing from his wife's ashes. "He sat on that bench crying," remembered Ramona, "and [talked about] how much he missed her. And it was just so sad." Sitting out in the country with them, Ryan went over and over Jessica's life, excoriating himself for the ways he had failed her. Friends saw him filled with regret for the months and years he had spent away from his family. "What the hell did I make all that money for?" he asked Philip Dunne, as James Tyrone in _Long Day's Journey into Night_ had asked, "What the hell was it I wanted to buy, I wonder."
Millard saw Ryan as a pillar: "He was sort of big daddy to everyone around him; that was the way Jessica felt toward him, that was the way his friends felt toward him. If you were in trouble, you could go to Robert." Yet Jessica always had been the emotional center of the family, and once she was gone Ryan and the children began to drift apart. Lisa saw her father the most; now that she was an adult, they could drink together, and as they stayed up late playing pool, he began to open up to her about his life in a way he never had. Cheyney kept tabs on his dad from Boston but was dismayed by what he saw. "I thought my father handled my mother's death very badly," he admitted, "and one of the reasons why it was such a traumatic event, I think for all of us, was because my father just did not deal with it.... He thought only of himself; he was drinking all the time. I never got the feeling once that he gave a thought to the impact of this on the three of us, and fairly soon into that year I just started to get sick of it, to feel that this was a completely dysfunctional situation."
Ryan returned to New York resolved to move out of the Dakota and buy a smaller unit back at 88 Central Park West, where the family had lived in the mid-1960s. He put out feelers that he wanted to lease the Dakota apartment and almost immediately had two prospective tenants: John Lennon and Yoko Ono, who had been living in the St. Regis Hotel on Fifth Avenue since they moved from England to the United States a year earlier. Ryan took Lisa with him when he went to meet the former Beatle; to her astonishment Lennon was excited to be shaking hands with Robert Ryan. "I guess he liked American westerns," she recalled. "And I'm just sitting there with my mouth hanging open, 'cause I'm meeting John Lennon." The two men chatted amiably about their family lines back in Ireland, and the deal went through. Lennon and Ono would move into the Ryans' old home in February 1973, and Ryan rented a unit down the street, where, according to Lisa, he re-created his late wife's room.
Around that time, Arvin Brown and his wife, Joyce, hosted Ryan at their oceanside home in Branford, Connecticut, and found him grateful for any company. He liked Joyce Brown and went out shopping with her around town. "In certain periods of his life, he had a funny kind of deference in his personality," said Arvin. "He would sort of go along with what everyone was doing, and just trot along. And it was always great fun, because he was a very famous man, and wherever he'd go and shop everyone would do double takes and whatnot, which he was always delighted by." Ryan filled them in on his children, praising them in a way he never would to their faces. "He was proud of what he felt was their honesty, and their integrity with themselves," Brown remembered. "Whatever concerns that he might have shared about their direction in life had nothing to do with what he felt was their character."
Late that year Ryan was invited to a dinner party and spent a pleasant evening talking to his old friend Maureen O'Sullivan. Born in County Roscommon, Ireland, and educated in a convent school, O'Sullivan was discovered by director Frank Borzage and became a star at MGM in the early 1930s, working steadily for ten years before she retired from show business in 1942 to concentrate on her family. With her husband, John Farrow (who had directed Ryan in the RKO potboiler _Back from Eternity_ ), she raised seven children; Mia Farrow had become a star in 1967 with _Rosemary's Baby_ (which was filmed at the Dakota), and Ryan had recently worked with her younger sister Tisa, one of the crime kids in René Clément's _And Hope to Die_. O'Sullivan, who resumed her career in the '50s, was known around Hollywood as a good mother and a faithful wife, until Farrow died in 1963. She and Ryan crossed paths briefly in January 1970, when she joined the cast of _The Front Page_ just as he was leaving it. After the dinner party, the two of them became an item, spotted together in restaurants and at the theater.
Coming so soon after Jessica's death, the relationship pained some of those closest to Ryan but warmed others; Robert Wallsten and Albert Hackett thought O'Sullivan a great tonic for their old friend. Few doubted her affection for Ryan, though some wondered if he returned it in equal measure or simply needed a woman's care. Cheyney, who wound up having more contact with O'Sullivan than his siblings, appreciated what his father saw in her and what she did for him: "I think he liked the fact that Maureen had this kind of Irish Catholic reputation about her," he recalled. O'Sullivan struck him as "a fairly reserved and a thoughtful person.... You know, my mother was [my father's] sole emotional connection to reality. He was the kind of guy that was gonna relate to those things through having a female partner."
Ryan had never forgotten the experience of filming _Billy Budd_ on the high seas, and before Christmas he traveled to Newport, Rhode Island, to shoot an ABC movie-of-the-week aboard a 1970 replica of the HMS _Rose_ , which had fired on American fortifications during the Revolutionary War. Based on an 1863 story by Edward Everett Hale, _The Man without a Country_ starred Cliff Robertson as army lieutenant Philip Nolan, charged with treason in 1807 for having conspired with former Vice President Aaron Burr to found a new nation in Texas and Mexico. "I wish I may never hear of the United States again!" Nolan shouts at his trial, prompting the judge to exile him for life. Over the years he's transferred from one navy ship to another, never to set foot on US soil again, and his crewmates are forbidden to tell him what's going on in his native land. Beau Bridges costarred as the career officer who befriends Nolan, and Ryan breezed through his cameo as a navy veteran who narrates part of Nolan's story. Beautifully written, the movie had a special resonance in a time when some thirty thousand Americans, having evaded the draft by going to Canada, now faced prosecution at home.
Since Ryan's cancer diagnosis two years earlier, he had become preoccupied with building up an estate, which meant doing pictures; yet Jessica had implored him to pursue theater work. A lovely compromise presented itself near the end of the year when John Frankenheimer, who had directed Ryan in _The Snows of Kilimajaro_ on CBS in 1960, offered him $25,000 (a sixth of what he now commanded, but a decent sum) to play the whiskey-soaked anarchist Larry Slade in a screen version of Eugene O'Neill's _The Iceman Cometh_. Since their experience doing live TV together, Frankenheimer had become a top movie director with smart, paranoid pictures such as _The Manchurian Candidate_ (1962), _Seven Days in May_ (1964), and _Seconds_ (1968). _The Iceman Cometh_ would be rehearsed for three weeks in LA and shot in sequence over seven weeks on the Fox lot, with a cast to include not only Lee Marvin and Jeff Bridges, but also Ryan's hero Fredric March. Ryan flew out to the coast the first week of January 1973, pleased with the new year.
"HAS THE ICEMAN COME YET?" a man calls upstairs to his wife. "No," she replies, "but he's breathing fast." O'Neill alludes to this dirty joke several times during _The Iceman Cometh_ — it's a favorite of his high-spirited traveling salesman, Hickey — but as the biblical conjugation of the title suggests, the iceman is also Death. Running four hours, the play is confined to Harry Hope's saloon, a Lower Manhattan dive where men come to drink themselves into an early grave. Hope owns the five-story building, an SRO hotel, and by drawing a curtain across the street-level bar he can turn the back section into a "hotel restaurant" and legally serve liquor at all hours. The play opens in early morning, as bartender Rocky (Tom Pedi) arrives and finds the usual assortment of drunks passed out on the tables. But Larry Slade, Ryan's character, is wide-awake and philosophically inclined. His dreams, he confesses to Rocky, "are all dead and buried behind me. What's before me is the comforting fact that death is a fine long sleep, and I'm damn tired, and it can't come too soon for me."
The picture was being produced by the American Film Theatre, a high-toned experiment in which faithful screen adaptations of great theater works would be exhibited at local movie houses on Monday and Wednesday nights, with patrons buying subscriptions for an entire "season" through American Express. Producer Eli Landau had already begun or completed adaptations of Edward Albee's _A Delicate Balance_ (with Katharine Hepburn), John Osborne's _Luther_ (with Stacy Keach), and Harold Pinter's _The Homecoming_ (with Cyril Cusack); _The Iceman Cometh_ would open the first season. The novel exhibition approach made running time less of an issue than usual, and though Frankenheimer pruned the play down with an editor, so many actors came to him during production with good arguments for restoring deleted dialogue that he wound up shooting close to the entire text at a length of four hours. Landau urged Frankenheimer to alleviate the play's claustrophobia by moving a few scenes out onto the street, but the director refused, arguing that the sense of enclosure was critical to O'Neill's take on illusion and reality.
_The Iceman Cometh_ offered a colorful ensemble of characters, the main antagonists being Larry, a former anarchist who's renounced politics, and Hickey, who drops in at the saloon every few months to dispense good cheer and free drinks. Frankenheimer quickly settled on Ryan as the perfect Larry: "He had a deep sadness inherent in most of these O'Neill characters." Casting Hickey was more difficult: Jason Robards was strongly identified with the role but had played it so many times that Frankenheimer opted for someone new, offering the part to Marlon Brando (who declined, saying he would never be able to memorize the lengthy soliloquies) and considering Gene Hackman before going with Lee Marvin. Fredric March, who had retired in 1970 to battle prostate cancer, returned for one last screen performance in the relatively small role of Harry Hope. Richard Dreyfuss, Keith Carradine, and John Savage all read for the part of Don Parritt, the young radical who arrives at the saloon looking for Larry and reveals their shared past. Savage even shot a screen test, though the role ultimately went to an uncertain Jeff Bridges.
As Bridges recalled, the offer came along when he was worn out from shooting _The Last American Hero_ with director Lamont Johnson (Ryan's old friend) and, more fundamentally, was unsure whether he wanted to keep acting or pursue a music career. Initially he turned down the role, but when Johnson heard that the young man had passed up the chance to work with Ryan, March, and Marvin, he phoned Bridges, called him a "stupid ass," and hung up on him. Bridges reversed himself, and by his own admission, the experience of making _The Iceman Cometh_ inspired him to take his craft much more seriously. He and Ryan had met the previous year, shooting _The Lolly Madonna War_ (ultimately released as _Lolly-Madonna XXX_ ), but shared no scenes together; in _The Iceman Cometh_ they went head to head in scenes that stretched out as long as nine minutes. "He was one of my favorite actors," said Bridges (who, like Stacy Keach, still remembered Ryan's chilling performance as Claggart in _Billy Budd_ ). "As an actor he stood alone for me."
Frankenheimer strived to create a good artistic environment. The ensemble of fifteen actors included such talents as Moses Gunn, Bradford Dillman, Sorrell Booke, and Clifton James, and Frankenheimer gave them all plenty of screen time; copying the Dutch masters, he used deep-focus photography to compose gloomy frames in which various barflies, stiff with drink, listened and sometimes reacted to a central character's extended speech. Ryan, March, and Marvin showed up whether they had scenes or not, just to watch. Arvin Brown took several phone calls from Ryan during the shoot: "Every time he called me it was to tell me about some brilliant thing that Freddy March had done on the set." March had been ill during rehearsals but showed up on the last day with his part down cold, and he delivered on camera. Evans Evans, who was married to Frankenheimer and played one of the three hookers who frequent the bar, recalled a collegial Friday-night dinner on the set that attracted not only the cast but the entire crew and that Ryan thoroughly enjoyed.
"As rehearsals progressed we became closer and closer," Frankenheimer later wrote in a letter to Lisa Ryan. "Naturally he told me of his sickness and your mother's death. Then the problem became not to let him personalize the role too much. But he didn't — always the pro — knowing just when to pull away."
According to Frankenheimer, Ryan mentored Bridges throughout the shoot, but just watching Ryan work could be a lesson in itself. One day he and Bridges were seated at a table together, waiting for the crew to set up one of their long and demanding dialogue scenes; these had to be played at their full length because Frankenheimer was shooting with two cameras, one for each actor and each with its own cues for zooming in and out. Years earlier, Ryan had confessed to one reporter that, for some reason, nothing fazed him like having to play a scene seated at a table. The assistant director announced they were ready, and Ryan lifted his palms from the table to reveal twin puddles of perspiration — the old flop sweat. When Bridges asked Ryan if he still got nervous after all these years, Ryan replied, "If I wasn't nervous, I'd be really nervous." Yet on-screen, Bridges is the one who looks nervous, giving the role his all but often giving too much; Ryan, ever the minimalist, pared his performance down to the bare essentials but made every reaction count. Spencer Tracy had upstaged Ryan in much the same fashion nearly twenty years earlier, in _Bad Day at Black Rock_.
Former anarchist Larry Slade (Ryan) confers with Don Parritt (Jeff Bridges) and Hugo Kalmar (Sorrell Booke) in _The Iceman Cometh_ (1973). "As an actor he stood alone for me," Bridges said of Ryan. _Franklin Jarlett Collection_
The evolving relationship between Larry and young Don Parritt was steeped in the anarchist politics of the 1910s but still relevant in the bitter era of the Black Panthers and the Weather Underground. O'Neill based Parritt on a real-life figure, Donald Vose, who betrayed his mother and her anarchist comrades to the police after the 1910 bombing of the _Los Angeles Times_ building; in _The Iceman Cometh_ , the guilt-ridden Parritt comes looking for Larry, his mother's old lover — and, possibly, his father — hoping to make an emotional connection that the older man steadfastly resists. Larry has given much of his life to the anarchist dream, but he can no longer maintain the fervor it demands. "I was born condemned to be one of those who has to see all sides of a question," he tells Parritt. "When you're damned like that, the questions multiply for you until in the end it's all question and no answer."
Cheyney Ryan visited LA during the shoot and spent some time on the set; he and his dad were sharing a car, and none of Cheyney's friends got up before noon, so he would hang around the Fox lot in the morning and watch the company work. As he later told writer Dwayne Epstein, Marvin showed up one day at 8 AM with a case of beer and proceeded to get hammered. "He got into a thing about what a big star he was," Cheyney recalled. "It was really unpleasant.... He said, 'Your father's not a big star anymore. I'm a big star. He used to be a big star and now I'm the big star.' This went on and on and on." Frankenheimer took Marvin aside later and read him the riot act about his drinking, just as Richard Brooks had on _The Professionals_ , and Marvin — who confessed to Frankenheimer that he was terrified to be working with Fredric March — promised to straighten up. "Bob did an awful lot toward calming Lee down," said Frankenheimer, "because Lee had tremendous respect for Bob Ryan."
Marvin may have been right about his and Ryan's relative stardom, but _The Iceman Cometh_ , with its philosophical contest between Larry and Hickey, gave the two actors a parity lacking in any of their previous collaborations. Hickey arrives at Harry Hope's saloon swearing off booze (though he sets up the drinks for everyone else) and urging his old friends to cast off their illusions and face the world. Larry already has expressed his feelings on the subject: "As the history of the world proves, the truth has no bearing on anything," he tells Rocky. "The lie of a pipe dream is what gives life to the whole misbegotten mad lot of us, drunk or sober." Marvin came into the project at a clear disadvantage: he hardly was known for his theatrical prowess, and he clearly was an offbeat choice to play such a seemingly cheery character. Ryan, however, had found in O'Neill the sort of weighty, tragic characters he had been chasing his entire career. When critic Charles Champlin visited the set, Ryan told him, "This is one I'll want to be remembered by."
RYAN WAS BOOKED into the Beverly Wilshire Hotel, but most nights he stayed at Phil Yordan's house on Benedict Canyon Drive. "He didn't want to be alone," Yordan recalled. "He was a very lonely man." As a houseguest, Ryan was quiet and undemanding: coffee in the morning, a ham sandwich and glass of milk at lunchtime. Yordan was in the process of negotiating a huge deal with Television Corporation of America for the sale of all his films and literary properties, as well as his half-interest in a forty-acre studio in Madrid; _Variety_ reported that he would join the company as a producer and listed as one of his upcoming projects a drama called _Riche_ , to be shot in Yugoslavia the following year with Ryan, Rod Taylor, and Claudia Cardinale. After _Iceman_ wrapped in early March, Ryan moved on to _The Outfit_ , a crime drama for MGM with Robert Duvall (who had become a star with _The Godfather_ ), Karen Black, and Joe Don Baker. Ryan played a mob boss, the sort of thing he could do in his sleep. "At that point his wife was gone and all he was interested in was creating an estate for his children," observed Yordan.
John Flynn, the forty-one-year-old director of _The Outfit_ , had apprenticed with Robert Wise on _Odds Against Tomorrow_ back in 1959, and like Wise he populated his story with familiar noir faces: Jane Greer, Marie Windsor, Timothy Carey, Elisha Cook Jr. An edgy score from Jerry Fielding accompanies the tense opening sequence, in which two mob assassins, disguised as a priest and his cab driver, arrive at the rural home of their target and wordlessly stalk him in his backyard, cutting him down with silencers on their pistols as a German shepherd fights to get off its chain. Paroled from prison, the victim's brother (Duvall) learns what happened from his girlfriend (Black) and rounds up an old pal (Baker) to get even with the mob kingpin responsible for the hit (Ryan).
Flynn had learned his lessons well: _The Outfit_ harked back to Wise's early, low-budget classics ( _The Set-Up, Born to Kill_ ) in its visual economy, inventive framing, and propulsive editing. All things considered, it was a more satisfying genre revival than René Clément's _And Hope to Die_. Joe Don Baker, a Texan who had learned his craft at the Actors Studio in New York, came to the project as a fan of Ryan's performances in _The Set-Up_ and _Odds against Tomorrow_. He found the elderly actor to be a quiet, modest man; the scuttlebutt around the set was that he was dying of cancer.
The private home doubling as the mobster's mansion was located on Sunset Boulevard, only a few blocks from the Ryans' old house in Holmby Hills. Ryan told _Variety_ that he had sold the place to George Axelrod for $175,000 back in 1962, at the bottom of the market; Axelrod, he reported, had recently sold it to Barbra Streisand for four times that amount. On the last day of shooting, producer Carter DeHaven threw a party in Ryan's honor, and the City of Los Angeles awarded him a plaque celebrating the completion of his eightieth picture.* Jane Russell attended, and so did Burt Lancaster, who had lined up Ryan to costar with him in a hush-hush project about the Kennedy assassination called _Executive Action_. Ryan got a laugh when he accepted the award, thanking the city but conspicuously omitting conservative mayor Sam Yorty. "Eighty pictures," Ryan marveled to Charles Champlin during the party. "And 70 of them were dogs. I mean, _dogs_."
Back in New York, Ryan made the rounds with Maureen O'Sullivan, who spent a good deal of time at his apartment at 88 Central Park West. "He felt that he should wait a decent length of time before he got married after his wife's death," remembered Albert Hackett. At one point Ryan and O'Sullivan paid him a visit, and Ryan did a little soft-shoe routine to a song playing on the phonograph. "I never saw him so well and so happy," said Hackett, "and I thought he was getting ready for the big moment when the year was up, and they were going to get married." Ryan also was plotting his return to musical theater after more than ten years: the _Hollywood Reporter_ soon would announce that he had signed to star in a musical version of the Jimmy Stewart drama _Shenandoah_ (1965), as a Virginia farmer who wants to keep his sons out of the Civil War. The show was scheduled to open on Broadway in March 1974.
_Executive Action_ , a speculative account of US industrialists plotting to assassinate President Kennedy, originated with Donald Sutherland and attorney Mark Lane, whose 1966 best seller _Rush to Judgment_ had raised serious questions about the Warren Report. Sutherland put Lane together with playwright Donald Freed to draft a screenplay, but when the actor failed to secure financing, the project was taken over by Edward Lewis, an executive producer on _The Iceman Cometh_ and a veteran of left-leaning Hollywood cinema ( _Spartacus, Lonely Are the Brave, Seven Days in May, Seconds_ ). Dalton Trumbo was brought in to rewrite the script, and though he professed skepticism initially, he was converted after seeing the uncut Abraham Zapruder film of the assassination, which had never been broadcast, and concluding that it showed the president being fired upon from two different locations.
With a paltry budget of about $500,000, Lewis persuaded Burt Lancaster to make the film for scale, and Lancaster sold Ryan on doing the same. Cheyney Ryan remembered his father praising the script, which at first he didn't know had been written by his old colleague Trumbo. "I originally bought the lone assassin theory because it's been American history, at least as far as we know," said Ryan when he was interviewed on the set for a making-of documentary. "But when I read the script, the machinery of the conspiracy was so convincing that I began to change my mind. I don't mean I have the answer, but the script itself made me want to do the picture." He always had an eye out for political provocations (for years he had wanted to make a picture about John Brown, the abolitionist who tried to launch a slave insurrection before the Civil War). _Executive Action_ also offered the comfort of familiar faces — not just Lancaster but Will Geer, whose association with Ryan stretched back twenty years to John Houseman's production of _Coriolanus_ , and who had recently gotten the last laugh on HUAC when he became a beloved figure on the hit CBS drama _The Waltons_.
Publicity materials would quote director David Miller saying that Ryan likened his _Executive Action_ character — Robert Foster, a Texas millionaire scheming to kill the president — to Montgomery, the murderous bigot in _Crossfire_. Both men are closet fascists, the millionaire couching his race hatred in the cool, clean arguments of social Darwinism. Strolling around the grounds of his estate with James Farrington (Lancaster), a shadowy paramilitary type, Foster sketches out a dystopian future of exploding minority populations clamoring for limited resources. That's why victory in Vietnam is crucial: "An all-out effort there will give us control of South Asia for years to come. And with proper planning, we can reduce the population to 550 million by the end of the century.... Not only will the nations affected be better off, but the techniques developed there can be used to reduce our own excess population: Puerto Ricans, Mexican Americans, poverty-prone whites, and so forth." One last time Ryan breathed life into an incendiary political picture by embracing the thing he loathed.
Unfortunately any parallels to _Crossfire_ end there, because _Executive Action_ is clumsy and inert, more like an illustrated lecture than a story. Trumbo opens in early June 1963, as the genteel Foster hosts a little conference of powerful right-wingers on his estate. Geer, clad in a cream-colored suit, plays Harold Ferguson, a folksy oil tycoon whom the others need to back their covert action against JFK (the reason for this is never really explained). The bespectacled Paulitz (Gilbert Green) predicts that in the next few months the president will back civil rights, endorse a nuclear test ban treaty, and begin a retreat from Vietnam. Farrington presents a slide show detailing past presidential assassination attempts and argues that their best chance of success is triangulated sniper fire during a presidential motorcade; a later slide show acquaints them with their chosen patsy, Lee Harvey Oswald.
Ryan gives his final performance, in _Executive Action_ (1973), as the right-wing oil man Robert Foster. Victory in Vietnam, Foster insists, "will give us control of South Asia for years to come. And with proper planning, we can reduce the population to 550 million by the end of the century." _Franklin Jarlett Collection_
These endless lecture sequences are interrupted periodically by desert scenes in which three snipers (commanded by bald, hawk-nosed Ed Lauter of _Lolly-Madonna XXX_ ) practice firing on a moving target. Miller incorporates archival footage to show Paulitz's predictions coming true: Kennedy makes his famous speech at American University in June 1963 supporting a test ban treaty, states the moral imperative for a civil rights bill in a speech from the Oval Office that same month, and tells a reporter at a press conference that he hopes to have a thousand US advisors out of Vietnam by the end of the year. An obviously phony newscast hammers home the point that Kennedy intends to pull out of Indochina — still a matter of historical debate — and the report so incenses Harry Ferguson that he finally gives the kill order on JFK.
Despite the conspiracy theory, Ryan was fascinated most by the plush train car being used as a set for _Executive Action_. It must have brought back memories of the New England whistle-stop campaign for Stevenson twenty years earlier. One could rent a furnished train car and travel across the country by connecting with various lines, and Ryan had decided that he and Maureen O'Sullivan would collect family and friends and travel by rail through the mountain country of Tennessee and West Virginia.
In Los Angeles, he visited Philip and Amanda Dunne, and one day he turned up at the doorstep of John and Evans Frankenheimer's house in Malibu. They invited him to stay over, and before long he was spending every weekend with them. "A lot of it was great reminiscing, just very relaxed," said Evans. "We would have lunch and then he would read, and then later he would wander off." He drank heavily, talking about Jessica, though otherwise his health appeared to be holding. "He honestly thought he had the cancer licked," said John Frankenheimer. "He never got over the fact that here he had the cancer and his wife nursed him through it, and after he was somewhat cured, then she came down with it. He couldn't understand it. He deeply, deeply missed his wife."
Onscreen Ryan looked pasty and tired, and after he began complaining of back pains, Miller hastened to complete his scenes so he could return to New York and look after himself. Larry Goebel, a cinema student at the University of Southern California, would write a letter to the _Los Angeles Times_ remembering his visit to the rail yard in Vernon, California, on the day Ryan shot his last scene with Burt Lancaster. When they were done, "the train's porter served martinis to the two stars. After one sip, Ryan went into one of the funniest drunk routines ever witnessed.... The entire crew was convulsed for five minutes. Shortly afterwards, he shook hands with everyone and left to catch a plane for New York." Against all odds, Ryan's last performance was pure comedy.
On Tuesday, July 3, Ryan was admitted to New York Hospital. Maureen O'Sullivan informed Cheyney, who drove in from Boston to assess the situation. Along the way he stopped off in Newport, Rhode Island, to visit Mia Farrow, who was shooting _The Great Gatsby_ with Robert Redford, and his sister, Lisa, who had accepted Farrow's invitation to be an extra in a party scene. When Cheyney arrived in Manhattan the next day, the information on his father was sketchy; he spent a few afternoons in the hospital with Ryan, watching sports on TV, and more time than he would have liked on the phone with John Lennon's people, trying to retrieve some air conditioners from the Dakota that his father wanted back. Ryan's mood was dark; at one point he told Cheyney, "I don't want to live anymore."
Ryan and Burt Lancaster on location for _Executive Action_ (1973). After completing their last scenes together, Ryan took one sip of a martini, performed a riotous drunk act for the crew's amusement, shook hands with everyone, and flew home to look after his health. _Franklin Jarlett Collection_
The following week the doctors came to Cheyney and O'Sullivan with grim news: the cancer had returned and spread to Ryan's lungs. Tim and Lisa were telephoned and began heading for New York, where they all were to confer with the doctors about the next phase of their father's treatment. Lisa and Cheyney visited Ryan on the morning of Wednesday, July 11. That afternoon his lungs hemorrhaged and he began to choke, but the staff managed to stabilize him. O'Sullivan arrived at 88 Central Park West that evening, distraught over Ryan's condition and unhappy with the care he was getting. Tim flew in from California that night; before he could see his father, though, the hospital phoned to notify them that Ryan had suffered another breathing attack and choked to death.
"I'VE HAD A GOOD SHOT AT LIFE," Ryan told a _Los Angeles Times_ reporter the previous summer, reflecting on his earlier cancer scare. "So what the hell do I have to complain about. My brother died when he was six, and I've thought about it my whole life. He never even got started. I've been lucky as hell with my career and my family. We were always close. Still are. How many men can say that?"
Tim, Cheyney, and Lisa were so shell-shocked that Millard and Ramona Lampell took care of the funeral arrangements. A private service was held on Monday, July 16, at Blessed Sacrament Church on Seventy-First Street, and descended into black comedy when the priest celebrating the funeral mass — a relative of Ryan's from Chicago — became confused by the English text, having performed the mass only in Latin, and made a mess of things. Jason Robards, Myrna Loy, Dore Schary, and John McGiver attended the service, and the guests were welcomed back to 88 Central Park West afterward for what Millard described as "a regular Irish, old-fashioned wake." He and Robards shot a few games of pool. The _Hollywood Reporter_ had reported that Ryan would be buried in Chicago, but in fact he was cremated and his ashes taken by the Lampells to be mixed with Jessica's on the grounds of their farm. His estate, estimated in the press at half a million dollars, was divided equally among the children.
The tributes that followed praised Ryan's personal qualities as much as his acting. "There should be a poem of farewell for Robert Ryan, who was a good man in a bad time," wrote Pete Hamill in the _New York Post_. Hamill reminded readers of Ryan's long years in the liberal trenches, founding Hollywood for SANE amid the Red Scare, but also pointed out Ryan's "refusal to make his life a performance. He saved his private life for himself, and his children, and his wife, Jessica. He took no punches at photographers. There were no drunken car wrecks. There were no messy divorces." In the _Los Angeles Times_ , Charles Champlin connected this same modesty to his onscreen brilliance: "He did what he did with a caring, self-effacing professionalism which illuminated the character rather than his private persona, and he did it so well that the art was, in its paradoxical way, invisible."
_Newsweek_ writer Paul Zimmerman would fashion Ryan's cultural epitaph when he wrote that the actor left behind "a lifetime of roles too small for his talent." This wasn't entirely accurate — Ryan had tackled great roles onstage and fallen short ( _Othello, Antony and Cleopatra_ ) — but the consensus view among critics was that he had never gotten his due. For several years he had served on the board of governors for the Motion Picture Academy, yet aside from _Crossfire_ , Oscar never had taken the least notice of his finest portrayals, not Stoker Thompson in _The Set-Up_ , nor Jim Wilson in _On Dangerous Ground_ , nor Ben Vandergroat in _The Naked Spur_ , nor Ty Ty Walden in _God's Little Acre_ , nor John Claggart in _Billy Budd_ , nor Deke Thornton in _The Wild Bunch_. Released in November 1973, on the heels of _The Outfit_ and _Executive Action, The Iceman Cometh_ became Ryan's final artistic testament, and attention finally was paid: the National Board of Review named him best actor of the year, and the National Society of Film Critics gave him a special award for "a consummate demonstration of acting skill at the end of a long distinguished career."
An even more fitting tribute followed a year later, when John Houseman hosted a performance of Sean O'Casey's _Juno and the Paycock_ at the Mark Taper Forum to benefit the Robert and Jessica Ryan Memorial Foundation. A $10,000 grant from Liz Harmon had established the foundation, which was to fund teacher training and instruction of children with learning disabilities, and to institutionalize the social studies program that Jessica and Marie Spottswood had worked so hard to create. Spottswood wrote that the program still was using the books and materials Jessica had written, praising "her able and courageous leadership in piloting the school through a most harassing and difficult period when rival radical factions among parents and teachers came close to destroying it. More than anyone else, she was responsible for Oakwood's survival." Even after the Ryans left Los Angeles, they had maintained their connection to the school.
For such a reserved man, Ryan could be surprisingly open with pen in hand; even on something as simple as an alumni questionnaire, he might hold forth on the state of the world and himself. "Our tendency to become a spectator nation has probably benefited me — nevertheless I deplore it," he wrote on one such form back in November 1956. "I look forward to a world at peace and to my children's lives. I am grateful for my work, my wife, and my children (not in that order)." He always aspired to be an actor, rather than a spectator, in both the political and the theatrical sense, though he would never see a world at peace, and ordering his personal and professional priorities would become more of a challenge than he could have imagined then. The characters Ryan created are all that remain of him now, men for whom shadow is a state of mind.
*Someone had miscounted, because _The Outfit_ was only his seventy-first theatrical feature.
Acknowledgments
I came to Ryan's work early for someone of my generation. As a freshman at Loyola Academy, a Jesuit school in Wilmette, Illinois, I began my four-year religion curriculum with a class that included a showing and discussion of the 1962 movie _Billy Budd_. My teacher stressed the Christlike sacrifice of young Billy, but this lesson was subverted by the fact that the most charismatic character in the movie was the evil master-of-arms, Mr. Claggart.
The actor, my teacher explained, was Robert Ryan, class of 1927; when I filed down the hallway that housed the school's class photos (passing Bill Murray, class of 1968) and found the portrait of seventeen-year-old Robert Ryan, I could barely reconcile it with the lined, hardened face I had seen on-screen. To begin, then, I thank Rev. James Arimond at Loyola for providing copies of Ryan's earliest published writings, in the school's newspaper and literary magazine, and other background material on his education.
His generosity has been matched by many others who supplied documents, articles, and photographs that inform this account: Dr. James Astman, headmaster of the Oakwood School; Deanna Chew of the La Jolla Playhouse; Sarah Hartwell of Dartmouth College Library; Eddie Muller and Alan K. Rode of the Film Noir Foundation; Vickie Ryan, the actor's daughter-in-law (who provided historical material on the Cheyneys and Cadwaladers); Cheryl S. Spiese and Jean L. Green of the Max Reinhardt Archive at SUNY–Binghamton; Tina Louise Happ of Pritzker Military Library in Chicago; and Vivian Teng of Cinema/Chicago. Above all, I want to thank Peter Jarlett, who provided photographs, audiotaped interviews, and transcriptions amassed by his late brother, Franklin Jarlett, author of _Robert Ryan: A Biography and Critical Filmography_ (1990). That book offers a more complete guide to Ry an's work in various media than I have, and I recommend it to those seeking more information.
I am much indebted to the fine research facilities where I conducted my work. In particular I want to thank Martin Gostanian at the Paley Center for Media in Los Angeles, Mark Quigley of the UCLA Film and Television Archive, Julie Graham and Amy Wong of UCLA Performing Arts Special Collections, Maxine Ducey and Mary Huelsbeck of the Wisconsin Center for Film and Theater Research, and Jenny Romero of the Margaret Herrick Library, Fairbanks Center for Motion Picture Study, Beverly Hills. I remember fondly my time researching this project with the Herrick Library's friendly and enthusiastic staff.
This book originated in an October 2009 story in the _Chicago Reader_ , "The Actor's Letter," and many _Reader_ colleagues contributed their expertise to that project or to this volume: Andrea Bauer, Michael Miner, Jonathan Rosenbaum, Mara Shalhoup, Alison True, Albert Williams, and Kiki Yablon. Dave Kehr, whose early film writing in the _Reader_ I much admired, urged me to write this book after reading "The Actor's Letter" and so must be credited with setting it in motion.
My thanks to Chris Linster of Quartet Digital Printing and to my friends and colleagues who read and offered their input on my book proposal and early drafts: Margaret Buchen, Lisa Dombrowski, Joseph C. Heinen, Jonathan Joe, Michael Phillips, Alan Rode, Martin Rubin, Parker Smathers at Wesleyan University Press, and above all my agent, Peter Riva of International Transactions.
I am blessed with family members who aided me in this project. My sister Ruth Ann Jones, a reference librarian, directed me toward countless revealing materials; my sister Julia Macintosh in the UK tracked down reviews of Ryan's 1967 residency at the Nottingham Playhouse; my sister-in-law, Kelsey Beson, translated a key Ryan interview from a French magazine; and my uncle Tom Jones in Louisville nabbed a story from the local _Courier-Journal_. My wife, Margaret, has been a constant source of love and encouragement as this project took over my life (and hers).
Many people were generous with their time in sharing their personal reminiscences of the Ryans and their children: Joe Don Baker, Phil Bauman, Harry Belafonte, Jeff Bridges, Arvin Brown, Rhonda Fleming, Evans Frankenheimer, Charles Haas, Andy Harmon, Toya Harrison, Seymour Hersh, Marsha Hunt, Stacy Keach, Ramona Lampell, Tina Louise, Mike Metzger, James Naughton, Rev. Lothar Nurnberger, Nehemiah Persoff, Priscilla Ulene, and Jacqueline White.
I'm especially grateful to Lisa Ryan, Cheyney Ryan, and Walker (formerly Timothy) Ryan for their candor and patience as I reconstructed their parents' story. Throughout the process they impressed me as people who shared their parents' desire for privacy but also felt that this story should be told.
_Appendix_
Robert Ryan Performances
_part one_ Stage Chronology
TOO MANY HUSBANDS (1940) Belasco Theater, Los Angeles, January 15, 1940. Director: Max Reinhardt. Producer: Lloyd D. Mitchell. Playwright: Somerset Maugham. Music: Bronislau (aka Bronislaw) Kaper. Cast: Robert Ryan, Maris Wrixon, Bruce Bennett, Arno Arno, Ernö Verebes, Adele Neff, Helene Hill, Martin Wessner, Millard Vincent, Ralph Freud, Ann Lee.
THE TIME OF YOUR LIFE (1941) Millpond Playhouse, Roslyn, Long Island, June 30 to July 12, 1941. Director: David Lowe. Producer: David Lowe. Playwright: William Saroyan. Cast: Robert Ryan (Joe), Jane Jeffreys, Harald Dyrenforth, James Murray, Cameron Mitchell, Jane Morrisey, Joel Steele, Neville Draper. _Note_ : Staged during the Ryans' summer repertory season at Millpond Playhouse; among the other productions that season were _The Barker_ (June 9, 1941), starring Robert Ryan, Jessica Cheyney, Edward Thompson, and Kenneth Forbes; _Petticoat Fever_ (June 16, 1941), starring Ryan and Cheyney; and _Angel Child_ (date unknown), starring Ryan, Jane Jeffreys, and Cameron Mitchell.
A KISS FOR CINDERELLA (1941) Maplewood Theatre, Maplewood, New Jersey, September 18 to 23, 1941. Playwright: J. M. Barrie. Producer: Cheryl Crawford. Cast: Luise Rainer, Robert Ryan (Policeman/Prince). _Note_ : Ryan and Rainer originated their roles at the Cape Playhouse in Dennis, Massachusetts, before the production moved to the Maplewood Theatre.
CLASH BY NIGHT (1941) Belasco Theatre, New York, December 27, 1941, to February 7, 1942. Director: Lee Strasberg. Producer: Billy Rose. Playwright: Clifford Odets. Cast: Seth Arnold, Tallulah Bankhead, Ralph Chambers, Lee J. Cobb, Stephan Eugene Cole, Harold Grau, John F. Hamilton, Katherine Locke, William Nunn, Robert Ryan (Joe W. Doyle), Joseph Schildkraut, Joseph Shattuck, Art Smith.
PETTICOAT FEVER (1949) La Jolla Playhouse, La Jolla, California, August 30 to September 4, 1949. Director: James Neilson. Playwright: Mark Reed. Cast: Robert Ryan, Ruth Warrick, Dorothy McGuire, Dan Tobin, Clifford Brooke, Chris-Pin Martin.
BORN YESTERDAY (1950) La Jolla Playhouse, La Jolla, California, July 4 to 9, 1950. Director: James Neilson. Playwright: Garson Kanin. Cast: Marie McDonald, Robert Ryan (Harry Brock), Tom Powers, Whit Bissell, Johnny Call, Paul Maxey, Louise Lorimer.
CORIOLANUS (1954) Phoenix Theatre, New York, New York, opened January 19, 1954. Director: John Houseman. Producers: T. Edward Hambleton, Norris Houghton. Playwright: William Shakespeare. Music: Alex North. Cast: Alan Napier, Robert Ryan (Caius Martius Coriolanus), Lou Polan, Joseph McCaulay, George Fells, Joseph Holland, John Randolph, Will Geer, Mildred Natwick, Lori March, Paula Laurence, John Emery, Jamie Smith, Gene Saks, Jack Klugman, Jerry Stiller, Terry Nardin.
TIGER AT THE GATES (1957) Ivar Theater, Los Angeles, January 29 to February 10, 1957. Director: Harold J. Kennedy. Producer: Harold J. Kennedy. Playwright: Jean Giraudoux. Cast: Robert Ryan, John Ireland, Marilyn Erskine, Ray Danton, Mary Astor, Marianne Stewart, Howard Wendell, Peg La Centra, Milton Parsons, Joel Ashley, Jon Poole, Gene Mekler.
MURDER IN THE CATHEDRAL (1960) Schoenberg Hall, University of California at Los Angeles, January 17 to January 31, 1960. Director: John Houseman. Playwright: T. S. Eliot. Cast: Robert Ryan (Thomas Beckett), John Hoyt, Alan Napier, Pippa Scott, Theodore Marcuse, Ruth Story.
ANTONY AND CLEOPATRA (1960) American Shakespeare Festival Theatre, Stratford, Connecticut, opened July 31, 1960. Director: Jack Landau. Cast: Robert Ryan (Antony), Katharine Hepburn, Douglas Watson, John Harkins, Donald Davis, Patrick Hines, Earle Hyman, Rae Allen, Anne Fielding, John Ragin, Morris Carnovsky, Will Geer, John Myhers, Stephen Strimpell, Clifton James, Claude Woolman, Sada Thompson, Richard Waring, Ted van Griethuysen, Clayton Corzatte.
MR. PRESIDENT (1962) St. James Theatre, New York, New York, October 20, 1962, to June 8, 1963. Director: Joshua Logan. Producer: Leland Hayward. Book: Howard Lindsay, Russel Crouse. Music and lyrics: Irving Berlin. Cast: Nanette Fabray, Robert Ryan (President Stephen Decatur Henderson), David Brooks, Wisa D'Orso, Charlotte Fairchild, Anita Gillette, Stanley Grover, Jack Haskell, John Cecil Holm, Jerry Strickler, Jack Washburn, John Aman.
OTHELLO (1967) Nottingham Playhouse, Nottingham, England, opened September 20, 1967. Director: Noel Willman. Cast: Robert Ryan (Othello), Christopher Hancock, Derek Woodward, John Neville, Terence Knapp, David Neal, Alan Dossor, Ronald Magill, Laurence Harrington, James O'Brien, Ann Bell, Ursula Smith, Christine Welch.
LONG DAY'S JOURNEY INTO NIGHT (1967) Nottingham Playhouse, Nottingham, England, opened September 27, 1967. Director: Michael Rudman. Cast: Robert Ryan (James Tyrone Sr.), Gillian Martell, Anthony Langdon, Alfred Bell, Ursula Smith.
OUR TOWN (1968) Mineola Theater, Mineola, New York, opened September 24, 1968. Director: Edward Hastings. Producers: Martha Scott, Alfred de Liagre Jr., Clifford Stevens. Cast: Henry Fonda, John Beal, Jo Van Fleet, John McGiver, Estelle Parsons, Robert Ryan (Mr. Webb), Katharine Winn.
THE FRONT PAGE (1968) Mineola Theater, Mineola, New York, October 5 to 20, 1968. Director: Leo Brady. Producers: Martha Scott, Alfred de Liagre Jr., Clifford Stevens. Cast: Robert Ryan, John Beal, Mark Bramhall, Henry Fonda, Anthony George, Anne Jackson, John McGiver, Estelle Parsons.
THE FRONT PAGE (1969) Ethel Barrymore Theatre, May 10 to July 5, 1969. Director: Harold J. Kennedy. Producers: Theater 1969 (Edward Albee, Richard Barr, Charles Woodward). Playwrights: Ben Hecht, Charles MacArthur. Cast: Robert Ryan (Walter Burns), Val Avery, Peggy Cass, Bert Convy, James Flavin, Conrad Janis, Harold J. Kennedy, John McGiver, Julia Meade, Doro Merande, Charles White, Tom Atkins, Bruce Blaine, Patrick Desmond, Walter Flanagan, Morison Gampel, Geoff Garland, Will Gregory, Rick Hagan, Scott Hagan, Katharine Houghton, Robert Milli, Don Porter, Ed Riley, Arnold Stang.
THE FRONT PAGE (1969–'70) Ethel Barrymore Theatre, October 18, 1969, to February 28, 1970. Directed by Harold J. Kennedy. Producers: Jay H. Fuchs, Jerry Schlossberg, Albert Zuckerman. Playwrights: Ben Hecht, Charles MacArthur. Cast: Val Avery, Peggy Cass, Bert Convy, Dody Goodman, Helen Hayes, Conrad Janis, John McGiver, Robert Ryan (Walter Burns), James Flavin, Harold J. Kennedy, Charles White, Bruce Blaine, Jack Collard, Patrick Desmond, Walter Flanagan, Joseph George, Will Gregory, Bob Larkin, Kendall March, Robert Milli, Robert Riesel, Ed Riley, Bernie West.
LONG DAY'S JOURNEY INTO NIGHT (1971) Promenade Theatre, New York, New York, April 21 to August 22, 1971. Director: Arvin Brown. Producers: Edgar Lansbury, Jay H. Fuchs, Stuart Duncan, Joseph Beruh. Cast: Robert Ryan (James Tyrone Sr.), Stacy Keach, Geraldine Fitzgerald, James Naughton, Paddy Croft.
_part two_ Film Chronology
THE GHOST BREAKERS (1940) Director: George Marshall. Producer: Arthur Hornblow Jr. Screenwriter: Walter DeLeon, from a play by Paul Dickey and Charles W. Goddard. Photography: Charles Lang. Editor: Ellsworth Hoagland. Music: Ernst Toch. Production and distribution: Paramount Pictures. Release date: June 7, 1940. Running time: 85 minutes. Cast: Bob Hope, Paulette Goddard, Richard Carlson, Paul Lukas, Willie Best, Robert Ryan (Intern, uncredited). Black and white.
QUEEN OF THE MOB (1940) Director: James Hogan. Screenwriters: J. Edgar Hoover, Walter R. Lipman, Horace McCoy, from Hoover's book _Persons in Hiding_. Photography: Theodor Sparkuhl. Editor: Arthur Schmidt. Production and distribution: Paramount Pictures. Release date: June 28, 1940. Running time: 61 minutes. Cast: Ralph Bellamy, J. Carrol Naish, Jeanne Cagney, Richard Denning, Hedda Hopper, Jack Carson, Billy Gilbert, Robert Ryan (Jim, uncredited). Black and white.
GOLDEN GLOVES (1940) Directors: Edward Dmytryk, Felix E. Feist. Producer: Carl Krueger. Screenwriters: Joe Ansen, Lewis R. Foster, Maxwell Shane. Photography: John L. Russell, Henry Sharp. Editors: William F. Claxton, Doane Harrison. Production and distribution: Paramount Pictures. Release date: August 2, 1940. Running time: 66 minutes. Cast: Richard Denning, Jeanne Cagney, J. Carrol Naish, Robert Paige, William Farley, Edward Brophy, Robert Ryan (Pete Wells). Black and white.
NORTH WEST MOUNTED POLICE (1940) Director: Cecil B. DeMille. Producer: Cecil B. DeMille. Screenwriters: Alan Le May, Jesse Laske Jr., C. Gardner Sullivan, from R. C. Fetherstonhaugh's novel _The Royal Canadian Mounted Police_. Photography: W. Howard Green, Victor Milner. Editor: Anne Bauchens. Music: Victor Young. Production and distribution: Paramount Pictures. Release date: October 21, 1940 (Canada); October 22, 1940 (US). Running time: 126 minutes. Cast: Gary Cooper, Madeleine Carroll, Paulette Goddard, Preston Foster, Robert Preston, George Bancroft, Akim Tamiroff, Lon Chaney Jr., George E. Stone, Regis Toomey, Robert Ryan (Constable Dumont). Black and white.
THE TEXAS RANGERS RIDE AGAIN (1940) Director: James Hogan. Screenwriters: William R. Lippman, Horace McCoy. Photography: Archie Stout. Editor: Arthur Schmidt. Production and distribution: Paramount Pictures. Release date: December 13, 1940. Running time: 68 minutes. Cast: Ellen Drew, John Howard, Akim Tamiroff, May Robson, Broderick Crawford, Charley Grapewin, Anthony Quinn, Robert Ryan (Eddie, uncredited). Black and white.
BOMBARDIER (1943) Director: Richard Wallace, Lambert Hillyer (uncredited). Producer: Robert Fellows. Screenwriter: John Twist. Story: John Twist, Martin Rackin. Photography: Nicholas Musuraca, Joseph F. Biroc (uncredited). Editor: Robert Wise. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: May 14, 1943. Running time: 99 minutes. Cast: Pat O'Brien, Randolph Scott, Anne Shirley, Eddie Albert, Walter Reed, Robert Ryan (Joe Connors), Barton MacLane, Leonard Strong, Richard Martin, Russell Wade, James Newill, John Miljan, Charles Russell. Black and white.
THE SKY'S THE LIMIT (1943) Director: Edward H. Griffith. Producer: David Hempstead. Screenwriters: Frank Fenton, Lynn Root, S. K. Lauren (uncredited), William T. Ryder (story, uncredited). Photography: Russell Metty. Editor: Roland Gross. Music: Harold Arlen. Production and distribution: RKO Radio Pictures. Release date: July 13, 1943. Running time: 89 minutes. Cast: Fred Astaire, Joan Leslie, Robert Benchley, Robert Ryan (Reginald Fenton), Elizabeth Patterson, Marjorie Gateson, Freddie Slack and His Orchestra. Black and white.
BEHIND THE RISING SUN (1943) Director: Edward Dmytryk. Producers: Edward Dmytryk (uncredited), Howard Hughes (uncredited). Screenwriter: Emmet Lavery, from the book by James R. Young. Photography: Russell Metty. Editor: Joseph Noriega. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: August 1, 1943. Running time: 88 minutes. Cast: Margo, Tom Neal, J. Carrol Naish, Robert Ryan (Lefty O'Doyle), Gloria Holden, Donald Douglas, George Givot. Black and white.
THE IRON MAJOR (1943) Director: Ray Enright. Producer: Robert Fellows. Screenwriters: Aben Kandel, Warren Duff. Story: Florence E. Cavanaugh. Photography: Robert De Grasse. Editors: Philip Martin Jr., Robert Wise. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: October 25, 1943. Running time: 85 minutes. Cast: Pat O'Brien, Ruth Warrick, Robert Ryan (Father Timothy "Tim" Donovan), Leon Ames, Russell Wade, Bruce Edwards, Richard Martin.
GANGWAY FOR TOMORROW (1943) Director: John H. Auer. Producer: John H. Auer. Screenwriter: Arch Oboler. Story: Aladar Laszlo. Photography: Nicholas Musuraca. Editor: George Crone. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: November 3, 1943. Running time: 69 minutes. Cast: Margo, John Carradine, Robert Ryan (Joe Dunham), Amelita Ward, William Terry, Harry Davenport, James Bell. Black and white.
TENDER COMRADE (1943) Director: Edward Dmytryk. Producer: David Hempstead. Screenwriter: Dalton Trumbo. Photography: Russell Metty. Editor: Roland Gross. Music: Leigh Harline. Production and distribution: RKO Radio Pictures. Release date: December 29, 1943. Running time: 102 minutes. Cast: Ginger Rogers, Robert Ryan (Chris Jones), Ruth Hussey, Patricia Collinge, Mady Christians, Kim Hunter, Jane Darwell, Richard Martin. Black and white.
MARINE RAIDERS (1944) Director: Harold Schuster. Producer: Robert Fellows. Screenwriters: Warren Duff, Jerome Odlum (uncredited). Story: Martin Rackin, Warren Duff. Photography: Nicholas Musuraca. Editor: Philip Martin Jr. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: June 30, 1944. Running time: 90 minutes. Cast: Pat O'Brien, Robert Ryan (Captain Dan Craig), Ruth Hussey, Frank McHugh, Barton MacLane, Richard Martin, Edmund Glover, Russell Wade, Robert Andersen, Michael St. Angel, Martha MacVicar (aka Martha Vickers), Harry Brown. Black and white.
TRAIL STREET (1947) Director: Ray Enright. Producer: Nat Holt. Screenwriters: Norman Houston, Gene Lewis, from the novel by William Corcoran. Photography: J. Roy Hunt. Editor: Lyle Boyer. Music: C. Bakaleinikoff. Production and distribution: RKO Radio Pictures. Release date: February 19, 1947. Running time: 84 minutes. Cast: Randolph Scott, Robert Ryan (Allen), Anne Jeffreys, George "Gabby" Hayes, Madge Meredith, Steve Brodie, Billy House, Virginia Sale, Harry Woods, Phil Warren, Harry Harvey, Jason Robards (Sr.). Black and white.
THE WOMAN ON THE BEACH (1947) Director: Jean Renoir. Producer: Jack J. Gross. Screenwriters: Frank Davis, Jean Renoir. Adaptation: Michael Hogan, from Mitchell Wilson's novel _None So Blind_. Photography: Leo Tover, Harry Wild. Ed itors: Lyle Boyer, Roland Gross. Music: Hanns Eisler. Production and distribution: RKO Radio Pictures. Release date: June 2, 1947. Running time: 71 minutes. Cast: Joan Bennett, Robert Ryan (Scott), Charles Bickford, Nan Leslie, Walter Sande, Irene Ryan, Glen Vernon, Frank Darien, Jay Norris. Black and white.
CROSSFIRE (1947) Director: Edward Dmytryk. Producer: Adrian Scott. Screenwriter: John Paxton, from Richard Brooks's novel _The Brick Foxhole_. Photography: J. Roy Hunt. Editor: Harry Gerstad. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: July 22, 1947. Running time: 86 minutes. Cast: Robert Young, Robert Mitchum, Robert Ryan (Montgomery), Gloria Grahame, Paul Kelly, Sam Levene, Jacqueline White, Steve Brodie, George Cooper, Richard Benedict, Richard Powers, William Phipps, Lex Barker, Marlo Dwyer.
BERLIN EXPRESS (1948) Director: Jacques Tourneur. Producer: Bert Granet. Screenwriter: Harold Medford. Story: Curt Siodmak. Photography: Lucien Ballard. Editor: Sherman Todd. Music: Frederick Hollander. Production and distribution: RKO Radio Pictures. Release date: May 1, 1948. Running time: 87 minutes. Cast: Merle Oberon, Robert Ryan (Robert Lindley), Charles Korvin, Paul Lukas, Robert Coote, Reinhold Schunzel, Roman Toporow, Peter Von Zerneck, Otto Waldis, Fritz Kortner, Michael Harvey, Richard Powers (aka Tom Keene). Black and white.
RETURN OF THE BAD MEN (1948) Director: Ray Enright. Producer: Nat Holt. Screenwriters: Charles O'Neal, Jack Natteford, Luci Ward. Story: Jack Natteford, Luci Ward. Photography: J. Roy Hunt. Editor: Samuel E. Beetley. Music: Paul Sawtell. Production and distribution: RKO Radio Pictures. Release date: July 17, 1948. Running time: 90 minutes. Cast: Randolph Scott, Robert Ryan (Sundance Kid), Anne Jeffreys, George "Gabby" Hayes, Jacqueline White, Steve Brodie, Richard Powers (aka Tom Keene), Robert Bray, Lex Barker, Walter Reed, Michael Harvey, Dean White, Robert Armstrong, Tom Tyler, Lew Harvey, Gary Gray, Walter Baldwin, Minna Gombell, Warren Jackson, Robert Clarke, Jason Robards (Sr.). Black and white.
THE BOY WITH GREEN HAIR (1948) Director: Joseph Losey. Producer: Stephen Ames, Adrian Scott (uncredited). Screenwriters: Ben Barzman, Alfred Lewis Levitt, from the story by Betsy Beaton. Photography: George Barnes. Editor: Frank Doyle. Music: Leigh Harline. Production and distribution: RKO Radio Pictures. Release date: November 16, 1948. Running time: 82 minutes. Cast: Pat O'Brien, Robert Ryan (Dr. Evans), Barbara Hale, Dean Stockwell, Richard Lyon, Walter Catlett, Samuel S. Hinds, Regis Toomey, Charles Meredith, David Clarke, Billy Sheffield, John Calkins, Teddy Infuhr, Dwayne Hickman, Eilene Janssen. Color.
ACT OF VIOLENCE (1948) Director: Fred Zinnemann. Producer: William H. Wright. Screenwriter: Robert L. Richards. Story: Collier Young. Photography: Robert Surtees. Editor: Conrad A. Nervig. Music: Bronislau (aka Bronislaw) Kaper. Production and distribution: Metro-Goldwyn-Mayer. Release date: December 21, 1948. Running time: 82 minutes. Cast: Van Heflin, Robert Ryan (Joe Parkson), Janet Leigh, Mary Astor, Phyllis Thaxter, Berry Kroeger, Taylor Holmes, Harry Antrim, Connie Gilchrist, Will Wright. Black and white.
CAUGHT (1949) Director: Max Ophuls (aka Max Ophüls). Producer: Wolfgang Reinhardt. Screenwriter: Arthur Laurents, from Libbie Block's novel _Wild Calendar_. Photography: Lee Garmes. Editor: Robert Parrish. Music: Frederick Hollander. Production: Enterprise Productions. Distribution: Metro-Goldwyn-Mayer. Release date: February 17, 1949. Running time: 88 minutes. Cast: James Mason, Barbara Bel Geddes, Robert Ryan (Smith Ohlrig), Frank Ferguson, Curt Bois, Ruth Brady, Natalie Schaefer, Art Smith. Black and white.
THE SET-UP (1949) Director: Robert Wise. Producer: Richard Goldstone. Screenwriter: Art Cohn, from the poem by Joseph Moncure March. Photography: Milton Krasner. Editor: Roland Gross. Music: C. Bakaleinikoff. Production and distribution: RKO Radio Pictures. Release date: March 29, 1949. Running time: 73 minutes. Cast: Robert Ryan (Stoker Thompson), Audrey Totter, George Tobias, Alan Baxter, Wallace Ford, Percy Helton, Hal Fieberling (aka Hal Baylor), Darryl Hickman, Kenny O'Morrison, James Edwards, David Clarke, Phillip Pine, Edwin Max. Black and white.
THE WOMAN ON PIER 13 (1949) Director: Robert Stevenson. Producer: Jack J. Gross. Screenwriters: Charles Grayson, Robert Hardy Andrews. Story: George W. George, George F. Slavin. Photography: Nicholas Musuraca. Editor: Roland Gross. Music: Leigh Harline. Production and distribution: RKO Radio Pictures. Release date: October 8, 1949. Running time: 73 minutes. Cast: Laraine Day, Robert Ryan (Brad Collins aka Frank Johnson), John Agar, Thomas Gomez, Janis Carter, Richard Rober, William Talman, Paul E. Burns, Paul Guilfoyle, G. Pat Collins, Fred Graham, Harry Cheshire, Jack Stoney. Black and white.
THE SECRET FURY (1950) Director: Mel Ferrer. Producer: Jack H. Skirball. Screenwriter: Lionel Houser. Story: Jack R. Leonard, James O'Hanlon. Photography: Leo Tover. Editor: Harry Marker. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: February 21, 1950. Running time: 85 minutes. Cast: Claudette Colbert, Robert Ryan (David McLean), Jane Cowl, Paul Kelly, Philip Ober, Elisabeth Risdon, Doris Dudley, Dave Barbour, Vivian Vance. Black and white.
BORN TO BE BAD (1950) Director: Nicholas Ray. Producer: Robert Sparks. Screenwriter: Edith Sommer, additional dialogue by Robert Soderberg, George Oppenheimer. Adaptation: Charles Schnee, from Anne Parish's novel _All Kneeling_. Photography: Nicholas Musuraca. Editor: Frederic Knudtson. Music: Frederick Hollander. Production and distribution: RKO Radio Pictures. Release date: August 27, 1950. Running time: 94 minutes. Cast: Joan Fontaine, Robert Ryan (Nick Bradley), Zachary Scott, Joan Leslie, Mel Ferrer, Harold Vermilyea, Virginia Farmer, Kathleen Howard, Dick Ryan, Bess Flowers, Joy Hallward, Hazel Boyne, Irvin Bacon, Gordon Oliver. Black and white.
HARD, FAST AND BEAUTIFUL (1951) Director: Ida Lupino. Producer: Collier Young. Screenwriter: Martha Wilkerson, from John R. Tunis's novel _American Girl_. Photography: Archie Stout. Editors: George C. Shrader, William Ziegler. Music: Roy Webb. Production: The Filmmakers. Distribution: RKO Radio Pictures. Release date: May 23, 1951. Running time: 78 minutes. Cast: Claire Trevor, Sally Forrest, Carleton G. Young, Robert Clarke, Kenneth Patterson, Marcella Cisney, Joseph Kearns, William Hudson, George Fisher, Ida Lupino (uncredited), Robert Ryan (uncredited). Black and white. _Note_ : Ryan and Lupino appear as extras in the Seabright tennis match sequence.
BEST OF THE BADMEN (1951) Director: William D. Russell. Producer: Herman Schlom. Screenwriters: Robert Hardy Andrews, John Twist. Story: Robert Hardy Andrews. Photography: Edward Cronjager. Editor: Desmond Marquette. Music: Paul Sawtell. Production and distribution: RKO Radio Pictures. Release date: August 9, 1951. Running time: 84 minutes. Cast: Robert Ryan (Jeff Clanton), Claire Trevor, Robert Buetel, Robert Preston, Walter Brennan, Bruce Cabot, John Archer, Lawrence Tierney, Barton MacLane, Tom Tyler, Robert J. Wilke, John Cliff, Lee MacGregor, Emmett Lynn, Carleton Young. Color.
FLYING LEATHERNECKS (1951) Director: Nicholas Ray. Producer: Edmund Grainger. Screenwriters: James Edward Grant, Beirne Lay Jr. (uncredited). Story: Kenneth Gamet. Photography: William E. Snyder. Editor: Sherman Todd. Music: Roy Webb. Production and distribution: RKO Radio Pictures. Release date: August 28, 1951. Running time: 102 minutes. Cast: John Wayne, Robert Ryan (Captain Carl "Griff" Griffin), Don Taylor, Janis Carter, Jay C. Flippen, William Harrigan, James Bell, Barry Kelley, Maurice Jara, Adam Williams, James Dobson, Carleton Young, Michael St. Angel, Brett King, Gordon Gebert. Color.
THE RACKET (1951) Directors: John Cromwell, Mel Ferrer (uncredited), Tay Garnett (uncredited), Nicholas Ray (uncredited), Sherman Todd (uncredited). Producer: Edmund Grainger. Screenwriters: William Wister Haynes, W. R. Burnett, from the play by Bartlett Cormack. Photography: George E. Diskant. Editor: Sherman Todd. Production and distribution: RKO Radio Pictures. Release date: December 12, 1951. Running time: 89 minutes. Cast: Robert Mitchum, Lizabeth Scott, Robert Ryan (Nick Scanlon), William Talman, Ray Collins, Joyce MacKenzie, Robert Hutton, Virginia Huston, William Conrad, Walter Sande, Les Tremayne, Don Porter, Walter Baldwin, Brett King, Richard Karlan, Tito Vuolo. Black and white.
ON DANGEROUS GROUND (1951) Director: Nicholas Ray. Producer: John Houseman. Screenwriter: A. I. Bezzerides. Adaptation: A. I. Bezzerides, Nicholas Ray, from Gerald Butler's novel _Mad with Much Heart_. Photography: George E. Diskant. Editor: Roland Gross. Music: Bernard Herrmann. Production and distribution: RKO Radio Pictures. Release date: December 17, 1951. Running time: 82 minutes. Cast: Ida Lupino, Robert Ryan (Jim Wilson), Ward Bond, Charles Kemper, Anthony Ross, Ed Begley, Ian Wolfe, Sumner Williams, Gus Schilling, Frank Ferguson, Cleo Moore, Olive Carey, Richard Irving, Pat Prest, A. I. Bezzerides (uncredited). Black and white.
CLASH BY NIGHT (1952) Director: Fritz Lang. Producer: Harriet Parsons. Screenwriter: Alfred Hayes, from the play by Clifford Odets. Photography: Nicholas Musuraca. Editor: George Amy. Music: Roy Webb. Production: Wald-Krasna Productions. Distribution: RKO Radio Pictures. Release date: June 16, 1952. Running time: 105 minutes. Cast: Barbara Stanwyck, Paul Douglas, Robert Ryan (Earl Pfeiffer), Marilyn Monroe, J. Carrol Naish, Silvio Minciotti, Keith Andes. Black and white.
BEWARE, MY LOVELY (1952) Director: Harry Horner. Producer: Collier Young. Screenwriter: Mel Dinelli, from his play _The Man_. Photography: George E. Diskant. Editor: Paul Weatherwax. Music: Leith Stevens. Production: The Filmmakers. Distribution: RKO Radio Pictures. Release date: August 7, 1952. Running time: 77 minutes. Cast: Ida Lupino, Robert Ryan (Howard Wilton), Taylor Holmes, Barbara Whiting, James Willmas, O. Z. Whitehead, Dee Pollock, Brad Morrow, Jimmy Mobley, Shelly Lynn Anderson, Ronnie Patterson, Jeanne Eggenweiler. Black and white.
HORIZONS WEST (1952) Director: Budd Boetticher. Producer: Albert J. Cohen. Screenwriter: Louis Stevens. Photography: Charles P. Boyle. Editor: Ted J. Kent. Music: Joseph Gershenson. Production: Universal-International Pictures. Distribution: Universal Pictures. Release date: October 11, 1952. Running time: 81 minutes. Cast: Robert Ryan (Dan Hammond), Julie Adams, Rock Hudson, Judith Braun, John McIntire, Raymond Burr, James Arness, Dennis Weaver, Frances Bavier, Tom Powers, John Hubbard, Rodolfo Acosta, Douglas Fowley, Walter Reed, Raymond Greenleaf, Dan Poore, Frank Chase, Mae Clarke. Color.
THE NAKED SPUR (1953) Director: Anthony Mann. Producer: William H. Wright. Screenwriter: Sam Rolfe, Harold Jack Bloom. Photography: William Mellor. Editor: George White. Music: Bronislau (aka Bronislaw) Kaper. Production and distribution: Metro-Goldwyn-Mayer. Release date: February 6, 1963. Running time: 91 minutes. Cast: James Stewart, Janet Leigh, Robert Ryan (Ben Vandergroat), Ralph Meeker, Millard Mitchell, Denver Pyle. Color.
CITY BENEATH THE SEA (1953) Director: Budd Boetticher. Producer: Albert J. Cohen. Screenwriters: Jack Harvey, Ramon Romero, from Harry E. Rieseberg's book _Port Royal: The Ghost City beneath the Sea_. Photography: Charles P. Boyle. Editor: Edward Curtiss. Music: Joseph Gershenson. Production: Universal-International Pictures. Distribution: Universal Pictures. Release date: March 11, 1953. Running time: 87 minutes. Cast: Robert Ryan (Brad Carlton), Mala Powers, Anthony Quinn, Suzan Ball, George Mathews, Karel Stepanek, Hilo Hattie, Lalo Rios, Woody Strode, John Warburton, Peter Mamakos, Barbara Morrison, LeRoi Antoine, Leon Lontoc, Marya Marco. Color.
INFERNO (1953) Director: Roy Ward Baker. Producer: William Bloom. Screenwriter: Francis M. Cockrell, from his story "The Waterhole." Photography: Lucien Ballard. Editor: Robert L. Simpson. Music: Paul Sawtell. Production and distribution: 20th Century-Fox. Release date: August 12, 1953. Running time: 83 minutes. Cast: Robert Ryan (Donald Whitley Carson III), Rhonda Fleming, William Lundigan, Larry Keating, Henry Hull, Carl Betz, Robert Burton. Color, 3-D.
ALASKA SEAS (1954) Director: Jerry Hopper. Producer: Mel Epstein. Screenwriters: Walter Doniger, Geoffrey Homes (aka Daniel Mainwaring). Story: Barrett Willoughby. Photography: William C. Mellor. Editor: Archie Marshek. Production and distribution: Paramount Pictures. Release date: January 27, 1954. Running time: 78 minutes. Cast: Robert Ryan (Matt Kelly), Jan Sterling, Brian Keith, Gene Barry, Richard Shannon, Ralph Dumke, Ross Bagdasarian, Fay Roope, Timothy Carey, Peter Coe, Jim Hayward, Aaron Spelling. Black and white.
ABOUT MRS. LESLIE (1954) Director: Daniel Mann. Producer: Hal B. Wallis. Screenwriters: Ketti Frings, Hal Kanter, from the novel by Viña Delmar. Photography: Ernest Laszlo. Editor: Warren Low. Music: Victor Young. Production and distribution: Paramount Pictures. Release date: June 27, 1954. Running time: 104 minutes. Cast: Shirley Booth, Robert Ryan (George Leslie), Marjie Millar, Alex Nicol, Sammy White, James Bell, Eilene Janssen, Philip Ober, Harry Morgan, Gale Page, Virginia Brissac, Ian Wolfe, Ellen Corby. Black and white.
HER TWELVE MEN (1954) Director: Robert Z. Leonard. Producer: John Houseman. Screenwriters: William Roberts, Laura Z. Hobson. Story: Louise Baker. Photography: Joseph Ruttenberg. Editor: George Boemler. Music: Bronislau (aka Bronislaw) Kaper. Production and distribution: Metro-Goldwyn-Mayer. Release date: August 11, 1954. Running time: 91 minutes. Cast: Greer Garson, Robert Ryan (Joe Hargrave), Barry Sullivan, Richard Haydn, Barbara Lawrence, James Arness, Rex Thompson, Tim Considine, David Stollery, Frances Bergen, Ian Wolfe. Color.
BAD DAY AT BLACK ROCK (1955) Director: John Sturgess. Producer: Dore Schary. Screenwriter: Millard Kaufman. Adaptation: Don McGuire, from Howard Breslin's story "Bad Time at Honda." Photography: William C. Mellor. Editor: Newell P. Kimlin. Music: André Previn. Production and distribution: Metro-Goldwyn-Mayer. Release date: January 7, 1955. Running time: 81 minutes. Cast: Spencer Tracy, Robert Ryan (Reno Smith), Anne Francis, Dean Jagger, Walter Brennan, John Ericson, Ernest Borgnine, Lee Marvin, Russell Collins, Walter Sande. Color, CinemaScope.
ESCAPE TO BURMA (1955) Director: Alan Dwan. Producer: Benedict Bogeaus. Screenwriters: Talbot Jennings, Hobart Donavan. Story: Kenneth Perkins. Photography: John Alton. Editors: James Leicester, Carlo Lodato. Music: Louis Forbes. Production: Benedict Bogeaus Production. Distribution: RKO Radio Pictures. Release date: April 9, 1955. Running time: 83 minutes. Cast: Barbara Stanwyck, Robert Ryan (Jim Brecan), David Farrar, Murvyn Vye, Lisa Montell, Robert Warwick, Reginald Denny, Robert Cabal, Peter Coe, Alex Montoya, Anthony Numkena. Color, Superscope.
HOUSE OF BAMBOO (1955) Director: Samuel Fuller. Producer: Buddy Adler. Screenwriters: Harry Kleiner, additional dialogue by Samuel Fuller. Photography: Joe MacDonald. Editor: James B. Clarke. Music: Leigh Harline. Production and distribution: 20th Century-Fox. Release date: July 1, 1955. Running time: 103 minutes. Cast: Robert Ryan (Sandy Dawson), Robert Stack, Shirley Yamaguchi, Cameron Mitchell, Brad Dexter, Sessue Hayakawa, Biff Elliott, Sandro Giglio, Elko Hanabusa. Color, CinemaScope.
THE TALL MEN (1955) Director: Raoul Walsh. Producers: William A. Bacher, William B. Hawks. Screenwriters: Sydney Boehm, Frank Nugent, from the novel by Clay Fisher. Photography: Leo Tover. Editor: Louis R. Loeffler. Music: Victor Young. Production and distribution: 20th Century-Fox. Release date: September 22, 1955. Running time: 122 minutes. Cast: Clark Gable, Jane Russell, Robert Ryan (Nathan Stark), Cameron Mitchell, Juan Garcia, Harry Shannon, Emile Meyer, Steve Darrell. Color, CinemaScope.
THE PROUD ONES (1956) Director: Robert D. Webb. Producer: Robert L. Jacks. Screenwriters: Edmund North, Joseph Petracca, from the novel by Verne Athanas. Photography: Lucien Ballard. Editor: Hugh S. Fowler. Music: Lionel Newman. Production and distribution: 20th Century-Fox. Release date: May 1956. Running time: 94 minutes. Cast: Robert Ryan (Marshal Cass Silver), Virginia Mayo, Jeffrey Hunter, Robert Middleton, Walter Brennan, Arthur O'Connell, Ken Clark, Rodolfo Acosta, George Mathews, Fay Roope, Edward Platt, Whit Bissell. Color, CinemaScope.
BACK FROM ETERNITY (1956) Director: John Farrow. Producer: John Farrow. Screenwriter: Jonathan Latimer, from the story by Richard Carroll. Photography: William Mellor. Editor: Eda Warren. Music: Franz Waxman. Production and distribution: RKO Radio Pictures. Release date: September 7, 1956. Running time: 100 minutes. Cast: Robert Ryan (Bill Lonagan), Anita Ekberg, Rod Steiger, Phyllis Kirk, Keith Andes, Gene Barry, Fred Clark, Beulah Bondi, Cameron Prud'Homme, Jesse White, Adele Mara, Jon Provost. Black and white, RKO-Scope.
MEN IN WAR (1956) Director: Anthony Mann. Producer: Sidney Harmon. Screenwriters: Philip Yordan, Ben Maddow (uncredited), from Van Van Praag's novel _Day Without End (Combat_ ). Photography: Ernest Haller. Editor: Richard C. Meyer. Music: Elmer Bernstein. Production: Security Pictures. Distribution: United Artists. Release date: September 7, 1956. Running time: 98 minutes. Cast: Robert Ryan (Lieutenant Benson), Aldo Ray, Robert Keith, Phillip Pine, Nehemiah Persoff, Vic Morrow, James Edwards, L. Q. Jones, Scott Marlowe, Adam Kennedy, Race Gentry, Walter Kelley, Anthony Ray, Robert Normand, Michael Miller, Victor Sen Yung. Black and white.
LONELYHEARTS (1958) Director: Vincent J. Donehue. Producer: Dore Schary. Screenwriter: Dore Schary, from Howard Teichmann's play and Nathanael West's original novel _Miss Lonelyhearts_. Photography: John Alton. Editors: Aaron Stell, John Faure. Music: Conrad Salinger. Production: Schary Productions. Distribution: United Artists. Release date: 1958. Running time: 100 minutes. Cast: Montgomery Clift, Robert Ryan (William Shrike), Myrna Loy, Dolores Hart, Jackie Coogan, Mike Kellin, Frank Maxwell, Frank Overton, Onslow Stevens, and introducing Maureen Stapleton. Black and white.
GOD'S LITTLE ACRE (1958) Director: Anthony Mann. Producer: Sidney Harmon. Screenwriter: Philip Yordan, from the novel by Erskine Caldwell. Photography: Ernest Haller. Editor: Richard C. Meyer. Music: Elmer Bernstein. Production: Security Pictures. Distribution: United Artists. Release date: August 13, 1958. Running time: 118 minutes. Cast: Robert Ryan (Ty Ty Walden), Aldo Ray, Buddy Hackett, Jack Lord, Fay Spain, Vic Morrow, Helen Westcott, Lance Fuller, Rex Ingram, Michael Landon, Russell Collins, Davis Roberts, Janet Brandt, and introducing Tina Louise. Black and white.
DAY OF THE OUTLAW (1959) Director: André de Toth. Producer: Sidney Harmon. Screenwriter: Philip Yordan, from the novel by Lee E. Wells. Photography: Russell Harlan. Editor: Robert Lawrence. Music: Alexander Courage. Production: Security Pictures. Distribution: United Artists. Release date: July 1959. Running time: 92 minutes. Cast: Robert Ryan (Blaise Starrett), Burl Ives, Tina Louise, Alan Marshal, Venetia Stevenson, David Nelson, Nehemiah Persoff, Jack Lambert, Frank DeKova, Lance Fuller, Elisha Cook Jr., Dabbs Greer, Betsey Jones-Moreland, Helen Westcott, Donald Elson, Robert Cornthwaite, Michael McGreevey. Black and white.
ODDS AGAINST TOMORROW (1959) Director: Robert Wise. Producer: Robert Wise. Screenwriters: Abraham Polonsky, Nelson Gidding, from the novel by William P. McGivern. Photography: Joseph Brun. Editor: Dede Allen. Music: John Lewis. Production: HarBel Productions. Distribution: United Artists. Release date: October 15, 1959. Running time: 96 minutes. Cast: Harry Belafonte, Robert Ryan (Earle Slater), Shelley Winters, Ed Begley, Gloria Grahame, Will Kuluva, Kim Hamilton, Mae Barnes, Richard Bright, Carmen De Lavallade, Lew Gallo, Wayne Rogers. Black and white.
ICE PALACE (1960) Director: Vincent Sherman. Producer: Henry Blanke, Harry Kleiner (uncredited). Screenwriter: Harry Kleiner, from the novel by Edna Ferber. Photography: Joseph F. Biroc. Editor: William H. Ziegler. Music: Max Steiner. Production and distribution: Warner Bros. Release date: January 2, 1960. Running time: 143 minutes. Cast: Richard Burton, Robert Ryan (Thor Storm), Martha Hyer, Jim Backus, Carolyn Jones, Ray Danton, Diane McBain. Karl Swenson, Shirley Knight, Barry Kelley, Sheridan Comerate, George Takei, Steve Harris. Color.
THE CANADIANS (1961) Director: Burt Kennedy. Producer: Herman E. Webber. Screenwriter: Burt Kennedy. Photography: Arthur Ibbetson. Editor: Douglas Robertson. Music: Douglas Gamley. Production: Associated Producers, 20th Century-Fox Productions. Distribution: 20th Century-Fox. Release date: March 11, 1961. Running time: 85 minutes. Cast: Robert Ryan (Inspector William Gannon), John Dehner, Torin Thatcher, Burt Metcalfe, John Sutton, Jack Creley, Scott Peters, Richard Alden, Teresa Stratas.
KING OF KINGS (1961) Director: Nicholas Ray. Producer: Samuel Bronston. Screenwriters: Philip Yordan, Ray Bradbury (uncredited narration). Photography: Franz F. Planer, Milton Krasner, Manuel Berenguer. Editors: Harold F. Kress, Renee Lichtig (uncredited). Music: Miklos Rozsa. Production: Samuel Bronston Productions (uncredited). Distribution: Metro-Goldwyn-Mayer. Release date: October 11, 1961. Running time: 168 minutes. Cast: Jeffrey Hunter, Siobhan McKenna, Hurd Hatfield, Ron Randell, Viveca Lindfors, Rita Gam, Carmen Sevilla, Brigid Bazlen, Harry Guardino, Rip Torn, Frank Thring, Guy Rolfe, Royal Dano, Robert Ryan (John the Baptist), Orson Welles (voice-over narration, uncredited). Color, Technirama.
THE LONGEST DAY (1962) Directors: Ken Annakin, Andrew Marton, Bernhard Wicki, Gerd Oswald (uncredited). Producer: Darryl F. Zanuck. Screenwriter: Cornelius Ryan, from his book, with additional scenes by Romain Gary, James Jones, David Pursall, Jack Seddon. Photography: Jean Bourgoin, Walter Wottitz. Editor: Samuel E. Beetley. Music: Maurice Jarre. Production and distribution: 20th Century-Fox. Release date: September 25, 1962. Running time: 178 minutes. Cast: John Wayne, Henry Fonda, Robert Mitchum, Sean Connery, Eddie Albert, Curd Jürgens, Richard Todd, Richard Burton, Peter Lawford, Rod Steiger, Irina Demick, Gert Fröbe, Edmond O'Brien, Kenneth More, Robert Ryan (Brigadier General James M. Gavin). Black and white.
BILLY BUDD (1962) Director: Peter Ustinov. Producer: Peter Ustinov. Screenwriters: Peter Ustinov, DeWitt Bodeen, Robert Rossen (uncredited), from Herman Melville's story "Billy Budd, Sailor" and Louis O. Coxe and Robert H. Chapman's play _Uniform of Flesh_. Photography: Robert Krasker. Editor: Jack Harris. Music: Antony Hopkins. Production: Anglo Allied. Distribution: Allied Artists Pictures. Release date: October 30, 1962. Running time: 123 minutes. Cast: Robert Ryan (John Claggart, Master of Arms), Peter Ustinov, Melvyn Douglas, Paul Rogers, John Neville, David McCallum, Ronald Lewis, Lee Montague, Thomas Heathcote, Ray McAnnally, Robert Brown, John Meillon, Cyril Luckham, Niall McGinnis, Victor Brooks, Barry Keegan, and introducing Terence Stamp. Black-and-white, CinemaScope.
THE INHERITANCE (1964) Director: Harold Mayer. Producer: Harold Mayer. Screenwriter: Millard Lampell. Photography: Edmund B. Gerard, Jesse Paley, Leonard Stark. Editor: Lawrence Silk. Music: George Kleinsinger. Production and distribution: Harold Mayer Productions. Release date: November 8, 1964. Running time: 58 min. Cast: Robert Ryan (narrator). Black and white.
THE CROOKED ROAD (1965) Director: Don Chaffey. Producer: David Henley. Screenwriters: J. Garrison, Don Chaffey. Adaptation: J. Garrison, from Morris L. West's novel _The Big Story_. Photography: Stephen Dade. Editor: Peter Tanner. Music: Bojan Adamic. Production: Argo Film Productions, Triglar Films, Trident Films. Distribution: Seven Arts Pictures. Release date: February 3, 1965. Running time: 92 minutes. Cast: Robert Ryan (Richard Ashley), Stewart Granger, Nadia Gray, Marius Goring, Katherine Woodville, George Coulouris, Robert Rietty, Milan Micic, Demeter Bitenc, Slobodan Dimitrijevic, Murray Kash, Vladimir Bacic, Niksa Stefani. Black and white.
THE SECRET AGENTS (1965) Directors: Christian-Jaque, Werner Klingler, Carlo Lizzani, Terence Young. Producers: Richard Hellman, Eugéne Tucherer. Screenwriters: Philippe Bouvard, Jacques Caborie, Christian-Jaque, Ennio De Concini, Jo Eisinger, Jacques Rémy. Photography: Richard Angst. Editor: Franco Fraticelli. Music: Robert Mellin, Gian Piero Reverberi. Production: American International Pictures, Eichberg-Film, Euro International Film, Fair Film, Franco London Films, Landau/Unger. Distribution: American International Pictures. Release date: June 23, 1965 (France); April 13, 1966 (US). Running time: 118 minutes. Cast: Henry Fonda, Vittorio Gassman, Annie Girardot, Robert Ryan (General Bruce), Bourvil, Peter van Eyck, Maria Grazia Bucella, Jacques Sernas, Robert Hossein. Black and white. _Note_ : Released in the United States as _The Dirty Game_ , with a running time of 87 minutes and no credit for director Werner Klingler, whose scenes were deleted.
BATTLE OF THE BULGE (1965) Director: Ken Annakin. Producers: Sidney Harmon, Milton Sperling, Philip Yordan. Screenwriters: Philip Yordan, Milton Sperling, John Melson. Photography: Jack Hildyard. Editor: Derek Parsons. Music: Benjamin Frankel. Production: United States Pictures, Cinerama Productions Corp., and Warner Bros. Distribution: Warner Bros. Release date: December 16, 1965. Running time: 167 minutes. Cast: Henry Fonda, Robert Shaw, Robert Ryan (General Grey), Dana Andrews, George Montgomery, Ty Hardin, Pier Angeli, Barbara Werle, Charles Bronson, Hans Christian Blech, Werner Peters, James MacArthur, Telly Savalas. Color, Ultra-Panavision.
THE PROFESSIONALS (1966) Director: Richard Brooks. Producer: Richard Brooks. Screenwriter: Richard Brooks, from Frank O'Rourke's novel _A Mule for the Marquesa_. Photography: Conrad Hall. Editor: Peter Zinner. Music: Maurice Jarre. Production: Pax Enterprises. Distribution: Columbia Pictures. Release date: November 2, 1966. Running time: 117 minutes. Cast: Lee Marvin, Robert Ryan (Hans Ehrengard), Woody Strode, Burt Lancaster, Jack Palance, Ralph Bellamy, Claudia Cardinale, Joe De Santis, Rafael Bertrand, Jorge Martinez de Hoyos, Marie Gomez, José Chavez, Carlos Romero, Vaughn Taylor. Color.
THE BUSY BODY (1966) Director: William Castle. Producer: William Castle. Screenwriter: Ben Starr, from the Donald Westlake novel. Photography: Harold Stine. Editor: Edwin H. Bryant. Music: Vic Mizzy. Production: William Castle Productions. Distribution: Paramount Pictures. Release date: March 12, 1966. Running time: 101 minutes. Cast: Sid Caesar, Robert Ryan (Charley Barker), Anne Baxter, Kay Medford, Jan Murray, Richard Pryor, Arlene Golonka, Charles McGraw, Ben Blue, Dom DeLuise, Bill Dana, Godfrey Cambridge, Marty Ingels, George Jessel. Color.
THE DIRTY DOZEN (1966) Director: Robert Aldrich. Producer: Kenneth Hyman. Screenwriters: Nunnally Johnson, Lukas Heller, from the novel by E. M. Nathanson. Photography: Edward Scaife. Editor: Michael Luciano. Music: Frank De Vol. Production: MKH and Seven Arts Productions. Distribution: Metro-Goldwyn-Mayer. Release date: June 15, 1967. Running time: 150 minutes. Cast: Lee Marvin, Ernest Borgnine, Charles Bronson, Jim Brown, John Cassavetes, Richard Jaeckel, George Kennedy, Trini Lopez, Ralph Meeker, Robert Ryan (Colonel Everett Dasher Breed), Telly Savalas, Donald Sutherland, Clint Walker, Robert Webber. Color.
HOUR OF THE GUN (1967) Director: John Sturges. Producer: John Sturges. Screenwriter: Edward Anhalt. Photography: Lucien Ballard. Editor: Ferris Webster. Music: Jerry Goldsmith. Production: Mirsch-Kappa Production. Distribution: United Artists. Release date: November 1, 1967. Running time: 100 minutes. Cast: James Garner, Jason Robards, Robert Ryan (Ike Clanton), Albert Salmi, Charles Aidman, Steve Ihnat, Michael Tolan, William Windom, William Schallert, Bill Fletcher, Karl Swenson, Austin Willis, Monte Markham, Richard Bull, Sam Melville, Frank Converse, Jon Voight, Robert Phillips. Color, Panavision.
CUSTER OF THE WEST (1968) Director: Robert Siodmak. Producer: Philip Yordan. Screenwriters: Bernard Gordon, Julian Halevy. Photography: Cecilio Paniagua. Editor: Peter Parasheles, Maurice Rootes. Music: Bernardo Segall. Production: Security Pictures and Cinerama Productions Corp. Distribution: Cinerama Releasing Corporation. Release date: November 9, 1967 (UK); January 24, 1968 (US). Running time: 143 minutes. Cast: Robert Shaw, Mary Ure, Ty Hardin, Jeffrey Hunter, Lawrence Tierney, Marc Lawrence, Kieron Moore, Charles Stalnaker, Robert Hall, Robert Ryan (Sergeant Patrick Mulligan). Color.
A MINUTE TO PRAY, A SECOND TO DIE (1968) Director: Franco Giraldi. Producer: Albert Band. Screenwriters: Louis Garfinkel, Ugo Liberatore, and Albert Band. Photography: Aiace Parolin. Editor: Alberto Gallitti. Music: Carlo Rustichelli. Production: American Broadcasting Company, Documento Film, Selmur Productions. Distribution: Cinerama Releasing Corporation. Release date: May 1, 1968. Running time: 118 minutes. Cast: Alex Cord, Arthur Kennedy, Robert Ryan (New Mexico Governor Lem Carter), Enzo Fiermonte, Renato Romano, Franco Lantieri, Giampiero Albertini, Mario Brega, Nicoletta Machiavelli. Color.
ANZIO (1968) Director: Edward Dmytryk. Producer: Dino De Laurentiis. Screenwriter: H. A. L. Craig. Adaptation: Frank De Felitta, Duilio Coletti, and Giuseppe Mangione, from the book by Wynford Vaughan-Thomas. Photography: Giuseppe Rotunno. Editor: Peter Taylor. Music: Riz Ortolani. Production: Columbia Pictures Corporation and Dino De Laurentiis Cinematografica. Distribution: Columbia Pictures. Release date: July 24, 1968. Running time: 117 minutes. Cast: Robert Mitchum, Peter Falk, Robert Ryan (General Carson), Earl Holliman, Mark Damon, Arthur Kennedy, Reni Santoni, Joseph Walsh, Thomas Hunter, Giancarlo Giannini, Patrick Magee, Arthur Franz. Color.
THE WILD BUNCH (1969) Director: Sam Peckinpah. Producer: Phil Feldman. Screenwriters: Walon Green, Sam Peckinpah. Story: Walon Green, Roy N. Sickner. Photography: Lucien Ballard. Editor: Louis Lombardo. Music: Jerry Fielding. Production and distribution: Warners Bros.–Seven Arts. Release date: June 18, 1969. Running time: 145 minutes. Cast: William Holden, Ernest Borgnine, Robert Ryan (Deke Thornton), Edmond O'Brien, Warren Oates, Jaime Sánchez, Ben Johnson, Emilio Fernández, Strother Martin, L. Q. Jones, Albert Dekker, Bo Hopkins, Dub Taylor, Jorge Russek, Alfonso Arau, Chano Urueta, Sonia Amelio, Aurora Clavel, Elsa Cárdenas. Color.
CAPTAIN NEMO AND THE UNDERWATER CITY (1969) Director: James Hill. Producer: Bertram Ostrer. Screenwriters: Pip Baker, Jane Baker, and R. Wright Campbell, from characters created by Jules Verne. Photography: Alan Hume. Editor: Bill Lewthwaite. Music: Walter Stott (aka Angela Morley). Production and distribution: Omnia Pictures Ltd., through Metro-Goldwyn-Mayer. Release date: July 1969 (UK); October 7, 1970 (US). Running time: 105 minutes. Cast: Robert Ryan (Captain Nemo), Chuck Connors, Nanette Newman, Luciana Paluzzi, John Turner, Bill Fraser, Kenneth Connor, Alan Cuthbertson, Christopher Hartstone. Color.
THE REASON WHY (1970) Director: Paul Leaf. Screenwriter: From the play by Arthur Miller. Production: Gino Giglio Co. Distribution: Pathé Contemporary Films. Release date: February 13, 1970. Running time: 13 minutes. Cast: Eli Wallach, Robert Ryan (Roger). Color.
LAWMAN (1971) Director: Michael Winner. Producer: Michael Winner. Screenwriter: Gerald Wilson. Photography: Robert Paynter. Editor: Freddie Wilson. Music: Jerry Fielding. Production: Scimitar Films. Distribution: United Artists. Release date: March 11, 1971 (UK); August 4, 1971 (US). Running time: 99 minutes. Cast: Burt Lancaster, Robert Ryan (Cotton Ryan), Lee J. Cobb, Robert Duvall, Sheree North, Albert Salmi, Richard Jordan, John McGiver, Ralph Waite, John Beck, William C. Watson, Walter Brooke, Robert Emhardt, Charles Tyner, J. D. Cannon, Joseph Wiseman, Richard Bull, John Hillerman. Color.
THE LOVE MACHINE (1971) Director: Jack Haley Jr. Producer: M. J. Frankovich. Screenwriter: Samuel Taylor, from the novel by Jacqueline Susann. Photography: Charles B. Lang. Editor: David Blewitt. Music: Artie Butler. Production: Frankovich Productions, Columbia Pictures Corporation. Distribution: Columbia Pictures. Release date: August 14, 1971. Running time: 109 minutes. Cast: Dyan Cannon, Robert Ryan (Gregory Austin), Jackie Cooper, David Hemmings, Shecky Greene, William Roerick, Maureen Arthur, Clinton Greyn, Sharon Farrell, Alexandra Hay, Eve Bruce, Greg Mullavey, Edith Atwater, Gene Baylos, Ben Lessy, Elizabeth St. Clair, Claudia Jennings, John Phillip Law, and introducing Jodi Wexler. In color.
AND HOPE TO DIE (1972) Director: René Clément. Producer: Serge Silberman. Screenwriter: Sébastien Japrisot, from David Goodis's novel _Black Friday_. Photography: Edmond Richard. Editor: Roger Dwyre. Music: Francis Lai. Production: Greenwich Film Productions. Distribution: 20th Century-Fox. Release date: September 15, 1972 (France); November 29, 1972 (US). Running time: 99 minutes. Cast: Jean-Louis Trintignant, Robert Ryan (Charley), Lea Massari, Aldo Ray, Jean Gaven, Tisa Farrow. Color.
LOLLY-MADONNA XXX (1973) Director: Richard Sarafian. Producer: Rodney Carr-Smith. Screenwriter: Rodney Carr-Smith and Sue Grafton, from Grafton's novel _The Lolly-Madonna War_. Photography: Philip H. Lathrop. Editor: Tom Rolf. Music: Fred Myrow. Production and distribution: Metro-Goldwyn-Mayer. Release date: February 21, 1973. Running time: 103 minutes. Cast: Rod Steiger, Robert Ryan (Pap Gutshall), Jeff Bridges, Scott Wilson, Katherine Squire, Joan Goodfellow, Tresa Hughes, Gary Busey, Randy Quaid, Season Hubley. Color.
THE OUTFIT (1973) Director: John Flynn. Producer: Carter DeHaven. Screenwriter: John Flynn, from the novel by Richard Stark (aka Donald E. Westlake). Photography: Bruce Surtees. Editor: Ralph E. Winters. Music: Jerry Fielding. Production and distribution: Metro-Goldwyn-Mayer. Release date: October 1973. Running time: 86 minutes. Cast: Robert Duvall, Karen Black, Joe Don Baker, Timothy Carey, Richard Jaeckel, Sheree North, Felice Orlandi, Marie Windsor, Jane Greer, Henry Jones, Joanna Cassidy, Tom Reese, Elisha Cook, Bill McKinney, Anita O'Day, Archie Moore, Tony Young, Roland La Starza, Edward Ness, Roy Roberts, Toby Andersen, Robert Ryan (Mailer). Color.
EXECUTIVE ACTION (1973) Director: David Miller. Producer: Edward Lewis. Screenwriter: Dalton Trumbo. Story: Donald Freed, Mark Lane. Photography: Robert Steadman. Editor: George Grenville. Music: Randy Edelman. Production: Wakeford/Orloff. Distribution: National General Pictures. Release date: November 7, 1973. Running time: 91 minutes. Cast: Burt Lancaster, Robert Ryan (Robert Foster), Will Geer, Gilbert Green, John Anderson, Paul Carr, Colby Chester, Walter Brooke, Ed Lauter. Color.
THE ICEMAN COMETH (1973) Director: John Frankenheimer. Producer: Ely Landau. Screenwriter: Thomas Quinn Curtiss, from the play by Eugene O'Neill. Photography: Ralph Woolsey. Editor: Harold F. Kress. Production and distribution: The American Film Theatre. Release date: November 10, 1973. Running time: 239 minutes. Cast: Lee Marvin, Fredric March, Ryan (Larry Slade), Jeff Bridges, Bradford Dillman, Sorrell Booke, Hildy Brooks, Nancy Juno Dawson, Evans Evans, Martyn Green, Moses Gunn, Clifton James, John McLiam, Stephen Pearlman, Tom Pedi, George Voskovec, Don McGovern, Bart Burns. Color.
_part three_ Notable Radio and Television Broadcasts
Ryan acted on radio in the late 1940s and early 1950s (on the CBS thriller anthology _Suspense_ and the Mutual Broadcasting System's Christian dramatic series _Family Theater_ ) and on television in the late 1950s and early 1960s (on such anthology series as _Goodyear Theatre, Alcoa Theatre, Zane Grey Theater_ , and _Kraft Suspense Theatre_ ). A more complete list of Ryan's TV appearances (including talk shows with Dick Cavett, David Frost, David Susskind, Johnny Carson, Jack Paar, and Steve Allen) can be found online at the Internet Movie Database. Franklin Jarlett's _Robert Ryan: A Biography and Critical Filmography_ has a detailed inventory of Ryan's episodic TV work, as well as his voluminous narrations and audio recordings. But following are Ryan's more significant radio and TV performances.
HOLLYWOOD FIGHTS BACK (1947) A live program protesting the House Un-American Activities Committee's investigation into communist subversion of the movie industry. Broadcast: October 26, 1947, ABC Radio. Cast: Charles Boyer, Judy Garland, Gene Kelly, Lauren Bacall, Joseph Cotten, Peter Lorre, June Havoc, John Huston, Danny Kaye, Marsha Hunt, Walter Wanger, Cornel Wilde, Melvyn Douglas, Richard Conte, Evelyn Keyes, Burt Lancaster, Paul Henreid, William Holden, Robert Ryan, Florence Eldridge, Myrna Loy, Robert Young, Lucille Ball, Van Heflin, Henry Morgan, Keenan Wynn, Humphrey Bogart, John Beal, Edward G. Robinson, Paulette Goddard, Norman Corwin, Audie Murphy, William Wyler, Fredric March, John Garfield, Deems Taylor, Harlow Shapley, Artie Shaw, Arthur Garfield Hayes, Elbert Thomas, Harley Kilgore, Archibald MacLeish, Claude Pepper, Glen Taylor, Vincent Price, John Rankin, J. Parnell Thomas. _Note_ : Archived at the Paley Center for Media, New York and Los Angeles.
CROSSFIRE (1948) A live radio adaptation of the RKO film, for the CBS anthology series _Suspense_. Broadcast: April 10, 1948, CBS Radio. Running time: 60 minutes. Cast: Robert Young, Robert Mitchum, Robert Ryan (Montgomery), George Cooper, William Phipps.
ADLAI STEVENSON CAMPAIGN RALLY (1952) A live program from Madison Square Garden, complete with Ryan's delayed entrance as master of ceremonies. Broadcast: October 29, 1952, WABD-TV, New York City. Running time: 30 minutes. Cast: Lauren Bacall, Robert Ryan, Kenny Delmar, George Hall, Richard Rodgers, Oscar Hammerstein, Al Capp, Channing Tobias, Mercedes McCambridge, Lew Parker, Al Kelly, Montgomery Clift, Tallulah Bankhead, Louis Calhern, Benay Venuta, Humphrey Bogart, Carl Sandburg, George Jessel, James A. Farley. _Note_ : Archived at the Paley Center for Media, New York and Los Angeles. Black and white.
THE ROBERT RYAN STORY (1953) A fanciful reenactment of Ryan's career for the NBC series _The Hollywood Story_ , with actors playing his mother, his wife, and his mentor, Max Reinhardt, among others. Broadcast: November 28, 1953, NBC Radio. Running time: 30 minutes. _Note_ : Archived at the Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
LINCOLN'S DOCTOR'S DOG (1955) A filmed episode of the NBC anthology series _Screen Directors Playhouse_. Director: H. C. Potter. Teleplay: Christopher Morley. Photography: James Wong Howe. Editor: G. E. Luckenbacher. Broadcast: December 14, 1955, NBC-TV. Running time: 30 minutes. Cast: Robert Ryan (Abraham Lincoln), Charles Bickford, Richard Long, Willis Bouchey, Howard Wendell, Johnny Lee, Paul Keast, Mack Williams, John Craven, Dennis King Jr. Black and white.
THE GREAT GATSBY (1958) A taped performance for the CBS anthology series _Playhouse 90_. Director: Franklin Schaffner. Producer: Martin Manulis. Adaptation: David Shaw, from the novel by F. Scott Fitzgerald. Music: Milton Anderson. Broadcast: June 26, 1958, CBS-TV. Running time: 88 minutes. Cast: Robert Ryan (Jay Gatsby), Rod Taylor, Jeanne Crain, Patricia Barry, Phillip Reed, Virginia Grey, Barry Atwater. Black and white. _Note_ : Archived at the Paley Center for Media, New York and Los Angeles.
30TH ANNUAL ACADEMY AWARDS A live program from the RKO Pantages Theatre in Hollywood; Ryan and actor Wendell Corey present the award for Best Costume Design. Director: Alan Handley. Producer: Jerry Wald. Broadcast: March 26, 1958, NBC-TV. Running time: 120 minutes. Black and white. _Note_ : Archived at the Paley Center for Media, New York and Los Angeles.
"A CALL FROM..." (1960) A documentary on the United Nations' World Refugee Year. Directors: Jack Orbison, William F. Wallace. Producer: Marsha Hunt. Writer: Robert Presnell Jr. Broadcast: February 10, 1960, KCOP-TV, Los Angeles. Running time: 60 minutes. Cast: Steve Allen, Harry Belafonte, Richard Boone, Spring Byington, Jeff Chandler, Bing Crosby, Burl Ives, Louis Jourdan, Phyllis Kirk, Paul Newman, David Niven, Robert Ryan, Jean Simmons, Joanne Woodward. Black and white. _Note_ : Archived at the UCLA Film and Television Archive, Los Angeles. Restored in 2009 as _A Call from the Stars_.
THE SNOWS OF KILIMANJARO (1960) A live performance from CBS Television City in Los Angeles, for the anthology series _Buick-Elektra Playhouse_. Director: John Frankenheimer. Producer: Gordon Duff. Teleplay: A. E. Hotchner, from the story by Ernest Hemingway. Broadcast: March 25, 1960, CBS-TV. Running time: 87 minutes. Cast: Robert Ryan (Harry Walters), Ann Todd, James Gregory, Liliane Montevecchi, Brock Peters, Janice Rule, Mary Astor, Clancy Cooper, Norma Crane, Clegg Hoyt, Albert Paulson, Frank Puglia. Black and white. _Note_ : Archived at the Paley Center for Media, New York and Los Angeles.
THE BELL TELEPHONE HOUR (1964) Ryan hosts this musical variety program, performing the narration to Aaron Copland's _A Lincoln Portrait_. Director: Sid Smith. Producer: Charles Andrews. Broadcast: February 11, 1964, NBC-TV. Running time: 60 minutes. Cast: Robert Ryan, Al Hirt, The Brothers Four, Joan Sutherland, Donald Voorhees, Suzanne Farrell, Patricia Neary, Conrad Ludlow. Color. _Note_ : Archived at the Paley Center for Media, New York and Los Angeles.
THE COMPLETE STORY: WORLD WAR I (1964–65) A twenty-six-episode documentary series, produced by CBS News and narrated by Ryan. Music: Morton Gould. Broadcast debut: September 22, 1964, CBS-TV. Total running time: 660 minutes. Black and white. _Note_ : Released on DVD by CBS DVD home video.
THE PRESIDENCY: A SPLENDID MISERY (1964) Ryan provides the voice of Abraham Lincoln for this historical documentary. Broadcast: September 23, 1964, CBS-TV. Total running time: 60 minutes. Cast: Dana Andrews, Ed Begley, Sidney Blackmer, Macdonald Carey, James Daly, Fredric March, E. G. Marshall, Herbert Marshall, Gary Merrill, Dan O'Herlihy, Jason Robards, Robert Ryan. Black and white.
GUILTY OR NOT GUILTY (1966) A filmed TV pilot that was ultimately broadcast on the NBC anthology series _Bob Hope Presents the Chrysler Theatre_. Broadcast: March 9, 1966, NBC-TV. Running time: 60 minutes. Directors: Roland Kibbee, David Lowell Rich. Producer: Richard Lewis. Teleplay: Evan Hunter, Roland Kibbee, Guthrie Lamb. Cast: Richard Beymer, Robert Duvall, Leif Erickson, Diana Hyland, Leslie Nielsen, Robert Ryan (Andrew Dixon), Pippa Scott. Color.
INAUGURAL EVENING AT FORD'S THEATRE (1968) A live program from Ford's Theatre in Washington, DC. Broadcast: January 30, 1968, CBS-TV. Running time: 59 minutes. Producer: Don Hewitt. Cast: Roger Mudd, Hubert Humphrey, Helen Hayes, Fredric March, Robert Ryan, Henry Fonda, Harry Belafonte, Nina Foch, Andy Williams, Richard Crenna, Patricia Brooks, Odetta, Carmen De Lavallade. Color. _Note_ : Archived at the Paley Center for Media, New York, and the UCLA Film and Television Archive, Los Angeles.
SIMON AND GARFUNKEL: SONGS OF AMERICA (1969) Ryan delivers a brief introduction to the program. Broadcast: November 30, 1969, CBS-TV. Running time: 53 minutes. Director: Charles Grodin. Producers: Paul Simon, Arthur Garfunkel, Mike Jackson. Photography: Abbot Mills, Peter Powell. Editors: Luke Bennett, Ellen Giffard. Cast: Paul Simon, Art Garfunkel, Charles Grodin. Color. _Note_ : Released as part of the Columbia/Legacy CD/DVD reissue of _Bridge Over Troubled Water_.
THE FRONT PAGE (1970) A taped performance of the play by Ben Hecht and Charles MacArthur, with some cast members from the 1969 Broadway production. Director: Alan Handley. Producer: Lewis Freedman. Production: Metromedia Producers Corporation, Plumstead Playhouse. Broadcast: January 31, 1970, Hughes Sports Network. Running time: 90 minutes. Cast: Robert Ryan (Walter Burns), George Grizzard, Helen Hayes, Vivian Vance, Estelle Parsons, Harold J. Kennedy, Susan Watson, John McGiver, Charles White. Color.
THE MAN WITHOUT A COUNTRY (1973) A TV movie, for the series _ABC Movie of the Week_. Director: Delbert Mann. Producer: Norman Rosemont. Screenwriter: Sidney Carroll, from the story by Edward Everett Hale. Photography: Andrew Laszlo. Editor: Gene Milford. Production: Norman Rosemont Enterprises, American Broadcasting Company. Broadcast: April 24, 1973, ABC-TV. Running time: 78 minutes. Cast: Cliff Robertson, Beau Bridges, Peter Strauss, Robert Ryan (Lieutenant Commander Vaughan), Walter Abel, Geoffrey Holder, John Cullum. Color.
Notes
_Introduction_
. Harold Kennedy, _No Pickle, No Performance: An Irreverent Theatrical Excursion from Tallulah to Travolta_ (Garden City, NY: Doubleday, 1978), 125.
. Martin Scorsese, "Scorsese Screens," _TCM Now Playing_ , November 2013, 11.
. Margaret McManus, "Robert Ryan Speaks Out on Reagan," _Bridgeport (Connecticut) Telegram_ , November 6, 1966.
. Millard Lampell, interview with Franklin Jarlett, March 10, 1987, private collection.
. Robert Ryan, as told to Jane Kesner Ardmore, "What Makes an Actor Tick?" (ca. 1957), Jane Ardmore Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
_one_ Inferno
. "Chicago Makes Kaiser's Wake Wild Bedlam." _Chicago Tribune_ , November 12, 1918, 1, 3.
. Robert Ryan, "The Full Text of Ryan's Letter," _Chicago Reader_ , October 29, 2009, <http://www.chicagoreader.com/chicago/actor-robert-ryans-letter-to-his-children/Content?oid=1223014>.
. _Illinois Political Directory 1899_ (Chicago: W. L. Bodine, 1899), 144.
. Ryan, "The Full Text of Ryan's Letter."
. Ibid.
. Ibid.
. Ibid.
. Ibid.
. Ibid.
. Jeanne Stein, "Robert Ryan: Unlike Most Handsome Actors He Was Willing to Be a Heavy," _Films in Review_ 9, no. 1 (January 1968): 9.
. William M. Tuttle Jr., _Race Riot: Chicago in the Red Summer of 1919_ (New York: Atheneum, 1970), 10.
. Ryan, "The Full Text of Ryan's Letter."
. Ibid.
. Ibid.
. _Loyola Prep_ , June 1926, 67.
. Ryan, "The Full Text of Ryan's Letter."
. Ibid.
. Ibid.
. Lupton A. Wilkinson, "These Fathers!" _Motion Picture_ , n.d., Robert Ryan clipping file, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Roger Biles, _Big City Boss in Depression and War: Mayor Edward J. Kelly of Chicago_ (DeKalb: Northern Illinois University Press, 1984), 9.
. Elmer Lynn Williams, _The Fix-It Boys: The Inside Story of the New Deal and the Kelly-Nash Machine_ (Chicago: Elmer Lynn Williams, 1940), 22.
. Jessica Ryan, "Campaign–'52, or A Camera's-Eye View from Two Odd Birds" (ca. 1970), private collection.
. "12 Dead, 16 Saved, in Tunnel," _Chicago Daily News_ , April 14, 1931, 1, 3.
. Ibid.
. "12 Rescued in Disaster," _Chicago Evening Post_ , April 14, 1931, 1, 3.
. Williams, _The Fix-It Boys_ , 62.
. "Coroner's Jury Goes Through Death Tunnel," _Chicago Daily News_ , April 15, 1931, 3.
. "Little Chance to Fix Tunnel Disaster Blame," _Chicago American_ , April 15, 1931, 1, 2.
. Tina Louise, telephone interview with author, July 15, 2012; Lisa Ryan, interview with author, San Francisco, April 17, 2011.
. Irv Kupcinet, "Kup's Column," _Chicago Sun-Times_ , April 7, 1949.
. J. M. Waldreck, "Ryan the Rip-Roaring Adventurer" n.p., n.d., Robert Ryan clipping file, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Stein, "Robert Ryan," 10.
. Wilkinson, "These Fathers!"
. Captain James McNamara (chairman, Maritime Industry Museum, Fort Schuyler, New York), telephone interview with author, July 1, 2012.
. Ryan, "The Full Text of Ryan's Letter."
. Jessica Ryan, "Campaign–'52."
. June Skinner Sawyers, _Chicago Portraits_ (Chicago: Loyola University Press, 1991), 140.
. Williams, _The Fix-It Boys_ , 31.
. Dartmouth College Class of 1932 newsletter (ca. March 1945), Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
. "Robert Ryan Dies of Cancer," _Newark Star-Ledger_ , July 12, 1973, 43.
. Wilkinson, "These Fathers!"
. Autographed program for _Dear Brutus_ , May 6, 1938, private collection.
. _Loyola Prep_ , 78.
. Stein, "Robert Ryan," 13.
. William Shakespeare, _King Lear_ , ed. G. Blakemore Evans. _The Riverside Shakespeare_ (Boston: Houghton Mifflin, 1974), 1.4.288–89. References are to act, scene, and line.
_two_ The Mysterious Spirit
. William Shakespeare, _Hamlet_ , ed. G. Blakemore Evans. _The Riverside Shakespeare_ (Boston: Houghton Mifflin, 1974), 1.5.106–109.
. Fredda Duddley, "Romancing with Ryan," _Photoplay_ , April 1944, 104.
. "Living the Life of Ryan," _Screen Guide_ , January 1950.
. "The Robert Ryan Story," NBC Radio's _The Hollywood Story_ (November 28, 1953), Wisconsin Center for Film and Theater Research, Madison, Wisconsin. Audio recording.
. George Wellworth and Alfred G. Brooks, ed., _Max Reinhardt: A Centennial Festschrift_ , 1873–1973 (Binghamton, NY: Max Reinhardt Archives, 1973), 129.
. Wellworth and Brooks, _Max Reinhardt_ , 129.
. Ibid., 1.
. Ibid., 5.
. Ibid., 75.
. "Robert Ryan's Advice to Would-Be Actors," _Salt Lake City Deseret News_ , November 30, 1951.
. Wellworth and Brooks, _Max Reinhardt_ , 110.
. Jeanne Stein, "Robert Ryan: Unlike Most Handsome Actors He Was Willing to Be a Heavy," Films in Review 9, no. 1 (January 1968): 13.
. Jessica Ryan, "Recollections of a Pioneer Grandmother" (ca. 1970), private collection.
. Jessica Ryan, "If School Keeps" (ca. 1970), private collection.
. William Shakespeare, _The Merchant of Venice_ , ed. G. Blakemore Evans. _The Riverside Shakespeare_ (Boston: Houghton Mifflin, 1974), 4.1.186–87. References are to act, scene, and line.
. Jessica Ryan, "Recollections of a Pioneer Grandmother."
. Robert Ryan, "I'm Gambling with My Career," _Movieland_ , August 1947, 42.
. Ibid.
. Helen Louise Walker, "Portrait of a Happy Man," _Movieland_ , December 1949, 76.
. Marsha Hunt, telephone interview with author, February 9, 2011.
. Edward Dmytryk, _It's a Hell of a Life But Not a Bad Living: A Hollywood Memoir_ (New York: Times Books, 1978), 48.
. Sidney Skolsky, "Tintypes," _Hollywood Citizen-News_ , June 24, 1954.
. Cheyney Ryan, _The Chickenhawk Syndrome: War, Sacrifice, and Personal Responsibility_ (Lanham, MD: Rowman and Littlefield, 2009), x–xi.
. "Acts of Birth: Robert Ryan," _Films and Filming_ , March 1971, 29.
. Robert Wallsten, interview with Franklin Jarlett, March 1986, private collection.
. Denis Brian, _Tallulah, Darling: A Biography of Tallulah Bankhead_ (New York: Macmillan, 1972), 93.
. Ibid., 114.
. Ibid., 120.
. Tallulah Bankhead, _Tallulah: My Autobiography_ (New York: Harper and Brothers, 1952), 245.
. Lee Israel, _Miss Tallulah Bankhead_ (New York: G. P. Putnam's Sons, 1972), 213.
. Joel Lobenthal, _Tallulah! The Life and Times of a Leading Lady_ (New York: Regan Books, 2004), 336.
. Brian, _Tallulah, Darling_ , 118.
. "Pare Lorentz Picks American Odysseus," _Brooklyn Eagle_ , July 14, 1942, 4.
. Patricia Bosworth, "Robert Ryan: In Search of Action," _New York Times_ , June 1, 1969, 1, 7.
_three_ Bombs Away
. John Houseman, _Run-Through_ (New York: Simon and Schuster, 1972), 484.
. Edward Dmytryk, _It's a Hell of a Life, but Not a Bad Living: A Hollywood Memoir_ (New York Times Books, 1978), 54.
. "How Much? RKO Wants to Know from Lorentz," _Variety_ , July 22, 1942, 3.
. Robert Ryan to A. J. Dickerson (mid-1945), Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
. "Movies Are Put in Essential Class by Draft Ruling," _New York Times_ , February 9, 1942, 1.
. "Special Draft Deferment Creates Furor in Hollywood," _Los Angeles Times_ , February 11, 1942, A2.
. Joe McCarthy, "Antic Arts: Robert Ryan" (ca. 1963), private collection.
. Pat O'Brien, _The Wind at My Back: The Life and Times of Pat O'Brien_ (Garden City, NY: Doubleday, 1964), 271.
. Joe McCarthy, "Antic Arts."
. Dmytryk, _It's a Hell of a Life_ , 56.
. Fredda Dudley, "Romancing with Ryan," _Photoplay_ , April 1944, 103.
. Jessica Ryan, "Marine Ryan," _Movieland_ , October 1944, 31.
_four_ You Know the Kind
. Jessica Ryan, "Marine Ryan," _Movieland_ , October 1944, 32.
. Robert Witty and Neil Morgan, _Marines of the Margharita_ (San Diego: Frye and Smith, 1970), 10.
. "Living the Life of Ryan," _Screen Guide_ , January 1950.
. Robert Ryan to commandant of the Marine Corps, August 25, 1944, National Personnel Records Center, St. Louis, Missouri.
. Robert Ryan to A. J. Dickerson (mid-1945), Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
. Dartmouth College Class of 1932 newsletter (ca. March 1945), Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
. Richard Brooks, _The Brick Foxhole_ (New York: Harper and Brothers, 1945), viii.
. Ibid., 29.
. Ibid.
. Douglass K. Daniels, _Tough as Nails: The Life and Films of Richard Brooks_ (Madison: University of Wisconsin Press, 2011), 34.
. Robert Ryan to A. J. Dickerson (mid-1945), Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
. Jessica Ryan, _The Man Who Asked Why_ (Garden City, NY: Doubleday, Doran, 1945), 7.
. U.S. Marine Corps report of medical survey for Pvt. Robert Ryan (September 8, 1945), National Archives and Records Administration, Washington, DC.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Jeanne Stein, "Robert Ryan: Unlike Most Handsome Actors He Was Willing to Be a Heavy," _Films in Review_ 9, no. 1 (January 1968): 19.
. Bert Cardullo, ed., _Jean Renoir: Interviews_ (Jackson, MS: University of Jackson Press, 2005), 173. From Rui Nogueira and Francois Truchaud, _Sight and Sound_ 37, no. 2 (Spring 1968): 25.
. Jean Renoir, _My Life and My Films_ (New York: Atheneum Publishers, 1974), 244.
. Rui Nogueira and Nicoletta Zalaffi, "A Bastard's Long Career: Meeting with Robert Ryan," _Cinema 70_ (April 1970): 50.
. Cardullo, _Jean Renoir_ , 173.
. John Paxton to Clay Steinman, Keith Kelly, and Mario Falsetto (ca. July 1977), John Paxton Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Robert Ryan, "I'm Gambling with My Career," _Movieland_ , August 1947.
. Robert Ryan, "My Role in Crossfire," _Daily Worker_ , July 20, 1947, Southern edition.
. Ryan, "I'm Gambling with My Career."
. N. Peter Rathvon memo to Dore Schary (February 12, 1947), Dore Schary Papers, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Dore Schary, _Heyday_ (Boston, Toronto: Little, Brown, 1979), 157.
. Edward Dmytryk, _On Screen Directing_ (Boston: Focal Point Press, 1984), 86.
. Lee Server, _Robert Mitchum: "Baby I Don't Care"_ (New York: St. Martin's Press, 2001), 135.
. Clay Steinman, Keith Kelly, and Mario Falsetto to John Paxton, June 29, 1977, John Paxton Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Ryan, "I'm Gambling with My Career."
_five_ We Will Succeed, You Will Not
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. "Jeep Hunt," _Photoplay_ , June 1947, 102.
. "A-Hunting We Will Go," _Screen Guide_ , November 1947, 65.
. Lynn Bowers, "Gentle Heel" n.p., n.d., Robert Ryan clipping file, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Bert Granet, "Berlin Express Diary," _Screen Writer_ , May 1948, 12.
. Herb Lightman, "The Story of Filming 'Berlin Express,'" _American Cinematographer_ , July 1948, 232.
. Bert Granet, letter draft n.p., n.d., Bert Granet Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Ibid.
. Charles Higham and Roy Moseley, _Princess Merle: The Romantic Life of Merle Oberon_ (New York: Coward-McCann, 1983), 183–84.
. Robert Ryan, "We're Not Quitting," _Screen Guide_ , May 1948, 84.
. Ibid., 55.
. Granet, "Berlin Express Diary," 12.
. Ryan, "We're Not Quitting," 85.
. Ibid., 85.
. Ibid., 84.
. Harold J. Kennedy, _No Pickle, No Performance: An Irreverent Theatrical Excursion from Tallulah to Travolta_ (Garden City, NY: Doubleday, 1978), 127.
. Advertisement, _Variety_ , July 30, 1947.
. Ryan, "We're Not Quitting," 87.
. Ibid.
. "RKO's Sensitive Pick, 'Crossfire,' Looks Well Over the Sales Hump," _Variety_ , October 15, 1947, 7.
. Elliot E. Cohen, "Letter to the Movie-Makers: The Film Drama as a Social Force," _Commentary_ , August 1947, 112.
. Robert Ryan, "My Role in Crossfire," _Daily Worker_ , July 20, 1947, Southern edition.
. Robert Ryan, "Don't Play It Safe" n.p., n.d., Robert Ryan clipping file, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Dore Schary, _Heyday: An Autobiography_ (Boston and Toronto: Little, Brown, 1979), 157–58.
. Jennifer Langdon, "Americanism on Trial," chap. 9 in _Caught in the Crossfire: Adrian Scott and the Politics of Americanism in 1940s Hollywood_ (Gutenberg-e.org), para. 33.
. Jessica Ryan, notes for "Campaign–'52" (ca. 1970), private collection.
. A. M. Sperber and Eric Lax, _Bogart_ (New York: William Morrow, 1994), 36.
. Larry Ceplar and Steven Englund, _The Inquisition in Hollywood: Politics in the Film Community, 1930–1960_ (Garden City, NY: Anchor Press/Doubleday, 1980), 273.
. Langdon, _Caught in the Crossfire_ , chap. 9, para. 26.
. ABC Radio's _Hollywood Fights Back_ , original radio broadcast, October 26, 1948. Old Time Radio Catalogue, otrcat.com.
. Edward Dmytryk, _Odd Man Out: A Memoir of the Hollywood Ten_ (Carbondale and Edwardsville: Southern Illinois University Press, 1996), 73.
. House Committee on Un-American Activities, _Hearings Regarding the Communist Infiltration of the Motion Picture Industry_ , 80th Cong., 1st sess., 1947, public law 601, sect. 121, subsect. Q(2). https://archive.org/stream/hearingsregardin1947aunit/hearingsregardin1947aunit_djvu.txt.
. Herman A. Lowe, "Big D.C. Whodunit: Who Killed That Red Probe? See Resumption in Dec.," _Variety_ , November 5, 1947, 3, 18.
. "Chatter—Hollywood," _Variety_ , October 29, 1947, 63.
. Schary, _Heyday_ , 369.
. "Bogart Terms Wash. Trip a 'Foolish' Move," _Variety_ , December 3, 1947, 5, 18.
. Higham and Moseley, _Princess Merle_ , 184.
. Louella O. Parsons, "In Hollywood with Louella O. Parsons" n.p., February 29, 1948, Robert Ryan clipping file, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Jessica Ryan, notes for "Campaign–'52."
. Jessica Ryan, "Campaign–'52, or A Camera's-Eye View from Two Odd Birds" (ca. 1970), private collection.
. Ibid.
. Jessica Ryan, notes for "Campaign–'52."
. "Portrait of a Happy Man," _Movieland_ , December 1949, 76.
. Philip Dunne, interview with Franklin Jarlett, June 12, 1987, private collection.
. Jessica Ryan, notes for "Campaign–'52."
. "Ryan for 'Glory,'" _Variety_ , November 5, 1947, 4.
. Norman Cousins, "Modern Man Is Obsolete," _Saturday Review_ , August 18, 1945, 8.
. Lisa Ryan, telephone interview with author, May 9, 2010.
. Dmytryk, _Odd Man Out_ , 99.
_six_ Caught
. Thomas Schatz, _The Genius of the System: Hollywood Filmmaking in the Studio Era_ (New York: Pantheon Books, 1988), 411–12.
. "Despite Rathvon's Balm to Aides, They're Still Uneasy on Hughes Buy," _Variety_ , May 19, 1948, 5.
. "No RKO Changes Due, Schary Assures Help," _Variety_ , June 9, 1948, 3.
. Arthur Noletti Jr., "Conversation with Fred Zinnemann," in _Fred Zinnemann: Interviews_ , ed. Gabriel Miller (Jackson: University Press of Mississippi, 2005), 116.
. Fred Zinnemann, _A Life in the Movies: An Autobiography_ (New York: Charles Scribner's Sons, 1992), 74.
. Robert Surtees, "The Story of Filming 'Act of Violence,'" _American Cinematographer_ (August 1948): 268.
. Dore Schary, _Heyday: An Autobiography_ (Boston and Toronto: Little, Brown, 1979), 171.
. "Hollywood's Economy Jitters," _Variety_ , July 14, 1948, 1.
. Betty Lasky, _RKO, the Biggest Little Major of Them All_ (Englewood Cliffs, NJ: Prentice Hall, 1984), 216–17.
. "RKO Closure Shrinks Backlog," _Variety_ , February 9, 1949, 5.
. Lisa Ryan, telephone interview with author, May 26, 2010.
. Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1994), 35.
. Arthur Laurents, _Original Story by: A Memoir of Broadway and Hollywood_ (New York: Knopf, 2000), 140.
. Ibid., 143.
. "Inside Stuff—Pictures," _Variety_ , August 4, 1948, 16.
. Robert Wallsten, interview with Franklin Jarlett, May 1986, private collection; also Jarlett, _Robert Ryan_ , 89.
. Lutz Bacher, _Max Ophuls in the Hollywood Studios_ (Brunswick, NJ: Rutgers University Press, 1996), 237.
. Joseph Moncure March, _The Wild Party / The Set-Up / A Certain Wildness_ (Freeport, ME: Bond Wheelwright, 1968), 53.
. Ibid., 153.
. Robert Ryan, interview with Tony Thomas (1960), Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. George Stevens Jr., _Conversations with the Great Moviemakers of Hollywood's Golden Age at the American Film Institute_ (New York: Knopf, 2006), 465.
. Richard C. Keenan, _The Films of Robert Wise_ (Lanham, MD: Scarecrow Press, 2007), 43–44.
. Robert Ryan, "The Role I Liked Best...," _Saturday Evening Post_ , July 15, 1950, 68.
. Ibid.
. Ibid.
. Arthur (Weegee) Fellig column, _Los Angeles Mirror_ , November 26, 1948.
. Jessica Ryan, notes for "Campaign–'52" (ca. 1970), private collection.
. Jarlett, _Robert Ryan_ , 40.
. Jessica Ryan, notes for "Campaign–'52."
. Philip Dunne, oral history, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California, 69.
. Jessica Ryan, notes for "Campaign–'52."
. Ibid.
. Michael Ciment, _Conversations with Losey_ (New York: Methuen, 1985), 81.
. Ben Barzman, "Pour Joe," _Positif_ , no. 293/4 (July–August 1985): 11, in David Caute, _Joseph Losey: A Revenge on Life_ (New York: Oxford University Press, 1994), 87.
. "After 150G Preparation, 'Boy' Goes sans Haircut," _Variety_ , September 1, 1948, 2.
. Ciment, _Conversations with Losey_ , 81.
. Bosley Crowther, review of _The Boy with Green Hair, New York Times Book Review_ , January 13, 1949, 26.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Ciment, _Conversations with Losey_ , 82.
. Patrick McGilligan, _Backstory 2: Interviews with Screenwriters of the 1940s and 1950s_ (Berkeley: University of California Press, 1991), 197.
. Bernard Eisenschitz, _Nicholas Ray: An American Journey_ (Berkeley: University of California Press, 1993), 121.
. Gregory G. Hewett, _The Heavy: The Somewhat Noir Life of Thomas Gomez, Hollywood's Quintessential Character Actor_ (forthcoming).
. RKO production file for _The Woman on Pier 13_ , UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
_seven. Learning by Doing_
. Jessica Ryan, notes for "Campaign–'52" (ca. 1970), private collection.
. Jessica Ryan to Dido and Jean Renoir, n.d., Jean Renoir Papers, UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Jessica Ryan, "If School Keeps" (ca. 1970), private collection.
. Erskine Johnson, "Lazy Newcomers Irk Ryan," _Los Angeles Mirror News_ , December 2, 1959.
. Bob Thomas, "Male Cheesecake! Robert Ryan Comments on New Trend," _Hollywood Citizen-News_ , July 11, 1949.
. Laraine Day, oral history, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California, 201.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Reba and Bonnie Churchill, "Ryan Goes Romantic" n.p., n.d., Robert Ryan clipping file, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Bernard Eisenschitz, _Nicholas Ray: An American Journey_ (Berkeley: University of California Press, 1993), 127.
. Ibid., 129.
. Patrick McGilligan, _Nicholas Ray: The Glorious Failure of an American Director_ (New York: HarperCollins, 2011), 176.
. Eisenschitz, _Nicholas Ray_ , 513.
. Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1994), 43.
. "Inside Stuff—Pictures," _Variety_ , December 7, 1949, 18.
. Eisenschitz, _Nicholas Ray_ , 130.
. Richard B. Jewell, _The_ RKO _Story_ (New York: Arlington House, 1982), 143.
. Michel Ciment, _Conversations with Losey_ (New York: Methuen, 1985), 79.
. Gerald Butler, _Mad with Much Heart_ (New York: Rinehart, 1946), 6.
. Butler, _Mad with Much Heart_ , 121.
. Lee Server, _Screenwriter: Words Become Pictures_ (Pittstown, NJ: Main Street Press, 1987), 40.
. Eisenschitz, _Nicholas Ray_ , 155.
. Joseph I. Breen to Harold Melniker, March 23, 1950, RKO production file for _On Dangerous Ground_ , UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Joseph I. Breen to Harold Melniker, March 20, 1950, RKO production file for _On Dangerous Ground_ , UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Eisenschitz, _Nicholas Ray_ , 156.
. Lamont Johnson, interview with Franklin Jarlett, August 17, 1986, private collection; also Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 62.
. Norma H. Goodhue, "Crowded Schools Do Good Job Despite Difficulties," _Los Angeles Times_ , February 26, 1950, 1, 3.
. Jessica Ryan, "If School Keeps."
. Ibid.
. Ibid.
. Peer J. Oppenheimer, "A Film Hero Fights for Better Schools," _New Haven Sunday Register_ , August 7, 1960, 12.
. Jessica Ryan, "If School Keeps."
. Ibid.
. Larry Ceplair and Steven Englund, _The Inquisition in Hollywood: Politics in the Film Community, 1930–1960_ (Garden City, NY: Anchor Press/Doubleday, 1980), 362–63.
. Greg Mitchell, _Tricky Dick and the Pink Lady: Richard Nixon vs. Helen Gahagan Douglas—Sexual Politics and the Red Scare_ (New York: Random House, 1998), 215.
. Jessica Ryan, notes for "Campaign–'52."
. Eisenschitz, _Nicholas Ray_ , 162.
. Nicholas Ray, _I Was Interrupted: Nicholas Ray on Making Movies_ (Berkeley: University of California Press, 1993), 106.
. Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 39.
. Ray, _I Was Interrupted_.
. Robert Ryan, open letter to parents in North Hollywood, January 1951, Oakwood School Archives, North Hollywood, California.
. Jessica Ryan, "If School Keeps."
. John Dewey, "My Pedagogic Creed," in _The Essential Dewey_ , vol. 1: _Pragmatism, Education, Democracy_ , ed. Larry A. Hickman and Thomas M. Alexander (Bloomington and Indianapolis: Indiana University Press, 1998), 230.
. Elsie M. Walker and David T. Johnson, _Conversations with Directors_ (Lanham, MD: Scarecrow Press, 2008), 209.
. Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 39.
. Robert Ryan, "How Do You Remember All Those Words?" (ca. 1957), Jane Ardmore Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. _Flying Leathernecks_ publicity, Lincoln Quarberg Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Extension of Remarks of Hon. Richard M. Nixon of California in the Senate of the United States (Monday, August 27, 1951), Lincoln Quarberg Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Jessica Ryan, "If School Keeps."
. Ibid.
. Howard McClay, "He Didn't Look Like an Actor—But," _Los Angeles Daily News_ , May 20, 1952.
_eight_ The Whiz Kids
. Louis Berg, "Gentle Irishman," _Los Angeles Times_ , February 4, 1951.
. Peter Bogdanovich, _Fritz Lang in America_ (New York: Praeger, 1969), 81.
. Rui Nogueira and Nicoletta Zalaffi, "A Bastard's Long Career: Meeting with Robert Ryan," _Cinema 70_ (April 1970): 50–51.
. Axel Madsen, _Stanwyck_ (New York: HarperCollins, 1994), 291.
. Jane Ellen Wayne, _Marilyn's Men: The Private Life of Marilyn Monroe_ (New York: St. Martin's Press, 1992), 49.
. Bogdanovich, _Fritz Lang in America_ , 82.
. Richard Buskin, _Blonde Heat: The Sizzling Screen Career of Marilyn Monroe_ (New York: Billboard Books, 2001), 96.
. Ibid., 95.
. Ella Smith, _Starring Miss Barbara Stanwyck_ (New York: Crown Publishers, 1984), 233.
. Lisa Ryan, e-mail to author, April 14, 2014.
. Louella O. Parsons, "Robert Ryan: Nice Man to Have Around the Movies," _Los Angeles Examiner_ , February 10, 1952.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Harriet Parsons, "Battle-Scarred Ryan Is Home," _Los Angeles Herald-Examiner_ , December 12, 1965, 1, 5.
. Jessica Ryan, "If School Keeps" (ca. 1970), private collection.
. Ibid., 68.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. "Films' Biggest Mystery—RKO," _Variety_ , February 20, 1952, 3, 12.
. "Hughes' Commie Blast Viewed as Cue to Exit RKO; Mayer Report Up Again," _Variety_ , April 9, 1952, 3, 29.
. Janet Leigh, _There Really Was a Hollywood_ (Garden City, NY: Doubleday, 1984), 159.
. Jane Morris, "He Makes Living His Business" n.p., December 1952, Robert Ryan alumni file, Dartmouth College Library, Hanover, New Hampshire.
. Michael Munn, _Jimmy Stewart: The Truth Behind the Legend_ (Fort Lee, NJ: Barricade Books, 2006), 215.
. Leigh, _There Really Was a Hollywood_ , 159.
. Nogueira and Zalaffi, "A Bastard's Long Career," 57–58.
. Jessica Ryan, "If School Keeps."
. Jessica Ryan, "Campaign–'52."
. Porter McKeever, _Adlai Stevenson: His Life and Legacy_ (New York: William Morrow, 1989), 215–16.
. Jessica Ryan, "Campaign–'52."
. Robert Ryan, as told to Dick Pine, "The Trouble with Me Is," _Movieland_ , June 1951, 21, 79.
. Jessica Ryan, "Campaign–'52."
. Jessica Ryan, notes for "Campaign–'52."
. Jessica Ryan, "Campaign–'52."
. Jessica Ryan, notes for "Campaign–'52."
. Ibid.
. Robert Wallsten, interview with Franklin Jarlett, May 1986, private collection.
. Jessica Ryan, notes for "Campaign–'52."
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Jessica Ryan, notes for "Campaign–'52."
. Roy Ward Baker, _The Director's Cut: A Memoir of 60 Years in Film and Television_ (London: Reynolds and Hearn, 2000), 82.
. Ibid., 83.
. Rhonda Fleming, e-mail to author, August 21, 2012.
. "Acts of Birth: Robert Ryan," _Films and Filming_ , March 1971, 28.
. M. Nichols, "Robert Ryan—Hero and Heel," _Coronet_ , January 1960, 16.
. John Houseman, _Front and Center_ (New York: Simon and Schuster, 1979), 423.
. Jessica Ryan, "If School Keeps."
. Charles Haas, interview with author, Studio City, California, April 20, 2011.
. Ibid.
. Jessica Ryan, "If School Keeps."
. Charles Haas, interview with author.
. Jeanne Stein, "Robert Ryan: Unlike Most Handsome Actors He Was Willing to Be a Heavy," _Films in Review_ (January 1968): 21.
. Albert Hackett, interview with Franklin Jarlett, August 1986, private collection; also Glenn Loney, "In the Words of Robert Ryan," _Cue_ , July 11, 1970, 11.
. Houseman, _Front and Center_ , 438.
_nine_ Rum, Rebellion, and Ryan
. John Houseman, _Front and Center_ (New York: Simon and Schuster, 1979), 436.
. William Shakespeare, _Coriolanus_ , in _The Riverside Shakespeare_ , ed. G. Blakemore Evans (Boston: Houghton Mifflin, 1974), 1.1.170–72, 82–84. References are to act, scene, and line.
. Shakespeare, _Coriolanus_ , 3.1.138–39.
. Denis Brian, _Tallulah, Darling: A Biography of Tallulah Bankhead_ (New York: Macmillan, 1972), 84–85.
. Howard McClay, "Shakespeare Summons Bob Ryan," _Los Angeles Daily News_ , March 24, 1954.
. Houseman, _Front and Center_ , 437.
. Robert Ryan, as told to Naomi Engelsman, "Backstage with Us Ryans," _Parents_ (September 1954): 130.
. George Shea, "Shakespeare on Treason," review of _Coriolanus, Wall Street Journal_ , January 21, 1954, 10.
. Brooks Atkinson, "Again, the Phoenix," review of _Coriolanus, New York Times Book Review_ , January 24, 1954, X1.
. Jessica Ryan, "If School Keeps" (ca. 1970), private collection.
. Ibid., 130.
. Howard Breslin, "Bad Time at Honda," _American_ (January 1947): 41, 136.
. Ibid., 138.
. Kenneth MacKenna, interoffice memo to Dore Schary, June 10, 1954, Dore Schary Papers, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Tom Weaver, _They Fought in the Creature Features_ (Jefferson, NC: McFarland, 1995), 163.
. "Acts of Birth: Robert Ryan," _Films and Filming_ , March 1971, 27.
. Glenn Lovell, _Escape Artist: The Life and Films of John Sturges_ (Madison: University of Wisconsin Press, 2008), 103.
. James Curtis, _Spencer Tracy_ (London: Hutchinson, 2011), 673.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Lisa Ryan, telephone interview with author, October 3, 2009.
. Erskine Johnson column, _Los Angeles Daily News_ , July 7, 1951.
. Ryan, "Backstage with Us Ryans," 128.
. Jessica Ryan, "If School Keeps."
. James Naughton, telephone interview with author, July 18, 2012.
. Millard Lampell, interview with Franklin Jarlett, March 10, 1987, private collection.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Roy Norr, "Cancer by the Carton," _Reader's Digest_ , September 1952, 739.
. John O'Hara, review of _Bad Day at Black Rock, Collier's_ , March 18, 1955.
. Robert Hatch, review of _Bad Day at Black Rock, Nation_ , February 19, 1955, 165.
. Samuel Fuller, _A Third Face: My Tale of Writing, Fighting, and Filmmaking_ (New York: Knopf, 2002), 317.
. Fuller, _A Third Face_ , 315.
. Rui Nogueira and Nicoletta Zalaffi, "A Bastard's Long Career: Meeting with Robert Ryan," _Cinema_ 70 (April 1970): 56.
. Lee Server, _Sam Fuller: Film Is a Battleground_ (Jefferson, NC: McFarland, 1994), 115.
. Fuller, _A Third Face_ , 316.
. Jessica Ryan to Dido and Jean Renoir, April 8, 1955, Jean Renoir Papers, UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. William Otterburn-Hall, "A Good Bad Man Is Hard to Find," _Louisville Courier-Journal and Times_ , June 7, 1970, E4.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Otterburn-Hall, "A Good Bad Man Is Hard to Find."
_ten_ The Gates of War
. Robert Ryan, as told to Jane Kesner Ardmore, "What Makes an Actor Tick?" (ca. 1957), Jane Ardmore Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Jane Morris, "He Makes Living His Business" n.p., December 1952. Robert Ryan alumni file, Dartmouth College Library, Hanover, New Hampshire.
. Philip Yordan, interview with Franklin Jarlett, October 27, 1986, private collection; also Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 48.
. Patrick McGilligan, _Backstory 2: Interviews with Screenwriters of the 1940s and 1950s_ (Berkeley: University of California Press, 1991), 361.
. Ibid., 181.
. Ibid., 182.
. Ibid., 357.
. Robert Ryan, as told to Jane Kesner Ardmore, "What Makes an Actor Tick?" (ca. 1957), Jane Ardmore Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Toya Harrison, interview with author, Studio City, California, April 20, 2011.
. Chalmers M. Roberts, "Adlai Calls for All-Out Final Drive," _Washington Post and Times Herald_ , October 21, 1956, A1.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. _Dartmouth Alumnus_ , December 1956, Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
. Jean Giraudoux, _Tiger at the Gates_ , trans. Christopher Fry (London: Samuel French, 1955), 25.
. Ibid., 11.
. Ibid., 48.
. Harold J. Kennedy, _No Pickle, No Performance: An Irreverent Theatrical Excursion from Tallulah to Travolta_ (Garden City, NY: Doubleday, 1978), 128.
. Ibid., 129.
. "Actor Robert Ryan's Mother Injured by Auto," _Los Angeles Times_ , February 2, 1957.
. Kennedy, _No Pickle, No Performance_ , 128.
. Richard L. Coe, "Here's One Not to Miss," review _of Men in War, Washington Post_ , March 9, 1957, D9.
. Philip Scheuer, "Suspense Pulses in 'Men in War,'" _Los Angeles Times_ , January 13, 1957, F1.
. "Expansion Program on at Oakwood," _San Fernando Valley Mirror-News_ , May 30, 1957.
. Lamont Johnson, interview with Franklin Jarlett, August 17, 1986, private collection; also Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland: 1990), 66.
. Marie Spottswood, "Such a Resource in One Individual," in _A Memorial Tribute for the Robert and Jessica Ryan Memorial_ , November 3, 1974, Oakwood School archives, North Hollywood, California.
. Robert Wallsten, interview with Franklin Jarlett, May 1986, private collection.
. Erskine Caldwell, _God's Little Acre_ (Athens: University of Georgia Press, 1995), 88.
. "Harmon-Mann Solicit No Seal for 'God's Little Acre,' Tell Why," _Variety_ , April 3, 1957, 10.
. Philip Yordan, interview with Franklin Jarlett, October 27, 1986, private collection.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Robert Ryan to Corey Ford, October 28, 1957, Dartmouth Boxing Club File, Dartmouth College, Hanover, New Hampshire.
. Albert Schweitzer, "Declaration of Conscience," _Saturday Review_ , May 18, 1957, 20.
. Norman Cousins, interview with Franklin Jarlett, June 1, 1987, private collection; also Jarlett, _Robert Ryan_ , 108.
. Joseph Wershba, "Outspoken Actor," _New York Post_ , March 7, 1963.
. Nathanael West, _Miss Lonelyhearts and Day of the Locust_ (New York: New Directions, 1962), 1.
. Jeanne Stein, "Robert Ryan: Unlike Most Handsome Actors He Was Willing to Be a Heavy," _Films in Review_ 9, no. 1 (January 1968): 22.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Patricia Bosworth, _Montgomery Clift: A Biography_ (New York: Harcourt Brace Jovanovich, 1978), 300.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. James Kotsilibas-Davis and Myrna Loy, _Myrna Loy: Being and Becoming_ (New York: Knopf, 1987), 287.
. Stanley Kaufmann, "Far East and Far Off," review of _Lonelyhearts, New Republic_ , February 2, 1959, 21.
. Dwight Macdonald, "No Art and No Box Office," _Esquire_ (March 1959): 66.
. Pat O'Brien, _The Wind at My Back_ (Garden City, NY: Doubleday, 1964), 309.
. "Public, Not Producers, Guilty of Typing Actors: Bob Ryan," _Variety_ , May 23, 1958.
. "Sees Participation Deal as 'Income Roulette,'" _Motion Picture Herald_ , October 24, 1959.
. Anthony Slide, ed., _De Toth on De Toth: Putting the Drama in Front of the Camera_ (London: Faber and Faber, 1996), 142.
. Philip Yordan, interview with Franklin Jarlett, October 27, 1986, private collection.
. Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 103.
_eleven_ Beautiful Creatures
. Andy Harmon, telephone interview with author, May 17, 2012.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Jessica Ryan, "Woman: The Mythless American" (ca. 1970), private collection.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Cheyney Ryan, telephone interview with author, September 26, 2009.
. Robert Ryan, "I Didn't Want to Play a Bigot," _Ebony_ , (November 1959): 68–69.
. Don Alpert, "Ryan: 1 in 50 Critics Knows," _Los Angeles Times_ , December 3, 1961.
. Harry Belafonte, telephone interview with author, October 10, 2012.
. Ibid.
. Ryan, "I Didn't Want to Play a Bigot," 69.
. Robert Wise, interview with Franklin Jarlett, February 18, 1986, private collection.
. Shelley Winters, _Shelley II: The Middle of My Century_ (New York: Simon and Schuster, 1989), 263–64.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Robert Ryan, "The Full Text of Ryan's Letter," _Chicago Reader_ , October 29, 2009, <http://www.chicagoreader.com/chicago/actor-robert-ryans-letter-to-his-children/Content?oid=1223014>.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Lamont Johnson, interview with Franklin Jarlett, August 17, 1986, private collection; also Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 48.
. Sidney Skolsky, "An Interview with Robert Ryan," n.d., n.p., Sidney Skolsky Papers, Margaret Herrick Library Special Collections, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Bill Becker, "Focus on the Forty-Ninth State," _New York Times_ , September 27, 1959, sec. 2, p. 7.
. Skolsky, "An Interview with Robert Ryan."
. John L. Scott, "Ryan Lifts Self by His 'Heels,'" _Los Angeles Times_ , October 25, 1959, E1.
. "Film Stars Join in Nuclear Plea," _New York Times_ , October 20, 1959, 45.
. Ibid.
. Ibid.
. John Houseman, _Final Dress_ (New York: Simon and Schuster, 1982), 196.
. T. S. Eliot, _Murder in the Cathedral_ (New York: Harcourt Brace, 1935), 83.
. Philip K. Scheuer, "Eliot Verse Play Well Staged, Acted," review of _Murder in the Cathedral, Los Angeles Times_ , January 21, 1960, C9.
. Eliot, _Murder in the Cathedral_ , 49.
. Cecil Smith, "At the UCLA Theater Group, the Ordinary Was a Rarity," _Los Angeles Times_ , May 14, 1981, Q7.
. Harold Hildebrand, "Robert Ryan Speaks Out," _Los Angeles Examiner_ , October 11, 1961.
. Marsha Hunt, telephone interview with author, February 9, 2011.
. "Directed by John Frankenheimer: The Seminar," _Museum of Television and Radio Seminar Series_ , Paley Center for Media, Los Angeles, January 18, 1996. Video recording.
. "A Conversation with John Frankenheimer," Museum of Television and Radio Seminar Series, Paley Center for Media, Los Angeles, September 24, 1977. Video recording.
. "Directed by John Frankenheimer: The Seminar."
. Ibid.
. Ibid.
. Alpert, "Ryan: 1 in 50 Critics Knows."
. Harold Hildebrand, "Ryan's 'Between' Films," _Los Angeles Examiner_ , October 8, 1961.
. Bernard Eisenschitz, _Nicholas Ray: An American Journey_ (Minneapolis: University of Minnesota Press, 1990), 369.
. C. Gregory Jensen, "The Actor Who Will Play Christ," _Family Weekly_ , December 25, 1960, 11–13.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Jesse Zunser, "Stratford: Ryan, Hepburn, et al.," _Cue_ , July 30, 1960, 8.
. Raymond J. Buck, "A Rare Kind of Movie Star," _Dartmouth Varsity_ , October 1960, 30.
. Judith Crist, "Katharine Hepburn Stars in 'Antony and Cleopatra,'" _New York Herald Tribune_ , August 1, 1960, 8.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Rich, review of _The Canadians, Variety_ , March 8, 1961, 6.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Norman Cousins, interview with Franklin Jarlett, June 1, 1987, private collection.
. Milton S. Katz, _Ban the Bomb: A History of SANE, the Committee for a Sane Nuclear Policy, 1957–1985_ (New York: Greenwood Press, 1986), 61.
. Mike Metzger, telephone interview with author, March 9, 2011.
. Peter Ustinov, _Dear Me_ (Boston: Atlantic Monthly Press, 1977), 311.
. Herman Melville, _Billy Budd and The Piazza Tales_ (New York: Barnes and Noble Classics, 2006), 40.
. Melville, _Billy Budd_ , 41–42.
. Louis O. Coxe and Robert Chapman, _Billy Budd_ (New York: Hill and Wang, 1962), 30, 33.
. Terence Stamp, audio commentary, _Billy Budd_ , Warner Bros., 2007, DVD.
. Hank Werba, "'Billy Budd' Budget Item: Dramamine," _Variety_ , June 28, 1961, 17.
_twelve_ The Longest Day
. _Milwaukee Journal_ , January 13, 1963.
. Lisa Ryan, telephone interview with author, May 4, 2010.
. Cheyney Ryan, telephone interview with author, October 17, 2013.
. Joseph Wershba, "Outspoken Actor," _New York Post_ , March 7, 1963.
. Marsha Hunt, telephone interview with author, February 9, 2011.
. "Actor Threatened with Bomb Attack," _Hollywood Citizen-News_ , February 5, 1962.
. Ibid.
. Philip Dunne, _Take Two: A Life in Movies and Politics_ (New York: Limelight Editions, 1992), 268.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Sue Chambers, "Robert Ryan—Heavy and Hero," _Milwaukee Journal_ , January 13, 1963.
. Dorothy Manners, "His Knees Knock in Broadway Tempo," _Los Angeles Herald-Examiner_ , July 8, 1962.
. Joshua Logan, _Movie Stars, Real People, and Me_ (New York: Delacorte Press, 1978), 173–74.
. Bob Lardine, "This Dove Is a Tough Bird," _New York Sunday News_ , August 27, 1967, 4.
. Logan, _Movie Stars, Real People, and Me_ , 177.
. Les Carpenter, "'Mr. Prez' Leaves D.C. Better Than When JFK Saw It," _Variety_ , October 17, 1962.
. Cornelia Otis Skinner, _Life with Lindsay and Crouse_ (Boston: Houghton Mifflin, 1985), 226.
. Jessica Ryan, _America—Dream or Nightmare?_ (ca. 1970), private collection.
. Harry Belafonte, telephone interview with author, October 10, 2012.
. Joe McCarthy, "Antic Arts: Robert Ryan" (ca. 1963), private collection.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. McCarthy, "Antic Arts: Robert Ryan."
. Robert Wallsten, interview with Franklin Jarlett, June 14, 1986, private collection; also Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 126.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Ibid.
. Harry Belafonte, telephone interview with author, October 10, 2012.
. "Establish Free Southern Theatre for Summer Sked in Jackson, Miss., Then Local Groups, Touring Units," _Variety_ , April 8, 1964, 79, 82.
. "November 22, 1963: In Search of an Answer," _Executive Action_ , Warner Bros., 2007, DVD.
. Lisa Ryan, interview with author, San Francisco, April 17, 2011.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Philip K. Scheuer, "Robert Ryan Lifts Veil on Yugoslavia," _Los Angeles Times_ , August 11, 1964.
. Ibid.
. Robert Ryan to chairman, Stars for SANE (ca. July 1964), Dore Schary Papers, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Millard Lampell, interview with Franklin Jarlett, March 10, 1987, private collection.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Scheuer, "Robert Ryan Lifts Veil on Yugoslavia."
. Harriet Parsons, "Battle-Scarred Ryan Is Home," _Los Angeles Herald-Examiner_ , December 12, 1965, 5. Parsons, daughter of gossip columnist Louella Parsons, earlier had produced Ryan's RKO feature _Clash by Night_ (1952).
. Lampell, interview with Franklin Jarlett.
. Philip Yordan, interview with Franklin Jarlett, October 27, 1986, private collection; also Jarlett, _Robert Ryan_ , 135.
. Lampell, interview with Franklin Jarlett; also Jarlett, _Robert Ryan_ , 90.
_thirteen_ One of the Boys
. Woody Strode and Sam Young, _Goal Dust: The Warm and Candid Memoirs of a Pioneer Black Athlete and Actor_ (Lanham, MD: Madison Books, 1990), 227.
. Douglass K. Daniel, _Tough as Nails: The Life and Films of Richard Brooks_ (Madison: University of Wisconsin Press, 2011), 166.
. Strode, _Goal Dust_ , 230.
. Phil Yordan, interview with Franklin Jarlett, October 27, 1986, private collection.
. Philip Dunne, interview with Franklin Jarlett, August 24, 1987, private collection; also Michael Winner, _Winner Take All: A Life of Sorts_ (London: Robson Books, 2004), 63.
. Gary Fishgall, _Against Type: The Biography of Burt Lancaster_ (New York: Scribner's, 1995), 244.
. Virginia E. Rodgers, "Robert Ryan: 'A Most Un-Actor-Like Actor,'" _New Bedford (Massachusetts) Standard-Times_ , July 22, 1965.
. Lisa Ryan, interview with author, San Francisco, April 17, 2011.
. Richard T. Stout, _People_ (New York: Harper and Row, 1970), 43.
. Jessica Ryan, notes for "Campaign–'52" (ca. 1970), private collection.
. Margaret McManus, "Robert Ryan Speaks Out on Reagan," _Bridgeport (Connecticut) Telegram_ , November 6, 1966.
. Jessica Ryan, notes for "Campaign–'52."
. Dwayne Epstein, _Lee Marvin: Point Blank_ (Tucson: Schaffer Press, 2013), 171.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Bert Reisfeld, _Wieder Wildwestfilme in Hollywood_ , radio broadcast (ca. November 1967), Margaret Herrick Library, Academy of Motion Picture Arts and Sciences, Beverly Hills, California. Audio recording.
. McManus, "Robert Ryan Speaks Out on Reagan."
. Ken Furie, "Robert Ryan on Everything but Birth Control," _Dartmouth_ , April 21, 1967.
. Jessica Ryan to Jean Renoir (ca. May 1967), Jean Renoir Papers, UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Philip Yordan, interview with Franklin Jarlett, October 27, 1986, private collection; also Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 142.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Philip Yordan, interview with Franklin Jarlett; also Jarlett, _Robert Ryan_ , 77.
. Robert Ryan to Dore Schary, August 5, 1967, Dore Schary Papers, Wisconsin Center for Film and Theater Research, Madison, Wisconsin.
. Furie, "Robert Ryan on Everything but Birth Control."
. Daniel Rosenthal, "Fifty Years at the Cutting Edge," _London Times_ , February 19, 1999, 35.
. Alvin Shuster, "Robert Ryan Plays _Othello_ Abroad for $150 a Week," _New York Times_ , October 18, 1967, 37.
. John Peter, "John Neville as Iago," review of _Othello, London Times_ , September 22, 1967, 7.
. John Peter, "Embattled Family," review of _Long Day's Journey into Night, London Times_ , September 28, 1967, 6.
. "Legitimate: Asides and Ad-Libs," _Variety_ , April 12, 1967, 70.
. Robert Ryan to Lisa Ryan (ca. November 1967), private collection.
. Bob Lardine, "This Dove Is a Tough Bird," _New York Sunday News_ , August 27, 1967, 4.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Millard Lampell, interview with Franklin Jarlett, March 10, 1987, private collection.
. Ibid.
. Lisa Ryan, interview with author, San Francisco, April 17, 2011.
. Lampell, interview with Franklin Jarlett; also Jarlett, _Robert Ryan_ , 144.
. Patricia Bosworth, "Robert Ryan: In Search of Action," _New York Times_ , June 1, 1969, sec. 2, pp. 1, 7.
. Hersh, telephone interview with author.
. Arthur Herzog, _McCarthy for President_ (New York: Viking Press, 1969), 10.
. "Unforeseen Eugene," _Time_ , March 22, 1968, 15.
. Hersh, telephone interview with author.
. Stout, _People_ , 177.
. Marshall Fine, _Bloody Sam: The Life and Films of Sam Peckinpah_ (New York: Donald I. Fine, 1991), 130.
. Movie Body Counts Boards, accessed September 10, 2012, http://moviebodycounts.proboards.com/index.cgi?board=finished&action=display&thread=629.
. Bob Thomas, _Golden Boy: The Untold Story of William Holden_ (New York: St. Martin's Press, 1983), 166.
. Robert Ryan to Jean Renoir (ca. August 1968), Jean Renoir Papers, UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Cheyney Ryan, telephone interview with author, September 26, 2009.
_fourteen_ My Good Bad Luck
. Jessica Ryan, "My Name Is Robert Ryan" (ca. 1969), private collection.
. Betty Friedan, _The Feminine Mystique: Twentieth Anniversary Edition_ (New York: Dell, 1983), 21.
. "Understanding and Coping with Anxiety—Rollo May," www.existentialanalysis.org.uk/assets/articles/Understanding_and_Coping_with_Anxiety_Rollo_May_transcription_Martin_Adams.pdf.
. Jessica Ryan, _Woman—The Mythless American_ (ca. 1970), private collection.
. Ibid.
. Ramona Lampell, telephone interview with author, May 18, 2012.
. Ibid.
. Millard Lampell, interview with Franklin Jarlett, March 10, 1987, private collection.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Lampell, interview with Franklin Jarlett.
. Harold J. Kennedy, _No Pickle, No Performance: An Irreverent Theatrical Excursion from Tallulah to Travolta_ (Garden City, NY: Doubleday, 1978), 136.
. Ibid.
. Ibid.
. Bob Lardine, "This Dove Is a Tough Bird," _New York Sunday News_ , August 27, 1967, 4.
. Lisa Ryan, interview with Franklin Jarlett, February 23, 1986, private collection.
. William MacAdams, _Ben Hecht: The Man Behind the Legend_ (New York: Charles Scribner's Sons, 1990), 107.
. Ben Hecht and Charles MacArthur, _The Front Page_ (New York: Samuel French, 1928), 30.
. Walter Kerr, "After 41 Years, It's Still Page One," _New York Times_ , May 25, 1969, D1.
. Patricia Bosworth, "Robert Ryan: In Search of Action," _New York Times_ , June 1, 1969.
. Lenore Rottenberg, "No Wonder," _New York Times_ , August 17, 1969, D23.
. Aljean Harmetz, "'Man Was a Killer Long Before He Served a God,'" _New York Times_ , August 31, 1969, D9.
. Darwin Willard, "Just a Movie," _New York Times_ , September 14, 1969, D36.
. Arnold M. Auerbach, "The Wildest Bunch of All," _New York Times_ , August 17, 1969, D11.
. Vincent Canby, "Violence and Beauty Mesh in 'Wild Bunch,'" _New York Times_ , June 26, 1969, 45.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 11, 2011.
. Joe McGinniss, _The Selling of the President_ (New York: Penguin Books, 1988), 240–41.
. Jessica Ryan, _America—Dream or Nightmare?: American Myths of Power, Aggression and Violence_ (ca. 1970), private collection.
. "Acts of Birth: Robert Ryan," _Films and Filming_ , March 1971, 26.
. William Otterburn-Hall, "A Good Bad Man Is Hard to Find," _Louisville Courier-Journal_ , June 7, 1970, E4.
. Arthur Miller, _Collected Plays 1964–1972_ , ed. Tony Kushner (New York: Library of America, 2012), 283.
. Ibid., 281.
. Ibid., 285.
. "The Harmony Game" (video documentary) on _Bridge over Troubled Water_ , Columbia/Legacy Video, 2011, CD/DVD.
. Otterburn-Hall, "A Good Bad Man Is Hard to Find."
. Michael Winner, _Winner Take All: A Life of Sorts_ (London: Robson Books, 2004), 143.
. Harry Belafonte, telephone interview with author, October 10, 2012.
. Jessica Ryan, "Campaign–'52, or A Camera's-Eye View from Two Odd Birds" (ca. 1970), private collection.
. Robert Wallsten, interview with Franklin Jarlett, May 1986, private collection.
. "Acts of Birth: Robert Ryan," _Films and Filming_ , March 1971, 26.
. Robert Ryan to Mr. Kemeny, March 23, 1971, Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
. Lisa Ryan, interview, "Lucy Talks Movies," www.podtech.net/home/3848/robert-ryan-and-dana-andrews-had-daughters.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Bob Thomas, "Robert Ryan Fights Back After Tragic Two Years," _Milwaukee Journal_ , August 25, 1972.
. James Naughton, telephone interview with author, July 18, 2012.
. Arvin Brown, interview with Franklin Jarlett, September 3, 1986, private collection; also Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland: 1990), 59.
. Eugene O'Neill, _Long Day's Journey into Night_ (New Haven: Yale University Press, 1956), 150.
. Brown, interview with Franklin Jarlett; also Jarlett, _Robert Ryan_ , 162.
. Naughton, telephone interview with author.
. Brown, interview with Franklin Jarlett.
. Walter Kerr, "Do the Tyrones Live Here?," _New York Times_ , May 2, 1971, D3.
. Arvin Brown, interview with Franklin Jarlett; also Franklin Jarlett, _Robert Ryan_ , 160.
. Naughton, telephone interview with author.
. Stacy Keach, telephone interview with author, July 18, 2012; Naughton, telephone interview with author.
. Arvin Brown, telephone interview with author, November 1, 2012.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Brown, interview with Franklin Jarlett.
. Robert Ryan, "The Next Time You Want to Do a Play" (ca. July 1971), private collection.
. Brown, interview with Franklin Jarlett; also Jarlett, _Robert Ryan_ , 162, 164, 87.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. "Ryan-Buddy Hackett for Mexican 'Quixote,'" _Variety_ , March 29, 1972, 38.
. Pete Hamill, "For Robert Ryan," _New York Post_ , July 13, 1973.
. Bob Thomas, "Robert Ryan Fights Back After Tragic Two Years," _Milwaukee Journal_ , August 25, 1972.
. Ramona Lampell, telephone interview with author, May 18, 2012.
_fifteen_ The Loneliest Place in Town
. Jean Renoir to Robert Ryan, May 25, 1972, Jean Renoir Papers, UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Mary Murphy, "Robert Ryan—A New Life on Borrowed Time," _Los Angeles Times_ , September 5, 1972, 1, 12.
. Robert Ryan to Jean Renoir, July 20, 1972, Jean Renoir Papers, UCLA Special Collections, University of California at Los Angeles, Los Angeles, California.
. Murphy, "Robert Ryan—A New Life on Borrowed Time."
. Franklin Jarlett, _Robert Ryan: A Biography and Critical Filmography_ (Jefferson, NC: McFarland, 1990), 164.
. Ramona Lampell, telephone interview with author, May 18, 2012.
. Jarlett, _Robert Ryan_ , 165.
. Millard Lampell, interview with Franklin Jarlett, March 10, 1987, private collection.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Lisa Ryan, interview with author, San Francisco, California, April 17, 2011.
. Arvin Brown, interview with Franklin Jarlett, September 3, 1986, private collection.
. Arvin Brown, telephone interview with author, November 1, 2012.
. Cheyney Ryan, interview with author, Eugene, Oregon, September 22, 2012.
. Edward Everett Hale, "The Man without a Country," _Atlantic_ , December 1863, <http://www.theatlantic.com/magazine/archive/1863/12/the-man-without-a-country/308751/?single_page=true>.
. Louis Sheaffer, _O'Neill: Son and Artist_ , vol. 2 (New York: Cooper Square Press, 1973), 492.
. Eugene O'Neill, _The Iceman Cometh_ (New York: Vintage International, 1999), 9.
. John Frankenheimer, interview with Franklin Jarlett, May 25, 1987, private collection; also Jarlett, _Robert Ryan_ , 165.
. "Jeff Bridges: The Dude Abides," Public Broadcasting System's _American Masters_ , January 12, 2011.
. Jeff Bridges, telephone interview with author, November 2, 2012.
. Arvin Brown, interview with Franklin Jarlett.
. Evans Frankenheimer, interview with Franklin Jarlett, May 30, 1987, private collection.
. John Frankenheimer to Lisa Ryan, December 14, 1973, private collection.
. Robert Ryan, "How Do You Remember All Those Words?" (ca. 1957), Jane Ardmore Papers, Margaret Herrick Library, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
. Bridges, telephone interview with author.
. O'Neill, _The Iceman Cometh_ , 23.
. Dwayne Epstein, _Lee Marvin: Point Blank_ (Tucson: Schaffner Press, 2013), 196.
. John Frankenheimer, interview with Franklin Jarlett; also Jarlett, _Robert Ryan_ , 168.
. O'Neill, _The Iceman Cometh_ , 9.
. Charles Champlin, "Robert Ryan: In Memoriam," _Los Angeles Times_ , July 12, 1973, sec. 4, p. 1.
. Philip Yordan, interview with Franklin Jarlett, October 27, 1986, private collection.
. "Phil Yordan into Tele Corp of America; Red Silverstein Sales Chief Beyond U.S.," _Variety_ , July 4, 1973, 14.
. Yordan, interview with Franklin Jarlett.
. Joe Don Baker, telephone interview with author, January 25, 2014.
. Champlin, "Robert Ryan."
. Albert Hackett, interview with Franklin Jarlett, August 1986, private collection.
. Sue Cameron, "Robert Ryan Signed for Broadway Star Role in 'Shenandoah,'" _Hollywood Reporter_ , June 29, 1973.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. "November 22, 1963: In Search of an Answer," _Executive Action_ , Warner Bros., 2007, DVD.
. Evans Frankenheimer, interview with Franklin Jarlett, May 30, 1987, private collection.
. John Frankenheimer, interview with Franklin Jarlett.
. Larry Goebel, "Robert Ryan," _Los Angeles Times_ , July 18, 1973, sec. 2, p. 6.
. Cheyney Ryan, interview with author, Eugene, Oregon, April 15, 2011.
. Mary Murphy, "Robert Ryan—A New Life on Borrowed Time," _Los Angeles Times_ , September 5, 1972, 1, 12.
. "Chatter: Broadway," _Variety_ , July 18, 1973, 109.
. Millard Lampell, interview with Franklin Jarlett, March 10, 1987, private collection.
. "Private Services for Robert Ryan, Burial in Chicago," _Hollywood Reporter_ , July 13, 1973.
. "Robert Ryan Estate Left to Children," _Los Angeles Times_ , July 22, 1973, sec. 1, p. 15.
. Pete Hamill, "For Robert Ryan," _New York Post_ , July 13, 1973.
. Champlin, "Robert Ryan."
. Paul D. Zimmerman, _Newsweek_ , November 12, 1973.
. "Film Critics Soc. Award for Ryan," _Hollywood Reporter_ , January 30, 1974.
. Marie Spottswood, "Such a Resource in One Individual," in _A Memorial Tribute for the Robert and Jessica Ryan Memorial_ , November 3, 1974, Oakwood School archives, North Hollywood, California.
. Dartmouth College, Twenty-five-Year Report Biographical Data Sheet (ca. 1956), Dartmouth Alumni File, Dartmouth College, Hanover, New Hampshire.
Selected Bibliography
_Books_
Bacher, Lutz. _Max Ophuls in the Hollywood Studios_. Brunswick, NJ: Rutgers University Press, 1996.
Bacon, Margaret Hope. _Mothers of Feminism: The Story of Quaker Women in America_. San Francisco: Harper and Row, 1986.
Baker, Roy Ward. _The Director's Cut: A Memoir of 60 Years in Film and Television_. London: Reynolds and Hearn, 2000.
Bankhead, Tallulah. _Tallulah: My Autobiography_. New York: Harper and Brothers, 1952.
Bernstein, Walter. _Inside Out: A Memoir of the Blacklist_. New York: Knopf, 1996.
Biles, Roger. _Big City Boss in Depression and War: Mayor Edward J. Kelly of Chicago_. DeKalb: Northern Illinois University Press, 1984.
——. "Edward J. Kelly: New Deal Machine Builder," in _The Mayors: The Chicago Political Tradition_ , ed. Paul M. Green and Melvin G. Holli. Carbondale and Edwardsville: Southern Illinois University Press, 1995.
Block, Libbie. _Wild Calendar_. New York: Knopf, 1946.
Bogdanovich, Peter. _Fritz Lang in America_. New York: Praeger, 1969.
Borgnine, Ernest. _Ernie: The Autobiography_. New York: Citadel Press, 2005.
Bosworth, Patricia. _Montgomery Clift: A Biography_. New York: Harcourt Brace Jovanovich, 1978.
Brian, Denis. _Tallulah, Darling: A Biography of Tallulah Bankhead_. New York: Macmillan, 1972.
Brooks, Richard. _The Brick Foxhole_. New York: Harper and Brothers, 1945.
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_Periodicals_
"Actor Robert Ryan's Mother Injured by Auto." _Los Angeles Times_ , February 2, 1957.
"Actor Threatened with Bomb Attack." _Hollywood Citizen-News_ , February 5, 1962.
"Actor to Build Vacation Home in Holderness." _Lakes Region Trader_ , November 6, 1968.
"Actor's Son Gets Jail Term, Fine." _Los Angeles Herald-Examiner_ , November 11, 1970.
"Acts of Birth: Robert Ryan." _Films and Filming_ , March 1971.
"A-Hunting We Will Go." _Screen Guide_ 5, no.3 (November 1947).
"Amateur Critics Lift Ryan to Stardom with 'Rave Cards.'" _New York Herald Tribune_ , May 28, 1944.
"At Home with Robert Ryan." _New York Sunday News_ , September 25, 1960.
"'B'Way for Peace' Show Realizes over $100,000." _Variety_ , January 24, 1968.
"Bob Ryan in Boxing Roles." _Boston Sunday Advertiser_ , December 8, 1957.
"Bob Ryan, KPFK Get Threats." _Los Angeles Herald-Examiner_ , February 5, 1962.
"Bob Ryan May Sell." _Variety_ , December 24, 1958.
"Brady, Hastings to Stage 'Town,' 'Page' for Plumstead, Mineola." _Variety_ , August 14, 1968.
"Broadway Hit Comes to TV." _Los Angeles Times_ , January 30, 1970.
"Caldwell Novel for UA Roster." _Variety_ , January 30, 1957.
"Camera Debut." _Los Angeles Times_ , August 28, 1952.
"'The Canadians' under Eady Plan." _Variety_ , November 30, 1960.
"Chatter: Broadway." _Variety_ , July 18, 1973.
"Chin-Up Guy." _Movie Stars Parade_ 4 no.5 (April 1944): 40.
"Coroner's Jury Goes through Death Tunnel." _Chicago Daily News_ , April 15, 1931.
"'Crossfire': Thriller with a Dynamite Punch." _Cue_ , July 26, 1947.
"Death Takes Mother of Robert Ryan." _Los Angeles Times_ , March 15, 1963.
"Establish Free Southern Theatre for Summer Sked in Jackson, Miss., Then Local Groups, Touring Units." _Variety_ , April 8, 1964.
"Expansion Program on at Oakwood." _San Fernando Valley Mirror-News_ , May 30, 1957.
"Extraordinary Properties on the Market." _Architectural Digest_ , March 1998.
"Film Critics Soc. Award for Ryan." _Hollywood Reporter_ , January 30, 1974.
"Film Stars Join in Nuclear Plea." _New York Times_ , October 20, 1959.
"Films' Biggest Mystery—RKO." _Variety_ , February 20, 1952.
"Guthrie Memorial Salute Nets 7-1/2 G from 2 SRO Perfs. at Carnegie Hall." _Variety_ , January 24, 1968.
"Harmon-Mann Solicit No Seal for 'God's Little Acre,' Tell Why." _Variety_ , April 3, 1957.
"Hollywoodites Join in Study of Actor Lore; Yen Stage Presentation." _Variety_ , February 26, 1958.
"Hughes' Commie Blast Viewed as Cue to Exit RKO; Mayer Report Up Again." _Variety_ , April 9, 1952.
"Hush-Filming on Kennedy Killing." _Variety_ , June 6, 1973.
"Inside Stuff—Pictures." _Variety_ , December 7, 1949.
"International: International Sound Track." _Variety_ , December 22, 1971.
"John Beal, Carol Teitel to Expand 'Journey' Sked." _Variety_ , July 14, 1971.
"John Ryan, Pioneer Joliet Merchant, Dies." _Chicago Tribune_ , May 27, 1919.
"Jug Bob Ryan's Son." _Variety_ , November 11, 1970.
"'Juno' Will Aid Ryan Memorial." _Hollywood Reporter_ , October 8, 1974.
"Just for Variety." _Variety_ , April 19, 1973.
"Just for Variety." _Variety_ , April 26, 1973.
"Just for Variety." _Variety_ , May 1, 1973.
"The Kelly-Nash Political Machine." _Fortune_ 14, no. 1 (August 1936): 47–126.
"Legitimate: Asides and Ad-Libs." _Variety_ , April 12, 1967.
"Legitimate: N.Y. Critics' Poll Results." _Variety_ , July 21, 1971.
"Leone Prepping $5,000,000 Saga." _Variety_ , January 10, 1968.
"Living the Life of Ryan." _Screen Guide_ , January 1950.
"Little Chance to Fix Tunnel Disaster Blame." _Chicago American_ , April 15, 1931.
"Long Lunches, Cocktails, Censorship but Spain Grows as Show Biz Center." _Variety_ , June 6, 1960.
"Many Top Names Asked to Explain." _Variety_ , May 21, 1952.
"Marie Spottswood: Guided Oakwood School Growth." _Los Angeles Times_ , June 26, 1987.
"Movieland Events." _Los Angeles Times_ , April 2, 1955.
"Movies Are Put in Essential Class by Draft Ruling." _New York Times_ , February 9, 1942.
"MPC, Plumstead Tie on Teleplay Specs with Xerox Backing." _Variety_ , July 16, 1969.
"'Mr. President' Spurns More Boston Orders." _Variety_ , August 1, 1962.
"N.Y. Critics Rave: 'Billy Budd' Break for Allied Artists." _Variety_ , November 7, 1962.
"New 'Front Page' to Cost $20,000." _Variety_ , April 9, 1969.
"One Minute Interviews." _Hollywood Citizen-News_ , November 5, 1953.
"Opinions of an Actor." _Hollywood Citizen-News_ , January 23, 1948.
"Parade in Loop Launches Movie about Marines." _Chicago Tribune_ , August 14, 1951.
"Pare Lorentz Picks American Odysseus." _Brooklyn Eagle_ , July 14, 1942.
"Phil Yordan into Tele Corp of America: Red Silverstein Sales Chief beyond U.S." _Variety_ , July 4, 1973.
"Plumstead Group Aims 3 Revivals for Next Season." _Variety_ , May 21, 1969.
"Posthumous Award Given to Actor Robert Ryan." _Boxoffice_ , February 4, 1974.
"Private Services for Robert Ryan, Burial in Chicago." _Hollywood Reporter_ , July 13, 1973.
"Public, Not Producers, Guilty of Typing Actors: Bob Ryan." _Variety_ , May 23, 1958.
"Radio Station, Actor's Home Guarded after Bomb Threat." _Los Angeles Times_ , February 6, 1962.
"Robert Ryan Estate Left to Children." _Los Angeles Times_ , July 22, 1973.
"Robert Ryan Pilot for NBC-TV Rolling at U." _Variety_ , October 21, 1964.
"Robert Ryan Speaks." _Los Angeles Daily News_ , October 22, 1947.
"Robert Ryan to Appear with British Repertory." _Hollywood Reporter_ , February 13, 1967.
"Robert Ryan's Advice to Would-Be Actors." _Salt Lake City Deseret News_ , November 30, 1951.
"Robert Ryan's Spiel." _Variety_ , January 22, 1964.
"Rugged Ryan." _Movie Play_ , January 1947.
"Ryan—Buddy Hackett for Mexican 'Quixote.'" _Variety_ , March 29, 1972.
"Ryan for 'Glory.'" _Variety_ , November 4, 1947.
"Ryan Project Will Benefit." _Los Angeles Times_ , October 23, 1974.
"Salute of the Week: Robert Ryan." _Cue_ , May 10, 1969.
"Saskatchewan Socialist Backing for Mounties' Film with Bob Ryan." _Variety_ , September 28, 1960.
"Savalas Inks 3-Film Deal with Scotia Int'l." _Variety_ , July 7, 1971.
"Search for 'Typical American' for Screen a Six Months' Job." _New York Herald Tribune_ , April 19, 1942.
"See 'Set-Up' Scoring Close Verdict over 'Champion' in Suit." _Variety_ , May 11, 1949.
"Sees Participation Deal as 'Income Roulette.'" _Motion Picture Herald_ , October 24, 1959.
"Set Guthrie Memorial at Carnegie Hall, N.Y." _Variety_ , January 10, 1968.
"Show Biz Names from Paris, B'Way and H'wood Perform at D.C. March." _Variety_ , September 4, 1963.
"Special Draft Deferment Creates Furor in Hollywood." _Los Angeles Times_ , February 11, 1942.
"Teachers Learn from Students at Hollywood Private School." _Pasadena Star-News_ , September 4, 1957.
"Threaten to Bomb Bob Ryan's Home." _Variety_ , February 5, 1962.
"Timothy Ryan, Contractor, Dies at 61; Rites Tomorrow." _Chicago Tribune_ , April 28, 1936.
"12 Dead, 16 Saved, in Tunnel." _Chicago Daily News_ , April 14, 1931.
"21 Rescued in Disaster." _Chicago Evening Post_ , April 14, 1931.
"UCLA Theatre Division Plans 3 Play Evenings." _Variety_ , July 20, 1960.
"Understanding and Coping with Anxiety—Rollo May." Audio recording, _Psychology Today_ , 1978. www.existentialanalysis.org.uk/assets/articles/Understanding_and_Coping_with_Anxiety_Rollo_May_transcription_Martin_Adams.pdf.
"Unforeseen Eugene." _Time_ , March 22, 1968.
"Unknown British Actor Playing Billy Budd." _Variety_ , June 7, 1961.
"Wald-Krasna Unit Still Stalled at RKO; Former Ill." _Variety_ , March 19, 1952.
"Youngstein's Shuttle; 7 Properties Shape." _Variety_ , September 30, 1964.
"Yugoslavia Sets Coproductions." _Variety_ , July 8, 1964.
Alpert, Don. "Ryan: 1 in 50 Critics Knows." _Los Angeles Times_ , December 3, 1961.
Auerbach, Arnold. "The Wildest Bunch of All." _New York Times_ , August 17, 1969.
Becker, Bill. "Focus on the Forty-Ninth State." _New York Times_ , September 27, 1959.
Benjamin, George. "What a Man!" _Modern Screen_ , October 1944.
Berg, Louis. "Gentle Irishman." _Los Angeles Times_ , February 4, 1951.
Bergstrom, Janet. "Oneiric Cinema: _The Woman on the Beach." Film History_ 11 (1999): 114–25.
Bosworth, Patricia. "Robert Ryan: In Search of Action." _New York Times_ , June 1, 1969.
Breslin, Howard. "Bad Time at Honda." _American Magazine_ , January 1947.
Brogan, Phil F. "Valley of No Return Doesn't Stop Brogan." _Bend Bulletin_ , November 12, 1958.
Buck, Raymond J. "A Rare Kind of Movie Star." _Dartmouth Varsity_ , October 1960.
Cameron, Sue. "Robert Ryan Signed for Broadway Star Role in 'Shenandoah.'" _Hollywood Reporter_ , June 29, 1973.
Carpenter, Les. "Lincoln Program Dramatic Opener for Famous Ford's Theatre, Wash." _Variety_ , February 7, 1968.
——. "'Mr. Prez' Leaves D.C. Better Than When JFK Saw It." _Variety_ , October 17, 1962.
Chambers, Sue. "Robert Ryan—Heavy and Hero." _Milwaukee Journal_ , January 13, 1963.
Champlin, Charles. "Robert Ryan: In Memoriam." _Los Angeles Times_ , July 12, 1973.
Churchill, Reba, and Bonnie Churchill. "Mister Velvet." _Movie Stars Parade_ , April 1948.
Coe, Richard. "Visitor Ryan." _Washington Post and Times Herald_ , October 20, 1956.
Cohen, Elliot E. "Letter to the Movie-Makers: The Film Drama as a Social Force." _Commentary_ 4, no. 2 (August 1947).
Cousins, Norman. "Modern Man Is Obsolete." _Saturday Review_ , August 18, 1945.
Crist, Judith. "Katharine Hepburn Stars in 'Anthony and Cleopatra.'" _New York Herald Tribune_ , August 1, 1960
Cuffs, John. "Robert Ryan: Villain Extraordinary." _Films and Filming_ , July 1961.
Dudley, Fredda. "Romancing with Ryan." _Photoplay_ , April 1944.
Fellig, Arthur (Weegee). Column for _Los Angeles Mirror_ , November 26, 1948.
Friedman, Lester D. "A Very Narrow Path: The Politics of Edward Dmytryk." _Literature/Film Quarterly_ 12, no. 4 (October 1984): 214–24. In Elsie M. Walker and David T. Johnson, _Conversations with Directors_. Lanham, MD: Scarecrow Press, 2008.
Funke, Lewis. "Robert Ryan's 'Journey.'" _New York Times_ , March 7, 1971.
Furie, Ken. "Robert Ryan on 'Everything but Birth Control.'" _Dartmouth_ , April 21, 1967.
Gent, George. "Ryan Sees Something of Himself in O'Neill's People." _New York Times_ , April 5, 1971.
Goebel, Larry. "Robert Ryan." _Los Angeles Times_ , July 18, 1973.
Granet, Bert. "Berlin Express Diary." _Screen Writer_ , May 1948.
Grant, Ila S. "Ila Finds That Movie Making Is Hard Work." _Bend Bulletin_ , November 22, 1958.
Greenberg, Abe. "Rugged Robert Ryan Speaks Up." _Hollywood Citizen-News_ , July 19, 1966.
Grenquist, Peter C. "'Stranger Than Fiction' Ryan Stops Changing." _Dartmouth_ , March 17, 1950.
Hamill, Pete. "For Robert Ryan." _New York Post_ , July 13, 1973.
Harmetz, Aljean. "'Man Was a Killer Long before He Served a God.'" _New York Times_ , August 31, 1969.
Harris, Radie. "Broadway Ballyhoo." _Hollywood Reporter_ , June 17, 1969.
Heyn, Howard C. "Friend of the Underdog: Bob Ryan Favors Stories about Hard-Knock People." Associated Press [ca. 1949]. Robert Ryan file, Margaret Herrick Library, Beverly Hills, California.
Hildebrand, Harold. "Robert Ryan Speaks Out." _Los Angeles Examiner_ , October 11, 1961.
——. "Ryan's 'Between' Films." _Los Angeles Examiner_ , October 8, 1961.
Hopper, Hedda. "Robert Ryan Awaits Stork's Third Visit." _Los Angeles Times_ , August 25, 1951.
Hunt, Gerry. "Maureen O'Sullivan Tells of Her Secret Romance with the Late Robert Ryan." _National Enquirer_ , December 23, 1973.
Jensen, C. Gregory. "The Actor Who Will Play Christ." _Family Weekly_ , December 25, 1960.
Johnson, Arnold [photog]. "Rugged!" _Movie Life_ , August 1948.
Johnson, Erskine. "Lazy Newcomers Irk Ryan." _Los Angeles Mirror News_ , December 2, 1959.
——. Untitled column. _Los Angeles Daily News_ , July 7, 1951.
Jones, J. R. "The Actor's Letter." _Chicago Reader_ , October 29, 2009.
Kerr, Walter. "After 41 Years, It's Still Page One." _New York Times_ , May 25, 1969.
——. "Do the Tyrones Live Here?" _New York Times_ , May 2, 1971.
Kupcinet, Irv. "Kup's Column." _Chicago Sun-Times_ , April 7, 1949, 45.
Lardine, Bob. "This Dove Is a Tough Bird." _New York Sunday News_ , August 27, 1967.
Lightman, Herb A. "The Story of Filming 'Berlin Express.'" _American Cinematographer_ 29, no. 7 (July 1948).
Lowe, Herman A. "Big D.C. Whodunit: Who Killed That Red Probe? See Resumption in Dec." _Variety_ , November 5, 1947.
Nichols, M. "Robert Ryan—Hero and Heel." _Coronet_ 47, no.16 (January 1960).
Macdonald, Dwight. "No Art and No Box Office." _Esquire_ , March 1959.
MacPherson, Virginia. "Why Shouldn't the Kids Be Actors?—Ryan." _Los Angeles Daily News_ , January 1, 1951.
Manners, Dorothy. "His Knees Knock in Broadway Tempo." _Los Angeles Herald-Examiner_ , July 8, 1962.
——. "Ryan's Courage Faces Sorrow." _Los Angeles Herald-Examiner_ , July 24, 1972.
McClay, Howard. "He Didn't Look Like an Actor—But." _Los Angeles Daily News_ , May 20, 1952.
——. "It's Always Nice to Have Something Going for You." _Los Angeles Daily News_ , June 30, 1953.
——. "Shakespeare Summons Bob Ryan." _Los Angeles Daily News_ , March 24, 1954.
McManus, Margaret. "Robert Ryan Speaks Out on Reagan." _Bridgeport (Connecticut) Telegram_ , November 6, 1966.
McMurtry, Charles. "Ex-Dartmouth Athlete Stars in First Movie Role." _Worcester Telegram_ , May 12, 1942.
Meyers, Joe. "Whatever Happened to Robert Ryan?" _Connecticut Post_ , August 28, 1994.
Moskowitz, Gene. "Serge Silberman: France to Quebec." _Variety_ , October 20, 1971.
Moss, Morton. "Buoyancy and Bounce." _Los Angeles Herald-Examiner_ , January 29, 1970.
Murphy, Mary. "Robert Ryan—A New Life on Borrowed Time." _Los Angeles Times_ , September 5, 1972.
Nogueira, Rui, and Nicoletta Zalaffi. "A Bastard's Long Career: Meeting with Robert Ryan." _Cinema_ 70 (April 1970). Translation by Kelsey Beson.
Oppenheimer, Peer J. "A Film Hero Fights for Better Schools." _New Haven Sunday Register_ , August 7, 1960.
Parsons, Harriet. "Battle-Scarred Ryan Is Home." _Los Angeles Herald-Examiner_ , December 12, 1965.
Parsons, Louella O. "In Hollywood with Louella O. Parsons." _Los Angeles Examiner_ , February 29, 1948.
——. "Robert Ryan: Nice Man to Have around the Movies." _Los Angeles Examiner_ , February 10, 1952.
Roberts, Chalmers M. "Adlai Calls for All-Out Final Drive." _Washington Post and Times Herald_ , October 21, 1956.
Rode, Alan K. "'First Is First and Second Is Nobody': The Philip Yordan Story." _Noir City Sentinel_ , November-December 2009.
Rodgers, Virginia E. "Robert Ryan: 'A Most Un-Actor-Like Actor.'" _New Bedford_ ( _Massachusetts) Standard-Times_ , July 22, 1965.
Ryan, Jessica. "Marine Ryan." _Movieland_ 4, no. 9 (October 1944). Ryan, Robert. "Acts of Birth." _Films and Filming_ , March 1971.
Ryan, Robert, as told to Naomi Engelsman. "Backstage with Us Ryans." _Parents' Magazine_ , September 1951.
Ryan, Robert. "'Front Page' Puts Him on Broadway." _Hollywood Citizen-News_ , May 22, 1969.
——. "I Didn't Want to Play a Bigot." _Ebony_ , November 1959.
——. "I'm Gambling with My Career." _Movieland_ , August 1947.
——. "'Just an Actor,' Says Robert Ryan." _Hollywood Citizen-News_ , February 18, 1970.
——. "My Role in Crossfire." _Worker_ , July 20, 1947. Southern edition.
——. "The Role I Liked Best..." _Saturday Evening Post_ , July 15, 1950.
——. "We're Not Quitting!" _Screen Guide_ , May 1948.
——. "What Is Proper Speech?" _New York Times_ , October 24, 1965.
Ryan, Robert, as told to Loney, Glenn. "In the Words of Robert Ryan." _Cue_ , January 11, 1970.
Ryan, Robert, as told to Pine, Dick. "The Trouble with Me Is..." _Movieland_ , June 1951.
Schary, Dore. "Letter from a Movie-Maker: 'Crossfire' as a Weapon against Anti-Semitism." _Commentary_ , October 1947.
Scheuer, Philip K. "Robert Ryan Lifts Veil on Yugoslavia." _Los Angeles Times_ , August 11, 1964.
Schumach, Murray. "Art's Call Heard by Robert Ryan." _New York Times_ , February 5, 1960.
Schweitzer, Albert. "Declaration of Conscience." _Saturday Review_ , May 18, 1957.
Scott, John L. "Good or Bad; Robert Ryan Plays Either." _Los Angeles Times_ , August 10, 1947.
——. "Play Features All-Star Cast." _Los Angeles Times_ , January 27, 1957.
——. "Ryan Lifts Self by His 'Heels.'" _Los Angeles Times_ , October 25, 1959.
Scott, Vernon. "Ryan Works Again Despite Cancer." _Los Angeles Herald-Examiner_ , August 27, 1972.
Shuster, Alvin. "Robert Ryan Plays _Othello_ Abroad for $150 a Week." _New York Times_ , October 18, 1967.
Skolsky, Sidney. "Tintypes." _Hollywood Citizen-News_ , October 16, 1947.
——. "Tintypes." _Hollywood Citizen-News_ , June 24, 1954; _New York Post_ , June 27, 1954.
——. "Tintypes." _Hollywood Citizen-News_ , October 8, 1959.
Smallwood, Frank. "Ryan '32 Tops Career as _Crossfire's_ Killer." _Dartmouth_ , January 28, 1948.
Smith, Cecil. "At the UCLA Theater Group, the Ordinary Was a Rarity." _Los Angeles Times_ , May 14, 1981.
——. "Medicine for 'Tired and Shabby' Theater." _Los Angeles Times_ , January 22, 1961.
Smith, Darr. Untitled column. _Los Angeles Daily News_ , February 17, 1949.
Stein, Jeanne. "Robert Ryan: Unlike Most Handsome Actors He Was Willing to Be a Heavy." _Films in Review_ 9, no. 1 (January 1968).
Stinson, Charles. "UCLA to Present Eliot Masterpiece." _Los Angeles Times_ , January 17, 1960.
Surtees, Robert. "The Story of Filming Act of Violence.'" _American Cinematographer_ , August 1948.
Thomas, Bob. "Male Cheesecake! Robert Ryan Comments on New Trend." _Hollywood Citizen-News_ , July 11, 1949.
——. "Tots' School Begun by Film Star Thrives." _Los Angeles Mirror News_ , July 24, 1958.
Vallee, William Lynch. "Movie Life of Robert Ryan." _Dartmouth Varsity_ , ca. 1949.
Walker, Helen Louise. "Portrait of a Happy Man." _Movieland_ , December 1949.
Werba, Hank. "'Billy Budd' Budget Item: Dramamine." _Variety_ , June 28, 1961.
——. "Dmytryk Repeats History as 'Anzio' Restaged Yanks' Liberation of Rome." _Variety_ , August 23, 1967.
Wershba, Joseph. "Outspoken Actor." _New York Post_ , March 7, 1963.
Wheatley, James H. "Boxing Team to 'Set-up' Is Circuit for Ryan '32." _Dartmouth_ , April 30, 1949.
Whitman, Alden. "Robert Ryan, Actor, Dies at 63." _New York Times_ , July 12, 1973.
Williams, Dick. "Over Hill, Dale, A-Golfing We Go with Slicer Robert Ryan." _Los Angeles Mirror_ , February 8, 1950.
Williams, Whitney. "Be Smart: Send Screen Stars to Japan for Personals, Protect U.S. Films—Bob Ryan's Advice." _Variety_ , March 30, 1955.
Zolotow, Sam. "'Front Page' Here as Benefit in May." _New York Times_ , January 8, 1969.
——. "New Group Forms at Theater on L.I." _New York Times_ , August 1, 1968.
——. "The Plumstead to Offer in Fall 3 Pulitzer Plays." _New York Times_ , May 21, 1969.
Zunser, Jesse. "Stratford: Ryan, Hepburn, et al." _Cue_ , July 30, 1960.
Zylstra, Freida. "From Chicago Sandhog to Hollywood Star: Robert Ryan." _Chicago Tribune_ , July 19, 1950.
_Unpublished Manuscripts_
Harmon, Sidney. "Robert Ryan."
Jarlett, Franklin. Interview transcriptions for _Robert Ryan: A Biography and Critical Filmography_ : Arvin Brown (September 3, 1986), Norman Cousins (June 1, 1987), Philip Dunne (June 12 and August 24, 1987), Evans Frankenheimer (May 30, 1987), John Frankenheimer (May 25, 1987), Albert Hackett (August 1986), John Houseman (spring 1986), Lamont Johnson (August 17, 1986), Harold J. Kennedy (March 20, 1986), Millard Lampell (March 10, 1987), Harold Mayer (August 1986), Cheyney Ryan (February 20 and March 15, 1986), Lisa Ryan (February 23, 1986, and September 1, 1987), Timothy Ryan (February 2 and March 20, 1986; June 6 and August 26, 1987), Virginie van Bark (March 5, 1986), Robert Wallsten (May 1986), Robert Wise (February 18, 1986), and Philip Yordan (October 27, 1986). In the case of discrepancies between the transcribed interviews and Jarlett's published text, priority was given to the transcriptions.
McCarthy, Joe. "Antic Arts: Robert Ryan." 1963.
Ryan, Jessica. "America—Dream or Nightmare?: American Myths of Power, Aggression and Violence," ca. 1970.
——. "Campaign–'52, or A Camera's-Eye View from Two Odd Birds," notes and text, ca. 1970.
——. "If School Keeps," ca. 1970.
——. "Recollections of a Pioneer Grandmother," ca. 1970.
——. "Woman: The Mythless American," outline and text, ca. 1970.
Ryan, Robert. "How Do You Remember All Those Words?," n.d. Jane Ardman Papers, Margaret Herrick Library, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
——. "The Next Time You Want to Do a Play." 1971.
——. Untitled essay on American culture. 1968.
——. "What Makes an Actor Tick?," n.d. Jane Ardman Papers, Margaret Herrick Library, Academy of Motion Picture Arts and Sciences, Beverly Hills, California.
_Selected Archival Resources_
DARTMOUTH COLLEGE LIBRARY, DARTMOUTH COLLEGE, HANOVER, NEW HAMPSHIRE.
Dartmouth Boxing Club file: (1) Robert Ryan to Corey Ford, July 24, 1957. (2) Robert Ryan to Corey Ford, October 28, 1957.
Robert Ryan alumnus file: (1) Robert Ryan to A. J. Dickerson, mid-1945. (2) Five excerpts from Dartmouth College Class of 1932 newsletter, 1940–1947. (3) Excerpt from Dartmouth College Class of 1932 newsletter, March 19, 1962. (4) Dartmouth alumni office biographical data sheet, ca. 1956. (5) Dartmouth alumni records office questionnaire, ca. September 1967. (6) Robert Ryan to Mr. Kemeny of Dartmouth College alumni office, March 23, 1971. (7) Dartmouth College alumni records office questionnaire, ca. 1972.
MARGARET HERRICK LIBRARY, ACADEMY OF MOTION PICTURES ARTS AND SCIENCES, BEVERLY HILLS, CALIFORNIA.
Audiovisual archive: (1) Robert Ryan, interview with Tony Thomas, audio recording, ca. 1960. (2) Robert Ryan, interview with Bert Reisfeld for German radio program _Wieder Wildwestfilme in Hollywood_ , ca. 1967.
Bert Granet Papers: Granet, unaddressed letter draft regarding production of _Berlin Express_ , n.d.
Hedda Hopper Papers: Robert Ryan, interview with Hedda Hopper for column promoting _King of Kings_ , June 14, 1960.
History of Cinema: Hollywood and the Production Code collection: Stephen S. Jackson to David Hopkins of Enterprise Productions regarding _Caught_ , February 12, 1948.
John Huston Papers: (1) SANE letter from Robert Ryan and Steve Allen to Huston, February 9, 1960. (2) SANE document, "Suggestions for Platform Hearings of the Democratic National Committee," July 7, 1960.
John Paxton Papers: (1) Clay Steinman to Paxton, June 29, 1977. (2) Clay Steinman and Keith Kelly, interview questions for Paxton, July 1, 1977. (3) Paxton to Clay Steinman and Keith Kelly, July 14, 1977. (4) Paxton to Clay Steinman, Keith Kelly, and Mario Falsetto, n.d.
John Sturges Papers: Sturges to cast and crew of _The Law and Tombstone_ (aka _Hour of the Gun_ ), October 15, 1966.
Lincoln Quarberg Papers: Congressional statement of Congressman Richard M. Nixon on the release of _Flying Leathernecks_ , August 27, 1951.
Motion Picture Association of America Production Code Administration Files: (1) Joseph I. Breen to William Gordon of RKO Radio Pictures regarding _The Woman on the Beach_ , March 28, 1945. (2) Joseph I. Breen to Harold Melniker of RKO Radio Pictures regarding _On Dangerous Ground_ , March 20, 1950; March 23, 1950; and May 2, 1950. (3) Geoffrey M. Shurlock to Hal Wallis of Paramount Pictures regarding _About Mrs. Leslie_ , June 17, 1954. (4) Joseph I. Breen to Hal Wallis regarding _About Mrs. Leslie_ , August 20, 1953. (5) Geoffrey M. Shurlock to Hal Wallis regarding _About Mrs. Leslie_ , August 24, 1953.
Sidney Skolsky Papers: Interview with Robert Ryan for "Tintypes" column to promote _The Ice Palace_ , ca. January 1960.
OAKWOOD SCHOOL, NORTH HOLLYWOOD, CALIFORNIA.
1. Robert Ryan to parents in North Hollywood and Studio City, California, January 1951.
2. Material pertaining to background of the Ryan Center, The Robert and Jessica Ryan Memorial, 1972–1975.
PALEY CENTER FOR MEDIA, LOS ANGELES, CALIFORNIA.
1. "Directed by John Frankenheimer: The Seminar." _Museum of Television and Radio Seminar Series_. Video recording, January 18, 1996.
2. "A Conversation with John Frankenheimer." _Museum of Television and Radio Seminar Series_. Video recording, September 24, 1997.
MAX REINHARDT ARCHIVES AND LIBRARY, SUNY–BINGHAMTON, BINGHAMTON, NEW YORK.
Four addresses to students and faculty at the Max Reinhardt School of the Theater: June 1938; June 27, 1938; July 12, 1938; and September 12, 1938.
UCLA FILM AND TELEVISION ARCHIVE, UNIVERSITY OF CALIFORNIA AT LOS ANGELES, LOS ANGELES, CALIFORNIA.
(1) _A Call from..._ (aka _A Call from the Stars_ ). Video recording, February 10, 1960. Television Collection, inventory number DVD 4142 T. (2) _It Could Be You_. Video recording, November 7, 1959. Television Collection, inventory no. DVD 10465.
UCLA LIBRARY SPECIAL COLLECTIONS, UNIVERSITY OF CALIFORNIA AT LOS ANGELES, LOS ANGELES, CALIFORNIA.
Jean Renoir Papers: (1) Jessica Ryan to Dido and Jean Renoir, ca. March 1948. (2) Jessica Ryan to Dido and Jean Renoir, April 8, 1955. (3) Jessica Ryan to Jean Renoir, ca. May 1967. (4) Robert Ryan to Jean Renoir, ca. August 1968. (5) Jean Renoir to Robert Ryan, May 25, 1972. (6) Robert Ryan to Jean Renoir, July 20, 1972.
WISCONSIN CENTER FOR FILM AND THEATER RESEARCH, MADISON, WISCONSIN.
Patrick McGilligan Papers: Interview with Dore Schary.
Robert Ryan clipping file: Autographed fan questionnaire in Ryan's hand, February 7, 1942.
Dore Schary Papers: (1) Peter Rathvon to Dore Schary regarding _Crossfire_ , February 12, 1947. (2) Research on anti-Japanese violence for _Bad Day at Black Rock_ , October 26, 1953. (3) Audience comments from preview _of Bad Day at Black Rock_ , Encino Theatre, Encino, California, October 14, 1954. (4) MGM summary of British reviews for _Bad Day at Black Rock_ , March 21, 1955. (5) Robert Ryan to Dore Schary, November 16, 1959. (6) Robert Ryan remarks to New York SANE event, Town Hall, New York City, March 10, 1963. (7) Robert Ryan to the chairman of "Stars for SANE," ca. 1965. (8) Robert Ryan to Dore Schary, August 5, 1967. (9) SANE brochure, "Some Things You Should Know." (10) Form letter from Robert Ryan and Steve Allen to Dore Schary, January 29, 1960. (11) SANE program for Steve Allen testimonial dinner, November 15, 1962.
Index
_About Mrs. Leslie_ , –, __ ,
Academy of Motion Picture Arts and Sciences, , –
acting of RR, _x_ , , –, ; attitude toward "heavy" roles, _x_ , , , , ; attitude toward his career, _xii–xiii_, , , , , , ; behavior on set, _ix_ , ; study of acting, –, –, , –; technique, –, , , , , –, , ; thoughts on acting, _xii_ , , –, –
_Act of Violence_ , , –, __ , , , , ,
_Alaska Seas_ , ,
Albee, Edward,
Aldrich, Robert,
Allen, Steve, , –,
American Civil Liberties Union, ,
American Friends Service Committee, , , , –, , ,
American International Pictures,
American Legion, ,
American Shakespeare Festival Theatre, –
Andes, Keith. ,
_And Hope to Die_ , –
Anthony, Susan B.,
Anticommunism, , , –, –. _See also_ Hollywood blacklist; House Un-American Activities Committee; Motion Picture Alliance for the Preservation of American Ideals; _Red Channels_
_Antony and Cleopatra,x_, –
_Anzio,xi_,
Astaire, Fred, ,
athletics of RR, ,
atomic bombing of Hiroshima, ,
Avala Film,
Axelrod, George, , –
Bacall, Lauren, , –, ,
_Back from Eternity_ ,
_Bad Day at Black Rock,x_, –, , , , , ,
"Bad Time at Honda," –
Baker, Joe Don,
Baker, Roy Ward, –
Ballard, Lucien, –, , ,
Bamberger, Henry, –,
Bankhead, Tallulah, –, , , –, ,
Barker, Lex,
_Battle of the Bulge,xi_, , , ,
_Bed of Roses_. See _Born to Be Bad_
Beery, Wallace,
Begley, Ed, ,
_Behind the Rising Sun_ , –, __ , , , , , ,
Belafonte, Harry, _xi_ , , –, __ , , , ; friendship with RR, , ,
Bel Geddes, Barbara, , –
Bennett, Joan, –, __ ,
Berlin, Irving,
_Berlin Express_ , –, __ , , , , ,
Bernstein, Elmer,
Berry, John,
_Best of the Badmen_ , , , , 205n
_Beware, My Lovely_ (working title: _Day without End_ ), –, , , ,
Beymer, Richard,
Bezzerides, A.I., ,
Bickford, Charles, , ,
_Billy Budd_ (film), –, __ , , , , , ,
_Billy Budd_ (novella), –
Boetticher, Budd, _ix_ ,
Bogart, Humphrey, , , , –,
_Bombardier,xiii_, –, ,
Bond, Ward,
Booke, Sorrell, , __
Booth, Shirley, –, __
Borgnine, Ernest, –, , ,
_Born to Be Bad_ (working title: _Bed of Roses_ ), –, ,
boxing of RR, , , , –, , –
_Boy with Green Hair, The_ , –, –, , ,
Brady, Leo,
Brando, Marlon,
Brecht, Bertolt, , ,
Breen, Joseph,
Brennan, Walter,
_Brick Foxhole, The_ , –, ,
Bridges, Beau,
Bridges, Jeff, , , –, __
Brodie, Steve, __
Bronston, Samuel, –
Brooks, Richard, _xiii_ , , , , –, ,
Brown, Arvin, –, –, , , –,
Brown, Marvin, , , __
Burnett, W.R.,
_Busy Body, The_ , –,
Cabeen, Ross, , , , , ,
Cabeen, Wendy, , ,
Caesar, Sid,
Cagney, James, , , , ,
Caldwell, Erskine,
_A Call from..._ , –
Camp Pendleton: U.S. Marine Corps training base at, , , , ,
_Canadians, The_ , ,
Cannes Film Festival, , –,
Cannon, Dyan,
_Captain Nemo and the Underwater City_ , –
Cardinale, Claudia, ,
_Cat Ballou_ , , ,
_Cat People_ , ,
_Caught,x_, , , , , , –
Cavanaugh, Frank,
Cermak, Anton, ,
_Champion_ , –
Champlin, Charles, ,
Chandler, Raymond,
Chaplin, Charles, , ,
Cheyney, George Washington
(grandfather-in-law), –,
Chicago: 9 race riot,
_City beneath the Sea_ , –
Clark, Mark V.,
_Clash by Night_ (film), –, _, _, ,
_Clash by Night_ (play), –, , , , ,
Clément, René, –
Clift, Montgomery, , , –
Cobb, Lee J., –, , ,
Cohen, Eliot,
Cohn, Art, –
Colbert, Claudette,
Collins, Judy,
Columbia Pictures, –, , ,
Committee for the First Amendment, , –, ,
Communist Party of America, , –, ,
Convy, Bert,
Conyers, John,
Coote, Robert, ,
_Coriolanus,x_, –, ,
Costello, Frank,
Coughlin, Charles,
Cousins, Norman, , –, , –, ,
_Cradle of Fear_. See _Crossfire_
Crawford, Joan, ,
critical assessment of RR, _ix_ , , , , , , , , –
Cromwell, John, –
_Crooked Road, The_ ,
_Crossfire_ (working title: _Cradle of Fear_ ), _ix, _, , , , –, , , ; and Hollywood blacklist, , ; impact of, , –, , –, , ; production of, –,
Crouse, Russel,
Cuban Missile Crisis, –
Curtis, Tony, ,
Curtiz, Michael, –
_Custer of the West_ , –
_Daily Worker_ , , , ,
Dakota Building, , , , ,
Daley, Richard J.,
Dartmouth College, , , , , , , ,
Day, Dorothy,
Day, Laraine, ,
_Day of the Outlaw_ ,
_Day without End_. See _Beware, My Lovely_
Dee, Frances,
DeHaven, Carter,
Depinet, Ned, , ,
_Desirable Woman_. See _The Woman on the Beach_
_Detective Story_ , ,
De Toth, André, xiii, –
Devi, Indra,
Dewey, John, ,
Dewey, Thomas, –
_Dirty Dozen, The,xi_, –, __ , , ,
_Dirty Game, The_. See _Secret Agents, The_
Dmytryk Edward, –, –, –, , 51n, , ; and _Crossfire_ , , , , , , ; and the Hollywood Ten, , , , , , –
Dodd, Thomas J., –
Domergue, Faith, ,
_Don Quixote_ ,
Donehue, Vincent,
Donovan, Timothy,
Douglas, Helen Gahagan, –, ,
Douglas, Kirk, –, ,
Douglas, Melvyn, –,
Douglas, Paul, –
Drazevic, Ratso,
Dunne, Amanda, –,
Dunne, Philip, , , , , , , , ,
Duvall, Robert, , ,
Dylan, Bob, –
_Ebony_ ,
education, RR's interest in, ,
education of RR, , , ,
Edwards, James,
Einfeld, Charlie, –
Eisenhower, Dwight D., , –
Eliot, T.S., , –
Emery, John,
Enterprise Studio, –,
_Escape to Burma_ ,
Essanay Film Manufacturing Company,
Ethical Culture Fieldston School, , , ,
Evans, Evans. _See_ Frankenheimer, Evans
_Executive Action_ , xiii, –, __ ,
_Exodus_ ,
Fabray, Nanette, , ,
Fairbanks, Douglas, , ,
_Farmer's Daughter, The_ , ,
Farrow, John,
Farrow, Mia, ,
Federal Theatre Project, ,
Fellig, Arthur "Weegee,"
_Feminine Mystique, The_ ,
film noir, _ix_ , ,
Finney, Albert,
Fitzgerald, F. Scott,
Fitzgerald, Geraldine, , –, ,
Flaherty, Robert,
Fleming, Rhonda, –
_Flying Leathernecks_ , –, , ,
Flynn, John,
Fonda, Henry, , , , , ,
Fontaine, Joan, , –,
Ford, John,
Ford's Theatre,
Francis, Anne,
Frankenheimer, Evans, ,
Frankenheimer, John, –, –,
Free Southern Theatre,
Friedan, Betty, ,
_Front Page, The,x_, , –, , ,
Fuller, Samuel, _ix, xiii_, , –
Gable, Clark, , , ,
_Gangway for Tomorrow_ ,
Garfunkel, Art, ,
Garner, James,
Garson, Greer,
Gavin, James M.,
Geer, Will, , , –
_Gentleman's Agreement_ , , –
Gerard, Bryson, –, , –,
Ginsberg, Allen,
_God's Little Acre_ (film), _x_ , , , –, __ , , , ,
_God's Little Acre_ (novel), –
_Golden Gloves_ , –, , ,
Goldwyn, Samuel,
Gomez, Thomas,
Goodrich, Frances. _See_ Hackett, Frances
Goodrich Grahame, Gloria, , , , ,
Grainger, Edmund, ,
Granet, Bert, –, ,
Granger, Stewart, –
Grant, Cary, , ,
_Great Gatsby, The_ (film),
_Great Gatsby, The_ (TV play), –
Greer, Jane,
Grodin, Charles,
Gross, Jack, ,
Group Theatre, , ,
Guardino, Harry,
Gunn, Moses,
Guthrie, Woody, , ,
Gwenn, Edmund,
Haas, Chuck, , –,
Hackett, Albert, , ,
Hackett, Buddy, –,
Hackett, Frances Goodrich, ,
Hamill, Pete, ,
_Hamlet_ , , ,
Harmon, Andy, , ,
Harmon, Elizabeth, –, , ,
Harmon, Sidney, , , ; and Oakwood School, –, , , , , –, , , ; and Security Pictures, –, , , , , –
Harrison, Toya,
Hayes, George "Gabby," ,
Hayes, Helen, __ ,
Hays, Lee,
Hayward, Leland,
Hayworth, Rita,
health of RR, , ; alcoholism, xi, , –, –, , –, ; cancer treatment, –, –, , –
Hecht, Ben, _x_ , , , –
Heflin, Van, –
Hepburn, Katharine, _x_ , , –, , , ,
Hersh, Seymour, –,
_Her Twelve Men_ , , ,
_Hitler's Children_ ,
Holden, William, ,
Holliday, Judy,
Hollywood blacklist, –, –, , , , , , . _See also_ Hollywood Ten
_Hollywood Fights Back_ ,
Hollywood Nineteen. _See_ Hollywood Ten
_Hollywood Reporter_ ,
Hollywood for SANE. _See_ National Committee for a Sane Nuclear Policy (SANE)
Hollywood Ten, , , –,
homes of RR: in California, , , , , , –, , ; in Chicago, , , ; the Dakota Building, , , , , ; on East Coast, , , –,
_Horizons West_ , –
_Hour of the Gun_ (working title: _The Law and Tombstone_ ), 75n,
Houseman, Joan, –,
Houseman, John, , , , , , , , , ; and _Coriolanus,x_, –, ; and _Murder in the Cathedral_ , –; and _Name, Age and Occupation_ , , ; and _On Dangerous Ground_ , ,
_House of Bamboo,x_, , –, __
House Un-American Activities Committee, , , , , , –,
Hubley, Season, –
Hudson, Rock,
Hughes, Howard, , –, –, , –; relationship with RR, , –, , ; and RKO Radio Pictures, –, –, , , ,
Humphrey, Hubert, , ,
Hunt, Marsha, , –,
Hunter, Jeffrey, , , –
Huston, John, , 70n, ,
_Iceman Cometh, The_ (film), –, __ , , ,
_Iceman Cometh, The_ (play),
_Ice Palace_ , –, ,
images of RR, _, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , _
_I Married a Communist_. See _The Woman on Pier _
Indrisano, Johnny,
_Inferno_ (working title: _The Waterhole_ ), , –,
_Inheritance, The_ , ,
investments of RR, , ,
_Iron Major, The_ ,
_It's All True_ ,
Jarrico, Paul,
John Birch Society,
Johnson, Lamont, –, , ,
Johnson, Lyndon, , ,
Johnston, Eric, ,
Kaufman, Millard, –,
Kazan, Elia, ,
Keach, Stacy, –, ,
Kefauver, Estes, ,
Kelly, Ed, , , , –
Kelly, Paul,
Kennedy, Harold J., _ix_ , , –, –, ,
Kennedy, Jacqueline,
Kennedy, John F., _xiii_ , , , –, ,
Kennedy, Robert F., , 239n
Killens, John Oliver,
King, Jr., Martin Luther, ,
_King of Kings_ , –, , ,
_Kiss for Cinderella, A_ ,
Klugman, Jack,
Knauer, Siegfried, ,
Koerner, Charles, , , , , ,
Korvin, Charles, –,
Kramer, Stanley, , ,
Krasna, Norman, ,
Krotona Institute of Theosophy,
Ladd, Alan,
Lampell, Millard, , , , –, , , ; on RR's personality, _xi_ , –, , , –,
Lampell, Ramona, –, , ,
Lancaster, Burt, , , , –, , , , __
Landon, Michael, ,
Lane, Mark,
Lang, Fritz, _ix_ , ,
Laurents, Arthur, –
Lauter, Ed, –,
_Law and Tombstone, The_. See _Hour of the Gun_
_Lawman_ , –
Lawson, John Howard,
Lee, Canada,
Leigh, Janet, , __ , –,
Lennon, John, ,
Levene, Sam, __ ,
Lewis, Edward, ,
Lewis, John,
Lewton, Val, ,
Lincoln, Abraham: RR's performances as, , –, –
_Lincoln's Doctor's Dog_ ,
Lindsay, Howard,
Lindsay, John,
Locke, Katharine, –
Logan, Joshua, –
_Lolly-Madonna XXX_ (working title: _The Lolly Madonna War_ ), , –, ,
_Lonelyhearts_ ,
_Long Day's Journey into Night,x_, , , , –,
_Longest Day, The_ , –,
Lord, Jack, ,
Lorentz, Pare, _ix_ , , , , __ ,
Losey, Joseph, _ix_ , , , , , –, ,
Louis, Joe,
Louise, Tina, –, ,
_Love Machine, The_ ,
Lowell, Robert,
Loy, Myrna, , –,
Loyola Academy,
Lukas, Paul,
Lundigan, William,
Lupino, Ida, _xii_ , , __ ,
MacArthur, Charles, _x_ , , –,
MacArthur, Douglas,
Maddow, Ben, –,
_Mad with Much Heart_ (novel), –
_Mad with Much Heart_ (film). See _On Dangerous Ground_
Mainwaring, Daniel,
Mann, Anthony, _ix, xiii_, , , –,
Mann, Daniel, –
Mansfield, Irving,
_Man without a Country, The_ , –
March, Fredric, , , ,
March, Joseph Moncure, , , –,
March on Washington,
Margo, ,
_Marine Raiders,_, , ,
Marshall, George, ,
Marvin, Lee, _xiii_ , , , –, –, , –
Mason, James,
Max Reinhardt School of the Theater, , ,
May, Rollo,
Mayer, Louis B., , ,
Mazurki, Mike, , __
McCallum, David,
McCambridge, Mercedes, ,
McCarthy, Eugene, _xi_ , , –, , 239n,
McGiver, John, , , ,
McGovern, George, ,
McHugh, Frank,
Meeker, Ralph, –
Melville, Herman, , , –
_Men in War_ , , –, _, _, –, ,
Mercury Theatre,
Metzger, Mike, –
MGM, , , , , , , , , ; and _Act of Violence_ , , , ; and _Bad Day at Black Rock_ , –
_Midsummer Night's Dream, A_ ,
military service of RR, _x_ , , , , –
Miller, Arthur,
Miller, David,
_Minute to Pray, A Second to Die, A_ ,
_Miss Lonelyhearts_ , –
Mitchell, Cameron, , ,
Mitchell, Millard,
Mitchum, Robert, _xiii_ , 51n, , , , , , ; and _Crossfire_ , , ,
_Moby Dick_ (film),
_Moby-Dick_ (novel),
Monroe, Marilyn, –, __ , ,
Morrow, Vic, , ,
Morse, Wayne,
Motion Picture Alliance for the Preservation of American Ideals, , ,
Motion Pictures Producers' Association, –,
Mott, Lucretia,
_Mr. President_ , –, , , ,
_Murder in the Cathedral,x_, –,
Murphy, George,
_Naked Spur, The_ , –, , , ,
_Name, Age and Occupation_ , –, –, __
Nash, Patrick,
National Committee for a Sane Nuclear Policy (SANE), _xi_ , –, , , , ; Hollywood for SANE, , ; internal schisms of, –, –; and presidential politics, –, –; RR's involvement in, _xi_ , , , , –
Naughton, James, , –,
Neal, Annie (grandmother-in-law), –,
Neville, John, , –,
Newman, Paul, ,
Nixon, Richard, , , , , , ,
Nottingham Repertory Theatre, –, –
Oakwood School, xi, , , ; administration of, –, ; conflicts among parents, , –; founding of, –, –, –, –; and Marie Spottswood, –, , ,
Oberon, Merle, –, __ , –, , ,
O'Brien, Pat, , , , __ , , ,
_Odds against Tomorrow,xi_, –, __ , , ,
Odets, Clifford, , , , ,
Odlum, Floyd, ,
Office of War Information, , ,
OK Corral, gunfight at, ,
_On Dangerous Ground_ (working title: _Mad with Much Heart),xii, _, , , , , , , ; production of, –
O'Neill, Eugene, _x_ , , , , , , –
Ono, Yoko,
Ophuls, Max, _ix_ , –
O'Sullivan, Maureen, , , –
Oswald, Lee Harvey, ,
_Othello_ , , –
_Our Town_ , ,
_Outfit, The_ , –,
_Outlaw, The_ , ,
Paramount Pictures, , , , , , , ,
Parsons, Harriet,
Parsons, Louella, ,
Pasadena Playhouse, ,
Paul, Alice,
Paxton, John, , , ,
Pearl Harbor, attack on, , ,
Peck, Gregory, 78n, , , ,
Peckinpah, Sam, _xiii_ , , –, –
Persoff, Nehemiah, ,
personality of RR, _ix, xii, xiii_, , –; Catholic faith, –, –, –; depressions, _xi_ , –, , ; experience of fame, , , –, –; movies, interest in, , ; outdoors, interest in, , ; pool, interest in, –; privacy, _ix, xi, xii_, , , ; seafaring, interest in, , –; Shakespeare, interest in, , , , , , –, –; writing, interest in, , , ,
personal relationships of RR: with brother, , , ; with children, , , –, , , ; with father, –, , , , –; with mother, –, –, ; with wife, _xiii_ , , , , , , –, , , –
Pine, Phillip, __
Plumstead Playhouse, –, ,
politics of RR, _x, xiii_; antinuclear activism, , –, –, –, ( _see also_ National Committee for a Sane Nuclear Policy SANE]); antiwar activism, –, , –; civil liberties activism, , –, ; civil rights activism, –, , ; Democratic Party politics, –, , ; McCarthy presidential campaign, _[xi_ , –, –; pacifist philosophy, –; Stevenson presidential campaigns, –,
Polonsky, Abraham, , –
Preminger, Otto,
Production Code Administration, , , , , ,
_Professionals, The,xii_, –, –, , , ,
Progressive Citizens of America, ,
_Protocols of the Elders of Zion_ ,
_Proud Ones, The_ , ,
Quinn, Anthony, , ,
_Racket, The_ , –, , ,
Radio-Keith-Orpheum, , , , ,
Rainer, Louise,
Randall, Tony, , ,
Rankin, John E.,
Rathvon, N. Peter, , , , , , –,
Ray, Aldo, –, , , ,
Ray, Nicholas, _ix, xii_, –, , –, –,
Reagan, Ronald, ,
_Reason Why, The_ , –
_Rebel without a Cause_ , , ,
_Red Badge of Courage, The_ ,
_Red Channels_ , , ,
Reinhardt, Max, _ix_ , , –, , , , ,
Reinhardt, Wolfgang,
Renoir, Dido, , , , ,
Renoir, Jean, _ix_ , –, __ , , –, ; friendship with Jessica Ryan, , , , ; friendship with RR, , ,
_Return of the Bad Men_ ,
Rivkin, Allen, ,
RKO Radio Pictures, _ix_ , , , –, , , –, , ; and Howard Hughes, , , , , –, ; and _Name, Age and Occupation_ , , ,
Robards, Jason, , –, ,
Robertson, Cliff,
Robson, Mark, ,
Rockefeller, Nelson,
Rogell, Sid, , ,
Rogers, Ginger, –,
Rogers, Paul,
Roosevelt, Franklin D.,
Rose, Billy, ,
Ross, Katharine,
_Rush to Judgment_ ,
Russell, Jane, , , , ,
Russell, Rosalind, , ,
Ryan, Cheyney (son), , _, _, , ; childhood of, , , , , –; education of, , , , ; on Jessica Ryan, , , –; political activism and public service of, , , , –, , –; relationship with RR, –, –, , , –; on RR's career, , , –, , , , , , ; on RR's personal life , , , , , , ; on RR's politics, , , , , ,
Ryan, Jessica Cadwalader (wife), , , , , , , , , ; acting of, , , , ; alcoholism and breakdown of, _xi_ , –, –; _America—Dream or Nightmare?_ , , –; conflicts with RR, , , , ; courtship by RR, –, ; death and burial, –; feminism of, –, –, –; and health of RR, , –, , ; images of, _, , , , , , ; Man Who Asked Why, The_, –; marriage to RR, , , , , , –, , ; and Oakwood School administration, , , –, , , ; and Oakwood School founding, –, –, , –; personality of, _xi, xiii_, , –, –, , , , ; and politics of RR, , , , –, ; Quaker philosophy of, , , ; relationship with Dido and Jean Renoir, , , , , –, ; _Woman—The Mythless American_ , , –; writing of, _xiii_ , , –, , , ,
Ryan, Larry (uncle), ,
Ryan, Lisa (daughter), _xii, _, –, __ , , , , –; childhood and education of, , , , , , , , , , ; on Jessica Ryan, , ; on RR's career, , , , –, , ; on RR's personal life, , , , , , , ,
Ryan, Mabel Bushnell (mother), , , , __ , , ,
Ryan, Thomas (uncle), , ,
Ryan, Timothy Aloysius (father), _, _, , , , ; business career of, , ; political career of, , , ; and sanitary district tunnel disaster, _xii_ , , –
Ryan, Timothy Cadwalader (son), __ , , _, , _, , , , , , , ; childhood of, , , , , , , ; education of, , , , , , ; relationship with RR, –,
Ryan, Timothy E. (great-uncle), ,
Ryan Company, , , , , –, ; and sanitary district tunnel disaster, _xii_ , –
Saks, Gene,
Samuel Goldwyn Studio, ,
SANE. _See_ National Committee for a Sane Nuclear Policy (SANE)
Sarafian, Richard,
Schaefer, George, ,
Schaffner, Franklin,
Schary, Dore, , , –, , –, , , ; and _Crossfire_ , , , , ; and Howard Hughes, , , ; and HUAC, , , , ; and RKO Radio Pictures, , , , ,
Schenck, Nicholas, ,
Schildkraut, Joseph, , ,
Schweitzer, Albert,
Scorsese, Martin, _ix, _
Scott, Adrian, , , , , , ; and the Hollywood Ten, , , , , ,
Scott, Martha, ,
Scott, Randolph, , , ,
Screen Actors Guild,
Screen Writers Guild, ,
_Secret Agents, The_ , , ,
_Secret Fury, The_ , ,
Security Pictures, Inc., , , –,
Seeger, Pete, ,
Selznick, David O., , ,
Selznick, Irene,
_Set-Up, The_ (film), , , –, __ , , , ; impact of, ,
_Set-Up, The_ (poem), , , –
Shaw, Robert, ,
Shenandoah,
Simon, Paul, ,
Sinatra, Frank, ,
_Sister Kenny_ , ,
_Sky's the Limit, The_ , ,
Smith, Solomon "Smith," xi, , __ ,
Smith, Williana, _xi_ , , __ , , ,
_Snows of Kilimanjaro, The_ , –, 192n,
Society of Friends, –
Sokoloff, Lisa,
Sokoloff, Vladimir, , , ,
Southern Christian Leadership Conference,
_Spartacus_ ,
Spock, Benjamin, ,
Spottswood, Marie, –, –, , __ ,
Stack, Robert, , __ ,
Stamp, Terence, , ,
Stanley, Kim,
Stanwyck, Barbara, –, __ ,
Steiger, Rod, , –
Stevenson, Adlai, –, , ,
Stevenson, Robert,
Stewart, James, , –,
Stiller, Jerry,
Stockwell, Dean, , 78n, ,
Strasberg, Lee, , ,
_Street with No Name, The_ ,
Streisand, Barbra,
Strode, Woody, ,
Students for a Democratic Society,
Sturges, John, ,
Sturges, Preston,
Surtees, Robert, –
Susann, Jacqueline,
Sutherland, Donald, ,
Swanson, Gloria,
_Tall Men, The_ , ,
Talman, William,
Taylor, Rod,
Teller, Edward,
_Tender Comrade_ , –, , ,
_Texas Rangers Ride Again, The_ ,
Thomas, J. Parnell, ,
_Tiger at the Gates,x_, –,
_Time of Your Life, The_ ,
Tito, Josip Broz,
Toporow, Roman, ,
Tourneur, Jacques, _ix_ , , ,
Tracy, Spencer, , , , ; and _Bad Day at Black Rock_ , , –, , ; influence on RR, ,
_Trail Street_ ,
Trintignant, Jean-Louis,
Truman, Harry S., , , –, –, ,
Trumbo, Dalton, , , , , –
Twentieth Century Fox, , , , , , , , , ; and _Gentleman's Agreement_ , ,
UCLA Extension, Professional Theatre Group of the, –
Ulene, Priscilla,
_Ulysses_ , ,
United Artists, –, –, , , , ,
United World Federalists, _xi_ , , , –, ,
Universal-International Pictures,
Universal Pictures, , ,
Ustinov, Peter, –, __
Vanguard Pictures,
_Vendetta_ , ,
Vietnam War, –, , –
Voice of America,
Vose, Donald,
Wald, Jerry, ,
Wallace, Henry, ,
Wallach, Eli,
Wallis, Hal,
Wallsten, Robert, , , , , ,
Walsh, Raoul,
Wanger, Walter,
Warner, Jack, , –,
Warner Bros., , , , , , ,
_Waterhole, The_. See _Inferno_
Wayne, John, _xiii_ , –, , –, ,
Welles, Orson, ,
Whitmore, James,
Widmark, Richard, , ,
_Wild Bunch, The,xii_, 75n, , –, __ , , –,
Wilder, Billy,
Wilder, Thornton, , ,
Williams, Elmer Lynn, , ,
Wilson, Scott, –
Winner, Michael, , ,
Winters, Shelley, –
Wise, Robert, _ix_ , –, , –,
_Woman on Pier, The_ (working title: _I Married a Communist_ ), –, , ,
_Woman on the Beach, The_ (working title: _Desirable Woman_ ), –, __ , , , , ,
work experience of RR, , , , , ,
Wyler, William,
Yordan, Philip, –, , , , , , –; and Security Pictures, –, , , , , –
Young, James R.,
Young, Robert,
Zanuck, Darryl F., , , –, ,
Zeller, Max,
Zinnemann, Fred, _ix_ , –,
A SERIES FROM WESLEYAN UNIVERSITY PRESS
_Edited by Lisa Dombrowski and Scott Higgins_
ORIGINATING EDITOR: Jeanine Basinger
_Anthony Mann_
by Jeanine Basinger
_It's the Pictures That Got Small_
_Hollywood Film Stars on 1950s Television_
by Christine Becker
_The South Korean Film Renaissance_
_Local Hitmakers, Global Provocateurs_
by Jinhee Choi
_The Art of Comedy_
_The Films of Frank Tashlin_
by Ethan de Seife
_The Films of Samuel Fuller_
_If You Die, I'll Kill You!_
by Lisa Dombrowski
_Kazan Revisited_
edited by Lisa Dombrowski
_The Lives of Robert Ryan_
by J.R. Jones
_Physical Evidence_
_Selected Film Criticism_
by Kent Jones
_The New Entrepreneurs_
_An Institutional History of_
_Television Anthology Writers_
by Jon Kraszewski
_Action Speaks Louder_
_Violence, Spectacle_ ,
_and the American Action Movie_
by Eric Lichtenfeld
_Hollywood Ambitions_
_Celebrity in the Movie Age_
by Marsha Orgeron
_Brutal Intimacy_
_Analyzing Contemporary French Cinema_
by Tim Palmer
_The Cinema of Errol Morris_
by David Resha
_Soul Searching_
_Black-Themed Cinema from the March on Washington_
_to the Rise of Blaxploitation_
by Christopher Sieving
_Paul on Mazursky_
by Sam Wasson
_A Splurch in the Kisser_
_The Movies of Blake Edwards_
by Sam Wasson
J.R. JONES is the film editor for the _Chicago Reader_ , where his work has appeared since 1996 and won multiple awards from the Association of Alternative Newsmedia. His writing has also appeared in the _Chicago Sun-Times, New York Press, Kenyon Review_ , Noir City, and _Da Capo Best Music Writing 2000_ , edited by Peter Guralnick. J.R. Jones lives in Chicago.
| {
"redpajama_set_name": "RedPajamaBook"
} | 8,983 |
'Bill and Ted Face the Music' Will Have the Duo Face Off Against Kid Cudi
Posted on Thursday, June 13th, 2019 by Hoai-Tran Bui
Bill & Ted Face the Music will be adding an actual musician to its increasingly impressive cast. The cast has added Kid Cudi in an undisclosed but "significant" role in the third installment of the time-traveling comedy franchise starring Keanu Reeves and Alex Winter as the two truly bodacious dudes.
Keanu Reeves has fought against ninja assassins, jealous boyfriends, sociopathic toys, and video game brawlers, but soon he and Winter will be facing off against middle age in Bill & Ted Face the Music — and apparently, Grammy Award winning artists.
The band is getting back together for the third installment of the Bill & Ted franchise, and they will likely find a worthy competitor in Scott Mescudi , better known by his stage name Kid Cudi, who has joined the growing cast of Bill & Ted Face the Music which includes Alex Winter, Keanu Reeves, William Sadler, Samara Weaving, and Brigette Lundy-Paine. Mescudi will have a "significant role" though details are being kept under wraps. Mescudi is best known for his musical career, but he's been acting regularly for the past decade through cameo appearances and sitcom roles, most recently appearing in an indie film Drunk Parents. But I would imagine, with a title like Face the Music, Mescudi will be playing a musician to rival Wyld Stallyns.
Bill & Ted Face the Music will be directed by Dean Parisot (Galaxy Quest, Red 2, Fun With Dick and Jane), from a screenplay by returning franchise writers Chris Matheson (Imagine That, A Goofy Movie) and Ed Solomon (Men in Black, Mosaic).
Here's the official synopsis:
Following 1989's Bill & Ted's Excellent Adventure and 1991's Bill & Ted's Bogus Journey, the stakes are higher than ever for William "Bill" S. Preston Esq. (Winter) and Theodore "Ted" Logan (Reeves). Yet to fulfill their rock and roll destiny, the now middle-aged best friends set out on a new adventure, when a visitor from the future warns them that only their song can save life as we know it and bring harmony to the universe. Along the way, they will be helped by their families, old friends and a few music legends.
Bill and Ted Face the Music opens August 21, 2020.
'Bill and Ted 3' Rounds Out Its Cast with Kristen Schaal and Holland Taylor
'Bill and Ted Face the Music' Adds '22 Jump Street' Scene Stealer Jillian Bell
'Bill and Ted Face the Music' Has Started Shooting in New Orleans
Sequel Bits: 'Space Jam', 'Saw', 'Ghostbusters', 'Coming 2 America', 'Gremlins', 'Iron Sky', 'The Boy', 'Bill and Ted'
Casting, Comedy, Sequels, Alex-Winter, Bill and Ted 3, Bill and Ted Face The Music, Keanu-Reeves, Kid Cudi | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,562 |
\section{Introduction}
The nucleon sea exhibits two interesting properties: flavour asymmetry
\cite{New,NA51,E866,Hermes} and quark-antiquark
asymmetry \cite{CCFR,NMC}.
While there have been many studies of the nucleon sea from both experiment
(see e.~g. \cite{New,NA51,E866,Hermes,CCFR,NMC}) and
theory (see e.~g. \cite{Thomas83,Holtmann,Speth,Kumano,Boros} and references therein),
the studies of the sea distributions of the other baryons
in the baryon octet predicted by the $SU(3)$ quark model are very few.
It is of interest to know whether the sea of the other members of the
baryon octet has the same properties
(flavour asymmetry and quark-antiquark asymmetry) as the nucleon sea.
Also, through the study of the quark sea of the
other members of the baryon octet,
we can improve our understanding of
the structure of the baryons and the non-perturbative properties of QCD.
Alberg {\it et al.} \cite{Alberg} pointed out that
the valence and sea quark distributions of the $\Sigma^{\pm}$
may exhibit large deviations from the $SU(3)$ predictions,
and these parton distributions
could be obtained from Drell-Yan experiments using charged hyperon
beams on proton and deuteron targets.
Alberg, Falter, and Henley \cite{Alberg2} studied the flavour asymmetry
in the $\Sigma^+$ sea employing the meson cloud model
and effective Lagrangian for the baryon-meson-baryon interaction,
and found large deviations from $SU(3)$.
Boros and Thomas \cite{BorosThomas} calculated the quark distributions of
$\Lambda$ and $\Sigma^{\pm}$ employing the MIT bag model.
It was found that the valence quark distributions are quite different from
the $SU(3)$ predictions and that the quark sea is flavour asymmetric.
More recently, Ma, Schmidt and Yang \cite{MaSY} showed that
there are significant differences between the predictions
of perturbative QCD and $SU(6)$ quark-diquark model for the flavor
and spin structure of the $\Lambda$ baryon's quark distributions near $x=1$.
In this letter we shall investigate the flavour asymmetry and quark-antiquark
asymmetry of the $\Sigma^+$ sea using the light-cone baryon-meson
fluctuation model (LCM) suggested by Brodsky and Ma \cite{BrodskyMa}.
The baryon-meson fluctuation (meson cloud) mechanism is very successful
in understanding on the flavour asymmetry and quark-antiquark
asymmetry of nucleon sea.
The various fluctuations can be described via corresponding
baryon-meson-nucleon Lagrangians \cite{Thomas83,Holtmann,Speth,Kumano}.
Recently, Brodsky and Ma \cite{BrodskyMa} proposed that
the baryon-meson fluctuation could be described
by using a light-cone two body wave function which is a function of
the invariant mass squared of the baryon-meson Fock state.
Compared to the commonly used effective Lagrangian method (ELM)
\cite{Thomas83,Holtmann,Speth,Kumano} for the description
of baryon-meson fluctuations, the LCM is relatively simple.
Furthermore our study \cite{Cao} showed that the LCM can produce
very similar results to the effective Lagrangian method
for a suitable choice of parameter.
The basic idea of the meson cloud model
(for recent reviews see Refs.~\cite{Speth,Kumano}) is that the nucleon
can be viewed as a bare nucleon surrounded by a mesonic cloud.
The nucleon wave function can be expressed in terms of bare nucleon and
virtual baryon-meson Fock states.
Although this model was developed mainly in the study of nucleon sea,
applying this model to the other baryons is straightforward.
For the $\Sigma^+$, the wave function can be written as
\begin{eqnarray}
|\Sigma^+\rangle_{\rm physical} = Z |\Sigma^+\rangle_{\rm bare}
+ \sum_{B M} \int dy \, d^2 {\bf k}_\perp \, \phi_{B M}(y,k_\perp^2)
\, |{B}(y, {\bf k}_\perp); {M}(1-y,-{\bf k}_\perp)\rangle,
\label{NMCM}
\end{eqnarray}
where $Z$ is the wave function renormalization constant,
$\phi_{BM}(y,k_\perp^2)$ is the wave function
of Fock state containing a baryon ($B=\Lambda$, $\Sigma^0$, $\Sigma^+$, $p$)
with longitudinal momentum fraction $y$, transverse momentum ${\bf k}_\perp$,
and a meson ($M=\pi^+$, $\pi^0$, ${\bar K^0}$) with momentum fraction $1-y$,
transverse momentum $-{\bf k}_\perp$.
Here we consider the most energetically-favoured fluctuations
in the baryon octet and meson octet.
The fluctuation $\Sigma^+ \rightarrow \Xi^0 K^+$ is neglected due to the
higher mass of $\Xi^0$ ($m_{\Xi}=1.32$ GeV while $m_\Lambda=1.12$ GeV,
$m_\Sigma=1.19$ GeV).
It would seem that the fluctuation $\Sigma^+ \rightarrow \Sigma^+ \eta$
is also important in the calculations of ${\bar d} - {\bar s}$ and ${\bar u} - {\bar s}$.
However, applying the common $\eta_8$-$\eta_1$ mixing scheme
\begin{eqnarray}
\eta={\rm cos} \, \theta \frac{1}{\sqrt{6}}(u {\bar u} + d {\bar d} -2 s {\bar s})
-{\rm sin} \, \theta \frac{1}{\sqrt{3}}(u {\bar u} + d {\bar d} + s {\bar s})
\end{eqnarray}
and assuming $SU(3)$ symmetry for the quark distributions in
the $\eta_8$ and $\eta_1$,
we find that compared to the fluctuation $\Sigma^+ \rightarrow \Lambda \pi^+$,
the contributions to the ${\bar d} -{\bar s}$ and ${\bar u}-{\bar s}$
from the fluctuation $\Sigma^+ \rightarrow \Sigma^+ \eta$
are suppressed by a factor of
$(\frac{1}{\sqrt{6}}{\rm cos} \theta -\frac{1}{\sqrt{3}} {\rm sin} \theta)^2
-(\frac{2}{\sqrt{6}}{\rm cos} \theta +\frac{1}{\sqrt{3}} {\rm sin} \theta)^2$
which is in the range of $-0.20 \sim -0.01$ for mixing angle in
the theoretically accepted range $\theta=-12^o \sim -20^o$
\cite{Schechter,Feldmann,CaoMixing,Donoghue,Burakovsky}.
The higher mass of the $\eta$ ($m_\eta=0.547$ GeV, $m_\pi=0.139$ GeV)
also suppresses the contribution from this fluctuation.
Thus we neglect this fluctuation in our calculation.
Provided that the lifetime of a virtual baryon-meson Fock state is much longer
than the strong interaction time in the Drell-Yan process,
the contribution from the virtual baryon-meson Fock states to the quark and
anti-quark sea of $\Sigma^+$ can be written as convolutions
\begin{eqnarray}
q(x)&=&\sum_{BM} \left[\int^1_x \frac{dy}{y} f_{BM}(y) q^B(\frac{x}{y})
+\int^1_x \frac{dy}{y} f_{MB}(1-y) q^M(\frac{x}{y}) \right],
\label{qBM} \\
{\bar q}(x)&=&\sum_{BM} \int^1_x \frac{dy}{y} f_{MB}(1-y) {\bar q}^M(\frac{x}{y}),
\label{qbarBM}
\end{eqnarray}
where $f_{B M}(y)=f_{MB}(1-y)$ is fluctuation function which gives
the probability for the $\Sigma^+$ to fluctuate into a virtual $BM$ state
\begin{eqnarray}
f_{BM}(y)=\int^\infty_0 d k_\perp^2 \left | \phi_{B M}(y, k_\perp^2)\right |^2.
\label{fBM}
\end{eqnarray}
A common practice in the evaluation of the wave function
$\phi_{B M}(y, {\bf k}^2_\perp)$ is to employ time-ordered perturbative theory
in the infinite momentum frame and the effective meson-baryon-nucleon
interaction Lagrangian \cite{Thomas83,Holtmann,Speth,Kumano}.
On the other hand, Brodsky and Ma \cite{BrodskyMa} suggested
that this wave function can also be described by using light-cone
two-body wave function which is a function of the
the invariant mass squared of the baryon-meson Fock state
\begin{eqnarray}
\phi_{BM}(y,{\bf k}_\perp)=A\, {\rm exp}
\left[\frac{1}{8\alpha^2}\left(\frac{m_B^2+{\bf k}_\perp^2}{y}
+\frac{m_M^2+{\bf k}_\perp^2}{1-y} \right)\right],
\label{phi}
\end{eqnarray}
where $\alpha$ is a phenomenological parameter which determines the shape of
the fluctuation function.
Compared to the effective Lagrangian method,
Eq.~(\ref{phi}) is quite simple.
Furthermore, our study on the $s$-${\bar s}$ asymmetry in the nucleon sea \cite{Cao}
showed that Eq.~(\ref{phi}) can provide similar results to
the effective Lagrangian method for $\alpha=1.0$ GeV.
Because the spin structure of the baryon-meson-baryon vertex is the same for
all members of the baryon octet (ignoring fluctuations to decuplet baryons),
we might expect that the value of $\alpha$ should be similar for all the members
of the baryon octet.
We will use $\alpha=0.3$ GeV and $1.0$ GeV in our calculation
as there is little constraint from experimental data or theoretical studies
on the $\Sigma$ sea.
The normalization $A$ in Eq.~(\ref{phi}) can be determined by the probability
for the corresponding fluctuation.
We adopt the result given in Ref.~ \cite{BorosThomas} for the probabilities
of the various fluctuations\footnote{Note the relationship between the
fluctuation functions for various iospin states:
$f_{\Sigma^0 \pi^+}=f_{\Sigma^+\pi^0}$ and the fluctuation functions
given in Ref.~\cite{BorosThomas} for a given type of fluctuation are defined as
the sum of all iospin states:
$f_{\Sigma \pi}=f_{\Sigma^0\pi^+}+f_{\Sigma^+ \pi^0}$.}:
\begin{eqnarray}
P_{\Lambda \pi^+}=3.2\%, \,\,\,\, P_{p {\bar K^0}}=0.4\%, \\
P_{\Sigma^0 \pi^+}=P_{\Sigma^+ \pi^0}=\frac{1}{2}P_{\Sigma \pi}=1.85\%.
\label{P}
\end{eqnarray}
In the baryon-meson fluctuation model, the non-perturbative contributions
to the quark and the anti-quark distributions
in the $\Sigma^+$ sea come from the quarks and anti-quarks of the baryons
($\Lambda, \, \Sigma^+,\, \Sigma^0$ and $p$) and mesons
($\pi^+, \, \pi^0$ and ${\bar K^0}$) in the virtual baryon-meson Fock states.
So we need the parton distributions of the involved baryons and mesons as input.
For the parton distribution in the pion, we employ the parameterization
given by Gl\"{u}ck, Reya, and Stratmann (GRS98) \cite{GRS98}
and we neglect the sea content in the meson, that is,
\begin{eqnarray}
{\bar d}^{\pi^+}=u^{\pi^+}&=&{\bar u}^{\pi^-}=d^{\pi^-}=\frac{1}{2} v^\pi, \\
{\bar u}^{\pi^0}=u^{\pi^0}&=&{\bar d}^{\pi^0}=d^{\pi^0}=\frac{1}{4} v^\pi, \\
v^\pi(x,\mu_{\rm NLO}^2) &=&1.052 x^{-0.495} (1 +0.357 \sqrt{x}) (1-x)^{0.365},
\label{vpion}
\end{eqnarray}
at scale $\mu_{\rm NLO}^2=0.34$ GeV$^2$.
For the ${\bar d}$ distribution in the ${\bar K^0}$ we relate it
to the $u$ distribution in the $K^+$
which are given in the GRS98 parameterization \cite{GRS98} also
\begin{eqnarray}
{\bar d}^{{\bar K^0}}(x,\mu_{\rm NLO}^2)=u^{K^+}(x,\mu_{\rm NLO}^2)
= 0.540 (1-x)^{0.17} v^\pi(x,\mu_{\rm NLO}^2),
\label{vK0bar}
\end{eqnarray}
at scale $\mu_{\rm NLO}^2=0.34$ GeV$^2$.
In order to investigate the quark-antiquark asymmetry via $d(x)-{\bar d}(x)$
in the $\Sigma^+$ sea, we also need to know the $d$-quark distribution in
the $\Lambda$, $\Sigma^+$ and $p$, for which
we use the parameterization for the $d$ quark distribution in the proton
given by Gl\"{u}ck, Reya, and Vogt (GRV98) \cite{GRV98},
\begin{eqnarray}
d^p(x,\mu_{\rm NLO}^2)=0.400 \, x^{-0.57} (1-x)^{4.09} (1+18.2 x),
\end{eqnarray}
at scale $\mu_{\rm NLO}^2=0.40$ GeV$^2$.
We evolve the distributions to the scale $Q^2 = 4$ GeV$^2$
using the program of Miyama and Kumano \cite{MiyamaK}
in which the evolution equation is solved numerically in a brute-force method.
We found that at $Q^2=4$ GeV$^2$ all parton distributions
$v^\pi(x, Q^2)$, ${\bar d}^{{\bar K^0}}(x, Q^2)$ and $d^p(x, Q^2)$
can be parameterized using the following form
\begin{eqnarray}
q(x, Q^2)=a \, x^b \, (1-x)^c\, (1+d \, \sqrt{x} +e \, x)
\label{qfit}
\end{eqnarray}
with the parameters given in Table 1.
We estimate the uncertainty in solving the evolution equations numerically
and parameterizating the parton distribution in the form of Eq.~(\ref{qfit})
to be about $2\%$ in the $x$-region which we are interested in
{\it i.e.} $x >10^{-3}$.
The effect of evolution from a lower scale to a higher
scale is to make the parton distribution more concentrated
in the small $x$ region. Thus we may expect that the $x$ position at which
an asymmetry exhibits a maximum will move to
smaller $x$ as we evolve to higher values of $Q^2$.
However, we do not expect the asymmetry
to ``evolve away" at a higher $Q^2$ scale if it exists
at a lower scale such as $\mu^2_{\rm NLO}$.
We investigate the flavour asymmetry in the $\Sigma^+$ sea
through calculating the differences between the antiquark distributions:
$x[{\bar d}(x)-{\bar u}(x)]$, $x[{\bar d}(x)-{\bar s}(x)]$ and $x[{\bar u}(x)-{\bar s}(x)]$
which are given by
\begin{eqnarray}
x\left[{\bar d}(x)-{\bar u}(x)\right] &=& x \left[ {\bar d}_{\Lambda \pi^+}(x)
+{\bar d}_{\Sigma^0 \pi^+}(x)
+{\bar d}_{p {\bar K^0}}(x) \right] \nonumber \\
&=&\int^1_x dy \, \frac{x}{y} \left[
f_{\Lambda \pi^+}(1-y) {\bar d}^{\pi^+}(\frac{x}{y})
+f_{\Sigma^0 \pi^+}(1-y) {\bar d}^{\pi^+}(\frac{x}{y}) \right.\nonumber \\
& &~~~~~~~~~ \left. + f_{p {\bar K^0}}(1-y) {\bar d}^{\bar K^0}(\frac{x}{y})
\right],
\label{xdmu}
\end{eqnarray}
\begin{eqnarray}
x\left[{\bar d}(x)-{\bar s}(x)\right]&=& x \left[{\bar d}_{\Lambda \pi^+}(x)
+{\bar d}_{\Sigma^0 \pi^+}(x)
+{\bar d}_{\Sigma^+ \pi^0}(x)
+{\bar d}_{p {\bar K^0}}(x) \right]\nonumber \\
&=&\int^1_x dy \, \frac{x}{y} \left[
f_{\Lambda \pi^+}(1-y) {\bar d}^{\pi^+}(\frac{x}{y})
+f_{\Sigma^0 \pi^+}(1-y) {\bar d}^{\pi^+}(\frac{x}{y}) \right.\nonumber \\
& &~~~~~~~~~~\left. +f_{\Sigma^+ \pi^0}(1-y) {\bar d}^{\pi^0}(\frac{x}{y})
+ f_{p {\bar K^0}}(1-y) {\bar d}^{\bar K^0}(\frac{x}{y})
\right],
\label{xdms}
\end{eqnarray}
\begin{eqnarray}
x\left[{\bar u}(x)-{\bar s}(x)\right]&=& x \, {\bar d}_{\Sigma^+ \pi^0}(x) \nonumber \\
&=&\int^1_x dy \, \frac{x}{y} f_{\Sigma^+ \pi^0}(1-y)
{\bar d}^{\pi^0}(\frac{x}{y}).
\label{xums}
\end{eqnarray}
$x [{\bar u}(x)-{\bar s}(x)]$ comes from
only fluctuation $\Sigma^+ \rightarrow \Sigma^+ \pi^o$,
while $x [{\bar d}(x)-{\bar u}(x)]$ and $x [{\bar d}(x)-{\bar s}(x)]$
come from also $\Sigma^+ \rightarrow \Lambda \pi^+$,
$\Sigma^+ \rightarrow \Sigma^0 \pi^+$ as well as $\Sigma^+ \rightarrow p {\bar K^0}$.
In Fig.~1 we present our results for $x [{\bar d}(x)-{\bar u}(x)]$
at the scales $\mu^2_{\rm NLO}=0.34$ GeV$^2$ and $Q^2=4$ GeV$^2$
with $\alpha=0.3$ GeV.
It can be found that the contribution from the fluctuation
$\Sigma^+ \rightarrow \Lambda \pi^+$ is about twice as large as that
from $\Sigma^+ \rightarrow \Sigma^0 \pi^+$, and both are much larger
than the contribution from $\Sigma^+ \rightarrow p {\bar K^0}$.
Under evolution the distributions move
from larger $x$ to smaller $x$ --
the $x$ position at which $x[{\bar d}(x)-{\bar u}(x)]$ exhibits a maximum shifts
from about $0.1$ to $0.06$
and the maximum decreases about $20\%$,
which coincides with our naive expection.
The numerical results for $x[{\bar d}(x)-{\bar u}(x)]$, $x[{\bar d}(x)-{\bar s}(x)]$
and $x[{\bar u}(x)-{\bar s}(x)]$ at $Q^2=4$ GeV$^2$
are given in Figs.~2 and 3 for $\alpha=0.3$ GeV and $1.0$ GeV respectively.
We can see that ${\bar d}(x) > {\bar u}(x) > {\bar s}(x)$, that is
the anti-quark distribution in the $\Sigma^+$ sea is flavour asymmetric.
As is well known, the nucleon sea is also asymmetric
and for the proton sea ${\bar d}>{\bar u}>{\bar s}$
\cite{Thomas83,Holtmann,Speth,Kumano,Boros}.
The main difference between the proton$(uud)$ and
$\Sigma^+(uus)$ is the replacement of a valance $d$ quark
by a valance $s$ quark.
Thus one may expect from $SU(3)$ symmetry that
the $\Sigma^+(uus)$ sea to be ${\bar s}>{\bar u}>{\bar d}$.
This prediction is opposite to our above conclusion (${\bar d}>{\bar u}>{\bar s}$)
from the light-cone baryon-meson fluctuation model.
If the $SU(3)$ symmetry breaking in the ${\bar u}$, ${\bar d}$ and ${\bar s}$
distributions in the $\Sigma^+$ sea has the same source as that for
the $u$, $d$, and $s$ quark masses,
we may expect that $x[{\bar d}(x)-{\bar u}(x)] < x[{\bar u}(x)-{\bar s}(x)]$
since the mass difference between the $u$ and $d$ quarks is far smaller than
that between the $u$ and $s$ quarks.
However, our calculations (see Figs.~2 and 3) show that
$x[{\bar d}(x)-{\bar s}(x)] > x[{\bar d}(x)-{\bar u}(x)] > x[{\bar u}(x)-{\bar s}(x)]$.
The relation $x[{\bar d}(x)-{\bar u}(x)] > x[{\bar u}(x)-{\bar s}(x)]$
is opposite to our above argument,
which implies that the dynamics responsible for the $SU(3)$ symmetry breaking
in the quark distributions of the $\Sigma^+$ sea, as calculated in our model,
are different from that responsible for the mass differences among the
$u$, $d$ and $s$ quarks.
Another interesting question concerning the $\Sigma^+$ sea is the
quark-antiquark asymmetry.
Although the perturbative sea created from gluon-splitting is symmetric
$q={\bar q}$ (in the leading twist approximation in perturbative calculation),
the non-perturbative sea, which may exist over a long time
and has a strong connection with the ``bare" $\Sigma^+$,
may be asymmetric $q\neq {\bar q}$.
Because of the existence of valance $u$ and $s$ quarks in the $\Sigma^+$,
it is difficult to measure the differences $u-{\bar u}$ and $s-{\bar s}$
in the $\Sigma^+$ sea.
The most likely experiment is to measure the difference
between $d$ and ${\bar d}$.
From the baryon-meson fluctuation model the $d(x)-{\bar d}(x)$
turns out to be:
\begin{eqnarray}
d(x)-{\bar d}(x)&=& d_{\Lambda \pi^+}(x)-{\bar d}_{\Lambda \pi^+}(x)
+d_{\Sigma^0 \pi^+}(x)-{\bar d}_{\Sigma^0 \pi^+}(x)
+d_{p {\bar K^0}}(x)-{\bar d}_{p {\bar K^0}}(x) \nonumber \\
&=& \int^1_x \frac{dy}{y}
\left\{
\left[
f_{\Lambda \pi^+}(y) + f_{\Sigma^0 \pi^+}(y) + f_{p {\bar K^0}}(1-y)
\right]
d^p(\frac{x}{y})
\right. \nonumber\\
& & ~~~~~~~
- \left[
f_{\Lambda \pi^+}(1-y) + f_{\Sigma^0 \pi^+}(1-y) \right]
{\bar d}^{\pi^+}(\frac{x}{y}) \nonumber \\
& & ~~~~~~~
\left.
- f_{p {\bar K^0}}(1-y) {\bar d}^{\bar K^0}(\frac{x}{y}) \right\}.
\label{dmd}
\end{eqnarray}
The numerical results at scales $\mu^2_{\rm NLO}$ and $Q^2=4$ GeV$^2$
are presented in Fig.~4.
Once again one can find that evolution ``pushes" the
distributions to the small $x$ region.
It can be seen that $d\neq{\bar d}$ in the $\Sigma^+$ sea.
However, the prediction for the behavior of $d(x)-{\bar d}(x)$
depends strongly on the value of $\alpha$
-- for $\alpha=0.3$~GeV $d(x) <{\bar d}(x)$ in the smaller $x$ region and
$d(x) >{\bar d}(x)$ in the larger $x$ region,
while for the $\alpha=1.0$~GeV $d(x) >{\bar d}(x)$ in the smaller $x$ region and
$d(x) < {\bar d}(x)$ in the larger $x$ region.
This result is similar to our earlier finding on the $s(x)-{\bar s}(x)$
in the nucleon sea \cite{Cao}
employing the same light-cone baryon-meson fluctuation model,
which suggests that $SU(3)$ symmetry in the sea holds in this case.
We turn to the discussion about $\alpha$-dependence in our calculation.
Comparing Figs.~2 and 3 one can find that
for the $x[{\bar d}(x)-{\bar u}(x)]$, $x[{\bar d}(x)-{\bar s}(x)]$ and $x[{\bar u}(x)-{\bar s}(x)]$
the shape and maximum of asymmetries are very similar
for different $\alpha$, while the $x$ position at which the asymmetries
exhibit maxima shifts slightly.
The calculations with $\alpha=0.3$~GeV peak at about $x\simeq 0.06$
while the calculations with $\alpha=1.0$~GeV peak at about $x\simeq 0.1$.
Thus the calculations for the flavour asymmetry are not very sensitive
to the value of $\alpha$, and
$x$ being about $0.08$ is a good region to study the flavor asymmetry
in the $\Sigma^+$ sea.
This observation is consistent with the prediction given in Ref.~\cite{Alberg}
-- the region $0.1\leq x \leq 0.2$ should be a good one to
measure the flavour asymmetry in the $\Sigma$ sea.
The calculation for the $d(x)-{\bar d}(x)$ (see Fig.~4) is much more
sensitive to the value of $\alpha$ than that for the flavour asymmetry
-- the calculations with $\alpha=0.3$ GeV and $1.0$ GeV even give
opposite predictions for the $x$-dependence of $d(x)-{\bar d}(x)$.
We may expect that further calculation \cite{CaoPreparation} on the nucleon sea
employing the light-cone baryon-meson fluctuation model
may provide useful constraints on the value of $\alpha$,
and thereby give more definite predictions on
the sea quark content in the $\Sigma^+$ baryon.
In summary, besides the nucleon sea
the studies on the sea quark content of the other members of the baryon octet
are interesting and important since it is helpful to our understanding of
both the structure of the octet baryons and non-perturbative QCD effects
such as $SU(3)$ symmetry breaking and flavour asymmetry.
We calculated the sea quark content of the $\Sigma^+$ baryon employing
the light-cone baryon-meson fluctuation model.
It was found that the $\Sigma^+$ sea is flavour asymmetric
(${\bar d} > {\bar u}> {\bar s}$) and quark-antiquark asymmetric
($q \not= {\bar q}$).
Our prediction for the flavor asymmetry, ${\bar d} > {\bar u} > {\bar s}$, is significantly
different from the $SU(3)$ prediction (${\bar d} < {\bar u} < {\bar s}$),
while our prediction for the $d$-${\bar d}$ asymmetry is consistent with the
$SU(3)$ prediction.
\section*{Acknowledgments}
This work was partially supported by the Massey Postdoctoral Foundation, New Zealand.
| {
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\section{Introduction}
Density functional theory (DFT) provides a very successful
description of nuclei all over the periodic table.
Based on relatively simple functionals, which are adjusted in a
phenomenological way to the properties of infinite nuclear matter and
a few finite nuclei, this theory allows a highly accurate
reproduction of many nuclear structure data, such as binding
energies, radii, deformation parameters of finite nuclei and their
dependence on mass number and isospin. In addition to these static
properties, one can use the nuclear response to external multipole
fields to investigate the dynamics of such systems. In the framework
of time dependent density functional theory, this response can be
calculated from the linearized Bethe Salpeter equation using an
effective interaction derived from the same functional.
A very successful scheme of this type is covariant density functional theory
(CDFT). It is based on Lorentz invariance, connecting in a
consistent way the spin and spatial degrees of freedom of the nucleus.
Therefore, it needs only a relatively small number of parameters which are
adjusted to reproduce a set of bulk properties of spherical closed-shell
nuclei. Numerous works have shown that observations involving both ground
state and excited state phenomena, can be nicely interpreted in a relativistic framework.
The most popular applications of this type are based on the Walecka
model~\cite{SW.86}, where the nucleus is described as a system of
Dirac nucleons interacting with each other via the exchange of
virtual mesons with finite mass and the electromagnetic field through
an effective Lagrangian. In the mean field approximation this yields
to various contributions to the nuclear self energy depending on the
quantum numbers of these mesons. Early investigations have shown that this simple ansatz is not able to describe the incompressibility of infinite nuclear matter nor the surface properties of finite nuclei such as nuclear deformations. For that reason, a medium dependence has been introduced by including nonlinear meson self-interaction terms in the Lagrangian~\cite{BB.77}.
Several very successful phenomenological RMF interactions of this type have been adopted, as for instance the popular set NL3~\cite{NL3}. Closer to the concept of density functional theory are models with an explicit density dependence for the meson nucleon couplings. This density dependence can be calculated from first principles in a microscopic Dirac-Brueckner scheme~\cite{FLW.95} or it can be adjusted in a completely phenomenological way to properties of finite nuclei~\cite{TW.99,DD-ME2}.
One of the advantages of density functional theory is the fact that
with a proper choice of the parameters the success of RMF for nuclear
ground states ensures also a good basis upon which one can apply
time-dependent density functional theory to study nuclear
excitations. In order to investigate the dynamic behavior of the
nuclear system, one considers oscillations around the self-consistent
static solution. This can be done by solving the time dependent
relativistic mean field equations (TDRMF)~\cite{VBR.95} or, in the
limit of small amplitudes, by using the relativistic random phase
approximation (RRPA)~\cite{RMG.01}. The corresponding eigen modes can
be determined either by diagonalizing the RRPA equation in an
appropriate basis or by solving the linear response equations in a
time-dependent external field. This requires a matrix inversion for
given frequency $\omega$.
These two methods lead in principle to exactly identical results.
There are, however, cases where one of them is clearly preferable.
The proper treatment of the coupling to the continuum is such a case,
which can be solved in a very elegant way, by the solution of the
Bethe Salpeter equation within the response formalism.
We recall that the spectrum of the Dirac equations has a discrete and
a continuous part. For the ground state properties of the nucleus,
one needs only the single particle wave functions of the occupied
orbitals in the Fermi sea. They are either determined by solving the
corresponding differential equations in $r$-space or by expansion in
an appropriate basis, given for for instance by a finite number of
eigenfunctions of a harmonic oscillator~\cite{GRT.90} or of a
Saxon-Woods potential in a finite box~\cite{ZMR.03}. For the bound
states both methods yield the same solutions with high accuracy.
However, this is no longer true for the states in the continuum. Here
we have, in the first case scattering solutions in $r$-space for each
energy with proper boundary conditions, while in the second case, a
finite number of discrete eigenstates which depend strongly on the
dimension of the expansion. They provide only a basis and have little
to do with physics.
These discrete eigenstates lead to a finite number of $ph$-configurations for
the solution of response equations. with a discrete spectrum. They provide us
with the so called \textit{spectral representation} of the response function
in contrast to the \textit{continuum representation}, where the exact
scattering states with the proper boundary conditions are used at each energy.
Self-consistent relativistic RPA (RRPA) calculations have a long history. The early investigations in the eighties~\cite{Fur.85,HG.89,SRM.89,DF.90} were based on the Walecka model with linear meson-nucleons couplings. They were able to describe the low-lying negative-parity excitations in $^{16}$O by the method of matrix diagonalization~\cite{Fur.85}, isoscalar giant resonances in light and medium nuclei~\cite{HG.89} by the solution of the linear response equation in the spectral representation, and the longitudinal response for quasi-elastic electron scattering with a proper treatment of the continuum.
The first RRPA calculations based on non-linear
models were carried out in the spectral
representation including only normal particle-hole ($ph$) pairs with
particles above the Fermi energy and holes in the Fermi sea. This
seemed to be a reasonable approximation, since the configurations
formed by particles in the Dirac sea and holes in the Fermi sea
($ah$-pairs) are more than 1.2 GeV away from the normal $ph$-pairs.
Indeed, a proper coupling to the Dirac sea and current conservation
was neglected in these investigations. They showed considerable
deviations from the results obtained form time-dependent
RMF-calculations with the same Lagrangian, particularly for isoscalar
excitations~\cite{VRL.99}. A fully self-consistent treatment
with current conservation requires the inclusion of a very large
number of $ah$-pairs connected with a considerable numerical effort.
Most of the very successful applications of RRPA theory based on
non-linear meson-nucleon coupling models in the last ten years have
been carried out in this way \cite{VWR.00,MGW.01,VPR.02,PNVR.05,NVR.05}.%
There are also relativistic continuum RPA calculations based on the
non-spectral representation of the response function using the single
particle Green's function in the continuum with proper boundary
conditions~\cite{SRM.89}. These calculations are done for
meson exchange forces with finite range. The early investigations
were based on linear models. Later on the method was
generalized to include non-linear coupling terms between the
mesons~\cite{Pie.00}. This leads to a a more sophisticated density
dependence which is crucial for a realistic description of giant
resonances in nuclei~\cite{Pie.00,Pie.01}.
Of course, because of the finite range of the effective force these models are
relatively complicated not only for static applications to triaxially deformed
or rotating nuclei, but also for investigations of nuclear dynamics, such as
the solution of the relativistic RPA or linear response equations for the
description of excited states. In particular one needs simpler forces for
applications going beyond the mean field approach such as Particle Vibrational
Coupling (PVC)~\cite{LRT.08} or configuration mixing
calculations in the framework of the Generator Coordinate Method
(GCM)~\cite{NVR.06}. Therefore over the years several
attempts have been made to develop relativistic point coupling (PC) models
with forces of zero range~\cite{MM.89}, in analogy to
non-relativistic Skyrme-functionals. but only recently parameter sets have
been found, which are comparable in quality to the density dependent
meson-exchange models~\cite{BMM.02,NVLR.08b}.
PC models do not contain mesonic degrees of freedom and are therefore
closer to the philosophy of the density functional theory. Their
essential advantage is of course the fact that the zero range of the
effective interaction reduces considerably the numerical effort in
practical applications. Because of their simplicity they are nowadays
much used in many complex calculations going beyond the mean field
approach~\cite{LRT.08,NVR.06}.
However, so far they have not been used much for the dynamic
investigations and it is only quite recently that a code has been
developed to diagonalize the RPA equations for relativistic Point
Coupling models~\cite{NVR.05} and it has been shown that this latter
approach reproduces excitation and collective phenomena, in
particular Giant Multipole Resonances, with a quality comparable to
that of standard finite-range forces.
This manuscript is devoted to an investigation of relativistic point
coupling models with an exact treatment of the coupling to the
continuum. The relativistic response equations are solved both in the
continuum and in the spectral representation and the corresponding
results are compared. We use the Lagrangian PC-F1~\cite{BMM.02},
which is capable of reproducing a wide range of experimental data.
The paper is organized in the following way: In Sec.~\ref{PCRMF}
we present the main characteristics of the point coupling RMF theory,
while the relativistic RPA equations are derived in Sec.~\ref{CRPA}.
The proper treatment of the continuum in connection with point
coupling models is discussed in Sec.~\ref{continuum} and in
Sec.~\ref{MGR} we finally present applications of this method for the
spectra of in spherical nuclei. In particular we calculate the
strength function of Isoscalar and Isovector Giant Resonances as well
as their contributions to their respective energy weighted sum rules.
The results are summarized in Sec.~\ref{summary}.
\section{Relativistic mean field theory of zero range.}
\label{PCRMF}
As in all the relativistic models, the nucleons are described as
point like Dirac particles. In contrast to the Walecka
model, however, where these particles interact by the
exchange of effective mesons with finite mass, point coupling
models~\cite{MM.89} neglect mesonic degrees of
freedom and consider only interactions with zero range. In principle,
these models are similar to the Nambu Jona-Lasinio model
\cite{NJL.61a} used extensively in hadron physics. There is, however,
an important difference: in order to obtain a satisfactory description
of the nuclear surface properties one needs gradient terms in the
Lagrangian simulating a finite range of the interaction.
A general point-coupling effective Lagrangian is constructed to be consistent
with the underlying symmetries of $QCD$ (e.g., Lorentz covariance, gauge
invariance, and chiral symmetry). It should in principle contain every
possible term, allowed by these symmetries, but at the same time should also
be described by the least possible number of parameters in order to give a quantitative solution.
In this work we use the point coupling Lagrangian introduced by
Buervenich et al. in Ref.~\cite{BMM.02}. It presents an expansion in
powers of the nucleon scalar, vector and isovector-vector densities.
The Lagrangian
\begin{equation}
\mathcal{L}= \mathcal{L}_{\mathrm{free}}+\mathcal{L}_{\mathrm{4f}%
}+\mathcal{L}_{\mathrm{hot}}+\mathcal{L}_{\mathrm{der}}+\mathcal{L}%
_{\mathrm{em}}%
\label{Lag-PC}%
\end{equation}
consists of the term for free nucleons:
\begin{equation}
\mathcal{L}_{\mathrm{free}}=\bar{\psi}(i\gamma_{\mu}\partial^{\mu}-m_{N})\psi,
\label{Lparts}%
\end{equation}
the term for normal four-fermion interactions
\begin{align}
\mathcal{L}_{\mathrm{4f}}= & -\frac{\alpha_{S}}{2}(\bar{\psi}\psi)(\bar
{\psi}\psi)-\frac{\alpha_{V}}{2}(\bar{\psi}\gamma_{\mu}\psi)(\bar{\psi}%
\gamma^{\mu}\psi)\label{L_4f}\\
& -\frac{\alpha_{TS}}{2}(\bar{\psi}\vec{\tau}\psi)(\bar{\psi}\vec{\tau}%
\psi)-\frac{\alpha_{TV}}{2}(\bar{\psi}\vec{\tau}\gamma_{\mu}\psi)(\bar{\psi
}\vec{\tau}\gamma^{\mu}\psi),\nonumber
\end{align}
the term for higher order terms leading in mean field approximation
to a density dependence
\begin{equation}
\mathcal{L}_{\mathrm{hot}}=-\frac{\beta_{S}}{3}(\bar{\psi}\psi)^{3}%
-\frac{\gamma_{S}}{4}(\bar{\psi}\psi)^{4}-\frac{\gamma_{V}}{4}[(\bar{\psi
}\gamma_{\mu}\psi)(\bar{\psi}\gamma^{\mu}\psi)]^{2}, \label{L_hot}%
\end{equation}
the term containing derivative terms which simulate in a simple way
the finite range of the forces:
\begin{align}
\mathcal{L}_{\mathrm{der}} & =-\frac{\delta_{S}}{2}(\partial_{\mu}\bar{\psi
}\psi)(\partial^{\mu}\bar{\psi}\psi)-\frac{\delta_{V}}{2}(\partial_{\mu}%
\bar{\psi}\gamma_{\nu}\psi)(\partial^{\mu}\bar{\psi}\gamma^{\nu}%
\psi)\nonumber\\
& -\frac{\delta_{TS}}{2}(\partial_{\mu}\bar{\psi}\vec{\tau}\psi
)(\partial^{\mu}\bar{\psi}\vec{\tau}\psi)\label{L_der} -\frac{\delta_{TV}}{2}(\partial_{\mu}\bar{\psi}\vec{\tau}\gamma_{\nu}%
\psi)(\partial^{\mu}\bar{\psi}\vec{\tau}\gamma^{\nu}\psi)
\end{align}
and finally the electro-magnetic part%
\begin{equation}
\mathcal{L}_{\mathrm{em}}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{e}{2}%
(1-\tau_{3})A_{\mu}\bar{\psi}\gamma^{\mu}\psi.
\end{equation}
In these equations, $\psi$ represents the nucleon spinors. The subscripts $S$
and $V$ are attributed to scalar and vector fields, while the subscript $T$ is
attributed to isovector fields. As usual, vectors in isospin space are denoted
by arrows, where symbols in bold indicate vectors in ordinary
three-dimensional coordinate space.
From this Lagrangian and the corresponding energy momentum tensor we
can
derive a relativistic energy density functional. It has the form:%
\begin{equation}
\mathcal{E}_{\mathrm{RMF}}[\hat{\rho},t]=\int d^{3}r~{H(\bm{r},t}),
\label{Energy}%
\end{equation}
where the energy density
\begin{equation}
H(\bm{r},t)=H_{\mathrm{kin}}(\bm{r},t)+H_{\mathrm{int}}%
(\bm{r},t)+H_{\mathrm{em}}(\bm{r},t) \label{energy_density}%
\end{equation}
consists of a kinetic part
\begin{equation}
H_{\mathrm{kin}}(\bm{r},t)=\sum_{i}^{A}\,{\bar{\psi}_{i}(\bm{r},t)\left(
\bm{\alpha}\bm{p}+\beta m-m\right) \psi_{i}(\bm{r},t)}, \label{kinetic}%
\end{equation}
an interaction part
\begin{align}
H_{\mathrm{int}}(\bm{r},t) & =\frac{\alpha_{S}}{2}\rho_{S}^{2}+\frac
{\beta_{S}}{3}\rho_{S}^{3}+\frac{\gamma_{S}}{4}\rho_{S}^{4}+\frac{\delta_{S}%
}{2}\rho_{S}\Delta\rho_{S}\nonumber\label{E12}\\
& +\frac{\alpha_{V}}{2}j_{\mu}j^{\mu}+\frac{\gamma_{V}}{4}(j_{\mu}j^{\mu
})^{2}+\frac{\delta_{V}}{2}j_{\mu}\triangle j^{\mu}\\
& +\frac{\alpha_{TV}}{2}\vec{j}_{TV}^{\mu}\cdot\vec{j}_{TV\mu}+\frac
{\delta_{TV}}{2}\vec{j}_{TV}^{\mu}\cdot\triangle(\vec{j}_{TV})_{\mu}\nonumber
\end{align}
and an electromagnetic part
\begin{equation}
H_{\mathrm{em}}(\bm{r},t)=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-F^{0\mu}\partial
_{0}A_{\mu}+eA_{\mu}j_{p}^{\mu}.
\end{equation}
The interaction part depends on the local densities:
\begin{align}
\rho_{S}(\bm{r},t) & =\sum_{i}^{A}\bar{\psi}_{i}(\bm{r},t)\psi
_{i}(\bm{r},t),\label{densities}\\
\rho_{V}(\bm{r},t) & =\sum_{i}^{A}\bar{\psi}_{i}(\bm{r},t)\gamma_{0}\psi
_{i}(\bm{r},t),\\
\rho_{TS}(\bm{r},t) & =\sum_{i}^{A}\bar{\psi}_{i}(\bm{r},t)\tau_{3}\psi
_{i}(\bm{r},t),\\
\rho_{TV}(\bm{r},t) & =\sum_{i}^{A}\bar{\psi}_{i}(\bm{r},t)\tau_{3}
\gamma_{0}\psi_{i}(\bm{r},t)
\end{align}
and currents
\begin{align}
j_{V}^{\mu}(\bm{r},t) & =\sum_{i}\bar{\psi}_{i}(\bm{r},t)\gamma^{\mu}%
\psi_{i}(\bm{r},t),\label{E13c}\\
\vec{j}_{TV}^{\mu}(\bm{r},t) & =\sum_{i}\bar{\psi}_{i}(\bm{r},t)\vec{\tau
}\gamma^{\mu}\psi_{i}(\bm{r},t).
\end{align}
As in all relativistic mean field models, the \textit{no-sea}
approximation is used in the calculations of the nuclear densities by
summing only over the single-particle states with energies in the
Fermi sea. Vacuum polarization effects are not taken into account
explicitly but only in a global way by the correct choice of the Lagrangian
parameters. All interactions in the Lagrangian
(\ref{Lag-PC}) are then expressed in terms of the corresponding local
densities
Many effects, which go beyond mean field, seem to be neglected on the
classical level, such as Fock-terms, vacuum polarization, short range
Brueckner correlations etc. However, the coupling constants of the method are
adjusted to experimental data, which, of course, contain all these effects and
many more. Therefore these effects are not neglected. On the contrary, they
are taken into account in an effective way. This concept of RMF methods is
therefore equivalent to that of density functional theory.
The time-dependent variational principle
\begin{equation}
\delta\int\left\{ i\langle\Phi(t)|\frac{\partial}{dt}|\Phi(t)\rangle
-E[\hat{\rho}(t)]\right\} \, dt=0 \label{TD_variation}%
\end{equation}
allows us to derive from the energy density functional $E[\hat{\rho}]$ an
equation of motion for the time-dependent relativistic single particle
density:
\begin{equation}
\label{density_matrix}\hat\rho(\mathbf{r},\mathbf{r^{\prime}},t)=\sum_{i}^{A}
|\psi_{i}(\mathbf{r},t)\rangle\langle\psi_{i}(\mathbf{r^{\prime}},t)|,
\end{equation}
which has the form
\begin{equation}
\label{TDRMF}i\partial_{t}\hat\rho(t)=[\hat{h}(\hat\rho(t)),\hat\rho(t)].
\end{equation}
The self energy, i.e. the single particle hamiltonian
$\hat{h}(\hat\rho(t))$ is obtained as the functional derivative of
the energy density functional with respect to the relativistic
density matrix:
\begin{equation}
\hat{h}=\frac{\delta E[\hat{\rho}]}{\delta\hat{\rho}}. \label{hph1}
\end{equation}%
This yields the Dirac hamiltonian:
\begin{equation}
\hat{h}=\bm{\alpha}[-i\bm{\nabla}-\bm{V}(\bm{r},t)]%
+V(\bm{r},t)+\beta(m+S(\bm{r},t))%
\label{Dirac-hamiltonian}
\end{equation}
with the self-consistent scalar and vector potentials
\begin{align}
S(\bm{r},t) & = \Sigma_{S}(\bm{r},t)+\vec{\tau}\cdot\vec{\Sigma}%
_{TS}(\bm{r},t),\\
V^{\mu}(\bm{r},t) & = \Sigma^{\mu}(\bm{r},t)+\vec{\tau}\cdot\vec{\Sigma
}^{\mu}_{T}(\bm{r},t).
\end{align}
The nucleon isoscalar-scalar, isovector-scalar, isoscalar-vector and
isovector-vector self-energies are density dependent and defined by the
following relations:
\begin{align}
\Sigma_{S} & = \alpha_{S}\rho_{S}+ \beta_{S}\rho_{S}^{2}+\gamma_{S}\rho
_{S}^{3}-\delta_{S}\Delta\rho_{S},\\
\vec{\Sigma}_{TS} & = \alpha_{TS}\rho_{TS}-\delta_{TS}\Delta\rho_{TS},\\
\Sigma^{\mu} & = \alpha_{V}\rho_{V}+\gamma_{V}\rho_{V}^{3}-\delta_{V}
\Delta\rho_{V} -eA^{\mu}\frac{1-\tau_{3}}{2},\\
\vec{\Sigma}^{\mu}_{T} & = \alpha_{TV}\rho_{TV}-\delta_{TV}\Delta\rho_{TV}.
\end{align}
Here we have neglected retardation effects, i.e. second derivatives with
respect to the time for the various densities.
In the static limit we have
\begin{equation}
\label{static}[\hat{h}(\hat\rho),\hat\rho]=0,
\end{equation}
thus the static density $\hat\rho_{0}$ is obtained from the solution of the
self-consistent Dirac equations upon all the nucleons with eigenvalues
$\varepsilon_{k}$ and eigenfunctions $\psi_{k}(r)$:
\begin{equation}
\label{Dirac}\hat{h}|\psi_{k}(\bm{r})\rangle=\varepsilon_{k}|\psi_{k}%
(\bm{r})\rangle.
\end{equation}
For spherical symmetry the spinors have the form:
\begin{equation}
|\mathcal{\psi}_{n\kappa\,m}(\bm{r})\rangle=\frac{1}{r}{\binom{f_{n\kappa
}(r)\mathcal{Y}_{\kappa\,m}(\Omega)}{ig_{n\kappa}(r)\mathcal{Y}_{\bar{\kappa
}\,m}(\Omega)}}. \label{wavefunctions}%
\end{equation}
The subscripts $n$, $\kappa$ and $m$ are principal and angular momentum
quantum numbers; $\kappa=\mp(j+\frac{1}{2})$ for $j=l\pm\frac{1}{2}$, where
$j$ and $l$ are the total and the orbital angular momenta of the nucleon. As
usual, $m$ is the $z$ component of the total angular momentum. The spherical
spinors $\mathcal{Y}_{\kappa\,m}(\Omega)$ are given in terms of spherical
harmonics $Y_{lm_{l}}(\Omega)$ and Pauli spinors $\chi_{m_{s}}$ as:
\begin{equation}
\mathcal{Y}_{\kappa\,m}(\Omega)=\sum_{m_{l}m_{s}}(\frac{1}{2}m_{s}%
lm_{l}|jm)Y_{lm_{l}}(\Omega)\chi_{m_{s}}, \label{harmonics}%
\end{equation}
while the functions $f_{i}(r)$ and $g_{i}(r)$ satisfy the static
radial Dirac equations:
\begin{equation}
\left(
\begin{array}
[c]{cc}%
V+S & -\partial_{r}+\frac{\kappa}{r}\\
\partial_{r}+\frac{\kappa}{r} & V-S-2m
\end{array}
\right) \left(
\begin{array}
[c]{c}%
f_{i}(r)\\
g_{i}(r)
\end{array}
\right) =\left(
\begin{array}
[c]{c}%
f_{i}(r)\\
g_{i}(r)
\end{array}
\right) \varepsilon_{i}. \label{Dirac-radial}
\end{equation}
\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.5}%
\begin{tabular}
[c]{|l|r@{.}l|}\hline ~~Coupling const.~~ &
\multicolumn{2}{c|}{PC-F1}\\\hline
~~~~~~~~~$\alpha_{S}$~~~ & ~~-14 & 935894~~~\\
~~~~~~~~~$\delta_{S}$~~~ & ~~ -0 & 634576\\\hline
~~~~~~~~~$\alpha_{V}$~~~ & 10 & 098025\\
~~~~~~~~~$\delta_{V}$~~~ & -0 & 180746\\\hline
~~~~~~~~~$\alpha_{TS}$ & 0 & 0\\
~~~~~~~~~$\delta_{TS}$ & 0 & 0\\\hline
~~~~~~~~~$\alpha_{TV}$ & 1 & 350268\\
~~~~~~~~~$\delta_{TV}$ & -0 & 063680\\\hline\hline
~~~~~~~~~$\beta_{S}$ & 22 & 994736\\
~~~~~~~~~$\gamma_{S}$ & -66 & 769116\\\hline ~~~~~~~~~$\gamma_{V}$ &
-8 & 917323\\\hline
\end{tabular}
\caption{The coupling constants in the parameter set PC-F1 resulting
from the fitting procedure in Ref.~\cite{BMM.02}. The units are
[fm$^{-2}$] for the constants $\alpha$ of the quadratic terms,
[fm$^{-4}$] for the constants $\delta$ of the derivative terms,
[fm$^{-5}$] for the constants $\beta$ of the cubic terms, and
[fm$^{-8}$] for the constants $\gamma$ of the quartic terms
in the Lagrangian.}%
\label{tab1}%
\end{table}
The point coupling Lagrangian used in this work contains eleven
coupling constants. Based on an extensive multi parameter $\chi^{2}$
minimization procedure, B\"urvenich et al.~\cite{BMM.02} have
adjusted the parameter set $PC$-$F1$ to reproduce ground state
properties of infinite nuclear matter and spherical doubly closed
shell nuclei. This set is listed in Table~\ref{tab1} and it has
been tested in the calculation of many ground state properties of
spherical and deformed nuclei all over the periodic table. The
results are very well comparable with reasonable effective
meson-exchange interactions.
The nuclear ground state is defined as the equilibrium point of the functional
(\ref{Energy}), thus, is associated with the density which minimizes
$E_{\mathrm{RMF}}[\hat{\rho}]$. Furthermore, small oscillations around this
equilibrium point correspond to the vibrational nuclear states. They are
usually described within the harmonic approximation, that is, using linear
response theory. In nuclear physics, this is the so called Random Phase
Approximation (RPA) which has been already mentioned in our discussion and
will be described in more detail in the next section.
\section{Relativistic RPA formalism}
\label{CRPA}
Under the influence of an external field $F(\omega)$ oscillating with
the frequency $\omega$ the nucleus is excited. The cross section of
this process is proportional to the strength function:
\begin{align}
S(\omega)&=-\frac{1}{\pi}\operatorname{Im}%
\sum_{\alpha\beta\alpha^{\prime}\beta^{\prime}}%
F_{\alpha\beta^{\ast}}R_{\alpha\beta\alpha^{\prime}\beta^{\prime}}(\omega)%
F_{\alpha^{\prime}\beta^{\prime}}\nonumber\\
&:=-\frac{1}{\pi}\operatorname{Im}R_{FF}(\omega), \label{strenghtfunction}
\end{align}
where $F_{\alpha\beta}$ is the operator inducing the reaction and
$R_{\alpha\beta\gamma\delta}(\omega)$ is the response function which,
in an arbitrary representation indicated by the Greek indices
$\alpha,\beta,\ldots$ (e.g. the ($\bm{r},s)$-representation) is
defined as:
\begin{equation}
\label{full_response}
R_{\alpha\beta\alpha^{\prime}\beta^{\prime}}(\omega)%
=\sum\limits_{\nu}\left\{\frac{\langle0|a_\beta^+a _\alpha|\nu\rangle \langle\nu|a_{\alpha^{\prime}}^{+}a _{\beta^{\prime}}|0\rangle}{\omega-E_{\nu}+E_{0}+i\eta}\right. -\left.\frac{\langle\nu|a_\beta^+a _{\alpha}|0\rangle
\langle0|a_{\alpha^{\prime}}^{+}a _{\beta^{\prime}}|\nu\rangle}{\omega+E_{\nu}-E_{0}+i\eta}\right\}.
\end{equation}
The imaginary part $i\eta$ is infinitesimal and is introduced in
order to fulfill the proper boundary conditions and to prevent
$R(\omega)$ from diverging at $\omega=E_{\nu}-E_{0}$. We use here the
response derived from the retarded Green's functions as defined in
Ref.~\cite{FW.71}
In the independent particle model, $|0\rangle$ is the Slater
determinant of the ground state, formed by the self-consistent
solutions of the Dirac equation~(\ref{Dirac}) and
$|\nu\rangle=a_{p}^{+}a_{h}|0\rangle$ are $ph$-states, while $E_{0}$
and $E_{\nu}$ are the corresponding energies. In the basis
$|k\rangle$, where the single particle
hamiltonian~(\ref{Dirac-hamiltonian}) is diagonal we obtain the free
response function:
\begin{equation}
R_{klk^{\prime}l^{\prime}}^{\,0}(\omega)=\frac{n_{k}-n_{l}}{\omega
-\varepsilon_{k}+\varepsilon_{l}+i\eta}\delta_{kk^{\prime}}\delta_{ll^{\prime
}} \label{R0}%
\end{equation}
with the occupation factors:
\begin{equation}
n_{k}=\langle0|a_{k}^{+}a_{k} |0\rangle=\left\{
\begin{array}
[c]{ll}%
1 & \text{for hole states with }\varepsilon_{k}\leq\varepsilon_{F}\\
0 & \text{for particle states with }\varepsilon_{k}>\varepsilon_{F}%
\end{array}
\right.
\end{equation}
The full response of Eq.~(\ref{full_response}) contains the transition densities:
\begin{equation}
\rho_{\alpha\beta}^{\nu}=\langle0|a_\beta^{+}a _\alpha|\nu\rangle.
\label{transition_density}%
\end{equation}
They can be deduced from the time-dependent density matrix in
Eq.~(\ref{density_matrix}), which is derived from the variational
principle in Eq.~(\ref{TD_variation}).
In the small amplitude limit one uses the linear response
approximation to obtain the full response $R(\omega)$ of
Eq.~(\ref{full_response}) as the solution of the \textit{linearized
Bethe-Salpeter equation}:
\begin{equation}
\label{Bethe-Salpeter}
R_{\alpha\beta\alpha^\prime\beta^\prime} (\omega)=R_{\alpha\beta\alpha^\prime\beta^\prime}^{\,0}(\omega)
+\sum_{\gamma\delta\gamma^\prime\delta^\prime}\,\,R^0_{\alpha\beta\gamma\delta}(\omega)
V_{\gamma\delta\gamma^\prime\delta^\prime}^{\text{ph}} R_{\gamma^\prime\delta^\prime\alpha^\prime\beta^\prime}(\omega).
\end{equation}
The relativistic residual interaction is found as the second
derivative of the energy density functional~(\ref{Energy}) with
respect to the density matrix
\begin{equation}
V_{\alpha\beta\alpha^{\prime}\beta^{\prime}}^{\text{ph}}=\frac{\delta
^{2}E[\hat{\rho}]}{\delta\hat{\rho}_{\alpha\beta}\delta\hat{\rho}%
_{\alpha^{\prime}\beta^{\prime}}}.%
\label{energy_variation}%
\end{equation}
Once again, we have neglected retardation and this effective
interaction has to be calculated at the static density.
In a short hand notation the response equation~(\ref{Bethe-Salpeter})
has the formal solution
\begin{equation}
R (\omega)=(1-R ^{\,0}(\omega)V ^{\text{ph}})^{-1}R^{\,0}(\omega)
\label{formal-solution}%
\end{equation}
or introducing the inverse of $R^{\,0}$ we have%
\begin{equation}
R (\omega)=\frac{1}{R^{\,0}(\omega)^{-1}-V ^{\text{ph}}}
\label{RPA-inversion}%
\end{equation}
The evaluation of the strength function (\ref{strenghtfunction})
requires therefore three steps. The starting point is the calculation
of the free response function $R^{\,0}(\omega)$. In the next step one
determines the interaction $V ^{\text{ph}}$ and finally one solves the response equation by the inversion
(\ref{formal-solution}). In details there are several methods to
proceed. In particular one can choose various basis sets to solve
these equations.
a) As we have seen in Eq. (\ref{R0}) the free response has a
particularly simple form in the basis of Dirac spinors (\textit{Dirac
basis}) diagonalizing the self-consistent mean field
equation~(\ref{Dirac}). This is in particular simple for cases where
the Dirac equation is solved in a discrete basis, as for instance the
oscillator basis~\cite{GRT.90} or in a Saxon Woods
basis~\cite{ZMR.03} determined by the solution of the Dirac equation
in a box with finite size. However, the simplicity in the calculation
of $R^{\,0} (\omega)$ is compensated by the computational effort
required in the next steps. First we have to calculate a large number
of matrix elements for the interaction~(\ref{energy_variation}) in
the basis of the corresponding $ph$-states and in a second step the
matrix $(1-R^{\,0}(\omega)V^{\text{ph}})$ has to be inverted for each
value of the frequency $\omega$. In general the number of single
particle states is rather large and this leads to a huge number of
$ph$-states, requiring considerable computational sources, not only
in memory but also in computer time. This is in particular a problem
in the case of deformed nuclei. By this reason this method can only
be used successfully for light spherical nuclei, where the number of
$ph$-states is limited.
b) The inversion is particular simple in the \textit{RPA-basis}.
Inserting expression~(\ref{R0}) into Eq.~(\ref{RPA-inversion}) we
find that the response function is equivalent to the resolvent of the
RPA matrix
\begin{equation}
R^{\,0}(\omega)^{-1}-V ^{\text{ph}}=\omega-\left(
\begin{array}
[c]{cc}%
A & B\\
-B^{\ast} & -A^{\ast}%
\end{array}
\right) \label{RPA-matrix}%
\end{equation}
where%
\begin{align}
A_{php^{\prime}h^{\prime}} & =(\varepsilon_{p}-\varepsilon_{h})\delta
_{pp^{\prime}}\delta_{hh^{\prime}}+V_{php^{\prime}h^{\prime}}^{\text{ph}%
},\text{\ }\\
B_{php^{\prime}h^{\prime}} & =V_{phh^{\prime}p^{\prime}}^{\text{ph}}%
\end{align}
Of course, the calculation of this matrix requires the same numerical effort
as the evaluation of $V ^{\text{ph}}$ in the Dirac basis discussed above.
However there exist standard routines for the diagonalization of the
RPA-matrix
\begin{equation}
\left(
\begin{array}
[c]{cc}%
A & B\\
-B^{\ast} & -A^{\ast}%
\end{array}
\right) \left(
\begin{array}
[c]{c}%
X\\
Y
\end{array}
\right) _{\mu}=\left(
\begin{array}
[c]{c}%
X\\
Y
\end{array}
\right) _{\mu}\Omega_{\mu}\label{RPA-diagonalization}%
\end{equation}
and this diagonalization has to be carried out only once, whereas the
inversion of the response equation has to be done for each value of
the frequency $\omega$. In the RPA-basis given by the eigenvectors
$|\mu\rangle$ the reduced response function defined in
Eq.~(\ref{reduced-response}) has a particular simple form
\begin{equation}
R_{cc^{\prime}}(\omega)={\sum\limits_{\mu>0}}\frac{\langle0|Q_{c}^{\dag}%
|\mu\rangle\langle\mu|Q_{c^{\prime}}|0\rangle}{\omega-\Omega_{\mu}+i\eta
}-\frac{\langle\mu|Q_{c}^{\dag}|0\rangle\langle0|Q_{c^{\prime}}|\mu\rangle
}{\omega+\Omega_{\mu}+i\eta}.
\label{Resp-RPA}%
\end{equation}
Using
\begin{equation}
\langle0|F|\mu\rangle=\sum\limits_{ph}F_{ph}(X_{ph}^{\mu}+Y_{ph}^{\mu})
\end{equation}
we find for $R_{FF}(\omega)$%
\begin{equation}
R_{FF}(\omega)={\sum\limits_{\mu>0}}\frac{|\langle0|F|\mu\rangle|^{2}}%
{\omega-\Omega_{\mu}+i\eta}-\frac{|\langle0|F|\mu\rangle|^{2}}{\omega
+\Omega_{\mu}+i\eta}%
\end{equation}
and for the strength function in Eq.~(\ref{strenghtfunction})%
\begin{align}
S(\omega+i\frac{\Delta}{2}) & =-\frac{1}{\pi}\operatorname{Im}R_{FF}%
(\omega+i\frac{\Delta}{2})\label{E54}\\
& ={\sum\limits_{\mu}}|\langle0|F|\mu\rangle|^{2}\frac{1}{2\pi}\frac{\Delta
}{(\omega-\Omega_{\mu})^{2}+\frac{1}{4}\Delta^{2}}\nonumber
\end{align}
Here $\Delta$ is a smearing parameter, which introduces a folding
with a Lorentzian and is introduced by numerical reasons.
c) In many cases the effective interaction
$V_{\alpha\beta\alpha^{\prime} \beta^{\prime}}^{\text{ph}}$
can formally be written as a sum of separable terms.
\begin{equation}
V_{\alpha\beta\alpha^{\prime}\beta^{\prime}}^{\text{ph}}=\sum_{c}
Q_{\alpha\beta}^{c}V_{c}^{\text{ph}}Q_{\alpha^{\prime}%
\beta^{\prime}}^{\dag\,c} \label{separable}%
\end{equation}
where $Q^{c}$ are single particle operators characterized by the
channel index $c.$ As discussed in Appendix~\ref{AppA}, this is
particular the case for the effective interaction of the relativistic
point coupling model PC-F1 used in the present investigation. Working
in the channels given by these operators $Q^{c}$ the numerical effort
can be simplified considerably.
We insert the effective interaction (\ref{separable}) into the
Bethe-Salpeter equation (\ref{Bethe-Salpeter}) and introducing the
reduced response function:
\begin{equation}
R_{cc^{\prime}}(\omega)=\sum_{\alpha\beta\alpha^{\prime}\beta^{\prime}%
}\,\,Q_{\alpha\beta}^{c\dag}R_{\alpha\beta\alpha^{\prime}\beta^{\prime}%
}(\omega)Q_{\alpha^{\prime}\beta^{\prime}}^{c^{\prime}},
\label{reduced-response}%
\end{equation}
equation (\ref{Bethe-Salpeter}) turns into the reduced Bethe Salpeter
equation
\begin{equation}
R _{cc^{\prime}}(\omega)
=R_{cc^{\prime}}^{\,0}(\omega)%
+\sum_{c^{\prime\prime}}%
R_{cc^{\prime\prime}}^{\,0}(\omega)V_{c^{\prime\prime}}^{\text{ph}%
}R _{c^{\prime\prime}c^{\prime}}(\omega). \label{reduced-response-eq}%
\end{equation}
which has the same formal solution as given in Eq.
(\ref{formal-solution}). In all cases, where one has a continuous
channel index $c$, as for instance the radial coordinate $r$, this is
an integral equation. In Eq. (\ref{reduced-response-eq}) the
interaction $V_{c }^{\text{ph}}$ is diagonal with respect to the
channel index $c$. This is not always the case. However, as we shall
see in Appendix~\ref{AppA}, the relativistic interaction PC-F1 can
be expressed to a large extent in this way. We have to allow only in
specific cases also for non-diagonal interactions
$V_{cc^{\prime}}^{\text{ph}}$, as for instance in the case of the
Coulomb force or in the case of derivative terms. This is a rather
simple extension of the present method and therefore, for the sake of
simplicity, we will restrict ourselves in the following to an
interaction diagonal in the cannel index $c$. If the external
operator $F$ in Eq. (\ref{strenghtfunction}) can be expressed by the
operators $Q_{c}$ as
\begin{equation}
F=\sum_{c}f_{c}Q_{c}%
\end{equation}
we finally obtain the strength function as%
\begin{equation}
S(\omega)=-\frac{1}{\pi}\operatorname{Im}R_{FF}=-\frac{1}{\pi}%
\operatorname{Im}\sum_{cc^{\prime}}f_{c}^{\ast}R_{cc^{\prime}} %
(\omega)f_{c^{\prime}} .%
\label{E55}%
\end{equation}
If $F$ cannot be expressed in terms of the operators $Q_{c}$ we
obtain $R_{FF}$ from the Bethe-Salpeter equation (\ref{Bethe-Salpeter}) as
\begin{equation}
R_{FF} (\omega) =R_{FF}^{\,0}(\omega) +\sum_{cc^{\prime}}R_{F c}^{\,0}(\omega)V_{c}(1-R ^{\,0}
(\omega)V ^{\text{ph}})_{cc^{\prime}}^{-1}R_{c^{\prime}F}^{\,0}(\omega).
\end{equation}
\section{Treatment of the continuum.}
\label{continuum}
As we have briefly discussed earlier, a proper treatment of the
continuum is not possible by using a discrete basis, because one
needs a tremendously large number of $ph$-states to fill up the
continuum with. Instead, it can only be properly taken into account
if one makes use of the more flexible linear response formalism in an
appropriate channel space.
Starting from Eq.~(\ref{reduced-response}) for the reduced response
function and using Eq.~(\ref{Resp-RPA}) we derive the following
expression for the reduced free response, which depends only on the
energy $\omega$ and the channel indices $c, c^{\prime}$:
\begin{equation}
R_{cc^{\prime}}^{0}(\omega)=%
\sum\limits_{ph}%
\frac{\langle h|Q_{c }^{+}|p\rangle%
\langle p|Q_{c^{\prime}}|h\rangle}%
{\omega-\varepsilon_{p}+\varepsilon_{h}}%
-\frac{\langle p|Q_{c}^{+}|h\rangle%
\langle h|Q_{c^{\prime}}|p\rangle}%
{\omega+\varepsilon_{p}-\varepsilon_{h}}%
\label{R0QQ}%
\end{equation}
where $h$ stands for occupied (hole) and $p$ for unoccupied
(particle) states. It is easy to show that the sum over $p$ can be
safely extended to run over the full space, since terms of the form
$\sum_{hh^{\prime}}$ vanish due to the cancellation of forward and
backward going parts. Using completeness we obtain:
\begin{eqnarray}
\label{QGQ}
R_{cc^{\prime}}^{0}(r,r';\omega) &=&
\sum_{h}\langle\,h|Q_{c}^{+} \frac{1}{\omega+\varepsilon_{h}-\hat{h}}Q_{c^{\prime}}-
Q_{c^{\prime}}\frac{1}{\omega-\varepsilon_{h}+\hat{h}}Q_{c}^{+}|h\rangle \nonumber \\
&=&
\sum_{h}\langle\,h|Q_{c}^{+}G(\omega+\varepsilon_{h})Q_{c^{\prime}}~+~Q_{c^{\prime}}G(-\omega+\varepsilon_{h})
Q_{c}^{+}|h\rangle.
\end{eqnarray}
Here, $\hat{h}$ is the Dirac hamiltonian (\ref{Dirac-hamiltonian}) and
$G(E)=1/(E-\hat{h})$ is the corresponding single particle Green's
function.
In this work we use relativistic zero range forces, thus it is
appropriate to work in coordinate space. The method described in the
following is a relativistic generalization of the method introduced
by Bertsch et al \cite{SB.75} for
non-relativistic zero range forces. In this case we solve the
response equation in $r$-space, which is considerably simpler than
the method introduced in Refs.~\cite{SRM.89} for finite range
forces.
In coordinate representation the indices $\alpha$,$\beta,\ldots$ in
Eq.~(\ref{full_response}) are abbreviations for the "coordinates"
$1=({\bm r}_1,d_1,s_1,t_1)$, where $s$ is the spin, $t$ the isospin
coordinate, and $d=1,2$ labels large and small components. Starting
from the energy density functional (\ref{Energy}) we find the
effective interaction in Eq.~(\ref{energy_variation}) to be of the
form (\ref{separable}):
\begin{equation}
V^{\text{ph}}(1,2)~=~%
{\displaystyle\sum\limits_{c}}
{\displaystyle\int\limits_{0}^{\infty}}
dr~Q_{c}^{(1)}(r)~\upsilon_{c}(r)~Q_{c}^{\dag(2)}(r)
\end{equation}
with the local channel operators $Q_{c}({r)}$ defined by
\begin{equation}
Q_{c}^{(1)}(r)=~\frac{\delta({r}-{r}_1)}{rr_{1}}\gamma^{(1)}_{D}
\left[ \sigma^{(1)}_{S}Y_{L} (\Omega_{1})\right]_{J} \tau^{(1)}_{T}
\label{channel}%
\end{equation}
where we distinguish the "coordinates" abbreviated by the upper index~(1) and the channel index ($r,c$) used in Eq.~(\ref{separable}). Due to this r-dependance, the dimension of the matrix $R^{0}_{cc^{\prime}}(r,r^{\prime};\omega)$ in the numerical applications will be the number of r-mesh points times eight, which represents the number of the covariant channels c, given in Table~\ref{tab6} of the Appendix~\ref{AppA}. This implies that all scalar, longitudinal, and transverse modes (isoscalar and isovector) are fully included and mixed by the matrix inversion of Eq.(\ref{formal-solution}).
This channel index has now a continuous part
given by the radial coordinate $r$ and a discrete part characterized
by the quantum numbers $c=(D,S,L,T)$ where the Dirac index $D$ runs
over three 2$\times$2 matrices $\gamma_D=$ $\gamma_{0},1,\gamma_{5}$
defined in Eq. (\ref{gamma-2}), $S=0,1$ is the spin, $L$ the orbital
angular momentum and $T=0,1$ the isospin. Further details are given
in Appendix~\ref{AppA}.
Inserting the channel operators (\ref{channel}) into
Eqs.~(\ref{reduced-response}) and (\ref{QGQ}) we obtain the reduced
free response function:
\begin{eqnarray}
\mathcal{R}_{cc^{\prime}}^{\,0}(r,r^{\prime};\omega)&=&{\sum\limits_{h\kappa}%
}\left\{ Q_{\kappa h}^{\ast c}Q_{\kappa h}^{c^{\prime}}\,\langle
h(r)|\gamma_{D}^{+}G_{\kappa} (r,r^{\prime};\omega+\varepsilon_{h}%
)\gamma_{D^{\prime}} |h(r^{\prime})\rangle\,\right.\\
&+&\left.Q_{h\kappa}^{\ast
c}Q_{h\kappa}^{c^{\prime}}\langle h(r^{\prime})|\gamma_{D^{\prime}}^{{}%
}G_{\kappa} (r^{\prime},r;-\omega+\varepsilon_{h})\gamma_{D }%
^{+}|h(r)\rangle\right\}.\nonumber
\label{E59}
\end{eqnarray}
The sum runs over all the occupied states (hole) states $h$ with the
2-dimensional radial Dirac spinor $\langle
h(r)|=(f^\ast_{h}(r)~g^\ast_{h}(r))$ in Eq. (\ref{Dirac-radial}) and
over all the quantum numbers $\kappa=(lj)$ compatible with the
selection rules in the reduced angular and isospin matrix elements
\begin{equation}
Q_{h\kappa}^{c}:{=~e}_{T_{c}}\langle\kappa_{h}||\left[ \sigma_{S_{c}}Y_{L_{c}} \right] _{J}||\kappa\rangle,
\end{equation}
where $e_{T_{c}}=1$ in the isoscalar channel ($T_{c}=0$) and
$e_{T_{c}}=\pm1$ (for protons or neutrons) in the isovector channel
($T_{c}=1$). The reduced matrix elements of the operator $\left[
\sigma_{S_{c}} Y_{L_{c}}^{{} }\right] _{J}$ contain integrations
over the orientation angles $\Omega$ and sums over the spin indices.
The matrix elements of the form $\langle
h|\gamma_{D}G(E)\gamma_{D^{\prime}}|h\rangle$ depend on $r$ and
$r^{\prime}$ and are obtained by summing over the Dirac indices
$d=1,2$ for large and small components.
The Green's function $G_{\kappa} (r,r^{\prime},E)$ describes the
propagation of a particle with the energy $E$ and the quantum numbers
$\kappa$ from $r$ to $r^{\prime}$. It can either be calculated by
\textit{spectral} or \textit{non-spectral} methods. In the
\textit{spectral }representation~\cite{BT.75} it is obtained as a
discrete sum
\begin{equation}
G_{\kappa} (r,r^{\prime};E)=\sum_{n}\frac{|n(r)\rangle\langle
n(r^{\prime})|}{E-\varepsilon_{n}}. \label{E49}%
\end{equation}
over a complete set of eigenstates $|n(r)\rangle$ of the radial Dirac
equation~(\ref{Dirac-radial}) with the quantum number $\kappa$ using
box boundary conditions (or an oscillator expansion). In this case
the continuum is discretized, in correspondence to the bound states
inside the potential. In principle, the radial quantum number $n$
runs over the whole single particle basis characterized by the
angular quantum number $\kappa$, but one can show that this is
identical to summing only over the unoccupied states, since the
hole-hole pairs in Eq.~(\ref{E59}) are not contributing, due to the
cancellation between forward and backward going part. Furthermore,
because of the no-sea approximation the states in the Dirac sea are
empty and therefore the sum over $n$ in Eq. (\ref{E49}) has also to
be extended over the negative energy states. This corresponds to the
sum over the $ah$-components discussed in the introduction. In
practical applications one has to restrict this infinite set by a
finite sum introducing an upper limit
$\epsilon_p-\epsilon_h<E^{ph}_{cut}$ in energy for the particle
states $p$ above the Fermi surface and a lower limit
$\epsilon_a-\epsilon_h > -E^{ah}_{cut}$ for the negative energy
solutions $a$ is introduced in order to make the - otherwise infinite
- sum, tractable. This leads to a discretized spectrum.
In the spectral representation the response function
$\mathcal{R}^{\,0}(\omega)$ has poles at the $ph$-energies
$\omega=\pm(\varepsilon_{p}-\varepsilon_{h})$ and the full response
function $\mathcal{R}(\omega)$ has poles at the eigenenergies
$\Omega_{\mu}$ of the RPA-equation (\ref{RPA-diagonalization}) in the
same restricted space. For real frequencies $\omega$ it is purely
real, and therefore the strength function vanishes everywhere apart
from these poles. For complex energies $\omega+i\Delta/2$, however,
these poles are shifted from the real axis and one obtains a
continuous spectrum, with the phenomenological width $\Delta$. This
procedure yields identical results as the diagonalization of the
RPA-matrix in (\ref{RPA-diagonalization}) along with a subsequent
folding with a Lorentzian as discussed in Eq. (\ref{E54}).
In the non-spectral or continuum approach~\cite{SB.75} the single
particle Green's function is constructed at each energy from two
linearly independent solutions of the Schroedinger equation with
different boundary conditions at $r=0$ and at $r\rightarrow\infty$.
In the relativistic case the Dirac-equation in $r$-space depending on
the quantum number $\kappa$ is a two-dimensional equation and
therefore the corresponding single particle Green's function is a
2$\times2$ matrix. Using the bracket notation of Dirac for the
2-dimensional spinors we can write \cite{Tam.92}:
\begin{equation}
G_{\kappa} (r,r^{\prime};E)=\left\{
\begin{array}
[c]{cc}%
|w _{\kappa}(r)\rangle\langle u^\ast_{\kappa}(r^{\prime})| & \text{for}%
\,\,r>r^{\prime}\\
|u _{\kappa}(r)\rangle\langle w^\ast_{\kappa}(r^{\prime})| & \text{for}%
\,\,r<r^{\prime}%
\end{array}
\right. \label{continuum-greens}%
\end{equation}
where $u(r)$ and $w(r)$ are two independent Dirac spinors~\cite{Tam.92}:
\begin{equation}
|u_{\kappa\,} (r)\rangle={\binom{f_{u}(r)}{g_{u}(r)},}\text{
\ \ \ \ \ \ \ \ }|w_{\kappa\,} (r)\rangle={\binom{f_{w}(r)}{g_{w}(r)}}
\label{scat_spinors}%
\end{equation}
normalized in such a way that the Wronskian
\begin{equation}
W=f_{w}(r)g_{u}(r)-g_{w}(r)f_{u}(r),
\end{equation}
which is independent of $r$, is normalized to unity. The solution
$u_{\kappa }(r)$ is regular at the origin and the solution
$w_{\kappa}(r)$ fulfills outgoing wave boundary conditions
\cite{Gre.90}. Further details are given in Appendix \ref{AppB}.
Provided that the free response function
$\mathcal{R}^{0}_{c,c^{\prime}}(r,r^{\prime};\omega)$ has been
properly derived, we are able to solve the reduced Bethe-Salpeter
equation~(\ref{reduced-response-eq})
\begin{eqnarray}
\mathcal{R}_{c,c^{\prime}}(r,r^{\prime};\omega)&=&
\mathcal{R}_{c,c^{\prime}}^{\,0}(r,r^{\prime};\omega)\\
&+&\sum_{c^{\prime\prime}}\int_{\,0}^{\infty}dr^{\prime\prime}\,
\mathcal{R}_{c,c^{\prime\prime}}^{0}(r,r^{\prime\prime};
\omega)\frac{\upsilon_{c^{\prime\prime}}(r^{\prime\prime})}{r^{\prime
\prime\,\,2}}\mathcal{R}_{c^{\prime\prime},c^{\prime}}(r^{\prime\prime
},r^{\prime};\omega).
\nonumber%
\label{E65}%
\end{eqnarray}
where the index $c^{\prime\prime}$ runs over the various discrete
channels given in Table~\ref{tab6}. Finally the strength function is
obtained as:
\begin{align}
\label{E68}
S(\omega)& =-\frac{1}{\pi}\operatorname{Im}\mathcal{R}_{FF}\nonumber\\
& =-\frac{1}{\pi}\operatorname{Im}%
{\displaystyle\iint\limits_{\,0}^{\infty}}
drdr^{\prime}F_{c}^{\ast}(r)\mathcal{R}_{cc^{\prime}} (r,r^{\prime}%
;\omega)F_{c^{\prime}} (r^{\prime}).
\end{align}
The sum rules are defined as moments of the strength function $S(\omega)$:
\begin{equation}
m_{k}=\int_{0}^{\infty}\omega^{k}S(\omega)\,d\omega.
\label{moments}%
\end{equation}
They are helpful to characterize the spectral distribution of the oscillator
strength. In particular they allow us to define the centroid energy by the
ratio%
\begin{equation}
E_{c}=\frac{m_{1}}{m_{0}}.%
\label{centroid-energy}%
\end{equation}
This quantity can be compared directly with experimental values. Of
course, in most experiments only a restricted energy range is
accessible and therefore one also has to restrict the integration in
Eq.~(\ref{moments}) to the same energy window.
Other important quantities are transition densities in various channels $c$
with respect to the operator $F$
\begin{equation}
\delta\rho_{c}(r;\omega)=\sum_{c^{\prime}}\int_{o}^{\infty}dr^{\prime}\mathcal{R}_{cc^{\prime}%
} (r,r^{\prime};\omega)F_{c^{\prime}} (r^{\prime}) \label{traden}%
\end{equation}
as for instance the neutron and proton transition densities:
\begin{equation}
\delta\rho(r)_{n,p}=\delta\rho_{T=0}(r;\omega)\pm\delta\rho_{T=1}(r;\omega)
\label{traden-pn}%
\end{equation}
\section{Applications}
\label{MGR}
In the previous section we briefly described how conventional RPA
methods treat the continuum part of the spectrum through the
introduction of a potential "wall" far from the nucleus. In the
credit side of this approach, general properties of collective
excitations can be very well reproduced, either by using finite range
or point coupling interactions (Nik{\v{s}}i{\'{c}}
et.al.~\cite{NVR.05}). Since CRPA can treat the coupling to the
continuum exactly, it is of interest to see how well this model does
in reproducing the properties of excited state in finite nuclei, in
particular the giant resonances.
The most prominent resonances are the Isoscalar Monopole Resonance
(ISGMR), which is a breathing of the nucleus as a whole, the
Isovector Dipole Resonance (IVGDR) which corresponds to a collective
excitation of the proton against the neutron density, and Isoscalar
Quadrupole Resonance (ISGQR). In addition we have the Isoscalar
Dipole Resonance (ISGDR) revealing the spurious state corresponding
to a translational motion of the nucleus. These modes show up in an
energy range of $10-30$ MeV and they exhaust a major portion of the
corresponding sum rules. In the next sections we
investigate the ISGMR, the IVGDR and the ISGDR in more detail.
\subsection*{Numerical details}
In the following, we perform several calculations using the
relativistic continuum RPA approach in $r$-space with Point Coupling
forces~\cite{BMM.02}. We select the doubly magic nuclei $^{16}$O,
$^{40}$Ca, $^{132}$Sn and $^{208}$Pb to investigate how the
collective excitation phenomena depend on an exact coupling to the
continuum.
In a first step, the ground state of the nucleus is determined by
solving the self-consistent RMF equations (\ref{Dirac-radial}) for
the parameter set PC-F1 given in Table~\ref{tab1}. The method we are
using is a fourth order Runge-Kutta in $r$-space (Dirac-mesh) where
nucleons move in a spherical box with radius $R_{D}=15$ fm and with a
mesh size $d_{D}=0.05$ fm.
Using the single particle wave functions and the corresponding
energies of this static solution, we determine the free response
$\mathcal{R}^{\,0}$ of Eq. (\ref{E59}) in the same box radius but
using a wider mesh in $r$-space (response-mesh). The size $d_{R}$ of
this mesh depends on the excitation mode; for the monopole modes we
use $d_{R}=0.15$ fm, while for the dipole a larger interval
$d_{R}=0.30$ fm is sufficient. Then we solve the Bethe salpeter
equation (\ref{E65}) to get the strength distribution $S(\omega)$.
At the same time, we perform similar calculations using the discrete
RPA approach, where the continuum is not treated exactly, aiming of
course to a more precise comparison with the CRPA results. For those
calculations, an energy cut-off is necessary, so that a feasible
diagonalization is achieved. In particular, we have used an energy
cut-off $|\epsilon_{p}-\epsilon_{h}|< E^{ph}_{cut}= 300$ MeV for the
configurations with particles above the Fermi sea and $|\epsilon_{a}
-\epsilon_{h}|<E^{ah}_{cut}=1500$ MeV for configurations with
anti-particles in the Dirac sea.
\subsection*{Isoscalar Giant Monopole Resonances}
Results for the isoscalar monopole strength distribution are attainable, once the corresponding external field
\begin{equation}
F_{L=0}^{T=0}=\sum_{i}^{A}r^{2}_{i}
\end{equation}
is used. In this case, the classical energy weighted sum rule $m_{1}(E0)$ becomes:
\begin{equation}
m_{1}(E0)=\frac{1}{2}\langle[F,[T,F]]\rangle=\frac{\hbar^{2}}{2m}\langle
\nabla^{2} F\rangle=\frac{2\hbar^{2}}{m}\langle\,r^{2}\rangle.
\end{equation}
The doubly magic spherical nucleus $^{208}$Pb is a particularly good
example in perform our calculations, since it has been used in the
literature to test numerous nuclear structure models in the past, in
particular applications of the random phase approximation
\cite{RSp.74,Pie.00,Pie.01,CG.04}.
In Fig.~\ref{fig1} we show the ISGMR strength distribution obtained
by continuum RPA (full red line) and compare it with the discrete
B(E0) values (blue) obtained by the spectral representation of the
response function for the same parameter set PC-F1~\cite{BMM.02}.
\begin{figure}[!t]
\centering
\includegraphics[width=350pt]{fig1.pdf}%
\caption{(Color online) (a) The isoscalar monopole spectrum in
$^{208}$Pb, calculated with the parameter set PC-F1. The red curve
corresponds to the strength distribution (units on the l.h.s.)
obtained by a non-spectral representation without smearing
($\Delta=0$), the blue lines give the discrete B(E0)-values (units on
the r.h.s.) obtained by the spectral representation with the same
force. The black arrow indicates the experimental centroid energy of
the resonance~\cite{YLC.04a}. (b) the neutron and proton transition
densities at the peak with the energy $E=14.40$
MeV.}%
\label{fig1}%
\end{figure}
Using the CRPA approach, we find for the calculated centroid energy
defined in Eq. (\ref{centroid-energy}) that $m_{1}/m_{0}=14.40\,$
MeV, which is rather close to the result $m_{1}/m_{0}=14.17$ MeV
deduced from discrete RPA~calculations as well as to the experimental
value $m_{1}/m_{0}=13.96\pm0.2$~MeV \cite{YLC.04a}.
In those two methods, no additional smearing $\Delta=0$ has been
used. This means that the observed width of the continuum RPA
strength corresponds entirely to the escape width which in the Pb
region is very small, due to the relatively high Coulomb and
centrifugal barriers in this heavy nucleus. In contrast, discrete RPA
provides no width at all. Otherwise, the agreement of these two
methods in this nucleus is excellent.
In the panel (b) of Fig.~\ref{fig1}, we give the neutron and proton
transition densities at the peak energy, as it is calculated in
Eq.~(\ref{traden-pn}). They emphasize the collective character of the
isoscalar breathing mode extended over the entire interior of the
nucleus with neutrons and protons always in phase.
In addition, the energy weighted sum rule obtained in CRPA using Eq.
(\ref{moments}) is $m_{1} (E0)=5.448\cdot10^{5}$
[MeV$\cdot$fm$^{4}$]. This result is in excellent agreement with the
DRPA calculation $m_{1}(E0)=5.446\cdot10^{5}$ [MeV$\cdot$fm$^{4}$] as
well as the classical value $m_{1}(E0)=4A\hbar/2m\langle
r^{2}\rangle=5.453\cdot10^{5}$ [MeV$\cdot$fm$^{4}$]. This shows that
the results obtained in the literature by relativistic RPA
calculations using the spectral method are very reliable for such
heavy nuclei \cite{VWR.00,RMG.01}.
\begin{figure}[t]
\centering
\includegraphics[width=350pt]{fig2.pdf}\caption{(Color online) The isoscalar
monopole strength distribution for doubly magic nuclei (a) in
$^{16}$O, (b) in $^{40}$Ca,
and (c) in $^{132}$Sn. Details are the same as in the panel (a)
of Fig. \ref{fig1}.}%
\label{fig2}%
\end{figure}
In Fig.~\ref{fig2} we show the E0 strength distributions for the
lighter doubly magic nuclei $^{16}$O, $^{40}$Ca, and $^{132}$Sn. As
in Fig.~\ref{fig1}, the smearing parameter $\Delta$ is zero, but now
the escape width is considerably larger for these nuclei.
Fig.~\ref{fig3} summarizes the results for the isoscalar monopole
strength distributions as a function of the mass number $A$. In
panel (a), we plot the centroid energies of both continuum RPA (red
dots) and discrete RPA (blue dots), together with the experimental
centroid energies taken from Ref.~\cite{YLC.04a}. We also show the
phenomenological $A$-dependence $\bar{E}_{1^{-}}\approx
31.2\,A^{-1/3}+20.6\,A^{-1/6}$ by the dashed line. It becomes clear
that CRPA can successfully reproduce collective excitations over the
known range of nuclei.
\begin{figure}[t]
\centering
\includegraphics[width=220pt]{fig3.pdf}
\caption{(Color online) (a) The ISGMR centroid energies as a function
of the mass number, (b) The experimental and theoretical width of the
ISGMR as a function of the mass number. Details are given in the text.}%
\label{fig3}%
\end{figure}
In panel (b) of Fig.~\ref{fig3} we show the escape width
$\Gamma^{\uparrow}$ of E0 resonances. The red values correspond to
the full width half maximum (FWHM) of the peak, using continuum RPA ,
while the experimental values are indicated in black. The evident
disagreement is not surprising, if we consider that only
$1p1h$-configurations are taken into account, i.e. the major part of
the width resulting from the coupling to more complicated
configurations such as $2p2h$ etc. is not described well in this
simple RPA approach. It has been shown in recent investigations of
the coupling to complex configurations within the framework of the
relativistic time-blocking approximation (RTBA) or the
relativistic quasiparticle-time-blocking approximation
(RQTBA)~\cite{LRT.08} that such couplings can be taken into account
successfully in a fully consistent way starting from one density
functional $E[\rho]$. So far, relativistic investigations of this
type have been carried out with discrete methods. At present,
investigations in this direction including the continuum properly go
beyond the scope of this paper.
\subsection*{Isovector Giant Dipole Resonances}
\label{IVGD}
Isovector Giant Dipole resonance is the most well studied collective
excitation and the first to be observed experimentally~\cite{BK.47}.
An external electromagnetic field of the form:
\begin{equation}
F_{L=1}^{T=1}=\frac{N}{A}\sum_{p=1}^{Z}r_{p}Y_{1M}(\Omega_{p})-
\frac{Z}{A}\sum_{n=1}^{N}r_{n}Y_{1M}(\Omega_{n})
\end{equation}
causes protons and neutrons to oscillate in opposite phases to each
other and this leads to a pronounced peak in the photoabsorption
cross section. This mode has been well studied in many
nuclei~\cite{Speth.91}.
With the increasing number of experiments in systems far from
stability and systems with large neutron excess, one has been able to
observe also low-lying E1 strength in the area of the neutron
emission threshold. It is called Pygmy Dipole Resonance PDR and can
be interpreted as a collective mode with dipole character where the
neutron skin oscillates against an isospin saturated proton-neutron
core. This mode has first been predicted in phenomenological models
~\cite{MDB.71} exhausting several percent of the electric
dipole sum rule. In recent years, it has been intensively
investigated both on the experimental side by the Darmstadt
group~\cite{RHK.02,ZBH.05} as well as on the theoretical side, using
discrete relativistic RPA calculations based on
NL3~\cite{VPR.01a}.
\begin{figure}[!t]
\centering
\includegraphics[width=300pt]{fig4.pdf}%
\caption{(Color online) (a) The isovector dipole strength
distribution in $^{208}$Pb. Details are essentially the same as in
the panel (a) of Fig. \ref{fig1}. However, in order to distinguish
the continuum (red curve) and the discrete (blue lines) calculations
we have used here a small smearing parameter $\Delta=10$ keV in the
continuum calculation. The black arrow indicates the theoretical
neutron emission threshold. (b) transition densities for neutrons and
(c) for protons
at the energy of the PDR (left) and at the GDR (right).}%
\label{fig4}%
\end{figure}
In Fig.~\ref{fig4} we show in panel (a) the results of the isovector
dipole strength E1 in the nucleus $^{208}$Pb using the CRPA approach.
The centroid energy at $13.32$ MeV is in excellent agreement with the
experimental excitation energy $E=13.3$~MeV \cite{RBK.93}. The energy
weighted sum rule~(\ref{moments}) is found as $m_{1}(E1)=916.28$
[MeV$\cdot$fm$^{2}$]. This result is in agreement with the DRPA
calculation, where we obtain $m_{1}(E1)=943.32$ [MeV$\cdot$fm$^{2}$]
and as usual somewhat (23.8 \%) larger than the classical
Thomas-Reiche-Kuhn sum
rule%
\begin{equation}
m_{\rm TRK}= \frac{9}{4\pi}\frac{\hbar^2}{2m}\frac{NZ}{A}=740.13~
[{\rm MeV}\cdot{\rm fm}^2].%
\label{TRK}
\end{equation}
In addition to the giant dipole resonance a smaller peak appears at
the energy region of the neutron emission threshold around
$E\sim7.5$~MeV, that corresponds to the pygmy resonance.
In panel (b) of Fig.~\ref{fig4} we give the transition densities
associated the low-lying peak at $E=7.66$ MeV and the GDR peak at
$E=12.9$ MeV. The higher peak has clearly an isovector character,
since the neutrons are oscillating against the protons over a large
radial range centered at the surface. The lower peak shows an
isoscalar core, where neutrons and protons oscillate in phase and a
pure neutron skin moving against the $T=0$ core. This is the typical
behavior of the pygmy mode.
\begin{figure}[!t]
\centering
\includegraphics[width=300pt]{fig5.pdf} \caption{(Color online) The E1 pygmy
resonance (PDR) in the nucleus $^{208}$Pb. The black arrow indicates the theoretical neutron emission threshold at
$7.58$~MeV. The red dashed lines are obtained by CRPA calculations below the threshold.}%
\label{fig5}%
\end{figure}
\begin{table}[!b]
\centering
\renewcommand{\arraystretch}{1.5}%
\begin{tabular}
[c]{|c|r@{.}l|r@{.}l|r@{.}l|r@{.}l|}\hline
~~ & \multicolumn{4}{c|}{CRPA} & \multicolumn{4}{c|}{DRPA}\\\hline
~~No.~~ & \multicolumn{2}{c|}{~E~} & \multicolumn{2}{c|}{~B(E1)~} &
\multicolumn{2}{c|}{~E~} & \multicolumn{2}{c|}{~B(E1)~}\\\hline
~1~ & ~~~6 & 90~~ & ~~0 & 19~~ & ~~7 & 12~~ & ~~0 & 23~~\\
~2~ & ~~~7 & 44~~ & ~~1 & 45~~ & ~~7 & 46~~ & ~~2 & 82~~\\
~3~ & ~~~\textit{7} & \textit{66}~~ & ~~\textit{1} & \textit{11}~~ & ~~7 &
69~~ & ~~0 & 40~~\\\hline
~$\Sigma$~ & \multicolumn{2}{c|}{} & ~~2 & 75~~ & \multicolumn{2}{c|}{} &
~~3 & 45~~\\\hline
\end{tabular}
\caption{Energies and B(E1) values for the three most dominant peaks
in the PDR area around the neutron threshold for the nucleus
$^{208}$Pb for continuum (CRPA) and discrete (DRPA) calculations.}
\label{tab2}%
\end{table}
Closer investigation of pygmy resonances have shown that this mode is
in the neighborhood of the neutron separation threshold, slightly
below for small and slightly above for large neutron excess (see for
instance Ref.~\cite{PNVR.05}). It is therefore of particular
importance to study this mode with a proper treatment of the
continuum, since in most of the previous investigations this has not
been possible. We show in Fig.~\ref{fig5} the details
of the PDR in the nucleus $^{208}$Pb. Above the theoretical neutron
separation threshold which is found at $E_{\mathrm{th}}=7.58$ MeV
(black arrow) we have a continuous red curve showing the E1 strength
distribution calculated with CRPA (units at the l.h.s) and also few
full blue vertical lines that correspond to the discrete poles of the
DRPA equations (\ref{E49}) (units at the r.h.s.) and with length
equal to the corresponding B(E1) values.
\begin{figure}[!t]
\centering
\includegraphics[width=250pt]{fig6.pdf}\caption{(Color online) The isovector
dipole strength distribution in $^{132}$Sn. Details are the same as
in panel (a) of Fig. \ref{fig4}.}%
\label{fig6}%
\end{figure}
In the same figure and below the threshold we have in both cases
discrete lines. The solid blue ones are again the eigen-solutions of
the DRPA-equation (\ref{RPA-diagonalization}). The solutions of the
CRPA equations lead in this region also to discrete poles. We show
them by dashed red lines at the pole of the full response function.
Numerically, the only way to determine the B(E1) values of these
poles in CRPA is by using very small imaginary parts
$\Delta\rightarrow0$ in the frequency $\omega+i\frac{1}{2}\Delta$ and
then determining the B(E1) values by simple integration over a small
interval around this pole.
By doing that, we finally observe that there are differences in the
details between the continuum and the discrete RPA calculations close
to the neutron separation threshold. In Table~\ref{tab2} we show for
both calculations the three most dominant peaks in the area of the
PDR around $7.5$ MeV. In the discrete calculations (DRPA) the
strength is concentrated in one peak at $E=7.46$ MeV, whereas in the
continuum calculations (CRPA) most of the strength in this region is
distributed over two peaks, one below the neutron threshold at
$E=7.44$ MeV and a sharp resonance slightly above the threshold at
$E=7.66$ MeV. The energy weighted strength in this area is 17.09
[e$^{2} $fm$^{2}$] (i.e. 1.86 \% of the total sum rule) for CRPA and
26.95 [e$^{2} $fm$^{2}$] (i.e. 2.85 \% of the total sum rule) for
DRPA.
\begin{figure}[!t]
\centering
\includegraphics[width=250pt]{fig7.pdf}\caption{(Color online) The E1 pygmy
resonance (PDR) in the nucleus $^{132}$Sn. Details are the same as in
Fig. \ref{fig5}. The arrow indicates the theoretical neutron emission
threshold at
$E_{\mathrm{th}}=7.13$ MeV.}%
\label{fig7}%
\end{figure}
In Fig.~\ref{fig6} we show the distribution of the isovector dipole
strength in the doubly magic nucleus $^{132}$Sn. Again, results using
continuum RPA equations (red curve) are compared with the solutions
obtained from the spectral representation (blue lines). As one can
see, there is excellent agreement between the two methods, as far as
the resonance position and the overall distribution is concerned.
Moreover, the energy weighted sum rule obtained in CRPA is given by
$m_{1}(E1)=563.60$ [MeV$\cdot$fm$^{2} $], which is in very good
agreement with the DRPA calculation $m_{1}(E1)=591.02$
[MeV$\cdot$fm$^{4}$] and 22,9 \% larger than the Thomas-Reiche-Kuhn
sum rule in Eq.~(\ref{TRK})
\begin{table}[!h]
\centering
\renewcommand{\arraystretch}{1.5}%
\begin{tabular}
[c]{|c|r@{.}l|r@{.}l|r@{.}l|r@{.}l|}\hline
~~ & \multicolumn{4}{c|}{CRPA} & \multicolumn{4}{c|}{DRPA}\\\hline
~~No.~~ & \multicolumn{2}{c|}{~E~} & \multicolumn{2}{c|}{~B(E1)~} &
\multicolumn{2}{c|}{~E~} & \multicolumn{2}{c|}{~B(E1)~}\\\hline
~1~ & ~~8 & 11~ & ~~0 & 03~ & ~~8 & 067 & ~~0 & 037\\
~2~ & ~~8 & 48~ & ~~0 & 02~ & ~~8 & 186 & ~~1 & 601\\
~3~ & ~~8 & 82~ & ~~1 & 44~ & ~~8 & 511 & ~~0 & 260\\\hline
~$\Sigma$~ & \multicolumn{2}{c|}{} & ~~1 & 490 & \multicolumn{2}{c|}{} &
~~1 & 898\\\hline
\end{tabular}
\caption{Energies and B(E1) values for the three most dominant peaks
in the PDR area above the neutron threshold for the nucleus
$^{132}$Sn for continuum (CRPA) and discrete (DRPA) calculations. The
units are MeV for the energies and [e$^{2}$fm$^{2}$] for the B(E1)
values. More details are given in the text}
\label{tab3}%
\end{table}
In addition, we find that the escape width in this nucleus is
considerably smaller in the E1 channel as compared to the E0 channel
in Fig.~\ref{fig2}. This has the following explanation: The selection
rules for $ph$-excitations with E0 character is $\Delta j=0$ and no
change in parity. It turns out that most of the $ph$-excitations
contributing to the strong peak in the resonance region have rather
small $\ell$ values for the particle configurations and therefore a
very low or no centrifugal barrier. This is different for the E1
resonance, where one has a change in parity and in addition changes of
$\Delta j=0,\pm 1$. In such a case, a large part of the contributing
$ph$-pairs have particles with larger $\ell$-values i.e. a strong
centrifugal barrier and hence the width becomes smaller.
\begin{figure}[!t]
\centering
\includegraphics[width=350pt]{fig8.pdf}%
\caption{(Color online) The isovector dipole strength distribution in
the nuclei $^{16}$O (a) and $^{40}$Ca (b). Details are the same as in
panel (a) of Fig. \ref{fig4}. The theoretical neutron separation
energies, indicated by black arrow are $E_{\mathrm{thr}}=11.33$ MeV
for $^{16}$O and $E_{\mathrm{thr}}=8.91$ MeV for $^{40}$Ca.}%
\label{fig8}%
\end{figure}
In Fig.~\ref{fig7} we show the region of the PDR in the doubly magic
nucleus $^{132}$Sn. As already found in Ref.~\cite{PNVR.05}, the
theoretical neutron emission threshold at $E=7.13$ MeV lies much
below the area of interest. As before, we calculate the B(E1) values
of the prominent peaks, for both discrete and continuum calculations
with the total strength to be in good agreement. In Table~\ref{tab3} we show in what extent each level contributes to the
total pygmy collective state. Finally, the energy weighted strength
$m_{1}$ in this area is 13.24 [e$^{2} $fm$^{2}$] (i.e. 2.35 \% of the
total sum rule) for CRPA and 20.45 [e$^{2} $fm$^{2}$] (i.e. 3.46 \%
of the total sum rule) for DRPA.
In Fig.~\ref{fig8} we show the electric dipole strength distribution
of the lighter nuclei $^{16}$O and $^{40}$Ca. The strength obtained
in CRPA calculations (red curves) are compared with the B(E1)-values
resulting from discrete DRPA calculations (blue lines). The position
of the corresponding peaks and poles with large strength are in
rather good agreement, as explained in Table~\ref{tab4}. We find,
however, that in the continuum calculations a much larger escape
width emerges, in particular for the nucleus $^{16}$O.
\begin{table}[!t]
\centering
\renewcommand{\arraystretch}{1.5}%
\begin{tabular}{|r|c|c|c|}
\hline
& ~~CRPA~~ & ~~DRPA~~ & ~Exp.~ \\
\hline
$^{16}$O &20.6279& 21.623&23.35$\pm$0.12 \cite{Va.93} \\
$^{40}$Ca &18.367&19.32&21.76$\pm$0.11 \cite{Ve.74} \\
$^{132}$Sn &14.503&14.78& \\
$^{208}$Pb &13.32&13.23&13.3$\pm$0.10 \cite{RBK.93} \\
\hline
\end{tabular}
\caption{Isovector dipole ($IVGDR$) excitation energies in [MeV] for
several spherical nuclei, calculated with both continuum and discrete
relativistic $RPA$ based on the point coupling force PC-F1.}
\label{tab4}
\end{table}
\subsection*{Isoscalar Giant Dipole Resonances}
Besides the distribution of the isovector dipole strength which is
dominated by the IVGDR in many experimental spectra, in recent years
there has also been considerable interest in measuring the isoscalar
dipole strength distribution~\cite{DGR.97,USI.04}. In a
similar way, one expects to find the ISGDR, which corresponds to a
compression wave going through the nucleus along a definite direction
and to learn from such experiments more about the nuclear
incompressibility. Relativistic calculations based on discrete
RPA~\cite{VWR.00,Pie.01} have shown that the resonance energy
of this mode is indeed closely connected to the incompressibility of
nuclear matter.
Along with this ISGDR resonance built on $3\hbar\omega$-excitations
above 20 MeV, calculations based on both relativistic~\cite{VWR.00}
and non-relativistic~\cite{CGB.00} RPA approaches have revealed a
low-lying isoscalar dipole strength in the region below and around 10
MeV. Experimental investigations with inelastic scattering of
$\alpha$-particles at small angles~\cite{CLY.01,USI.04} have also
found isoscalar dipole strength in this region. This strength has
been attributed in Ref. \cite{VPR.02} to an exotic mode of a
toroidal motion predicted already in early theoretical investigations
on multipole expansions of systems with currents~\cite{DC.75}
and investigated also by semiclassical methods~\cite{BMS.93}
On the theoretical point of view, there is further interest in the
isoscalar dipole mode, characterized by the quantum numbers
($J^{\pi}=1^{-},T=0$), because it contains the Goldstone mode
connected with the violation of translational symmetry in the mean
field solutions. This mode corresponds to the center of mass motion
of the entire nucleus. Because of the missing restoring force, this
mode has vanishing excitation energy. It is one of the essential
advantages of the RPA approximation, that it preserves translational
symmetry and therefore it has an eigenvalue at zero energy with the
eigenfunction given by the $ph$-matrix elements of the linear
momentum operator.
\begin{figure}[!t]
\centering
\includegraphics[width=235pt]{fig9.pdf}\caption{(Color online) (a)
Spurious E1 isovector strength distribution in $^{208}$Pb obtained by
CRPA calculations with two different values of the radial mesh size
$\delta r$. (b) the position of the spurious E1-state as a function
of the radial
mesh size}%
\label{fig9}%
\end{figure}
Since the ISGDR is expected to be a $3\hbar\omega$-excitation it is
usually associated with the external field derived in Ref.~\cite{SG.81}
\begin{equation}
F_{L=1}^{T=0}~=~\sum_i^A (r_i^{3} -\eta r_i ) Y_{1\mu}(\Omega_i),
\label{ISGDR-op}%
\end{equation}
where the factor $\eta=\frac{5}{3}\langle r^{2}\rangle$ is used to extract the spurious center of mass motion.
In the upper part of Fig.~\ref{fig9} we display the distribution of
the isoscalar dipole strength in $^{208}$Pb, calculated with the
operator (\ref{ISGDR-op}) for $\eta=0$, that is, we take no action for the spurious state. We therefore observe a huge peak close to zero energy, which dominates the spectrum and corresponds to the spurious translational mode.
It turns out that the position of this spurious state is an extremely
sensitive object which strongly depends on the numerics of the
model. Of course the optimal would be to calculate the spurious state
at exactly zero energy. Therefore this excitation mode presents an
ideal benchmark for numerical efficiency of the RPA or the linear
response equations. Detailed studies have shown that the exact
separation of the spurious state requires a fully self-consistent
solution~\cite{Pie.00}; a fact which was not given in most of the
older applications with Skyrme or Gogny forces. In many cases, only
few of the different terms in the residual interaction had been taken
into account in RPA calculations.
\begin{figure}[!t]
\centering
\includegraphics[width=300pt]{fig10.pdf}\caption{(Color online) The isoscalar
dipole strength distribution in $^{208}$Pb. Details are the same as
in the panel (a) of Fig. \ref{fig4}.}%
\label{fig10}%
\end{figure}
In addition, the configuration space must be full. Indeed, the
discussed drawback of the conventional spectral representation in a
truncated $ph$-configuration space affects the position of the
spurious state. Therefore, the convergence to zero eigenvalue of the
spurious translational mode occurs very slowly and only in extremely
large configuration space. In relativistic applications this is
translated to including also large spectrum in the Dirac
sea~\cite{DF.90,RMG.01}. As a consequence, in the spectral
representation, one has to take into account many configuration with
particles in the Dirac and holes in the Fermi sea, which complicates
the numerical applications considerably and inevitably decreases the
efficiency of the method.
Fortunately, using the continuum RPA approach, one is free from such
constraints and limitations, since the entire configuration space is
automatically included. The results in Fig.~\ref{fig9} obtained with
the operator (\ref{ISGDR-op}) for $\eta=0$ show clearly the spurious
state dominating the entire spectrum (see the scale). Its position is
not precisely at zero energy, rather it depends on the mesh size used
for the solution of the continuum response equation (the response
mesh). In panel (a) of Fig.~\ref{fig9} we present two calculations
with different mesh-sizes, where in panel (b) we show how the
spurious state moves to zero energy as we use a finer radial
interval. For the ideal case of an infinitesimal mesh, the strength
connected with the spurious state would be completely separated from
the rest of the spectrum.
In Fig.~\ref{fig10} we show results obtained with the full operator
(\ref{ISGDR-op}), i.e. with $\eta=\frac{5}{3}\langle r^2\rangle$, in
a scale increased by three orders of magnitude. Obviously this
procedure removes the spurious state with high precision. We also did
not observe any influence of the isoscalar mode in the isovector
channel due to isospin mixing. In this context we have to remember,
that the isospin mixing introduced on the mean field level is
corrected on the RPA level to a large extend~\cite{MW.69}.
The main part of the remaining isoscalar dipole spectrum in
Fig.~\ref{fig10} is located at $E\approx25$ MeV. This "exotic" mode
is best described as a "hydrodynamical density oscillation", in which
the volume of the nucleus remains constant and the state can be
visualized as a compression wave oscillating back and forth through
the nucleus~\cite{VPR.02}.
\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.5}%
\begin{tabular}
[c]{|l|r@{.}l|r@{.}l|}\hline & \multicolumn{2}{c|}{~~Low[MeV]} &
\multicolumn{2}{c|}{~~High[MeV]}\\\hline
CRPA & ~~~~10 & 97 & ~~~~25 & 05\\
Hamamoto \textit{et al}~\cite{HSZ.98} &
\multicolumn{2}{c|}{$\sim$\,14~~~} &
23 & 4\\
Col\'o \textit{et al}~\cite{CVB.00} & 10 & 9 & 23 & 9\\
Vretenar \textit{et al}.~\cite{VWR.00} & 10 & 4 & 26 & \\
Piekarewicz~\cite{Pie.01} & \multicolumn{2}{c|}{$\sim$\,~8~~~} & 24 & 4\\
Shlomo, Sanzhur~\cite{SS.02} & \multicolumn{2}{c|}{$\sim$\,15~~~} &
\multicolumn{2}{c|}{$\sim$\,25~~~}\\\hline Uchida \textit{et
al}.~\cite{USI.04} & 12 & 7 $\pm$ 0.2 & 22 & 4 $\pm$ 0.5\\\hline
\end{tabular}
\caption{Self-consistent (relativistic and non-relativistic) RPA
calculations performed for the ISGDR in $^{208}$Pb, compared with the
most recent experimental data. The two columns refer to the centroid
energies of both the low- and high-energy sides of the ISGDR mode.}%
\label{tab5}%
\end{table}
Moreover, Fig.~\ref{fig10} shows an additional mode in the region of
$10-15$ MeV that exhausts roughly $20\%$ of the total sum rule. This
peak does not correspond to a compression mode, but as discussed in
Ref.~\cite{VPR.02} rather to a kind of toroidal motion. The toroidal
dipole mode is understood as a transverse zero-sound wave and its
experimental observation would invalidate the hydrodynamical picture
of the nuclear medium, since there is no restoring force for such
modes in an ideal fluid.
In conclusion, continuum RPA calculations manage not only to predict
the existence of the toroidal and the compression mode, but also to
achieve a reasonable agreement of the corresponding centroid energies
to other models focusing on the same problem, as well as to recent
experimental data~\cite{USI.04,DGR.97}. In Table~\ref{tab5}, these
results are presented for the case of the well studied nucleus
$^{208}$Pb.
\section{Conclusions}
\label{summary}
Starting from a point coupling Lagrangian, we have used the
non-spectral relativistic RPA approach to examine the corresponding
excitation spectra and we have compared the results with spectral
calculations based on the same Lagrangian. This non-spectral method
has several advantages. The coupling to the continuum is treated
consistently using the relativistic single particle Green's function
at the appropriate energy. In this way, complicated sums over
unoccupied states are avoided. This is particularly important for
relativistic applications since the Dirac sea is now automatically
treated properly and the unphysical transitions from holes in the
Fermi sea to particles in the Dirac sea is avoided as long as we
restrict our investigations to positive energies.
The ground state phenomena are calculated using the same Lagrangian
by a self-consistent solution of the relativistic mean field
equations in $r$-space. The residual particle-hole interaction used
in the RPA calculations is derived in a fully self-consistent way
from the second derivative of the corresponding energy density
functional. In this way no additional parameters are required and one
is able to reproduce the collective properties, namely the multipole
giant resonances for various doubly closed shell spherical nuclei
over the entire periodic table.
The calculations are carried out by using a new relativistic
continuum RPA program for point-coupling models, that includes all
the terms in the Lagrangian, in particular the two-body interactions
with zero range, the density dependent parts with all the
rearrangement terms, the derivative terms, the various
current-current terms and the Coulomb interaction. As applications
the nuclei $^{16}$O, $^{40}$Ca, $^{90}$Zr, $^{132}$Sn,
and $^{208}$Pb have been investigated proving that a hight level of
accuracy is achieved, as compared to the discrete methods. Comparing
calculations with spectral and non-spectral representations of the
response function for the same Lagrangian we find, that in general
the spectra are well reproduced within the spectral approximation, if
an appropriate phenomenological smearing parameter is used and if a
sufficiently large number of $ph$-configurations is taken into
account in the latter case. We find, however, differences in
neighborhood of the neutron threshold, where the coupling to the
continuum is not properly reproduced in the spectral method.
As compared to the discrete case the non-spectral representation has
the advantage of (i) a precise treatment of the coupling to the
continuum and a fully consistent determination of the escape width
without a phenomenological smearing parameter, (ii) a faster
evaluation of the cross section, because one needs for fixed energy
only two scattering solutions instead of the thousands of
$ph$-configurations in the discrete case and (iii) a proper treatment
of the Dirac sea without any further $ah$-configurations.
Relativistic CRPA describes very well the position of resonances in
doubly magic spherical nuclei. Provided that proper pairing
correlations are taken into account, a similar method can also be
applied in open-shell nuclei. This requires the development of the
relativistic continuum quasiparticle random phase approximation
(CQRPA). This approach accounts on equal footing for the influence of
the residual particle-hole ($ph$) as well as the particle-particle
($pp$) correlations. In analogy to non-relativistic calculations~\cite{KLL.98,HS.01,Mat.01,KSG.02} this can be achieved
on the basis of relativistic CRPA theory developed in this manuscript
either by treating the pairing correlations in the BCS approach for
nuclei far from the drip lines where no level in the continuum is
occupied, or in the Hartree-Bogoliubov approximation valid for all
nuclei up to the drip line. Investigations in this direction are in
progress.
Of course, the present approach is based on the RPA and includes only
$1p1h$-configurations. Therefore only the escape width of the
resonances can be reproduced properly. For heavy nuclei the decay
width resulting from a coupling to more complex configurations is
very important. In fact, such couplings have been introduced
successfully in the relativistic scheme using the spectral
representation in Refs.~\cite{LRT.08}. On the non-relativistic
side, such techniques have also been used in the context of the
non-spectral representation without~\cite{KTT.97,KST.04} and
with~\cite{LT.07} pairing. So far, however, fully self-consistent
relativistic applications including complex configurations with a
proper treatment of the continuum are still missing.
Helpful discussions with G. Lalazissis, E. Litvinova, T.
Nik\v{s}i\'{c}, N. Paar, V. Tselyaev, and D. Vretenar are gratefully
acknowledged. This research has been supported the Gesellschaft f\"ur
Schwerionenforschung (GSI), Darmstadt, the Bundesministerium f\"{u}r
Bildung und Forschung, Germany under project 06 MT 246 and by the DFG
cluster of excellence \textquotedblleft Origin and Structure of the
Universe\textquotedblright\ (www.universe-cluster.de).
\begin{appendix}
\section{The effective interaction in density dependent point-coupling models}
\label{AppA}
In Eq. (\ref{energy_variation}) the effective interaction for RPA
calculations is defined as the second derivative of the energy
functional with respect to the density matrix:
\begin{equation}
V_{\alpha\beta\alpha^{\prime}\beta^{\prime}}^{\text{ph}}=\frac{\delta
^{2}E[\hat{\rho}]}{\delta\hat{\rho}_{\alpha\beta}\delta\hat{\rho}%
_{\alpha^{\prime}\beta^{\prime}}}.
\end{equation}
In coordinate representation the indices $\alpha$,$\beta,\dots$ are
an abbreviation for the "coordinates" $1=({\bm r_1},s_1,d_1,t_1)$,
where $s$ is the spin and $t$ the isospin coordinate, and $d=1,2$ is
the Dirac-index for large and small components. Starting from the
energy density functional (\ref{Energy}) and neglecting for the
moment the Coulomb force, we find the density dependent zero range
force
\begin{equation}
V^{\text{ph}}(1,2)~=~\sum\limits_{c}\Gamma_{c}^{(1)}~\delta({\bm r}%
_{1}-{\bm r}_{2})\upsilon _{c}({\bm r}_{1})\Gamma_{c}^{\dag(2)}%
\label{Veff}%
\end{equation}
where the \textit{vertices} $\Gamma_{c}$ are 8$\times$8 matrices
acting on the indices $s,d,t$ and reflect the different covariant
structures of the fields including spin and isospin degrees of
freedom. We express the 4$\times4$ Dirac matrices as a direct product
of spin matrices $\sigma$ and 2$\times2$ matrices $\gamma_{D}$ acting
on large and
small components%
\begin{equation}
\gamma_{0}=\left(
\begin{array}
[c]{cc}%
1 & 0\\
0 & -1
\end{array}
\right) ,~~~1=\left(
\begin{array}
[c]{cc}%
1 & 0\\
0 & 1
\end{array}
\right) ,~~~\gamma_{5}=\left(
\begin{array}
[c]{cc}%
0 & 1\\
1 & 0
\end{array}
\right)%
\label{gamma-2}%
\end{equation}
and the spin matrices $\sigma_{S=0}=1$ and $\sigma_{S=1}=\sigma_\mu$
with the spherical coordinates of the Pauli spin matrices. In this
way we obtain the vertices
$\Gamma_{c}=\gamma_D\times\sigma_S\times\tau_T$ as direct products of
2-dimensional Dirac-, spin- and isospin matrices (see also the second
column of Table~\ref{tab6}).
Finally, in Eq.~(\ref{Veff}) the quantities $\upsilon_{c}({\bm r}) $
describe the strengths of all the various parts of the interaction
derived in a consistent way from the Lagrangian. The ones derived
from the four-fermion terms (\ref{L_4f}) are constants. Furthermore,
due to a density dependence of the higher order terms (\ref{L_hot})
as well as the corresponding rearrangement terms, $\upsilon_{c}({\bm
r})$ depends on the static density and therefore on the coordinate
${\bm r}$. In addition, because of the derivative terms (\ref{L_der}), they also contain Laplace operators. Summarizing, we
have:
\begin{equation}%
\begin{array}
[c]{ll}%
\text{ \ \ \ }c & \text{ \ \ }\upsilon_{c}(r) =\\
\text{scalar:} & \text{ \ \
}\alpha_{S}+2\beta_{S}\rho_{S}({r})+3\gamma
_{S}\rho_{S}^{2}({r})+\delta_{S}\Delta\\
\text{time-like vector:} & \text{ \ \ }\alpha_{V}+3\gamma_{V}\rho_{V}^{2}%
({r})+\delta_{V}\Delta\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\text{space-like vector:} & \text{ \ }-\alpha_{V}-\gamma_{V}\rho_{V}^{2}%
({r})-\delta_{V}\Delta
\end{array}
\label{VphC}%
\end{equation}
In the isovector case the constants $\alpha_{S}$, $\alpha_{V}$,
$\delta_{S}$ and $\delta_{V}$ are replaced by $\alpha_{TS}$,
$\alpha_{TV}$, $\delta_{TS}$ and $\delta_{TV}$. As we see in
Table~\ref{tab1} the corresponding values $\beta_{TS}=$
$\gamma_{TS}=\gamma_{TV}$ vanish.
For spherical nuclei, the densities and currents in the Lagrangian
depend only on the radial coordinate~$r$. Therefore we expand the
$\delta$-function in Eq. (\ref{Veff}) in terms of spherical harmonics
\begin{equation}%
\delta({\bm r}_1-{\bm r}_2)~=~\frac{\delta(r_1 - r_2)}{r_1 r_2}\sum_L
Y_{L}(\Omega_{1})\cdot Y_{L}(\Omega_{2}).
\label{spher_harmon}%
\end{equation}
Combining spin ($S$) and orbital ($L$) degrees of freedom we find by
re-coupling to total angular momentum~$J$
\begin{equation}%
(\mathbf{\sigma}^{(1)}_S\cdot\mathbf{\sigma}^{(2)}_S) (Y_{L}(1)\cdot
Y_{L}(2))= \sum_J[\mathbf{\sigma} _S Y _{L}]_{J}^{(1)}
\cdot\lbrack\mathbf{\sigma} _SY _{L}]_{J}^{(2)}
\end{equation}
Inserting this expression into Eq.~(\ref{VphC}) we obtain for the
interaction a sum (or integral) of separable terms (channels)
\begin{equation}
V^{\text{ph}}(1,2)~=~\sum\limits_{c}\int\limits_{0}^{\infty}dr~Q_{c}^{(1)}%
(r)~\upsilon _{c}(r)~Q_{c}^{\dag(2)}(r)%
\label{Veff1}%
\end{equation}
Each channel is characterized by a continuous parameter $r$ and the
discrete numbers $c=(D,S,L,J,T)$. The corresponding channel operators
$Q_{c}^{(1)}(r)$ are local single particle operators
\begin{equation}
Q_{c}^{(1)}(r)~=~\frac{\delta(r-r_1)}{rr_1}\gamma^{(1)}_{D} \left[
\sigma^{(1)}_{S}Y_L (\Omega_{1})\right] _{J}\tau^{(1)}_{T}
\label{channel-operator}%
\end{equation}
and the upper indices (1) and (2) in Eq.~(\ref{Veff1}) indicate that
these operators act on the "coordinates"
$1=(r_{1}\Omega_{1}s_{1}d_{1}t_{1})$ and
$2=(r_{2}\Omega_{2}s_{2}d_{2}t_{2})$.
The total angular momentum is a good quantum number and for fixed $J$
the sum over $c$ in Eq.~\ref{Veff1} runs only over specific numbers
$c=(D,S,L,T)$ determined by the selection rules. We concentrate in
this manuscript on states with natural parity, i.e.
$\pi=(-)^{L}=(-)^{J}$. Considering that $S=0$ for the scalar and the
time-like vector and that $S=1$ for the space-like vector we
therefore have
\begin{displaymath}
L=\left\{\begin{array}{cc} J & \text{for $S=0$} \\
J \pm 1 & \text{for $S=1$}
\end{array}\right.
\end{displaymath}
Finally we have eight discrete channels. Their quantum numbers are
shown in Table~\ref{tab6}.
\begin{table}[h]
\renewcommand{\arraystretch}{1.5}\centering
\begin{tabular}
[c]{|c|r@{$\,\otimes\,$}c@{$\,\otimes\,$}c|c|c|l|c|}\hline%
~c~&~$\Gamma_c=\gamma_{D}$ & $\sigma_{S}$ & $\tau_{T}$~ & ~~$D$~~ &
~~$S$~~ & ~$L$~~ & ~~$T$~~\\\hline%
~1~& $\gamma_{0}$ & 1 & 1 & $S$ & 0 & ~$J$ & 0\\
~2~& 1 & 1 & 1 & $V$ & 0 & ~$J$ & 0\\
~3~& $\gamma_{5}$ & $\sigma$ & 1 & $V$ & 1 & ~$J-1$ & 0\\
~4~& $\gamma_{5}$ & $\sigma$ & 1 & $V$ & 1 & ~$J+1$ & 0\\
~5~& $\gamma_{0}$ & 1 & $\tau_{3}$ & $S$ & 0 & ~$J$ & 1\\
~6~& 1 & 1 & $\tau_{3}$ & $V$ & 0 & ~$J$ & 1\\
~7~& $\gamma_{5}$ & $\sigma$ & $\tau_{3}$ & $V$ & 1 & ~$J-1$ & 1\\
~8~& $\gamma_{5}$ & $\sigma$ & $\tau_{3}$ & $V$ & 1 & ~$J+1$ & 1\\%
\hline
\end{tabular}
\caption{Vertices and quantum numbers of the different channels in
Eq.~(\ref{Veff})}%
\label{tab6}%
\end{table}
An essential feature of the effective interaction $(\ref{VphC})$ is
that it contains derivative terms in the form of Laplacians $\Delta$
(retardation effects are neglected). In spherical coordinates, they
contain radial derivatives as well as angular derivatives. The latter
can be expressed by the angular momentum operators acting on
spherical harmonics $Y_{LM}$. Therefore we obtain:
\begin{equation}
\Delta=r^{2}\overleftarrow{\partial}_{r}\frac{1}{r^{2}}\overrightarrow
{\partial}_{r}+\frac{L(L+1)-2}{r^{2}}. \label{E73}%
\end{equation}
Here the radial derivatives $\overleftarrow{\partial}_{r}$ and
$\overrightarrow{\partial}_{r}$ act on the right and on the left side
in Eq.~(67),
i.e. on $\mathcal{R}_{c^{\prime}c}^{\,0}(r^{\prime}r)$ and on
$\mathcal{R}_{cc^{\prime\prime}}(r,r^{\prime \prime})$. Since the
integration is discretized $r\rightarrow r_{n}=$ $nh$ the operator
$\overrightarrow{\partial}_{r}$ is represented by a matrix in $r
$-space as for instance by the tree-point formula:
\begin{equation}
\hat{\partial}_{nn^{\prime}}=\frac{1}{2h}(\delta_{n^{\prime},n+1}%
-\delta_{n^{\prime},n-1}).
\end{equation}
This means that the term $\upsilon_{c}(r)$ in Eq. (\ref{E65}) is no
more diagonal in the coordinate $r$ and it must be replaced by a
matrix $\upsilon_{c}(r,r^{\prime})$.
The term which leads to off-diagonal terms in channel space is the
Coulomb interaction. It brakes isospin symmetry and therefore it will
be described by the general form
$\upsilon_{cc^{\prime}}(r,r^{\prime})$. In particular, we will
have
\begin{equation}
V_{\text{C}}(1,2)=(\frac{1}{2}(1-\tau_{3} ))^{(1)}\frac{\alpha
}{|\mathbf{r}_{1}\mathbf{-r}_{2}|}(\frac{1}{2}(1-\tau_{3} ))^{(2)}%
\end{equation}
and the $r$ dependance can be written as:
\begin{equation}
\frac{\alpha}{|\mathbf{r}_{1}\mathbf{-r}_{2}|}=\sum\limits_{L}\upsilon
_{\text{C}}(r,r^{\prime})Y_{L} (\Omega)\cdot Y_{L} (\Omega^{\prime})
\end{equation}
with
\begin{equation}
\upsilon_{\text{C}}(r,r^{\prime})=\frac{4\pi\alpha}{2L+1}\cdot\frac{r_{<}^{L}%
}{r_{>}^{L+1}},
\end{equation}
and $r_{<}$ and $r_{>}$ are the smaller and the greater of $r$ and
$r^{\prime}$. This leads to a matrix $\upsilon_{cc^{\prime}}(r,r^{\prime})$ in
Eq. (\ref{E65}) as shown in Table~\ref{tab7}.%
\begin{table}[h]
\centering\renewcommand{\arraystretch}{1.5}%
\begin{tabular}
[c]{|c|cccccc|}%
\hline%
&~~$\beta$ & $1$ & $\mbox{\boldmath$\alpha$}$ &
$\beta\vec{\tau}$ & $\vec{\tau }$ &
$\mbox{\boldmath$\alpha$}\vec{\tau}$\\%
\hline%
$\beta$ &~~0 & 0 & 0 & 0 & 0 & 0\\%
1 &~~0 & $+\frac{1}{4}\upsilon_{C}$ & 0 & 0 & $-\frac{1}{4}\upsilon_{C}$ & 0\\%
$\mbox{\boldmath$\alpha$}$ &~~0 & 0 & - $\frac{1}{4}\upsilon_{C}$ & 0 &
0 & $-\frac{1}{4}\upsilon_{C}$\\%
$\beta\vec{\tau}$ &~~0 & 0 & 0 & 0 & 0 & 0\\%
$\vec{\tau}$ &~~0 & $-\frac{1}{4}\upsilon_{C}$ & 0 & 0 & $+\frac{1}{4}%
\upsilon_{C}$ & 0\\%
$\mbox{\boldmath$\alpha$}\vec{\tau}$ &~~0 & 0 &
$-\frac{1}{4}\upsilon_{C}$ &
0 & 0 & $-\frac{1}{4}\upsilon_{C}$\\%
\hline
\end{tabular}
\caption{The structure of the channel matrix $\upsilon_{cc^{\prime}%
}(r,r^{\prime}$) for the Coulomb interaction.}%
\label{tab7}%
\end{table}
\section{The continuum representation for the Green's function}
\label{AppB} In a non-spectral or continuum approach the relativistic single
particle Green's function $G_{\kappa} (r,r^{\prime};E)$ obeys the
equation:
\begin{equation}
\left( E-\hat{h}_{\kappa} (r)\right)%
G_{\kappa}(r,r^{\prime};E)=\delta(r-r^{\prime}),
\end{equation}
where $\hat{h}_{\kappa} (r)$ is the radial Dirac-operator of
Eq.~(\ref{Dirac-radial}) depending on the quantum number $\kappa=(lj)$. This
Green's function can be constructed at each energy $E$ from two linearly
independent solutions
\begin{eqnarray}
|u(r)\rangle&=&{\binom{f_u(r)}{g_u(r)}},\qquad%
|w(r)\rangle={\binom{f_w(r)}{g_w(r)}}\\%
\langle u^*(r)|&=&(f_u(r)\, g_u(r)),%
\langle w^*(r)|=(f_w(r)\,g_w(r))%
\end{eqnarray}
of the Dirac equation with the same energy $E$
\begin{equation}
\left( E-\hat{h}_{\kappa} (r)\right) |u(r)\rangle=0,\text{
\ \ \ \ }\left( E-\hat{h}_{\kappa}(r)\right) |w(r)\rangle=0,
\end{equation}
but with different boundary conditions.
The functions $\ u(r)$ and $w(r)$ are normalized in such a way that the Wronskian is equal to:
\begin{equation}
W=\left\vert
\begin{array}
[c]{cc}%
f_{w}(r) & f_{u}(r)\\
g_{w}(r) & g_{u}(r)
\end{array}
\right\vert =f_{w}(r)g_{u}(r)-g_{w}(r)f_{u}(r)=1.
\end{equation}
Of course these scattering solutions depend on the energy $E$ and on
the quantum number $\kappa$, i.e. we have $|u_{\kappa}(r,E)\rangle$
and $|w_{\kappa}(r,E)\rangle$. The Dirac-equation in $r$-space is a
two-dimensional equation and therefore the corresponding single
particle Green's function is a 2$\times2$ matrix. Using the bracket
notation of Dirac for the 2-dimensional spinors and following
Ref.~\cite{Tam.92} we can express this Green's function as:
\begin{equation}
G_{\kappa} (r,r^{\prime};E)=\left\{
\begin{array}
[c]{cc}%
|w _{\kappa}(r;E)\rangle\langle u^\ast_{\kappa}(r^{\prime};E)| & \text{~~for}%
\,\,r>r^{\prime}\\
|u _{\kappa}(r;E)\rangle\langle w^\ast_{\kappa}(r^{\prime};E)| & \text{~~for}%
\,\,r<r^{\prime}%
\end{array}
\right. \label{continuum-greens-A}%
\end{equation}
with
\begin{equation}
G _\kappa(r^{\prime},r;E)= G^\top_\kappa(r,r^{\prime};E)%
\label{EB7}
\end{equation}%
The solution $u_{\kappa}(r)$ is regular at the
origin, i.e. following
Ref.~\cite{Gre.90} we have for $E>V+S$ in the limit $r\rightarrow0$:%
\begin{equation}
u(r)\rightarrow r{\binom{j_{l} (kr)}{\frac{\kappa}{|\kappa|}\frac
{E-V-S}{k}j_{\tilde{l}}(kr)}\rightarrow\binom{\frac{r}{(2l+1)!!}(kr)^{l}%
}{\frac{\kappa}{|\kappa|}\frac{r(E-V-S)}{k(2\tilde{l}+1)!!}(kr)^{\tilde{l}}}},
\end{equation}
with $k^{2}=(E-V-S)(E-V+S+2m)>0$ and $j_{l}(z)$ is a spherical Bessel
function of the first kind. The wave function $w_{\kappa}(r)$
represents at large distances for $E>0$ an outgoing wave, i.e. we
have for $r\rightarrow\infty$
\begin{equation}
w(r)\rightarrow{\binom{rh_{l}^{(1)}(kr)}{\frac{\kappa}{|\kappa|}\frac
{ikr}{E+2m}h_{\tilde{l}}^{(1)}(kr)}\rightarrow\binom{1}{\frac{\kappa}%
{|\kappa|}\frac{ik}{E+2m}}}e^{ikr},
\end{equation}
where $h_{l}^{(1)}(z)$ is the spherical Hankel function of the first kind and
for $E<0$ an exponentially decaying state, i.e. we have for $r\rightarrow
\infty$
\begin{equation}
w(r)\rightarrow{\binom{r\sqrt{\frac{2Kr}{\pi}}K_{l+\frac{1}{2}}(Kr)}%
{\frac{-Kr}{E+2m}\sqrt{\frac{2Kr}{\pi}}K_{\tilde{l}+\frac{1}{2}}%
(Kr)}\rightarrow\binom{1}{\frac{-K}{E+2m}}}e^{-Kr},
\end{equation}
where $K^{2}=(V-S-E)(E-V+S+2m)>0$ and $j_{l}(z)$ and $K_{l+1/2}(z)$
are modified spherical Bessel functions \cite{AS.70}. For $E<0$ the
two scattering solutions are both real. This absence of any imaginary
term will eventually give no contribution to the cross section of
Eq.~(\ref{strenghtfunction}). We have to keep in mind, however, that
at energies that correspond to eigen energies of a bound state, the
solutions $u_{\kappa}(r,E)$ and $w_{\kappa}(r,E)$ coincide up to a
factor, which means that the Wronskian vanishes at this energy. This
corresponds to a pole in the response function on the real energy
axis. By adding a small imaginary part to the energy $E\rightarrow
E+i\Delta$ we obtain a sharp peak in the strength distribution.
\section{The free response function in $r$-space}
The reduced free response\begin{equation}
R_{cc^{\prime}}^{0}(\omega)=%
\sum\limits_{ph}%
\frac{\langle h|Q_{c }^{+}|p\rangle%
\langle p|Q_{c^{\prime}}|h\rangle}%
{\omega-\varepsilon_{p}+\varepsilon_{h}}%
-\frac{\langle p|Q_{c}^{+}|h\rangle%
\langle h|Q_{c^{\prime}}|p\rangle}%
{\omega+\varepsilon_{p}-\varepsilon_{h}}%
\label{R0QQ1}%
\end{equation}
depends on the energy E and the channel indices $c,c^{\prime}$. The
operators $Q_{c}$ given by Eq. (\ref{channel-operator}) are
characterized by the channel index $c=(r,DSLT)$. Each single
particle matrix element of the form $\langle p|Q_{c}|h\rangle$ in Eq.
(\ref{R0QQ1}) separates into an angular, an isospin and a radial
part.
\begin{equation}
\langle p|Q_{c}|h\rangle =%
\langle p|\tau_{T}|h\rangle%
\langle\kappa_{p}||\left[\sigma_{S}Y_{L}\right]_{J}||\kappa_{h}\rangle%
\langle p|\gamma_{D}|h\rangle_{r}.%
\label{QCph}
\end{equation}
Since we consider in this paper only $ph$-RPA in the same nucleus,
the particle states have the same isospin as the hole states and thus
the isospin matrix element $\langle p|\tau_{T}|h\rangle$ is simply a
phase $\pm1$.
Considering that this channel operator has a $\delta$-function in the
radial coordinate, the radial matrix elements $\langle
p|\gamma_{c}|h\rangle_{r}=\langle p(r)|\gamma_{D}|h(r)\rangle$ then
depend on $r$. They are found as
sums over the large and small components in the radial spinors
$|h(r)\rangle$ and $|p(r)\rangle$ for fixed values of $r$.
The angular matrix elements depend on the quantum numbers $\kappa$ of
particle and hole states, and, of course, on the channel quantum
numbers $S$ and $L$. In particular, we find for $S=0$:
\begin{equation}
\left\langle lj||Y_{J}||l^{\prime}j^{\prime}\right\rangle =\frac
{1+(-)^{l+l^{\prime}+J}}{2}\frac{\hat{\jmath}\hat{\jmath}^{\prime}\hat{J}%
}{\sqrt{4\pi}}(-)^{j-\frac{1}{2}}\left(
\begin{array}
[c]{ccc}%
j & J & j^{\prime}\\
-\frac{1}{2} & 0 & \frac{1}{2}%
\end{array}
\right)%
\label{yjj}%
\end{equation}
while for $S=1$, it is
\begin{eqnarray}
\label{syjj}
\langle lj||\left[\sigma Y_{L}\right]_{J}||l^{\prime}j^{\prime}\rangle &=&\frac{1+(-)^{l+l^{\prime}+L}}{2}
\frac{\hat{\jmath}\hat{\jmath}^{\prime}\hat{L}\hat{J}}{\sqrt{4\pi}}\left[(-)^{j^{\prime}+\frac{1}{2}}
\left(\begin{array}[c]{ccc} 1 & L & J\\0 & 0 & 0\end{array}\right)
\left(\begin{array} [c]{ccc} j & J & j^{\prime}\\\frac{1}{2} & 0 & -\frac{1}{2}\end{array}\right)\right. \nonumber\\
&-&\left.\sqrt{2}(-)^{l^\prime}\left(\begin{array} [c]{ccc}1 & L & J\\-1 & 0 & 1\end{array}\right)
\left(\begin{array} [c]{ccc}j & J & j^{\prime}\\\frac{1}{2} & -1 & \frac{1}{2}\end{array} \right)\right].
\end{eqnarray}
Using for the angular and isospin part the abbreviation
\begin{equation}
Q_{ph}^{c}=\langle\kappa_{p}||\left[ \sigma_{S}Y_{L}\right]
_{J}||\kappa_{h}\rangle\langle p|\tau_{T}|h\rangle,%
\label{red-me}
\end{equation}
we obtain for the reduced response function of Eq. (\ref{R0QQ}) in
$r$-space:
\begin{equation}
\mathcal{R}_{cc^{\prime}}^{0}(r,r^{\prime};\omega) =%
\sum\limits_{ph}\left\{%
Q_{ph}^{\ast c}Q_{ph}^{c^{\prime}}%
\frac{\langle h|\gamma_{c}^{+}|p\rangle_{r}%
\langle p|\gamma_{c^{\prime}}|h\rangle_{r^{\prime}}}%
{\omega-\varepsilon_{p}+\varepsilon_{h}}%
- Q_{hp}^{\ast c}Q_{hp}^{c^{\prime}}%
\frac{\langle h|\gamma_{c^{\prime}}|p\rangle_{r^{\prime}}%
\langle p|\gamma_{c}^{+}|h\rangle_{r}}%
{\omega+\varepsilon_{p}-\varepsilon_{h}}%
\right\}%
\end{equation}%
As in Eq.~(\ref{QGQ}) we extend the sum over $p$ over the full space
and use completeness in the radial wave functions:
\begin{eqnarray}
\mathcal{R}_{cc^{\prime}}^{\,0}(r,r^{\prime};\omega)&=&{\sum\limits_{h\kappa}}\left\{Q_{\kappa h}^{\ast c}Q_{\kappa h}^{c^{\prime}}\,\langle h(r)|\gamma_{c}^{+}G_{\kappa}(r,r^{\prime};\omega+\varepsilon_{h})
\gamma_{c^{\prime}}|h(r^{\prime})\rangle\right. \nonumber \\
&+&\left.Q_{h\kappa}^{\ast c}Q_{h\kappa}^{c^{\prime}}\langle h(r^{\prime})|\gamma_{c}G_{\kappa}(r^{\prime},r;-\omega+\varepsilon_{h})
\gamma_{c^{\prime}}^{+}|h(r)\rangle\right\}.
\end{eqnarray}
Since the angular matrix elements depend only on the quantum numbers
$\kappa$ the sum over $p$ is here replaced by a sum over the quantum
numbers $\kappa$, which is restricted by the selection rules of the
reduced matrix elements (\ref{red-me}). Having the exact form of the
Green's function for the static radial Dirac equation
(\ref{Dirac-radial}), one can finally construct the non-spectral or
continuum reduced response function (\ref{E59}):
\begin{eqnarray}
\label{E83}
\mathcal{R}_{cc^{\prime}}^{\,0}(rr^{\prime};\omega)
&=&\sum_{h\kappa}\left\{Q_{\kappa h}^{\ast c}Q_{\kappa h}^{c^{\prime}}
\gamma_{hw}^{c}(r;\omega+\varepsilon_{h})\gamma_{uh}^{c^{\prime}}(r^{\prime};\omega+\varepsilon_{h})\right.\\
&&\qquad-\left.Q_{h\kappa}^{\ast c}Q_{h\kappa}^{c^{\prime}} \gamma_{hw}^{c}(r;\omega-\varepsilon_{h})
\gamma_{uh}^{c^{\prime}}(r^{\prime};\omega-\varepsilon_{h})\right\}\mathrm{~~for~~}r>r^{\prime}\nonumber\\
&=&
\sum_{h\kappa}\left\{Q_{\kappa h}^{\ast c}Q_{\kappa h}^{c^{\prime}}
\gamma_{hu}^c(r;\omega+\varepsilon_{h})\gamma_{wh}^{c^{\prime}}(r;\omega+\varepsilon_{h})\right.\nonumber\\
&&\qquad-\left.Q_{h\kappa }^{\ast c}Q_{h\kappa}^{c^{\prime}}\gamma_{hu}^{c}(r;\omega-\varepsilon_{h})
\gamma_{wh}^{c^{\prime}}(r^{\prime};\omega-\varepsilon_{h})\right\}\mathrm{~~for~~}r<r^{\prime}\nonumber
\end{eqnarray}
where the Dirac matrix elements depend on the coordinate $r$:
\begin{eqnarray}
\gamma_{hw}^{c}(r;E) &=&\langle h|\gamma_{c}|w(E)\rangle_{r},\\
\gamma_{hu}^{c}(r;E) &=&\langle h|\gamma_{c}|u(E)\rangle_{r},\\
\gamma_{uh}^{c}(r;E) &=&\langle u^\ast(E)|\gamma_{c}|h\rangle_{r},\\
\gamma_{wh}^{c}(r;E) &=&\langle w^\ast(E)|\gamma_{c}|h\rangle_{r}.
\end{eqnarray}
Using Eq.~(\ref{EB7}) we find
\begin{equation}
\mathcal{R}_{c^{\prime}c}^{\,0}(r^{\prime},r;\omega)=
\mathcal{R}_{cc^{\prime}}^{\,0}(r,r^{\prime};\omega)
\end{equation}
It becomes clear now that the undeniable advantage of the
non-spectral approach as compared to the spectral one, is the fact
that the sum over the unoccupied states (particle states) is replaced
by a sum over the quantum number $\kappa$, which is restricted by the
selection rules for the reduced matrix elements $Q_{\kappa h}^{c}$.
For each $\kappa$, one has to determine only the pairs of the
scattering wave functions $|u\rangle$ and $|w\rangle$ for the forward
and backward term. In particular the sum over $\kappa$ does not have
to be extended over the states in the Dirac sea as in the spectral
representation (for details see Ref. \cite{RMG.01}). Therefore, not
only the size of the configuration space is significantly reduced,
but, more notably, the particle-hole as well as the antiparticle-hole
basis is taken into account fully and without any approximation.
\end{appendix}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,520 |
Its Cochin…n time to GO BERSERK….School of engineering, CUSAT presents BERSERK – a rock competition for all the brave bands, daring to mix the passion for metal, zeal and the addictive elements of new age rock. The event is held as part of Dhishna 2K11, which is to be held on February 25th to 28th at CUSAT Campus.
Providing an opportunity to mark their Signature on the rock scene, BERSERK hopes to blend all styles of rock and hail originality from bands waiting to make it big. For metal lovers craving for some soaring melodies and epic energy, Berserk is definitely going to let the punters witness some cool live acts and dry heaving with sheer excitement.
Berserk 11, a one night rock music extravaganza, featuring some of South India's leading bands. Cochin University, has always had a tradition of promoting quality rock music, and has been hosts to Asia's leading act Motherjane, Kochi's very own Whitesugar and more. Berserk is an 'open-to-all' rock music competition with an aim to promote the brand of music as well as find the next the big show-stopper. Berserk is also the first of its kind to give equal importance to all genres of rock music and thereby encouraging bands to produce their own compositions.
The winners of Berserk will be given a sum of Rs. 25, 000 and the runner up rs.10, 000 along with fabulous goodies. Moreover there are other prizes to be won including the best guitarist, vocalist and drummer. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,617 |
Pianist Joseph Kalichstein was born in Tel Aviv, Israel. In 1969, he won the Leventritt International Piano Competition. He has appeared with many of the world's major orchestras, including the Berlin, London, Boston, and Chicago Symphony Orchestras, the Cleveland Orchestra, the New York Philharmonic, and London Symphony. He has played solo recitals and chamber appearances in annual tours of Europe, North America, the Far East, and Australia. Kalichstein has made regular appearances at the La Jolla, Aspen, Ravinia, Santa Fe, Tanglewood, and Verbier summer festivals. He has made recordings for Audiofon, Erato, Koch International, Nimbus, RCA, and Vanguard. He is a founding member (1977) of the Kalichstein-Laredo-Robinson Trio, which was named 2002 Musical America Ensemble of the Year. He has been a chamber music consultant to the Kennedy Center since 1997 and he is artistic director of the Fortas Chamber Music Concerts at the Kennedy Center.
Kalichstein has studied in Israel with Joshua Shor. He received his bachelor's and master's degrees from Juilliard, where he studied with Edward Steuermann and Ilona Kabos. In the summer of 1969, he studied with Vladimir Ashkenazy.
Kalichstein has been a piano faculty member at Juilliard since 1983 and since 2003 has served as the Edwin S. and Nancy A. Marks Chair in Chamber Music Studies. | {
"redpajama_set_name": "RedPajamaC4"
} | 290 |
Thar she blows — just in time for New Year's.
Mariners along the East River spotted a whale touring Manhattan's shoreline Saturday morning.
The whale was seen spouting up near E. 87th St. around 10 a.m., officials said.
"#Harbor spotted another whale in the E. River this morning right next to Gracie Mansion," NYPD Chief of Special Operations Division Harry Wedin tweeted. "Even the wildlife want to ring in #NYE2017 in #NYC."
The United States Coast Guard put out a warning to mariners in the area to steer clear of the seafaring mammal.
This is the second time in two months that a whale was seen in the city's waterways, officials said.
In November, a humpback whale nicknamed "Gotham" was repeatedly spotted near the George Washington Bridge.
The group Gotham Whale, which tracks whales in the areas around New York Harbor, said the massive creature found its way to the Rockaways then out into the ocean in the beginning of December, officials said. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,662 |
EU Campaign to Challenge Stereotypes About Women and Science Uses Lots of Stereotypes, Little Science
Neetzan Zimmerman
Filed to: Bad ideas
Science: It's a Girl Thing
The outrage du jour concerns a just-unveiled EU Commission campaign aimed at getting girls interested in science.
A teaser promo for the "Science: It's a Girl Thing!" campaign has unleashed a firestorm of negative reactions from people around the world who see it as perpetuating female stereotypes, rather than overturning them.
On his blog, noted biologist PZ Myers summarizes the teaser's many flaws thus:
Serious man sits at microscope. Fashionable, slender girls slink in on ridiculous high heels and vogue to shots of bubbling flasks, splashes of makeup, twirling skirts, and giggling hot chicks. Seriously, this is not how you get women excited about science, by masquerading it as an exercise shallow catwalking. This is a campaign that perpetuates myths about women's preferences. The lab is not a place where you strut in 3″ heels.
Twitter users have taken to using the hashtag #sciencegirlthing to express their indignation. The Guardian's Higher Education Twitter account notes that some are using the hashtag #realwomenofscience instead to encourage other users to follow female Twitterers who are already making a difference in the world of science.
The British current affairs magazine New Statesman says that the problem with the campaign is that it is a victim of its own attempt to target an entire gender.
It's like this: "We're trying to overcome stereotypes. Yet we're targeting a whole gender - women in general. We need to find a way to appeal to the whole of womenkind. Yet we don't want to use stereotypes. Yet we need to appeal to a whole gender. Yet we don't want to use stereotypes."
It's difficult. Solution? Don't do it. This kind of campaign insults women who are interested in science already and can more than hold their own with the boys. They're the ones we need to think about.
Unless the backlash inspired the EU Commission to reevaluate its message, the campaign will proceed to roll out as planned, ultimately reaching 27 EU members states and sticking around for the next three years.
[diagram via James Monk] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,299 |
package org.openengsb.core.services.internal.security;
import java.util.Collection;
import org.apache.shiro.SecurityUtils;
import org.apache.shiro.mgt.DefaultSecurityManager;
import org.apache.shiro.realm.Realm;
public class OpenEngSBSecurityManager extends DefaultSecurityManager {
public OpenEngSBSecurityManager() {
super();
}
public OpenEngSBSecurityManager(Collection<Realm> realms) {
super(realms);
}
public OpenEngSBSecurityManager(Realm singleRealm) {
super(singleRealm);
}
public void init() {
SecurityUtils.setSecurityManager(this);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,953 |
package com.mastertheboss.quarkus.main;
import javax.inject.Inject;
import com.mastertheboss.quarkus.model.Ticket;
import com.mastertheboss.quarkus.service.TicketService;
import io.quarkus.runtime.QuarkusApplication;
import io.quarkus.runtime.annotations.QuarkusMain;
@QuarkusMain
public class TicketMain implements QuarkusApplication {
@Inject
TicketService service;
@Override
public int run(String... args) {
if(args.length<2) {
System.out.println("Usage: mvn quarkus:dev -Dquarkus.args=\"<name> <seat>\"");
return 1;
}
Ticket ticket = new Ticket();
ticket.name=args[0];
ticket.seat=args[1];
service.createTicket(ticket);
return 0;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,502 |
\section{Introduction}
Blazars are a peculiar class of active galactic nuclei (AGNs) which dominate the observable $\gamma$-ray Universe because of their extreme properties and abundant population. The blazar properties are a result of non-thermal emitting plasma traveling towards the observer causing relativistic amplification of flux. This leads to an amplification of low energy photons in the medium to intense levels via inverse Compton process, making blazars valuable sources to understand the physics of an AGN. The Third {{\it Fermi }}\/--LAT Catalog of High-Energy Sources \citep[3FHL][]{ajello17}, which encompasses seven years of observations made by the Large area telescope (LAT) aboard the {\it Fermi Gamma-ray Space Telescope} \citep{atwood09}, contains more than 1500 sources detected at $>10$\,GeV, the vast majority of which ($\approx$ 1160) are blazars \citep{ajello17}.
Innovative scientific results can be obtained using the blazar data collected by the {{\it Fermi }}\/ LAT in the $\gamma$-ray regime, provided the redshift ($z$) of the observed blazar source is known. These are not only limited to blazar physics such as, understanding their basic emission processes \citep[e.g. ][]{ghisellini17} or their evolution with redshift \citep{ajello14}, but also to other areas of study, like understanding the extragalactic background light (EBL), which encompasses all the radiation emitted by stars and galaxies and reprocessed radiation from interstellar dust, and its evolution with $z$ \citep{ackermann12,dominguez13}. Out of the confirmed blazar sources reported in the 3FHL catalog a redshift measurement of only $\approx$50\,\% sources is present \citep{ajello17}. To overcome this limitation, extensive optical spectroscopic campaigns, targeting those 3FHL objects still lacking redshift and classification, must be performed.
Besides being used for redshift determination, optical spectroscopy campaigns of blazars are also essential to distinguish between blazar sub-classes, namely BL Lacs (BLL) and flat spectrum radio quasars (FSRQs). FSRQs are generally high redshift objects with average luminosity larger than that of the BLL \citep{padovani92,paiano17}. As a result, the emission lines in the BLL spectra are weak or absent and the lines in FSRQs are extremely prominent. This is seen by the difference in the equivalent width (EW) of the lines where generally, FSRQ have lines with EW$>5$\r{A} and BLL have lines with EW$<5$ \r{A}\citep{urry95,ghisellini17}. The blazar sources not classified as FSRQ or BLL are listed as blazar candidates of uncertain type (BCU) in the 3FHL catalog, and constitute $\approx 25\,\%$ of the reported blazar sample \citep{ajello17}. Obtaining a spectroscopically complete classification of the blazars observed by {{\it Fermi }}\/ LAT in the $\gamma$-ray regime is essential to validate claims of different cosmological evolution of the two classes \citep{ajello12,ajello14}.
The ground based telescopes used in the spectroscopy campaigns are generally of the 4--m,8--m and 10--m class type. While the 10--m and 8--m class telescopes are shown to be significantly more effective in obtaining redshift measurements for blazars \citep[60--80\% versus 25--40\% success rate, see, e.g.][]{paiano17,marchesi18}, even 4--m class telescopes have proven to be useful for effectively distinguishing between the two different blazar subclasses \citep[see ][]{shaw13,massaro14,paggi14,landoni15,ricci15,marchesini16,alvarez16a,alvarez16b}.
This work is part of a larger spectroscopic follow-up campaign to classify the BCUs in the 3FHL catalog and measure their redshift.
{The first part of the campaign took place in the second half of 2017, when we observed 28 sources in seven nights of observations at the 4--m telescope at Kitt Peak National Observatory (KPNO). The results of this work are reported in \citet{marchesi18}: we classified 27 out of 28 sources as BL Lacs, while the remaining object was found to be a FSRQ. Furthermore, we measured a redshift for 3 sources and set a lower limit on $z$ for other four objects; the farthest object in our KPNO sample has $z>$0.836.
The spectroscopic campaign will then continue with seven nights of observations at the 4--m telescope at Cerro Tololo Interamerican Observatory (CTIO) in Chile and five nights of observations at the 8--m Gemini-N and Gemini-S telescopes (to be performed in 2019). In this work, we report the results} of the observations made during the first four nights at CTIO. Our source sample contains 23 BCUs in the 3FHL catalog without a redshift measurement. The paper is organized as follows: Section~\ref{sample_sel} reports the criteria used in sample selection, Section~\ref{obs} describes the methodology used for the source observation and spectral extraction procedures, Section~\ref{spectral} lists the results of this work, both, for each individual source and also in general terms, while Section~\ref{conclusion} reports the conclusions inferred from this spectroscopic campaign.
\section{Sample Selection}\label{sample_sel}
We selected the 23 objects in our sample among the BCUs in the 3FHL catalog, using the following three criteria.
\begin{itemize}
\item {\bf The object should have an measured optical magnitude measurement}, and it should be V$\le$19.5. Based on previous works, sources with magnitude V$>$19.5 require more than two hours of observations to obtain an acceptable signal-to-noise ratio (S/N), therefore significantly reducing the number of sources that one can observe in a night.
\item The 3FHL source should be bright in the hard $\gamma$-ray spectral regime ($f_{\rm 50-150 GeV}>$10$^{-12}$ erg s$^{-1}$ cm$^{-2}$). Selecting 3FHL objects bright in the 50--150\,GeV band ensures that the completeness of the 3FHL catalog evolves to lower fluxes as more optical observations are performed.
\item The target should be observable from Cerro Tololo with an altitude above the horizon $\delta$$>$40\,\degree (i.e., with airmass $<$1.5): this corresponds to a declination range -80\degree$<$Dec$<$20\degree. The target should also be observable in October, when the observations take place (i.e., it should have R.A.$\geq$09h0m00s and R.A.$\leq$0h30m00s).
\end{itemize}
A total of 77 3FHL sources satisfy all these criteria. Our 23 sources were selected among these 77 objects with the goal of covering a wide range of optical magnitudes (V=[16--19.5]) and, consequently, of potential redshifts and luminosities. {In Figure \ref{fig:hist} we show the normalized V-band magnitude distribution of our sources, compared with the one of the overall population of 173 3FHL BCUs still lacking a redshift measurement and having available magnitude information. We also plot the magnitude distribution of the 28 sources studied in \citet{marchesi18}, where we sampled a larger number of bright sources (V$<$16) which all turned out to be featureless BL Lacs.}
The sources used in our sample and their properties are listed in Table~\ref{tab:sample}.
\section{Observations and Data Analysis}
\label{obs}
All the sources in our sample were observed using the 4\,$m$ Blanco telescope located at the Cerro Tololo Inter-American Observatory (CTIO) in Chile. The spectra were obtained using the COSMOS spectrograph with the Red grism and the 0.9$^{\prime\prime}$ slit. This experimental setup corresponds to a dispersion of $\sim$ 4\,\AA\ pixel$^{-1}$, over a wavelength range $\lambda$=[5000--8000]\,\AA, and a spectral resolution R$\sim$2100. The data were taken with the slit aligned along the parallactic angle.
The seeing was 1.3$^{\prime\prime}$ during the first and third night, 1$^{\prime\prime}$ during the second night and 2.2$^{\prime\prime}$ in the last night, respectively; all four nights were photometric.
All spectra reported here are obtained by combining at least three individual observations of the source with varying exposure times. This allows us to reduce both instrumental effects and cosmic ray contribution. The data reduction is done following a standard procedure: the final spectra are all bias-subtracted, flat-normalized and corrected for bad pixels. { We normalize the flat-field to remove any wavelength dependent variations that could be present in the flat-field source but not in the observed spectrum. This is done by fitting a cubic spline function on the calibration spectrum and taking a ratio of the flat-field to the derived fit \citep[see response function in ][]{iraf_doc}. We choose an order $>$5 for the cubic spline function fit with a $\chi^2$ value less than 1 to account for all variable features in the flat-field }An additional visual inspection is also done on the combined spectra to remove any artificial features that may still be present. This data reduction and spectral extraction is done using the IRAF pipeline \citep{tody86}.
The wavelength calibration for each source is done using the Hg-Ne lamp: we took a lamp spectrum after each observation of a source, to avoid potential shifts in the pixel-$\lambda$ calibration due to changes in the telescope position during the night. Finally, all spectra were flux-calibrated using a spectroscopic standard, which were observed using the same 0.9$^{\prime\prime}$ slit used in the rest of the analysis,
and then corrected for the Galactic reddening using the extinction law by \citet{cardelli89} and the $E(B-V)$ value based on the \cite{schlafly11} measurements, as reported in the NASA/IPAC Infrared Science Archive.\footnote{\url{http://irsa.ipac.caltech.edu/applications/DUST/}}
\section{Spectral Analysis}
\label{spectral}
{ To visually enhance the spectral features of our sources, in Figure \ref{fig:spec} we report the normalized spectra of the objects in our sample. These normalized spectra are obtained by dividing the flux-calibrated spectra using a continuum fit \citep[an approach similar to the one reported in ][]{landoni18}.
The continuum is taken to be a power-law unless the optical shape is more complex, in which case the preferred fit is described in ~\ref{individual_src}. The S/N of the normalized spectrum is then measured in a minimum of five individual featureless regions in the spectrum with a width of $\Delta\lambda\approx40$\,\r{A}.} The spectral analysis results for each source, including the computed S/N, are reported in Table~\ref{tab:redshift}.
To find a redshift measurement, each spectrum was visually inspected for any absorption or emission feature. Any potential feature that matched known atmospheric lines\footnote{\url{https://www2.keck.hawaii.edu/inst/common/makeewww/Atmosphere/atmabs.txt}} was not taken into consideration. To test the reliability of any potential feature, its existence was verified in each of the individual spectral files used to obtain the final combined spectrum shown in Fig~\ref{fig:spec}. { For example, the broad emission feature seen in the spectrum of 3FHL J0935.2-1735 around 5633\,\r{A} is not found in the individual files and is thus considered to be an artifact.} The verified features are then matched with common blazar lines, such as the Mg II doublet lines (2797\,\r{A} and 2803\,\r{A}) or O III line (5007\,\r{A}), to compute the redshift.
All the sources in our sample were classified as BLL based on their spectral properties. Out of the 23 sources, we were able to determine a redshift measurement for 3 sources, a lower limit on the redshift for 2 of them and a tentative redshift measurement for 3 of them. The remaining 15 sources in our sample were found to be featureless. { Details for some of the sources for which a spectral feature or redshift is found are given in Sec~\ref{individual_src}. These features are also listed in Table~\ref{tab:redshift} with the derived redshift measurement.}
\subsection{Comments on Individual sources}
\label{individual_src}
{\bf3FHL J0936.4-2109:} This BCU is associated with the X-ray source 1RXS J093622.9-211031. The optical spectrum of this source shows the presence of two absorption features at 6176\,\r{A} and 6160\,\r{A}. If they are associated with the Mg II doublet, a redshift measurement of 1.1974 and 1.1976 is obtained respectively. Corresponding to this $z$ value, other typical features observed in blazars, either in emission or in absorption (e.g., the Ca II doublet, the G-band, O II or O III features) will fall out of our observed wavelength range of $5000$\r{A}$-8200$\r{A}. We report a tentative lower limit of the redshift as $z>1.197$ for this BLL.
{\bf3FHL J1030.6-2029:} This source is associated with the radio source NVSS J103040-203032. Its optical spectrum shows the presence of the Mg II doublet at 5579\,\r{A} and 5591\,\r{A} respectively. This gives a redshift lower limit of $z>0.995$.
{\bf3FHL J1042.8+0055:} This source is associated with the X-ray source RBS 0895. A redshift value of 0.73 exists in the literature, \citep{mnras90}, however the authors flagged it as an uncertain measurement. We were not able to detect any absorption or emission lines in our optical spectrum, so we classify this source as a BLL.
{\bf3FHL J1155.5-3418:} This source is associated with the radio source NVSS J115520-341718. The Mg II doublet is identified in the optical spectrum of the source at 5174\,\r{A} and 5185\,\r{A} allowing us to measure the lower limit of the redshift as $z>0.849$.
{\bf3FHL J1212.1-2328:} This source is associated with the radio source PMN J1212-2327. We obtain an optical spectrum with S/N of 102.8 and detect an emission feature at 8345\,\r{A} with an equivalent width of 0.8\,\r{A}. If associated to the O III line, we derive a redshift $z$=0.666.
{\bf3FHL J1223.5-3033:} This source is associated with the radio source NVSS J122337-303246. We see possible absorption features at 5245\,\r{A}, 5256\,\r{A}, 5577\,\r{A} and 6341\,\r{A}. If 5245\,\r{A} and 5256\,\r{A} absorption features are associated with the Mg II line, a redshift of $0.875$ is measured. However we were not able to detect the presence of any other features and also identify the features at 5577\,\r{A} and 6341\,\r{A} to confirm the redshift measurement with certainty. This source is thus classified as a BLL and a tentative lower limit of $z$$>$0.875 is reported.
{\bf3FHL J1433.5-7304:} This source is associated with the X-ray source 1RXS J143343.2-730433. One emission feature (H$_\alpha$) and four absorption features (G-band, Mg I,Na and Ca+Fe ) are detected in the spectrum. This gives us a redshift measurement of $z = 0.200$.
{\bf3FHL J1439.4-2524:} This source is associated with the radio source NVSS J143934-252458. We detect two strong absorption lines at 6008\,\r{A} and 6115\,\r{A} and an absorption line at 6835\,\r{A} close to an atmospheric feature (6845\,\r{A}) in its optical spectrum. If these lines are associated with the Mg I, Ca+Fe and NaD absorption features respectively, a redshift of $z=0.16$ is derived.
{\bf3FHL J1605.0-1140:} The IR counterpart of this source is WISE J160517.53-113926.8. The optical spectrum shows the presence of an emission feature at 6801\,\r{A} with equivalent width of 7.044\,\r{A}. This feature can be associated with the O II or O III line giving a redshift of 0.824 or 0.358 respectively, however due to no significant detection of any other emission or absorption features and a low S/N measurement, the redshift of this source cannot be measured with certainty.
\section{Conclusion}
{ In this work, we present the results the optical spectroscopic campaign directed towards rendering the 3FHL a spectroscopically complete sample using the COSMOS spectrograph mounted on the 4\,$m$ Blanco telescope at CTIO in Chile.} We observed 23 extragalactic sources classified as BCU (blazars of uncertain classification) in the 3FHL catalog.
All the objects in our source sample are classified as BLL based on their observed optical spectrum. In the 3FHL catalog, out of the already classified 901 blazars $\approx 84.1\%$ sources are classified as BLL. Moreover out of the 28 sources observed by \cite{marchesi18}, 27 are identified as BLL denoting that our results are not surprising.
Out of the 23 BLL in our sample we find a reliable { redshift} measurement for 3 sources, a reliable { redshift} constraint for 2 sources, a tentative { redshift} constraint for 3 sources and a featureless spectrum with no { redshift} measurement for the remaining 15 sources. Combining our results with the results of \cite{marchesi18}, our optical spectroscopic campaign reports a redshift measurement for $\approx23.5\%$ of the observed BLL sources using 4\,$m$ telescopes. This measurement is in line with the expected consistency of $10-35\%$, obtained for redshift determination of pure BLL using using 4\,$m$ telescopes \citep{landoni15,ricci15,alvarez16a,pena17}. Moreover, our work combined with \cite{marchesi18} also classifies, as either BLL or FSRQs, 51 blazars of previously uncertain classification.
The third and fourth part of our spectroscopic campaign will include observations from the 4\,$m$ CTIO telescope and 8\,$m$ Gemini-N and Gemini-S telescope respectively\footnote{{\it Fermi} Guest Investigator Program Cycle 11, ID:111128, PI: S. Marchesi.}. Additionally we also aim to extend the campaign by inducing follow up observations\footnote{{\it Swift} Cycle 14, prop ID 1417063 PI: M. Ajello}, similar to \cite{kaur18}, using the Swift X-ray telescope. These follow up observations in the X-ray regime will help us confirm the classification of the blazar sources contributing to the spectral completion of the 3FHL catalog.
\label{conclusion}
\section{Acknowledgements}
A.D. acknowledges funding support from NSF through grant AST-1715256. S.M. acknowledges support from NASA contract 80NSSC17K0503. The authors thank Alberto Alvarez and Sean Points for the help provided during the observing nights at CTIO. This work made use of the TOPCAT software (Taylor 2005) for the analysis of data tables.
\begingroup
\renewcommand*{\arraystretch}{1.8}
\begin{table*}
\centering
\scalebox{0.65}{
\begin{tabular}{lcccccccc}
\hline
\hline
3FHL Name & Counterpart & R.A. & Dec & E(B-V) & mag & Obs Date & Exposure & continuum slope\\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\\
\hline
3FHL J0002.1$-$6728 & SUMSS J000215$-$672653 & 00:02:15.21 & $-$67:26:52.91 & 0.0253 & 18.6 & June 1 2018 & 5400 & $-1.44$\\
3FHL J0935.2$-$1735 & NVSS J093514$-$173658 & 09:35:14.77 & $-$17:36:58.30 & 0.0643 & 17.8 & June 1 2018 & 3900 & $-0.12$ \\
3FHL J0936.4$-$2109 & 1RXS J093622.9$-$211031 & 09:36:23.08 & $-$21:10:39.00 & 0.0574 & 18.5 & June 2,3 2018 & 5100 & $-0.28$ \\
3FHL J1030.6$-$2029 & NVSS J103040$-$203032 & 10:30:40.46 & $-$20:30:32.70 & 0.0469 & 18.4 &June 3 2018 & 3300 & $-1.91$\\
3FHL J1042.8$+$0055 & RBS 895 & 10:43:03.84 & $+$00:54:20.43 & 0.0419 & 19.3 & June 4 2018 & 5600 & -1.03\\
3FHL J1130.5$-$7801 & SUMSS J113032$-$780105 & 11:30:32.92 & $-$78:01:05.20 & 0.1921 & 17.6 & June 2 2018 & 3400 & $-1.07$\\
3FHL J1155.5$-$3418 & NVSS J115520$-$341718 & 11:55:20.43 & $-$34:17:18.30 & 0.0702 & 16.8 & June 1 2018 & 2400 & $-1.10$\\
3FHL J1212.1$-$2328 & PMN J1212$-$2327 & 12:12:04.54 & $-$23:27:42.00 & 0.0656 & 18.2 & June 1 2018 & 4500 & $-0.77$\\
3FHL J1223.5$-$3033 & NVSS J122337$-$303246 & 12:23:37.32 & $-$30:32:46.10 & 0.0593 & 17.2 & June 2 2018 & 3400 & $-2.15$\\
3FHL J1229.7$-$5304 & AT20G J122939$-$530332 & 12:29:39.93 & $-$53:03:32.20 & 0.1293 & 17.8 & June 3 2018 & 2300 & $-0.44$\\
3FHL J1315.9$-$0732 & WISE J131552.98$-$073301.9 & 13:15:53.00 & $-$07:33:02.07 & 0.0352 & 18.2 & June 4 2018 & 4500 & $-0.87$\\
3FHL J1433.5$-$7304 & GALEX J143343.0$-$730437 & 14:33:42.81 & $-$73:04:36.84 & 0.1592 & 17.9 & June 1 2018 & 4000 & $-0.81$\\
3FHL J1439.4$-$2524 & NVSS J143934$-$252458 & 14:39:34.66 & $-$25:24:59.10 & 0.0862 & 16.2 & June 3 2018 & 2800 & $-0.01$\\
3FHL J1605.0$-$1140 & WISE J160517.53$-$113926.8 & 16:05:17.53 & $-$11:39:26.83 & 0.2584 & 18.7 & June 4 2018 & 5400 & $-0.35$\\
3FHL J1612.3$-$3100 & NVSS J161219$-$305937 & 16:12:19.95 & $-$30:59:37.80 & 0.2003 & 18.1 & June 2 2018 & 3600 & $-1.11$\\
3FHL J1640.1$+$0629 & NVSS J164011$+$062827 & 16:40:11.06 & $+$06:28:27.70 & 0.0695 & 18.6 & June 2 2018 & 3800 & $-1.71$\\
3FHL J1842.4$-$5841 & 1RXSJ184230.6$-$584202 & 18:42:29.67 & $-$58:41:57.19 & 0.0848 & 17.5 & June 1 2018 & 3600 & $-1.67$\\
3FHL J1924.2$-$1548 & NVSS J192411$-$154902 & 19:24:11.82 & $-$15:49:02.10 & 0.1491 & 17.7 & June 3 2018 & 3600 & $-1.35$\\
3FHL J2034.9$-$4200 & SUMSS J203451$-$420024 & 20:34:51.06 & $-$42:00:37.60 & 0.0360 & 17.2 & June 2,4 2018 & 3900 & $-0.62$\\
3FHL J2041.7$-$7319 & SUMSS J204201$-$731911 & 20:42:01.85 & $-$73:19:13.01 & 0.0544 & 18.2 & June 4 2018 & 3400 & $-4.47$\\
3FHL J2240.3$-$5240 & SUMSS J224017$-$524111 & 22:40:17.64 & $-$52:41:13.07 & 0.0118 & 16.7 & June 4 2018 & 1950 & $-5.84$\\
3FHL J2321.8$-$6437 & PMN J2321$-$6438 & 23:21:42.17 & $-$64:38:06.90 & 0.02 & 17.4 & June 4 2018 & 2800 & $-0.06$\\
3FHL J2339.2$-$7404 & 1RXS J233919.8$-$740439 & 23:39:20.88 & $-$74:04:36.12 & 0.0262 & 16.1 & June 4 2018 & 1500 & $-0.65$\\
\hline
\hline
\hline
\end{tabular}}\caption{List of sources and their properties sorted in the order of increasing R.A. (Right ascension) values. (1): 3FHL catalog \citep{ajello17} name for the source. (2): optical, IR, X-ray or radio counterpart of the source. (3) Right ascension. (4) Declination. (5) $E(B-V)$ value obtained using the measurements of \citet{schlafly11} and the NASA/IPAC Infrared Science Archive online tool. (6) V band magnitude. (7) Date of observation. (8) Exposure time (in seconds).(9)Slope of continuum fit obtained from the observed fits file }
\label{tab:sample}
\end{table*}
\endgroup
\begingroup
\renewcommand*{\arraystretch}{1.8}
\begin{table*}
\centering
\scalebox{0.65}{
\begin{tabular}{cccccc}
\hline
\hline
Source & S/N & Spectral line & Observed $\lambda$ (\r{A}) & line type & redshift\\
& & Rest frame $\lambda$ (\r{A})& & &\\
\hline
3FHL J0002.1$-$6728 & 41.4 & & & & \\
3FHL J0935.2$-$1735 & 51.5 & & & & \\
3FHL J0936.4$-$2109 & 27.2 & Mg II(2797) & 6176 & absorption & $>1.197^*$\\
& & Mg II(2803) & 6160 & absorption & \\
3FHL J1030.6$-$2029 & 29.3 & Mg II(2797) & 5579 & absorption & $>0.995$\\
& & Mg II(2803) & 5591 & absorption & \\
3FHL J1042.8$+$0055 & 46.6 & & & & \\
3FHL J1130.5$-$7801 & 72.2 & & & & \\
3FHL J1155.5$-$3418 & 42.7 & Mg II(2797) & 5174 & absorption & $>0.849$\\
& & Mg II(2803) & 5185 & absorption & \\
3FHL J1212.1$-$2328 & 102.8 & O III(5007) & 8345 & emission & $0.666$\\
3FHL J1223.5$-$3033 & 46.5 & Mg II(2797) & 5245 & absorption & $>0.875^*$\\
& & Mg II(2803) & 5256 & absorption & \\
3FHL J1229.7$-$5304 & 78.6 & & & & \\
3FHL J1315.9$-$0732 & 60.8 & & & & \\
3FHL J1433.5$-$7304 & 64.9 & G-band(4304) & 5165 & absorption & $0.200$\\
& & Mg I(5175) & 6209 & absorption & \\
& & Ca+Fe(5269) & 6340 & absorption & \\
& & Na (5895) & 7074 & absorption & \\
& & H$\_${$\alpha$}(6562) & 7876 & absorption & \\
3FHL J1439.4$-$2524 & 82.7 & Mg I(5175) & 6008 & absorption & $0.16$\\
& & Ca+Fe(5269) & 6115 & absorption & \\
& & NaD(5892) & 6835 & absorption & \\
3FHL J1605.0$-$1140 & 17.2 & O II(3727) & 6801 & emission & $0.358^*$\\
& & (or) O III(5007) & 6801 & emission & $0.824^*$\\
3FHL J1612.3$-$3100 & 75.4 & & & & \\
3FHL J1640.1$+$0629 & 83.1 & & & & \\
3FHL J1842.4$-$5841 & 32.7 & & & & \\
3FHL J1924.2$-$1548 & 64.4 & & & & \\
3FHL J2034.9$-$4200 & 33.4 & & & & \\
3FHL J2041.7$-$7319 & 70.1 & & & & \\
3FHL J2240.3$-$5240 & 71.2 & & & & \\
3FHL J2321.8$-$6437 & 33.7 & & & & \\
3FHL J2339.2$-$7404 & 45.5 & & & & \\
\hline
\hline
\hline
\end{tabular}}\caption{Results obtained from spectral analysis discussed in Section~\ref{spectral}. The redshift measurement values marked with a $^*$ are tentative $z$ measurements.}
\label{tab:redshift}
\end{table*}
\endgroup
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{percent_norm_all_sample}
\caption{{Normalized V-band magnitude distribution of the sources analyzed in this work (red dashed line), compared with the distribution of the 173 3FHL BCUs lacking of redshift and having magnitude information (blue solid line). The magnitude distribution of the objects analyzed in \citet{marchesi18} using KPNO is also shown for comparison.}}
\label{fig:hist}
\end{figure*}
\begin{figure*}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J0002}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J0935}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J0936}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1030}
\end{minipage}
\vspace{0.5cm}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1042}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1130}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1155}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1212}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1223}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1229}
\end{minipage}
\caption{Optical spectra of the observed candidates after performing flux calibration and dereddening. The bottom panel displays the normalized spectra where the atmospheric features are denoted by $\otimes$ while the absorption or emission features are labeled as per the lines they signify.}
\label{fig:spec}
\end{figure*}
\begin{figure*}
\setcounter{figure}{1}
\vspace{0.5cm}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1315}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1433}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1439}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1605}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1612}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1640}
\end{minipage}
\vspace{0.5cm}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1842}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1924}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J2034}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J2041}
\end{minipage}
\caption{Continued from Fig~\ref{fig:spec}}
\end{figure*}
\begin{figure*}
\setcounter{figure}{1}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J2240}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J2321}
\end{minipage}
\vspace{0.5cm}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J2339}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\end{minipage}
\caption{Continued from Fig~\ref{fig:spec}}
\end{figure*}
\begin{figure*}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J0936_zoom.eps}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1030_zoom.eps}
\end{minipage}
\vspace{0.5cm}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1155_zoom.eps}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1212_zoom.eps}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1223_zoom.eps}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1433_zoom.eps}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1439_zoom.eps}
\end{minipage}
\begin{minipage}[b]{.5\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{plots/J1605_zoom.eps}
\end{minipage}
\caption{The zoomed spectra of selected sources from Fig~\ref{fig:spec} are shown above to highlight absorption and emission features }
\end{figure*}
\bibliographystyle{aasjournal}
| {
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{"url":"http:\/\/yang.amp.i.kyoto-u.ac.jp\/lab\/collo_old\/heisei-13\/010621.html","text":"Colloquim\n\nReduction of quantum systems with symmetry, continuous and discrete\n\n\u001b$B4d0f\u001b(B \u001b$BIRMN\u001b(B\n\n2001\u001b$BG\/\u001b(B6\u001b$B7n\u001b(B21\u001b$BF|\u001b(B13\u001b$B;~\u001b(B30\u001b$BJ,\u001b(B Reduction of dynamical systems is closely related with symmetry. The purpose of this talk is to show that Fourier analysis both on compact Lie groups and on finite groups serve as reduction procedure for quantum systems on an equal footing. The reduction procedure is applied to systems of many identical particles lying in${\\bf R}^3$which admit the action of a rotation group$SO(3)\\$ and of a symmetric group or a permutation group.","date":"2021-01-18 07:51:16","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\": 0.6445318460464478, \"perplexity\": 2326.404153739688}, \"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-2021-04\/segments\/1610703514423.60\/warc\/CC-MAIN-20210118061434-20210118091434-00368.warc.gz\"}"} | null | null |
{"url":"https:\/\/codereview.stackexchange.com\/questions\/2763\/reduce-the-code-in-a-where-clause-without-using-dynamic-sql","text":"# Reduce the code in a WHERE clause without using dynamic SQL\n\nHow can I write the following T-SQL query part in a shorter way without using dynamic SQL?\n\nWHERE\n( (@Max IS NULL OR @Type <> 'Products')\nOR (@Max IS NOT NULL AND @Type = 'Products'\nAND ProductCount > @Max ) )\n\nAND ( (@Min IS NULL OR @Type <> 'Products')\nOR (@Min IS NOT NULL AND @Type = 'Products'\nAND ProductCount < @Min ) )\n\nAND ( (@Max IS NULL OR @Type <> 'Vendors')\nOR (@Max IS NOT NULL AND @Type = 'Vendors'\nAND VendorCount > @Max ) )\n\nAND ( (@Min IS NULL OR @Type <> 'Vendors' )\nOR (@Min IS NOT NULL AND @Type = 'Vendors'\nAND VendorCount < @Min ) )\n\nAND ( (@Max IS NULL OR @Type <> 'Order')\nOR (@Max IS NOT NULL AND @Type = 'Order'\nAND OrderCount > @Max ) )\n\nAND ( (@Min IS NULL OR @Type <> 'Order')\nOR (@Min IS NOT NULL AND @Type = 'Order'\nAND OrderCount < @Min ) )\n\n\u2022 By \"shorter\", do you mean \"shorter\" in terms of text length, or in terms of efficiency? The first really doesn't matter-long queries can be efficient and short ones inefficient. Is the query running poorly for you?\n\u2013\u00a0Todd\nJun 1 '11 at 6:13\n\u2022 @Todd No performance issue with current query, i just want make it \"shorter\" in terms of text length Jun 1 '11 at 6:15\n\u2022 maybe because you are not using it? it's definition is for posting code for peer review. Jun 1 '11 at 6:28\n\u2022 Maybe your question will inspire some people to visit the site slightly more often, and some others to come up with further interesting questions. :) Jun 1 '11 at 7:37\n\u2022 What is the purpose of the Query? What does the rest of the query look like?\n\u2013\u00a0Malachi\nJul 21 '14 at 15:03\n\nYou will have to test this carefully, but the following query should work:\n\nWHERE\n(\n@Max IS NULL\nOR @Type = 'Products' AND ProductCount > @Max\nOR @Type = 'Vendors' AND VendorCount > @Max\nOR @Type = 'Order' AND OrderCount > @Max\n)\nAND\n(\n@Min IS NULL\nOR @Type = 'Products' AND ProductCount < @Min\nOR @Type = 'Vendors' AND VendorCount < @Min\nOR @Type = 'Order' AND OrderCount < @Min\n)\n\n\nSomething like this\n\nYou can rely on the NULL comparison to always give false (strictly: unknown) if @Max or @Min is NULL for the relevant CASE\n\nWHERE\nCASE @Type\nWHEN 'Products' THEN ProductCount\nWHEN 'Vendors' THEN VendorCount\nWHEN 'Order' THEN OrderCount\nEND > @Max\nOR\nCASE @Type\nWHEN 'Products' THEN ProductCount\nWHEN 'Vendors' THEN VendorCount\nWHEN 'Order' THEN OrderCount\nEND < @Min\n\n\u2022 Pedant's corner: the NULL comparison gives UNKNOWN, not FALSE. putting a NOT in front of it will still exclude those rows.\n\u2013\u00a0Damien_The_Unbeliever\nJun 1 '11 at 8:21\n\u2022 @Damien_The_Unbeliever: yes, yes, I thought about that. But didn't want to deprive anyone of their fun ;-)\n\u2013\u00a0gbn\nJun 1 '11 at 8:23\n\nHere's another stab at it, based on gbn's CASE idea, but using BETWEEN to avoid repeating the cases:\n\nWHERE\nCASE @Type\nWHEN 'Products' THEN ProductCount\nWHEN 'Vendors' THEN VendorCount\nWHEN 'Orders' THEN OrderCount\nEND BETWEEN IFNULL(@Min,0) AND IFNULL(@Max,99999999)\n\n\nNote: IFNULL in MySQL should be replaced by ISNULL in TSQL\n\nI am just going to step through this so that it is a little easier to understand for someone happening upon this question in the future.\n\nThis is what we are starting with\n\nWHERE\n( (@Max IS NULL OR @Type <> 'Products')\nOR (@Max IS NOT NULL AND @Type = 'Products'\nAND ProductCount > @Max ) )\n\nAND ( (@Min IS NULL OR @Type <> 'Products')\nOR (@Min IS NOT NULL AND @Type = 'Products'\nAND ProductCount < @Min ) )\n\nAND ( (@Max IS NULL OR @Type <> 'Vendors')\nOR (@Max IS NOT NULL AND @Type = 'Vendors'\nAND VendorCount > @Max ) )\n\nAND ( (@Min IS NULL OR @Type <> 'Vendors' )\nOR (@Min IS NOT NULL AND @Type = 'Vendors'\nAND VendorCount < @Min ) )\n\nAND ( (@Max IS NULL OR @Type <> 'Order')\nOR (@Max IS NOT NULL AND @Type = 'Order'\nAND OrderCount > @Max ) )\n\nAND ( (@Min IS NULL OR @Type <> 'Order')\nOR (@Min IS NOT NULL AND @Type = 'Order'\nAND OrderCount < @Min ) )\n\n\nI am going to first take out all the <> conditions because we are already checking in the or statements for @Type = {type} and I am going to take out the check for @Max IS NOT NULL because if it were NULL we wouldn't hit that condition anyway.\n\nWHERE\n( @Max IS NULL\nOR (@Type = 'Products'\nAND ProductCount > @Max ) )\n\nAND ( @Min IS NULL\nOR (@Type = 'Products'\nAND ProductCount < @Min ) )\n\nAND ( @Max IS NULL\nOR (@Type = 'Vendors'\nAND VendorCount > @Max ) )\n\nAND ( @Min IS NULL\nOR (@Type = 'Vendors'\nAND VendorCount < @Min ) )\n\nAND ( @Max IS NULL\nOR (@Type = 'Order'\nAND OrderCount > @Max ) )\n\nAND ( @Min IS NULL\nOR (@Type = 'Order'\nAND OrderCount < @Min ) )\n\n\nnow we have @Max IS NULL and @Min IS NULL checks that we could combine so we aren't repeating ourselves.\n\nWHERE\n(\n@Max IS NULL\nOR\n(@Type = 'Products' AND ProductCount > @Max)\nOR\n(@Type = 'Vendor' AND VendorCount > @Max)\nOR\n(@Type = 'Order' AND OrderCount > @Max)\n)\nAND\n(\n@Min IS NULL\nOR\n(@Type = 'Products' AND ProductCount < @Min)\nOR\n(@Type = 'Vendor' AND VendorCount < @Min)\nOR\n(@Type = 'Order' AND OrderCount < @Min)\n)\n\n\nThis is the Final Solution that @Peter Lang came to. I use Parenthesis to make sure that the where clause is being interpreted by the RDBMS the way that I want them interpreted, if they aren't interpreted the way I think they will be it can lead to weird results that sometimes are hard to spot.\n\nAlways double check your returned data to make sure you are getting what you want.","date":"2022-01-16 20:40:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.36982259154319763, \"perplexity\": 13878.472648604149}, \"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-2022-05\/segments\/1642320300010.26\/warc\/CC-MAIN-20220116180715-20220116210715-00704.warc.gz\"}"} | null | null |
Sterno è il nome commerciale (di proprietà della società statunitense The Sterno Group) di una gelatina combustibile a base alcolica usata nell'ambito del catering per il riscaldamento delle pietanze e nel campeggio in sostituzione dei fornelli da campeggio a gas.
Il materiale, inventato intorno al 1900, è di aspetto gelatinoso ed è costituito da etanolo, metanolo, acqua e un agente gelificante basato su un ossido anfotero; è studiato per essere privo di odore e bruciare direttamente nella propria confezione (una confezione da 200 g brucia per circa 2 ore). Presenta un caratteristico colore rosa.
Cocktail
Lo Sterno fu utilizzato ai tempi della grande depressione e fino al secondo dopoguerra per produrre un drink chiamato canned heat (cui ad esempio fa riferimento il brano del 1928 Canned Heat Blues del bluesman Tommy Johnson), squeeze o pink lady, che risulta tossico per la presenza di metanolo. Consumato in particolare da alcolizzati poveri e senzatetto, fu messo in relazione a numerose intossicazioni e morti per avvelenamento, tra cui 31 persone a Filadelfia nel 1963.
Riferimenti cinematografici
Nel film Andromeda (The Andromeda Strain), film di fantascienza del 1971 diretto da Robert Wise, tratto dall'omonimo romanzo di Michael Crichton del 1969, Jackson (interpretato da George Mitchell), il paziente anziano, è un bevitore di Sterno.
Note
Voci correlate
Fornello da campeggio
Altri progetti
Collegamenti esterni
Combustibili
Materiali per la cucina | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,816 |
\section{Introduction}
In this paper we describe our submission to the WMT20 Metrics shared task. Our work is based on the {\sc Comet}\footnote{\textbf{C}rosslingual \textbf{
O}ptimized \textbf{M}etric for \textbf{E}valuation of \textbf{T}ranslation hosted at: \url{https://github.com/Unbabel/COMET}} framework, as presented in \citet{comet}, and extended here to evaluation of MT output at segment, document and system-level, forming the basis of our submissions to the corresponding task tracks. Recently, automatic evaluation of MT has followed most other sub-fields in NLP with a notable interest in leveraging the power of large, pre-trained language models. Metrics such as {\sc Bert Regressor} \citep{shimanaka2019machine}, {\sc Bertscore} \citep{ZhangBERTScore}, {\sc Bleurt} \citep{sellam-etal-2020-bleurt} and our more recent {\sc Comet} \citep{comet}, all build upon developments in language modelling to generate automatic metrics with high correlation with human judgement. Our MT evaluation models follow a similar strategy, specifically utilizing the most recent iterations of the XLM-RoBERTa model presented in \citet{conneau2019unsupervised}.
The uniqueness of our approach comes from our inclusion of the source text as input which was demonstrated in \citet{takahashi-etal-2020-automatic} and \citet{comet} to be beneficial to the model. In our contribution to the shared task, we demonstrate methods of further exploiting information in the source text as well as a technique to fully harness the power of pre-trained language models to further improve the prediction accuracy of our evaluation framework when more than one reference translation is available.
For the shared task, we utilize two primary types of models built using the {\sc Comet} framework, namely; the Estimator models, which regress directly on human scores of MT quality such as Direct Assessment; and the {\sc Comet-rank} (base) model used to rank MT outputs and systems.
In addition to the models themselves, we also make the following research contributions:
\begin{enumerate}
\item We introduce a method for handling multiple references at inference time and for optimizing the utility of information from all available text inputs
\item We propose a simple technique for calculating a document-level score from a weighted average of segment-level scores
\end{enumerate}
We demonstrate that our {\sc Comet} framework trained models achieve state-of-the-art results or are competitive on all settings introduced in the WMT19 Metrics shared task, outperforming, in some cases, more recently proposed metrics such as {\sc Bertscore} \citep{ZhangBERTScore}, {\sc Bleurt} \citep{sellam-etal-2020-bleurt} and {\sc Prism} \citep{thompson-post-2020-automatic}.
\section{The {\sc Comet} Framework}
As outlined in \citet{comet}, the {\sc Comet} framework allows for training of specialized evaluation metrics that correlate well with different types of human-generated quality scores. The general structure of the framework consists of a cross-lingual encoder that produces a series of token-level vector embeddings for source, hypothesis and reference inputs, a pooling layer which converts the various token-level representations into segment-level vectors for each input, and a predictive neural network that generates a quality score. The latter model can either be trained to regress directly on a score to produce predictions of segment-level quality, or can be trained as a ranker to differentiate MT systems. In our contribution to the shared task, we introduce two varieties of models built on the {\sc Comet} framework that are extensions of the models evaluated in \citet{comet}.
\section{{\sc Comet} Models}
\subsection{Estimator Models}
Our Estimators generally follow the architecture proposed in \citet{comet}, that is to say we encode segment-level representations using XLM-RoBERTa and pass these outputs through a feed-forward regressor. As in \citet{comet}, we train three versions of this basic estimator model against different types of human judgement; \textit{Human-mediated Translation Edit Rate} (HTER) \cite{Snover06astudy}, a proprietary implementation of \textit{Multidimensional Quality Metric} (MQM) \cite{mqm} and (in-line with the present task) \textit{Direct Assessments} (DA) \cite{graham-etal-2013-da}. The hyper-parameters used for these models are exactly as described in \citet{comet}, excluding the following alterations: we use XLM-RoBERTa large instead of base and we increase the feed-forward hidden sizes (from 2304 in the first layer and 1152 in the second to 3072 and 1536 hidden units, respectively). We also keep the embedding layer frozen and apply a layer-wise learning rate decay (as proposed in \citet{howard2018universal}) by which each transformer layer has a learning rate scaled at 0.95 times the rate of the layer above. By doing this, we hope that our metric generalizes better to new language pairs introduced this year.
\subsection{Translation Ranking Model}
While for the Estimators using a larger pretrained encoder seems to improve performance we found that for the Translation Ranking Model, larger pretrained encoders lead to training instability and an overall worse performance. For that reason we decided to keep the model proposed in \cite{comet} without any alteration.
\section{Corpora}
Below we provide an outline of the various datasets used to train our models:
\subsection{HTER Corpora}
Our HTER corpus is a concatenation of two publicly available corpora: the QT21 corpus
and the APE-QUEST corpus.
While the QT21 corpus contains segments from the information technology and life sciences domains \cite{specia-etal_MTSummit:2017}, the APE-QUEST contains segments from the legal domain \cite{ive-etal-2020-post}. Concatenation of these two corpora gives a total of 211K tuples with source sentence, corresponding human-generated reference, MT hypothesis, and post-edited MT (PE). With regard to the language pairs in each corpus, QT21 covers: English to German (en-de), Latvian (en-lt) and Czech (en-cs), and
German to English (de-en); while APE-QUEST covers: English-Dutch (en-nl), English-French (en-fr), English-Portuguese (en-pt). Finally, the HTER score is obtained by calculating the translation edit rate (TER) \cite{Snover06astudy} between the MT hypothesis and the corresponding PE. By doing this, we were able to create a large HTER corpus covering several language pairs and different domains.
\subsection{MQM Corpus}
Our MQM corpus is an extension of the proprietary corpus presented in \citet{comet}. This internal data consists of customer support chat messages translated using a domain adapted MT model and their corresponding references (consisting of post-edited translations from earlier iterations of the MT systems). The data was then MQM-annotated according to the guidelines set out in \citet{burchardt}. Our final corpus contains 27K tuples from English into 15 different languages and/or dialects: German (en--de), Spanish (en--es), Latin-American Spanish (en--es-latam), French (en--fr), Italian (en--it), Japanese (en--ja), Dutch (en--nl), Portuguese (en--pt), Brazilian Portuguese (en--pt-br), Russian (en--ru), Swedish (en--sv), Turkish (en--tr), Polish (en--pl), simplified Chinese (en--zh-CN), and Taiwanese Chinese (en--zh-TW).
\subsection{DA Corpora}
Every year, since 2008, the WMT News Translation shared task organizers collect human judgements in the form of DAs. Since 2017, due to a lack of annotators, these scores are mapped to relative rankings ({\small{DA}}RR). We take advantage of this data in two ways: 1) we use the scores directly in order to train an estimator model, 2) as in \citet{comet}, we use the {\small{DA}}RR to train a translation ranking system. The collective corpora of 2017, 2018 and 2019 contain a total of 24 language pairs, including low-resource languages such as English-Gujarati (en-gu) and English-Kazakh (en-kk). For the purposes of this paper we use the data from 2017 and 2018 to train and the data from 2019 to validate. Later, for participation in the 2020 shared task, we intend to include the data from 2019 in our training corpus.
\section{Segment-Level Task}
\label{sec:seg-lvl-task}
\begin{table*}[!ht]
\centering
\caption{Segment-level Kendall's Tau ($\tau$) correlations for language pairs from English-to-other for the WMT19 Metrics {\footnotesize DA}RR corpus.}
\label{tab:english-to-x2019-seg}
\begin{tabular}{lccccccccc}
\hline
& \textbf{en-cs} & \textbf{en-de} & \textbf{en-fi} & \textbf{en-gu} & \textbf{en-kk} & \textbf{en-lt} & \textbf{en-ru} & \multicolumn{1}{c|}{\textbf{en-zh}} & \\
Nº Tuples & 27178 & 99840 & 31820 & 11355 & 18172 & 17401 & 24334 & \multicolumn{1}{c|}{18658} & \textbf{avg.} \\\specialrule{1.5pt}{1pt}{1pt}
{\sc Bleu}& 0.364 & 0.248 & 0.395 & 0.463 & 0.363 & 0.333 & 0.4691 & \multicolumn{1}{c|}{0.235} & 0.410 \\
{\sc chrF} & 0.444 & 0.321 & 0.518 & 0.548 & 0.510 & 0.438 & 0.548 & \multicolumn{1}{c|}{0.241} & 0.510 \\
{\sc Bertscore} (F1) & 0.486 & 0.350 & 0.526 & 0.559 & 0.534 & 0.464 & 0.581 & \multicolumn{1}{c|}{0.350} & 0.550 \\
{\sc Prism} & 0.580 & 0.416 & 0.590 & - & 0.529 & 0.555 & 0.581 & \multicolumn{1}{c|}{0.373} & 0.518 \\ \hline
{\sc Comet-mqm} (large) & 0.595 & 0.405 & 0.594 & 0.580 & 0.546 & 0.607 & 0.693 & \multicolumn{1}{c|}{0.400} & 0.553 \\
{\sc Comet-hter} (large) & 0.610 & 0.427 & 0.610 & 0.587 & 0.569 & 0.615 & 0.707 & \multicolumn{1}{c|}{0.405} & 0.566 \\
{\sc Comet-da} (large) & \textbf{0.618} & \textbf{0.435} & 0.620 & \textbf{0.617} & 0.585 & 0.619 & \textbf{0.711} & \multicolumn{1}{c|}{0.427} & 0.579 \\
{\sc Comet-rank} (base) & 0.603 & 0.427 & \textbf{0.664} & 0.611 & \textbf{0.693} & \textbf{0.665} & 0.580 & \multicolumn{1}{c|}{\textbf{0.449}} & \textbf{0.587} \\
\end{tabular}
\end{table*}
\begin{table*}[!ht]
\centering
\caption{Segment-level Kendall's Tau ($\tau$) correlations on language pairs with English as a target for the WMT19 Metrics {\footnotesize DA}RR corpus.}
\label{tab:x-to-english2019-seg}
\begin{tabular}{lcccccccc}
\hline
& \textbf{de-en} & \textbf{fi-en} & \textbf{gu-en} & \textbf{kk-en} & \textbf{lt-en} & \textbf{ru-en} & \multicolumn{1}{c|}{\textbf{zh-en}} & \\
Nº Tuples & 85365 & 32179 & 20110 & 9728 & 21862 & 39852 & \multicolumn{1}{c|}{31070} & \textbf{avg.} \\ \specialrule{1.5pt}{1pt}{1pt}
{\sc Bleu}& 0.054 & 0.236 & 0.194 & 0.276 & 0.249 & 0.115 & \multicolumn{1}{c|}{0.321} & 0.206 \\
{\sc chrF} & 0.123 & 0.292 & 0.240 & 0.323 & 0.304 & 0.177 & \multicolumn{1}{c|}{0.371} & 0.261 \\
{\sc Bertscore} (F1) & 0.191 & 0.354 & 0.292 & 0.351 & 0.381 & 0.221 &\multicolumn{1}{c|}{0.433} & 0.318 \\
{\sc Bleurt} (large-512) & 0.174 & 0.374 & 0.313 & 0.372 & 0.388 & 0.220 &\multicolumn{1}{c|}{ 0.436} & 0.325 \\
{\sc Prism} & 0.189 & 0.366 & 0.320 & 0.362 & 0.382 & 0.220 & \multicolumn{1}{c|}{0.434} & 0.325 \\ \hline
{\sc Comet-mqm} (large) & 0.191 & 0.360 & 0.289 & 0.346 & 0.373 & 0.213 & \multicolumn{1}{c|}{0.426} & 0.314 \\
{\sc Comet-hter} (large) & 0.193 & 0.351 & 0.286 & 0.340 & 0.375 & 0.209 & \multicolumn{1}{c|}{0.429} & 0.312 \\
{\sc Comet-da} (large) & \textbf{0.220} & 0.368 & 0.316 & \textbf{0.378} & \textbf{0.405} & \textbf{0.231} & \multicolumn{1}{c|}{\textbf{0.462}} & \textbf{0.340} \\
{\sc Comet-rank} (base) & 0.202 & \textbf{0.399} & \textbf{0.341} & 0.358 & \textbf{0.407} & 0.180 & \multicolumn{1}{c|}{0.445} & 0.333
\end{tabular}
\end{table*}
At segment-level, we take each of our Estimator models trained to predict MQM, HTER and DA and predict segment-level scores on the {\small{DA}}RR data from WMT19. We then generate pairwise rankings based on these predicted scores. For each language pair we apply the formulation of Kendall's Tau ($\tau$) from the shared task \cite{metrics-2019-results} as follows:
\begin{equation}
\label{eq:kendall}
\tau = \frac{\textit{Concordant} - \textit{Discordant}}{\textit{Concordant} + \textit{Discordant}}
\end{equation}
\textit{Concordant} here being the number of times a metric assigns a higher score to the ``better'' hypothesis $h^+$ and \textit{Discordant}, the number of times a metric assigns a higher score to the ``worse'' hypothesis $h^-$, or that the evaluation was otherwise equal.
\section{Document-Level Task}
\label{sec:doc-lvl-task}
In the WMT2019 News Translation the organizers introduced a document-level translation task \cite{barrault-etal-2019-findings} for en-de and en-cs. This means that for those language pairs we are able to obtain document-level direct assessments. We can compute a score taking into account an entire document and correlate it with the human evaluation also carried out at document-level.
For our document-level submission we propose the generation of a document-level score as a weighted average of the predicted scores for each segment composing that document (hereinafter called micro-average score), where the same is weighted by segment length.
To calculate this score at inference time we pass the entire document (divided into segments) through the model as a single batch. This has the added effect of reducing inference time.
\section{System-level Task}
Following previous years, the metric used in the System-level Task will be Pearson's $r$ correlation score. The correlation is calculated between the average of all DA human z-scores for a given system and language pair, and the average of the corresponding scores predicted by a given metric. Because the goal of some metrics is to maximize the correlation with human judgements (i.e. {\sc Bleu}), while for others is to minimize that correlation (i.e. HTER), the value reported is its absolute value.
\subsection{Robustness to high-permorming systems}
One important finding from WMT19 is the general deterioration of metrics' performance when considering only the top $n$ MT systems \cite{metrics-2019-results}. Previously, we showed robustness of our metrics in this scenario in terms of Kendall's Tau at segment-level \citep{comet}. \citet{mathur2020tangled} show that at system-level, Pearson correlation is highly influenced by outliers and that performances for metrics such as {\sc Bleu} drop significantly when considering only the top systems. To address this, we propose a system-level pairwise comparison measured with the same Kendall's Tau formulation used for segment-level analysis outlined in section \ref{sec:seg-lvl-task} above. By doing this, we are not only better handling possible outliers, but emulating a real world application of these metrics: In most cases (both in academia and industry), we want a metric that can successfully differentiate between two systems, even if those systems are very close in terms of performance, which is often the case.
\section{Quality Estimation as a Metric}
Given the clear parallels between the {\sc Comet} framework and modern approaches to Quality Estimation such as \citet{kepler2019unbabels}, we used our framework to participate in the ``QE as a Metric'' track of the shared task by removing the reference at input and proportionately reducing the dimensions of the feed-forward network to accommodate the reduced input.
\section{Multi-Reference Handling}
\label{sec:multi-ref}
In this year's shared task we are provided with a second human-generated reference for German-to-English (de-en), Russian-English (ru-en) and Chinese-to-English (zh-en). Given that our base framework currently supports the input of only one single reference, we introduce a method of leveraging information from a second reference at inference time.
During standard training of our models, we input source, hypothesis and reference in that order, resulting in a concatenation of embeddings as detailed further in \citet{comet}. During training, with a probability of $p=0.5$ we switch the positions of source and reference, such that the system receives the reference as the source and vice versa. This has two primary effects on our model. Firstly, during fine-tuning of the underlying language model, the source embeddings are aligned with the target language embedding space resulting in more useful source embeddings. Secondly, it forces the model to treat source and reference as interchangeable inputs, allowing it to handle switching of inputs at inference time without excessively hindering the model's predictive ability. Finally, at inference time we embed source $\bm{s}$, hypothesis $\bm{h}$, reference $\bm{r}$ and the alternative reference $\bm{\hat{r}}$. These embeddings are then passed to the feed-forward neural network in the following six permutations: $[\bm{s};\bm{h};\bm{r}]$, $[\bm{r};\bm{h};\bm{s}]$, $[\bm{s};\bm{h};\bm{\hat{r}}]$, $[\bm{\hat{r}};\bm{h};\bm{s}]$, $[\bm{r};\bm{h};\bm{\hat{r}}]$ and $[\bm{\hat{r}};\bm{h};\bm{r}]$.
Six passes through the feed-forward gives us six predictions. Final, aggregated scores are achieved by taking the mean of the six predictions and multiplying it by 1 minus the standard deviation ($\sigma$). The intuition being that $1-\sigma$ gives something of an idea of confidence of the model at the segment-level and that scaling the mean prediction to penalize lower confidence might align better with human judgement.
\begin{table*}[!ht]
\centering
\caption{Kendall's Tau ($\tau$) correlation and standard deviation ($\sigma$) across all language pairs for the top 5 high-performing systems.}
\label{tab:kendall-sys}
\begin{tabular}{lcc}
Model & \multicolumn{1}{l}{Avg. Kendall (all)} & \multicolumn{1}{l}{Avg. Kendall (en)} \\ \specialrule{1.5pt}{1pt}{1pt}
{\sc Bleu} & 0.387$\pm$0.366 & 0.257$\pm$0.395 \\
{\sc chrF} & 0.387$\pm$0.463 & 0.343$\pm$0.513 \\
{\sc Bertscore} (F1) & 0.453$\pm$0.267 & 0.429$\pm$0.279 \\
{\sc Bleurt} & - & 0.571$\pm$0.355 \\
{\sc Prism} & 0.52$\pm$0.270 & 0.514$\pm$0.279 \\ \hline
{\sc Comet-mqm} (large) & 0.587$\pm$0.277 & 0.543$\pm$0.276 \\
{\sc Comet-hter} (large) & 0.547$\pm$0.325 & 0.486$\pm$0.363 \\
{\sc Comet-da} (large) & \textbf{0.653}$\pm$\textbf{0.233} & \textbf{0.629}$\pm$\textbf{0.269} \\
{\sc Comet-rank} (base) & 0.547$\pm$0.256 & 0.543$\pm$0.276 \\
\end{tabular}
\end{table*}
\section{Experimental Results}
\label{sec:length}
Below we present results of our various {\sc Comet} models on WMT19 evaluation sets as described above. Segment-level and document-level results are outlined in the corresponding tables within the body of the paper, the remaining tables for other task results are contained in the Appendices hereto.
\subsection{Segment-level Task}
Our segment-level results on the shared task test sets for WMT19 are detailed in tables \ref{tab:english-to-x2019-seg} and \ref{tab:x-to-english2019-seg}. We note that for all language pairs out of English (Table \ref{tab:english-to-x2019-seg}) both our DA Estimator and our {\sc Comet-rank} (base) outperform prior metrics, in some cases by a significant margins. The same can be said in most language pairs into English, where we consistently perform at the level competitive with or exceeding prior metric performance in this task. Table \ref{tab:not-english2019-seg} (contained in the appendices) further illustrates performance of our models on non-English language pairs. We note that in all settings our {\sc Comet} models outperform state-of-the-art for these language pairs.
\subsection{System-level Task}
System-level results are outlined in tables \ref{tab:english-to-x2019-sys}, \ref{tab:x-to-english2019-sys} and \ref{tab:not-english2019-sys} in the appendix. In most language pairs we outperform the best metrics in terms of correlation with human judgement. For those language pairs for which our metrics are outperformed by others, we note that ours are at least competitive with other, recent metrics.
An unexpected result is that at system-level our {\sc Comet-rank} (base) does not perform as well as our Estimators, regardless of its strong segment-level results. We believe that this is an artifact of training directly on {\small{DA}}RR data. Since in WMT shared tasks, the DA rating scale employed is defined at the 0-25-50-75-100 point margins, the minimum required difference between two hypothesis to produce {\small{DA}}RR judgement is 25 points \cite{metrics-2019-results}. All other segments are discarded, as within that range the notion of which hypothesis is better becomes ambiguous. As a result we believe that our ranker model learns to successfully discriminate
less ambiguous examples and struggles to correctly assign a score otherwise.
\subsubsection{Robustness to high-performing systems}
As outlined above, we also complement our evaluation at system-level with an analysis of metric performance in terms of the pairwise ranking of the top five performing systems from each language pair. For each setting we output the Kendall's Tau (that is to say the formulation outlined in section \ref{sec:seg-lvl-task} above) and report the mean and standard deviation of results across language pairs in table \ref{tab:kendall-sys}.
In both settings we note that our DA Estimator (large) model significantly outperforms other metrics both in terms of mean and standard deviation. This strongly suggests that not only do we perform well in terms of system-level Pearson but that at a practical level, our model can much more successfully differentiate high-performing systems.
\subsection{Document-level Task}
Table \ref{fig:micro-vs-macro} compares the micro-averaging against a simple unweighted average. From table \ref{fig:micro-vs-macro} we can observe that micro-averaging outperforms macro-averaging by a small margin. Table \ref{tab:doc-level} summarizes our results for the Document-level Task using our segment-level Estimators with micro-averaging. In this task, the HTER Estimator shows generally superior performance on average surpassing our best performing segment-level model, the DA Estimator. An important conclusion to draw from the strong document-level correlations noted here is that a model trained to generate segment-level scores, can also perform well as a document-level metric.
\begin{table}[!ht]
\centering
\caption{Pearson correlation ($r$) between Document-level DAs and micro average segment-level scores for English-to-German and English-to-Czech.}
\label{tab:doc-level}
\begin{tabular}{lccc}
\hline
& \textbf{en-cs} & \textbf{en-de} & \\
Nº Documents & 1115 & \multicolumn{1}{c|}{2355} & \textbf{avg.} \\ \specialrule{1.5pt}{1pt}{1pt}
{\sc Comet-mqm} (large) & 0.638 & \multicolumn{1}{c|}{0.516} & 0.577 \\
{\sc Comet-hter} (large) & 0.655 & \multicolumn{1}{c|}{\textbf{0.558}} & \textbf{0.607} \\
{\sc Comet-da} (large) & \textbf{0.667} & \multicolumn{1}{c|}{0.528} & 0.598 \\
\end{tabular}
\end{table}
\begin{table}[!ht]
\centering
\caption{Pearson correlations ($r$) and adequacy (as reported in \citet{freitag-bleu-paraphrase-references-2020}) for segment-level DA using our DA Estimator (large) model on WMT19 Metrics shared task test data for en-de. We show Pearson's $r$ for the single reference scenario using the corresponding reference (`1-ref') and the multi-reference scenario where the reference is combined with the original in the manner outlined in section \ref{sec:multi-ref} above (`2-ref').}
\label{tab:multiref}
\begin{tabular}{llll}
\hline
\textbf{Reference} & \textbf{Adequacy} & \textbf{$r$ (1-ref)} & \textbf{$r$ (2-ref)} \\
\specialrule{1.5pt}{1pt}{1pt}
WMT & 85.3 & 0.523 & - \\
AR & 86.7 & 0.539 & 0.555 \\
WMTp & 81.8 & 0.470 & 0.529 \\
ARp & 80.8 & 0.476 & 0.537 \\
\end{tabular}
\end{table}
\subsection{Multi-Reference Handling}
\label{sec:multi-ref-res}
Additional references were obtained for two language pairs: en-de and de-en. For the former, we conducted experiments using 3 additional references from \citet{freitag-bleu-paraphrase-references-2020}: AR reference (an additional high quality reference translation), ARp reference (a paraphrased-as-much-as-possible version of AR), and WMTp reference (a paraphrased-as-much-as-possible version of the original WMT reference); for the latter, we use the alternative reference given in the WMT19 News shared task test set. Conveniently, \citet{freitag-bleu-paraphrase-references-2020} also offer a notion of the quality of the extra references for en-de by providing human-generated adequacy assessments for each. In table \ref{tab:multiref} we show the performance of our DA Estimator (large) model with each reference, either as a single reference or combined in the manner described in section \ref{sec:multi-ref} above with the original reference.
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\newcommand{\raisebox{2pt}{\tikz{\draw[red,solid,line width=1.2pt](0,0) -- (5mm,0);}}}{\raisebox{2pt}{\tikz{\draw[red,solid,line width=1.2pt](0,0) -- (5mm,0);}}}
\newcommand{\raisebox{2pt}{\tikz{\draw[brown,solid,line width=1.2pt](0,0) -- (5mm,0);}}}{\raisebox{2pt}{\tikz{\draw[brown,solid,line width=1.2pt](0,0) -- (5mm,0);}}}
\begin{figure}[ht!]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tikzpicture}
\begin{axis}[
ybar,
enlargelimits=0.35,
title= {Segment-level Kendall's Tau ($\tau$)},
symbolic x coords={AR ref.,ARp ref.,WMTp ref.},
y tick label style={/pgf/number format/precision=5},
xtick=data,
ticklabel style = {font=\small},
]
\addplot coordinates {
(AR ref., 0.4483173077)
(ARp ref.,0.4416266026)
(WMTp ref.,0.4344751603)
};
\addplot coordinates {
(AR ref.,0.4507612179)
(ARp ref.,0.4426282051)
(WMTp ref.,0.4363982372)
};
\addplot coordinates {
(AR ref.,0.4420472756)
(ARp ref.,0.4420472756)
(WMTp ref.,0.4420472756)
};
\end{axis}
\end{tikzpicture}
}
\caption{Performance impact of using different kinds of references in combination with the original WMT English-to-German reference. In {\raisebox{2pt}{\tikz{\draw[blue,solid,line width=1.2pt](0,0) -- (5mm,0);}}} we observe the Kendall-Tau $\tau$ ranking correlation achieved by our multi-reference Estimator model (section \ref{sec:multi-ref}). In {\raisebox{2pt}{\tikz{\draw[red,solid,line width=1.2pt](0,0) -- (5mm,0);}}} we present the Kendall-Tau $\tau$ ranking correlation of our ``one-reference'' Estimator model using the alternative reference. Finally, for comparison, in {\raisebox{2pt}{\tikz{\draw[brown,solid,line width=1.2pt](0,0) -- (5mm,0);}}} we show the Kendall-Tau $\tau$ ranking correlation of our ``one-reference'' Estimator model using the original reference.}
\label{fig:multi-ref-kendall-en-de}
\end{figure}
While we lack data to draw any statistically significant conclusions, there is a strong suggestion from these results of a positive correlation between reference quality and utility to the predictive model.
For de-en, using an alternative reference did not offer any gain in Pearson's $r$. We note that when using it alone we only achieve $r$=0.34 compared to using the original reference which achieves $r$=0.42. We speculate, based on our observations above, that this might be due to the alternative reference being of lower quality.
These results potentially show that for approaches such as {\sc Comet}, quality is more important than quantity, and that lower-quality additional references can potentially hurt rather than help improve the correlations obtained using only one single high-quality reference.
With regard to the Kendall Tau measured at segment-level, by looking at Figure \ref{fig:multi-ref-kendall-en-de} (en-de), we see no significant differences in using the multi-reference technique. This suggests that having a higher Pearson's $r$ score does not necessarily guarantee a better Kendall's Tau.
We note that by design, with an approach such as {\sc Comet} that is based on a meaning-representation of references, extra references are expected to provide only minor additional value, especially versus lexical-based metrics such as {\sc Bleu} \cite{papineni-etal-2002-bleu}. Whereas the adequacy of the reference(s) is (again by design) expected to have a more significant impact on the performance of the model. Our initial results seem to strongly support this hypothesis.
\section{Conclusions}
In this paper we present {\sc Comet}, Unbabel's contribution to the WMT 2020 Metrics shared task. We leverage the framework outlined in \citet{comet} to demonstrate state-of-the-art or otherwise competitive levels of correlation with human judgements in all tasks and introduce a novel method of making optimal use of alternative references and demonstrate that the quality of the reference used is relevant to the success of our framework. Further investigation of the latter, in particular how to better leverage different kinds of references, represent an interesting direction for future work.
\section{Acknowledgments}
We are grateful to Fabio Kepler, Daan Van Stigt, Miguel Vera, and the reviewers, for their valuable feedback and discussions. This work was supported in part by the P2020 Program through projects MAIA and Unbabel4EU, supervised by ANI under contract numbers 045909 and 042671, respectively.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,712 |
\section{Local Retrograde Halo Tidal Streams}
If the Galactic halo has been formed partly
\citep{Searle1978} or entirely \citep[e.g.,][]{Majewski1993}
from accretion of smaller systems then one might expect groups of
stars with halo-like motions having strong velocity coherence
\citep[e.g.,][]{Helmi1999,Meza2005} near the Sun. Claims for nearby
halo moving groups date back at least to \citet{Eggen1959} and include
several retrograde candidates -- including, in particular, the widely
recognized Kapteyn Group -- among the halo groups long discussed by
\citet[e.g.,][]{Eggen1965,Eggen1996a,Eggen1996b}.
\citet{Majewskietal1994,Majewskietal1996} analyzed nearby halo stars
(selected by asymmetric drift) towards the North Galactic Pole (NGP)
and claimed strong organization of their phase space distribution into
a few clumps, including a prominent retrograde moving group initially
identified via proper motions in \citet{Majewski1992}. A possible
association of horizontal branch stars to this retrograde feature has
been identified by \citet{Kinman2007}. \citet{Eggen1996b} suggests an
association of the \citet{Majewski1992} retrograde group to Kapteyn's
group.
More recently, interest in the notion that the globular cluster
$\omega$Centauri (``$\omega$Cen") may be the remnant core of a tidally
disrupted satellite galaxy \citep[e.g.,][]{Lee1999,Majewski2000b,Bekki2003} --- a notion inspired by (1) internal chemical and age
distributions belying multiple $\omega$Cen stellar populations, (2)
the example of the similarly massive cluster M54 located in/near/as
the ``core" of the disrupting Sagittarius galaxy \citep{Majewski2000b}
and apparent chemical similarities of M54+Sagittarius to $\omega$Cen
\citep{Carretta2010}, and (3) the unusual, low-inclination, retrograde
orbit of $\omega$Cen itself \citep{Dinescu2002} --- has led to several
$\omega$Cen tidal disruption simulations; these models generally
produce retrograde-moving, $\omega$Cen debris relatively near the
solar circle \citep{Dinescu2002,Tsuchiya2003,Tsuchiya2004,Mizutani2003,Bekki2003}. This has prompted searches for retrograde
halo stars possibly shed by the ``cluster" and led to suggestions that
an ``$\omega$Cen" signal is present among local metal-weak stars
\citep{Dinescu2002,Mizutani2003,Meza2005}. In fact,
\citet[e.g.,][]{Eggen1978} speculated a connection between his
Kapteyn's star group and $\omega$Cen \citep[see also][]{Kotoneva2005},
a dubious connection \citep{Proust1988} before the breadth of
$\omega$Cen's metallicity distribution function was fully recognized,
but more recently bolstered by detailed chemical analysis of Kapteyn
group stars \citep{Wylie2010}.
Despite these recent claims for stripped $\omega$Cen stars in the {\it
solar neighborhood}, searches for extratidal stars near $\omega$Cen
itself have been less promising. While the photometric search by
\citet{Leon2000} seemed to suggest a ``significant" pair of tidal
tails extending from $\omega$Cen, these results were cast in doubt
when the substantial foreground differential reddening was assessed
\citep{Law2003}.\footnote{\citet{Leon2000} themselves warned that dust
extinction might be influencing their results.} An expansive
spectroscopic search for $\omega$Cen stars beyond its tidal radius by
\citet{DaCosta2008} reveals only six candidates among more
than 4,000 stars selected from the $\omega$Cen giant branch in the
color-magnitude diagram; these authors suggest that this meager haul
is consistent with models where most stripping took place long ago,
and with the lost stars now widely distributed about the Galaxy.
Stronger support for the tidally-disrupted dwarf galaxy model, and for
confidently linking solar neighborhood candidate members with
$\omega$Cen itself, would come from actually being able to trace
debris along the satellite's orbit, and, eventually, from $\omega$Cen
continuously to the solar neighborhood.
Here we report detection of a kinematically coherent ``tidal debris"
signature spanning $\gtrsim$$60^{\circ}$ of Galactic longitude in a
large radial velocity (RV) survey of giant stars mostly ($\sim95\%$)
within $\sim$$5$ kpc of the Sun. Stars within this dynamically
coherent group show a specific chemical marker thought to be unique to
$\omega$Cen, as well as distances and velocities consistent with
models of $\omega$Cen tidal debris. Though still mostly only a few
kiloparsecs away, these extended tidal debris stars provide a start at
tracing the $\omega$Cen stream and provide crucial, though still
crude, dynamical constraints on the disruption of the closest known
dwarf galaxy to the Sun. At minimum, these discovered debris stars
suggest that $\omega$Cen may be a principal source of local retrograde
stars.
\section{Substructure in the Southern GGSS}
Our analysis uses the Grid Giant Star Survey (GGSS), a
partially-filled, all-sky search for giant stars using the Washington
$M,T_2$+$DDO51$ photometric selection technique described in
\citet{PaperI}. The GGSS had as one goal the
identification of bright but distant, metal-poor giants suitable for
the Astrometric Grid of the (now defunct) Space Interferometry
Mission. To this end, a specific subset of potential SIM Grid stars
was drawn from the GGSS: the four most distant giant stars with
$M<13.5$ in each of the 1302 evenly spaced, $\sim0.4$-$0.6$ deg$^2$
GGSS fields, where photometric distances were estimated by assigning
absolute magnitudes to stars from their position in the
($M-T_2,M-DDO51$) diagram \citep[which separates dwarf and giant stars and,
within these luminosity classes, sorts by metallicity;][]{PaperI}. This
particular selection biases this ``SIM Grid sample" to more metal poor
giants, because these are farther at a given apparent magnitude. An
echelle resolution study of 774 Grid candidates selected in this way
verified 100\% of them to have giant branch surface gravities, and
indicated a median [Fe/H]$\sim$$-0.7$ and distance of $\sim$2 kpc
\citep{Bizyaev2006}, but with long tails to more distant and
metal-poor stars. Thus, the sample contains a mix of old thin disk,
Intermediate Population II (IPII) thick disk and a smaller fraction of
halo stars in mean proportions varying by Galactic latitude. Our
initial analysis here relies on $R$$\sim$$2,600$ spectroscopy of a
larger sample of 3318 GGSS ``SIM Grid" giants from our southern
observing campaign (sky distribution shown in Fig.\ 1a); further
discussion of the GGSS is given in \citet{Patterson2001} and
S. Majewski (in preparation).
In 1999 a spectroscopic follow-up campaign for these candidate
Astrometric Grid stars began at Las Campanas Observatory using the
Modular Spectrograph on the Swope 1-m telescope (and the DuPont 2.5-m
for 68 stars). The typical set-up used the 600 line/mm grating to
sample a $\sim2000$\AA\ spectral region including both H$\alpha$ and
H$\beta$ at 1.0\AA\ pixel$^{-1}$ (in a small fraction of cases the Ca
infrared triplet region was observed; details of both setups are in
\citet{Majewskietal2004}. The $S/N$ of the spectra exceeds 10 in almost
all cases and reaches $>$$50$ in some cases. RV cross-correlation of
the spectra used the methodology discussed in \citet{Majewskietal2004}.
Typical RV errors are 5-10 km~s$^{-1}$, estimated a variety of ways
including repeat measures of many stars and comparison to echelle
resolution RVs.
\begin{figure}[th]
\epsscale{0.70}
\includegraphics[angle=0,scale=0.42]{f1.pdf}
\caption{The distribution of the GGSS giant stars used in this study
in (a) Galactic coordinates, (b) $v_{\rm GSR}~$ versus longitude, and (c)
$v_{b}$ = $v_{\rm GSR}~$ $/ \cos{b}$ versus longitude (for only $|b|<$60\degr
in panel c). The curve and green shading in (c) highlight the arc
of stars we believe contains $\omega$Cen debris. Stars in this
sequence are marked with green or red points in panel (a). Grey
shadings show several other potential halo substructures. The
position of the $\omega$Cen core in all panels is shown by the large
green cross. Panel (d) is the same as panel (c) but showing the
``probability distribution" of $\omega$Cen tidal debris based on our
suite of models. Superposed red and blue points in all panels
represent stars having echelle spectra with red designating those
stars that follow the $\omega$Cen [Ba/Fe]-[Fe/H] patterns and blue
those that do not.}
\end{figure}
Figure 1b shows the distribution of Galactic Standard of Rest (GSR)
RVs ($v_{GSR}$) for the ``SIM Grid" stars, assuming a solar motion in
right-handed Galactic coordinates of $(+10.0,+225.3,+7.2)$
km~s$^{-1}$. For stars moving predominantly in Galactic planar orbits
(e.g., disk stars {\it and} putative $\omega$Cen debris), the
approximate ``planar RV" (i.e., the observed RV the star would have
were it on the Galactic equator) is given by $v_b=v_{GSR}/\cos(b)$
(this latitude normalization breaks down at high latitude, where a
star's $Z$ motion dominates the observed RV, so Fig.\ 1c is limited to
stars with $|b|<60^{\circ}$). In this planar projection stellar
populations more clearly sort by their relative asymmetric drifts, and
Figure 1c for the most part shows the expected longitudinal
distribution of $v_{GSR}$ for a predominantly thin disk/IPII mix of
stars.
However, stars not following disk kinematics are evident, including a
number having retrograde velocities. Among stars with halo-like
velocities (and particularly among likely retrograde stars), outlier
groupings (e.g., at [$l$, $v_{b}$] $\sim$ [$280^{\circ}$, 300
km~s$^{-1}$]) or thin, coherent strands of stars (e.g., from
[$20^{\circ}$, $-125$ km~s$^{-1}$] to [$300^{\circ}$, $-275$
km~s$^{-1}$]) can be seen in Figures 1b and/or 1c (where we have
highlighted some interesting features with shading). Such cold and
coherent RV trends with sky position are characteristic of long tidal
streams, such as the Sagittarius system
\citep[e.g.,][]{Majewskietal2004, Law2005}; in this case, however, the
substructure is found among relatively nearby giants and therefore
corresponds to stars with a much broader sky distribution than typical
for more distant streams (Fig.\ 1a). Such substructure among the GGSS
giants is not surprising given that they probe distances similar to
the mostly main sequence stars in the
\citet{Majewskietal1994,Majewskietal1996} study, which also showed
significant halo substructure (but in only a single pencil beam).
Moreover, a new ``all-sky" study of bright M giants by A. Sheffield
(in preparation) and probing comparable distances shows analogous,
though even more striking, [$l$, $v_{b}$] coherences among halo-like
stars (most likely because M giants probe typically younger tidal
streams). Together, the GGSS giants, Sheffield M giants,
\citet{Majewskietal1994,Majewskietal1996} subdwarfs, and
\citeauthor{Kinman2007} horizontal branch stars point to the high
degree of substructure in the halo even at the solar circle. Indeed,
the degree of velocity coherence and substructure of the local halo
does not differ much from that seen in the distant halo
\citep{Majewski2004}.
\section{Tidal Debris Model and the $\omega$Centauri Connection}
Are there stars in the GGSS sample that can be associated with
$\omega$Cen? The studies of solar neighborhood $\omega$Cen debris
mentioned in \S1 were able to make use of the expected ($E, L_z$)
distributions to trawl for the best local representatives.
Unfortunately, our giant stars are far enough away that most available
proper motions (and therefore complete space velocity determinations)
are unreliable. Therefore we winnow our search to those [$l$,
$v_{b}$] ranges expected to be populated by any realistic model of
$\omega$Cen tidal disruption. To do so we create a suite of N-body
simulations of satellites undergoing tidal disruption along
$\omega$Cen-like orbits in the static Milky Way (MW) potential given
by \citet{Johnston1995}, with 30,000 particles representing the
parent satellite in an initially Plummer configuration. To account
for observational uncertainties that prohibit us from accurately
knowing the true position, space motion, and therefore orbits of both
$\omega$Cen and the Sun, we create a grid of models spanning ranges of
uncertainty around typical mean values for each critical interaction
parameter \citep[from][]{Dinescu1999}: $\omega$Cen distances of
[4.9,5.1,5.3] kpc, Galactic Cartesian velocities (right-handed system
in the Galactic rest frame) of $V_x$ = $[42,53,64,75,86]$ km~s$^{-1}$,
$V_y$ = $[-52,-43,-34,-25,-16]$ km~s$^{-1}$, and $V_z$ = $[-6,4,14]$
km~s$^{-1}$, a solar Galactocentric distance of $[7.0, 7.75, 8.5]$
kpc, and, to set the MW mass scale, a local circular velocity of
$[220, 254]$ km~s$^{-1}$, where the latter value is that suggested by
\citet{Reid2009}. Within this grid we adopt satellites of three
different initial masses, [3e7, 3e8, 3e9] M$_{\odot}$, and evolved for
$\sim0.4$ Gyr, ensuring that each model orbit places the satellite at
the current ($l$,$b$) position and RV of $\omega$Cen.
These 4050 simulations produce a variety of tidal streams from which
we can establish the range of possible [$l$, $v_{b}$] distributions;
in fact, the sum of these models (Fig.~1d) gives us something like a
``probability distribution function" (PDF) of where the last 0.4 Gyr
of $\omega$Cen tidal debris might most likely lie (Fig.\ 1d).
Comparing this to the GGSS distribution, and ignoring regions
dominated by disk stars, we call attention to a particularly striking
match of the PDF to a relatively tight sequence of likely retrograde
stars over $l \sim280$-$360^{\circ}$, highlighted with green shading
in Figure 1c. That these stars form a coherent velocity structure
orbiting in a near-Galactic planar orbit is demonstrated by the fact
that, despite their broad sky distribution (green and red points,
Fig.\ 1a), they show a coherent, string-like configuration in planar,
$v_{b}$ projection.
A 2nd-order polynomial fit to the Figure 1c sequence with iterative
rejection settles on 35 stars in the feature with an observed RV
dispersion of only $\sim$40 km~s$^{-1}$. Most interestingly, the
sequence passes through the ($l$, $v_{b}$) position of $\omega$Cen,
also shown.
\begin{figure}[t]
\includegraphics[angle=0,scale=0.42]{f2.pdf}
\caption{(Top panel) The distribution of [Ba/Fe]-[Fe/H] for MW stars
(blue points) and a characteristic locus (blue line) from data by
\citet{Fulbright2002}, \citet{Johnson2002}, and \citet{Reddy2003,Reddy2006},
overlaid with the same for $\omega$Cen stars (magenta points and
line) from \citet{Francois1988}, \citet{Norris1995}, and
\citet{Smith2000}. (Bottom panel) The distribution of barium
abundances for the ten stars following the retrograde sequence lying
within the ``$\omega$Cen PDF region" and containing the position of
$\omega$Cen shown in Fig. 1 (red points) versus those stars lying
outside the ``$\omega$Cen PDF region" (blue). The colored lines are
those shown in the top panel.}
\end{figure}
To test whether $\omega$Cen stars might fall among those GGSS stars
lying within this retrograde feature, high-resolution
($R$$\sim$55,000), echelle spectra for eight of them, as well as a
control sample of four other halo GGSS stars (including two very
extreme $v_{b}$ stars at similar longitudes), were obtained using the
Sandiford echelle spectrometer \citep{McCarthy1993} on the McDonald
2.1-m Struve telescope. Given the modest wavelength coverage and
$S/N$$\sim$25-50 of the spectra, iron abundances were derived from a
set of unblended \ion{Fe}{1} lines using measured equivalent widths. The
stellar parameters $T_{\rm eff}$ and $\log{g}$ were taken from the
analysis of GGSS stars by \citet{Bizyaev2006}, with determinations
of the microturbulence velocities ($\xi$) set by the \ion{Fe}{1} lines
measured for this study.
The well-defined \ion{Ba}{2} line at 5854\AA\ was used as an s-process
abundance indicator because (1) it is the most easily detectable in
these generally mediocre spectra, and (2) the distribution of [Ba/Fe]
-- [Fe/H] for $\omega$Cen is quite distinctive (Fig.~2a) --- indeed it
is unique among all star systems studied to date in the extreme
overabundances of s-process elements characterizing its more
metal-rich stellar population (see Fig.~11 of the Geisler et al.~2007 review).
Figure 2b shows the derived [Ba/Fe]--[Fe/H] pattern for the
12 GGSS stars; abundances and other data derived for these stars are
given in Table 1. The quoted velocities are from the original
medium-resolution spectra and have typical random errors of $\sim$10
km~s$^{-1}$; photometric uncertainties are about 0.01 mag. Abundance
uncertainties (shown in Fig. 2b) were set by the sensitivities of \ion{Fe}{1}
and \ion{Ba}{2} abundances to changes in stellar parameters of $\pm$100K in
$T_{\rm eff}$, $\pm$0.3 dex in $\log{g}$, and $\pm$0.5 km~s$^{-1}$ in
$\xi$. As is vividly demonstrated, the Table 1 stars most likely to
be kinematically associated with $\omega$Cen (Fig.\ 1d) clearly follow
the characteristic $\omega$Cen trend in [Ba/Fe] versus [Fe/H], while
two stars least likely to be kinematically associated with $\omega$Cen
lie along the MW trend. The combination of kinematical consistency
and possession of the hallmark barium abundance trends for the former
group of stars are strong evidence that an extended part of the
$\omega$Cen tidal debris stream has been found.\footnote{The two Table
1 stars with $\omega$Cen chemistry and extreme $v_b$, off the main
``green trend" in Fig. 1c, are plausibly associated with older
$\omega$Cen tidal debris wraps, shown with faint probability in
Fig.\ 1d, or not currently part of our 0.4 Gyr-long models (see
Fig.\ 4).}
It is worth noting that within this rather small sample of $\omega$Cen
stream candidates is a carbon-rich star, G1358-16.167. This red giant
exhibits strong Swan C$_{\rm 2}$ bands (e.g., at $\lambda$5165\AA;
Fig.\ 3) and is strongly barium-enhanced. Such C-rich halo giants at
this modestly low metallicity are relatively rare and constitute only
about 1-2\% of halo giants. However, $\omega$Cen has at least five
known C-rich giants \citep[see Table 3 in][which both subgiants and
giants]{Bartkevicius1996} whereas the only other globular clusters
known to harbor C-rich giants are M22 (with two --- as well as a
spread in heavy-element abundances and s-process enrichment, similar
to, but not to the degree of, $\omega$Cen), M2 (with one), and M55
(with one --- \citet*{Smith1982}). Given the likely association of
most Table 1 stars to $\omega$Cen it is not too surprising that one is
found to be C-rich; this observation only strengthens the tie to their
chemically peculiar parent stellar system.
\begin{figure}[t]
\includegraphics[angle=0,scale=0.34]{f3.pdf}
\caption{Example of a C$_{\rm 2}$ Swan band in the spectrum of G1358-16.167.}
\end{figure}
Based on the [Fe/H]-[$\alpha$/Fe] and age-metallicity distributions
for $\omega$Cen given by \citet{Stanford2006} --- adopting a 4 Gyr
$\omega$Cen age span --- and the derived [Fe/H] and 2MASS photometric
data for the ten good GGSS $\omega$Cen candidates we estimate their
distances using matching \citet{Dotter2008} isochrones. With these
distances we can place the stars in their Galactic planar positions
relative to $\omega$Cen and the Sun (Fig.\ 4). Superposed on this
distribution we plot the model debris and satellite orbit from one of
several $\omega$Cen models in our grid (the example model parameters
are given in the figure legend) that provide a reasonable match to the
positions and RVs of the GGSS stars. This model, based
on a satellite orbit with peri-/apo-Galactica limits of (1 kpc)/(7
kpc), respectively, not only demonstrates how the stars of interest
very plausibly trace $\omega$Cen debris, but also how it might be
possible for $\omega$Cen tidal debris to reach the solar neighborhood,
as suggested by the various claims for this discussed in \S1.
In fact, though, as stated earlier, available proper motions for the
GGSS $\omega$Cen stars are typically of low quality, the UCAC
astrometry \citep{Zacharias2010} does hint at further tantalizing
connections to previous claims for nearby $\omega$Cen debris: The GGSS
$\omega$Cen stars with the smallest derived ($E, L_z$) uncertainties
happen to fall in, or quite near, the ``$\omega$Cen debris
expectation" box defined in ($E, L_z$) by \citet[][using the same
gravitational potential]{Dinescu2002}, whereas the weighted mean $L_z$ of all
ten GGSS $\omega$Cen stars, $-179\pm135$ km~s$^{-1}$ kpc, matches
extremely well the ``$\omega$Cen peak" identified within the stellar
sample explored by \citet[][their Fig.\ 9]{Meza2005}.
\begin{figure}[t]
\includegraphics[angle=0,scale=0.35]{f4.jpg}
\caption{The Galactic X-Y positions of Table 1 stars with respect to
the Sun ($\odot$) and $\omega$Cen (large green cross). Overlaid is
an N-body simulation of $\omega$Cen tidal disruption and the
associated orbit over the past 0.4 Gyr with the parameters given in
the legend. The model and Table 1 stars with $\omega$Cen-like
chemistry are color-coded by $v_b$, which match well in general at
the computed positions of the Table 1 stars. The two black points
are the Table 1 stars that appear to be kinematically, spatially and
chemically unassociated with $\omega$Cen.}
\end{figure}
\section{Some Implications}
In the GGSS sample of bright giant stars within $\sim$5 kpc across the
southern 2/3 of the celestial sphere and biased toward metal-poor
stars we have shown evidence that those having halo-like velocities,
and particularly those with retrograde velocities, show a highly
substructured, rather than random, distribution. This result is a
further demonstration that even in the inner halo, at the position of
the solar circle, the Galactic halo is not well-mixed, but shows the
signature of multiple minor accretion events. We identify one group
of stars kinematically and chemically consistent with being
$\omega$Cen debris, which we use to get a rough constraint on the
$\omega$Cen tidal stream in the inner Galaxy, a model demonstrating
how $\omega$Cen stars can reach the solar neighborhood. We have
probed with high resolution spectroscopy less than half of the several
dozen stars we associate with $\omega$Cen [$l$,$v_{b}$] features, yet
all of the best candidates are found to be chemically consistent with
being from $\omega$Cen; we can project that most likely a large
fraction of the fuller sample of ``$\omega$Cen" candidates are
authentic. Considering that only a few dozen clearly retrograde stars
are found in the southern GGSS at all (Fig.\ 1b-c), our results
suggest that $\omega$Cen tidal debris is a primary contributor of
retrograde stars near the Sun, and likely the overwhelmingly dominant
contributor in the inner two Galactic quadrants.
\acknowledgements We gratefully acknowledge support by NASA/JPL
contracts 1201670 and 1228235, and the David and Lucile Packard
Foundation to build the GGSS, as well as NSF grants AST-0307851 and
AST-0807945. This work would not have been possible without the
generous access to the Swope telescope for development of the SIM
Astrometric Grid granted by Carnegie Observatories Directors Augustus
Oemler and Wendy Freedman.
| {
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{"url":"https:\/\/www.zbmath.org\/?q=an%3A0807.62090","text":"# zbMATH \u2014 the first resource for mathematics\n\nAutoregressive conditional density estimation. (English) Zbl\u00a00807.62090\nSummary: R. F. Engle\u2019s ARCH model [Econometrica 50, 987-1007 (1982; Zbl 0491.62099)] is extended to permit parametric specifications for conditional dependence beyond the mean and variance. The suggestion is to model the conditional density with a small number of \u201cparameters\u201d, and then model these parameters as functions of the conditioning information.\nThis method is applied to two data sets. The first application is to the monthly excess holding yield on U.S. Treasury securities, where the conditional density used is a Student\u2019s $$t$$ distribution. The second application is to the U.S. Dollar\/Swiss Franc exchange rate, using a new \u201cskewed Student $$t$$\u201d conditional distribution.\n\n##### MSC:\n 62P20 Applications of statistics to economics 91B84 Economic time series analysis\nFull Text:","date":"2021-09-18 19:42:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5075529217720032, \"perplexity\": 2626.9372180314435}, \"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\/1631780056572.96\/warc\/CC-MAIN-20210918184640-20210918214640-00377.warc.gz\"}"} | null | null |
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Ebbinghof is a locality in the municipality Schmallenberg in the district Hochsauerlandkreis in North Rhine-Westphalia, Germany.
The village has 23 inhabitants and lies in the north of the municipality of Schmallenberg at a height of around 420 m. Ebbinghof borders on the villages of Altenhof, Bad Fredeburg, Berghausen, Gleidorf, Obringhausen, Schmallenberg and Wormbach.
The village used to belong to the municipality of Wormbach in Amt Schmallenberg until the end of 1974.
External links
Hawerland.de: Ebbinghof ]
References
Villages in North Rhine-Westphalia
Schmallenberg | {
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It could be hit by the escalating US-China trade wars paired with US Fed hikes.
The Singapore dollar will remain 'range-bound' against the USD in the near-term despite headwinds from global trade developments as the Monetary Authority of Singapore (MAS) pushed to modestly appreciate the SGD nominal effective exchange rate, Fitch Solutions said. With this, the exchange rate could go around $1.35 against USD for 2018, the agency said.
The firm noted that the SGD is managed against a basket of currencies of Singapore's major trading partners. Moreover, they believe that MAS will keep the relative strength of the SGD as it targets to safeguard medium-term price stability over the coming months.
However, Fitch Solutions thinks that the further escalation of the US-China trade wars will be a headwind to the SGD, with the city-state's export-oriented manufacturing sector already hit.
"In addition, continued interest rate hikes by a hawkish US Federal Reserve over the coming quarters will likely put broad upside pressure on the US dollar," they explained.
On a longer-term view, the firm thinks that the SGD is likely to get hurt from the combination of a strong external balance sheet and subdued inflationary pressure due to the government's prudent monetary and fiscal policy framework.
"Singapore's small and open economy could slow significantly, which as a result would see the SGD undergo a sharp depreciation," Fitch solutions said. | {
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\section{Introduction}
The dynamic behavior of an elastic material is classically described by continuum mechanics in terms of a deformation mapping that satisfies the second-order, nonlinear, hyperbolic partial differential equations of elastodynamics subject to given initial and boundary conditions. The precise equations are given by Newton's second law of motion. For hyperelastic materials, the internal forces are obtainable as the first variation of an elastic energy that depends in a local but nonlinear way on the deformation gradient.
At the same time, on a microscopic level, crystalline solids consist of many atoms, e.g. on a part of a Bravais lattice, and can be described directly by their interaction. The interatomic forces can effectively be modeled in terms of classical interaction potentials. Using again Newton's second law of motion, we arrive at a very high dimensional system of ordinary differential equations.
The classical connection between atomistic and continuum models of nonlinear elasticity is provided by the Cauchy-Born rule: The continuum stored energy function associated to a macroscopic affine map is given by the energy per unit volume of a crystal which is homogeneously deformed with the same affine mapping. In particular, this entails the assumption that there are no fine scale oscillations on the atomistic scale. We will call this function the Cauchy-Born energy density in the following. Note though, that it is not clear a priori whether the Cauchy-Born hypothesis is true or not.
In the previous work \cite{braun16static}, Schmidt and the author rigorously discuss existence and convergence of solutions as well as the Cauchy-Born rule in the case of elastostatics. We also refer to the introduction of \cite{braun16static} for a more exhaustive account of recent mathematical progress in this field with an emphasis on static equilibrium problems.
Our aim in this work is to establish a rigorous link between atomistic models and the corresponding Cauchy-Born continuum models for the elastodynamic behavior of crystalline solids accounting for body forces, boundary values, and initial conditions. We will prove such a connection in the asymptotic regime where the interatomic distance $\varepsilon$ goes to $0$ and will even consider long times and large deformations.
In more detail, we will show that as long as the continuum solution exists and satisfies certain stability conditions, there are solutions of the corresponding atomistic initial-boundary value problems with lattice spacing $\varepsilon$ that converge to the continuum solution uniform in time as $\varepsilon \to 0$. Or to look at it from the other direction: We will give sufficient conditions on the body forces, initial conditions, and boundary conditions in the atomistic model such that there are solutions that follow a continuum solution as $\varepsilon \to 0$ and thus obey the Cauchy-Born rule.
Recent publications (\cite{emingstatic}, \cite{emingdynamic}, \cite{ortnertheil13}) already provide results of this type for small displacements on a flat torus and for the full space problem with a far-field condition, respectively. But naturally the question arises if such an analysis is possible for a material occupying a general finite domain, on the boundary of which there might also be prescribed time-dependent boundary values. To cite Ericksen \cite[p.~207]{Ericksen08} ``Cannot someone do something like this for a more realistic case, say zero surface tractions on part of the boundary and given displacements on the remainder?'' In \cite{braun16static}, Schmidt and the author have given a positive answer to this question for the static problem with given displacements on the full boundary. Implicitly, results under mixed or pure traction boundary conditions are largely included. Here, we want to extend these results to the dynamic case. For this we will make use of some of the estimates proven in \cite{braun16static}.
Such a treatment of arbitrary domains and general displacement boundary conditions is of interest not only from a theoretical perspective but also with a view to specific situations that are of interest in applications, that can use our results as a starting point. Besides discussing elastic behavior, our results can also be used to discuss questions of stability and even the onset of instabilities, since the stability assumptions we make are designed to be quite sharp.
While equilibrium situations play an important role in mechanics, in many situations the material behavior decisively depends on inertial effects and, as a consequence, static or quasistatic descriptions are insufficient. Mathematically, the dynamic equations are considerably more challenging. Already in the continuum description, we have to discuss a quasi-linear, second-order, hyperbolic system under time-dependent boundary conditions. While the problems have natural energies that are conserved, they are of limited use in the analysis since level sets are typically unbounded in relevant norms and can contain regions where there is a loss of stability.
In view of the recent results by other authors mentioned above, it should be pointed out that the treatment of the boundary value case is not a straightforward extension of the previous results. Let us just mention some of the difficulties. An important but subtle point in our main theorem is the condition posed on the atomistic boundary data. The atomistic boundary values can not be arbitrary but have to be chosen in a precise range to reflect the continuum boundary values while at the same time ensuring that there are no surface effects, so that the Cauchy-Born rules holds up to the boundary. Indeed, for arbitrary atomistic boundary values, one expects surface effects and a failure of the Cauchy-Born rule. A precise and rigorous mathematical treatment of such surface relaxation effects is currently still out of reach (but cf.\ \cite{theilsurface11}). In order to allow for as many atomistic boundary conditions (and body forces) as possible, we consider general convergence rates $\varepsilon^\gamma$ in our main theorem, Theorem \ref{thm:atomisticwavethm}, and only restrict $\gamma$ as much as necessary. While smaller $\gamma$ will lead to a larger variety of atomistic boundary values, the maximal $\gamma = 2$ gives optimal convergence rates. Furthermore, there are several more technical problems. Most importantly, certain methods which are available on the flat torus or on the whole space do not translate to our setting. E.g., quasi-interpolations do not preserve boundary conditions. Instead, we use a different and more robust approach to the residual estimates that works in all cases and does not require any more regularity than the dynamical result in \cite{ortnertheil13}.
For the atomistic equations of elastodynamics, besides considering boundary conditions, there is an additional open question of considerable importance. Is it possible to establish the existence of atomistic solutions that satisfy the Cauchy-Born rule and their link to the continuum solutions not just for small times and deformations close to a stable affine configuration but for long times and large deformations?
In \cite{emingdynamic}, E and Ming only prove a short time result. An extension is not obvious, since their methods were restricted to small displacements. In \cite{ortnertheil13}, Ortner and Theil are indeed aware of this restriction that also applies to their results. They proposed that one could indeed extend the results to long times if one were to establish an atomistic version of the Gårding inequality.
The need for such an inequality can be understood as follows. For small displacements it is sufficient to stay close enough to a given stable affine deformation to ensure that the second variation of the potential energy is positive uniformly in $H^1_0$. For large deformations this is, in general, false. Even if at each point the gradient corresponds locally to a stable affine deformation, the second variation can still be negative globally. A Gårding inequality helps to work around this situation. It states that one can still get uniform positivity in $H^1_0$ from the local stability if one is willing to add a large constant times a lower order term (more precisely, the square of the $L^2$-norm). While a Gårding inequality alone is not sufficient to treat large deformations for the static equations, this inequality is key to the dynamic equations. In the continuum case the Gårding inequality is indeed part of a well-established theory. In the discrete case such a Gårding inequality is more subtle. Already the question of what constitutes a `locally stable deformation' requires a deeper analysis in the atomistic case, as we will discuss in Section \ref{sec:preparations}. In particular, the local stability assumption from the continuum case turns out to be insufficient. Additionally, in the continuous case the continuity of the coefficients and the deformation gradient is a crucial assumption for the Gårding inequality. This assumption has to be replaced by a more quantified version that is adapted to the discrete nature of the atomistic problem. In this spirit we will indeed establish an atomistic Gårding inequality, Theorem \ref{thm:discreteGårding}.
To give a short overview, we will start by giving a precise description of our models in Section \ref{sec:models}. Next we will shortly discuss in Section \ref{sec:preparations} the different concepts of stability and cite some important results about the stability constants. A crucial ingredient in the proof of our main theorem will be the observation that a smooth solution of the Cauchy-Born continuum equations solves the atomistic equations up to a small residuum. Still in Section \ref{sec:preparations}, this is made rigorous with the residual estimates that we can cite from \cite{braun16static}. In Section \ref{sec:cont}, we will use existing short time results, as well as different ideas about hyperbolic regularity, optimal elliptic regularity under weak assumptions, and a fine discussion of compositions and products in Sobolev spaces, to prove a result on the maximal existence time of solutions to the continuum equations of elastodynamics.
But the main results in this work can be found in Section \ref{sec:atom} where we will state and prove the atomistic Gårding inequality, Theorem \ref{thm:discreteGårding}, as well as our main theorem, Theorem \ref{thm:atomisticwavethm}. It states that as long as the continuum solutions exists and is atomistically stable, there exists a solution to the atomistic equations close to it. We also quantify the required conditions on the atomistic body forces, boundary values, and initial values in relation to their continuum counterparts.
Lastly, in the appendices we collect and prove a few technical lemmata concerning elliptic and hyperbolic regularity as well as the multiplication of many Sobolev functions. Some of these results might already be (implicitly) known to experts in the field but they do not seem to be available in the literature.
\section{The Models} \label{sec:models}
\subsection{The Continuum Model}
We consider a bounded, open set $\Omega \subset \mathbb{R}^d$, a time interval $[0,T)$, deformations $y \colon \Omega \times [0,T) \to \mathbb{R}^d$, an energy density $W_{\rm cont} \colon \mathbb{R}^{d \times d} \to (-\infty,\infty]$, initial positions $h_0 \in H^1(\Omega; \mathbb{R}^d)$, initial velocities $h_1 \in L^2(\Omega;\mathbb{R}^d)$, a body force $f \in L^2(\Omega \times [0,T); \mathbb{R}^d)$, and Dirichlet boundary data $g \in L^2([0,T); H^1(\Omega ; \mathbb{R}^d))$. We then consider the potential energy
\[
E(y;f)(t) = \int\limits_\Omega W_{\rm cont}(\nabla y (x,t)) - y(x,t)f(x,t) \,dx,
\]
whenever it is well-defined.
In the static case the relevant deformations are the local minimizers of the potential energy. In the dynamic case, we use Newton's second law of motion where the forces are given by the first variation of the potential energy. The reference body is assumed to have constant density $\rho$. By choice of units we can just take $\rho \equiv 1$. That means we are looking for (weak) solutions to the initial boundary value problem
\[ \left\{ \begin{array}{r c l l}
\ddot{y}(x,t)-\divo (DW_{\rm cont}(\nabla y (x,t))) &=& f(x,t) & \text{in}\ \Omega \times (0,T), \\
y(x,t) & = & g(x,t) & \text{on}\ \partial\Omega \times (0,T),\\
y(x,0) & = & h_0(x) & \text{in}\ \Omega,\\
\dot{y}(x,0) & = & h_1(x) & \text{in}\ \Omega.
\end{array} \right. \]
The assumptions on $W_{\rm cont}$ should be weak enough to be consistent with typical interatomic interaction potentials, e.g., Lennard-Jones potentials. Therefore, we should not assume global (quasi-)convexity or growth at infinity and $W_{\rm cont}$ should be allowed to have singularities. Still one can solve the problem quite generally, as long as the energy density and all the data are sufficiently smooth and as long as the energy density is well-behaved at the initial datum.
\subsection{The Atomistic Model}
We will mostly use the same notation as in \cite{braun16static}. We consider the reference lattice $\varepsilon \mathbb{Z}^d$, where $\varepsilon > 0$ is the lattice spacing. This partitions $\mathbb{R}^d$ into the cubes $\{z\} + \big(-\frac{\varepsilon}{2},\frac{\varepsilon}{2}\big]^d$ with $z \in \varepsilon \mathbb{Z}^d$. Given $x \in \mathbb{R}^d$, we then define $\hat{x}\in \varepsilon \mathbb{Z}^d$ to be the midpoint of the corresponding cube and $Q_\varepsilon(x)$ the cube itself.
The actual position of the atoms are described by a deformation map $y \colon (\Omega \cap \varepsilon \mathbb{Z}^d) \times [0,T) \to \mathbb{R}^d$. We want to look at a general finite range interaction model, i.e., there is a finite set $\mathcal{R} \subset \mathbb{Z}^d \backslash \{0\}$ denoting the possible interactions. We will always assume that $\spano_{\mathbb{Z}} \mathcal{R}=\mathbb{Z}^d$ and $\mathcal{R} = - \mathcal{R}$. In the energy, the atom marked by $x,\tilde{x} \in \varepsilon \mathbb{Z}^d$ then can only interact directly if there is a $z\in \varepsilon \mathbb{Z}^d$ with $x,\tilde{x} \in z + \varepsilon \mathcal{R}$. Furthermore, we assume our system to be translationally invariant such that the interaction can only depend on the matrix of differences $D_{\mathcal{R},\varepsilon} y (x) = (\frac{y(x+\varepsilon \rho)-y(x)}{\varepsilon})_{\rho \in \mathcal{R}}$ with $x \in \varepsilon \mathbb{Z}^d$, where we already use the natural scaling that has an optimal interatomic distance on scale $\varepsilon$. The site potential $W_{\rm atom} \colon (\mathbb{R}^d)^\mathcal{R} \to (-\infty,\infty]$ is then assumed to be independent of $\varepsilon$. Compare \cite{BLL:02} for a detailed discussion of this scaling.
As a mild symmetry assumption on $W_{\rm atom}$, we will assume throughout that
\[ W_{\rm atom} (A) = W_{\rm atom} (T(A))\]
for all $A \in (\mathbb{R}^d)^\mathcal{R}$, where
\[T(A)_{\rho} = -A_{-\rho}.\]
This is indeed a quite weak assumption. In a typical situation this just means that we have partitioned the overall energy in such a way, that the site potential is invariant under a point reflection at that atom combined with the natural relabeling.
In particular, if $W_{\rm atom}$ is sufficiently smooth, we have
\[D^k W_{\rm atom} ((B \rho)_{\rho \in \mathcal{R}})[T(A_1), \dotsc, T(A_k)] = D^k W_{\rm atom} ((B \rho)_{\rho \in \mathcal{R}})[A_1, \dotsc, A_k].\]
Letting $R_{\rm max} = \max \{\lvert \rho \rvert \colon \rho \in \mathcal{R}\}$ and $R_0 = \max\{R_{\rm max}, \frac{\sqrt{d}}{4}\}$, the discrete gradient $D_{\mathcal{R},\varepsilon}y$ is surely well-defined on the discrete 'semi-interior'
\[\sinto_\varepsilon \Omega = \{x \in \Omega \cap \varepsilon \mathbb{Z}^d \colon \dist(x,\partial \Omega) > \varepsilon R_0\}.\]
Additionally, the definition of $R_0$ implies that
\[\Omega_\varepsilon = \bigcup_{z \in \into_\varepsilon \Omega} Q_\varepsilon(z) \subset \Omega\]
which will be used later on.
The energy is then defined by a sum over $\sinto_\varepsilon \Omega$, such that any variations should only change these gradients. Hence, variations should only be allowed on the full discrete interior
\[\into_\varepsilon \Omega = \{x \in \Omega \cap \varepsilon \mathbb{Z}^d \colon \dist(x,\partial \Omega) > 2\varepsilon R_0\}\]
and the boundary values should be prescribed on the boundary layer $\partial_\varepsilon \Omega = \Omega \cap \varepsilon \mathbb{Z}^d \backslash \into_\varepsilon \Omega$. Indeed, we consider a boundary datum $g_{\rm atom} \colon \partial_\varepsilon \Omega \times [0,T) \to \mathbb{R}^d$ and define the set of admissible deformations for a fixed time as
\[ \mathcal{A}_\varepsilon (\Omega, g) = \{ y \colon \Omega \cap \varepsilon \mathbb{Z}^d \to \mathbb{R}^d \colon y(x)=g(x) \text{ for all } x \in \partial_\varepsilon \Omega\}.\]
Given a body force $f_{\rm atom} \colon \into_\varepsilon \Omega \times [0,T) \to \mathbb{R}^d$ we will look at the potential energy
\[
E_\varepsilon(y;f_{\rm atom})(t) = \varepsilon^d \Big( \sum\limits_{x \in \sinto_\varepsilon \Omega} W_{\rm atom}(D_{\mathcal{R},\varepsilon} y (x,t)) -\sum\limits_{x \in \varepsilon \mathbb{Z}^d \cap \Omega} y(x,t)f_{\rm atom}(x,t) \Big).
\]
The scaling ensures that affine deformations have an energy independent of $\varepsilon$ (up to lower order terms) and, more generally, sufficiently smooth deformations have a finite and non-trivial energy in the limit, cf.\ \cite{BLL:02}.
For later use, we define the atomistic (semi-)norms
\begin{align*}
\lVert y \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega}^2 &= \varepsilon^d \sum_{x \in \into_\varepsilon \Omega} \lvert y(x) \rvert^2 \\
\lVert y \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}^2 &= \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} y(x) \rvert^2.
\end{align*}
Note in particular the norm equivalency
\[\sup\limits_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} y(x) \rvert \leq \varepsilon^{-\frac{d}{2}} \lVert y \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)} \]
which will play a crucial role later on.
Given $g \colon \partial_\varepsilon \Omega \to \mathbb{R}^d$, $y \colon \Omega \cap \varepsilon \mathbb{Z}^d$ minimizes $\lVert y \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}$ under the constraint $y(x)=g(x)$ for all $x \in \partial_\varepsilon \Omega$ if and only if $(y,u)_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}=0$ for all $u \in \mathcal{A}_\varepsilon(\Omega,0)$ and $y(x)=g(x)$ for all $x \in \partial_\varepsilon \Omega$. Thus, for every $g \colon \partial_\varepsilon \Omega \to \mathbb{R}^d$ there is precisely one such $y$, it depends linearly on $g$ and is the unique solution to $\divo_{\mathcal{R},\varepsilon} D_{\mathcal{R},\varepsilon} y = 0$ with boundary values $g$. We write $y=T_\varepsilon g$. Accordingly, we define the semi-norm
\[ \lVert g \rVert_{\partial_\varepsilon \Omega, 0} = \lVert T_\varepsilon g \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}.\]
This norm on the boundary does not play such an important role in the dynamic case as in the static case, but we will later define the more important $\lVert g \rVert_{\partial_\varepsilon \Omega, dyn}$ in the same spirit.
In the static case one is interested in finding local minimizers. In the dynamic case we additionally have an initial configuration $h_{0,{\rm atom}} \colon (\Omega \cap \varepsilon \mathbb{Z}^d) \to \mathbb{R}^d$ and initial velocities $h_{1,{\rm atom}} \colon (\Omega \cap \varepsilon \mathbb{Z}^d) \to \mathbb{R}^d$ such that the compatibility conditions $h_{0,{\rm atom}} \in \mathcal{A}_\varepsilon (\Omega, g_{\rm atom}(\cdot,0))$ and $h_{1,{\rm atom}} \in \mathcal{A}_\varepsilon (\Omega, \dot{g}_{\rm atom}(\cdot,0))$ hold true. At last, let us assume that all atoms have the same mass $m_\varepsilon = \varepsilon^d \rho$. This scaling ensures that the macroscopic reference body has a finite positive mass density $\rho$. As remarked before, we can assume $\rho=1$. Note that the scaling of the potential energy and the masses only affects the scaling of time. With our choice the time will not be rescaled and remains a macroscopic quantity. For the body forces, this scaling corresponds to a macroscopic acceleration of each atom (e.g. through gravity).
Again we apply Newton's second law of motion and arrive at the initial boundary value problem
\[ \left\{ \begin{array}{r c l l}
\ddot{y}(x,t) -\divo_{\mathcal{R},\varepsilon} \big( DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y (x,t))\big) &=& f_{\rm atom}(x,t) & \text{in}\ \into_\varepsilon \Omega \times (0,T), \\
y(x,t) & = & g_{\rm atom}(x,t) & \text{on}\ \partial\Omega_\varepsilon \times [0,T),\\
y(x,0) & = & h_{0,{\rm atom}}(x) & \text{in}\ \Omega \cap \varepsilon \mathbb{Z}^d,\\
\dot{y}(x,0) & = & h_{1,{\rm atom}}(x) & \text{in}\ \Omega \cap \varepsilon \mathbb{Z}^d
\end{array} \right. \]
where $DW_{\rm atom}(M) = \big( \frac{\partial W_{\rm atom}(M)}{\partial M_{i\rho}} \big)_{\substack{1 \leq i \leq d \\ \rho \in \mathcal{R}}}$ for $M = (M_{i\rho})_{\substack{1 \leq i \leq d \\ \rho \in \mathcal{R}}} \in \mathbb{R}^{d \times \mathcal{R}} \cong (\mathbb{R}^d)^\mathcal{R}$ and we write
\[\divo_{\mathcal{R},\varepsilon} M(x) = \sum\limits_{\rho \in \mathcal{R}} \frac{M_\rho (x) - M_\rho(x-\varepsilon \rho)}{\varepsilon}\]
for any $M \colon \Omega \cap \varepsilon \mathbb{Z}^d \to \mathbb{R}^{d \times \mathcal{R}} \cong (\mathbb{R}^d)^\mathcal{R}$. There are, of course, no actual derivatives in space involved here. These are just our short notations for the finite difference operators.
\subsection{The Cauchy-Born Rule}
As described in detail in the introduction, it is a fundamental problem to identify the correct $W_{\rm cont}$ that should be taken for the continuous equation so that one can hope for atomistic solutions close by as $\varepsilon$ becomes small enough. The classical ansatz to resolve this question by applying the Cauchy-Born rule, leads to setting $W_{\rm cont} = W_{\rm CB}$, where in our setting the Cauchy-Born energy density has the simple mathematical expression
\[W_{\rm CB}(A) := W_{\rm atom} ((A\rho)_{\rho \in \mathcal{R}}).\]
In the following we will only consider $W_{\rm cont} = W_{\rm CB}$, where $W_{\rm atom}$ is given. One of our main goals is to justify this choice rigorously.
\section{Preparations} \label{sec:preparations}
\subsection{Stability}
First we want to discuss the question of stability. We will only give a short summary. All proofs can be found in \cite{braun16static} and even more details in \cite{braunphdthesis}.
For the continuous equations the correct notion of stability for a $A\in\mathbb{R}^{d \times d}$ is given by the positivity of the Legendre-Hadamard stability constant, which for any nonempty, open, bounded $U \subset \mathbb{R}^n$ is given by
\begin{align*}
\lambda_{\rm LH}(A) &:= \inf\limits_{u \in H^1_0(U;\mathbb{R}^d) \backslash \{0\}} \frac{\int_U D^2W_{\rm CB}(A)[\nabla u (x), \nabla u(x)] \,dx}{\int_U \lvert \nabla u (x) \rvert^2 \,dx}\\
&= \inf\limits_{\xi, \eta \in \mathbb{R}^d\backslash \{0\}} \frac{D^2W_{\rm CB}(A)[\xi \otimes \eta,\xi \otimes \eta]}{\lvert \xi \rvert^2 \lvert \eta \rvert^2}
\end{align*}
While this condition is the correct notion to ensure existence and uniqueness of solutions to the continuous equation, it turns out that it is to weak to guarantee that there is a solution to the atomistic problem close by.
For fixed $\varepsilon>0$, a tensor $K \in \mathbb{R}^{(d \times \mathcal{R})\times (d \times \mathcal{R})}$, and a bounded, open, and non-empty $\Omega \subset \mathbb{R}^d$ we set
\[ \lambda_\varepsilon(K, \Omega) = \inf\limits_{\substack{
y \in \mathcal{A}_\varepsilon(\Omega,0)\\
y \neq 0}}
\frac{ \varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} K [D_{\mathcal{R},\varepsilon} y (x),D_{\mathcal{R},\varepsilon} y (x)]}{\varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} y (x)\rvert^2}.\]
We then define the atomistic stability constant by
\[\lambda_{\rm atom}(K)=\lim_{\varepsilon \to 0}\lambda_\varepsilon (K, \Omega) = \inf\limits_{\varepsilon > 0} \lambda_\varepsilon (K, \Omega).\]
In particular, we are interested in the cases $K=D^2W_{\rm atom}(A)$ for $A \in \mathbb{R}^{d \times \mathcal{R}}$ and $K=D^2W_{\rm atom}((A\rho)_{\rho \in \mathcal{R}})$ for $A \in \mathbb{R}^{d \times d}$. In either case we will just write $\lambda_{\rm atom}(A)$.
The limit in the definition exists, converges to the infimum as claimed, and is independent of $\Omega$, cf.\ \cite[Prop. 3.2]{braun16static}. Furthermore, one can show that looking at larger and larger boxes with periodic boundary conditions gives the same quantity, cf.\ \cite[Prop. 3.1]{braun16static}. In this way the notion of stability here can be shown to be equivalent to the notion in \cite{hudsonortner}. The only difference is a choice of equivalent norms.
One can also give a characterization in spirit of the Legendre-Hadamard condition
\begin{align*}
\lambda_{\rm atom}(K)= &\inf \bigg\{\frac{K[\xi \otimes c(k),\xi \otimes c(k)]}{\lvert \xi \rvert^2 (\lvert c(k) \rvert^2 + \lvert s(k) \rvert^2)}\\ &+ \frac{K[\xi \otimes s(k), \xi \otimes s(k)]}{\lvert \xi \rvert^2 (\lvert c(k) \rvert^2 + \lvert s(k) \rvert^2)} \colon\\
&\qquad \xi \in \mathbb{R}^d \backslash \{0\}, k \in [0,2\pi)^d\backslash \{0\} \bigg\},
\end{align*}
where $c(k)_\rho = \cos(\rho k) -1$ and $s(k)_\rho = \sin(\rho k)$, cf.\ \cite[Cor. 3.7]{braun16static}. In the case $K=D^2W_{\rm atom}((A\rho)_{\rho \in \mathcal{R}})$ it is very easy to see, by looking at the liminf as $k \to 0$ instead of the full infimum, that $\lambda_{\rm atom}(A) \leq C \lambda_{\rm LH}(A)$ for $A \in \mathbb{R}^{d \times d}$. The constant $C$, again, is just a consequence of the choice of equivalent norms. Since here $k$ is indeed a wave number, this is a quite intuitive property: For a crystalline material to be stable, it has to be stable by perturbations on all wave lengths. In contrast, for the continuous equations the stability under long wave length perturbations is sufficient, which corresponds to the long wave length limit $k \to 0$.
We also note, that the stability constants $\lambda_{\rm atom}(A)$ and $\lambda_{\rm LH}(A)$ depend continuously on $A$ as long as $W_{\rm atom} \in C^2$.
In \cite{braun16static} we also give criteria for atomistic stability that are simpler to check but not as sharp. Additionally, we discuss examples analytically and, in particular, give an example that $\lambda_{\rm atom}(A) < 0 < \lambda_{\rm LH}(A)$ can indeed occur.
\subsection{Residual Estimates}
Here we just want to state the crucial residual estimate as well as two results on approximations. These results have been proven in \cite{braun16static}. To avoid stronger regularity assumptions, it is important to not just estimate the residuum at the atom sites by using the continuum equations there. Instead, one uses the continuum equations at every point and gets rid of certain error terms by averaging. Additionally, the norms in the error terms can be improved with a regularization of the continuum solution.
\begin{prop} \label{prop:ell2residuum}
Let $V \subset \mathbb{R}^{d \times \mathcal{R}}$ be open and $W_{\rm atom} \in C^4_b(V)$. Let $f \in L^2(\Omega;\mathbb{R}^d)$ and set
\[ \tilde{f}(x) = \fint_{Q_\varepsilon(x)} f(a)\,da\]
for $x \in \into_\varepsilon \Omega$. Furthermore let $\varepsilon \in (0,1]$ and $y \in C^{3,1}(\mathbb{R}^d;\mathbb{R}^d)$ with
\begin{align*}
\co \{D_{\mathcal{R},\varepsilon} y (\hat{x}+ \varepsilon \sigma), (\nabla y (x) \rho)_{\rho \in \mathcal{R}}\} \subset V
\end{align*}
for all $x \in \Omega_\varepsilon$ and $\sigma \in \mathcal{R} \cup \{0\}$. Then we have
\begin{align*}
\big\lVert -\tilde{f} &- \divo_{\mathcal{R},\varepsilon} \big( DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y)\big) \big\rVert_{\ell_\varepsilon^2(\into_\varepsilon \Omega)}\\
&\leq \lVert -f - \divo DW_{\rm CB}(\nabla y)\rVert_{L^2(\Omega_\varepsilon; \mathbb{R}^d)}+ C \varepsilon^2 \Big\lVert \lVert \nabla^4 y \rVert_{L^\infty(B_{\varepsilon R}(x))}\\
&+ \lVert \nabla^3 y \rVert_{L^\infty(B_{\varepsilon R}(x))}^\frac{3}{2} + \lVert \nabla^2 y \rVert_{L^\infty(B_{\varepsilon R}(x))}^3 + \varepsilon\lVert \nabla^3 y \rVert_{L^\infty(B_{\varepsilon R}(x))}^2 \Big\rVert_{L^2(\Omega_\varepsilon)},
\end{align*}
where $\Omega_\varepsilon = \bigcup_{z \in \into_\varepsilon \Omega} Q_\varepsilon(z)$, $R=2R_{\rm max}+\frac{3\sqrt{d}}{2}$ and $C = C(d,\mathcal{R}, \lVert D^2 W_{\rm atom} \rVert_{C^2(V)}) >0$.
\end{prop}
These residual estimates are particularly strong if we combine them with the following two approximation results:
\begin{prop} \label{prop:approximation1}
For any $R>0$, $k,d \in \mathbb{N}$, $p\geq 1$, there is a $C=C(R,d,p)>0$ such that for any $U \subset \mathbb{R}^d$ measurable and $y \in W^{k,p}(U + B_{(R+1)\varepsilon}(0);\mathbb{R}^d)$ we have
\begin{align*}
\Big\lVert \lVert \nabla^k (y \ast \eta_\varepsilon) \rVert_{L^\infty(B_{\varepsilon R}(\cdot))} \Big\rVert_{L^p(U)} \leq C \lVert \nabla^k y \rVert_{L^{p}(U+B_{(R+1)\varepsilon}(0))},
\end{align*}
where $\eta_\varepsilon$ is the standard scaled smoothing kernel.
\end{prop}
\begin{prop} \label{prop:approximation2}
Let $d \in \{1,2,3,4\}$, $\Omega \subset \mathbb{R}^d$ open and bounded with Lipschitz boundary, $V \subset \mathbb{R}^{d \times \mathcal{R}}$ be open and $W_{\rm atom} \in C^5_b(V)$. Then, there is a $C>0$ such that for all $\varepsilon \in (0,1]$ and all $y \in H^4(\Omega + B_{\varepsilon}(0);\mathbb{R}^d)$
with
\begin{align*}
\inf_{x\in\Omega}\inf_{t \in [0,1]} \dist ((1-t)(\nabla y (x) \rho)_{\rho \in \mathcal{R}}+t(\nabla (y \ast \eta_\varepsilon) (x) \rho)_{\rho \in \mathcal{R}}, V^c)>0,
\end{align*}
we have
\begin{align*}
\lVert \divo & DW_{\rm CB}(\nabla y(x)) - \divo DW_{\rm CB}(\nabla (y \ast \eta_\varepsilon)(x)) \rVert_{L^2(\Omega)}\\
& \leq C \varepsilon^2 \big( \lVert \nabla^2 y \rVert_{L^4(\Omega+B_{\varepsilon}(0))} \lVert \nabla^3 y \rVert_{L^4(\Omega+B_{\varepsilon}(0))} + \lVert \nabla^4 y \rVert_{L^2(\Omega+B_{\varepsilon}(0))} \big)
\end{align*}
where $\eta_\varepsilon$ is the standard scaled smoothing kernel.
\end{prop}
\section{Continuum Elastodynamics} \label{sec:cont}
Before we can discuss the atomistic equations, we have to discuss the continuous Cauchy-Born problem. We are not only interested in existence and uniqueness, but also in the maximal existence interval and higher order regularity. The most important part of the result is the short time existence, already contained in \cite{dafermoshrusa85}. But, since we want to discuss the atomistic equations for long times, it is important that extend this short time result to a result about the maximal existence time. This will still require a considerable amount of work. Furthermore, there are two key regularity theorems that are only stated and used in \cite{dafermoshrusa85}. Their proofs were left out by the authors. Since we will also need these statements in our proof directly, we will prove them in the appendix. Theorem \ref{thm:additionalregularity} is about additional regularity for solutions of second order hyperbolic equations, while Theorem \ref{thm:optimalsobolevregularity} is about higher order elliptic regularity under very weak assumptions on the coefficients.
If we want to achieve higher regularity for such a second order hyperbolic initial-boundary-value problem, compatibility conditions on $f,g,h_0,h_1$ are crucial. We say that $f,g,h_0,h_1$ satisfy the compatibility conditions of order $m$, if
\[ u_k := \frac{\partial^k}{\partial t^k} (y-g) |_{t=0} \in H^1_0(\Omega;\mathbb{R}^d)\]
for all $k \in \{0,\dotsc, m-1\}$ as computed formally using the equation in terms of $f,g,h_0,h_1$. This can be written explicitly and, even though the expressions are quite nasty, we still want to do so in order to be able to discuss some regularity issues in more detail.
If $m-1 > \frac{d}{2}$, $h_0 \in H^m(\Omega; \mathbb{R}^d)$, $h_1 \in H^{m-1}(\Omega; \mathbb{R}^d)$, $\partial_t^{k-2} f(\cdot,0) \in H^{m-k}(\Omega; \mathbb{R}^d)$ for $2 \leq k \leq m-1$, $\partial_t^{k} g(\cdot,0) \in H^{m-k}(\Omega; \mathbb{R}^d)$ for $0 \leq k \leq m-1$ and $W_{\rm CB} \in C^{2m-2}$ on an open set containing $\{\nabla h_0(x) \colon x \in \Omega\}$, then we define $u_0(x) = h_0(x) -g(x,0)$, $u_1(x) = h_1(x) - \dot{g}(x,0)$,
\[u_2(x) = f(x,0) - \ddot{g}(x,0)+ \divo \big( DW_{\rm CB}(\nabla h_0(x)) \big)\]
and recursively, for $3 \leq k \leq m-1$,
\begin{align*}
(u_k(x)&)_i = \partial_t^{k-2} f_i(x,0) - \partial_t^{k} g_i(x,0)\\
&+\sum_{j,q,r=1}^d D^{E_{ij}+E_{qr}}W_{\rm CB}(\nabla h_0(x)) \Big( \partial_{x_r} \partial_{x_j} \partial_t^{k-2} g_q(x,0) + \partial_{x_r} \partial_{x_j} (u_{k-2})_q(x)\Big) \\
&+\sum_{j,q,r=1}^d \sum_{n=1}^{k-2} \sum_{\substack{
\beta \in \mathbb{N}_0^{d \times d}\\
1 \leq \vert \beta \rvert \leq n}} \sum_{s=1}^n \sum_{p_s(n,\beta)} \frac{(k-2)!}{(k-2-n)!} D^{\beta+E_{ij}+E_{qr}}W_{\rm CB}(\nabla h_0(x)) \\
&\quad \cdot \Big( \prod_{l=1}^s \frac{(\partial_t^{\gamma_l} \nabla g(x,0) + \nabla u_{\gamma_l}(x))^{\lambda_l}}{\lambda_l! (\gamma_l!)^{\lambda_l}} \Big) \Big( \partial_{x_r} \partial_{x_l} \partial_t^{k-2-n} g_q(x,0) + \partial_{x_r} \partial_{x_l} (u_{k-2-n})_q(x)\Big),
\end{align*}
where
\begin{align*}
p_s(n,\beta)&=\Big\{ (\lambda_1,\dots,\lambda_s;\gamma_1,\dots,\gamma_s) \colon \lambda_l \in \mathbb{N}_0^{d \times d}, \gamma_l \in \mathbb{N}_0,\\
&\quad 0< \gamma_1 < \dots < \gamma_s, \lvert \lambda_l \rvert > 0, \sum_{l=1}^s \lambda_l = \beta, \sum_{l=1}^s \gamma_l \lvert \lambda_l \rvert = n \Big\},
\end{align*}
and $E_{ij}$ is the matrix with $(E_{ij})_{ij}=1$ and zeros everywhere else.
The following result shows that $u_k \in H^1_0(\Omega;\mathbb{R}^d)$ is only a condition on the boundary values and not on the regularity.
\begin{prop} \label{prop:compatibilitycond1}
If $m-1 > \frac{d}{2}$, $h_0 \in H^m(\Omega; \mathbb{R}^d)$, $h_1 \in H^{m-1}(\Omega; \mathbb{R}^d)$, $\partial_t^{k-2} f(\cdot,0) \in H^{m-k}(\Omega; \mathbb{R}^d)$ for $2 \leq k \leq m-1$, $\partial_t^{k} g(\cdot,0) \in H^{m-k}(\Omega; \mathbb{R}^d)$ for $0 \leq k \leq m-1$ and $W_{\rm CB} \in C^{m}$ on an open set containing $\{\nabla h_0(x) \colon x \in \Omega\}$, then $u_k \in H^{m-k}(\Omega; \mathbb{R}^d)$ for all $0 \leq k \leq m-1$.
\end{prop}
\begin{proof}
This is clear for $k=0,1$. For $k=2$ this follows directly from the arguments in Lemma \ref{lem:composition}. If $k \geq 3$, inductively, the same arguments show $D^{\beta+E_{ij}+E_{qr}}W_{\rm CB} \circ \nabla h_0 \in H^{m-1}$ and then one can apply Lemma \ref{lem:prodofsobolev} with $M = m-1$ and $N=0 + \sum \gamma_j \lvert \lambda_j \rvert +(k-l-1) = k-1$ to estimate the product and gives the desired result. Actually, for this to be completely true, we would need the stronger assumption $W_{\rm CB} \in C^{2m-2}$ so that $D^{\beta + E_{ij}+ E_{qr}}W_{\rm CB} \circ \nabla h_0 \in H^{m-1}$. To reduce this assumption to $W_{\rm CB} \in C^{m}$, we note that in the application of Lemma \ref{lem:prodofsobolev} we only take the $\alpha$-th derivative of $v=D^{\beta+E_{ij}+E_{qr}}W_{\rm CB} \circ \nabla h_0$ with $0 \leq \lvert \alpha \rvert \leq m-k =M-N$ and then have to know that $D^\alpha v \in L^q$ for a certain $q$ formerly coming from the Sobolev embedding of $H^{m-1-\lvert \alpha \rvert}$. Now we have to prove this estimate differently.
From Corollary \ref{cor:FaadiBruno} we know that
\[ \lvert D^\alpha v(x) \rvert \leq C\sum_{r=1}^{\lvert \alpha \rvert} \lvert D^{r+2+ \lvert \beta \rvert}W_{\rm CB}(\nabla h_0(x)) \rvert \sum_{\substack{l_1, \dots, l_r\geq 1 \\ l_1+ \dots+ l_r = \lvert \alpha \rvert}} \prod_{j=1}^r \lvert D^{(l_j+1)}h_0(x)\rvert,\]
if $W_{\rm CB} \in C^m$ and $h_0 \in C^{1+\lvert \alpha \rvert}$. But of course this extends to $h_0 \in H^m$ once we estimate the product on the right hand side suitably. These estimates, which also give the desired integrability of $D^\alpha v$ follow along the lines of the proof of Lemma \ref{lem:prodofsobolev}.
\end{proof}
If we already have a solution and use it as a starting point, then the compatibility conditions are automatically satisfied and the $u_k$ are indeed directly given by $y-g$.
\begin{prop} \label{prop:compatibilitycond2}
Let $m\in\mathbb{N}$ with $m > \frac{d}{2} + 1$, $\delta>0$, let $\Omega \subset \mathbb{R}^d$ be an open, bounded set with $\partial \Omega$ of class $C^{m}$, $V \subset \mathbb{R}^{d \times d}$ open, $W_{\rm CB} \in C_b^{m+1}(V)$,
\begin{align*}
&f \in C^{m-1}(\overline{\Omega}\times [-\delta,\delta]; \mathbb{R}^d),\\
&g \in C^{m+1}(\overline{\Omega}\times [-\delta,\delta]; \mathbb{R}^d) \text{, and}\\
&y \in \bigcap_{k=0}^m C^k\big([-\delta,\delta]; H^{m-k}(\Omega;\mathbb{R}^d)\big) \ \text{with}\\
&\overline{\{\nabla y(x,t) \colon x \in \Omega, t \in [-\delta,\delta]\}} \subset V_{\rm LH}.
\end{align*}
Furthermore let $y$ be a solution of the equations. If we now set $h_0 = y(0)$ and $h_1=\partial_t y(0)$, then we have
\[ u_k = \frac{\partial^k}{\partial t^k} (y-g) |_{t=0} \in H^1_0(\Omega;\mathbb{R}^d)\]
for all $k \in \{0, \dotsc, m-1\}$.
\end{prop}
\begin{proof}
Since $y-g \in C^{m-1}\big([-\delta,\delta]; H^1(\Omega;\mathbb{R}^d)\big)$, $H^1_0(\Omega;\mathbb{R}^d)$ is a closed subspace of $H^1(\Omega;\mathbb{R}^d)$ and $y(t)-g(t) \in H^1_0(\Omega;\mathbb{R}^d)$ for all $t \in [- \delta, \delta]$, we clearly find
\[\frac{\partial^k}{\partial t^k} (y-g) |_{t=0} \in H^1_0(\Omega;\mathbb{R}^d).\]
Let us now proof $u_k = \frac{\partial^k}{\partial t^k} (y-g) |_{t=0}$ by induction over $k$. By definition this is true for $k=0,1$. $k=2$ follows from the equation. If now $3 \leq k \leq m-1$, we have to show that the recursion formula for the $u_k$ also holds for the derivatives of $y-g$. Clearly we have
\begin{align*}
\partial_t^k (y-g) &= \partial_t^{k-2} \Big( f - \partial_t^2 g + \divo (DW_{\rm CB}(\nabla y))\Big)\\
&= \partial_t^{k-2} f - \partial_t^k g + \partial_t^{k-2} \divo (DW_{\rm CB}(\nabla y)).
\end{align*}
If $y$ were smooth the last term can be written explicitly with the chain rule, the Leibniz rule, as well as the generalized Faà di Bruno formula, Lemma \ref{lem:multivariateFaaDiBruno}. We first get
\[\divo (DW_{\rm CB}(\nabla y))_i = \sum_{j,q,r=1}^d D^{E_{ij}+E_{qr}}W_{\rm CB}(\nabla y) \partial_{x_r} \partial_{x_j} y_q, \]
then
\begin{align*}
\partial_t^{k-2} \divo (DW_{\rm CB}(\nabla y))_i = \sum_{j,q,r=1}^d \sum_{n=0}^{k-2} \binom{k-2}{n} \partial_t^n(D^{E_{ij}+E_{qr}}W_{\rm CB}(\nabla y)) \partial_t^{k-2-n} \partial_{x_r} \partial_{x_j} y_q,
\end{align*}
and finally
\begin{align*}
(\partial_t^k (y-g))_i &= \partial_t^{k-2} f_i - \partial_t^{k} g_i\\
&+\sum_{j,q,r=1}^d D^{E_{ij}+E_{qr}}W_{\rm CB}(\nabla y) \partial_{x_r} \partial_{x_j} \partial_t^{k-2} y_q \\
&+\sum_{j,q,r=1}^d \sum_{n=1}^{k-2} \sum_{\substack{
\beta \in \mathbb{N}_0^{d \times d}\\
1 \leq \vert \beta \rvert \leq n}} \sum_{s=1}^n \sum_{p_s(n,\beta)} \frac{(k-2)!}{(k-2-n)!} D^{\beta+E_{ij}+E_{qr}}W_{\rm CB}(\nabla y) \\
&\quad \cdot \Big( \prod_{l=1}^s \frac{(\partial_t^{\gamma_l} \nabla y)^{\lambda_l}}{\lambda_l! (\gamma_l!)^{\lvert \lambda_l \rvert}} \Big) \partial_{x_r} \partial_{x_l} \partial_t^{k-2-n} y_q,
\end{align*}
where
\begin{align*}
p_s(n,\beta)&=\Big\{ (\lambda_1,\dots,\lambda_s;\gamma_1,\dots,\gamma_s) \colon \lambda_l \in \mathbb{N}_0^{d \times d}, \gamma_l \in \mathbb{N}_0,\\
&\quad 0< \gamma_1 < \dots < \gamma_s, \lvert \lambda_l \rvert > 0, \sum_{l=1}^s \lambda_l = \beta, \sum_{l=1}^s \gamma_l \lvert \lambda_l \rvert = n \Big\}.
\end{align*}
Due to the bounds discussed in Proposition \ref{prop:compatibilitycond1} and Lemma \ref{lem:prodofsobolev} this still holds under the given weaker regularity assumption on $y$. Inductively, we thus have proven the claim.
\end{proof}
In the following, for $V \subset \mathbb{R}^{d \times d}$ open and $W_{\rm CB} \in C^2(V)$ we write
\[V_{\rm LH} = \{A \in V \colon \lambda_{\rm LH}(A)>0\},\]
which is again an open set, since $\lambda_{\rm LH}$ is continuous.
Let us start with a local existence result.
\begin{thm} \label{thm:localcontwave}
Let $m\in\mathbb{N}$ with $m > \frac{d}{2} +2$, $T_0>0$, let $\Omega \subset \mathbb{R}^d$ be an open, bounded set with $\partial \Omega$ of class $C^{m}$, $V \subset \mathbb{R}^{d \times d}$ open and $W_{\rm CB} \in C_b^{m+1}(V)$. Given a body force $f$, initial data $h_0,h_1$ and boundary values $g$ such that
\begin{align*}
&f \in C^{m-1}(\overline{\Omega}\times [0,T_0]; \mathbb{R}^d)\\
&g \in C^{m+1}(\overline{\Omega}\times [0,T_0]; \mathbb{R}^d)\\
&h_0 \in H^m (\Omega; \mathbb{R}^d)\\
&\overline{\{\nabla h_0(x) \colon x \in \Omega\}} \subset V_{\rm LH} \\
&h_1 \in H^{m-1} (\Omega; \mathbb{R}^d)
\end{align*}
and such that the compatibility conditions of order $m$ are satisfied (see above).
Then, for all sufficiently small $T \in (0,T_0]$ the problem has a unique solution
\[ y \in \bigcap_{k=0}^m C^k\big([0,T]; H^{m-k}(\Omega;\mathbb{R}^d)\big)\]
and we have
\[\overline{\{\nabla y(x,t) \colon x \in \Omega, t \in [0,T]\}} \subset V_{\rm LH}.\]
\end{thm}
\begin{proof}
This follows from \cite[Thm.~5.1]{dafermoshrusa85} by setting $\textbf{u}^0 = h_0-g(\cdot,0)$, $\textbf{u}^1 = h_1-\dot{g}(\cdot,0)$,
\[ \textbf{g}(x,t,u,p,M)_k = f(x,t)_k - \ddot{g}(x,t)_k + \sum \frac{\partial^2 W_{\rm CB}}{\partial a_{ki} \partial a_{lj}}(M+\nabla g(x,t)) \frac{\partial^2 g_l}{\partial x_i \partial x_j}(x,t)\]
and
\[ (\textbf{A}_{ij})_{kl} (x,t,u,p,M) = \chi (M+\nabla g(x,t))\frac{\partial^2 W_{\rm CB}}{\partial a_{ki} \partial a_{lj}}(M+\nabla g(x,t)) + (1-\chi(M+\nabla g(x,t))) \delta_{kl} \delta_{ij},\]
where $\chi \colon \mathbb{R}^{d \times d} \to [0,1]$ is a smooth cutoff with $\chi(M)=1$ for $M \in W_1$ and $\chi(M)=0$ for $M \notin W_2$ and $W_1,W_2$ are open sets such that
\[\overline{\{\nabla h_0(x) \colon x \in \Omega\}} \subset\subset W_1 \subset\subset W_2 \subset\subset V_{\rm LH}.\]
We then set $y = u+g$ and reduce the existence time $T$ enough to ensure $\nabla y(x,t) \in W_1$ for all $(x,t) \in \overline{\Omega} \times [0,T]$.
\end{proof}
We can use this local result to construct a maximal solution.
\begin{thm} \label{thm:maxcontwave}
Let $m\in\mathbb{N}$ with $m > \frac{d}{2} +2$, $T_0>0$, let $\Omega \subset \mathbb{R}^d$ be an open, bounded set with $\partial \Omega$ of class $C^{m}$, $V \subset \mathbb{R}^{d \times d}$ open and $W_{\rm CB} \in C^{m+1}(V)$. Given a body force $f$, initial data $h_0,h_1$ and boundary values $g$ such that
\begin{align*}
&f \in C^{m-1}(\overline{\Omega}\times [0,T_0]; \mathbb{R}^d)\\
&g \in C^{m+1}(\overline{\Omega}\times [0,T_0]; \mathbb{R}^d)\\
&h_0 \in H^m (\Omega; \mathbb{R}^d)\\
&\overline{\{\nabla h_0(x) \colon x \in \Omega\}} \subset V_{\rm LH} \\
&h_1 \in H^{m-1} (\Omega; \mathbb{R}^d)
\end{align*}
and such that the compatibility conditions of order $m$ are satisfied.
Then there are unique $T_{\rm cont} > 0$ and
\[ y \in \bigcap_{k=0}^m C^k\big([0,T_{\rm cont}); H^{m-k}(\Omega;\mathbb{R}^d)\big),\]
such that
\[\{\nabla y(x,t) \colon x \in \overline{\Omega}, t \in [0,T_{\rm cont})\} \subset V_{\rm LH},\]
$y$ is a solution on $[0,T_{\rm cont})$ and at least one of the following conditions is true:
\begin{enumerate}[label=(\roman*)]
\item $T_{\rm cont}=T_0$,
\item $\liminf_{t \to T_{\rm cont}}\dist \big(V_{\rm LH}^c, \{\nabla y(x,t) \colon x \in \overline{\Omega}\}\big) = 0$,
\item $\limsup_{t \to T_{\rm cont}} \lVert y(t) \rVert_{H^m(\Omega;\mathbb{R}^d)} + \lVert \partial_t y(t) \rVert_{H^{m-1}(\Omega;\mathbb{R}^d)} = \infty$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $T_{\rm cont}$ be the supremum of all $0<T \leq T_0$ such that there is a
\[ y \in \bigcap_{k=0}^m C^k\big([0,T]; H^{m-k}(\Omega;\mathbb{R}^d)\big)\]
with
\[\overline{\{\nabla y(x,t) \colon x \in \Omega, t \in [0,T]\}} \subset V_{\rm LH}\]
that solves the problem on $[0,T]$. Theorem \ref{thm:localcontwave} ensures that there is at least one such $T$. If we take any of these solutions and $t_0 \in (0,T)$ then $h_0=y(t_0)$, $h_1= \partial_t y(t_0)$ as well as the translated $f$ and $g$ can be used in Theorem \ref{thm:localcontwave} to show existence and uniqueness in some $[t_0, t_0+\delta]$, $\delta>0$. This is possible since the new $u_k$ satisfies $u_k = \frac{\partial^k}{\partial t^k} (y-g) |_{t=t_0} \in H^1_0$ by Proposition \ref{prop:compatibilitycond2}.
The uniqueness ensures in particular, that all these $y$ are equal pairwise on the intersection of their existence intervals. Therefore, we have a
\[ y \in \bigcap_{k=0}^m C^k\big([0,T_{\rm cont}); H^{m-k}(\Omega;\mathbb{R}^d)\big)\]
with
\[\{\nabla y(x,t) \colon x \in \overline{\Omega}, t \in [0,T_{\rm cont})\} \subset V_{\rm LH}\]
that solves the problem on $[0,T_{\rm cont})$.
Now assume $T_{\rm cont} < T_0$,
\[\liminf_{t \to T_{\rm cont}}\dist \big(V_{\rm LH}^c, \{\nabla y(x,t) \colon x \in \overline{\Omega}\}\big)>0\]
and
\[\limsup_{t \to T_{\rm cont}} \lVert y(t) \rVert_{H^m(\Omega;\mathbb{R}^d)} + \lVert \partial_t y(t) \rVert_{H^{m-1}(\Omega;\mathbb{R}^d)} < \infty.\]
Then
\[y \in L^\infty(0,T_{\rm cont}; H^m(\Omega;\mathbb{R}^d)) \cap W^{1,\infty}(0,T_{\rm cont}; H^{m-1}(\Omega;\mathbb{R}^d)).\]
We claim that
\[\partial_t^k y \in L^\infty(0,T_{\rm cont}; H^{m-k}(\Omega;\mathbb{R}^d))\]
for $0 \leq k \leq m$. We already know this $k=0,1$. For $2 \leq k \leq m$ we represent the derivatives of $y$ as we did in Proposition \ref{prop:compatibilitycond2} and then argue inductively as in the proof of Proposition \ref{prop:compatibilitycond1}.
In particular, the limit $\tilde{h}_k := \lim_{t \to T_{\rm cont}} \partial_t^k y (t)$ exists strongly in $H^{m-k-1}(\Omega;\mathbb{R}^d)$ and weakly in $H^{m-k}(\Omega;\mathbb{R}^d)$ for $0 \leq k \leq m-1$. Since $H^{m-1}(\Omega;\mathbb{R}^d) \hookrightarrow C^1(\overline{\Omega};\mathbb{R}^d)$ we also have the convergence $y(t) \to \tilde{h}_0 $ in $C^1(\overline{\Omega};\mathbb{R}^d)$. In particular,
\[\overline{\{\nabla \tilde{h}_0(x) \colon x \in \Omega\}} \subset V_{\rm LH}.\]
Now we want to use the local existence result, Theorem \ref{thm:localcontwave}, with shifted $f,g$ and initial conditions $\tilde{h}_0,\tilde{h}_1$. All we have to do, is to check that the compatibility conditions of order $m$ are satisfied. For $k=0$ or $k=1$, we clearly have
\[u_k = \tilde{h}_k - \partial_t^k g(\cdot,T_{\rm cont}) = \lim_{t \to T_{\rm cont}} \partial_t^k( y(\cdot,t) - g(\cdot,t)) \in H^1_0.\]
For $2 \leq k \leq m-1$, we know that $\partial_t^k(y-g)(t)$ converges to $\tilde{h}_k - \partial_t^k g(\cdot,T_{\rm cont})$ strongly in $H^{m-k-1}(\Omega;\mathbb{R}^d)$ and weakly in $H^{m-k}(\Omega;\mathbb{R}^d)$. Therefore, $\tilde{h}_k - \partial_t^k g(\cdot,T_{\rm cont})\in H^1_0(\Omega;\mathbb{R}^d)$. Now we just have to argue inductively that $u_k = \tilde{h}_k - \partial_t^k g(\cdot,T_{\rm cont})$. If this is already true for all $l<k$, we know in particular that $\partial_t^l(y-g)(t) \rightharpoonup u_l$ in $H^{m-l}(\Omega;\mathbb{R}^d)$. Expressing $\partial_t^k(y-g)(t)$ with the equation in terms of $\partial_t^l (y-g)$, $0 \leq l \leq k-2$ as in Proposition \ref{prop:compatibilitycond2}, we can thus conclude that $\partial_t^k(y-g)(t) \to u_k$ at least in some weaker sense, e.g. in $H^{m-k-1}(\Omega;\mathbb{R}^d)$. To see this one needs to combine the arguments in Proposition \ref{prop:compatibilitycond1} with the statement on weak-to-strong continuity in Lemma \ref{lem:prodofsobolev} with $M=m-1$, $N=k-1$, $L=m-k-1$. Therefore, $f(\cdot, T_{\rm cont} + \cdot)$, $g(\cdot, T_{\rm cont} + \cdot)$, $\tilde{h}_0$, and $\tilde{h}_1$ satisfy the compatibility conditions of order $m$.
Hence, we can use Theorem \ref{thm:localcontwave} to find a $\delta>0$ and an extension of $y$ to $[0, T_{\rm cont}+\delta]$, such that
\[ y \in \bigcap_{k=0}^m C^k\big([T_{\rm cont},T_{\rm cont}+\delta]; H^{m-k}(\Omega;\mathbb{R}^d)\big),\]
$y$ is a solution of the equation on $(T_{\rm cont},T_{\rm cont}+\delta)$ with $y(T_{\rm cont})= \tilde{h}_0$ and $\dot{y}(T_{\rm cont})= \tilde{h}_1$. Here, $\dot{y}(T_{\rm cont})$ is to be understood in terms of the values on $[T_{\rm cont},T_{\rm cont}+\delta]$ alone. Furthermore, we have
\[\overline{\{\nabla y(x,t) \colon x \in \Omega, t \in [T_{\rm cont},T_{\rm cont}+\delta]\}} \subset V_{\rm LH}.\]
We have to take a closer look at what happens in $T_{\rm cont}$. We clearly have
\[u_k = \lim_{t \to T_{\rm cont}^+} \partial_t^k( y(\cdot,t) - g(\cdot,t))\]
strongly in $H^{m-k}(\Omega;\mathbb{R}^d)$. But we already saw that $u_k = \tilde{h}_k - \partial_t^k g(\cdot,T_{\rm cont})$ for $0 \leq k \leq m-1$. So the weak derivatives are continuous, which directly implies the strong differentiability
\[ y \in \bigcap_{k=0}^{m-1} C^k\big([0,T_{\rm cont}+\delta]; H^{m-k-1}(\Omega;\mathbb{R}^d)\big).\]
Furthermore, we have one more strong derivative outside of $T_{\rm cont}$ which extends to the entire interval including $T_{\rm cont}$ as a weak derivative. By continuity it is bounded on $[T_{\rm cont},T_{\rm cont} + \delta]$ and we have already shown the boundedness on $[0,T_{\rm cont})$. Therefore,
\[ y \in \bigcap_{k=0}^{m} W^{k,\infty}\big([0,T_{\rm cont}+\delta]; H^{m-k}(\Omega;\mathbb{R}^d)\big).\]
Additionally, by compactness and identification $\partial_t^k y$ is continuous in $H^{m-k}(\Omega;\mathbb{R}^d)$ with respect to the weak topology for all $0 \leq k \leq m-1$.
Now, we want to use the ideas of \cite{strauss66} to get the missing additional regularity. The key is to use that $y$ solves an equation.
Clearly $v:= \partial_t^{m-1} (y-g)$ satisfies $v \in L^\infty(0,T_{\rm cont}+\delta; H^1_0(\Omega;\mathbb{R}^d))$ with weak derivative $\partial_t v \in L^\infty(0,T_{\rm cont}+\delta; L^2(\Omega;\mathbb{R}^d))$. We claim that it also has a weak second derivative in $L^\infty(0,T_{\rm cont}+\delta; H^{-1}(\Omega;\mathbb{R}^d))$. To that end, we calculate
\begin{align*}
\partial_t^{m-1} (DW_{\rm CB} (\nabla y)_{ij}) &= D^2 W_{\rm CB} (\nabla y)[\nabla v, E_{ij}] + D^2 W_{\rm CB} (\nabla y)[\partial_t^{m-1} \nabla g, E_{ij}]\\
&+ \sum_{\substack{
\beta \in \mathbb{N}_0^{d \times d}\\
2 \leq \vert \beta \rvert \leq m-1}}
\sum_{s=1}^{m-1} \sum_{p_s(m-1,\beta)} (m-1)! D^{\beta+E_{ij}}W_{\rm CB}(\nabla y) \prod_{j=1}^s \frac{(\partial_t^{\gamma_j}\nabla y)^{\lambda_j}}{\lambda_j! \gamma_j!^{\lvert \lambda_j \rvert}}
\\
&=: D^2 W_{\rm CB} (\nabla y)[\nabla v, E_{ij}]+ R_{ij}.
\end{align*}
We can now use Lemma \ref{lem:prodofsobolev} with $M=m-2$ and $N = \sum \lvert \lambda_j \rvert (\gamma_j -1) = m-1-\lvert \beta \rvert \leq m-3$ to see that
\[\prod_{j=1}^s (\partial_t^{\gamma_j}\nabla y)^{\lambda_j} \in L^\infty (0,T_{\rm cont}+\delta; H^1(\Omega;\mathbb{R}^d)).\]
Since
\[D^{\beta+E_{ij}}W_{\rm CB}(\nabla y) \in L^\infty (0,T_{\rm cont}+\delta; W^{1,\infty}(\Omega;\mathbb{R}^d)),\]
we obtain
\[R \in L^\infty (0,T_{\rm cont}+\delta; H^1(\Omega;\mathbb{R}^{d \times d}))\]
and
\[F:= \partial_t^{m-1}f - \partial_t^{m+1}g + \divo R \in L^\infty (0,T_{\rm cont}+\delta; L^2(\Omega;\mathbb{R}^d)).\]
Defining $A(t) : H^1_0(\Omega;\mathbb{R}^d) \to H^{-1}(\Omega;\mathbb{R}^d)$ by
\[A(t)u = -\divo(D^2 W_{\rm CB} (\nabla y(\cdot,t))[\nabla u]),\]
we can use a weak formulation (in time and space) of the equation to see that indeed $\partial_t^2 v$ exists as a weak derivative in $L^\infty (0,T_{\rm cont}+\delta; H^{-1}(\Omega;\mathbb{R}^d))$ and satisfies
\[\partial_t^2 v(t) + A(t)v(t) = F(t).\]
Let us look more precisely at $A$. Since $\nabla y \in C([0,T_{\rm cont}+\delta] \times \overline{\Omega}; \mathbb{R}^{d \times d})$, the coefficients $D^2 W_{\rm CB} (\nabla y(x,t))$ are uniformly bounded, uniformly continuous and have a positive, uniform Legendre-Hadamard constant. Therefore, it is well known that $A(t)$ satisfies a Gårding-inequality uniformly in time, see Theorem \ref{thm:contGårding}. I.e., there are $\lambda_1 >0$, $\lambda_2 \in \mathbb{R}$ such that
\[\langle A(t)v,v\rangle_{H^{-1},H^1_0} \geq \lambda_1 \lVert v \rVert_{H^1_0} - \lambda_2 \lVert v \rVert_{L^2}\]
for all $t$ and all $v \in H^1_0(\Omega;\mathbb{R}^d)$. Given $v_1,v_2 \in H^1_0(\Omega;\mathbb{R}^d)$, $\langle A(t)v_1,v_2\rangle_{H^{-1},H^1_0}$ has weak derivative $\langle A'(t)v_1,v_2\rangle_{H^{-1},H^1_0}$, where
\[A'(t)u = -\divo(D^3 W(\nabla y (\cdot,t))[\partial_t \nabla y(\cdot,t),\nabla u]).\]
Since $D^3 W(\nabla y)[\partial_t \nabla y] \in L^\infty([0,T_{\rm cont}+\delta] \times \Omega; \mathbb{R}^{d \times d \times d \times d})$, we see that $A'$ is bounded with values in $L(H^1_0(\Omega;\mathbb{R}^d), H^{-1}(\Omega;\mathbb{R}^d))$. Therefore, we can use Theorem \ref{thm:additionalregularity} to conclude that
\[\partial_t^m y \in C([0,T_{\rm cont}+\delta];L^2(\Omega;\mathbb{R}^d))\]
and
\[\partial_t^{m-1} y \in C([0,T_{\rm cont}+\delta];H^1(\Omega;\mathbb{R}^d)).\]
For $1 \leq k \leq m-2$, taking $k$ time derivatives in the equation we find
\[\partial_t^k y = (A(t) + \lambda \Id)^{-1} (-\partial_t^{k+2} y + \lambda \partial_t^k y + \partial_t^k f + \divo S).\]
Here
\begin{align*}
S &= \partial_t^k (DW_{\rm CB}(\nabla y)) - D^2 W_{\rm CB}(\nabla y)[\partial_t^k \nabla y]\\
&= \sum_{\substack{
\beta \in \mathbb{N}_0^{d \times d}\\
2 \leq \vert \beta \rvert \leq k}}
\sum_{s=1}^{k} \sum_{p_s(k,\beta)} k! D^{\beta+E_{ij}}W_{\rm CB}(\nabla y) \prod_{j=1}^s \frac{(\partial_t^{\gamma_j}\nabla y)^{\lambda_j}}{\lambda_j! \gamma_j!^{\lvert \lambda_j \rvert}}
\end{align*}
and $A(t) + \lambda \Id \colon H^{m-k} \cap H^1_0 \to H^{m-k-2}$ is invertible for $\lambda$ large enough because of Theorem \ref{thm:optimalsobolevregularity}, where we use that $D^2W_{\rm CB}(\nabla y) \in L^{\infty}([0,T_{\rm cont}+\delta],H^{m-1}(\Omega;\mathbb{R}^d))$ according to Lemma \ref{lem:composition}. Theorem \ref{thm:optimalsobolevregularity} also gives a time independent bound on $\lVert (A(t) + \lambda \Id)^{-1} \rVert_{L(H^{m-k-2}, H^{m-k})}$.
According to Lemma \ref{lem:composition}, $B \mapsto D^3 W_{\rm CB} \circ B$ is a bounded map from $H^{m-2}$ to $H^{m-2}$. Therefore, we can use Lemma \ref{lem:prodofsobolev} with $M=m-2$ to see that $A'(t) \colon H^{m-k} \to H^{m-k-2}$ is well defined with
\[\lVert A'(t) \rVert_{L(H^{m-k}, H^{m-k-2})} \leq C\]
uniform in $t$. Since
\[(A(t) + \lambda \Id)^{-1} - (A(s) + \lambda \Id)^{-1} = -(A(t) + \lambda \Id)^{-1}(A(t)-A(s))(A(s) + \lambda \Id)^{-1},\]
we obtain
\[\lVert (A(t) + \lambda \Id)^{-1} - (A(s) + \lambda \Id)^{-1} \rVert_{L(H^{m-k-2},H^{m-k}\cap H^1_0)} \leq C \lvert t-s \rvert.\]
Using that $\partial_t^\gamma \nabla y$ is weakly continuous in $H^{m-1-\gamma}$, we can use Lemma \ref{lem:prodofsobolev} and its additional statement with $M=m-2$, $N=k-\lvert \beta \rvert$, $L=m-k-1 < M-N$ and $\lambda_k = \gamma_j - 1$ to find that $S$ is (strongly) continuous with values in $H^{m-k-1}$.
Putting all of this together we find inductively, starting at $k=m$ and $k=m-1$, that
\[\partial_t^k y \in C([0,T_{\rm cont}+\delta]; H^{m-k}(\Omega;\mathbb{R}^d))\]
for $1 \leq k \leq m$.
For $k=0$ we can no longer use the theory for linear systems in divergence form. Instead, we look at the operator
\[(A(t)u)_i = - \sum_{k,j,l} (D^2 W_{\rm CB} (\nabla y(\cdot,t)))_{ijkl} \frac{\partial^2 u_k}{\partial x_j \partial x_l} .\]
Now we have
\[y = (A(t) + \lambda \Id)^{-1} (\partial_t^{2} y + \lambda y - f).\]
But Theorem \ref{thm:optimalsobolevregularity} also holds in non-divergence form, hence
\[(A(t) + \lambda \Id)^{-1} \colon H^{m-2}(\Omega;\mathbb{R}^d) \to H^{m}(\Omega;\mathbb{R}^d) \cap H^1_0(\Omega;\mathbb{R}^d)\]
is well defined and bounded independently of $t$ since $m-2>\frac{d}{2}$ gives a bound on $D^2W_{\rm CB}(\nabla y(\cdot;t))$ in $W^{1, \infty}$. The continuity then follows along the same lines as for $k \geq 1$.
Having established the additional regularity, we have a contradiction to the definition of $T_{\rm cont}$. This proves the existence of a $T_{\rm cont}$ with the desired properties. But due to the local uniqueness of solutions, any smaller $T$ cannot satisfy either one of (i), (ii) or (iii). Therefore, $T_{\rm cont}$ is unique.
\end{proof}
\section{Atomistic Elastodynamics} \label{sec:atom}
The main theorem of this paper is the existence of a solution to the atomistic equations (for $\varepsilon$ small enough), for as long as the corresponding solution to the Cauchy-Born continuum equations exists and is atomistically stable. But before we state and prove the theorem, let us prove two auxiliary theorems that are already interesting on their own. In both cases we will prove more general versions than what we will actually need for the main theorem.
We start with a theorem on local existence and uniqueness.
\begin{thm} \label{thm:discretelocalexistence}
Let $d \in \{1,2,3,4\}$, $V \subset \mathbb{R}^{d \times \mathcal{R}}$, $W_{\rm atom} \in C^2_b(V)$ and set $V_{\rm atom} = \{A \in V \colon \lambda_{\rm atom}(A)>0\}$.
Let $\varepsilon_0 >0$, $C_1>0$, $r_0>0$ and $\gamma \in \big[\frac{d}{2},2 \big]$, such that $4C_1 \varepsilon_0^{\gamma-\frac{d}{2}} \leq r_0$. Furthermore, let $0<\varepsilon \leq \varepsilon_0$, $T_0>0$ and fix $f_{\rm atom}$, $g_{\rm atom}$, $y_{\rm ref}$ with
\begin{align*}
f_{\rm atom}(x,\cdot) &\in L^2((0,T_0); \mathbb{R}^d) \quad \text{for all } x \in \into_\varepsilon \Omega,\\
g_{\rm atom}(x,\cdot) &\in H^2((0,T_0); \mathbb{R}^d) \quad \text{for all } x \in \partial_\varepsilon \Omega,\\
y_{\rm ref}(x,\cdot) &\in H^2((0,T_0); \mathbb{R}^d) \quad \text{for all } x \in \Omega \cap \varepsilon \mathbb{Z}^d,
\end{align*}
such that
\[\dist (D_{\mathcal{R},\varepsilon} y_{\rm ref} (x,t), V_{\rm atom}^c) > r_0\]
in $\sinto_\varepsilon \Omega \times [0,T_0]$ and
\[\sup_t \lVert y_{\rm ref}(t) -g_{\rm atom}(t) \rVert_{\partial_\varepsilon \Omega, 0} \leq C_1 \varepsilon^{\gamma}.\]
Then there exists a time $T>0$ which may depend on all the previous quantities, including $\varepsilon$, such that the following holds:\\
Given any $t_0 \in [0,T_0)$ and $h_{{\rm atom},0} \in \mathcal{A}_\varepsilon (\Omega, g_{\rm atom}(\cdot,t_0))$, $h_{{\rm atom},1} \in \mathcal{A}_\varepsilon (\Omega, \dot{g}_{\rm atom}(\cdot,t_0))$, such that
\begin{align*}
\lVert h_{{\rm atom},1}-\dot{y}_{\rm ref}(\cdot,t_0) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2 &+ \lVert h_{{\rm atom},0} - y_{\rm ref}(\cdot,t_0) \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}^2 \leq C_1^2 \varepsilon^{2\gamma},
\end{align*}
there is a unique solution $y \in H^2((t_0,\min\{t_0+T,T_0\}); \mathbb{R}^d)^{\Omega \cap \varepsilon \mathbb{Z}^d}$ to the discrete initial-boundary-value problem on $[t_0,\min\{t_0+T,T_0\}]$.
\end{thm}
\begin{proof}
This is basically the Picard-Lindelöf Theorem. But we want to quantify the dependence on the initial conditions. We look at the set
\begin{align*}
K_{T,b,z_0} = \Big\{ (z_1,z_2) &\colon z_1,z_2 \in C([t_0,\min\{ t_0+T,T_0\}];\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d))\\
& z_1(t) \in \mathcal{A}_\varepsilon(\Omega;0), z_2(t) \in \mathcal{A}_\varepsilon(\Omega;0)\\
& \sup_t \lVert z(t) - z^0 \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)}\leq b \Big\},
\end{align*}
with the metric induced by $\lVert z \rVert = \sup_t \lVert z(t) \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)}$.
Here we substituted
\[z(t)=\begin{pmatrix} y(t) - y_{\rm ref}(t) - T_\varepsilon (g_{\rm atom}(t) - y_{\rm ref}(t)) \\ \dot{y}(t) - \dot{y}_{\rm ref}(t) - T_\varepsilon (\dot{g}_{\rm atom}(t) - \dot{y}_{\rm ref}(t)) \end{pmatrix}\]
and
\[z^0=\begin{pmatrix} h_{{\rm atom},0} - y_{\rm ref}(t_0) - T_\varepsilon (g_{\rm atom}(t_0) - y_{\rm ref}(t_0)) \\ h_{{\rm atom},1} - \dot{y}_{\rm ref}(t_0) - T_\varepsilon (\dot{g}_{\rm atom}(t_0) - \dot{y}_{\rm ref}(t_0)) \end{pmatrix}.\]
The equation can be written as $\dot{z}(t) = F(t,z(t))$, where $F_1(t,z_1,z_2) = z_2$ and
\begin{align*}
F_2&(t,z_1,z_2)(x) = f_{\rm atom}(x,t) - \ddot{y}_{\rm ref}(x, t) - T_\varepsilon (\ddot{g}_{\rm atom}(t) - \ddot{y}_{\rm ref}(t)) \\
&\quad +\divo_{\mathcal{R},\varepsilon} \big( DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y_{\rm ref} (x,t) + D_{\mathcal{R},\varepsilon} T_\varepsilon (g_{\rm atom}(t) - y_{\rm ref}(t))(x) + D_{\mathcal{R},\varepsilon} z_1(x))\big)
\end{align*}
for $x \in \into_\varepsilon \Omega$, but $F_2(t,z_1,z_2)(x)=0$ for $x \in \partial_\varepsilon \Omega$.
Since we do not even claim strong differentiability, it is best to look at the fixed point equation of
\[ G(z)(t) = z^0 + \int_{t_0}^t F(s,z(s))\,ds.\]
Clearly,
\[\sup_t \lVert y_{\rm ref}(t) -g_{\rm atom}(t) \rVert_{\partial_\varepsilon \Omega, 0} \leq C_1 \varepsilon^{\gamma}\]
implies
\begin{align*}
\lvert D_{\mathcal{R},\varepsilon} T_\varepsilon(g_{\rm atom} - y_{\rm ref}) (x,t) \rvert &\leq \varepsilon^{- \frac{d}{2}} \lVert D_{\mathcal{R},\varepsilon} T_\varepsilon(g_{\rm atom} - y_{\rm ref}) (t) \rVert_{h^1_\varepsilon}\\
&=\varepsilon^{- \frac{d}{2}} \lVert g_{\rm atom} - y_{\rm ref} (t) \rVert_{\partial_\varepsilon \Omega, 0}\\
&\leq C_1 \varepsilon^{\gamma - \frac{d}{2}}
\end{align*}
uniformly in $x$ and $t$. Now, if $0<b\leq \frac{\varepsilon r_0}{8 \lvert \mathcal{R} \rvert^{\frac{1}{2}}}$
then for any $z \in K_{T,b,z_0}$
\begin{align*}
&\lvert D_{\mathcal{R},\varepsilon} T_\varepsilon(g_{\rm atom} - y_{\rm ref}) (x,t)+ D_{\mathcal{R},\varepsilon}z_1(x,t) \rvert\\
&\quad \leq \lvert D_{\mathcal{R},\varepsilon} T_\varepsilon(g_{\rm atom} - y_{\rm ref}) (x,t) \rvert + \lvert D_{\mathcal{R},\varepsilon} (z_1(x,t)-z_1^0(x)) \rvert + \lvert D_{\mathcal{R},\varepsilon}z_1^0(x) \rvert \\
&\quad \leq C_1 \varepsilon^{\gamma - \frac{d}{2}} + \frac{2 b \lvert \mathcal{R}\rvert^{\frac{1}{2}}}{\varepsilon}+ \lvert D_{\mathcal{R},\varepsilon}z_1^0(x) \rvert \\
&\quad \leq 3 C_1 \varepsilon^{\gamma - \frac{d}{2}} + \frac{2 b \lvert \mathcal{R}\rvert^{\frac{1}{2}}}{\varepsilon}\\
&\quad \leq r_0.
\end{align*}
Therefore $F(s,z(s))$ is well defined. Furthermore,
\begin{align*}
\sup\limits_t \sup\limits_{x \in \Omega \cap \varepsilon \mathbb{Z}^d} &\lvert G(z)(x,t)-z^0(x) \rvert\\ &\leq \sup\limits_{t,x} \int_{t_0}^t \lvert F_1(s,z(s)) \rvert + \lvert F_2(s,z(s)) \rvert \,ds\\
&\leq bT + T C_1 \varepsilon^{\gamma - \frac{d}{2}} + T \lVert T_\varepsilon ( \dot{g}_{\rm atom} - \dot{y}_{\rm ref})\rVert_{L^\infty(0,T_0;\ell^\infty)} + T\frac{2 \lvert \mathcal{R}\rvert}{\varepsilon} \lVert DW_{\rm atom}\rVert_\infty\\
&\quad + \sqrt{T} \big( \lVert f \rVert_{L^2(0, T_0; \ell^\infty)} + \lVert \ddot{y}_{\rm ref} \rVert_{L^2(0, T_0; \ell^\infty)} + \big\lVert T_\varepsilon (\ddot{g}_{\rm atom} - \ddot{y}_{\rm ref}) \big\lVert_{L^2(0,T_0; \ell^\infty)}\big).
\end{align*}
In particular, for $T$ small enough
\[\sup\limits_t \sup\limits_{x \in \Omega \cap \varepsilon \mathbb{Z}^d} \lvert G(z)(x,t)-z^0(x) \rvert \leq b.\]
Since $G(z)$ also has the correct boundary values, $G \colon K_{T,b,z_0} \to K_{T,b,z_0}$ is well defined. Given $z,\tilde{z} \in K_{T,b,z_0}$ we calculate
\begin{align*}
\sup\limits_t &\sup\limits_{x \in \Omega \cap \varepsilon \mathbb{Z}^d} \lvert G(z)(x,t) - G(\tilde{z})(x,t) \rvert
\leq T \sup\limits_t \lVert F(t,z(t)) - F(t,\tilde{z}(t)) \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)}\\
&\leq T\big( \sup_t \lVert z_2(t) - \tilde{z}_2(t) \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)} + \lvert \mathcal{R}\rvert^{\frac{3}{2}} \frac{4}{\varepsilon^2} \lVert D^2W_{\rm atom} \rVert_\infty \sup_t \lVert z_1(t) - \tilde{z}_1(t) \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)}\big)\\
&\leq T\big(1+\lvert \mathcal{R}\rvert^{\frac{3}{2}} \frac{4}{\varepsilon^2} \lVert D^2W_{\rm atom} \rVert_\infty \big) \sup_t \lVert z(t) - \tilde{z}(t) \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)}\\
&\leq \frac{1}{2}\sup_t \lVert z(t) - \tilde{z}(t) \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)},
\end{align*}
if we also require
\[T \leq \frac{1}{2+2\lvert \mathcal{R}\rvert^{\frac{3}{2}} \frac{4}{\varepsilon^2} \lVert D^2W_{\rm atom} \rVert_\infty}.\]
Now we can use the Banach fixed point theorem. If $b$ and $T$ satisfy the constraints above, then $G$ has a unique fixed point $z \in K_{T,b,z_0}$. Setting $y = z_1 + y_{\rm ref} + T_\varepsilon (g_{\rm atom}(t) - y_{\rm ref}(t))$, we have
\[y \in H^2((t_0,\min\{ t_0+T,T_0\});\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d))\]
and $y$ solves the discrete initial-boundary-value problem in the absolutely continuous sense on $[t_0,\min\{t_0+T,T_0\}]$.
Now conversely, if $y$ is any solution in $H^2((t_0,\min\{ t_0+T,T_0\});\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d))$ that satisfies
\[D_{\mathcal{R},\varepsilon} y(x,t) \in V\]
for all $t$ and $x \in \sinto_\varepsilon \Omega$, we can substitute back to $z$ and calculate
\begin{align*}
&\lVert z(t) - z^0 \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)} \leq \int_{t_0}^t \lVert z(s) - z^0 \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)}\,ds + \frac{2 \mathcal{R} \lVert DW_{\rm atom} \rVert_\infty}{\varepsilon} (t-t_0)\\
&\qquad+ \sqrt{t-t_0} \big( \lVert f \rVert_{L^2(0, T_0; \ell^\infty)} + \lVert \ddot{y}_{\rm ref} \rVert_{L^2(0, T_0; \ell^\infty)} + \big\lVert T_\varepsilon (\ddot{g}_{\rm atom} - \ddot{y}_{\rm ref}) \big\lVert_{L^2(0,T_0; \ell^\infty)} \big)\\
&\leq \int_{t_0}^t \lVert z(s) - z^0 \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)}\,ds +C_1\big( (t-t_0)+ \sqrt{t-t_0} \big)
\end{align*}
Using Grönwall's inequality we thus get
\begin{align*}
\lVert z(t) - z^0 \rVert_{\ell^\infty(\Omega \cap \varepsilon \mathbb{Z}^d)} &\leq C_1(t-t_0 + \sqrt{t-t_0})e^{t-t_0}\\
&\leq 2C_1\sqrt{T}e^T\\
&\leq b
\end{align*}
if we additionally assume $T \leq 1$ and
$T \leq \frac{b}{2 C_1 e}$.
Therefore, $z \in K_{T,b,z_0}$, and the uniqueness of the solution follows.
\end{proof}
Although this lemma already gives us a local solution, the time $T$ depends heavily on $\varepsilon$ and is not necessarily bounded from below as $\varepsilon$ goes to $0$. One of our main goals is to show existence on an $\varepsilon$-independent time interval. Actually, we even want to go one step further. We will show that the atomistic solution exists as long as the solution to the continuous problem exists and is atomistically stable.
As mentioned in the introduction establishing an atomistic Gårding inequality is key to provide control of the stability of solutions for long times and large deformations. There are some differences to the continuous Gårding inequality (Theorem \ref{thm:contGårding}). Unsurprisingly, we need to require atomistic stability. Due to the discreteness of the problem we also need to track the variation of the coefficients and the dependence on $\varepsilon$ more explicitly.
\begin{thm} \label{thm:discreteGårding}
Let $d \in \mathbb{N}$, $\Omega \subset \mathbb{R}^d$ open and bounded, and $\lambda_1, \Lambda, \varepsilon_0>0$. Consider a family $A_\varepsilon \colon \sinto_\varepsilon \Omega \to \mathbb{R}^{d \times \mathcal{R} \times d \times \mathcal{R}}$, for $0< \varepsilon \leq \varepsilon_0$, with $\lambda_{\rm atom} (A_\varepsilon(x)) \geq \lambda_1$ for all $x \in \sinto_\varepsilon \Omega$ and $0< \varepsilon \leq \varepsilon_0$. Assume also that $ \sup_\varepsilon \lVert A_\varepsilon \rVert_\infty \leq \Lambda$ and that there are $r_\varepsilon \geq \varepsilon$ such that
\[\sup_{0<\varepsilon \leq \varepsilon_0} \sup_{\substack{x,x' \in \sinto_\varepsilon \Omega \\ \lvert x-x' \rvert \leq 2 r_\varepsilon+ 2 \varepsilon R_{\rm max}}} \lvert A_\varepsilon(x) - A_\varepsilon(x') \rvert \leq \frac{\lambda_1}{4},\]
then there is a $\lambda_2=\lambda_2(\lambda_1, \Lambda, d, \mathcal{R})$, such that
\[ \varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x)[D_{\mathcal{R},\varepsilon} u (x),D_{\mathcal{R},\varepsilon} u (x)] \geq \frac{\lambda_1}{2} \lVert u \rVert^2_{h^1_\varepsilon(\sinto_\varepsilon \Omega)} - \frac{\lambda_2}{r_\varepsilon^2} \lVert u \rVert^2_{\ell_\varepsilon^2(\into_\varepsilon \Omega)}\]
for all $u \in \mathcal{A}_\varepsilon(\Omega,0)$ and $0< \varepsilon \leq \varepsilon_0$.
\end{thm}
\begin{rem}
In this paper we will only use the theorem in the case where $r_\varepsilon$ is independent of $\varepsilon$. This corresponds to $A_\varepsilon$ only changing on the macroscopic scale. We will still prove the more general version since the theorem has some interest itself.
\end{rem}
\begin{proof}
By the definition of atomistic stability we have
\[ \varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(z) [D_{\mathcal{R},\varepsilon} u (x),D_{\mathcal{R},\varepsilon} u (x)] \geq \lambda_1 \lVert u \rVert^2_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}\]
for every $z \in \sinto_\varepsilon \Omega$, every $\varepsilon>0$, and every $u \in \mathcal{A}_\varepsilon(\Omega,0)$.
Now, choose countable many $z_j \in \mathbb{R}^d$ and $\eta_j \in C_c^\infty(\mathbb{R}^d;[0,1])$ such that $\sum_j \eta_j^2(x) = 1$ for every $x \in \mathbb{R}^d$, $\supp \eta_j \subset B_{r_\varepsilon}(z_j)$, $\lvert \nabla \eta_j \rvert \leq \frac{C(d)}{r_\varepsilon}$, and the decomposition is locally finite in the sense that
\[ \lvert \{j \colon B_{r_\varepsilon}(z_j) \cap B_R(x) \neq \emptyset \} \rvert \leq C(d) \big(1+\frac{R}{r_\varepsilon}\big)^d\]
for all $x \in \mathbb{R}^d$ and $R>0$. Whenever $B_{r_\varepsilon + \varepsilon R_{\rm max}}(z_j)\cap \sinto_\varepsilon \Omega \neq \emptyset$ fix a point $x_{j,\varepsilon} \in B_{r_\varepsilon + \varepsilon R_{\rm max}}(z_j)\cap \sinto_\varepsilon \Omega$. By assumption we then have $\lvert A_\varepsilon (x_{j, \varepsilon}) - A_\varepsilon(x) \rvert \leq \frac{\lambda_1}{4}$ for every $x \in B_{r_\varepsilon + \varepsilon R_{\rm max}}(z_j)\cap \sinto_\varepsilon \Omega$.
Now, since
\[(D_{\mathcal{R},\varepsilon} (\eta_j u) (x))_\rho = \eta_j(x)(D_{\mathcal{R},\varepsilon} u(x))_\rho + u(x+\varepsilon \rho) (D_{\mathcal{R},\varepsilon} \eta_j (x))_\rho \]
for any $\delta>0$ we can calculate with Young's inequality
\begin{align*}
\varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} &A_\varepsilon(x) [D_{\mathcal{R},\varepsilon} u (x),D_{\mathcal{R},\varepsilon} u (x)] \\
&= \varepsilon^d \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x) [\eta_j(x) D_{\mathcal{R},\varepsilon} u (x),\eta_j(x) D_{\mathcal{R},\varepsilon} u (x)] \\
&\geq \varepsilon^d \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x) [D_{\mathcal{R},\varepsilon} (\eta_j u) (x),D_{\mathcal{R},\varepsilon} (\eta_j u) (x)]\\
&\qquad - \delta \varepsilon^d \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} n_j^2(x) \lvert D_{\mathcal{R},\varepsilon} u (x) \rvert^2\\
&\qquad - \Lambda (1+ \frac{\Lambda}{\delta})\varepsilon^d \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} \sum_\rho \lvert u(x+\varepsilon \rho) \rvert^2 \Big\lvert \frac{\eta_j(x+\varepsilon \rho) - \eta_j(x)}{\varepsilon}\Big\rvert^2\\
&\geq \varepsilon^d \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x_{j,\varepsilon}) [D_{\mathcal{R},\varepsilon} (\eta_j u) (x),D_{\mathcal{R},\varepsilon} (\eta_j u) (x)] - \frac{\lambda_1}{4} \lvert D_{\mathcal{R},\varepsilon} (\eta_j u) (x) \rvert^2 \\
&\qquad - \delta \varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} u (x) \rvert^2\\
&\qquad - \Lambda (1+ \frac{\Lambda}{\delta})\varepsilon^d \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} \sum_\rho \lvert u(x+\varepsilon \rho) \rvert^2 \Big\lvert \frac{\eta_j(x+\varepsilon \rho) - \eta_j(x)}{\varepsilon}\Big\rvert^2.
\end{align*}
Using the atomistic stability at $x_{j,\varepsilon}$, we can continue in the spirit to find
\begin{align*}
\varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} &A_\varepsilon(x) [D_{\mathcal{R},\varepsilon} u (x),D_{\mathcal{R},\varepsilon} u (x)] \\
&\geq \varepsilon^d \frac{3}{4} \lambda_1 \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} (\eta_j u) (x) \rvert^2\\
&\qquad - \delta \varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} u (x) \rvert^2\\
&\qquad - \Lambda (1+ \frac{\Lambda}{\delta})\varepsilon^d \sum_j \sum\limits_{x \in \sinto_\varepsilon \Omega} \sum_\rho \lvert u(x+\varepsilon \rho) \rvert^2 \Big\lvert \frac{\eta_j(x+\varepsilon \rho) - \eta_j(x)}{\varepsilon}\Big\rvert^2\\
&\geq \varepsilon^d (\frac{3}{4} \lambda_1-2 \delta) \sum\limits_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} u (x) \rvert^2 \\
&\qquad - \Big(\Lambda(1 + \frac{\Lambda}{\delta}) + \frac{3}{4} \lambda_1 (1+ \frac{3 \lambda_1}{4\delta}) \Big) \lVert u \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2 C(d, \mathcal{R}) \frac{1}{r_\varepsilon^2} \Big(1+ \frac{\varepsilon R_{\rm max}}{r_\varepsilon}\Big)^d
\end{align*}
Now, choosing $\delta = \frac{\lambda_1}{8}$ and using $r_\varepsilon \geq \varepsilon$, we indeed get
\begin{align*}
\varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} &A_\varepsilon(x) [D_{\mathcal{R},\varepsilon} u (x),D_{\mathcal{R},\varepsilon} u (x)] \\
&\geq \frac{\lambda_1}{2}\varepsilon^d \sum\limits_{x \in \sinto_\varepsilon \Omega} \lvert D_{\mathcal{R},\varepsilon} u (x) \rvert^2 - C(\lambda_1, \Lambda, d, \mathcal{R}) \frac{1}{r_\varepsilon^2} \lVert u \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2.
\end{align*}
\end{proof}
Let us make some last preparations for our main theorem. We will show that there are atomistic solutions close to the extended and regularized reference configuration
\[y_{\rm ref} = \eta_\varepsilon \ast (E y_{\rm cont}) \]
where $y_{\rm cont}$ is a solution of the continuous problem, $\eta_\varepsilon(x)$ denotes the standard scaled mollifying kernel, and $E$ denotes the Stein extension which is an extension operator for all Sobolev spaces requiring only very little regularity of the boundary, cf. \cite[Chapter VI]{steinsingint}.
The conditions that we will pose on the time-dependent atomistic boundary conditions can be formulated much easier with the following norm. Given $g \colon \partial_\varepsilon \Omega \times [0,T_0]$, such that $g(x, \cdot) \in H^2(0,T_0)$ for all $x \in \partial_\varepsilon \Omega$, we look at the (quadratic) functional
\begin{align*}
\mathcal{F}(z) &= \lVert z(0) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2 + \lVert z(0) \rVert_{h^1_\varepsilon(\sinto_\varepsilon\Omega)}^2 + \lVert \dot{z}(0) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2\\
&\quad + \int_0^{T_0} \lVert z(\tau) \rVert_{h^1_\varepsilon(\sinto_\varepsilon\Omega)}^2 + \lVert \dot{z}(\tau) \rVert_{h^1_\varepsilon(\sinto_\varepsilon\Omega)}^2 + \lVert \ddot{z}(\tau) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2 \,d\tau
\end{align*}
for $z \colon \Omega \cap \varepsilon \mathbb{Z}^d \times [0,T_0]$, such that $z(x, \cdot) \in H^2(0,T_0)$ for all $x \in \Omega \cap \varepsilon \mathbb{Z}^d$ and $z |_{\partial_\varepsilon \Omega \times [0, T_0]} = g$. Clearly the functional is lower semi-continuous and coercive in $H^2$ and thus has a minimizer. By strict convexity this minimizer is unique and it is also given as the unique solution to
\begin{align*}
0&=(z(0), w(0))_{\ell^2_\varepsilon}^2 + (z(0), w(0))_{h^1_\varepsilon}^2 + (\dot{z}(0), \dot{w}(0))_{\ell^2_\varepsilon}^2\\
&\quad + \int_0^{T_0} (z(\tau), w(\tau))_{h^1_\varepsilon}^2 + (\dot{z}(\tau), \dot{w}(\tau))_{h^1_\varepsilon}^2 + (\ddot{z}(\tau), \ddot{w}(\tau))_{\ell^2_\varepsilon}^2 \,d\tau
\end{align*}
for all $w \in H^2$ with $w |_{\partial_\varepsilon \Omega} = 0$. In particular, the mapping $K_\varepsilon$ that maps $g$ to this minimizer is linear. Furthermore, $\lVert g \rVert_{\partial_\varepsilon \Omega,dyn} := \big(\mathcal{F}(K_\varepsilon g) \big)^{\frac{1}{2}}$ is a norm. Besides dominating the norms used in its definition, we will also use that
\[\lVert K_\varepsilon g \rVert_{L^\infty(0,T_0; h^1_\varepsilon)} \leq \lVert g \rVert_{\partial_\varepsilon \Omega,dyn}\]
and
\[\lVert K_\varepsilon g \rVert_{W^{1,\infty}(0,T_0; \ell^2_\varepsilon)} \leq C(T) \lVert g \rVert_{\partial_\varepsilon \Omega,dyn}.\]
We will then require
\[\lVert y_{\rm ref}-g_{\rm atom} \rVert_{\partial_\varepsilon \Omega,dyn} \leq C_g \varepsilon^\gamma, \]
in our main theorem below for some convergence rate $\gamma \in (\frac{d}{2},2]$.
While this specific norm is mainly chosen to satisfy certain inequalities in the proof, it is not at all surprising. The terms at the starting time are obviously required by the convergence estimate we want to prove uniformly in time (see below). The terms controlling the $h^1_\varepsilon$-norm are crucial. Among other things, they ensure the uniform convergence of the gradients. Therefore, at the boundary, the atomistic boundary conditions enforce not only the correct asymptotic boundary values but also the correct asymptotic (normal) derivative and thus suppress surface relaxation effects. This is important for the Cauchy-Born rule to hold near the boundary. At last, a difference in the second time derivatives has a similar effect as a difference in the body forces and thus, unsurprisingly, we want both terms to be small in the same norm.
\begin{thm} \label{thm:atomisticwavethm}
Let $d \in \{2,3\}$ and $m \in \mathbb{N}$, $m \geq 4$. Let $T_0>0$ and let $\Omega \subset \mathbb{R}^d$ be an open, bounded set with $\partial \Omega$ of class $C^m$. Let $V \subset \mathbb{R}^{d \times \mathcal{R}}$ be open and $W_{\rm atom} \in C_b^{m+1}(V)$. Let $f$ be a continuous body force, $h_0,h_1$ initial data and $g$ boundary values such that
\begin{align*}
&f \in C^{m-1}(\overline{\Omega}\times [0,T_0]; \mathbb{R}^d)\\
&g \in C^{m+1}(\overline{\Omega}\times [0,T_0]; \mathbb{R}^d)\\
&h_0 \in H^m (\Omega; \mathbb{R}^d)\\
&\{(\nabla h_0(x)\rho)_{\rho \in \mathcal{R}} \colon x \in \overline{\Omega}\} \subset V \cap \{A \colon \lambda_{\rm atom}(A)>0\} \\
&h_1 \in H^{m-1} (\Omega; \mathbb{R}^d)
\end{align*}
and such that the compatibility conditions of order $m$ are satisfied. Furthermore, assume that the unique solution of the Cauchy-Born problem $y_{\rm cont}$ from Theorem \ref{thm:localcontwave} exists until $T_0$ and satisfies
\[ y_{\rm cont} \in \bigcap_{k=0}^m C^k\big([0,T_0]; H^{m-k}(\Omega;\mathbb{R}^d)\big),\]
as well as
\[\{(\nabla y_{\rm cont}(x,t) \rho)_{\rho \in \mathcal{R}} \colon x \in \overline{\Omega}, t \in [0,T_0]\} \subset V \cap \{A \colon \lambda_{\rm atom}(A)>0\}.\]
Now let $C_g, C_f, C_h>0$ and $\gamma \in (\frac{d}{2}+ \frac{1}{m-1},2]$. Then there is an $\varepsilon_0>0$ such that the following holds for every $0< \varepsilon \leq \varepsilon_0$.
Given atomistic data $f_{\rm atom} \colon \into_\varepsilon \Omega \times [0,T_0] \to \mathbb{R}^d$, $g_{\rm atom} \in H^2((0,T_0); \mathbb{R}^d)^{\partial_\varepsilon \Omega}$, $h_{{\rm atom},0} \in \mathcal{A}_\varepsilon (\Omega, g_{\rm atom}(\cdot,0))$, and $h_{{\rm atom},1} \in \mathcal{A}_\varepsilon (\Omega, \dot{g}_{\rm atom}(\cdot,0))$ with
\[\lVert y_{\rm ref}-g_{\rm atom} \rVert_{\partial_\varepsilon \Omega,dyn} \leq C_g \varepsilon^\gamma, \]
\[\lVert h_{\rm atom, 1}- \dot{y}_{\rm ref}(0) \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} + \lVert h_{\rm atom, 0}- y_{\rm ref}(0) \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} + \lVert h_{\rm atom, 0}- y_{\rm ref}(0) \rVert^2_{h^1_\varepsilon(\into_\varepsilon \Omega)} \leq C_h^2 \varepsilon^{2\gamma},\]
\[\lVert f_{\rm ref}-f_{\rm atom} \rVert_{L^2(0,T_0; \ell^2_\varepsilon(\into \Omega))} \leq C_f \varepsilon^\gamma,\]
where
\[f_{\rm ref} = \tilde{f} + \ddot{y}_{\rm ref} - \ddot{\tilde{y}}_{\rm cont}.\]
Then there is a unique $y \in H^2((0,T_0); \mathbb{R}^d)^{\Omega \cap \varepsilon \mathbb{Z}^d}$ that solves the atomistic equations with body force $f_{\rm atom}$ boundary values $g_{\rm atom} $ and initial conditions $h_{{\rm atom},0}, h_{{\rm atom},1}$. Furthermore, we have the convergence estimate
\begin{align*}
&\lVert \dot{y} - \dot{y}_{\rm ref} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} +\lVert y - y_{\rm ref} \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}+ \lVert y- y_{\rm ref} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\qquad \leq C e^{Ct} (C_g + C_h + C_f + \varepsilon^{2-\gamma}) \varepsilon^{\gamma}
\end{align*}
for some $C=C(\mathcal{R}, V, W_{\rm atom}, y_{\rm cont}, \Omega, m, \gamma)>0$.
\end{thm}
\begin{rem}
Remember that
\[\tilde{f}(x)= \fint_{Q_\varepsilon(x)} f(z)\,dz.\]
If $y_{\rm cont} \in H^2(0,T;C^{1,\gamma-1}(\Omega; \mathbb{R}^d))$, the more natural choice $f_{\rm ref} = \tilde{f}$ suffices since then
\[\lVert \ddot{y}_{\rm ref} - \ddot{\tilde{y}}_{\rm cont} \rVert_{L^2(0,T_0; \ell^2_\varepsilon(\into \Omega))} \leq C(y_{\rm cont}) \varepsilon^\gamma.\]
This condition is automatically satisfied if $m \geq 6$.
\end{rem}
\begin{proof}
First let us prove that $Ey_{\rm cont}$ and $y_{\rm ref}$ inherit the atomistic stability from $y_{\rm cont}$ as long as we stay in or close to $\Omega$.
Given $R>0$ and $x \in \Omega + B_{\varepsilon R}(0)$, take $x' \in \Omega$ with $\lvert x-x'\rvert \leq R \varepsilon$. Then we directly see
\begin{align*}
\lvert \nabla Ey_{\rm cont}(x) - \nabla y_{\rm cont}(x') \rvert &\leq \lVert \nabla^2 Ey_{\rm cont} \rVert_{L^\infty} R \varepsilon \\
&\leq C(\Omega) R \varepsilon \lVert \nabla^2 y_{\rm cont} \rVert_{L^\infty}
\end{align*}
since $y_{\rm cont} \in H^4(\Omega)$, which embeds into $W^{2, \infty}$ and even $C^2$ for $d \leq 3$. It immediately follows that
\[\lvert \nabla y_{\rm ref}(x) - \nabla y _{\rm cont}(x')\rvert \leq C(\Omega) (R+1) \varepsilon \lVert \nabla^2 y_{\rm cont} \rVert_{L^\infty}\]
and
\begin{align*}
\lvert D_{\mathcal{R},\varepsilon} y_{\rm ref}(x) - (\nabla y _{\rm cont}(x')\rho)_{\rho \in \mathcal{R}}\rvert &= \Big(\sum_\rho \Big\lvert \int_0^1 \nabla y_{\rm ref}(x+s\varepsilon \rho)\rho - \nabla y_{\rm cont}(x')\rho \,ds \Big\rvert^2\Big)^{\frac{1}{2}}\\
&\leq C(\Omega, \mathcal{R}, R) \varepsilon \lVert \nabla^2 y_{\rm cont} \rVert_{L^\infty}\\
&= C(\Omega, \mathcal{R}, R, y_{\rm cont}) \varepsilon
\end{align*}
Since the stability constant is continuous, the set $\{A \in V \colon \lambda_{\rm atom}(A)>0\}$ is open. On the other hand, $\{(\nabla y_{\rm cont}(x,t) \rho)_{\rho \in \mathcal{R}} \colon x \in \overline{\Omega}, t \in [0,T]\}$ is compact. Therefore,
\[\{(\nabla y_{\rm cont}(x,t) \rho)_{\rho \in \mathcal{R}} \colon x \in \overline{\Omega}, t \in [0,T]\} + \overline{B_{\varepsilon C}(0)} \subset \{A \in V \colon \lambda_{\rm atom}(A)>0\} \]
for all $\varepsilon \leq \varepsilon_0$ if $\varepsilon_0 = \varepsilon_0(\mathcal{R}, \Omega, y_{\rm cont}, R, V, W_{\rm atom})$ is chosen small enough.
For a time dependent atomistic deformation we define the norm-energy
\begin{align*}
\mathcal{E}(t) =\lVert \dot{u} \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} +\lVert u \rVert^2_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}+ \lVert u \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)},
\end{align*}
where $u = y- y_{\rm ref} - K_\varepsilon (g_{\rm atom}- y_{\rm ref})$. Note that this energy is well-defined and continuous on $[a,b]$ if $u \in H^2((a,b); \mathbb{R}^d)^{\Omega \cap \varepsilon \mathbb{Z}^d}$.
For $B>0$ to be defined later, let $T_\varepsilon$ be the supremum of all times $T \leq T_0$ such that a solution $y$ exists on $[0,T)$ and
\[\mathcal{E}(t) \leq B^2 \varepsilon^{2\gamma}\]
for $t \in [0,T]$.
Note that
\[\sup_t \lVert y_{\rm ref}(t) - g_{\rm atom}(t)\rVert_{\partial_\varepsilon\Omega,0} \leq \lVert y_{\rm ref} - g_{\rm atom}\rVert_{\partial_\varepsilon \Omega,dyn} \leq C_g \varepsilon^\gamma.\]
Choosing $\varepsilon_0$ so small that
\[4(\max\{C_g,C_h\} +1) \varepsilon_0^{\gamma - \frac{d}{2}} \leq \inf_{x,t} \dist(D_{\mathcal{R},\varepsilon} y_{\rm ref}(x,t), V_{\rm atom}^c)\]
we can apply the local result, Theorem \ref{thm:discretelocalexistence}. If furthermore $B> \sqrt{2C_g^2 + 2C_h^2}$, which will be the case in our choice of $B$, then we indeed see that $T_\varepsilon > 0$. The uniqueness part of Theorem \ref{thm:discretelocalexistence} implies that all such solutions agree on the intersection of their domains of definition. Putting these solutions together we thus have a $y$ on $(0,T_\varepsilon)$ such that for every $0<T<T_\varepsilon$ it holds that $y \in H^2(0,T)$ and $y$ is a solution of the problem. If we choose $\varepsilon_0$ even smaller, such that
\[4(\sqrt{2B^2+2C_g^2} +1) \varepsilon_0^{\gamma - \frac{d}{2}} \leq \inf_{x,t} \dist(D_{\mathcal{R},\varepsilon} y_{\rm ref}(x,t), V_{\rm atom}^c)\]
we can again apply Theorem \ref{thm:discretelocalexistence} with $t_0 \in (0,T_\varepsilon)$ and initial conditions $y(t_0), \dot{y}(t_0)$, since
\begin{align*}
\lVert \dot{y}(t_0) - \dot{y}_{\rm ref}(t_0) \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} +\lVert y(t_0) - y_{\rm ref}(t_0) \rVert^2_{h^1_\varepsilon(\sinto_\varepsilon \Omega)} &\leq 2 \mathcal{E}(t) + 2 C_g^2 \varepsilon^{2 \gamma}\\
&\leq (2 B^2 + 2 C_g^2) \varepsilon^{2 \gamma}.
\end{align*}
We thus get a solution on $(t_0, \max\{t_0 + T_{\rm loc}, T_0\})$ for some $T_{\rm loc}$ independent of $t_0$. Again by uniqueness all solutions fit together. Therefore, $y \in H^2(0,T_\varepsilon)$ and $y$ is a solution of the problem on $(0,T_\varepsilon)$. Additionally, we know that $T_\varepsilon = T_0$ or the solution exists on a larger intervall than $(0,T_\varepsilon)$. In the second case we must have $\mathcal{E}(T_\varepsilon) = B^2 \varepsilon^{2\gamma}$. To ensure that we are in the first case it thus suffices to estimate the energy on $[0,T_\varepsilon]$. This is what we will do in the rest of the proof.
The energy bound implies
\[\lVert D_{\mathcal{R},\varepsilon} y- D_{\mathcal{R},\varepsilon} y_{\rm ref} \rVert_\infty \leq (C_g +B)\varepsilon^{\gamma - \frac{d}{2}}.\]
Choosing $\varepsilon_0$ even smaller, now also depending on $C_g$, $B$ and $\gamma$, by continuity of the stability constant, we can find a $\lambda_0=\lambda_0(y_{\rm cont}, V, W_{\rm atom})>0$ such that
\[\lambda_{\rm atom}(M)\geq \lambda_0 \quad \text{and} \quad M \in V\]
for all $M$ with $\lvert M - D_{\mathcal{R},\varepsilon} y_{\rm ref} \rvert \leq (C_g + B) \varepsilon_0^{\gamma - \frac{d}{2}}$ for any $x,t$. In particular, we see that this is true for $M = D_{\mathcal{R},\varepsilon} y$ or $M=s D_{\mathcal{R},\varepsilon} y + (1-s)D_{\mathcal{R},\varepsilon} y_{\rm ref}$, $s\in [0,1]$ as long as $t < T_\varepsilon$.
Setting
\[A_\varepsilon = \int_0^1 D^2 W_{\rm atom}\big(D_{\mathcal{R},\varepsilon}y_{\rm ref} + s(D_{\mathcal{R},\varepsilon}y -D_{\mathcal{R},\varepsilon}y_{\rm ref} )\big) \,ds,\]
we see that for $\lvert x-x'\rvert \leq 2r + 2 \varepsilon R_{\rm max}$
\begin{align*}
\lvert A_\varepsilon(x) - A_\varepsilon(x') \rvert &\leq \lVert D^3 W_{\rm atom} \rVert_\infty \big( \lVert D^2y_{\rm ref} \rVert_\infty \lvert x-x' \rvert + 2 (B+C_g) \varepsilon^{\gamma - \frac{d}{2}}\big) \\
&\leq C ( r + \varepsilon + (B + C_g) \varepsilon^{\gamma - \frac{d}{2}}).
\end{align*}
If again $\varepsilon_0$ is small enough we can therefore use the atomistic Gårding inequality from Theorem \ref{thm:discreteGårding} with $r=r(y_{\rm cont},W_{\rm atom}, \lambda_0)$ small enough and independent of $\varepsilon$ to get
\begin{align*}
\mathcal{E}(t)&\leq \lVert u \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} + \lVert \dot{u} \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} + \max\{2 , \frac{\lambda_0}{2}\} \lVert u \rVert^2_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}\\
&\leq C \lVert u \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} + \max\{\frac{4}{\lambda_0} , 1\} \lVert \dot{u} \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} \\
&\quad + \max\{\frac{4}{\lambda_0} , 1\} \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x,t)[D_{\mathcal{R},\varepsilon}u]^2\\
&\leq C \lVert u \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} + \max\{\frac{8}{\lambda_0} , 2\} \Big( \frac{1}{2}\lVert \dot{u} \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} +\frac{1}{2} \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x,t)[D_{\mathcal{R},\varepsilon}u ]^2 \Big)
\end{align*}
for some $C=C(y_{\rm cont}, W_{\rm atom}, \lambda_0, \mathcal{R})$. If we rewrite this in terms of the initial conditions and take absolute values, we get
\begin{align*}
\mathcal{E}(t) &\leq C \Big( \lVert h_{\rm atom, 1}- \dot{y}_{\rm ref}(0) - \frac{\partial}{\partial t}K_\varepsilon(y- y_{\rm ref})(0) \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\quad + \lVert h_{\rm atom, 0}- y_{\rm ref}(0) - K_\varepsilon(y- y_{\rm ref})(0) \rVert^2_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\quad + \lVert h_{\rm atom, 0}- y_{\rm ref}(0) - K_\varepsilon(y- y_{\rm ref})(0) \rVert^2_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}\\
&\quad+ \Big\lvert \int_0^t (u, \dot{u})_{\ell^2_\varepsilon} \,d\tau \Big\rvert \\
&\quad+ \Big\lvert \int_0^t (\dot{u}, \ddot{u})_{\ell^2_\varepsilon} + \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x,\tau)[D_{\mathcal{R},\varepsilon}u, D_{\mathcal{R},\varepsilon}\dot{u} ]\\
&\quad+ \frac{1}{2}\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} \dot{A}_\varepsilon(x,\tau)[D_{\mathcal{R},\varepsilon}u]^2 \,d\tau \Big\rvert\Big).
\end{align*}
Using our assumptions at $t=0$ and for the boundary conditions we can continue by
\begin{align*}
\mathcal{E}(t) &\leq C \Big( (C_g^2+ C_h^2) \varepsilon^{2 \gamma} + \int_0^t \mathcal{E}(\tau) \,d\tau + \Big\lvert \int_0^t \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} \dot{A}_\varepsilon(x,\tau)[D_{\mathcal{R},\varepsilon}u]^2 \,d\tau \Big\rvert \\
&\quad + \Big\lvert \int_0^t \big(\dot{u}, \frac{\partial^2}{\partial t^2}K_\varepsilon (g_{\rm atom} - y_{\rm ref})\big)_{\ell^2_\varepsilon} \,d\tau \Big\rvert\\
&\quad + \Big\lvert \int_0^t \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x,\tau)[D_{\mathcal{R},\varepsilon}K_\varepsilon (g_{\rm atom} - y_{\rm ref}), D_{\mathcal{R},\varepsilon}\dot{u} ] \,d\tau \Big\rvert\\
&\quad + \Big\lvert \int_0^t (\dot{u}, \ddot{y} - \ddot{y}_{\rm ref})_{\ell^2_\varepsilon} + \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_\varepsilon(x,\tau)[D_{\mathcal{R},\varepsilon}(y - y_{\rm ref}), D_{\mathcal{R},\varepsilon}\dot{u} ] \,d\tau \Big\rvert \Big)\\
&=: C \big( (C_g^2+ C_h^2) \varepsilon^{2 \gamma} + \int_0^t \mathcal{E}(\tau) \,d\tau \big) + I_1 + I_2 + I_3 + I_4
\end{align*}
Clearly,
\[ I_2 \leq C\big(\int_0^t \mathcal{E}(\tau) \,d\tau + C_g^2 \varepsilon^{2 \gamma}\big).\]
For $I_4$ we can use the estimates from the static case. Indeed, partial summation gives
\begin{align*}
I_4 &\leq C\Big( \int_0^t \mathcal{E}(t) \,d\tau + \int_0^t \lVert \ddot{y} - \ddot{y}_{\rm ref} - \divo_{\mathcal{R},\varepsilon} (A_\varepsilon (x, \tau) D_{\mathcal{R},\varepsilon}(y-y_{\rm ref})) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2 \,d\tau \Big) \\
&= C\Big( \int_0^t \mathcal{E}(t) \,d\tau\\
&\quad + \int_0^t \lVert \ddot{y} - \ddot{y}_{\rm ref} - \divo_{\mathcal{R},\varepsilon} (DW_{\rm atom} (D_{\mathcal{R},\varepsilon}y) -DW_{\rm atom} (D_{\mathcal{R},\varepsilon} y_{\rm ref})) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2 \,d\tau \Big)
\end{align*}
As we showed at the beginning of this proof, we have
\begin{align*}
\co \{D_{\mathcal{R},\varepsilon} y_{\rm ref} (\hat{x}+ \varepsilon \sigma), (\nabla y_{\rm ref} (x) \rho)_{\rho \in \mathcal{R}}\} \subset V
\end{align*}
for all $x\in \Omega_\varepsilon$ and $\sigma \in \mathcal{R} \cup \{0\}$. We are therefore in a position to apply Proposition \ref{prop:ell2residuum}.
\begin{align*}
\lVert \ddot{y} - &\ddot{y}_{\rm ref} - \divo_{\mathcal{R},\varepsilon}(DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y)- DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y_{\rm ref})) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&= \lVert f_{\rm atom} - \ddot{y}_{\rm ref} + \divo_{\mathcal{R},\varepsilon} DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y_{\rm ref}) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\leq \lVert f_{\rm atom} - \ddot{y}_{\rm ref} + \ddot{\tilde{y}}_{\rm cont} -\tilde{f} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\quad + \lVert -\ddot{\tilde{y}}_{\rm cont} +\tilde{f}+ \divo_{\mathcal{R},\varepsilon} DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y_{\rm ref}) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\leq \lVert f_{\rm atom} - f_{\rm ref} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} + \lVert -\ddot{y}_{\rm cont} +f+ \divo DW_{\rm CB}(\nabla y_{\rm ref}) \rVert_{L^2(\Omega_\varepsilon; \mathbb{R}^d)}\\
&\quad + C \varepsilon^2 \Big\lVert \lVert \nabla^4 y_{\rm ref} \rVert_{L^\infty(B_{\varepsilon R}(x))} + \lVert \nabla^3 y_{\rm ref} \rVert_{L^\infty(B_{\varepsilon R}(x))}^\frac{3}{2} + \lVert \nabla^2 y_{\rm ref} \rVert_{L^\infty(B_{\varepsilon R}(x))}^3 \\
&\quad +\varepsilon\lVert \nabla^3 y_{\rm ref} \rVert_{L^\infty(B_{\varepsilon R}(x))}^2 \Big\rVert_{L^2(\Omega_\varepsilon)},
\end{align*}
where $C$ and $R$ just depend on $d, \mathcal{R}$ and $\lVert D^2 W_{\rm atom} \rVert_{C^2(V)}$. Now, remember that $y_{\rm ref} = \eta_\varepsilon \ast (E y_{\rm cont})$. Hence, we can apply Proposition \ref{prop:approximation1} and Proposition \ref{prop:approximation2} and get
\begin{align*}
\lVert \ddot{y} - &\ddot{y}_{\rm ref} - \divo_{\mathcal{R},\varepsilon}(DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y)- DW_{\rm atom}(D_{\mathcal{R},\varepsilon} y_{\rm ref})) \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\leq \lVert f_{\rm atom} - f_{\rm ref} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}\\
&\quad + C \varepsilon^2 \big( \lVert \nabla^4 Ey_{\rm cont}\rVert_{L^2(\Omega_\varepsilon + B_{(R+1)\varepsilon}(0))} + \lVert \nabla^3 Ey_{\rm cont}\rVert_{L^3(\Omega_\varepsilon + B_{(R+1)\varepsilon}(0))}^{\frac{3}{2}}\\
&\quad + \lVert \nabla^2 Ey_{\rm cont}\rVert_{L^6(\Omega_\varepsilon + B_{(R+1)\varepsilon}(0))}^3 + \varepsilon \lVert \nabla^3 Ey_{\rm cont}\rVert_{L^4(\Omega_\varepsilon + B_{(R+1)\varepsilon}(0))}^2 \\
&\quad + \lVert \nabla^2 Ey_{\rm cont} \rVert_{L^4(\Omega+B_{\varepsilon}(0))} \lVert \nabla^3 Ey_{\rm cont} \rVert_{L^4(\Omega+B_{\varepsilon}(0))} + \lVert \nabla^4 Ey_{\rm cont} \rVert_{L^2(\Omega+B_{\varepsilon}(0))} \big)\\
&\leq \lVert f_{\rm atom} - f_{\rm ref} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} +C\varepsilon^2 \lVert y_{\rm cont} \rVert_{H^4(\Omega; \mathbb{R}^d)} (1 + \lVert y_{\rm cont} \rVert_{H^4(\Omega; \mathbb{R}^d)}^2),
\end{align*}
where in the last step we used standard embedding theorems with $d \in \{2,3\}$, as well as the fact that $E$ is a continuous extension operator on all Sobolev spaces. Hence, we find
\begin{align*}
I_4 \leq C\big(\int_0^t \mathcal{E}(t) \,d\tau + C_f^2 \varepsilon^{2\gamma} + \varepsilon^4 \big).
\end{align*}
Now let us look at the nonlinearity $I_1$. Evaluating the time derivative, we see that we can control it in terms of (some power of) the energy. But the resulting estimates are not good enough in $\varepsilon$. The idea is to improve the estimates with a specific scheme of partial integrations in time. Indeed, it turns out that the estimates improve by $\varepsilon^{\gamma-\frac{d}{2}}$ with each step. For this let us extend the definition of $A_\varepsilon = A_{\varepsilon,2}$ to
\[A_{\varepsilon,k} = \int_0^1 D^k W_{\rm atom}\big(D_{\mathcal{R},\varepsilon}y_{\rm ref} + s(D_{\mathcal{R},\varepsilon}y -D_{\mathcal{R},\varepsilon}y_{\rm ref} )\big)\,ds.
\]
Furthermore, let us write for $k \geq 2$
\begin{align*}
B_k(t) &= \int_0^1 \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} \dot{A}_{\varepsilon,k-1}[D_{\mathcal{R},\varepsilon}u]^{k-1} s^{k-3} \,ds,\\
C_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon}u]^{k-1}[D_{\mathcal{R},\varepsilon} \dot{y}_{\rm ref}] s^{k-3} \,ds,\\
D_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon}u]^{k-1}[D_{\mathcal{R},\varepsilon} \dot{u}] s^{k-2} \,ds,\\
E_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon}u]^{k-1}[D_{\mathcal{R},\varepsilon} (K_\varepsilon (g_{\rm atom}-y_{\rm ref}))^\cdot] s^{k-2} \,ds,\\
F_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon}u]^{k} s^{k-2} \,ds.
\end{align*}
In this notation, we have
\[I_1 = C \Big\lvert \int_0^t B_3(\tau) \,d\tau \Big\rvert\]
and, for $3 \leq k \leq m+1$, by partial integration in time,
\[\int_0^t B_{k}(\tau) + (k-1) D_{k-1}(\tau) \,d\tau = F_{k-1}(t) - F_{k-1}(0), \]
as well as
\[B_k(t) = C_k(t) + D_k(t) + E_k(t)\]
by evaluating the time derivative. We claim to have relatively good estimates on the $C_k$, $E_k$, and $F_k$. At the same time we will prove estimates on the $D_k$ that get better with increasing $k$. Due to the two equations above this is sufficient. We just need to control all the $C_k$, $E_k$, and $F_k$, as well as $D_{m+1}$. Since
\[\lvert D_{\mathcal{R},\varepsilon} u \rvert \leq \varepsilon^{-\frac{d}{2}} \lVert u \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)} \leq B \varepsilon^{\gamma-\frac{d}{2}},\]
we have the following estimates:
\begin{align*}
\Big\lvert \int_0^t C_k(\tau)\,d\tau \Big\rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-3} \lVert y_{\rm cont} \rVert_{C^1(0,t; H^3(\Omega))}\int_0^t \mathcal{E}(\tau) \,d\tau, \\
\Big\lvert \int_0^t E_k(\tau)\,d\tau \Big\rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-2} \Big(\int_0^t \mathcal{E}(\tau) \,d\tau + C_g^2 \varepsilon^{2\gamma} \Big), \\
\lvert F_k(0) \rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-2} (C_g^2 + C_h^2) \varepsilon^{2 \gamma},\\
\lvert F_k(t) \rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-2} \mathcal{E}(t).
\end{align*}
Furthermore,
\begin{align*}
\Big\lvert \int_0^t D_{m+1}(\tau)\,d\tau \Big\rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{m-1} \int_0^t \lVert u \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)} \lVert \dot{u} \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)} \,d\tau \\
&\leq C \varepsilon^{-1} (B \varepsilon^{\gamma - \frac{d}{2}})^{m-1} \int_0^t \lVert u \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)} \lVert \dot{u} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)} \,d\tau \\
&\leq C \varepsilon^{-1} (B \varepsilon^{\gamma - \frac{d}{2}})^{m-1} \int_0^t \lVert u \rVert_{h^1_\varepsilon(\sinto_\varepsilon \Omega)}^2 + \lVert \dot{u} \rVert_{\ell^2_\varepsilon(\into_\varepsilon \Omega)}^2 \,d\tau \\
&\leq C \varepsilon^{-1} (B \varepsilon^{\gamma - \frac{d}{2}})^{m-1} \int_0^t \mathcal{E}(\tau) \,d\tau.
\end{align*}
Choosing $\varepsilon_0$ small enough, such that $B \varepsilon^{\gamma-\frac{d}{2}} \leq 1$, we can combine these estimates from $k=3$ up to $k=m+1$ to get
\[I_1 \leq C \Big( \int_0^t \mathcal{E}(\tau) \,d\tau + (C_g^2 + C_h^2)\varepsilon^{2 \gamma} + B \varepsilon^{\gamma-\frac{d}{2}} \mathcal{E}(t) + B^{m-1} \varepsilon^{(m-1)(\gamma - \frac{d}{2})-1}\int_0^t \mathcal{E}(\tau) \,d\tau\Big)\]
for some $C=C(y_{\rm cont}, W_{\rm atom}, V, \mathcal{R}, \Omega, m)$. Choosing $\varepsilon_0$ even smaller, we can ensure that $C B \varepsilon^{\gamma-\frac{d}{2}} \leq \frac{1}{3}$ and $B^{m-1} \varepsilon^{(m-1)(\gamma - \frac{d}{2})-1} \leq 1$, since $\gamma > \frac{d}{2} + \frac{1}{m-1}$ by assumption. Therefore,
\[I_1 \leq \frac{1}{3} \mathcal{E}(t) + C \Big( \int_0^t \mathcal{E}(\tau) \,d\tau + (C_g^2 + C_h^2)\varepsilon^{2 \gamma}\Big).\]
The additional error term $I_3$ coming from the boundary conditions can be handled in a similar way. We now set
\begin{align*}
B_k(t) &= \int_0^1 \varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} \dot{A}_{\varepsilon,k-1}[D_{\mathcal{R},\varepsilon} K_\varepsilon (g_{\rm atom}-y_{\rm ref})][D_{\mathcal{R},\varepsilon}u]^{k-2} s^{k-3} \,ds,\\
C_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon} K_\varepsilon (g_{\rm atom}-y_{\rm ref})][D_{\mathcal{R},\varepsilon}u]^{k-2}[D_{\mathcal{R},\varepsilon} \dot{y}_{\rm ref}] s^{k-3} \,ds,\\
D_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon} K_\varepsilon (g_{\rm atom}-y_{\rm ref})][D_{\mathcal{R},\varepsilon} u]^{k-2}[D_{\mathcal{R},\varepsilon} \dot{u}] s^{k-2} \,ds,\\
E_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon} K_\varepsilon (g_{\rm atom}-y_{\rm ref})][D_{\mathcal{R},\varepsilon}u]^{k-2}\\
&\qquad [D_{\mathcal{R},\varepsilon} (K_\varepsilon (g_{\rm atom}-y_{\rm ref}))^\cdot] s^{k-2} \,ds,\\
F_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon} K_\varepsilon (g_{\rm atom}-y_{\rm ref})][D_{\mathcal{R},\varepsilon}u]^{k-1} s^{k-2} \,ds\\
G_k(t) &= \int_0^1\varepsilon^d \sum_{x \in \sinto_\varepsilon \Omega} A_{\varepsilon,k}[D_{\mathcal{R},\varepsilon} (K_\varepsilon (g_{\rm atom}-y_{\rm ref}))^\cdot][D_{\mathcal{R},\varepsilon}u]^{k-1} s^{k-2} \,ds.
\end{align*}
In analogy to before, we have
\[I_3 = C \Big\lvert \int_0^t D_2(\tau) \,d\tau \Big\rvert\]
and, for $3 \leq k \leq m+1$,
\[\int_0^t B_{k}(\tau) + (k-2) D_{k-1}(\tau) + G_{k-1}(\tau) \,d\tau = F_{k-1}(t) - F_{k-1}(0), \]
as well as
\[B_k(t) = C_k(t) + D_k(t) + E_k(t).\]
Again, we have the estimates
\begin{align*}
\Big\lvert \int_0^t C_k(\tau)\,d\tau \Big\rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-3} \lVert y_{\rm cont} \rVert_{C^1(0,t; H^3(\Omega))}\Big(\int_0^t \mathcal{E}(\tau) \,d\tau + C_g^2 \varepsilon^{2 \gamma} \Big), \\
\Big\lvert \int_0^t E_k(\tau)\,d\tau \Big\rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-2} C_g^2 \varepsilon^{2\gamma}, \\
\Big\lvert \int_0^t G_k(\tau)\,d\tau \Big\rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-2} \Big(\int_0^t \mathcal{E}(\tau) \,d\tau + C_g^2 \varepsilon^{2 \gamma} \Big), \\
\lvert F_k(0) \rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-2} (C_g^2 + C_h^2) \varepsilon^{2 \gamma},\\
\lvert F_k(t) \rvert &\leq C (B \varepsilon^{\gamma - \frac{d}{2}})^{k-2} (\mathcal{E}(t) +C_g^2 \varepsilon^{2 \gamma}).
\end{align*}
Furthermore,
\begin{align*}
\Big\lvert \int_0^t D_{m+1}(\tau)\,d\tau \Big\rvert &\leq C \varepsilon^{-1} (B \varepsilon^{\gamma - \frac{d}{2}})^{m-1} \Big( \int_0^t \mathcal{E}(\tau) \,d\tau +C_g^2 \varepsilon^{2 \gamma} \Big).
\end{align*}
As before this implies
\[I_3 \leq \frac{1}{3} \mathcal{E}(t) + C \Big( \int_0^t \mathcal{E}(\tau) \,d\tau + (C_g^2 + C_h^2)\varepsilon^{2 \gamma}\Big)\]
Overall we proved
\begin{align*}
\mathcal{E}(t) &= 3 (\mathcal{E}(t) - \frac{2}{3} \mathcal{E}(t))\\
&\leq C \Big( (C_f^2 + C_g^2+ C_h^2 + \varepsilon^{4-2\gamma}) \varepsilon^{2 \gamma} + \int_0^t \mathcal{E}(\tau) \,d\tau\Big)
\end{align*}
for some $C=C(y_{\rm cont}, W_{\rm atom}, V, \mathcal{R}, \Omega, m, \gamma)$, all $\varepsilon \leq \varepsilon_0(y_{\rm cont}, W_{\rm atom}, V, \mathcal{R}, \Omega, m, \gamma, B)$ and $t \in [0, T_\varepsilon)$.
Grönwall's inequality then yields
\begin{align*}
\mathcal{E}(t) &\leq C (C_f^2 + C_g^2+ C_h^2 + \varepsilon^{4-2\gamma}) \varepsilon^{2 \gamma} e^{Ct}\\
&\leq\frac{B^2}{2} \varepsilon^{2 \gamma},
\end{align*}
where we have finally chosen $B:=\Big(2C (C_f^2 + C_g^2+ C_h^2 + 1) e^{CT_0}\Big)^{\frac{1}{2}}$. In particular, with $C\geq 1$ we satisfy the condition $B > \sqrt{2 C_g^2 + 2C_h^2}$, that we required at the beginning.
In particular, $\mathcal{E}(T_\varepsilon) \leq \frac{B^2}{2} \varepsilon^{2 \gamma}$ and therefore $T_\varepsilon = T_0$ for $\varepsilon \leq \varepsilon_0$.
The convergence estimate immediately follows from the energy estimate and the estimate we assumed for the boundary conditions. Uniqueness follows directly from the local uniqueness in Theorem \ref{thm:discretelocalexistence}.
\end{proof}
\section*{Acknowledgments}
This work is part of my PhD thesis \cite{braunphdthesis}. I would like to express my gratitude to my advisor Bernd Schmidt for his guidance and support. Also, I am grateful that I had the opportunity to discuss different aspects of this work with Dirk Blömker, Christoph Ortner, and Florian Theil.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 850 |
\section{Introduction}\label{sec:1}
{\it Biharmonic maps} $\phi:(M,g)\to(N,h)$ between Riemannian
manifolds are critical points of the {\it bienergy functional}
$$
E_2(\phi)=\frac{1}{2}\int_{M}\,|\tau(\phi)|^2\,v_g,
$$
where $\tau(\phi)=\trace\nabla d\phi$ is the tension field of $\phi$
that vanishes for harmonic maps (see \cite{ES}). The Euler-Lagrange
equation corresponding to $E_2$ is given by the vanishing of the
{\it bitension field}
$$
\tau_2(\phi)=-J^{\phi}(\tau(\phi))=-\Delta\tau(\phi) -\trace
R^N(d\phi,\tau(\phi))d\phi,
$$
where $J^{\phi}$ is formally the Jacobi operator of $\phi$ (see
\cite{J1}). The operator $J^{\phi}$ is linear, thus any harmonic map
is biharmonic. We call {\it proper biharmonic} the non-harmonic
biharmonic maps. Geometric and analytic properties of proper
biharmonic maps were studied, for example, in \cite{PBAFSO, TL,RM}.
The submanifolds with non-harmonic (non-minimal) biharmonic
inclusion map are called {\it proper biharmonic submanifolds}.
Initially encouraged by the non-existence results for proper
biharmonic submanifolds in non-positively curved space forms (see,
for example, \cite{CMO2, BYC_ISH,D,HV}), the study of proper
biharmonic submanifolds in spheres constitutes an important research
direction in the theory of proper biharmonic submanifolds.
The present paper is organized as follows.
Section \ref{sec:2} is devoted to the main examples of proper
biharmonic submanifolds in spheres and to their geometric
properties, mainly regarding the type and the order in the sense of
Chen. Also, it gathers the most recent classification results for
such submanifolds (for detailed proofs see \cite{B}).
In Section \ref{sec:3} we present a series of new results concerning
geometric properties of proper biharmonic constant mean curvature
submanifolds in spheres. We begin with some identities which hold
for proper biharmonic submanifolds with parallel mean curvature
vector field (Propositions~\ref{prop: prop||MC} and \ref{prop:
prop||MC_from_Gauss_eq}). We then obtain some necessary conditions
that must be fulfilled by proper biharmonic constant mean curvature
submanifolds (Corollary~\ref{cor: conseq_type}) and we end this
section with a refinement, for hypersurfaces, of a result on the
estimate of the mean curvature of proper biharmonic submanifolds in
spheres (Theorem~\ref{th: raf_tip_hyper}).
The fourth section presents two open problems concerning the
classification of proper biharmonic hypersurfaces and the mean
curvature of proper biharmonic submanifolds in spheres.
In the last section we briefly present an interesting link between
proper biharmonic hypersurfaces and $II$-minimal hypersurfaces in
spheres.
Other results on proper biharmonic submanifolds in spaces of
non-constant sectional curvature can be found, for example, in
\cite{CMP, YJCHS, FO, FLMO, I, OU2, OU1}.
\section{Proper biharmonic submanifolds in spheres}\label{sec:2}
The attempt to obtain classification results for proper biharmonic
submanifolds in spheres was initiated with the following
characterization theorem.
\begin{theorem}[\cite{O2}]\label{th: bih subm S^n}
{\rm (i)} The canonical inclusion $\phi:M^m\to\mathbb{S}^n$ of a
submanifold $M$ in an $n$-dimensional unit Euclidean sphere is
biharmonic if and only if
\begin{equation}\label{caract_bih_spheres}
\left\{
\begin{array}{l}
\ \Delta^\perp H+\trace B(\cdot,A_H(\cdot))-mH=0
\\ \mbox{} \\
\ 4\trace A_{\nabla^\perp_{(\cdot)}H}(\cdot)+m\grad(\vert H
\vert^2)=0,
\end{array}
\right.
\end{equation}
where $A$ denotes the Weingarten operator, $B$ the second
fundamental form, $H$ the mean curvature vector field,
$\nabla^\perp$ and $\Delta^\perp$ the connection and the Laplacian
in the normal bundle of $M$ in $\mathbb{S}^n$, and $\grad$ denotes
the gradient on $M$.
If $M$ is a submanifold with parallel mean curvature vector field in
$\mathbb{S}^n$, then $M$ is biharmonic if and only if $\trace
B(\cdot,A_H(\cdot))=mH$.
{\rm (ii)} A hypersurface $M$ with nowhere zero mean curvature
vector field in $\mathbb{S}^{m+1}$ is biharmonic if and only if
\begin{equation}\label{eq: caract_bih_hipersurf_spheres}
\left\{
\begin{array}{l}
\Delta^\perp H-(m-|A|^2)H=0
\\ \mbox{} \\
\ 2A(\grad (|H|))+m|H|\grad(|H|)=0.
\end{array}
\right.
\end{equation}
If $M$ is a non-zero constant mean curvature hypersurface in
$\mathbb{S}^{m+1}$, then $M$ is proper biharmonic if and only if
$|A|^2=m$.
\end{theorem}
We note that the compact minimal, i.e. $H=0$, hypersurfaces with
$|A|^2=m$ in $\mathbb{S}^{m+1}$ are the Clifford tori
$\mathbb{S}^k(\sqrt{k/m})\times\mathbb{S}^{m-k}(\sqrt{(m-k)/m})$,
$1\leq k\leq m-1$ (see \cite{CCK}).
Before presenting the basic examples of proper biharmonic
hypersurfaces in spheres, together with some of their geometric
properties, we recall the following definition (see, for example,
\cite{BYC1}), which shall be used throughout the paper.
\begin{defi}
An isometric immersion of a compact manifold $M$ in
$\mathbb{R}^{n}$, $\varphi:M\to \mathbb{R}^{n}$, is called of {\it
k-type} if its spectral decomposition contains exactly $k$ non-zero
terms, excepting the center of mass $\varphi_0=\frac{1}{\mathrm{Vol}
(M)}\int_M\varphi\,v_g$. More precisely,
$$
\varphi=\varphi_0+\sum_{t=p}^q \varphi_t,
$$
where $\Delta\varphi_t=\lambda_t\varphi_t$ and
$0<\lambda_1<\lambda_2<\cdots\uparrow\infty$.
The pair $[p,q]$ is called {\it the order of the immersion}
$\varphi:M\to \mathbb{R}^{n}$.
\end{defi}
\newpage
\subsection{The main examples of proper biharmonic submanifolds in spheres}\label{sec:2.1}
\subsubsection*{The hypersphere $\mathbb{S}^{m}(1/{\sqrt
2})\subset\mathbb{S}^{m+1}$} \quad
\noindent Consider $\mathbb{S}^{m}(a)=\Big\{(x^1,\ldots, x^{m},
x^{m+1},b)\in\mathbb{R}^{m+2}: |x|=a\Big\}\subset\mathbb{S}^{m+1}$,
where $a^2+b^2=1$. If $H$ is the mean curvature vector field of
$\mathbb{S}^{m}(a)$ in $\mathbb{S}^{m+1}$, one gets $\nabla^\perp
H=0$, $|H|=\frac{|b|}{a}$ and $|A|^2=m\frac{b^2}{a^2}$.
Theorem \ref{th: bih subm S^n} implies that $\mathbb{S}^{m}(a)$ is
proper biharmonic in $\mathbb{S}^{m+1}$ if and only if $a=1/{\sqrt
2}$ (see \cite{CMO1}).
\subsubsection*{The generalized Clifford torus $\mathbb{S}^{m_1}(1/{\sqrt 2})
\times\mathbb{S}^{m_2}(1/{\sqrt 2})\subset\mathbb{S}^{m+1}$}\quad
\noindent The generalized Clifford torus,
$M=\mathbb{S}^{m_1}(1/{\sqrt 2})\times\mathbb{S}^{m_2}(1/{\sqrt
2})$, $m_1+m_2=m$, $m_1\neq m_2$, was the first example of proper
biharmonic submanifold in $\mathbb{S}^{m+1}$ (see \cite{J1}).
Consider
\begin{eqnarray*}
M&=&\Big\{(x^1,\ldots,x^{m_1+1}, y^1,\ldots,
y^{m_2+1})\in\mathbb{R}^{m+2}: |x|=a_1,
|y|=a_2\Big\}\\
&=&\mathbb{S}^{m_1}(a_1)\times\mathbb{S}^{m_2}(a_2)\subset\mathbb{S}^{m+1},
\end{eqnarray*}
where $a_1^2+a_2^2=1$. Then $\nabla^\perp H=0$,
$|H|=\frac{1}{a_1a_2m}|a_2^2m_1-a_1^2m_2|$ and
$|A|^2=\left(\frac{a_2}{a_1}\right)^2m_1+\left(\frac{a_1}{a_2}\right)^2m_2$.
From Theorem \ref{th: bih subm S^n} it follows that $M$ is proper
biharmonic in $\mathbb{S}^{m+1}$ if and only if $a_1=a_2=1/{\sqrt
2}$ and $m_1\neq m_2$ (see, also, \cite{CMO2}).
Inspired by these basic examples, two methods for constructing
proper biharmonic submanifolds of codimension higher than one in
$\mathbb{S}^n$ were given.
\begin{theorem}[\cite{CMO2}]\label{th: rm_minim}
Let $M$ be a minimal submanifold of\,
$\mathbb{S}^{n-1}(a)\subset\mathbb{S}^n$. Then $M$ is proper
biharmonic in $\mathbb{S}^{n}$ if and only if $a=1/{\sqrt 2}$.
\end{theorem}
\begin{remark}
(i) This result, called the {\it composition property}, proved to be
quite useful for the construction of proper biharmonic submanifolds
in spheres. For instance, it implies the existence of closed
orientable embedded proper biharmonic surfaces of arbitrary genus in
$\mathbb{S}^4$ (see \cite{CMO2}).
(ii) All minimal submanifolds of $\mathbb{S}^{n-1}(1/{\sqrt
2})\subset\mathbb{S}^n$ are pseudo-umbilical, i.e. $A_H=|H|^2\Id$,
with pa\-rallel mean curvature vector field in $\mathbb{S}^n$ and
$|H|=1$.
(iii) Denote by $\phi:\mathbb{S}^{m}(1/{\sqrt
2})\to\mathbb{S}^{m+1}$ the inclusion of $\mathbb{S}^{m}(1/{\sqrt
2})$ in $\mathbb{S}^{m+1}$ and by
$\mathbf{i}:\mathbb{S}^{m+1}\to\mathbb{R}^{m+2}$ the canonical
inclusion. Let $\varphi:\mathbb{S}^{m}(1/{\sqrt
2})\to\mathbb{R}^{m+2}$, $\varphi=\mathbf{i}\circ \phi$, be the
inclusion of $\mathbb{S}^{m}(1/{\sqrt 2})$ in $\mathbb{R}^{m+2}$.
Then
\begin{equation}\label{eq: spectr_decomp_min}
\varphi=\varphi_0+\varphi_p,
\end{equation}
where $\varphi_0, \varphi_p:\mathbb{S}^{m}(1/{\sqrt
2})\to\mathbb{R}^{m+2}$, $ \varphi_0(x,1/\sqrt 2)=(0,1/\sqrt 2)$,
$\varphi_p(x,1/\sqrt 2)=(x,0)$ and $ \Delta\varphi_p=2m\varphi_p$.
Thus $\mathbb{S}^{m}(1/{\sqrt 2})$ is a $1$-type submanifold of
$\mathbb{R}^{m+2}$ with center of mass in $\varphi_0=(0,1/\sqrt 2)$
and eigenvalue $\lambda_p=2m$, which is the first eigenvalue of the
Laplacian on $\mathbb{S}^{m}(1/{\sqrt 2})$, i.e. $p=1$.
Moreover, it is not difficult to verify that all minimal
submanifolds in $\mathbb{S}^{m}(1/{\sqrt
2})\subset\mathbb{S}^{m+1}$, as submanifolds in $\mathbb{R}^{m+2}$,
have the spectral decomposition given by \eqref{eq:
spectr_decomp_min}.
\end{remark}
Non pseudo-umbilical examples were also produced by proving the
following {\it product composition property}.
\begin{theorem}[\cite{CMO2}]\label{th:hipertor}
Let $M_1^{m_1}$ and $M_2^{m_2}$ be two minimal submanifolds of
$\mathbb{S}^{n_1}(a_1)$ and $\mathbb{S}^{n_2}(a_2)$, respectively,
where $n_1+n_2=n-1$, $a_1^2+a_2^2=1$. Then $M_1\times M_2$ is proper
biharmonic in $\mathbb{S}^n$ if and only if $a_1=a_2=1/{\sqrt 2}$
and $m_1\neq m_2$.
\end{theorem}
\begin{remark}
(i) The proper biharmonic submanifolds of $\mathbb{S}^n$ constructed
as above are not pseudo-umbilical, but they still have parallel mean
curvature vector field, thus constant mean curvature, and
$|H|=\frac{|m_2-m_1|}{m_1+m_2}\in (0,1)$.
(ii) Let $\varphi:\mathbb{S}^{m_1}(1/{\sqrt 2})
\times\mathbb{S}^{m_2}(1/{\sqrt 2})\to\mathbb{R}^{m+2}$ be the
inclusion of $\mathbb{S}^{m_1}(1/{\sqrt 2})
\times\mathbb{S}^{m_2}(1/{\sqrt 2})$ in $\mathbb{R}^{m+2}$,
$m_1<m_2$, $m_1+m_2=m$. Then
\begin{equation}\label{eq: spectr_decomp_Cliff}
\varphi=\varphi_p+\varphi_q,
\end{equation}
where $\varphi_p, \varphi_q:\mathbb{S}^{m_1}(1/{\sqrt 2})
\times\mathbb{S}^{m_2}(1/{\sqrt 2})\to\mathbb{R}^{m+2}$,
$\varphi_p(x,y)=(x,0)$, $\varphi_q(x,y)=(0,y)$ and $
\Delta\varphi_p=2m_1\varphi_p$, $ \Delta\varphi_q=2m_2\varphi_q$.
Thus $\mathbb{S}^{m_1}(1/{\sqrt 2}) \times\mathbb{S}^{m_2}(1/{\sqrt
2})$ is a $2$-type submanifold of $\mathbb{R}^{m+2}$ with
eigenvalues $\lambda_p=2m_1$ and $\lambda_q=2m_2$, and it is
mass-symmetric, i.e. it has center of mass in the origin.
Since the eigenvalues of the torus $\mathbb{S}^{m_1}(1/{\sqrt 2})
\times\mathbb{S}^{m_2}(1/{\sqrt 2})$ are obtained as the sum of
eigenvalues of the spheres $\mathbb{S}^{m_1}(1/{\sqrt 2})$ and
$\mathbb{S}^{m_2}(1/{\sqrt 2})$, we conclude that $p=1$. Also,
$q=2$, i.e. $\mathbb{S}^{m_1}(1/{\sqrt 2})
\times\mathbb{S}^{m_2}(1/{\sqrt 2})$ has order $[1,2]$ in
$\mathbb{R}^{m+2}$, if and only if $m_2\leq 2(m_1+1)$. Note that
this holds, for example, for $\mathbb{S}^{1}(1/{\sqrt 2})
\times\mathbb{S}^{2}(1/{\sqrt 2})\subset\mathbb{S}^4$.
Moreover, it can be easily proved that all proper biharmonic
submanifolds in $\mathbb{S}^{m+1}$ obtained by means of the product
composition property, as submanifolds in $\mathbb{R}^{m+2}$, have
the spectral decomposition given by \eqref{eq: spectr_decomp_Cliff}.
\end{remark}
\subsubsection*{Other examples of proper biharmonic immersed submanifolds in
spheres}\quad
\noindent In \cite{S1} and \cite{AEMS} the authors studied the
proper biharmonic Legendre immersed surfaces and the proper
biharmonic $3$-di\-men\-sional anti-invariant immersed submanifolds
in Sasakian space forms. They obtained the explicit representations
of such submanifolds in the unit Euclidean $5$-dimensional sphere
$\mathbb{S}^5$.
\begin{theorem}[\cite{S1}]\label{th:Chen_T^2 in S^5}
Let $\phi:M^2\to\mathbb{S}^5$ be a proper biharmonic Legendre
immersion. Then the position vector field
$\varphi=\mathbf{i}\circ\phi=\varphi(u,v)$ of $M$ in $\mathbb{R}^6$
is given by
$$
\varphi(u,v)=\frac{1}{\sqrt 2}(e^{iu},ie^{-iu}\sin\sqrt 2 v,
ie^{-iu}\cos\sqrt 2 v),
$$
where $\mathbf{i}:\mathbb{S}^5\to\mathbb{R}^6$ is the canonical
inclusion.
\end{theorem}
\begin{remark}
The map $\phi$ is a full proper biharmonic Legendre embedding of a
$2$-dimensional flat torus $\mathbb{R}^2/\Lambda$ into
$\mathbb{S}^5$, where the lattice $\Lambda$ is generated by
$(2\pi,0)$ and $(0,\sqrt{2}\pi)$. It has constant mean curvature
$|H|=1/2$, it is not pseudo-umbilical and its mean curvature vector
field is not parallel. Moreover, $\varphi=\varphi_p+\varphi_{q}$,
where
$$
\varphi_{p}(u,v)=\frac{1}{\sqrt 2}(e^{iu},0,0)
$$
$$
\varphi_{q}(u,v)=\frac{1}{\sqrt 2}(0,ie^{-iu}\sin\sqrt 2 v,
ie^{-iu}\cos\sqrt 2 v)
$$
and $ \Delta \varphi_{p}=\varphi_{p}$, $ \Delta
\varphi_{q}=3\varphi_{q}$. Thus $\varphi$ is a $2$-type immersion in
$\mathbb{R}^6$ with eigenvalues $1$ and $3$. In this case, $p=1$ and
$q=3$, i.e. $\varphi$ is a $[1,3]$-order immersion in
$\mathbb{R}^6$.
\end{remark}
\begin{theorem}[\cite{AEMS}]
Let $\phi:M^3\to\mathbb{S}^5$ be a proper biharmonic anti-invariant
immersion. Then the position vector field
$\varphi=\mathbf{i}\circ\phi=\varphi(u,v,w)$ of $M$ in
$\mathbb{R}^6$ is given by
$$
\varphi(u,v,w)=\frac{1}{\sqrt 2} e^{iw}(e^{iu},ie^{-iu}\sin\sqrt 2
v, ie^{-iu}\cos\sqrt 2 v).
$$
\end{theorem}
\begin{remark}
The map $\phi$ is a full proper biharmonic anti-invariant immersion
from a $3$-dimen\-sional flat torus $\mathbb{R}^3/\Lambda$ into
$\mathbb{S}^5$, where the lattice $\Lambda$ is generated by
$(2\pi,0,0)$, $(0,\sqrt{2}\pi,0)$ and $(0,0,2\pi)$. It has constant
mean curvature $|H|=1/3$, is not pseudo-umbilical, but its mean
curvature vector field is parallel. Moreover,
$\varphi=\varphi_{p}+\varphi_{q}$, where
$$
\varphi_{p}(u,v,w)=\frac{1}{\sqrt 2} e^{iw}(e^{iu},0,0)
$$
$$
\varphi_{q}(u,v,w)=\frac{1}{\sqrt 2} e^{iw}(0,ie^{-iu}\sin\sqrt 2 v,
ie^{-iu}\cos\sqrt 2 v)
$$
and $\Delta \varphi_{p}=2\varphi_{p}$, $\Delta
\varphi_{q}=4\varphi_{q}$. Thus $\varphi$ is a $2$-type submanifold
of $\mathbb{R}^6$ with eigenvalues $2$ and $4$. It is easy to verify
that $\varphi$ is a $[2,4]$-order immersion in $\mathbb{R}^6$.
Since the immersion $\phi$ has parallel mean curvature vector field,
one could ask weather its image arises by means of the product
composition property. Indeed, it can be proved that, up to an
orthogonal transformation of $\mathbb{R}^6$ which commutes with the
usual complex structure, $\phi$ covers twice the proper biharmonic
submanifold $\mathbb{S}^1(1/\sqrt 2)\times \mathbb{S}^1(1/2)\times
\mathbb{S}^1(1/2)\subset\mathbb{S}^5$.
\end{remark}
\subsection{Classification results}
Some of the techniques used in order to obtain non-existence results
in the case of non-positively curved space forms were adapted to the
study of proper biharmonic submanifolds in spheres. Thus, in order
to approach the classification problem for proper biharmonic
hypersurfaces in spheres, the study was divided according to the
number of principal curvatures. For submanifolds of higher
codimension, supplementary conditions on the mean curvature vector
field were imposed. This leaded to a series of rigidity results,
which we enumerate below.
\subsubsection{Proper biharmonic hypersurfaces}
First, if $M$ is a proper biharmonic umbilical hypersurface in
$\mathbb{S}^{m+1}$, i.e. all its principal curvatures are equal,
then it is an open part of $\mathbb{S}^m(1/\sqrt{2})$.
Afterwards, proper biharmonic hypersurfaces with at most two
distinct principal curvatures were considered.
\begin{theorem}[\cite{BMO1}]\label{th: curb_med_const_2_curv_princ}
Let $M$ be a hypersurface with at most two distinct principal
curvatures in $\mathbb{S}^{m+1}$. If $M$ is proper biharmonic in
$\mathbb{S}^{m+1}$, then it has constant mean curvature.
\end{theorem}
By using this result, the classification of such hypersurfaces was
obtained.
\begin{theorem}[\cite{BMO1}]\label{th: classif_hypersurf_2_curv_princ}
Let $M^m$ be a hypersurface with at most two distinct principal
curvatures in $\mathbb{S}^{m+1}$. Then $M$ is proper biharmonic if
and only if it is an open part of $\mathbb{S}^{m}(1/\sqrt{2})$ or of
$\mathbb{S}^{m_1}(1/\sqrt{2})\times \mathbb{S}^{m_2}(1/\sqrt{2})$,
$m_1+m_2=m$, $m_1\neq m_2$.
\end{theorem}
Then followed the case of biharmonic hypersurfaces with at most
three distinct principal curvatures. In order to solve this problem,
the following property of proper biharmonic hypersurfaces in spheres
was needed.
\begin{proposition}[\cite{BMO1}]\label{th: curb_scal_hyp}
Let $M$ be a proper biharmonic hypersurface with constant mean
curvature $|H|$ in $\mathbb{S}^{m+1}$, $m\geq 2$. Then $M$ has
positive constant scalar curvature $ s=m^2(1+|H|^2)-2m$.
\end{proposition}
First a non-existence result was obtained.
\begin{theorem}[\cite{BMO3}]\label{th: non-exist_isoparam_3princ}
There exist no compact proper biharmonic hypersurfaces of constant
mean curvature and with three distinct principal curvatures
everywhere in the unit Euclidean sphere.
\end{theorem}
The proof relies on the fact that such hypersurfaces are
isoparametric, i.e. have constant principal curvatures with constant
multiplicities, and then, on the explicit expressions of the
principal curvatures.
We note that, in \cite{IIU}, the authors classified the
isoparametric proper biharmonic hypersurfaces in spheres.
\begin{theorem}[\cite{IIU}]
Let $M^m$ be an isoparametric hypersurface in $\mathbb{S}^{m+1}$.
Then $M$ is proper biharmonic if and only if it is an open part of
$\mathbb{S}^{m}(1/\sqrt{2})$ or of
$\mathbb{S}^{m_1}(1/\sqrt{2})\times \mathbb{S}^{m_2}(1/\sqrt{2})$,
$m_1+m_2=m$, $m_1\neq m_2$.
\end{theorem}
Compact proper biharmonic hypersurfaces in $\mathbb{S}^4$ were fully
classified.
\begin{theorem}[\cite{BMO3}]\label{th: hyper_S4}
The only compact proper biharmonic hypersurfaces in $\mathbb{S}^4$
are the hypersphere $\mathbb{S}^3(1/\sqrt{2})$ and the torus
$\mathbb{S}^1(1/\sqrt{2})\times\mathbb{S}^2(1/\sqrt{2})$.
\end{theorem}
The proof uses the fact that a proper biharmonic hypersurface in
$\mathbb{S}^4$ has constant mean curvature, and thus constant scalar
curvature, and a result in \cite{CHA1}.
\subsubsection{Proper biharmonic submanifolds of
codimension higher than one}
In higher codimension, it was proved that the proper biharmonic
pseudo-umbilical submanifolds, of dimension different from four, in
spheres have constant mean curvature. This result leaded to the
classification of proper biharmonic pseudo-umbilical submanifolds of
codimension two.
\begin{theorem}[\cite{BMO1}]\label{th: classif_pseudo_umb_codim2}
Let $M^m$ be a pseudo-umbilical submanifold in $\mathbb{S}^{m+2}$,
$m\neq 4$. Then $M$ is proper biharmonic in $\mathbb{S}^{m+2}$ if
and only if it is minimal in $\mathbb{S}^{m+1}(1/\sqrt{2})$.
\end{theorem}
Surfaces with parallel mean curvature vector field in $\mathbb{S}^n$
were also investigated.
\begin{theorem}[\cite{BMO1}]\label{th: classif_surf_parallelH}
Let $M^2$ be a surface with parallel mean curvature vector field in
$\mathbb{S}^n$. Then $M$ is proper biharmonic in $\mathbb{S}^n$ if
and only if it is minimal in $\mathbb{S}^{n-1}(1/\sqrt{2})$.
\end{theorem}
The above two results allowed the classification of proper
biharmonic constant mean curvature surfaces in $\mathbb{S}^4$.
\begin{theorem}[\cite{BO}]
The only proper biharmonic constant mean curvature surfaces in
$\mathbb{S}^4$ are the minimal surfaces in
$\mathbb{S}^3(1/\sqrt{2})$.
\end{theorem}
\begin{proof}
The key of the proof is to show that $\nabla^\perp H=0$, in order to
be able to apply Theorem \ref{th: classif_surf_parallelH}.
We assume that $\nabla^\perp H\neq 0$ and consider $\{E_1,E_2\}$
tangent to $M$ and $\{E_3, E_4=\frac{H}{|H|}\}$ normal to $M$, such
that $\{E_1,E_2,E_3,E_4\}$ constitutes a local orthonormal frame
field on $\mathbb{S}^4$. Using the connection $1$-forms w.r.t.
$\{E_1,E_2,E_3,E_4\}$ and the tangent part of the biharmonic
equation \eqref{caract_bih_spheres}, we get $A_4=0$, where $A_4$ is
the shape operator in direction of $E_4$.
Then we identify two cases:\\
(i) If $A_3=|H|\Id$, then $M$ is pseudo-umbilical and, by Theorem
\ref{th: classif_pseudo_umb_codim2}, it is minimal in
$\mathbb{S}^3(1/\sqrt{2})$. This implies that $\nabla^\perp
H=0$, and we have a contradiction.\\
(ii) If $A_3\neq|H|\Id$, then the Gauss and Codazzi equations lead
us to a contradiction and we conclude.
\end{proof}
\section{Properties of proper biharmonic submanifolds in spheres}\label{sec:3}
We begin this section by presenting some general properties of
proper biharmonic submanifolds with parallel mean curvature vector
field in spheres, which are consequences of
\eqref{caract_bih_spheres} and of the Codazzi and Gauss equations,
respectively.
\begin{proposition}\label{prop: prop||MC}
Let $M$ be a proper biharmonic submanifold with parallel mean
curvature vector field in $\mathbb{S}^n$. Then
\begin{itemize}
\item[(i)] $|A_H|^2=m|H|^2$, and it is constant,
\item[(ii)] $\trace \nabla A_H=0$,
\item[(iii)] $\langle\trace (\nabla^\perp B)(X,\cdot, A_H(\cdot)),H\rangle=\langle\trace (\nabla^\perp B)(\cdot,X, A_H(\cdot)),H\rangle=0$, for
all $X\in C(TM)$.
\end{itemize}
\end{proposition}
\begin{proposition}\label{prop: prop||MC_from_Gauss_eq}
Let $M$ be a proper biharmonic submanifold with parallel mean
curvature vector field in $\mathbb{S}^n$. Let $p$ be an arbitrary
point on $M$ and consider $\{e_i\}_{i=1}^m$ to be an orthonormal
basis of eigenvectors for $A_H$ in $T_pM$ . Denote by
$\{a_i\}_{i=1}^m$ the eigenvalues of $A_H$ at $p$. Then, at $p$,
\begin{itemize}
\item[(i)] $m|H|^2=\displaystyle{\sum_{i=1}^m a_i=\sum_{i=1}^m (a_i)^2}$,
\item[(ii)] $\displaystyle{(2m-1)m|H|^2=\frac{1}{2}\sum_{i,j=1}^m(a_i+a_j)(K_{ij}+|B(e_i,e_j)|^2)}$,
\item[(iii)] $\displaystyle{(m-1+m|H|^2)m|H|^2=\sum_{i,j=1}^m a_i
a_j(K_{ij}+|B(e_i,e_j)|^2)}$,
\end{itemize}
where $K_{ij}$ denotes the sectional curvature of the $2$-plane
tangent to $M$ generated by $e_i$ and $e_j$.
\end{proposition}
For what concerns proper biharmonic constant mean curvature
submanifolds in spheres, a partial classification result was
obtained.
\begin{theorem}[\cite{O3}]\label{th: classif_bih const mean}
Let $M$ be a proper biharmonic submanifold with constant mean
curvature in $\mathbb{S}^n$. Then $|H|\in(0,1]$. Moreover, if
$|H|=1$, then $M$ is a minimal submanifold of a hypersphere
$\mathbb{S}^{n-1}(1/\sqrt{2})\subset\mathbb{S}^n$.
\end{theorem}
Also, the properties regarding the type of the main examples
previously presented are not casual. In fact, Theorem \ref{th:
classif_bih const mean} was extended by establishing a general link
between compact proper biharmonic constant mean curvature
submanifolds in spheres and finite type submanifolds in the
Euclidean space.
\begin{theorem}[\cite{BMO1}]\label{th: tip_subv} Let $M^m$ be a compact constant
mean curvature, $|H|\in(0,1]$, sub\-manifold in $\mathbb{S}^n$.
Then $M$ is proper biharmonic if and only if
either
\begin{itemize}
\item[(i)] $|H|=1$ and $M$ is a 1-type
submanifold of $\mathbb{R}^{n+1}$ with eigenvalue $\lambda=2m$ and
center of mass of norm equal to $1/\sqrt{2}$,
\end{itemize}
or
\begin{itemize}
\item[(ii)] $|H|\in (0,1)$ and $M$ is a mass-symmetric $2$-type
submanifold of $\mathbb{R}^{n+1}$ with eigenvalues
$\lambda_{p}=m(1-|H|)$ and $\lambda_{q}=m(1+|H|)$.
\end{itemize}
\end{theorem}
This can be further used in order to obtain some necessary
conditions that compact proper biharmonic submanifolds with constant
mean curvature in spheres must fulfill.
\begin{corollary}\label{cor: conseq_type}
Let $M^m$ be a compact proper biharmonic constant mean curvature,
$|H|\in(0,1)$, sub\-manifold in $\mathbb{S}^n$. Then
\begin{itemize}
\item[(i)] $\lambda_1\leq m(1-|H|)$, where $\lambda_1$ is the first non-zero eigenvalue
of the Laplacian on $M$,
\item[(ii)] if $\rm{Ricci}(X,X)\geq cg(X,X)$, for all $X\in C(TM)$,
where $c>0$, we have $c\leq (m-1)(1-|H|)$.
\end{itemize}
\end{corollary}
\begin{proof}
(i) From Theorem \ref{th: tip_subv} it follows that the inclusion
map of $M$ in $\mathbb{R}^{n+1}$, $\varphi:M\to\mathbb{R}^{n+1}$,
decomposes as $\varphi=\varphi_p+\varphi_q$, where $\Delta
\varphi_p=\lambda_p\varphi_p$, $\Delta
\varphi_q=\lambda_q\varphi_q$, $\lambda_p=m(1-|H|)$ and
$\lambda_q=m(1+|H|)$. Conclusively, $m(1-|H|)$ is a non-zero
eigenvalue of the Laplacian on $M$, and thus $\lambda_1\leq
m(1-|H|)$.
(ii) The condition $\rm{Ricci}(X,X)\geq cg(X,X)$, for all $X\in
C(TM)$, implies, by a well-known result of Lichnerowicz (see
\cite{BGM}), that $\lambda_1\geq \frac{m}{m-1}c$. This, together
with (i), leads to the conclusion.
\end{proof}
We shall need the following result in order to obtain a refinement
of Theorem \ref{th: classif_bih const mean}.
\begin{theorem}[\cite{HZL}]\label{th: Haizhong Li}
Let $M$ be a compact hypersurface with constant normalized scalar
curvature $r=\frac{s}{m(m-1)}$ in $\mathbb{S}^{m+1}$. If
\begin{itemize}
\item[(i)] $r\geq 1$,
\item[(ii)] the squared norm $|B|^2$ of the second fundamental
form of $M$ satisfies
\begin{equation}\label{eq: eval_Li}
m(r-1)\leq |B|^2\leq(m-1)\frac{m(r-1)+2}{m-2}+\frac{m-2}{m(r-1)+2},
\end{equation}
\end{itemize}
then either $ |B|^2=m(r-1)$ and $M$ is a totally umbilical
hypersurface; or
$$
|B|^2=(m-1)\frac{m(r-1)+2}{m-2}+\frac{m-2}{m(r-1)+2}
$$
and $M=\mathbb{S}^1(\sqrt{1-c^2})\times\mathbb{S}^{m-1}(c)$, with
$c^2=\frac{m-2}{mr}$.
\end{theorem}
We get the following theorem.
\begin{theorem}{\label{th: raf_tip_hyper}}
Let $M^m$, $m\geq 4$, be a compact proper biharmonic constant mean
curvature hypersurface in $\mathbb{S}^{m+1}$. Then
$|H|\in(0,\frac{m-2}{m}]\cup\{1\}$. Moreover,
\begin{itemize}
\item[(i)] $|H|=1$ if and only if $M=\mathbb{S}^m(1/\sqrt{2})$,
\end{itemize}
and
\begin{itemize}
\item[(ii)] $|H|=\frac{m-2}{m}$ if and only if
$M=\mathbb{S}^1(1/\sqrt{2})\times\mathbb{S}^{m-1}(1/\sqrt{2})$.
\end{itemize}
\end{theorem}
\begin{proof}
Since $M$ is proper biharmonic with constant mean curvature $|H|$,
Theorem \ref{th: bih subm S^n} implies that
\begin{equation}\label{eq: normB^2}
|B|^2=|A|^2=m.
\end{equation}
We shall denote, for convenience, $t=m|H|^2-1$.
Suppose that $|H|\in(\frac{m-2}{m},1)$, which is equivalent to
$t\in\left(\frac{(m-4)(m-1)}{m},m-1\right)$. By using Proposition
\ref{th: curb_scal_hyp}, we obtain that
\begin{equation}\label{eq: r}
r=1+\frac{t}{m-1}.
\end{equation}
Condition (i) of Theorem \ref{th: Haizhong Li} is equivalent to
$t\geq 0$, which is satisfied. Also, using \eqref{eq: normB^2},
since $t< m-1$, the first inequality of \eqref{eq: eval_Li} is
satisfied. The second inequality of \eqref{eq: eval_Li} becomes
$$
0\leq mt^2-(m^2-6m+4)t-(m-4)(m-1)
$$
and it is satisfied since $t>\frac{(m-4)(m-1)}{m}$. We are now in
the hypotheses of Theorem \ref{th: Haizhong Li} and we get $r=2$,
i.e. $|H|=1$, or $r=\frac{2(m-2)}{m}$, i.e. $|H|=\frac{m-2}{m}$,
thus we have a contradiction. Conclusively, we obtain
$|H|\in(0,\frac{m-2}{m}]\cup\{1\}$.
Case (i) is given by Theorem \ref{th: classif_bih const mean}. It
can also be proved by using Theorem \ref{th: Haizhong Li}.
For (ii), as we have already seen, if
$M=\mathbb{S}^1(1/\sqrt{2})\times\mathbb{S}^{m-1}(1/\sqrt{2})$, then
$|H|=\frac{m-2}{m}$. Conversely, if $|H|=\frac{m-2}{m}$, then
$r=\frac{2(m-2)}{m}$, and we are in the hypotheses of Theorem
\ref{th: Haizhong Li}, thus we conclude.
\end{proof}
\section{Open problems}
In view of all the above results the following conjectures were
proposed.
\begin{conjecture}[\cite{BMO1}]
The only proper biharmonic hypersurfaces in $\mathbb{S}^{m+1}$ are
the open parts of hyperspheres $\mathbb{S}^{m}(1/\sqrt{2})$ or of
generalized Clifford tori $\mathbb{S}^{m_1}(1/\sqrt{2})\times
\mathbb{S}^{m_2}(1/\sqrt{2})$, $m_1+m_2=m$, $m_1\neq m_2$.
\end{conjecture}
\begin{conjecture}[\cite{BMO1}]
Any proper biharmonic submanifold in $\mathbb{S}^n$ has constant
mean curvature.
\end{conjecture}
\section{Further remarks}
There is an interesting link between the proper biharmonic
hypersurfaces in $\mathbb{S}^{m+1}$ and the $II$-minimal
hypersurfaces. We briefly recall here the notion of $II$-minimal
hypersurfaces (see \cite{SHSV}). We denote by $\mathcal{E}$ the set
of all hypersurfaces in a semi-Riemannian manifold $(N,h)$ for which
the first, as well as the second, fundamental form is a
semi-Riemannian metric. The critical points of the area functional
of the second fundamental form
$$
\mathrm{Area}_{II}:\mathcal{E}\to\mathbb{R}, \qquad
\mathrm{Area}_{II}(M)=\int_M \sqrt{|\mathrm{det} A|}\,v_g
$$
are called $II$-minimal. According to \cite{SHSV}, we have
\begin{proposition}
Let $\mathbb{S}^m(a)$ be the hypersphere of radius $a\in(0,1)$ in
$\mathbb{S}^{m+1}$. The following are equivalent
\begin{itemize}
\item[(i)] $\mathbb{S}^m(a)$ is proper biharmonic,
\item[(ii)] $\mathbb{S}^m(a)$ is $II$-minimal,
\item[(iii)] $a=1/\sqrt{2}$.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $M=\mathbb{S}^{m_1}(a_1)\times \mathbb{S}^{m_2}(a_2)$,
$a_1\in(0,1)$, $a_1^2+a_2^2=1$, be the generalized Clifford torus in
$\mathbb{S}^{m+1}$, $m_1+m_2=m$. The following are equivalent
\begin{itemize}
\item[(i)] $M$ is proper biharmonic,
\item[(ii)] $M$ is $II$-minimal and non-minimal,
\item[(iii)] $a_1=a_2=1/\sqrt{2}$ and $m_1\neq m_2$.
\end{itemize}
\end{proposition}
\section*{Acknowledgements}
The authors would like to thank Professors I. Dimitric and S.
Verpoort for helpful suggestions and discussions.
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\section{Introduction}
\label{Introduction}
Superconducting qubits are a leading platform for quantum computing~\cite{zhang2020,arute2019}. This has been driven, in part, by improvements in coherence times over five orders of magnitude since the realization of coherent dynamics in a cooper pair box~\cite{nakamura1999}. However, further improving coherence times remains crucial for enhancing the scope of noisy superconducting quantum processors as well as the long term challenge of building a fault tolerant quantum computer. Recent advances~\cite{hong2020,kandala2020,hashim2020,Foxen2020} in two-qubit gate control have placed their fidelities at the cusp of their coherence limit, implying that improvements in coherence could directly drive gate fidelities past the fault tolerant threshold. In this context, coherence stability and its impact on multi-qubit device performance is also an important theme, since superconducting qubits have been shown to display large temporal fluctuations in their energy relaxation times $T_1$\cite{muller2015,klimov_fluctuations_2018,kandala2019,burnett2019,schlor2019}. This places additional challenges for benchmarking the coherence properties of these devices~\cite{burnett2019},
and also for error mitigation strategies such as zero noise extrapolation~\cite{kandala2019}.
The fluctuations of qubit $T_1$ are often attributed to resonant couplings with two level systems (TLSs) that have been historically studied in the context of amorphous solids \cite{phillips1972,muller_towards_2019} and their low temperature properties. More recently, TLSs have attracted renewed interest due to their effect on the coherence properties of superconducting quantum circuits\cite{martinis2005,grabovskij2012strain,Barends2013,klimov_fluctuations_2018,lisenfeld2019electric,burnett2019,schlor2019}, and are attributed to defects in amorphous materials at surfaces, interfaces, and the Josephson junction tunnel barrier. Frequency resolved measurements of $T_1$ in flux and stress tunable devices\cite{Barends2013,klimov_fluctuations_2018,lisenfeld2019electric} have also displayed fluctuations, suggesting an environment of TLSs with varying coupling strengths around the qubit frequency. The variability of $T_1$ over time is explained\cite{muller_towards_2019,klimov_fluctuations_2018}, at least in part, by temporal fluctuations in this frequency environment, associated with the spectral diffusion of the TLSs \cite{black_spectral_1977, phillips1972}.
Furthermore, two-qubit gates that involve frequency excursions~\cite{kandala2020,stehlik2021,klimov_fluctuations_2018} can also interact with TLS near the qubit frequency leading to additional incoherent error. The fluctuations in TLS peak positions, therefore, can also introduce fluctuations in two qubit fidelity. Spectroscopy of defect TLS is, therefore, central to understanding the short and long time $T_1$ and gate fidelity of qubits.
\begin{figure}
\centering
\includegraphics[width=90mm, clip,trim = 10.0mm 85.0mm 0.0mm 7.0mm]{Fig1v5.pdf}
\caption{(a) Map of a 20 qubit device including the qubit number, two-qubit connectivity and the qubit frequency. (b) Box and whisker plot of $T_1$'s measured daily
over approximately nine months.
The box covers the first to third quartile, the vertical line indicates the median and the whiskers are drawn to the maximum or minimum values that fall within 1.5 times the interquartile range. All other points are outliers.}
\label{fig:Figure1}
\end{figure}
Single Josephson junction transmons with fixed frequency couplings represent a successful device architecture achieving networks of over 60 qubits~\cite{zhang2020} with all microwave control and state of the art device coherence. The single junction configuration offers advantages such as reduced sensitivity to flux noise, while preserving the transmon charge insensitivity and also reducing system complexity with fewer control inputs. However, there is little TLS spectroscopy of single junction transmons because of the limited tunability, despite the central importance of understanding the TLS environment both for device and process characterization.
\begin{figure}
\centering
\includegraphics[width=8.5 cm, clip,trim = 0.0mm 5.0mm 0.0mm 0.0mm]{Fig2_final.pdf}
\caption{(a) Schematic of qubit $T_1$ response to shift in qubit frequency (red traces), at different times, $t$. The qubit frequency is tuned by an amount $\Delta\omega_q$ by an off-resonant tone placed $|\Delta_{qs}|$ above the qubit frequency $\omega_q$. The dependence of the frequency shift on the off-resonant tone amplitude $\Omega_s$ is depicted by the blue trace. The $T_1$ dips are indicative of the qubit coming into resonance with a strongly coupled TLS, at the frequency $\omega_{TLS}$.The TLS frequency shifts in time due to spectral diffusion, schematically indicated by changes in the $T_1$ dips at different time snapshots. (b) Schematic of a Ramsey pulse sequence used to calibrate $\Delta \omega_q$ as a function of $\Omega_s$ (top); and (bottom) schematic pulse sequence for the relaxation time spectroscopy. For each $\Omega_s$ (i.e., $\Delta \omega_q$), the $|1\rangle$ occupation is measured at a fixed time (i.e., 50 $\mu$s in this work). (c) Measured $\Delta\omega_q$ as a function of normalized DAC amplitude, $\Omega_s$ using the AC Stark shifted Ramsey technique. Solid line is a quadratic fit functionally consistent with a perturbative model. (inset) For an arbitrary case, calculated qubit frequency dependence on $\Delta_{qs}$ for fixed $\Omega_s$. Negative and positive qubit shifts can be produced and large shifts can be induced depending on $\Delta_{qs}$.}
\label{fig:Figure2}
\end{figure}
In this work, we introduce an all-microwave technique for the fast spectroscopy of TLSs in single junction transmon qubits that requires no additional hardware resources. In contrast to flux based approaches to TLS spectroscopy, we employ off-resonant microwave tones to drive AC-Stark shifts of the fundamental qubit transition and spectrally resolve qubit relaxation times. Dips in relaxation times serve as a probe of the frequency location of a strongly coupled TLS. We use repeated frequency sweeps to probe the time dynamics of the relaxation probabilities including tracking the spectral diffusion of strongly coupled TLS.
Across 10 qubits, we observe strong correlations between the long time mean, averaged over several months $\langle{T_1}\rangle_{T}$, and the short time mean, averaged around the local qubit frequency $\langle{T_1}\rangle_{\omega,t}$
This strong correlation
suggests a quasi-ergodic behavior of the TLS spectral diffusion in the nearby frequency neighborhood of the qubit. In contrast, there is lower correlation between $\langle{T_1}\rangle_{T}$ and $T_1$ measured over a single day. The $\langle{T_1}\rangle_{\omega,t}$ can provide, therefore, a more rapid estimate of long time behavior.
\section{Device and spectroscopy technique}
\label{section:Technique}
The experiments reported in this letter were performed on IBM Q Almaden, a 20 qubit processor based off single junction transmons and fixed couplings. The device topology is shown in Fig. \ref{fig:Figure1} (a), and qubit frequencies are around $\sim$5 GHz. Fig. \ref{fig:Figure1} (b) depicts the characteristic spread of the qubit $T_1$s and their mean, from $\sim$250 measurements over 9 months. The
base plate (to which the device was mounted) temperature of the dilution refrigerator was typically $\sim$13 mK
excepting several temperature excursions to $\sim1 $ K, which were not observed to have any significant effects on the $T_1$ correlations discussed later.
Several qubits on the device display mean $T_1$s exceeding 100 $\mu$s. However, the large spread in individual qubit $T_1$s highlights the challenge for rapid benchmarking of device coherence, since any single $T_1$ measurement can disagree substantially from its longtime mean.
We study the spectral dynamics of these $T_1$ times by employing off-resonant microwave tones to induce an effective frequency shift $\Delta\omega_{q}$ in single junction transmons by the AC Stark effect.
Shifting the qubit frequency into resonance with a defect TLS induces a faster relaxation time, which in turn is used to detect the frequency location of the TLS \cite{simmonds_decoherence_2004}, as depicted in Fig. \ref{fig:Figure2} (a).
The Stark shift can be described analytically by a Duffing oscillator model \cite{magesan2020, Schneider_PhysRevA.97.062334}
\begin{equation}
\Delta\omega_{q} = \frac{\delta_{q}\Omega_{s}^{2}}{2\Delta_{qs}(\delta_{q}+\Delta_{qs})}
\end{equation}
where $\delta_{q}$ is the qubit anharmonicity, $\Omega_{s}$ is the drive amplitude and $\Delta_{qs}= \omega_{q} - \omega_{s}$ is the detuning between the qubit frequency and the Stark tone.
As seen from the expression above, the magnitude and sign of the Stark shift can be manipulated by the detuning and the drive amplitude of the Stark tone, Fig. \ref{fig:Figure2} (c). Very large frequency shifts can be obtained by driving close to the transmon transitions, but this typically leads to undesired excitations/leakage out the two-state manifold. In this work, we obtain Stark shifts of 10's of MHz, with modest drive amplitudes and a fixed detuning $\Delta_{qs}$ of $\pm$ 50 MHz. The frequency shifts are experimentally measured using a modified Ramsey sequence \cite{ramsey_molecular_1950}, schematically shown in Fig. \ref{fig:Figure2} (b), and display good agreement with the perturbative model in the low-drive limit, Fig. \ref{fig:Figure2} (c).
\begin{figure}
\centering
\includegraphics[width=85mm, clip,trim = 0.0mm 0.0mm 0.0mm 0.0mm]{Fig2_redone_v18.pdf}
\caption{ (a) Measured probability of being in the $|1\rangle$ state, $P_1$, at 50 $\mu$s wait time with varying $\Delta \omega_q$ and tone detuned 50 MHz above ${\omega_{q}/2\pi}$. (b) Example of Ramsey measurements used to extract frequency shifts ${\Delta\omega_{q}/2\pi}$ from pulse amplitudes. The two curves result from starting the Ramsey oscillations with a $X_{\pi/2}$ or $Y_{\pi/2}$. (c) and (d) are $T_1$ measurement with Stark shifts $\Delta{\omega_{q}/2\pi}$ = -18.9 and -5.3 MHz, respectively. (e) $T_1$ measurement with no Stark shift (i.e., no Stark tone).}
\label{fig:Figure3}
\end{figure}
We focus on the spectrally resolved $T_1$ measurements in Fig. \ref{fig:Figure3} that we use as a probe of defect TLS transition frequencies. However, instead of measuring the entire $T_1$ decay, we use the excited state probability, $P_1$, after a fixed delay time as a measure of $T_1$. This speeds up the spectral scans significantly. Our experiments are performed at a repetition rate of 1kHz, but our scheme can be further accelerated with reset techniques~\cite{egger2018}, which can be crucial for probing faster TLS dynamics. For an effective frequency sweep, we run an amplitude sweep with off-resonant pulses at fixed detuning ($\pm$ 50 MHz) and duration (delay time of 50 $\mu$s), after exciting the qubit with an initial $\pi$ pulse. The pulsed Stark sequence enables faster spectroscopy by circumventing the need to re-calibrate the $\pi$, $\pi/2$ pulses at every frequency. The off-resonant pulses have Gaussian-square envelopes with a 2$\sigma$ rise-fall profile, where $\sigma=10$ ns. This pulse sequence is shown in Fig. \ref{fig:Figure2} (b). The amplitude points in the sweep are then related to Stark shifts by Ramsey sequences. Fig. \ref{fig:Figure3} shows representative data of such a sweep on qubit 19 (Q$19$) with distinctive dips in $P_1$ that we attribute to strongly coupled TLS at their transition frequencies. $T_1$ measurements at Stark tone amplitudes corresponding to high/low $P_1$ points, as seen in the bottom panel of Fig. \ref{fig:Figure3}, explicitly show the substantial variation in $T_1$ as a function of frequency and the consistent tracking of $P_1$ with $T_1$.
Variations in $P_1$ can potentially be caused by sources other than TLS. In our experiments, $P_1$ is spectrally resolved to $\sim\pm$25 MHz around the individual qubit frequencies. The narrow frequency range combined with measuring non-neighbor sets of qubits simultaneously avoids strong $P_1$ suppression from resonances with neighboring qubits, the coupling bus or common low-Q parasitic microwave modes. Control experiments show that time insensitive features in the $P_1$ fingerprint are robust to choice of the Stark tone detuning, ruling out a power dependence for the power range used in this work.
Finally, while a recent report \cite{yan2016} modeled their broadband $T_1$ scatter as arising from quasi-particle fluctuations, this is not sufficient to explain the sharp frequency dependent $P_1$ features depicted, for instance in Fig. \ref{fig:Figure3}. Furthermore, recent experiments on our qubits suggest a quasi-particle limit to $T_1$ that exceeds several milliseconds.\cite{kurter2020}
\section{TLS dynamics and correlations between $P_1(\omega,t)$ and $\langle T_1\rangle_T$}
\label{section:TLS dynamics}
We repeat the line traces of Fig. \ref{fig:Figure3} for both positive and negative 50 MHz detuning, approximately once every 3-4 hours, extended over hundreds of hours for all the qubits. A representative example of the cumulative scans is shown in Fig. \ref{fig:Figure4} for Q15. Spectroscopy of the other qubits is shown in the supplemental information, appendix \ref{other spectroscopy}. The TLS dynamics around the qubit frequency are qualitatively similar to previous TLS spectroscopy using flux or stress tunable devices \cite{muller_towards_2019}.
In the case of Q15, Fig. \ref{fig:Figure4}, there are prominent dips in relaxation probability around positive 1 MHz, negative 5-10 MHz, and negative 15-20 MHz. The spectral diffusion of the positions of the $T_1$ dips can vary between order of 1 MHz to 10 MHz over the 272 hours of measurement providing a qualitative measure of linewidths. A more quantitative discussion of linewidths can be found in appendix \ref{line widths}. The background is covered by an ensemble of smaller dips of relaxation, Fig. \ref{fig:Figure3}, that also dynamically evolve, with features that are larger than the sampling noise in the measurement.
\begin{figure}
\centering
\includegraphics[width=85mm, clip,trim = 0.0mm 2.0mm 0.0mm 0.0mm]{Spectroscopy2XaxisM15.pdf}
\caption{Time dependence of the energy relaxation spectroscopy for Q$15$ using $\Delta\omega_s = \pm 50$MHz and varying $\Omega_s$ to sweep $\Delta\omega_{q}$. The $P_1$ is measured at 50 $\mu$s.}
\label{fig:Figure4}
\end{figure}
As discussed previously, $T_1$ fluctuations introduce uncertainty in the coherence benchmarking, stability of multi-qubit circuit performance and process optimization of superconducting qubit devices.
In this context of better predictors, we examine if the long time averages ($T \sim$ 9 months) $\langle T_1 \rangle_{T} $ and $\langle P_1 \rangle_{T} $ are correlated with the frequency neighborhood of the qubit $\langle T_1 \rangle_{\omega,t} $ and $\langle P_1 \rangle_{\omega,t} $, respectively. The averaged relaxation probabilities and $T_1$s are defined as
\begin{equation}
\langle P_1 \rangle_{T} = \frac{1}{N}\sum_{i=1}^{N}P_1(\omega_q,\tau, T_i)
\label{def4}
\end{equation}
\begin{equation}
\langle T_1 \rangle_{T} = \frac{1}{N}\sum_{i=1}^{N}T_1(\omega_q, T_i)
\label{def4}
\end{equation}
\begin{equation}
\langle P_1 \rangle_{\omega,t}=\frac{1}{n}\sum_{i=1}^n \frac{1}{2\Delta\omega}\sum_{-\Delta \omega}^{\Delta \omega}P_1(\omega_q+ \omega_j,\tau,t_i)d\omega_j,
\label{def3}
\end{equation}
where definitions of variables can be found in table \ref{table:paperdefs}.
We compare $\langle P_1 \rangle_{\omega,t}$ to $\langle P_1 \rangle_T$ from the daily $T_1$ measurements over $T_{\mathrm{max}} \sim$ 9 months evaluated at $\tau = 53 ~\mu s$, shown in Fig. \ref{fig:Figure1}. The $\langle P_1 \rangle_{\omega,t}$ are calculated for a $T_1$ delay time of $\tau = 50 ~\mu$s for 10 qubits in the device for the first time slice and a cutoff frequency $\Delta \omega/2\pi = $5 MHz. A qualitatively close agreement for all 10 qubits is observed, see Fig. \ref{fig:Fig4} (a).
\begin{table}[h!]
\begin{tabular}{||c |c ||}
\hline
\makecell{Symbol}&\makecell{Definition} \\
\hline\hline
$\omega_q$ & Qubit frequency \\
\hline
$\omega_j$ & Qubit Stark shift \\
\hline
$d\omega_j$ & Frequency bin size at the \\
& Stark shifted frequency location \\
\hline
$\Delta \omega$ & Span of qubit Stark shift \\
\hline
$N$ & Number of $T_1$ measurements \\
& for $\sim$ 9 month time series\\
\hline
$n$ & Number of spectroscopy \\
& time slices\\
\hline
$T_i$ & Time stamp for the \\
& $\sim$ 9 month $T_1$ time series \\
\hline
$t_i$ & Time slice \\
& for the spectroscopy \\
\hline
$\tau$ & Decay time at \\
& which $P_1$ was evaluated \\
\hline
$P_1(\omega_q+\omega_j,\tau,t_i)$ & Probability of $|1\rangle$ at $\tau$ for\\
& time slice $t_i$ and frequency shift $\omega_j$ \\
\hline
$\langle P_1 \rangle_{T}$ & Probability of $|1\rangle$ at $\tau$\\
& averaged over $\sim$9 month time series \\
\hline
$\langle P_1 \rangle_{\omega,t}$ & Probability of $|1\rangle$ at $\tau$ averaged over \\
& frequency and spectroscopy time \\
\hline
\end{tabular}
\caption{List of symbols}
\label{table:paperdefs}
\end{table}
A $\langle T_1 \rangle_{\omega,t}$ can also be estimated for each $\langle P_1 \rangle_{\omega,t}$ at $\tau$ = 50 $\mu$s by assuming an exponential decay. The approximate equivalence of $\langle T_1 \rangle_{\omega,t}$ and $\langle T_1 \rangle_T$ is seen in the scatter plot of Fig \ref{fig:Fig4} (a) inset. Furthermore, the poorer correlation between $\langle T_1 \rangle_T$ and a single instance of $T_1$ measurements, is also shown by larger scatter, as seen in Fig \ref{fig:Fig4} (a) inset.
To quantify these correlations we use a Pearson $R$ measure across the ten odd-labeled qubits,
\begin{equation}
R =
\frac{\sum\limits_{i=0}^{9}(\langle P_1\rangle_{T,Q_i}-\overline{\langle P_1\rangle_{T}})(\langle P_1\rangle_{\omega,t,Q_i}-\overline{\langle P_1\rangle}_
{\omega,t})}{\sqrt{\sum\limits_{i=0}^{9}(\langle P_1\rangle_{T,Q_i}-\overline{\langle P_1\rangle_{T}})^2\sum\limits_{i=0}^{9}(\langle P_1\rangle_{\omega,t,Q_i}-\overline{\langle P_1\rangle}_{\omega,t})^2}}.
\end{equation}
The Pearson correlation is a normalized covariance between two variables reflecting a linear correlation from 1 to -1, where $R =$ 1 (-1) represents a $100\%$ positive (negative) correlation and $R =$ 0 indicates no correlation. For a single frequency sweep that takes $\sim$20 minutes, we obtain 0.76 $<$ $R$ $<$ 0.84 correlation between $\langle T_1\rangle_{T}$ and $\langle T_1\rangle_{\omega,t}$ for 0.5 MHz $<$ $\Delta \omega$ $<$ 5MHz. Using the $P_1$ values without assuming an exponential dependence leads to stronger correlations of 0.87 $<$ $R$ $<$ 0.91. Both of these are substantially stronger than the correlation found between the representative instance of $T_1$ and $\langle T_1\rangle_{T}$, which was $R =$ 0.29. We note this instance of $R$ can have a large spread, as seen by simulations in Appendix \ref{PearsonCorVstd}.
A better estimate of the $\langle T_{1} \rangle$ for the qubit and $R$ for the device comes from averaging multiple measurements. We show the evolution of $R(T_i)$ using the $T_1$ time series for each qubit and employing an updated $\langle T_1 \rangle_{T_i}$ for each qubit, Fig. \ref{fig:Fig4} (b). The $R(T_i)$ exceeds $R$ $\sim$ 0.8 after $\sim$10 measurements, corresponding to a time exceeding 100 hours at order of 10 separate measurements.
An underlying challenge to achieving a similar $R$ faster from the time series is autocorrelation of the $T_1$ values. Evidence of autocorrelation can be seen for example in long term drifts in the average and short term correlations between $T_1$, inset of Fig. \ref{fig:Fig4} (b). First, to provide guidance on how many uncorrelated measurements, $N$, one requires to achieve $R\sim$0.8, we simulate the evolution of $R(N)$ for a 20\% standard deviation for the individual qubit $T_1$s. We find that $\sim$10 uncorrelated $T_1$ measurements are required for $\langle R \rangle$ to exceed 0.8 for 10 qubit devices, see the dependence and details in appendix \ref{PearsonCorVstd}. On shorter time scales, our experimental data shows evidence of stronger correlation frustrating faster determination of $\langle T_1 \rangle$ and that the fastest $R\sim$ 0.8 can be obtained on order of 1-2 days, see appendices \ref{autocorrelation} and \ref{Rspec}.
We conclude that this instance shows the promise of using the spectroscopy method for faster characterization of devices than single frequency $T_1$ measurements.
It is important to note that our calculations of $\langle T_1\rangle_{\omega,t}$ employ an equal weighting of $P_1$ associated with every frequency bin. However, it is not \textit{a priori} clear that equal weighting is a representative choice over the $\Delta \omega$ range. For example, how evenly does the spectral diffusion of each TLS contribute to the $T_1$ of the qubit? The strong correlation of $\langle T_1\rangle_{\omega,t}$ with $\langle T_1\rangle_{T}$ with equal weighting suggests that an ergodic-like sampling of the TLSs near the qubit frequency is a reasonable first approximation. Furthermore, the strong correlation of $\langle T_1 \rangle_T$ to $\langle T_1 \rangle_{\omega,t}$ using only the $P_1(\omega,\tau,t)$ spectrum around the qubit is consistent with a leading hypothesis that the $\langle T_1 \rangle_T$ is dominated by TLS behavior rather than other stochastic or static contributions.
\begin{figure*}
\centering
\includegraphics[width=\textwidth,height=5cm, clip,trim = 0.0mm 72.0mm 0.0mm 0.0mm]{Fig5new15v3.pdf}
\caption{(a) Comparison of $\langle P_1 \rangle_{\omega,t}$ at $50 \mu$s and $\langle P_1 \rangle_T$ averaged for $\sim$9 months and evaluated at a $\tau$ of 53 $\mu$s decay time. $\langle P_1\rangle_{\omega,t}$, is averaged over $\Delta \omega/2\pi = 5$ MHz after a single measurement that took $\sim$20 minutes. (inset) A scatter plot using $\langle T_1 \rangle_T$'s averaged over 9 months of measurement as the dependent variable and $\langle T_1 \rangle_{\omega,t}$ or $T_1$s from a single day. The line is a guide to the eye showing a 1:1 correlation. (b) The Pearson $R$ dependence on time averaging of the $T_1$'s of the odd numbered qubits up to T. (inset) The $T_1$ time series for Q13. (c) Pearson correlation, $R$, dependence on time slice averaging and frequency range, $\Delta \omega$, of the odd numbered qubits.}
\label{fig:Fig4}
\end{figure*}
\section{Correlation dependence on frequency and measurement time}
\label{section:Correlation dependence}
A natural question for obtaining $\langle T_1\rangle_{\omega,t}$ is the optimal frequency range $\Delta \omega$ and choice of $dt$ and $n$ autocorrelated samples to be employed to obtain an accurate measure of $\langle T_1 \rangle_T$. Since the optimum choices are not known \textit{a priori}, we evaluate and plot $R$ versus $\Delta \omega$ and $t_i$ in Fig. \ref{fig:Fig4} (c), with equal frequency bin weighting of $P_1$. While this choice produces a reasonably good first approximation for correlation across the entire range, the plot displays several unexplained features (e.g., non-monotonic dependence on $\Delta \omega$) indicating the unsurprising insufficiency of these two parameters (i.e., $\Delta \omega$ and $t$) alone to weight the frequency contribution of all the qubits and approach $R\sim1 $.
\section{Discussion: implications for process characterization}
\label{process}
The strong correlation between $\langle T_1 \rangle_{\omega,t}$ and $\langle T_1 \rangle_{T}$ suggests that long time $T_1$ averages might be estimated relatively rapidly using spectroscopy. This is in contrast to overcoming correlation times in $T_1$ at a single $\omega_q$ to obtain a representative $\langle T_1 \rangle_T$ for the qubit.
Identification of $\Delta \omega$ and total $t$, in this study, were made with pre-knowledge of what $\langle T_1 \rangle_T$ was. However, both dependencies appear to be relatively weak suggesting that a heuristic choice for a single $\Delta \omega$ and $dt_i$ might be sufficient to obtain useful estimates (i.e., $R > 0.8$) of $\langle T_1 \rangle_{T}$ for new processes when using this simple equal weighting approach (i.e., in contrast to different frequency spans for each qubit or weighted averaging over frequency). For a single value, the optimal choice for $\Delta \omega$ and $dt_i$ might then depend on some representative measure of the inhomogeneous TLS linewidth, a measure of the range of frequencies the TLSs sample for all the qubits in the device. Since we find good correlations using a relatively small $\Delta \omega \sim$ 10 MHz for the $\sim$ 9 month time series (e.g., similar magnitude of spectral diffusion as the TLS at -7.5 MHz in under 272 hours, Fig. \ref{fig:Figure4}), rather than a larger $\Delta \omega$ due to extended evolution over months (e.g., proportional to a time dependence such as $\sqrt{t}$, expected in some limits \cite{herzog_transient_1956, klimov_fluctuations_2018}), we speculate that the representative TLS linewidth is also bounded and might therefore be estimated from the short time TLS spectral diffusion, rather than growing indefinitely from spectral diffusion processes. Notably, Klauder et al. calculate that dipole coupled ensembles that are proposed for TLS spectral diffusion \cite{black_spectral_1977}, will produce a truncated linewidth \cite{Klauder_PhysRev.125.912}.
\section{Conclusion}
In this work, we probe the temporal and spectral dynamics of superconducting qubit relaxation times. We study these dynamics in high coherence, single-junction transmons by developing a technique for energy relaxation spectroscopy of defect TLSs via the AC Stark effect. Our technique requires no additional hardware resources and can be easily sped up further by integration with reset schemes. Autocorrelation of $T_1$ frustrates rapid characterization of the long-time average $\langle T_1 \rangle_T$ and therefore accurate characterization of devices. Our analysis of the dynamics identifies a strong correlation between $\langle T_1 \rangle_T$ and its short time average over the local frequency span, $\langle T_1 \rangle_{\omega,t}$. The strong correlation of $\langle T_1 \rangle_T$ with $\langle T_1 \rangle_{\omega,t}$ is also consistent with a TLS dominated $T_1$ that quasi-ergodically samples the qubit local frequency neighborhood in contrast to static or uncorrelated stochastic processes. This work opens up several new promising directions for rapid process characterization and evaluation of device stability.
\section{Acknowledgements}
We acknowledge technical support on the IBMQ\_Almaden device from the IBM Quantum deployment team. Additional insightful discussions, suggestions and assistance came from Nick Bronn, Andrew Cross, Oliver Dial, Doug McClure, Easwar Magesan, Hasan Nayfeh, James Raferty, Martin Sandberg, Srikanth Srinivasan, Neereja Sundaresan, Jerry Tersoff and Jerry Chow.
MC also acknowledges support from Princeton Plasma Physics Laboratory through the Department of Energy Laboratory Directed Research and Development program and contract number DE-AC02-09CH11466 to complete parts of the analysis and manuscript.
| {
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} | 265 |
Gage Cartographics worked with the Wilderness Society's California office to assess biodiversity and species richness. We looked at potential areas for conservation designation and assessed their importance as compared to other public lands in the state. Methods included landscape analysis and big-data processing using R. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,202 |
\section{Introduction}
\vspace*{-2mm}
Content-Based Image Retrieval (CBIR) on very large datasets typically relies on hashing for efficient approximate nearest neighbor search; see e.g. \cite{Wang:2016up} for a review.
Early methods such as (LSH) \cite{Gionis:1999wf} were data-independent, but Data-dependent methods (either supervised or unsupervised) have shown better performance.
Recently, Deep hashing methods using CNNs have had much success over traditional methods, see e.g. Hashnet \citep{Cao:2017vm}, DADH \citep{Li:2018tj}.
Most supervised hashing techniques rely on a pairwise binary similarity matrix $S=\{s_{ij}\}$, whereby $s_{ij}=1$ for images i and j taken from the same class, and $0$ otherwise.
A richer set of affinity is possible using semantic relations, for example in the form of class hierarchies.
\cite{Yan:2017gh} consider the semantic hierarchy for non-deep hashing, minimizing inner product distance of hash codes from the distance in the semantic hierarchy.
In the SHDH method \citep{Wang:2017vc}, the pairwise similarity matrix is defined from such a hierarchy according to a weighted sum of weighted Hamming distances.
In Unsupervised Semantic Deep Hashing (USDH, \cite{Jin:2018vo}), semantic relations are obtained by looking at embeddings on a pre-trained VGG model on Imagenet. The goal of the semantic loss here is simply to minimize the distance between binarized hash codes and their pre-trained embeddings, i.e. neighbors in hashing space are neighbors in pre-trained feature space.
This is somewhat similar to our notion of semantic similarity except for using a pre-trained embedding instead of a pre-labeled semantic hierarchy of relations.
\cite{Zhe:2019vi} consider class-wise Deep hashing, in which a clustering-like operation is used to form a loss between samples both from the same class and different levels from the hierarchy.
Recently \cite{Barz:2018vt} explored image retrieval using semantic hierarchies to design an embedding space, in a two step process.
Firstly they directly find embedding vectors of the class labels on a unit hypersphere, using a linear algebra based approach, such that the distances of these embeddings are similar to the supplied hierarchical similarity.
In the second stage, they train a standard CNN encoder model to regress images towards these embedding vectors. They do not consider hashing in their work.
\section{Formulation}
We also make use of hierarchical relational distances in a similar way to constrain our embeddings. However compared to our work, \cite{Barz:2018vt} consider continuous representations and require the embedding dimension to equal the number of classes, whereas we learn compact quantized hash codes of arbitrary length, which are more practical for real world retrieval performance.
Moreover, we do not directly find fixed target embeddings for the classes, but instead require that the neural network embeddings will be learned in conjunction with the network weights, to best match the similarities derived from the labels.
And unlike \cite{Zhe:2019vi}, in our weakly supervised SHREWD method, we do not require explicit class membership, only relative semantic distances to be supplied.
Let $(x, y)$ denote a training example pair consisting of an image and some (possibly weakly) supervised target y, which can be a label, tags, captions etc. The embeddings are defined as $\hat{z} = f_\theta (x)$ for a deep neural network $f$ parameterized by weights $\theta$. Instead of learning to predict the target $y$, we assume that there exists an estimate of similarity between targets, $d(y, y')$. The task of the network is then to learn this similarity by attempting to match $\left\Vert \hat z - \hat z'\right\Vert $ with $ d(y, y')$ under some predefined norm in the embedding space.
While in this work we use class hierarchies to implicitly inform our loss function via the similarity metric $d$, in general our formulation is weakly supervised in the sense that these labels themselves are not directly required as targets. We could equally well replace this target metric space with any other metric based on for instance web-mined noisy tag distances in a word embedding space such as GloVe or word2vec, as in \cite{Frome:2013ux}, or ranked image similarities according to recorded user preferences.
In addition to learning similarities between images, it is important to try to fully utilize the available hashing space in order to facilitate efficient retrieval by using the Hamming distance to rank most similar images to a given query image. Consider for example a perfect ImageNet classifier. We could trivially map all 1000 class predictions to a 10-bit hash code, which would yield a perfect mAP score. The retrieval performance of such a ``mAP-miner" model would however be poor, because the model is unable to rank examples both within a given class and between different classes \citep{Ding:2018wj}. We therefore introduce an empirical Kullback-Leibler (KL) divergence term between the embedding distribution and a (near-)binary target distribution, which we add as an additional loss term. The KL loss serves an additional purpose in driving the embeddings close to binary values in order to reduce the information loss due to binarizing the embeddings.
We next describe the loss function, $L({\theta})$, that we minimize in order to train our CNN model. We break down our approach into the following 3 parts:
\begin{equation}
L({\theta}) = L_{sim} + \lambda_1 L_{KL} + \lambda_2 L_{cls}
\end{equation}
$L_{cls}$ represents a traditional categorical cross-entropy loss on top of a linear layer with softmax placed on the non-binarized latent codes. The meaning and use of each of the other two terms are described in more detail below.
Similar to \cite{Barz:2018vt} we consider variants with and without the $L_{cls}$, giving variants of the algorithm we term SHREWD (weakly supervised, no explicit class labels needed) and SHRED (fully supervised).
\subsection{Semantic Similarity loss}
In order to weakly supervise using a semantic similarity metric, we seek to find affinity between the normalized distances in the learned embedding space and normalized distances in the semantic space. Therefore we define
\begin{equation}
L_{sim} = \frac{1}{B^2} \sum\limits_{b, b'=1}^{B} \left\vert \frac{1}{\tau_z} \left\Vert \hat z_b - \hat z_{b'}\right\Vert_M - \frac{1}{\tau_y}d\left(y_b, y_{b'}\right)\right\vert w_{b b'} ,
\end{equation}
where $B$ is a minibatch size, $\left\Vert \ldots \right\Vert_M$ denotes Manhattan distance (because in the end we will measure similarity in the binary space by Hamming distance), $d\left(y_b, y_{b'}\right)$ is the given ground truth similarity and $w_{b b'}$ is an additional weight, which is used to give more weight to similar example pairs (e.g. cat-dog) than distant ones (e.g. cat-moon).
$\tau_z$ and $\tau_y$ are normalizing scale factors estimated from the current batch.
We use a slowly decaying form for the weight, $w_{b b'} = \gamma ^\rho / \left(\gamma + d\left(y_b, y_{b'}\right) \right)^\rho$, with parameter values $\gamma = 0.1$ and $\rho=2$.
\subsection{KL-divergence based distribution matching loss}
Our empirical loss for minimizing the KL divergence $KL(p||q) \doteq \int dz p(z) \log (p(z) / q(z))$ between the sample embedding distribution $p(z)$ and a target distribution $q(z)$ is based on the Kozachenko-Leonenko estimator of entropy \cite{KL1}, and can be defined as
\begin{equation}
L_{KL} = \frac{1}{B} \sum\limits_{b=1}^B \left[\log \left(\nu(\hat z_b ; z)\right) - \log\left(\nu(\hat z_b ; \hat z)\right) \right],
\end{equation}
where $\nu(\hat z_b ; z)$ denotes the distance of $\hat z_b$ to a nearest vector $z_{b'}$, where $z$ is a sample (of e.g. size $B$) of vectors from a target distribution. We employ the beta distribution with parameters $\alpha = \beta = 0.1$ as this target distribution, which is thus moderately concentrated to binary values in the embedding space.
The result is that our embedding vectors will be regularized towards uniform binary values, whilst still enabling continuous backpropagation though the network and giving some flexibility in allowing the distance matching loss to perform its job. When quantized, the resulting embeddings are likely to be similar to their continuous values, meaning that the binary codes will have distances more similar to their corresponding semantic distances.
\section{Experimental results}
\paragraph{Metrics}
As discussed in section 2, the mAP score can be a misleading metric for retrieval performance when using class information only. Similarly to other works such as \cite{Deng:2011fw,Barz:2018vt}, we focus on measuring the retrieval performance taking semantic hierarchical relations into account by the mean Average Hierarchical Precision (mAHP).
However more in line with other hashing works, we use the hamming distance of the binary codes for ranking the retrieved results.
\paragraph{CIFAR-100}
We first test on CIFAR-100 \cite{Krizhevsky:2009tr} using the same semantic hierarchy and Resnet-110w architecture as in \cite{Barz:2018vt}, where only the top fully connected layer is replaced to return embeddings at the size of the desired hash length.
See Tables~\ref{cifar1},\ref{cifar2} for comparisons with previous methods, an ablation study, and effects of hash code length.
\begin{table}[t]
\begin{center}
\begin{tabular}{lccc}
\\ \hline
\multicolumn{1}{c}{\bf Method} &\multicolumn{1}{c}{\bf mAP} &\multicolumn{1}{c}{\bf mAHP@250} &\multicolumn{1}{c}{\bf Classification accuracy}
\\ \hline
DeViSE \citep{Frome:2013ux}\textsuperscript\dag & 0.5016 & 0.7348 & 74.66\% \\
\cite{Sun:2017wl}\textsuperscript\dag & 0.6202 & 0.7950 & \textbf{76.96\%} \\
\cite{Barz:2018vt}\textsuperscript\dag, $L_{CORR}$ & 0.5900 & 0.8290 & 75.03\% \\
\cite{Barz:2018vt}\textsuperscript\dag, $L_{CORR+CLS}$ & 0.6107 & 0.8329 & 76.60\% \\
\cite{Zhe:2019vi}\textsuperscript\ddag & 0.8259\textsuperscript\ddag & 0.8667\textsuperscript\ddag & n/a \\
\hline
$L_{sim}$ only & 0.2204 & 0.7007 & 10.01\% \\
$L_{cls}$ only & 0.5647 & 0.8124 & 73.00\% \\
$L_{sim} + L_{cls}$ only & 0.5292 & 0.8188 & 69.68\% \\
$L_{KL} + L_{cls}$ only & 0.3010 & 0.6215 & 69.25\% \\
\hline
SHREWD [Ours] $L_{KL} + L_{sim}$ & 0.6514 & 0.8690 & 70.79\%\\
SHRED [Ours] $L_{KL} + L_{sim} + L_{cls}$ & {0.7049} & \textbf{0.8802} & 72.77\%
\\ \hline
\end{tabular}
\caption{
Retrieval Performance and Ablation Study on CIFAR-100, 64 bit hash codes. $\dag$ indicates non-quantized embedding codes. $\ddag$ mAHP@2500 measured with this method, so not equivalent.
Note that while $L_{cls}$ performs best on supervised classification, $L_{sim}$ allows for better retrieval performance, however this is degraded unless $L_{KL}$ is also included to regularize towards binary embeddings. For measuring classification accuracy on methods that don't include $L_{cls}$, we measure using a linear classifier with the same structure as in $L_{cls}$ trained on the output of the first network.\label{cifar1}
}
\end{center}
\end{table}
\begin{table}[t]
\begin{center}
\begin{tabular}{lccc}
\\ \hline
\multicolumn{1}{c}{\bf Code length} &\multicolumn{1}{c}{\bf mAP result} &\multicolumn{1}{c}{\bf mAHP result} &\multicolumn{1}{c}{\bf Classification accuracy}
\\ \hline
16 bits & 0.3577 & 0.7478 & 65.65\% \\
32 bits & 0.5114 & 0.8202 & 65.00\% \\
64 bits & 0.6514 & 0.8690 & 70.79\% \\
128 bits & \textbf{0.6857} & \textbf{0.8760} & 70.29\% \\
\hline
\end{tabular}
\caption{Effect of hash code length on CIFAR-100 for SHREWD\label{cifar2}}
\end{center}
\end{table}
\paragraph{ILSVRC 2012}
We also evaluate on the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2012 dataset. For similarity labels, we use the same tree-structured WordNet hierarchy as in \cite{Barz:2018vt}. We use a standard Resnet-50 architecture with a fully connected hashing layer as before.
Retrieval results are summarized in Table~\ref{imagenet}.
We compare the resulting Hierarchical Precision scores with and without $L_{KL}$, for binarized and continuous values in Figure~\ref{HPgraphImagenet}.
We see that our results improve on the previously reported hierarchical retrieval results whilst using quantized embeddings, enabling efficient retrieval.
\begin{table}[t]
\begin{center}
\begin{tabular}{lccc}
\\ \hline
\multicolumn{1}{c}{\bf Method} &\multicolumn{1}{c}{\bf mAP} &\multicolumn{1}{c}{\bf mAHP@250} &\multicolumn{1}{c}{\bf Classification accuracy}
\\ \hline
\cite{Barz:2018vt} (floating point embeddings) & 0.3037 & 0.7902 & 48.97\%\\
SHREWD [Ours] (64 bit binary) & 0.4604 & 0.8676 & ---\\
SHREWD [Ours] (128 bit binary hash codes) & 0.5067 & 0.8674 & ---\\
SHREWD [Ours] (floating point embeddings) & --- & 0.8733 & 63.28\% \\
SHRED [Ours] (64 bit binary) & \textbf{0.5594} & \textbf{0.8885} & --- \\
\hline
\end{tabular}
\caption{Performance on ILSVRC 2012, floating point vs quantized hash codes (NB classifier is only trained by using floating point embeddings as features)
\label{imagenet}}
\end{center}
\end{table}
\begin{figure}[H]
\begin{center}
\includegraphics[keepaspectratio=true,scale=0.45]{kl_ablation_cifar}
\includegraphics[keepaspectratio=true,scale=0.45]{kl_ablation}
\end{center}
\caption{Hierarchical precision @k for CIFAR-100 (left) and ILSVRC-2012 (right) for 64-bit SHREWD. We see a substantial drop in the precision after binarization when not using the KL loss. Also binarization does not cause as severe a drop in precision when using the KL loss.
}
\label{HPgraphImagenet}
\end{figure}
\section{Conclusions}
We approached Deep Hashing for retrieval, introducing novel combined loss functions that balance code binarization with equivalent distance matching from hierarchical semantic relations.
We have demonstrated new state of the art results for semantic hierarchy based image retrieval (mAHP scores) on CIFAR and ImageNet with both our fully supervised (SHRED) and weakly-supervised (SHREWD) methods.
| {
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} | 4,031 |
The Victoria's Secret Fashion Show is an annual fashion show sponsored by Victoria's Secret, an American premium brand of lingerie and sleepwear. Victoria's Secret uses the show to promote and market its goods in high-profile settings. The show features some of the world's leading fashion models, such as current Victoria's Secret Angels Adriana Lima, Alessandra Ambrosio, Miranda Kerr, Doutzen Kroes, Behati Prinsloo, Candice Swanepoel, Erin Heatherton, Lily Aldridge, and Lindsay Ellingson.
The show featured performances by Rihanna, Bruno Mars, and Justin Bieber.
Fashion show segments
Segment 1: Circus (Choreographed by Christopher Harrison)
Segment 2: Dangerous Liaisons
Segment 3: Calendar Girls
Special Performance
Segment 4: PINK Ball
Segment 5: Silver Screen Angels
Segment 6: Angels In Bloom
Finale
Adriana Lima and Candice Swanepoel led the finale.
Index
Victoria's Secret
2012 in fashion | {
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} | 9,910 |
La stazione Progress-2 (in russo Станция Прогресс-2) è una base antartica permanente russa (precedentemente sovietica) nella Terra della principessa Elisabetta nel territorio antartico australiano.
Localizzata ad una latitudine di 69°22'S e ad una longitudine di 76°23'E in una zona libera dai ghiacci ad un'altitudine di 15,5 metri.
La base è stata inaugurata nel gennaio 1989 inglobando alcune strutture della stazione Progress.
La popolazione estiva è di 77 persone, che si riducono a 20 durante l'inverno australe.
La base effettua studi scientifici relativi al monitoraggio ambientale e geologico. Si occupa inoltre dello studio della glaciologia marina e terrestre e della sismologia. Nella stazione opera anche una stazione meteorologica.
Note
Voci correlate
Basi scientifiche in Antartide
Stazione Progress
Altri progetti
Collegamenti esterni
Progress-2
Russia in Antartide
Unione Sovietica in Antartide | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,465 |
HomeMeraj Ahsan QureshiWorld Health Day 2022
To honor the working of WHO, World Health day is celebrated on 7th of April every year. This can also be considered as anniversary of WHO. This started by the contributions of China and Brazil, who formed independent international organization which should be free from any government influence.
Back in 1946, it was in New York where the constitution for World Health Organization was approved and it came into force on 7 April 1948 when 61 countries signed for its inception. The first World Health Day was observed on 22 July but then the date was changed to the day of WHO's formation to encourage student and general public participation.
A theme is chosen each year to priorities the concern for WHO and highlights the issues along with its remediation. With the ongoing pandemic, increase in the incidents due to disease and a health impacts due to pollution, the theme of this year World Health Day 2022 is "Our Planet, Our Health". This call from WHO and other partners is to present a unique way to recover from COVID-19 pandemic, that puts a healthy individual to put his efforts to create societies focused on wellbeing.
This World Health Day 2022 will throw light on the contribution of health workers, the nurses and midwives that have made the healthcare system which is present today. They are the frontline workers whose work often goes unnoticed but they are important for the everyday wellbeing of a patient
Previous few decades, in the region of Americas, with the enhancement of health services and controlled environment emissions, economic progress and few other factors led to improvement in health sector. However, it is estimated that one million early deaths per year are attributable to known preventable environmental risks.
The environmental risks including air pollution, contaminated water, chemical spillage, inappropriate solid waste management and undesirable impacts of climate change are the most pressing environment and public health threats in region. The threats are mostly compounded by weak or lack of governance and inequalities in distribution of health facilities as well as limited leadership.
The theme of this year is powerful reminder to all of us that the resolution of issues mentioned above are beyond the consideration of the health sector and, as a consequence, an effective response will demand whole of government and health sectors. So we need to stand up for the problems which we are facing and make people aware so as to make the change. The change which not only needed for us but for the upcoming generations, so that they can live a healthy life.
https://www.thequint.com/fit/world-health-day-date-history-significance-and-theme#read-more
https://www.paho.org/en/events/virtual-commemoration-world-health-day-april-7th-2022-our-planet-our-health
About the Author: Meraj Ahsan Qureshi is a young, motivated environmentalist with the passion to create awareness and change the environment.
#lockdowns #globalpandemic #healthiertomorrow #worldpandemic #outbreak #pandemic #ventilators #climateandhealth
Meraj Ahsan Qureshi World Health Day | {
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{"url":"https:\/\/www.vtmarkets.com\/analysis\/3956\/","text":"English\nEurope\nMiddle East\nAsia\n\n# Daily market analysis\n\n###### February 17, 2021\n\nDaily Market Analysis\n\nMarket Focus\n\nUS bond yields surged to the highest in a year, while American stocks climbed to records as optimism over the economic recovery continued to ripple through markets. The yield on the benchmark 10-year Treasury note increased as much as nine basis points to 1.30%, the highest since Feb 2020. Global bonds extended the worst start to a year since 2013.\n\nThe S&P 500, Nasdaq, and Dow Jones all set records Tuesday before easing from the highs. The MSCI benchmark for emerging and developed market stocks snapped an 11-session winning streak. The reflation trade is powering assets tired to economic growth and price pressure, including commodities and cyclical stocks. At the same time, investors are riding a wave of speculative euphoria from penny stocks to Bitcoin amid abundant policy support.\n\nAccording to Eric Freedman, chief investment officer at US Bank Wealth management, \u201cWe certainly have data that suggests that being more glass half full than glass half empty remains the right posture. People are looking at parts of the world as well as sectors that have been underperforming and saying, \u2018hey this is maybe the next part of the market that heads higher.\u2019\u201d\n\nMarket Wrap\n\nMain Pairs Movement\n\nEURUSD trades near the 1.2100 level, down for the day, amid renewed demand for the greenback. US equities are struggling around their opening levels while Treasury yields are surging.\n\nAUDUSD eased from 0.7804, with bulls still in charge despite the poor performance across Wall Street.\n\nThe Loonie pair has witnessed a slide during the Asian session, losing almost 60 pips, and touched a fresh multi-week low at 1.2610. Nevertheless, thanks to the solid crude oil prices, the pair modestly recovered its loss and is later seen trading around the high of 1.2690s.\n\nThe DXY captures some positive traction on Tuesday and has surpassed the 90.50 level. The reflation trade has been gathering pace which has seen investors move over to riskier asset classes in the past number of trading session where US oil prices rallied to fresh cycle highs.\n\nTechnical Analysis:\n\nUSDJPY (Four-hour Chart)\n\nUSDJPY successfully broke above the key resistance at 105.68 and advanced towards the 106.00 zone in the American session. The 10-year US bond is up nearly 4% and the USD Index has reclaimed 90.50 threshold, both supported the greenback to overwhelm its JPY rival. From a technical perspective, the bullish trend of the pair is supported by the 21-Day SMAVG and MACD histogram. However, because the RSI has topped the 70 overbought threshold, it is inferable that the USDJPY pair might not advance above the psychological resistance at 106.00 in the short-term. On the upside, if USDJPY can penetrate 106.00, then the next resistance would be around 106.17, which is a price zone last seen in Oct 2020. On the flip side, if the current bullish trend is reversed, the most immediate support would be 105.68, then 105.30, followed by 105.07.\n\nResistance: 105.96, 106.17\n\nSupport: 105.68, 105.30, 105.07\n\nGBPUSD (Four-hour Chart)\n\nEver since the greenback found demand in the early US session, the GBPUSD pair has remained confined between a compacted region between the high 1.3800s and low 1.3900s. A stronger than expected NY Empire State Manufacturing Index Survey and a surging US 10-Year bond are fueling fresh demands for the greenback. The sterling continues to be supported by the drop in UK covid cases. From the 4-hour chart, we can see that the Cable pair is still on a bullish run, indicated by the 15-Day SMAVG; and at the same time, because today\u2019s pullback has cleared some room in the pair\u2019s RSI reading, it is likely that GBPUSD can resume its prior rally. However, before reclaiming the surging momentum, the bulls must first find acceptance above the most immediate resistance near 1.3939.\n\nResistance: 1.3939, 1.4025\n\nSupport: 1.3881, 1.3849, 1.3787\n\nXAUUSD (Four-hour Chart)\n\nWith a broad based risk-on sentiment and a strengthened USD, the precious metal plummeted below $1800 and is currently trading around$1794. The Gold has been on the back foot for nearly a week, and the current market is still pretty much biased against the bulls as indicated by the 29ish RSI reading. Although the RSI is implying an upward correction is likely, the precious metal can still dive down further towards the $1778 support if the bulls remain unable to carry the sell-off weight of XAUUSD. Moreover, with both 60-Day SMAVG and MACD histogram signaling a bearish trend, it would not be prudent to place any long position of the yellow metal until the Gold price is rejected solidly in its most immediate support level at$1789 and regains some momentum back on top over the \\$1800 threshold.\n\nResistance: 1808.50, 1818.24, 1829.73\n\nSupport: 1789.28, 1778.28\n\nEconomic Data\n\n Currency Data Time (TP) Forecast GBP CPI (YoY) (Jan) 15.00 0.6% EUR ECB Monetary Policy Statement 16.00 \u2013 USD Core Retail Sales (MoM) (Jan) 21.30 1.0% USD PPI (MoM) (Jan) 21.30 0.4% USD Retail Sales (MoM) (Jan) 21.30 1.1% CAD Core CPI (MoM) (Jan) 21.30 \u2013","date":"2022-10-05 04:49:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.23759804666042328, \"perplexity\": 11666.538082784524}, \"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-2022-40\/segments\/1664030337537.25\/warc\/CC-MAIN-20221005042446-20221005072446-00639.warc.gz\"}"} | null | null |
<?php
namespace Symfony\Component\HttpFoundation;
/**
* Response represents an HTTP response in JSON format.
*
* @author Igor Wiedler <igor@wiedler.ch>
*/
class JsonResponse extends Response
{
protected $data;
protected $callback;
/**
* Constructor.
*
* @param mixed $data The response data
* @param integer $status The response status code
* @param array $headers An array of response headers
*/
public function __construct($data = array(), $status = 200, $headers = array())
{
parent::__construct('', $status, $headers);
$this->setData($data);
}
/**
* {@inheritDoc}
*/
static public function create($data = array(), $status = 200, $headers = array())
{
return new static($data, $status, $headers);
}
/**
* Sets the JSONP callback.
*
* @param string $callback
*
* @return JsonResponse
*/
public function setCallback($callback = null)
{
if ($callback) {
// taken from http://www.geekality.net/2011/08/03/valid-javascript-identifier/
$pattern = '/^[$_\p{L}][$_\p{L}\p{Mn}\p{Mc}\p{Nd}\p{Pc}\x{200C}\x{200D}]*+$/u';
$parts = explode('.', $callback);
foreach ($parts as $part) {
if (!preg_match($pattern, $part)) {
throw new \InvalidArgumentException('The callback name is not valid.');
}
}
}
$this->callback = $callback;
return $this->update();
}
/**
* Sets the data to be sent as json.
*
* @param mixed $data
*
* @return JsonResponse
*/
public function setData($data = array())
{
// root should be JSON object, not array
if (is_array($data) && 0 === count($data)) {
$data = new \ArrayObject();
}
// Encode <, >, ', &, and " for RFC4627-compliant JSON, which may also be embedded into HTML.
$this->data = json_encode($data, JSON_HEX_TAG | JSON_HEX_APOS | JSON_HEX_AMP | JSON_HEX_QUOT);
return $this->update();
}
/**
* Updates the content and headers according to the json data and callback.
*
* @return JsonResponse
*/
protected function update()
{
if ($this->callback) {
// Not using application/javascript for compatibility reasons with older browsers.
$this->headers->set('Content-Type', 'text/javascript', true);
return $this->setContent(sprintf('%s(%s);', $this->callback, $this->data));
}
$this->headers->set('Content-Type', 'application/json', false);
return $this->setContent($this->data);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,415 |
Pearson, a California native, has always enjoyed riding sand tracks and was looking forward to the Southwick course. "I didn't grow up around a lot of sand tracks but I have always adapted well to sandy terrain. It was raining hard today because of the storm and the wet course just added to the challenge," remarked Pearson.
Pearson got a great start in the first moto and was sitting in 4th in the opening lap. She challenged Strong for 3rd place in moto one before suffering a crash. Pearson quickly recovered only allowing her teammate, Whitmore, to pass. "I got nervous once I realized I was closing in on 3rd and I rode tight. That's what caused me to crash the first time. Then, once I recovered, I had to ditch my goggles and gloves because of the amount of mud on them and that led to a crash a few turns later," remarked Pearson. "It was such a frustrating first moto but I was able to manage a 7th place finish." Meanwhile, Strong went on to finish 4th on her KTM while Whitmore put in an impressive ride to climb from her 11th place start to 5th at the moto end.
Moto two was a whole new story. Pearson started in 3rd and remained in podium position throughout the entire moto. She managed to hold off a hard charge from Whitmore for the seven laps. As the two ladies battled, they picked up their lap times and began to catch the 2nd place ride of Ashley Fiolek. "Sarah and I were able to push each other at the end of the race and close in on Ashley. It was so exciting once we had her in our view. That's the first time I have been able to run in a podium spot this season and it was a great feeling to be competitive in that position," stated Pearson. She went on to finish the race in 3rd place to earn her first top three this season. Her 7-3 results tied her for 3rd overall with Whitmore who had 5-4 moto scores. The podium spot was given to Pearson who had the better second moto finish, allowing the KTM rider to celebrate her first career podium.
"It was a tough day, but also the best day of my career and I am so happy right now. I have been working so hard this year and I finally have my bike dialed in perfect. I am really looking forward to the final two rounds and hoping to get two more podium finishes," said Pearson. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,708 |
package org.lwjgl.vulkan;
import javax.annotation.*;
import org.lwjgl.system.*;
import static org.lwjgl.system.Checks.*;
import static org.lwjgl.system.JNI.*;
import static org.lwjgl.system.MemoryUtil.*;
/**
* One of the states that contributes to the combinatorial explosion of pipeline state objects that need to be created, is the vertex input binding and attribute descriptions. By allowing them to be dynamic applications may reduce the number of pipeline objects they need to create.
*
* <p>This extension adds dynamic state support for what is normally static state in {@link VkPipelineVertexInputStateCreateInfo}.</p>
*
* <h5>VK_EXT_vertex_input_dynamic_state</h5>
*
* <dl>
* <dt><b>Name String</b></dt>
* <dd>{@code VK_EXT_vertex_input_dynamic_state}</dd>
* <dt><b>Extension Type</b></dt>
* <dd>Device extension</dd>
* <dt><b>Registered Extension Number</b></dt>
* <dd>353</dd>
* <dt><b>Revision</b></dt>
* <dd>2</dd>
* <dt><b>Extension and Version Dependencies</b></dt>
* <dd><ul>
* <li>Requires support for Vulkan 1.0</li>
* <li>Requires {@link KHRGetPhysicalDeviceProperties2 VK_KHR_get_physical_device_properties2} to be enabled for any device-level functionality</li>
* </ul></dd>
* <dt><b>Contact</b></dt>
* <dd><ul>
* <li>Piers Daniell <a target="_blank" href="https://github.com/KhronosGroup/Vulkan-Docs/issues/new?body=[VK_EXT_vertex_input_dynamic_state]%20@pdaniell-nv%250A%3C%3CHere%20describe%20the%20issue%20or%20question%20you%20have%20about%20the%20VK_EXT_vertex_input_dynamic_state%20extension%3E%3E">pdaniell-nv</a></li>
* </ul></dd>
* </dl>
*
* <h5>Other Extension Metadata</h5>
*
* <dl>
* <dt><b>Last Modified Date</b></dt>
* <dd>2020-08-21</dd>
* <dt><b>IP Status</b></dt>
* <dd>No known IP claims.</dd>
* <dt><b>Contributors</b></dt>
* <dd><ul>
* <li>Jeff Bolz, NVIDIA</li>
* <li>Spencer Fricke, Samsung</li>
* <li>Stu Smith, AMD</li>
* </ul></dd>
* </dl>
*/
public class EXTVertexInputDynamicState {
/** The extension specification version. */
public static final int VK_EXT_VERTEX_INPUT_DYNAMIC_STATE_SPEC_VERSION = 2;
/** The extension name. */
public static final String VK_EXT_VERTEX_INPUT_DYNAMIC_STATE_EXTENSION_NAME = "VK_EXT_vertex_input_dynamic_state";
/**
* Extends {@code VkStructureType}.
*
* <h5>Enum values:</h5>
*
* <ul>
* <li>{@link #VK_STRUCTURE_TYPE_PHYSICAL_DEVICE_VERTEX_INPUT_DYNAMIC_STATE_FEATURES_EXT STRUCTURE_TYPE_PHYSICAL_DEVICE_VERTEX_INPUT_DYNAMIC_STATE_FEATURES_EXT}</li>
* <li>{@link #VK_STRUCTURE_TYPE_VERTEX_INPUT_BINDING_DESCRIPTION_2_EXT STRUCTURE_TYPE_VERTEX_INPUT_BINDING_DESCRIPTION_2_EXT}</li>
* <li>{@link #VK_STRUCTURE_TYPE_VERTEX_INPUT_ATTRIBUTE_DESCRIPTION_2_EXT STRUCTURE_TYPE_VERTEX_INPUT_ATTRIBUTE_DESCRIPTION_2_EXT}</li>
* </ul>
*/
public static final int
VK_STRUCTURE_TYPE_PHYSICAL_DEVICE_VERTEX_INPUT_DYNAMIC_STATE_FEATURES_EXT = 1000352000,
VK_STRUCTURE_TYPE_VERTEX_INPUT_BINDING_DESCRIPTION_2_EXT = 1000352001,
VK_STRUCTURE_TYPE_VERTEX_INPUT_ATTRIBUTE_DESCRIPTION_2_EXT = 1000352002;
/** Extends {@code VkDynamicState}. */
public static final int VK_DYNAMIC_STATE_VERTEX_INPUT_EXT = 1000352000;
protected EXTVertexInputDynamicState() {
throw new UnsupportedOperationException();
}
// --- [ vkCmdSetVertexInputEXT ] ---
/**
* Unsafe version of: {@link #vkCmdSetVertexInputEXT CmdSetVertexInputEXT}
*
* @param vertexBindingDescriptionCount the number of vertex binding descriptions provided in {@code pVertexBindingDescriptions}.
* @param vertexAttributeDescriptionCount the number of vertex attribute descriptions provided in {@code pVertexAttributeDescriptions}.
*/
public static void nvkCmdSetVertexInputEXT(VkCommandBuffer commandBuffer, int vertexBindingDescriptionCount, long pVertexBindingDescriptions, int vertexAttributeDescriptionCount, long pVertexAttributeDescriptions) {
long __functionAddress = commandBuffer.getCapabilities().vkCmdSetVertexInputEXT;
if (CHECKS) {
check(__functionAddress);
}
callPPPV(commandBuffer.address(), vertexBindingDescriptionCount, pVertexBindingDescriptions, vertexAttributeDescriptionCount, pVertexAttributeDescriptions, __functionAddress);
}
/**
* Set the vertex input state dynamically for a command buffer.
*
* <h5>C Specification</h5>
*
* <p>To <a target="_blank" href="https://www.khronos.org/registry/vulkan/specs/1.3-extensions/html/vkspec.html#pipelines-dynamic-state">dynamically set</a> the vertex input attribute and vertex input binding descriptions, call:</p>
*
* <pre><code>
* void vkCmdSetVertexInputEXT(
* VkCommandBuffer commandBuffer,
* uint32_t vertexBindingDescriptionCount,
* const VkVertexInputBindingDescription2EXT* pVertexBindingDescriptions,
* uint32_t vertexAttributeDescriptionCount,
* const VkVertexInputAttributeDescription2EXT* pVertexAttributeDescriptions);</code></pre>
*
* <h5>Description</h5>
*
* <p>This command sets the vertex input attribute and vertex input binding descriptions state for subsequent drawing commands when the graphics pipeline is created with {@link #VK_DYNAMIC_STATE_VERTEX_INPUT_EXT DYNAMIC_STATE_VERTEX_INPUT_EXT} set in {@link VkPipelineDynamicStateCreateInfo}{@code ::pDynamicStates}. Otherwise, this state is specified by the {@link VkGraphicsPipelineCreateInfo}{@code ::pVertexInputState} values used to create the currently active pipeline.</p>
*
* <p>If the bound pipeline state object was also created with the {@link VK13#VK_DYNAMIC_STATE_VERTEX_INPUT_BINDING_STRIDE DYNAMIC_STATE_VERTEX_INPUT_BINDING_STRIDE} dynamic state enabled, then {@link VK13#vkCmdBindVertexBuffers2 CmdBindVertexBuffers2} can be used instead of {@code vkCmdSetVertexInputEXT} to dynamically set the stride.</p>
*
* <h5>Valid Usage</h5>
*
* <ul>
* <li>The <a target="_blank" href="https://www.khronos.org/registry/vulkan/specs/1.3-extensions/html/vkspec.html#features-vertexInputDynamicState">{@code vertexInputDynamicState}</a> feature <b>must</b> be enabled</li>
* <li>{@code vertexBindingDescriptionCount} <b>must</b> be less than or equal to {@link VkPhysicalDeviceLimits}{@code ::maxVertexInputBindings}</li>
* <li>{@code vertexAttributeDescriptionCount} <b>must</b> be less than or equal to {@link VkPhysicalDeviceLimits}{@code ::maxVertexInputAttributes}</li>
* <li>For every {@code binding} specified by each element of {@code pVertexAttributeDescriptions}, a {@link VkVertexInputBindingDescription2EXT} <b>must</b> exist in {@code pVertexBindingDescriptions} with the same value of {@code binding}</li>
* <li>All elements of {@code pVertexBindingDescriptions} <b>must</b> describe distinct binding numbers</li>
* <li>All elements of {@code pVertexAttributeDescriptions} <b>must</b> describe distinct attribute locations</li>
* </ul>
*
* <h5>Valid Usage (Implicit)</h5>
*
* <ul>
* <li>{@code commandBuffer} <b>must</b> be a valid {@code VkCommandBuffer} handle</li>
* <li>If {@code vertexBindingDescriptionCount} is not 0, {@code pVertexBindingDescriptions} <b>must</b> be a valid pointer to an array of {@code vertexBindingDescriptionCount} valid {@link VkVertexInputBindingDescription2EXT} structures</li>
* <li>If {@code vertexAttributeDescriptionCount} is not 0, {@code pVertexAttributeDescriptions} <b>must</b> be a valid pointer to an array of {@code vertexAttributeDescriptionCount} valid {@link VkVertexInputAttributeDescription2EXT} structures</li>
* <li>{@code commandBuffer} <b>must</b> be in the <a target="_blank" href="https://www.khronos.org/registry/vulkan/specs/1.3-extensions/html/vkspec.html#commandbuffers-lifecycle">recording state</a></li>
* <li>The {@code VkCommandPool} that {@code commandBuffer} was allocated from <b>must</b> support graphics operations</li>
* </ul>
*
* <h5>Host Synchronization</h5>
*
* <ul>
* <li>Host access to {@code commandBuffer} <b>must</b> be externally synchronized</li>
* <li>Host access to the {@code VkCommandPool} that {@code commandBuffer} was allocated from <b>must</b> be externally synchronized</li>
* </ul>
*
* <h5>Command Properties</h5>
*
* <table class="lwjgl">
* <thead><tr><th><a target="_blank" href="https://www.khronos.org/registry/vulkan/specs/1.3-extensions/html/vkspec.html#VkCommandBufferLevel">Command Buffer Levels</a></th><th><a target="_blank" href="https://www.khronos.org/registry/vulkan/specs/1.3-extensions/html/vkspec.html#vkCmdBeginRenderPass">Render Pass Scope</a></th><th><a target="_blank" href="https://www.khronos.org/registry/vulkan/specs/1.3-extensions/html/vkspec.html#VkQueueFlagBits">Supported Queue Types</a></th></tr></thead>
* <tbody><tr><td>Primary Secondary</td><td>Both</td><td>Graphics</td></tr></tbody>
* </table>
*
* <h5>See Also</h5>
*
* <p>{@link VkVertexInputAttributeDescription2EXT}, {@link VkVertexInputBindingDescription2EXT}</p>
*
* @param commandBuffer the command buffer into which the command will be recorded.
* @param pVertexBindingDescriptions a pointer to an array of {@link VkVertexInputBindingDescription2EXT} structures.
* @param pVertexAttributeDescriptions a pointer to an array of {@link VkVertexInputAttributeDescription2EXT} structures.
*/
public static void vkCmdSetVertexInputEXT(VkCommandBuffer commandBuffer, @Nullable @NativeType("VkVertexInputBindingDescription2EXT const *") VkVertexInputBindingDescription2EXT.Buffer pVertexBindingDescriptions, @Nullable @NativeType("VkVertexInputAttributeDescription2EXT const *") VkVertexInputAttributeDescription2EXT.Buffer pVertexAttributeDescriptions) {
nvkCmdSetVertexInputEXT(commandBuffer, remainingSafe(pVertexBindingDescriptions), memAddressSafe(pVertexBindingDescriptions), remainingSafe(pVertexAttributeDescriptions), memAddressSafe(pVertexAttributeDescriptions));
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,786 |
The Little Things ist ein US-amerikanischer Krimi-Thriller von John Lee Hancock, in dem Denzel Washington und Rami Malek als Polizisten mit gegensätzlichem Charakter und Vorgehensweise gemeinsam einen Serienmörder jagen.
Warner Bros. Pictures veröffentlichte den Film am 29. Januar 2021 in den US-Kinos und gleichzeitig auf seiner Streaming-Plattform HBO Max. In Deutschland erschien der Film am 8. Juli 2021 in den Kinos sowie bei einigen Streaminganbietern als exklusiver Download.
Handlung
Der ausgebrannte Sheriff Deputy Joe "Deke" Deacon aus dem kalifornischen Kern County muss sich mit dem jungen Kollegen Baxter vom Los Angeles County Sheriff's Department zusammenraufen, um einen Serienkiller zu stoppen.
Eines Nachts im Jahr 1990 fährt ein Mädchen auf einer Autobahn und wird von einem Autofahrer verfolgt. Nachdem sie an einer verlassenen Tankstelle angehalten hat, erregt sie die Aufmerksamkeit eines vorbeifahrenden Lastwagenfahrers und entkommt ihrem Verfolger.
Einige Zeit später wird Joe "Deke" Deacon, stellvertretender Sheriff von Kern County in Bakersfield, zum Los Angeles County Sheriffs Department geschickt, um Beweise für einen kürzlich begangenen Mord zu sammeln. Deacon, ein ehemaliger Detective des Sheriffs von L.A., begleitet den kürzlich ernannten leitenden Detective Jimmy Baxter zum Schauplatz eines neuen Mordes in L.A. Deacon bemerkt Ähnlichkeiten zwischen der Vorgehensweise bei diesem Mord und der Vorgehensweise bei einem alten Serienmordfall, den er nicht lösen konnte.
In dieser Nacht wird eine Frau namens Ronda Rathbun beim Joggen von einem Auto verfolgt und am nächsten Morgen als vermisst gemeldet. Baxter erfährt von Revierleiter Farris, dass Deacon sich scheiden ließ und aufgrund seiner Besessenheit von dem ungelösten Fall einen Herzinfarkt erlitt. Baxter wird geraten, Deacon nicht weiter einzubeziehen, aber Deacon nimmt Urlaub, um bei der Lösung von Baxters Fall zu helfen.
In der nächsten Nacht entdeckt die Polizei die Leiche eines weiteren Opfers, das unter einer Brücke angespült wurde. Baxter erfährt, dass die Vorgehensweise mit der des früheren Mordes und der anderen übereinstimmt: Die Opfer waren allesamt Prostituierte, die erstochen wurden. Deacon beginnt, gegen Albert Sparma zu ermitteln, einen Verdächtigen, der in einem Reparaturgeschäft in der Nähe der Morde arbeitet. Deacon nimmt die Verfolgung von Sparma auf, wird aber ausgebremst, so dass er Sparma zum Verhör festnimmt. Während des Verhörs verhöhnt Sparma die Ermittler und wird freigelassen, nachdem er Deacon zu einem Wutausbruch provoziert hat. Die junge Frau, die in der Wüste verfolgt wurde, wird befragt, hat aber Sparma in Handschellen auf der Polizeiwache gesehen, was ihre Eignung als objektive Zeugin, ihn als Verdächtigen zu identifizieren, beeinträchtigt.
Das FBI wird innerhalb einer Woche die Ermittlungen übernehmen, so dass Deacon und Baxter weniger Zeit haben, den Fall zu lösen. Farris teilt Baxter mit, dass Sparma acht Jahre zuvor einen Mord gestanden hat, den er unmöglich begangen haben kann, und daher ein unwahrscheinlicher Verdächtiger ist. Sparmas Fingerabdruck ähnelt dem des Mörders, ist aber keine eindeutige Übereinstimmung. Baxter und Deacon gehen zu Sparmas Wohnung und führen eine illegale Durchsuchung durch, finden aber nichts Belastendes. Als Deacon die Wohnung durchsucht, wird sein tragbarer Polizeiscanner aktiviert, und er entkommt nur knapp aus Sparmas Wohnung, nachdem Sparma eine Meldung über einen angeblichen verwundeten Beamten an dieser Adresse abgesetzt hat. Die Polizei trifft ein und Deacon entkommt, während Baxter sieht, wie Sparma die Tortur beobachtet.
Während Baxter Sparma überwacht, stellt er den Verdächtigen allein und verlangt, Rathbuns Aufenthaltsort zu erfahren. Sparma bietet ihm an, ihn dorthin zu fahren, wo er angeblich Rathbuns Leiche versteckt hat. Baxter willigt vorsichtig ein, während Deacon ihm heimlich folgt. Sparma bringt Baxter in ein abgelegenes Gebiet in der Wüste und lässt ihn mehrere Löcher graben, bevor er Baxter erzählt, dass er nie jemanden getötet hat. Baxter ist skeptisch und gräbt weiter. Sparma beginnt, ihn zu verspotten, bis Baxter ausrastet und ihm mit der Schaufel ins Gesicht schlägt, was ihn tötet. Als Deacon eintrifft, zeigt eine Rückblende, dass er versehentlich den einzigen Überlebenden seines letzten Mordfalls erschossen hat und dass Farris und Dunigan, der Gerichtsmediziner, geholfen haben, den Fall zu vertuschen. Deacon weist Baxter an, Sparma in der Wüste zu vergraben. Deacon verbringt die Nacht damit, alles in Sparmas Wohnung einzusammeln und sein Fahrzeug zu entsorgen. Als er am nächsten Morgen in die Wüste zurückkehrt, stellt er fest, dass Baxter Sparma nicht begraben hat, sondern immer noch nach dem Opfer sucht und verzweifelt glaubt, Sparma sei der Mörder. Deacon (dessen Wachträume von den früheren Opfern heimgesucht werden) rät Baxter, den Fall zu vergessen, da er ihn sonst ein Leben lang verfolgen werde.
Später erhält Baxter zu Hause einen Umschlag von Deacon, in dem sich eine rote Haarspange befindet, die derjenigen gleicht, die Ronda Rathbun trug, als sie entführt wurde. Zurück in Kern County verbrennt Deacon alles, was er in der Wohnung gesammelt hat, zusammen mit einer brandneuen Packung Haarspangen, in der die rote fehlt.
Produktion
Im März 2019 wurde bekannt, dass John Lee Hancock Regie führen und das Drehbuch schreiben und dass Denzel Washington eine der Hauptrollen spielen würde. Im Mai 2019 wurde Rami Malek für eine weitere Hauptrolle gecastet. Im August wurde erstmals Jared Leto als möglicher Antagonist genannt.
Die Dreharbeiten begannen Anfang September 2019 in Los Angeles und endeten im Dezember 2019. Bis einschließlich Oktober 2019 wurden die Nebenrollen besetzt.
Synchronisation
Die deutschsprachige Synchronisation entstand bei der FFS Film- & Fernseh-Synchron GmbH in Berlin unter Dialogbuch von Tobias Neumann und der Dialogregie von Tobias Meister. Nach dem Tod von Washingtons Standardstimme Leon Boden wurde dessen Part erstmals mit Sven Brieger besetzt.
Rezeption
Altersfreigabe
In den USA erhielt der Film von der MPAA ein R-Rating, was einer Freigabe ab 17 Jahren entspricht.
Kritiken
Insgesamt stieß der Film bei den Kritikern auf geteiltes Echo. Etwa 44 % der bislang 263 Kritiken sind positiv gestimmt.
Auszeichnungen
Golden Globe Awards 2021
Nominierung als Bester Nebendarsteller (Jared Leto)
Screen Actors Guild Awards 2021
Nominierung als Bester Nebendarsteller (Jared Leto)
Weblinks
The Little Things – Offizieller Trailer von Warner Bros. DE bei YouTube (Video)
The Little Things – Drehbuch zum Film (PDF, englisch)
Einzelnachweise
Filmtitel 2021
US-amerikanischer Film
Kriminalfilm
Thriller | {
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{"url":"https:\/\/labs.tib.eu\/arxiv\/?author=Matthias%20Schott","text":"\u2022 ### High magnetic fields for fundamental physics(1803.07547)\n\nVarious fundamental-physics experiments such as measurement of the birefringence of the vacuum, searches for ultralight dark matter (e.g., axions), and precision spectroscopy of complex systems (including exotic atoms containing antimatter constituents) are enabled by high-field magnets. We give an overview of current and future experiments and discuss the state-of-the-art DC- and pulsed-magnet technologies and prospects for future developments.\n\u2022 ### Design, Construction and Performance Tests of a Prototype MicroMegas Chamber with Two Readout Planes in a Common Gas Volume(1610.09539)\n\nOct. 29, 2016 physics.ins-det\nIn this paper, the design and the performance of a prototype detector based on MicroMegas technology with two detection planes in a common gas volume is discussed. The detector is suited for the forward region of LHC detectors, addressing the high-rate environment and limited available space. Each detection plane has an active area of 9x9 cm^2 with a two-dimensional strip readout and is separated by a common gas region with a height of 14 mm. A micro-mesh, working as a cathode, is placed in the middle of the common gas volume separating it into two individual cells. This setup allows for an angle reconstruction of incoming particles with a precision of 2 mrad. Since this design reduces the impact of multiple scattering effects by the reduced material budget, possible applications for low energy beam experiments can be envisioned. The performance of the prototype detector has been tested with a 4.4 GeV electron beam, provided by the test beam facility at DESY.\n\u2022 ### Determination of the muonic branching ratio of the W boson and its total width via cross-section measurements at the Tevatron and LHC(1607.05084)\n\nAug. 16, 2016 hep-ph, hep-ex\nThe total $W$-boson decay width $\\Gamma_W$ is an important observable which allows testing of the standard model. The current world average value is based on direct measurements of final state kinematic properties of $W$-boson decays, and has a relative uncertainty of 2%. The indirect determination of $\\Gamma_W$ via the cross-section measurements of vector-boson production can lead to a similar accuracy. The same methodology leads also to a determination of the leptonic branching ratio. This approach has been successfully pursued by the CDF and D0 experiments at the Tevatron collider, as well as by the CMS collaboration at the LHC. In this paper we present for the first time a combination of the available measurements at hadron colliders, accounting for the correlations of the associated systematic uncertainties. Our combination leads to values of $\\textrm{BR}(W\\rightarrow\\mu\\nu)=(10.72 \\pm 0.16)\\%$ and $\\Gamma_W = 2113 \\pm 31$ MeV, respectively, both compatible with the current world averages.\n\u2022 ### Determination of the Transverse Momentum of W Bosons in Hadronic Collisions via Forward Folding Techniques(1512.03276)\n\nDec. 10, 2015 hep-ex\nThe measurement of the transverse momentum of W bosons in hadron collisions provides not only an important test of QCD calculations, but also is an important input for the precision measurement of the W boson mass. While the measurement of the Z boson transverse momentum is experimentally well under control, the available unfolding techniques for the W boson final states lead generically to relatively large uncertainties. In this paper, we present a new methodology to estimate the W boson transverse momentum spectrum, significantly improving the systematic uncertainties of current approaches.\n\u2022 ### Construction of two large-size four-plane micromegas detectors(1511.03884)\n\nNov. 12, 2015 physics.ins-det\nWe report on the construction and initial performance studies of two micromegas detector quadruplets with an area of 0.5 m$^2$. They serve as prototypes for the planned upgrade project of the ATLAS muon system. Their design is based on the resistive-strip technology and thus renders the detectors spark tolerant. Each quadruplet comprises four detection layers with 1024 readout strips and a strip pitch of 415 $\\mu$m. In two out of the four layers the strips are inclined by $\\pm$1.5$^{\\circ}$ to allow for the measurement of a second coordinate. We present the detector concept and report on the experience gained during the detector construction. In addition an evaluation of the detector performance with cosmic rays and test-beam data is given.\n\u2022 ### Development and Study of a Micromegas Pad-Detector for High Rate Applications(1506.01164)\n\nJune 3, 2015 physics.ins-det\nIn this paper, the design and the performance of two prototype detectors based on Micromegas technology with a pad readout geometry is discussed. In addition, two alternative implementations of a spark-resistent protection layer on top of the readout pads have been tested to optimize the charge-up behavior of the detector under high rates. The prototype detectors consist of 500 pads with a size of 5x4 mm, each connected to one independent readout channel, and cover an active area of 10x10 cm. The design of these prototypes and its associated readout infrastructure was developed in such a way that it can be easily adapted for large-size detector concepts.\n\u2022 ### Diboson Production in Proton-Proton Collisions at $\\sqrt{s}=7$ TeV(1406.7731)\n\nAug. 11, 2014 hep-ex\nThis review article summarizes results on the production cross section measurements of electroweak boson pairs ($WW$, $WZ$, $ZZ$, $W\\gamma$ and $Z\\gamma$) at the Large Hadron Collider (LHC) in $pp$ collisions at a center-of-mass energy of $\\sqrt{s}=7$ \\TeV. The two general-purpose detectors at the LHC, ATLAS and CMS, recorded an integrated luminosity of $5fb^{-1}$ in 2011, which offered the possibility to study the properties of diboson production to high precision. These measurements test predictions of the Standard Model (SM) in a new energy regime and are crucial for the understanding and the measurement of the SM Higgs boson and other new particles. In this review, special emphasis is drawn on the combination of results from both experiments and a common interpretation with respect to state-of-the-art SM predictions.\n\u2022 ### Signal Characteristics of a Resistive-Strip Micromegas Detector with an Integrated Two-Dimensional Readout(1406.6871)\n\nJune 26, 2014 physics.ins-det\nIn recent years, micropattern gaseous detectors, which comprise a two-dimensional readout structure within one PCB layer, received significant attention in the development of precision and cost-effective tracking detectors in medium and high energy physics experiments. In this article, we present for the first time a systematic performance study of the signal characteristics of a resistive strip micromegas detector with a two-dimensional readout, based on test-beam and X-ray measurements. In particular, comparisons of the response of the two independent readout-layers regarding their signal shapes and signal reconstruction efficiencies are discussed.\n\u2022 ### Review of single vector boson production in pp collisions at $\\sqrt{s} = 7$ TeV(1405.1160)\n\nJune 18, 2014 hep-ph, hep-ex\nThis review summarises the main results on the production of single vector bosons in the Standard Model, both inclusively and in association with light and heavy flavour jets, at the Large Hadron Collider in proton-proton collisions at a center-of-mass energy of 7 TeV. The general purpose detectors at this collider, ATLAS and CMS, each recorded an integrated luminosity of $\\approx 40\\,{\\rm pb^{-1}}$ and $5\\,{\\rm fb^{-1}}$ in the years 2010 and 2011, respectively. The corresponding data offer the unique possibility to precisely study the properties of the production of heavy vector bosons in a new energy regime. The accurate understanding of the Standard Model is not only crucial for searches of unknown particles and phenomena but also to test predictions of perturbative Quantum-Chromo-Dynamics calculations and for precision measurements of observables in the electroweak sector. Results from a variety of measurements in which single W or Z bosons are identified are reviewed. Special emphasis in this review is given to interpretations of the experimental results in the context of state-of-the-art predictions.\n\u2022 ### Development and Performance of spark-resistant Micromegas Detectors(1111.0426)\n\nNov. 2, 2011 hep-ex, physics.ins-det\nThe Muon ATLAS MicroMegas Activity (MAMMA) focuses on the development and testing of large-area muon detectors based on the bulk-Micromegas technology. These detectors are candidates for the upgrade of the ATLAS Muon System in view of the luminosity upgrade of Large Hadron Collider at CERN (sLHC). They will combine trigger and precision measurement capability in a single device. A novel protection scheme using resistive strips above the readout electrode has been developed. The response and sparking properties of resistive Micromegas detectors were successfully tested in a mixed (neutron and gamma) high radiation field, in a X-ray test facility, in hadron beams, and in the ATLAS cavern. Finally, we introduced a 2-dimensional readout structure in the resistive Micromegas and studied the detector response with X-rays.\n\u2022 ### First Results of the Full-Scale OSQAR Photon Regeneration Experiment(1110.0774)\n\nOct. 6, 2011 hep-ex\nRecent intensive theoretical and experimental studies shed light on possible new physics beyond the standard model of particle physics, which can be probed with sub-eV energy experiments. In the second run of the OSQAR photon regeneration experiment, which looks for the conversion of photon to axion (or Axion-Like Particle), two spare superconducting dipole magnets of the Large Hadron Collider (LHC) have been used. In this paper we report on first results obtained from a light beam propagating in vacuum within the 9 T field of two LHC dipole magnets. No excess of events above the background was detected and the two-photon couplings of possible new scalar and pseudo-scalar particles could be constrained.\n\u2022 ### Determination of Integrated Luminosity via W and Z Boson Production with the ATLAS Detector(1108.2230)\n\nAug. 10, 2011 hep-ex\nThe possibility to determine the recorded integrated luminosity via the measurements of the W and Z boson production cross-sections with the ATLAS detector is discussed. The current results based on 2010 data are briefly summarized. Special attention is drawn to theoretical uncertainties of the measurement. The latter give a large contribution to the systematic uncertainties of the measurements. An outlook on the expected precision of an analysis based on 1fb-1 is given and the implications on a possible luminosity determination are discussed.\n\u2022 ### Z boson transverse momentum spectrum from the lepton angular distributions(1002.1850)\n\nFeb. 9, 2010 hep-ex\nIn view of recent discussions concerning the possibly limiting energy resolution systematics on the measurement of the Z boson transverse momentum distribution at hadron colliders, we propose a novel measurement method based on the angular distributions of the decay leptons. We also introduce a phenomenological parametrization of the transverse momentum distribution that adapts well to all currently available predictions, a useful tool to quantify their differences.","date":"2020-04-07 03:15:03","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\": 0.6956679224967957, \"perplexity\": 1123.6107948126555}, \"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-2020-16\/segments\/1585371665328.87\/warc\/CC-MAIN-20200407022841-20200407053341-00207.warc.gz\"}"} | null | null |
Documents mixing prose text with machine-executable code have been around for decades (at least as early as Knuth's tangle and weave programs to support literate programming in the early 1980s).
But the idea of mingling blocks of prose with functional blocks of documents is ancient, and characteristic of all of Claudius Ptolemy's works.
I'm experimenting with ways to represent Ptolemy's notions of "functional documents" using contemporary systems like Jupyter notebooks. | {
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction} \label{sec:introduction}
Given a reduction $\leq_R$ (such as Turing reduction $\leq_T$, or hyperarithmetic reduction $\leq_h$), we say that a perfect set $P$ is {\em $\leq_R$-pointed} if there is a perfect tree $S\subseteq 2^{<\omega}$ so that $[S]=P$ and for any $x\in P$, $S\leq_R x$, where $[S]=\{x\in 2^{\omega}\mid \forall n (x\upharpoonright n \in S)\}$. A perfect set $P$ is {\em uniformly $\leq_R$-pointed} if there is a perfect tree $S\subseteq 2^{<\omega}$ so that $[S]=P$ and for any $x\in P$, $S$ is $\leq_R$ reducible to $x$ with a fixed reduction. Sometimes we simplify $\leq_T$-pointed as pointed.
Andrew Marks made the following conjecture.
\begin{conjecture}[Marks \cite{PLutz21}]\label{conjecture: marks}
Assume that ${\mathrm{ZF}}+\mathrm{DC}+{\mathrm{AD}}$. Given any function $f:2^{\omega}\to 2^{\omega}$, there is a pointed set $[S]$ so that, restricted to $[S]$, either $f$ is constant or $f$ is injective.
\end{conjecture}
Some remarkable conclusions were derived from Conjecture \ref{conjecture: marks}. For example, Patrick Lutz \cite{{PLutz21}} showed that the conjecture implies that no nonprincipal ultrafilter on the Turing degrees is strictly below Martin measure in the Rudin-Keisler order.
The goal of this paper is to give a counterexample to the conjecture.
\bigskip
We assume that readers have some knowledge of descriptive set theory, recursion theory and algorithmic randomness theory.
\subsection{Set theory}
Our major reference of set theory is \cite{Jech03}. ${\mathrm{ZF}}$ is Zermelo-Fraenkel axiom system. $\mathrm{DC}$ is the axiom of dependent choice. ${\mathrm{AD}}$ is the axiom of determinacy. Throughout the paper, we work within ${\mathrm{ZF}}$.
A tree $S\subseteq 2^{<\omega}$ is a set downward closed. $[T]$ is the collection of infinite paths trough $T$. Given any $x\in \omega^{\omega}$ and natural number $n$, we use $x\upharpoonright n$ to denote an initial segment of $x$ with length $n$. In other words, $x\upharpoonright n$ is a finite string $\sigma\in \omega^{<\omega}$ of length $n$ so that for any $i<n$, $\sigma(i)=x(i)$.
\subsection{Recursion theory}
Our major references of recursion theory are \cite{Sacks90} and \cite{Ler83}.
We use $\leq_T$ to denote Turing reduction and $\leq_h$ to denote hyperarithmetic reduction. We use $\Phi^x$ denote a Turing machine with oracle $x$. Sometimes we also say that $\Phi^x$ is a recursive functional. We fix an effective enumeration $\{\Phi_e^x\}_{e\in \omega}$ of recursive functionals.
Given a real $x$, its corresponded Turing degree ${\mathbf{x}}$ is a set of reals defined as $\{y\mid y\equiv_T x\}$. We say ${\mathbf{x}}\leq \mathbf{y}$ if $x\leq_T y$. We use $\mathcal{D}$ to denote the set of Turing degrees. An {\em upper cone} of Turing degrees is a set $\{\mathbf{y}\mid \mathbf{y}\geq \mathbf{x} \}$ for some fixed $\mathbf{x}$.
\begin{theorem}[Jockusch and Simpson \cite{JoSi76}]\label{theorem: uniformly subset}
Any $\leq_T$ pointed set $P$ has a uniformly $\leq_T$-pointed subset.
\end{theorem}
Kleene's $\mathcal{O}$, which is a standard $\Pi^1_1$-complete set, is as defined in \cite{Sacks90}. $\omega_1^{\mathrm{CK}}$ is the least non-recursive ordinal and $\omega_1^x$ is the least ordinal not recursive in $x$.
We say a set $A$ ranges Turing degrees cofinally if for any real $x$, there is some $y\geq_T x$ in $A$. We use $x'$ to denote the Turing jump relative to $x$. More generally, if $\alpha<\omega_1^x$, then $x^{(\alpha)}$ is that $\alpha$-th Turing jump of $x$.
\begin{theorem}[Martin \cite{Martin76}]\label{theorem: friedman's conjecture}
For any $\Delta^1_1$ set $A\subseteq 2^{\omega}$, if $A$ has a nonhyperarithmetic real, then it has a $\leq_h$-pointed subset.
\end{theorem}
\subsection{Algorithmic randomness theory}
For the classical algorithmic randomness theory, see \cite{Niesbook09} and \cite{DH10}. A prefix-free Turing machine $M$ is a Turing machine so that for any input $\sigma\in 2^{<\omega}$, if $M(\sigma)$ halt, then $M(\tau)$ does not for any $\tau$ extending $\sigma$. A universal prefix-free Turing machine is a prefix-free Turing machine coding all the others.
A real $x$ is left-r.e. if there is a recursive increasing sequence rationals $\{q_s\}_{s\in \omega}$ so that $\lim_{s\to infty} q_s=x$.
\begin{theorem}[Chaitin \cite{Cha75}]\label{theorem: chaitin theorem}
Given any universal prefix-Turing machine $U$, $\Omega_U^x=\sum_{U(\sigma)\downarrow}2^{-|\sigma|}$ is a left-r.e. Martin-L\" of random real relative to $x$.
\end{theorem}
Sometimes, we simply say random instead of Martin-L\" of random, and $x$-random instead of Martin-L\" of random relative to $x$.
The following theorem is called van Lambalgen theorem.
\begin{theorem}[van Lambalgen \cite{van90}]\label{theorem: van lambalgen}
For any random reals $x$ and $y$, $x$ is $y$-random if and only if $y$ is $x$-random if and only if $x\oplus y$ is random.
\end{theorem}
\begin{theorem}[Ku\v{c}era \cite{Ku86}]\label{theorem: kucera}
If $x$ is random and left-r.e., then $x\equiv_T \emptyset'$, the Turing jump of $\emptyset$.
\end{theorem}
For the facts of higher randomness theory, see \cite{CY15book}. A real $r$ is $\Pi^1_1$-random if it does not belong to any $\Pi^1_1$-null set.
\section{The counterexample}
We present two methods to show that $\Omega^x_U$ is a counterexample to the Conjecture \ref{conjecture: marks}.
The first method is an application of higher randomness theory.
\begin{theorem}\label{theorem: main 1}
For any $\leq_T$-pointed set $[S]$ and universal prefix-free Turing machine $U$, there is a real $z\in [S]$ so that $\{y\in [S] \mid \Omega_U^y=\Omega_U^z\}$ has a $\leq_h$-pointed subset. So $\Omega^x_U$ is a counterexample to Conjecture \ref{conjecture: marks}.\footnote{Lutz proves that a corrected version of Conjecture \ref{conjecture: marks}. I.e. for any $f$, if for any pointed set $[S]$, $f([S])$ ranges Turing degrees cofinally, then $f$ is injective on some pointed set.}
\end{theorem}
\begin{proof}
Clearly $\Omega^x$ cannot be constant over any $\leq_T$-pointed set. Let $z\equiv_T \mathcal{O}^{(\mathcal{O}^S)}$ in $[S]$. Such a real $z$ exists since $[S]$ is a $\leq_T$-pointed set. Then $\Omega^z$ is $\Delta^1_1(\mathcal{O}^S)$- and so $\Pi^1_1(S)$-random (see \cite{CNY08}). Also note that $$\Omega^z\oplus S\leq_h z.$$ Let $$A=\{y\in [S]\mid \Omega_U^y=\Omega_U^z\}.$$ It is clear that $A$ is $\Delta^1_1(\Omega_U^z\oplus S)$ and $z\in A$. Since $\Omega_U^z$ is $\Pi^1_1(S)$ random, we have that $\omega_1^{\Omega_U^z\oplus S}=\omega_1^S$ by Sacks \cite{Sacks69}. So we have that $ z\not\leq_h \Omega^z\oplus S$ and so $$\Omega_U^z\oplus S<_h z.$$ Also since $[S]$ is pointed, we have that $$\Omega^z_U\oplus S=\Omega^y_U\oplus S\leq_h y$$ for any $y\in A$. Then by Theorem \ref{theorem: friedman's conjecture} relative to $\Omega_U^z\oplus S$, we have that $A$ has a $\leq_h$-pointed subset.
\end{proof}
After we provided the counterexample, Lutz asked the following question.
\begin{question}[Lutz \cite{PLutzemail}]
Given any $\leq_T$-pointed set $[S]$, are there two different reals $x\equiv_T y$ in $[S]$ so that $\Omega_U^x=\Omega_U^y$?
\end{question}
We give a positive answer to the question via classical randomness theory and so the second way to refute Conjecture \ref{conjecture: marks}. The basic ideas are from \cite{DHMN} and \cite{RS:08}.
Fix a pointed set $[S]$ and universal prefix-free Turing machine $U$. Clearly there is an $S$-recursive function $f:2^{<\omega} \to 2^{<\omega}$ so that $\hat{f}: x\mapsto \bigcup_n f(x\upharpoonright n)$ is a homeomorphism from $2^{\omega}$ to $[S]$. Let $\hat{S}_1\subset 2^{<\omega}$ be an $S$-recursive tree so that
\begin{itemize}
\item $\mu([\hat{S}_1])>0$; and
\item $[\hat{S}_1]$ only contains $S$-random reals.
\end{itemize}
Set $$r=\inf\{\Omega_U^{\hat{f}(x)}\mid x\in [\hat{S}_1]\}.$$ Clearly $r$ is left-r.e. relative to $S$ and so $r\oplus S\equiv_T (S)'$. Since $[\hat{S}_1]$ is compact, there is a real $x_0\in [\hat{S}_1]$ so that $r=\Omega_U^{\hat{f}(x_0)}$.\footnote{This can be proved by a standard technique in recursion theory but may not be so clear to the readers not familiar with it. The proof of Theorem 6.1 in \cite{DHMN} contains more details. } Since $[S]$ is pointed, we have that $$\hat{f}(x_0)\equiv_T S\oplus x_0.$$ So $r$ is random relative to $S\oplus x_0$ and so to $S$. By van Lambalgen Theorem \ref{theorem: van lambalgen} and Theorem \ref{theorem: kucera} relative to $S$, we have that $x_0$ is random relative to $S\oplus r\equiv_T (S)'$. Since $r$ is left-r.e. relative to $S$, the set $$F^U_r=\{y \mid \Omega_U^{\hat{f}(y)}=r\}$$ is $\Pi^0_2(S)$ and so measurable. Since $x_0\in F^U_r$ is $S'$-random, $F^U_r$ must have positive measure.
So there must be some reals $y_0,y_1\in F^U_r$ so that $y_0\equiv_T y_1$ (otherwise, $F^U_r$ must be null). Then $\hat{f}(y_0)\oplus S\equiv_T y_0\oplus S\equiv_T y_1 \oplus S\equiv_T \hat{f}(y_1)\oplus S$ since $f\leq_T S$ and $\hat{f}$ is a homeomorphism. By the pointedness of $S$, we have that $\hat{f}(y_0)\geq_T S$ and $\hat{f}(y_1)\geq_T S$. Thus $$\hat{f}(y_0) \equiv_Ty_0\oplus S\equiv_T y_1 \oplus S\equiv_T \hat{f}(y_1).$$
So $$f(y_0)\equiv_T f(y_1),\Omega^{f(y_0)}_U=\Omega^{f(y_1)}_U \mbox{ but } f(y_0)\neq f(y_1).$$
Hence we have the following theorem.
\begin{theorem}\label{theorem: classcial proof}
Given any universal prefix-free Turing machine $U$ and any pointed set $[S]$, there are two different reals $z_0$ and $z_1$ in $[S]$ so that $\Omega^{z_0}_U=\Omega^{z_1}_U$ and $z_0\equiv_T z_1$.
\end{theorem}
\section{On degree invariantness}
A function $f:2^{\omega}\to 2^{\omega}$ is {\em degree invariant} if $\forall x\forall y(x\equiv_T y \implies f(x)\equiv_T f(y))$. The following question is open to us.
\begin{question}\label{question: degree invaraint}
Is it consistent with ${\mathrm{ZF}}+\mathrm{DC}$ that there is a degree invariant Borel function $f$ so that for any $x$, $f(x)$ is random relative to $x$?
\end{question}
Clearly Question \ref{question: degree invaraint} is related to Martin's conjecture.\footnote{\cite{PLutz21} contains an up-to-date survey concerning Martin's conjecture.} One may wonder whether Chaitin's $\Omega$ can be served as a solution to the question. But in \cite{DHMN}, it has been shown that $\Omega_U^x$ is not degree invariant for any universal prefix-free Turing machine $U$. The following results says that $\Omega_U^x$ fails to be degree invariant in a very strong sense.
\begin{proposition}\label{proposition: turing incomparable}
Given any universal prefix-free Turing machine $U$ and pointed set $[S]$, there are two reals $z_0,z_1\in [S]$ so that $z_0\equiv_T z_1$ but $\Omega^{z_0}_U$ is Turing incomparable with $\Omega^{z_1}_U$.
\end{proposition}
\begin{proof}
The proof is based on the proof of Theorem 6.7 in \cite{DHMN} via a pushout-pullback method. We follow the notations in the proof of Theorem \ref{theorem: classcial proof}.
By Theorem \ref{theorem: uniformly subset}, we may assume that $[S]$ is uniformly pointed. So we may assume that there is a recursive function $\Phi$ so that for any $x\in [S]$, $\Phi^x=S$. Then there is another recursive functional $\Psi$ so that for any $x\in [S]$, $\Psi^x=f^{-1}(x)$. Now let $V$ be another universal prefix-free Turing machine so that for any $\sigma\in 2^{<\omega}$ and real $x$, $V^x(0\sigma)=U^x(\sigma)$ and $V^x(1\sigma)=M^{\Psi^x}(\sigma)$, where $M^{z}$ is a prefix-free Turing machine so that $\Omega_{M }^z=\sum_{M^z(\sigma)\downarrow}2^{-|\sigma|}=z$. Then for any real $x\in [S]$, $$\Omega_{V}^x=\frac{\Omega_U^x+\hat{f}^{-1}(x)}{2}.$$
By replacing $U$ with $V$ in the proof of Theorem \ref{theorem: classcial proof}, there is another left-r.e real $r_0$ relative to $S$ so that $$F^V_{r_0}=\{y\in 2^{\omega}\mid \Omega_V^{\hat{f}(y)}=r_0\}$$ has positive measure. Let $z\geq_T S'$ and $y \in F^V_{r_0}$ be any real random relative to $z$, then $$\Omega_U^{\hat{f}(y )}=2 \Omega_V^{\hat{f}(y_0)}-\hat{f}^{-1}(\hat{f}(y ))=2r_0-y .$$
Since $r_0\leq_T S'\leq_T z$ and $y$ is $z$-random, we have that $\Omega_U^{\hat{f}(y )}$ must also be $z$-random. So the set $$G_z^U=\{y\mid \Omega_U^{\hat{f}(y)}\mbox{ is }z\mbox{-random}\}$$ has positive measure.
Now let $$\hat{F}^U_r=\{y^*\mid \exists y\in F^U_r (y\equiv_T y^*)\}\mbox{ and } \hat{G}^U_z=\{y^*\mid \exists y\in G^U_z(y\equiv_T y^*)\}.$$
Then both the sets have measure 1 and so there must be $s_0\in \hat{F}^U_r$ and $s_1\in \hat{G}^U_r$ so that $s_0\equiv_T s_1$. Then we have that $$\hat{f}(s_0) \equiv_Ts_0\oplus S\equiv_T s_1 \oplus S\equiv_T \hat{f}(s_1).$$ Since $z\geq_T S'\geq_T r$, we have that $\Omega_U^{\hat{f}(s_1)}$ must be $r$-random and so Turing incomparable with $r=\Omega_U^{\hat{f}(s_0)}$ by van Lambalgen Theorem \ref{theorem: van lambalgen}. Set $z_0=f(s_0)$ and $z_1=f(s_1)$. They are as required.
\end{proof}
By combining Theorem \ref{theorem: classcial proof}, Proposition \ref{proposition: turing incomparable} and Borel determinacy \cite{Martin75}, we have the following conclusion.
\begin{corollary}
Assuming ${\mathrm{ZF}}+\mathrm{DC}$, there is a real $x$ so that for any universal prefix-free Turing machine $U$ and any real $y\geq_T x$, there are three different reals $z_0\equiv_T z_1 \equiv_T z_2\equiv_T y$ so that $\Omega_U^{z_0}= \Omega_U^{z_1}$ is Turing incomparable with $\Omega_U^{z_2}$.
\end{corollary}
\begin{proof}
We follow the notations in the proof of Proposition \ref{proposition: turing incomparable}. Given any pointed set $[S]$ and universal prefix-free Turing machine $U$, by the proofs of Theorem \ref{theorem: classcial proof} and Proposition \ref{proposition: turing incomparable} (we assume $z\geq_T S'\geq_T r$), both $F^U_r$ and $G^U_z$ have positive measure. Then it is clear that the Borel set $$\{y\in F^U_r \mid \exists y^*\equiv_T y(y^*\neq y\wedge y^*\in F^U_r)\}$$ must also have positive measure. So there must be three different reals $y_0\equiv_T y_1\equiv_T y_2$ so that $y_0,y_1\in F^U_r$ and $y_2\in G^U_z$. So $f(y_0)\equiv_T f(y_1)\equiv_T f(y_2)\geq_T S$ are three different reals so that $\Omega_U^{f(y_0)}= \Omega_U^{f(y_1)}=r$ is Turing incomparable with $\Omega_U^{f(y_2)}$. Thus the set
\begin{multline*}B_U=\{y\mid \mbox{There are three different reals }z_0\equiv_T z_1 \equiv_T z_2\equiv_T y\mbox{ so that }\\ \Omega_U^{z_0}= \Omega_U^{z_1}\mbox{ is Turing incomparable with }\Omega_U^{z_2}\}\end{multline*} is a Borel set of Turing degrees that ranges Turing degrees cofinally. By the Borel determinacy, $B_U$ contains an upper cone of Turing degrees. But there are only countably many such Turing machines. So $$\bigcap\{B_U\mid U\mbox{ is a universal prefix-free Turing machine}\}$$ contains an upper cone of Turing degrees.
\end{proof}
\bigskip
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,981 |
\section{Introduction}
The main motivation of the present work was to study the effect of
tropomyosin/troponin (Tm/Tn) binding on the bending stiffness of actin
filaments. The semiflexible filaments with a length of 400\,\AA\ bind along the
groove between the two twisted strands of the actin filament
(Figure~\ref{bin}). The binding is influenced by the presence of Ca$^{++}$.
This effect is of great biological significance, since it plays a central role
in the regulation of the myosin-actin coupling in muscles.
\begin{figure}
\narrowtext \epsfxsize=0.9\columnwidth \epsfbox{fig1.eps}
\vspace{0.2truecm}
\caption{Model of Ca$^{++}$ mediated Tm/Tn binding to actin filaments.
Tm/Tn is known to bind along the groove of the actin filaments.}\label{bin}
\end{figure}
Actin, when polymerized in vitro, forms semiflexible macromolecules of contour
lengths $L$ up to some 30 $\mu$m and with a persistence length $L_p$ of at
least some $\mu$m. The system we are interested in is a semidilute solution of
these macromolecules, i.e.\ an entangled network, where the single actin
filaments interact strongly but the free volume is still much larger than the
excluded volume. As a consequence of the great extension and large persistence
length of the molecules, as compared to their lateral diameter $a$ (some nm),
the semidilute regime is unusually large. We probe the samples with scattering
wavelengths $\lambda=2\pi/k$ that are somewhat shorter than the mesh size
$\xi$ of
the network, which, in turn, is shorter than the persistence length $L_p$;
i.e.\ the condition
\begin{equation}\label{skaltrenn}
a\ll\lambda < \xi \ll L_p,\, L
\end{equation}
is fulfilled for all of our samples. An electron micrograph of a typical
system under study is reproduced in Figure~\ref{mic}.
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig2.eps} \vspace{0.2truecm}
\caption{Electron micrograph of a \protect $0.4 \frac{\rm mg}{\rm ml}$
actin solution polymerized in vitro. The bar indicates the length of 1
$\mu$m. We probe the samples with scattering wavelengths \protect
$\lambda=2\pi/k$ that are somewhat shorter than the mesh size \protect $\xi$
of the network, which, in turn, is shorter than the persistence length
\protect $L_p$. The global motion of all polymers is strongly hindered by the
entanglement.}\label{mic}
\end{figure}
The bending stiffness of a filament-like macromolecule may be inferred from an
analysis of its conformational dynamics. Therefore various experimental
techniques have been applied to investigate the dynamical properties of actin
\cite{jan94}. Solutions and gels were probed by high sensitivity rheology
using torsional \cite{mul91} and magnetic bead rheometers \cite{zie94}. K\"as
et al.\ \cite{kas93} and others \cite{ott93} analyzed single labeled filaments
by microfluorescence microscopy combined with dynamic image processing. There
have also been several attempts to establish QELS as a quantitative method to
probe the static and dynamical properties of semiflexible macromolecules (for
critical discussions of the literature see e.g.\ Refs.\
\cite{son91,ara91,har95}). But in contrast to the dynamic properties of
flexible polymers the dynamics of semiflexible polymers is not yet very well
understood. This is mainly a consequence of the difficulties caused by the
rigid constraint of a (virtually) unextensible contour. Models that try to
represent the unextensible contour honestly \cite{ara85,ara91} have to deal
with considerable technical difficulties. On the other hand, models that relax
the constraint like the various modified versions of the so called
Harris-Hearst-Beals-model \cite{har66} (for a short summary see \cite{har94})
allow for artificial stretching modes and predict a gaussian probability
distribution for the spacial distances of the contour elements. As the most
promising model with relaxed constraint we regard the model described in
Refs.\
\cite{lag91,har94,har95}. We will neither relax the constraint, nor do we need
all the machinery developed by Arag\'on and Pecora in Refs.\
\cite{ara85,ara91}. (However, one of our results, Eq.\ \r{res}, may be
obtained by their method \cite{krotp}.) In the special case, we are interested
in, we profit from some simplifications brought about by the separation of
length scales, Eq.\ \r{skaltrenn}, which in turn gives rise to a hierarchy of
time scales: QELS measures the temporal decay of configurational correlations
of the filaments. For long semiflexible actin filaments in entangled networks
the internal configurational dynamics dominates over the center of mass and
rotational motion of the molecules in time intervals typically probed by QELS.
In addition, we can restrict ourselves to the weakly bending rod limit, where
one can approximately take the undulations to be transverse to the mean contour
\cite{mae84,son91}, which is virtually fixed for the relevant time scales. The
decay rate of the dynamic structure factor becomes thus accessible to a simple
physical interpretation in terms of the bending modes only, i.e.\ the local
bending modes may be studied ``in isolation". This is a great practical
advantage compared to the general case.
But the time decay of the structure factor also depends on the strength of
hydrodynamic correlations. To obtain quantitative results it is therefore
necessary to take into account the screening of the hydro\-dynamic
self-interaction of the filaments in semidilute solutions (or their finite
length in the dilute case). As explained in Section~\ref{str} this is most
simply achieved by means of a concentration dependent renormalized friction
coefficient, $\zeta_{\perp}$.
In contrast to the expression for the dynamic structure factor for intermediate
times, which takes a simple form only after some approximations, its initial
slope, resulting from the quasi-free Brownian fluctuations about the
equilibrium configuration may be calculated exactly if Eq.\ \r{skaltrenn}
holds. The result is compared with previous experimental data in
Section~\ref{ins}. The good agreement of the theoretical prediction with the
available data confirms our hypothesis that the intrinsic rodlike structure of
polymers is a significant detail, which influences the dynamic properties.
Moreover, the microscopic hydrodynamic diameter $a$ of the filament, which
enters the dynamic structure factor as a parameter, also appears in the
expression for the initial decay rate and may thus be determined by a
measurement of the initial slope. Section~\ref{mat} lists materials and
methods, and the new experimental results are presented in Section~\ref{exp}.
Finally we discuss possible refinements to our approach in Section~\ref{dis}.
\section{Theoretical Background of QELS from Semiflexible Polymers}\label{the}
\subsection{Stretched Exponential Decay of the Dynamic Structure
Factor}\label{str}
The dynamic structure factor of a chain of length $L=N\Delta s$ with $N$
segments located at $\bm r_n$ ($n=1,\dots ,N$) is defined by
\begin{equation}\label{def} g(\bm{k},t)=\frac1N \sum_{n,m}\langle
\exp[i\bm{k}(\bm{r}_n(t)-\bm{r}_m(0))] \rangle \, .
\end{equation}
The brackets $\langle \dots \rangle$ denote the ensemble average over all chain
conformations, and \bm k is the scattering wave vector. On the typical length
scales probed by QELS ($0.1\sim 1$ $\mu$m) the conformational dynamics of actin
filaments are dominated by the bending undulations. Consequently, we take the
bending energy, given by the contour integral $\int_Lds$ over the local
curvature multiplied by the bending modulus $\kappa$,
\begin{equation}\label{ene} E[\bm{r}_s]=\frac{\kappa}2 \int_L
ds\Bigl(\frac{\partial^2 \bm{r}}{\partial s^2}\Bigr)^2 \, ,
\end{equation} to
be the only relevant energetic term in a canonical description. This implies
that the static mean square end-to-end-distance is given by the well known
Kratky-Porod formula \cite{kra49} with the persistence length $L_p
=\kappa/k_BT$. The time decay of the structure factor $g(\bm{k},t)/g(\bm{k},0)$
is not easily calculated in the most general case. However, the particular case
under study allows for simplifications, which enable us to adapt some of the
ideas common in the theory of flexible polymers \cite{gen67,doi86}. We start
from the Langevin equation for a single polymer
\begin{equation}\label{lan}
\frac{\partial}{\partial t} \bm{r}_s(t)= \int_L ds'\bm H_{\perp}(\bm r_s,\bm
r_{s'}) \left( -\frac{\delta}{\delta \bm{r}_{s'}} E[\bm{r}_{s'}]
+\bm{f}_{s'}\right)\, ,
\end{equation}
where $\bm{f}_s$ is the stochastic force (white noise) and $\bm H_{\perp}$ is
an effective mobility matrix, which takes into account the solvent-mediated
self-interaction of the filament. This can be understood in analogy to the
usual Oseen tensor. However, the effective reduction of the degrees of freedom
due to the rigid constraint of a fixed contour length requires that the local
longitudinal motion be projected out. In the weakly bending rod limit this is
practically achieved by a suitable choice of coordinates: longitudinal
distances are kept fixed, and the bending undulations are described by
transverse coordinates (cf.\ Eq.~\r{mob} below). This is how one would address
the problem in analogy to the classical Zimm model for flexible polymers, and
how it was attempted previously in Ref.~\cite{far93}. We shall neglect the
center of mass and rotational motion, which are slow compared to the internal
dynamics of the molecule as a consequence of the scale separation Eq.\
\r{skaltrenn}, the entanglement and hydrodynamic screening.
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig3.eps} \vspace{0.2truecm}
\caption{Flexible filaments interact mostly with themselves, whereas
semiflexible filaments interact with each other even at quite low
concentrations. The perturbation of a single semiflexible filament by its
surrounding may be modeled by a screening of the hydrodynamic
self-interaction.}\label{sem}
\end{figure}
For the following it is important to realize that there is a profound
difference between the hydrodynamic interaction of flexible polymers and
semiflexible polymers in semidilute solutions. A flexible polymer is coiled and
is thus far more likely to interact with itself than with surrounding polymers.
On the other hand, semiflexible filaments like actin are much more stretched
and strongly interact with each other even
at quite low concentrations (Figure~\ref{sem}).
As the scattering vector $\bm k$ in our experiments is large enough in
magnitude to resolve single actin filaments (see Eq.\ \r{skaltrenn}), we are
interested in the dynamics of a single filament and use a mean field
approximation to model the hydrodynamic interaction with the surrounding,
i.e.\ we introduce a screening of the hydrodynamic self-interaction along a
single polymer. This is achieved by use of the (preaveraged) screened
transverse mobility matrix,
\begin{equation}\label{mob}
\bm H_{\perp}(\bm r) =\frac{e^{- r/\Lambda}}{8\pi\eta r}\left(\bm 1 -
\frac{|\bm{r}\rangle\langle\bm{r}|}{r^2}\right) ,
\end{equation}
where $\bm r:=\bm r_s-\bm r_{s'}$, $\Lambda$ is the screening length and $\eta$
the solvent viscosity. Beyond this length $\Lambda$, the hydrodynamic
self-interaction of a filament is weak and correlations decay rapidly. The
explicit form of the projector is a consequence of modelling the actin filament
as a straight rod on length scales smaller than $\Lambda$ with respect to the
hydrodynamic interaction. This should be a good approximation for $\xi \ll
L_p$.
To simplify the calculation we use a kinetic coefficient $\zeta_{\perp}^{-1}$,
\begin{equation}\label{sim}
(\zeta_{\perp}/L)^{-1}=\frac{\log \Lambda/a}{4\pi\eta} \, ,
\end{equation}
in place of the tensor $\bm H_{\perp}$. $\zeta_{\perp}$ is a renormalized
friction, obtained by taking the terms in parentheses in Eq.\ \r{lan} out of
the integral and averaging over all segment positions $s$ for a rigid rod of
diameter $a$. Replacing the mobility matrix by a simple coefficient amounts to
setting the effective friction for all modes equal to the friction of the
dominant mode of wavelength $\Lambda$. In this approximation interactions
between different modes are neglected and the mobility for the very short
wavelength modes is supposedly slightly overestimated. The latter should not
profit as much by correlations over distances larger than their wavelength, as
is implied by Eq.\ \r{mob}. This approximation will allow for a simple
expression [Eq. \r{res} below] for the structure factor and at the same time
captures the main effect of the hydrodynamic self-interaction of the single
polymers as well as their mutual interaction. A theoretical basis for the
above ansatz was already worked out by Muthukumar and Edwards \cite{mut83} and
will be explained in more detail in a forthcoming paper \cite{krotp}.
$\Lambda$, $a$ and the persistence length $L_p$ are the three characteristic
length scales in terms of which the dynamics of a semiflexible polymer in
semidilute solution is characterized. The microscopic cutoff parameter $a$
takes care of the finite thickness of the filament. Its value may be determined
experimentally by the QELS method as described in the next subsection. We
suppose that the hydrodynamic screening length $\Lambda$ may be identified (up
to a numerical factor) with the mesh size $\xi$ of the actin network,
\begin{equation}\label{mes}
\Lambda \simeq \xi \sim c_a^{-\frac12}.
\end{equation}
(Basically the same relation $\Lambda\sim c^{-1/2}$ can be derived within the
effective medium approach for rods \cite{mut83}.) It is through Eq.\ \r{mes}
that the actin concentration $c_a$ ultimately enters Eq.\ \r{res} for the
structure factor. The scaling law for the mesh size $\xi$ in Eq.\ \r{mes}
should be valid in the semidilute regime, when $\xi \leq L_p$ and the solution
appears as a random network of almost rodlike segments. This was indeed
confirmed experimentally \cite{sch89}. The dimensionless quantity $\Lambda/a$,
which could be called the ``hydrodynamic aspect ratio" of the polymer, is
actually the only characteristic parameter of the filament entering expression
\r{res} for the structure factor besides the persistence length $L_p$ (cf.\
Eq.\ \r{dec} below).
Detailed calculations \cite{krotp} show that the relaxation time of the
$p^{th}$ mode is $\tau_p=\frac{\zeta_{\perp}/L}{\kappa}\left(\frac L{\pi p}
\right)^4$, as one would guess from dimensional analysis, and that in the
intermediate time regime\footnote{The second condition is actually
modified by entanglement. This may indeed be exploited in future high
precision measurements, as is explained in \protect\cite{krotp}, but is of
minor importance here.}\label{taux},
\begin{equation}\label{tim}
(kL_p)^{-4/3}\gamma_k^{-1}\ll t\ll \tau_1,
\end{equation}
the structure factor factor is to a good approximation given by a
stretched exponential
\begin{equation}\label{res}
g(\bm{k},t)=g(\bm{k},0)
\exp\left( -\frac{2\Gamma(1/4)}{9\pi} (\gamma_k t)^{3/4} \right).
\end{equation}
The decay rate,
\begin{equation}\label{dec}
\gamma_k=\frac{k_BT}{\zeta_{\perp}/L}k^{8/3}L_p^{-1/3},
\end{equation}
depends on the filament stiffness through the persistence length $L_p$ and on
the actin concentration $c_a$ (or the mesh size) through the renormalized
friction coefficient $\zeta_{\perp}$. It should be compared with the initial
decay rate given below in Eq.\ \r{ini} for the dilute case. Both are expected
to play an important role in many dynamical problems, i.e.\ they are supposed
to constitute the characteristic time scales of motion.
Some remarks have to be made on the derivation and the domain of validity of
Eq.~\r{res}. First of all it is important to realize that the local motion of
the monomers is neither strictly isotropic (like in the random coil limit) nor
strictly transverse to the mean contour (as was assumed in Eq.\ \r{mob}). The
averaging over the different orientations can be performed in both limits
\cite{krotp}. Upon neglecting the anisotropy of the local segment motion, one
obtains the simple stretched exponential from of the dynamic structure factor,
Eq. \r{res}. In the opposite limit -- i.e.\ for local segment motion strictly
transverse with respect to the mean contour -- averaging over the different
orientations is performed after calculating the anisotropic dynamic structure
factor for an individual macromolecule, whose mean orientation and center of
mass position is assumed to be fixed in space and time. In our case this is
realized by the time scale separation mentioned above and by the fact that the
macromolecules are imbedded in a network. It is found that the resulting
dynamic structure factor is very similar to the stretched exponential form
obtained by neglecting the anisotropy of the segment motion in the averaging
procedure. We propose to use the stretched exponential form also in the
intermediate regime, where the segment motion is neither strictly transverse
nor isotropic. This allows for a simple fit to the experimental data and has
the advantage that the relevant physical mechanisms are not obscured by
complicated numerical analysis. But, as a consequence of the approximations
made, the numerical value of the prefactor in the exponential of Eq.~\r{res}
should be taken with some precaution. Unfortunately, the value obtained for
the persistence length by Eq.~\r{res} is very sensitive to this prefactor as
well as to all the experimental parameters entering the exponent because the
persistence length enters the expression for the decay rate, Eq.\ \r{dec}, as
$L_p^{1/3}$. In addition the actual prefactor in the scaling law, Eq.~\r{mes},
for $\Lambda$ is not known. Hence the method described here is not capable of
producing very accurate absolute values for the bending modulus so far. On the
other hand, relative changes in the stiffness, which may be caused by diverse
chemical or physical mechanisms are readily detected.
\subsection{Initial Slope of the Dynamic Structure Factor}\label{ins}
Although the dynamic structure factor $g(\bm k,t)$ is a complicated object, and
several physical assumptions and approximations enter its explicit calculation,
its initial slope may be evaluated exactly in the case of Eq.\ \r{skaltrenn}.
The general scheme of computation may be found in Ref.\ \cite{doi86} (see also
Ref.\ \cite{sch84} for a summary of predictions derived from various
semiflexible models). In the semiflexible case caution has to be paid to the
rigid constraint of constant contour length. It causes an effective reduction
of the degrees of freedom of the local quasi-free Brownian fluctuations about
the equilibrium configuration, which determine the initial slope for large
scattering vectors and moderate chain stiffness. (For very stiff molecules the
time regime for these quasi-free fluctuations diminishes \cite{son91}, but
actin is far from this limit as can be inferred from a comparison of the
characteristic time scales $\tau_p$, $\gamma_k$ and $\gamma_k^{(0)}$.) For the
detailed calculation of the initial decay rate,
\begin{equation}\label{gde}
\gamma_k^{(0)} :=-\left. \frac{d}{dt}\log g(\bm k,t)\right|_{t=0}\, ,
\end{equation}
for the case of a semiflexible polymer in semidilute solution we refer the
reader to Ref.\ \cite{krotp}. Here we give the asymptotic result for a dilute
solution and for scattering wave vectors $\bm{k}$ of modulus $k$ larger than
the inverse persistence length $L_p^{-1}$ but much smaller than the inverse of
the microscopic cutoff length $a$ introduced in Eq. \r{sim}:
\begin{equation}\label{ini}
\gamma_k^{(0)}= \frac{k_BT}{6\pi^2\eta}k^3\left(\frac56 -\log ka\right) \, .
\end{equation}
The deviation from ideal scaling is rather weak, hence it is useful to express
Eq.\ \r{ini} as a ``quasi-scaling law" $\gamma_k^{(0)}\sim k^{z(k)}$ with an
effective dynamic exponent (Figure~\ref{zvk}) given by
\begin{equation}\label{qsc}
z(k)=3\frac{6 \log ka -3}{6 \log ka -5} \, .
\end{equation}
It is quite striking that the microscopic cutoff $a$ appears here.
Hydrodynamic screening does not substantially alter the result of Eq.\ \r{ini}
in the large wave vector regime, but flattens the increase of
$\gamma_k^{(0)}/k^3$ for small wave vectors. However, the theory is not valid
for small scattering vectors, because scattering vectors smaller than the
inverse mesh size $\xi^{-1}$ do not resolve single filaments but average over
several molecules. Moreover, as a concequence of local fluctuations in the
mesh size there is scattering in the experimental data in the crossover region
$k\approx \xi^{-1}$.
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig4.eps}
\caption{Effective exponent $z(k)$ of the ``quasi-scaling"
law for the initial decay rate of the dynamic structure factor of actin. $z$
was predicted to be a universal number, $z=3$, for all flexible polymers by
classical scale invariant models. This was never observed experimentally. We
argue that the (intrinsic) semiflexibility of all real polymers is
responsible for the discrepancy. The dashed line was computed for a
semidilute solution of actin filaments ($c_a=0.16\, \frac{\rm mg}{\rm ml}$).
The solid line corresponds to the dilute case, Eq.\ \protect\r{qsc}. In the
typical $k$ intervall probed by QELS ($5\sim 30$ $\mu\rm m^{-1}$)
$z(k)\approx 2.7$ for actin.}\label{zvk}
\end{figure}
Figure~\ref{cor} shows a comparison of the computed initial decay rate
$\gamma_{k}^{(0)}$ for a solution of $0.16\, \frac{\rm mg}{\rm ml}$ actin
(dashed line) with previously measured data \cite{schdr,sch89}. A simple least
square fit of Eq.\ \r{ini} to the data gives $a = 5.4$ nm. $a$ represents an
effective hydrodynamic diameter of the filament to be compared with two times
the cross sectional radius of gyration $r_{\perp}^g$. The remarkable agreement
of $a$ with the value of $2 \, r_{\perp}^g=5.16 \pm 0.3$ nm determined by
different methods \cite{bre91,ege84} strongly supports the above ideas.
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig5.eps}
\caption{Correction to the classical prediction $\gamma_k^{(0)} \sim k^3$ for
the initial decay rate of the dynamic structure factor. The theoretical
predictions for dilute solutions (solid line) and semidilute solutions
(dashed line) are compared with experimental data of Schmidt
\protect\cite{schdr,sch89}. Also included is the prediction for gaussian
chains from Ref.\ \protect\cite{doi86}. The experimental data and the dashed
line both correspond to the same actin concentration $c_a=0.16\, \frac{\rm
mg}{\rm ml}$. The positions of the theoretical curves depend on the
effective hydrodynamic diameter $a$ of the filament [see Eq.\
\protect\ref{ini}]. It was used as fit parameter and found to be $a=5.4$
nm.}\label{cor}
\end{figure}
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig6.eps} \vspace{0.2truecm}
\caption{A real polymer is not a fractal (i.e.\ not a gaussian chain,
as often assumed for computational convenience) but rodlike on small scales.
This affects the dynamical properties of polymers.}\label{rod}
\end{figure}
Finally we would like to note that in classical neutron scattering experiments
with more or less flexible synthetic polymers the ratio of wavelength to
persistence length $\lambda/L_p$ is not very different from what is
encountered in light scattering from large biomolecules. So one expects
similar results in both cases. It is well known \cite{gensc,doi86} that there
is sometimes poor agreement between the experiments with synthetic polymers and
the classical theoretical predictions for the initial slope of $g(\bm{k},t)$
derived from scale invariant models. The excellent agreement achieved now with
actin networks provides strong evidence that the usual scale invariant polymer
models are not capable of describing quantitatively the dynamic properties of
real polymers, which, after all, are semiflexible at heart (Figure~\ref{rod}).
\section{Materials and Methods}\label{mat}
Actin was prepared from rabbit muscles according to Pardee and Spudich
\cite{par82} with an additional gel filtration step as suggested by
MacLean-Fletcher and Pollard \cite{mac80} using a Sephacryl S-300 HR column.
Tropomyosin/Troponin were prepared from the residue of rabbit muscle acetone
powder left after the actin extraction \cite{spu71}, and separated into
tropomyosin and troponin by hydroxyl apatite column chromatography
\cite{eis74}. The purity of the proteins was checked by SDS polyacrylamide gel
electrophoresis \cite{lam70} stained with Commassie Blue, and estimated to be
of at least 95\% purity. Actin was tested for its ability to polymerize by low
shear viscometry with the falling ball capillary apparatus as described by
Pollard and Cooper \cite{pol82}, and by fluorescence increase of 5\%
NBD-labeled actin \cite{det81}. Functionality of tropomyosin/troponin in the
presence and absence of Ca$^{++}$ was characterized by an actin binding test:
they were added to F-actin in varying concentration ratios, centrifuged at 100
000 g for 1 h, and analyzed by SDS gel electrophoresis.
Actin was stored in a buffer containing 2 mM imidazole, 0.2 mM ATP, 0.2 mM DTT,
0.2 mM CaCl$_2$ and 0.05 vol.\% NaN$_3$. For the polymerization of actin, a
buffer with 2 mM imidazole, 0.5 mM ATP, 2 mM MgCl$_2$, 100 mM KCl, 0.2 mM DTT
and 0.2 mM CaCl$_2$ was used. The buffers where adjusted to a pH of 7.4. For
the experiment without Ca$^{++}$ the CaCl$_2$ was left out and 1 mM EGTA was
added. The molar ratio of actin:tropomyosin:troponin was 7:1:1.
For QELS measurements, all solutions were freed from dust by sterile
filtration, mixed, and stored overnight at $4^{\circ}$ C to achieve an
equilibrium polymerisation state. Samples to be inspected by electron
microscopy were adsorbed on glow discharged carbon coated formvar films on
copper grids, and negatively stained with uranyl acetate.
The experimental setup for QELS has been described in detail previously
\cite{sch89,pie92}. We used the correlator ALV 3000 (ALV Langen) with 1024
linear channels to calculate the dynamic structure factor. The light source was
an Innova 70-4 argon-ion laser from Coherent with 200 mW for the 488 nm line.
\section{Results}\label{exp}
We now turn to a discussion of the experimental data and their analysis in
terms of the theory described above.
\subsection*{Stretched Exponential}
Figure~\ref{fit} shows a fit of the theoretical dynamic structure factor, Eq.\
\r{res}, to experimental data for a scattering angle of $90^{\circ}$
corresponding to $k= 24.2 \, \mu \rm m^{-1}$. Note that there is only one free
parameter, $\gamma_k$. An excellent fit of the experimental curves is obtained
in the time domain $10^{-5}\sim 10^{-2}$ s, for which the condition, Eq.
\r{tim}, is approximately fulfilled (cf.\ Footnote \protect\ref{taux}), whereas
a simple exponential decay is clearly ruled out. Hence the theory is well
suited to interpret our data within the present experimental accuracy.
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig7.eps} \vspace{0.2truecm}
\caption{Fit of theoretical dynamic structure factor Eq.\
\protect\r{res} to experimental data for $k= 24.2 \, \mu {\rm m}^{-1}$.
Clearly Eq.\ \protect\r{res} describes very well the experimental situation
in the time interval $10^{-5} \sim 10^{-2}$ s, wheras the simple exponential
fit is ruled out.}\label{fit}
\end{figure}
\subsection*{Bending Modulus}
As pointed out in Section~\ref{str} the decay rate $\gamma_k$, Eq.\ \r{dec}, is
determined by the bending modulus $\kappa$ and also by the screening length
$\Lambda$. In order to check the effect of the screening length on the decay
rate, we measured the dynamic structure factor for various actin concentrations
$c_a$. The screening length was taken to be equal to the mesh size $\xi$, which
was found previously \cite{sch89} to obey the scaling law $\xi\, [\mu \rm m]
= 0.35\sqrt{c_a \, [\frac{\rm mg}{\rm ml}]}$. The results are shown in
Figure~\ref{con}. As discussed in Section~\ref{ins}, the theory only applies
to scattering vectors large enough to resolve single filaments. Scattering
vectors $k$ smaller than the inverse mesh size $\xi^{-1}$ average over several
filaments. Hence, the decay of the structure factor obeys Eq.\ \r{dec} only
for large scattering vectors $k$. The onset of the deviation may be taken as a
lower bound for the mesh size.
Although the concentration was increased by a factor of 8 the derived values
for the bending modulus $\kappa$ are fairly consistent. They agree within a
standard deviation of 35\% for the entire range of scattering vectors and
concentrations. However, for a single scattering vector and fixed concentration
the deviations are considerably smaller, e.g.\ for $k=24.2 \, \mu \rm m^{-1}$
($90^{\circ}$) and $0.4 \, \frac{\rm mg}{\rm ml}$ the data are reproducible
within 5\%. Considering the discussion at the end of Section \ref{str} and the
dependence of the value of the bending modulus $\kappa$ derived from Eq.\
\ref{res} on the hydrodynamic aspect ratio $\Lambda/a$ -- and hence on our
choice of the screening length $\Lambda$ -- we are presently not able to
determine an accurate absolute value for $\kappa$. (An average value of $9.5
\cdot 10^{-27}$ Jm was obtained for $\kappa$.) On the other hand, relative
changes in the stiffness can be resolved rather precisely.
Methods to determine $\Lambda$
experimentally are discussed in Section~\ref{dis}.
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig8.eps} \vspace{0.2truecm}
\caption{Values for the
bending modulus $\kappa$ obtained for various actin concentrations $c_a$ in
the semidilute regime using Eq.\ \protect\r{res} and $\Lambda =\xi$. Only
large wave vectors $k\gg \xi^{-1}$ resolve single filaments and are
accessible to our theory.}\label{con}
\end{figure}
\subsection*{Effect of Tm/Tn Binding}
Figure~\ref{sti} shows the effect of the tropomyosin/troponin complex (Tm/Tn)
on the decay of the dynamic structure factor for actin solutions with a
concentration of $c_a= 0.3 \, \frac{\rm mg}{\rm ml}$ ($\simeq7.1\, \mu$M) at
$10^{\circ}$ C. Since Ca$^{++}$ is known to regulate the coupling of the
Tm/Tn-complex to actin, experiments were performed with and without Ca$^{++}$.
Figure~\ref{sti} clearly shows the appreciable decrease of the decay rate in
the presence of Tm/Tn. The reduction of Ca$^{++}$ appears to decrease the
stiffness slightly but the effect is too weak to be considered significant with
the present accuracy of measurement. The experiments were repeated several
times with different actin preparations and at two different temperatures
($10^{\circ}$ C and $25^{\circ}$ C) and the same absolute value of $\kappa$
as well as the same degree of stiffening by Tm/Tn was always observed.
The results of the measurements at $10^{\circ}$C are summarized in
Table~\ref{tro}.
$$\vbox{
\begin{table}[tb]
\begin{tabular}{|l|*{3}{r@{.}l|}}
\nobreak
scattering vector $k$ [$\mu\rm m^{-1}$] & {17}&1 & {24}&2 & {29}&6 \\
\hline
$\kappa$ for actin [$10^{-27}$ Jm] & 9&9 & 8&7 & 8&6 \\
\hline
$\kappa$ for actin + Tm/Tn $-{\rm Ca}^{++}$ & {11}&4 & {12}&9 & {14}&7 \\
\hline
$\kappa$ for actin + Tm/Tn $+{\rm Ca}^{++}$ & {12}&1 & {15}&9 & {13}&9 \\
\end{tabular}
\vspace{0.2truecm}
\caption{Summary of values obtained for the bending modulus $\kappa$ [Jm]
of actin in the presence of Tm/Tn with and without Ca$^{++}$ at different
scattering angles in comparison to pure actin.}\label{tro}
\end{table}}$$
To exclude preparation artifacts and in order to check whether Tm/Tn has some
effect on the mesh size, which would in turn affect the hydrodynamic screening
length $\Lambda$ and thus the derived value of $\kappa$, the actin network was
examined by negative-staining EM for all samples. We could not observe an
effect of Tm/Tn on the network structure. In summary, we find that in the
presence of Ca$^{++}$ Tm/Tn causes an increase of the bending modulus of
F-actin by about 50\%.
\begin{figure}[tb]
\narrowtext \hspace{0.005\columnwidth} \epsfxsize=0.9\columnwidth
\epsfbox{fig9.eps}
\vspace{0.2truecm}
\caption{Effect of Tm/Tn binding with Ca$^{++}$ (0.2 mM uppermost curve) and
without Ca$^{++}$ (closely below) in comparison with the pure actin sample
(lower curve) observed at a scattering angle of $90^{\circ}$, temperature
$T=10^{\circ}$ C and actin concentration $c_a=0.3 \, \frac{\rm mg}{\rm
ml}$.}\label{sti}
\end{figure}
\section{General Discussion}\label{dis}
The present analysis of the dynamic structure factor of semidilute entangled
actin solutions shows that QELS is a reliable tool for the study of the
internal dynamics of semiflexible polymers. However, as may be seen from Eq.\
\r{dec}, the interpretation of experimental data is not entirely
straightforward. The expression for the dynamic structure factor \r{res} is
ambiguous with respect to the microscopic source of an observed change in the
decay rate. Changes may be caused by variations in the intrinsic stiffness of
the filament as well as by variations of the mesh size of the network,
which in turn affect the screening length $\Lambda$. As experiments with
$\alpha-$actinin show \cite{gottp}, even a local change in the mesh size, as
caused by such crosslinking proteins, results in a corresponding change of the
decay rate over the whole range of scattering vectors. This is exactly what
one would expect from Eq.\ \r{dec}, if the main contribution to the scattered
light is attributed to the crosslinked clusters (seen in EMs), which contain
most of the filaments.
The main uncertainty in the quantitative predictions of the QELS method
described above presently arises from the uncertainty in the absolute value of
the hydrodynamic screening length $\Lambda$. Setting it equal to the mesh size
seems to be reasonable, but perhaps one could do better. There are various
methods to determine $\Lambda$. The QELS method itself as described above
could be used, if $\kappa$ was known. Direct information on $\Lambda$ should be
provided by the autocorrelation of a labeled filament in solution, which may be
investigated by fluorescence microscopy in the case of actin. Another approach
exploits reptation. In the same manner as the transversal friction, Eq.\
\r{sim}, one may introduce the kinetic coefficient of the longitudinal center
of mass diffusion of semiflexible filaments in entagled networks
$\zeta_{\|}=2\pi \eta L/\log(\Lambda/a)$.
There are also effects from the center of mass and rotational motion, which we
have neglected so far. Because of the polydispersity of the actin solution,
some filaments shorter than the mesh size ($L<\xi$) that are free to rotate and
diffuse are always present. Their hydrodynamic correlation is not disturbed,
nor are they hindered by entanglement. The center of mass and rotational
dynamics of those short filaments may to a good approximation be described by
the theory for rigid rod molecules \cite{doi86} (for a more complete treatment
including bending motions see \cite{son91}). For the time being, the accuracy
of the experimental data does not allow for a quantitative analysis of this
small effect.
Finally we want to compare our findings with results obtained by the
fluorescence technique. The two methods can be considered complementary: They
probe adjacent wavevector and time regimes. The fluorescence technique is more
direct. QELS, on the other hand, avoids problems associated with potential
perturbations from fluorescence markers (for the case of actin and phalloidin
see Ref.\ \cite{bre91}) and gives much better statistics. For the time being,
a significant discrepancy remains concerning the effect of Tm/Tn. Tm/Tn was
seen to cause a softening of the filaments with the fluorescence technique
\cite{kas93}. This contradicts the QELS data given above as well as some recent
direct measurements \cite{koj94,isa95}. We do not have a fully convincing
explanation for this discrepancy. A stiffening of actin by Tm/Tn binding seems
very probable, since other proteins binding along actin filaments are also
known to enhance the stiffness of the filaments, e.g.\ talin \cite{rud93}, a
protein involved in the membrane binding of actin in cells.
However, one may also think of a more intriguing scenario in terms of torsional
modes and other nonlinear or higher order derivative contributions to the
configurational energy of the filament. No thorough theoretical treatment of
all those possible contributions has been given so far, and it is not known,
which of them are actually relevant. A first principles analysis of the various
linear modes of actin (in vacuo) has only been published very recently
\cite{avr95}. Especially for F-actin local torsional modes and additional
``groove swinging" and axial slipping motions are thought to be important
because of its double-stranded structure. Very strong evidence for local
torsional excitations was provided by high resolution electron microscopy
studies \cite{bre91} showing that the twist angle is not fixed but may
fluctuate by $\pm 10^{\circ}$. Torsional fluctuations of some 100 nm length
have been identified. The presence of torsional modes may affect the dynamic
structure factor in two ways: First, they cause ``crankshaft" motions of bent
chains and thus provide an additional mechanism contributing to the time decay
of the correlation function. Second, they could lead to a scale dependent
renormalized bending stiffness of actin as was recently observed for the
railway track model \cite{eve94}. Though we did not see any systematic
$k$dependence of the bending modulus with the QELS method, such effects may
well occur in other $k$regimes. The more ``exotic" motions of actin are of
great biological interest, since they could play an active role in the
conformational changes of the actin-myosin-complex during ATP-cleavage. It is
hoped that future QELS studies with enhanced precision and larger ranges of
scattering vectors and measurement times will help to clarify this important
point.
\vspace{1cm}
Acknowledgment: This work was supported by the Deutsche
Forschungsgemeinschaft (Sa 246/24-1 and Fr 850-2) and in part by the National
Science Foundation under Grant No PHY 89-04035. One of the authors (E.
Sackmann) gratefully acknowledges the hospitality encountered during his stay
at the Institute for Theoretical Physics, UCSB, under the directorship of J.
Langer. We are grateful to Toni Maggs, Kurt Kremer and Joseph K\"as for
enlightening discussions and for making available unpublished results. We thank
Irene Sprenger, who contributed to this work by making many EM-micrographs, and
our biochemical laboratory staff for the preparation of the proteins.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 169 |
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FreshToku
Salt VS Pepper
I don't know how much it has to do with nostalgia, but the Ride Vendor might be my second-favourite bike in the franchise. It's got a really nice design and being a literal vending machine is just amazing.
Absolute favourite is probably due to nostalgia and happens to be from Ryuki, though I think it'll probably be commented on by Fish Sandwich when they get there.
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Fish Sandwich
The Immortal King Tasty
Location: Every diner you've ever been to.
Ryuki 17-18:
Episode 17 has a lot of team building going on for Shinji and Tezuka, while also following up on Ren's wavering resolve, something Tezuka's predictions (which, of course, are always right) said are going to end badly, which, since I wasn't clear about that last time, is actually the reason he's been following Ren around. It's a good time as always, and since it's an episode by Kobayashi, Ryuki even gets to do a cool thing where he kinda kicks off of Raia to get up to a Monster. It's not the absolute most awesome moment he's had, but I feel the need to stress that the action scenes when she's writing almost always have at least one bit like this. It's not even just that I want to see Shinji look cool (though that's definitely part of it), it's that having the hero use quick thinking and guile to turn the tables on a powerful opponent is a central part of any interesting superhero fight, and this show almost always nails that.
Episode 18 debuts our third new Rider in a really short span, but I think I'll save talking about him for next time. What's most interesting here is that it's the first time we actually get to see Shirou recruiting somebody, which is inherently super neat. There's also a truly epic example of dramatic irony where Kitaoka makes an offhand sarcastic remark wondering if there wasn't anyone less ill-mannered available to be a Rider after meeting Gai for the first time. Oh, man, if only he knew.
Den-O 17-18:
These episodes lean more heavily into the drama than any Den-O arc yet and are also completely amazing. The overarching plot finally starts moving after spending a considerable amount of time barely budging, and in the process we also see the growth of the main characters starting to take root. Ryoutarou proves to be more proactive than he's given credit for (which means, among other things, more Plat Form, guaranteeing this arc gets a thumbs up from me), and it becomes evident how much the Imagin have actually started to care about him. Especially Momotaros, who's really a total sweetheart even by this point.
I struggle with how to accurately get across how excellent this arc is, because so much of it is in little moments and details I don't want to sit here listing. It just loses so much merely being described, you know? Talking about the comedy especially is pretty much pointless, when it would take me so much longer to attempt to explain why something is funny than it would to just watch the gag and laugh.
OOO 17-18:
Firstly, describing why Birth is so awesome is no trouble at all. He's part of that really cool little era we unfortunately seem to be out of now where secondary Riders were specifically designed to contrast with the leads more than they compared, with a neat, pared down version of the series' main gimmick to boot. Now, as I've established, OOO kicks the s*** out of other shows when it comes to making gimmicks work, so of course, the relationship between OOO and Birth is insanely strong in that regard. Cell Medals were an established part of the show from the beginning, so having them be Birth's power source versus Core Medals is as natural as it gets. It also fits the obvious high tech bent of his design that after 800 years of scientific progress, it's possible to have a belt that can get comparable power to OOO out of way less. But of course, OOO gives off more of a mystical vibe to his powers anyway, so the contrast also works that way. There's a lot of believability to Birth's existence within the setting. You see stuff like his weapons that are magnetized to attract Medals, and it just all makes sense.
Secondly, the guy under the suit ain't too shabby himself. I think it's pretty indisputable that Date is a huge fan favorite. I mean, it's literally a fact that certain elements of OOO's plot were reworked because people liked him. And it's not hard to see why. He's cool enough on his own merits even today, but I'll refer you back to my point a few posts ago about having a huge chip on your shoulder and a grudge against the main character getting you a belt. At the time especially, nobody else, besides maybe Ibuki I guess, was anything like this guy. He's lighthearted. He's chill. He's nice. He doesn't hate OOO's guts for some contrived reason. Yet, thanks to the nuanced characterization of this show, there's still drama to be had between him and Eiji, because they're two completely different kinds of nice guy, and that's before mentioning the whole issue of Ankh also being after Medals. Birth and OOO also have THE worst teamwork in battles. It's honestly kind of stressful how often their attacks nearly hit each other because they never bother to coordinate at all. It's actually pretty clever. I feel like this is the "realistic" version of what a new Rider showing up would be like, you know? I don't think any other show has bothered to show Riders that have to learn how to deal with each other's presence on such a basic level like this before or since.
And thirdly, the Yummy this arc has the gimmick of splitting into two Yummies midway through the first episode. Which is a super smart card for the show to play right here, because that means there's an extra Yummy that's allowed to actually die before the second part, giving Birth a chance to show his skills without undercutting the moment with the usual routine of the monster suddenly escaping. OOO also gets to do his Rider Kick one more time, although I still don't feel totally satisfied, because it was in combination with Birth's finisher.
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Gai's master plan comes together here, and I have to give him some credit, it IS way more fun to have half a dozen Riders in one room than the usual two or three. Unfortunately for him though, he just can't compete with Ouja when it comes to being a total sack of crap.
I also have to give credit to Kanzaki, because the battle between the Riders DOES get way more heated the second this guy shows up. Which sucks for... well, all the other Riders (except Gai, who was ironically pretty happy about it), but as someone watching the show? I mean, does anybody who's seen it not like Ouja? Asakura is the perfect anti-Shinji to spice up the plot at this point. Just when it starts to seem like things are looking up a bit with Ren almost mellowing out and Tezuka showing up, here comes this complete madman who for the next stretch of the show will cause Shinji to question his entire goal as a Rider while smacking around just about everyone in the process. He's a big deal, and you couldn't ask for a better villain.
Let me say this to start! Zeronos is one of my favorite secondary Riders out there, so I couldn't be happier about being back at this point in Den-O. It's already started to hit its stride, but now that Yuuto's here there's no looking back. Also, Yuuto showing up means Deneb showing up. I rest my case.
The plot here does a great job at the usual routine of building a new status quo now that there's new main characters. Yuuto looks down on Ryoutarou for obvious reasons, but it's interesting to note that by this point in the story, Ryoutarou has a personal stake in things, and is clearly starting to become more assertive, even if just a little. Yuuto himself is also in a bit more of a glass house than he realizes, too. Sure, on the surface he seems like your fairly typical jerky, too cool rival type, but this is Den-O, so the truth is a little more nuanced than that. You can kind of tell from stuff like the way he throws childish temper tantrums around Deneb that his behavior is less due to any legitimate gripes he might have, and more to do with the fact that's he's secretly a little on the immature side. Deneb is even characterized as something of a doting parent to drive this home. It's a great dynamic, and the best part is, none of this is even the reason I love Zeronos. I'll get to all the other stuff that's great about him as it comes up.
Firstly, TaJaDor is freakin' awesome. As far as form debuts go this actually isn't the most impressive set of episodes ever, but make no mistake, they're more than up to OOO's high standards. Naturally there's some extra significance to a Combo made with Ankh's own Medals, and the story capitalizes on that well enough. You get to see how turbulent the partnership he and Eiji have is, but it also highlights that, at the very least, neither of them doubts the other's dedication to their respective goals.
Secondly, Date formally takes Gotou under his wing here, and if Shinji and Tezuka are a legendary duo, than I lack the words to describe what a great team these two make.
And thirdly, how great is Time Judged All? I mean, this probably could've been worked into the first point, but still. It's awesome, and it stands out even compared to OOO's other insert songs, which are no slouches to begin with.
Last edited by Fish Sandwich; 06-15-2019 at 04:47 PM..
A lot of things are going on at once in these episodes. Ren finally snaps out of the funk he's been in for a while, Shinji learns something that causes him to question the morality of stopping the Riders from fighting, and Kanzaki tries to get Tezuka to be a little more cooperative. This is an extremely interesting stretch of the show. It puts Shinji in more difficult position then he's ever been in, while also teasing the idea of him losing the only real ally he has.
I didn't know a thing about Ryuki's plot going in originally, so I remember being legitimately worried about that last bit. I was probably just really gullible or something, but it goes to show how completely absorbing I found this show. It trains you with Scissors and Zolda to assume other Riders are all jerks, so everything about Tezuka immediately screamed "too good to be true". I figured there had to be some cruel twist at some point, especially since the two Riders after him are also huge jerks. But no, for a fifth of the entire series now, he's been there, and he's been reliable, and that's why he's my favorite. He brings something to the show that nobody else can.
Ryutaros decided to save me the trouble of having to summarize the plot this time:
These are hands down some of my favorite episodes of the show. The actual showdown between Zeronos and Den-O Gun Form in the middle is the obvious reason why. It's classic. The choreography is super tense and frenetic, AND a significant chunk of it involves bike action. In a fight between my two favorite characters. It's like they made this just for me!
It sticks out in my mind like crazy to this day when I think of Den-O (or cool Rider fight scenes in general), but seeing the episodes again, it certainly helps that they're supported by such a great plot. So to speak, anyway. I mean, the actual Monster of the Week stuff this time is there and it's fine, but what I mean by "great plot" is all the extremely well done, chracter-driven drama that's really at the center here.
We're still feeling out Yuuto's character, with lots of details about him established but not explained, like the limited number of times he can become Zeronos, creating all kinds of interesting roads for future drama without forgetting to have moments to show off his actual personality too, like him only deciding to "waste" a transformation to fight Den-O after Ryutaros' careless fighting style starts running the risk of hurting Hana. Ryutaros himself is also amazing here. It's the craziest he's been at since his debut, but what makes him endearing to me is the show's commitment to writing him as an actual child. He's not quite capable of genuine malice the same way he's not quite capable of understanding why running around shooting at people is a big deal. He's also somewhat lacking in authority figures who can correct his misbehavior. It's three-dimensional, you know? Look at how violent his crayon drawings are, he's clearly rather troubled. Fortunately, Ryoutarou, awesome guy that he is, actually decides to take responsibility for the brat here, refusing to make excuses to Yuuto for not having Ryutaros under control, and trying more than anyone else usually does to explain why the way he acts is a problem. I honestly forgot how much Ryoutarou had grown by this point in the show. I'm not even halfway through, and he already feels totally different from how he did at the start.
On an unrelated note, one other revelation I've had going back to this show is that it's unfortunate so many of the episode titles involve very slick, natural sounding wordplay that you'd have to be some kind of super genius to translate without losing the joke.
Firstly, it's Nobuhiro Mouri's turn to write an arc, and he comes out of the gate pretty strong. The plot here has a very memorable twist to it, with the premise being a Yummy born of a desire to do good. Or rather, to beat up people who do bad. There are quite a few layers to why this one works. More than anything, I think it's actually quite clever to show how weird and creepy it would be to directly apply the logic of fictional heroes and villains to stuff like random litterers on the street. It stays very true to OOO's thematic spirit that desire is never quite black and white, and Mouri shows a strong grasp on the characters while he's at it. Eiji's guile, and his unique brand of optimism, are on full display here, once again showing that he's way, WAY more intellectual than the average goodhearted hero. Gotou's character arc also makes enough genuine progress to disqualify these episodes from being filler, so they're a win any way you wanna look at it.
Secondly, The Grasshopper Yummy is a great mythology gag. Showa references of any kind are more than welcome to me, but the thematic callback to Kamen Rider's origins as a story about a monster who becomes a hero is awesome, and just on that alone, he's the most memorable monster in the whole show to me.
And thirdly, the way the Medals are used here struck me as extremely elegant. Mouri got stuck writing at a point in the show where OOO has barely any toys to play with (TaJaDor is literally his only Combo right now), but he uses that to his advantage. There's a great gag where Eiji is asking Ankh for different Medals during in the middle of a fight only to react dejectedly when told they don't have what he wants, which almost seems kind of meta. But good use is made of what it is available, including some individual use of the red Medals. Special mention goes to Condor getting used specifically because Eiji lost his only Grasshopper Medal earlier in the episode. Kicking things slightly harder isn't a particularly interesting power compared to OOO's usual fare, so using it in the context of a sort of last resort actually makes it still stick out and seem cool.
Damienthathedge
Avatar by: @autorun__exe
So I'm watching Kabuto and loving it. But I need to talk about episodes 21 - 22 and how absolutely stunning they were.
When people joked that Kagami is the main character of Kabuto they really weren't kidding. Everything that he strived for in 20 episodes before culminated in this two parter, and I loved every moment of it.
Kagami's sheer determination of wanting to protect this kid, making him try and tame a Zecter that absolutely bodied him and yet eventually managed to transform into Gatack was amazing. And sure the kid being a worm was something that could've been seen from a mile away but the ending wasn't something I saw coming in the slightest.
I was fully expecting Kagami's first fight to end with Kabuto ending it due to Kagami being unable to kill the worm. But no, he shows him a Moonbow and tries to hand him his telescope before being denied and delivering a Rider Kick. the Worm then sacrificing himself for Kagami is while kind of cliche for Kamen Rider nowdays is something that really does show the Worms do possess the memories of the ones they copied.
Seriously that two parter is fucking good.
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Episode 23 hit me like a truck the first time I watched it, and it definitely holds up on a repeat viewing. Shinji shouting Tezuka's name toward the end is burned into my memory quite thoroughly, and the proper reveal of Raia's backstory further cemented his place as my favorite Rider in the whole show.
Episode 24 begins yet another trip into the Inoue Zone, which means, you guessed it, an ineffectual Shinji (which is fine because he's going through some crap right now), some good laughs, and more focus on the ORE Journal crew. In this case, Reiko (who, for the record, I've always really liked) decides to get the scoop on Asakura from the man himself. You can probably guess this isn't going to end well. This is the first time Inoue gets to play with Ouja, and, perhaps because his character is a perfect fit for Inoue's darker habits, it's the first episode of his where I felt a significant disconnect from the other episodes. Asakura goes on a brief rant about eating mud at one point that just doesn't seem like something Kobayashi would ever write. Namely because it lacks all subtlety. That's not always a bad thing, though.
I've never considered myself a huge fan of Sieg, so going back to these episodes I was taken aback by how great they are. They tell a really complete, self-contained little story that stands on its own far more than the usual Monster of the Week plots Den-O has, that are clearly just there to prop up the other parts of the show before anything else. Here, the tale of this strange bird monster and the baby he calls his brother is very much the focus, and that's definitely a good thing. The pacing is impeccable, and Den-O's insanely strong characterization mixed with its also quite remarkable attention to detail elevate it a considerable degree.
Once again, there are too many individual moments to list them all, but they add up to an extremely entertaining set of episodes. Ryoutarou getting interrogated by the police is definitely a big highlight, but what I'm more getting at is all the ~really~ tiny bits and pieces Den-O always remembers to get in there. Stuff like Ryutaros turning his head away from Yuuto when he walks by like a pouting child, that show the characters behaving in a consistent and characterful way even when it has nothing to do with what's actually going on. Every single scene on the Den-Liner especially is staged in such an elaborate, well thought out way that goes so far beyond the requirements of the story that it's almost baffling.
Episode 24 also has yet another healthy dose of screentime for Plat Form, which I know isn't any kind of objective reason for liking an arc, but I'm telling you guys, any time it gets to do something, the arc is good.
Firstly, the show digs into Date's backstory for the first time here, in the process adding a lot of depth to a character who could've been mistaken for someone more shallow to this point. One of my favorite scenes is actually in 23 when him and Eiji are just getting some oden. It's emblematic of (part of) what makes OOO great, because in the middle of a show where the most insane, unrealistic crap happens constantly (the arc straight up ends with Eiji using his octopus legs to drill a hole through a giant sea monster), we also get this very natural conversation between our two heroes that informs us a lot of their characters in a pretty short amount of time, in particular showing that Date sort of sees through Eiji in a way that nobody else, including Eiji himself, really does.
Secondly, the Victim of the Week here suffers from abuse to such a comical extent it's honestly hard not to root for her even when she's going a little crazy.
And thirdly, even after the token apology it would later get, I'm honestly still a little upset all these years later that Shout Out doesn't play during ShaUTa's debut. They had a perfect streak going! Why make a sweet insert song for every Combo and then NOT use one?!
Originally Posted by Damienthathedge
When people joked that Kagami is the main character of Kabuto they really weren't kidding.
Wait, were you guys all just joking? Because I consider this a fact on the same level as Asumu obviously being the focus of Hibiki. It's a huge part of the reason Tendou is characterized the way he is. Gattack was basically Cross-Z before it was cool.
Stronger Than You
Location: nyet
Sieg is easily my favorite Imagin just because he's so different from how all the other Imagin are handled, the train crew included. An amnesiac prince who can't even remember what the wish he fulfilled was, let alone who was actually his host. It helps his character really stand out, and his brief form change insanely memorable (And my favorite Den-O form).
And I may joke about Kagami, but I honestly do feel he's the main character of Kabuto. The first half of the story is literally structured around him, and even when the story places a greater emphasis on Tendou (Rather than Kagami's interactions with Tendou), he's still arguably the focal point of the series.
Why pretend we're all the same, when some of us shine brighter~?
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Remember when I said being unsubtle isn't always a bad thing? Well, Inoue has the honor of writing what is, far and away, the most abhorrent thing Asakura does in the entire series, which is saying something. It's so brutal I almost felt physically ill watching the episode knowing what was about to go down. It's a super impactful moment and completely destroys whatever sympathy you might have towards Ouja in a way that really crystallizes his role in the show. There's absolutely no way you can spin him as any kind of anti-hero. He's as much of a monster as the actual Monsters.
The only thing going against the episode is that everybody is arguably taking stupid pills for it to happen, but that's Inoue for you. Once things are back to normal in 26, everybody gets right back on track, naturally. Like, it was actually jarring how much smarter everyone suddenly was again. One part I really loved was when Kitaoka and Shinji go chasing after a guy who was threatening Kitaoka over a trial to protect him from a Mirror Monster. Any hopes of our favorite Super Lawyer acting selfless are dashed when he says he's only doing this because it would look suspicious if the guy mysteriously disappears on the way back from his house. He also later muses that he should just come up with an alibi instead after the endeavor proves to be more trouble than he expected, so you know he's not pulling a Ren and only pretending not to care.
It's up to Shinji to do all the caring, getting out of the rut he's been stuck in by deciding to double down on protecting people instead of racking his head about the Rider battle. It's a great episode for him, because there genuinely hasn't been a lot of Rider on Monster action since Ouja showed up, and consequently it feels super refreshing to see Shinji back in his element. He also gets his first smart hero moment in a good while too. One of the funniest gags in the whole show happens when Zolda starts loaning Ryuki his cards (because Kitaoka has a busted arm), only for Shinji (who was understandably excited about having sweet shoulder cannons) to be let down when it turns out that the gear still just goes to Zolda. He gets a bit upset about Kitaoka using him as glorified vending machine, but then shuts up mid-sentence because the idea of using Zolda's contract Monster as a shield pops into his head. Not super impressive, but it once again shows that Shinji can and does use his brain when he fights. The episode also finds an excuse to bring up that he makes good gyoza again, which I was definitely not expecting to see after the one time that was shown in 13, but that's how you know Kobayashi doesn't mess around.
These episodes do a great job of ironing out the relationship between Ryoutarou and Yuuto. Compared to the standards at the time, they were hardly bitter rivals or anything (the obligatory Rider on Rider fight was entirely on Ryutaros, which I always thought was a clever workaround), but they definitely don't like each other as people all that much. Ryoutarou is pretty skeptical that Yuuto is who he claims to be, and Yuuto of course thinks Ryoutarou is an unreliable wimp. By the end of this arc, that's started to turn around, after they each see a different side of each other. It definitely helps Ryoutarou's case especially that he's been run so ragged he's going around with a limp for most of the episode again, and still manages to get the job done. He's seriously made of iron or something.
What's that? Why was he so beat up? Well, finding that out requires a different kind of ticket than usual, according to Owner at the end of 26, although perplexingly, we're actually going to be waiting until after 27 to take that trip.
Also, I'm pretty sure episode 25 didn't actually have Den-O anywhere in it. Just Zeronos. I mean, sometimes I complain about episodes of Kamen Rider that don't have the base form, but that's taking it to the next level. At least they balance it out with some more Plat Form in 26.
Firstly, this is the most engaging Victim of the Week plot in the show so far. A story about a boxer forced into retirement due to all the injuries he's accumulated through his fights is pretty heavy and interesting on its own, but then OOO decides to kick it up a notch by drawing a direct parallel to the way Eiji has been constantly ignoring his own health due to his compulsive need to help people, even as he's constantly falling over from using Combos and in bandages from getting beat up by monsters. It's honestly quite impressive how much the comparison lands, and I appreciate the way the very real world problem of punch drunk syndrome is explicitly the focal point. It emphasizes the idea that Eiji is a "real" person who absolutely can not keep doing what he does without paying the price. It's easy to see these things in a show and just think "wow, that's guy's cool and heroic", like I've been doing with Ryoutarou constantly, so I appreciate how bold it was of OOO to reframe that and instead show how having that kind of determination can actually be really, really scary. This idea isn't something the show will just forget about after (or before!) this point, either.
Secondly, Eiji's messed up relationship with Ankh is the other focus here, and I actually think it's portrayed even better than in TaJaDor's debut. Ankh gets to look way more suspicious than usual, and when Eiji shows faith that he isn't doing anything too sketchy, it's pointedly NOT because he thinks Ankh is some kind of good person. Instead he comes to that conclusion based entirely on what he logically knows or can assume about how Ankh operates, which is probably the best summation of their dynamic yet.
And thirdly, it's really freakin' amazing that a show that goes out of its way to subvert stereotypical saccharine storytelling like this still manages to be legitimately upbeat and fun.
Two pretty memorable episodes this time. One features an audience surrogate child getting scared straight by Ren after he sees him transform and decides being a Kamen Rider sounds cool. It's a pretty on the nose plot, but Ryuki is a perfect show to do something like this in. It's also written by Kobayashi, which means, you guessed it, Shinji's gyoza comes up again! Apparently that's more of a subplot than I remember. Oh, and it also means the episode is in safe hands and delivered in a way that really sells it. I love how Ren's plan to fake getting beat up by a monster gets interrupted by him genuinely getting beat up by other Riders. All around, it's a great reminder that the fight between the Riders isn't fun and games for those taking part in it.
But man is it ever fun for the spectators! The other episode is the one where Odin shows up to make a small correction, which I'm pretty sure is one of the most famous bits of Ryuki? I don't know how to go about proving that, though. Just like how Shinji doesn't know how to go about preventing the same tragedies from repeating even when given a chance to stop them. The whole Time Vent plot was an extremely clever twist on the normal concept of a recap episode, showing the events of the series up to this point play out again in a fashion that reminds me of a Twilight Zone episode or something. The whole context of the episode is that Shinji is completely powerless and can't even remember he remembers what's going to happen most of the time, which is kind of horrifying, and gives things a very different atmosphere than the usual clip show where people sit there reminiscing about things in a laid-back manner. Actually, I keep comparing it to a clip show, but I'm not even sure that's fair. It might just be a total coincidence the episode functions as a summary of the story, because I have to imagine restaging a bunch of old scenes again probably ruins the low-effort part that usually motivates such things, even if most of the Rider action is recycled. At any rate, I love this episode to bits because it's one I think of whenever I think of how captivating this show was to me at the time (and obviously still is). It seriously felt like anything could happen by this point.
Den-O 27:
So, in case I need to explain this to anyone, Den-O made the choice to have episodes of the TV show directly set up the usual summer movie, which, to this day, hasn't been done before or since. Sure, plenty of them nowadays do have a specific place in the show's canon (W even acknowledged it in-series), but Den-O is still one of a kind in having a direct prologue episode that aired a day after (yes, after) the film came out. It literally goes right up to the start of the movie, then ends with what is essentially a trailer. Part of me doesn't like this whole idea because it feels like such shameless promotion, but then, the whole point of Kamen Rider is shameless promotion, so what's the big deal, right?
The episode itself is also crazy sharp. The amount of energy it has and the way it all flows together into one big entertaining romp is peak Den-O to the point where even without an ending, it manages to stand on its own. I can't stress that enough. Watching it made me realize all over again just how much Den-O had found its groove by this point. This show is on a roll, and it ain't stopping anytime soon.
Firstly, how cool is it that OOO was the show that got the honor of airing Kamen Rider's 1000th episode? The show is already riding high anyway, and then a golden opportunity to throw a huge party like this comes along. It's perfect.
Secondly, Shouji Yonemura got the honor of writing this arc, and judging by what he wrote, I think he was pretty jacked about that.
And thirdly, these episodes are completely insane. Ankh is desperately trying to pretend things are still normal, but even he gets caught up in the wild meta antics of Kougami's efforts to make the ultimate Kamen Rider fan film. He's not screwing around, either. You will never be as hardcore of a Rider fan as Kougami. He can watch every Rider show ever at the same time and his ideas for the movie are all heavily inspired directly by the OG show, so you can tell he respects the roots. Mixed in with this crazy premise is the equally crazy premise of a Shocker grunt that wants to prove the glory of all the mooks in Rider history by defeating OOO, and the end result is a pretty spectacular two-part celebration of the franchise that knows not to take things too seriously.
King of the Rolex
Location: Broken Arrow , OK
Originally Posted by Fish Sandwich
I'm honestly really sad and glad no other Rider series has done it the way Den-O did it's summer movie integration. I'm sad, because imo no other summer movie has ever felt so connected to the show proper as Den-O's summer movie what with the build up with Yuuto "kidnapping" Ryotaro for some purpose in the episodes prior to episode 27, the fact that MOTW being a lackey of the movie villain, and that the movie has tangible effects on show proper beyond a throwaway shot/line. I'm glad however because that wait for the dvd and subsequent subs would suuuuuuuuuuuuuuuuck.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 410 |
Q: SQL Argument data type int is invalid for argument 1 of charindex function I'm trying to convert MySQL query to MsSql query but I'm having trouble. Here is my query:
MySQL
SELECT *,
(SELECT count(books.id)
FROM books
WHERE books.status = 1
AND FIND_IN_SET(categories.id, books.multiple_category_id)) AS book_count
FROM categories, books
WHERE categories.parent_id=0
AND categories.status=1 ;
SQL I've tried
SELECT *,
(SELECT count(books.id)
FROM books
WHERE books.status = 1
AND CHARINDEX(categories.id, books.multiple_category_id) > 0) AS book_count
FROM categories, books
WHERE categories.parent_id=0
AND categories.status=1 ;
The errors I'm getting are:
Argument data type int is invalid for argument 1 of charindex function.
For reference.
http://sqlfiddle.com/#!3/4ed19/3
does anyone have any ideas?
Thanks in advance!
A: Use this query. I just changed categories.id to CAST(categories.id AS VARCHAR). Because the id is integer in categories table.
SELECT *,
(SELECT count(books.id)
FROM books
WHERE books.status = 1
AND CHARINDEX(CAST(categories.id AS VARCHAR), books.multiple_category_id) > 0) AS book_count
FROM categories,
books
WHERE categories.parent_id=0
AND categories.status=1 ;
A: Here is another way to do it
SELECT *,
(SELECT Count(books.id)
FROM books
WHERE books.status = 1
AND ',' + books.multiple_category_id + ',' LIKE '%,' + cast(categories.id as varchar(50))+ ',%') AS book_count
FROM categories,
books
WHERE categories.parent_id = 0
AND categories.status = 1;
Note : There is a Cartesian product between categories and books table in your query. I guess you don't need books in From clause. Stop storing comma separated values in column it violates the First Normal Form
A: MS-SQL's CHARINDEX function requires VARCHARs or NVARCHARs as parameters, so your categories.id should be converted to VARCHAR. Something like this:
CHARINDEX(CAST(categories.id AS VARCHAR(16)), books.multiple_category_id ) > 0 )
However, I would consider refactoring both tables and your query:
1. Query refactoring
As it is written, you can have false matches. E.g. categories.id = 1 and your multiple_category_id is '11,12'. An alternative is to split your string is to have your condition like this:
AND EXISTS (SELECT 1 FROM dbo.SplitString(books.multiple_category_id) WHERE Token = categories.id)
Also, * should be replaced with actually needed columns (in this case, it will bring all the columns in both tables). Something like this:
2. Table refactoring
Instead of parsing strings you should define an X table like this:
categoryXbook
categoryId FK references category
bookId FK references book
PRIMARY KEY (categoryId, bookId)
so, that you can do your query using simple JOINs instead of strings searches which can be slower (no indexes can be used).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,708 |
Grigora I Ora Perase (The Time Passed Quickly) is an album by popular Greek artist Eleftheria Arvanitaki and it was released in 2006. On it, Eleftheria performs the songs that the composer wrote based on poetry mainly by Sappho. The album sold 15,000 copies in Greece and was certified Gold 4 weeks after its release.
Track listing
"Grigora I Ora Perase" (Instrumental)
"Athanati Afroditi"
"Theos Mou Fainetai"
"O Adonis"
"Polles Fores"
"Os Astra Gyro Vriskontai"
"Na 'Cha Pethanei"
"Pyretos Kryfos"
"Ti Thelo Ti"
"Grigora I Ora Perase"
"Afroditi"
"Ilisos"
"Iridanos"
"Einai Poly Noris"
"To Teleftaio Taxidi"
References
2006 albums
Eleftheria Arvanitaki albums
Greek-language albums
Universal Music Greece albums
Adaptations of works by Sappho | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,226 |
\section{Historical remarks}
Since this talk is about a very old issue, it seems appropriate to start
with a few historical remarks to put the problem and its proposed solution
by Tomboulis into context.
The confinement problem in lattice Yang-Mills theory was a hot issue in
the late 1970's and early 1980's. Center vortices were identified by
several authors as crucial objects (\cite{yoneya, thooft, mp}). 't~Hooft
proposed a confinement criterion inspired by these vortices; unlike the
earlier criterion proposed by Wilson \cite{wilson} and its modification
by Polyakov \cite{polyakov} it did not involve infinitely heavy quark
sources but (sourceless) central electric flux in a torus.
A little later it was proven that 't Hooft's confinement criterion implies
confinement in the sense of Wilson and Polyakov \cite{bs,ty}.
About the same time Tomboulis \cite{tombprl} came up with a charming idea
how to prove that lattice Yang-Mills theory based on a nonabelian
(compact, semisimple) gauge group has a nonzero string tension in
't~Hooft's sense at all values of the bare coupling constant: he proposed
to link by rigorous inequalities lattice Yang-Mills theory to the
solution of an approximate Renormalization invented earlier by Migdal and
Kadanoff (MK RG) \cite{mk, jose}.
It was proven a little later that in 4 dimensions the MK RG drives
lattice $SU(N)$ Yang-Mills theory, but also compact lattice QED to the
strong coupling fixed point \cite{ito}. This signals confinement for
these models, and is therefore misleading for the abelian model, which is
known to have a deconfining transition \cite{guth, fs}.
This fact raised problems for Tomboulis's approach, because it was not
clear how his inequalities would distinguish between the groups $SU(N)$
and $U(1)$, especially since in his short letter there were no details
given concerning the proof of the crucial inequalities. In fact there
even remained room for doubt as to the existence of confinement in this
sense in the $SU(N)$ lattice models or the analogous question of mass
generation in $2D$ $O(N)$ spin models (see for instance \cite{seiler} and
references given there). In spite of these efforts as well as the efforts
by others, such as K.~R.~Ito \cite{itoconf}, who tried to prove
mathematically the correctness of the common expectations, neither
confinement for arbitrarily weak bare coupling nor its absence could be
established (nor could the analogous $2D$ problem be definitely settled).
The problem remains an important open question to this day.
In 2007 Tomboulis \cite{tomb07} revived his old idea (with some
modifications) and published a paper providing details about the
purported proof. The following remarks, while critical of his work,
should nevertheless not diminish his credit for having revived interest
in this old, important but neglected and unsolved problem. A more
detailed discussion can be found in our paper \cite{is}.
\section{Sketch of Tomboulis's strategy}
The goal of Tomboulis's strategy, for simplicity for the gauge group
$SU(2)$, is to establish the spreding of central magnetic flux on a
torus $\Lambda$ of dimensions $L_1\times L_2\times L_3 \times L_4$:
\be
\frac{Z^{(-1)}_\Lambda}{Z_\Lambda}\ge
\exp\left[-cL_2 L_3 e^{-\alpha L_1 L_4}\right]
\quad {\rm for} \quad L_1 L_4 \gg \log (L_2 L_3)\,,
\label{ineq}
\ee
where $Z^{(-1)}_\Lambda$ has twisted boundary conditions in the (12)
direction. (\ref{ineq})is supposed to follow from
\be
\frac{Z_\Lambda^{(-)}}{Z_\Lambda}\ge
\frac{Z^{(-)}_{MKT}(n)}{Z_{MKT}(n)}\,.
\label{MKTineq}
\ee
Here $Z_{MKT}(n)$, $Z^{(-)}_{MKT}(n)$ are the partition functions under
the $n$-fold iteration of the `MKT' decimation which is Tomboulis's
modification of the MK RG.
If we assume for a moment that inequality (\ref{MKTineq}) holds {\it and}
the MKT iteration leads eventually into the strong coupling regime,
inequality (\ref{ineq}) follows, and this implies electric flux string
formation and confinement in the sense that 't Hooft's string tension
$\sigma_{tH}$ satisfies
\be
\sigma_{tH}>0 \quad \forall g^2\,,
\ee
where $g$ denotes the bare coupling constant.
One question that arises immediately is whether Ito's result, establishing
flow to the strong coupling fixed point, also holds for Tomboulis's
modification, which depends on an additional parameter $r=1-\epsilon$,
$\epsilon>0$. We found that $r<1$ has the same effect as increasing the
dimension and therefore for weak coupling the flow actually goes towards
the {\it weak coupling fixed point}.
\section{The fundamental issue}
As remarked before, the MK RG in 4$D$ shows {\it no structural difference}
between abelian (such as $U(1)$) and nonabelian (such as $SU(N)$) models:
the flow is always attracted by the strong coupling fixed point.
This was already pointed out in the seminal paper \cite{jose}, where
this insight was actually traced to Wilson's 1976 Carg\`ese lectures;
as remarked, a proof of this fact was given by Ito \cite{ito}.
This means that the original comparison argument given by Tomboulis
{\it has to fail} for $U(1)$, because the $4D$ $U(1)$ model has vanishing
string tension for sufficiently weak coupling. In fact, any similar
argument that does not explicitly make use of the nonabelian nature of the
gauge group has to fail.
\section{Technical points}
{\it (a) The parameter $r$}.
The MKT decimation proceeds as follows: one starts with the character
expansion of the plaquette coupling function (Gibbs factor)
\be
f(U)\equiv \exp A_p(U)=
F_0\left[1+\sum_{j\neq0}(2j+1)c_j(\beta)\chi_j(U)\right]\,.
\ee
In essence the decimation amounts to alternating raising the coupling
function to the power $2^{D-2}$ and raising the Fourier coefficients
$c_j$ to the power $4r$. Explicitly
\be
f^{(n)}(U)\mapsto \frac{f^{(n)}(U)^4}{\int f^{(n)}(U)^4 dU}\equiv
g^{(n)}(U)\,,
\ee
\be
g^{(n)}(U)=1+\sum_{j>0} (2j+1)c_j(n) \chi_j(U)
\ee
\be
f^{(n+1)}(U)=1+\sum_{j>0} (2j+1)c_j(n)^{4r} \chi_j(U)\,.
\ee
Equality of the two exponents $2^{D-2}$ and $4r$ would mean that one is
working in the critical dimension $D_c$ and one finds easily
\be
D_c=4+\frac{\ln r}{\ln 2}<4\,,
\ee
so that with $r<1$ in $4D$ one is {\it above} the critical dimension and
has to expect a phase transition. This is indeed the case; we have run
the iteration for $r=0.9$ and two close values of $\beta\equiv 2/g^2$ and
found a bifurcation of the flow: for $\beta=4.79$ the flow is attracted
to the weak coupling fixed point, whereas for $\beta=4.80$ is flows to the
strong coupling fixed point. The fact that for weak coupling the flow
converges to the weak coupling fixed point can also be seen in a simple
Gaussian approximation.
\begin{figure}[htb]
\includegraphics[width=.5\columnwidth]{mktc4.80.eps}
\includegraphics[width=.5\columnwidth]{mktc4.79.eps}
\caption{Evolution of $c_j/c_0$ under Tomboulis' modified MK RG with
$r=0.9$. $\beta=4.80$ (left plot), $\beta=4.79$ (right plot); lines drawn
to guide the eye.
}
\label{plot}
\end{figure}
{\it (b) Existence of a common interpolation parameter $\alpha^\ast$ for
$Z$ and $Z^{(-)}$.}
This is an essential point in Tomboulis's strategy. He has to find for all
$n$ an $\alpha(n)^\ast\le 1-\delta$ such that
\be
\frac{Z^{(-)}_\Lambda}{Z_\Lambda}=
\frac{Z_\Lambda^{(-)} (\{\alpha^*(n)c_{j}(n)\}}
{Z_\Lambda (\{\alpha^*(n)c_{j}(n)\})}
\ee
His argument (in Appendix C of \cite{tomb07}), based on the {\it implicit
function theorem}, is flawed. He introduces a certain function
$\Psi(\lambda,t)$ in terms of interpolated partition functions and
is looking for a $t(\lambda)$ such that
\be
\Psi(\lambda,t)=0\,.
\ee
There is a solution $t_0$ at $\lambda=0$, but the sought after $\alpha^*$
would emerge from $t(1)$. Tomboulis is able to show that
$\frac{\partial}{\partial t}\Psi(\lambda,t)\neq 0$, so by the implicit
function theorem there is a solution near $\lambda=0$. But the information
is not sufficient to allow the extension to $\lambda=1$, as shown
by a simple counterexample due to T. Kanazawa \cite{kana}:
\be
\Psi(\lambda,t)\equiv e^{-t}-1+2\lambda
\ee
which has the solution $t(\lambda)=-\log(1-2\lambda)$.
\section{Can the problems be fixed?} The choice of the parameter $r$ is
very subtle, because one has to make sure of two things: (1) the
decimation has to run into the strong coupling fixed point and (2) $r$
has to be kept away from 1, as is stressed in \cite{tomb07}. This second
issue is not discussed in \cite{tomb07} in a quantitative way, while the
issue (1) is not addressed at all. Tomboulis hinted orally at the option
of making $r$ dependent on $n$, the number of the iterations, but exactly
how this would have to be done remains unclear.
In this respect the case of $U(1)$ is instructive: for $r=1$ the common
interpolation parameter $\alpha^*$ cannot exist, because it would imply
the existence of a nonvanishing string tension at all values of the bare
coupling, in contradiction with proven facts (\cite{guth,fs}).
Quite generally, we think that any strategy based on a Migdal-Kadanoff
type decimation is very unlikely to succeed, because these hierarchical
approximations do not show any structural difference between abelian (like
$U(1)$) and nonabelian (like $SU(2)$) models.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 101 |
Home Tags Hocus Pocus 2 movie
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Hocus Pocus 2: Unlimited Magic And Witches, Sequel Who is coming and much more..
SUBHAN MATANIA - Wednesday, 20 May 2020, 03:24 EDT 0
Hocus Pocus is! Following a decade that the sequel for the show, Hocus Pocus two is believed to arrive in displays! Will witches and... | {
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} | 2,036 |
package de.dkiefner.sample.fastlane;
import android.support.test.runner.AndroidJUnit4;
import org.junit.Test;
import org.junit.runner.RunWith;
import de.dkiefner.sample.fastlane.login.LoginFixture;
import static android.support.test.espresso.Espresso.onView;
import static android.support.test.espresso.assertion.ViewAssertions.matches;
import static android.support.test.espresso.matcher.ViewMatchers.withId;
import static android.support.test.espresso.matcher.ViewMatchers.withText;
@RunWith(AndroidJUnit4.class)
public class MainActivityTest extends BaseTest {
@Test
public void thatUsernameIsShown_whenLoginWasSuccessful() {
// given
LoginFixture loginFixture = new LoginFixture();
String expectedUsername = loginFixture.getUsername();
// when
loginFixture.load();
// then
onView(withId(R.id.username)).check(matches(withText(expectedUsername)));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,518 |
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GOVERNMENT OF INDIA STATIONERY OFFICE
Indentor's corner
Indentors Information
Govt of India Sty. office - Eastern Zone
Regional Sty. Depot New Delhi - Northern Zone
Regional Sty. Depot Chennai - Southern Zone
Regional Sty. Depot Mumbai - Western Zone/ Central Zone
Employees' corner
Leave Form
GPF Form
Staff Seniority List
Directory of Officers and Office in Charge (O.I.C.)
1 Particulars for any arrangement for consultation with or representation by the members of the public in relation to the formulation of policy or implementation there of [(Section 4(1)(b)(vii)] of RTI Act, 2005 Publicity Band Public Interface 22/11/2019
2 Budget allocated to each agency including all plans proposed expenditure and reports on disbursements made etc. [Section 4(1)(b)(xi)] of RTI Act, 2005 Budget and Programme 22/11/2019
3 The names, designations and other particulars of the Public Information Officers. [(Section 4(1)(b)(xvi)] of RTI Act, 2005 Name and Designation of PIO 22/11/2019
5 Such other information as may be prescribed. [Section 4(1)(b)(xvii)] of RTI Act, 2005 E-Governance 22/11/2019
6 A directory of its officers and employees. [Section 4(1)(b)(ix)] of RTI Act, 2005 A Directory of Officers and Employees 22/11/2019
7 A Statement of the Boards, Councils, Committees and other bodies consisting of two or more persons constituted as its part of the public Authority [Section 4(1)(b)(viii)] of RTI Act, 2005 Board, Counsils , Committee 22/11/2019
8 The powers and duties of its officers and employees. [Section 4(1)(b)(ii)] of RTI Act, 2005 Power & Duties of its Officers & Employees 22/11/2019
9 The particulars of its organisation, functions and duties [Section 4(1)(b)(i)] of RTI Act, 2005 Organisation, Functions and Duties 22/11/2019
Copyright & 2018 All Rights Reserved
The site designed by Department of Publication through Velocis Systems Pvt. Ltd. & hosted by National Informatics Centre (NIC), Department of Electronics and Information Technology, Ministry of Communications & Information Technology, Government of India. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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A Mini Survey of Digital Humanities in European Research Libraries
2018-2022 Strategy
LIBER's Digital Humanities and Digital Cultural Heritage working group is preparing a survey for the entire LIBER community, to gain a fuller understanding of the DH-world of European research libraries.
We've been gathering input for this survey in many ways, including by conducting a mini-survey of our own working group members. Their answers both provide us with input about how to organise the bigger survey and serve as institutional Digital Humanities' use cases. This post describes the most striking outcomes of our small survey.
The survey was opened after our working group meeting on 10 April and closed just after LIBER's 2018 Annual Conference in July. In total, we received 22 replies. Five libraries requested we keep their input within the working group. Their answers are represented anonymously in the graphs below. The other 17 libraries have agreed to share their stories, and these can be downloaded via LIBER's Zenodo Community.
We asked the libraries to describe one DH activity in this survey. The survey is therefore not a comprehensive description of what libraries are working on in this field but it does give an insight into their organisation. In this summary, we don't describe all answers. Instead we've opted for a user-friendly overview.
Word cloud of activities mentioned in the survey.
How Are DH Activities Organised?
Of the 22 libraries who responded, eight have been running a DH activity for under a year. Half have been active between 1-5 years and only three libraries have had a DH activity for more than five year. Most (13 libraries) have a team of 2-5 people working on DH, with teams of 6-10 people (six libraries) following closely. A majority of the libraries (16) conduct the activity as part of a policy, while six libraries do it as an ad-hoc activity. Looking at budget, over half have dedicated funding for the activity. Nine libraries are doing this without any dedicated budget.
Do you have dedicated budget for this activity?
Collections & Activity
The activities described in the survey vary but all libraries are using digital collections with varying licenses. The libraries are both using collections to which they have a license and collections which they have digitised or curated themselves. Digitised collections created by the library are most popular. Looking at licensing, there is a strong preference for openly available collections with a CC-license or in the public domain.
What kind of collections are you using?
How is the data you are using licensed?
Who Are We Working With & How?
Since DH is, by definition a field where you work with others, we naturally asked the libraries to tell us how they found the researchers they are working with and which activities they are working on. As you see a lot of the libraries are working with people they already knew.
However, when asking how aware academics are of the DH activities in the library, most libraries said 'Somewhat aware' which we described as 'They know the library does something, but not sure what'.
This might also make sense when looking at what the 22 libraries are mostly doing: providing access to digital collections. However, skills training and advice is also often mentioned. We'd like to explore this further in the larger survey.
How did you find/build a relationship with the researchers working in this activity?
What would be the main topics describing your relationship with the researchers working in the activity?
How aware are academics in your institution of the DH activities the library is active in?
Skill-building
We also asked about the skills of library staff and any skill gaps. Nineteen libraries identified a skills gap. Of these, 16 said they were missing hard skills such as programming or tools. Three libraries missed soft skills such as communication and project management.
Half of the libraries subsequently offered training to their librarians, such as Library Carpentry workshops. These events were often organised within the scope of the activity or belonged to a personal development programme. Only two libraries indicated it was part of a library-wide training programme. Unfortunately the remaining libraries which did not offer training said this was because of lack of budget.
Communication and Outreach
As shown above, not a single library indicated the academics in their institutions are very aware of what the library is doing with DH. However, we communicate a lot! Or at least, on a lot of platforms. That raises the question: why aren't we reaching academics more? And why do libraries mostly work with academics who they already know?
If the library promotes the activity, where does it do so?
Will you help?
This small overview gives us some interesting insights into what is happening in this small selection of libraries and raises some questions about the choices we make, but we want to know more!
The team of the LIBER Digital Humanities and Digital Cultural Heritage Working Group is assembling a larger survey in the coming months and we would be very happy if you could help us by filling it out.
Watch this space, subscribe to our mailing list or — even better — join our working group!
Lotte Wilms
Lotte is co-chair of LIBER's Digital Humanities & Digital Cultural Heritage Working Group. | {
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Food Rescue app saves 235,000 pounds in first year for Hunger Network of Greater Cleveland
Karin Connelly Rice | Monday, November 11, 2019
When volunteers register with the Food Rescue app, they can list their preferences and the app will alert them when a rescue is waiting. Kevin Kopanski
Food Rescue volunteer Tom Cardello picks up produce from a local grocer. Bob Perkoski
Food Rescue donors deliver to 73 nonprofit agencies in Cuyahoga County helping the region's hungry. Bob Perkoski
Food Rescue volunteerTom Cardello says he's done at least 120 rescues this year. Bob Perkoski
Hunger Network's Food Rescue program volunteers are eager to make rescues. Kevin Kopanski
Bob Perkoski
Hunger Network's Food Rescue program Bob Perkoski
Tom Cardello officially retired from the IT department at Sherwin-Williams in February 2018, and he was just beginning to look at volunteer opportunities to help fill his days.
As a veteran, serving in the U.S. Navy from 1979 to 1983 on the USS Nimitz, Cardello first turned to the VA Medical Center of Greater Cleveland (where he continues to volunteer today). While there, he heard about the Hunger Network of Greater Cleveland's recent launch of Food Rescue. The program lets volunteers and food donors download an app (for both Apple and Android cellphones) and sign up to receive alerts when a donation needs to be picked up and delivered to food pantries, youth programs, transitional housing, and other organizations that help feed the hungry.
Retiree Tom Cardello volunteers for the Hunger Network's Food Rescue program picking up and delivering donations every Thursday and Friday.Cardello jumped at the opportunity. He's been picking up and delivering donations on Thursdays and Fridays, plus any time he's needed the rest of the week, ever since Food Rescue launched a year ago this month.
Hunger Network launched Food Rescue, modeled after a program in Pittsburgh called 412 Food Rescue, to help reduce the amount of food that goes to waste in the United States, which is estimated to be 40%.
"Access to fresh, healthy food is a right, not a privilege," says Stacy Soulimiotis, Hunger Network's program director. "With only 10% of food donations nationwide being fresh produce, it is imperative that we find innovative ways to provide healthier options to food insecure individuals. One in five Cuyahoga County residents are food insecure. Food Rescue is an innovative way to address that issue and supplement fresh, healthy food to those we serve who wouldn't otherwise have access."
Since the Food Rescue app's launch Nov. 30 last year, Hunger Network has seen 544 volunteers download it, enrolled 42 food donors, rescued 235,000 pounds of food (worth $587,500) from those donors, and delivered it to 73 nonprofit agencies in Cuyahoga County helping the region's hungry.
Additionally, 127,600 pounds of carbon dioxide emissions have been mitigated. "The food cannot properly decompose in landfills, therefore it emits methane emissions, "says Soulimiotis. "Food is the No. 1 material in landfills, therefore it's the leading contributor of rising greenhouse gas emissions."
Food donors include grocers, restaurants, caterers, and bakeries. "Our biggest donor is Giant Eagle, as they are focused on minimizing food waste," says Soulimiotis. Other donors include Perfectly Imperfect Produce, Bruegger's Bagels, On the Rise Artisan Bread and Pastries, and Cleveland Vegan. "And now we are rescuing catering from movie sets, among other smaller donors," she says.
When volunteers register with the Food Rescue app, they can list their preferences—preferred neighborhoods, times of day, or any place, any time—and the app will alert them when a rescue is waiting. The app also lets the volunteers indicate what size cars they drive, so larger orders will go to people with larger cars.
Food Rescue donors deliver to 73 nonprofit agencies in Cuyahoga County helping the region's hungry.The app also instructs rescuers on "all of the details to get to the door, who to talk to, and details to get them to the right place," Soulimiotis says.
Volunteers are eager to make rescues, she says. "A lot of them will do multiple rescues a day. There are some who do multiple days. They really love it, not seeing that food go to waste."
Shelli Smith, who works "very part-time" at Cleveland Rocks and Beads in Cleveland Heights, started volunteering in February. She saw a plea for volunteers come across her Facebook feed and decided to answer the call, she says.
"It came up in my feed, and it sounded like a cool and doable thing, and I thought, 'Let's give it a try'," Smith says. "I went crazy in the beginning, and I had days with multiple deliveries." Now things have settled down, and she does regular pickups from the Giant Eagle in Beachwood, taking them to Edwins Leadership and Restaurant Institute, a nonprofit organization at Shaker Square that trains formerly incarcerated adults in culinary and hospitality training.
Not only has she gotten to meet a lot of the people who help the community, she's gotten to see a lot more of the city, the lifelong Clevelander and Shaker Heights resident says. "One thing I like about it is I have gone to so many places in the city I have never seen before," she says.
Smith fondly remembers when Brassica in Shaker Height's Van Aken District was training its staff before it opened, and she answered a rescue to pick up entire meals from the training and drop them off at a downtown Cleveland drop-in center for the homeless.
"Food is my comfort zone—whenever there's a meal involved with cooking for 100 people, that's me," she says. "Food Rescue is just a no-brainer. Why wouldn't you do this?" Smith says she now has two friends who regularly make rescues for the Hunger Network.
Cardello says he finds someone else often picks up rescues before he even has a chance to respond. "As soon as I see a rescue pop up, within seconds it's claimed," he says, adding he's done at least 120 rescues this year. "When I first got involved, there were only a handful of volunteers, and I was pretty busy, doing rescues four or five times a week."
Although he drives a Honda Accord, Cardello says he has had some large loads. "I get anywhere from one box to 13 boxes, and I've had my entire vehicle loaded—front seat, back seat, and trunk," he says. "They should have put [that load] out for a larger vehicle, but they put it out for mine."
Smith, too, has sometimes eyed a rescue and doubted that she could fit the delivery in her Toyota Prius. But she's always succeeded.
For Cardello, he says he gets a lot of satisfaction in volunteering for the Hunger Network. "What makes it all worthwhile is you see directly the payoff from your work," he says. "You see the benefit of what it does directly, you see the recipients of the food—parents with kids, grandparents with kids, the elderly, sick people. It's unlike writing a check. When I do Food Rescue, I can see first-hand the benefits of my work."
Read more articles by Karin Connelly Rice.
Karin Connelly Rice enjoys telling people's stories, whether it's a promising startup or a life's passion. Over the past 20 years she has reported on the local business community for publications such as Inside Business and Cleveland Magazine. She was editor of the Rocky River/Lakewood edition of In the Neighborhood and was a reporter and photographer for the Amherst News-Times. At Fresh Water she enjoys telling the stories of Clevelanders who are shaping and embracing the business and research climate in Cleveland.
Health + Wellness, Local Food Economy, Social Change
Shaker Heights, Downtown, AsiaTown/St. Clair Superior | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/using-expm-of-matlab-to-plot-state-responses.856422\/","text":"# Homework Help: Using expm of matlab to plot state responses\n\nTags:\n1. Feb 9, 2016\n\n### shuh\n\n1. The problem statement, all variables and given\/known data\nThe state space model of a nonlinear system is $$x'_1(t) = 2x^2_2(t) - 50$$ $$x'_2(t) = -x_1(t) - 3x_2(t) + u(t)$$ where x_1(t) and x_2(t) are the states, and u(t) is the input. The output of the system is x_2(t).\n\nFind the zero input response (u(t) = 0) of this system linearized at the equilibrium point (-15, 5) with initial states (-14.5, 5). Use Matlab (expm.m) to plot these state responses from 0 to 5s.\n\n2. Relevant equations\n\nState Space Modeling, Matlab\n\n3. The attempt at a solution\n\nThe bold part is where I have an issue with this problem. Generally, if you want to find response to initial conditions, you use initial function, not expm. Expm simply takes a matrix and exponentiates it.\n\nAnyways, matlab code for generating response to initial condition is:\n\n\/\/State Matrix\nA = [ 0 20 ; -1 -3]\nB = [0 ; 1]\nC = [0 1]\nD = 0\nx0 = [ -14.5 ; 5]\n\nsys = ss(A,B,C,d)\ninitial(sys, x0)\n\nAnd you get a beautiful plot looking like the following:\n\nHow, and especially why on earth would you use expm function to generate such plot?\n\n2. Feb 12, 2016\n\n### jasonRF\n\nDo you know what a matrix exponential is, and how it can be used to solve a system of linear, constant coefficient first order equations? If not, then either look in your textbook or google it. Even the Wikipedia page on the matrix exponential has some applications that should help here.\n\nWhy would you do this? Because it will give you an idea of one way to solve such a system of ODEs analytically. Numerically, I do not know what the 'initial' command in matlab does (might just numerically integrate the equations - probably doesn't do the matrix exponential). I would think that anyone who learns about state space analysis should learn about matrix exponentials at some point. Perhaps that is the real reason 'why' you would do this - so you learn more than just how to use a few Matlab commands.\n\nenjoy,\n\njason","date":"2018-06-22 19:45:27","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\": 0.5786167979240417, \"perplexity\": 704.7304076490743}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267864776.82\/warc\/CC-MAIN-20180622182027-20180622202027-00276.warc.gz\"}"} | null | null |
Q: What is the name for someone who gives his subordinates a morale boost? What do you call someone who, merely by their presence, gives a morale boost to the people he is responsible for? What about someone who gains power from being supported by their troop?
If possible I would like to have word with military connotation, but a more "civilian" approach is also interesting.
EDIT: Jez gave a lot of interesting examples, but I am also interested in nouns, and all of Jez's example can't be "translated" into nouns that easily.
A: In terms of giving a morale boost, one could refer to an inspirational or totemic leader. As a noun, a leader could simply be described as an inspiration, although this is a broad term that can refer to any inspiration. If someone gains power/morale from being supported by their troop, they can be said to have been rallied by the troop, and in return they can rally their troop. As you suggested, the leader could also be motivated by their troops, or motivate their troops.
In a civilian as opposed to a military context, there are all sorts of positive adjectives one might wish to assign a good leader who boosts their people's morale. Some of these could be competent, charismatic, and charming. A great leader may become famous and/or celebrated.
A: Besides the previously-mentioned inspiring, motivating, and charismatic, consider
*
*compelling, "forceful". Example: Under her compelling leadership they succeeded where none had before.
*influential, "Having considerable influence". Example: He was influential in leading the team to success.
*stimulating, "Having a manner that stimulates (encourages into action; arouses... to functional activity)". Example: Under her stimulating leadership, they found a new explanation.
A literal term for "someone who gains power from being supported by their troop" might be symbiont, a noun for a participant in symbiosis, "a relationship with mutual benefit between individuals or organisms". Of such a leader, one might use terms energised ("invigorated, made energetic; supplied with energy") if the leader feels inspired by the troops, or idolized or trusted if the leader gains political or military power via loyalty of troops.
A: I've head some people being described as talismanic which you could convert into the noun talisman to describe the individual? It's always used in the sense of an individual bringing good luck to a group, or who is always somehow linked to success of that group (but not necessarily being the leader of the group..)
A couple of examples where people are described as such:
*
*"But the week after setting the record with a 4-0 demolition of
Perth, they lost talisman Thomas Broich to injury and Sydney FC
brought them crashing back down to Earth."(Road to the Final,
Football Federation Australia
news)
*"Former Cardiff City target Rickie Lambert has been described as a
"talisman" for promotion chasers Southampton." ('Talisman' Lambert
ready for Bluebirds test, Wales
Online)
I'd also swear there's a reference in my copy of "The Junior Officers' Reading Club" by Patrick Hennessy to a word like what you're after but I can't find it (the reference I'm thinking of, not my copy).
It is quite possible I'm thinking of the term old man (slang for the unit commander) as I've always read it in an affectionate/morale boosting sense. Will see if I can dig up some references to that effect (for argument's sake, I'm sure there's some in Starship Troopers..) but thought I'd post in lieu just incase this gives people ideas.
A: "Role model" is surely a (current) term for somebody who "inspires" their followers in some way. Whether it is a term which causes the "role model" to reciprocate the "follower's" feelings is beyond my understanding. It's a modern term which has been ill-defined. "Role model" could have both civilian (usually) and military connotations.
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{"url":"https:\/\/socratic.org\/questions\/what-is-the-covalent-compound-formula-for-phosphoric-acid","text":"# What is the covalent compound formula for phosphoric acid?\n\n##### 1 Answer\nJan 12, 2017\n\nFor $\\text{phosphoric acid}$?\n\n#### Explanation:\n\n${H}_{3} P {O}_{4}$ $\\equiv$ $O = P {\\left(O H\\right)}_{3}$; we have a $\\text{phosphorus V}$ centre here. Note that this is a DIACID in water.","date":"2019-12-12 00:26:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 5, \"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\": 0.5321419835090637, \"perplexity\": 5507.262818643936}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540534443.68\/warc\/CC-MAIN-20191212000437-20191212024437-00426.warc.gz\"}"} | null | null |
Something In The Water
Leave Carrie Underwood Alone Over This Gay Marriage Controversy
The controversy stems from Carrie Underwood performing at the Georgia Dome in Atlanta on January 2nd at the evangelical "Passion" conference. The song she performed was her uplifting, critically-acclaimed, faith-based song "Something in the Water" that reached #1 on both the Hot Country Songs and Hot Christian Songs charts in 2014.
American Family Association, Carrie Underwood, Louie Giglio, Something In The Water, Wesley Wildmon
Album Review – Pokey LaFarge's "Something in the Water"
Some 70 years behind the times and yet still cooler than the rest of us, Pokey LaFarge is like the musical equivalent of the Austin Powers character brought out of cryogenic freeze to do battle with the forces of bad music by reminding the world of a time when popular songs still embodied taste, composition, and a timeless charisma instead of the diarrhetic pap dictated by the fickle tastes of 15-year-olds.
JD McPherson, Jimmy Sutton, Pokey LaFarge, Review, Something In The Water
Pokey LaFarge Announces New Album "Something In The Water"
Trigger News 8 Comments
Country folky throwback Pokey LaFarge has announced the release of his latest album and his first with Rounder Records called "Something In The Water." The old-school throwback St. Louis singing and strumming song man signed with Rounder in November of 2014 after releasing six albums since his self-released debut in 2006.
JD McPherson, Jimmy Sutton, Pokey LaFarge, Something In The Water
"Something In The Water" Becomes Bro-Country's Coffin Nail
But what "Something In The Water" had that no other song that could offer battle to Bro-Country had previously was substance, and one of the most powerful performances we've heard from a country artist in the last few years. This is what was needed to defeat Bro-Country. It wasn't going to take pandering. Leadership is what was needed, and an exhibition of raw talent that could not be denied.
bro-country, Carrie Underwood, Connie Smith, Florida Georgia Line, Maddie & Tae, Miranda Lambert, Sam Hunt, Somethin' Bad, Something In The Water, Taylor Swift
Equal Time – The Best in Mainstream Country Music in 2014
Independent music fans love to say "90% of what they play on the radio is crap!" Well then it would stand to reason that 10% actually has some value. And in the interest of pragmatism and inclusiveness that is vital to the charge of Saving Country Music, it is important to not ignore when Music Row and mainstream artists get it right, but to celebrate these moments.
Brett Eldredge, Bridges, Caitlyn Smith, Carrie Underwood, Dave Grohl, Dierks Bentley, Dirt, Eric Paslay, Everything To You, Florida Georgia Line, Garth Brooks, Jon Pardi, Kellie Pickler, Maddie & Tae, Man Against Machine, Mary Sarah, Riser, Something In The Water, The Grohl Sessions Vol. 1, The Mavericks, Tim McGraw, Zac Brown Band
Review – Carrie Underwood's "Something In The Water"
A wide, sweeping undertaking, "Something In The Water" sees Carrie Underwood carve out the sweet spot for her voice and make an inspiring and faith-based composition the vessel to illustrate the mighty ferocity of her God-given vocal prowess, along with instilling the moments with an elegance and grace that in unison swell to achieve one awe-inspiring performance height.
Amazing Grace, Carrie Underwood, Miranda Lambert, Something In The Water, Taylor Swift | {
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\section{Introduction}
In the last decade the spectre of quantum computing has begun to materialize. Recently, there have been a number of claims of hardware that represents some form of a quantum advantage \cite{arute2019quantum, zhong2020quantum, arrazola2021quantum, wu2021strong}. While it is still debated whether \say{Quantum Supremacy} has been reached \cite{pednault2019leveraging, huang2020classical, pan2021simulating}, there is no doubt that these advances represent substantial improvements in quantum computing hardware. There are a number of challenges when working with these devices, such as noise \cite{steane1998space}, decoherence \cite{pellizzari1995decoherence} and even cosmic rays \cite{mcewen2021resolving}. In addition to these common problems, the supposed \say{killer app} of near term quantum hardware, quantum machine learning, faces additional challenges \cite{mcclean2018barren}. Parallel to the developments of quantum computing in the past decade, deep reinforcement learning (RL) has achieved a number of impressive results. From reaching superhuman performance in games such as Chess \cite{silver2018general}, Poker \cite{brown2019superhuman} and Dota 2 \cite{berner2019dota}, to robotic control \cite{haarnoja2018learning} and chip design \cite{mirhoseini2021graph}.
Quantum Machine Learning (QML) seeks to apply the potential advantages of quantum computing for machine learning problems. Quantum machine learning presents a number of significant (both polynomial and exponential) theoretical speedups \cite{biamonte2017quantum, huang2021provably, liu2021rigorous}. QML algorithms have been developed for supervised learning \cite{schuld2018supervised, havlivcek2019supervised, perez2020data}, unsupervised learning \cite{otterbach2017unsupervised, kerenidis2018q, wiebe2015quantum}, and reinforcement learning \cite{chen2020variational, lockwood2020reinforcement, pmlr-v148-lockwood21a, jerbi2021variational}. A shared problem among many of these techniques is the optimization routine. Gradient \cite{cerezo2020cost} and gradient-free \cite{arrasmith2020effect} optimization techniques decrease in efficacy exponentially as the number of qubits grows. Additionally, independent of these phenomena, the presence of noise also induces exponential difficulties optimization \cite{wang2021noise}. While there has been some work in mitigating the effects of barren plateaus \cite{pesah2020absence, grant2019initialization, larocca2021diagnosing}, it remains a pervasive problem in optimizing QML models.
In this work, we propose a reinforcement learning based approach to the problems of optimizing QML systems. Specifically, we train a deep reinforcement learning agent to minimize the loss of random quantum variational circuits of random sizes with random objectives. There have been several previous applications of reinforcement learning to aid with some of the challenges of optimizing QML systems \cite{sordal2019deep, yao2020noise, yao2020policy, wauters2020reinforcement, khairy2020learning, baum2021experimental}. However, many of these works are limited in their applicable problem space, e.g. to only combinatorial/QAOA \cite{farhi2014quantum} routines or only to certain ansatzs/circuit structures. In this work, we shift towards a more general setup. Specifically, we create an ansatz, depth, qubit number, and cost function agnostic training routine (specified up to a maximum in each of these categories). We find that this optimizer can be used to effectively augment gradient based routines in noisy circuit simulations, increasing the performance across a variety of tasks without increasing the complexity of the circuit sampling.
\section{Background}
\subsection{Reinforcement Learning}
Deep Reinforcement learning (RL) is one of the three main branches of contemporary deep learning. The goal of RL is to have an agent learn to interact with an environment so as to maximize a reward signal \cite{sutton2018reinforcement}. The framework of RL is often formalized as a Markov Decision Process (MDP) with states $\mathcal{S}$, actions $\mathcal{A}$, and rewards $R$. The objective of this RL optimization problem is $J(\pi) = \max_\pi \sum_{t=0}^H \left [ \mathbbm{E}_{(s_t, a_t) \sim \pi} r(s_t, a_t) \right ] $ with horizon (the number of timesteps in the environment) $H$, state at time t $s_t$, action at time t $a_t$, and reward function $r$. In other words, the goal is to find the policy, $\pi$, which maximizes the expected return. In this work, we employ entropy maximizing RL algorithms, a state of the art class of algorithms that have shown to be especially robust \cite{eysenbach2021maximum}. Experiments were conducted with other SotA model free algorithm such as Proximal Policy Optimization (PPO) \cite{schulman2017proximal} and Twin Delayed Deep Deterministic Policy Gradient (TD3) \cite{fujimoto2018addressing}, however, we found them to be consistently outperformed by entropy based methods. These entropy maximizing algorithms have a slightly modified objective function, $J(\pi) = max_\pi \sum_{t=0}^H \left [ \mathbbm{E}_{(s_t, a_t) \sim \pi} r(s_t, a_t) + \alpha \mathcal{H}(\pi(\cdot | s_t)) \right ] $ \cite{haarnoja2018soft}, i.e. these algorithms seek to maximize the expected reward and the expected entropy of the policy. At $\alpha = 0$, this is the same as the previous objective. Note that for continuous functions, $\mathcal{H}(\pi(\cdot | s_t)) = - \int_\mathcal{A} \pi(a|s_t) log \pi(a|s_t) da$.
In order to maximize this objective, we utilize Soft-Actor Critic (SAC) \cite{haarnoja2018app}. SAC is a model free, off-policy, actor critic algorithm. The algorithm is composed of five total neural networks, one policy network, two Q networks and two target Q networks. The Q networks use neural networks to approximate the Q function \cite{watkins1992q}, which is an estimation of the expected reward given a state action pair. In the case of maximum entropy RL this takes the form of $Q_\theta(s_t,a_t) = \mathbbm{E}_\theta \left [r_t + \gamma \; log \int_\mathcal{A} e^{Q(s_{t+1}, a)} da \right ]$. The Q functions are updated via the Soft Mean Squared Bellman Error: \begin{equation*}
\begin{aligned}
\mathcal{L}_{SMSBE}(\theta) = [ Q_\theta(s_t, a_t) - (r(s_t, a_t) + \gamma (Q_{\theta^\prime}(s_{t+1}, a_{t+1}) \\ - \alpha \; log(\pi(a_{t+1}|s_{t+1})))) ]^2
\end{aligned}
\end{equation*}The target Q networks (with parameters denoted by $\theta^\prime$) serve to prevent premature numerical overestimation of the Q value and are updated via Polyak averaging \cite{polyak1992acceleration}. The policy network is updated in a similar manner to DDPG \cite{lillicrap2015continuous}, via estimation of the gradient from the Q function: $\nabla_{\phi} \alpha \; log(\pi_\phi(s_t, a_t)) + \nabla_{a_t} \alpha \; log(\pi_\phi(s_t, a_t)) - \nabla_{a_t} Q_\theta(s_t, a_t)$. Additionally, the temperature parameter $\alpha$ can be automatically adjusted.
\subsection{Quantum Machine Learning}
Quantum machine learning is built upon both advancements in quantum computing and in machine learning. Quantum computing advantages often stem from the ability of quantum computers to represent and operate on information that scales exponentially with the number of qubits. In this work, we focus on the Quantum Variational Circuit (QVC) as the machine learning model \cite{benedetti2019parameterized}. QVCs are a type of quantum circuit with learnable parameters. Any number of QVC setups are possible, in this work we consider arbitrary structure QVCs with the gate set $\left \{CNOT, H, R_x, R_y, R_z \right \}$. Note that this is a universal gateset, meaning any circuit can be represented via these gates. The Pauli rotations gates, $R_x(\theta), R_y(\theta), R_z(\theta)$, rotate around the specified axis $\theta$ radians, $R_\nu (\theta) = e^{-i\frac{\theta}{2}\sigma_\nu}$, where $\nu = X, Y, Z$. The controlled NOT (CNOT) gate is a two qubit gate that can induce entanglement in qubits. The aforementioned $\theta$ are the learnable parameters \cite{mcclean2016theory}. The measurement operator (from which the cost function is calculated) we utilize is the Pauli $\hat{Z}$ operator, or the `computational basis'.
The gradients of these quantum circuits can be calculated using the parameter shift rule \cite{schuld2019evaluating}. The rotation gates, $R_\alpha(\theta) = e^{-i\frac{\theta}{2}\sigma_\alpha}$, can be differentiated via $\pdv{}{\theta_i} = \frac{f(\theta_i + s) - f(\theta_i - s)}{2 sin(s)}$ given $s \in \mathbbm{R}, s \neq k\pi, k \in \mathbbm{Z}$ \cite{mari2021estimating} where $f(\theta) = \langle 0 | U^\dagger (\theta) \hat{Z} U(\theta) |0 \rangle$ and $U(\theta)$ is composed of these single qubit rotation gates. A common choice for s is $\pi/2$ \cite{bergholm2018pennylane}. Gradients can also be calculated in simulations using adjoint differentiation \cite{plessix2006review, luo2020yao} which requires no parameter perturbations (hence making it substantially faster for classical simulations), but is not feasible on quantum hardware.
\section{Approach}
To work with this QVC optimization problem, we must reconceptualize it into an environment compatible with RL agents. To this end we must consider the how states, actions, and rewards can be represented and numerically fed to the agent's neural network. Here we discuss our approach to each of these. First we have the state space problem, i.e. how do we convert the information from the QVC into an useful format for a RL agent to work with? Additionally, how can we effectively convey information about the structure, inputs, etc. that may vary? In this work we utilize two distinct approaches to convert the QVC information into inputs to the RL agent. Both encoding techniques rely on no simulation/statevector information and are fully compatible with any future or existing hardware. The first approach, which we call \say{feature} encoding, is inspired by the FLIP \cite{sauvage2021flip} algorithm. This encoding takes the QVC and returns a matrix with dimensions $\left [max \: parameters, 8 \right ]$, where each row contains the following information about the parameterized gate: current circuit error, current parameter value, gate type, qubit number, qubit layer, max qubits, max depth and input type. This input is then flatten and processed by a MLP in the RL agent. The second encoding scheme we call \say{block} encoding and is inspired by \cite{fosel2021quantum}. Block encoding takes the QVC and returns a 3D array with dimensions $\left [ max \; qubits, max \; depth, 5 \right ]$. Similar to other RL encoding schemes \cite{silver2018general}, each sheet/plane in this array represents information about a certain feature. The first 3 sheets contain the parameters of the associated Pauli rotation gates (i.e. sheet 1 corresponds with $R_x$ gates, and so on), the fourth represents the input type and the final layer is the current error. This matrix is then fed into a CNN for the agent to process. Prior to training the agent, a maximum number of parameters must be enforced (due to the static size of the weight matrices in the RL neural networks) but can be arbitrarily large. Now we consider the action space. The output of the RL agent is a vector that has the same length as the maximum number of parameters, representing the new value for each possible parameter. Not all parameters would necessarily be used, in cases where the number of parameters are less than the maximum the output is simply clipped to match this size. Finally, we consider the reward function. Due to the brittle nature of RL, constructing an effective reward function can be challenging and important for the agent to succeed \cite{henderson2018deep}. Our reward function is the negative mean squared error between the QVC with the parameters provided by the RL agent and the target value (which is randomly chosen during training). This is not required to be the loss function used in deployment, and we conduct experiments with a variety of cross entropy based losses by simply feeding this loss into the agent as the current error.
With the basics of the RL agent established, we can detail the training setup. At the beginning of a training routine, the maximum qubits, depth, and optimization timesteps are specified. Implicit within these is the maximum number of parameters (max qubits times max depth). The values used in this work (and provided in the pretrained examples) are 20, 20, and 150, respectively. This limits the number of parameters to be at most 400. At the beginning of each environment iteration a circularly entangled circuit is created with random Pauli rotation gates, random depth and a random number of qubits. An input type (either ground state or equal superposition state) is then selected. Finally a readout cost function is chosen. The possible functions are a product of the Z readout values on each qubit, the Z readout on the first qubit, and the sum of Z readout values on each qubit. A target value is then selected and the agent learns to maximize the negative mean squared error between the target value and the output of QVC.
The agents were then trained using 2 RTX 6000 24 GB GPUs with 12 vCPUs. With this, the total training time was approximately 150 hours. The evaluation time was done on the same hardware and took approximately 70 hours (largely dominated by some slower noisy simulations). The circuit simulation environment code was generated with TensorFlow-Quantum \cite{broughton2020tensorflow} and Stable Baselines 3 \cite{stable-baselines3} was used as the RL package.
While the agent is trained to directly optimize the circuit, given the scope and complexity of the environment (with no restrictions on the hyperparameters of the circuit), the agent is not meant to be used in this capacity. Rather, the agent is meant to augment traditional approaches by providing an alternative set of parameters at each optimization iteration, as outlined in the following algorithm. Essentially, each optimization step is done by taking the parameters that minimize the loss the most out of the two RL agents and gradient descent as shown in the algorithm below.
\begin{algorithm}[h]
\While{$iteration < max \; iterations$}{
$L$ = $\mathcal{L} \left ( f \left ( \theta \right )\right )$ \\
$\theta_g$ = $\theta - \nabla L$ \\
$\theta_{MLP} = SAC_{MLP} \left ( L \right )$ \\
$\theta_{CNN} = SAC_{CNN} \left ( L \right )$ \\
$L_g$ = $\mathcal{L} \left ( f \left ( \theta_g \right )\right )$ \\
$L_{MLP}$ = $\mathcal{L} \left ( f \left ( \theta_{MLP} \right )\right )$ \\
$L_{CNN}$ = $\mathcal{L} \left ( f \left ( \theta_{CNN} \right )\right )$ \\
$\theta = \min_L \left \{ \theta_g, \theta_{MLP}, \theta_{CNN} \right \}$ \\
}
return $\theta$
\caption{Augmented Algorithm}
\end{algorithm}
\section{Results}
We evaluate our RL techniques on six different problems, four classical and two quantum. The four classical problems include two binary classification problems, one multi-class classification and one regression (the Boston Housing Dataset). The classification problems can be visualized in Figure \ref{fig:class}. The two binary classification problems utilize 2 qubits, blobs uses 7 qubits, and the regression problem makes use of 13 qubits. The two quantum problems are the optimization routines of the Variational Quantum Eigensolver (VQE) \cite{peruzzo2014variational} and the Quantum Approximate Optimization Algorithm (QAOA) \cite{farhi2014quantum}. For VQE and QAOA we consider 3 different problems sizes of 5, 10, and 20 qubits. Note that the other hyperparameters for VQE and QAOA are constant. The QAOA problem is MAX-CUT and the random graph has regularity two and $p = 10$. The VQE problem generates random Hamiltonians that are decomposed into 10 Pauli sums and utilizes 5 layers of a hardware efficient ansatz \cite{kandala2017hardware}. The results are presented in three tables, each experiment was repeated 3 times (with different random initializations) and the $\pm$ indicates one standard deviation. Note that one cannot compare the numerical values across tables as the problems are randomly generated for each table. Each column represents an optimization technique to the same problem.
The SAC MLP and SAC CNN columns represent using just the specified RL agent for every optimization step (MLP corresponds with feature encoding and CNN with block). These results of these optimizers used solo can be found in Appendix A.
For all experiments the gradient descent optimizer is Adam \cite{kingma2014adam}
. In Table \ref{tab:exp1}, we show the results for simulations with zero noise (shot or depolarizing). The gradient descent makes use of adjoint differentiation to enable larger simulations. Table \ref{tab:exp2} shows the results for simulations with only shot noise. This table is shorter as the larger simulations are less feasible to conduct while using parameter shift differentiation techniques. The noisy simulation results are shown in Table \ref{tab:exp3}. In addition to shot noise, these circuits are simulated with depolarizing noise, modifying the density operator via $\rho \rightarrow \left ( 1 - p \right ) \rho + \frac{p}{4^n - 1} \sum_i P_i \rho P_i $ where $p$ is the probability and $P_i$ are the Pauli gates. In these experiments $p = 0.075$. The results tend to show consistent advantages for the augmented optimizer in the \say{real world} regimes (i.e. at least noise from expectation approximation) and inconsistent relative performance in completely noiseless simulations (achieving the best results in approximately 1/3 of the experiments).
\begin{figure}[h]
\centering
\subfloat[Binary Classification of Circles]{\includegraphics[width=0.33\textwidth]{circle_fig.png}\label{fig:circ}}
\hfill
\subfloat[Binary Classification of Moons]{\includegraphics[width=0.33\textwidth]{moon_fig.png}\label{fig:moon}}
\hfill
\subfloat[Multi-Class Classification of Blobs]{\includegraphics[width=0.33\textwidth]{blob_fig.png}\label{fig:blob}}
\caption{Classical Data Classification Problems}
\label{fig:class}
\end{figure}
\begin{table}
\centering
\begin{tabular}{|l|c|c|}
\hline
Evaluation & Gradient Descent & Augmented \\
\hline
Circle Train Loss & $ \mathbf{0.5834 \pm 2 * 10^{-6}} $ & $ 0.5834 \pm 5 * 10^{-6}$ \\
\hline
Circle Validation Loss & $ 0.5761 \pm 0.0001$ & $ \mathbf{0.57048 \pm 0.0017} $ \\
\hline
Moons Train Loss & $ 0.354218 \pm 0.000379 $ & $ \mathbf{0.3538 \pm 3 * 10^{-5} }$ \\
\hline
Moons Validation Loss & $ 0.36714 \pm 0.0042747 $ & $ \mathbf{0.3586 \pm 0.00143}$ \\
\hline
Blobs Train Loss & $ \mathbf{ 1.5677 \pm 0.03122} $ & $ 1.578 \pm 0.01315 $ \\
\hline
Blobs Validation Loss & $ \mathbf{1.56367 \pm 0.02969} $ & $ 1.5798 \pm 0.0155 $ \\
\hline
Regression Train Loss& $ \mathbf{0.0254 \pm 0.00502} $ & $ 0.0719 \pm 0.0182 $ \\
\hline
Regression Validation & $ \mathbf{0.0291 \pm 0.00513} $ & $0.0747 \pm 0.0131 $ \\
Loss & & \\
\hline
5 Qubit QAOA Cost & $ -1.50 \pm 8 * 10^{-7} $ & $ \mathbf{ -1.50 \pm 9 * 10^{-7}} $ \\
\hline
10 Qubit QAOA Cost & $\mathbf{ -2.9998 \pm 0.000205} $ & $ -2.9997 \pm 0.000262 $ \\
\hline
20 Qubit QAOA Cost & $\mathbf{ -7.8989 \pm 0.0877} $ & $ -7.667 \pm 0.461 $ \\
\hline
5 Qubit VQE Cost & $ -2.51396 \pm 0.1517 $ & $ \mathbf{-2.5595 \pm 0.1471} $ \\
\hline
10 Qubit VQE Cost & $ \mathbf{-1.2362 \pm 0.0109} $ & $ -1.2345 \pm 0.0195 $ \\
\hline
20 Qubit VQE Cost & $ -0.05 \pm 0.000838 $ & $\mathbf{ -0.05 \pm 0.000827} $ \\
\hline
\end{tabular} \caption{Noiseless} \label{tab:exp1}
\end{table}
\begin{table}
\centering
\begin{tabular}{|l|c|c|}
\hline
Evaluation & Gradient Descent & Augmented \\
\hline
Circle Train Loss & $\mathbf{0.5674 \pm 0.00032}$ & $0.5676 \pm 0.0003$ \\
\hline
Circle Validation Loss & $0.6087 \pm 0.001076$ & $\mathbf{0.605 \pm 0.0005565}$ \\
\hline
Moons Train Loss & $0.344 \pm 0.0002$ & $\mathbf{0.3439 \pm 0.00042}$ \\
\hline
Moons Validation Loss & $0.3844 \pm 0.000683$ & $\mathbf{0.3835 \pm 0.0008}$ \\
\hline
5 Qubit QAOA Cost & $-1.561 \pm 0.0061$ & $\mathbf{-1.5627 \pm 0.0034}$ \\
\hline
10 Qubit QAOA Cost & $\mathbf{-4.8 \pm 0.2382}$ & $-4.687 \pm 0.2756$ \\
\hline
5 Qubit VQE Cost & $\mathbf{-3.575 \pm 0.198}$ & $-3.5626 \pm 0.076$ \\
\hline
10 Qubit VQE Cost & $-1.0922 \pm 0.0813$ & $\mathbf{-1.1215 \pm 0.0599}$ \\
\hline
\end{tabular} \caption{Only shot noise}\label{tab:exp2}
\end{table}
\begin{table}
\centering
\begin{tabular}{|l|c|c|}
\hline
Evaluation & Gradient Descent & Augmented \\
\hline
Circle Train Loss & $0.6739 \pm 0.000303$ & $\mathbf{0.6724 \pm 0.00012}$ \\
\hline
Circle Validation Loss & $0.656 \pm 0.000278$ & $\mathbf{0.6547 \pm 0.00106}$ \\
\hline
Moons Train Loss & $0.644 \pm 0.005$ & $\mathbf{0.6347 \pm 0.00307}$ \\
\hline
Moons Validation Loss & $0.6406 \pm 0.0058$ & $\mathbf{0.631 \pm 0.0043}$ \\
\hline
5 Qubit QAOA Cost & $-1.52 \pm 0.00998$ & $\mathbf{-1.5773 \pm 0.01062}$ \\
\hline
10 Qubit QAOA Cost & $-4.805 \pm 0.683$ & $\mathbf{-5.2257 \pm 0.0889}$ \\
\hline
5 Qubit VQE Cost & $ -0.1634 \pm 0.0297 $ & $\mathbf{-0.1716 \pm 0.0242} $ \\
\hline
10 Qubit VQE Cost & $-1.0683 \pm 0.06495$ & $\mathbf{-1.10984 \pm 0.0272}$ \\
\hline
\end{tabular} \caption{Shot and depolarizing noise}\label{tab:exp3}
\end{table}
We also briefly present the present the potential of this augmented approach to aid with the barren plateaus problem. We consider a toy example of two layers of $R_X, R_Y$ rotations on 6 qubits. Traditional gradient descent approaches are unable to succeed (even with 10000 shots to reduce shot noise). However, we present two approaches that may help to alleviate this problem. First is the exact same algorithm as above, these results can be seen in Figure \ref{fig:bp}. We are also able to achieve better results in this case by keeping slightly more history and instead of performing gradient descent on $\theta_t$ we also perform gradient descent on $\theta_{t-1}$ and simply add this into the greedy selection of minima for the next $\theta$. This is able to achieve the results shown in Figure \ref{fig:bp1}.
\begin{figure}
\centering
\includegraphics[scale=0.5]{p.png}
\caption{Loss vs. Iterations of 6 Qubit System with Global Cost Function}
\label{fig:bp}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5]{p2.png}
\caption{Loss vs. Iterations of 6 Qubit System with Global Cost Function With Rolling Gradients}
\label{fig:bp1}
\end{figure}
\section{Discussion}
Here it is important to highlight and clarify a potential confusion. It may not be clear as to why the augmented optimizer is not upper bounded by the other optimizers and does not always perform as well. This is primarily because it does not necessarily explore the same region of the cost landscape. At every iteration, the augmented optimizer takes the biggest step in the negative direction, hence it does not follow the same trajectories as the other optimizers. As the RL optimizers' predictions are dependent on the current parameters, these differing steps result in different inputs to the agent and therefore different output. This is likely why the gradient descent approach out performs the augmented optimizer in completely noise free situation, i.e. the greedy augmented approach leads to differing local minima. Secondly, although the RL agents are trained as solo optimizers (without any augmentation), they seem to consistently be the worst performing. This is likely because the evaluations are so different from the training. Given the variety of hyperparameters of current QVC systems (ansatz, cost function, etc.) not only is it impossible to directly train on all combinations, but the goal is to be somewhat generalizable to situations not directly in the training data. Thus, excellent performance on a limited set of tasks is traded for worse performance on a larger set of problems.
\subsection{Future Work}
There are a number of potential future directions for this line of work. A clear extension is to expand training and validation for larger systems, up to and beyond 30 qubit simulations which would require substantially more computational power. Another important step would be to verify and experiment with these optimizers on actual quantum hardware. Additionally, this approach can be compared and combined with other gradient (and non-gradient) based optimizers to provide further insight into this approach. Finally, the potential to alleviate barren plateaus seems substantial but is limited to a toy problem. Expanding on this analysis is important and potentially very impactful.
\section{Conclusion}
In this work, we presented and experimented with an approach to train and evaluate deep reinforcement learning to optimizer quantum variational circuits. These agents have the potential to take advantage of recent advancements in deep learning and reinforcement learning to aid with the difficult task of quantum circuit optimization. We trained (and provided) models on large systems of up to 20 qubit, 400 parameter quantum variational circuits. We analyzed these agents capabilities to augment existing gradient based approaches to optimization on a variety of quantum machine learning tasks of various sizes and various levels of noise. The empirical findings suggest the augmentation can help even in the absence of noise, and consistently helps in the presence of shot or depolarizing noise. Our work is indicative that contemporary deep learning research can help to alleviate some challenges of working with and optimizing on current and near term quantum hardware.
\newpage
\section*{References}
\bibliographystyle{iopart-num}
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{"url":"https:\/\/computationalnonlinear.asmedigitalcollection.asme.org\/electronicpackaging\/article\/144\/3\/031015\/1136901\/Experimental-and-Numerical-Analysis-of-Data-Center","text":"## Abstract\n\nAn increasingly common power saving practice in data center thermal management is to swap out air cooling unit blower fans with electronically commutated plug fans, Although, both are centrifugal blowers. The blade design changes: forward versus backward curved with peak static efficiencies of 60% and 75%, respectively, which results in operation power savings. The side effects of which are not fully understood. Therefore, it has become necessary to develop an overall understanding of backward curved blowers and compare the resulting flow, pressure, and temperature fields with forwarding curved ones in which the induced fields are characterized, compared, and visualized in a reference data center which may aid data center planning and operation when making the decisions of which computer room air handler (CRAH) technology to be used. In this study, experimental and numerical characterization of backward curved blowers is introduced. Then, a physics-based computational fluid dynamics model is built using the 6sigmaroom tool to predict\/simulate the measured fields. Five different scenarios were applied at the room level for the experimental characterization of the cooling units and another two scenarios were applied for comparison and illustration of the interaction between different CRAH technologies. Four scenarios were used to characterize a CRAH with backward curved blowers, during which a CRAH with forwarding curved was powered off. An alternate arrangement was examined to quantify the effect of possible flow constraints on the backward curved blower's performance. Then parametric and sensitivity of the baseline modeling are investigated and considered. Different operating conditions are applied at the room level for experimental characterization, comparison, and illustration of the interaction between different CRAH technologies. The measured data is plotted and compared with the computational fluid dynamics (CFD) model assessment to visualize the fields of interest. The results show that the fields are highly dependent on CRAH technology. The tile to CRAH airflow ratios for the flow constraints of scenarios 1, 2, 3, and 4 are 85.5%, 83.9%, 61%, and 59%, respectively. The corresponding leakage ratios are 14.5%, 16%, 38.9%, and 41%, respectively. Furthermore, the validated CFD model was used to investigate and compare the airflow pattern and plenum pressure distribution. Lastly, it is notable that a potential side effect of backward curved technology is the creation of an airflow dead zone.\n\n## 1 Introduction\n\nThe amount of energy consumed by data center (DC) cooling is rapidly increasing and becoming a major concern for DC operators and managers due to increasing demand for DC services such as artificial intelligence, image processing, machine learning, and internet usage. Current trends seek to maximize data center utilization, reduce the energy consumed by information technology (IT) equipment and cooling infrastructure, and utilize cooling systems more efficiently. To that end, practical measurement tools and predictive models are needed to accurately capture the temperature, pressure, and flow fields of cooling and heating sources. A few examples of approaches that are often considered for reducing cooling energy consumption are direct and indirect evaporative cooling, wet-bulb economizer systems, data center best practices, and the implementation of more efficient cooling units such as electronically commutated (EC) plug fans [13].\n\nIn the literature, Hannman et al. [4] proposed an empirical-based measurement method to investigate the energy efficiency of best practices. In a different study, Radmehr et al. [5] used an averaged face velocity Pitot tube array to measure the volumetric airflow of the cooling unit and the perforated tiles in a raised-floor data center. The authors emphasized the integration of accurate facility transport measurements into data center infrastructure management tools based on computational fluid dynamics (CFD) for further predictions and management. The mismatch between the measured data and the CFD results was investigated by Iyengar et al. [6]. Their results showed that the maximum difference between the measurements and predictions was found at the air path between the exhaust of the server's rack and the intake side of the air cooling unit for a small data center cell.\n\nFluctuations in the tile airflow measurements in raised-floor data centers are induced by the turbulent residual component. Other factors that can cause variations in the margin of error include room size, perforated tile location, IT equipment inventory, plenum geometry, and obstructions. The variations in the flow measurements can be \u00b110% or more when impacted by the aforementioned factors. Therefore, significant flow measurement variations can make compact modeling and validation difficult. Nonetheless, compact modeling is widely used to predict and simulate the momentum transport through perforated tiles and pressure field variations. Large errors are expected when numerical and experimental errors are combined. Samadiani et al. [7] discussed floor grills modeling in their study. They added details to their six different models of the DC, including chiller pipelines and different tile open area ratios. They validated their numerical results with experimentally measured data. In a different study, a comparison between experimental and computational results was conducted by Abdelmaksoud et al. [8] to demonstrate the importance of considering the grill geometry details. Their results showed that tile modeling should account for opening patterns that affect the overall air-jet behavior. In another study by Garimella et al. [9] a summary of challenges and future trends for electronics thermal management was introduced as defined by experts in this field representing a wide range of industries. The authors showed that air cooling remains resilient as a data center cooling technique. Nada and Said [10] presented a CFD-based investigation of data center thermal management and airflow with different plenum depths, tiles openness, and rack power densities. They recommended 25\u201330% for the tile perforation and 60\u2009cm as a suitable value for the plenum depth. Moazamigoodarzi et al. [11] presented CFD-based airflow and temperature field characteristic methodologies for three different architectures of DC cooling: room, row, and rack-based cooling. Their results showed a 29% reduction in the cooling power of row and rack cooling over the room. An additional reduction was accomplished by adding enclosures at the row and rack-level architectures. In another study, Lim and Chang [12] used CFD simulation to examine the effect of the servers' air outflow angles on their temperature fields. Their validated numerical results showed that an exhaust angle of 60\u2009deg leads to the suppression of hot spots. The authors introduced two ventilation efficiency indices at a server level to pinpoint hot spots as well. Song et al. [13] proposed deflectors in cold aisle containment to improve the airflow distribution uniformity. Their experimentally validated numerical simulation proved the feasibility of applying such deflectors for airflow uniformity and hot spot elimination.\n\nComputational fluid dynamics modeling was used to examine the effect of under-floor plenum obstructions on the thermal performance and tile flow delivery [1416]. The researchers characterized the impact of underfloor blockages on DC performance and established guidelines in the form of a plenum color code. Moreover, they experimentally validated the guidelines in a different DC and presented a comparison of the numerical and experimental results. The results showed the severe impact of critically located blockages on reducing tile airflow by 19%. In a different study, Farazan Roknaldin [17] used the existing fan macromodel to develop a macromodel for a blower by introducing a systematic approach. The CFD results of this study were in alignment with the experimental results.\n\nStavreva and Serafimov [18] employed a CFD technique to investigate the velocity, temperature, and pressure fields of a dense data center. They studied and presented the impact of airflow distribution on energy efficiency. The results of their simulation found a potential method to improve the data center's energy efficiency, which was to adopt upgraded working conditions. Additionally, by using their CFD technique the authors discovered operational weak points within the dense DC.\n\nHuang et al. [19] numerically studied three different levels of air-cooling using a CFD technique: underfloor, rack level, and row-level systems. Among the available cooling performance indices evaluated, the authors used the index of mixing, return temperature index, KT, \u03b2, and \u03b7r to compare different cooling levels. The authors revealed that rack-level cooling showed the best performance with 0.0011 and 0.0082 values for the index of mixing and KT, respectively. Moreover, Yann et al. [20] introduced and analyzed a new concept of an in-rack-cold-aisle (IR-CA) system. The authors studied seven different rack intake cross-sectional areas with an additional partition plane placed at the rack inlet. Their results revealed the optimal thermal distribution was obtained for the IR-CA case with the partition plane among the rest of the studied cases. In another study, using air to bypass the computer room air handler (CRAH) unit was proposed by Erden et al. [21]. The authors used tile fans to induce a fraction of room air into the pressurized underfloor plenum. Experimental verification of the flow network model and the thermodynamic model was introduced. An optimum bypass fraction was determined based on their analysis of the combined power consumption of the chiller and CRAH fans. Among the available turbulent models, Wibron et al. [22] examined a more advanced turbulence model's performance for data center CFD modeling. The authors considered the standard $k\u2010\u03b5$ model, the Reynolds stress model, and the detached eddy simulation using experimentally validated CFD. Both the Reynolds stress model and detached eddy simulation models had an advantage over the standard $k\u2010\u03f5$ model, which failed to predict a low-velocity regime.\n\nWan et al. [23] surveyed and provided an overview of the current research that tends to improve the efficiency of air-cooling. Recently, Jin et al. [24] reviewed and summarized the aspects of airflow performance metrics and thermal optimization. The authors categorized the data center's thermal environment into three levels: room, rack, and server environments. The data center's airflow importance was emphasized. Based on their summary, the interactions of different factors such as location\/CRACs model, plenum height, and perforated tile area openness percent need to be further investigated. In addition, Gong et al. [25] introduced a comprehensive review of the state of the art of thermal performance evaluation in data centers. They focused on metric characteristics and application levels (room, row, rack, and server levels). The authors discussed the advantages and limitations of major metrics and proposed an evaluation criterion for DC designers and operators to use the most appropriate indices for DC optimization. Huang et al. [26] investigated the effect of perforated tile arrangement on the airflow and temperature fields under nonuniform rack heat loads using numerical simulation. Under proper tile arrangement and openness ratio, they reported a 2.1\u2009\u00b0C reduction in the maximum rack outlet temperature.\n\nIn this study, common data centers tools are used to measure the airflow rate, temperature fields above the tiles, and the pressure differential across the raised floor. A custom velocity\/temperature sensor grid was designed and built to measure the velocity and temperature fields at the intake of the cooling units for different operating conditions. The measured data is plotted, presented, and compared with a physics-based CFD model used later to visualize the fields of interest. The study aims to characterize and compare the flow, pressure, and temperature fields for different CRAH technologies using experimental measurements along with CFD visualizations. The results of this study can help to develop an understanding of different fields induced by different cooling units in a raised floor data center environment, to capture the hidden risk and key features of the produced momentum transfer by different blower technologies, and thus to guide capacity and installation planning of different units. The variations in airflow and pressure field measurements provide an important characteristic of backward curved blowers. The CFD visualization proves that these variations can be caused by the presence of a vortex, and in doing so, can also guide the installation of air driving technology to avoid or minimize the effect of vortices. To the best of the authors' knowledge, this is the first report of its kind that provides a detailed experimental and numerical analysis showing the effect and interaction of different CRAH technologies.\n\n## 2 Experimental Characterization\n\n### 2.1 Physical Domain Description.\n\nA plane view of the air-cooled raised-floor data center laboratory, which is located at the State University of New York (SUNY) at Binghamton is illustrated in Fig. 1. There is a slab to slab height of 4.37\u2009m (14.33\u2009ft), which is broken down as follows: 0.91\u2009m (3\u2009ft) under-floor plenum depth and 3.45\u2009m (11.3\u2009ft) from the ceiling down to the raised floor. The room air is returned to two chilled-water cooling units. A traditional arrangement of hot aisle\/cold aisle is used in the laboratory. The DC has an aisle pitch of 14\u2009ft. The center to center distance between cold aisles varies between seven and nine raised floor panels, each of which measures 2\u2009ft \u00d7 2\u2009ft. Furthermore, the perforated tiles in the cold aisles (A, C, and D) have a 22% open area. IT racks are placed in the white space of the real facility forming four cold aisles A, B, C, and D as follows: Aisle A is the network aisle that servers the other aisles and the data logging systems. Aisle B has no deployed IT equipment inside the IT racks. Aisle C is the main compute aisle where most of the IT load is concentrated. Finally, aisle D is mainly equipped with storage devices. Due to the low number of infrastructure blockages (chiller pipes), the underfloor plenum can be considered empty for the purposes of this study. A typical modern DC has modular air conditioning units and underfloor cool air distribution. The cooling cycle starts at the internally housed air handlers (i.e., CRAH1 and CRAH2). First, heat is extracted from the warm intake air thereby lowering its temperature, and the resulting cold air is supplied to the DC through the perforated tiles, after which it is forced across the IT equipment to carry away the produced heat.\n\nFig. 1\nFig. 1\nClose modal\n\n### 2.2 Computer Room Air Handler Technology (Forward and Backward Curved Blowers).\n\nBackward curved blowers, which are driven by EC motors, are rapidly replacing traditional belt-driven forward curved blowers. The primary reasons are their increased operational efficiency, which is a result of avoiding the power losses associated with belt drives, and their reduced mechanical complexity, which increases the lifespan of the bearings. EC motors also provide a less complex method of controlling variable speed drive for the blower's wheel. Notably, backward curved blowers provide an airflow delivery pattern that is different from forwarding curved blowers: the change in blower design improves the peak static efficiency from 60% for forward curved blowers (per original equipment manufacturer (OEM)) to more than 75% for backward curved blowers.\n\nConventional forward curved blowers were previously studied [27]. The results of that study led to the installation of an EC driven backward curved blower unit (CRAH2) of identical dimensions and capacity. Both types are shown in Figs. 2(b) and 2(d). This entire unit has three EC plug fans and is also lowered into the plenum. It is attached to a separate chiller line and provides cool air directly to the plenum to support the deployment of additional IT. It is important to note that Liebert chilled water (CW) down-flow models equipped with EC driven backward curved blowers can be operated with the fans in the fully raised position or lowered into the floor. The lower position provides these blowers with reduced air resistance and thus increased efficiency, but they also work well when not lowered. They are more efficient than forward curved blowers that have a bottom discharge with a mostly vertical downward velocity vector. Additionally, backward curved blowers gain more efficiency in the fully lowered position since air delivery is a combination of radial and tangential velocity. A summary of each cooling unit's specifications is provided in Table 1, as reported by the OEM technical manual.\n\nFig. 2\nFig. 2\nClose modal\nTable 1\n\nSummary of cooling unit's specifications\n\nSpecificationCRAH 1CRAH 2\nOverall dimensions (mm)Both are identical with [H \u00d7 W \u00d7 D] of [1930 \u00d7 3099 \u00d7 899]\nBlowersForward curvedBackward curved\nStatic efficiency (%)6075\nNominal flow rate (CFM)16,50017,300\nNominal cooling capacity (Ton)3232\nModelCW14DCVCCW14DC1A\nScoop availabilityAvailableNot Available\nBlower driving mechanismAll blowers are driven by variable speed motorEach blower is driven by an independent EC motor\nMaximum power (kW)10.239.5\nSpecificationCRAH 1CRAH 2\nOverall dimensions (mm)Both are identical with [H \u00d7 W \u00d7 D] of [1930 \u00d7 3099 \u00d7 899]\nBlowersForward curvedBackward curved\nStatic efficiency (%)6075\nNominal flow rate (CFM)16,50017,300\nNominal cooling capacity (Ton)3232\nModelCW14DCVCCW14DC1A\nScoop availabilityAvailableNot Available\nBlower driving mechanismAll blowers are driven by variable speed motorEach blower is driven by an independent EC motor\nMaximum power (kW)10.239.5\n\nFigure 2 illustrates a closer look at both blowers. In forward curved blowers, the air enters through the sides and is driven out in a downward direction by the in-place rotating blades. In backward curved blowers, the air enters through the top and is driven outward in all directions by the rotating blades.\n\n### 2.3 Experimental Setup.\n\nAn experimental characterization of the backward curved blower was conducted based on CRAH2 in the DC laboratory. The flow and pressure fields induced by CRAH1 and CRAH2 are compared. The tile airflow resulting from the interaction of both CRAH technologies is analyzed. Table 2 summarizes all of the studied scenarios and operating conditions. It is worth mentioning that scenarios 1-4 were used to characterize CRAH2, therefore, CRAH1 was powered off during these scenarios. Scenarios 6 and 7 were used to investigate the interaction of different CRAH technologies. An alternate arrangement was examined to quantify the effect of possible flow constraints on the backward curved blower's performance. Although Alissa et al. [27] previously presented experimental and analytical evaluations of CRAH1, the operating conditions in scenario 5 were conducted for comparison and model validation.\n\nTable 2\n\nSummary of tested scenarios and operating conditions\n\nCRAH1CRAH2Aisle C doors\nScenario #(blowers status)(blowers status)(opened\/closed)Perforated tiles\n1OFFONOpenedAll tiles opened\n2OFFONClosedAll tiles opened\n3OFFONOpenedBlocked tiles in Aisles A and D\n4OFFONClosedBlocked tiles in Aisles A and D\n5ONOFFOpenedAll tiles opened\n6ON at 50% variable frequency drive (VFD)ON at 50% VFDOpenedAll tiles opened\n7ON at 100% VFDON at 100% VFDOpenedAll tiles opened\nCRAH1CRAH2Aisle C doors\nScenario #(blowers status)(blowers status)(opened\/closed)Perforated tiles\n1OFFONOpenedAll tiles opened\n2OFFONClosedAll tiles opened\n3OFFONOpenedBlocked tiles in Aisles A and D\n4OFFONClosedBlocked tiles in Aisles A and D\n5ONOFFOpenedAll tiles opened\n6ON at 50% variable frequency drive (VFD)ON at 50% VFDOpenedAll tiles opened\n7ON at 100% VFDON at 100% VFDOpenedAll tiles opened\n\nFor all of the tested scenarios, most of the IT equipment in aisles C and D are powered off. Hence, heat dissipation in the facility can be broken as follows:\n\n\u2022 25\u2009light emitting diode F8B lighting fixtures. Each has three bulbs with heat dissipation of 144 W, therefore, their share in the total heat dissipation can be approximated as 3.6\u2009kW.\n\n\u2022 6\u201315\u2009kW dissipated from a uninterruptible power supply unit (Galaxy EPS 6000, 300 KAV) depending on the battery's charge status and operation.\n\n\u2022 1.32\u2009kW dissipated from a single-phase AC, 208-240 V, and 24 Amps PDUs.\n\n\u2022 42\u2009kW dissipated from IT equipment that is kept running at idling computational load. These ITEs are kept running to not disturb other experiments and are considered during the data analysis and numerical model validation.\n\n### 2.4 Tools and Measurement Methodology\n\n#### 2.4.1 Flow Hood.\n\nAs shown in Fig. 3, a flow hood (ADM-850 L) multimeter was used to report the airflow rate of the perforated tiles (locations shown in Fig. 1) for each test scenario with a measurement accuracy of\u2009\u00b13% of reading and\u2009\u00b17 CFM from 100 to 2000 CFM, as reported by the vendor technical manual. The device was equipped with an electronic micrometer to compensate for the additional flow impedance by the hood. The airflow balance provided backpressure airflow measurement, therefore, the error caused by the hood was eliminated. More information can be found in previous studies [28,29].\n\nFig. 3\nFig. 3\nClose modal\n\n#### 2.4.2 Velocity Temperature Sensor Grid.\n\nTo measure the air velocity and temperature at the intake side of the cooling units, a custom velocity\/temperature sensor grid was designed and built. As illustrated in Fig. 3, the sensor grid was installed on a wooden frame. A 6\u2009\u00d7\u20096 array of AccuSenseTM UAS1200 sensors was installed on the grid mesh. These sensors are a hot wire anemometer and thermistor that measure the air velocity and temperature simultaneously. The sensor's measurement ranges are 0.15\u20135\u2009m\/s (30\u2013400\u2009ft\/min) and 0\u201370\u2009\u00b0C (32\u2013158\u2009\u00b0F) with an accuracy of\u2009\u00b15% of the reading and\u2009\u00b11\u2009\u00b0C for velocity and temperature, respectively. An ATM2400 data acquisition hub was used to attach all 36 sensors via a USB port. Simplicity, mobility, minimum disturbance of airflow or straightening effect, and remote collection of data features were factors considered in the design of this custom build.\n\n#### 2.4.3 Cooling Unit Intake Flow Measurement Methodology.\n\nThe airflow of the cooling unit was measured using the velocity\/temperature sensor grid (VTSG). The intake side of the cooling unit was divided into five zones, as illustrated in Fig. 4. Each zone measured 24\u2033 wide. The VTSG depth was designed to be equal to CRAH unit depth. A traverse duct was used to restrict the normal component of return air velocity, as demonstrated in Fig. 4. This measurement methodology was used to report both air temperature and normal velocity components for the available CRAH technology (CRAH1 and CRAH2). Moreover, the reported velocities were used to estimate the nominal airflow capacity for each CRAH unit. On top of that, the reported data from the VTSG was used for the qualitative and quantitative analysis and illustration of the flow and temperature fields at the cooling unit's intake.\n\nFig. 4\nFig. 4\nClose modal\n\n### 2.5 Results and Discussion\n\n#### 2.5.1 Characterizing CRAH2 Flow Field (Backward Curved Blower).\n\nTo experimentally characterize CRAH2, four scenarios were considered, which are summarized in Table 2. In these scenarios, CRAH1 was powered off and the intake side was covered to prevent backflow through the unit, as represented in Fig. 5(a). For scenarios 1 and 2, the doors of aisle C were opened and closed, respectively. This helped depict the effect of the flow resistance added by the cold aisle containment (CAC). Meanwhile, all of the perforated tiles were opened in both scenarios. For scenarios 3 and 4, the doors of aisle C were opened and closed, respectively. Meanwhile, the tiles of aisles A and D were blocked in both scenarios. CRAH2 intake flow was measured for all scenarios. Then, the total airflow rate was obtained by integrating all zones over the small square area (3 \u00d7 3 in2) in the VTSG grid. Simultaneously, the airflow delivered to the aisle through the tiles was measured. The raised floor leakage through the seams and holes was estimated to be equal to the difference between the CRAH and tile airflows, and the measured raised floor gap was included in the CFD model. The model was thus able to report the leakage through the raised floor seems and pluming holes based on the specified gap size. According to Table 2 and for CRAH2 characterization the applied four-flow constraints are shown in Fig. 5(b).\n\nFig. 5\nFig. 5\nClose modal\n\nFigure 6(a) presents the measured airflow rate at the return face of the CRAH unit. A similar reading was noted for scenarios 1\u20133, which was approximately 18,400 CFM. For scenario 4 the drop in the airflow rate was insignificant, approximately 211 CFM which counts for 1% of the total airflow. Based on this, it can be inferred that the containment doors had no effect on the cooling unit's airflow rate for scenarios 1 and 2. In scenario 3, when the tiles of aisles A and D were blocked while those of aisle C were opened, the floor leakage and voids reduced the effect of the flow constraint on the cooling unit's airflow rate. In scenario 4, the cooling unit airflow rate dropped insignificantly by 211 CFM due to the introduction of CAC compared to scenario 3. Next, Fig. 6(b) presents the total tile airflow rate. The tile airflow rate readings for scenarios 1 and 2 dropped by \u223c205 CFM due to the introduction of CAC. From scenarios 3 and 4, the tile's airflow rate dropped by approximately 437 CFM.\n\nFig. 6\nFig. 6\nClose modal\n\nAn important consideration for the measurement procedure in this study is the uncertainty of the hot wire sensors, which was about 5%. That amount of error corresponds to approximately 900 CFM. While the measurement procedure was useful for obtaining the cooling unit's airflow rate within a 5% margin of error, this uncertainty meant that the variation in room pressure could have been slightly overestimated.\n\nA comparison of the individual tiles is demonstrated in Fig. 7. It can be inferred from this figure that containment in aisle C reduced the received flow and increased the airflow rates in the other aisles. The additional barrier to the air path in aisle C caused the total tile delivery to drop by 568 CFM. The reduced lower airflow was redirected to aisles A and D, while 205 CFM escaped through floor leakage paths. The amount of floor leakage was estimated to be 2658 and 2974 CFM for scenarios 1 and 2, respectively. These results indicate that floor leakage increased due to CAC. By restraining the flow to one aisle, in this case, aisle C, floor leakage increased significantly. For scenarios 3 and 4, the amount of leakage was estimated to be 7167 and 7495 CFM, respectively.\n\nFig. 7\nFig. 7\nClose modal\n\nThe tile to CRAH airflow ratios for the flow constraint in scenarios 1, 2, 3, and 4 were 85.5%, 83.9%, 61%, and 59%, respectively. The corresponding leakage ratios were 14.5%, 16%, 38.9%, and 41%, respectively. Furthermore, it can be noted from Figs. 7(b) and 7(d) that tiles c12, c14, and c11 always received a lower airflow rate than the other tiles.\n\nFigure 8 illustrates the various plots of successive measurements over 2\u2009min with a time-lapse of 15\u2009s between successive measurements for the perforated tiles of aisle C in scenario 2. The variation in measurements was plotted. It can be noted that tiles c14, c13, and c12 had higher variation. Those variations were caused by pressure wake under those tiles for which a vortex was suspected.\n\nFig. 8\nFig. 8\nClose modal\n\n#### 2.5.2 Backward and Forward Curved CRAH Technology Comparison of Tile's Airflow Delivery.\n\nTo demonstrate a comparison of the airflow delivered through perforated tiles provided by a standalone cooling unit. Figure 9 sets individual perforated tiles side by side for scenarios 1 and 5, where only one cooling unit was on duty and the other one was powered off with the intake side covered, as shown earlier in Fig. 5(a). For both scenarios, all of the tiles were open and the CAC doors of aisle C were open. Thus, it can be assumed that the airflow resistance in each scenario is similar. It can be noted that the tile airflow delivery was always higher when CRAH2 was on duty, except for tiles c13, c14, c15, c16, c17, and c18. This was related to the airflow pattern out of the unit and to the spatial location of these tiles with respect to the cooling unit, where they were in close proximity to high static pressure near the wall. This is explained in more detail in the numerical section.\n\nFig. 9\nFig. 9\nClose modal\n##### 2.5.2.1 Velocity field and profile at the intake side of the cooling units.\n\nThe air velocity contours and profiles were analyzed to investigate the effect of different CRAH technology on the air velocity across the width and length of the CRAH intake. In addition, part of these measurements is used later to validate the CFD model. Figure 10(a) demonstrates the velocity\/temperature sensor distribution at the intake side of the cooling units. Figures 10(b) and 10(c) illustrate the air velocity contours of CRAH2 and CRAH1, respectively, for scenarios 1 and 5. It can be inferred from the figures that at CRAH1's intake the air velocity was lower and more uniform than at CRAH2's intake. Additionally, the figures show the high normal component of air velocity at the front and side edges of CRAH2, which could have been the result of many factors, such as the momentum transfer provided by different CRAH technology, sensor movement during the test, or the edge effect, which is higher for backward curved blowers. Furthermore, the zonal approach provided more flexibility when comparing the air velocity across the width of the cooling units, as illustrated in Fig. 11(a). Each zone (refer to Fig. 10(a)) was divided into seven series. The air velocities measured across each series were plotted, as shown in Figs. 11(b)11(g). The figure illustrates the edge effect, which is similar to the one previously discussed. This edge effect caused a decrease in the air velocity of zone 1, whilst the effect was reversed for zone 5 (Fig. 12).\n\nFig. 10\nFig. 10\nClose modal\nFig. 11\nFig. 11\nClose modal\nFig. 12\nFig. 12\nClose modal\n##### 2.5.2.2 Interaction of different CRAH technology.\n\nFor insight into the interaction of the two different CRAH technologies, the flow, temperature, and pressure fields for scenarios 6 and 7 are compared. Figure 13 illustrates the tile's airflow rates in these scenarios, in which both CRAHs were on duty but with different supply capacities. The tiles airflow in aisles A and D exhibited similar behavior for both scenarios, whilst that of aisle C varied between each scenario. The results show that tile c8 reported a higher airflow rate compared to the adjacent tiles (c7 and c9) for scenario 7, whereas its airflow rate was lower for scenario 6. Other examples of this type of variation in aisle C between the scenarios can be found as well. Furthermore, the measured pressure differentials between the plenum and the room at different spatial locations for scenarios 1, 5, 6, and 7 are illustrated in Fig. 14. The results showed pressure maldistribution for all single CRAH scenarios with more uniformity noted when both CRAHs were on duty. Given the complex and turbulent nature of the airflow distribution in the underfloor plenum, the next part of the study will use a validated CFD model to visualize the flow, temperature, and pressure fields for the different CRAH technologies.\n\nFig. 13\nFig. 13\nClose modal\nFig. 14\nFig. 14\nClose modal\n\n## 3 Numerical Modeling\n\nComputational modeling is needed to reduce the time, cost, and risk associated with full-scale experiments. Therefore, the entire physical domain of the ES2 data center laboratory was replicated in a physics-based CFD model to visualize the flow, temperature, and pressure fields induced by the two different standalone CRAH technologies as well as the interaction of the two. In addition, the visualization illustrates the impact of different CRAH technologies on the data center environment and the possible formation of hot spots at the intake side of IT equipment. A grid independence test was conducted by varying the mesh size and then a chosen number of grids was selected to quantify the confidence and predictive accuracy of the CFD model [30,31].1 Adiabatic walls with no radiation heat transfer were assumed for the model. Air was assumed to be incompressible and have constant properties. Lastly, the wall roughness and body forces were considered negligible. Therefore, the classical Navier\u2013Stokes, and energy governing equations were numerically solved\n\nMass conservation\n$\u2207 . V\u00af=0$\n(1)\nMomentum conservation\n$\u03c1DV\u00afDt= \u2212 \u2207P+\u03bc \u22072 V\u00af+ \u03c1g\u00af$\n(2)\nEnergy conservation\n$DT\u00afDt= \u03b1 \u22072 T+1\u03c1cp\u03d5$\n(3)\n\nGiven the turbulent nature of the flow in raised-floor data centers, the standard k\u03b5 model coupled with RANS equations was selected to simulate the flow fields. The selection of such a model was based on the available literature that confirmed the feasibility of employing it in data center simulations [32].\n\n### 3.1 Model Validation.\n\nThe quantified confidence and predictive accuracy of the ES2 data center CFD model have been developed and refined over several years under the supervision of a commercial software developer [29]. Furthermore, to ensure the model predictivity and accuracy for this study, the lumped fan model was adopted to model the fans due to its simplicity and the fact that it is computational cost effective. The geometry and specifications of the fans were imported from the OEM technical manuals and defined in the CFD model. The experimental operating conditions, summarized in Table 1 above, were imposed exactly in the CFD model. Thereafter, the modeling results were compared to the experimental ones. The comparison considered pressure, tile airflow rates, and temperature measurements at the intake side of the cooling unit, with an additional assessment of thermal IR images. It is worth mentioning that for all of the tested scenarios the supply air temperature was held constant at 68\u2009\u00b0F (20\u2009\u00b0C) and a constant thermal load was maintained.\n\nFigure 15 presents a comparison of individual tile's average measured and CFD predicted airflow rates for scenarios 2 and 4. The CFD model and the experiment agreed with a reasonable mismatch and a maximum absolute error of 8% in tiles' airflow rate. On the pressure side, Fig. 16 depicts the agreement between the average pressures reported by the pressure sensors at the preselected spatial coordinates in the DC room. Although the maximum calculated absolute error was 16.2%, the figures show good agreement between the CFD results and the experiment. As a final step of model validation, experimental quantitative and qualitative measures of air temperature at the intake side of the cooling unit were compared with the ones resulting from CFD simulation, as displayed in Figs. 17(a) and 17(b). In addition, Fig. 17(c) and 17(d) represent a thermal IR image for scenario 1. It can be inferred from the figure that there is good agreement between the predicted and actual thermal images as well as between the predicted and measured air temperatures. This validation of the physics-based CFD model highlights the importance of the digital twin for predicting the effect of any changes in the data center room.\n\nFig. 15\nFig. 15\nClose modal\nFig. 16\nFig. 16\nClose modal\nFig. 17\nFig. 17\nClose modal\n\n### 3.2 Modeling Results and Discussion.\n\nHaving established the level of agreement between the CFD model and experimental measurements, the pressure fields induced by the different CRAH technologies can be visualized and assessed. To that end, the pressure distribution across a horizontal plane in the plenum for all of the tested scenarios is illustrated in Fig. 18. Comparing the results for scenarios 1 and 2, it is apparent that the higher plenum pressure in scenario 2 indicates a more resistive airflow path due to the doors being closed. A similar pattern is evident between scenarios 3 and 4.\n\nFig. 18\nFig. 18\nClose modal\n\nIn addition, in the results for scenarios 1 and 5, the difference in the pressure fields induced by each CRAH technology is notable. The resulting fields were highly related to the direction of the CRAH outlet jets. For the forward curved blowers, the pressure increased as the distance from the CRAH increased vertically, with respect to the cooling unit.\n\nThe simulation results for the airflow patterns out of the cooling units and into the floors voids demonstrate the differences in the air outlet jets for each CRAH technology. These results are presented side by side in Figs. 19(a) and 19(b), wherein the airflow patterns are colored based on velocity. Each blower technology moved the air in significantly different ways. The forward blowers directed air straight out from the unit. The backward blowers directed air biased to the blowing direction of the fans. In the case of the backward blowers, there was also a dead zone that did not receive any air, which is circled in white in Fig. 19(b). Based on airflow patterns and pressure fields, different CRAH technology will behave differently in larger data center rooms. In the future, additional CFD models could be developed to further investigate, analyze and compare different CRAH technology deployment in hyper-scale data centers (Fig. 20).\n\nFig. 19\nFig. 19\nClose modal\nFig. 20\nFig. 20\nClose modal\n\nFurthermore, to have a straightforward comparison between different CRAH technologies, two additional scenarios were modeled. In these scenarios, CRAH2 was placed at the location of CRAH1 and CRAH1 at the location of CRAH2, where only one cooling unit was on duty and the other one was powered off. Pressure fields of both cooling technologies at different locations are illustrated in Fig. 21. It can be inferred that swapping the locations of CRAH2 (Figs. 21(a) and 21(b)) with CRAH1 (Figs. 21(c) and 21(d)) has an impact on the pressure distribution since it is highly dependent on the air outlet jets for each CRAH technology as demonstrated in Fig. 19.\n\nFig. 21\nFig. 21\nClose modal\n\n## 4 Conclusions\n\nThe main objective of this study was to develop an overall and improved understanding of the role of backward curved blowers and to compare the resulting flow, pressure, and temperature fields with those from forwarding curved blowers. Although backward curved blowers are 30% more energy efficient, there are other important aspects of their airflow patterns that distinguish them from forwarding curved blowers. Most notably, the airflow bias toward the blowing direction can create a dead zone wherein no air is delivered to the IT.\n\nIn this study, a CFD model was validated against experimental measurements while comparing the airflow, pressure, and temperature fields of two different CRAH blower technologies when used to cool the same data center geometry and IT load. The fields induced by backward curved blowers were characterized and compared with those of forwarding curved blowers in a reference data center laboratory. It was concluded that the resulting fields were highly related to the direction of airflow from each of the examined blower technologies. Initial CFD modeling used 22% open perforated tiles to match those currently in the data center. Further simulations using tiles with higher open ratios showed that the open ratio of the perforated tiles would have no effect on the overall observations. However, the simulations did show that perforated tiles with higher open ratios would lower raised floor leakage due to the lower pressure differential between the plenum and the white space.\n\nThe major experimental and simulation results of this study can be summarized as follows:\n\n\u2022 Containment doors constrain tile's airflow rate but do not affect the cooling unit measured intake airflow rate. Results showed a drop of 568 CFM in the total tile airflow delivery due to the introduction of an additional barrier to the air path through the aisle by containment. Thus, the floor leakage increased as well.\n\n\u2022 The procedure used for measuring the actual airflow rate of cooling units is useful in practice considering the small (5%) margin of error compared to OEM data. However, this uncertainty may result in room pressure over-estimation.\n\n\u2022 Restraining the flow to one aisle (Aisle C here) results in a significant increase of the floor leakage, which was estimated to be 7167 and 7495 CFM for scenarios 3 and 4, respectively. The tile to CRAH airflow ratios for the flow constraint scenarios 1, 2, 3, and 4 were 85.5%, 83.9%, 61%, and 59%, respectively. The corresponding leakage ratios were 14.5%, 16%, 38.9%, and 41%, respectively.\n\n\u2022 With respect to the different airflow patterns out from the cooling units and the spatial location of tiles with respect to the unit, the airflow delivered through perforated tiles was mostly higher for CRAH2, except for specific tiles.\n\n\u2022 Considering the air velocity uniformity at the cooling unit intake side, the results showed that CRAH1 had a more uniform air velocity, which was related to how the air entered and exited each blower type.\n\n\u2022 Finally, the simulation results obtained from the physics-based CFD model showed an 8% and 16.2% mismatch from the experimental airflow and pressure measurements, respectively. The CFD model qualitatively predicted the temperature field.\n\n\u2022 Different blowers move the air in significantly different ways. The forward blowers directed the air straight out from the unit, while the backward blower's airflow was biased to the blowing direction of the fans. There was also a dead zone that did not get any air. Based on airflow patterns and pressure fields, different CRAH technology will behave differently in larger data center rooms.\n\n## Acknowledgment\n\nWe would like to acknowledge Vertiv and Future Facilities Ltd. We would also like to thank the ES2 Partner Universities for their support and advice.\n\n## Funding Data\n\n\u2022 NSF IUCRC (Award No. IIP-1738793 and MRI Award No. CNS1040666; Funder ID: 10.13039\/100000001).\n\n## Nomenclature\n\nACU =\n\nair cooling unit\n\nCFD =\n\ncomputational fluid dynamics\n\nCRAC =\n\ncompute room air conditioner\n\nCRAH =\n\ncomputer room air handler\n\nDC =\n\ndata center\n\nEC =\n\nelectronically commutated\n\nFNM =\n\nflow network model\n\nIOM =\n\nindex of mixing\n\nOEM =\n\noriginal equipment manufacturer\n\nSAT =\n\nsupply air temperature\n\nTDM =\n\nthermodynamic model\n\n## References\n\n1.\nGreenberg\n,\nS.\n,\nMills\n,\nE.\n,\nTschudi\n,\nB.\n,\nRumsey\n,\nP.\n, and\nMyatt\n,\nB.\n,\n2006\n, \u201c\nBest Practices for Data Centers: Lessons Learned From Benchmarking 22 Data Centers\n,\u201d\nProceedings of the ACEEE Summer Study on Energy Efficiency in Buildings in Asilomar, ACEEE\n, Pacific Grove, CA, Aug. 3, pp.\n76\n87\n.https:\/\/www.researchgate.net\/publication\/237375801_Best_Practices_for_Data_Centers_Lessons_Learned_from_Benchmarking_22_Data_Centers\n2.\nSchmidt\n,\nR.\n,\nBeaty\n,\nD.\n, and\nDietrich\n,\nJ.\n,\n2007\n, \u201c\nIncreasing Energy Efficiency in Data Centers\n,\u201d\nASHRAE J.\n,\n49\n(\n12\n), p.\n18\n.https:\/\/www.proquest.com\/openview\/6d91807833f6a455ba1007d136cd8b61\/1?pq-origsite=gscholar&cbl=41118\n3.\nPatankar\n,\nS. V.\n,\n2010\n, \u201c\nAirflow and Cooling in a Data Center\n,\u201d\nASME J. Heat Transfer-Trans. ASME\n,\n132\n(\n7\n), p. 073001.10.1115\/1.4000703\n4.\nHamann\n,\nH. F.\n,\nSchappert\n,\nM.\n,\nIyengar\n,\nM.\n,\nvan Kessel\n,\nT.\n, and\nClaassen\n,\nA.\n,\n2008\n, \u201c\nMethods and Techniques for Measuring and Improving Data Center Best Practices\n,\u201d\n11th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems\n,\nIEEE\n, Orlando, FL, May 28\u201331, pp.\n1146\n1152\n.10.1109\/ITHERM.2008.4544390\n5.\n,\nA.\n,\nSchmidt\n,\nR. R.\n,\nKarki\n,\nK. C.\n, and\nPatankar\n,\nS. V.\n,\n2005\n, \u201c\nDistributed Leakage Flow in Raised-Floor Data Centers\n,\u201d\nASME\nPaper No. IPACK2005-73273.10.1115\/IPACK2005-73273\n6.\nIyengar\n,\nM.\n,\nSchmidt\n,\nR. R.\n,\nHamann\n,\nH.\n, and\nVanGilder\n,\nJ.\n,\n2007\n, \u201c\nComparison Between Numerical and Experimental Temperature Distributions in a Small Data Center Test Cell\n,\u201d\nASME\nPaper No. IPACK2007-33508.10.1115\/IPACK2007-33508\n7.\n,\nE.\n,\nRambo\n,\nJ.\n, and\nJoshi\n,\nY.\n,\n2010\n, \u201c\nNumerical Modeling of Perforated Tile Flow Distribution in a Raised-Floor Data Center\n,\u201d\nASME J. Electron. Packag.\n,\n132\n(\n2\n), p.\n021002\n.10.1115\/1.4001589\n8.\nAbdelmaksoud\n,\nW. A.\n,\nKhalifa\n,\nH. E.\n,\nDang\n,\nT. Q.\n,\n,\nB.\n,\nSchmidt\n,\nR. R.\n, and\nIyengar\n,\nM.\n,\n2010\n, \u201c\nExperimental and Computational Study of Perforated Floor Tile in Data Centers\n,\u201d\n12th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems\n,\nIEEE\n, Las Vegas, NV, June 2\u20135, pp.\n1\n10\n.10.1109\/ITHERM.2010.5501413\n9.\nGarimella\n,\nS. V.\n,\nPersoons\n,\nT.\n,\nWeibel\n,\nJ. 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Heat Mass Transfer\n,\n174\n, p.\n121291\n.10.1016\/j.ijheatmasstransfer.2021.121291\n16.\n,\nM. I.\n,\nKhalili\n,\nS.\n,\nKhatabi\n,\nM.\n,\nSammakia\n,\nB. G.\n,\nSeymour\n,\nM.\n,\nTipton\n,\nR.\n, and\nAlissa\n,\nH. A.\n,\n2019\n, \u201c\nNumerical Investigation of Novel Underfloor Air-Directors Effect on Data Center Performance\n,\u201d 18th IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (\nITherm\n),\nIEEE\n, Las Vegas, NV, May 28\u201331, pp.\n886\n896\n.10.1109\/ITHERM.2019.8757464\n17.\nRoknaldin\n,\nF.\n,\nSahan\n,\nR. A.\n, and\nSun\n,\nX. H.\n,\n2002\n, May, \u201c\nA Simplified CFD Model for the Radial Blower\n,\u201d\nITherm Eighth Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems (Cat. No. 02CH37258)\n,\nIEEE\n, San Diego, CA, May 31\u2013June 1, pp.\n600\n604\n.10.1109\/ITHERM.2002.1012509\n18.\nStavreva\n,\nS.\n, and\nSerafimov\n,\nM.\n,\n2014\n, \u201c\nComputational Fluid Dynamics (CFD) Analysis for Predicting the Airflow in a Data Centre\n,\u201d\nTEM J.\n,\n3\n(\n3\n), p.\n235\n19.\nHuang\n,\nZ.\n,\nDong\n,\nK.\n,\nSun\n,\nQ.\n,\nSu\n,\nL.\n, and\nLiu\n,\nT.\n,\n2017\n, \u201c\nNumerical Simulation and Comparative Analysis of Different Airflow Distributions in Data Centers\n,\u201d\nProcedia Eng.\n,\n205\n, pp.\n2378\n2385\n.10.1016\/j.proeng.2017.09.854\n20.\nYuan\n,\nX.\n,\nXu\n,\nX.\n,\nLiu\n,\nJ.\n,\nPan\n,\nY.\n,\nKosonen\n,\nR.\n, and\nGao\n,\nY.\n,\n2020\n, \u201c\nImprovement in Airflow and Temperature Distribution With an In-Rack UFAD System at a High-Density Data Center\n,\u201d\nBuild. Environ.\n,\n168\n, p.\n106495\n.10.1016\/j.buildenv.2019.106495\n21.\nErden\n,\nH. S.\n,\nKoz\n,\nM.\n,\nYildirim\n,\nM. T.\n, and\nKhalifa\n,\nH. E.\n,\n2017\n, \u201c\nExperimental Demonstration and Flow Network Model Verification of Induced CRAH Bypass for Cooling Optimization of Enclosed-Aisle Data Centers\n,\u201d\nIEEE Trans. Compon., Packag. Manuf. Technol.\n,\n7\n(\n11\n), pp.\n1795\n1803\n.10.1109\/TCPMT.2017.2737878\n22.\nWibron\n,\nE.\n,\nLjung\n,\nA. L.\n, and\nLundstr\u00f6m\n,\nT. S.\n,\n2018\n, \u201c\nComputational Fluid Dynamics Modeling and Validating Experiments of Airflow in a Data Center\n,\u201d\nEnergies\n,\n11\n(\n3\n), p.\n644\n.10.3390\/en11030644\n23.\nWan\n,\nJ.\n,\nGui\n,\nX.\n,\nKasahara\n,\nS.\n,\nZhang\n,\nY.\n, and\nZhang\n,\nR.\n,\n2018\n, \u201c\nAirflow Measurement and Management for Improving Cooling and Energy Efficiency in Raised-Floor Data Centers: A Survey\n,\u201d\nIEEE Access\n,\n6\n, pp.\n48867\n48901\n.10.1109\/ACCESS.2018.2866840\n24.\nJin\n,\nC.\n,\nBai\n,\nX.\n, and\nYang\n,\nC.\n,\n2019\n, \u201c\nEffects of Airflow on the Thermal Environment and Energy Efficiency in Raised-Floor Data Centers: A Review\n,\u201d\nSci. Total Environ.\n,\n695\n, p.\n133801\n.10.1016\/j.scitotenv.2019.133801\n25.\nGong\n,\nX.\n,\nZhang\n,\nZ.\n,\nGan\n,\nS.\n,\nNiu\n,\nB.\n,\nYang\n,\nL.\n,\nXu\n,\nH.\n, and\nGao\n,\nM.\n,.\n2020\n, \u201c\nA Review on Evaluation Metrics of Thermal Performance in Data Centers\n,\u201d\nBuild. 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Eng.\n,\n10\n, pp.\n60\n68\n.10.1016\/j.jobe.2017.01.002","date":"2023-03-27 16:11:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 5, \"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\": 0.3778698444366455, \"perplexity\": 3018.5720170622512}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296948673.1\/warc\/CC-MAIN-20230327154814-20230327184814-00602.warc.gz\"}"} | null | null |
Vinaya Pitaka (2): The Analysis of Nun' Rules (Bhikkhuni-vibhanga)
by I. B. Horner | 2014 | 66,469 words | ISBN-13: 9781921842160
Buddhism Theravada
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The English translation of the Bhikkhuni-vibhanga: the second part of the Suttavibhanga, which itself is the first book of the Pali Vinaya Pitaka, one of the three major 'baskets' of Therevada canonical literature. It is a acollection of rules for Buddhist nuns. The English translation of the Vinaya-pitaka (second part, bhikkhuni-vibhanga) contain...
Nuns' Expiation (Pācittiya) 24
Bi-Pc.24.1.1 BD.3.290 … at Sāvatthī in the Jeta Grove in Anāthapiṇḍika's monastery. Now at that time[1] nuns, having entrusted robes[2] to the hands of (other) nuns, set out on a tour of the country with (only) the inner and the upper robes. Those robes, deposited for a long time, became soiled; nuns dried them in the sun. Nuns spoke thus to these nuns: "Ladies, whose are these robes that are soiled?" Then these nuns told this matter to the nuns. Those who were modest nuns … spread it about, saying: "How can nuns, having entrusted robes to the hands of (other) nuns, set out on a tour of the country with (only) the inner and the upper robes?" …
"Is it true, as is said, monks, that nuns … with (only) the inner and the upper robes?"
"It is true, lord."
The enlightened one, the lord, rebuked them, saying:
"How, monks, can nuns … with (only) the inner and the upper robes? It is not, monks, for pleasing those who are not (yet) pleased … this rule of training:
"Whatever nun should miss going about in an outer cloak for five days,[3] there is an offence of expiation."
Bi-Pc.24.2.1 Whatever means: … nun is to be understood in this case.
BD.3.291 Should miss going about in an outer cloak for five days means: if on the fifth day she neither dresses in nor puts on nor dries in the sun the five robes, (but) lets the fifth day pass, there is an offence of expiation.
Bi-Pc.24.2.2 If she thinks that five days are passed when they are passed, there is an offence of expiation.[4] If she is in doubt as to whether five days are passed, there is an offence of expiation. Vin.4.282 If she thinks that five days are not passed when they are passed, there is an offence of expiation. If she thinks that five days are passed when they are not passed, there is an offence of wrong-doing. If she is in doubt as to whether five days are not passed, there is an offence of wrong-doing. If she thinks that five days are not passed when they are not passed, there is no offence.
Bi-Pc.24.2.3 There is no offence if, on the fifth day, she dresses in or puts on or dries the five robes in the sun; if she is ill; if there are accidents; if she is mad, if she is the first wrong-doer.
Cf. Monks' Bu-NP.2 (BD.2.12).
Merely called cīvara here. The sikkhāpada makes it clear that the saṅghāti, outer cloak, is meant; Vin-a.652 says that this is the case with the monks' cīvara mentioned in Bu-NP.2. At some time the nuns came to be allowed five robes, mentioned below. For these see BD.2, Introduction, p.xix. It is therefore quite possible to say here that the nuns went with "only" their inner and upper robes, if we think of these with the outer cloak as constituting the regular set of three robes, to which the other two were merely added as extras for the nuns.
pañcāhikaṃ, what consists of five days
Some material left out here. These clauses should state that the offence also depends on her not dressing in, putting on or drying the five robes. Vin-a.929 says that for each robe there is an offence, thus for the five (robes) there are five (offences). | {
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