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\section{INTRODUCTION}
Early studies on highly dynamic motion such as jumping and running in legged robots are largely influenced by the heuristic control implemented on Raibert's hoppers \cite{raibert1986legged}. The hopper is able to achieve the dynamic behaviors through simple composition of a set of simple controllers that control hopping height, speed, and posture separately. This is possible due to their prismatic leg design which differs from most humanoid robots with human-like morphology.
Despite its success, heuristic control is quite limited since it needs large amount of work on parameter tuning. More recently, model based approaches are becoming more and more popular. The spring loaded inverted pendulum (SLIP) model is a well recognized template model for running and jump.
An approximated SLIP model has been used to represent the translational motion, robust running and jump are planned with model predictive control \cite{mordatch2010robust}.
3D-SLIP model has been used to generate high speed running motion \cite{wensing2013high} and long jump \cite{wensing2014development}. However, the SLIP model only considers the point mass dynamics and the angular momentum is ignored. It is therefore difficult to consider motions involving body rotation, such as a twisting jump.
\begin{figure}
\centering
\begin{subfigure}{0.8\columnwidth}
\includegraphics[width=0.9\linewidth]{figs/SLIDER_JumpUp_45D_trans.png}
\label{fig:intro_1}
\end{subfigure}
\begin{subfigure}{0.8\columnwidth}
\vspace{0.3cm}
\includegraphics[width=0.9\linewidth]{figs/ANYmal_flight.jpg}
\label{fig:intro_2}
\end{subfigure}\hfi
\caption{Highly dynamic jump motions can be generated for both bipedal and quadrupedal robots using the LL-SRBM. Top: SLIDER robot. Bottom: ANYmal robot.}
\vspace{-5mm}
\end{figure}
Compared to humanoid robots, quadrupedal robots are often built with lower degree of freedom (DoF) legs and point feet. This equips them with lightweight legs which are often ignored during the motion planning phase or the control phase without introducing significant modeling error.
With the assumption of massless legs, dynamic gaits including trotting, pronking, bounding and pacing have been demonstrated on Mini Cheetah through online convex model predictive control \cite{di2018dynamic,kim2019highly,katz2019mini}. The convex formulation is made possible due to a small angle assumption for roll and pitch angle of the Single Rigid Body Model (SRBM) and predefined footstep locations. Unfortunately, these assumptions do not hold for generating versatile and highly dynamic motions.
Recently, a more versatile and acrobatic motion generation framework, the kino-dynamic planner, is proposed in \cite{chignoli2021mit}. However, the kino-dynamic planner is complex and computationally inefficient. Therefore, a good trade-off between model complexity and computation efficiency is needed.
At the same time, the robustness of the controller is important for guaranteeing the successful execution of the dynamic motion. The robot experiences a large impact when landing; these large contact forces and fast changing velocities can easily make the robot unstable. Roboticists have made various efforts to reduce the effect of the impact. One direction is changing the reference trajectory in real time. \cite{corberes:hal-03034022} shows a quadrupedal robot walking with 50 Hz re-planning with a slightly simplified SRBM. But no results have been shown with more dynamic motions such as jumping. In \cite{halm2019modeling, 9341246} the impacts are explicitly modelled, however the simplified model and large uncertainties of state estimation make the control more difficult. \cite{yang2021impact} presents a feedback controller in an impact invariant space. Although this approach is not affected by the rapid change of velocities, it can not reduce the effects caused by the impact. To reduce the effect of the impact, good contact detection with a soft landing is a promising direction. External force sensors are commonly used to monitor contacts for robot arms \cite{haddadin2017robot}. But in practice force signals are too noisy for legged robots. The average spatial velocity, first introduced by \cite{4209816}, is a synthetic representation of all link velocities and an indicator for the kinetic energy of the robot. We will rely on the average spatial velocity for contact detection.
\subsection{Contributions}
In this paper, we aim to improve the performance of highly dynamic jumps with the LL-SRBM and the robustness of the tracking controller with contact detection. Additionally, we show that the proposed planning and control methods can be applied to both bipedal and quadrupedal robots. The twisting jump and forward jump are demonstrated in simulation and real robot experiments. Here, we would like to highlight the contributions of the paper:
\begin{itemize}
\item {A general framework that can generate highly dynamic jumping motions for both bipedal and quadrupedal robots. Each layer of the framework is generalized for both types of robots, including the lump leg concept. }
\item {The Lumped Leg Single Rigid Body Model (LL-SRBM) provides a computationally efficient way to optimize the shape of the robot's inertia while planning the jump motion. By taking the motion of legs into consideration, the jump motion generated by the LL-SRBM achieves better performance (e.g. faster twisting jump, more stable forward jump etc.) than the motion generated by using the SRBM. }
\item {Safe landing is ensured by using a novel contact detection method and switching to force control in landing phase. We propose to use the norm of the average spatial velocity to detect contacts. By taking account of all link states, the average spatial velocity is more robust to noises and general enough for detecting impulses from all directions. After the contact is detected, contact force tracking control is switched on for a period to achieve soft landing. We cannot use force control for the whole jump process since the LL-SRBM is still an approximated model for planning.}
\end{itemize}
\section{SYSTEM OVERVIEW}
The highly dynamic jumping behaviour is achieved via a hierarchical structure which includes the motion planner and the whole body controller as shown in Fig. \ref{fig:control_hierarchy}.
The high-level user selects the motion (e.g. twisting jump, forward jump, etc.) and inputs to the motion planner with initial and final states and contact positions for non-flight phases. The trajectory optimization with LL-SRBM outputs the reference centroidal and foot trajectories to the whole body controller. The whole body controller computes optimal joint torques for the robot to track these task space targets while considering all physical constraints. The contact detection runs after the robot takes off and monitors the average spatial velocity. Once the contact is detected, the whole body controller switches to force control for a short time to achieve a soft landing. After landing, trajectory tracking is reinstated to achieve the desired steady finishing pose.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{figs/control_hierarchy.pdf}
\caption{The planning and control framework.}
\label{fig:control_hierarchy}
\vspace{-5mm}
\end{figure}
\section{The Lump Leg Single Rigid Body Model}
We are interested in generating high fidelity dynamic jump motions while keeping the model simple enough. The novel aspect of LL-SRBM model is that it abstracts the robot as a single rigid body for the base and point masses for the legs, as illustrated in Fig. \ref{fig:LLSRBM}. The point mass of the leg is located at the CoM of the leg at the default configuration and the distance between the point mass and the leg contact location is proportional to the leg length. Compared to SRBM, the LL-SRBM models the leg dynamics as well, which is important for computing the centroidal inertia, as show in Fig. \ref{fig:inertia_shape_illustrate}. The full Newton-Euler dynamics of the LL-SRBM with multiple contacts with the environment can be written as:
\begin{equation}
\label{eq1}
\begin{bmatrix}
\dot{\bm{H}} = \sum^{n}_{i=1}\bm{f}_{i} + m\bm{g} \\
\dot{\bm{L}} = \sum^{n}_{i=1}(\bm{p}_{i} - \bm{r})\times \bm{f}_{i}
\end{bmatrix}
\end{equation}
where $\bm{H}$ and $\bm L$ are the linear and angular momentum of the model, $\bm f_{i}$ is the $i$th ground reaction force acting on $\bm p_{i}$, $n$ forces are assumed. $\bm r$ is the position of the CoM, $m$ and $\bm g$ are the mass of the model and the gravity vector.
The linear momentum is directly related to the translational velocity of the CoM:
\begin{equation}
\label{eq2}
\dot{\bm{r}} = \bm{H}/m \\
\end{equation}
\begin{figure}[t]
\centering
\begin{subfigure}{0.49\columnwidth}
\includegraphics[width=1.1\linewidth]{figs/SingleRigidBodyModel.png}
\caption{}
\label{fig:model_schema_1}
\end{subfigure}
\begin{subfigure}{0.49\columnwidth}
\includegraphics[width=1.1\linewidth]{figs/SingleRigidBodyModel_quadruped.png}
\caption{}
\label{fig:model_schema_2}
\end{subfigure}\hfi
\caption{The LL-SRBM is a general model for bipedal and quadrupedal robots, it takes the leg dynamics into consideration. It also unifies planar, line and point feet with the distributed Ground Reaction Force representation. (a) shows LL-SRBM applied to a bipedal robot with planar feet. The ground reaction forces are located at four corners of the foot with arbitrary directions. (b) shows LL-SRBM applied to a quadrupedal robot with point feet. For both figures $\mathcal{W}$ and $\mathcal{B}$ stand for the world frame and the body frame.}
\label{fig:LLSRBM}
\vspace{-5mm}
\end{figure}
For LL-SRBM, the CoM is computed as
\begin{equation}
\label{eq2-1}
\bm{r} = \frac{\bm{r}_B\cdot m_B+\sum_i{\bm{r}_l^i \cdot m_l^i}}{m_B+\sum_i{m_l^i}}
\end{equation}
where $\bm{r}_B$ and $m_B$ are CoM and mass of the body, $\bm{r}_l^i$ and $m_l^i$ are the position and mass of the $i$th leg.
Angular momentum of the LL-SRBM, $\bm L$, can be expressed as:
\begin{equation}
\label{eq3}
\bm L = \bm{I}_{W}\bm{\omega}
\end{equation}
where $\bm I_{W}$ represents the centroidal inertia of the model in the world frame and $\bm{\omega}$ is the angular velocity of the body. The centroidal inertia matrix, also called the centroidal composite rigid body inertia (CCRBI) matrix in \cite{4209816} can be computed as:
\begin{equation}
\label{eq4}
\bm{I}_W =\bm{R}_B\bm{I}_G\bm{R}_B^{T}
=\bm{R}(\bm{q}_{B})\bm{I}_G\bm{R}^{T}(\bm{q}_{B})
\end{equation}
where $\bm{R}(\bm{q}_{B})$ is a transformation from local to world frame as given in the Appendix. $\bm{I}_G \in \mathbb{R}^{6 \times 6}$ is the centroidal inertia matrix at CoM frame and can be computed as:
\begin{equation}
\label{eq5}
\bm{I}_G = \bm{I}_B + \sum_i{\bm{I}_l^i} = \bm{I}_B + \sum_i{m_i(\bm{r}-\bm{p}_i+\Delta\bm{p}_i)^2}
\end{equation}
where $\bm{I}_B$ is the inertia of the single rigid body projected to the CoM, and $\Delta\bm{p}_i$ is the offset between the leg's contact location and the position of leg mass, the second term represents the inertia of legs projected to CoM.
In \cite{di2018dynamic} the Euler angles are used to represent the orientation, however this approach suffers from the problem of gimbal lock, so we prefer quaternion representation for the orientation. The dynamics of the quaternion $\bm{q}_{B}$ can be expressed as:
\begin{equation}
\label{eq6}
\dot{\bm{q}}_{B} = \frac{1}{2}\bm{q}_{B} \circ \bm{\omega} = \bm{Q}(\bm{q}_{B})\bm{\omega}
\end{equation}
where $\circ$ represents the quaternion product and $\bm{Q}(\bm{q}_{B}) \in \mathbb{R}^{4 \times 3}$ is the corresponding matrix representation.
By associating equation (\ref{eq3}), (\ref{eq4}), (\ref{eq6}), we can get the dynamics of orientation represented by quaternion:
\begin{equation}
\label{eq7}
\dot{\bm{q}}_{B} = \frac{1}{2}\bm{Q}(\bm{q}_{B})\bm I_{W}^{-1}(\bm{q}_{B})\bm{L}
\end{equation}
So the complete dynamics of the LL-SRBM can be summarized as:
\begin{equation}
\label{eq8}
\left. \begin{bmatrix}
\dot{\bm{r}} = \bm{H}/m \\
\dot{\bm{q}}_{B} = \frac{1}{2}\bm{Q}(\bm{q}_{B})\bm I_{W}^{-1}(\bm{q}_{B})\bm{L}\\
\dot{\bm{H}} = \sum^{n}_{i=1}\bm{f}_{i} + m\bm{g} \\
\dot{\bm{L}} = \sum^{n}_{i=1}(\bm{p}_{i} - r)\times \bm{f}_{i} \\
\end{bmatrix}\right.
\end{equation}
with
\begin{equation}
\bm{r} = \frac{\bm{r}_B \cdot m_B+\sum_i{\bm{r}_l^i \cdot m_l^i}}{m_B+\sum_i{m_l^i}}
\tag{\ref{eq2-1}}
\end{equation}
\begin{equation}
\bm{I}_W =\bm{R}(\bm{q}_{B})(\bm{I}_B + \sum_i{m_i(\bm{r}-\bm{p}_i+\Delta\bm{p}_i)^2})\bm{R}^{T}(\bm{q}_{B})
\end{equation}
\section{TRAJECTORY OPTIMIZATION}
This section describes the mathematical formulation of trajectory optimization and how we formulate trajectory optimization into a Nonlinear Programming (NLP) problem.
\subsection{Problem Formulation}
\begin{figure}[t]
\label{fig:foot_schematic}
\centering
\begin{subfigure}{0.48\columnwidth}
\includegraphics[width=1.0\linewidth]{figs/Twist_jump_Inertia.png}
\caption{}
\label{fig:model_ill_1}
\end{subfigure}
\begin{subfigure}{0.48\columnwidth}
\includegraphics[width=1.0\linewidth]{figs/Twist_jump_NoInertia.png}
\caption{}
\label{fig:model_ill_2}
\end{subfigure}\hfi
\caption{The shape of the centroidal inertia of the robot is affected by the leg configuration, which can be modelled by LL-SRBM. (a) and (b) show two different leg configurations when ANYmal is performing a twisting jump. Compared with the centroidal inertia in (b), the centroidal inertia in (a) has smaller values on \textit{y} and \textit{z} axes.}
\label{fig:inertia_shape_illustrate}
\vspace{-5mm}
\end{figure}
The trajectory optimization \cite{kelly2017introduction} is a powerful tool to find optimal trajectories for complex tasks that contain nonlinear dynamics and involves multiple phases. In this paper, we employ the \textit{multiple shooting method} to transcribe our trajectory optimization problem into a NLP formulation. In the formulation, multiple phases are considered since we are focusing on versatile dynamic jumps that contains multiple pre-defined contact phases: \textit{takeoff}, \textit{flight phase} and \textit{post-landing phase}.
The system has been discretised into $N$ segments and therefore $N+1$ knots exists. Since $m$ phases are assumed, each phase gets $N/m$ segments.
Because timing is important for generating highly dynamic motions, we do not fix the time for each phase $\Delta T$. However we keep the same time interval $dt$ between knots inside each phase to reduce the computation cost.
For knot $k$, the state $\boldsymbol{x}[k]$ and control $\boldsymbol{u}[k]$ are defined as:
\begin{align*}
\boldsymbol{x}[k] =& [
\boldsymbol{r}[k],
\boldsymbol{q}_B[k],
\boldsymbol{H}[k],
\boldsymbol{L}[k]]
\\
\boldsymbol{u}[k] =& [
\boldsymbol{f}_{1}[k], ...,
\boldsymbol{f}_{N_l}[k],
\boldsymbol{p}_{1}[k], ...,
\boldsymbol{p}_{N_l}[k]
] \\
\boldsymbol{\xi}[k] =& [\boldsymbol{x}[k], \boldsymbol{u}[k], dt[k]]
\end{align*}
where $dt[k]$ is the time interval from knot $k$ to $k+1$, $N_l$ is the leg number, $\boldsymbol{f}_{1}[k], ..., \boldsymbol{f}_{N_l}[k]$ are the ground reaction forces acting on the feet. For bipedal robots with planar feet, the left foot and right foot collect four corresponding corner forces:
$\boldsymbol{f}_{L}[k]=[\boldsymbol{f}_{L_1}[k], \boldsymbol{f}_{L_2}[k], \boldsymbol{f}_{L_3}[k], \boldsymbol{f}_{L_4}[k]]$,
$\boldsymbol{f}_{R}[k]=[\boldsymbol{f}_{R_1}[k], \boldsymbol{f}_{R_2}[k], \boldsymbol{f}_{R_3}[k], \boldsymbol{f}_{R_4}[k]]$ as shown in Fig. \ref{fig:model_schema_1}.
$\boldsymbol{p}_{L}[k]$ and $\boldsymbol{p}_{R}[k]$ are the center position of the left foot and right foot respectively. For quadrupedal robots, $ \boldsymbol{u}[k] = [
\boldsymbol{f}_{1}[k],
\boldsymbol{f}_{2}[k],
\boldsymbol{f}_{3}[k],
\boldsymbol{f}_{4}[k],
\boldsymbol{p}_{1}[k],
\boldsymbol{p}_{2}[k],
\boldsymbol{p}_{3}[k],
\boldsymbol{p}_{4}[k]
]$, as represented by Fig. \ref{fig:model_schema_2}.
Given the open parameters $\boldsymbol{\xi}[k]$, the complete trajectory optimization problem can be formulated as follows:
\begin{align}
\label{eq:complete formulation}
\min_{\boldsymbol{\xi}} \sum_{k=0}^{N}l(\boldsymbol{\xi}[k]) + \phi(\boldsymbol{\xi}[N]) \tag*{(cost function)} \\
\textrm{s.t.} \quad {\boldsymbol{x}}[k+1] = f(\boldsymbol{x}[k], \boldsymbol{u}[k]) \tag*{(LL-SRBM dynamics)} \\
\boldsymbol{x}[0] = \boldsymbol{x}_{ini}^{} \tag*{(initial states)}
\\
\boldsymbol{x}[N] = \boldsymbol{x}_{fin}^{} \tag*{(final states)}
\\
t_{min} \leq dt[k] \leq t_{max} \tag*{(time limits)} \\
\lVert \boldsymbol{q}_B[k] \rVert = 1 \tag*{(quaternion norm)}
\end{align}
\begin{align}
l_{min} \leq \lVert\boldsymbol{r}[k] - \boldsymbol{p}_{i}[k] \rVert \leq l_{max} \quad (i\in \{1, ..., N_l\})
\tag*{(kinematics limits)}
\end{align}
\quad if foot in {contact}:
\begin{align}
{0} \leq {f}_{i}^z \leq {f_{max}^z} \quad (i\in\{1,...,N_l\}) \tag*{(ground force limit)}
\\
\boldsymbol{P} \bm{f}_{i} \leq \boldsymbol{0} \quad (i\in\{1,...,N_l\}) \tag*{(friction cone)}
\\
\boldsymbol{p}_{i}[k] = \boldsymbol{p}_{i}^* \quad (i\in\{1,...,N_l\}) \tag*{(given contacts)}
\end{align}
\quad if foot in the {air}:
\begin{align}
\bm{f}_{i} = \boldsymbol{0} \quad (i\in\{1,...,N_l\}) \tag*{(no contact force)}
\end{align}
\subsection{Cost Function}
The items inside the cost function can be categorized into 4 groups: \textit{control inputs smoothness, energy consumption, time penalization} and \textit{final state target}.
\textit{Control Inputs Smoothness}: To generate a smooth motion, the difference of ground reaction forces and foot movements between adjacent nodes are minimized:
\begin{equation}
\sum_{k=0}^{N}\sum_{i=0}^{N_l}\big(
\dot{\boldsymbol{p}}_{i}^2[k]
+ \dot{\boldsymbol{f}}_{i}^2[k] \big)
\end{equation}
\textit{Energy Consumption}: We minimize squared momentum to achieve minimal energy consumption:
\begin{equation}
\sum_{k=0}^{N} (\boldsymbol{H}^2[k] + \boldsymbol{L}^2[k])
\end{equation}
\textit{Time Penalization}: We want the optimization to generate a minimum-time trajectory to finish a task while respecting the constraints. Also, this cost term encourages the robot to change the shape of the centroidal inertia in the \textit{flight phase} to achieve a fast motion:
\begin{equation}
\sum_{i=0}^{N}(dt^2[i])
\end{equation}
\textit{Final State Target}: Though final state targets are formulated as constraints, we find that putting final orientation target into cost function can speed up the solving process. This is especially the case in tasks requiring large change of orientation, like twisting jump. So we put the final orientation targets into the cost function:
\begin{equation}
(\boldsymbol{q}_B[N]-\boldsymbol{q}_{B}^{fin})^2 + (\dot{\boldsymbol{q}}_B[N]-\dot{\boldsymbol{q}}_{B}^{fin})^2
\end{equation}
\subsection{Kinematics Constraint}
The kinematics constraint bounds the distance between CoM and the feet to be within $[l_{min},\, l_{max}]$,
\begin{align}
l_{min} \leq \lVert\boldsymbol{r}[k] - \boldsymbol{p}_{i}[k] \rVert \leq l_{max} \quad (i\in \{1,...,N_l\})
\end{align}
For the bipedal robot SLIDER, $l_{min}=$ 0.35~m and $l_{max}=$ 0.75~m. For the quadrupedal robot ANYmal, $l_{min}=$ 0.31~m and $l_{max}=$ 0.6~m.
\subsection{Contact Constraint}
We have pre-defined the contact sequence as \textit{takeoff phase}, \textit{flight phase} and \textit{landing phase}. We pre-define the start and target position of the foot, namely:
\begin{align}
\boldsymbol{p}_{i}[k] = \boldsymbol{p}_{i}^{ini}
\quad (i\in\{1,...,N_l\})
\tag*{(takeoff phase)}
\\
\boldsymbol{p}_{i}[k] = \boldsymbol{p}_{i}^{fin}
\quad (i\in\{1,...,N_l\}) \tag*{(post-landing phase)}
\end{align}
\subsection{Ground Reaction Force Constraint}
In \cite{chignoli2021mit} the actuator limits are introduced in the planner by approximating the configuration-dependent reaction forces. Although joint angles of the robot are not included in LL-SRBM, we can still approximate the ground reaction force limit using the default configuration by:
\begin{equation}
\bm{f}_{max} = \bm{J}^{\mathrm{T}}(\bm{q}_0)\bm{\tau}_{max}
\end{equation}
where $\bm{J}$ is the jacobian matrix, $\bm{q}_0$ represents the joint coordinates at the default configuration, $\bm{\tau}_{max}$ is the maximum joint torque vector.
In \textit{takeoff phase} and \textit{landing phase} both feet need to be in contact with the ground, so all the ground reaction forces would be 0 or pushing against the ground, therefore the ground force limit is:
\begin{align}
{0} \leq {f}_{i}^z \leq {f^z_{max}} \quad (i\in\{1,...,N_l\})
\end{align}
We also ensure no slippage of foot contact points. The tangential forces are constrained to remain inside the Coulomb friction cone defined by the friction coefficient $\mu$. We approximate the friction cone by the friction pyramid, which is a common approach to make the constraints linear and speed up the computation. The friction cone constraint is given by:
\begin{align}
-\mu f_{i}^{z} \leq f_{i}^{x} \leq \mu f_{i}^{z}\\
-\mu f_{i}^{z} \leq f_{i}^{y} \leq \mu f_{i}^{z}
\end{align}
where $f_{i}^{x}$, $f_{i}^{y}$, $f_{i}^{z}$ are components of the ground reaction force $\bm{f}_{i} \, (i\in\{1,...,N_l\})$.
If all feet is in the air, which is the case of \textit{flight phase}, we enforce all ground reaction force to be exactly 0:
\begin{align}
\bm{f}_{i} = \boldsymbol{0} \quad (i\in\{1,...,N_l\})
\end{align}
\section{WHOLE-BODY CONTROL and CONTACT DETECTION}
The whole-body controller takes responsibility of computing the joint torques to achieve the desired motions while respecting a set of constraints. In the paper, the tasks of interest are the CoM position, the pelvis orientation, the angular momentum of the robot, the foot positions and orientations. Each task is comprised of a desired acceleration as a feed-forward term and a state feedback term to stabilize the trajectory. Generally, the task for the linear motion can be expressed as:
\begin{equation*}
\bm{J}_{\mathrm{T}}\ddot{\bm{q}} = \ddot{\bm{x}}^{\mathrm{cmd}} - \dot{\bm{J}}_{\mathrm{T}}\dot{\bm{q}},
\end{equation*}
\begin{equation*}
\ddot{\bm{x}}^{\mathrm{cmd}} = \ddot{\bm{x}}^{\mathrm{des}} + \bm{K}_{\mathrm{P}}^{\mathrm{pos}}(\bm{x}^{\mathrm{des}} - \bm{x}) + \bm{K}_{\mathrm{D}}^{\mathrm{pos}}(\dot{\bm{x}}^{\mathrm{des}} - \dot{\bm{x}}),
\end{equation*}
where $\bm{J}_{\mathrm{T}}$ is the translational Jacobian for the task, $\bm{x}$ is the actual position of the link, and the superscript $\mathrm{des}$ indicates the desired motion.
For the task of angular motion, the command can be formulated as:
\begin{equation*}
\bm{J}_{\mathrm{R}}\ddot{\bm{q}} = \dot{\bm{\omega}}^{\mathrm{cmd}} - \dot{\bm{J}}_{\mathrm{R}}\dot{\bm{q}},
\end{equation*}
\begin{equation*}
\dot{\bm{\omega}}^{\mathrm{cmd}} = \dot{{\bm \omega}}^{\mathrm{des}} + \bm{K}_{\mathrm{P}}^{\mathrm{ang}}(\mathrm{AngleAxis}(\bm{R}^{\mathrm{des}}\bm{R}^{\mathrm{T}})) + \bm{K}_{\mathrm{D}}^{\mathrm{ang}}(\bm{\omega}^{\mathrm{des}} - \bm{\omega}),
\end{equation*}
where $\bm{J}_{\mathrm{R}}$ is the rotational Jacobian for the task, $\bm{R}$ and $\bm{R}^{\mathrm{des}}$ denote the actual and desired orientation of the pelvis link respectively,
$\mathrm{AngleAxis}()$ maps a rotation matrix to the corresponding axis-angle representation,
$\bm {\omega} \in \mathbb{R}^3$ is the angular velocity of the link.
For the CoM task and angular momentum task, the centroidal momentum matrix \cite{orin2013centroidal} is used as the task jacobian.
For balancing or walking, angular momentum task are often defined as a damping task that damps out excess angular momentum. However for highly dynamic motion such as twisting jump, angular momentum varies a lot during the process and it plays a vital role to achieve the jump motion. In this case, the reference angular momentum is needed and it comes from our motion planner. Since the single rigid body model is used in the motion planner, its orientation and associated angular momentum are both well defined.
\subsection{QP formulation}
Inspired by \cite{Herzog_2015}, the full dynamics can be decomposed into the underactuated part and actuated part:
\begin{equation*}
\begin{bmatrix}
\bm{M}_f \\ \bm{M}_a
\end{bmatrix} \bm{\ddot{q}} +
\begin{bmatrix}
\bm{H}_f \\ \bm{H}_a
\end{bmatrix} =
\begin{bmatrix}
\bm{0} \\ \bm{S}_a
\end{bmatrix} \bm\tau +
\begin{bmatrix}
\bm{J}^{\mathrm{T}}_f \\ \bm{J}^{\mathrm{T}}_a
\end{bmatrix} \bm{f},
\end{equation*}
where $\bm{M}$, $\bm{H}$, $\bm{S}_a$, $\bm {\tau}$, $\bm{J}$ and $\bm{f}$ are the mass matrix, Coriolis force matrix and gravitation force vector, the actuator selection matrix, joint torques vector, the stacked contact Jocabian and reaction force vector. The subscript, $f$ and $a$, indicates the floating part and actuated part respectively. The weighted sum formulation is applied, in which one QP problem is solved at each control loop.
The formulation of the QP problem can be written as
\begin{align}
\min_{\ddot{\bm{q}},\, \bm{f}} \quad & \frac{1}{2}\| \bm{A}\ddot{\bm{q}} + \dot{\bm{A}}\dot{\bm{q}} - \bm{B}^{\mathrm{cmd}} \|_{\bm{W}_1}^2 + \frac{1}{2}\| \bm{f}-\bm{f}_{\mathrm{des}} \|^2_{\bm{W}_2}\\
\textrm{s.t.} \quad & \bm{M}_f\bm{\ddot{q}} - \bm{J}^{\mathrm{T}}_f\bm{f} = - \bm{H}_f \tag*{(floating base dynamics) }\\
& \bm{P}\bm{f} \leq \bm{0}\tag*{(friction cone)}\\
& \bm{S}_a^{-1}(\bm{M}_a \ddot{\bm{q}} + \bm{H}_a - \bm{J}^{\mathrm{T}}_a\bm{f}) \in [\bm{\tau}_{min},\, \bm\tau_{max}]\tag*{(input limits)}
\end{align}
where $\bm{A}$ is a stack of the Jacobian matrices for the tasks of interest, $\bm{B}^{\mathrm{cmd}}$ is a stack of the commanded accelerations and $\bm{W}_i \, (i = 1, \,2)$ are the weighting matrices, $\bm{P}$ denotes the linearized friction cone matrix. Similar to~\cite{apgar2018fast}\cite{kuindersma2016optimization} , the unilateral contact constraint is treated as a soft constraint by simply assigning a large weight on the desired zero acceleration. It is reported in \cite{feng2014optimization} that this gives a better stability.
The output torque commands $\bm \tau$ at each control iteration is computed by
\begin{equation}
\bm\tau = \bm{S}_a^{-1}(\bm{M}_a \bm{\ddot{q}} + \bm{H}_a - \bm{J}^{\mathrm{T}}_a\mathbf{f})
\end{equation}
Since the SRBM is an approximate model, we set $\bm{W}_2 = \bm{0}$ to track the desired trajectory in the jump motion, except the landing phase. In the landing phase, in order to achieve soft landing, we switch to tracking desired contact forces by increasing $\bm{W}_2$ while decreasing the weights for centroidal momentum control to be $\bm{0}$. After a short safe landing phase (around 0.2 second), we switch back to trajectory tracking control in order to achieve the desired steady pose. This strategy of switching to force control during landing is similar to \cite{chignoli2021mit}, which is particular critical for real robot experiments in the existing of trajectory tracking and state estimation errors.
\subsection{Contact detection}
According to \cite{orin2013centroidal}, the spatial average velocity of a robot is defined as
\begin{equation*}
\mathbf{v}_G=\begin{bmatrix}
\bm{v}_{\mathrm{com}} \\
\bm{\omega}_G
\end{bmatrix} = \bm{I}_f^{-1}\bm{h}_G
\end{equation*}
where $\bm{h}_G \in \mathbb{R}^6$ is the centroidal momentum of the robot, $\bm{I}_f \in \mathbb{R}^{6 \times 6}$ is the centroidal inertia matrix of the robot, $\bm{v}_{\mathrm{com}}$ is the center of mass velocity, $\bm{\omega}_{G}$ denotes the average angular velocity of the robot. Please refer to \cite{orin2013centroidal} for the details of $\bm{h}_G$ and $\bm{I}_f$.
We use the change rate of $\| \mathbf{v}_G \|_2$ to judge the contact. During the flight phase, the gravity is the only external force that changes the state of the robot. The norm of $\bm{v}_{\mathrm{com}}$ will increase under the gravity while the robot is dropping down from the peak. When contacts happen, the velocities will decrease. In order to respond to all kinds of contact, we choose to use $\| \mathbf{v}_G \|_2$ to detect contacts which is a comprehensive metric including all the changes of velocities of each link of a robot. When the numerical derivatives of $\| \mathbf{v}_G \|_2$ in a sampling window are all less than a threshold, we tell the controller that the robot gets contact. In practice, the proposed method is more reliable than using force signals that are very noisy.
\section{EXPERIMENT RESULTS}
\subsection{Robot Platforms}
SLIDER is a knee-less bipedal robot designed by the Robot Intelligence Lab at Imperial College London \cite{9341143, wang2018clawar}. It is 1.2~m tall and has 10 actuated joints with a total weight of 16~Kg. Most of its weight is concentrated in the pelvis. The prismatic knee joint design is an unique feature of this robot that differentiates it from many other robots with anthropomorphic design.
Also the sliding joint has a relatively large range of motion. The overall light weight and large range of leg motion make the robot suitable for agile locomotion.
The quadrupedal robot ANYmal is made by ANYbotics. It has 12 SEAs (Series Elastic Actuators) and weighs approximately 35~Kg. When ANYmal standing still in the default configuration, it is about 0.5~m tall.
\begin{figure}[t]
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Twist_anymal_inertia_11.png}
\end{subfigure}
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Twist_anymal_inertia_22.png}
\end{subfigure}\hfi
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Twist_anymal_inertia_33.png}
\end{subfigure}\hfi
\medskip
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Twist_anymal_Noinertia_11.png}
\end{subfigure}
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Twist_anymal_Noinertia_22.png}
\end{subfigure} \hfi
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Twist_anymal_Noinertia_33.png}
\end{subfigure} \hfi
\caption{Snapshots of the ANYmal robot performing a $90^{\circ}$ twisting jump with inertia shaping and without inertia shaping. First row: twisting jump with inertia shaping. Second row: twisting jump without inertia shaping.}
\label{fig:jumpTwist_snapshot}
\end{figure}
\begin{figure}[t]
\centering
\begin{subfigure}{0.46\columnwidth}
\includegraphics[width=1.1\linewidth]{figs/Twist_jump_pos.pdf}
\end{subfigure}
\begin{subfigure}{0.46\columnwidth}
\includegraphics[width=1.1\linewidth]{figs/Twist_jump_angMom.pdf}
\end{subfigure}
\begin{subfigure}{0.46\columnwidth}
\includegraphics[width=1.1\linewidth]{figs/Twist_jump_ori.pdf}
\end{subfigure}
\begin{subfigure}{0.46\columnwidth}
\includegraphics[width=1.1\linewidth]{figs/Twist_jump_angVel.pdf}
\end{subfigure} \hfi
\caption{The measured trajectories of ANYmal $90^{\circ}$ twisting jump in simulation, with measured trajectories generated by LL-SRBM and SRBM repectively. In each plot the vertical dashed lines split the entire trajectory into three phases: \textit{takeoff}, \textit{flight} and \textit{landing}. The full lines show the measured trajectories with LL-SRBM and the dashed lines show measured trajectories with SRBM.}
\label{fig:jumpTwist_plot}
\vspace{-5mm}
\end{figure}
\subsection{Implementation}
The trajectory optimization framework is implemented in CasADi \cite{andersson2019casadi} with Python using the interior-point solver IPOPT \cite{wachter2006implementation}. We write the whole body controller with C++ using the Pinocchio library \cite{pinocchioweb} to compute full rigid body dynamics and qpOASES \cite{ferreau2014qpoases} to solve the QP problem. For SLIDER robot the whole body controller runs at 1k~Hz to track the desired trajectory. For ANYmal the whole-body controller is running at 400~Hz in simulation and real experiments as well. For both robots the simulation is done in Gazebo with ODE as the physics engine.
\subsection{Twisting Jump}
We experimented the twisting jump on both ANYmal and SLIDER robots and compared the performance between the trajectories generated by LL-SRBM and SRBM. In the trajectory optimization we simply provide a linear interpolation of the orientation as the initial guess. Also we assign a large weight to the \textit{final state target} and \textit{time penalization} in the cost function to encourage the solver to find a trajectory that reaches the target pose in a minimum time.
As shown in Fig. \ref{fig:jumpTwist_snapshot}, the foot motion generated with LL-SRBM has a big difference compared to the one generated with SRBM. In the flight phase, the feet try to keep close to the central rotating axis with the LL-SRBM generated trajectories. The SRBM generated foot trajectories just do an interpolation from the start to the goal position. This interesting behaviour emerges as LL-SRBM takes leg dynamics into consideration in the planning and tries to maximize the angular velocity in flight phase by modifying the shape of inertia with the leg motion. Figure \ref{fig:jumpTwist_plot} shows the measured trajectories of the twisting jump experiments. It can be seen clearly that in flight phase the base angular velocity in experiments with LL-SRBM does not drop as much as the base angular velocity in experiments with SRBM. The robot reaches the goal orientation earlier with LL-SRBM. To accomplish the twisting jump for SLIDER, we have added two joints for the robot, one yaw joint per leg. SLIDER can perform a $90^{\circ}$ twisting jump and the motion can be seen in the accompany video.
\begin{figure}[b]
\vspace{-5mm}
\centering
\includegraphics[width=0.9\linewidth]{figs/Average_Spatial_velocity.png}
\caption{The norm of average spatial velocity when ANYmal performed a twisting jump. A wooden board was put under the robot while it was flying. A big drop of the value after getting contact when the robot was dropping down.}
\label{fig:twist_average_spatial}
\end{figure}
\begin{figure*}[t]
\centering
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=1.0\linewidth]{figs/Anymal_twist_real_2.png}
\caption{}
\end{subfigure
\hspace{0.01cm}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=1.0\linewidth]{figs/Anymal_twist_real_3.png}
\caption{}
\end{subfigure}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=1.0\linewidth]{figs/Anymal_twist_real_4.png}
\caption{}
\end{subfigure
\hspace{0.01cm}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=1.0\linewidth]{figs/Anymal_twist_real_5.png}
\caption{}
\end{subfigure}
\caption{Snapshots of ANYmal performing a twisting jump with $30^{\circ}$. We also put a wooden board with a height of 25~mm under the robot when it was in the \textit{flight} phase to test the contact detection and the robustness of our controller. From left to right: $\text{(a)}$: takeoff phase, $\text{(b)}$: flight phase, $\text{(c)}$: contact is detected. $\text{(d)}$: the robot is stable after a soft landing.}
\label{fig:twistJump_real}
\vspace{-5mm}
\end{figure*}
In real experiments, we demonstrate the effectiveness of the proposed approaches by twisting jumps on unknown objects, as shown in Fig \ref{fig:twistJump_real}. A wooden board with a height of 25~mm was put under ANYmal when it was in the flight phase. Figure \ref{fig:twist_average_spatial} shows the norm of the average spatial velocity. Even though collected from real robot experiments, the norm of the average spatial velocity is smooth and can be used as a reliable judgement for contact detection. The robot successfully detected the contact event and switched to a soft landing.
\subsection{Forward Jump}
In the trajectory optimization, we assign a large weight to the \textit{final state target} and \textit{energy consumption}. For ANYmal, a forward jump motion of 30~cm is generated with LL-SRBM and SRBM, as show in Fig. \ref{fig:jumpForward_snapshot_anymal}. It can be seen that for trajectories generated with LL-SRBM model, the feet stretch out more in the air than trajectories generated with SRBM. This is because the optimizer tries to increase the inertia around $y$ axis to reduce the change of base pitch angle. The stretched feet in the flight phase also help to prevent early touchdown of the robot.
We also tried jump onto a box for SLIDER with LL-SRBM. The maximum height of the box SLIDER can jump on is 35~cm, with a forward jump length of 20~cm, as shown in Fig. \ref{fig:jumpUp_snapshot_SLIDER}. Considering that SLIDER is 1.2~m high, the robot can jump onto a box equal to 30\% of its total height. Compared with the anthropomorphic robot design which has knees, SLIDER's straight legs have a larger range of motion, allowing SLIDER to reach the same jump height with a relatively smaller CoM height. Because there are no knee joints on legs of SLIDER, the feet can go all the way up until the ankles touch the pelvis. That is the reason why SLIDER can jump onto a high box.
\begin{figure}[t]
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Forward_jump_inertia_1.png}
\end{subfigure}
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Forward_jump_inertia_2.png}
\end{subfigure}\hfi
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Forward_jump_inertia_3.png}
\end{subfigure}\hfi
\medskip
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Forward_jump_Noinertia_1.png}
\end{subfigure}
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Forward_jump_Noinertia_2.png}
\end{subfigure} \hfi
\begin{subfigure}{0.155\textwidth}
\includegraphics[width=1.02\linewidth]{figs/Forward_jump_Noinertia_3.png}
\end{subfigure} \hfi
\caption{Snapshots of the ANYmal robot performing a forward jump of 30~cm with inertia shaping and without inertia shaping. First row: forward jump with inertia shaping. Second row: forward jump without inertia shaping.}
\label{fig:jumpForward_snapshot_anymal}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\centering
\begin{subfigure}{0.2\textwidth}
\includegraphics[width=0.9\linewidth, height=3.5cm]{figs/SLIDER_JumpUp_0.png}
\caption{}
\end{subfigure}
\begin{subfigure}{0.2\textwidth}
\includegraphics[width=0.9\linewidth,height=3.5cm]{figs/SLIDER_JumpUp_1.png}
\caption{}
\end{subfigure}\hfi
\medskip
\begin{subfigure}{0.2\textwidth}
\includegraphics[width=0.9\linewidth, height=3.5cm]{figs/SLIDER_JumpUp_2.png}
\caption{}
\end{subfigure}
\begin{subfigure}{0.2\textwidth}
\includegraphics[width=0.9\linewidth, height=3.5cm]{figs/SLIDER_JumpUp_3.png}
\caption{}
\end{subfigure} \hfi
\caption{Snapshots of SLIDER performing a forward jump onto a 35~cm box using LL-SRBM. $\text{(a)}\sim\text{(b)}$: takeoff phase, (c): flight phase, (d): landing phase.}
\label{fig:jumpUp_snapshot_SLIDER}
\vspace{-5mm}
\end{figure}
\subsection{Computation Time}
We recorded the computation times of trajectory optimizations for various dynamic jump motions. For the bipedal robot we used 60 knots and each knot has 59 variables (26 for states and 33 for control inputs). For the quadrupedal robot we used 50 knots and each knots has 53 variables (26 for states and 27 for control inputs). Although the complexity of LL-SRBM has increased compared with SRBM, the computation time with LL-SRBM only increase by an average of 33.6\% with respect to that with SRBM, as shown in Table \ref{tab:table_1}.
\begin{table}[h]
\caption{Solve Times for Planners.}
\label{table_2}
\begin{center}
\setlength\tabcolsep{1.5pt}
\begin{tabular}{|c|c|c|c|c|}
\hline
&\thead{quadruped \\ forward jump} & \thead{quadruped \\twist jump} & \thead{biped \\ forward jump} & \thead{biped \\twist jump} \\
\hline
\textbf{SRBM} & 1.50s & 2.45s & 6.33s & 5.44s\\
\hline
\textbf{LL-SRBM} & 1.53s & 2.56s & 9.54s & 7.37s\\
\hline
\end{tabular}
\end{center}
\vspace{-5mm}
\label{tab:table_1}
\end{table}
\section{CONCLUSIONS}
This paper proposes an unified model with inertia shaping for planning highly dynamic jumps of legged robots. This model allows the motion planner to improve the jumping performance by actively changing the centroidal inertia. In the meanwhile, this paper also proposes a novel contact detection method using the norm of average spatial velocity. The twisting jump and forward jump experiments on bipedal robot SLIDER and quadrupedal robot ANYmal show the improved jump performance after using the proposed model and the robustness of the controller to unforeseen impacts. In the future, we are interested in speeding up the computation to re-plan the jump motion with the proposed LL-SRBM.
\section*{APPENDIX}
\section*{Quaternion to Rotation Matrix Conversion}\label{appendix1}
Given a quaternion $\boldsymbol{q} =q_w+q_x \boldsymbol{i}+q_y \boldsymbol{j}+q_z \boldsymbol{k}$ which represents the orientation of the single rigid body in world frame, the equivalent rotation matrix representation can be derived as:
\begin{equation*}
\boldsymbol{R}=
\left[\begin{array}{ccc}
1-2\left(q_y^{2}+q_z^{2}\right) & 2 q_x q_y-2 q_w q_z & 2 q_w q_y+2 q_x q_z \\
2 q_x q_y+2 q_w q_z & 1-2\left(q_x^{2}+q_z^{2}\right) & 2 q_y q_z-2 q_w q_x \\
2 q_x q_z-2 q_w q_y & 2 q_w q_x+2 q_y q_z & 1-2\left(q_x^{2}+q_y^{2}\right)
\end{array}\right]
\end{equation*}
\section*{Derivative of Quaternion}
Followed by \cite{graf2008quaternions}, given a rigid body with quaternion $\boldsymbol{q} =q_w+q_x \boldsymbol{i}+q_y \boldsymbol{j}+q_z \boldsymbol{k}$ and with angular velocity $\bm{\omega}$, the derivative of the quaternion $\dot{\boldsymbol{q}}$ can be calculated as:
$$\dot{\boldsymbol{q}}=\frac{1}{2} \boldsymbol{q} \circ {\bm{\omega}}$$
$$\left[\begin{array}{c}
\dot{q}_{x} \\
\dot{q}_{y} \\
\dot{q}_{z} \\
\dot{q}_{w}
\end{array}\right]
= \frac{1}{2} \left[\begin{array}{c}
{q}_{x} \\
{q}_{y} \\
{q}_{z} \\
{q}_{w}
\end{array}\right]
\circ
\left[\begin{array}{c}
{\omega}_{x} \\
{\omega}_{y} \\
{\omega}_{z} \\
0
\end{array}\right]
$$
$$\left[\begin{array}{c}
\dot{q}_{x} \\
\dot{q}_{y} \\
\dot{q}_{z} \\
\dot{q}_{w}
\end{array}\right]=\frac{1}{2}\left[\begin{array}{cccc}
q_{w} & -q_{z} & q_{y} & q_{x} \\
q_{z} & q_{w} & -q_{x} & q_{y} \\
-q_{y} & q_{x} & q_{w} & q_{z} \\
-q_{x} & -q_{y} & -q_{z} & q_{w}
\end{array}\right]\left[\begin{array}{l}
w_{x} \\
w_{y} \\
w_{z} \\
0
\end{array}\right]$$
\section*{ACKNOWLEDGMENT}
This work is supported by the CSC Imperial Scholarship, EPSRC UK RAI Hubs NCNR (EP/R02572X/1), FAIR-SPACE(EP/R026092/1). The authors would like to thank Digby Chappell for helpful discussions.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,955
|
There has been lots of talking, scoffing, and downright outrage as of late about the recent news that certain film studios will begin distributing their movies via at home service providers (DirecTV launches a VOD service this week) a mere weeks after the films are released in theatres. This is of course very detrimental to the movie theatres themselves. When a film is released the Studios and Theatres have a contract in place for the rental of the film, per se. The key part of this agreement is the percentages allotted to the Studio and the Theatre on a week by week basis from ticket sales. The Studio makes the bulk of the money in the first few weeks, leaving the Theatre to hope that the film is popular, with longevity, because as each week passes they see more of the profit. The opening weekend to a Theatre is not as important as the fourth week of release–except that opening weekend and word-of-mouth, critical reviews, and social media now determine whether a film even survives to the fourth week.
. People can rent the film, for around ten dollars, and watch it at their leisure. Amazon-On-Demand is available in every city across the country, as is iTunes. Instantly this small film that will only see a "big" screen in two cities for a couple weeks has the ability to reach millions of people. For an independent filmmaker this is an amazing opportunity–and it is cheap!
Are Theatre Chains up in arms about these indies playing on demand? No. They were not going to show that film in their megaplex anyways; they are reserved for the arthouse cinemas. The Theatre Chains are angered over the "BIG ONES", like Warner Bros., Sony, Fox, etc., putting their films on-demand in homes. These are the movies with marketing budgets, that will open on over 1,000 screens on opening day and expect to rake in upwards of $20 million dollars on opening weekend (if they are lucky in this economic climate). The Theatre Chains do not want to lose out on the possibility of a big screen success going to Cox, DirecTV or Time Warner Communications and the Studios, or Amazon-On-Demand and iTunes.
This is completely understandable. Their business model is dependent upon people going to the cinema. If people know that in four weeks time they can watch the movie at home, for one flat fee, regardless of how many people are in the room, then many will wait. It is similar to DVD purchases (better termed 'media' since releases are available in a variety of formats for purchase now). The most common phrase I hear when telling people about a great film coming to theatres is, "I will watch it on DVD." As much as this hurts to hear it is the reality of the market. Movies come out on media faster and faster now, especially if they proved unsuccessful at the box office. It is now April and a film that was released in February is scheduled for media release in May. I remember when it took a year, sometimes longer, to buy the VHS and/or DVD of a film and now it happens so fast one cannot believe it. Considering how many movies are not worth the high ticket prices at theatres today who can blame people for waiting, especially when it is only a few months. But you cannot condemn the studios for wanting to make the leap to on-demand, as it is good business practice. Imagine if you could watch the newest release from Warner Bros. pictures on your Ipad in-between classes at college? Or while on a road trip with your family put in the latest Disney movie, currently in theatres? The possibilities for on-demand entertainment are endless, and the Studios are hurting for profits also. Attendance is down, ticket prices are up, and film budgets are not getting any smaller. The product is also hurting drastically but well, that is another topic for another day.
What is a Theatre to do then?
Why not start your own video-on-demand service? AMC, Regal, Loews, you have the money to do this, and the Studios are warming up to the idea so run with it. Take the money you are wasting on converting theatres to 3D and start your own at home video-on-demand service. Make it available through Roku players, Blu-Rays, on Computer Systems (a Netflix-type system), and see just how fast it grows. Partner with Fandango.
and MovieTickets.com to advertise the on-demand feature as well as buying tickets. Sure, you may not make as much money as you do at the theatre but you just may make more on those films that are costing you tons of money to run and seeing no return. You could even expand to include more independent films, opening you up to another market of filmgoers. It is time to stop allowing yourselves to be treated like the unwanted step-child. There are great possibilities here. Change is inevitable. You can't stay open all night, but your on-demand service sure could. Take advantage of the opportunities that are ahead with on-demand and maybe in the process, with the extra revenue, you could lower the price of your concessions and introduce some more healthy options for moviegoers–just a thought.
This idea may make people angry, or cringe at the thought of not going to the cinema to see the newest release. I am with you as I will always be a dedicated theatre patron. I am also aware of the changing habits of people and with how quickly they are adapting to the new distribution models of entertainment. The thought of watching a film on an Ipod is outrageous to me; yet I see it happening on airplanes all the time. Movies will always exist, but the ways in which people watch them is changing and will continue to do so. It is time to step-up Theatre Chains and pave the way for this new distribution model, instead of being told what you will concede to by those who rely on you to show their product. You have power, it is time to use it.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,897
|
"""
Support for Sesame, by CANDY HOUSE.
For more details about this platform, please refer to the documentation
https://home-assistant.io/components/lock.sesame/
"""
from typing import Callable
import voluptuous as vol
import homeassistant.helpers.config_validation as cv
from homeassistant.components.lock import LockDevice, PLATFORM_SCHEMA
from homeassistant.const import (
ATTR_BATTERY_LEVEL, CONF_EMAIL, CONF_PASSWORD,
STATE_LOCKED, STATE_UNLOCKED)
from homeassistant.helpers.typing import ConfigType
REQUIREMENTS = ['pysesame==0.1.0']
ATTR_DEVICE_ID = 'device_id'
PLATFORM_SCHEMA = PLATFORM_SCHEMA.extend({
vol.Required(CONF_EMAIL): cv.string,
vol.Required(CONF_PASSWORD): cv.string
})
def setup_platform(
hass, config: ConfigType,
add_entities: Callable[[list], None], discovery_info=None):
"""Set up the Sesame platform."""
import pysesame
email = config.get(CONF_EMAIL)
password = config.get(CONF_PASSWORD)
add_entities([SesameDevice(sesame) for sesame in
pysesame.get_sesames(email, password)],
update_before_add=True)
class SesameDevice(LockDevice):
"""Representation of a Sesame device."""
def __init__(self, sesame: object) -> None:
"""Initialize the Sesame device."""
self._sesame = sesame
# Cached properties from pysesame object.
self._device_id = None
self._nickname = None
self._is_unlocked = False
self._api_enabled = False
self._battery = -1
@property
def name(self) -> str:
"""Return the name of the device."""
return self._nickname
@property
def available(self) -> bool:
"""Return True if entity is available."""
return self._api_enabled
@property
def is_locked(self) -> bool:
"""Return True if the device is currently locked, else False."""
return not self._is_unlocked
@property
def state(self) -> str:
"""Get the state of the device."""
if self._is_unlocked:
return STATE_UNLOCKED
return STATE_LOCKED
def lock(self, **kwargs) -> None:
"""Lock the device."""
self._sesame.lock()
def unlock(self, **kwargs) -> None:
"""Unlock the device."""
self._sesame.unlock()
def update(self) -> None:
"""Update the internal state of the device."""
self._sesame.update_state()
self._nickname = self._sesame.nickname
self._api_enabled = self._sesame.api_enabled
self._is_unlocked = self._sesame.is_unlocked
self._device_id = self._sesame.device_id
self._battery = self._sesame.battery
@property
def device_state_attributes(self) -> dict:
"""Return the state attributes."""
attributes = {}
attributes[ATTR_DEVICE_ID] = self._device_id
attributes[ATTR_BATTERY_LEVEL] = self._battery
return attributes
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,951
|
\section{Trihedral Hecke algebras}\label{section:funny-algebra}
As before, $k,l,m,n$ etc. will be non-negative integers,
and $e$ will denote the level.
We are now going to introduce the trihedral Hecke algebras.
The reader possibly spots the
analogies with the dihedral Hecke algebra right away, but, for
completeness, we have also listed some of
them in \fullref{subsec:dihedral-group}.
\subsection{Some color conventions}\label{subsec:our-color-code}
Throughout we will use the set of primary colors
$\Bset\Rset\Yset=\{{\color{myblue}b},{\color{myred}r},{\color{myyellow}y}\}$, the elements of which are {\color{myblue}blue} ${\color{myblue}b}$,
{\color{myred}red} ${\color{myred}r}$ and {\color{myyellow}yellow} ${\color{myyellow}y}$, the set of
secondary colors $\Gset\Oset\Pset=\{{\color{mygreen}g},{\color{myorange}o},{\color{mypurple}p}\}$, the elements of which are
{\color{mygreen}green} ${\color{mygreen}g}=\{{\color{myblue}b},{\color{myyellow}y}\}$, {\color{myorange}orange} ${\color{myorange}o}=\{{\color{myyellow}y},{\color{myred}r}\}$ and
{\color{mypurple}purple} ${\color{mypurple}p}=\{{\color{myblue}b},{\color{myred}r}\}$, and the color {\color{mycream}white} $\emptyset$.
We also use dummy colors ${\color{dummy}\textbf{u}},{\color{dummy}\textbf{v}}\in\Gset\Oset\Pset$, and from now on ${\color{dummy}\textbf{u}},{\color{dummy}\textbf{v}}$, etc. will
always denote arbitrary elements in $\Gset\Oset\Pset$.
Moreover, we fix a cyclic ordering, and its inverse, of the secondary colors:
\begin{gather}\label{eq:color-tensor}
\xy
(0,0)*{
\raisebox{.1cm}{$\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=.2cm, column
sep=.1cm, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&] {
{\color{mypurple}p} \& \& {\color{myorange}o} \\
\& {\color{mygreen}g} \& \\};
\path[thick, myyellow, ->] ($(m-2-2)+(.1,.15)$) edge ($(m-1-3)+(-.1,-.3)$);
\path[thick, densely dashed, myred, ->] (m-1-3) edge (m-1-1);
\path[thick, densely dotted, myblue, ->] ($(m-1-1)+(.1,-.3)$) edge ($(m-2-2)+(-.1,.15)$);
\end{tikzpicture}$}};
(0,-7)*{\text{{\tiny$\rho^{\phantom{-}}\!\!\!\!\colon {\color{mygreen}g}\mapsfrom{\color{mypurple}p}\mapsfrom{\color{myorange}o}\mapsfrom{\color{mygreen}g}$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\raisebox{.1cm}{$\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=.2cm, column
sep=.1cm, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&] {
{\color{mypurple}p} \& \& {\color{myorange}o} \\
\& {\color{mygreen}g} \& \\};
\path[thick, myyellow, <-] ($(m-2-2)+(.1,.15)$) edge ($(m-1-3)+(-.1,-.3)$);
\path[thick, densely dashed, myred, <-] (m-1-3) edge (m-1-1);
\path[thick, densely dotted, myblue, <-] ($(m-1-1)+(.1,-.3)$) edge ($(m-2-2)+(-.1,.15)$);
\end{tikzpicture}$}};
(0,-7)*{\text{{\tiny$\rho^{-1}\colon{\color{mygreen}g}\mapsfrom{\color{myorange}o}\mapsfrom{\color{mypurple}p}\mapsfrom{\color{mygreen}g}$}}};
\endxy
\end{gather}
Note that we usually read from right to left, i.e.
we use the operator notation.
The action of $\rho$ on $\Gset\Oset\Pset$ can be read off
from \eqref{eq:color-tensor}: $\rho({\color{mygreen}g})={\color{myorange}o}$, $\rho({\color{myorange}o})={\color{mypurple}p}$
and $\rho({\color{mypurple}p})={\color{mygreen}g}$, and $\rho^{k-l}$ only depends on $(k-l)\bmod 3$,
for any $k,l$.
\subsection{The trihedral Hecke algebra of level \texorpdfstring{$\infty$}{infty}}\label{subsec:definition}
In this section and in \fullref{subsec:quotient-algebra}, we work over $\C_{\vpar}=\mathbb{C}(\varstuff{v})$,
with $\varstuff{v}$ being a generic parameter.
\subsubsection{The underlying Coxeter group}\label{subsec:weyl-group}
Let $\algstuff{W}$ be the Coxeter group of affine type $\typea{2}$, generated by
three reflections that we denote by ${\color{myblue}b},{\color{myred}r},{\color{myyellow}y}$, i.e.
\begin{gather*}
\begin{tikzpicture}[anchorbase, xscale=.4, yscale=.55]
\draw [thick] (0,0) to (.4,.4) node[right] {\text{{\tiny$3$}}} to (1,1);
\draw [thick] (0,0) to (-.4,.4) node[left] {\text{{\tiny$3$}}} to (-1,1);
\draw [thick] (1,1) to (0,1) node[above] {\text{{\tiny$3$}}} to (-1,1);
\node at (0,0) {$\bullet$};
\node at (0,-.3) {$\text{{\tiny${\color{myblue}b}$}}$};
\node at (1,1) {$\bullet$};
\node at (1.25,1.25) {$\text{{\tiny${\color{myred}r}$}}$};
\node at (-1,1) {$\bullet$};
\node at (-1.25,1.25) {$\text{{\tiny${\color{myyellow}y}$}}$};
\node at (0,.675) {\text{{\tiny$\typeat{2}$}}};
\end{tikzpicture}
\rightsquigarrow
\algstuff{W}
=
\left\langle
{\color{myblue}b},{\color{myred}r},{\color{myyellow}y}\mid
{\color{myblue}b}^2={\color{myred}r}^2={\color{myyellow}y}^2=1,
\;
{\color{myblue}b}{\color{myred}r}{\color{myblue}b}={\color{myred}r}{\color{myblue}b}{\color{myred}r},
\,
{\color{myblue}b}{\color{myyellow}y}{\color{myblue}b}={\color{myyellow}y}{\color{myblue}b}{\color{myyellow}y},
\,
{\color{myred}r}{\color{myyellow}y}{\color{myred}r}={\color{myyellow}y}{\color{myred}r}{\color{myyellow}y}
\right\rangle.
\end{gather*}
In order to simplify the notation,
we identify the vertices in the Coxeter diagram of $\algstuff{W}$ with
the corresponding reflections.
Moreover, let ${\color{mygreen}g}$, ${\color{myorange}o}$ and ${\color{mypurple}p}$ be the maximal proper parabolic subsets,
and let $\algstuff{W}_{{\color{mygreen}g}}, \algstuff{W}_{{\color{myorange}o}}$ and $\algstuff{W}_{{\color{mypurple}p}}$
be the corresponding
standard parabolic subgroups of $\algstuff{W}$, which
are all isomorphic to the (finite) type $\mathsf{A}_2$ Weyl group.
Furthermore, let
\begin{gather}\label{eq:sec-Weyl}
w_{{\color{mygreen}g}}={\color{myblue}b}{\color{myyellow}y}{\color{myblue}b}={\color{myyellow}y}{\color{myblue}b}{\color{myyellow}y}\in\algstuff{W}_{{\color{mygreen}g}},
\quad
\quad
w_{{\color{myorange}o}}={\color{myred}r}{\color{myyellow}y}{\color{myred}r}={\color{myyellow}y}{\color{myred}r}{\color{myyellow}y}\in\algstuff{W}_{{\color{myorange}o}},
\quad
\quad
w_{{\color{mypurple}p}}={\color{myblue}b}{\color{myred}r}{\color{myblue}b}={\color{myred}r}{\color{myblue}b}{\color{myred}r}\in\algstuff{W}_{{\color{mypurple}p}}
\end{gather}
denote the longest elements
in these parabolic
subgroups.
\subsubsection{The trihedral Hecke algebra}\label{subsec:def-sl3alg-1}
We define now the trihedral Hecke algebra
of level $\infty$.
\begin{definition}\label{definition:funny-alg1}
Let $\subquo$ be the associative, unital ($\C_{\vpar}$-)algebra generated by three elements
$\theta_{{\color{mygreen}g}}$, $\theta_{{\color{myorange}o}}$, $\theta_{{\color{mypurple}p}}$
subject to the following relations.
\begin{gather}\label{eq:first-rel}
\theta_{{\color{mygreen}g}}^2=\vfrac{3}\theta_{{\color{mygreen}g}},
\quad\quad
\theta_{{\color{myorange}o}}^2=\vfrac{3}\theta_{{\color{myorange}o}},
\quad\quad
\theta_{{\color{mypurple}p}}^2=\vfrac{3}\theta_{{\color{mypurple}p}},
\end{gather}
\begin{gather}\label{eq:second-rel}
\theta_{{\color{mygreen}g}}\theta_{{\color{myorange}o}}\theta_{{\color{mygreen}g}}=\theta_{{\color{mygreen}g}}\theta_{{\color{mypurple}p}}\theta_{{\color{mygreen}g}},
\quad\quad
\theta_{{\color{myorange}o}}\theta_{{\color{mygreen}g}}\theta_{{\color{myorange}o}}=\theta_{{\color{myorange}o}}\theta_{{\color{mypurple}p}}\theta_{{\color{myorange}o}},
\quad\quad
\theta_{{\color{mypurple}p}}\theta_{{\color{mygreen}g}}\theta_{{\color{mypurple}p}}=\theta_{{\color{mypurple}p}}\theta_{{\color{myorange}o}}\theta_{{\color{mypurple}p}}.
\end{gather}
Here, $\vfrac{3}$ is the $\varstuff{v}$-factorial from \eqref{eq:qnumbers-typeAD}.
\end{definition}
Let $\hecke=\hecke(\typeat{2})$ denote the Hecke algebra
of affine type $\typea{2}$, see e.g. \cite[Section 2]{So}.
Recall that $\hecke$ can be defined as
the associative, unital ($\C_{\vpar}$)-algebra
generated by $\theta_{{\color{myblue}b}},\theta_{{\color{myred}r}}$ and $\theta_{{\color{myyellow}y}}$
subject to
\begin{gather}\label{eq:quadratic}
\theta_{{\color{myblue}b}}^2=\vnumber{2}\theta_{{\color{myblue}b}},
\quad\quad
\theta_{{\color{myyellow}y}}^2
=\vnumber{2}\theta_{{\color{myyellow}y}},
\quad\quad
\theta_{{\color{myred}r}}^2=\vnumber{2}\theta_{{\color{myred}r}},
\\
\label{eq:cubic}
\begin{aligned}
(\theta_{w_{{\color{mygreen}g}}}
=)&\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}-\theta_{{\color{myblue}b}}\\
=&\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}-\theta_{{\color{myyellow}y}},
\end{aligned}
\quad\quad
\begin{aligned}
(\theta_{w_{{\color{myorange}o}}}
=)&\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}-\theta_{{\color{myred}r}}\\
=&\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}-\theta_{{\color{myyellow}y}},
\end{aligned}
\quad\quad
\begin{aligned}
(\theta_{w_{{\color{mypurple}p}}}
=)&\theta_{{\color{myblue}b}}\theta_{{\color{myred}r}}\theta_{{\color{myblue}b}}-\theta_{{\color{myblue}b}}\\
=&\theta_{{\color{myred}r}}\theta_{{\color{myblue}b}}\theta_{{\color{myred}r}}-\theta_{{\color{myred}r}}.
\end{aligned}
\end{gather}
For any $w\in\algstuff{W}$,
let $\theta_w$ be the
corresponding Kazhdan--Lusztig (KL for short) basis element of
$\hecke$, e.g. the expression $\theta_{w_{{\color{dummy}\textbf{u}}}}$
in \eqref{eq:cubic}.
(Note that $\theta_w$ is denoted $\underline{H}_w$ in \cite[Section 2]{So},
while the standard basis is denoted $H_w$ therein.)
\begin{lemma}\label{lemma:quotient-of-affine}
The algebra homomorphism given by
\[
\theta_{{\color{mygreen}g}}\mapsto\theta_{w_{{\color{mygreen}g}}},
\quad\quad
\theta_{{\color{myorange}o}}\mapsto\theta_{w_{{\color{myorange}o}}},
\quad\quad
\theta_{{\color{mypurple}p}}\mapsto\theta_{w_{{\color{mypurple}p}}},
\]
defines an embedding $\subquo\hookrightarrow\hecke$ of algebras.
\end{lemma}
\begin{proof}
By \eqref{eq:quadratic}, \eqref{eq:cubic}
and the identity $\vnumber{2}^3-\vnumber{2}=\vnumber{2}\vnumber{3}$, we obtain
\[
\theta_{w_{{\color{mygreen}g}}}^2=\vfrac{3}\theta_{w_{{\color{mygreen}g}}},
\quad\quad
\theta_{w_{{\color{myorange}o}}}^2=\vfrac{3}\theta_{w_{{\color{myorange}o}}},
\quad\quad
\theta_{w_{{\color{mypurple}p}}}^2=\vfrac{3}\theta_{w_{{\color{mypurple}p}}}.
\]
This shows that \eqref{eq:first-rel} holds in $\hecke$.
Proving \eqref{eq:second-rel} is harder.
Let us indicate how to prove
$\theta_{w_{{\color{mygreen}g}}}\theta_{w_{{\color{myorange}o}}}\theta_{w_{{\color{mygreen}g}}}
=\theta_{w_{{\color{mygreen}g}}}\theta_{w_{{\color{mypurple}p}}}\theta_{w_{{\color{mygreen}g}}}$.
(The other two follow by exchanging colors.)
By \eqref{eq:cubic},
this is equivalent to proving
\begin{gather}\label{eq:triple}
\left(\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}-\theta_{{\color{myblue}b}}\right)
\left(\theta_{{\color{myblue}b}}\theta_{{\color{myred}r}}\theta_{{\color{myblue}b}}-\theta_{{\color{myblue}b}}\right)
\left(\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}-\theta_{{\color{myblue}b}}\right)
=
\left(\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}-\theta_{{\color{myyellow}y}}\right)
\left(\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}-\theta_{{\color{myyellow}y}}\right)
\left(\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}-\theta_{{\color{myyellow}y}}\right).
\end{gather}
By \eqref{eq:quadratic}, the right-hand side in \eqref{eq:triple} is equal to
\begin{gather}\label{eq:triple-lhs-rewritten}
\begin{gathered}
\vnumber{2}^2(\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}
\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}
-\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}
-\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}
-\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}
+2\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}} +
\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}-\theta_{{\color{myyellow}y}})
\\
\stackrel{\eqref{eq:cubic}}{=}
\vnumber{2}^2\left(\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}
\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}
-\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}
-\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}
-\vnumber{3}\theta_{w_{{\color{mygreen}g}}}
+\theta_{{\color{myyellow}y}}\theta_{{\color{myred}r}}\theta_{{\color{myyellow}y}}\right).
\end{gathered}
\end{gather}
Similarly, the left-hand side in \eqref{eq:triple} is equal to
\begin{gather}\label{eq:triple-rhs-rewritten}
\vnumber{2}^2\left(\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myred}r}}
\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}
-\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myred}r}}\theta_{{\color{myblue}b}}
-\theta_{{\color{myblue}b}}\theta_{{\color{myred}r}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}
-\vnumber{3}\theta_{w_{{\color{mygreen}g}}}
+\theta_{{\color{myblue}b}}\theta_{{\color{myred}r}}\theta_{{\color{myblue}b}}\right).
\end{gather}
One can obtain \eqref{eq:triple-rhs-rewritten}
from \eqref{eq:triple-lhs-rewritten} by systematically
using \eqref{eq:cubic}
and the fact that $w_{{\color{mygreen}g}},w_{{\color{myorange}o}},w_{{\color{mypurple}p}}$ have two equivalent expressions each.
For example, by \eqref{eq:cubic},
we have $\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}=
\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}+\theta_{{\color{myblue}b}}-\theta_{{\color{myyellow}y}}$.
Using this to rewrite the first term in \eqref{eq:triple-lhs-rewritten} and
carefully continuing in this way yields the
claimed equality.
Finally, using an appropriate integral form, $\hecke$ specializes
to $\mathbb{C}[\algstuff{W}]$ for $\varstuff{v}=1$. Moreover, recall that $\mathbb{C}[\algstuff{W}]$ has a faithful
representation $\algstuff{P}_{1}$, which is induced by the regular $\algstuff{W}$-action on the
set of alcoves obtained from the hyperplane
arrangement associated to $\algstuff{W}$, and
that $\algstuff{P}_{1}$ can be
$\varstuff{v}$-deformed to $\algstuff{P}_{\varstuff{v}}$, cf. \cite[Section 4 and Lemma 4.1]{So}. The
$\varstuff{v}$-deformation $\algstuff{P}_{\varstuff{v}}$
stays faithful: Each standard basis element $H_w\in\mathbb{C}[\algstuff{W}]$
is mapped to a different $\mathbb{C}$-linear operator by $\algstuff{P}_{1}$,
so each KL basis element $\theta_w\in\hecke$
is mapped to a different $\C_{\vpar}$-linear operator by
$\algstuff{P}_{\varstuff{v}}$, due to the particular form of the change
of basis
\[
\theta_w\in H_w+
{\scriptstyle\sum_{w^{\prime}\leq_{\mathrm{B}} w}}\,
\varstuff{v}\mathbb{Z}[\varstuff{v}]H_{w^{\prime}}.
\]
Here $\leq_{\mathrm{B}}$ is the Bruhat order, see e.g. \cite[Claim 2.3]{So}.
By pulling back $\algstuff{P}_{\varstuff{v}}$ to $\subquo$ along the
algebra homomorphism in this lemma, injectivity of the latter follows from the
faithfulness of the representation.
\end{proof}
\subsubsection{The trihedral Kazhdan--Lusztig combinatorics}\label{subsec:def-sl3alg-2}
We are going to define a quotient of $\subquo$.
In order to do that, we first have to introduce certain elements.
For any $k,l,{\color{dummy}\textbf{u}}$, let
\begin{gather}\label{eq:KLelement}
\rklx{k,l}=\rklx{k,l}(\theta)=\theta_{{\color{dummy}\textbf{u}}_{k+l}}\cdots\theta_{{\color{dummy}\textbf{u}}_1}\theta_{{\color{dummy}\textbf{u}}_0},
\end{gather}
where ${\color{dummy}\textbf{u}}_i$ for all
$0\leq i\leq k+l$ is given by ${\color{dummy}\textbf{u}}_0={\color{dummy}\textbf{u}}$, ${\color{dummy}\textbf{u}}_{i+1}=\rho({\color{dummy}\textbf{u}}_i)$
for (any) $k$ values of
$i$, and ${\color{dummy}\textbf{u}}_{i+1}=\rho^{-1}({\color{dummy}\textbf{u}}_i)$ for the remaining values
of $i$. Note that
\[
\rklx{0,0}=\theta_{\color{dummy}\textbf{u}}\quad\text{for any}\; {\color{dummy}\textbf{u}}.
\]
Moreover, by convention, $\rklx{k,l}=0$ in case $k$ or $l$ are negative.
We call ${\color{dummy}\textbf{u}}$ the (right) starting color of $\rklx{k,l}$.
The fact that $\rklx{k,l}$ is well-defined is established by the following lemma.
\begin{lemma}\label{lemma:well-defined-paths}
For any $k,l,{\color{dummy}\textbf{u}}$,
the element $\rklx{k,l}$ only depends on
$k$ and $l$, not on the chosen sequence ${\color{dummy}\textbf{u}}_{k+l},\cdots,{\color{dummy}\textbf{u}}_{1},{\color{dummy}\textbf{u}}_{0}={\color{dummy}\textbf{u}}$.
\end{lemma}
\begin{proof}
We claim that there is a normal form, i.e. any
word representing $\rklx{k,l}$ is
equivalent to the word $\theta_{{\color{dummy}\textbf{u}}_{k+l}}\cdots\theta_{{\color{dummy}\textbf{u}}_{0}}$
such that ${\color{dummy}\textbf{u}}_0={\color{dummy}\textbf{u}}$ and
\begin{gather}\label{eq:normal-form}
{\color{dummy}\textbf{u}}_r=\rho({\color{dummy}\textbf{u}}_{r-1}),\text{ for all }1\leq r\leq k,
\quad\quad
{\color{dummy}\textbf{u}}_r=\rho^{-1}({\color{dummy}\textbf{u}}_{r-1}),\text{ for all }k+1\leq r\leq k+l,
\end{gather}
which is clear if $l=0$.
Otherwise, any word representing $\rklx{k,l}$ involves $k$
counterclockwise rotations and $l$ clockwise rotations of $\Gset\Oset\Pset$.
Hence, if such a word is not in normal form, then
we will find a subsequence of the form
\begin{gather*}
\theta_{{\color{dummy}\textbf{u}}_{i}}\theta_{\rho^{-1}({\color{dummy}\textbf{u}}_{i})}\theta_{{\color{dummy}\textbf{u}}_{i}}
\stackrel{\eqref{eq:second-rel}}{=}
\theta_{{\color{dummy}\textbf{u}}_{i}}\theta_{\rho({\color{dummy}\textbf{u}}_{i})}\theta_{{\color{dummy}\textbf{u}}_{i}},
\end{gather*}
which we rewrite as above.
We can then continue recursively until we get \eqref{eq:normal-form}.
\end{proof}
Similarly, we can define
\begin{gather}\label{eq:left-right-relations}
\kly{k,l}=\rklx{k,l}\quad\text{such that}\; {\color{dummy}\textbf{v}}=\rho^{k-l}({\color{dummy}\textbf{u}}),
\end{gather}
for $k,l,{\color{dummy}\textbf{v}}$.
\fullref{lemma:well-defined-paths} implies,
mutatis mutandis, that
$\kly{k,l}$ is also independent of the chosen sequence
${\color{dummy}\textbf{v}}={\color{dummy}\textbf{u}}_{k+l},\dots,{\color{dummy}\textbf{u}}_0$.
\begin{remark}\label{remark-KL-combinatorics}
We can view $\varstuff{X}$ and $\varstuff{Y}$ as acting via counterclockwise
respectively clockwise rotation of \eqref{eq:color-tensor}.
By \fullref{lemma:well-defined-paths},
we can view the elements $\rklx{k,l}$ as being associated to
$\varstuff{X}^k\varstuff{Y}^l$ (after fixing a starting color ${\color{dummy}\textbf{u}}$), because its
definition involves $k$ times the application of $\rho$ and
$l$ times that of $\rho^{-1}$. \fullref{lemma:well-defined-paths}
then translates into the equality
$\varstuff{X}\varstuff{Y}=\varstuff{Y}\varstuff{X}$.
\end{remark}
\begin{example}\label{example-KL-combinatorics}
Let us fix ${\color{mygreen}g}$ as a starting color. Then
\begin{gather*}
\,
\xy
(0,2.5)*{\rklg{2,0}=\theta_{\color{mypurple}p}\theta_{\color{myorange}o}\theta_{\color{mygreen}g}};
(0,-2.5)*{\text{$\leftrightsquigarrow\varstuff{X}^2\varstuff{Y}^0$}};
\endxy
,
\quad
\xy
(0,2.5)*{\rklg{1,1}=\theta_{\color{mygreen}g}\theta_{\color{mypurple}p}\theta_{\color{mygreen}g}
=\theta_{\color{mygreen}g}\theta_{\color{myorange}o}\theta_{\color{mygreen}g}
};
(0,-2.5)*{\text{$\leftrightsquigarrow\varstuff{X}^1\varstuff{Y}^1=\varstuff{Y}^1\varstuff{X}^1$}};
\endxy
,
\quad
\xy
(0,2.5)*{\rklg{0,2}=\theta_{\color{myorange}o}\theta_{\color{mypurple}p}\theta_{\color{mygreen}g}};
(0,-2.5)*{\text{$\leftrightsquigarrow\varstuff{X}^0\varstuff{Y}^2$}};
\endxy
,
\end{gather*}
where we think of the color changes
${\color{mygreen}g}\mapsfrom{\color{mypurple}p}\mapsfrom{\color{myorange}o}\mapsfrom{\color{mygreen}g}$ as corresponding to
multiplication by $\varstuff{X}$, and the
color changes ${\color{mygreen}g}\mapsfrom{\color{myorange}o}\mapsfrom{\color{mypurple}p}\mapsfrom{\color{mygreen}g}$ as corresponding to
multiplication by $\varstuff{Y}$.
\end{example}
Recall that $d^{k,l}_{m,n}$ denote the numbers from
\fullref{section:sl3-stuff}, coming from the
representation theory of $\mathfrak{sl}_{3}$. For each pair $m,n$, we define
three colored KL basis elements:
\begin{gather}\label{eq:the-expressions}
\begin{gathered}
\RKLg{m,n}=
{\textstyle\sum_{k,l}}\,
\vnumber{2}^{-k-l}\,d^{k,l}_{m,n}\,\rklg{k,l},
\quad\quad
\RKLo{m,n}=
{\textstyle\sum_{k,l}}\,
\vnumber{2}^{-k-l}\,d^{k,l}_{m,n}\,\rklo{k,l},
\\
\RKLp{m,n}=
{\textstyle\sum_{k,l}}\,
\vnumber{2}^{-k-l}\,d^{k,l}_{m,n}\,\rklp{k,l}.
\end{gathered}
\end{gather}
Note that the three sums are finite,
because $d^{k,l}_{m,n}=0$ unless $k+l\leq m+n$, as mentioned after \eqref{eq:L-vs-L}.
Moreover, by convention, $\RKLx{k,l}=0$ in case $k$ or $l$ are negative.
Furthermore, by \eqref{eq:L-vs-L} and \fullref{lemma:central-character},
$d^{k,l}_{m,n}=0$ if $k-l \not\equiv m-n\bmod 3$.
This implies that, for any $m,n,{\color{dummy}\textbf{u}}$,
all terms $\rklx{k,l}$ of $\RKLx{m,n}$ in \eqref{eq:the-expressions}
have the same left-most factor $\theta_{\color{dummy}\textbf{v}}$, where
${\color{dummy}\textbf{v}}=\rho^{m-n}({\color{dummy}\textbf{u}})$, by \eqref{eq:left-right-relations}. Therefore, we can also define
\begin{gather}\label{eq:left-right-KL}
\KLy{m,n}=\RKLx{m,n} \quad\text{such that}\; {\color{dummy}\textbf{v}}=\rho^{m-n}({\color{dummy}\textbf{u}}).
\end{gather}
We call $\RKLg{m,n}, \RKLo{m,n}$ and $\RKLp{m,n}$
the (right) colored KL elements.
As before,
\[
\RKLx{0,0}=\theta_{\color{dummy}\textbf{u}}\quad\text{for any}\; {\color{dummy}\textbf{u}}.
\]
\begin{example}\label{example-KL-combinatorics-2}
For a fixed ${\color{dummy}\textbf{u}}$, the element $\RKLx{m,n}$ (or alternatively $\KLx{m,n}$) is associated
to the orthogonal polynomial $\pxy{m,n}$
from \fullref{subsec:opolys}, cf. \fullref{example-KL-combinatorics-2}.
For example, fix ${\color{mygreen}g}$ as a starting color. Then
\begin{gather*}
\,
\xy
(0,2.5)*{\RKLg{2,0}=\vnumber{2}^{-2}\theta_{\color{mypurple}p}\theta_{\color{myorange}o}\theta_{\color{mygreen}g}-\vnumber{2}^{-1}\,\theta_{\color{mypurple}p}\theta_{\color{mygreen}g}};
(0,-2.5)*{\text{$\leftrightsquigarrow\pxy{2,0}=\varstuff{X}^2-\varstuff{Y}$}};
\endxy
,
\;\;
\xy
(0,2.5)*{\RKLg{1,1}=\vnumber{2}^{-2}\theta_{\color{mygreen}g}\theta_{\color{mypurple}p}\theta_{\color{mygreen}g}
-\,\theta_{\color{mygreen}g}
};
(0,-2.5)*{\text{$\leftrightsquigarrow\pxy{1,1}=\varstuff{X}\varstuff{Y}-1$}};
\endxy
,
\;\;
\xy
(0,2.5)*{\RKLg{0,2}=\vnumber{2}^{-2}\theta_{\color{myorange}o}\theta_{\color{mypurple}p}\theta_{\color{mygreen}g}-\vnumber{2}^{-1}\,\theta_{\color{myorange}o}\theta_{\color{mygreen}g}};
(0,-2.5)*{\text{$\leftrightsquigarrow\pxy{0,2}=\varstuff{Y}^2-\varstuff{X}$}};
\endxy
.
\end{gather*}
Similarly for the other colors.
\end{example}
As we will see, \fullref{proposition:cat-the-algebra} identifies the colored KL elements with the Grothendieck
classes of the indecomposables in a certain
$2$-full $2$-subcategory of singular Soergel bimodules. In particular, the
next lemma and proposition need some notions from categorification
which we only recall in \fullref{sec:A2-diagrams}. Consequently,
we postpone their proofs until the end of \fullref{sec:A2-diagrams}.
\begin{lemma}\label{lemma:multiplication}
For all $m,n,{\color{dummy}\textbf{u}},{\color{dummy}\textbf{v}}$, we have
\begin{gather}\label{eq:multiplication}
\theta_{{\color{dummy}\textbf{u}}}\RKLy{m,n}=
\begin{cases}
\vfrac{3}\RKLy{m,n}, &\text{if } \rho^{m-n}({\color{dummy}\textbf{u}})={\color{dummy}\textbf{v}},\\
\vnumber{2}\left(\RKLy{m+1,n}+ \RKLy{m-1,n+1} + \RKLy{m,n-1}\right), &\text{if } \rho^{m+1-n}({\color{dummy}\textbf{u}})={\color{dummy}\textbf{v}},
\\
\vnumber{2}\left(\RKLy{m,n+1}+ \RKLy{m+1,n-1} + \RKLy{m-1,n}\right), &\text{if } \rho^{m-(n+1)}({\color{dummy}\textbf{u}})={\color{dummy}\textbf{v}},
\end{cases}
\end{gather}
where terms with negative indices are zero.
\end{lemma}
By \eqref{eq:left-right-KL} and \fullref{lemma:multiplication}, we also have
\begin{gather}\label{eq:multiplication2}
\RKLy{m,n}\theta_{{\color{dummy}\textbf{u}}}=
\begin{cases}
\vfrac{3}\RKLx{m,n}, &\text{if } {\color{dummy}\textbf{u}}={\color{dummy}\textbf{v}},\\
\vnumber{2}\left(\RKLx{m+1,n}+ \RKLx{m-1,n+1} + \RKLx{m,n-1}\right), &\text{if }\rho({\color{dummy}\textbf{u}})={\color{dummy}\textbf{v}},\\
\vnumber{2}\left(\RKLx{m,n+1}+ \RKLx{m+1,n-1} + \RKLx{m-1,n}\right), &\text{if }\rho^{-1}({\color{dummy}\textbf{u}})={\color{dummy}\textbf{v}},
\end{cases}
\end{gather}
where again terms with negative indices are zero. Moreover,
there are also the evident versions of \eqref{eq:multiplication}
and \eqref{eq:multiplication2} using $\KLy{m,n}$ instead of $\RKLy{m,n}$.
\begin{example}\label{example-KL-combinatorics-3}
The reader should compare \eqref{eq:multiplication}
and \eqref{eq:multiplication2} with the recursion formulas
from \fullref{lemma:recursion}. This is no coincidence, keeping
\fullref{remark-KL-combinatorics} and \fullref{example-KL-combinatorics-2} in mind. For example,
one can easily check directly that
\begin{gather*}
\RKLo{0,1}\theta_{\color{mygreen}g}=
(\vnumber{2}^{-1}\theta_{\color{mygreen}g}\theta_{\color{myorange}o})\theta_{\color{mygreen}g}=
\vnumber{2}(\vnumber{2}^{-2}\theta_{\color{mygreen}g}\theta_{\color{myorange}o}\theta_{\color{mygreen}g}-\theta_{\color{mygreen}g})
+\vnumber{2}\theta_{\color{mygreen}g}
=
\vnumber{2}(\RKLg{1,1}+\underbrace{\RKLg{-1,1}}_{=0}+\RKLg{0,0}),
\\
\theta_{\color{mygreen}g}\RKLo{0,1}=
\theta_{\color{mygreen}g}(\vnumber{2}^{-1}\theta_{\color{mygreen}g}\theta_{\color{myorange}o})
=
\vfrac{3}(\vnumber{2}^{-1}\theta_{\color{mygreen}g}\theta_{\color{myorange}o})
=
\vfrac{3}\RKLo{0,1},
\end{gather*}
and similarly for right or left multiplication by $\theta_{\color{myorange}o}$ or $\theta_{\color{mypurple}p}$.
\end{example}
\begin{proposition}\label{proposition:two-bases}
Each of the four sets
\begin{gather*}
\basisH=
\{1\}
\cup
\{\rklx{k,l}\mid (k,l)\in X^+,\, {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\},
\;\;
\basisC=
\{1\}
\cup\{\RKLx{m,n}\mid (m,n)\in X^+,\, {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\}
\\
\Hbasis=
\{1\}\cup
\{\klx{k,l}\mid (k,l)\in X^+,\, {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\},
\;\;
\Cbasis=
\{1\}\cup
\{\KLx{m,n}\mid (m,n)\in X^+,\, {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\}
\end{gather*}
is a basis of $\subquo$.
\end{proposition}
As for \fullref{lemma:multiplication},
the proof of \fullref{proposition:two-bases} is postponed until \fullref{sec:A2-diagrams}.
As we will see, the bases $\basisH$ and $\Hbasis$ could be called
Bott--Samelson bases.
Following \cite{KaLu}, we can define
left, right and two-sided
cells for $\subquo$. We have chosen to work with the basis $\basisC$.
\begin{definition}\label{definition:cells-first}
We define a left preorder on $\basisC$
by declaring $\RKLx{m,n}\geq_{\Lcell}\RKLy{m^{\prime},n^{\prime}}$
if there exists an element
$\algstuff{Z}\in\basisC$ such that $\RKLx{m,n}$ appears as a summand of
$\algstuff{Z}\RKLy{m^{\prime},n^{\prime}}$, when the latter is written as a linear combination
of elements in $\basisC$.
This preorder gives rise to
an equivalence relation by declaring
$\RKLx{m,n}\sim_{\Lcell}\RKLy{m^{\prime},n^{\prime}}$ whenever
$\RKLx{m,n}\geq_{\Lcell}\RKLy{m^{\prime},n^{\prime}}$ and
$\RKLy{m^{\prime},n^{\prime}}\geq_{\Lcell}\RKLx{m,n}$.
The equivalence classes of $\sim_{\Lcell}$ are called left cells.
Similarly, right multiplication gives rise to a
right preorder $\geq_{\Rcell}$, a right equivalence relation $\sim_{\Rcell}$ and
right cells $\mathsf{R}$. Multiplication on both
sides, gives rise to a two-sided preorder $\geq_{\Tcell}$,
a two-sided equivalence relation $\sim_{\Tcell}$ and
two-sided cells $\mathsf{J}$.
\end{definition}
Clearly, $1\in\subquo$ forms
a cell $\{1\}$ on its own, which is left, right and two-sided at once, and always the lowest cell.
We call $\{1\}$ the trivial cell. The other cells are as follows.
\begin{proposition}\label{proposition:cells}
The non-trivial cells for the algebra $\subquo$ are
\begin{gather*}
\mathsf{L}_{{\color{dummy}\textbf{u}}}=\left\{\RKLx{m,n}\mid (m,n)\in X^+\right\},
\quad\quad
{}_{{\color{dummy}\textbf{u}}}\mathsf{R}=\left\{\KLx{m,n}\mid (m,n)\in X^+\right\},
\quad\quad \hbox{ for ${\color{dummy}\textbf{u}}\in \Gset\Oset\Pset$,}
\\
\mathsf{J}=
\left\{\RKLx{m,n}\mid (m,n)\in X^+,{\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\}
=
\left\{\KLx{m,n}\mid (m,n)\in X^+,{\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\},
\end{gather*}
where $\mathsf{L}_{{\color{dummy}\textbf{u}}}$, ${}_{{\color{dummy}\textbf{u}}}\mathsf{R}$ and $\mathsf{J}$
are left, right and two-sided cells
respectively.
\end{proposition}
\begin{proof}
Fix ${\color{mygreen}g}$ as a starting color. Applying \eqref{eq:multiplication2} to
$\theta_{{\color{dummy}\textbf{u}}}\RKLg{m,n}$, yields $\RKLg{m+1,n}\geq_{\Lcell}\RKLg{m,n}$ for
${\color{dummy}\textbf{u}}$ being chosen such we can apply the middle cases.
We also obtain
$\RKLg{m,n}\geq_{\Lcell}\RKLg{m+1,n}$, by
applying \eqref{eq:multiplication2} to $\theta_{{\color{dummy}\textbf{v}}}\RKLg{m+1,n}$
for appropriate ${\color{dummy}\textbf{v}}$. Thus, we have
$\RKLg{m,n}\sim_{\Lcell}\RKLg{m+1,n}$.
Similarly, we deduce $\RKLg{m,n}\sim_{\Lcell}\RKLg{m,n-1}$,
$\RKLg{m,n}\sim_{\Lcell}\RKLg{m,n+1}$ and $\RKLg{m,n}\sim_{\Lcell}\RKLg{m-1,n}$.
Thus, for fixed $m$ we get that all $\RKLg{m,\underline{\phantom{a}}}$ are in the same left cell,
and similarly for fixed $n$ we get that all $\RKLg{\underline{\phantom{a}},n}$ are in the same left cell.
We can also deduce that $\RKLg{m,n}\sim_{\Lcell}\RKLg{m-1,n+1}$ and $\RKLg{m,n}\sim_{\Lcell}\RKLg{m+1,n-1}$.
In summary, all $\RKLg{m,n}$ belong to the same
left cell.
Since left multiplication will never change the rightmost
color of a word, we conclude
that $\mathsf{L}_{{\color{mygreen}g}}$ is indeed a left cell.
Analogously, one can show that $\mathsf{L}_{{\color{myorange}o}}$ and
$\mathsf{L}_{{\color{mypurple}p}}$ are left cells, and,
mutatis mutandis, that
${}_{{\color{dummy}\textbf{u}}}\mathsf{R}$ is a right cell, for ${\color{dummy}\textbf{u}}$.
Finally, the statement about two-sided cell follows from \eqref{eq:multiplication}.
\end{proof}
\subsection{The quotient of level \texorpdfstring{$e$}{e}}\label{subsec:quotient-algebra}
\subsubsection{Its definition}\label{subsec:quotient-algebra-1}
We are now ready to define interesting,
finite-dimensional quotients of $\subquo$, which are compatible
with the cell structure.
\begin{definition}\label{definition:the-quotient-defined}
For fixed level $e$, let $\killideal{e}$
be the two-sided ideal in $\subquo$ generated by
\begin{gather*}
\left\{
\RKLx{m,n} \mid m+n=e+1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset
\right\}
=
\left\{
\phantom{\RKLx{m,n}}\hspace*{-.75cm}\KLx{m,n} \mid m+n=e+1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset
\right\}.
\end{gather*}
We define the the trihedral Hecke algebra of level $e$ as
\[
\subquo[e]=\subquo/\killideal{e}
\]
and we call $\killideal{e}$ the vanishing ideal of level $e$.
\end{definition}
\begin{remark}\label{remark:small-quotient}
We point out that $\subquo[e]$ is the trihedral analog of the
so called small quotient in the dihedral case, cf. \fullref{remark:dihedral-group2}.
\end{remark}
\begin{proposition}\label{proposition:dimension}
Each of the two sets
\begin{gather*}\label{eq:cell-basis-2}
\begin{aligned}
\basisC[e]=
\{1\}\cup
&
\left
\{\RKLx{m,n} \mid 0\leq m+n\leq e,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset
\right\}
,
\\
\Cbasis[e]=
\{1\}\cup
&
\left
\{\KLx{m,n} \mid 0\leq m+n\leq e,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset
\right\},
\end{aligned}
\end{gather*}
is a basis of $\subquo[e]$.
Thus, we have
$\dim_{\C_{\vpar}}\subquo[e] = 3\tfrac{(e+1)(e+2)}{2}+1=3t_e+1$.
\end{proposition}
\begin{proof}
By \fullref{lemma:multiplication}, $\subquo$ is an $\mathbb{N}$-filtered
algebra with $\subquo\cong {\textstyle\bigcup_{i\in\mathbb{N}}}(\subquo)_i$
such that $(\subquo)_0=\{1\}$ and, for any $i\in\mathbb{Z}_{\geq 1}$, we have
\[
(\subquo)_{i}=\C_{\vpar}\left\{
\RKLx{m,n} \mid 0\leq m+n\leq i-1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset
\right\}.
\]
Note that $\RKLx{0,0}=\theta_{{\color{dummy}\textbf{u}}}$ has filtration degree $1$, so
the multiplication rule in
\fullref{lemma:multiplication} is compatible with this filtration.
Since $\killideal{e}$ is generated by
homogeneous elements, $\subquo[e]$ is also an $\mathbb{N}$-filtered algebra.
In order to prove finite-dimensionality, consider the associated $\mathbb{N}$-graded algebra
\[
\catstuff{E}(\subquo[e])=
{\textstyle\bigoplus_{i\in\mathbb{N}}}\,
(\subquo[e])_i/(\subquo[e])_{i-1},
\]
where $(\subquo[e])_{-1}=\{0\}$, by convention. Note that
\[
\RKLx{m,n}\equiv\rklx{m,n}\bmod(\subquo[e])_{m+n},
\]
for all $m,n$. We have $\catstuff{E}(\subquo[e])_{e+2}=\{0\}$,
by \fullref{lemma:multiplication}, and
$\catstuff{E}(\subquo[e])_i=\{0\}$ for
all $i\geq e+3$, also by \fullref{lemma:multiplication}.
The first statement follows, since
$\left\{\RKLx{m,n}\mid m+n=i-1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset \right\}$ is,
by \fullref{proposition:two-bases}, a
basis of $\catstuff{E}(\subquo[e])_i$, for all $1\leq i\leq e+1$.
The dimension
formula is then clear.
The version with $\KLx{m,n}$ can be shown verbatim.
\end{proof}
\begin{corollary}\label{corollary:cells}
The non-trivial cells for the algebra $\subquo[e]$ are
as in \fullref{proposition:cells}, but intersected with the
bases from \fullref{proposition:dimension}.
In particular, the non-trivial left and right cells have each
cardinality $t_e=\tfrac{(e+1)(e+2)}{2}$. The non-trivial
two-sided cell is the disjoint
union of them all, so it has cardinality $3t_e$.
\end{corollary}
\begin{example}\label{example:cells}
The left cells correspond to the generalized type $\mathsf{A}$ Dynkin diagrams
$\graphA{e}$ in \fullref{subsec:gen-D-list}, which are
cut-offs of the positive Weyl chamber as in \eqref{eq:weight-picture}, such that
the basis elements of the left cells correspond to the vertices of the diagram.
The prototypical examples to keep in mind are
\[
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\text{\tiny$\RKLg{0,0}$}$} to (1,1) node[right, black] {$\text{\tiny$\RKLg{1,0}$}$};
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) node[left, black] {$\text{\tiny$\RKLg{0,1}$}$};
\draw [thick, densely dashed, myred] (1,1) to (-1,1);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}};
(0,-14)*{\text{{\tiny $\mathsf{L}^{{\color{mygreen}g}}$ for $e=1$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\text{\tiny$\RKLg{0,0}$}$} to (1,1) node[right, black] {$\text{\tiny$\RKLg{1,0}$}$} to (0,2);
\draw [thick, myyellow] (0,2) to (-2,2) node[left, black] {$\text{\tiny$\RKLg{0,2}$}$};
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) node[left, black] {$\text{\tiny$\RKLg{0,1}$}$} to (0,2);
\draw [thick, densely dotted, myblue] (0,2) node[above, black] {$\text{\tiny$\RKLg{1,1}$}$} to (2,2) node[right, black] {$\text{\tiny$\RKLg{2,0}$}$};
\draw [thick, densely dashed, myred] (2,2) to (1,1) to (-1,1) to (-2,2);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}};
(0,-14)*{\text{{\tiny $\mathsf{L}^{{\color{mygreen}g}}$ for $e=2$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\text{\tiny$\RKLg{0,0}$}$} to (1,1) node[right, black] {$\text{\tiny$\RKLg{1,0}$}$} to (0,2) to (1,3) node[above, black] {$\text{\tiny$\RKLg{2,1}$}$} to (3,3) node[right, black] {$\text{\tiny$\RKLg{3,0}$}$} node[above, black] {$\text{\tiny$\RKLg{1,1}$}$};
\draw [thick, myyellow] (0,2) to (-2,2) node[left, black] {$\text{\tiny$\RKLg{0,2}$}$} to (-3,3) node[left, black] {$\text{\tiny$\RKLg{0,3}$}$};
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) node[left, black] {$\text{\tiny$\RKLg{0,1}$}$} to (0,2) to (-1,3) to (-3,3);
\draw [thick, densely dotted, myblue] (0,2) to (2,2) node[right, black] {$\text{\tiny$\RKLg{2,0}$}$} to (3,3);
\draw [thick, densely dashed, myred] (1,1) to (-1,1) to (-2,2) to (-1,3) node[above, black] {$\text{\tiny$\RKLg{1,2}$}$} to (1,3) to (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\draw [<-] (.35,2.15) to (2.5,3.4);
\end{tikzpicture}};
(0,-14)*{\text{{\tiny $\mathsf{L}^{{\color{mygreen}g}}$ for $e=3$}}};
\endxy
\]
where we also display the associated colored KL basis elements.
The starting (rightmost) color is indicated by $\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$.
The color of any vertex is the color of the
leftmost $\theta_{{\color{dummy}\textbf{u}}}$ in any of the terms of the corresponding
colored KL basis element. The arrows of the same type, emanating from a given vertex, indicate
the terms which appear on the right-hand
side of the multiplication rule in \eqref{eq:multiplication2}.
\end{example}
\subsubsection{Trihedral simples}\label{subsec:semisimplicity}
Next, we classify all simple representations of $\subquo[e]$ on
($\C_{\vpar}$-)vector spaces,
cf. \eqref{eq:the-simples}. To this end, note that
the ideal $\killideal{e}$
defining $\subquo$ is built such that we can use
Koornwinder's Chebyshev polynomials and their roots
as in \fullref{subsec:roots-poly} below.
First, the one-dimensional representations of $\subquo[e]$ are easy
to define, since they correspond to characters. Each such character
\[
\algstuff{M}_{\lambda_{\color{mygreen}g},\lambda_{\color{myorange}o},\lambda_{\color{mypurple}p}}
\colon\subquo[e]\to\C_{\vpar}
\]
is completely determined by its value on the generators
\[
\theta_{\color{mygreen}g}\mapsto \lambda_{\color{mygreen}g},\;
\theta_{\color{myorange}o}\mapsto \lambda_{\color{myorange}o},\;
\theta_{\color{mypurple}p}\mapsto \lambda_{\color{mypurple}p}.
\]
Therefore, we can identify $\algstuff{M}_{\lambda_{\color{mygreen}g},\lambda_{\color{myorange}o},\lambda_{\color{mypurple}p}}$ with a triple
$(\lambda_{\color{mygreen}g},\lambda_{\color{myorange}o},\lambda_{\color{mypurple}p})\in\C_{\vpar}^{3}$.
\begin{lemma}\label{lemma:further-restrictions}
The following table
\begin{gather}\label{eq:the-one-dims}
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep={.85cm,between origins}, column
sep={4.0cm,between origins}, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&] {
e\equiv 0\bmod 3 \& e\not\equiv 0\bmod 3 \\
\begin{gathered}\algstuff{M}_{0,0,0},\,\algstuff{M}_{\vfrac{3},0,0},
\\
\algstuff{M}_{0,\vfrac{3},0},\,\algstuff{M}_{0,0,\vfrac{3}}\end{gathered} \& \algstuff{M}_{0,0,0} \\};
\draw[densely dashed] ($(m-1-1.south west)+ (-.9,0)$) to (m-1-2.south east);
\draw[densely dashed] ($(m-1-2.north west) + (-.3,0)$) to ($(m-1-2.north west) + (-.3,-2.75)$);
\end{tikzpicture}
\end{gather}
gives a complete, irredundant list of one-dimensional
$\subquo[e]$-representations.
\end{lemma}
\begin{proof}
Let us first check which
triples $(\lambda_{\color{mygreen}g},\lambda_{\color{myorange}o},\lambda_{\color{mypurple}p})$ give a well-defined character of $\subquo[e]$:
by \eqref{eq:first-rel}, we see that $\lambda_{\color{dummy}\textbf{u}}$ has to be zero or $\vfrac{3}$.
Moreover, \eqref{eq:second-rel} implies that either
all $\theta_{\color{dummy}\textbf{u}}$ act by zero, precisely one
of them acts by $\vfrac{3}$, or
all of them act by $\vfrac{3}$.
Further restrictions are imposed by requiring
the representation to vanish
on $\vanideal{e}$.
Let us now give the details. The representation $\algstuff{M}_{0,0,0}$ vanishes on $\vanideal{e}$,
because, by definition, $\RKLg{m,n}$, $\RKLo{m,n}$, $\RKLp{m,n}$ have no constant term for all $m,n$,
since their starting color is always $\theta_{\color{dummy}\textbf{u}}$.
The representation $\algstuff{M}_{\vfrac{3},\vfrac{3},\vfrac{3}}$ does not vanish
on $\vanideal{e}$, since all polynomials $\pxy{m,n}$ have a unique term
of highest degree. This follows
from the representation theory of of $\mathfrak{sl}_{3}$, since $\varstuff{X}^{m}\varstuff{Y}^{n}$
has a unique highest summand. This coefficient of this term contributes a maximal power of $\varstuff{v}$ when
evaluated, which cannot be canceled by the coefficients of other terms,
e.g.
\[
\varstuff{X}^{2}-\varstuff{Y}
\leftrightsquigarrow
\vnumber{2}^{-2}\theta_{\color{mypurple}p}\theta_{\color{myorange}o}\theta_{\color{mygreen}g}-\vnumber{2}^{-1}\theta_{\color{mypurple}p}\theta_{\color{mygreen}g}
\mapsto
\vnumber{2}^{-2}\vfrac{3}\vfrac{3}\vfrac{3}-\vnumber{2}^{-1}\vfrac{3}\vfrac{3}\neq 0.
\]
Thus, $\algstuff{M}_{\vfrac{3},\vfrac{3},\vfrac{3}}$ is not a representation
of $\subquo[e]$.
When $e\equiv 0\bmod 3$, there are three more characters, namely
\[
\algstuff{M}_{\vfrac{3},0,0},\quad\algstuff{M}_{0,\vfrac{3},0},\quad\algstuff{M}_{0,0,\vfrac{3}}.
\]
To see this note that,
for $m+n=e+1$ and $e\equiv 0\bmod 3$, we have $m+n\equiv 1 \bmod 3$.
Hence, $m\equiv n\equiv 0,1\bmod 3$ is impossible in this case.
By \fullref{lemma:no-const-term}, this means that $\pxy{m,n}$ does not have a non-zero constant term.
It follows that all terms in $\RKLg{m,n}$, $\RKLo{m,n}$, $\RKLp{m,n}$
contain a factor $\theta_{\color{dummy}\textbf{v}}\theta_{\color{dummy}\textbf{u}}$ for some ${\color{dummy}\textbf{u}}\neq{\color{dummy}\textbf{v}}$.
For any of the three
$\algstuff{M}_{\vfrac{3},0,0}$, $\algstuff{M}_{0,\vfrac{3},0}$ or $\algstuff{M}_{0,0,\vfrac{3}}$, we therefore have
$\theta_{\color{dummy}\textbf{v}}\theta_{\color{dummy}\textbf{u}}\mapsto 0$. This shows that
$\RKLg{m,n},\RKLo{m,n},\RKLp{m,n}\mapsto 0$.
The three corresponding one-dimensional representations are clearly non-isomorphic.
\end{proof}
Let us now study the simple representations of dimension three, which
depend on a complex number $z\in\mathbb{C}$.
To this end, we define three matrices
\begin{gather}\label{eq:three-dims}
\begin{gathered}
\Mg=
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
\vnumber{3} & \overline{z}& z\\
0 & 0 & 0\\
0& 0 & 0
\end{pmatrix}
}
,
\quad
\Mo=
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
0 & 0 & 0\\
z& \vnumber{3} & \overline{z}\\
0& 0 & 0
\end{pmatrix}
}
,
\\
\Mp=
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
0 & 0 & 0\\
0 & 0 & 0\\
\overline{z} & z & \vnumber{3}
\end{pmatrix}
}
.
\end{gathered}
\end{gather}
Let $\Mt=\Mg+\Mo+\Mp$.
Next, we use the explicit description
of the elements in $\vanset{e}$, cf. \fullref{remark:level-vanishing}.
\begin{lemma}\label{lemma:three-dim-rep1}
The matrices $\Mg,\Mo,\Mp$ define a representation $\algstuff{M}_z$ of $\subquo[e]$
on $\C_{\vpar}^3$, such that
\[
\theta_{{\color{mygreen}g}} \mapsto \Mg,\quad \theta_{{\color{myorange}o}} \mapsto \Mo,\quad \theta_{{\color{mypurple}p}} \mapsto \Mp,
\]
if and only if $(z,\overline{z})\in\vanset{e}$.
\end{lemma}
\begin{proof}
Two short calculations show that
$\algstuff{M}_z({\color{dummy}\textbf{u}})$ respects the
relations \eqref{eq:first-rel} and \eqref{eq:second-rel}.
The fact that $\algstuff{M}_z$ vanishes on $\vanideal{e}$
if and only if $(z,\overline{z})\in\vanset{e}$,
follows by the proof of \fullref{lemma:level-vanishing}, as we defined
$\RKLg{m,n}$, $\RKLo{m,n}$ and $\RKLp{m,n}$
in terms of $\pxy{m,n}$. Note that in the calculation of $\algstuff{M}_z(\RKLx{m,n})$ the positive powers of $\vnumber{2}$,
due to \eqref{eq:three-dims}, cancel against the negative powers of $\vnumber{2}$, which
appear in \eqref{eq:the-expressions}, up to an overall factor $\vnumber{2}$.
\end{proof}
Recall that $\mathtt{\zeta}=\exp(2\pi\mathsf{i}\neatfrac{1}{3})$.
\begin{lemma}\label{lemma:three-dim-rep2}
Let
$(z,\overline{z}),(z^{\prime},
\overline{z}^{\prime})\in\vanset{e}$, $z\neq z^\prime$.
\smallskip
\begin{enumerate}
\setlength\itemsep{.15cm}
\renewcommand{\theenumi}{(\ref{lemma:three-dim-rep2}.a)}
\renewcommand{\labelenumi}{\theenumi}
\item\label{lemma:three-dim-rep2-a} $\algstuff{M}_z\cong\algstuff{M}_{z^{\prime}}$ as representations
of $\subquo[e]$ if and only if
$z^{\prime}=\mathtt{\zeta}^{\pm 1}z$.
\renewcommand{\theenumi}{(\ref{lemma:three-dim-rep2}.b)}
\renewcommand{\labelenumi}{\theenumi}
\item\label{lemma:three-dim-rep2-b} $\algstuff{M}_z$ is simple if and only if $z\neq 0$.\qedhere
\end{enumerate}
\end{lemma}
\begin{proof}
\textit{\ref{lemma:three-dim-rep2-a}.}
Suppose that $z^{\prime}=\mathtt{\zeta}^{\pm 1}z$. Then we have
the following base change between $\Mt$ and $\Mt[z^{\prime}]$:
\[
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
\vnumber{3} & \overline{z}^{\prime}& z^{\prime}\\
z^{\prime} & \vnumber{3} & \overline{z}^{\prime}\\
\overline{z}^{\prime}& z^{\prime} & \vnumber{3}
\end{pmatrix}
}
=
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
\mathtt{\zeta}^{\mp 1} & 0 & 0\\
0 & 1 & 0 \\
0 & 0 & \mathtt{\zeta}^{\pm 1}
\end{pmatrix}
}
\!
{\scriptstyle
\begin{pmatrix}
\vnumber{3} & \overline{z}& z\\
z & \vnumber{3} & \overline{z}\\
\overline{z} & z & \vnumber{3}
\end{pmatrix}
}
\!
{\scriptstyle
\begin{pmatrix}
\mathtt{\zeta}^{\pm 1} & 0 & 0\\
0 & 1 & 0 \\
0 & 0 & \mathtt{\zeta}^{\mp 1}
\end{pmatrix}
}
.
\]
This shows that $\algstuff{M}_z\cong\algstuff{M}_{z^{\prime}}$ as $\subquo[e]$-representations.
To see the converse, we compute the
eigenvalues and eigenvectors of $\Mt$:
\begin{gather}\label{eq:eigen-things}
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=.5em, column
sep=1em, text height=1.0ex, text depth=0.25ex, ampersand replacement=\&] {
\vnumber{2}\left(z+\overline{z}+\vnumber{3}\right)
\&
\vnumber{2}\left(\mathtt{\zeta}^{-1} z+\mathtt{\zeta} \overline{z}+\vnumber{3}\right)
\&
\vnumber{2}\left(\mathtt{\zeta} z+\mathtt{\zeta}^{-1}\overline{z}+\vnumber{3}\right)
\\
(1,1,1)\in\C_{\vpar}^3
\&
(1, \mathtt{\zeta},\mathtt{\zeta}^{-1})\in\C_{\vpar}^3
\&
(1,\mathtt{\zeta}^{-1}, \mathtt{\zeta})\in\C_{\vpar}^3
\\};
\draw[densely dotted] ($(m-1-2.north west) + (-.15,0)$) to ($(m-1-2.north west) + (-.15,-1.75)$);
\draw[densely dotted] ($(m-1-3.north west) + (-.15,0)$) to ($(m-1-3.north west) + (-.15,-1.75)$);
\end{tikzpicture}
\end{gather}
Since $\varstuff{v}$ is generic, these are three non-zero eigenvalues with
three linearly independent eigenvectors showing that
$\Mt$ can be diagonalized.
Now suppose $\algstuff{M}_z\cong\algstuff{M}_{z^{\prime}}$. Then $\Mt$
and $\Mt[z^{\prime}]$ must have the same eigenvalues, so
the above implies that one of the following three triples of equations must hold.
\begin{gather*}
z+\overline{z}=z^{\prime}+\overline{z}^{\prime}
\quad\text{and}\quad
\mathtt{\zeta} z + \mathtt{\zeta}^{-1}\overline{z}=\mathtt{\zeta} z^{\prime} + \mathtt{\zeta}^{-1}\overline{z}^{\prime}
\quad\text{and}\quad
\mathtt{\zeta}^{-1}z + \mathtt{\zeta} \overline{z}=\mathtt{\zeta}^{-1} z^{\prime} + \mathtt{\zeta} \overline{z}^{\prime},
\\
z+\overline{z}=\mathtt{\zeta} z^{\prime}+\mathtt{\zeta}^{-1}\overline{z}^{\prime}
\quad\text{and}\quad
\mathtt{\zeta} z + \mathtt{\zeta}^{-1}\overline{z}=\mathtt{\zeta}^{-1} z^{\prime} + \mathtt{\zeta}\overline{z}^{\prime}
\quad\text{and}\quad
\mathtt{\zeta}^{-1}z + \mathtt{\zeta} \overline{z}=z^{\prime} +\overline{z}^{\prime},
\\
z+\overline{z}=\mathtt{\zeta}^{-1} z^{\prime}+\mathtt{\zeta}\overline{z}^{\prime}
\quad\text{and}\quad
\mathtt{\zeta} z + \mathtt{\zeta}^{-1}\overline{z}=z^{\prime} + \overline{z}^{\prime}
\quad\text{and}\quad
\mathtt{\zeta}^{-1}z + \mathtt{\zeta} \overline{z}=\mathtt{\zeta} z^{\prime} + \mathtt{\zeta}^{-1} \overline{z}^{\prime}.
\end{gather*}
One easily checks that these are satisfied
if and only if
$z=z^{\prime}$ (top triple),
$z=\mathtt{\zeta} z^{\prime}$ (middle triple)
or $z=\mathtt{\zeta}^{-1} z^{\prime}$ (bottom triple).
\newline
\noindent\textit{\ref{lemma:three-dim-rep2-b}.}
In case $z=0$, one clearly has
\[
\algstuff{M}_0\cong\algstuff{M}_{\vfrac{3},0,0}\oplus\algstuff{M}_{0,\vfrac{3},0}\oplus\algstuff{M}_{0,0,\vfrac{3}},
\]
where the one-dimensional representations were defined in \eqref{eq:the-one-dims}.
Now suppose that $z\neq 0$ and that $\algstuff{M}_z$ is reducible.
Then it must have a subrepresentation of dimension one or two.
The explicit description of
the eigenvalues and eigenvectors of $\Mt$
from \eqref{eq:eigen-things} shows that this is impossible.
To see this, first note that
the restriction of $\Mt$
to the vector space underlying the potential subrepresentation
would be diagonalizable as well.
Secondly, in case the eigenvalues in \eqref{eq:eigen-things}
are all distinct, at least one eigenvector therein is also an eigenvector
for the restriction.
However, applying $\Mg$, $\Mo$ and $\Mp$ to
any of the three eigenvectors in \eqref{eq:eigen-things} gives three linear independent vectors, which
shows that no subrepresentation can exist in case of distinct eigenvalues.
Thirdly, assume that
two of the three eigenvalues in \eqref{eq:eigen-things} coincide.
Then there must exist a linear combination of the corresponding two eigenvectors which is an eigenvector
for the restriction. Applying $\Mg$, $\Mo$ and
$\Mp$ to that eigenvector would give
three linear independent vectors, as can easily be checked. We get a contradiction again.
Finally, since $z\neq 0$, not all eigenvalues
in \eqref{eq:eigen-things} can be equal, so we are done.
\end{proof}
Recall from the proof
of \fullref{lemma:level-vanishing} the functions $Z$
and $\overline{Z}$, which map $D$ bijectively onto
the discoid $\mathsf{d}_{3}$, and which determine $\vanset{e}$.
If $e\not\equiv 0\bmod 3$, then
$Z(\sigma,\tau)\neq 0$ for all $(\sigma,\tau)$ as in \eqref{eq:zeros2}.
By \fullref{lemma:three-dim-rep2}, this implies
that the total number of pairwise
non-isomorphic $\algstuff{M}_z$ is
equal to $\neatfrac{t_e}{3}$.
If $e\equiv 0\bmod 3$, then that number
is equal to $\neatfrac{(t_e-1)}{3}$, because
$Z(\sigma,\tau)=0$ if and only if $2k+l=e=k+2l$ if and only if
$k=l=\neatfrac{e}{3}$.
Summarized, we have the following
non-isomorphic, simple
$\subquo[e]$-representations:
\begin{gather}\label{eq:the-simples}
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=1em, column
sep=1em, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{a}
\&
e\equiv 0\bmod 3
\&
e\not\equiv 0\bmod 3
\\
\text{one-dim.}
\&
\begin{gathered}
\algstuff{M}_{0,0,0},\,\algstuff{M}_{\vfrac{3},0,0},
\\
\algstuff{M}_{0,\vfrac{3},0},\,\algstuff{M}_{0,0,\vfrac{3}}
\end{gathered}
\&
\algstuff{M}_{0,0,0}
\\
\text{quantity}
\&
4
\&
1
\\
\text{three-dim.}
\&
\algstuff{M}_z,
\;
(z,\overline{z})\in\vanset{e}^{\mathtt{\zeta}}-\{(0,0)\}
\&
\algstuff{M}_z,
\;
(z,\overline{z})\in\vanset{e}^{\mathtt{\zeta}}
\\
\text{quantity}
\&
\neatfrac{(t_e-1)}{3}
\&
\neatfrac{t_e}{3}
\\};
\draw[densely dashed] ($(m-1-1.south west)+ (-.75,0)$) to ($(m-1-3.south east)+ (.4,0)$);
\draw[densely dashed] ($(m-3-1.south west)+ (-.25,-.25)$) to ($(m-3-3.south east)+ (1.25,-.25)$);
\draw[densely dashed] ($(m-1-3.north west) + (-.6,0)$) to ($(m-1-3.north west) + (-.6,-7.2)$);
\draw[densely dashed] ($(m-1-2.north west) + (-1.4,0)$) to ($(m-1-2.north west) + (-1.4,-7.2)$);
\end{tikzpicture}
\end{gather}
Here $\vanset{e}^{\mathtt{\zeta}}$ denotes the set of $\Z/3\Z$-orbits in $\vanset{e}$
under the action $(z,\overline{z})\mapsto (\mathtt{\zeta} z,\mathtt{\zeta}^{-1}\overline{z})$.
\begin{example}\label{example:three-dims}
By \eqref{eq:the-simples}, the three-dimensional simple
representation of $\subquo[e]$ are indexed by the $\Z/3\Z$-orbits of points
in the interior of $\mathsf{d}_{3}$ (cf. \fullref{example:plot-zeros}.), e.g.:
\[
\begin{tikzpicture}[anchorbase, scale=.6, tinynodes]
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (0,-3) to (0,3);
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (-3,0) to (3,0);
\draw[thick, white, fill=mygreen, opacity=.2] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\draw[thin, densely dotted, ->, >=stealth] (-3.5,0)
to (-3.35,0) node [above] {$-3$}
to (3.2,0) node [above] {$3$}
to (3.5,0) node[right] {$x$};
\draw[thin, densely dotted, ->, >=stealth] (0,-3.5)
to (0,-3.2) node [right] {$-3$}
to (0,3.2) node [right] {$3$}
to (0,3.5) node[above] {$y$};
\draw[thick] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\node at (3,3) {$\mathbb{C}$};
\node[myblue] at (0,0) {$\bullet$};
\node[myblue] at (2,0) {$\bullet$};
\node[myblue] at (-1,1.73) {$\bullet$};
\node[myblue] at (-1,-1.73) {$\bullet$};
\node[myblue] at (-.77,.64) {$\bullet$};
\node[myblue] at (-.77,-.64) {$\bullet$};
\node[myblue] at (-.17,.98) {$\bullet$};
\node[myblue] at (-.17,-.98) {$\bullet$};
\node[myblue] at (.94,.34) {$\bullet$};
\node[myblue] at (.94,-.34) {$\bullet$};
\draw[very thin, densely dashed, myblue] (2,0) to (.94,.34) to (-.17,.98) to (-1,1.73) to (-.77,.64) to (-.77,-.64) to (-1,-1.73) to (-.17,-.98) to (.94,-.34) to (2,0);
\node[myblue] at (2.9,2) {case $e=3$};
\node[myblue] at (2.9,1.5) {$\#(\vanset{3}{-}\{0,0\})=9$};
\node[myblue] at (2.9,1) {$t_e-1=9$};
\draw[ultra thick, mygreen, ->] (2,.2) to [out=110, in=0] (-.8,1.73);
\draw[ultra thick, mygreen, ->] (-1.1,1.63) to [out=225, in=135] (-1.1,-1.63);
\draw[ultra thick, mygreen, ->] (-.8,-1.73) to [out=0, in=250] (2,-.2);
\draw[ultra thick, myorange, ->] (.8,.37) to (-.62,.64);
\draw[ultra thick, myorange, ->] (-.72,.54) to (-.19,-.88);
\draw[ultra thick, myorange, ->] (-.1,-.88) to (.89,.28);
\draw[ultra thick, mypurple, ->] (-.19,.88) to (-.72,-.54);
\draw[ultra thick, mypurple, ->] (-.62,-.64) to (.8,-.37);
\draw[ultra thick, mypurple, ->] (.9,-.23) to (-.1,.88);
\end{tikzpicture}
\]
Here the arrows indicate the $\Z/3\Z$-symmetry.
\end{example}
We are now ready to provide a classification of simple
$\subquo[e]$-representations.
\begin{theorem}\label{theorem:classification-simples}
The table \eqref{eq:the-simples} gives a complete, irredundant list of
simple $\subquo[e]$-representa\-tions.
Furthermore, the algebra $\subquo[e]$ is semisimple.
\end{theorem}
\begin{proof}
By \fullref{proposition:dimension}
the algebra $\subquo[e]$ is of dimension
$3\tfrac{(e+1)(e+2)}{2}+1$.
From the representation theory of finite-dimensional algebras we thus have
\begin{gather}\label{eq:ineq-semisimple}
\dim_{\C_{\vpar}}\subquo[e]
=
3\tfrac{(e+1)(e+2)}{2}+1
=
3t_{e}+1
\geq
{\textstyle\sum_{\algstuff{M}}}\,(\dim_{\C_{\vpar}}\algstuff{M})^2,
\end{gather}
where the sum is taken over any set
of pairwise non-isomorphic, simple
$\subquo[e]$-representations $\algstuff{M}$. If equality holds in
\eqref{eq:ineq-semisimple} for such a set, then that set
is complete and $\subquo[e]$ is semisimple.
\newline
\noindent\textit{Case $e\not\equiv 0\bmod 3$.} We
use the data
from \eqref{eq:the-simples} in
\eqref{eq:ineq-semisimple} and obtain
\[
{\textstyle\sum_{\algstuff{M}}}\,(\dim_{\C_{\vpar}}\algstuff{M})^2
=
\neatfrac{t_e}{3}
\cdot
3^2
+
1\cdot
1^2
=
3t_{e}+1
,
\]
which shows both statements.
\newline
\noindent\textit{Case $e\equiv 0\bmod 3$.}
Similarly, we compute
\[
{\textstyle\sum_{\algstuff{M}}}\,(\dim_{\C_{\vpar}}\algstuff{M})^2
=
\neatfrac{(t_e-1)}{3}
\cdot 3^2+4\cdot 1^2
=
3(t_e-1)+4=
3t_{e}+1
,
\]
which again shows both statements.
\end{proof}
\subsection{Generalizing dihedral Hecke algebras}\label{subsec:dihedral-group}
We finish this section by listing some analogies to the dihedral case.
The crucial link between the dihedral and the trihedral case is the following:
The $\mathfrak{sl}_{2}$-version of the polynomial $\pxy{m,n}$ from \fullref{subsec:opolys}
is the Chebyshev polynomial $\pxy[\varstuff{X}]{m}$ (normalized and of the second kind).
Using the convention that the $\pxy[\varstuff{X}]{m}$ are zero for negative subscripts,
they satisfy the recursion relation
\[
\pxy[\varstuff{X}]{0}=1,
\quad
\pxy[\varstuff{X}]{1}=\varstuff{X},
\quad
\varstuff{X}\pxy[\varstuff{X}]{m}=
\pxy[\varstuff{X}]{m+1}+\pxy[\varstuff{X}]{m-1}.
\]
Here, $\varstuff{X}$ corresponds to the
fundamental representation of $\mathfrak{sl}_{2}$.
The analog of the discoid $\mathsf{d}_{3}$ from \eqref{eq:deltoid}
is the interval $\mathsf{d}_{2}=[-2,2]$, whose boundary is the
pair of primitive, complex second roots of unity, multiplied by $2$. (Note the evident
$\Z/2\Z$-symmetry of $\mathsf{d}_{2}$.)
\begin{dihedral}\label{remark:dihedral-group1}
Let $\dihquo=\hecke(\typei[\infty])=\hecke(\typeat{1})$ denote the dihedral
Hecke algebra of the infinite dihedral group, i.e.
the Weyl group of affine type $\typea{1}$,
and $\hecke(\typei)$ the dihedral Hecke algebra of dimension $2(e+2)$,
which is of finite Coxeter type $\typei$.
The first analogy of our story to the dihedral case
is provided by \fullref{lemma:quotient-of-affine}, the difference
being that the trihedral Hecke algebra
is a proper subalgebra of $\hecke$.
The entries of the change-of-basis matrix from the
(colored) KL basis to the Bott--Samelson basis of $\dihquo$ are precisely the
coefficients of the polynomials $\pxy[\varstuff{X}]{m}$, see for example \cite[Section 2.2]{El2}.
\end{dihedral}
\begin{dihedral}\label{remark:dihedral-group-cells}
By \fullref{proposition:cells},
all non-trivial cells of $\subquo$ are infinite, and there
are three non-trivial left and right
cells, one for each ${\color{dummy}\textbf{u}}\in\Gset\Oset\Pset$, whose disjoint union
forms the only non-trivial two-sided cell. This is
another analogy to the dihedral case:
the algebra $\dihquo$ has two non-trivial left and right cells, one for each of its
Coxeter generators, whose disjoint union forms
the only non-trivial two-sided cell.
\end{dihedral}
\begin{dihedral}\label{remark:dihedral-group2}
Let $\dihquo[e]$ denote the small quotient of $\hecke(\typei)$,
obtained by killing the top cell. \fullref{subsec:quotient-algebra-1} provides the
third analogy: $\dihquo[e]$ can be obtained as a quotient of $\dihquo[\infty]$
by the ideal generated by the two elements related to the irreducible
$\mathfrak{sl}_{2}$-module $\algstuff{L}_{e+1}$ under the quantum Satake correspondence; the non-trivial
left cells of $\dihquo[e]$ have order $e+1$ and $\dim_{\C_{\vpar}}\dihquo[e]=2(e+2)-1=2e+3$.
\end{dihedral}
\begin{dihedral}\label{remark:dihedral-group3}
\fullref{theorem:classification-simples}
provides another analogy to the dihedral case:
$\dihquo[e]$ is semisimple over $\mathbb{C}$, and all of its
simples are either one- or two-dimensional,
with the number of their isomorphism classes
depending on whether $e\equiv 0\bmod 2$ or
$e\equiv 1\bmod 2$.
Analogously to \eqref{eq:three-dims}, the two-dimensional simples
can be defined by matrices whose off-diagonal, non-zero entries are the roots of the
Chebyshev polynomials $\pxy[\varstuff{X}]{e+1}$, i.e.
its (colored) KL generators are send to
\scalebox{.8}{$\begin{psmallmatrix}
\vnumber{2} & z\\
0 & 0
\end{psmallmatrix}$} and
\scalebox{.8}{$\begin{psmallmatrix}
0 & 0\\
\overline{z} & \vnumber{2}
\end{psmallmatrix}$},
where $z=\overline{z}$ is a root of $\pxy[\varstuff{X}]{e+1}$.
\end{dihedral}
\section{Some \texorpdfstring{$\mathfrak{sl}_{3}$}{sl3} combinatorics}\label{section:sl3-stuff}
This section is mostly a collection of known
results, formulated in our notation.
\subsection{Quantum \texorpdfstring{$\mathfrak{sl}_{3}$}{sl3}}\label{subsec:qslthree}
Throughout, $m,n,k,l$ will denote non-negative integers.
\subsubsection{Some conventions}\label{subsec:weights-etc}
We always use the following conventions when working with $\mathfrak{sl}_{3}$.
Denote by
$\varepsilon_1,\varepsilon_2,\varepsilon_3$
the standard basis vectors of $\mathbb{R}^3$. We endow
$\mathbb{R}^3$ with the usual symmetric bilinear form
$(\varepsilon_i,\varepsilon_j)=\delta_{ij}$ and
let $E=\{(x_1,x_2,x_3)\in\mathbb{R}^3\mid x_1+x_2+x_3=0\}$ be the
Euclidean subspace of $\mathbb{R}^3$ (with induced
symmetric bilinear form).
We also fix two simple roots $\alpha_1=\varepsilon_1-\varepsilon_2$
and $\alpha_2=\varepsilon_2-\varepsilon_3$,
and coroots $\alpha_1^\vee$ and $\alpha_2^\vee$
such that
$\langle\alpha_i,\alpha_j^{\vee}
\rangle=(\alpha_i,\alpha_j)=a_{ij}$, for $i,j=1,2$,
are the entries of
the (usual) Cartan matrix
$\begin{psmallmatrix}
2 & -1 \\
-1 & 2
\end{psmallmatrix}$ of $\mathfrak{sl}_{3}$.
The (integral) weights
are $X=\{\lambda \in E\mid
\langle\lambda,\alpha_1^{\vee}\rangle\in\mathbb{Z}\text{ and }
\langle\lambda,\alpha_2^{\vee}\rangle\in\mathbb{Z}\}$. The
dominant (integral) weights are
$X^+=\{\lambda \in E\mid
\langle\lambda,\alpha_1^{\vee}\rangle\in\mathbb{N}\text{ and }
\langle\lambda,\alpha_2^{\vee}\rangle\in\mathbb{N}\}$.
We identify $X=\mathbb{Z}^2$ and $X^+=\mathbb{N}^2$, cf. \eqref{eq:weight-picture},
with $X^+$ also called the
positive Weyl chamber. We also use the
fundamental weights $\omega_{1},\omega_{2}\in E$
(which are characterized by
$\langle\omega_{i},\alpha_j^{\vee}\rangle=\delta_{i,j}$),
and $\lambda=(m,n)\in X^+$ for us
means $\lambda=m\omega_{1}+n\omega_{2}$.
The following picture summarizes our root and weight conventions for $e=3$:
\begin{gather}\label{eq:weight-picture}
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw[thin, white, fill=mygray, opacity=.2] (0,0) to (5,5) to (-5,5) to (0,0);
\draw[thin, densely dashed, mygray] (0,0) to (7,7);
\draw[thin, densely dashed, mygray] (0,0) to (-7,7);
\node at (0,6.75) {\text{{\tiny $X^+=\mathbb{N}^{2}$}}};
\draw[thick, densely dotted, myblue] (4,3) node [right] {\text{{\tiny $e=3$}}} to (-4,3);
\draw[thick, densely dotted, myred] (-5,4) node [left] {\text{{\tiny $e+1=4$}}} to (5,4);
\draw[thick, densely dotted, myyellow] (6,5) node [right] {\text{{\tiny $e+2=5$}}} to (-6,5);
\draw[thick, myorange, ->] (0,0) to (3,1.5) node [right] {\text{{\tiny $\alpha_1$}}};
\draw[thick, myorange, ->] (0,0) to (-3,1.5) node [left] {\text{{\tiny $\alpha_2$}}};
\node at (0,0) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (0,2) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (3,3) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (-3,3) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (0,4) {$\scalebox{.7}{\text{{\color{myred}$\bigstar$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (-2,4) {$\scalebox{.7}{\text{{\color{myred}$\bigstar$}}}$};
\node at (4,4) {$\scalebox{.7}{\text{{\color{myred}$\bigstar$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$};
\node at (2,4) {$\scalebox{.7}{\text{{\color{myred}$\bigstar$}}}$};
\node at (-4,4) {$\scalebox{.7}{\text{{\color{myred}$\bigstar$}}}$};
\node at (-5,5) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (-3,5) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (-1,5) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (1,5) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (3,5) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (5,5) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (-6,6) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (-4,6) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (-2,6) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (0,6) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (2,6) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (4,6) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\node at (6,6) {$\scalebox{.7}{\text{{\color{myyellow}$\blacktriangle$}}}$};
\draw[thin, ->] (-5.5,4.75) node [left] {\text{{\tiny $(2,2)$}}} to [out=0, in=165] (-.2,4.2);
\draw[thin, ->] (5.5,3.75) node [right] {\text{{\tiny $(1,2)$}}} to [out=180, in=15] (-.8,3.2);
\draw[thin, ->] (-5.5,2.75) node [left] {\text{{\tiny $(0,2)$}}} to [out=0, in=165] (-2.2,2.2);
\end{tikzpicture}
\end{gather}
We have also indicated an example of a cut-off, denoting its weights by $\scalebox{.7}{\text{{\color{myblue}$\blacktriangledown$}}}$, which
depend on the level $e$, i.e. the integral
points $X^+(e)=\{\lambda \in X^+\mid
\langle\lambda,\alpha^{\vee}_{1}\rangle\leq
e\text{ and }\langle\lambda,\alpha^{\vee}_{2}\rangle\leq
e\text{ and }\langle\lambda,\alpha^{\vee}_{1}+\alpha^{\vee}_{2}\rangle\leq e\}$. Such
cut-offs play an important role in our paper. Moreover, we
usually quotient by data associated to
the line $e+1$ as illustrate by the symbols $\scalebox{.7}{\text{{\color{myred}$\bigstar$}}}$ in \eqref{eq:weight-picture}.
\subsubsection{Generic quantum \texorpdfstring{$\mathfrak{sl}_{3}$}{sl3}}\label{subsec:generic}
Let $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$ denote the quantum enveloping ($\Cq$-)algebra
associated to $\mathfrak{sl}_{3}$. We refer the reader to \cite[Chapters 4-7]{Ja}
(whose conventions we silently adopt using
the root and weight setting from above)
for details.
We denote by $\sltcat=\sltcatpre$ the category
of finite-dimensional (left)
$\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules (of type $1$, cf. \cite[Section 5.2]{Ja}).
Recall that $\sltcat$ is semisimple with a
complete set of pairwise non-isomorphic, irreducible
$\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules parametrized by (the integral part of) the
positive Weyl chamber
\[
\left\{
\algstuff{L}_{m,n}\mid (m,n)\in X^+
\right\}.
\]
The subscripts $m,n$ indicate the highest weight
$m\omega_1+n\omega_2$ of the
irreducible module, by which it is uniquely determined. Moreover,
$\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$ is a Hopf algebra, so we can tensor
$\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules and take duals. Thus, if $\GG{\underline{\phantom{a}}}$ denotes the (additive)
Grothendieck group, then
\[
\left\{
[\algstuff{L}_{m,n}]=\GG{\algstuff{L}_{m,n}}\in\GG{\sltcat}\mid (m,n)\in X^+
\right\}
\]
is a $\mathbb{Z}$-basis of $\GG{\sltcat}$, and
$\GG{\sltcat}$ is a ring. Extending the scalars to $\mathbb{C}$, we get a $\mathbb{C}$-algebra:
\[
\GGc{\sltcat}=\GG{\sltcat}\otimes_\mathbb{Z}\mathbb{C}.
\]
Throughout the paper, we will use
notations similar to $\GGc{\underline{\phantom{a}}}$, indicating scalar extensions.
\begin{remark}\label{remark:generic-quantum-group}
Since $\varstuff{q}$ is generic, we can
identify $\GG{\sltcat}$
with the corresponding Grothendieck ring of the
category of complex, finite-dimensional representations
of $\mathfrak{sl}_{3}$, cf. \cite[Theorems 5.15 and 5.17]{Ja}.
This means that all our calculations
below follow from standard results in the representation theory of $\mathfrak{sl}_{3}$.
\end{remark}
The two $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules
\begin{gather}\label{eq:the-variables}
\varstuff{X}=\algstuff{L}_{1,0},
\quad
\quad
\varstuff{Y}=\algstuff{L}_{0,1}
\end{gather}
are called the fundamental representations
of $\mathfrak{sl}_{3}$. Note that they are dual, i.e. $\varstuff{X}^*\cong\varstuff{Y}$ as
$\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules. More generally, we have
$(\algstuff{L}_{m,n})^*\cong\algstuff{L}_{n,m}$ for all $m,n\in\mathbb{N}$.
In the following we write $\varstuff{X}^k=\varstuff{X}^{\otimes k}$,
$\varstuff{Y}^l=\varstuff{Y}^{\otimes l}$ and $\varstuff{X}\varstuff{Y}=\varstuff{X}\otimes\varstuff{Y}$ for short, and below
we will consider these as variables in some polynomial ring.
\begin{remark}\label{remark:sl3-cat-GG}
Every $\algstuff{L}_{m,n}$ appears
as a direct summand of a suitable tensor product
of $\varstuff{X}$ and $\varstuff{Y}$. Moreover,
$\sltcat$ is braided monoidal, so $\varstuff{X}\varstuff{Y}\cong\varstuff{Y}\varstuff{X}$
as $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules.
Hence, the Grothendieck group of $\sltcat$ is a commutative ring and
\[
\left\{
[\varstuff{X}^k\varstuff{Y}^l]
\mid
(k,l)\in X^+
\right\}
\]
is an alternative basis of $\GG{\sltcat}$ and $\GGc{\sltcat}$.
\end{remark}
Using the above, in particular \fullref{remark:sl3-cat-GG},
we define $d^{k,l}_{m,n}\in\mathbb{Z}$ as follows:
\begin{gather}\label{eq:L-vs-L}
[\algstuff{L}_{m,n}]
=
{\textstyle\sum_{k,l}}\,
d^{k,l}_{m,n}\cdot
[\varstuff{X}^{k}\varstuff{Y}^{l}].
\end{gather}
Clearly, $d^{k,l}_{m,n}=0$ unless $k+l\leq m+n$.
Thus, the sum in \eqref{eq:L-vs-L} is finite.
Note that the $d^{k,l}_{m,n}$ can be computed inductively,
cf. \fullref{example:sl3-polys}.
Moreover, we have $d^{k,l}_{m,n}=d^{l,k}_{n,m}$ and
$d_{m,n}^{m,n}=1=d_{k,l}^{k,l}$.
\begin{definition}\label{definition:color-code}
For later usage, let us define colors associated
to the $\algstuff{L}$'s:
\begin{gather}\label{eq:color-code}
\chi_c(\algstuff{L}_{m,n})
=
\begin{cases}
{\color{mygreen}g}, & \text{if }m+2n\equiv 0 \bmod 3,\\
{\color{myorange}o}, & \text{if }m+2n\equiv 1 \bmod 3,\\
{\color{mypurple}p}, & \text{if }m+2n\equiv 2 \bmod 3.\\
\end{cases}
\end{gather}
We call $\chi_c(\algstuff{L}_{m,n})$
the central character of $\algstuff{L}_{m,n}$.
\end{definition}
(To explain our choice of name: The center of $\mathrm{SU}_3$ is
$\Z/3\Z$. The generator of $\Z/3\Z$
can act on any irreducible $\mathrm{SU}_3$-module
by multiplication with a primitive, complex, third root of unity. This is what is
encoded by $\chi_c$.)
Observe that $\chi_c(\varstuff{X})={\color{myorange}o}$ and $\chi_c(\varstuff{Y})={\color{mypurple}p}$, while
the representation theory of $\mathfrak{sl}_{3}$ immediately gives that tensoring with $\varstuff{X}$
changes the central character by adding $1 \bmod 3$, while tensoring
with $\varstuff{Y}$ adds $2 \bmod 3$. Thus:
\begin{lemmaqed}\label{lemma:central-character}
All irreducible summands of $\varstuff{X}^k\varstuff{Y}^l$ have
central character $\chi_c(\algstuff{L}_{k,l})$.
\end{lemmaqed}
\subsubsection{The semisimplified root of unity case}\label{subsec:non-generic}
Let $\algstuff{U}_{\qqpar}(\mathfrak{sl}_{3})$ be the specialization
of (the integral form of) $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$ obtained by
specializing $\varstuff{q}$ to $\varstuff{\eta}$, see e.g. \cite{Lu3}, \cite{APW} for details.
Its category $\sltcat[\varstuff{\eta}]=\sltcatpre[\varstuff{\eta}]$ of
finite-dimensional (left) $\sltcat[\varstuff{\eta}]$-modules
(of type $1$) is far from being semisimple. However,
it has a semisimple quotient $\slqmod$, which is roughly obtained
by killing the so-called tilting modules
of quantum dimension zero. We refer
to \cite{AP} for details, but all the reader needs to know
for our purposes is that all $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules $\algstuff{L}_{m,n}$ with
$0\leq m+n\leq e$ can also be regarded as irreducible $\algstuff{U}_{\qqpar}(\mathfrak{sl}_{3})$-modules.
Moreover,
\[
\left\{
[\algstuff{L}_{m,n}]\phantom{\varstuff{X}^{k}}\!\!\!\mid 0\leq m+n\leq e
\right\},
\quad
\left\{
[\varstuff{X}^{k}\varstuff{Y}^{l}]\mid 0\leq k+l\leq e
\right\},
\]
are bases of $\GG{\slqmod}$ and $\GGc{\slqmod}$, and
the quantum fusion product endows
$\slqmod$ with the structure of a monoidal category, so
$\GG{\slqmod}$ is a ring and $\GGc{\slqmod}$ is an algebra.
Note also that the rank of $\GG{\slqmod}$, respectively the dimension of $\GGc{\slqmod}$, is equal to
the triangular number
\[
t_e=\tfrac{(e+1)(e+2)}{2},
\]
which follows from the fact that $\slqmod$ is only supported
on a triangular cut-off of the positive Weyl chamber of $\mathfrak{sl}_{3}$,
cf. \eqref{eq:weight-picture}.
Said otherwise, since $\slqmod$ is semisimple, we have
\[
\GGc{\slqmod}\cong\mathbb{C}^{t_e}
\]
as vector spaces.
\subsection{Chebyshev-like polynomials for \texorpdfstring{$\mathfrak{sl}_{3}$}{sl3}}\label{subsec:opolys}
We now recall certain polynomials introduced in the context of orthogonal polynomials
by Koornwinder \cite{Ko}, but phrased in a more convenient way for our purposes.
\subsubsection{The $\mathfrak{sl}_{3}$-polynomials}\label{subsec:def-poly}
Consider the polynomial ring $\mathbb{Z}[\varstuff{X},\varstuff{Y}]$,
in which $\varstuff{X}$ and $\varstuff{Y}$ from \eqref{eq:the-variables} are treated
as formal variables.
\begin{definition}\label{definition:sl3-polys}
For each $m,n$ we define
$\pxy{m,n}\in\mathbb{Z}[\varstuff{X},\varstuff{Y}]$ by
\begin{gather}\label{eq:the-c-poly}
\pxy{m,n}=
{\textstyle\sum_{k,l}}\,
d_{m,n}^{k,l}
\cdot
\varstuff{X}^k\varstuff{Y}^l,
\end{gather}
with $d_{m,n}^{k,l}\in\mathbb{Z}$ as in \eqref{eq:L-vs-L}.
\end{definition}
For fixed $e$, we often consider all the polynomials
$\pxy{m,n}$ with $m+n=e+1$ together, cf. \fullref{example:sl3-polys}.
\begin{example}\label{example:sl3-polys}
The first few of these polynomials are $\pxy{0,0}=1$ and:
\[
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes, row sep=.0cm, column
sep=.0cm, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&, font=\scriptsize,
nodes={anchor=south},
row 3/.style={nodes={minimum height=1cm}},
row 4/.style={nodes={minimum height=1.5cm}},
row 5/.style={nodes={minimum height=1.5cm}},
] {
e=0 \&
\pxy{1,0}=\varstuff{X},
\;\;
\pxy{0,1}=\varstuff{Y},
\\
e=1 \&
\pxy{2,0}
=
\varstuff{X}^{2}-\varstuff{Y},
\;\;
\pxy{1,1}
=
\varstuff{X}\varstuff{Y}-1,
\;\;
\pxy{0,2}
=
\varstuff{Y}^{2}-\varstuff{X},
\\
e=2 \&
\begin{gathered}
\pxy{3,0}
=
\varstuff{X}^{3}-2\varstuff{X}\varstuff{Y}+1,
\;\;
\pxy{2,1}
=
\varstuff{X}^{2}\varstuff{Y}-\varstuff{Y}^{2}-\varstuff{X},
\\
\pxy{1,2}
=
\varstuff{X}\varstuff{Y}^{2}-\varstuff{X}^{2}-\varstuff{Y},
\;\;
\pxy{0,3}
=
\varstuff{Y}^{3}-2\varstuff{X}\varstuff{Y}+1,
\end{gathered}
\\
e=3 \&
\begin{gathered}
\pxy{4,0}
=
\varstuff{X}^{4}-3\varstuff{X}^{2}\varstuff{Y}+\varstuff{Y}^{2}+2\varstuff{X},
\;\;
\pxy{3,1}
=
\varstuff{X}^{3}\varstuff{Y}-2\varstuff{X}\varstuff{Y}^{2}-\varstuff{X}^{2}+2\varstuff{Y},
\\
\pxy{2,2}
=
\varstuff{X}^{2}\varstuff{Y}^{2}-\varstuff{X}^{3}-\varstuff{Y}^{3},
\\
\pxy{1,3}
=
\varstuff{X}\varstuff{Y}^{3}-2\varstuff{X}^{2}\varstuff{Y}-\varstuff{Y}^{2}+2\varstuff{X},
\;\;
\pxy{0,4}
=
\varstuff{Y}^{4}-3\varstuff{X}\varstuff{Y}^{2}+\varstuff{X}^{2}+2\varstuff{Y},
\end{gathered}
\\
e=4 \&
\begin{gathered}
\pxy{5,0}
=
\varstuff{X}^{5}-4\varstuff{X}^{3}\varstuff{Y}+3\varstuff{X}\varstuff{Y}^{2}+3\varstuff{X}^{2}-2\varstuff{Y},
\;\;
\pxy{4,1}
=
\varstuff{X}^{4}\varstuff{Y}-3\varstuff{X}^{2}\varstuff{Y}^{2}-\varstuff{X}^{3}+\varstuff{Y}^{3}+4\varstuff{X}\varstuff{Y}-1,
\\
\pxy{3,2}
=
\varstuff{X}^{3}\varstuff{Y}^{2}-\varstuff{X}^{4}-2\varstuff{X}\varstuff{Y}^{3}
+\varstuff{X}^{2}\varstuff{Y}+2\varstuff{Y}^{2}-\varstuff{X},
\;\;
\pxy{2,3}
=
\varstuff{X}^{2}\varstuff{Y}^{3}-\varstuff{Y}^{4}-2\varstuff{X}^{3}\varstuff{Y}
+\varstuff{X}\varstuff{Y}^{2}+2\varstuff{X}^{2}-\varstuff{Y},
\\
\pxy{1,4}
=
\varstuff{X}\varstuff{Y}^{4}-3\varstuff{X}^{2}\varstuff{Y}^{2}-\varstuff{Y}^{3}+\varstuff{X}^{3}+4\varstuff{X}\varstuff{Y}-1,
\;\;
\pxy{0,5}
=
\varstuff{Y}^{5}-4\varstuff{X}\varstuff{Y}^{3}+3\varstuff{X}^{2}\varstuff{Y}+3\varstuff{Y}^{2}-2\varstuff{X}.
\end{gathered}
\\
};
\draw[thin, densely dotted] ($(m-1-1)+(-.5,-.25)$) to ($(m-1-1)+(13.1,-.25)$);
\draw[thin, densely dotted] ($(m-1-1)+(-.5,-.8)$) to ($(m-1-1)+(13.1,-.8)$);
\draw[thin, densely dotted] ($(m-1-1)+(-.5,-1.8)$) to ($(m-1-1)+(13.1,-1.8)$);
\draw[thin, densely dotted] ($(m-1-1)+(-.5,-3.35)$) to ($(m-1-1)+(13.1,-3.35)$);
\draw[thin, densely dotted] ($(m-1-1)+(.5,.25)$) to ($(m-1-1)+(.5,-4.8)$);
\end{tikzpicture}
\]
Note that the ones with $m+n=e+1$ correspond to the $e+1$-line in \eqref{eq:weight-picture}.
\end{example}
By convention, $\pxy{m,n}$ and $\algstuff{L}_{m,n}$ with negative subscripts $m$ or $n$ are zero.
\begin{lemma}\label{lemma:recursion}
We have the following Chebyshev-like recursion relations
\begin{gather*}
\pxy{m,n}=\pxy[\varstuff{Y},\varstuff{X}]{n,m},
\\
\varstuff{X}\pxy{m,n}
=
\pxy{m+1,n}+\pxy{m-1,n+1}+\pxy{m,n-1},
\\
\varstuff{Y}\pxy{m,n}=
\pxy{m,n+1}+\pxy{m+1,n-1}+\pxy{m-1,n}.
\end{gather*}
Together with the starting conditions for $e=0,1$ as in \fullref{example:sl3-polys}, these
recursion relations determine the polynomials
$\pxy{m,n}$ for all $m,n$.
\end{lemma}
\begin{proof}
The relation $\pxy{m,n}=\pxy[\varstuff{Y},\varstuff{X}]{n,m}$
boils down to $\varstuff{X}\cong\varstuff{Y}^*$. Moreover,
by standard results in the representation theory of $\mathfrak{sl}_{3}$, we obtain
\begin{align}
\varstuff{X}\otimes\algstuff{L}_{m,n}
&\cong
\algstuff{L}_{m+1,n}\oplus\algstuff{L}_{m-1,n+1}\oplus\algstuff{L}_{m,n-1},
\label{eq:sl3-thingy-a}
\\
\varstuff{Y}\otimes\algstuff{L}_{m,n}
&\cong
\algstuff{L}_{m,n+1}\oplus\algstuff{L}_{m+1,n-1}\oplus\algstuff{L}_{m-1,n},
\label{eq:sl3-thingy-b}
\end{align}
which proves the two recursions.
\end{proof}
\begin{lemma}\label{lemma:no-const-term}
The polynomial $\pxy{m,n}$ has a non-zero
constant term if and only if $m\equiv n\bmod 3$ and $m\not\equiv 2\bmod 3$.
This constant term is equal to
$1$ if $m\equiv n\equiv 0\bmod 3$, and equal to $-1$ if $m\equiv n\equiv 1\bmod 3$.
\end{lemma}
\begin{proof}
The claim follows inductively from \fullref{example:sl3-polys}
and \fullref{lemma:recursion}.
\end{proof}
\subsubsection{Their complex roots}\label{subsec:roots-poly}
The following definition will be crucial for us.
\begin{definition}\label{definition:the-main-ideal}
For fixed level $e$, let $\vanideal{e}$ be the ideal generated by
\[
\left\{
\pxy{m,n}\mid m+n=e+1
\right\}
\subset\mathbb{Z}[\varstuff{X},\varstuff{Y}].
\]
We call $\vanideal{e}$ the vanishing ideal of level $e$. Associated
to it is the vanishing set of level $e$
\[
\vanset{e}=
\left\{
(\alpha,\beta)\in\mathbb{C}^2\mid
p(\alpha,\beta)=0\;\text{for all}\; p\in \vanideal{e}
\right\}\subset\mathbb{C}^2
\]
which we consider as a complex variety.
\end{definition}
Since
$\varstuff{X}$ and $\varstuff{Y}$ generate
$\slqmod$, we have
\begin{gather}\label{eq:roots1}
\GGc{\slqmod}\cong\mathbb{C}[\varstuff{X},\varstuff{Y}]/\vanideal{e}\cong\mathbb{C}^{t_e},
\end{gather}
as vector spaces, where $t_e=\tfrac{(e+1)(e+2)}{2}$ denotes the triangular number.
Note that the left isomorphism in \eqref{eq:roots1} is actually an isomorphism of algebras,
which follows from the explicit form of the
fusion rules for $\slqmod$ (which can be deduced from e.g. \cite[Corollary 8]{Saw}
or \cite[Proposition 3.2.2]{Sch}).
Using this, we can compute $\#\vanset{e}$, the number of points in $\vanset{e}$, i.e.
the number of common roots of the polynomials in $\vanideal{e}$.
\begin{lemma}\label{lemma:level-vanishing}
We have $\#\vanset{e}=t_e$.
\end{lemma}
Before we prove \fullref{lemma:level-vanishing}, let us fix some notation for
complex numbers: $\mathsf{i}$ denotes $\sqrt{-1}$
(in the positive upper half-plane),
$\mathtt{\zeta}=\exp(2\pi\mathsf{i}\neatfrac{1}{3})$ and $\overline{z}$
will denote the complex conjugate of a complex number $z\in\mathbb{C}$.
\begin{proof}
By \eqref{eq:roots1} and a corollary of Hilbert's Nullstellensatz
\cite[Corollary I.7.4]{Fu}, we immediately see that
$\#\vanset{e}\leq t_e$,
To see the equality, consider the following functions, due to \cite{Ko}:
\begin{align*}
Z\colon\mathbb{C}^2\to\mathbb{C},\quad
Z(\sigma,\tau)=
&\exp(\mathsf{i}\sigma)+\exp(-\mathsf{i}\tau)+\exp(\mathsf{i}(-\sigma+\tau)),
\\
E^-_{a,b}\colon\mathbb{C}^2\to\mathbb{C},\quad
E^-_{a,b}(\sigma,\tau)=
&\exp(\mathsf{i}(a\sigma+b\tau))-\exp(\mathsf{i}((a+b)\sigma-b\tau))
\\
&+\exp(\mathsf{i}(-(a+b)\sigma+a\tau))-\exp(\mathsf{i}(-b\sigma-a\tau))
\\
&+\exp(\mathsf{i}(b\sigma-(a+b)\tau))-\exp(\mathsf{i}(-a\sigma+(a+b)\tau)),
\end{align*}
where $a,b\in\mathbb{N}$.
The functions $Z$ and
$E^-_{a,b}$ are clearly $2\pi$-periodic
in both variables, i.e. they define
functions on a $2$-torus $\mathrm{T}^2$. As one easily checks, $Z$ is invariant
and $E^-_{a,b}$ is antiinvariant
under the reflections $(\sigma,\tau)\mapsto (-\sigma+\tau,\tau)$
and $(\sigma,\tau)\mapsto (\sigma, \sigma-\tau)$, which generate
the symmetric group $S_3$. The fundamental domain of the quotient
$\mathrm{T}^2/S_3$ is equal to
\[
D=
\left\{
(\sigma,\tau)\mid 0\leq \sigma+\tau\leq 2\pi,\;
\neatfrac{\sigma}{2}\leq \tau\leq 2\sigma
\right\}.
\]
Note that all zeros of $E^-_{1,1}$ lie on the boundary of $D$. Therefore
$Z$ and
$E^-_{m+1,n+1}/E^-_{1,1}$
define functions on the interior of $D$.
As explained in \cite{Ko}, $Z$ and
its complex conjugate $\overline{Z}$ map $D$ bijectively onto
the ($3$-cusps) discoid $\mathsf{d}_{3}=\{z=(x,y)\in\mathbb{C}\mid -z^2\overline{z}^2+4z^3+\overline{z}^3-18z\overline{z}+27\geq 0\}$
bounded by the deltoid curve $\mathsf{d}=\{z=2\exp(\mathsf{i} t)+\exp(-2\mathsf{i} t)\mid t\in[0,2\pi[\}$
(also called Steiner's hypocycloid):
\begin{gather}\label{eq:deltoid}
\begin{tikzpicture}[anchorbase, scale=.6, tinynodes]
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (0,-3) to (0,3);
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (-3,0) to (3,0);
\draw[thick, white, fill=mygreen, opacity=.2] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\draw[thin, densely dotted, ->, >=stealth] (-3.5,0)
to (-3.35,0) node [above] {$-3$}
to (3.2,0) node [above] {$3$}
to (3.5,0) node[right] {$x$};
\draw[thin, densely dotted, ->, >=stealth] (0,-3.5)
to (0,-3.2) node [right] {$-3$}
to (0,3.2) node [right] {$3$}
to (0,3.5) node[above] {$y$};
\draw[thick] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\node at (-2,3) {\scalebox{.85}{$3\exp(2\pi\mathsf{i}\neatfrac{1}{3})$}};
\node at (-2,-3) {\scalebox{.85}{$3\exp(2\pi\mathsf{i}\neatfrac{2}{3})$}};
\node at (3,3) {$\mathbb{C}$};
\node at (5.6,1.75) {\scalebox{.85}{$\mathsf{d}= \begin{gathered} \{z=2\exp(\mathsf{i} t)+\exp(-2\mathsf{i} t)
\\
\mid t\in[0,2\pi[
\} \end{gathered}$}};
\draw[thin, ->] (2.9,1.75) to [out=180, in=45] (.9,.75);
\node at (-4.5,1.75) {\scalebox{.85}{$\begin{gathered}
\mathsf{d}_{3}=\{z=(x,y)\in\mathbb{C} \\
\mid -z^2\overline{z}^2{+}4z^3{+}\\
\overline{z}^3{-}18z\overline{z}{+}27{\geq} 0\}
\end{gathered}$}};
\draw[thin, ->] (-3,1.75) to [out=0, in=180] (-.5,.75);
\draw[thin, densely dashed, opacity=.5, myorange] (0,0) circle (3cm);
\node at (0,-4) {The disciod $\mathsf{d}_{3}=\mathsf{d}_{3}(\mathfrak{sl}_{3})$ bounded by the deltoid curve $\mathsf{d}$};
\end{tikzpicture}
\end{gather}
The discoid $\mathsf{d}_{3}$ has a $\Z/3\Z$-symmetry,
given by $(z,\overline{z})\mapsto (\mathtt{\zeta}^{\pm 1}z,\mathtt{\zeta}^{\mp 1}\overline{z})$, and
its singularities are the primitive, complex third roots of unity multiplied by $3$.
For any $a,b\in\mathbb{N}$, the zeros of $E^-_{a,b}$ are
known, cf. \cite[Section 7.1]{EP}. However, let us give an independent proof.
\newline
\noindent\textit{\setword{`\fullref{lemma:level-vanishing}.Claim'}{claim-section-sl3}.}
Let $a,b\in\mathbb{N},a+b=s\geq 2$.
Then $E^-_{a,b}(\sigma,\tau)=0$ if
\begin{equation}\label{eq:zeros}
\left(\sigma,\tau\right)=
\left(\neatfrac{2\pi(2c+d+3)}{3s},\neatfrac{2\pi (c+2d+3)}{3s}\right),
\quad\text{with}\; c,d\in\mathbb{N}.
\end{equation}
\noindent\textit{Proof of \ref{claim-section-sl3}.}
We have
\[
\mathtt{\zeta}^{\sneatfrac{a(2c+d)+b(c+2d)}{s}}
=
\mathtt{\zeta}^{\sneatfrac{a(2c+d)+b(c+2d)-3(a+b)(c+d)}{s}}
=
\mathtt{\zeta}^{\sneatfrac{-b(2c+d)-a(c+2d)}{s}},
\]
using that $a+b=s$, $(2c+d)+(c+2d)=3(c+d)$ and $\mathtt{\zeta}^3=1$. Similarly, we obtain
\begin{gather*}
\mathtt{\zeta}^{\sneatfrac{(a+b)(2c+d+3)-b(c+2d+3)}{s}}
=\mathtt{\zeta}^{\sneatfrac{(a+b)(2c+d+3)-b(c+2d+3)+3(a+b)(c+d+2)}{s}}
\\
=\mathtt{\zeta}^{\sneatfrac{2(a+b)(2c+d+3)+a(c+2d+3)}{s}}
=\mathtt{\zeta}^{\sneatfrac{-(a+b)(2c+d+3)+a(c+2d+3)}{s}}.
\end{gather*}
\begin{gather*}
\mathtt{\zeta}^{\sneatfrac{b(2c+d+3)-(a+b)(c+2d+3)}{s}}
=\mathtt{\zeta}^{\sneatfrac{b(2c+d+3)-(a+b)(c+2d+3)-3(a+b)(c+d+2)}{s}}
\\
=\mathtt{\zeta}^{\sneatfrac{-a(2c+d+3)-2(a+b)(c+2d+3)}{s}}
=\mathtt{\zeta}^{\sneatfrac{-a(2c+d+3)+(a+b)(c+2d+3)}{s}}.
\end{gather*}
This gives $E^-_{a,b}(\sigma,\tau)=0$ for
$(\sigma,\tau)$ as in \eqref{eq:zeros}, and completes the proof of \ref{claim-section-sl3}.
\medskip
Next, for any $m,n$, we have
\[
\pxy[Z(\sigma,\tau),\overline{Z(\sigma,\tau)}]{m,n}
=
E^-_{m+1,n+1}(\sigma,\tau)/E^-_{1,1}(\sigma,\tau).
\]
Let $(\sigma,\tau)$ be as
in \eqref{eq:zeros} with $a=m+1$ and
$b=n+1$, and assume $(\sigma,\tau)$ is in
the interior of $D$.
Then we have $\pxy[Z(\sigma,\tau),\overline{Z(\sigma,\tau)}]{m,n}=0$
by \ref{claim-section-sl3}.
To make the connection with our notation from before,
let $m+n=e+1$ and $k=c,l=d$. By the above, for all
\begin{equation}\label{eq:zeros2}
\left(\sigma,\tau\right)=
\left(\neatfrac{2\pi(2k+l+3)}{3(e+3)},\neatfrac{2\pi (k+2l+3)}{3(e+3)}\right)
\quad\text{with}\; 0\leq k+l\leq e,
\end{equation}
we have $\pxy[Z(\sigma,\tau),\overline{Z(\sigma,\tau)}]{m,n}=0$.
Thus, $\#\vanset{e}\geq\#\left\{(k,l)\in X^+\mid 0\leq k+l\leq e \right\} = t_e$.
Since we already know that $\#\vanset{e}\leq t_e$, equality must hold.
\end{proof}
\begin{remark}\label{remark:level-vanishing}
Applying $Z$ to \eqref{eq:zeros2} gives the
precise form of the elements of $\vanset{e}$:
\[
\vanset{e}=
\left\{
(\alpha,\beta)\in\mathbb{C}^2
\mid
\alpha=Z(\sigma,\tau),
\beta=\overline{Z(\sigma,\tau)}
\right\}
\]
for $(\sigma,\tau)$ as in \eqref{eq:zeros2}. For a fixed level, the common
roots of the polynomials $\pxy{m,n}$ all lie in the interior of the
discoid from \eqref{eq:deltoid}.
\end{remark}
\begin{example}\label{example:plot-zeros}
The polynomials for $e=1,2,3$
are given in \fullref{example:sl3-polys}.
The first (or $\varstuff{X}$) entries of their common zeros are
\begin{gather*}
e=1\colon
\{
\text{roots of }
(X-1)(X^2+X+1)
\}
,
\\
e=2\colon
\{
\text{roots of }
(X^2-X-1)(X^4+X^3+2X^2-X+1)
\}
,
\\
e=3\colon
\{
\text{roots of }
X(X-2)(X^2 + 2X + 4)(X^6 - X^3 + 1)
\}
.
\end{gather*}
The second (or $\varstuff{Y}$)
entries are the complex conjugates.
Plotted to $\mathbb{C}$ one gets
\[
\begin{tikzpicture}[anchorbase, scale=.6, tinynodes]
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (0,-3) to (0,3);
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (-3,0) to (3,0);
\draw[thick, white, fill=mygreen, opacity=.2] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\draw[thin, densely dotted, ->, >=stealth] (-3.5,0)
to (-3.35,0) node [above] {$-3$}
to (3.2,0) node [above] {$3$}
to (3.5,0) node[right] {$x$};
\draw[thin, densely dotted, ->, >=stealth] (0,-3.5)
to (0,-3.2) node [right] {$-3$}
to (0,3.2) node [right] {$3$}
to (0,3.5) node[above] {$y$};
\draw[thick] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\node at (3,3) {$\mathbb{C}$};
\node[myorange] at (1,0) {$\bullet$};
\node[myorange] at (-.5,.87) {$\bullet$};
\node[myorange] at (-.5,-.87) {$\bullet$};
\draw[very thin, densely dashed, myorange] (1,0) to (-.5,.87) to (-.5,-.87) to (1,0);
\node[mypurple] at (1.62,0) {$\bullet$};
\node[mypurple] at (-.62,0) {$\bullet$};
\node[mypurple] at (-.81,1.4) {$\bullet$};
\node[mypurple] at (-.81,-1.4) {$\bullet$};
\node[mypurple] at (.31,.54) {$\bullet$};
\node[mypurple] at (.31,-.54) {$\bullet$};
\draw[very thin, densely dashed, mypurple] (1.62,0) to (.31,.54) to (-.81,1.4) to (-.62,0) to (-.81,-1.4) to (.31,-.54) to (1.62,0);
\node[myblue] at (0,0) {$\bullet$};
\node[myblue] at (2,0) {$\bullet$};
\node[myblue] at (-1,1.73) {$\bullet$};
\node[myblue] at (-1,-1.73) {$\bullet$};
\node[myblue] at (-.77,.64) {$\bullet$};
\node[myblue] at (-.77,-.64) {$\bullet$};
\node[myblue] at (-.17,.98) {$\bullet$};
\node[myblue] at (-.17,-.98) {$\bullet$};
\node[myblue] at (.94,.34) {$\bullet$};
\node[myblue] at (.94,-.34) {$\bullet$};
\draw[very thin, densely dashed, myblue] (2,0) to (.94,.34) to (-.17,.98) to (-1,1.73) to (-.77,.64) to (-.77,-.64) to (-1,-1.73) to (-.17,-.98) to (.94,-.34) to (2,0);
\node[myorange] at (2.75,2) {inner is $e=1$};
\node[mypurple] at (2.75,1.5) {middle is $e=2$};
\node[myblue] at (2.75,1) {outer is $e=3$};
\end{tikzpicture}
\]
Letting $e\gg 0$, these approximate the deltoid curve
$\mathsf{d}$ (layer-wise).
\end{example}
\section{Trihedral \texorpdfstring{$2$}{2}-representation theory}\label{sec:2-reps}
Keeping all notations from the previous sections,
we are now going to explain the $2$-re\-presentation theory of
the trihedral Soergel bimodules. Again, we have collected the
analogies to the dihedral case at the end in \fullref{subsec:dihedral-group-cat}.
\medskip
\noindent\textbf{Background.}
\medskip
Let us briefly recall some terminology and results from $2$-re\-presentation theory as in e.g.
\cite{MM3} or \cite{MM5}, where we also need the graded setup as in \cite[Section 3]{MT1}.
\subsubsection{\texorpdfstring{$\mathbb{N}_{[\varstuff{v}]}$}{Nv}-representation theory}\label{subsec:decat-story-a}
We start with the decategorified picture. Recall that $\varstuff{v}$
denotes a generic parameter, $\N_{\intvpar}=\mathbb{N}[\varstuff{v},\varstuff{v}^{-1}]$,
$\Z_{\intvpar}=\mathbb{Z}[\varstuff{v},\varstuff{v}^{-1}]$ and $\C_{\vpar}=\mathbb{C}(\varstuff{v})$.
Following various authors, see e.g. \cite[Section 1]{EK1}, \cite[Chapter 3]{EGNO} or \cite{KM1}
and the references therein, we define:
\begin{definition}\label{definition:two-bases-integral}
A pair $(\algstuff{P},\posbasis)$ of an associative, unital
($\C_{\vpar}$-)algebra $\algstuff{P}$
and a finite basis $\posbasis$ with $1\in\posbasis$
is called a $\N_{\intvpar}$-algebra
if
\[
\algstuff{x}\algstuff{y}\in\N_{\intvpar}\posbasis
\]
holds for all $\algstuff{x},\algstuff{y}\in\posbasis$.
\end{definition}
\begin{definition}\label{definition:pos-integral-modules}
Let $(\algstuff{M},\posmodbasis)$ be a pair
of a (left) $(\algstuff{P},\posbasis)$-representation $\algstuff{M}$ and a choice of a finite basis
$\posmodbasis$ for it. We call $(\algstuff{M},\posmodbasis)$
a $\N_{\intvpar}$-representation if
\[
\algstuff{M}(\algstuff{z})\algstuff{m}\in\N_{\intvpar}\posmodbasis
\]
holds for all $\algstuff{z}\in\posbasis,\algstuff{m}\in\posmodbasis$.
\end{definition}
\begin{example}\label{example:transitive}
These $\N_{\intvpar}$-algebras and $\N_{\intvpar}$-representations
arise naturally as the decategorification
of $2$-categories and $2$-representations, which will be recalled in the next section.
\end{example}
Abusing notation, we sometimes write $\algstuff{P}$ instead of
$(\algstuff{P},\posbasis)$ and $\algstuff{M}$ instead of $(\algstuff{M},\posmodbasis)$.
\begin{definition}\label{definition:eq-integral-modules}
Two $\N_{\intvpar}$-representations $\algstuff{M},\algstuff{M}^{\prime}$ are
$\N_{\intvpar}$-equivalent, denoted by
$\algstuff{M}\cong_{+}\algstuff{M}^{\prime}$, if there exist a bijection
$\posmodbasis\to\posmodbasis[\algstuff{M}^{\prime}]$
such that the induced linear map $\algstuff{M}\to\algstuff{M}^{\prime}$
is an isomorphism of $\algstuff{P}$-representations.
\end{definition}
\begin{example}\label{example:eq-integral-modules}
$\algstuff{M}\cong_{+}\algstuff{M}^{\prime}$ implies
$\algstuff{M}\cong\algstuff{M}^{\prime}$ (meaning that the are
isomorphic as $\algstuff{P}$-representa\-tions over $\C_{\vpar}$),
but not vice versa:
First of all, $\algstuff{M}$ might be isomorphic over
$\C_{\vpar}$ to a $\algstuff{P}$-representation $\algstuff{M}^{\prime}$ that is
not a $\N_{\intvpar}$-re\-presentation. For example, consider the
($\C_{\vpar}$-)group algebra of any finite group with
its basis given by the group elements.
Its regular representation is a $\N_{\intvpar}$-representation on this basis, and over $\C_{\vpar}$ this
representation decomposes into simple modules.
However, most simple modules are not $\N_{\intvpar}$-representations
and the decomposition can usually not be obtained
via base change matrices with entries from $\N_{\intvpar}$.
Secondly, even if $\algstuff{M}\cong\algstuff{M}^{\prime}$ are
two isomorphic $\N_{\intvpar}$-representations,
they may not be $\N_{\intvpar}$-equivalent. For example, the dihedral
Hecke algebra of type $\typei[12]$ has two
$\N_{\intvpar}$-representations,
associated to the type $\mathsf{E}_6$ Dynkin, which are
isomorphic over $\C_{\vpar}$ but not $\N_{\intvpar}$-equivalent
(c.f. \cite[Theorem II(iii)]{MT1}).
\end{example}
\subsubsection{Cells}\label{subsec:background-decat}
For any $\N_{\intvpar}$-algebra $\algstuff{P}$ one can define
cell theory as in \fullref{definition:cells-first}, e.g.
$\algstuff{x}\geq_{\Lcell}\algstuff{y}$ for $\algstuff{x},\algstuff{y}\in\posbasis$
if there exists an element
$\algstuff{z}\in\posbasis$ such that $\algstuff{x}$ appears as a summand of
$\algstuff{z}\algstuff{y}$, when the latter is written as a linear combination
of elements in $\posbasis$. We
hence obtain (left, right and two-sided) cells $\mathsf{L}$,
$\mathsf{R}$ and $\mathsf{J}$, and we can write $\mathsf{L}^{\prime}\geq_{\Lcell}\mathsf{L}$ etc.
See also e.g. \cite{KM1}
(incorporating $\varstuff{v}$)
for details. The same notions be can defined for any
$\N_{\intvpar}$-representation $\algstuff{M}$, e.g.
$\algstuff{m}\geq_{\Lcell}\algstuff{n}$ for
$\algstuff{m},\algstuff{n}\in\posmodbasis$ if there exists some $\algstuff{z}\in\posbasis$ such that
$\algstuff{m}$ appears in $\algstuff{M}(\algstuff{z})\algstuff{n}$ with non-zero coefficient
when written in terms of $\posmodbasis$.
\begin{definition}\label{definition:two-bases-integral-modules}
We call a $\N_{\intvpar}$-representation
$\algstuff{M}$ transitive
if all basis elements belong to the same $\sim_{\Lcell}$ equivalence class.
\end{definition}
\begin{remark}\label{remark:transitive}
Consider the graph with vertices given by
$\posmodbasis$ and with an oriented edge from $\algstuff{n}$ to
$\algstuff{m}$ whenever $\algstuff{m}\geq_{\Lcell}\algstuff{n}$. Transitivity of $\algstuff{M}$ means that this graph
is strongly connected.
\end{remark}
Similarly, we can also define the notion of a transitive $\Z_{\intvpar}$-representation associated to a strongly
connected graph. (Note that $\algstuff{m}\geq_{\Lcell}\algstuff{n}$ also makes sense over $\Z_{\intvpar}$.)
\begin{definition}\label{definition:cell-module}
Fix $\mathsf{L}$. Let $\amod[M](\geq_{\Lcell})$, respectively
$\amod[M](>_{\Lcell})$, be the $\N_{\intvpar}$-representations spanned by
all $\algstuff{x}\in\posbasis$ in the union of all left cells $\mathsf{L}^{\prime}\geq_{\Lcell}\mathsf{L}$,
respectively $\mathsf{L}^{\prime}>_{\Lcell}\mathsf{L}$. (These are well-defined by
\cite[Proposition 1]{KM1}.) We call
$\amod[C]_{\mathsf{L}}=\amod[M](\geq_{\Lcell})/\amod[M](>_{\Lcell})$
the (left) cell module for $\mathsf{L}$.
\end{definition}
By definition, all cell modules are transitive $\N_{\intvpar}$-representations.
\begin{example}\label{example:cell-module-sym-group}
Coming back to \fullref{example:eq-integral-modules}: There is only one
left (right, two-sided) cell for the group algebra of a finite group.
The associated cell module is the regular representation.
However, on a different basis this might change considerably:
The Hecke algebras for (finite) Coxeter groups
are $\N_{\intvpar}$-algebras, where the
KL basis plays the role of the basis $\posbasis$, see \cite{KaLu}. Their
cell modules are Kazhdan--Lusztig's original cell modules.
In the case of the symmetric group, these cell modules are the simple modules,
but in general cell modules are not simple (since most simples are
not $\N_{\intvpar}$-representations).
\end{example}
\begin{example}\label{example:cell-module}
Decategorifications of cell $2$-representations, which will be recalled
below, are key examples of cell modules.
\end{example}
Given any cell module $\amod[C]_{\mathsf{L}}$, the results in \cite[Section 8]{KM1} show that there
exists a unique, maximal two-sided cell, called apex, which does not annihilate $\amod[C]_{\mathsf{L}}$.
The same is true for general transitive $\N_{\intvpar}$-representations by \cite[Section 9.2]{KM1}.
Thus, we can restrict the study of transitive $\N_{\intvpar}$-representations to a given apex.
\subsubsection{\texorpdfstring{$2$}{2}-representations of finitary \texorpdfstring{$2$}{2}-categories}\label{subsec:background-1}
Let $\algstuff{R}$ be a ring. An additive, $\algstuff{R}$-linear, ($\mathbb{Z}$-)graded $2$-category $\twocatstuff{C}$
(with the same grading conventions as in \fullref{convention:grading}),
which is idempotent complete and Krull--Schmidt, is called
graded finitary if:
\smallskip
\begin{enumerate}[label=$\blacktriangleright$]
\setlength\itemsep{.15cm}
\item It has finitely many objects, and all identity $1$-morphisms are indecomposable.
\item The $2$-hom spaces are free of finite $\algstuff{R}$-rank in each degree, and their grading is bounded from below.
\item Consider the $2$-subcategory of $\twocatstuff{C}$
having the same objects and $1$-morphisms, but only degree-preserving $2$-morphisms. Its
split Grothendieck group is a free $\Z_{\intvpar}$-module, with $\varstuff{v}$ corresponding to
the grading shift, which we assume to be of finite $\Z_{\intvpar}$-rank.
\end{enumerate}
\smallskip
(Note that the last point above implies that a graded finitary $2$-category has only
finitely many equivalence classes of indecomposable $1$-morphisms up to grading shift.)
Similarly, a graded locally finitary $2$-category is as above, but relaxing the
condition on the Grothendieck group by requiring it to be of countable $\Z_{\intvpar}$-rank.
We also use graded finitary categories (having graded
hom-spaces which are free of finite $\algstuff{R}$-rank), which are
the objects of a $2$-category $\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}}$ with
$1$-morphisms being additive, $\algstuff{R}$-linear, degree-preserving functors and
$2$-morphisms being homogeneous natural
transformations of degree-zero.
Let $(\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}})^{\star}$ denote the
$2$-category obtained from $\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}}$ by adding formal shifts
to the $1$-morphisms. Its $2$-hom spaces are given by
\[
\twocatstuff{H}\mathrm{om}_{(\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}})^{\star}}(\obstuff{i},\obstuff{j})
=
{\textstyle \bigoplus_{s\in\mathbb{Z}}}\,
\twocatstuff{H}\mathrm{om}_{\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}}}(\obstuff{i}\{s\},\obstuff{j}).
\]
\begin{example}\label{example:graded-finitary}
All $2$-categories in \fullref{sec:A2-diagrams} become graded
(locally) finitary after taking their Karoubi envelope.
\end{example}
\begin{example}\label{example:graded-finitary-2}
Let $\algstuff{B}$ be a graded $\algstuff{R}$-algebra which is
free of finite $\algstuff{R}$-rank.
The category of free, finite $\algstuff{R}$-rank, graded (left) $\algstuff{B}$-representations
is a prototypical object of $\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}}$. For example,
the graded representation categories of the quiver algebras $\zig[e]$ in \fullref{subsec:quiver} below
are objects of $\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}}$.
\end{example}
A graded finitary $2$-representation of $\twocatstuff{C}$ is an additive, $\algstuff{R}$-linear
$2$-functor
\[
\twocatstuff{M}\colon
\twocatstuff{C}
\to
(\twocatstuff{A}^{\mathrm{f}}_{\mathrm{gr}})^{\star}
\]
which is degree-preserving and commutes with shifts as in \cite[Definition 3.4]{MT1}.
\begin{example}\label{example:graded-finitary-3}
The principal $2$-representation
$\twocatstuff{P}_{\obstuff{i}}=\twocatstuff{C}(\obstuff{i},\underline{\phantom{a}})$,
where $\obstuff{i}$ is an object of $\twocatstuff{C}$, is a graded finitary
$2$-representation of $\twocatstuff{C}$.
\end{example}
Graded finitary $2$-representations of $\twocatstuff{C}$ form a
graded $2$-category (in the sense of \fullref{convention:grading}),
see \cite{MM3} for details, which can be adapted to the graded setting. In particular, there exists
a well-defined notion of equivalence between such $2$-representations.
For simplicity, we say \textit{$2$-representation} instead of
\textit{graded finitary $2$-representation} etc. from now on, i.e. we omit the
\textit{graded finitary}.
\subsubsection{\texorpdfstring{$2$}{2}-cells}\label{subsec:background-2}
As in the case of $\N_{\intvpar}$-algebras, one can define cells and cell $2$-representations of
finitary $2$-categories: Let $\morstuff{X}$ and $\morstuff{Y}$ be
indecomposable $1$-morphisms in a finitary $2$-category $\twocatstuff{C}$.
Set $\morstuff{X}\geq_{\Lcell}\morstuff{Y}$ if $\morstuff{X}$ is isomorphic
to a direct summand of $\morstuff{Z}\morstuff{Y}$, up to a degree shift, for some
indecomposable $1$-morphism $\morstuff{Z}$. Similarly one defines $\geq_{\Rcell}$ and $\geq_{\Tcell}$.
The equivalence classes for these are
called the respective cells, denoted by $\mathsf{L}$, $\mathsf{R}$ or $\mathsf{J}$.
All these notions can be defined in a similar way
for $2$-representations as well.
A finitary $2$-representation $\twocatstuff{M}$ is transitive (see \cite[Section 3.1]{MM5},
or \cite[Definition 3.6]{MT1} in the graded setup), if $\twocatstuff{M}$ is
supported on one $\obstuff{i}\in\twocatstuff{C}$, and if all indecomposable objects
$\obstuff{O},\obstuff{P}\in\twocatstuff{M}(\obstuff{i})$ are in the same $\sim_{\Lcell}$-equivalence class. A
transitive $2$-representation is simple transitive, see \cite[Section 3.5]{MM5} (or \cite[Definition 3.6]{MT1}
in the graded setup), if it does not have any non-zero, proper $\twocatstuff{C}$-invariant ideals.
\begin{remark}\label{remark:graded-finitary-3}
By \cite[Section 4]{MM5}, any $2$-representation has a weak Jordan--H{\"o}lder series
with simple transitive subquotients, which are unique up to
permutation and equivalence. Therefore, it is natural to ask for the classification
of simple transitive $2$-representations. Moreover, by \cite[Section 3]{MM5}, any
transitive $2$-representation has a unique maximal $\twocatstuff{C}$-stable
ideal which one can quotient by to get
a simple transitive $2$-representation, called the simple transitive quotient.
\end{remark}
Every (graded) finitary $2$-category comes with a natural class of simple transitive $2$-representations:
\begin{definition}\label{definition:2-cell-module}
Fix $\mathsf{L}$.
Then there exists $\obstuff{i}\in\twocatstuff{C}$ such that all
$1$-morphisms in $\mathsf{L}$ start
at $\obstuff{i}$.
Let $\twocatstuff{M}(\geq_{\Lcell})$ be the $2$-representations of $\twocatstuff{C}$ spanned by
the additive closure of all indecomposable $1$-morphisms $\morstuff{F}$, in
$\coprod_{\obstuff{j}\in\twocatstuff{C}}\twocatstuff{P}_{\obstuff{i}}(\obstuff{j})$, which belong to the union of all
left cells $\mathsf{L}^{\prime}\geq_{\Lcell}\mathsf{L}$.
Let $\twocatstuff{Z}(\geq_{\Lcell})$ be the unique,
proper two-sided $2$-ideal in $\twocatstuff{M}(\geq_{\Lcell})$.
(All of this is well-defined by
\cite[Section 3.3 and Lemma 3]{MM5}.)
We call $\twocatstuff{C}_{\mathsf{L}}=\twocatstuff{M}(\geq_{\Lcell})/\twocatstuff{Z}(\geq_{\Lcell})$
the cell $2$-representation for $\mathsf{L}$.
\end{definition}
Note that cell $2$-representations are always simple transitive.
\begin{example}\label{example:simple-cell}
In case of Soergel bimodules for the symmetric group,
these exhaust all simple transitive $2$-representations and categorify
the simple modules \cite{MM5}. However, both these facts are false in general, as
the example of dihedral Soergel bimodules shows, see e.g. \cite{KMMZ}, \cite{MT1}.
\end{example}
\begin{remark}\label{remark:might-be-bigger}
On the decategorified level, the cell representation
is obtained as the quotient of $\algstuff{M}(\geq_{\Lcell})$ by $\algstuff{M}(>_{\Lcell})$,
cf. \fullref{definition:cell-module}. On the level of $2$-representations, the proper maximal
two-sided $2$-ideal $\twocatstuff{Z}(\geq_{\Lcell})$ strictly
contains the two-sided $2$-ideal generated by the $2$-subrepresentation $\twocatstuff{M}(>_{\Lcell})$ in general.
\end{remark}
Again, there is a unique, maximal
two-sided cell, called $2$-apex,
which does not annihilate a given cell $2$-representation.
The same works for general transitive $2$-representations.
See \cite[Section 3.2]{CM1} for more details.
\subsubsection{(Co)algebra \texorpdfstring{$1$}{1}-morphisms}\label{subsec:background-3}
An algebra $1$-morphism in $\twocatstuff{C}$
is a triple $(\morstuff{A},\mu,\eta)$, where
$\morstuff{A}$ is a $1$-morphism and $\mu\colon\morstuff{A}\morstuff{A}\to\morstuff{A}$
and $\eta\colon\mathbbm{1}\to\morstuff{A}$ are $2$-morphisms satisfying
the usual axioms for the multiplication and unit of an
algebra.
Furthermore, there are compatible notions of module $1$-morphism
over an algebra $1$-morphism $\morstuff{A}$, and
of $2$-homomorphism between these. In this way, we get the $2$-categories
$\modtwocat{\twocatstuff{C}}(\morstuff{A})$
(or $(\morstuff{A})\modtwocat{\twocatstuff{C}}$)
of right (or left) $\morstuff{A}$-module $1$-morphisms in $\twocatstuff{C}$.
By post-composition $\modtwocat{\twocatstuff{C}}(\morstuff{A})$
becomes a left $2$-representation of $\twocatstuff{C}$. Similarly, by pre-composition
$(\morstuff{A})\modtwocat{\twocatstuff{C}}$
becomes a right $2$-representation of $\twocatstuff{C}$.
One defines coalgebra $1$-morphisms $(\morstuff{C},\delta,\varepsilon)$ in $\twocatstuff{C}$
and their respective comodule $2$-categories, which are also $2$-representations of $\twocatstuff{C}$, dually.
Finally, there are also compatible notions of bimodule $1$-morphism over an algebra $1$-morphism and $2$-homomorphism
between bimodule $1$-morphisms. By definition,
a Frobenius $1$-morphism $\morstuff{F}$ in $\twocatstuff{C}$ is
an algebra $1$-morphism which is also a coalgebra $1$-morphism, such that
the comultiplication $2$-morphism is a $2$-homomorphism between $\morstuff{F}$--$\morstuff{F}$-bimodule $1$-morphisms.
We refer to \cite{MMMT1} or \cite[Chapter 7]{EGNO} for further details.
\begin{remark}\label{remark:alg-objects}
Suppose that $\twocatstuff{C}$ is additionally fiat (meaning that it
has a certain involution \cite[Section 2.4]{MM1}). Then
\cite[Theorem 9]{MMMT1} asserts that, for any simple transitive $2$-representation $\twocatstuff{M}$ of $\twocatstuff{C}$,
there exists a simple algebra $1$-morphism $\morstuff{A}$ in $\overline{\twocatstuff{C}}$
(the projective abelianization of $\twocatstuff{C}$, as introduced in \cite[Section 3.2]{MMMT1})
such that $\twocatstuff{M}$ is equivalent (as a $2$-representation of $\twocatstuff{C}$)
to the subcategory of projective objects of $\modtwocat{\overline{\twocatstuff{C}}}(\morstuff{A})$.
Hence, the classification of simple transitive $2$-representations of $\twocatstuff{C}$ is equivalent to
the classification of simple algebra $1$-morphisms in $\overline{\twocatstuff{C}}$.
Or, dually, to the classification of cosimple coalgebra $1$-morphisms in $\underline{\twocatstuff{C}}$, the
injective abelianization of $\twocatstuff{C}$.
The fiat $2$-categories $\twocatstuff{C}$ in this paper are special, because they are closely related to
semisimple $2$-categories by the quantum Satake correspondence, and
the simple algebra $1$-morphisms which we study
in this paper all belong to $\twocatstuff{C}$.
\end{remark}
\subsection{Decategorified story}\label{subsec:decat-story}
\subsubsection{Trihedral transitive \texorpdfstring{$\mathbb{N}_{[\varstuff{v}]}$}{Nv}-representations}\label{subsec:Z-reps}
\makeautorefname{subsection}{Sections}
From
\fullref{subsec:definition}
and \ref{subsec:quotient-algebra},
in particular
the connection to the representation theory of $\mathfrak{sl}_{3}$,
the following is evident.
\makeautorefname{subsection}{Section}
\makeautorefname{proposition}{Propositions}
\begin{propositionqed}\label{proposition:two-bases-integral}
The trihedral Hecke algebras are $\N_{\intvpar}$-algebras, i.e.
for the basis $\basisC$ and $\basisC[e]$ from \fullref{proposition:two-bases}
an \ref{proposition:dimension} we have
\[
\algstuff{x}\algstuff{y}\in\N_{\intvpar}\basisC
\quad\text{and}\quad
\algstuff{x}^{\prime}\algstuff{y}^{\prime}\in\N_{\intvpar}\basisC[e]
\]
for all $\algstuff{x},\algstuff{y}\in\basisC$ and
$\algstuff{x}^{\prime},\algstuff{y}^{\prime}\in\basisC[e]$.
The same holds for the left colored KL bases.
\end{propositionqed}
\makeautorefname{proposition}{Proposition}
This is our starting point for studying $\N_{\intvpar}$-representations of the trihedral
Hecke algebras. From now on, we fix the right colored KL bases for $\subquo$ and $\subquo[e]$,
as in \fullref{proposition:two-bases-integral}.
\begin{example}\label{example:int-valued-0}
Most of the three-dimensional $\subquo[e]$-representations
in \eqref{eq:the-simples} are not $\N_{\intvpar}$-re\-presentations
(for any choice of basis).
For $e>1$, the one-dimensional representations $\algstuff{M}_{\vfrac{3},0,0}$, $\algstuff{M}_{0,\vfrac{3},0}$ and $\algstuff{M}_{0,0,\vfrac{3}}$
are also not $\N_{\intvpar}$-representations, e.g.
by \fullref{example-KL-combinatorics-2}, the action of $\RKLg{1,1}$ on
$\algstuff{M}_{\vfrac{3},0,0}$ is given by
\[
\RKLg{1,1}=\vnumber{2}^{-2}\theta_{\color{mygreen}g}\theta_{\color{mypurple}p}\theta_{\color{mygreen}g}
-\,\theta_{\color{mygreen}g}
\mapsto
-\vfrac{3}.
\]
Thus, $\algstuff{M}_{\vfrac{3},0,0}$ is not an $\N_{\intvpar}$-representation.
\end{example}
Our next goal is to define
several families of
$\N_{\intvpar}$-representations of the trihedral
Hecke algebras.
Recall from \fullref{proposition:cells} and \fullref{corollary:cells} that
the trihedral algebras have one trivial and one non-trivial two-sided
cell, both of which can be the apex of a transitive $\N_{\intvpar}$-representation.
For the trivial cell there is only one such representation:
\begin{example}\label{example:int-valued}
The simple $\algstuff{M}_{0,0,0}$
(cf. \eqref{eq:the-simples})
is a transitive $\N_{\intvpar}$-re\-pre\-sen\-ta\-tion of $\subquo[\infty]$, which also descends to $\subquo[e]$ for any $e$,
and its apex is the trivial cell. By \eqref{eq:the-one-dims} and \fullref{example:int-valued-0}, there are
no other transitive $\N_{\intvpar}$-representations whose apex is the
trivial cell.
\end{example}
From now on we will only consider transitive $\N_{\intvpar}$-representations
whose apex is the unique, non-trivial two-sided cell.
For this purpose, we consider tricolored graphs, denoted by $\boldsymbol{\Gamma}$ etc., fixing certain conventions as follows.
\subsubsection{Graph-theoretic recollections}\label{subsec:three-colored-graphs}
For us a graph $\boldsymbol{\Gamma}$ is an undirected, connected, finite graph without loops,
but possibly with multiple edges. We will also need
graphs with directed edges and we indicate these by adding the superscript ${\underline{\phantom{a}}}^\varstuff{X}$
or ${\underline{\phantom{a}}}^\varstuff{Y}$.
We call $\boldsymbol{\Gamma}=(\boldsymbol{\Gamma},V=\{{\color{mygreen}G},{\color{myorange}O},{\color{mypurple}P}\},E=\{{\color{myblue}B},{\color{myred}R},{\color{myyellow}Y}\})$
tricolored, with colors ${\color{mygreen}g},{\color{myorange}o},{\color{mypurple}p}$, if $V$ and $E$ can be partitioned into three disjoint sets
${\color{mygreen}G},{\color{myorange}O},{\color{mypurple}P}$ and ${\color{myblue}B},{\color{myred}R},{\color{myyellow}Y}$ such that
\[
\left(
\begin{gathered}
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) to (3,0);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\end{tikzpicture}
\in{\color{myyellow}Y}\Rightarrow
\\
\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}\in{\color{mygreen}G}\text{ and }\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}\in{\color{myorange}O}
\end{gathered}
\right),
\quad\quad
\left(
\begin{gathered}
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, densely dashed, myred] (0,0) to (3,0);
\node at (0,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (3,0) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\in{\color{myred}R}\Rightarrow
\\
\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}\in{\color{myorange}O}\text{ and }\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}\in{\color{mypurple}P}
\end{gathered}
\right),
\quad\quad
\left(
\begin{gathered}
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, densely dotted, myblue] (0,0) to (3,0);
\node at (0,0) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (3,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\end{tikzpicture}
\in{\color{myblue}B}\Rightarrow
\\
\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}\in{\color{mypurple}P}\text{ and }\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}\in{\color{mygreen}G}
\end{gathered}
\right).
\]
(We usually denote a tricolored graph simply by $\boldsymbol{\Gamma}$, suppressing the tricoloring.)
The vertices of any tricolored graph $\boldsymbol{\Gamma}$ can be ordered
such that the adjacency matrix $A(\boldsymbol{\Gamma})$ is of the following form.
\begin{gather}\label{eq:ad-matrix}
A(\boldsymbol{\Gamma})=\!\!
\raisebox{.25cm}{$\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=.5em, column
sep=.5em, text height=.5ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{a}
\& {\color{mygreen}G}
\& {\color{myorange}O}
\& {\color{mypurple}P}
\\
{\color{mygreen}G}
\& 0
\& \hspace*{.01cm} A^{\mathrm{T}}\hspace*{.01cm}
\& C
\\
{\color{myorange}O}
\& A
\& 0
\& B^{\mathrm{T}}
\\
{\color{mypurple}P}
\& \,C^{\mathrm{T}}
\& \,\,B\,\,
\& 0
\\
};
\draw[densely dotted] (m-2-2.south west) to (m-2-4.south east);
\draw[densely dotted] (m-3-2.south west) to ($(m-3-4.south east)+(-.05,0)$);
\draw[densely dotted] ($(m-2-3.north west)+(0,.15)$) to (m-4-3.south west);
\draw[densely dotted] ($(m-2-3.north east)+(0,.15)$) to (m-4-3.south east);
\draw[thick] ($(m-2-2.north west)+(-.075,.15)$) to [out=255, in=90] ($(m-2-2.north west)+(-.2,-1.05)$) to [out=270, in=105] ($(m-2-2.north west)+(-.075,-2.25)$);
\draw[thick] ($(m-2-4.north east)+(.05,.15)$) to [out=285, in=90] ($(m-2-4.north east)+(.175,-1.05)$) to [out=270, in=75] ($(m-2-4.north east)+(.05,-2.25)$);
\end{tikzpicture}$}
,
\quad\quad
A(\Gg^{\varstuff{X}})=A(\Gg^{\varstuff{Y}})^{\mathrm{T}}=\!\!
\raisebox{.25cm}{$
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=.5em, column
sep=.5em, text height=.5ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{a}
\& {\color{mygreen}G}
\& {\color{myorange}O}
\& {\color{mypurple}P}
\\
{\color{mygreen}G}
\& 0
\& \,\,\,0\,\,\,
\& C
\\
{\color{myorange}O}
\& A
\& 0
\& 0
\\
{\color{mypurple}P}
\& 0
\& \,\,B\,\,
\& 0
\\
};
\draw[densely dotted] (m-2-2.south west) to (m-2-4.south east);
\draw[densely dotted] (m-3-2.south west) to ($(m-3-4.south east)+(-.05,0)$);
\draw[densely dotted] ($(m-2-3.north west)+(0,.15)$) to (m-4-3.south west);
\draw[densely dotted] ($(m-2-3.north east)+(0,.15)$) to (m-4-3.south east);
\draw[thick] ($(m-2-2.north west)+(-.075,.15)$) to [out=255, in=90] ($(m-2-2.north west)+(-.2,-1.05)$) to [out=270, in=105] ($(m-2-2.north west)+(-.075,-2.25)$);
\draw[thick] ($(m-2-4.north east)+(.05,.15)$) to [out=285, in=90] ($(m-2-4.north east)+(.175,-1.05)$) to [out=270, in=75] ($(m-2-4.north east)+(.05,-2.25)$);
\end{tikzpicture}$}.
\end{gather}
Here $A,B,C$ are matrices with
entries in $\mathbb{N}$, encoding the connections
${\color{mygreen}G}\rightarrow{\color{myorange}O}$ (matrix $A$),
${\color{myorange}O}\rightarrow{\color{mypurple}P}$ (matrix $B$),
and ${\color{mypurple}P}\rightarrow{\color{mygreen}G}$ (matrix $C$).
We will always consider vertex-orderings of this form.
Moreover, $\boldsymbol{\Gamma}$ has two associated
directed graphs $\Gg^{\varstuff{X}}$ and $\Gg^{\varstuff{Y}}$
whose adjacency matrices are $A(\Gg^{\varstuff{X}})$
and $A(\Gg^{\varstuff{Y}})$ as in \eqref{eq:ad-matrix}.
They have the same vertex sets as $\boldsymbol{\Gamma}$,
but their edges are oriented according to \eqref{eq:color-tensor}.
We write $i\in\boldsymbol{\Gamma}$ ($i\in{\color{mygreen}G}$ etc.) meaning that
$i$ is a (${\color{mygreen}g}$-colored etc.) vertex of $\boldsymbol{\Gamma}$.
Furthermore, we denote by $S_{\boldsymbol{\Gamma}}$ the spectrum
of $\boldsymbol{\Gamma}$, i.e. the multiset of eigenvalues of $A(\boldsymbol{\Gamma})$, and we use similar notations for $\Gg^{\varstuff{X}}$ and $\Gg^{\varstuff{Y}}$.
\begin{example}\label{example:triangle0}
Our main examples of tricolored graphs are all displayed in \fullref{subsec:gen-D-list}.
Their spectra play an important role for us.
\end{example}
\begin{example}\label{example:triangle1}
The simplest examples, which are, however, fundamental for this paper, are the
generalized type $\mathsf{A}$ Dynkin diagrams, e.g.:
\begin{gather*}
\graphA{1}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, mygreen] {\tiny{\text 1}}
to (1,1) node[below, myorange] {\tiny{\text 1}};
\draw [thick, densely dotted, myblue] (0,0)
to (-1,1) node[below, mypurple] {\tiny{\text 1}};
\draw [thick, densely dashed, myred] (1,1) to (-1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,\quad\quad
\graphA{1}^{\varstuff{X}}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, directed=.55] (0,0) node[below, mygreen] {\tiny{\phantom{1}}} to (1,1);
\draw [thick, densely dotted, myblue, rdirected=.55] (0,0) to (-1,1);
\draw [thick, densely dashed, myred, directed=.55] (1,1) to (-1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,\quad\quad
\graphA{1}^{\varstuff{Y}}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, rdirected=.55] (0,0) node[below, mygreen] {\tiny{\phantom{1}}} to (1,1);
\draw [thick, densely dotted, myblue, directed=.55] (0,0) to (-1,1);
\draw [thick, densely dashed, myred, rdirected=.55] (1,1) to (-1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=1.65em, column
sep=1em, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&,font=\scriptsize] {
A=
\begin{pmatrix}
1\\
\end{pmatrix}
\&
B=
\begin{pmatrix}
1\\
\end{pmatrix}
\&
C=
\begin{pmatrix}
1\\
\end{pmatrix}
\\};
\end{tikzpicture}
\end{gather*}
\begin{gather*}
\graphA{2}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, mygreen] {\tiny{\text 1}}
to (1,1) node[below, myorange] {\tiny{\text 1}}
to (0,2) node[below, mygreen] {\tiny{\text 2}};
\draw [thick, myyellow] (0,2)
to (-2,2) node[below, myorange] {\tiny{\text 2}};
\draw [thick, densely dotted, myblue] (0,0)
to (-1,1) node[below, mypurple] {\tiny{\text 1}}
to (0,2);
\draw [thick, densely dotted, myblue] (0,2)
to (2,2) node[below, mypurple] {\tiny{\text 2}};
\draw [thick, densely dashed, myred] (1,1) to (-1,1) to (-2,2);
\draw [thick, densely dashed, myred] (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,\quad\quad
\graphA{2}^{\varstuff{X}}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, directed=.55] (0,0) node[below, mygreen] {\tiny{\phantom{1}}} to (1,1);
\draw [thick, myyellow, rdirected=.55] (1,1) to (0,2);
\draw [thick, myyellow, directed=.55] (0,2) to (-2,2);
\draw [thick, densely dotted, myblue, rdirected=.55] (0,0) to (-1,1);
\draw [thick, densely dotted, myblue, directed=.55] (-1,1) to (0,2);
\draw [thick, densely dotted, myblue, rdirected=.55] (0,2) to (2,2);
\draw [thick, densely dashed, myred, directed=.55] (1,1) to (-1,1);
\draw [thick, densely dashed, myred, rdirected=.55] (-1,1) to (-2,2);
\draw [thick, densely dashed, myred, rdirected=.55] (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,\quad\quad
\graphA{2}^{\varstuff{Y}}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, rdirected=.55] (0,0) node[below, mygreen] {\tiny{\phantom{1}}} to (1,1);
\draw [thick, myyellow, directed=.55] (1,1) to (0,2);
\draw [thick, myyellow, rdirected=.55] (0,2) to (-2,2);
\draw [thick, densely dotted, myblue, directed=.55] (0,0) to (-1,1);
\draw [thick, densely dotted, myblue, rdirected=.55] (-1,1) to (0,2);
\draw [thick, densely dotted, myblue, directed=.55] (0,2) to (2,2);
\draw [thick, densely dashed, myred, rdirected=.55] (1,1) to (-1,1);
\draw [thick, densely dashed, myred, directed=.55] (-1,1) to (-2,2);
\draw [thick, densely dashed, myred, directed=.55] (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=1.65em, column
sep=1em, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&,font=\scriptsize] {
A=
\begin{pmatrix}
1 & 1 \\
0 & 1 \\
\end{pmatrix}
\&
B=
\begin{pmatrix}
1 & 1\\
1 & 0\\
\end{pmatrix}
\&
C=
\begin{pmatrix}
1 & 0\\
1 & 1\\
\end{pmatrix}
\\};
\end{tikzpicture}
\end{gather*}
\begin{gather*}
\graphA{3}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, mygreen] {\tiny{\text 1}}
to (1,1) node[below, myorange] {\tiny{\text 1}}
to (0,2) node[below, mygreen] {\tiny{\text 2}}
to (1,3) node[below, myorange] {\tiny{\text 3}}
to (3,3) node[below, mygreen] {\tiny{\text 3}};
\draw [thick, myyellow] (0,2)
to (-2,2) node[below, myorange] {\tiny{\text 2}}
to (-3,3) node[below, mygreen] {\tiny{\text 4}};
\draw [thick, densely dotted, myblue] (0,0)
to (-1,1) node[below, mypurple] {\tiny{\text 1}}
to (0,2)
to (-1,3) node[below, mypurple] {\tiny{\text 3}}
to (-3,3);
\draw [thick, densely dotted, myblue] (0,2)
to (2,2) node[below, mypurple] {\tiny{\text 2}}
to (3,3);
\draw [thick, densely dashed, myred] (1,1) to (-1,1) to (-2,2) to (-1,3) to (1,3) to (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,\quad\quad
\graphA{3}^{\varstuff{X}}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, directed=.55] (0,0) node[below, mygreen] {\tiny{\phantom{1}}} to (1,1);
\draw [thick, myyellow, rdirected=.55] (1,1) to (0,2);
\draw [thick, myyellow, directed=.55] (0,2) to (1,3);
\draw [thick, myyellow, rdirected=.55] (1,3) to (3,3);
\draw [thick, myyellow, directed=.55] (0,2) to (-2,2);
\draw [thick, myyellow, rdirected=.55] (-2,2) to (-3,3);
\draw [thick, densely dotted, myblue, rdirected=.55] (0,0) to (-1,1);
\draw [thick, densely dotted, myblue, directed=.55] (-1,1) to (0,2);
\draw [thick, densely dotted, myblue, rdirected=.55] (0,2) to (-1,3);
\draw [thick, densely dotted, myblue, directed=.55] (-1,3) to (-3,3);
\draw [thick, densely dotted, myblue, rdirected=.55] (0,2) to (2,2);
\draw [thick, densely dotted, myblue, directed=.55] (2,2) to (3,3);
\draw [thick, densely dashed, myred, directed=.55] (1,1) to (-1,1);
\draw [thick, densely dashed, myred, rdirected=.55] (-1,1) to (-2,2);
\draw [thick, densely dashed, myred, directed=.55] (-2,2) to (-1,3);
\draw [thick, densely dashed, myred, rdirected=.55] (-1,3) to (1,3);
\draw [thick, densely dashed, myred, directed=.55] (1,3) to (2,2);
\draw [thick, densely dashed, myred, rdirected=.55] (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,\quad\quad
\graphA{3}^{\varstuff{Y}}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, rdirected=.55] (0,0) node[below, mygreen] {\tiny{\phantom{1}}} to (1,1);
\draw [thick, myyellow, directed=.55] (1,1) to (0,2);
\draw [thick, myyellow, rdirected=.55] (0,2) to (1,3);
\draw [thick, myyellow, directed=.55] (1,3) to (3,3);
\draw [thick, myyellow, rdirected=.55] (0,2) to (-2,2);
\draw [thick, myyellow, directed=.55] (-2,2) to (-3,3);
\draw [thick, densely dotted, myblue, directed=.55] (0,0) to (-1,1);
\draw [thick, densely dotted, myblue, rdirected=.55] (-1,1) to (0,2);
\draw [thick, densely dotted, myblue, directed=.55] (0,2) to (-1,3);
\draw [thick, densely dotted, myblue, rdirected=.55] (-1,3) to (-3,3);
\draw [thick, densely dotted, myblue, directed=.55] (0,2) to (2,2);
\draw [thick, densely dotted, myblue, rdirected=.55] (2,2) to (3,3);
\draw [thick, densely dashed, myred, rdirected=.55] (1,1) to (-1,1);
\draw [thick, densely dashed, myred, directed=.55] (-1,1) to (-2,2);
\draw [thick, densely dashed, myred, rdirected=.55] (-2,2) to (-1,3);
\draw [thick, densely dashed, myred, directed=.55] (-1,3) to (1,3);
\draw [thick, densely dashed, myred, rdirected=.55] (1,3) to (2,2);
\draw [thick, densely dashed, myred, directed=.55] (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=1.65em, column
sep=1em, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&,font=\scriptsize] {
A=
\begin{pmatrix}
1 & 1 & 0 & 0\\
0 & 1 & 0 & 1\\
0 & 1 & 1 & 0\\
\end{pmatrix}
\&
B=
\begin{pmatrix}
1 & 1 & 0\\
1 & 0 & 1\\
0 & 1 & 1\\
\end{pmatrix}
\&
C=
\begin{pmatrix}
1 & 0 & 0\\
1 & 1 & 1\\
0 & 1 & 0\\
0 & 0 & 1\\
\end{pmatrix}
\\};
\end{tikzpicture}
\end{gather*}
\vspace*{.1cm}
(The matrices $A,B,C$ are given with respect to the ordering of the vertices as indicated in the unoriented graphs.)
The vertices of these graphs can be identified with the cut-offs of the positive Weyl chamber
of $\mathfrak{sl}_{3}$, cf. \eqref{eq:weight-picture}, where e.g the vertex with label
$4$ in $\graphA{3}$ corresponds to the $\mathfrak{sl}_{3}$-weight $(0,3)$.
Moreover, the spectra of these graphs are:
\begin{gather*}
S_{\graphA{1}^{\varstuff{X}}}
=\left\{
\text{roots of }
(X-1)
(X^2+X+1)
\right\}
,
\\
S_{\graphA{2}^{\varstuff{X}}}
=
\left\{
\text{roots of }
(X^2-X-1)
(X^4+X^3+2X^2-X+1)
\right\},
\\
S_{\graphA{3}^{\varstuff{X}}}=
\left\{
\text{roots of }
X
(X-2)
(X^2 + 2X + 4)
(X^6 - X^3 + 1)
\right\}.
\end{gather*}
The reader should compare these to \fullref{example:plot-zeros}.
\end{example}
Next, recall that an oriented graph $\boldsymbol{\Gamma}^{\mathrm{or}}$ is called strongly connected,
if there is a path from $i$ to $j$ for any $i,j\in\boldsymbol{\Gamma}^{\mathrm{or}}$. Further,
we say that $\boldsymbol{\Gamma}^{\mathrm{or}}$
is quasi regular if, for all $i,j\in\boldsymbol{\Gamma}^{\mathrm{or}}$,
the number of two-step paths $i\rightarrow\underline{\phantom{a}}\leftarrow j$ going first with and then
against the orientation is the same as the number of two-step paths
$i\leftarrow\underline{\phantom{a}}\rightarrow j$ going first against and then
with the orientation.
\begin{example}\label{example:weakly-regular}
Recall that an oriented graph is called
weakly regular if the numbers of incoming and outgoing edges
agree at each vertex, counting $r$ parallel edges $r^2$ times
(e.g. a vertex with two incoming parallel edges must have two outgoing parallel edges or four outgoing single edges). By considering $i=j$, we
see that any quasi regular graph is weakly regular, with the latter being
a local condition which one easily checks.
(In particular, each vertex is of even degree.) However, the converse is
not true as e.g.
\begin{gather*}
\boldsymbol{\Gamma}^{\mathrm{or}}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, directed=.55] (0,0) to (1,1);
\draw [thick, myyellow, directed=.55] (0,0) to (1,-1);
\draw [thick, densely dotted, myblue, directed=.55] (-1,1) to (0,0);
\draw [thick, densely dotted, myblue, directed=.55] (-1,-1) to (0,0);
\draw [thick, densely dashed, myred, directed=.55] (1,1) to (-1,1);
\draw [thick, densely dashed, myred, directed=.55] (1,-1) to (-1,-1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,-1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,-1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\end{gather*}
is weakly regular, but not quasi regular.
\end{example}
By convention, we call $\boldsymbol{\Gamma}$ as above strongly connected, respectively quasi regular,
if $\Gg^{\varstuff{X}}$ and $\Gg^{\varstuff{Y}}$ are both strongly connected, respectively quasi regular.
\begin{definition}\label{definition:admissible}
A graph $\boldsymbol{\Gamma}$ is called admissible if it admits a tricoloring,
such that $\boldsymbol{\Gamma}$ is strongly connected and quasi regular.
\end{definition}
\begin{example}\label{example:triangle2}
All of our main examples from \fullref{subsec:gen-D-list}
are admissible.
\end{example}
\begin{lemma}\label{lemma:weakly-regular}
The matrices $A,B,C$ in \eqref{eq:ad-matrix}, which are blocks of $A(\boldsymbol{\Gamma})$, satisfy
\begin{gather}\label{eq:main-transposes}
A^{\mathrm{T}}A=
CC^{\mathrm{T}},
\quad\quad
AA^{\mathrm{T}}=
B^{\mathrm{T}}B,
\quad\quad
C^{\mathrm{T}}C=
BB^{\mathrm{T}}
\end{gather}
if and only if $\boldsymbol{\Gamma}$ is quasi regular.
\end{lemma}
In particular, $AA^{\mathrm{T}}$, $A^{\mathrm{T}}A$,
$BB^{\mathrm{T}}$, $B^{\mathrm{T}}B$,
$CC^{\mathrm{T}}$ and $C^{\mathrm{T}}C$ have the same non-zero eigenvalues
for any quasi regular graph $\boldsymbol{\Gamma}$.
\begin{proof}
Assume that $\boldsymbol{\Gamma}$ is quasi regular.
Then, in $\boldsymbol{\Gamma}^{\varstuff{X}}$, the entries of $A^{\mathrm{T}}A$ count the number of two-step paths
${\color{mygreen}G}\rightarrow{\color{myorange}O}\leftarrow{\color{mygreen}G}$, while the entries of $CC^{\mathrm{T}}$
count the number of two-step paths
${\color{mygreen}G}\leftarrow{\color{mypurple}P}\rightarrow{\color{mygreen}G}$.
A similar statement holds
for the other colors respectively matrix equations in \eqref{eq:main-transposes}.
Hence, all equations in \eqref{eq:main-transposes}
hold if and only if
$\boldsymbol{\Gamma}$ is quasi regular.
\end{proof}
\subsubsection{Some trihedral \texorpdfstring{$\mathbb{N}_{[\varstuff{v}]}$}{Nv}-represenations}\label{subsec:three-colored-graphs-reps}
We denote by $\C_{\vpar}\{{\color{mygreen}G},{\color{myorange}O},{\color{mypurple}P}\}$
the free ($\C_{\vpar}$-)vector space on the vertex set of $\boldsymbol{\Gamma}$.
\begin{definition}\label{definition:n-modules}
We define a $\subquo$-representation
\[
\algstuff{M}_{\boldsymbol{\Gamma}}\colon\subquo\to\algstuff{E}\mathrm{nd}_{\C_{\vpar}}(\C_{\vpar}\{{\color{mygreen}G},{\color{myorange}O},{\color{mypurple}P}\})
\]
by associating the following matrices to the generators $\theta_{{\color{mygreen}g}}$, $\theta_{{\color{myorange}o}}$, $\theta_{{\color{mypurple}p}}$:
\begin{gather}\label{eq:main-matrices}
\begin{gathered}
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{mygreen}g}})=
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
\vnumber{3}\mathrm{Id} & A^{\mathrm{T}} & C \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
}
,\quad\quad
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{myorange}o}})=
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
0 & 0 & 0 \\
A & \vnumber{3}\mathrm{Id} & B^{\mathrm{T}} \\
0 & 0 & 0
\end{pmatrix}
}
,
\\
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{mypurple}p}})
=
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
C^{\mathrm{T}} & B & \vnumber{3}\mathrm{Id}
\end{pmatrix}
}
.
\end{gathered}
\end{gather}
Here $A,B,C$ are as in
\eqref{eq:ad-matrix}.
\end{definition}
Note that we have
\[
\Mt[\boldsymbol{\Gamma}]
=
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{mygreen}g}})
+
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{myorange}o}})
+
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{mypurple}p}})
=
\vnumber{2}\left(
\vnumber{3}\mathrm{Id}
+
A(\Gg)
\right).
\]
\begin{remark}\label{remark:spectrum-matrices}
The three-dimensional simple $\subquo[e]$-representations $\algstuff{M}_z$
in \eqref{eq:three-dims} are similar to the $\algstuff{M}_{\boldsymbol{\Gamma}}$ in \eqref{eq:main-matrices}.
In $\algstuff{M}_{\boldsymbol{\Gamma}}$ the complex entry $z$ of $\algstuff{M}_z$ has been replaced by $\mathbb{N}$-matrices $A,B,C$
which have these complex numbers as eigenvalues, as we will see in \fullref{corollary:poly-killed} below.
However, in $\algstuff{M}_{\boldsymbol{\Gamma}}$ the matrices $A,B,C$ need not be equal, whereas in $\algstuff{M}_z$ we only have one complex number.
\end{remark}
We always choose $\{{\color{mygreen}G},{\color{myorange}O},{\color{mypurple}P}\}$ as a basis.
Recalling the setup from \fullref{subsec:three-colored-graphs} we get:
\begin{lemma}\label{lemma:n-modules}
$\algstuff{M}_{\boldsymbol{\Gamma}}$ is well-defined
if and only if $\boldsymbol{\Gamma}$ is quasi regular.
\end{lemma}
\begin{proof}
By direct computation, one immediately sees that \eqref{eq:first-rel} always holds,
irrespective of $A,B$ and $C$.
Furthermore, note that $\algstuff{M}_{\boldsymbol{\Gamma}}$ preserves the relations in \eqref{eq:second-rel} if and only if
the equations in \eqref{eq:main-transposes} hold. The claim then follows from \fullref{lemma:weakly-regular}.
\end{proof}
From now on we assume that $\boldsymbol{\Gamma}$ is quasi regular whenever we write $\algstuff{M}_{\boldsymbol{\Gamma}}$.
Proving that these are $\N_{\intvpar}$-representations
is hard and follows from categorification. However, if we drop the positivity condition, then the following is clear
by noting that the scalars $\vnumber{2}^{-k-l}$ appearing in the
definition of the colored KL elements cancel against the positive powers of $\vnumber{2}$ in \eqref{eq:main-matrices}.
\begin{lemmaqed}\label{lemma:trans-graphs}
$\algstuff{M}_{\boldsymbol{\Gamma}}$ is a transitive $\Z_{\intvpar}$-representation if and only if
$\boldsymbol{\Gamma}$ is admissible.
\end{lemmaqed}
\begin{example}\label{example:small-tri}
Take $e=2$ and the graph $\graphA{2}$ as
in \fullref{subsec:gen-D-list}. Fix ${\color{mygreen}g}$ as a starting color. Then the
six non-trivial, colored KL basis elements of
$\subquo[2]$ act on $\algstuff{M}_{\graphA{2}}$ via matrices whose
entries are all in $\N_{\intvpar}$.
For $\RKLg{0,0}=\theta_{{\color{mygreen}g}}$,
$\RKLg{1,0}=\vnumber{2}^{-1}\theta_{{\color{myorange}o}}\theta_{{\color{mygreen}g}}$
and $\RKLg{0,1}=\vnumber{2}^{-1}\theta_{{\color{mypurple}p}}\theta_{{\color{mygreen}g}}$
this is immediately clear. For the other basis elements, one can check the claim by calculation. For example,
$\RKLg{2,0}=\vnumber{2}^{-2}\theta_{{\color{mypurple}p}}\theta_{{\color{myorange}o}}\theta_{{\color{mygreen}g}}-
\vnumber{2}^{-1}\theta_{{\color{mypurple}p}}\theta_{{\color{mygreen}g}}$, since $\pxy{2,0}=\varstuff{X}^2-\varstuff{Y}$, so
\[
\algstuff{M}_{\graphA{2}}(\RKLg{2,0})=
\vnumber{2}
{\scriptstyle
\scalebox{0.85}{$
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & \vnumber{3} & 1 & 1 & 1 & 1 \\
\vnumber{3} & 0 & 1 & 0 & 1 & 0 \\
\end{pmatrix}$}}.
\]
The matrices associated to $\RKLg{1,1}$ and $\RKLg{0,2}$ can be computed similarly. The fact that we get a
$\N_{\intvpar}$-representation is non-trivial, because the expressions for the $\RKLg{m,n}$
in terms of the $\rklx{k,l}$ have negative coefficients.
\end{example}
The following can be proved as in the dihedral case \cite[Section 5.4]{MT1}.
\begin{lemmaqed}\label{lemma:find-all-transitives-2}
Let $\boldsymbol{\Gamma}$ and $\boldsymbol{\Gamma}^{\prime}$ be two admissible graphs. Then
$\algstuff{M}_{\boldsymbol{\Gamma}}\cong_{+}\algstuff{M}_{\boldsymbol{\Gamma}^{\prime}}$
if and only if $\boldsymbol{\Gamma}$ and $\boldsymbol{\Gamma}^{\prime}$ are
isomorphic as tricolored graphs.
\end{lemmaqed}
\begin{example}\label{example:find-all-transitives-2}
For the graphs from \fullref{subsec:gen-D-list} we get the following.
The graph $\graphA{e}$ allows three
non-isomorphic tricolorings in case $e\equiv 0\bmod 3$, but only one otherwise.
The graph $\graphD{e}$ can always be tricolored in three non-isomorphic ways, while the
graph $\graphC{e}$ admits only one tricoloring up to isomorphism. Finally,
in type $\mathsf{E}$ there are always three non-isomorphic tricolorings except for
the graph $\graphE{5}$ which has only one such tricoloring up to isomorphism.
Thus, \fullref{lemma:find-all-transitives-2} gives us the corresponding $\Z_{\intvpar}$-representations
which are not $\N_{\intvpar}$-equivalent.
\end{example}
\begin{lemma}\label{lemma:find-all-transitives}
Let $\algstuff{M}$ be a transitive $\N_{\intvpar}$-representation of
$\subquo[\infty]$ which satisfies
\begin{gather*}
\algstuff{M}(\theta_{{\color{dummy}\textbf{u}}})\algstuff{m}
=
a\algstuff{m}
+
\N_{\intvpar}
\left(\posmodbasis[\algstuff{M}]{-}\{\algstuff{m}\}\right)
\;\Rightarrow\;
a\in\{0,\vfrac{3}\},
\quad\text{for all }{\color{dummy}\textbf{u}},m,
\\
\text{and}\quad
a=\vfrac{3}\text{ only if }\algstuff{M}(\theta_{{\color{dummy}\textbf{u}}})\algstuff{m}
=
a\algstuff{m}.
\end{gather*}
Then there exists an admissible
graph $\boldsymbol{\Gamma}$ with $\algstuff{M}\cong_{+}\algstuff{M}_{\boldsymbol{\Gamma}}$.
\end{lemma}
\begin{proof}
Recall that $\algstuff{M}$ has a fixed basis $\posmodbasis[\algstuff{M}]$
on which all elements of the colored KL basis act by
matrices with entries in $\N_{\intvpar}$, that $\theta_{{\color{dummy}\textbf{u}}}^2=\vfrac{3}\theta_{{\color{dummy}\textbf{u}}}$
and that the trace of an idempotent
matrix is equal to its rank (which thus holds
for $\vfrac{3}^{-1}\algstuff{M}(\theta_{{\color{dummy}\textbf{u}}})$). In particular,
the assumption implies that for each generator
$\theta_{{\color{dummy}\textbf{u}}}$ there is an ordering of $\posmodbasis[\algstuff{M}]$ such that
\begin{gather}\label{eq:funny-matrix}
\algstuff{M}(\theta_{{\color{dummy}\textbf{u}}})=
\begin{pmatrix}
\vfrac{3}\mathrm{Id} & D \\
0 & 0
\end{pmatrix}
\end{gather}
for some matrix $D$ with entries in $\N_{\intvpar}$. The rest of the proof now follows along the lines of
\cite[Corollary 5.5]{Zi1} or \cite[Section 4.3]{KMMZ}:
\smallskip
\begin{enumerate}[label=$\blacktriangleright$]
\setlength\itemsep{.15cm}
\item First observe that each $\algstuff{m}$ is a $\vfrac{3}$-eigenvector
of some $\theta_{\color{dummy}\textbf{u}}$, since otherwise
$\algstuff{M}(\theta_{{\color{mygreen}g}})+\algstuff{M}(\theta_{{\color{myorange}o}})+\algstuff{M}(\theta_{{\color{mypurple}p}})$ would have a zero row
by \eqref{eq:funny-matrix}, which contradicts the transitivity.
\item Secondly, $\algstuff{m}$ is not a $\vfrac{3}$-eigenvector for all the
$\theta_{\color{dummy}\textbf{u}}$. To see this, assume the contrary. Then, by transitivity,
$\algstuff{M}$ has to be one-dimensional with all $\theta_{\color{dummy}\textbf{u}}$ acting by $\vfrac{3}$.
However, as in \fullref{lemma:further-restrictions}, this contradicts
the fact that $\algstuff{M}$ is a $\subquo[e]$-representation.
\item Finally, $\algstuff{m}$ is not a common $\vfrac{3}$-eigenvector of two of the
$\theta_{\color{dummy}\textbf{u}}$. Assume on the contrary that $\theta_{\color{mygreen}g}$ and $\theta_{\color{myorange}o}$ had such a common
eigenvector. Then $\algstuff{M}(\theta_{{\color{mygreen}g}})\algstuff{m}=\algstuff{M}(\theta_{{\color{myorange}o}})\algstuff{m}=\vfrac{3}\algstuff{m}$ and $\algstuff{M}(\theta_{{\color{mypurple}p}})\algstuff{m}=0$.
This contradicts \eqref{eq:second-rel}.
\end{enumerate}
\smallskip
\makeautorefname{lemma}{Lemmas}
Together with \fullref{lemma:weakly-regular}, \ref{lemma:n-modules} and \ref{lemma:trans-graphs},
this proves the claim. \makeautorefname{lemma}{Lemma}
\end{proof}
\subsubsection{The classification problem \fullref{problem:classification}}\label{subsec:Z-reps-2}
Back to the polynomials $\pxy{m,n}$ from \fullref{definition:sl3-polys}.
Observe that quasi regularity implies that
\begin{gather}\label{eq:weakly-regular}
A(\Gg^{\varstuff{X}})A(\Gg^{\varstuff{Y}})
=
{\scriptstyle
\begin{pmatrix}
CC^{\mathrm{T}} & 0 & 0 \\
0 & AA^{\mathrm{T}} & 0 \\
0 & 0 & BB^{\mathrm{T}}
\end{pmatrix}
}
\stackrel{\eqref{eq:main-transposes}}{=}
{\scriptstyle
\begin{pmatrix}
A^{\mathrm{T}}A & 0 & 0 \\
0 & B^{\mathrm{T}}B & 0 \\
0 & 0 & C^{\mathrm{T}}C
\end{pmatrix}
}
=
A(\Gg^{\varstuff{Y}})A(\Gg^{\varstuff{X}}).
\end{gather}
Thus, we can formulate the following classification problem.
\newline
\begin{problem}\label{problem:classification}
Classify all admissible graphs $\boldsymbol{\Gamma}$ such that
\begin{gather*}
\pxy[A(\Gg^{\varstuff{X}}),A(\Gg^{\varstuff{Y}})]{m,n}=0, \quad\text{for all }m+n=e+1.
\end{gather*}
(In other words, classify all admissible graphs $\boldsymbol{\Gamma}$ such
that $z\in S_{\boldsymbol{\Gamma}^{\varstuff{X}}}$ only if $(z,\overline{z})\in\vanset{e}$.)
\end{problem}
\begin{proposition}\label{proposition:poly-killed}
A graph $\boldsymbol{\Gamma}$ is a solution of \fullref{problem:classification}
if and only if $\algstuff{M}_{\boldsymbol{\Gamma}}$
descends to a transitive $\Z_{\intvpar}$-representation of $\subquo[e]$.
\end{proposition}
\begin{proof}
Recall that admissible graphs are always strongly connected. Thus,
the claim about transitivity is clear and it remains to check the other claims.
To this end, fix $m,n$.
Observe that $\pxy[A(\Gg^{\varstuff{X}}),A(\Gg^{\varstuff{Y}})]{m,n}$
has at most one non-zero block matrix entry in each of the ${\color{mygreen}G}$-, ${\color{myorange}O}$- and ${\color{mypurple}P}$-rows
(as indicated in \eqref{eq:ad-matrix}), since
$A(\Gg^{\varstuff{X}})$ and $A(\Gg^{\varstuff{Y}})$ just permute the ${\color{mygreen}G}$-, ${\color{myorange}O}$- and ${\color{mypurple}P}$-blocks,
and multiply them by $A,B,C$ or their transpose. Let us denote these
block matrix entries by $N^{{\color{mygreen}G}}_{m,n}$, $N^{{\color{myorange}O}}_{m,n}$ and $N^{{\color{mypurple}P}}_{m,n}$.
Let us fix ${\color{mygreen}g}$ as a starting color, the other two cases work verbatim.
In this case an easy calculation yields:
\begin{gather*}
\algstuff{M}_{\boldsymbol{\Gamma}}(\RKLg{m,n})
=
\begin{cases}
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
N^{{\color{mygreen}G}}_{m,n}\cdot\vnumber{3}\mathrm{Id}
&
N^{{\color{mygreen}G}}_{m,n}\cdot A^{\mathrm{T}}
&
N^{{\color{mygreen}G}}_{m,n}\cdot C
\\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
}
,&
\text{if }
m+2n \equiv 0 \bmod 3,
\\
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
0 & 0 & 0 \\
N^{{\color{myorange}O}}_{m,n}\cdot\vnumber{3}\mathrm{Id}
&
N^{{\color{myorange}O}}_{m,n}\cdot A^{\mathrm{T}}
&
N^{{\color{myorange}O}}_{m,n}\cdot C
\\
0 & 0 & 0
\end{pmatrix}
}
,&
\text{if }
m+2n \equiv 1 \bmod 3,
\\
\vnumber{2}
{\scriptstyle
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
N^{{\color{mypurple}P}}_{m,n}\cdot\vnumber{3}\mathrm{Id}
&
N^{{\color{mypurple}P}}_{m,n}\cdot A^{\mathrm{T}}
&
N^{{\color{mypurple}P}}_{m,n}\cdot C
\end{pmatrix}
}
,&
\text{if }
m+2n \equiv 2 \bmod 3.
\end{cases}
\end{gather*}
As in the proof of \fullref{lemma:three-dim-rep1},
we note that in the calculation of $\algstuff{M}_{\boldsymbol{\Gamma}}(\RKLg{m,n})$
the positive powers of $\vnumber{2}$, due to \eqref{eq:main-matrices},
cancel against the negative powers of $\vnumber{2}$,
which appear in \eqref{eq:the-expressions}, up to an
overall factor $\vnumber{2}$. We see that $\algstuff{M}_{\boldsymbol{\Gamma}}(\RKLg{m,n})$
vanishes if and only if $\pxy[A(\Gg^{\varstuff{X}}),A(\Gg^{\varstuff{Y}})]{m,n}=0$.
\end{proof}
Our main examples of solutions of \fullref{problem:classification} are the graphs from \fullref{subsec:gen-D-list}.
Indeed, as can be seen in \fullref{subsec:gen-D-spectra}, their spectra are such that
\fullref{proposition:poly-killed} applies:
\begin{corollary}\label{corollary:poly-killed}
The generalized $\ADE$ Dynkin diagrams from \fullref{subsec:gen-D-list}
give transitive $\Z_{\intvpar}$-representations $\algstuff{M}_{\boldsymbol{\Gamma}}$
for the associated level $e$.
\end{corollary}
\makeautorefname{lemmaqed}{Lemmas}
By \fullref{lemma:find-all-transitives-2},
\ref{lemma:find-all-transitives} and \fullref{proposition:poly-killed}, classifying all
$\Z_{\intvpar}$-representations of $\subquo[e]$ boils down to
\fullref{problem:classification}. We have already seen that the generalized
$\ADE$ Dynkin diagrams give solutions of \fullref{problem:classification}.
So two questions remain: whether these are all solutions and whether these are
$\N_{\intvpar}$-representations (transitivity is clear because
the graphs are strongly connected).
\makeautorefname{lemmaqed}{Lemma}
We do not have a complete answer to these questions. However, we are able to prove:
\begin{proposition}\label{proposition:typeA-D-decat}
Let ${\color{mygreen}g},{\color{myorange}o},{\color{mypurple}p}$ indicate the starting color. Then
we have (at least) the following transitive $\N_{\intvpar}$-representations
of $\subquo[e]$.
\begin{gather}\label{eq:the-transitives}
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=1em, column
sep=1em, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{a}
\&
e\equiv 0\bmod 3
\&
e\not\equiv 0\bmod 3
\\
\text{$\N_{\intvpar}$-reps.}
\&
\begin{gathered}
\algstuff{M}_{\graphA{e}^{{\color{mygreen}g}}},\,
\algstuff{M}_{\graphA{e}^{{\color{myorange}o}}},\,
\algstuff{M}_{\graphA{e}^{{\color{mypurple}p}}},
\\
\algstuff{M}_{\graphD{e}^{{\color{mygreen}g}}},\,
\algstuff{M}_{\graphD{e}^{{\color{myorange}o}}},\,
\algstuff{M}_{\graphD{e}^{{\color{mypurple}p}}}
\end{gathered}
\&
\algstuff{M}_{\graphA{e}^{{\color{mygreen}g}}}
\\
\text{quantity}
\&
6
\&
1
\\
\\};
\draw[densely dashed] ($(m-1-1.south west)+ (-.75,0)$) to ($(m-1-3.south east)+ (.6,0)$);
\draw[densely dashed] ($(m-1-3.north west) + (-.3,0)$) to ($(m-1-3.north west) + (-.3,-4.2)$);
\draw[densely dashed] ($(m-1-2.north west) + (-.85,0)$) to ($(m-1-2.north west) + (-.85,-4.2)$);
\end{tikzpicture}
\end{gather}
Moreover, the representations $\algstuff{M}_{\graphA{e}}$
are the cell modules of $\subquo[e]$.
\end{proposition}
\makeautorefname{lemmaqed}{Lemmas}
\begin{proof}
Except for the claim about positivity,
this is clear by \fullref{corollary:poly-killed}, \fullref{lemma:trans-graphs},
\ref{lemma:find-all-transitives-2} and the construction.
For example, if $e\not\equiv 0\bmod 3$, then there is only one type
$\mathsf{A}$ representation up to $\N_{\intvpar}$-equivalence, since all tricolorings of $\graphA{e}$ give isomorphic
tricolored graphs. To see this, note that a tricoloring of the lowest
triangle fixes the tricoloring of the whole graph,
and that there are six choices. When $e\not\equiv 0\bmod 3$, they all give isomorphic tricolored graphs,
as can be easily seen. When $e\equiv 0\bmod 3$, we get three different
isomorphism classes of tricolored graphs,
which are determined by the color of the corner vertices. Note that there
is one more vertex with that color, e.g. ${\color{mygreen}g}$, than vertices with one
of the other two colors, e.g. ${\color{myorange}o}$ resp. ${\color{mypurple}p}$. This explains why tricolored graphs whose
corner vertices have different colors, are non-isomorphic. Finally, for
any fixed color of the corner vertices, there are
two tricolored graphs, which are isomorphic by a $\Z/2\Z$-symmetry in the
bisector of the angle at that vertex.
Positivity
follows from \fullref{theorem:typeA-D-cat}
which we proof later on.
\end{proof}
\makeautorefname{lemmaqed}{Lemma}
In contrast to simples, the
transitive $\N_{\intvpar}$-representations of $\subquo[e]$ can
get arbitrarily large as $e$ grows, c.f. \eqref{eq:the-simples} and \eqref{eq:the-transitives}. We only know
their complete classification for small values of $e$.
\subsubsection{Classification for small levels}\label{subsec:decat-story-b}
\begin{theorem}\label{theorem:low-level-classification}
Let $e\in\{0,1,2,3\}$.
An admissible graph $\boldsymbol{\Gamma}$ provides a solution
to \fullref{problem:classification} if and only if
$\boldsymbol{\Gamma}$ is a generalized $\ADE$ Dynkin
diagram of the corresponding level or
\begin{gather}\label{eq:special-solution}
\boldsymbol{\Gamma}=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow, double] (0,0) to (1,1);
\draw [thick, densely dotted, myblue, double] (0,0) to (-1,1);
\draw [thick, densely dashed, myred, double] (1,1) to (-1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,
\quad
\text{for }e=3,
\end{gather}
called special solution. (Note the double edges.)
\end{theorem}
\begin{proof}
We do the hardest case where $e=3$ and omit the others, all of which
can be proven similarly. In this case, the vanishing ideal $\vanideal{3}$ is
generated by
\[
\left\{
\pxy{4,0}
,
\pxy{3,1}
,
\pxy{2,2}
,
\pxy{1,3}
,
\pxy{0,4}
\right\}
\subset\mathbb{Z}[\varstuff{X},\varstuff{Y}]
\]
with the polynomials as in \fullref{example:sl3-polys}.
We proceed as follows: consider the polynomials
\begin{gather}\label{eq:new-cp}
\left\{
\varstuff{Y}^{4}\pxy{4,0}
,
\varstuff{Y}^{2}\pxy{3,1}
,
\varstuff{Y}^{3}\pxy{2,2}
,
\varstuff{Y}\pxy{1,3}
,
\varstuff{Y}^{2}\pxy{0,4}
\right\}
\subset\mathbb{Z}[\varstuff{X},\varstuff{Y}],
\end{gather}
which are now polynomials in the variables $\varstuff{x}=\varstuff{X}\varstuff{Y}$ and $\varstuff{y}=\varstuff{Y}^3$.
Clearly, any solution of \fullref{problem:classification} gives an admissible graph $\boldsymbol{\Gamma}$
such that $(\varstuff{x}=A(\boldsymbol{\Gamma}^{\varstuff{X}})A(\boldsymbol{\Gamma}^{\varstuff{Y}}),\varstuff{y}=A(\boldsymbol{\Gamma}^{\varstuff{Y}})^3)$
is annihilated by the polynomials in \eqref{eq:new-cp}.
Hence, one can solve \fullref{problem:classification} by first classifying
all solutions of \eqref{eq:new-cp} in terms of
$\varstuff{x}$ and $\varstuff{y}$ and then checking which ones give solutions of
\fullref{problem:classification} in terms of $A(\boldsymbol{\Gamma}^{\varstuff{X}})$ and $A(\boldsymbol{\Gamma}^{\varstuff{Y}})$.
To this end, we can use the theory of Gr{\"o}bner bases, for the
lexicographical ordering on monomials induced by $\varstuff{x}<\varstuff{y}$,
to rewrite \eqref{eq:new-cp}. This shows that $\boldsymbol{\Gamma}$ solves
\fullref{problem:classification} if and only if $\varstuff{x}$ and $\varstuff{y}$ satisfy
\begin{gather}\label{eq:groebner-2}
\varstuff{x}^3-5\varstuff{x}^2+4\varstuff{x}=0
\;\;\&\;\;
\varstuff{x}\varstuff{y}-\varstuff{y}-2\varstuff{x}^2+2\varstuff{x}=0
\;\;\&\;\;
\varstuff{y}^2-\varstuff{y}-5\varstuff{x}^2+6\varstuff{x}=0.
\end{gather}
We observe further that the polynomial
$\varstuff{x}^3-5\varstuff{x}^2+4\varstuff{x}$ evaluated at $A(\boldsymbol{\Gamma}^{\varstuff{X}})A(\boldsymbol{\Gamma}^{\varstuff{Y}})$ vanishes
if and only if it vanishes evaluated at
the symmetric matrix $A^{\mathrm{T}}A$, cf. \fullref{lemma:weakly-regular}
and \eqref{eq:weakly-regular}.
Thus, it suffices to solve the first equation in \eqref{eq:groebner-2}
for $\varstuff{x}=A^{\mathrm{T}}A$, and then to check
whether the candidate solutions one obtains satisfy \fullref{problem:classification}.
The upshot is that the first equation in \eqref{eq:groebner-2} is then
an equation in one symmetric matrix $A^{\mathrm{T}}A$
with entries from $\mathbb{N}$.
In order to check which matrices $A^{\mathrm{T}}A$ are annihilated by
$\varstuff{x}^3-5\varstuff{x}^2+4\varstuff{x}$ we first note that the complex roots of the polynomial
$\varstuff{x}^3-5\varstuff{x}^2+4\varstuff{x}=\varstuff{x}(\varstuff{x}-1)(\varstuff{x}-4)$
are $0,1,4$, and that $\begin{psmallmatrix}0 & A^{\mathrm{T}}\\A & 0\end{psmallmatrix}$
is the adjacency matrix of the connected, bicolored subgraph of $\boldsymbol{\Gamma}$
obtained by erasing ${\color{mypurple}P}$ (and all edges with a vertex in ${\color{mypurple}P}$).
Moreover, the eigenvalues of
$\begin{psmallmatrix}0 & A^{\mathrm{T}}\\A & 0\end{psmallmatrix}$
are the square roots of the eigenvalues of $A^{\mathrm{T}}A$
(this linear algebra fact follows from e.g. \cite[Theorem 3]{Si-block-matrices})
and hence, have to be $0,\pm 1,\pm 2$. Therefore,
$\begin{psmallmatrix}0 & A^{\mathrm{T}}\\A & 0\end{psmallmatrix}$ has
to be the adjacency matrix of a finite or affine
type $\ADE$ Dynkin diagram, by the classification in
\cite{Sm1}, \cite[Section 3.1.1]{BH1},
with its Perron--Frobenius eigenvalue being $1$ or $2$, provided it is not zero.
Furthermore, again by the classification in
\cite{Sm1}, \cite[Section 3.1.1]{BH1}, the only
connected graph such that $\begin{psmallmatrix}0 & A^{\mathrm{T}}\\A & 0\end{psmallmatrix}$
has Perron--Frobenius eigenvalue $1$ is of
finite type $\typea{2}$, all those with Perron--Frobenius
eigenvalue $2$ correspond to affine types. But for finite type $\typea{2}$ we
get $A=A^{\mathrm{T}}=\begin{psmallmatrix}1\end{psmallmatrix}$
which by \eqref{eq:main-transposes}
and strong connectivity
implies $B=B^{\mathrm{T}}=C=C^{\mathrm{T}}=\begin{psmallmatrix}1\end{psmallmatrix}$, and thus,
\eqref{eq:groebner-2} is not satisfied.
Hence, we only need to consider affine type $\ADE$ Dynkin diagrams
whose only eigenvalues are $0,\pm 1,-2$ in addition to $2$.
Using the list of eigenvalues from \cite[Section 3.1.1]{BH1},
we obtain the following possibilities for the associated
bicolored graph.
\begin{gather*}
\typeat{1}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (2,0) to [out=192, in=348] (0,0);
\draw [thick, myyellow] (2,0) to [out=168, in=12] (0,0);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\end{tikzpicture}
,
\quad
\typeat{3}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (2,-.7) to (0,-.7) to (0,.7) to (2,.7) to (2,-.7);
\node at (0,-.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,-.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (0,.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\end{tikzpicture}
,
\quad
\typeat{5}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (3,0) to (2,-1.2) to (0,-1.2) to (-1,0) to (0,1.2) to (2,1.2) to (3,0);
\node at (0,-1.2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,-1.2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (0,1.2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,1.2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (3,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\end{tikzpicture}
,
\quad
\typedt{4}^{{\color{mygreen}G}=1}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (-1,-.7) to (0,0);
\draw [thick, myyellow] (1,-.7) to (0,0);
\draw [thick, myyellow] (-1,.7) to (0,0);
\draw [thick, myyellow] (1,.7) to (0,0);
\node at (-1,-.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,-.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\end{tikzpicture}
\\
\typedt{4}^{{\color{mygreen}G}=4}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (-1,-.7) to (0,0);
\draw [thick, myyellow] (1,-.7) to (0,0);
\draw [thick, myyellow] (-1,.7) to (0,0);
\draw [thick, myyellow] (1,.7) to (0,0);
\node at (-1,-.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-1,.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,-.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\end{tikzpicture}
,
\quad
\typedt{5}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (-1,-.7) to (0,0);
\draw [thick, myyellow] (3,-.7) to (2,0);
\draw [thick, myyellow] (-1,.7) to (0,0);
\draw [thick, myyellow] (3,.7) to (2,0);
\draw [thick, myyellow] (0,0) to (2,0);
\node at (-1,-.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-1,.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,-.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (3,.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\end{tikzpicture}
,
\quad
\typeet{6}^{{\color{mygreen}G}=3}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (-2,0) to (2,0);
\draw [thick, myyellow] (0,0) to (0,1.4);
\node at (-2,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (0,.7) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,1.4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\end{tikzpicture}
,
\quad
\typeet{6}^{{\color{mygreen}G}=4}
=
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (-2,0) to (2,0);
\draw [thick, myyellow] (0,0) to (0,1.4);
\node at (-2,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-1,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,.7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (0,1.4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\end{tikzpicture}
\end{gather*}
($\typeat{2}$, which has eigenvalues $-1,-1,2$,
is ruled out since it does not allow a bicoloring.) The same holds for the other colors, of course.
One can now write down all candidates solutions for the bicolored subgraphs:
\begin{center}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{c||c|c|c|c|c|c|c|c|c|c}
$\boldsymbol{\Gamma}_{{\color{mygreen}g},{\color{myorange}o}}$ & $\typeat{1}$ & $\typeat{1}$ & $\typeat{3}$ & $\typeat{5}$ & $\typeat{5}$ & $\typeat{5}$ & $\typeat{5}$ & $\typeat{5}$ & $\typedt{4}^{{\color{mygreen}G}=1}$ & $\typedt{4}^{{\color{mygreen}G}=4}$ \\
\hline
$\boldsymbol{\Gamma}_{{\color{myorange}o},{\color{mypurple}p}}$ & $\typeat{1}$ & $\typedt{4}^{{\color{myorange}O}=1}$ & $\typeat{3}$ & $\typeat{5}$ & $\typeat{5}$ & $\typedt{5}$ & $\typedt{5}$ & $\typeet{6}^{{\color{myorange}O}=3}$ & $\typedt{4}^{{\color{myorange}O}=4}$ & $\typeat{1}$ \\
\hline
$\boldsymbol{\Gamma}_{{\color{mypurple}p},{\color{mygreen}g}}$ & $\typeat{1}$ & $\typedt{4}^{{\color{mypurple}P}=4}$ & $\typeat{3}$ & $\typeat{5}$ & $\typedt{5}$ & $\typeat{5}$ & $\typedt{5}$ & $\typeet{6}^{{\color{mypurple}P}=4}$ & $\typeat{1}$ & $\typedt{4}^{{\color{mypurple}P}=1}$ \\
\hline\hline
\eqref{eq:groebner-2}? & \eqref{eq:special-solution} & $\graphD{3}^{{\color{mypurple}p}}$ & $\graphC{3}$ & no & no & no & no & $\graphA{3}^{{\color{mypurple}p}}$ & $\graphD{3}^{{\color{myorange}o}}$ & $\graphD{3}^{{\color{mygreen}g}}$
\end{tabular}
\end{center}
\begin{center}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{c||c|c|c|c|c|c|c|c|c}
$\boldsymbol{\Gamma}_{{\color{mygreen}g},{\color{myorange}o}}$ & $\typedt{5}$ & $\typedt{5}$ & $\typedt{5}$ & $\typedt{5}$ & $\typedt{5}$ & $\typeet{6}^{{\color{mygreen}G}=3}$ & $\typeet{6}^{{\color{mygreen}G}=3}$ & $\typeet{6}^{{\color{mygreen}G}=4}$ & $\typeet{6}^{{\color{mygreen}G}=4}$ \\
\hline
$\boldsymbol{\Gamma}_{{\color{myorange}o},{\color{mypurple}p}}$ & $\typeat{5}$ & $\typeat{5}$ & $\typedt{5}$ & $\typedt{5}$ & $\typeet{6}^{{\color{myorange}O}=3}$ & $\typeet{6}^{{\color{myorange}O}=4}$ & $\typeet{6}^{{\color{myorange}O}=4}$ & $\typeat{5}$ & $\typedt{5}$ \\
\hline
$\boldsymbol{\Gamma}_{{\color{mypurple}p},{\color{mygreen}g}}$ & $\typeat{5}$ & $\typedt{5}$ & $\typeat{5}$ & $\typedt{5}$ & $\typeet{6}^{{\color{mypurple}P}=4}$ & $\typeat{5}$ & $\typedt{5}$ & $\typeet{6}^{{\color{mypurple}P}=3}$ & $\typeet{6}^{{\color{mypurple}P}=3}$ \\
\hline\hline
\eqref{eq:groebner-2}? & no & no & no & no & no & $\graphA{3}^{{\color{myorange}o}}$ & no & $\graphA{3}^{{\color{mygreen}g}}$ & no
\end{tabular}
\end{center}
Here we have indicated whether all equations in
\eqref{eq:groebner-2} are satisfied or not.
The remaining possibilities
give solutions to \fullref{problem:classification}.
\end{proof}
The solution \eqref{eq:special-solution} is not on the list of
generalized $\ADE$ Dynkin diagrams, and we do not know whether this is an exception
for $e=3$ or whether there exist more solutions which are not
generalized $\ADE$ Dynkin diagrams for $e>3$.
\begin{example}\label{example:classification}
For $e=0,1,2,3$ the list of transitive $\N_{\intvpar}$-representations
given in \eqref{eq:the-transitives} is almost complete. Adding a
representation $\algstuff{M}_{\graphC{3}}$ to this list, for any color by
\fullref{lemma:find-all-transitives-2}, and a representation
for the special solution \eqref{eq:special-solution} completes the list, where
one can check by hand that these are $\N_{\intvpar}$-representations.
\end{example}
\subsection{Categorified story}\label{subsec:cat-story}
Recall that $\GGc{\subcatquo[e]}\cong\subquo[e]$, see \fullref{proposition:cat-the-quoalgebra}
(excluding $e=0$), and
assume that we have a transitive $2$-representation $\twocatstuff{M}$ of $\subcatquo[e]$.
Then $\GGc{\twocatstuff{M}}$ is a transitive $\N_{\intvpar}$-representation of $\subquo[e]$.
So, by the discussion in \fullref{subsec:decat-story}, the
classification of simple transitive $2$-representations of $\subcatquo[e]$
boils down to \fullref{problem:classification} together with the construction of the
corresponding $2$-representations (i.e. their categorification). We are going to explain this construction
for types $\mathsf{A}$ and $\mathsf{D}$.
\subsubsection{Satake and \texorpdfstring{$2$}{2}-representations}\label{subsec:cat-story-0}
Let us first sketch how \fullref{theorem:q-satake} gives rise to a correspondence between
the simple transitive $2$-representations of $\slqmodgop$
and those of $\subcatquo[e]$. We will discuss the details in the sections below.
Recall that there is a bijection
between the equivalence classes of
simple transitive $2$-re\-presentations of $\slqmodgop$
and the Morita equivalence classes of
simple algebra $1$-morphisms in $\slqmodgop$, cf. \cite[Theorem 9]{MMMT1}.
For $\subcatquo[e]$ the
situation is more complicated, because it is
additive, but not abelian. However,
it is still true that, if $\morstuff{A}$ is an indecomposable
algebra $1$-morphism in $\subcatquo[e]$,
then $\modcat{\subcatquo[e]}(\morstuff{A})$
is a transitive $2$-representation of $\subcatquo[e]$. By
taking its simple quotient, as
described in \fullref{remark:graded-finitary-3}, we get a simple transitive
$2$-representation associated to $\morstuff{A}$.
As we will see, any algebra $1$-morphism in $\morstuff{A}$ in $\slqmod$ gives rise to
an algebra $1$-morphism $\morstuff{A}^{{\color{dummy}\textbf{u}}}$ in $\slqmodgop[e]({\color{dummy}\textbf{u}},{\color{dummy}\textbf{u}})$. (Without loss of generality,
we will concentrate on the case ${\color{dummy}\textbf{u}}={\color{mygreen}g}$ below.)
The Satake $2$-functor from \fullref{lemma:Satake} transports $\morstuff{A}^{{\color{mygreen}g}}$
to an algebra $1$-morphism in $\Subcatquo[e]({\color{mygreen}g},{\color{mygreen}g})$ (where we keep the same notation). Biinduction,
which means gluing white outer regions to the diagrams which
define the multiplication and unit $2$-morphisms (see also \fullref{example:clasps}), then gives an algebra $1$-morphism
$\morstuff{B}^{{\color{mygreen}g}}=\morstuff{B}(\morstuff{A}^{{\color{mygreen}g}})\in\subcatquo[e]=\Subcatquo[e](\emptyset,\emptyset)$.
As we will show, the algebra $1$-morphism also has to be translated
(which would correspond to shifting the grading if the algebra $1$-morphism were given by a graded bimodule),
so that the final degree of the unit and multiplication $2$-morphisms becomes zero.
The fact that $\morstuff{B}^{{\color{mygreen}g}}$ is associative
and unital, follows almost immediately from the associativity and unitality
of $\morstuff{A}^{{\color{mygreen}g}}$. (For a detailed proof we refer to \cite[Section 7.3]{MMMT1}.)
By construction, $\morstuff{B}^{{\color{mygreen}g}}$ is indecomposable if $\morstuff{A}$ is simple.
Thus, given a simple transitive $2$-representation $\twocatstuff{M}$
of $\slqmodgop$, let $\morstuff{A}$
be the corresponding simple algebra $1$-morphism in
$\slqmodgop$ and take $\twocatstuff{M}_{{\color{mygreen}g}}$ to be the simple quotient
(as recalled in \fullref{remark:graded-finitary-3}) of
$\modcat{\subcatquo[e]}(\morstuff{B}^{{\color{mygreen}g}})$.
\begin{remark}\label{remark:nothing-new}
Note that all simple algebra $1$-morphisms in
$\slqmodgop$ arise via coloring from simple algebra
$1$-morphisms in $\slqmod$. So our first task below is to
recall the latter, which were already known, see e.g. \cite{Sch}.
However, we present a self-contained construction in this paper.
As a service to the reader, we also recall the proof of the
known classification of their simple module $1$-morphisms
in $\slqmod$.
\end{remark}
\subsubsection{Type \texorpdfstring{$\mathsf{A}$}{A} \texorpdfstring{$2$}{2}-representations}\label{subsec:cat-story-b}
The object $\obstuff{A}^{\graphA{e}}=\algstuff{L}_{0,0}\cong\mathbb{C}$ is clearly
a simple algebra object in $\slqmod$, because it is the identity
object. Thus, coloring gives us a simple algebra
$1$-morphism $\morstuff{A}^{\graphA{e}^{{\color{mygreen}g}}}$
in $\slqmodgop$. After applying the Satake $2$-functor,
we get $\elfunctor[e](\morstuff{A}^{\graphA{e}^{{\color{mygreen}g}}})={\color{mygreen}g}$,
which is the identity $1$-morphism in $\Subcatquo[e]({\color{mygreen}g},{\color{mygreen}g})$.
Therefore, we have $\morstuff{B}^{\graphA{e}^{{\color{mygreen}g}}}=\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset \{-3\}\in\subcatquo[e]$, which is an indecomposable
$1$-morphism in $\subcatquo[e]$. Recall that $\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset\cong
\emptyset{\color{myyellow}y}{\color{mygreen}g}{\color{myyellow}y}\emptyset$, by \fullref{lemma:clasps-well-defined}. We have translated
$\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset$ by $-3$, so that the final degree of the unit and multiplication $2$-morphisms
below becomes zero.
Note further that $\morstuff{B}^{\graphA{e}}$ is a Frobenius $1$-morphism, and its
(co)unit and (co)multiplication $2$-morphisms (with their respective unshifted degrees) are given by
\[
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0) to (3,0) to (3,-3.5) to (-1,-3.5)
to (-1,0) to (-.5,0);
\fill[mygreen, opacity=0.8] (.25,0) to [out=270, in=180] (1,-.75) to [out=0, in=270] (1.75,0) to (.25,0);
\fill[myblue, opacity=0.3] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0) to (1.75,0) to [out=270, in=0] (1,-.75) to [out=180, in=270] (.25,0) to (-.5,0);
\draw[ystrand, rdirected=.5] (.25,0) to [out=270, in=180] (1,-.75) to [out=0, in=270] (1.75,0);
\draw[bstrand, rdirected=.5] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0);
\end{tikzpicture}
};
(0,10)*{\text{{\tiny unit}}};
(0,-10)*{\text{{\tiny degree $3$}}};
\endxy
\quad,\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to (3,0) to (3,3.5) to (-1,3.5)
to (-1,0) to (-.5,0);
\fill[mygreen, opacity=0.8] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0) to (.25,0);
\fill[myblue, opacity=0.3] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to (1.75,0) to [out=90, in=0] (1,.75) to [out=180, in=90] (.25,0) to (-.5,0);
\draw[ystrand, directed=.5] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0);
\draw[bstrand, directed=.5] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0);
\end{tikzpicture}
};
(0,10)*{\text{{\tiny counit}}};
(0,-10)*{\text{{\tiny degree $-3$}}};
\endxy
\quad,\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (4,0) to (4,3.5) to (4.5,3.5) to (4.5,0) to (4,0);
\draw[very thin, densely dotted, fill=white] (-2,0) to (-2,3.5) to (-2.5,3.5) to (-2.5,0) to (-2,0);
\draw[very thin, densely dotted, fill=white] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0) to (.25,0);
\fill[myblue, opacity=0.3] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=180, in=0] (1.75,0) to [out=90, in=0] (1,.75) to [out=180, in=90] (.25,0) to [out=180, in=0] (-.5,0);
\fill[mygreen, opacity=0.8] (2.5,0) to (3.25,0) to [out=90, in=-45] (2.75,1.25) to [out=135, in=270] (1.75,3.5)
to (.25,3.5) to [out=270, in=45] (-.75,1.25) to [out=225, in=90] (-1.25,0) to (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0);
\fill[myblue, opacity=0.3] (3.25,0) to [out=90, in=-45] (2.75,1.25) to [out=135, in=270] (1.75,3.5) to (4,3.5) to (4,0) to (3.25,0);
\fill[myblue, opacity=0.3] (-1.25,0) to [out=90, in=225] (-.75,1.25) to [out=45, in=270] (.25,3.5) to (-2,3.5) to (-2,0) to (-1.25,0);
\draw[bstrand, rdirected=.5] (4,0) to (4,3.5);
\draw[bstrand, directed=.5] (-2,0) to (-2,3.5);
\draw[bstrand, rdirected=.5] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0);
\draw[ystrand, rdirected=.5] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0);
\draw[ystrand, directed=.5] (-1.25,0) to [out=90, in=225] (-.75,1.25) to [out=45, in=270] (.25,3.5);
\draw[ystrand, rdirected=.5] (3.25,0) to [out=90, in=-45] (2.75,1.25) to [out=135, in=270] (1.75,3.5);
\end{tikzpicture}
};
(0,10)*{\text{{\tiny multiplication}}};
(0,-10)*{\text{{\tiny degree $-3$}}};
\endxy
\quad,\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (4,0) to (4,-3.5) to (4.5,-3.5) to (4.5,0) to (4,0);
\draw[very thin, densely dotted, fill=white] (-2,0) to (-2,-3.5) to (-2.5,-3.5) to (-2.5,0) to (-2,0);
\draw[very thin, densely dotted, fill=white] (.25,0) to [out=270, in=180] (1,-.75) to [out=0, in=270] (1.75,0) to (.25,0);
\fill[myblue, opacity=0.3] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0) to [out=180, in=0] (1.75,0) to [out=270, in=0] (1,-.75) to [out=180, in=270] (.25,0) to [out=180, in=0] (-.5,0);
\fill[mygreen, opacity=0.8] (2.5,0) to (3.25,0) to [out=270, in=45] (2.75,-1.25) to [out=-135, in=90] (1.75,-3.5)
to (.25,-3.5) to [out=90, in=-45] (-.75,-1.25) to [out=-225, in=270] (-1.25,0) to (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0);
\fill[myblue, opacity=0.3] (3.25,0) to [out=270, in=45] (2.75,-1.25) to [out=-135, in=90] (1.75,-3.5) to (4,-3.5) to (4,0) to (3.25,0);
\fill[myblue, opacity=0.3] (-1.25,0) to [out=270, in=-225] (-.75,-1.25) to [out=-45, in=90] (.25,-3.5) to (-2,-3.5) to (-2,0) to (-1.25,0);
\draw[bstrand, directed=.5] (4,0) to (4,-3.5);
\draw[bstrand, rdirected=.5] (-2,0) to (-2,-3.5);
\draw[bstrand, directed=.5] (.25,0) to [out=270, in=180] (1,-.75) to [out=0, in=270] (1.75,0);
\draw[ystrand, directed=.5] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0);
\draw[ystrand, rdirected=.5] (-1.25,0) to [out=270, in=-225] (-.75,-1.25) to [out=-45, in=90] (.25,-3.5);
\draw[ystrand, directed=.5] (3.25,0) to [out=270, in=45] (2.75,-1.25) to [out=-135, in=90] (1.75,-3.5);
\end{tikzpicture}
};
(0,10)*{\text{{\tiny comultiplication}}};
(0,-10)*{\text{{\tiny degree $3$}}};
\endxy
\]
By construction,
the corresponding simple transitive
$2$-representation $\twocatstuff{M}_{\graphA{e}^{{\color{mygreen}g}}}$
of $\subcatquo[e]$ is equivalent to a cell $2$-representation
and decategorifies to $\algstuff{M}_{\graphA{e}^{{\color{mygreen}g}}}$ from \fullref{definition:n-modules}.
Similarly for the secondary colors ${\color{myorange}o}$ and ${\color{mypurple}p}$.
\subsubsection{Type \texorpdfstring{$\mathsf{D}$}{D} \texorpdfstring{$2$}{2}-representations}\label{subsec:cat-story-c}
Let $e\equiv 0 \bmod 3$.
In this case, the decomposition of the algebra $1$-morphism into simple $1$-morphisms in $\slqmod$ is given by
\[
\obstuff{A}^{\graphD{e}}\cong \algstuff{L}_{0,0}\oplus\algstuff{L}_{e,0}\oplus\algstuff{L}_{0,e}.
\]
In order to define the multiplication and unit $2$-morphisms of
$\obstuff{A}^{\graphD{e}}$ in $\slqmod$,
recall from the representation theory of $\mathfrak{sl}_{3}$ that
\[
(\algstuff{L}_{e,0})^{\ast}
\cong
(\mathrm{Sym}^{e}(\mathbb{C}^{3}))^{\ast}
\cong
\mathrm{Sym}^{e}((\mathbb{C}^{3})^{\ast})
\cong
\algstuff{L}_{0,e},
\]
where ${}^{\ast}$ means the dual module and $\mathrm{Sym}^{e}$ the $e^{\mathrm{th}}$ symmetric power.
Note that, by using e.g. \cite[Proposition 2.11]{BZ1}, we have similar isomorphisms in the quantum
case as well.
Thus, in order to delineate the monoidal subcategory
generated by the quantum
symmetric powers, we can use the symmetric web categories
described in \cite{RT1}, \cite{TVW1}, after adding the duals as in \cite{QS1}.
These symmetric web categories are built from certain labeled,
trivalent graphs, and we need the
monoidal subcategory of it generated by the objects $e,e^{\ast}$
and the morphisms
\[
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (0,0) node [above] {$e$} to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) node [above] {$e^{\ast}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (0,0) node [above] {$e^{\ast}$} to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) node [above] {$e$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (2,0) node [below] {$e$} to [out=90, in=0] (1,1) to [out=180, in=90] (0,0) node [below] {$e^{\ast}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (2,0) node [below] {$e^{\ast}$} to [out=90, in=0] (1,1) to [out=180, in=90] (0,0) node [below] {$e$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-1,1) node [above] {$e$} to (0,0);
\draw[dstrand, Ymarked=.55] (1,1) node [above] {$e$} to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,-1) node [below] {$e^{\ast}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (-1,-1) node [below] {$e$} to (0,0);
\draw[dstrand, Xmarked=.55] (1,-1) node [below] {$e$} to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,1) node [above] {$e^{\ast}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (-1,1) node [above] {$e^{\ast}$} to (0,0);
\draw[dstrand, Xmarked=.55] (1,1) node [above] {$e^{\ast}$} to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,-1) node [below] {$e$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-1,-1) node [below] {$e^{\ast}$} to (0,0);
\draw[dstrand, Ymarked=.55] (1,-1) node [below] {$e^{\ast}$} to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,1) node [above] {$e$};
\end{tikzpicture}
\]
(our reading conventions are still from bottom to top),
subject to some relations which are all stated e.g. in \cite[Section 2.1]{RW1}.
(We stress that some of the
cited papers work with $\mathrm{gl}$ instead of $\mathrm{sl}$, and
we also semisimplify according to the cut-off as in \eqref{eq:weight-picture}.
In diagrammatic terms this amounts to a slightly different web calculus where
e.g. an edge of label $2e$ in \cite[Section 2.1]{RW1} is identified
in our notation with an
edge of label $e^{\ast}$ with the orientation reversed.)
These are basically thick, but uncolored versions
of the webs which we met in \fullref{subsec:quotient-category}, and the
object $e$ corresponds to the $e^{\mathrm{th}}$ quantum symmetric power
of $\algstuff{L}_{1,0}$, which is $\algstuff{L}_{e,0}$, and $e^{\ast}$ to its dual,
which is $\algstuff{L}_{0,e}$.
Hence, we can use the diagrammatic calculus of symmetric webs to
describe the intertwiners in $\sltcat[\varstuff{q}]$ that we need. So far, we have
assumed that $\varstuff{q}$ is a generic parameter. By putting it equal to a primitive, complex
$2(e+3)^{\mathrm{th}}$ root of unity $\varstuff{\eta}$, we get a projection onto $\sltcat[e]$, and we can use the
specialized relations of the symmetric web calculus.
To be absolutely clear, we do not claim that the symmetric web calculi are equivalent
to the monoidal subcategories in question. All we
need is that the functor from the web calculus to $\sltcat[e]$ is full, which is true.
We use the following shorthand:
\[
\algstuff{L}_{0,0}\leftrightsquigarrow\emptyset,
\quad
\algstuff{L}_{e,0}\leftrightsquigarrow e,
\quad
\algstuff{L}_{0,e}\leftrightsquigarrow e^{\ast},
\]
and $\algstuff{L}_{e,0}\otimes\algstuff{L}_{0,e}\leftrightsquigarrow ee^{\ast}$ etc.,
where we omit the $\otimes$ symbol.
\begin{proposition}\label{proposition:typeD-object}
$\obstuff{A}^{\graphD{e}}$ has the structure of a Frobenius object in $\slqmod$ with
unit $\iota\colon\emptyset\to\obstuff{A}^{\graphD{e}}$, $\iota(1)=1$,
counit $\epsilon\colon\obstuff{A}^{\graphD{e}}\to\emptyset$, $\epsilon(1)=1$ and multiplication
$m\colon\obstuff{A}^{\graphD{e}}\otimes\obstuff{A}^{\graphD{e}}\to\obstuff{A}^{\graphD{e}}$ given by
\begin{gather}\label{eq:typeD-mult-table}
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep={1.0cm,between origins}, column
sep={1.0cm,between origins}, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{a} \&
\emptyset\emptyset \& \emptyset e \& \emptyset e^{\ast} \&
e\emptyset \& ee \& ee^{\ast} \&
e^{\ast}\emptyset \& e^{\ast}e \& e^{\ast}e^{\ast}
\\
\emptyset
\&
\emptyset \& {\color{mygray}0} \& {\color{mygray}0} \&
{\color{mygray}0} \& {\color{mygray}0} \& \begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=90, in=0] (.5,.5) to [out=180, in=90] (0,0);
\node at (0,-.4) {$e$};
\node at (1.1,-.4) {$e^{\hspace*{-.04cm}\ast}$};
\node at (0,1) {$\phantom{e}$};
\end{tikzpicture} \&
{\color{mygray}0} \& \begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (0,0) to [out=90, in=180] (.5,.5) to [out=0, in=90] (1,0);
\node at (.1,-.4) {$e^{\hspace*{-.04cm}\ast}$};
\node at (1,-.4) {$e$};
\node at (0,1) {$\phantom{e}$};
\end{tikzpicture} \& {\color{mygray}0}
\\
e
\&
{\color{mygray}0} \& \begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (.25,0) to [out=90, in=270] (-.25,1);
\node at (.25,-.4) {$e$};
\node at (-.25,1.15) {$e$};
\end{tikzpicture} \& {\color{mygray}0} \&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.25,0) to [out=90, in=270] (.25,1);
\node at (-.25,-.4) {$e$};
\node at (.25,1.15) {$e$};
\end{tikzpicture} \& {\color{mygray}0} \& {\color{mygray}0} \&
{\color{mygray}0} \& {\color{mygray}0} \& \begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (0,0) to (0,.5);
\node at (-.35,-.9) {$e^{\hspace*{-.04cm}\ast}$};
\node at (.6,-.9) {$e^{\hspace*{-.04cm}\ast}$};
\node at (0,.65) {$e$};
\end{tikzpicture}
\\
e^{\ast}
\&
{\color{mygray}0} \& {\color{mygray}0} \& \begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (.25,-1) to [out=90, in=270] (-.25,0);
\node at (.35,-1.4) {$e^{\hspace*{-.04cm}\ast}$};
\node at (-.15,.1) {$e^{\hspace*{-.04cm}\ast}$};
\end{tikzpicture} \&
{\color{mygray}0} \& \begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,.5);
\node at (-.5,-.9) {$e$};
\node at (.5,-.9) {$e$};
\node at (.1,.6) {$e^{\hspace*{-.04cm}\ast}$};
\end{tikzpicture} \& {\color{mygray}0} \&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-.25,-1) to [out=90, in=270] (.25,0);
\node at (-.15,-1.4) {$e^{\hspace*{-.04cm}\ast}$};
\node at (.35,.1) {$e^{\hspace*{-.04cm}\ast}$};
\end{tikzpicture} \& {\color{mygray}0} \& {\color{mygray}0}
\\
};
\draw[densely dashed] ($(m-1-1.south west)+ (-.2,-.3)$) to ($(m-1-3.south east)+ (7.25,-.3)$);
\draw[densely dashed] ($(m-1-1.south west)+ (-.2,-2.0)$) to ($(m-1-3.south east)+ (7.25,-2.0)$);
\draw[densely dashed] ($(m-1-1.south west)+ (-.2,-3.8)$) to ($(m-1-3.south east)+ (7.25,-3.8)$);
\draw[densely dashed] ($(m-1-1.east)+ (.25,.35)$) to ($(m-3-1.east)+ (.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (1.25,.35)$) to ($(m-3-1.east)+ (1.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (2.25,.35)$) to ($(m-3-1.east)+ (2.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (3.25,.35)$) to ($(m-3-1.east)+ (3.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (4.25,.35)$) to ($(m-3-1.east)+ (4.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (5.25,.35)$) to ($(m-3-1.east)+ (5.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (6.25,.35)$) to ($(m-3-1.east)+ (6.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (7.25,.35)$) to ($(m-3-1.east)+ (7.25,-2.4)$);
\draw[densely dashed] ($(m-1-1.east)+ (8.25,.35)$) to ($(m-3-1.east)+ (8.25,-2.4)$);
\end{tikzpicture}
\end{gather}
The comultiplication $\Delta\colon\obstuff{A}^{\graphD{e}}\to\obstuff{A}^{\graphD{e}}\otimes\obstuff{A}^{\graphD{e}}$
is given by transposing \eqref{eq:typeD-mult-table} and turning the symmetric webs upside down.
\end{proposition}
We omit the edge labels (which are always $e$ or $e^{\ast}$) from now on,
and also sometimes silently identify $e\emptyset=e$ etc.
\begin{proof}
First, we observe that the relations in the symmetric web calculus are invariant under
horizontal and vertical reflections, which
reduces the number of cases we need to verify. For example,
checking the unitality of $\obstuff{A}^{\graphD{e}}$ boils down
to checking the commutativity of
\[
\xymatrix@C+=1.0cm@L+=6pt{
e
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (.25,0) to [out=90, in=270] (-.25,1);
\end{tikzpicture}}
\ar[dr]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (0,0) to (0,1);
\end{tikzpicture}
}
&
e\emptyset
\ar[d]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.25,0) to [out=90, in=270] (.25,1);
\end{tikzpicture}}
\\
&
e
}
\]
which follows directly from the symmetric web calculus.
Next, we show that $m$ and $\Delta$ are (co)associative.
Up to symmetries and trivial compositions, we need
to check that
\[
\xymatrix@C+=1.0cm@L+=6pt{
eee
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,.5);
\draw[dstrand, Xmarked=.7] (1,-.5) to (1,.5);
\end{tikzpicture}}
\ar[d]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,.5);
\draw[dstrand, Xmarked=.7] (-1,-.5) to (-1,.5);
\end{tikzpicture}}
&
e^{\ast}e
\ar[d]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (0,0) to [out=90, in=180] (.5,.5) to [out=0, in=90] (1,0);
\end{tikzpicture}}
\\
ee^{\ast}
\ar[r]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=90, in=0] (.5,.5) to [out=180, in=90] (0,0);
\end{tikzpicture}}
&
\emptyset
}
\quad\quad
\xymatrix@C+=1.0cm@L+=6pt{
eee^{\ast}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,.5);
\draw[dstrand, Ymarked=.55] (1,-.5) to (1,.5);
\end{tikzpicture}}
\ar[d]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=90, in=0] (.5,.5) to [out=180, in=90] (0,0);
\draw[dstrand, Xmarked=.7] (-.5,0) to [out=90, in=270] (-.5,1);
\end{tikzpicture}}
&
e^{\ast}e^{\ast}
\ar[d]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (0,0) to (0,.5);
\end{tikzpicture}}
\\
e\emptyset
\ar[r]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (0,0) to [out=90, in=270] (0,1);
\end{tikzpicture}}
&
e
}
\quad\quad
\xymatrix@C+=1.0cm@L+=6pt{
ee^{\ast}e
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=90, in=0] (.5,.5) to [out=180, in=90] (0,0);
\draw[dstrand, Xmarked=.7] (1.5,0) to [out=90, in=270] (1.5,1);
\end{tikzpicture}}
\ar[d]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (0,0) to [out=90, in=180] (.5,.5) to [out=0, in=90] (1,0);
\draw[dstrand, Xmarked=.7] (-.5,0) to [out=90, in=270] (-.5,1);
\end{tikzpicture}}
&
\emptyset e
\ar[d]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (0,0) to [out=90, in=270] (0,1);
\end{tikzpicture}}
\\
e\emptyset
\ar[r]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (0,0) to [out=90, in=270] (0,1);
\end{tikzpicture}}
&
e
}
\]
commute. The leftmost case is just an isotopy in the symmetric web calculus.
The other two cases follow by observing that we have
\begin{gather}\label{eq:sym-web-rels}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
=
1
\quad
\text{and}
\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (0,-3) to (0,-2);
\draw[dstrand, Ymarked=.55] (0,-2) to [out=180, in=270] (-.75,-1) to [out=90, in=180] (0,0);
\draw[dstrand, Ymarked=.55] (0,-2) to [out=0, in=270] (.75,-1) to [out=90, in=0] (0,0);
\draw[dstrand, Xmarked=.7] (0,0) to (0,1);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (0,-3) to (0,1);
\end{tikzpicture}
\quad
\text{and}
\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.7] (0,-3) to (0,-2);
\draw[dstrand, Xmarked=.55] (0,-2) to [out=180, in=270] (-.75,-1) to [out=90, in=180] (0,0);
\draw[dstrand, Xmarked=.55] (0,-2) to [out=0, in=270] (.75,-1) to [out=90, in=0] (0,0);
\draw[dstrand, Ymarked=.7] (0,0) to (0,1);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (0,-3) to (0,1);
\end{tikzpicture}
\end{gather}
by specializing the relations for symmetric powers
in \cite[Equations (12), (14) and (15)]{RW1}. (As recalled above,
a label $m+n=2e$ in their picture corresponds to $e^{\ast}$
in our notation and all $m+n=2e$ in \cite[Equations (12), (14) and (15)]{RW1}
are then to be replaced by $e$.) Here it is crucial that $\varstuff{\eta}$ is a
$2(e+3)^{\mathrm{th}}$ root of unity. For example, $\qqnumber{e{+}1}=\qqnumber{2}$ in this case,
which implies that $\qbinq{e{+}2}{e}=\qqnumber{e{+}1}\qqnumber{2}^{-1}=1$.
Next, the relations in \eqref{eq:sym-web-rels} give
\begin{gather}\label{eq:sym-rel-1}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\draw[dstrand, Ymarked=.55] (0,3) to [out=270, in=180] (1,2) to [out=0, in=270] (2,3);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (0,3) to (0,0);
\draw[dstrand, Ymarked=.55] (2,0) to (2,3);
\end{tikzpicture}
\quad
\text{and}
\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\draw[dstrand, Xmarked=.55] (0,3) to [out=270, in=180] (1,2) to [out=0, in=270] (2,3);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (0,3) to (0,0);
\draw[dstrand, Xmarked=.55] (2,0) to (2,3);
\end{tikzpicture}
\quad
\text{and}
\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,1) to (2,0);
\draw[dstrand, Ymarked=.55] (1,1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,3) to (1,2);
\draw[dstrand, Ymarked=.55] (2,3) to (1,2);
\draw[dstrand, Ymarked=.55] (1,1) to (1,2);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (0,3) to (0,0);
\draw[dstrand, Ymarked=.55] (2,3) to (2,0);
\end{tikzpicture}
\quad
\text{and}
\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (1,1) to (2,0);
\draw[dstrand, Xmarked=.55] (1,1) to (0,0);
\draw[dstrand, Xmarked=.55] (0,3) to (1,2);
\draw[dstrand, Xmarked=.55] (2,3) to (1,2);
\draw[dstrand, Xmarked=.55] (1,1) to (1,2);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.55] (0,3) to (0,0);
\draw[dstrand, Xmarked=.55] (2,3) to (2,0);
\end{tikzpicture}
\end{gather}
which are needed to show associativity.
For $\obstuff{A}^{\graphD{e}}$ to be Frobenius, we additionally need to check that
\[
\xymatrix@C+=1.0cm@L+=6pt{
ee
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,-.5) to [out=90, in=270] (-1,.5);
\draw[dstrand, Xmarked=.7] (0,-.5) to (0,.5);
\end{tikzpicture}}
\ar[d]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,.5);
\end{tikzpicture}}
&
e\emptyset e
\ar[d]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (1,-.5) to [out=90, in=270] (.5,.5);
\draw[dstrand, Xmarked=.7] (0,-.5) to (0,.5);
\end{tikzpicture}}
\\
e^{\ast}
\ar[r]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-.5,.5) to (0,0);
\draw[dstrand, Ymarked=.55] (.5,.5) to (0,0);
\draw[dstrand, Xmarked=.7] (0,0) to (0,-.5);
\end{tikzpicture}}
&
ee
}
\;\;\;\;
\xymatrix@C+=1.0cm@L+=6pt{
ee^{\ast}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,.5) to (0,0);
\draw[dstrand, Xmarked=.7] (.5,.5) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,-.5);
\draw[dstrand, Ymarked=.5] (1,-.5) to (1,.5);
\end{tikzpicture}}
\ar[d]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=90, in=0] (.5,.5) to [out=180, in=90] (0,0);
\end{tikzpicture}}
&
e^{\ast}e^{\ast}e^{\ast}
\ar[d]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (0,0) to (0,.5);
\draw[dstrand, Ymarked=.55] (-1,-.5) to (-1,.5);
\end{tikzpicture}}
\\
\emptyset
\ar[r]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=270, in=0] (.5,-.5) to [out=180, in=270] (0,0);
\end{tikzpicture}}
&
e^{\ast}e
}
\;\;\;\;
\xymatrix@C+=1.0cm@L+=6pt{
ee^{\ast}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (-.5,.5) to (0,0);
\draw[dstrand, Ymarked=.55] (.5,.5) to (0,0);
\draw[dstrand, Xmarked=.7] (0,0) to (0,-.5);
\draw[dstrand, Ymarked=.5] (-1,.5) to (-1,-.5);
\end{tikzpicture}}
\ar[d]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=90, in=0] (.5,.5) to [out=180, in=90] (0,0);
\end{tikzpicture}}
&
eee
\ar[d]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Xmarked=.7] (-.5,-.5) to (0,0);
\draw[dstrand, Xmarked=.7] (.5,-.5) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,.5);
\draw[dstrand, Ymarked=.55] (1,.5) to (1,-.5);
\end{tikzpicture}}
\\
\emptyset
\ar[r]_{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[dstrand, Ymarked=.55] (1,0) to [out=270, in=0] (.5,-.5) to [out=180, in=270] (0,0);
\end{tikzpicture}}
&
e^{\ast}e
}
\]
commute, which follows from \eqref{eq:sym-rel-1}.
The other diagrammatic equations, which prove the
compatibility between the multiplication and the comultiplication, are immediate.
\end{proof}
\begin{proposition}\label{proposition:typeD}
The rank of the module category associated to $\obstuff{A}^{\graphD{e}}$ is equal to $\tfrac{t_e-1}{3}+3$,
the number of nodes of the graph $\graphD{e}$.
\end{proposition}
Note here that $t_e=\tfrac{(e+1)(e+2)}{2}\equiv 1\bmod 3$ since $e\equiv 0\bmod 3$, by assumption.
\begin{proof}
We first recall some
facts. By \cite[Section 3.3]{BK1} or \cite[Lemma 3.2.1]{Sch}, the so-called twist
of $\obstuff{A}^{\graphD{e}}$ is the identity morphism.
(This is false when $e\not\equiv 0 \bmod 3$.) Note also that $\obstuff{A}^{\graphD{e}}$ is
simple as an algebra $1$-morphism since it cannot have a proper, non-zero two-sided ideal, because any ideal containing
$e$ or $e^{\ast}$ has to contain $ee^{\ast}=e^{\ast}e$, which is isomorphic to the unit object
by e.g. \eqref{eq:sym-web-rels} and \eqref{eq:sym-rel-1}.
Moreover, let $\qdim(\obstuff{O})$ denote the quantum
dimension of $\obstuff{O}\in\slqmod$, which is defined by specializing the generic quantum dimension
at the primitive, complex root of unity $\varstuff{\eta}$, see e.g. \cite[Section 4.7]{EGNO}.
By Weyl's character formula \cite[Theorem 5.15]{Ja} and its specialization, we have
\[
\qdim(\algstuff{L}_{m,n})=\qqnumber{2}^{-1}\qqnumber{m+1}\qqnumber{n+1}\qqnumber{m+n+2}.
\]
Hence, we get $\qdim(\obstuff{A}^{\graphD{e}})=3\neq 0$.
By \cite[Lemma 1.20]{K-O}, this
implies that
$\obstuff{A}^{\graphD{e}}$ is rigid,
as defined in \cite[Definition 1.11]{K-O}.
Therefore, $\modcat{\slqmod}(\obstuff{A}^{\graphD{e}})$ is a semisimple
category with finitely many isomorphism classes of simples,
by \cite[Theorem 3.3]{K-O}. Furthermore, any simple module in
$\modcat{\slqmod}(\obstuff{A}^{\graphD{e}})$ is a direct
summand of $\morstuff{F}(\obstuff{O})$ for a certain $\obstuff{O}\in\slqmod$,
by \cite[Lemma 3.4]{K-O}. Here $\morstuff{F}\colon\slqmod\to
\modcat{\slqmod}(\obstuff{A}^{\graphD{e}})$ is the free functor
defined by $\morstuff{F}(\obstuff{O})=\obstuff{O}\otimes\obstuff{A}^{\graphD{e}}$.
By \cite[Lemma 1.16]{K-O}, this functor is biadjoint to the forgetful functor
$\morstuff{G}\colon\modcat{\slqmod}(\obstuff{A}^{\graphD{e}})\to
\slqmod$. As a last ingredient recall that
\[
\algstuff{L}_{m,n}\otimes\algstuff{L}_{e,0}\cong\algstuff{L}_{e{-}m{-}n,m},
\quad
\algstuff{L}_{m,n}\otimes\algstuff{L}_{0,e}\cong\algstuff{L}_{n,e{-}m{-}n},
\]
hold in $\slqmod$ by e.g. \cite[Corollary 8]{Saw}.
It is now easy to determine the simples in $\modcat{\slqmod}(\obstuff{A}^{\graphD{e}})$.
Let us write $e=3r$.
\smallskip
\begin{enumerate}[label=$\blacktriangleright$]
\setlength\itemsep{.15cm}
\item Assume that $(m,n)\neq(r,r)$. Then
\[
\morstuff{GF}(\algstuff{L}_{m,n})\cong\algstuff{L}_{m,n}\oplus
\algstuff{L}_{e{-}m{-}n,m}\oplus\algstuff{L}_{n,e{-}m{-}n}.
\]
These three summands form a three element orbit of the cut-off of the positive
Weyl chamber under the rotation by $\tfrac{2\pi}{3}$.
Therefore, we have
\[
\dim_{\mathbb{C}}\algstuff{H}\mathrm{om}_{\obstuff{A}^{\graphD{e}}}(\morstuff{F}(\algstuff{L}_{m,n}),
\morstuff{F}(\algstuff{L}_{m,n}))=\dim_{\mathbb{C}}
\algstuff{H}\mathrm{om}_{\slqmod}(\algstuff{L}_{m,n},\morstuff{GF}(\algstuff{L}_{m,n}))=1.
\]
By the categorical version of Schur's lemma, see e.g.
\cite[Lemma 1.5.2]{EGNO}, $\morstuff{F}(\algstuff{L}_{m,n})$ is a
simple $\obstuff{A}^{\graphD{e}}$-module object.
Note further that
$\morstuff{GF}(\algstuff{L}_{m,n})\cong\morstuff{GF}(\algstuff{L}_{e-m-n,m})\cong\morstuff{GF}(\algstuff{L}_{n,e-m-n})$. Thus, we have
\[
\dim_{\mathbb{C}}\algstuff{H}\mathrm{om}_{\obstuff{A}^{\graphD{e}}}(\morstuff{F}(\algstuff{L}_{m,n}),\morstuff{F}(\algstuff{L}_{e-m-n,m}))=
\dim_{\mathbb{C}}\algstuff{H}\mathrm{om}_{\obstuff{A}^{\graphD{e}}}(\morstuff{F}(\algstuff{L}_{m,n}),\morstuff{F}(\algstuff{L}_{n,e-m-n}))=1,
\]
by adjointness of
$\morstuff{F}$ and $\morstuff{G}$. Thus,
$\morstuff{F}(\algstuff{L}_{m,n})
\cong\morstuff{F}(\algstuff{L}_{e-m-n,m})
\cong\morstuff{F}(\algstuff{L}_{n,e-m-n})$.
Finally, $\morstuff{F}(\algstuff{L}_{m,n})\not\cong\morstuff{F}(\algstuff{L}_{m^{\prime},n^{\prime}})$,
if $(m^{\prime},n^{\prime})\not\in\{(m,n),(e-m-n,m),(n,e-m-n)\}$, because
in that case $\morstuff{GF}(\algstuff{L}_{m,n})\not\cong\morstuff{GF}(\algstuff{L}_{m^{\prime},n^{\prime}})$.
\item Assume that $(m,n)=(r,r)$. Then
\[
\morstuff{GF}(\algstuff{L}_{r,r})\cong
\algstuff{L}_{r,r}\oplus\algstuff{L}_{r,r}\oplus\algstuff{L}_{r,r}.
\]
Here $\algstuff{L}_{r,r}$ is the fixed point in the cut-off of the positive
Weyl chamber under the rotation by $\tfrac{2\pi}{3}$.
Therefore, we have
\begin{gather}\label{eq:dim-argument}
\dim_{\mathbb{C}}\algstuff{H}\mathrm{om}_{\obstuff{A}^{\graphD{e}}}(\morstuff{F}(\algstuff{L}_{r,r}),\morstuff{F}(\algstuff{L}_{r,r}))=\dim_{\mathbb{C}}
\algstuff{H}\mathrm{om}_{\slqmod}(\algstuff{L}_{r,r}, \morstuff{GF}(\algstuff{L}_{r,r}))= 3.
\end{gather}
This shows that $\morstuff{F}(\algstuff{L}_{r,r})$ decomposes
into three simples, each of which is mapped to $\algstuff{L}_{r,r}$ by
$\morstuff{G}$, but which are
pairwise non-isomorphic as $\obstuff{A}^{\graphD{e}}$-module objects.
(Otherwise, the dimension in \eqref{eq:dim-argument} would be $5$ or $9$.)
\end{enumerate}
\smallskip
The statement now follows by counting.
\end{proof}
\begin{example}\label{example:hom-formula}
In case $e=3$ (the case illustrated on the left in
\fullref{fig:typeD} below),
we have ten simple objects in $\slqmod$:
\smallskip
\begin{enumerate}[label=$\blacktriangleright$]
\setlength\itemsep{.15cm}
\item $\algstuff{L}_{0,0},\algstuff{L}_{3,0},\algstuff{L}_{0,3}$, which have quantum dimension $\qqnumber{1}=1$.
\item $\algstuff{L}_{1,0},\algstuff{L}_{2,1},\algstuff{L}_{0,2}$ and $\algstuff{L}_{0,1},\algstuff{L}_{1,2},\algstuff{L}_{2,0}$, which have quantum dimension
$\qqnumber{3}=2$.
\item $\algstuff{L}_{1,1}$, which has quantum dimension $\qqnumber{2}\qqnumber{4}=3$.
\end{enumerate}
\smallskip
In contrast, the simple $\obstuff{A}^{\graphD{3}}$-module objects are:
\smallskip
\begin{enumerate}[label=$\blacktriangleright$]
\setlength\itemsep{.15cm}
\item $\algstuff{L}_{0,0}\oplus\algstuff{L}_{3,0}\oplus\algstuff{L}_{0,3}$, which have quantum dimension $3\qqnumber{1}=3$.
\item $\algstuff{L}_{1,0}\oplus\algstuff{L}_{2,1}\oplus\algstuff{L}_{0,2}$ and $\algstuff{L}_{0,1}\oplus\algstuff{L}_{1,2}\oplus\algstuff{L}_{2,0}$,
which have quantum dimension
$3\qqnumber{3}=6$.
\item Three non-isomorphic
copies of $\algstuff{L}_{1,1}$, which still have quantum dimension $3$.
\end{enumerate}
(The reader should compare this with the $\mathbb{Z}/3\mathbb{Z}$-symmetry in \fullref{fig:typeD}
and the identification respectively splitting of the vertices illustrated therein.)
\end{example}
\begin{remark}\label{remark:hom-formula}
The above classification is consistent
with the following analog of \eqref{eq:ineq-semisimple}.
Let
\[
\qdim(\slqmod)=
{\textstyle\sum_{0\leq m+n\leq e}}\,\qdim(\algstuff{L}_{m,n})^2.
\]
Since $\obstuff{A}^{\graphD{e}}$ is rigid, we have
\begin{gather}\label{eq:semisimpleincat}
\qdim(\obstuff{A}^{\graphD{e}})\qdim(\slqmod)=
{\textstyle\sum_{\obstuff{S}}}\; \qdim(\obstuff{S})^2
\end{gather}
where we sum over a complete set of pairwise
non-isomorphic, simple $\obstuff{A}^{\graphD{e}}$-module objects $\obstuff{S}$.
The formula in \eqref{eq:semisimpleincat} follows e.g. from \cite[Example 7.16.9(ii)]{EGNO}.
Note that in this example $\qdim(\obstuff{O})$ equals the Perron--Frobenius dimension of $\obstuff{O}$
as used in \cite[Example 7.16.9(ii)]{EGNO}
because $\obstuff{A}^{\graphD{e}}$ is rigid.
\end{remark}
By \fullref{proposition:typeD-object}, we see that $\obstuff{A}^{\graphD{e}}$
can be regarded as a Frobenius algebra $1$-morphism in $\slqmodgop$. Hence,
we get a Frobenius algebra $1$-morphism $\algstuff{B}^{\graphD{e}^{{\color{mygreen}g}}}\{-3\}$ in $\subcatquo[e]$.
\begin{remark}\label{remark:hard-clasps}
In this case it would be hard to write down explicitly the diagrams which define the structural
$2$-morphisms of $\algstuff{B}^{\graphD{e}^{{\color{mygreen}g}}}$, i.e. unit, multiplication, counit and comultiplication. The reason is
that the symmetric web calculus suppresses the clasps corresponding to
$\algstuff{L}_{e,0}$ and $\algstuff{L}_{0,e}$, i.e. the quantum symmetrizers and
antisymmetrizers on $e$ strands in Kuperberg's web calculus \cite{Kup}, cf. \fullref{example:clasps}.
\end{remark}
By \fullref{proposition:typeD}, $\twocatstuff{M}_{\graphD{e}^{{\color{mygreen}g}}}$ does not
correspond to the cell $2$-representation
if $e\equiv 0 \bmod 3$, and, by construction, it categorifies $\algstuff{M}_{\graphD{e}^{{\color{mygreen}g}}}$
from \fullref{definition:n-modules}. Similarly for ${\color{myorange}o}$ and ${\color{mypurple}p}$.
\subsubsection{Some simple transitive \texorpdfstring{$2$}{2}-representations}\label{subsec:cat-story-d}
Note that an equivalence of simple transitive $2$-re\-presentations decategorifies
to a $\N_{\intvpar}$-equivalence of transitive $\N_{\intvpar}$-representations. Hence,
the following is the summary of the above and \fullref{lemma:find-all-transitives-2}:
\begin{theoremqed}\label{theorem:typeA-D-cat}
Let us indicate by
${\color{mygreen}g},{\color{myorange}o},{\color{mypurple}p}$ the starting color. Then
we have (at least) the following simple transitive $2$-representations
of $\subcatquo[e]$.
\begin{gather}\label{eq:the-transitives-2}
\begin{tikzpicture}[baseline=(current bounding box.center),yscale=0.6]
\matrix (m) [matrix of math nodes, row sep=1em, column
sep=1em, text height=1.5ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{a}
\&
e\equiv 0\bmod 3
\&
e\not\equiv 0\bmod 3
\\
\text{$2$-reps.}
\&
\begin{gathered}
\twocatstuff{M}_{\graphA{e}^{{\color{mygreen}g}}},\,
\twocatstuff{M}_{\graphA{e}^{{\color{myorange}o}}},\,
\twocatstuff{M}_{\graphA{e}^{{\color{mypurple}p}}},
\\
\twocatstuff{M}_{\graphD{e}^{{\color{mygreen}g}}},\,
\twocatstuff{M}_{\graphD{e}^{{\color{myorange}o}}},\,
\twocatstuff{M}_{\graphD{e}^{{\color{mypurple}p}}}
\end{gathered}
\&
\twocatstuff{M}_{\graphA{e}^{{\color{mygreen}g}}}
\\
\text{quantity}
\&
6
\&
1
\\};
\draw[densely dashed] ($(m-1-1.south west)+ (-.75,0)$) to ($(m-1-3.south east)+ (.75,0)$);
\draw[densely dashed] ($(m-1-2.north west) + (-.9,0)$) to ($(m-1-2.north west) + (-.9,-4.2)$);
\draw[densely dashed] ($(m-1-2.north west) + (3.2,0)$) to ($(m-1-2.north west) + (3.2,-4.2)$);
\end{tikzpicture}
\end{gather}
Moreover, the $2$-representations $\twocatstuff{M}_{\graphA{e}}$
are the cell $2$-representations of $\subcatquo[e]$, and
all of these decategorify
to the corresponding $\subquo[e]$-representations in \eqref{eq:the-transitives}.
\end{theoremqed}
We note here that the case $e=0$ is not included
in our discussion above, but \fullref{theorem:typeA-D-cat} holds
in this case as well (but types $\mathsf{A}$ and $\mathsf{D}$ coincide), which can
be checked directly.
\subsection{Trihedral zigzag algebras}\label{subsec:quiver}
We are going to describe a weak categorification of the
$\N_{\intvpar}$-representations $\algstuff{M}_{\graphA{\infty}}$
and $\algstuff{M}_{\graphA{e}}$
from \fullref{subsec:decat-story}
using a certain quiver algebra. (By weak categorification
we mean the same as e.g. in \cite[Definition 1]{KMS1}.)
Below we let $\somevert{i},\somevert{j},\somevert{k}$
always be different elements in $\{\somevert{x},\somevert{y},\somevert{z}\}$.
Moreover, we write $\somevert{i}_{m,n}$ for the
idempotent at a given vertex labeled by $(m,n)$, and
a path from $\somevert{i}_{m,n}$ to
$\somevert{j}_{m^{\prime},n^{\prime}}$ is denoted by
$\pathx{i}{j}=\somevert{j}_{m^{\prime},n^{\prime}}
\circ\pathx{i}{j}\circ\somevert{i}_{m,n}$
(abusing notation,
we omit the idempotents) with compositions $\pathx{j}{k}\circ\pathx{i}{j}=\pathxx{i}{j}{k}$
and $\pathx{i}{j}\circ\loopy{k}=\pathx{i}{j}\loopy{k}$ etc.
\subsubsection{The trihedral zigzag algebra of level \texorpdfstring{$\infty$}{infty}}\label{subsec:quiver-algebra}
We work over $\mathbb{C}$ in this section.
\begin{definition}\label{definition:quiver}
Let $\zig[\ast]$ be the path algebra of the following quiver.
\begin{gather*}
\xy
(0,0)*{
\begin{tikzcd}[ampersand replacement=\&,row sep=large,column sep=scriptsize,arrows={shorten >=-.5ex,shorten <=-.5ex},labels={inner sep=.05ex}]
\dvert{y_{0,2}}
\arrow[out=90,in=120,loop,distance=1.25em,swap]{}{\loopy{x}}
\arrow[out=150,in=180,loop,distance=1.25em,swap]{}{\loopy{y}}
\arrow[out=210,in=240,loop,distance=1.25em,swap]{}{\loopy{z}}
\arrow[xshift=.6ex,<-]{dr}{\pathy{z}{y}}
\arrow[xshift=-.6ex,->,swap]{dr}{\pathx{y}{z}}
\&
\phantom{.}
\&
\dvert{x_{1,1}}
\arrow[out=30,in=60,loop,distance=1.25em,swap]{}{\loopy{x}}
\arrow[out=77,in=102,loop,distance=1.25em,swap]{}{\loopy{y}}
\arrow[out=120,in=150,loop,distance=1.25em,swap]{}{\loopy{z}}
\arrow[xshift=-.6ex,->,swap]{dl}{\pathy{x}{z}}
\arrow[xshift=.6ex,<-]{dl}{\pathx{z}{x}}
\arrow[yshift=-.4ex,<-]{ll}{\pathy{z}{x}}
\arrow[yshift=.4ex,->,swap]{ll}{\pathx{x}{z}}
\arrow[xshift=.6ex,<-]{dr}{\pathy{y}{x}}
\arrow[xshift=-.6ex,->,swap]{dr}{\pathx{x}{y}}
\&
\phantom{.}
\&
\dvert{z_{2,0}}
\arrow[out=60,in=90,loop,distance=1.25em,swap]{}{\loopy{x}}
\arrow[out=0,in=30,loop,distance=1.25em,swap]{}{\loopy{y}}
\arrow[out=300,in=330,loop,distance=1.25em,swap]{}{\loopy{z}}
\arrow[xshift=-.6ex,->,swap]{dl}{\pathy{z}{y}}
\arrow[xshift=.6ex,<-]{dl}{\pathx{y}{z}}
\arrow[yshift=-.4ex,<-]{ll}{\pathy{x}{z}}
\arrow[yshift=.4ex,->,swap]{ll}{\pathx{z}{x}}
\\
\phantom{.}
\&
\dvert{z_{0,1}}
\arrow[out=150,in=180,loop,distance=1.25em,swap,pos=0.575]{}{\loopy{x}}
\arrow[out=195,in=225,loop,distance=1.25em,swap]{}{\loopy{y}}
\arrow[out=240,in=270,loop,distance=1.25em,swap]{}{\loopy{z}}
\arrow[xshift=.6ex,<-]{dr}{\pathy{x}{z}}
\arrow[xshift=-.6ex,->,swap]{dr}{\pathx{z}{x}}
\&
\phantom{.}
\&
\dvert{y_{1,0}}
\arrow[out=0,in=30,loop,distance=1.25em,swap,pos=0.475]{}{\loopy{x}}
\arrow[out=315,in=345,loop,distance=1.25em,swap]{}{\loopy{y}}
\arrow[out=270,in=300,loop,distance=1.25em,swap]{}{\loopy{z}}
\arrow[xshift=-.6ex,->,swap]{dl}{\pathy{y}{x}}
\arrow[xshift=.6ex,<-]{dl}{\pathx{x}{y}}
\arrow[yshift=-.4ex,<-]{ll}{\pathy{z}{y}}
\arrow[yshift=.4ex,->,swap]{ll}{\pathx{y}{z}}
\&
\phantom{.}
\\
\phantom{.}
\&
\phantom{.}
\&
\dvert{x_{0,0}}
\arrow[out=195,in=225,loop,distance=1.25em,swap]{}{\loopy{x}}
\arrow[out=255,in=285,loop,distance=1.25em,swap]{}{\loopy{y}}
\arrow[out=315,in=345,loop,distance=1.25em,swap]{}{\loopy{z}}
\&
\phantom{.}
\&
\phantom{.}
\\
\end{tikzcd}};
(-37.5,38)*{\rotatebox{40}{{\Huge$\vdots$}}};
(0,38)*{\rotatebox{0}{{\Huge$\vdots$}}};
(37.5,38)*{\rotatebox{-40}{{\Huge$\vdots$}}};
(0,-31)*{\text{{\tiny living on the $\graphA{\infty}$ graph}}};
\endxy
\end{gather*}
We view $\zig[\ast]$ as being graded by putting paths $\pathx{i}{j}$
and loops $\loopy{i}$ in degree $2$.
\end{definition}
\begin{definition}\label{definition:quiver-infty}
Let $\zig[\infty]$ be the quotient algebra of $\zig[\ast]$ by the following
relations.
\smallskip
\begin{enumerate}[label=(\alph*)]
\setlength\itemsep{.15cm}
\renewcommand{\theenumi}{(\ref{definition:quiver-infty}.a)}
\renewcommand{\labelenumi}{\theenumi}
\item \label{enum:quiver-1} \textbf{Leaving a triangular face is zero.} Any oriented path of length two
between non-adjacent vertices is zero.
\renewcommand{\theenumi}{(\ref{definition:quiver-infty}.b)}
\renewcommand{\labelenumi}{\theenumi}
\item \label{enum:quiver-2} \textbf{The relations of the cohomology ring of
the variety of full flags in $\mathbb{C}^3$.} $\loopy{i}\loopy{j}=\loopy{j}\loopy{i}$,
$\loopy{x}+\loopy{y}+\loopy{z}=0$,
$\loopy{x}\loopy{y}+\loopy{x}\loopy{z}+\loopy{y}\loopy{z}=0$
and $\loopy{x}\loopy{y}\loopy{z}=0$.
\renewcommand{\theenumi}{(\ref{definition:quiver-infty}.c)}
\renewcommand{\labelenumi}{\theenumi}
\item \label{enum:quiver-3} \textbf{Sliding loops.}
$\pathx{i}{j}\loopy{i}=-\loopy{j}\pathx{i}{j}$,
$\pathx{i}{j}\loopy{j}=-\loopy{i}\pathx{i}{j}$ and
$\pathx{i}{j}\loopy{k}=\loopy{k}\pathx{i}{j}=0$.
\renewcommand{\theenumi}{(\ref{definition:quiver-infty}.d)}
\renewcommand{\labelenumi}{\theenumi}
\item \label{enum:quiver-4} \textbf{Zigzag.}
$\pathxx{i}{j}{i}=\loopy{i}\loopy{j}$.
\renewcommand{\theenumi}{(\ref{definition:quiver-infty}.e)}
\renewcommand{\labelenumi}{\theenumi}
\item \label{enum:quiver-5} \textbf{Zigzig equals
zag times loop.} $\pathxx{i}{j}{k}=\pathx{i}{k}\loopy{i}=-\loopy{k}\pathx{i}{k}$.
\end{enumerate}
\smallskip
We call $\zig[\infty]$ the trihedral zigzag algebra of
level $\infty$.
\end{definition}
The relations \ref{enum:quiver-1} to \ref{enum:quiver-5}
are homogeneous with respect to the degree defined in
\fullref{definition:quiver}, which endows $\zig[\infty]$ with
the structure of a graded algebra, and
we write $\qdim[\varstuff{v}](\underline{\phantom{a}})\in\mathbb{N}[\varstuff{v}]$ for
the graded dimension, viewing $\varstuff{v}$ as a
variable of degree $1$. Note that $\zig[\infty]$ is zero in all odd degrees, by definition.
\subsubsection{Some basic properties}\label{subsec:quiver-algebra-props}
\begin{proposition}\label{proposition:quadratic}
$\zig[\infty]$ is quadratic, i.e. it is generated in degree $2$ and the relations
are generated in degree $4$.
\end{proposition}
\begin{proof}
All relations except $\loopy{x}+\loopy{y}+\loopy{z}=0$ and
$\loopy{x}\loopy{y}\loopy{z}=0$ are of degree $4$.
The degree two relation shows that our
presentation is redundant: we could give a presentation with
fewer generators and no degree two relation. We prefer our
presentation, which is more symmetric and therefore easier
to write down. But one could get rid of the degree two relation by using only two degree two loops per vertex.
Thus, up to base change, remains to show that
$\loopy{x}\loopy{y}\loopy{z}=0$ is a consequence of
degree $4$ relations, which can be done as follows:
\[
\loopy{x}\loopy{y}\loopy{z}
\stackrel{\ref{enum:quiver-4}}{=}
\pathxx{x}{y}{x}\loopy{z}
\stackrel{\ref{enum:quiver-3}}{=}
0.
\]
This finishes the proof that $\zig[\infty]$ is quadratic.
\end{proof}
\begin{lemma}\label{lemma:quiver-homs}
Let $\obstuff{S}=\{(m{\pm1},n), (m{\pm}1,n{\mp}1), (m,n{\pm}1)\}$.
\begin{gather}\label{eq:basis-quiver}
\begin{aligned}
\algstuff{H}\mathrm{om}_{\mathbb{C}}(\somevert{i}_{m,n},\somevert{i}_{m^{\prime},n^{\prime}})
&=
\begin{cases}
\mathbb{C}
\{\somevert{i}_{m,n},\loopy{i},\loopy{j},\loopy{i}\loopy{j},\loopy{i}\loopy{k},
\loopy{i}^2\loopy{j}\}
, &\text{if }(m,n)=(m^{\prime},n^{\prime}),
\\
0, &\text{else},
\end{cases}
\\
\algstuff{H}\mathrm{om}_{\mathbb{C}}(\somevert{i}_{m,n},\somevert{j}_{m^{\prime},n^{\prime}})
&=
\begin{cases}
\mathbb{C}
\{\pathx{i}{j},\pathx{i}{j}\loopy{i}\}
, &\text{if }(m^{\prime},n^{\prime})\in\obstuff{S},
\\
0, &\text{else}.
\end{cases}
\end{aligned}
\end{gather}
Moreover, the non-trivial graded dimensions are
$\qdim[\varstuff{v}](\algstuff{H}\mathrm{om}_{\mathbb{C}}(\somevert{i}_{m,n},
\somevert{i}_{m,n}))=\varstuff{v}^3\vfrac{3}$ and
$\qdim[\varstuff{v}](\algstuff{H}\mathrm{om}_{\mathbb{C}}(\somevert{i}_{m,n},
\somevert{j}_{m^{\prime},n^{\prime}}))=\varstuff{v}\vnumber{2}$, when $(m^{\prime},n^{\prime})\in\obstuff{S}$.
\end{lemma}
\begin{proof}
This is clear for the trivial hom-spaces by
\ref{enum:quiver-1}. So let us focus on the non-trivial ones.
To this end, we first consider
homogeneous linear combinations of loops of degree $2,4,6$:
\begin{gather}\label{eq:loopy-rels}
\begin{gathered}
\loopy{x}+\loopy{y}=-\loopy{z},
\\
\loopy{x}^2+\loopy{x}\loopy{y}=-\loopy{x}\loopy{z},
\quad\quad
\loopy{x}\loopy{y}+\loopy{y}^2=-\loopy{y}\loopy{z},
\quad\quad
\loopy{x}\loopy{z}+\loopy{y}\loopy{z}=-\loopy{z}^2,
\\
\loopy{x}^2\loopy{z}=\loopy{x}\loopy{y}^2=\loopy{y}\loopy{z}^2
=
-\loopy{x}^2\loopy{y}=-\loopy{y}^2\loopy{z}=-\loopy{x}\loopy{z}^2.
\end{gathered}
\end{gather}
These relations follow immediately from \ref{enum:quiver-2},
and show that the endomorphism space of any vertex is spanned by the ones in \eqref{eq:basis-quiver}.
Next, we consider all homogeneous elements of degree $4$ in $\zig[\ast]$.
The ones that are composites of two paths leaving a
triangular face are zero by \ref{enum:quiver-1},
and the remaining ones are linear combinations of those
appearing in \ref{enum:quiver-3}, \ref{enum:quiver-4}, \ref{enum:quiver-5}.
The homogeneous elements of degree $6$ that are not annihilated by
\ref{enum:quiver-1} are
\begin{gather*}
\pathxxx{i}{j}{k}{i}
\stackrel{\ref{enum:quiver-5}}{=}
\pathxx{i}{k}{i} \loopy{i}
\stackrel{\ref{enum:quiver-4}}{=}
\loopy{i}^2\loopy{k}
\stackrel{\eqref{eq:loopy-rels}}{=}
-
\loopy{i}^2\loopy{j}
\stackrel{\ref{enum:quiver-4}}{=}
-
\pathxx{i}{j}{i} \loopy{i}
\stackrel{\ref{enum:quiver-5}}{=}
-
\pathxxx{i}{k}{j}{i},
\\
\pathxxx{i}{j}{i}{j}
\stackrel{\ref{enum:quiver-4}}{=}
\pathx{i}{j}
\loopy{i}\loopy{j}
\stackrel{\ref{enum:quiver-2}}{=}
\pathx{i}{j}
(-\loopy{i}\loopy{k}-\loopy{j}\loopy{k})
\stackrel{\ref{enum:quiver-3}}{=}
0,
\\
\pathxxx{i}{j}{i}{k}
\stackrel{\ref{enum:quiver-4}}{=}
\pathx{i}{k}\loopy{i}\loopy{j}
\stackrel{\ref{enum:quiver-3}}{=}
0,
\end{gather*}
including versions obtained by changing sources and targets.
Finally, we claim that
all homogeneous elements of degree $>6$ in $\zig[\ast]$
are zero in $\zig[\infty]$. For paths leaving a face or composites of only loops,
there is nothing to show by \ref{enum:quiver-1}
and \ref{enum:quiver-2}. For paths of length four around one
triangular face, we get
\begin{gather*}
\pathxxxx{i}{j}{k}{i}{j}
\stackrel{\ref{enum:quiver-5}}{=}
\pathxxx{i}{k}{i}{j}\loopy{i}
\stackrel{\ref{enum:quiver-4}}{=}
\pathx{i}{j}\loopy{i}^2\loopy{k}
\stackrel{\ref{enum:quiver-3}}{=}
0,
\\
\pathxxxx{i}{j}{k}{i}{k}
\stackrel{\ref{enum:quiver-5}}{=}
\pathxxx{i}{k}{i}{k}\loopy{i}
\stackrel{\ref{enum:quiver-4}}{=}
\pathx{i}{k}\loopy{i}^2\loopy{k}
\stackrel{\eqref{eq:loopy-rels}}{=}
-\pathx{i}{k}\loopy{i}^2\loopy{j}
\stackrel{\ref{enum:quiver-3}}{=}
0,
\\
\pathxxxx{i}{j}{i}{j}{i}
\stackrel{\ref{enum:quiver-4}}{=}
\pathxx{i}{j}{i}\loopy{i}\loopy{j}
\stackrel{\ref{enum:quiver-4}}{=}
\loopy{i}^2\loopy{j}^2
\stackrel{\eqref{eq:loopy-rels}}{=}
0,
\\
\pathxxxx{i}{j}{i}{k}{i}
\stackrel{\ref{enum:quiver-4}}{=}
\pathxx{i}{k}{i}\loopy{i}\loopy{j}
\stackrel{\ref{enum:quiver-4}}{=}
\loopy{i}^2\loopy{j}\loopy{k}
\stackrel{\ref{enum:quiver-2}}{=}
0.
\end{gather*}
Again, there are analogous relations obtained by changing sources and targets.
Altogether this shows that the sets in
\eqref{eq:basis-quiver} span the hom-spaces.
To show linear independence, we consider the following linear map, which on monomials in the path algebra is given by:
\[
\mathrm{tr}\colon
\zig[\infty]\to\mathbb{C},
\quad
\mathrm{tr}(a)
=
\begin{cases}
1, &\text{if }a\in\{\loopy{x}^2\loopy{z},\loopy{x}\loopy{y}^2,\loopy{y}\loopy{z}^2\},
\\
-1, &\text{if }a\in\{\loopy{x}^2\loopy{y},\loopy{y}^2\loopy{z},\loopy{x}\loopy{z}^2\},
\\
0, &\text{else}.
\end{cases}
\]
Note that $\mathrm{tr}$ is well-defined, which can be
checked by showing that the relations are preserved. It is also easy to see
that $\mathrm{tr}$ is a non-degenerate and symmetric trace form, e.g.
\begin{gather*}
\mathrm{tr}(\pathx{x}{y}\cdot\pathxx{y}{z}{x})
=
\mathrm{tr}(\pathxxx{x}{y}{z}{x})
\stackrel{\ref{enum:quiver-5}}{=}
\mathrm{tr}(\pathxx{x}{y}{x}\loopy{x})
\stackrel{\ref{enum:quiver-4}}{=}
\mathrm{tr}(\loopy{x}^2\loopy{y})
\\
\stackrel{\eqref{eq:loopy-rels}}{=}
\mathrm{tr}(-\loopy{x}\loopy{y}^2)
\stackrel{\ref{enum:quiver-4}}{=}
-\mathrm{tr}(\loopy{y}\pathxx{y}{x}{y})
\stackrel{\ref{enum:quiver-5}}{=}
\mathrm{tr}(\pathxxx{y}{z}{x}{y})
=
\mathrm{tr}(\pathxx{y}{z}{x}\cdot\pathx{x}{y})
\end{gather*}
We can now write down sets of morphisms
which are dual to the ones from \eqref{eq:basis-quiver} with
respect to $\mathrm{tr}$, e.g.:
\begin{align*}
(
\somevert{i}_{m,n},
\loopy{x},
\loopy{y},
\loopy{x}\loopy{y},
\loopy{x}\loopy{z},
\loopy{x}^2\loopy{z}
)
&
\leftrightsquigarrow
(
\loopy{x}^2\loopy{z},
\loopy{x}\loopy{z},
\loopy{x}\loopy{y},
\loopy{y},
\loopy{x},
\somevert{i}_{m,n}
),
\\
(
\pathx{i}{j}, \pathx{i}{j}\loopy{i}
)
&
\leftrightsquigarrow
(
\pm \pathx{j}{i}\loopy{j}, \pm \pathx{j}{i}
).
\end{align*}
Since these sets span the corresponding hom-spaces, we are done.
\end{proof}
The following result follows immediately from the proof of \fullref{lemma:quiver-homs}.
\begin{corollary}\label{corollary:frob}
$\zig[\infty]$ is a positively graded, symmetric Frobenius algebra.
\end{corollary}
\subsubsection{The quotient of level \texorpdfstring{$e$}{e}}\label{subsec:quotient-algebra-zig}
\begin{definition}\label{definition:quiver-e}
For fixed level $e$, let $\zigideal{e}$
be the two-sided ideal in $\zig[\infty]$ generated by
\begin{gather*}
\left\{
\somevert{i}_{m,n} \mid m+n\geq e+1
\right\}.
\end{gather*}
We define the trihedral zigzag algebra of level $e$ as
\[
\zig[e]=\zig[\infty]/\zigideal{e}
\]
and we call $\zigideal{e}$ the vanishing zigzag ideal of level $e$.
\end{definition}
Clearly, $\zig[e]$ has a basis given
by the elements in \eqref{eq:basis-quiver} for $m+n\leq e$.
Thus, $\zig[e]$ is a finite-dimensional, positively graded algebra, which is a symmetric Frobenius
algebra by \fullref{corollary:frob}. By the proof of
\fullref{proposition:quadratic} it is also quadratic,
as long as $e\neq 0$.
\subsubsection{Weak categorification}\label{subsec:quiver-algebra-weak}
Following ideas from
\cite{KS1}, \cite{AT1} and
\cite[Section 2]{MT1}, we let
$\lpro$, respectively $\rpro$, denote the
left, respectively right, ideals in $\zig[\infty]$
generated by $\somevert{i}_{m,n}$. These
are indecomposable, graded projective $\zig[\infty]$-modules, and
all indecomposable, graded projective left, respectively right, $\zig[\infty]$-modules
are of this form, up to grading shifts.
By the above, $\lpro\otimes\rpro$ is a biprojective
$\zig[\infty]$-bimodule, i.e. it is
projective as a left and as a right $\zig[\infty]$-module.
Therefore,
\begin{gather*}
\thetaf{{\color{mygreen}g}}(\underline{\phantom{a}})
=
{\textstyle\bigoplus_{\somevert{x}_{m,n}}}
(\lpro[\somevert{x}_{m,n}]\otimes \rpro[\somevert{x}_{m,n}]\{-3\})
\otimes_{\zig[\infty]}\underline{\phantom{a}},
\quad\;
\thetaf{{\color{myorange}o}}(\underline{\phantom{a}})
=
{\textstyle\bigoplus_{\somevert{y}_{m,n}}}
(\lpro[\somevert{y}_{m,n}]\otimes\rpro[\somevert{y}_{m,n}]\{-3\})
\otimes_{\zig[\infty]}\underline{\phantom{a}},
\\
\thetaf{{\color{mypurple}p}}(\underline{\phantom{a}})
=
{\textstyle\bigoplus_{\somevert{z}_{m,n}}}
(\lpro[\somevert{z}_{m,n}]\otimes\rpro[\somevert{z}_{m,n}]\{-3\})
\otimes_{\zig[\infty]}\underline{\phantom{a}},
\end{gather*}
define endofunctors on the category $\zigmod$ of
finite-dimensional, graded projective (left) $\zig$-modules.
Here $\otimes$ denotes $\otimes_{\mathbb{C}}$.
To state the weak categorification we denote by
$\twocatstuff{E}\mathrm{nd}(\zigmod)$ the category of endofunctors on $\zigmod$.
Considering $\subquo$ as a one-object category with the formal
object $\ast$ and morphisms being
its elements,
we get the following.
\begin{lemma}\label{lemma:endo-functors}
The functor $\subquo\to\twocatstuff{E}\mathrm{nd}(\zigmod)$ given by $\ast\mapsto\zigmod$ and
\[
\theta_{{\color{mygreen}g}}
\mapsto
\thetaf{{\color{mygreen}g}}(\underline{\phantom{a}}),
\quad\quad
\theta_{{\color{myorange}o}}
\mapsto
\thetaf{{\color{myorange}o}}(\underline{\phantom{a}}),
\quad\quad
\theta_{{\color{mypurple}p}}
\mapsto
\thetaf{{\color{mypurple}p}}(\underline{\phantom{a}})
\]
is well-defined.
Moreover, decategorification gives the transitive
$\N_{\intvpar}$-representation $\algstuff{M}_{\graphA{\infty}}$ of $\subquo$.
\end{lemma}
\begin{proof}
Let us first show that
$\thetaf{{\color{mygreen}g}}\thetaf{{\color{mygreen}g}}\cong \thetaf{{\color{mygreen}g}}^{\oplus \vfrac{3}}$, with
the superscript $\oplus \vfrac{3}$ meaning six degree-shifted copies of $\thetaf{{\color{mygreen}g}}$, with $\varstuff{v}$
corresponding to a degree-shift by one. Similar arguments show that the same holds for
$\thetaf{{\color{myorange}o}}$ and $\thetaf{{\color{mypurple}p}}$, of course. Note that $\thetaf{{\color{mygreen}g}}\thetaf{{\color{mygreen}g}}$ is given by
tensoring with the bimodule
\[
{\textstyle\bigoplus_{\somevert{x}_{m,n}}}\,
\lpro[\somevert{x}_{m,n}]\otimes\rpro[\somevert{x}_{m,n}]\otimes_{\zig[\infty]}
\lpro[\somevert{x}_{m,n}]\otimes\rpro[\somevert{x}_{m,n}]\{-6\},
\]
with all other direct summands being zero.
By definition, this is isomorphic to
\begin{gather*}
{\textstyle\bigoplus_{\somevert{x}_{m,n}}}\,
\lpro[\somevert{x}_{m,n}]\otimes \mathrm{End}_{\zig[\infty]}(\somevert{x}_{m,n})\otimes\rpro[\somevert{x}_{m,n}]\{-6\}\cong
{\textstyle\bigoplus_{\somevert{x}_{m,n}}}\,
\left(\lpro[\somevert{x}_{m,n}]\otimes
\rpro[\somevert{x}_{m,n}]\{-3\}\right)^{\oplus\vnumber{3}!},
\end{gather*}
where the displayed isomorphism follows from \fullref{lemma:quiver-homs}.
Next, we show that
$\thetaf{{\color{mygreen}g}}\thetaf{{\color{myorange}o}}\thetaf{{\color{mygreen}g}}\cong\thetaf{{\color{mygreen}g}}\thetaf{{\color{mypurple}p}}\thetaf{{\color{mygreen}g}}$.
Again, similar arguments show the
analogous result in the remaining cases. The functor
$\thetaf{{\color{mygreen}g}}\thetaf{{\color{myorange}o}}\thetaf{{\color{mygreen}g}}$
is given by tensoring with
\begin{gather}\label{eq:gogbimodule1}
{\textstyle\bigoplus}\,
\lpro[\somevert{x}_{m,n}]\otimes\rpro[\somevert{x}_{m,n}]\otimes_{\zig[\infty]} \lpro[\somevert{y}_{m^{\prime},n^{\prime}}]\otimes\rpro[\somevert{y}_{m^{\prime},n^{\prime}}] \otimes_{\zig[\infty]} \lpro[\somevert{x}_{m^{\prime\prime},n^{\prime\prime}}]\otimes\rpro[\somevert{x}_{m^{\prime\prime},n^{\prime\prime}}]\{-9\},
\end{gather}
where the direct sum is over all neighboring
pairs $(m,n), (m^{\prime},n^{\prime})$ and $(m^{\prime},n^{\prime}),
(m^{\prime\prime},n^{\prime\prime})$,
i.e. $(m^{\prime},n^{\prime})\in \{(m\pm 1,n), (m,n\pm 1), (m\pm 1, n\mp 1)\}$
and $(m^{\prime\prime},n^{\prime\prime})\in\{(m,n), (m\pm 1,n\pm 1),
(m\pm 2, n\mp 1), (m\pm 1, n\mp 2)\}$. This is isomorphic to
\begin{gather}\label{eq:gogbimodule2}
\begin{gathered}
{\textstyle\bigoplus}\,
\lpro[\somevert{x}_{m,n}]\otimes\algstuff{H}\mathrm{om}_{\zig[\infty]}
(\somevert{x}_{m,n}, \somevert{y}_{m^{\prime},n^{\prime}})\otimes
\algstuff{H}\mathrm{om}_{\zig[\infty]}(\somevert{y}_{m^{\prime},n^{\prime}},
\somevert{x}_{m^{\prime\prime},n^{\prime\prime}})\otimes\rpro[\somevert{x}_{m^{\prime\prime},n^{\prime\prime}}] \{-9\}
\cong
\\
{\textstyle\bigoplus}\,\left(
\lpro[\somevert{x}_{m,n}]\otimes
\rpro[\somevert{x}_{m^{\prime\prime},n^{\prime\prime}}]\{-7\}\right)^{\oplus \vnumber{2}^2},
\end{gathered}
\end{gather}
where the superscript $\oplus \vnumber{2}^2$ should be interpreted as before.
The isomorphism displayed in
\eqref{eq:gogbimodule2} follows from \fullref{lemma:quiver-homs}.
The functor $\thetaf{{\color{mygreen}g}}\thetaf{{\color{myorange}o}}\thetaf{{\color{mygreen}g}}$ is
given by tensoring with the $\zig[\infty]$-bimodule obtained from the
one in \eqref{eq:gogbimodule1} by replacing
$\somevert{y}_{m^{\prime},n^{\prime}}$ with $\somevert{z}_{m^{\prime},n^{\prime}}$, which is
also isomorphic to the $\zig[\infty]$-bimodule in \eqref{eq:gogbimodule2}.
Although the neighboring pairs change when
we replace $\somevert{y}_{m^{\prime},n^{\prime}}$
with $\somevert{z}_{m^{\prime},n^{\prime}}$,
their total number is equal by symmetry.
Thus, the final number of direct summands in \eqref{eq:gogbimodule2} is the
same in both cases. This finishes the
proof that $\thetaf{{\color{mygreen}g}}\thetaf{{\color{myorange}o}}\thetaf{{\color{mygreen}g}}\cong\thetaf{{\color{mygreen}g}}\thetaf{{\color{mypurple}p}}\thetaf{{\color{mygreen}g}}$.
\end{proof}
\begin{proposition}\label{proposition:endo-functors}
The functor from \fullref{lemma:endo-functors}
descends to a functor $\subquo[e]\to\twocatstuff{E}\mathrm{nd}(\zig[e])$.
Moreover, decategorification gives the transitive
$\N_{\intvpar}$-representation $\algstuff{M}_{\graphA{e}}$ of $\subquo[e]$.
\end{proposition}
\begin{proof}
Recall that the
$\N_{\intvpar}$-representation
$\algstuff{M}_{\graphA{e}}$
of $\subquo[e]$ from \fullref{definition:n-modules} satisfies
\begin{gather*}
\Mt[\boldsymbol{\Gamma}]
=
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{mygreen}g}})
+
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{myorange}o}})
+
\algstuff{M}_{\boldsymbol{\Gamma}}(\theta_{{\color{mypurple}p}})
=
\vnumber{2}\left(
\vnumber{3}\mathrm{Id}
+
A(\Gg)
\right).
\end{gather*}
Up to natural isomorphism, the same holds for the functors
$\thetaf{{\color{mygreen}g}},\thetaf{{\color{myorange}o}}$ and $\thetaf{{\color{mypurple}p}}$ when they are
applied to $\lpro[\somevert{x}_{m,n}],
\lpro[\somevert{y}_{m,n}]$ and $\lpro[\somevert{z}_{m,n}]$, by
\fullref{lemma:quiver-homs}. The proof uses exactly the same sort of arguments
as the proof of \fullref{lemma:endo-functors}. We therefore omit further details.
The statement then follows from \fullref{corollary:poly-killed}.
\end{proof}
\begin{remark}\label{remark:zig-zag-zog}
The quiver algebra defined in \cite{Gr1} is (in the
case of $\graphA{e}$ graphs) a subalgebra of $\zig[e]$.
We do not know any further connection between these two algebras.
In fact, up to certain scalars, the defining relations
of the quiver algebra in \fullref{definition:quiver-infty} are the ones of the quiver algebra
underlying the cell $2$-representations
of $\subcatquo[e]$. Those scalars can be computed
for small values of $e$, but we have not been able
to compute them for general $e$, unfortunately.
However, even without the correct scalars, we
thought that the trihedral zigzag algebra,
and its connection to \cite{Gr1},
was too nice to exclude it from this paper.
One also wonders whether any of the constructions
involving the zigzag algebras in \cite{HK1} have an analogue for the
trihedral zigzag algebras.
\end{remark}
\subsection{Generalizing dihedral \texorpdfstring{$2$}{2}-representation theory}\label{subsec:dihedral-group-cat}
\begin{dihedral}\label{remark:dihedral-group4}
Yet another analogy to the dihedral case $\dihquo[e]$ is provided
by \fullref{problem:classification} and \fullref{proposition:poly-killed}:
One can define transitive $\N_{\intvpar}$-representations of $\dihquo$
analogously to the $\N_{\intvpar}$-representations $\algstuff{M}_{\boldsymbol{\Gamma}}$, where
in the dihedral case $\boldsymbol{\Gamma}$ is any connected, bicolored graph. These
descend to the finite-dimensional $\dihquo[e]$ if and only if
$\pxy[A(\boldsymbol{\Gamma})]{e+1}=0$, where
$\pxy[\underline{\phantom{a}}]{e+1}$ is the Chebyshev
polynomial as in \fullref{remark:dihedral-group2}.
(This follows from \cite{KMMZ} and, a bit more directly, from \cite{MT1}.)
In that case, the analog of \fullref{problem:classification} has a
well-understood answer, namely $\boldsymbol{\Gamma}$ has to be of $\ADE$
Dynkin type.
\end{dihedral}
\begin{dihedral}\label{remark-zigzag}
In the dihedral case, the classification of
simple transitive $2$-representation is an
$\ADE$-type classification
(assuming gradeability), cf. \cite{KMMZ} and \cite{MT1}.
This follows from the classification of graphs recalled in
\fullref{remark:dihedral-group4} and the associated $2$-representations, which,
in analogy to \fullref{subsec:cat-story}, can be constructed using algebra $1$-morphisms
in the $\mathfrak{sl}_2$ analog of $\slqmod$ (cf. \cite[Section 7]{MMMT1}).
\end{dihedral}
\begin{dihedral}\label{remark-zigzag-real}
The quiver underlying the cell
$2$-representa\-tions in the dihedral case is the zigzag algebra from \cite{HK1}, which could be
presented as in our setup, although this is never done in the literature,
using two loops $\loopy{x},\loopy{y}$ at each vertex, subject to the relations of the cohomology ring of the variety of full flags in $\mathbb{C}^2$, i.e.
$\loopy{x}\loopy{y}=\loopy{y}\loopy{x}=0$, $\loopy{x}+\loopy{y}=0$.
(To make the connection with \cite{HK1}, note that this cohomology ring is
isomorphic to $\mathbb{C}[X]/(X^2)$.)
The same holds for all other simple transitive $2$-representations (in the dihedral case)
with the zigzag algebra for the corresponding graph, see \cite{MT1}. Thus, \fullref{subsec:quiver} can
be seen as the trihedral version of this. We think it would be interesting to work out the trihedral quiver algebras
for other simple transitive $2$-representations of $\subcatquo[e]$.
\end{dihedral}
\section{Introduction}\label{sec:intro}
\subsection*{Non-negative integral representation theory}\label{subsec:intro-first}
In pioneering work \cite{KaLu}, Kazhdan--Lusztig defined
their celebrated bases of Hecke algebras
for Coxeter groups. Crucially, on these bases the structure
constants of the algebras belong to
$\mathbb{N}=\mathbb{Z}_{\geq 0}$. This started a program to study $\mathbb{N}$-algebras,
which have a fixed basis with non-negative
integral structure constants, see e.g.
\cite{Lu2}, \cite{EK1}, where these algebras are called $\mathbb{Z}_+$-rings.
As proposed by the work of Kazhdan--Lusztig,
for $\mathbb{N}$-algebras it makes sense to study and classify
$\mathbb{N}$-representations, i.e. representations with a fixed basis on which the
fixed bases elements of the algebra act
by non-negative integral matrices, see e.g. \cite{EK1}.
The first examples are the so-called cell representations, which
were originally defined for Hecke algebras \cite{KaLu}, but can be defined for
all $\mathbb{N}$-algebras (and even $\mathbb{R}_{\geq 0}$-algebras, see
\cite{KM1}). As it turns out, $\mathbb{N}$-representations are interesting from various
points of view, with applications and connections to e.g. graph theory,
conformal field theory, fusion/modular tensor categories and subfactor theory.
\medskip
Categorical analogs of $\mathbb{N}$-algebras are
monoidal categories, which we consider as one-object $2$-categories,
or $2$-categories. These decategorify
to $\mathbb{N}$-algebras, because the isomorphism classes
of the indecomposable $1$-morphisms
form naturally a $\mathbb{N}$-basis. For example,
Hecke algebras of Coxeter groups are categorified by Soergel
bimodules \cite{So0} such that indecomposable bimodules decategorify to the
Kazhdan--Lusztig basis elements \cite{EW}.
The categorical incarnation of
$\mathbb{N}$-representation theory is $2$-re\-presentation theory.
Any $2$-re\-presentation decategorifies
naturally to a $\mathbb{N}$-representation, with the $\mathbb{N}$-basis given by
the isomorphism classes of the indecomposable $1$-morphisms. However, not
all $\mathbb{N}$-representations can be obtained in this way.
In $2$-representation theory, the
simple transitive $2$-representations play the role of the simple
representations \cite{MM5}. Although
their decategorifications need not be simple
as complex representations, they are the ``simplest'' $2$-representations,
as attested e.g. by the categorical
Jordan--H{\"o}lder theorem \cite{MM5}.
This naturally motivates the problem of classification of
simple transitive $2$-representations of $2$-categories. Just as the cell representations
form a natural class of $\mathbb{N}$-representations of any
$\mathbb{N}$-algebra, cell $2$-representations form a
natural class of simple transitive $2$-representations of any
finitary $2$-category (i.e. $2$-categories
with certain finiteness conditions \cite{MM1}).
A crucial difference is that cell $2$-representations are always simple
transitive, while cell representations are usually not simple.
\medskip
In this paper, we restrict our attention
to certain subquotients of the Hecke
algebra of affine type $\mathsf{A}_2$,
which we call \textit{trihedral Hecke algebras}, and their categorification by
subquotients of Soergel bimodules of affine
type $\mathsf{A}_2$, which we call
\textit{trihedral Soergel bimodules}.
These should have $2$-representations indexed by
\textit{tricolored generalized
$\ADE$ Dynkin diagrams} with \textit{trihedral zigzag algebras} making their appearance.
As we explain below, we think of these as
rank three analogs of dihedral
Hecke algebras, dihedral Soergel bimodules and zigzag algebras, respectively.
Finally, \cite{AT1} established a relation between dihedral
Soergel bimodules and the non-semisimple category of tilting modules
of quantum $\mathfrak{sl}_2$ at roots of unity. Based on that result and on
\cite{RiWi-tilting-p-canonical}, we expect there to be an interesting
relation between trihedral Soergel bimodules and a non-semisimple,
full subcategory of tilting modules of $\mathfrak{sl}_3$ at roots of
unity (or in prime characteristic).
\subsection*{The dihedral story}\label{subsec:intro-second}
For finite Coxeter types, the classification
of the simple transitive $2$-re\-presentations of Soergel bimodules
is only partially known, see e.g. \cite{KMMZ}, \cite{MMMZ}, \cite{Zi1}.
There are two exceptions:
For Coxeter type $\mathsf{A}$, the cell $2$-representations exhaust the
simple transitive $2$-representations of Soergel bimodules \cite{MM5},
so the classification problem has been solved.
For Coxeter type $\typei$, which is the type of the dihedral group
with $2(e+2)$ elements, there also
exists a complete classification of simple
transitive $2$-representations \cite{KMMZ}, \cite{MT1}
(for $e=10,16$ or $28$ the classification
is only known under the additional assumption of gradeability),
which is completely different from the one for type $\mathsf{A}$.
In this case, the simple transitive $2$-representations of rank greater than one
are classified by bicolored $\ADE$ Dynkin diagrams,
with the cell $2$-representations
being the ones corresponding to Dynkin diagrams
of type $\mathsf{A}$. The others, corresponding to Dynkin types $\mathsf{D}$
and $\mathsf{E}$, are not equivalent to cell
$2$-representations and revealed interesting new features
in $2$-representation theory, e.g. the two bicolorings of type $\mathsf{E}_7$
give non-equivalent $2$-representations which categorify the same $\mathbb{N}$-representation
\cite{MT1}. For completeness, we remark that
there are precisely two rank-one $2$-representations
corresponding to the highest and the lowest two-sided cells,
which categorify the trivial and the sign representation of the Hecke algebra.
We note that going to the small quotient $\twocatstuff{D}_{e}$ by annihilating the highest cell
avoids that we have to worry about the categorical analog of the trivial representation.
\medskip
This case is particularly interesting because
of Elias' quantum Satake correspondence \cite{El2}, \cite{El1} between
$\slqmod(\mathfrak{sl}_{2})$ and $\twocatstuff{D}_{e}$.
Here $\slqmod(\mathfrak{sl}_{2})$ denotes the semisimple quotient of the monoidal category of
finite-dimensional quantum $\mathfrak{sl}_{2}$-modules
(of type $1$), where the quantum parameter $\varstuff{\eta}$ is a primitive, complex
$2(e+2)^{\mathrm{th}}$ root of unity. This correspondence is given by
a nice, but slightly technical $2$-functor,
so we omit further details at this stage.
Note that, when $\varstuff{q}$ is generic, the quantum
Satake correspondence also exists,
but between the whole category of finite-dimensional
quantum $\mathfrak{sl}_{2}$-modules $\slqmod[\varstuff{q}](\mathfrak{sl}_{2})$
and Soergel bimodules of the infinite
dihedral type $\typei[\infty]$, which coincides with affine
type $\mathsf{A}_{1}$.
One consequence of Elias' Satake
correspondence is
a precise relation between the simple
transitive $2$-representations of
$\slqmod(\mathfrak{sl}_{2})$ and $\twocatstuff{D}_{e}$.
However, the corresponding $2$-representations
are not equivalent, because $\slqmod(\mathfrak{sl}_{2})$ is semisimple
while $\twocatstuff{D}_{e}$ is not.
\medskip
Let us explain this in a bit more detail.
Equivalence classes of simple transitive $2$-re\-presentations of
finitary $2$-categories (or graded versions of them) correspond
bijectively to Morita equivalence classes of simple
algebra $1$-morphisms in the abelianizations of these $2$-categories.
This was initially proved for
semisimple tensor categories \cite{Os1} and later
generalized to certain finitary
$2$-categories with duality \cite{MMMT1}.
Kirillov--Ostrik \cite{K-O} classified
the simple algebra $1$-morphisms
in $\slqmod(\mathfrak{sl}_{2})$ up to Morita equivalence,
under some natural assumptions, in terms of
$\ADE$ Dynkin diagrams. From their results,
via the quantum Satake correspondence, we can get all
indecomposable algebra $1$-morphisms, up to Morita
equivalence, in $\twocatstuff{D}_{e}$. (The latter is additive but
not abelian, which is why we get
indecomposable instead of simple algebra $1$-morphisms.)
Given an $\ADE$ Dynkin diagram $\boldsymbol{\Gamma}$ and the
corresponding algebra $1$-morphism $\morstuff{A}^{\boldsymbol{\Gamma}}$,
the category underlying the $2$-representation of $\slqmod(\mathfrak{sl}_{2})$ is equivalent to
the category of $\morstuff{A}^{\boldsymbol{\Gamma}}$-mo\-dules
in $\slqmod(\mathfrak{sl}_{2})$. The quiver of this category
is trivial: its vertices coincide with those of $\boldsymbol{\Gamma}$, but it has no edges
because $\slqmod(\mathfrak{sl}_{2})$ is semisimple. However,
the quiver underlying the corresponding simple transitive
$2$-representation of $\twocatstuff{D}_{e}$
is the so-called doubled quiver of type $\boldsymbol{\Gamma}$,
which has two oppositely oriented edges between each pair of adjacent vertices.
Its quiver algebra, the zigzag algebra, was for example
studied by Huerfano--Khovanov \cite{HK1}.
It has very nice properties and shows up in
various mathematical contexts nowadays.
\medskip
Kirillov--Ostrik's classification can be seen
as a quantum version of the McKay correspondence between
finite subgroups of $\mathrm{SU}(2)$
and $\ADE$ Dynkin diagrams. The vertices of such a Dynkin diagram $\boldsymbol{\Gamma}$
correspond to the simple $\morstuff{A}^{\boldsymbol{\Gamma}}$-modules
in $\slqmod(\mathfrak{sl}_{2})$. These module categories decategorify
to $\mathbb{N}$-representations of the
Grothendieck group of $\slqmod(\mathfrak{sl}_{2})$, the
so-called Verlinde algebra,
which were classified by Etingof--Khovanov \cite{EK1}. The Verlinde algebra is
isomorphic to a polynomial algebra in one
variable quotient by
the ideal generated by the $(e+1)^{\mathrm{th}}$
Chebyshev polynomial $\pxy[\varstuff{X}]{e+1}$
(normalized and of the second kind). Thus,
Etingof--Khovanov basically classified all non-negative integer matrices which are
killed by $\pxy[\varstuff{X}]{e+1}$. (Note that not all of them come from
$2$-representations of $\slqmod(\mathfrak{sl}_{2})$,
because some correspond to
graphs which are not Dynkin diagrams of type $\ADE$.)
Similarly, the Hecke algebra $\hecke(\typei)$ of Coxeter type $\typei$
can be obtained as a quotient of
the Hecke algebra $\hecke(\typeat{1})$ of affine type
$\mathsf{A}_{1}$, where $\varstuff{v}$ is a generic parameter
(the decategorification of the grading within the Soergel
$2$-category). Let $\theta_s,\theta_t$ denote
the Kazhdan--Lusztig generators corresponding to
the simple reflections, in
both $\hecke(\typei)$ and $\hecke(\typeat{1})$. Furthermore,
let $\theta_{w_{0}}$ be the Kazhdan--Lusztig
basis element in $\hecke(\typei)$ for the longest
word in the dihedral group. Then there are two ways to write
$\theta_{w_{0}}$ as a linear combination
of alternating products of $\theta_{s}$ and
$\theta_{t}$, which only differ by the choice of the
fixed final Kazhdan--Lusztig generator in each product.
The coefficients in both linear
combinations are precisely the coefficients of $\pxy[\varstuff{X}]{e+1}$.
(This observation is implicit in \cite{Lu1}.)
Then $\hecke(\typei)$ is obtained from $\hecke(\typeat{1})$
by declaring both these linear combinations
to be equal to each other. By
declaring them to be equal to zero,
we obtain the small quotient $\algstuff{D}_{e}$
of $\hecke(\typei)$, which is precisely the
algebra that corresponds to the Verlinde algebra under
the quantum Satake correspondence.
Moreover, one can show that these algebras
have a very similar $\mathbb{N}$-representation theory.
To conclude, one could say that Elias' quantum
Satake correspondence \cite{El2}, \cite{El1} categorifies the relation
between the Verlinde algebra and the small dihedral
quotient, while the results from
\cite{KMMZ}, \cite{MT1}, \cite{MMMT1}
categorify the relations between their $\mathbb{N}$-representations.
\subsection*{The trihedral story}\label{subsec:intro-third}
Now let us get to the topic of this paper. Elias
also defined a quantum Satake correspondence between
$\slqmod[e]=\slqmod[e](\mathfrak{sl}_3)$ and
the $2$-category of Soergel bimodules of affine type $\mathsf{A}_{2}$ \cite{El1}.
In this paper, we study certain
subquotients of these Soergel bimodules, depending on a choice
of a primitive, complex $2(e+3)^{\mathrm{th}}$
root of unity $\varstuff{\eta}$, and their $2$-representation theory.
Our construction uses the quantum Satake
correspondence with $\slqmod[e]$, whose
Grothendieck group is isomorphic to a polynomial algebra in
two variables quotient by the ideal generated by a set
of polynomials $\pxy{m,n}$, for $m+n=e+1, m,n\in\mathbb{N}$.
These polynomials were introduced to the field of orthogonal polynomials by
Koornwinder \cite{Ko} and they generalize the Chebyshev polynomials.
To the best of our knowledge, these subquotients are new
and have not been studied before.
\medskip
In fact, even their decategorifications,
which are certain subquotients of the Hecke algebra $\hecke(\typeat{2})$
of affine type $\mathsf{A}_{2}$, seem to be
new. For each $e\in\mathbb{N}$, we call the corresponding
subquotient the trihedral Hecke algebra of level $e$
and denoted it by $\subquo[e]$.
These algebras have their own Kazhdan--Lusztig
combinatorics and interesting $\mathbb{N}$-representations.
We see the trihedral Hecke algebras as rank
three analogs of the small quotients of the
dihedral Hecke algebras.
There are many similarities, but also some differences.
For example, as far as we can tell,
the trihedral Hecke algebras are not deformations of any
group algebra. But they are
semisimple algebras and the classification of their
irreducible representations
runs in parallel to the analogous classification for
dihedral Hecke algebras, and their $\mathbb{N}$-representation theory
has also a very similar behavior.
\medskip
Now to the categorified story: In the trihedral case, the quantum Satake correspondence
for $\varstuff{q}$ being generic only gives a
$2$-subcategory of the affine type $\mathsf{A}_{2}$ Soergel $2$-category. We call this
the $2$-category of trihedral Soergel bimodules of
level $\infty$ and denote it by $\subcatquo[\infty]$.
The $2$-category $\subcatquo[\infty]$
admits quotients
$\subcatquo[e]$,
the trihedral Soergel bimodules of
level $e$, which via the
quantum Satake correspondence for $\varstuff{\eta}$ is related to $\slqmod[e]$.
The corresponding decategorifications are
the trihedral Hecke algebras $\subquo[\infty]$
and $\subquo[e]$.
\medskip
Coming back to representation theory, people have studied the
$\mathbb{N}$-representations of the Grothendieck group of $\slqmod[e]$,
as they arise in conformal field theory and
the study of fusion/modular tensor
categories, see e.g. \cite{Ga1}, \cite{EP},
\cite{Sch} and related works. This time, four
families of graphs play an important role and, by analogy with
the $\mathfrak{sl}_{2}$ case, their types are called
$\mathsf{A}$, conjugate $\mathsf{A}$, $\mathsf{D}$ and $\mathsf{E}$, although they are
not Dynkin diagrams. Their adjacency matrices, which
are non-negative integral matrices,
are annihilated by Koornwinder's polynomials,
just as in \cite{EK1}.
Furthermore, the type $\mathsf{A}$ graphs can be seen
as a cut-off of the positive Weyl chamber of $\mathfrak{sl}_{3}$, just as the usual
type $\mathsf{A}$ Dynkin diagrams can be seen as cut-offs for $\mathfrak{sl}_{2}$.
Finally, the type $\mathsf{D}$ graphs for
$\mathfrak{sl}_{3}$ come from a $\Z/3\Z$-symmetry of
these cut-offs, just as the type $\mathsf{D}$ Dynkin
diagrams come from a $\Z/2\Z$-symmetry.
Simple algebra $1$-morphisms in $\slqmod[e]$
and the corresponding simple transitive $2$-representations
have also been studied e.g. in \cite{Sch}
and are closely related to these $\ADE$
type graphs. Via the quantum Satake correspondence, we therefore
get indecomposable algebra $1$-mor\-phisms in $\subquo[e]$
and the corresponding simple transitive
$2$-representations of the trihedral Soergel bimodules.
Since we are not familiar with some of the
ingredients in the construction of algebra $1$-morphisms in \cite{Sch}, we
have given an alternative construction, using the symmetric
$\mathfrak{sl}_{3}$-web calculus,
as in \cite{RT1}, \cite{TVW1}.
For this reason, our construction so
far only works for types $\mathsf{A}$ and $\mathsf{D}$, so
we restrict our attention to those two types. Almost by construction, the
cell $2$-representations are equivalent to the simple transitive
$2$-representations of type $\mathsf{A}$. The ones
of type $\mathsf{D}$ have a different rank and are therefore inequivalent.
For the other types, we have no
conjectures at all, and we are not even sure whether they
correspond to $2$-representations.
\medskip
Computing the quiver algebras explicitly proved to be much
harder this time. We define type $\mathsf{A}$ quiver algebras which, up to scalars,
are the ones underlying the cell $2$-representations of $\subcatquo[e]$.
These algebras are the trihedral analogs of the zigzag algebras
of type $\mathsf{A}$, e.g. the
endomorphisms algebras of their vertices
are the cohomology rings of the full flag variety of flags in
$\mathbb{C}^3$, instead of the flags in $\mathbb{C}^2$ as in the dihedral case.
For this reason, we call them trihedral zigzag algebras. The type $\mathsf{D}$
trihedral zigzag algebras can be obtained
from these by using the $\Z/3\Z$-symmetry, just as the
$\mathsf{D}$ dihedral zigzag algebras can be obtained from the type $\mathsf{A}$
via a $\Z/2\Z$-symmetry,
but we have not worked out the details.
Finally, let us stress that our trihedral
zigzag algebras are different from Grant's
\cite{Gr1} higher zigzag
algebras, which are only subalgebras of the
trihedral zigzag algebras of type $\mathsf{A}$, although both
underlying graphs come from a cut-off of the positive Weyl chamber of $\mathfrak{sl}_{3}$.
\subsection*{The Nhedral story}\label{subsec:intro-fourth}
We expect that our story generalizes to
$\mathfrak{sl}_N$ for arbitrary $N\geq 2$:
the Soergel bimodules of affine
type $\mathsf{A}_{N-1}$ are known, the quantum
Satake correspondence is conjectured to exist,
the analogs of Koornwinder's
Chebyshev polynomials are also known, and the corresponding
generalized $\ADE$ type graphs appear in the mathematical
physics literature on fusion algebras or
the classification of subgroups of quantum $\mathrm{SU}(N)$, see e.g. \cite{DFZ}, \cite{Oc}.
We expect that there exist \textit{$N$hedral
algebras} and \textit{$N$hedral Soergel bimodules} of level $e$
(where $\varstuff{\eta}$ would be a primitive, complex $2(e+N)^{\mathrm{th}}$ root of unity),
and \textit{$N$hedral zigzag algebras} of $\ADE$-type quivers
such that the endomorphism algebra
of every vertex is isomorphic to the cohomology ring
of the full flag variety of $\mathbb{C}^N$.
\begin{remarkcolor}\label{remark:colors}
We use colors in this paper (we recommend to read the paper in color),
and the colors which we need are blue $\,\tikz[baseline=-.05,scale=0.25]{\draw[myblue,fill=myblue] (0,0) rectangle (1,1);}\,$,
red $\,\tikz[baseline=-.05,scale=0.25]{\draw[myred,fill=myred] (0,0) rectangle (1,1);}\,$, yellow $\,\tikz[baseline=-.05,scale=0.25]{\draw[myyellow,fill=myyellow] (0,0) rectangle (1,1);}\,$, green $\,\tikz[baseline=-.05,scale=0.25]{\draw[mygreen,fill=mygreen] (0,0) rectangle (1,1);}\,$, orange $\,\tikz[baseline=-.05,scale=0.25]{\draw[myorange,fill=myorange] (0,0) rectangle (1,1);}\,$
and purple $\,\tikz[baseline=-.05,scale=0.25]{\draw[mypurple,fill=mypurple] (0,0) rectangle (1,1);}\,$ which will appear as indicated by the
preceding boxes.
\end{remarkcolor}
\begin{qconventions}\label{remark:qparamters}
The notation $\varstuff{v}$ will mean a generic parameter which plays the role of
the decategorification of the grading which we will meet in \fullref{sec:A2-diagrams}.
In contrast, $\varstuff{q}$ will also denote a generic parameter,
but it will turn up on the categorified
level as our quantum parameter.
Moreover, $\varstuff{\eta}$ will be a primitive, complex $2(e+3)^{\mathrm{th}}$
root of unity $\varstuff{\eta}^{2(e+3)}=1$
which is a specialization of $\varstuff{q}$, but never of $\varstuff{v}$.
Here $e\in\mathbb{N}=\mathbb{Z}_{\geq 0}$ will usually be arbitrary, but fixed, and is called the level.
The ground field will always be $\C_{\vpar}=\mathbb{C}(\varstuff{v})$,
$\Cq=\mathbb{C}(\varstuff{q})$ or $\mathbb{C}=\mathbb{C}(\varstuff{\eta})$, if not stated otherwise.
Sometimes, instead of working over a ground field,
we will work over rings as e.g. $\Z_{\intvpar}=\mathbb{Z}[\varstuff{v},\varstuff{v}^{-1}]$ or
semirings as e.g. $\N_{\intvpar}=\mathbb{N}[\varstuff{v},\varstuff{v}^{-1}]$ and their quantum counterparts.
(It will be clear from the notation whether we work
with $\varstuff{v}$ or $\varstuff{q}$. Moreover, a subscript $[\underline{\phantom{a}}]$
will always indicate that we are in the case of (semi)rings rather than fields.)
In this context, we use the
$\varstuff{v}$-numbers, factorials and binomials, where $s\in\mathbb{Z},t\in\mathbb{Z}_{\geq 1}$
\begin{gather}\label{eq:qnumbers-typeAD}
\vnumber{s}
=
\tfrac{\varstuff{v}^s - \varstuff{v}^{-s}}{\varstuff{v}^{\phantom{1}} - \varstuff{v}^{-1}},
\quad
\vfrac{t}
=
\vnumber{t} \vnumber{t{-}1} \dots \vnumber{1},
\quad
\vbinn{s}{t}
=
\tfrac{\vnumber{s} \vnumber{s{-}1}
\dots
\vnumber{s{-}t{+}1}}{\vnumber{t} \vnumber{t{-}1} \dots \vnumber{1}},
\end{gather}
all of which are in $\Z_{\intvpar}$.
By convention, $\vnumber{0}!=1=\vbin{s}{0}$.
Note that $\vnumber{0}=0=\vbin{0\leq s<t}{t}$
and $\vnumber{-s}=-\vnumber{s}$. Similarly with $\varstuff{q}$ or $\varstuff{\eta}$ instead of $\varstuff{v}$.
\end{qconventions}
\section*{Appendix: Generalized \texorpdfstring{$\ADE$}{ADE} Dynkin diagrams}\label{section:gen-D}
\renewcommand{\thesection}{App}
\renewcommand{\theequation}{\thesection-\arabic{equation}}
\renewcommand{\thefigure}{App-\arabic{figure}}
\setcounter{theoremm}{0}
\setcounter{equation}{0}
\setcounter{figure}{0}
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In this appendix we have listed certain solutions to \fullref{problem:classification}. Following \cite{Zu}, we call
these the generalized $\ADE$ Dynkin diagrams. The graphs below depend on the level $e$, which is
the same as e.g. in \fullref{section:sl3-stuff} and is indicated as a subscript.
\subsection{The list}\label{subsec:gen-D-list}
\makeautorefname{figure}{Figures}
The following are the generalized $\ADE$ Dynkin diagrams.
There are three infinite families, displayed in \fullref{fig:typeA}, \ref{fig:typeD}
and \ref{fig:typeC}, \makeautorefname{figure}{Figure} and a finite number of exceptions,
displayed in \fullref{fig:typeE}.
\begin{figure}[ht]
\[
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphA{0}$} to (0,0);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphA{1}$} to (1,1);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1);
\draw [thick, densely dashed, myred] (1,1) to (-1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphA{2}$} to (1,1) to (0,2);
\draw [thick, myyellow] (0,2) to (-2,2);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2);
\draw [thick, densely dotted, myblue] (0,2) to (2,2);
\draw [thick, densely dashed, myred] (2,2) to (1,1) to (-1,1) to (-2,2);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphA{3}$} to (1,1) to (0,2) to (1,3) to (3,3);
\draw [thick, myyellow] (0,2) to (-2,2) to (-3,3);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2) to (-1,3) to (-3,3);
\draw [thick, densely dotted, myblue] (0,2) to (2,2) to (3,3);
\draw [thick, densely dashed, myred] (1,1) to (-1,1) to (-2,2) to (-1,3) to (1,3) to (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphA{4}$} to (1,1) to (0,2) to (1,3) to (3,3) to (4,4);
\draw [thick, myyellow] (0,2) to (-2,2) to (-3,3) to (-2,4) to (0,4) to (1,3);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2) to (-1,3) to (-3,3) to (-4,4);
\draw [thick, densely dotted, myblue] (0,2) to (2,2) to (3,3) to (2,4) to (0,4) to (-1,3);
\draw [thick, densely dashed, myred] (1,1) to (-1,1) to (-2,2) to (-1,3) to (1,3) to (2,2) to (1,1);
\draw [thick, densely dashed, myred] (-4,4) to (-2,4) to (-1,3);
\draw [thick, densely dashed, myred] (4,4) to (2,4) to (1,3);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (4,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,4) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-4,4) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\dots
\]
\caption{The infinite family of (generalized) type $\mathsf{A}$, indexed by $e\in\mathbb{N}$. The graph of type $\graphA{e}$
can be obtained by cutting off the $\mathfrak{sl}_{3}$-weight lattice at level $e+1$, as in \eqref{eq:weight-picture}.}
\label{fig:typeA}
\end{figure}
\begin{figure}[ht]
\[
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphA{3}$} to (1,1) to (0,2) to (1,3) to (3,3);
\draw [thick, myyellow] (0,2) to (-2,2) to (-3,3);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2) to (-1,3) to (-3,3);
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\draw [thick, densely dashed, myred] (1,1) to (-1,1) to (-2,2) to (-1,3) to (1,3) to (2,2) to (1,1);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
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\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
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\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
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\draw [->, mypurple] (1.7,2.1) to (-.8,2.9);
\end{tikzpicture}
\rightsquigarrow
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphD{3}$} to (1,1);
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\draw [thick, myyellow] (2,2) to (1,1);
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\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphA{6}$} to (1,1) to (0,2) to (1,3) to (3,3) to (4,4);
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\draw [thick, myyellow] (4,4) to (3,5);
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\draw [thick, myyellow] (0,4) to (1,5);
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\draw [thick, densely dotted, myblue] (-4,4) to (-3,5);
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\node at (-3,5) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
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\node at (3,5) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
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\node at (-2,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-5,5) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,6) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,5) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (4,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (4,6) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
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\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-4,6) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,5) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,4) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,6) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (5,5) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\draw [thick, mypurple, ->] (1.8,6) to [out=180, in=30] (-3.8,4.2);
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\end{tikzpicture}
\rightsquigarrow
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphD{6}$} to (1,1) to (0,2) to (1,3);
\draw [thick, myyellow] (0,2) to (-2,2);
\draw [thick, myyellow] (-2,4) to (1,3) to (0,4);
\draw [thick, myyellow] (2,4) to (1,3);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2) to (-1,3);
\draw [thick, densely dotted, myblue] (0,2) to (2,2);
\draw [thick, densely dotted, myblue] (-2,4) to (-1,3) to (0,4);
\draw [thick, densely dotted, myblue] (2,4) to (-1,3);
\draw [thick, densely dashed, myred] (1,1) to (-1,1) to (-2,2) to (-1,3);
\draw [thick, densely dashed, myred] (1,3) to (2,2) to (1,1);
\draw [thick, densely dashed, myred, double] (-1,3) to (1,3);
\draw [thick, myyellow] (0,6) to [out=0, in=90] (2.5,4) to [out=270, in=0] (1,3);
\draw [thick, densely dotted, myblue] (0,6) to [out=180, in=90] (-2.5,4) to [out=270, in=180] (-1,3);
\draw [thick, densely dotted, myblue] (0,6) to [out=0, in=90] (3,4) to [out=270, in=0] (2,2);
\draw [thick, myyellow] (0,6) to [out=180, in=90] (-3,4) to [out=270, in=180] (-2,2);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (-2,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,6) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\dots
\]
\caption{The infinite family of (generalized) type $\mathsf{D}$, indexed by $e\equiv 0\bmod 3$ and $e\neq 0$.
The graph of type $\graphD{e}$ comes from the $\Z/3\Z$-symmetry of the graph of type $\graphA{e}$ with
the fixed points splitting into three copies, cf. \fullref{example:hom-formula}.
(Note the double edges.) By convention, $\graphA{0}=\graphD{0}$.
}\label{fig:typeD}
\end{figure}
\begin{figure}[ht]
\[
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphC{1}\cong\graphA{1}$} to (1,1);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1);
\draw [thick, densely dashed, myred] (1,1) to (-1,1);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
,
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphC{2}\cong\graphA{2}$} to (1,1) to (0,2) to (-2,2);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2) to (2,2);
\draw [thick, densely dashed, myred] (2,2) to (1,1) to (-1,1) to (-2,2);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (0,1.35) {\text{\tiny$\graphC{1}$}};
\node at (0,2.35) {\text{\tiny$\phantom{3}$}};
\end{tikzpicture}
,
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphC{3}$} to (1,1) to (0,2) to (-2,2) to [out=270, in=180] (0,0);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2) to (2,2) to [out=270, in=0] (0,0);
\draw [thick, densely dashed, myred] (2,2) to (1,1) to (-1,1) to (-2,2) to [out=20, in=160] (2,2);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (0,2.35) {\text{\tiny$\phantom{3}$}};
\end{tikzpicture}
,
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphC{4}$} to (2,2) to [out=90, in=0] (0,4) to (-4,4);
\draw [thick, myyellow] (0,4) to (-1,3) to (0,2) to (2,2);
\draw [thick, densely dotted, myblue] (0,0) to (-2,2) to [out=90, in=180] (0,4) to (4,4);
\draw [thick, densely dotted, myblue] (0,4) to (1,3) to (0,2) to (-2,2);
\draw [thick, densely dashed, myred] (4,4) to (2,2) to [out=200, in=340] (-2,2) to (-4,4);
\draw [thick, densely dashed, myred] (-2,2) to (-1,3) to (1,3) to (2,2);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-4,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (4,4) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-.05,2.7) {\text{\tiny$\graphC{3}$}};
\node at (0,4.75) {\text{\tiny$\phantom{3}$}};
\end{tikzpicture}
,
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphC{5}$} to (2,2) to [out=90, in=0] (0,4) to (-4,4) to [out=270, in=180] (0,0);
\draw [thick, myyellow] (0,4) to (-1,3) to (0,2) to (2,2);
\draw [thick, densely dotted, myblue] (0,0) to (-2,2) to [out=90, in=180] (0,4) to (4,4) to [out=270, in=0] (0,0);
\draw [thick, densely dotted, myblue] (0,4) to (1,3) to (0,2) to (-2,2);
\draw [thick, densely dashed, myred] (4,4) to (2,2) to [out=200, in=340] (-2,2) to (-4,4) to [out=20, in=160] (4,4);
\draw [thick, densely dashed, myred] (-2,2) to (-1,3) to (1,3) to (2,2);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-4,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (4,4) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (0,4.75) {\text{\tiny$\phantom{3}$}};
\end{tikzpicture}
\cdots
\]
\caption{The infinite family of conjugate type $\mathsf{A}$, indexed by $e\in\mathbb{N}$.
The graph of type $\graphC{e}$ comes from an iterative procedure on the graph of type $\graphA{e}$.
By convention, $\graphA{0}=\graphC{0}$.
}\label{fig:typeC}
\end{figure}
\begin{figure}[ht]
\[
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (-1,3) to (2,2) to (1,1);
\draw [thick, myyellow] (-1,3) to (-2,2) to (1,1);
\draw [thick, myyellow] (0,4) to (-1,3);
\draw [thick, myyellow] (0,0) node[below, black] {$\graphE{5}$} to (1,1);
\draw [thick, myyellow] (2,2) to (3,3);
\draw [thick, myyellow] (-2,2) to (-3,1);
\draw [thick, densely dotted, myblue] (1,3) to (2,2) to (-1,1);
\draw [thick, densely dotted, myblue] (1,3) to (-2,2) to (-1,1);
\draw [thick, densely dotted, myblue] (0,4) to (1,3);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1);
\draw [thick, densely dotted, myblue] (-2,2) to (-3,3);
\draw [thick, densely dotted, myblue] (2,2) to (3,1);
\draw [thick, densely dashed, myred] (1,3) to (-1,3) to (-1,1);
\draw [thick, densely dashed, myred] (1,3) to (1,1) to (-1,1);
\draw [thick, densely dashed, myred] (1,1) to (3,1);
\draw [thick, densely dashed, myred] (-1,3) to (-3,3);
\draw [thick, densely dashed, myred] (-3,1) to (-1,1);
\draw [thick, densely dashed, myred] (3,3) to (1,3);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-2,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-3,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (3,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (3,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-3,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphE{9_1}$} to (1,1) to (0,3);
\draw [thick, myyellow] (-4,0) to (-3,1) to (0,3);
\draw [thick, myyellow] (4,0) to (5,1) to (0,3);
\draw [thick, myyellow, double] (0,3) to (4,3);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,3);
\draw [thick, densely dotted, myblue] (4,0) to (3,1) to (0,3);
\draw [thick, densely dotted, myblue] (-4,0) to (-5,1) to (0,3);
\draw [thick, densely dotted, myblue, double] (0,3) to (-4,3);
\draw [thick, densely dashed, myred] (-3,1) to (-5,1);
\draw [thick, densely dashed, myred] (1,1) to (-1,1);
\draw [thick, densely dashed, myred] (5,1) to (3,1);
\draw [thick, densely dashed, myred] (-5,1) to (4,3);
\draw [thick, densely dashed, myred] (-1,1) to (4,3);
\draw [thick, densely dashed, myred] (3,1) to (4,3);
\draw [thick, densely dashed, myred] (5,1) to (-4,3);
\draw [thick, densely dashed, myred] (1,1) to (-4,3);
\draw [thick, densely dashed, myred] (-3,1) to (-4,3);
\draw [thick, densely dashed, myred] (-4,3) to [out=10, in=180] (0,3.5) to [out=0, in=170] (4,3);
\node at (0,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-4,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (4,0) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (5,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (4,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-5,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (3,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-4,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,-1) node[below, black] {$\graphE{9_2}$} to (1,0) to (0,2);
\draw [thick, myyellow] (1,0) to (0,2);
\draw [thick, myyellow] (1,0) to (0,4);
\draw [thick, myyellow] (1,0) to (0,6);
\draw [thick, myyellow] (0,2) to (-3,1) to (0,6);
\draw [thick, myyellow] (0,2) to (-3,3) to (0,4);
\draw [thick, myyellow] (0,4) to (-3,5) to (0,6);
\draw [thick, densely dotted, myblue] (0,-1) to (-1,0) to (0,2);
\draw [thick, densely dotted, myblue] (-1,0) to (0,2);
\draw [thick, densely dotted, myblue] (-1,0) to (0,4);
\draw [thick, densely dotted, myblue] (-1,0) to (0,6);
\draw [thick, densely dotted, myblue] (0,2) to (3,1) to (0,6);
\draw [thick, densely dotted, myblue] (0,2) to (3,3) to (0,4);
\draw [thick, densely dotted, myblue] (0,4) to (3,5) to (0,6);
\draw [thick, densely dashed, myred] (-3,1) to (-1,0) to (1,0) to (3,1);
\draw [thick, densely dashed, myred] (-3,3) to (-1,0);
\draw [thick, densely dashed, myred] (1,0) to (3,3);
\draw [thick, densely dashed, myred] (-3,5) to (-1,0);
\draw [thick, densely dashed, myred] (1,0) to (3,5);
\draw [thick, densely dashed, myred] (-3,1) to (3,1);
\draw [thick, densely dashed, myred] (-3,3) to (3,3);
\draw [thick, densely dashed, myred] (-3,5) to (3,5);
\node at (0,-1) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,6) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,0) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-3,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-3,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-3,5) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,0) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (3,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (3,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (3,5) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\]
\[
\begin{tikzpicture}[anchorbase, xscale=.7, yscale=1]
\draw [thick, myyellow] (1,1) to (0,2) to (1,3) to (3,3) to (4,4);
\draw [thick, myyellow] (0,2) to (-2,2) to (-3,3) to (-2,4) to (0,4) to (1,3);
\draw [thick, myyellow] (1,3) to (0,2.5) to [out=180, in=20] (-2,2);
\draw [thick, myyellow] (0,2.5) to (-.35,3.25) to (0,4);
\draw [thick, densely dotted, myblue] (-1,1) to (0,2) to (-1,3) to (-3,3) to (-4,4);
\draw [thick, densely dotted, myblue] (0,2) to (2,2) to (3,3) to (2,4) to (0,4) to (-1,3);
\draw [thick, densely dotted, myblue] (-1,3) to (0,2.5) to [out=0, in=160] (2,2);
\draw [thick, densely dotted, myblue] (0,2.5) to (.35,3.25) to (0,4);
\draw [thick, densely dashed, myred] (-1,1) to (-2,2) to (-1,3) to (1,3) to (2,2) to (1,1);
\draw [thick, densely dashed, myred] (-4,4) to (-2,4) to (-1,3);
\draw [thick, densely dashed, myred] (4,4) to (2,4) to (1,3);
\draw [thick, densely dashed, myred] (-1,3) to (-.35,3.25) to (.35,3.25) to (1,3);
\draw [thick, densely dashed, myred] (2,2) to [out=200, in=0] (0,1.5) to [out=180, in=340] (-2,2);
\node at (0,.5) {$\graphE{9_3}$};
\node at (0,2) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-3,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,2.5) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (4,4) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-.35,3.25) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,4) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-4,4) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (.35,3.25) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphE{9_4}$} to (2,2);
\draw [thick, myyellow] (0,1) to (2,2) to (0,3) to (-3,5) to (0,4) to (2,2) to (0,6) to (-3,5);
\draw [thick, myyellow] (0,3) to (2,7) to (0,4) to (2,7) to (0,6);
\draw [thick, densely dotted, myblue] (0,0) to (-2,2);
\draw [thick, densely dotted, myblue] (0,1) to (-2,2) to (0,3) to (3,5) to (0,4) to (-2,2) to (0,6) to (3,5);
\draw [thick, densely dotted, myblue] (0,3) to (-2,7);
\draw [thick, densely dotted, myblue] (0,4) to (-2,7);
\draw [thick, densely dotted, myblue] (-2,7) to (0,6);
\draw [thick, densely dashed, myred, double] (2,2) to (-2,2);
\draw [thick, densely dashed, myred] (2,2) to (3,5) to (2,7) to (-2,7) to (-3,5) to (-2,2);
\draw [thick, densely dashed, myred] (3,5) to (-3,5);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (0,1) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,3) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,6) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-3,5) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (3,5) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-2,7) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\;\;,\;\;
\begin{tikzpicture}[anchorbase, xscale=.35, yscale=.5]
\draw [thick, myyellow] (0,0) node[below, black] {$\graphE{21}$} to (1,1) to (0,2);
\draw [thick, myyellow] (1,3) to (0,2) to (-2,2);
\draw [thick, myyellow] (0,8) to (-1,7) to (0,6);
\draw [thick, myyellow] (-1,5) to (0,6) to (2,6);
\draw [thick, myyellow] (-1,5) to (2,4) to (1,3) to (-2,4) to (-1,5);
\draw [thick, myyellow] (-5,3) to (-4,4) to (-1,5);
\draw [thick, myyellow] (-2,2) to (-4,4);
\draw [thick, myyellow] (5,5) to (4,4) to (1,3);
\draw [thick, myyellow] (2,6) to (4,4);
\draw [thick, myyellow] (-5,3) to (-2,4);
\draw [thick, myyellow] (5,5) to (2,4);
\draw [thick, densely dotted, myblue] (0,0) to (-1,1) to (0,2);
\draw [thick, densely dotted, myblue] (-1,3) to (0,2) to (2,2);
\draw [thick, densely dotted, myblue] (0,8) to (1,7) to (0,6);
\draw [thick, densely dotted, myblue] (1,5) to (0,6) to (-2,6);
\draw [thick, densely dotted, myblue] (1,5) to (-2,4) to (-1,3) to (2,4) to (1,5);
\draw [thick, densely dotted, myblue] (5,3) to (4,4) to (1,5);
\draw [thick, densely dotted, myblue] (2,2) to (4,4);
\draw [thick, densely dotted, myblue] (-5,5) to (-4,4) to (-1,3);
\draw [thick, densely dotted, myblue] (-2,6) to (-4,4);
\draw [thick, densely dotted, myblue] (5,3) to (2,4);
\draw [thick, densely dotted, myblue] (-5,5) to (-2,4);
\draw [thick, densely dashed, myred] (-1,1) to (-2,2) to (-1,3) to (1,3) to (2,2) to (1,1) to (-1,1);
\draw [thick, densely dashed, myred] (-1,7) to (-2,6) to (-1,5) to (1,5) to (2,6) to (1,7) to (-1,7);
\draw [thick, densely dashed, myred] (-5,3) to (-5,5);
\draw [thick, densely dashed, myred] (-1,3) to (-1,5);
\draw [thick, densely dashed, myred] (1,3) to (1,5);
\draw [thick, densely dashed, myred] (5,3) to (5,5);
\draw [thick, densely dashed, myred] (-5,3) to (-1,3);
\draw [thick, densely dashed, myred] (-5,5) to (-1,5);
\draw [thick, densely dashed, myred] (5,3) to (1,3);
\draw [thick, densely dashed, myred] (5,5) to (1,5);
\node at (0,0) {$\scalebox{1.5}{\text{{\color{mygreen}$\star$}}}$};
\node at (0,2) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-2,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (2,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (-4,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (4,4) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,6) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (0,8) {$\scalebox{1.05}{\text{{\color{mygreen}$\bullet$}}}$};
\node at (1,1) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-2,2) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (1,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,5) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (2,6) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,7) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (5,5) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-5,3) {$\scalebox{.7}{\text{{\color{myorange}$\blacksquare$}}}$};
\node at (-1,1) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (2,2) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-1,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (1,5) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-2,6) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (1,7) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (-5,5) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\node at (5,3) {$\scalebox{.7}{\text{{\color{mypurple}$\blacklozenge$}}}$};
\end{tikzpicture}
\]
\caption{The finite exceptional family of (generalized) type $\mathsf{E}$, indexed as indicated and
denoted by $\graphE{e}$. (Note that there are four for $e=9$.)
}\label{fig:typeE}
\end{figure}
All the above graphs exist for color variations as well. Note further that
we have also indicated a starting vertex $\star$ in case such a choice is essential,
i.e. in case different tricolorings give non-isomorphic tricolored graphs.
We point out that the above list was obtained from \cite[Section 4]{Oc} by excluding the ones
that are not tricolorable.
\subsection{The spectra}\label{subsec:gen-D-spectra}
The spectra of the graphs from \fullref{subsec:gen-D-list} are
known, cf. \cite[Section 2]{EP}. Let us sketch
how they look like. To this end, recall vanishing set $\vanset{e}$ of level $e$
and the discoid $\mathsf{d}_{3}$ from \fullref{subsec:roots-poly}.
\newline
\noindent\textit{\setword{`\fullref{subsec:gen-D-spectra}.Claim$\mathsf{A}$'}{A-spectra}.}
$z\in S_{\graphA{e}^{\varstuff{X}}}$ if and only if $(z,\overline{z})\in\vanset{e}$.
\newline
\noindent\textit{Proof(Sketch) of \ref{A-spectra}.}
Observe that $\graphA{e}^{\varstuff{X}}$ and $\graphA{e}^{\varstuff{Y}}$
are the graphs encoding the action of $[\varstuff{X}\otimes\underline{\phantom{a}}\,]$,
respectively of $[\varstuff{Y}\otimes\underline{\phantom{a}}\,]$, on $\GGc{\slqmod}$,
and the claim follows.
\newline
\noindent\textit{\setword{`\fullref{subsec:gen-D-spectra}.Claim$\mathsf{D}\mathsf{E}$'}{ADE-spectra}.}
We have, without counting multiplicities of the zero eigenvalue, $S_{\boldsymbol{\Gamma}^{\varstuff{X}}}\subset S_{\graphA{e}^{\varstuff{X}}}$
for any $\boldsymbol{\Gamma}$ as in \fullref{subsec:gen-D-list}.
\newline
\noindent\textit{Proof(Sketch) of \ref{ADE-spectra}.}
For graphs of type $\graphD{e}$ this holds by virtue of their construction, using the $\Z/3\Z$-symmetry of the $\graphA{e}$ graphs.
In fact, one can get the eigenvalues of $A(\graphD{e})$ from the ones of $A(\graphA{e})$
by deleting two out of every three eigenvalues and adding
two additional eigenvalues $0$, e.g. for $e=3$:
\[
\begin{tikzpicture}[anchorbase, scale=.6, tinynodes]
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (0,-3) to (0,3);
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (-3,0) to (3,0);
\draw[thick, white, fill=mygreen, opacity=.2] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\draw[thin, densely dotted, ->, >=stealth] (-3.5,0)
to (-3.35,0) node [above] {$-3$}
to (3.2,0) node [above] {$3$}
to (3.5,0) node[right] {$x$};
\draw[thin, densely dotted, ->, >=stealth] (0,-3.5)
to (0,-3.2) node [right] {$-3$}
to (0,3.2) node [right] {$3$}
to (0,3.5) node[above] {$y$};
\draw[thick] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\node at (3,3) {$\mathbb{C}$};
\node[myblue] at (0,0) {$\bullet$};
\node[myblue] at (.25,.25) {$1$};
\node[myblue] at (2,0) {$\bullet$};
\node[myblue] at (-1,1.73) {$\bullet$};
\node[myblue] at (-1,-1.73) {$\bullet$};
\node[myblue] at (-.77,.64) {$\bullet$};
\node[myblue] at (-.77,-.64) {$\bullet$};
\node[myblue] at (-.17,.98) {$\bullet$};
\node[myblue] at (-.17,-.98) {$\bullet$};
\node[myblue] at (.94,.34) {$\bullet$};
\node[myblue] at (.94,-.34) {$\bullet$};
\draw[very thin, densely dashed, myblue] (2,0) to (.94,.34) to (-.17,.98) to (-1,1.73) to (-.77,.64) to (-.77,-.64) to (-1,-1.73) to (-.17,-.98) to (.94,-.34) to (2,0);
\node[myblue] at (2.75,1.5) {spectrum of $\graphA{3}^{\varstuff{X}}$};
\end{tikzpicture}
\quad
\rightsquigarrow
\quad
\begin{tikzpicture}[anchorbase, scale=.6, tinynodes]
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (0,-3) to (0,3);
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (-3,0) to (3,0);
\draw[thick, white, fill=mygreen, opacity=.2] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\draw[thin, densely dotted, ->, >=stealth] (-3.5,0)
to (-3.35,0) node [above] {$-3$}
to (3.2,0) node [above] {$3$}
to (3.5,0) node[right] {$x$};
\draw[thin, densely dotted, ->, >=stealth] (0,-3.5)
to (0,-3.2) node [right] {$-3$}
to (0,3.2) node [right] {$3$}
to (0,3.5) node[above] {$y$};
\draw[thick] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\node at (3,3) {$\mathbb{C}$};
\node[mypurple] at (0,0) {$\bullet$};
\node[mypurple] at (.25,.25) {$3$};
\node[mypurple] at (2,0) {$\bullet$};
\node[mypurple] at (-1,1.73) {$\bullet$};
\node[mypurple] at (-1,-1.73) {$\bullet$};
\draw[very thin, densely dashed, mypurple] (2,0) to [out=170, in=315] (-1,1.73) to [out=290, in=70] (-1,-1.73) to [out=45, in=190] (2,0);
\node[mypurple] at (2.75,1.5) {spectrum of $\graphD{3}^{\varstuff{X}}$};
\end{tikzpicture}
\]
The case of $\graphC{e}$ can be shown similarly (precisely which eigenvalues of $S_{\graphA{e}}$ also
belong to $S_{\graphC{e}}$ depends on $e\bmod 3$), with a prototypical
example given by:
\[
\begin{tikzpicture}[anchorbase, scale=.6, tinynodes]
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (0,-3) to (0,3);
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (-3,0) to (3,0);
\draw[thick, white, fill=mygreen, opacity=.2] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\draw[thin, densely dotted, ->, >=stealth] (-3.5,0)
to (-3.35,0) node [above] {$-3$}
to (3.2,0) node [above] {$3$}
to (3.5,0) node[right] {$x$};
\draw[thin, densely dotted, ->, >=stealth] (0,-3.5)
to (0,-3.2) node [right] {$-3$}
to (0,3.2) node [right] {$3$}
to (0,3.5) node[above] {$y$};
\draw[thick] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\node at (3,3) {$\mathbb{C}$};
\node[myblue] at (-.80,0) {$\bullet$};
\node[myblue] at (.56,0) {$\bullet$};
\node[myblue] at (2.27,0) {$\bullet$};
\node[myblue] at (-1.12,1.95) {$\bullet$};
\node[myblue] at (-1.12,-1.95) {$\bullet$};
\node[myblue] at (-.28,.48) {$\bullet$};
\node[myblue] at (-.28,-.48) {$\bullet$};
\node[myblue] at (.40,.69) {$\bullet$};
\node[myblue] at (.40,-.69) {$\bullet$};
\node[myblue] at (-.50,1.32) {$\bullet$};
\node[myblue] at (-.50,-1.32) {$\bullet$};
\node[myblue] at (-.90,1.09) {$\bullet$};
\node[myblue] at (-.90,-1.09) {$\bullet$};
\node[myblue] at (1.40,.23) {$\bullet$};
\node[myblue] at (1.40,-.23) {$\bullet$};
\draw[very thin, densely dashed, myblue] (.56,0) to (-.28,.48) to (-.28,-.48) to (.56,0);
\draw[very thin, densely dashed, myblue] (2.27,0) to (1.40,.23) to (.40,.69) to (-.50,1.32) to (-1.12,1.95) to (-.90,1.09) to (-.80,0) to (-.90,-1.09) to (-1.12,-1.95) to (-.50,-1.32) to (.40,-.69) to (1.40,-.23) to (2.27,0);
\node[myblue] at (2.75,1.5) {spectrum of $\graphA{4}^{\varstuff{X}}$};
\end{tikzpicture}
\quad
\rightsquigarrow
\quad
\begin{tikzpicture}[anchorbase, scale=.6, tinynodes]
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (0,-3) to (0,3);
\draw[thin, marked=.0, marked=.166, marked=.333, marked=.666, marked=.833, marked=1.0, white] (-3,0) to (3,0);
\draw[thick, white, fill=mygreen, opacity=.2] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\draw[thin, densely dotted, ->, >=stealth] (-3.5,0)
to (-3.35,0) node [above] {$-3$}
to (3.2,0) node [above] {$3$}
to (3.5,0) node[right] {$x$};
\draw[thin, densely dotted, ->, >=stealth] (0,-3.5)
to (0,-3.2) node [right] {$-3$}
to (0,3.2) node [right] {$3$}
to (0,3.5) node[above] {$y$};
\draw[thick] (3,0) to [out=170, in=315] (-1.5,2.5) to [out=290, in=70] (-1.5,-2.5) to [out=45, in=190] (3,0);
\node at (3,3) {$\mathbb{C}$};
\node[mypurple] at (-.80,0) {$\bullet$};
\node[mypurple] at (.56,0) {$\bullet$};
\node[mypurple] at (2.27,0) {$\bullet$};
\node[mypurple] at (-1.12,1.95) {$\bullet$};
\node[mypurple] at (-1.12,-1.95) {$\bullet$};
\node[mypurple] at (-.28,.48) {$\bullet$};
\node[mypurple] at (-.28,-.48) {$\bullet$};
\node[mypurple] at (.40,.69) {$\bullet$};
\node[mypurple] at (.40,-.69) {$\bullet$};
\draw[very thin, densely dashed, mypurple] (.56,0) to (-.28,.48) to (-.28,-.48) to (.56,0);
\draw[very thin, densely dashed, mypurple] (2.27,0) to (.40,.69) to (-1.12,1.95) to (-.80,0) to (-1.12,-1.95) to (.40,-.69) to (2.27,0);
\node[mypurple] at (2.75,1.5) {spectrum of $\graphC{4}^{\varstuff{X}}$};
\end{tikzpicture}
\]
For the exceptional type $\mathsf{E}$ graphs the claim can be checked case-by-case.
\newline
In particular, the spectra of the generalized $\ADE$ Dynkin diagrams
are all inside $\mathsf{d}_{3}$.
\begin{example}\label{example:triangle3}
The spectra of $\graphA{1}^{\varstuff{X}},\graphA{2}^{\varstuff{X}}$ and $\graphA{3}^{\varstuff{X}}$
are given in \fullref{example:triangle1}. (Again, these should be
compared to \fullref{example:plot-zeros}.)
Additionally we have
\begin{gather*}
S_{\graphD{3}^{\varstuff{X}}}
=S_{\graphC{3}^{\varstuff{X}}}
=
\left\{
\text{roots of }
X^3
(X-2)
(X^2 + 2X + 4)
\right\}
\end{gather*}
Forgetting the multiplicity of zero, we get the inclusion of the corresponding spectra.
\end{example}
\begin{example}\label{example:sce}
The graphs $\graphD{3}^{\varstuff{X}}$ and $\graphC{3}^{\varstuff{X}}$ have the same
spectrum, cf. \fullref{example:triangle3}.
However, they are not isomorphic as graphs, e.g. $\graphD{3}$ has a double-edge and $\graphC{3}^{\varstuff{X}}$ does not.
Both observations are true in general for $\graphD{e}$ and $\graphC{e}$.
\end{example}
\subsection{Zuber's classification problem and \texorpdfstring{\fullref{problem:classification}}{CP}}\label{subsec:gen-D-zuber}
Zuber (based on joint work with Di Francesco \cite{DFZ} and
Petkova \cite{PZ}) introduced the notion of a generalized $\ADE$
Dynkin diagram.
These graphs appear in various
disguises in the literature, e.g.
in conformal field
theories, integrable lattice models, topological field theories
for $3$-manifold invariants and subfactor theory.
Zuber wrote down a list of six axioms
which these graphs should satisfy, see \cite[Section 1.2]{Zu},
and asked for the classification of such graphs.
In \cite{Oc}, Ocneanu argued that Zuber's classification problem is related to the classification problem
of the so-called quantum subgroups of $\mathrm{SU}(N)$. He also proposed a list of graphs which should solve
Zuber's classification problem. The ones that are tricolorable are the graphs
that we reproduced in \fullref{subsec:gen-D-list}.
However, we already saw in \fullref{theorem:low-level-classification}
that we get solutions which are not on Ocneanu's list,
so we do not know whether \fullref{problem:classification}
and Zuber's classification problem are the same or not, or even how they are related.
\section{Trihedral Soergel bimodules}\label{sec:A2-diagrams}
The purpose of this section is to categorify the
trihedral Hecke algebras $\subquo$ and $\subquo[e]$
from \fullref{section:funny-algebra}, where $e$ still denotes the level.
As before, we have collected some analogies to the
dihedral case at the end of the section, cf. \fullref{subsec:dihedral-SB}
\subsection{Bott--Samelson bimodules for affine \texorpdfstring{$\typea{2}$}{A2}}\label{subsec:sbim}
First, we recall the diagrammatic $2$-category
$\Adiag$ from \cite[Section 3.3]{El1}. We
call it the ($2$-category of)
singular Bott--Samelson bimodules of affine type $\typea{2}$.
\subsubsection{\texorpdfstring{$2$}{2}-categorical conventions}\label{subsubsec:2-cat-conv}
For generalities and terminology on $2$-categories, we
refer for example to \cite{Le1} or \cite{McL1}.
\begin{convention}\label{convention:generated}
We use $2$-categories given by generators and relations.
This means that
$1$-morphisms are obtained by
compositions $\circ$ of the generating $1$-morphisms, and
$2$-morphisms are obtained by horizontal $\circ_{h}$ and vertical
$\circ_{v}$ compositions of the $2$-generators whenever this makes sense.
(In particular, the interchange law leads to additional relations in our
$2$-categories, called height relations.)
Relations are supposed to hold between $2$-morphisms.
Details about such $2$-categories can be found e.g. in \cite[Section 2.2]{Ro1}.
\end{convention}
\begin{convention}\label{convention:reading}
We read $1$-morphisms from right to left, using the operator-notation, and
$2$-morphisms from bottom to top and right to left.
These conventions are illustrated in \fullref{definition:ssbim-free} below.
Note that we usually omit the $1$-morphisms in the pictures,
and we will simplify diagrams by drawing them in a more
topological fashion, using e.g. \fullref{example:more-gens}.
\end{convention}
\begin{convention}\label{convention:grading}
A ($\mathbb{Z}$-)graded $2$-category for us is a $2$-category
whose $2$-hom spaces are ($\mathbb{Z}$-)graded, meaning that the
$2$-generators have a given degree, the relations are homogeneous
and the degree is additive under horizontal and vertical
composition. Moreover, $1$-morphisms are formal shifts
of generating $1$-morphisms, indicated by $\{a\}$ for $a\in\mathbb{Z}$, so
there is a formal $\mathbb{Z}$-action on $1$-morphisms
such that
$\{k\}(\morstuff{F}\{a\})=\morstuff{F}\{a+k\}$ for all $k\in\mathbb{Z}$. Finally,
a $2$-morphism
$\twomorstuff{f}\colon\morstuff{F}\{0\}\to\morstuff{G}\{0\}$,
homogeneous of degree $d$, is of degree $d-a+b$ seen as
a $2$-morphism $\twomorstuff{f}\colon\morstuff{F}\{a\}\to\morstuff{G}\{b\}$.
For more information on
such $2$-categories, see e.g. \cite[Section 5.1]{Lau}.
\end{convention}
\subsubsection{The definition of \texorpdfstring{$\Adiag$}{singSbim}}\label{subsubsec:def-adiag}
Let $\C_{\intqpar}=\mathbb{C}[\varstuff{q},\varstuff{q}^{-1}]$ and $\Rbim=\C_{\intqpar}[\alpha_{\bc},\alpha_{\rc},\alpha_{\yc}]$,
where $\alpha_{\bc},\alpha_{\rc},\alpha_{\yc}$ are formal variables.
We define an action of the affine Weyl group $\algstuff{W}$ from
\fullref{subsec:weyl-group} on $\Rbim$:
\begin{gather}\label{eq:sl3-exotic-action}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep={0.5cm,between origins}, column
sep={2.25cm,between origins}, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{{\color{myblue}b}} \& \alpha_{\bc} \& \alpha_{\rc} \& \alpha_{\yc}\\
{\color{myblue}b} \& -\alpha_{\bc} \& \alpha_{\bc}+\alpha_{\rc} \& \varstuff{q}^{-1}\alpha_{\bc}+\alpha_{\yc}\\
{\color{myred}r} \& \alpha_{\bc}+\alpha_{\rc} \& -\alpha_{\rc} \& \varstuff{q}\alpha_{\rc}+\alpha_{\yc}\\
{\color{myyellow}y} \& \alpha_{\bc}+\varstuff{q}\alpha_{\yc} \& \alpha_{\rc}+\varstuff{q}^{-1}\alpha_{\yc} \& -\alpha_{\yc}\\
};
\draw[densely dashed] ($(m-2-1.north)+(-.15,0)$) edge ($(m-2-4.north)+(1.05,0)$);
\draw[densely dashed] ($(m-3-1.north)+(-.15,0)$) edge ($(m-3-4.north)+(1.05,0)$);
\draw[densely dashed] ($(m-4-1.north)+(-.15,0)$) edge ($(m-4-4.north)+(1.05,0)$);
\draw[densely dashed] ($(m-1-1.east)+(1.1,.2)$) edge ($(m-4-1.east)+(1.1,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(3,.2)$) edge ($(m-4-1.east)+(3,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(5.4,.2)$) edge ($(m-4-1.east)+(5.4,-.3)$);
\end{tikzpicture}
\end{gather}
One easily checks that \eqref{eq:sl3-exotic-action} is well-defined. This also
gives rise to an action of the secondary colors on $\Rbim$
by using \eqref{eq:sec-Weyl} (recalling that e.g. ${\color{mygreen}g}=\{{\color{myblue}b},{\color{myyellow}y}\}$). Thus, we can define:
\begin{definition}\label{definition:thin-invariant-subrings}
For any ${\color{dummy}c}\in\Bset\Rset\Yset$ and ${\color{dummy}\textbf{u}}\in\Gset\Oset\Pset$, let $\Rbim^{{\color{dummy}c}}$ and $\Rbim^{{\color{dummy}\textbf{u}}}$
denote the subrings of $\Rbim$ consisting of all ${\color{dummy}c}$-invariant and ${\color{dummy}\textbf{u}}$-invariant elements,
respectively.
\end{definition}
Recall that we always use ${\color{dummy}\textbf{u}},{\color{dummy}\textbf{v}}\in\Gset\Oset\Pset$ as secondary dummy colors,
and we also use the primary dummy colors ${\color{dummy}c},{\color{dummy}d}\in\Bset\Rset\Yset$ from now on.
Moreover, identifying our colors with proper subsets of $\Bset\Rset\Yset$,
including the empty subset, we say that
two of them are compatible if one is a subset of the
other, e.g. as the colors connected by an edge below.
\begin{gather}\label{eq:color-compatible}
\begin{tikzpicture}[baseline=(current bounding box.center), tinynodes]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep=.02cm, column
sep=.01cm, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
{\color{mygray}\scalebox{.99}{$p$}} \& \scalebox{.99}{${\color{mygreen}g}$}\, \& {\color{mygray}\scalebox{.99}{$o$}} \\
\,\scalebox{.99}{${\color{myblue}b}$} \& {\color{mygray}\scalebox{.99}{$r$}} \& \scalebox{.99}{${\color{myyellow}y}$} \\
\& \scalebox{.99}{$\emptyset$} \& \\
};
\draw[thin] ($(m-2-1.north)+(0,-.2cm)$) to ($(m-1-2.south)+(-.05,.1cm)$);
\draw[thin] ($(m-2-3.north)+(0,-.2cm)$) to ($(m-1-2.south)+(.05,.1cm)$);
\draw[thin] ($(m-3-2.north)+(-.05,-.2cm)$) to ($(m-2-1.south)+(0,.1cm)$);
\draw[thin] ($(m-3-2.north)+(.05,-.2cm)$) to ($(m-2-3.south)+(0,.1cm)$);
\end{tikzpicture}
,\quad\quad
\begin{tikzpicture}[baseline=(current bounding box.center), tinynodes]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep=.02cm, column
sep=.01cm, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
{\color{mygray}\scalebox{.99}{$p$}} \& {\color{mygray}\scalebox{.99}{$g$}} \& \scalebox{.99}{${\color{myorange}o}$} \\
{\color{mygray}\scalebox{.99}{$b$}} \& \scalebox{.99}{${\color{myred}r}$} \& \scalebox{.99}{${\color{myyellow}y}$} \\
\& \scalebox{.99}{$\emptyset$}\& \\
};
\draw[thin] ($(m-2-3.north)+(.035,-.2cm)$) to ($(m-1-3.south)+(.035,.1cm)$);
\draw[thin] ($(m-3-2.north)+(0,-.2cm)$) to ($(m-2-2.south)+(0,.1cm)$);
\draw[thin] ($(m-3-2.north)+(.05,-.2cm)$) to ($(m-2-3.south)+(0,.1cm)$);
\draw[thin] ($(m-2-2.north)+(.05,-.2cm)$) to ($(m-1-3.south)+(0,.1cm)$);
\end{tikzpicture}
,\quad\quad
\begin{tikzpicture}[baseline=(current bounding box.center), tinynodes]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep=.02cm, column
sep=.01cm, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
\scalebox{.99}{${\color{mypurple}p}$} \& {\color{mygray}\scalebox{.99}{$g$}} \& {\color{mygray}\scalebox{.99}{$o$}} \\
\scalebox{.99}{${\color{myblue}b}$} \& \scalebox{.99}{${\color{myred}r}$} \& {\color{mygray}\scalebox{.99}{$y$}} \\
\& \scalebox{.99}{$\emptyset$} \& \\
};
\draw[thin] ($(m-2-1.north)+(-.035,-.2cm)$) to ($(m-1-1.south)+(-.035,.1cm)$);
\draw[thin] ($(m-3-2.north)+(-.05,-.2cm)$) to ($(m-2-1.south)+(0,.1cm)$);
\draw[thin] ($(m-3-2.north)+(0,-.2cm)$) to ($(m-2-2.south)+(0,.1cm)$);
\draw[thin] ($(m-2-2.north)+(-.05,-.2cm)$) to ($(m-1-1.south)+(0,.1cm)$);
\end{tikzpicture}
\end{gather}
\begin{example}\label{example:colors-compatible}
The color ${\color{myblue}b}$ is compatible with $\emptyset$, ${\color{mygreen}g}$ and ${\color{mypurple}p}$, but not with ${\color{myred}r}$, ${\color{myyellow}y}$ or ${\color{myorange}o}$.
\end{example}
We will define the $2$-category of singular Soergel bimodules
as a quotient of the following $2$-category, which we view as a free version
of it.
\begin{definition}\label{definition:ssbim-free}
Let $\ADiag$ be the $2$-category defined as follows.
\medskip
\noindent\textit{\setword{`Objects of $\ADiag$'}{sbim-objects}.}
The objects are proper subsets of $\Bset\Rset\Yset=\{{\color{myblue}b},{\color{myyellow}y},{\color{myred}r}\}$,
including the empty subset $\emptyset$.
The one-element subsets are identified with
${\color{myblue}b},{\color{myyellow}y},{\color{myred}r}$, the two-element subsets are
identified with ${\color{mygreen}g},{\color{myorange}o},{\color{mypurple}p}$, using
the color conventions from \fullref{subsec:our-color-code}.
\medskip
\noindent\textit{\setword{`$1$-morphisms of $\ADiag$'}{sbim-morphisms}.}
By definition, there is one generating $1$-morphism
for each pair of distinct compatible colors.
Namely, including all other
compatible variations using the conventions from \eqref{eq:color-compatible}
and writing e.g. ${\color{myblue}b}\emptyset={\color{myblue}b}\circ\emptyset$ for short:
\begin{gather*}
\xy
(0,0)*{
\emptyset{\color{myblue}b}\colon\emptyset\leftarrow{\color{myblue}b},
\quad
{\color{myblue}b}\emptyset\colon{\color{myblue}b}\leftarrow\emptyset,
\quad
{\color{myblue}b}{\color{mygreen}g}\colon{\color{myblue}b}\leftarrow{\color{mygreen}g},
\quad
{\color{myyellow}y}{\color{mygreen}g}\colon{\color{myyellow}y}\leftarrow{\color{mygreen}g},
\quad
{\color{mygreen}g}{\color{myblue}b}\colon{\color{mygreen}g}\leftarrow{\color{myblue}b},
\quad
{\color{mygreen}g}{\color{myyellow}y}\colon{\color{mygreen}g}\leftarrow{\color{myyellow}y},
\quad
\text{etc.}};
(0,-4.5)*{\text{{\tiny compatible as in \eqref{eq:color-compatible}}}};
\endxy
\end{gather*}
\smallskip
\noindent\textit{\setword{`$2$-morphisms of $\ADiag$'}{sbim-two-morphisms}.}
The $2$-morphisms are generated by
two kinds of $2$-generators.
The first kind are
cups, caps and crossings given as follows.
\begin{gather}\label{eq:gens-sbim-1}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (3,0) to (3,2) to (-1,2) to (-1,0) to (0,0);
\fill[myblue, opacity=0.3] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\draw[bstrand, directed=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0);
\node at (-.5,-.375) {$\emptyset$};
\node at (1,2.3) {$\emptyset$};
\node at (1,-.375) {${\color{myblue}b}$};
\node at (2.5,-.375) {$\emptyset$};
\end{tikzpicture}
\colon
\begin{matrix}
\emptyset
\\
\Uparrow
\\
\emptyset{\color{myblue}b}\emptyset
\end{matrix}};
(-3.6,-8)*{\text{{\tiny degree $1$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\fill[myblue, opacity=0.3] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (3,0) to (3,2) to (-1,2) to (-1,0) to (0,0);
\draw[bstrand, directed=.999] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\node at (-.5,-.375) {${\color{myblue}b}$};
\node at (1,2.3) {${\color{myblue}b}$};
\node at (1,-.375) {$\emptyset$};
\node at (2.5,-.375) {${\color{myblue}b}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny degree $-1$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (0,2);
\fill[myyellow, opacity=0.3] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (3,2) to (3,0) to (-1,0) to (-1,2) to (0,2);
\draw[rstrand, directed=.999] (2,2) to [out=270, in=0] (1,1) to [out=180, in=270] (0,2);
\node at (-.5,2.3) {${\color{myyellow}y}$};
\node at (1,-.375) {${\color{myyellow}y}$};
\node at (1,2.3) {${\color{myorange}o}$};
\node at (2.5,2.3) {${\color{myyellow}y}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny degree $2$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (3,2) to (3,0) to (-1,0) to (-1,2) to (0,2);
\fill[myyellow, opacity=0.3] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (0,2);
\draw[rstrand, directed=.999] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2);
\node at (-.5,2.3) {${\color{myorange}o}$};
\node at (1,-.375) {${\color{myorange}o}$};
\node at (1,2.3) {${\color{myyellow}y}$};
\node at (2.5,2.3) {${\color{myorange}o}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny degree $-2$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mypurple, opacity=0.8] (3,2) to (2,2) to [out=270, in=0] (1,1) to [out=0, in=90] (2,0) to (3,0) to (3,2);
\draw[very thin, densely dotted, fill=white] (-1,2) to (0,2) to [out=270, in=180] (1,1) to [out=180, in=90] (0,0) to (-1,0) to (-1,2);
\fill[myblue, opacity=0.3] (1,1) to [out=0, in=90] (2,0) to (0,0) to [out=90, in=180] (1,1);
\fill[myred, opacity=0.3] (1,1) to [out=0, in=270] (2,2) to (0,2) to [out=270, in=180] (1,1);
\draw[rstrand, directed=.999] (2,0) to [out=90, in=0] (1,1) to [out=180, in=270] (0,2);
\draw[bstrand, directed=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=270] (2,2);
\node at (-.5,2.3) {$\emptyset$};
\node at (-.5,-.375) {$\emptyset$};
\node at (1,2.3) {${\color{myred}r}$};
\node at (1,-.375) {${\color{myblue}b}$};
\node at (2.5,2.3) {${\color{mypurple}p}$};
\node at (2.5,-.375) {${\color{mypurple}p}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny degree $0$}}};
\endxy
\end{gather}
(We frame $\emptyset$-colored regions for readability.)
The generators displayed in \eqref{eq:gens-sbim-1} are all generators up to
colored variations: each strand separates
two regions colored by subsets of $\Bset\Rset\Yset$ that differ by a primary color,
which is used to color that strand.
The strands are oriented such that the region colored by
the smaller subset of $\Bset\Rset\Yset$ lies to their left.
The second kind of $2$-generators are decorations
of the regions by polynomials in $\Rbim$ that are invariant under the parabolic subgroup corresponding
to the color of the region, i.e.
\begin{gather}\label{eq:gens-sbim-2}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (0,0) to (2,0) to (2,2) to (0,2) to (0,0);
\node at (1,1) {$\polybox{p}$};
\node at (1,-.5) {$p{\in}\Rbim^{\emptyset}{=}\Rbim$};
\end{tikzpicture}
,\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myblue, opacity=0.3] (0,0) to (2,0) to (2,2) to (0,2) to (0,0);
\node at (1,1) {$\polybox{p}$};
\node at (1,-.5) {$p\in\Rbim^{{\color{myblue}b}}$};
\end{tikzpicture}
,\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myred, opacity=0.3] (0,0) to (2,0) to (2,2) to (0,2) to (0,0);
\node at (1,1) {$\polybox{p}$};
\node at (1,-.5) {$p\in\Rbim^{{\color{myred}r}}$};
\end{tikzpicture}
,\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (0,0) to (2,0) to (2,2) to (0,2) to (0,0);
\node at (1,1) {$\polybox{p}$};
\node at (1,-.5) {$p\in\Rbim^{{\color{myyellow}y}}$};
\end{tikzpicture}
,\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (2,0) to (2,2) to (0,2) to (0,0);
\node at (1,1) {$\polybox{p}$};
\node at (1,-.5) {$p\in\Rbim^{{\color{mygreen}g}}$};
\end{tikzpicture}
,\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,0) to (2,0) to (2,2) to (0,2) to (0,0);
\node at (1,1) {$\polybox{p}$};
\node at (1,-.5) {$p\in\Rbim^{{\color{myorange}o}}$};
\end{tikzpicture}
,\quad
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mypurple, opacity=0.8] (0,0) to (2,0) to (2,2) to (0,2) to (0,0);
\node at (1,1) {$\polybox{p}$};
\node at (1,-.5) {$p\in\Rbim^{{\color{mypurple}p}}$};
\end{tikzpicture}
\end{gather}
The polynomials are allowed to move around as long as they do not cross any strand.
\medskip
\noindent\textit{\setword{`Grading on $\ADiag$'}{sbim-grading}.}
We endow $\ADiag$ with the structure of a graded $2$-category
by giving the generators from \eqref{eq:gens-sbim-1} and
\eqref{eq:gens-sbim-2}
the following degree.
\begin{enumerate}[label=$\blacktriangleright$]
\setlength\itemsep{.15cm}
\item Clockwise cups and caps between $\emptyset$ and ${\color{dummy}c}$ have degree $1$, while
their anticlockwise counterparts have degree $-1$.
\item Clockwise cups and caps between ${\color{dummy}c}$ and a compatible ${\color{dummy}\textbf{u}}$ have degree $2$, while
their anticlockwise counterparts have degree $-2$.
\item Crossings are of degree $0$.
\item Homogeneous polynomials are graded by twice their polynomial degree, i.e. the
formal variables $\alpha_{\bc},\alpha_{\rc},\alpha_{\yc}$ are of degree $2$.
\end{enumerate}
We have indicate some of these in \eqref{eq:gens-sbim-1}.
\end{definition}
\begin{example}\label{example:2-cat-con-a}
In general, a $1$-morphism is a finite
string of generating $1$-morphisms, which are
indicated by their source and target, e.g.
${\color{myyellow}y}{\color{myorange}o}{\color{myred}r}{\color{mypurple}p}{\color{myblue}b}\emptyset\colon{\color{myyellow}y}\leftarrow{\color{myorange}o}\leftarrow{\color{myred}r}\leftarrow{\color{mypurple}p}\leftarrow{\color{myblue}b}\leftarrow\emptyset$.
(By convention, we identify the objects ${\color{dummy}c},{\color{dummy}\textbf{u}}$ with the
identity $1$-morphisms on them.)
Furthermore,
\[
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-6,0) to (-8,0) to (-8,2) to (-6,2) to (-6,0);
\draw[very thin, densely dotted, fill=white] (-2,0) to (-4,0) to (-4,2) to (-2,2) to (-2,0);
\draw[very thin, densely dotted, fill=white] (4,0) to (6,0) to (6,2) to (4,2) to (4,0);
\fill[myyellow, opacity=0.3] (-4,0) to (-4,2) to (-6,2) to (-6,0) to (0-4,0);
\fill[myred, opacity=0.3] (0,0) to (0,2) to (-2,2) to (-2,0) to (0,0);
\fill[myblue, opacity=0.3] (2,0) to (4,0) to (4,2) to (2,2) to (2,0);
\fill[myblue, opacity=0.3] (6,0) to (8,0) to (8,2) to (6,2) to (6,0);
\fill[mygreen, opacity=0.8] (8,0) to (8,2) to (10,2) to (10,0) to (8,0);
\fill[mypurple, opacity=0.8] (0,0) to (0,2) to (2,2) to (2,0) to (0,0);
\draw[ystrand, directed=.999] (-6,0) to (-6,2);
\draw[ystrand, directed=.999] (-4,2) to (-4,0);
\draw[ystrand, directed=.999] (8,0) to (8,2);
\draw[rstrand, directed=.999] (-2,0) to (-2,2);
\draw[rstrand, directed=.999] (2,2) to (2,0);
\draw[bstrand, directed=.999] (0,0) to (0,2);
\draw[bstrand, directed=.999] (4,2) to (4,0);
\draw[bstrand, directed=.999] (6,0) to (6,2);
\end{tikzpicture}
\]
is an example of the
coloring of facets
and strands.
\end{example}
\begin{example}\label{example:more-gens}
As usual, one can define sideways crossings, e.g.
\[
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (1,1) to [out=0, in=270] (2,2) to (0,2) to [out=270, in=180] (1,1);
\fill[myred, opacity=0.3] (3,2) to (2,2) to [out=270, in=0] (1,1) to [out=0, in=90] (2,0) to (3,0) to (3,2);
\fill[myblue, opacity=0.3] (-1,2) to (0,2) to [out=270, in=180] (1,1) to [out=180, in=90] (0,0) to (-1,0) to (-1,2);
\fill[mypurple, opacity=0.8] (1,1) to [out=0, in=90] (2,0) to (0,0) to [out=90, in=180] (1,1);
\draw[rstrand, directed=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=270] (2,2);
\draw[bstrand, directed=.999] (0,2) to [out=270, in=180] (1,1) to [out=0, in=90] (2,0);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-2,4) to (-2,0) to [out=270, in=180] (-1,-1) to [out=0, in=270] (0,0) to [out=90, in=180] (1,1) to [out=180, in=270] (0,2) to (0,4) to (-2,4);
\fill[myred, opacity=0.3] (5,-2) to (5,4) to (0,4) to (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to [out=90, in=180] (3,3) to [out=0, in=90] (4,2) to (4,-2) to (5,-2);
\fill[myblue, opacity=0.3] (-3,4) to (-2,4) to (-2,0) to [out=270, in=180] (-1,-1) to [out=0, in=270] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,-2) to (-3,-2) to (-3,4);
\fill[mypurple, opacity=0.8] (2,-2) to (2,0) to [out=90, in=0] (1,1) to [out=0, in=270] (2,2) to [out=90, in=180] (3,3) to [out=0, in=90] (4,2) to (4,-2) to (2,-2);
\draw[rstrand, directed=.999] (2,0) to [out=90, in=0] (1,1) to [out=180, in=270] (0,2);
\draw[rstrand] (2,0) to (2,-2);
\draw[rstrand] (0,2) to (0,4);
\draw[bstrand, directed=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=270] (2,2);
\draw[bstrand] (2,2) to [out=90, in=180] (3,3) to [out=0, in=90] (4,2) to (4,-2);
\draw[bstrand] (-2,4) to (-2,0) to [out=270, in=180] (-1,-1) to [out=0, in=270] (0,0);
\end{tikzpicture}};
(0,-14.5)*{\text{{\tiny degree $1$}}};
\endxy
,\quad\quad
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (1,1) to [out=0, in=90] (2,0) to (0,0) to [out=90, in=180] (1,1);
\fill[myred, opacity=0.3] (3,2) to (2,2) to [out=270, in=0] (1,1) to [out=0, in=90] (2,0) to (3,0) to (3,2);
\fill[myblue, opacity=0.3] (-1,2) to (0,2) to [out=270, in=180] (1,1) to [out=180, in=90] (0,0) to (-1,0) to (-1,2);
\fill[mypurple, opacity=0.8] (1,1) to [out=0, in=270] (2,2) to (0,2) to [out=270, in=180] (1,1);
\draw[rstrand, directed=.999] (2,0) to [out=90, in=0] (1,1) to [out=180, in=270] (0,2);
\draw[bstrand, directed=.999] (2,2) to [out=270, in=0] (1,1) to [out=180, in=90] (0,0);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-2,-2) to (-2,0) to [out=90, in=180] (-1,1) to [out=180, in=270] (-2,2) to [out=90, in=0] (-3,3) to [out=180, in=90] (-4,2) to (-4,-2) to (-2,-2);
\fill[myred, opacity=0.3] (3,4) to (2,4) to (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0) to [out=90, in=0] (-1,1) to [out=180, in=90] (-2,-2) to (3,-2) to (3,4);
\fill[myblue, opacity=0.3] (-5,-2) to (-5,4) to (0,4) to (0,2) to [out=270, in=0] (-1,1) to [out=180, in=270] (-2,2) to [out=90, in=0] (-3,3) to [out=180, in=90] (-4,2) to (-4,-2) to (-5,-2);
\fill[mypurple, opacity=0.8] (2,4) to (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0) to [out=90, in=0] (-1,1) to [out=0, in=270] (0,2) to (0,4) to (2,4);
\draw[rstrand, directed=.999] (-2,0) to [out=90, in=180] (-1,1) to [out=0, in=270] (0,2);
\draw[rstrand] (-2,0) to (-2,-2);
\draw[rstrand] (0,2) to (0,4);
\draw[bstrand, directed=.999] (0,0) to [out=90, in=0] (-1,1) to [out=180, in=270] (-2,2);
\draw[bstrand] (-2,2) to [out=90, in=0] (-3,3) to [out=180, in=90] (-4,2) to (-4,-2);
\draw[bstrand] (2,4) to (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0);
\end{tikzpicture}};
(0,-14.5)*{\text{{\tiny degree $1$}}};
\endxy
\]
Note that these are of degree $1$.
\end{example}
\begin{remark}\label{remark:q-sneaks-in}
The $2$-category $\ADiag$ depends on $\varstuff{q}$, since the quantum parameter is
in the definition of the rings $\Rbim$, cf. \eqref{eq:sl3-exotic-action}.
\end{remark}
Before we can go on, we need some algebraic notions.
\subsubsection{An interlude on Frobenius extensions}\label{subsubsec:frob}
The relations of $\Adiag$ actually come from
a cube of Frobenius extensions.
(For details on Frobenius extensions see e.g. \cite{ESW}.)
\begin{definition}\label{definition:frob-extensions}
A (commutative) Frobenius extension is an extension of
commutative rings $\algstuff{R}^{\prime}\subset\algstuff{R}$ with
$\algstuff{R}$ being a
free $\algstuff{R}^{\prime}$-bimodule of finite rank,
together with a $\algstuff{R}^{\prime}$-bilinear trace map
$\partial\colon\algstuff{R}\to\algstuff{R}^{\prime}$ which
gives rise to a non-degenerate bilinear pairing
\[
\langle\cdot,\cdot\rangle
\colon
\algstuff{B}\times\algstuff{B}^{\star}
\to
\algstuff{R}^{\prime}.
\]
Moreover, for a Frobenius extension there exist
two $\algstuff{R}^{\prime}$-bases
$\algstuff{B},\algstuff{B}^{\star}$ of $\algstuff{R}$,
such that for any $\algstuff{x}\in\algstuff{B}$ there is precisely one
element $\algstuff{x}^{\star}\in\algstuff{B}^{\star}$ satisfying
\[
\langle \algstuff{x},\algstuff{x}^{\prime}\rangle=\partial(\algstuff{x}\algstuff{x}^{\prime})
=
\delta_{\algstuff{x}^{\prime},\algstuff{x}^{\star}}.
\]
The elements $\algstuff{x}$ and
$\algstuff{x}^{\star}$, respectively the bases $\algstuff{B}$ and $\algstuff{B}^{\star}$
are called dual to each other.
The number of elements $\#\algstuff{B}=\#\algstuff{B}^{\star}$ is called the
rank.
Such an extension is called graded if $\algstuff{R},\algstuff{R}^{\prime}$ are
graded rings, $\algstuff{R}$ is graded as an
$\algstuff{R}^{\prime}$-bimodule, $\algstuff{B},\algstuff{B}^{\star}$ consist
of homogeneous elements, and $\partial$ is a homogeneous map.
\end{definition}
Note that the dual elements $\algstuff{x},\algstuff{x}^{\star}$ satisfy
$\deg(\algstuff{x})+\deg(\algstuff{x}^{\star})=-\deg(\partial)$.
\begin{definition}\label{definition:thin-demazure-action}
We let $\partial_{{\color{dummy}c}}\colon\Rbim\to\Rbim^{{\color{dummy}c}}$
be defined via the formula
$\partial_{{\color{dummy}c}}(f)=\sneatfrac{f-{\color{dummy}c}(f)}{\alpha_{\duc}}$.
We call these the primary Demazure operators.
Similarly, we define
\begin{gather*}
\begin{gathered}
\partial_{{\color{mygreen}g}}^{{\color{myblue}b}}=\varstuff{q}\partial_{{\color{myblue}b}}\partial_{{\color{myyellow}y}}
\colon\Rbim^{{\color{myblue}b}}\to\Rbim^{{\color{mygreen}g}},
\quad\quad
\partial_{{\color{mygreen}g}}^{{\color{myyellow}y}}=\partial_{{\color{myyellow}y}}\partial_{{\color{myblue}b}}
\colon\Rbim^{{\color{myyellow}y}}\to\Rbim^{{\color{mygreen}g}},
\quad\quad
\partial_{{\color{myorange}o}}^{{\color{myred}r}}=\varstuff{q}^{-1}\partial_{{\color{myred}r}}\partial_{{\color{myyellow}y}}
\colon\Rbim^{{\color{myred}r}}\to\Rbim^{{\color{myorange}o}},
\\
\partial_{{\color{myorange}o}}^{{\color{myyellow}y}}=\partial_{{\color{myyellow}y}}\partial_{{\color{myred}r}}
\colon\Rbim^{{\color{myyellow}y}}\to\Rbim^{{\color{myorange}o}},
\quad\quad
\partial_{{\color{mypurple}p}}^{{\color{myblue}b}}=\partial_{{\color{myblue}b}}\partial_{{\color{myred}r}}
\colon\Rbim^{{\color{myblue}b}}\to\Rbim^{{\color{mypurple}p}},
\quad\quad
\partial_{{\color{mypurple}p}}^{{\color{myred}r}}=\partial_{{\color{myred}r}}\partial_{{\color{myblue}b}}
\colon\Rbim^{{\color{myred}r}}\to\Rbim^{{\color{mypurple}p}},
\end{gathered}
\end{gather*}
which we call the mixed Demazure operators.
Finally, we define
\begin{gather*}
\begin{gathered}
\partial_{{\color{mygreen}g}}=\varstuff{q}\partial_{{\color{myblue}b}}\partial_{{\color{myyellow}y}}\partial_{{\color{myblue}b}}
=\partial_{{\color{myyellow}y}}\partial_{{\color{myblue}b}}\partial_{{\color{myyellow}y}}
\colon\Rbim\to\Rbim^{{\color{mygreen}g}},
\quad
\partial_{{\color{myorange}o}}=\varstuff{q}^{-1}\partial_{{\color{myred}r}}\partial_{{\color{myyellow}y}}\partial_{{\color{myred}r}}
=\partial_{{\color{myyellow}y}}\partial_{{\color{myred}r}}\partial_{{\color{myyellow}y}}
\colon\Rbim\to\Rbim^{{\color{myorange}o}},
\\
\partial_{{\color{mypurple}p}}=\partial_{{\color{myblue}b}}\partial_{{\color{myred}r}}\partial_{{\color{myblue}b}}
=\partial_{{\color{myred}r}}\partial_{{\color{myblue}b}}\partial_{{\color{myred}r}}
\colon\Rbim\to\Rbim^{{\color{mypurple}p}},
\end{gathered}
\end{gather*}
which we call the secondary Demazure operators.
\end{definition}
Note that the action on the linear terms
determines the whole action since
we have the twisted Leibniz rule
$\partial_{{\color{dummy}c}}(fg)=\partial_{{\color{dummy}c}}(f)g+{\color{dummy}c}(f)\partial_{{\color{dummy}c}}(g)$.
Moreover, a straightforward calculation (cf. \cite[(3.9)]{El1}) yields
\begin{gather*}
\varstuff{q}\partial_{{\color{myblue}b}}\partial_{{\color{myyellow}y}}\partial_{{\color{myblue}b}}
=\partial_{{\color{myyellow}y}}\partial_{{\color{myblue}b}}\partial_{{\color{myyellow}y}},
\quad\quad
\varstuff{q}^{-1}\partial_{{\color{myred}r}}\partial_{{\color{myyellow}y}}\partial_{{\color{myred}r}}
=\partial_{{\color{myyellow}y}}\partial_{{\color{myred}r}}\partial_{{\color{myyellow}y}},
\quad\quad
\partial_{{\color{myblue}b}}\partial_{{\color{myred}r}}\partial_{{\color{myblue}b}}
=\partial_{{\color{myred}r}}\partial_{{\color{myblue}b}}\partial_{{\color{myred}r}},
\end{gather*}
showing that the mixed Demazure operators are well-defined.
(The careful reader might additionally want
to check that the primary Demazure operators
are well-defined by checking that $\partial_{{\color{dummy}c}}(f)$ is a
${\color{dummy}c}$-invariant polynomial.)
\begin{remark}\label{remark:grading}
Recalling that the root variables are of degree $2$,
one easily observes that the
primary, mixed and secondary Demazure operators
are homogeneous of degree $-2,-4$ and $-6$, respectively.
\end{remark}
\begin{lemma}\label{lemma:frob-extensions}
We have Frobenius extensions
\[
\partial_{{\color{dummy}c}}\colon\Rbim\to\Rbim^{{\color{dummy}c}},
\quad\quad
\partial_{{\color{dummy}\textbf{u}}}^{{\color{dummy}c}}\colon\Rbim^{{\color{dummy}c}}\to\Rbim^{{\color{dummy}\textbf{u}}},
\quad\quad
\partial_{{\color{dummy}\textbf{u}}}
\colon\Rbim\to\Rbim^{{\color{dummy}\textbf{u}}},
\]
of rank $2,3$ and $6$, respectively,
which are compatible in the sense that $\partial_{{\color{dummy}c}}=\partial_{{\color{dummy}\textbf{u}}}^{{\color{dummy}c}}\partial_{{\color{dummy}\textbf{u}}}$.
\end{lemma}
\begin{proof}
One can prove this lemma by computing explicit dual bases.
(Note that this requires $2$ and $3$ to be invertible.)
We do not need them here and omit the calculations.
\end{proof}
\begin{definition}\label{definition:comult-elements}
Choose any pairs of dual bases $\algstuff{B}_{\duc},\algstuff{B}_{\duc}^{\fdual}$ of
$\partial_{{\color{dummy}c}}\colon \Rbim\to \Rbim^{{\color{dummy}c}}$, $\algstuff{B}^{\duc}_{\tduc},(\algstuff{B}^{\duc}_{\tduc})^\fdual$
of $\partial^{{\color{dummy}c}}_{{\color{dummy}\textbf{u}}}\colon \Rbim^{{\color{dummy}c}}\to \Rbim^{{\color{dummy}\textbf{u}}}$ and $\algstuff{B}_{\tduc},\algstuff{B}_{\tduc}^{\fdual}$ of $\partial^{{\color{dummy}\textbf{u}}}
\colon\Rbim\to\Rbim^{{\color{dummy}\textbf{u}}}$. Let
\begin{gather}\label{eq:comult-elements}
\Frobel{{\color{dummy}c}}{\phantom{{\color{dummy}c}}}={\textstyle \sum_{a\in \algstuff{B}_{\duc}}} a\otimes a^{\star},
\quad\quad
\Frobel{{\color{dummy}\textbf{u}}}{{\color{dummy}c}}={\textstyle \sum_{a\in \algstuff{B}^{\duc}_{\tduc}}} a\otimes a^{\star},
\quad\quad
\Frobel{{\color{dummy}\textbf{u}}}{\phantom{{\color{dummy}c}}}={\textstyle \sum_{a\in \algstuff{B}_{\tduc}}} a\otimes a^{\star},
\end{gather}
where $a^{\star}$ denotes the basis element dual to $a$.
\end{definition}
Note that the elements $\Frobel{\underline{\phantom{a}}}{\underline{\phantom{a}}}$ are
well-defined, i.e. do not depend on the choice
of dual bases (see e.g. \cite[Section 2.4]{El1}).
\begin{definition}\label{definition:frob-elements}
We define the following elements $\frobel{\underline{\phantom{a}}}{\underline{\phantom{a}}}$ in $\Rbim$.
\begin{gather}\label{eq:frob-elements}
\begin{gathered}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep={0.5cm,between origins}, column
sep={2.25cm,between origins}, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{{\color{myblue}b}} \& {\color{myblue}b},{\color{myyellow}y} \& {\color{myblue}b} \& {\color{myyellow}y} \& \emptyset\\
{\color{mygreen}g} \& \varstuff{q}^{-1}\alpha_{\bc}+\alpha_{\yc} \& \alpha_{\yc}\frobel{{\color{mygreen}g}}{{\color{myblue}b},{\color{myyellow}y}} \& \alpha_{\bc}\frobel{{\color{mygreen}g}}{{\color{myblue}b},{\color{myyellow}y}}
\& \alpha_{\bc}\alpha_{\yc}\frobel{{\color{mygreen}g}}{{\color{myblue}b},{\color{myyellow}y}}
\\
};
\draw[densely dashed] ($(m-2-1.north)+(-.15,0)$) edge ($(m-2-5.north)+(1.05,0)$);
\draw[densely dashed] ($(m-1-1.east)+(1.0,.2)$) edge ($(m-2-1.east)+(1.0,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(3.2,.2)$) edge ($(m-2-1.east)+(3.2,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(5.3,.2)$) edge ($(m-2-1.east)+(5.3,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(7.7,.2)$) edge ($(m-2-1.east)+(7.7,-.3)$);
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep={0.5cm,between origins}, column
sep={2.25cm,between origins}, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{{\color{myblue}b}} \& {\color{myred}r},{\color{myyellow}y} \& {\color{myred}r} \& {\color{myyellow}y} \& \emptyset\\
{\color{myorange}o} \& \varstuff{q}\alpha_{\rc}+\alpha_{\yc} \& \alpha_{\yc}\frobel{{\color{myorange}o}}{{\color{myred}r},{\color{myyellow}y}} \& \alpha_{\rc}\frobel{{\color{myorange}o}}{{\color{myred}r},{\color{myyellow}y}}
\& \alpha_{\bc}\alpha_{\yc}\frobel{{\color{myorange}o}}{{\color{myred}r},{\color{myyellow}y}}
\\
};
\draw[densely dashed] ($(m-2-1.north)+(-.15,0)$) edge ($(m-2-5.north)+(1.05,0)$);
\draw[densely dashed] ($(m-1-1.east)+(1.0,.2)$) edge ($(m-2-1.east)+(1.0,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(3.2,.2)$) edge ($(m-2-1.east)+(3.2,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(5.3,.2)$) edge ($(m-2-1.east)+(5.3,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(7.7,.2)$) edge ($(m-2-1.east)+(7.7,-.3)$);
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep={0.5cm,between origins}, column
sep={2.25cm,between origins}, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
\phantom{{\color{myblue}b}} \& {\color{myblue}b},{\color{myred}r} \& {\color{myblue}b} \& {\color{myred}r} \& \emptyset\\
{\color{mypurple}p} \& \alpha_{\bc}+\alpha_{\rc} \& \alpha_{\rc}\frobel{{\color{mypurple}p}}{{\color{myblue}b},{\color{myred}r}} \& \alpha_{\bc}\frobel{{\color{mypurple}p}}{{\color{myblue}b},{\color{myred}r}}
\& \alpha_{\bc}\alpha_{\rc}\frobel{{\color{mypurple}p}}{{\color{myblue}b},{\color{myred}r}}
\\
};
\draw[densely dashed] ($(m-2-1.north)+(-.15,0)$) edge ($(m-2-5.north)+(1.05,0)$);
\draw[densely dashed] ($(m-1-1.east)+(1.0,.2)$) edge ($(m-2-1.east)+(1.0,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(3.2,.2)$) edge ($(m-2-1.east)+(3.2,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(5.3,.2)$) edge ($(m-2-1.east)+(5.3,-.3)$);
\draw[densely dashed] ($(m-1-1.east)+(7.7,.2)$) edge ($(m-2-1.east)+(7.7,-.3)$);
\end{tikzpicture}
\end{gathered}
\end{gather}
This is to be read as e.g.
$\frobel{{\color{mygreen}g}}{{\color{myblue}b},{\color{myyellow}y}}=\varstuff{q}^{-1}\alpha_{\bc}+\alpha_{\yc}$
and
$\frobel{{\color{mygreen}g}}{\emptyset}=\frobel{{\color{mygreen}g}}{\phantom{\emptyset}}=\alpha_{\bc}\alpha_{\yc}(\varstuff{q}^{-1}\alpha_{\bc}+\alpha_{\yc})$ etc.
\end{definition}
\subsubsection{The continuation of definition of \texorpdfstring{$\Adiag$}{singSbim}}\label{subsubsec:def-adiag-second}
\begin{definition}\label{definition:ssbim}
Let $\Adiag$ be the $2$-quotient of the additive, $\C_{\intqpar}$-linear
closure of $\ADiag$ defined as follows.
\medskip
\noindent\textit{\setword{`Relations of $\Adiag$'}{sbim-rel}.}
(We only give the relations for one choice of compatible colors and comment
on the others choices, where `Var.: comp. color.' means that the analogous relation
holds for other compatible colorings in the sense of \eqref{eq:color-compatible}.)
First, polynomial multiplication, i.e. polynomial decorations
on a facet multiply, and isotopy relations:
\begin{gather}\label{eq:iso-rel}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-1,0) to (-1,1.5) to [out=90, in=180] (-.5,2) to [out=0, in=90] (0,1.5) to [out=270, in=180] (.5,1) to [out=0, in=270] (1,1.5) to (1,3) to (-2,3) to (-2,0) to (-1,0);
\fill[mygreen, opacity=0.8] (-1,0) to (-1,1.5) to [out=90, in=180] (-.5,2) to [out=0, in=90] (0,1.5) to [out=270, in=180] (.5,1) to [out=0, in=270] (1,1.5) to (1,3) to (2,3) to (2,0) to (-1,0);
\draw[bstrand] (-1,0) to (-1,1.5) to [out=90, in=180] (-.5,2) to [out=0, in=90] (0,1.5) to [out=270, in=180] (.5,1) to [out=0, in=270] (1,1.5) to (1,3);
\draw[bstrand, directed=.5] (0,1.4) to (0,1.39);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (0,0) to (0,3) to (-1,3) to (-1,0) to (0,0);
\fill[mygreen, opacity=0.8] (0,0) to (0,3) to (1,3) to (1,0) to (0,0);
\draw[bstrand, directed=.55] (0,0) to (0,3);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (1,0) to (1,1.5) to [out=90, in=0] (.5,2) to [out=180, in=90] (0,1.5) to [out=270, in=0] (-.5,1) to [out=180, in=270] (-1,1.5) to (-1,3) to (-2,3) to (-2,0) to (1,0);
\fill[mygreen, opacity=0.8] (1,0) to (1,1.5) to [out=90, in=0] (.5,2) to [out=180, in=90] (0,1.5) to [out=270, in=0] (-.5,1) to [out=180, in=270] (-1,1.5) to (-1,3) to (2,3) to (2,0) to (1,0);
\draw[bstrand] (1,0) to (1,1.5) to [out=90, in=0] (.5,2) to [out=180, in=90] (0,1.5) to [out=270, in=0] (-.5,1) to [out=180, in=270] (-1,1.5) to (-1,3);
\draw[bstrand, directed=.5] (0,1.4) to (0,1.39);
\end{tikzpicture}
,\;\;\;\;
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-2,2) to [out=90, in=180] (0,3.5) to [out=0, in=90] (2,2) to (2,-2) to (2.5,-2) to (2.5,4) to (-3,4) to (-3,0) to [out=270, in=180] (-2.5,-.5) to [out=0, in=270] (-2,0) to [out=90, in=180] (-1,1) to [out=180, in=270] (-2,2);
\fill[myorange, opacity=0.8] (0,0) to [out=270, in=0] (-2,-1.5) to [out=180, in=270] (-4,0) to (-4,4) to (-4.5,4) to (-4.5,-2) to (1,-2) to (1,2) to [out=90, in=0] (.5,2.5) to [out=180, in=90] (0,2) to [out=270, in=0] (-1,1) to [out=0, in=90] (0,0);
\fill[myyellow, opacity=0.3] (0,0) to [out=270, in=0] (-2,-1.5) to [out=180, in=270] (-4,0) to (-4,4) to (-3,4) to (-3,0) to [out=270, in=180] (-2.5,-.5) to [out=0, in=270] (-2,0) to [out=90, in=180] (-1,1) to [out=0, in=90] (0,0);
\fill[myred, opacity=0.3] (-2,2) to [out=90, in=180] (0,3.5) to [out=0, in=90] (2,2) to (2,-2) to (1,-2) to (1,2) to [out=90, in=0] (.5,2.5) to [out=180, in=90] (0,2) to [out=270, in=0] (-1,1) to [out=180, in=270] (-2,2);
\draw[ystrand, directed=.999] (-2,0) to [out=90, in=180] (-1,1) to [out=0, in=270] (0,2);
\draw[ystrand] (-2,0) to [out=270, in=0] (-2.5,-.5) to [out=180, in=270] (-3,0) to (-3,4);
\draw[ystrand] (0,2) to [out=90, in=180] (.5,2.5) to [out=0, in=90] (1,2) to (1,-2);
\draw[rstrand, directed=.999] (0,0) to [out=90, in=0] (-1,1) to [out=180, in=270] (-2,2);
\draw[rstrand] (0,0) to [out=270, in=0] (-2,-1.5) to [out=180, in=270] (-4,0) to (-4,4);
\draw[rstrand] (-2,2) to [out=90, in=180] (0,3.5) to [out=0, in=90] (2,2) to (2,-2);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (3,2) to (2,2) to [out=270, in=0] (1,1) to [out=0, in=90] (2,0) to (3,0) to (3,2);
\fill[myorange, opacity=0.8] (-1,2) to (0,2) to [out=270, in=180] (1,1) to [out=180, in=90] (0,0) to (-1,0) to (-1,2);
\fill[myyellow, opacity=0.3] (1,1) to [out=0, in=270] (2,2) to (0,2) to [out=270, in=180] (1,1);
\fill[myred, opacity=0.3] (1,1) to [out=0, in=90] (2,0) to (0,0) to [out=90, in=180] (1,1);
\draw[ystrand, directed=.999] (2,2) to [out=270, in=0] (1,1) to [out=180, in=90] (0,0);
\draw[rstrand, directed=.999] (0,2) to [out=270, in=180] (1,1) to [out=0, in=90] (2,0);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (0,0) to [out=270, in=180] (2,-1.5) to [out=0, in=270] (4,0) to (4,4) to (4.5,4) to (4.5,-2) to (-1,-2) to (-1,2) to [out=90, in=180] (-.5,2.5) to [out=0, in=90] (0,2) to [out=270, in=180] (1,1) to [out=180, in=90] (0,0);
\fill[myorange, opacity=0.8] (2,2) to [out=90, in=0] (0,3.5) to [out=180, in=90] (-2,2) to (-2,-2) to (-2.5,-2) to (-2.5,4) to (3,4) to (3,0) to [out=270, in=0] (2.5,-.5) to [out=180, in=270] (2,0) to [out=90, in=0] (1,1) to [out=0, in=270] (2,2);
\fill[myyellow, opacity=0.3] (0,0) to [out=270, in=180] (2,-1.5) to [out=0, in=270] (4,0) to (4,4) to (3,4) to (3,0) to [out=270, in=0] (2.5,-.5) to [out=180, in=270] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\fill[myred, opacity=0.3] (2,2) to [out=90, in=0] (0,3.5) to [out=180, in=90] (-2,2) to (-2,-2) to (-1,-2) to (-1,2) to [out=90, in=180] (-.5,2.5) to [out=0, in=90] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2);
\draw[ystrand, directed=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=270] (2,2);
\draw[ystrand] (0,0) to [out=270, in=180] (2,-1.5) to [out=0, in=270] (4,0) to (4,4);
\draw[ystrand] (2,2) to [out=90, in=0] (0,3.5) to [out=180, in=90] (-2,2) to (-2,-2);
\draw[rstrand, directed=.999] (2,0) to [out=90, in=0] (1,1) to [out=180, in=270] (0,2);
\draw[rstrand] (2,0) to [out=270, in=180] (2.5,-.5) to [out=0, in=270] (3,0) to (3,4);
\draw[rstrand] (0,2) to [out=90, in=0] (-.5,2.5) to [out=180, in=90] (-1,2) to (-1,-2);
\end{tikzpicture}
};
(-10,-14.5)*{\text{{\tiny Var.: comp. color.}}};
\endxy
\end{gather}
Then various relations involving circles, called
circle removals:
\\
\noindent\begin{tabularx}{0.99\textwidth}{XXX}
\begin{equation}\hspace{-9.5cm}\label{eq:circle-first}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,-1.5) to (-.5,1.5) to (2.5,1.5) to (2.5,-1.5) to (-.5,-1.5);
\fill[myblue, opacity=0.3] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0);
\draw[bstrand, directed=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0);
\draw[bstrand] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,-1.5) to (-.5,1.5) to (2.5,1.5) to (2.5,-1.5) to (-.5,-1.5);
\node at (1,0) {$\polybox{\alpha_{\bc}}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny Var.: comp. color.,}}};
(0,-10.5)*{\text{{\tiny using
$\alpha_{\rc}$ or $\alpha_{\yc}$.}}};
\endxy
\end{equation} &
\begin{equation}\hspace{-9.5cm}\label{eq:circle-secondary}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myblue, opacity=0.3] (-.5,-1.5) to (-.5,1.5) to (2.5,1.5) to (2.5,-1.5) to (-.5,-1.5);
\fill[white] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0);
\fill[mypurple, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0);
\draw[rstrand, directed=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0);
\draw[rstrand] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myblue, opacity=0.3] (-.5,-1.5) to (-.5,1.5) to (2.5,1.5) to (2.5,-1.5) to (-.5,-1.5);
\node at (1,0) {$\polybox{\frobel{{\color{mypurple}p}}{{\color{myblue}b}}}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny Var.: comp. color.,}}};
(0,-10.5)*{\text{{\tiny using $\frobel{{\color{dummy}\textbf{u}}}{{\color{dummy}c}}$.}}};
\endxy
\end{equation} &
\begin{equation}\hspace{-9.5cm}\label{eq:circle-primary}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-.5,-1.5) to (-.5,1.5) to (2.5,1.5) to (2.5,-1.5) to (-.5,-1.5);
\fill[white] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0);
\fill[myyellow, opacity=0.3] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to [out=270, in=0] (1,-1) to [out=180, in=270] (0,0);
\draw[bstrand, rdirected=.999] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0);
\draw[bstrand] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\node at (1,0) {$\polybox{p}$};
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-.5,-1.5) to (-.5,1.5) to (2.5,1.5) to (2.5,-1.5) to (-.5,-1.5);
\node at (1,0) {$\polybox{\partial_{{\color{mygreen}g}}^{{\color{myyellow}y}}(p)}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny Var.: comp. color.,}}};
(0,-10.5)*{\text{{\tiny using
$\partial_{{\color{dummy}c}}^{\phantom{{\color{dummy}c}}}$ or $\partial_{{\color{dummy}\textbf{u}}}^{{\color{dummy}c}}$.}}};
\endxy
\end{equation}
\end{tabularx}\\
(Note that there is also a variation of \eqref{eq:circle-primary}
with a circular $\emptyset$-region in the middle bounded by a primary colored region outside.)
Moreover, we have polynomial sliding and neck cutting relations, i.e.
\\
\noindent\begin{tabularx}{0.99\textwidth}{XX}
\begin{equation}\hspace{-7cm}\label{eq:poly-move}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (0,0) to (0,3) to (-2,3) to (-2,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to (0,3) to (2,3) to (2,0) to (0,0);
\draw[rstrand, directed=.55] (0,0) to (0,3);
\node at (-1,1.5) {$\polybox{p}$};
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (0,0) to (0,3) to (-2,3) to (-2,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to (0,3) to (2,3) to (2,0) to (0,0);
\draw[rstrand, directed=.55] (0,0) to (0,3);
\node at (1,1.5) {$\polybox{p}$};
\end{tikzpicture}
,\,p\in\Rbim^{{\color{myorange}o}}
};
(0,-8)*{\text{{\tiny Var.: comp. color.,}}};
(0,-10.5)*{\text{{\tiny for
$p\in\Rbim^{{\color{dummy}c}}$ or $p\in\Rbim^{{\color{dummy}\textbf{u}}}$.}}};
\endxy
\end{equation} &
\begin{equation}\hspace{-8cm}\label{eq:neck-cut}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myred, opacity=0.3] (-1,0) to (3,0) to (3,3) to (-1,3) to (-1,0);
\fill[white] (.5,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.5,0) to (.5,0);
\fill[white] (.5,3) to [out=270, in=180] (1,2.25) to [out=0, in=270] (1.5,3) to (.5,3);
\fill[mypurple, opacity=0.8] (.5,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.5,0) to (.5,0);
\fill[mypurple, opacity=0.8] (.5,3) to [out=270, in=180] (1,2.25) to [out=0, in=270] (1.5,3) to (.5,3);
\draw[bstrand] (.5,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.5,0);
\draw[bstrand] (1.5,3) to [out=270, in=0] (1,2.25) to [out=180, in=270] (.5,3);
\draw[bstrand, directed=.5] (1.1,.75) to (1.11,.75);
\draw[bstrand, directed=.5] (.91,2.25) to (.9,2.25);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myred, opacity=0.3] (-1,0) to (.5,0) to (.5,3) to (-1,3) to (-1,0);
\fill[myred, opacity=0.3] (3,0) to (1.5,0) to (1.5,3) to (3,3) to (3,0);
\fill[mypurple, opacity=0.8] (.5,0) to (.5,3) to (1.5,3) to (1.5,0) to (.5,0);
\draw[bstrand, directed=.2, directed=.9] (.5,0) to (.5,3);
\draw[bstrand, directed=.2, directed=.9] (1.5,3) to (1.5,0);
\node at (1,1.5) {$\polybox{\Frobel{{\color{mypurple}p}}{{\color{myred}r}}}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny Var.: comp. color.,}}};
(0,-10.5)*{\text{{\tiny using
$\Frobel{{\color{dummy}c}}{\phantom{{\color{dummy}c}}}$ or $\Frobel{{\color{dummy}\textbf{u}}}{{\color{dummy}c}}$.}}};
\endxy
\end{equation}
\end{tabularx}\\
The notation in the neck cutting relations \eqref{eq:neck-cut} indicates that one has put the tensor factors
of the various summands of the $\Frobel{\underline{\phantom{a}}}{\underline{\phantom{a}}}$ in the
corresponding regions (i.e. left tensor factors in the leftmost region and
right tensor factors in the rightmost region), cf. \fullref{example:more-relations}.
Next, Reidemeister-like relations:
\\
\noindent\begin{tabularx}{0.99\textwidth}{XXX}
\begin{equation}\hspace{-9.0cm}\label{eq:rm-first}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-1,0) to [out=90, in=225] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.75,3) to (-1.75,0) to (-1,0);
\fill[myblue, opacity=0.3] (-1,0) to [out=90, in=225] (-.5,.75) to [out=315, in=90] (0,0) to (-1,0);
\fill[myblue, opacity=0.3] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myyellow, opacity=0.3] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to [out=270, in=45] (-.5,.75);
\fill[mygreen, opacity=0.8] (0,0) to [out=90, in=315] (-.5,.75) to [out=45, in=270] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[ystrand, directed=.55] (0,0) to [out=90, in=270] (-1,1.5) to [out=90, in=270] (0,3);
\draw[bstrand, directed=.55] (-1,0) to [out=90, in=270] (0,1.5) to [out=90, in=270] (-1,3);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-1.75,0) to (-1.75,3) to (-1,3) to (-1,0) to (-1.75,0);
\fill[myblue, opacity=0.3] (-1,0) to (-1,3) to (0,3) to (0,0) to (-1,0);
\fill[mygreen, opacity=0.8] (0,0) to (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[ystrand, directed=.55] (0,0) to (0,3);
\draw[bstrand, directed=.55] (-1,0) to (-1,3);
\end{tikzpicture}
};
(0,-8)*{\text{{\tiny Var.: comp. color.}}};
(0,-10.5)*{\text{{\tiny $\phantom{\frobel{{\color{dummy}\textbf{u}}}{{\color{dummy}c}}}$}}};
\endxy
\end{equation} &
\begin{equation}\hspace{-9.0cm}\label{eq:rm-second}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-1,0) to [out=90, in=225] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.75,3) to (-1.75,0) to (-1,0);
\fill[myorange, opacity=0.8] (-1,0) to [out=90, in=225] (-.5,.75) to [out=315, in=90] (0,0) to (-1,0);
\fill[myorange, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[white] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to [out=270, in=45] (-.5,.75);
\fill[myred, opacity=0.3] (0,0) to [out=90, in=315] (-.5,.75) to [out=45, in=270] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[ystrand, directed=.55] (0,3) to [out=270, in=90] (-1,1.5) to [out=270, in=90] (0,0);
\draw[rstrand, directed=.55] (-1,0) to [out=90, in=270] (0,1.5) to [out=90, in=270] (-1,3);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-1.75,0) to (-1.75,3) to (-1,3) to (-1,0) to (-1.75,0);
\fill[myorange, opacity=0.8] (-1,0) to (-1,3) to (0,3) to (0,0) to (-1,0);
\fill[myred, opacity=0.3] (0,0) to (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[ystrand, directed=.2, directed=.9] (0,3) to (0,0);
\draw[rstrand, directed=.2, directed=.9] (-1,0) to (-1,3);
\node at (-.5,1.5) {$\polybox{\partial\Frobel{{\color{myorange}o}}{\phantom{.}}}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny Var.: comp. color.,}}};
(0,-10.5)*{\text{{\tiny using $\partial\Frobel{{\color{dummy}\textbf{u}}}{\phantom{.}}$.}}};
\endxy
\end{equation} &
\begin{equation}\hspace{-9.0cm}\label{eq:rm-third}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myred, opacity=0.3] (-1,0) to [out=90, in=225] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.75,3) to (-1.75,0) to (-1,0);
\draw[very thin, densely dotted, fill=white] (-1,0) to [out=90, in=225] (-.5,.75) to [out=315, in=90] (0,0) to (-1,0);
\draw[very thin, densely dotted, fill=white] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[mypurple, opacity=0.8] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to [out=270, in=45] (-.5,.75);
\fill[myblue, opacity=0.3] (0,0) to [out=90, in=315] (-.5,.75) to [out=45, in=270] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[rstrand, directed=.55] (-1,3) to [out=270, in=90] (0,1.5) to [out=270, in=90] (-1,0);
\draw[bstrand, directed=.55] (0,0) to [out=90, in=270] (-1,1.5) to [out=90, in=270] (0,3);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myred, opacity=0.3] (-1.75,0) to (-1.75,3) to (-1,3) to (-1,0) to (-1.75,0);
\fill[myblue, opacity=0.3] (0,0) to (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[very thin, densely dotted, fill=white] (-1,0) to [out=90, in=270] (-1.32,1.5) to [out=90, in=270] (-1,3) to (0,3) to [out=270, in=90] (.32,1.5) to [out=270, in=90] (0,0) to (-1,0);
\draw[rstrand, directed=.55] (-1,3) to [out=270, in=90] (-1.32,1.5) to [out=270, in=90] (-1,0);
\draw[bstrand, directed=.55] (0,0) to [out=90, in=270] (.32,1.5) to [out=90, in=270] (0,3);
\node at (-.475,1.5) {$\frobel{{\color{mypurple}p}}{{\color{myblue}b},{\color{myred}r}}$};
\end{tikzpicture}};
(0,-8)*{\text{{\tiny Var.: comp. color.,}}};
(0,-10.5)*{\text{{\tiny using $\frobel{{\color{dummy}\textbf{u}}}{{\color{dummy}c},{\color{dummy}d}}$.}}};
\endxy
\end{equation}
\end{tabularx}\\
where the notation $\partial\Frobel{{\color{myorange}o}}{\phantom{.}}$ in \eqref{eq:rm-second} means
\begin{gather*}
\partial\Frobel{{\color{myorange}o}}{\phantom{.}}=
\partial_{{\color{myyellow}y}}(\Frobel{{\color{myorange}o}}{{\color{myred}r}}(1))\otimes\Frobel{{\color{myorange}o}}{{\color{myred}r}}(2)
=
\Frobel{{\color{myorange}o}}{{\color{myyellow}y}}(1)\otimes\partial_{{\color{myred}r}}(\Frobel{{\color{myorange}o}}{{\color{myyellow}y}}(2))
\end{gather*}
which is to be read again in the corresponding regions.
Finally, the square relations, which we exemplify
by the case in which ${\color{myyellow}y}\emptyset{\color{myblue}b}$ is at the bottom and ${\color{myblue}b}\emptyset{\color{myyellow}y}$
is at the top:
\begin{gather}\label{eq:square}
\xy
(0,0)*{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted] (-3,0) to (-3,3);
\draw[very thin, densely dotted] (-1,3) to (1,3);
\draw[very thin, densely dotted] (-1,0) to (1,0);
\draw[very thin, densely dotted] (3,0) to (3,3);
\fill[myyellow, opacity=.3] (3,3) to (1,3) to (0,2.25) to (1,1.5) to (3,3);
\fill[myyellow, opacity=.3] (-3,0) to (-1,0) to (0,.75) to (-1,1.5) to (-3,0);
\fill[myblue, opacity=.3] (-3,3) to (-1,3) to (0,2.25) to (-1,1.5) to (-3,3);
\fill[myblue, opacity=.3] (3,0) to (1,0) to (0,.75) to (1,1.5) to (3,0);
\fill[mygreen, opacity=.8] (0,2.25) to (-1,1.5) to (0,.75) to (1,1.5) to (0,2.25);
\draw[ystrand, directed=.65] (-3,0) to (1,3);
\draw[ystrand, rdirected=.4] (-1,0) to (3,3);
\draw[bstrand, directed=.4] (1,0) to (-3,3);
\draw[bstrand, rdirected=.65] (3,0) to (-1,3);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted] (-3,0) to (-3,3);
\draw[very thin, densely dotted] (-1,3) to (1,3);
\draw[very thin, densely dotted] (-1,0) to (1,0);
\draw[very thin, densely dotted] (3,0) to (3,3);
\fill[myyellow, opacity=.3] (-3,0) to (-2.75,.25) to [out=45, in=45] (-.75,.25) to (-1,0) to (-3,0);
\fill[myyellow, opacity=.3] (1,3) to (.75,2.75) to [out=225, in=225] (2.75,2.75) to (3,3) to (1,3);
\fill[myblue, opacity=.3] (-3,3) to (-1,3) to (3,0) to (1,0) to (-3,3);
\draw[ystrand, directed=.55] (-3,0) to (-2.75,.25) to [out=45, in=45] (-.75,.25) to (-1,0);
\draw[ystrand, rdirected=.55] (1,3) to (.75,2.75) to [out=225, in=225] (2.75,2.75) to (3,3);
\draw[bstrand, directed=.4] (1,0) to (-3,3);
\draw[bstrand, rdirected=.65] (3,0) to (-1,3);
\end{tikzpicture}
+
\varstuff{q}^{-1}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted] (-3,0) to (-3,3);
\draw[very thin, densely dotted] (-1,3) to (1,3);
\draw[very thin, densely dotted] (-1,0) to (1,0);
\draw[very thin, densely dotted] (3,0) to (3,3);
\fill[myyellow, opacity=.3] (3,3) to (1,3) to (-3,0) to (-1,0) to (3,3);
\fill[myblue, opacity=.3] (1,0) to (.75,.25) to [out=135, in=135] (2.75,.25) to (3,0) to (1,0);
\fill[myblue, opacity=.3] (-3,3) to (-2.75,2.75) to [out=315, in=315] (-.75,2.75) to (-1,3) to (-3,3);
\draw[ystrand, directed=.65] (-3,0) to (1,3);
\draw[ystrand, rdirected=.4] (-1,0) to (3,3);
\draw[bstrand, directed=.45] (1,0) to (.75,.25) to [out=135, in=135] (2.75,.25) to (3,0);
\draw[bstrand, rdirected=.65] (-3,3) to (-2.75,2.75) to [out=315, in=315] (-.75,2.75) to (-1,3);
\end{tikzpicture}};
(0,-8.5)*{\text{{\tiny Var.: comp. color.; in case ${\color{myyellow}y}\emptyset{\color{myred}r}$ replace $\varstuff{q}^{-1}\rightsquigarrow\varstuff{q}$;
in case ${\color{mypurple}p}\emptyset{\color{myblue}b}$ replace $\varstuff{q}^{-1}\rightsquigarrow 1$.}}};
\endxy
\end{gather}
(We stress that \eqref{eq:square} is not invariant under color change.)
\end{definition}
\begin{definition}\label{definition:regular}
The $2$-category of regular Bott--Samelson bimodules is defined as
\[
\adiag=\Adiag(\emptyset,\emptyset),
\]
i.e. the $2$-full $2$-subcategory of $\Adiag$ generated by diagrams
whose left- and rightmost color is $\emptyset$. Note that $\adiag$
has only one object, namely $\emptyset$.
The $2$-category of maximally singular Bott--Samelson bimodules is defined as
\[
\aDiag={\textstyle\bigoplus_{{\color{dummy}\textbf{u}},{\color{dummy}\textbf{v}}\in\Gset\Oset\Pset}}\,\Adiag({\color{dummy}\textbf{u}},{\color{dummy}\textbf{v}}),
\]
i.e. the $2$-full $2$-subcategory of $\Adiag$ generated by diagrams whose
left- and rightmost colors are secondary.
\end{definition}
Note that we can always extend scalars to
e.g. $\Cq=\mathbb{C}(\varstuff{q})$ and we indicate this by changing the
subscript $[\varstuff{q}]$ to $\varstuff{q}$.
\begin{remark}\label{remark:cat-affine-a2-scalars}
$\Adiag$ is an additive, $\C_{\intqpar}$-linear, graded $2$-category, which is,
however, not idempotent closed.
This is remedied by considering its Karoubi envelope $\Kar{\Adiag}$, which we
take as the definition of the $2$-category of
singular Soergel bimodules of affine type $\typea{2}$.
\end{remark}
Thus, we have:
\smallskip
\begin{enumerate}[label=$\blacktriangleright$]
\setlength\itemsep{.15cm}
\item The $2$-category of singular
Bott--Samelson bimodules, whose notation contains an $\boldsymbol{s}$.
\item The $2$-category of regular
Bott--Samelson bimodules, whose notation has no $\boldsymbol{s}$.
\item The $2$-category of maximally
singular Bott--Samelson bimodules,
whose notation contains an $\boldsymbol{m}$.
As we will see, the degree-zero
part of this $2$-subcategory,
for a fixed choice of shifts of the $1$-morphisms, is semisimple.
\item The corresponding $2$-categories
of singular, regular and maximally singular Soergel bimodules are the Karoubi envelopes of these, by definition.
\item Various scalar extensions of these, indicated by subscripts.
\end{enumerate}
We use similar notations throughout, e.g. for scalar extensions
of $2$-functors.
\begin{remark}\label{remark:cat-affine-a2}
By \cite[Theorem A.1]{El1}, the decategorification
of $\Kar{\Adiag[\qpar]}$, via the split Grothendieck group, is isomorphic to the
affine $\typea{2}$ Hecke algebroid. As explained for example in \cite[Section 2.3]{Wi-sing-soergel}
(under the name Schur algebroid), this is a multi object version of the affine
Hecke algebra $\hecke$ from \fullref{subsec:def-sl3alg-1}. Moreover,
the $2$-full $2$-subcategory $\Kar{\adiag[\qpar]}$ decategorifies to
$\hecke$, see e.g. \cite[Theorem 3.17]{ElWi-soergel-calculus}.
\end{remark}
\subsubsection{Examples and further comments}\label{subsubsec:further-comments}
\begin{example}\label{example:q-demazure-action}
In accordance with \eqref{eq:color-compatible}, we have the following Frobenius extensions
\[
\begin{tikzpicture}[baseline=(current bounding box.center), tinynodes]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep=.02cm, column
sep=.01cm, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
{\color{mygray}\scalebox{.8}{$\Rbim^{p}$}} \& \scalebox{.8}{$\Rbim^{{\color{mygreen}g}}$}\, \& {\color{mygray}\scalebox{.8}{$\Rbim^{o}$}} \\
\,\scalebox{.8}{$\Rbim^{{\color{myblue}b}}$} \& {\color{mygray}\scalebox{.8}{$\Rbim^{r}$}} \& \scalebox{.8}{$\Rbim^{{\color{myyellow}y}}$} \\
\& \scalebox{.8}{$\Rbim$} \& \\
};
\draw[thin] ($(m-2-1.north)+(0,-.225cm)$) to ($(m-1-2.south)+(-.05,.075cm)$);
\draw[thin] ($(m-2-3.north)+(0,-.225cm)$) to ($(m-1-2.south)+(.05,.075cm)$);
\draw[thin] ($(m-3-2.north)+(-.05,-.225cm)$) to ($(m-2-1.south)+(0,.075cm)$);
\draw[thin] ($(m-3-2.north)+(.05,-.225cm)$) to ($(m-2-3.south)+(0,.075cm)$);
\end{tikzpicture}
\quad\quad
\begin{tikzpicture}[baseline=(current bounding box.center), tinynodes]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep=.02cm, column
sep=.01cm, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
{\color{mygray}\scalebox{.8}{$\Rbim^{p}$}} \& {\color{mygray}\scalebox{.8}{$\Rbim^{g}$}} \& \scalebox{.8}{$\Rbim^{{\color{myorange}o}}$} \\
{\color{mygray}\scalebox{.8}{$\Rbim^{b}$}} \& \scalebox{.8}{$\Rbim^{{\color{myred}r}}$} \& \scalebox{.8}{$\Rbim^{{\color{myyellow}y}}$} \\
\& \scalebox{.8}{$\Rbim$}\& \\
};
\draw[thin] ($(m-2-3.north)+(.035,-.225cm)$) to ($(m-1-3.south)+(.035,.075cm)$);
\draw[thin] ($(m-3-2.north)+(0,-.225cm)$) to ($(m-2-2.south)+(0,.075cm)$);
\draw[thin] ($(m-3-2.north)+(.05,-.225cm)$) to ($(m-2-3.south)+(0,.075cm)$);
\draw[thin] ($(m-2-2.north)+(.05,-.225cm)$) to ($(m-1-3.south)+(0,.075cm)$);
\end{tikzpicture}
\quad\quad
\begin{tikzpicture}[baseline=(current bounding box.center), tinynodes]
\matrix (m) [matrix of math nodes, nodes in empty cells, row sep=.02cm, column
sep=.01cm, text height=1.6ex, text depth=0.25ex, ampersand replacement=\&] {
\scalebox{.8}{$\Rbim^{{\color{mypurple}p}}$} \& {\color{mygray}\scalebox{.8}{$\Rbim^{g}$}} \& {\color{mygray}\scalebox{.8}{$\Rbim^{o}$}} \\
\scalebox{.8}{$\Rbim^{{\color{myblue}b}}$} \& \scalebox{.8}{$\Rbim^{{\color{myred}r}}$} \& {\color{mygray}\scalebox{.8}{$\Rbim^{y}$}} \\
\& \scalebox{.8}{$\Rbim$} \& \\
};
\draw[thin] ($(m-2-1.north)+(-.035,-.225cm)$) to ($(m-1-1.south)+(-.035,.075cm)$);
\draw[thin] ($(m-3-2.north)+(-.05,-.225cm)$) to ($(m-2-1.south)+(0,.075cm)$);
\draw[thin] ($(m-3-2.north)+(0,-.225cm)$) to ($(m-2-2.south)+(0,.075cm)$);
\draw[thin] ($(m-2-2.north)+(-.05,-.225cm)$) to ($(m-1-1.south)+(0,.075cm)$);
\end{tikzpicture}
\]
with the corresponding trace maps going upwards. Moreover,
\begin{gather}\label{eq:ranks}
\partial(\alpha_{\bc})=2,
\quad\quad
\partial_{{\color{mygreen}g}}^{{\color{myblue}b}}(\frobel{{\color{mygreen}g}}{{\color{myblue}b}})=3,
\quad\quad
\partial_{{\color{mygreen}g}}(\frobel{{\color{mygreen}g}}{})=6,
\end{gather}
as an easy calculation shows. Similar results hold for other colors.
Note that the numbers in \eqref{eq:ranks}, which follow from
\eqref{eq:circle-first}, \eqref{eq:circle-secondary} and \eqref{eq:circle-primary}, are precisely the ranks of the corresponding Frobenius extensions.
\end{example}
\begin{example}\label{example:some-relations}
When working with $\Adiag$, it is important to remember
that the polynomial $2$-generators
of a given facet are invariant under the action of the parabolic subgroup
which corresponds to the color of that region. For example, $\frobel{{\color{mygreen}g}}{{\color{myyellow}y}}$ is an element of
$\Rbim^{{\color{myyellow}y}}$, and applying $\partial_{{\color{mygreen}g}}^{{\color{myyellow}y}}$ to it
will make it additionally ${\color{myblue}b}$-invariant. In fact,
\[
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\fill[white] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\fill[myyellow, opacity=0.3] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\fill[white] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0) to [out=270, in=0] (1,-.75) to [out=180, in=270] (.25,0);
\fill[mygreen, opacity=0.8] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0) to [out=270, in=0] (1,-.75) to [out=180, in=270] (.25,0);
\draw[bstrand, directed=.999] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0);
\draw[bstrand] (.25,0) to [out=270, in=180] (1,-.75) to [out=0, in=270] (1.75,0);
\draw[bstrand, rdirected=.999] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0);
\draw[bstrand] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0);
\end{tikzpicture}
\stackrel{\eqref{eq:circle-secondary}}{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\fill[white] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\fill[myyellow, opacity=0.3] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\draw[bstrand, rdirected=.999] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0);
\draw[bstrand] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0);
\node at (1,0) {$\polybox{\frobel{{\color{mygreen}g}}{{\color{myyellow}y}}}$};
\end{tikzpicture}
\stackrel{\eqref{eq:circle-primary}}{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\node at (1,0) {$\polybox{\partial_{{\color{mygreen}g}}^{{\color{myyellow}y}}(\frobel{{\color{mygreen}g}}{{\color{myyellow}y}})}$};
\end{tikzpicture}
=
3\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\end{tikzpicture}
\]
We also get
\[
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\fill[white] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\fill[myyellow, opacity=0.3] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\fill[white] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0) to [out=270, in=0] (1,-.75) to [out=180, in=270] (.25,0);
\fill[myorange, opacity=0.8] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0) to [out=270, in=0] (1,-.75) to [out=180, in=270] (.25,0);
\draw[bstrand, rdirected=.999] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0);
\draw[bstrand] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0);
\draw[rstrand, directed=.999] (.25,0) to [out=90, in=180] (1,.75) to [out=0, in=90] (1.75,0);
\draw[rstrand] (.25,0) to [out=270, in=180] (1,-.75) to [out=0, in=270] (1.75,0);
\end{tikzpicture}
\stackrel{\eqref{eq:circle-secondary}}{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\fill[white] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\fill[myyellow, opacity=0.3] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0) to [out=270, in=0] (1,-1.5) to [out=180, in=270] (-.5,0);
\draw[bstrand, rdirected=.999] (-.5,0) to [out=90, in=180] (1,1.5) to [out=0, in=90] (2.5,0);
\draw[bstrand] (-.5,0) to [out=270, in=180] (1,-1.5) to [out=0, in=270] (2.5,0);
\node at (1,0) {$\polybox{\frobel{{\color{myorange}o}}{{\color{myyellow}y}}}$};
\end{tikzpicture}
\stackrel{\eqref{eq:circle-primary}}{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\node at (1,0) {$\polybox{\partial_{{\color{mygreen}g}}^{{\color{myyellow}y}}(\frobel{{\color{myorange}o}}{{\color{myyellow}y}})}$};
\end{tikzpicture}
=
\qnumber{3}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,-2) to (-1,2) to (3,2) to (3,-2) to (-1,-2);
\end{tikzpicture}
\]
which we will need below.
\end{example}
\begin{example}\label{example:more-relations}
We have
\[
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-1,0) to [out=90, in=225] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.75,3) to (-1.75,0) to (-1,0);
\fill[mygreen, opacity=0.8] (-1,0) to [out=90, in=225] (-.5,.75) to [out=315, in=90] (0,0) to (-1,0);
\fill[mygreen, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[white] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to [out=270, in=45] (-.5,.75);
\fill[myblue, opacity=0.3] (0,0) to [out=90, in=315] (-.5,.75) to [out=45, in=270] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[ystrand, directed=.55] (0,3) to [out=270, in=90] (-1,1.5) to [out=270, in=90] (0,0);
\draw[bstrand, directed=.55] (-1,0) to [out=90, in=270] (0,1.5) to [out=90, in=270] (-1,3);
\end{tikzpicture}
\stackrel{\eqref{eq:rm-second}}{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-1.75,0) to (-1.75,3) to (-1,3) to (-1,0) to (-1.75,0);
\fill[mygreen, opacity=0.8] (-1,0) to (-1,3) to (0,3) to (0,0) to (-1,0);
\fill[myblue, opacity=0.3] (0,0) to (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[ystrand, directed=.2, directed=.9] (0,3) to (0,0);
\draw[bstrand, directed=.2, directed=.9] (-1,0) to (-1,3);
\node at (-.5,1.5) {$\polybox{\partial\Frobel{{\color{mygreen}g}}{\phantom{.}}}$};
\end{tikzpicture}
=
{\textstyle \sum_{\algstuff{x}\in\algstuff{B}_{{\color{mygreen}g}}^{{\color{myblue}b}}}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-4,0) to (-4,3) to (-1,3) to (-1,0) to (-4,0);
\fill[mygreen, opacity=0.8] (-1,0) to (-1,3) to (0,3) to (0,0) to (-1,0);
\fill[myblue, opacity=0.3] (0,0) to (0,3) to (3,3) to (3,0) to (0,0);
\draw[ystrand, directed=.55] (0,3) to (0,0);
\draw[bstrand, directed=.55] (-1,0) to (-1,3);
\node at (-2.65,1.5) {$\polybox{\partial_{{\color{myyellow}y}}(\algstuff{x})}$};
\node at (1.5,1.5) {$\polybox{\algstuff{x}^{\star}}$};
\end{tikzpicture}
=
{\textstyle \sum_{\algstuff{y}\in\algstuff{B}_{{\color{mygreen}g}}^{{\color{myyellow}y}}}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-4,0) to (-4,3) to (-1,3) to (-1,0) to (-4,0);
\fill[mygreen, opacity=0.8] (-1,0) to (-1,3) to (0,3) to (0,0) to (-1,0);
\fill[myblue, opacity=0.3] (0,0) to (0,3) to (3,3) to (3,0) to (0,0);
\draw[ystrand, directed=.55] (0,3) to (0,0);
\draw[bstrand, directed=.55] (-1,0) to (-1,3);
\node at (-2.5,1.5) {$\polybox{\algstuff{y}}$};
\node at (1.65,1.5) {$\polybox{\partial_{{\color{myblue}b}}(\algstuff{y}^{\star})}$};
\end{tikzpicture}
\]
More generally, cf. \cite[(3.15d)]{El1}, the
relations in
\fullref{definition:ssbim} imply
\[
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-1,0) to [out=90, in=225] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.75,3) to (-1.75,0) to (-1,0);
\fill[mygreen, opacity=0.8] (-1,0) to [out=90, in=225] (-.5,.75) to [out=315, in=90] (0,0) to (-1,0);
\fill[mygreen, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[white] (-.5,.75) to [out=135, in=270] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to [out=270, in=45] (-.5,.75);
\fill[myblue, opacity=0.3] (0,0) to [out=90, in=315] (-.5,.75) to [out=45, in=270] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.75,3) to (.75,0) to (0,0);
\draw[ystrand, directed=.55] (0,3) to [out=270, in=90] (-1,1.5) to [out=270, in=90] (0,0);
\draw[bstrand, directed=.55] (-1,0) to [out=90, in=270] (0,1.5) to [out=90, in=270] (-1,3);
\node at (-.5,1.5) {\text{{\tiny$p$}}};
\end{tikzpicture}
=
{\textstyle \sum_{\algstuff{x}\in\algstuff{B}_{{\color{mygreen}g}}^{{\color{myblue}b}}}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-4,0) to (-4,3) to (-1,3) to (-1,0) to (-4,0);
\fill[mygreen, opacity=0.8] (-1,0) to (-1,3) to (0,3) to (0,0) to (-1,0);
\fill[myblue, opacity=0.3] (0,0) to (0,3) to (3,3) to (3,0) to (0,0);
\draw[ystrand, directed=.55] (0,3) to (0,0);
\draw[bstrand, directed=.55] (-1,0) to (-1,3);
\node at (-2.65,1.5) {$\polybox{\partial_{{\color{myyellow}y}}(p\algstuff{x})}$};
\node at (1.5,1.5) {$\polybox{\algstuff{x}^{\star}}$};
\end{tikzpicture}
=
{\textstyle \sum_{\algstuff{y}\in\algstuff{B}_{{\color{mygreen}g}}^{{\color{myyellow}y}}}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-4,0) to (-4,3) to (-1,3) to (-1,0) to (-4,0);
\fill[mygreen, opacity=0.8] (-1,0) to (-1,3) to (0,3) to (0,0) to (-1,0);
\fill[myblue, opacity=0.3] (0,0) to (0,3) to (3,3) to (3,0) to (0,0);
\draw[ystrand, directed=.55] (0,3) to (0,0);
\draw[bstrand, directed=.55] (-1,0) to (-1,3);
\node at (-2.5,1.5) {$\polybox{\algstuff{y}}$};
\node at (1.65,1.5) {$\polybox{\!\partial_{{\color{myblue}b}}(p\algstuff{y}^{\star})\!\!}$};
\end{tikzpicture},
p\in\Rbim.
\]
As usual, similar relations hold for other colors.
\end{example}
\subsection{The trihedral Soergel bimodules of level \texorpdfstring{$\infty$}{infty}}\label{subsec:ubsbim}
\subsubsection{The definition}\label{subsec:cat-thealgebra-def}
We first consider a $2$-subcategory categorifying
$\subquo$.
\begin{definition}\label{definition:the-subquo-cat}
Let $\subcatquo$ be
the additive closure of
the $2$-full $2$-subcategory of $\adiag$,
whose $1$-morphisms are generated by the color strings that have at least one secondary color and
have $\emptyset$ as the left- and rightmost color but nowhere else in the string.
Its scalar extension is denoted by $\subcatquo[\infty]=\subcatquo[\infty,\varstuff{q}]$.
\end{definition}
\begin{example}\label{example:the-subquo-cat}
The prototypical $1$-morphisms of $\subcatquo$ are $\emptyset$ and
all compatible color variation of
\[
\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset,
\quad
\emptyset{\color{myyellow}y}{\color{mygreen}g}{\color{myyellow}y}\emptyset,
\quad
\emptyset{\color{myyellow}y}{\color{mygreen}g}{\color{myblue}b}\emptyset,
\quad
\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myyellow}y}\emptyset,
\quad
\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myyellow}y}{\color{myorange}o}{\color{myyellow}y}\emptyset,
\quad
\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myyellow}y}{\color{myorange}o}{\color{myred}r}\emptyset,
\quad
\text{etc.}
\]
All other $1$-morphism in $\subcatquo$ are direct sums of these, e.g.
$\emptyset{\color{myyellow}y}{\color{mygreen}g}{\color{myyellow}y}{\color{myorange}o}{\color{myyellow}y}\emptyset\oplus\emptyset{\color{myyellow}y}{\color{myorange}o}{\color{myyellow}y}\emptyset$.
\end{example}
\subsubsection{Some lemmas}\label{subsec:cat-thealgebra-lemmas}
We note the following lemma, which follows
directly from \eqref{eq:rm-first}.
\begin{lemmaqed}\label{lemma:clasps-well-defined}
The following diagrams commute in $\Adiag$.
\begin{gather}\label{eq:clasps-well-defined}
\xymatrix@C+=1.3cm@L+=6pt{
\emptyset{\color{myblue}b}{\color{mygreen}g}
\ar@<-4pt>@/_/[rr]|{\,\twomorstuff{id}_{\emptyset{\color{myblue}b}{\color{mygreen}g}}\,}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-2,3) to (-2,1.5) to (-1,1.5);
\fill[myyellow, opacity=0.3] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myblue, opacity=0.3] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (1,3) to (1,1.5) to (0,1.5);
\draw[ystrand, directed=.999] (0,1.5) to [out=90, in=270] (-1,3);
\draw[bstrand, directed=.999] (-1,1.5) to [out=90, in=270] (0,3);
\end{tikzpicture}}
&
\emptyset{\color{myyellow}y}{\color{mygreen}g}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-2,3) to (-2,1.5) to (-1,1.5);
\fill[myblue, opacity=0.3] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myyellow, opacity=0.3] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (1,3) to (1,1.5) to (0,1.5);
\draw[ystrand, directed=.999] (-1,1.5) to [out=90, in=270] (0,3);
\draw[bstrand, directed=.999] (0,1.5) to [out=90, in=270] (-1,3);
\end{tikzpicture}}
&
\emptyset{\color{myblue}b}{\color{mygreen}g}
}
,\quad\quad
\xymatrix@C+=1.3cm@L+=6pt{
\emptyset{\color{myyellow}y}{\color{mygreen}g}
\ar@<-4pt>@/_/[rr]|{\,\twomorstuff{id}_{\emptyset{\color{myyellow}y}{\color{mygreen}g}}\,}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-2,3) to (-2,1.5) to (-1,1.5);
\fill[myblue, opacity=0.3] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myyellow, opacity=0.3] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (1,3) to (1,1.5) to (0,1.5);
\draw[ystrand, directed=.999] (-1,1.5) to [out=90, in=270] (0,3);
\draw[bstrand, directed=.999] (0,1.5) to [out=90, in=270] (-1,3);
\end{tikzpicture}}
&
\emptyset{\color{myblue}b}{\color{mygreen}g}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-2,3) to (-2,1.5) to (-1,1.5);
\fill[myyellow, opacity=0.3] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myblue, opacity=0.3] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (1,3) to (1,1.5) to (0,1.5);
\draw[ystrand, directed=.999] (0,1.5) to [out=90, in=270] (-1,3);
\draw[bstrand, directed=.999] (-1,1.5) to [out=90, in=270] (0,3);
\end{tikzpicture}}
&
\emptyset{\color{myyellow}y}{\color{mygreen}g}
}
\end{gather}
In particular, $\emptyset{\color{myblue}b}{\color{mygreen}g}\cong\emptyset{\color{myyellow}y}{\color{mygreen}g}$.
The same holds for color variations with compatible colors.
\end{lemmaqed}
The following, where we
silently use \fullref{lemma:clasps-well-defined}, should be
compared to \fullref{lemma:quotient-of-affine}.
\begin{lemma}\label{lemma:sts-decomp}
In $\Kar{\subcatquo[\infty]}$,
the $1$-morphism $\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset\cong\emptyset{\color{myyellow}y}{\color{mygreen}g}{\color{myyellow}y}\emptyset$ is
isomorphic to the indecomposable direct summand of
$\emptyset{\color{myblue}b}\emptyset{\color{myyellow}y}\emptyset{\color{myblue}b}\emptyset$ or of $\emptyset{\color{myyellow}y}\emptyset{\color{myblue}b}\emptyset{\color{myyellow}y}\emptyset$ which
corresponds to the word $w_{{\color{mygreen}g}}={\color{myblue}b}{\color{myyellow}y}{\color{myblue}b}={\color{myyellow}y}{\color{myblue}b}{\color{myyellow}y}\in\algstuff{W}_{{\color{mygreen}g}}$.
The same holds for all compatible color variations.
\end{lemma}
Recall that $\adiag[\qpar]$
decategorifies to $\hecke(\typeat{2})$
(cf. \fullref{remark:cat-affine-a2}), such that
the indecomposable $1$-morphisms in $\adiag[\qpar]$ decategorify
to the KL basis elements of the affine type $\typea{2}$ Weyl group $\algstuff{W}$.
Since $\subcatquo[\infty]$ is a $2$-full $2$-subcategory of $\adiag[\qpar]$, its indecomposable
$1$-morphisms are also indecomposable as $1$-morphisms of the latter. Therefore, $\Kar{\subcatquo[\infty]}$
decategorifies to a subalgebra of $\hecke(\typeat{2})$, with a basis consisting of a
particular subset of the KL basis elements.
\begin{proof}
By \eqref{eq:square}, we have
\begin{gather}\label{eq:byb-decomp}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-3.5,2) to (-3.5,-2) to (-2.5,-2) to (-2.5,2) to (-3.5,2);
\draw[very thin, densely dotted, fill=white] (-1.5,2) to (-1.5,-2) to (-.5,-2) to (-.5,2) to (-1.5,2);
\draw[very thin, densely dotted, fill=white] (.5,2) to (.5,-2) to (1.5,-2) to (1.5,2) to (.5,2);
\draw[very thin, densely dotted, fill=white] (2.5,2) to (2.5,-2) to (3.5,-2) to (3.5,2) to (2.5,2);
\fill[myyellow, opacity=0.3] (-.5,2) to (-.5,-2) to (.5,-2) to (.5,2);
\fill[myblue, opacity=0.3] (-2.5,2) to (-2.5,-2) to (-1.5,-2) to (-1.5,2);
\fill[myblue, opacity=0.3] (1.5,2) to (1.5,-2) to (2.5,-2) to (2.5,2);
\draw[ystrand, directed=.55] (-.5,-2) to (-.5,2);
\draw[ystrand, directed=.55] (.5,2) to (.5,-2);
\draw[bstrand, directed=.55] (-2.5,-2) to (-2.5,2);
\draw[bstrand, directed=.55] (-1.5,2) to (-1.5,-2);
\draw[bstrand, directed=.55] (1.5,-2) to (1.5,2);
\draw[bstrand, directed=.55] (2.5,2) to (2.5,-2);
\node at (-3,-.4) {$\emptyset$};
\node at (-2,-.4) {${\color{black}b}$};
\node at (-1,-.4) {$\emptyset$};
\node at (0,-.4) {${\color{black}y}$};
\node at (1,-.4) {$\emptyset$};
\node at (2,-.4) {${\color{black}b}$};
\node at (3,-.4) {$\emptyset$};
\end{tikzpicture}
=
\varstuff{q}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-3.5,2) to (-3.5,-2) to (-2.5,-2) to (-2.5,2) to (-3.5,2);
\draw[very thin, densely dotted] (-1.5,-2) to (-.5,-2);
\draw[very thin, densely dotted] (.5,-2) to (1.5,-2);
\draw[very thin, densely dotted] (-1.5,2) to (-.5,2);
\draw[very thin, densely dotted] (.5,2) to (1.5,2);
\draw[very thin, densely dotted, fill=white] (2.5,2) to (2.5,-2) to (3.5,-2) to (3.5,2) to (2.5,2);
\fill[mygreen, opacity=0.8] (-.5,-.8) to (-.5,.8) to (0,.75) to (.5,.8) to (.5,-.8);
\fill[myyellow, opacity=0.3] (-.5,2) to (-.5,.8) to (0,.75) to (.5,.8) to (.5,2) to (-.5,2);
\fill[myyellow, opacity=0.3] (-.5,-2) to (-.5,-.8) to (0,-.75) to (.5,-.8) to (.5,-2) to (-0.5,-2);
\fill[myblue, opacity=0.3] (-2.5,-2) to (-2.5,2) to (-1.5,2) to [out=270, in=160] (-.5,.8) to (-0.5,-.8) to [out=200, in=90] (-1.5,-2) to (-2.5,-2);
\fill[myblue, opacity=0.3] (2.5,-2) to (1.5,-2) to [out=90, in=350] (.5,-.8) to (.5,.8) to [out=10, in=270] (1.5,2) to (2.5,2) to (2.5,-2);
\draw[ystrand, directed=.55] (-.5,-2) to (-.5,2);
\draw[ystrand, directed=.55] (.5,2) to (.5,-2);
\draw[bstrand, directed=.55] (-2.5,-2) to (-2.5,2);
\draw[bstrand, directed=.85] (-1.5,2) to [out=270, in=180] (0,.75) to [out=0, in=270] (1.5,2);
\draw[bstrand, directed=.85] (1.5,-2) to [out=90, in=0] (0,-.75) to [out=180, in=90] (-1.5,-2);
\draw[bstrand, directed=.55] (2.5,2) to (2.5,-2);
\node at (-3,-.4) {$\emptyset$};
\node at (-1.5,-.4) {${\color{black}b}$};
\node at (0,-.4) {${\color{black}g}$};
\node at (1.5,-.4) {${\color{black}b}$};
\node at (3,-.4) {$\emptyset$};
\end{tikzpicture}
-
\varstuff{q}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-3.5,2) to (-3.5,-2) to (-2.5,-2) to (-2.5,2) to (-3.5,2);
\draw[very thin, densely dotted] (-1.5,-2) to (-.5,-2);
\draw[very thin, densely dotted] (.5,-2) to (1.5,-2);
\draw[very thin, densely dotted] (-1.5,2) to (-.5,2);
\draw[very thin, densely dotted] (.5,2) to (1.5,2);
\draw[very thin, densely dotted, fill=white] (2.5,2) to (2.5,-2) to (3.5,-2) to (3.5,2) to (2.5,2);
\fill[myyellow, opacity=0.3] (.5,2) to [out=270, in=0] (0,1.5) to [out=180, in=270] (-.5,2) to (.5,2);
\fill[myyellow, opacity=0.3] (-.5,-2) to [out=90, in=180] (0,-1.5) to [out=0, in=90] (.5,-2) to (-.5,-2);
\fill[myblue, opacity=0.3] (-2.5,-2) to (-2.5,2) to (-1.5,2) to [out=270, in=180] (0,.75) to [out=0, in=270] (1.5,2) to (2.5,2) to (2.5,-2) to (1.5,-2) to [out=90, in=0] (0,-.75) to [out=180, in=90] (-1.5,-2) to (-2.5,-2);
\draw[ystrand, directed=.85] (0.5,2) to [out=270, in=0] (0,1.5) to [out=180, in=270] (-.5,2);
\draw[ystrand, directed=.85] (-0.5,-2) to [out=90, in=180] (0,-1.5) to [out=0, in=90] (.5,-2);
\draw[bstrand, directed=.55] (-2.5,-2) to (-2.5,2);
\draw[bstrand, directed=.85] (-1.5,2) to [out=270, in=180] (0,.75) to [out=0, in=270] (1.5,2);
\draw[bstrand, directed=.85] (1.5,-2) to [out=90, in=0] (0,-.75) to [out=180, in=90] (-1.5,-2);
\draw[bstrand, directed=.55] (2.5,2) to (2.5,-2);
\node at (-3,-.4) {$\emptyset$};
\node at (0,-.4) {${\color{black}b}$};
\node at (3,-.4) {$\emptyset$};
\end{tikzpicture}
\end{gather}
It is not hard to check, using the relations in
\fullref{definition:ssbim},
that \eqref{eq:byb-decomp} gives a
decomposition into orthogonal idempotents.
Note that the first idempotent on the right-hand side
shows that $\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset$ is a direct summand of
$\emptyset{\color{myblue}b}\emptyset{\color{myyellow}y}\emptyset{\color{myblue}b}\emptyset$, as indicated in \eqref{eq:byb-decomp}, i.e.
$\emptyset{\color{myblue}b}\emptyset{\color{myyellow}y}\emptyset{\color{myblue}b}\emptyset\cong\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset\oplus\emptyset{\color{myblue}b}\emptyset$.
This decomposition decategorifies to
\[
\theta_{w_{{\color{mygreen}g}}}
\stackrel{\eqref{eq:cubic}}{=}
\theta_{{\color{myblue}b}}\theta_{{\color{myyellow}y}}\theta_{{\color{myblue}b}}=\theta_{{\color{myblue}b}{\color{myyellow}y}{\color{myblue}b}} + \theta_{{\color{myblue}b}},
\]
and it then follows from \cite[Theorem A.1]{El1} that the idempotents on
the right-hand side of \eqref{eq:byb-decomp} are primitive.
This shows the lemma in
case of ${\color{myblue}b}$, ${\color{myyellow}y}$ and ${\color{mygreen}g}$. The other cases are analogous.
\end{proof}
\begin{lemma}\label{lemma:remove-white}
We have ${\color{myblue}b}\emptyset{\color{myblue}b}\cong{\color{myblue}b}\{+1\}\oplus{\color{myblue}b}\{-1\}$ in $\Adiag$.
A similar result holds for all compatible color variations, keeping the color $\emptyset$.
\end{lemma}
Note that \fullref{lemma:remove-white} is only true for primary
colors, since $\emptyset$ is never compatible to a secondary color.
\begin{proof}
This follows from the following diagram.
\begin{gather*}
\xymatrix@C+=2.5cm@L+=6pt{
& {\color{myblue}b}\{+1\}
\ar[rd]|{\tfrac{1}{2}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,2) to [out=270, in=180] (0,1) to [out=0, in=270] (.5,2) to (-.5,2);
\fill[myblue, opacity=0.3] (-.5,2) to [out=270, in=180] (0,1) to [out=0, in=270] (.5,2) to (1.5,2) to (1.5,0) to (-1.5,0) to (-1.5,2) to (-.5,2);
\draw[bstrand] (-.5,2) to [out=270, in=180] (0,1) to [out=0, in=270] (.5,2);
\draw[bstrand, directed=.55] (.1,1) to (.11,1);
\end{tikzpicture}}
&
\\
{\color{myblue}b}\emptyset{\color{myblue}b} \ar[ru]|{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,0) to [out=90, in=180] (0,1) to [out=0, in=90] (.5,0) to (-.5,0);
\fill[myblue, opacity=0.3] (-.5,0) to [out=90, in=180] (0,1) to [out=0, in=90] (.5,0) to (1.5,0) to (1.5,2) to (-1.5,2) to (-1.5,0) to (-.5,0);
\draw[bstrand] (-.5,0) to [out=90, in=180] (0,1) to [out=0, in=90] (.5,0);
\draw[bstrand, directed=.55] (-.1,1) to (-.11,1);
\node at (0,.4) {$\alpha_{\bc}$};
\end{tikzpicture}}
\ar[rd]|{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,0) to [out=90, in=180] (0,1) to [out=0, in=90] (.5,0) to (-.5,0);
\fill[myblue, opacity=0.3] (-.5,0) to [out=90, in=180] (0,1) to [out=0, in=90] (.5,0) to (1.5,0) to (1.5,2) to (-1.5,2) to (-1.5,0) to (-.5,0);
\draw[bstrand] (-.5,0) to [out=90, in=180] (0,1) to [out=0, in=90] (.5,0);
\draw[bstrand, directed=.55] (-.1,1) to (-.11,1);
\end{tikzpicture}}
& & {\color{myblue}b}\emptyset{\color{myblue}b}
\\
& {\color{myblue}b}\{-1\}
\ar[ru]|{\tfrac{1}{2}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-.5,2) to [out=270, in=180] (0,1) to [out=0, in=270] (.5,2) to (-.5,2);
\fill[myblue, opacity=0.3] (-.5,2) to [out=270, in=180] (0,1) to [out=0, in=270] (.5,2) to (1.5,2) to (1.5,0) to (-1.5,0) to (-1.5,2) to (-.5,2);
\draw[bstrand] (-.5,2) to [out=270, in=180] (0,1) to [out=0, in=270] (.5,2);
\draw[bstrand, directed=.55] (.1,1) to (.11,1);
\node at (0,1.6) {$\alpha_{\bc}$};
\end{tikzpicture}}
&
\\
}
\end{gather*}
Observing that $\partial_{{\color{myblue}b}}(\alpha_{\bc})=2$, $\partial_{{\color{myblue}b}}(\alpha_{\bc}^2)=0$ and
$\Frobel{{\color{myblue}b}}{\phantom{.}}=\tfrac{1}{2}(\alpha_{\bc}\otimes 1+1\otimes\alpha_{\bc})$,
and using the relations in \fullref{definition:ssbim} , one can check that the left and right column in this diagram define
mutual inverses: In this setup $2$-morphisms correspond to matrices of
diagrams, such that composition corresponds to matrix multiplication. The two $2$-morphisms
above corresponding to the column matrices are
each other's inverses with respect to this composition.
Similar arguments can be used for all other compatible color variations.
\end{proof}
\begin{remark}\label{remark:no-white-regions}
Using \fullref{lemma:remove-white} we simplify
our diagrams and do not illustrate
$\emptyset$ colored regions in the middle, if not necessary. For example,
$\theta_{\color{mygreen}g}\theta_{\color{mygreen}g}$ should be thought
of as corresponding to $\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset$
instead of $\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset$.
However, the appearing grading shift in this simplification
is exactly the reason for the scaling by powers of
$\vnumber{2}^{-1}$ in \fullref{section:funny-algebra}.
\end{remark}
\begin{definitionnoqed}\label{definition:the-gog-gpg-iso}
Define the following $2$-morphisms in $\Adiag$ (and similar ones for other colors).
\begin{gather}\label{eq:o-and-u-cross}
\begin{gathered}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.5,3) to (-1.5,1.5) to (-1,1.5);
\fill[mypurple, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myorange, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.5,3) to (.5,1.5) to (0,1.5);
\draw[dstrand, Xmarked=.999] (0,1.5) to [out=90, in=270] (-1,3);
\draw[dstrand, Xmarked=.999] (0,3) to [out=270, in=90] (-1,1.5);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-3,0) to (-2,0) to [out=90, in=225] (-1.25,2) to [out=135, in=270] (-2,4) to (-3,4) to (-3,0);
\fill[myyellow, opacity=0.3] (0,0) to (0,1.1) to (-1.25,2) to [out=225, in=90] (-2,0) to (0,0);
\fill[myblue, opacity=0.3] (0,4) to (0,2.9) to (-1.25,2) to [out=135, in=270] (-2,4) to (0,4);
\fill[myorange, opacity=0.8] (0,0) to (2,0) to (2,1.1) to (1,1) to (0,1.1) to (0,0);
\fill[myred, opacity=0.3] (2,1.1) to (1,1) to (0,1.1) to (0,2.9) to (1,3) to (2,2.9) to (2,1.1);
\fill[mypurple, opacity=0.8] (0,4) to (2,4) to (2,2.9) to (1,3) to (0,2.9) to (0,0);
\fill[myyellow, opacity=0.3] (2,0) to (2,1.1) to (3.25,2) to [out=315, in=90] (4,0) to (2,0);
\fill[myblue, opacity=0.3] (2,4) to (2,2.9) to (3.25,2) to [out=45, in=270] (4,4) to (2,4);
\fill[mygreen, opacity=0.8] (5,0) to (4,0) to [out=90, in=315] (3.25,2) to [out=45, in=270] (4,4) to (5,4) to (5,0);
\draw[ystrand, directed=.525] (-2,4) to [out=270, in=180] (1,1) to [out=0, in=270] (4,4);
\draw[rstrand, directed=.55] (0,0) to (0,4);
\draw[rstrand, rdirected=.55] (2,0) to (2,4);
\draw[bstrand, rdirected=.525] (-2,0) to [out=90, in=180] (1,3) to [out=0, in=90] (4,0);
\end{tikzpicture}
+\varstuff{q}^{-1}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myblue, opacity=0.3] (-2,4) to [out=270, in=180] (1,2.25) to [out=0, in=270] (4,4) to (2,4) to [out=270, in=0] (1,3) to [out=180, in=270] (0,4) to (-2,4);
\fill[mypurple, opacity=0.8] (0,4) to (0,4) to [out=270, in=180] (1,3) to [out=0, in=270] (2,4);
\fill[myorange, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\fill[myyellow, opacity=0.3] (-2,0) to [out=90, in=180] (1,1.75) to [out=0, in=90] (4,0) to (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0) to (-2,0);
\fill[mygreen, opacity=0.8] (-2,0) to [out=90, in=180] (1,1.75) to [out=0, in=90] (4,0) to (5,0) to (5,4) to (4,4) to [out=270, in=0] (1,2.25) to [out=180, in=270] (-2,4) to (-3,4) to (-3,0) to (-2,0);
\draw[bstrand, rdirected=.525] (-2,0) to [out=90, in=180] (1,1.75) to [out=0, in=90] (4,0);
\draw[rstrand, directed=.55] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0);
\draw[rstrand, rdirected=.55] (0,4) to [out=270, in=180] (1,3) to [out=0, in=270] (2,4);
\draw[ystrand, directed=.525] (-2,4) to [out=270, in=180] (1,2.25) to [out=0, in=270] (4,4);
\end{tikzpicture}
\\
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.5,3) to (-1.5,1.5) to (-1,1.5);
\fill[myorange, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[mypurple, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.5,3) to (.5,1.5) to (0,1.5);
\draw[dstrand, Xmarked=.999] (-1,1.5) to [out=90, in=270] (0,3);
\draw[dstrand, Xmarked=.999] (-1,3) to [out=270, in=90] (0,1.5);
\end{tikzpicture}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-3,0) to (-2,0) to [out=90, in=225] (-1.25,2) to [out=135, in=270] (-2,4) to (-3,4) to (-3,0);
\fill[myblue, opacity=0.3] (0,0) to (0,1.1) to (-1.25,2) to [out=225, in=90] (-2,0) to (0,0);
\fill[myyellow, opacity=0.3] (0,4) to (0,2.9) to (-1.25,2) to [out=135, in=270] (-2,4) to (0,4);
\fill[mypurple, opacity=0.8] (0,0) to (2,0) to (2,1.1) to (1,1) to (0,1.1) to (0,0);
\fill[myred, opacity=0.3] (2,1.1) to (1,1) to (0,1.1) to (0,2.9) to (1,3) to (2,2.9) to (2,1.1);
\fill[myorange, opacity=0.8] (0,4) to (2,4) to (2,2.9) to (1,3) to (0,2.9) to (0,0);
\fill[myblue, opacity=0.3] (2,0) to (2,1.1) to (3.25,2) to [out=315, in=90] (4,0) to (2,0);
\fill[myyellow, opacity=0.3] (2,4) to (2,2.9) to (3.25,2) to [out=45, in=270] (4,4) to (2,4);
\fill[mygreen, opacity=0.8] (5,0) to (4,0) to [out=90, in=315] (3.25,2) to [out=45, in=270] (4,4) to (5,4) to (5,0);
\draw[ystrand, rdirected=.525] (-2,0) to [out=90, in=180] (1,3) to [out=0, in=90] (4,0);
\draw[rstrand, directed=.55] (0,0) to (0,4);
\draw[rstrand, rdirected=.55] (2,0) to (2,4);
\draw[bstrand, directed=.525] (-2,4) to [out=270, in=180] (1,1) to [out=0, in=270] (4,4);
\end{tikzpicture}
+\varstuff{q}^{\phantom{-1}}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myyellow, opacity=0.3] (-2,4) to [out=270, in=180] (1,2.25) to [out=0, in=270] (4,4) to (2,4) to [out=270, in=0] (1,3) to [out=180, in=270] (0,4) to (-2,4);
\fill[myorange, opacity=0.8] (0,4) to (0,4) to [out=270, in=180] (1,3) to [out=0, in=270] (2,4);
\fill[mypurple, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\fill[myblue, opacity=0.3] (-2,0) to [out=90, in=180] (1,1.75) to [out=0, in=90] (4,0) to (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0) to (-2,0);
\fill[mygreen, opacity=0.8] (-2,0) to [out=90, in=180] (1,1.75) to [out=0, in=90] (4,0) to (5,0) to (5,4) to (4,4) to [out=270, in=0] (1,2.25) to [out=180, in=270] (-2,4) to (-3,4) to (-3,0) to (-2,0);
\draw[bstrand, directed=.525] (-2,4) to [out=270, in=180] (1,2.25) to [out=0, in=270] (4,4);
\draw[rstrand, directed=.55] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0);
\draw[rstrand, rdirected=.55] (0,4) to [out=270, in=180] (1,3) to [out=0, in=270] (2,4);
\draw[ystrand, rdirected=.525] (-2,0) to [out=90, in=180] (1,1.75) to [out=0, in=90] (4,0);
\end{tikzpicture}
\end{gathered}
\hspace{3.1cm}
\raisebox{-1.55cm}{\hfill\ensuremath{\blacktriangle}}
\hspace{-3.1cm}
\end{gather}
\end{definitionnoqed}
\begin{lemma}\label{lemma:thecrossings-2}
The following diagrams commute in $\Adiag$.
\begin{gather}\label{eq:clasps-well-defined-2}
\xymatrix@C+=1.3cm@L+=6pt{
{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}
\ar@<-4pt>@/_/[rr]|{\,\twomorstuff{id}_{{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}}\,}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.5,3) to (-1.5,1.5) to (-1,1.5);
\fill[mypurple, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myorange, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.5,3) to (.5,1.5) to (0,1.5);
\draw[dstrand, Xmarked=.999] (0,1.5) to [out=90, in=270] (-1,3);
\draw[dstrand, Xmarked=.999] (0,3) to [out=270, in=90] (-1,1.5);
\end{tikzpicture}}
&
{\color{mygreen}g}{\color{myyellow}y}{\color{myorange}o}{\color{myyellow}y}{\color{mygreen}g}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.5,3) to (-1.5,1.5) to (-1,1.5);
\fill[myorange, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[mypurple, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.5,3) to (.5,1.5) to (0,1.5);
\draw[dstrand, Xmarked=.999] (-1,1.5) to [out=90, in=270] (0,3);
\draw[dstrand, Xmarked=.999] (-1,3) to [out=270, in=90] (0,1.5);
\end{tikzpicture}}
&
{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}
}
,\quad\quad
\xymatrix@C+=1.3cm@L+=6pt{
{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}
\ar@<-4pt>@/_/[rr]|{\,\twomorstuff{id}_{{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}}\,}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.5,3) to (-1.5,1.5) to (-1,1.5);
\fill[myorange, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[mypurple, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.5,3) to (.5,1.5) to (0,1.5);
\draw[dstrand, Xmarked=.999] (-1,1.5) to [out=90, in=270] (0,3);
\draw[dstrand, Xmarked=.999] (-1,3) to [out=270, in=90] (0,1.5);
\end{tikzpicture}}
&
{\color{mygreen}g}{\color{myyellow}y}{\color{myorange}o}{\color{myyellow}y}{\color{mygreen}g}
\ar[r]^{
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=135, in=270] (-1,3) to (-1.5,3) to (-1.5,1.5) to (-1,1.5);
\fill[mypurple, opacity=0.8] (-1,3) to [out=270, in=135] (-.5,2.25) to [out=45, in=270] (0,3) to (-1,3);
\fill[myorange, opacity=0.8] (-1,1.5) to [out=90, in=225] (-.5,2.25) to [out=315, in=90] (0,1.5) to (-1,1.5);
\fill[mygreen, opacity=0.8] (0,1.5) to [out=90, in=315] (-.5,2.25) to [out=45, in=270] (0,3) to (.5,3) to (.5,1.5) to (0,1.5);
\draw[dstrand, Xmarked=.999] (0,1.5) to [out=90, in=270] (-1,3);
\draw[dstrand, Xmarked=.999] (0,3) to [out=270, in=90] (-1,1.5);
\end{tikzpicture}}
&
{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}
}
\end{gather}
In particular, ${\color{mygreen}g}{\color{myyellow}y}{\color{myorange}o}{\color{myyellow}y}{\color{mygreen}g}\cong{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}$.
A similar result holds for other colors.
\end{lemma}
\begin{proof}
We only prove that the left diagram commutes. To this end,
we write $\twomorstuff{f}_1$ and
$\varstuff{q}^{-1}\twomorstuff{f}_2$ for the two summands on the right-hand side of the top
equality in \eqref{eq:o-and-u-cross}, and similarly $\twomorstuff{g}_1$ and $\varstuff{q}\twomorstuff{g}_2$ for the bottom
equality. Using \eqref{eq:square}, followed by \eqref{eq:rm-first} and
\cite[Claim 3.14]{El1}, we get $\twomorstuff{g}_1\circ_{v}\twomorstuff{f}_1
=\twomorstuff{id}_{{\color{mygreen}g}{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}{\color{mygreen}g}}+\twomorstuff{h}$.
Moreover, by first using \eqref{eq:rm-first} and \eqref{eq:rm-third}
(and $\partial_{{\color{myblue}b}}\frobel{{\color{myorange}o}}{{\color{myred}r},{\color{myyellow}y}}=-\qnumber{2}^{2}$),
and then \eqref{eq:circle-primary} and
\fullref{example:more-relations}, we get
$\twomorstuff{g}_1\circ_{v}\varstuff{q}^{-1}\twomorstuff{f}_2+
\varstuff{q}\twomorstuff{g}_2\circ_{v}\twomorstuff{f}_1=-\qnumber{2}^{2}\twomorstuff{h}$.
Finally, $\varstuff{q}\twomorstuff{g}_2\circ_{v}\varstuff{q}^{-1}\twomorstuff{f}_2=\qnumber{3}\twomorstuff{h}$
follows from \fullref{example:some-relations}, and we are done
since $\qnumber{3}=\qnumber{2}^{2}-1$.
\end{proof}
\subsubsection{Categorifying \texorpdfstring{$\subquo$}{Tinfty}}\label{subsec:cat-thealgebra}
\begin{proposition}\label{proposition:cat-the-algebra}
The assignment given by
\[
\theta_{{\color{mygreen}g}}\mapsto[\emptyset{\color{myblue}b}{\color{mygreen}g}{\color{myblue}b}\emptyset]=[\emptyset{\color{myyellow}y}{\color{mygreen}g}{\color{myyellow}y}\emptyset],
\quad\quad
\theta_{{\color{myorange}o}}\mapsto[\emptyset{\color{myred}r}{\color{myorange}o}{\color{myred}r}\emptyset]=[\emptyset{\color{myyellow}y}{\color{myorange}o}{\color{myyellow}y}\emptyset],
\quad\quad
\theta_{{\color{mypurple}p}}\mapsto[\emptyset{\color{myblue}b}{\color{mypurple}p}{\color{myblue}b}\emptyset]=[\emptyset{\color{myred}r}{\color{mypurple}p}{\color{myred}r}\emptyset],
\]
defines an isomorphism
$\subquo\xrightarrow{\cong}\GGcv{\Kar{\subcatquo[\infty]}}$
of algebras.
Under this isomorphism, the elements
of the basis $\basisC$ (or of $\Cbasis$) from \fullref{proposition:two-bases}
correspond to a complete set of indecomposables in
$\Kar{\subcatquo[\infty]}$ (up to grading shift).
\end{proposition}
\makeautorefname{lemma}{Lemmas}
\begin{proof}
This follows now directly from \fullref{remark:cat-affine-a2}, and
\fullref{lemma:quotient-of-affine}, \ref{lemma:clasps-well-defined}
and \ref{lemma:sts-decomp}.
\end{proof}
\makeautorefname{lemma}{Lemma}
\subsection{The \texorpdfstring{$2$}{2}-quotient of level \texorpdfstring{$e$}{e}}\label{subsec:quotient-category}
The quotient $\subquo[e]$ of $\subquo$
from \fullref{definition:the-quotient-defined} is defined
by killing certain elements
which correspond to the irreducibles
$\algstuff{L}_{m,n}$ for $m+n=e+1$
in the representation
category of $\mathfrak{sl}_{3}$. We follow
the same strategy on the categorified level.
\subsubsection{From \texorpdfstring{$\mathfrak{sl}_{3}$}{sl3} to singular bimodules: the generic case}\label{subsubsec:webs-to-soergel}
Recall from \fullref{subsec:generic} that $\sltcat$
denotes the category of finite-dimensional $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-modules
(with $\Cq$ being the ground field).
The central character \eqref{eq:color-code} allows us to view
$\sltcat$ as a $2$-category $\sltcatgop$:
\begin{definition}\label{definition:colored-webs-1}
For ${\color{dummy}\textbf{u}}$, let $\sltcat^{{\color{dummy}\textbf{u}}}$ denote the full subcategory
of $\sltcat$ generated by the irreducibles with central character ${\color{dummy}\textbf{u}}$.
\end{definition}
Note that the subcategories
$\sltcat^{{\color{dummy}\textbf{u}}}$ are not monoidal. However,
by \fullref{lemma:central-character}, tensoring with $\varstuff{X}$ or $\varstuff{Y}$ defines functors between them
\begin{gather}\label{eq:some-colored-endos}
\xy
(0,0)*{
\fuf{{\color{dummy}\textbf{u}}}{\rho({\color{dummy}\textbf{u}})}=\varstuff{X}\otimes\underline{\phantom{a}}
\colon
\sltcat^{{\color{dummy}\textbf{u}}}\to\sltcat^{\rho({\color{dummy}\textbf{u}})},
\quad\quad
\fudf{{\color{dummy}\textbf{u}}}{\rho^{-1}({\color{dummy}\textbf{u}})}=\varstuff{Y}\otimes\underline{\phantom{a}}
\colon
\sltcat^{{\color{dummy}\textbf{u}}}\to\sltcat^{\rho^{-1}({\color{dummy}\textbf{u}})},};
\endxy
\end{gather}
which the reader should compare to \eqref{eq:color-tensor}. We will (reading
right to left) depict them by
\[
\fuf{{\color{mygreen}g}}{{\color{myorange}o}}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to (0,2) to (-1,2) to (-1,0) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) node [below] {$\varstuff{X}$} to (0,2) node [above] {$\varstuff{X}$};
\end{tikzpicture}
\colon
\sltcat^{{\color{mygreen}g}}\to\sltcat^{{\color{myorange}o}},
\quad\quad
\fudf{{\color{myorange}o}}{{\color{mygreen}g}}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,0) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[mygreen, opacity=0.8] (0,0) to (0,2) to (-1,2) to (-1,0) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) node [below] {$\varstuff{Y}$} to (0,2) node [above] {$\varstuff{Y}$};
\end{tikzpicture}
\colon
\sltcat^{{\color{myorange}o}}\to\sltcat^{{\color{mygreen}g}},
\quad\quad
\fudf{{\color{myorange}o}}{{\color{mygreen}g}}\circ\fuf{{\color{mygreen}g}}{{\color{myorange}o}}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[mygreen, opacity=0.8] (-2,0) to (-2,2) to (-1,2) to (-1,0) to (-2,0);
\fill[myorange, opacity=0.8] (0,0) to (0,2) to (-1,2) to (-1,0) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) node [below] {$\varstuff{X}$} to (0,2) node [above] {$\varstuff{X}$};
\draw[dstrand, Ymarked=.55] (-1,0) node [below] {$\varstuff{Y}$} to (-1,2) node [above] {$\varstuff{Y}$};
\end{tikzpicture}
\]
etc. The orientation is such that the color on the
left-hand side comes directly after the color on the right-hand side in the cyclic ordering
determined by $\rho$ in \eqref{eq:color-tensor}.
(We omit the $\varstuff{X}$ and the $\varstuff{Y}$
in the pictures from now on.)
\begin{definition}\label{definition:colored-webs-2}
We define $\sltcatgop$ to be the additive, $\Cq$-linear closure of the
$2$-category whose objects are the categories $\sltcat^{{\color{dummy}\textbf{u}}}$,
whose $1$-morphisms are composites of the functors in \eqref{eq:some-colored-endos},
and whose $2$-morphisms are natural transformations.
\end{definition}
A natural transformation between composites of
the functors from \eqref{eq:some-colored-endos} is the same as
an $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-equivariant map, see e.g. \cite[Proposition 2.5.4]{EGNO}. Therefore, we define
\begin{gather}\label{eq:basic-webs}
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2.5,0) to (2.5,-2) to (-.5,-2) to (-.5,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
\colon
\begin{matrix}
\varstuff{Y}\varstuff{X}
\\
\rotatebox{90}{$\hookrightarrow$}
\\
\Cq
\end{matrix},
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-1) to (0,0) to (1,1) to (1.5,1) to (1.5,-1) to (0,-1);
\fill[myorange, opacity=0.8] (0,0) to (-1,1) to (1,1) to (0,0);
\fill[mypurple, opacity=0.8] (0,1) to (0,0) to (-1,1) to (-1.5,1) to (-1.5,-1) to (0,-1);
\draw[dstrand, Ymarked=.55] (-1,1) to (0,0);
\draw[dstrand, Ymarked=.55] (1,1) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,-1);
\end{tikzpicture}
\colon
\begin{matrix}
\varstuff{X}\fu
\\
\rotatebox{90}{$\hookrightarrow$}
\\
\varstuff{Y}
\end{matrix},
\end{aligned}
\quad\quad
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2.5,0) to (2.5,2) to (-.5,2) to (-.5,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\draw[dstrand, Xmarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\end{tikzpicture}
\colon
\begin{matrix}
\Cq
\\
\rotatebox{90}{$\twoheadrightarrow$}
\\
\varstuff{Y}\varstuff{X}
\end{matrix},
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,1) to (0,0) to (1,-1) to (1.5,-1) to (1.5,1) to (0,1);
\fill[myorange, opacity=0.8] (0,0) to (-1,-1) to (1,-1) to (0,0);
\fill[mypurple, opacity=0.8] (0,1) to (0,0) to (-1,-1) to (-1.5,-1) to (-1.5,1) to (0,1);
\draw[dstrand, Xmarked=.55] (-1,-1) to (0,0);
\draw[dstrand, Xmarked=.55] (1,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,1);
\end{tikzpicture}
\colon
\begin{matrix}
\varstuff{Y}
\\
\rotatebox{90}{$\twoheadrightarrow$}
\\
\varstuff{X}\fu
\end{matrix},
\end{aligned}
\quad\quad
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2.5,0) to (2.5,-2) to (-.5,-2) to (-.5,0) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
\colon
\begin{matrix}
\varstuff{X}\varstuff{Y}
\\
\rotatebox{90}{$\hookrightarrow$}
\\
\Cq
\end{matrix},
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-1) to (0,0) to (1,1) to (1.5,1) to (1.5,-1) to (0,-1);
\fill[myorange, opacity=0.8] (0,1) to (0,0) to (-1,1) to (-1.5,1) to (-1.5,-1) to (0,-1);
\fill[mypurple, opacity=0.8] (0,0) to (-1,1) to (1,1) to (0,0);
\draw[dstrand, Xmarked=.55] (-1,1) to (0,0);
\draw[dstrand, Xmarked=.55] (1,1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,-1);
\end{tikzpicture}
\colon
\begin{matrix}
\varstuff{Y}\fud
\\
\rotatebox{90}{$\hookrightarrow$}
\\
\varstuff{X}
\end{matrix},
\end{aligned}
\quad\quad
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2.5,0) to (2.5,2) to (-.5,2) to (-.5,0) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\draw[dstrand, Ymarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\end{tikzpicture}
\colon
\begin{matrix}
\Cq
\\
\rotatebox{90}{$\twoheadrightarrow$}
\\
\varstuff{X}\varstuff{Y}
\end{matrix},
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,1) to (0,0) to (1,-1) to (1.5,-1) to (1.5,1) to (0,1);
\fill[myorange, opacity=0.8] (0,1) to (0,0) to (-1,-1) to (-1.5,-1) to (-1.5,1) to (0,1);
\fill[mypurple, opacity=0.8] (0,0) to (-1,-1) to (1,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (-1,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (1,-1) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,1);
\end{tikzpicture}
\colon
\begin{matrix}
\varstuff{X}
\\
\rotatebox{90}{$\twoheadrightarrow$}
\\
\varstuff{Y}\fud
\end{matrix},
\end{aligned}
\end{gather}
to be the corresponding inclusions respectively projections, which
are well-defined up to scalars.
We do this in all color variations.
In this way we can view $\sltcatgop$ as being generated by the diagrams as in \eqref{eq:basic-webs}.
Fixing scalars appropriately (which we will do below), it is not hard to see that we get
\\
\noindent\begin{tabularx}{0.99\textwidth}{XXX}
\begin{equation}\hspace{-9.75cm}\label{eq:basic-webs-rels-isoa}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-1.5) to (0,0) to [out=90, in=180] (.5,1) to [out=0, in=90] (1,0) to [out=270, in=180] (1.5,-1) to [out=0, in=270] (2,0) to (2,1.5) to (2.5,1.5) to (2.5,-1.5) to (0,-1.5);
\fill[myorange, opacity=0.8] (0,-1.5) to (0,0) to [out=90, in=180] (.5,1) to [out=0, in=90] (1,0) to [out=270, in=180] (1.5,-1) to [out=0, in=270] (2,0) to (2,1.5) to (-.5,1.5) to (-.5,-1.5) to (0,-1.5);
\draw[dstrand, Xmarked=.55] (0,-1.5) to (0,0) to [out=90, in=180] (.5,1) to [out=0, in=90] (1,0) to [out=270, in=180] (1.5,-1) to [out=0, in=270] (2,0) to (2,1.5);
\end{tikzpicture}
{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (0,3) to (.5,3) to (.5,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to (0,3) to (-.5,3) to (-.5,0) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,3);
\end{tikzpicture}
{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-1.5) to (0,0) to [out=90, in=0] (-.5,1) to [out=180, in=90] (-1,0) to [out=270, in=0] (-1.5,-1) to [out=180, in=270] (-2,0) to (-2,1.5) to (.5,1.5) to (.5,-1.5) to (0,-1.5);
\fill[myorange, opacity=0.8] (0,-1.5) to (0,0) to [out=90, in=0] (-.5,1) to [out=180, in=90] (-1,0) to [out=270, in=0] (-1.5,-1) to [out=180, in=270] (-2,0) to (-2,1.5) to (-2.5,1.5) to (-2.5,-1.5) to (0,-1.5);
\draw[dstrand, Xmarked=.55] (0,-1.5) to (0,0) to [out=90, in=0] (-.5,1) to [out=180, in=90] (-1,0) to [out=270, in=0] (-1.5,-1) to [out=180, in=270] (-2,0) to (-2,1.5);
\end{tikzpicture}
\end{equation} &
\begin{equation}\hspace{-9.65cm}\label{eq:basic-webs-rels-isob}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2,1) to (1,1) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to (-1,1) to (1,1) to (0,0);
\fill[mygreen, opacity=0.8] (-1.5,1) to (-1,1) to (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2,1) to (2.5,1) to (2.5,-2) to (-1.5,-2) to (-1.5,1);
\draw[dstrand, Ymarked=.55] (-1,1) to (0,0);
\draw[dstrand, Ymarked=.55] (1,1) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2,1);
\end{tikzpicture}
{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,1) to (0,.5) to [out=180, in=90] (-.5,0) to [out=270, in=0] (-1,-.75) to [out=180, in=270] (-2,1) to (0,1);
\fill[mypurple, opacity=0.8] (0,.5) to [out=0, in=90] (.5,0) to [out=270, in=0] (-1,-1.5) to [out=180, in=270] (-3,1) to (-2,1) to [out=270, in=180] (-1,-.75) to [out=0, in=270] (-.5,0) to [out=90, in=180] (0,.5);
\fill[mygreen, opacity=0.8] (0,1) to (0,.5) to [out=0, in=90] (.5,0) to [out=270, in=0] (-1,-1.5) to [out=180, in=270] (-3,1) to (-3.5,1) to (-3.5,-2) to (1,-2) to (1,1) to (0,1);
\draw[dstrand, Xmarked=.55] (0,.5) to [out=180, in=90] (-.5,0) to [out=270, in=0] (-1,-.75) to [out=180, in=270] (-2,1);
\draw[dstrand, Xmarked=.55] (0,.5) to [out=0, in=90] (.5,0) to [out=270, in=0] (-1,-1.5) to [out=180, in=270] (-3,1);
\draw[dstrand, Xmarked=.85] (0,.5) to (0,1);
\end{tikzpicture}
\end{equation}
&
\begin{equation}\hspace{-9.65cm}\label{eq:basic-webs-rels-isoc}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2,-1) to (1,-1) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to (-1,-1) to (1,-1) to (0,0);
\fill[mygreen, opacity=0.8] (-1.5,-1) to (-1,-1) to (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2,-1) to (2.5,-1) to (2.5,2) to (-1.5,2) to (-1.5,-1);
\draw[dstrand, Xmarked=.55] (-1,-1) to (0,0);
\draw[dstrand, Xmarked=.55] (1,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2,-1);
\end{tikzpicture}
{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[myorange, opacity=0.8] (0,-1) to (0,-.5) to [out=180, in=270] (-.5,0) to [out=90, in=0] (-1,.75) to [out=180, in=90] (-2,-1) to (0,-1);
\fill[mypurple, opacity=0.8] (0,-.5) to [out=0, in=270] (.5,0) to [out=90, in=0] (-1,1.5) to [out=180, in=90] (-3,-1) to (-2,-1) to [out=90, in=180] (-1,.75) to [out=0, in=90] (-.5,0) to [out=270, in=180] (0,-.5);
\fill[mygreen, opacity=0.8] (0,-1) to (0,-.5) to [out=0, in=270] (.5,0) to [out=90, in=0] (-1,1.5) to [out=180, in=90] (-3,-1) to (-3.5,-1) to (-3.5,2) to (1,2) to (1,-1) to (0,-1);
\draw[dstrand, Ymarked=.55] (0,-.5) to [out=180, in=270] (-.5,0) to [out=90, in=0] (-1,.75) to [out=180, in=90] (-2,-1);
\draw[dstrand, Ymarked=.55] (0,-.5) to [out=0, in=270] (.5,0) to [out=90, in=0] (-1,1.5) to [out=180, in=90] (-3,-1);
\draw[dstrand, Ymarked=.85] (0,-.5) to (0,-1);
\end{tikzpicture}
\end{equation}
\end{tabularx}
\\
\noindent\begin{tabularx}{0.99\textwidth}{XXX}
\begin{equation}\hspace{-10.0cm}\label{eq:basic-webs-rels-a}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2.5,0) to (2.5,-2) to (-.5,-2) to (-.5,0) to (0,0);
\fill[mygreen, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2.5,0) to (2.5,2) to (-.5,2) to (-.5,0) to (0,-0);
\fill[myorange, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\draw[dstrand, Xmarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\end{tikzpicture}
{=}
\qnumber{3}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (2.5,-2) to (2.5,2) to (-.5,2) to (-.5,-2) to (2.5,-2);
\end{tikzpicture}
\end{equation} &
\begin{equation}\hspace{-9.7cm}\label{eq:basic-webs-rels-b}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=45, in=270] (.5,1) to [out=90, in=315] (0,2) to (0,3) to (1.5,3) to (1.5,-1) to (0,-1) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to [out=135, in=270] (-.5,1) to [out=90, in=225] (0,2) to [out=315, in=90] (.5,1) to [out=270, in=45] (0,0);
\fill[mypurple, opacity=0.8] (0,0) to [out=135, in=270] (-.5,1) to [out=90, in=225] (0,2) to (0,3) to (-1.5,3) to (-1.5,-1) to (0,-1) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to [out=135, in=270] (-.5,1) to [out=90, in=225] (0,2);
\draw[dstrand, Xmarked=.55] (0,0) to [out=45, in=270] (.5,1) to [out=90, in=315] (0,2);
\draw[dstrand, Ymarked=.55] (0,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,2) to (0,3);
\end{tikzpicture}
{=}
{-}\qnumber{2}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (0,3) to (1.5,3) to (1.5,-1) to (0,-1) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to (0,3) to (-1.5,3) to (-1.5,-1) to (0,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,-1) to (0,3);
\end{tikzpicture}
\end{equation} &
\begin{equation}\hspace{-9.15cm}\label{eq:basic-webs-rels-c}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (1,-1) to (1,3) to (1.5,3) to (1.5,-1) to (1,-1);
\fill[mygreen, opacity=0.8] (-1,-1) to (-1,3) to (-1.5,3) to (-1.5,-1) to (-1,-1);
\fill[myorange, opacity=0.8] (-1,.2) to (1,.4) to (1,1.6) to (-1,1.8) to (-1,.2);
\fill[mypurple, opacity=0.8] (-1,.2) to (1,.4) to (1,-1) to (-1,-1) to (-1,.2);
\fill[mypurple, opacity=0.8] (1,1.6) to (1,3) to (-1,3) to (-1,1.8) to (1,1.6);
\draw[dstrand, Ymarked=.55] (1,-1) to (1,0);
\draw[dstrand, Xmarked=.55] (1,0) to (1,2);
\draw[dstrand, Ymarked=.55] (1,2) to (1,3);
\draw[dstrand, Xmarked=.55] (-1,-1) to (-1,0);
\draw[dstrand, Ymarked=.55] (-1,0) to (-1,2);
\draw[dstrand, Xmarked=.55] (-1,2) to (-1,3);
\draw[dstrand, Ymarked=.55] (-1,.2) to (1,.4);
\draw[dstrand, Xmarked=.55] (-1,1.8) to (1,1.6);
\end{tikzpicture}
{=}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (1,-1) to (1,3) to (1.5,3) to (1.5,-1) to (1,-1);
\fill[mygreen, opacity=0.8] (-1,-1) to (-1,3) to (-1.5,3) to (-1.5,-1) to (-1,-1);
\fill[mypurple, opacity=0.8] (-1,-1) to (-1,3) to (1,3) to (1,-1) to (-1,-1);
\draw[dstrand, Ymarked=.55] (1,-1) to (1,3);
\draw[dstrand, Xmarked=.55] (-1,-1) to (-1,3);
\end{tikzpicture}
{+}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (-1,3) to (-1.5,3) to (-1.5,-1) to (-1,-1) to [out=90, in=180] (0,0) to [out=0, in=90] (1,-1) to (1.5,-1) to (1.5,3) to (1,3) to [out=270, in=0] (0,2) to [out=180, in=270] (-1,3);
\fill[mypurple, opacity=0.8] (-1,-1) to [out=90, in=180] (0,0) to [out=0, in=90] (1,-1) to (-1,-1);
\fill[mypurple, opacity=0.8] (-1,3) to [out=270, in=180] (0,2) to [out=0, in=270] (1,3) to (-1,3);
\draw[dstrand, Xmarked=.55] (-1,-1) to [out=90, in=180] (0,0) to [out=0, in=90] (1,-1);
\draw[dstrand, Ymarked=.55] (-1,3) to [out=270, in=180] (0,2) to [out=0, in=270] (1,3);
\end{tikzpicture}
\end{equation}
\end{tabularx}\\
together with those obtained by varying the orientation and the colors, and the vertical
mirrors of \eqref{eq:basic-webs-rels-isob} and \eqref{eq:basic-webs-rels-isoc}.
The following result is a consequence of \cite[Theorem 6.1]{Kup}.
\begin{lemmaqed}\label{lemma:webs-and-slt}
The $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$-equivariant maps/diagrams from \eqref{eq:basic-webs}
together with the relations \eqref{eq:basic-webs-rels-isoa} to \eqref{eq:basic-webs-rels-c}
give a generator-relation $2$-presentation of $\sltcatgop$.
\end{lemmaqed}
Following \cite[Section 3]{El1} we define a Satake $2$-functor.
\begin{definition}\label{definition:slt-soergel}
For ${\color{dummy}\textbf{u}}$ let $\elfunctor\colon\sltcatgop\to\aDiag[\qpar]$ be the
$2$-functor defined as follows. On objects by $\elfunctor(\sltcat^{{\color{dummy}\textbf{u}}})={\color{dummy}\textbf{u}}$,
on $1$-morphisms by $\elfunctor(\fuf{{\color{dummy}\textbf{u}}}{\rho({\color{dummy}\textbf{u}})})=\rho({\color{dummy}\textbf{u}}){\color{dummy}c}{\color{dummy}\textbf{u}}$ and
$\elfunctor(\fudf{{\color{dummy}\textbf{u}}}{\rho^{-1}({\color{dummy}\textbf{u}})})=\rho^{-1}({\color{dummy}\textbf{u}}){\color{dummy}c}{\color{dummy}\textbf{u}}$, and on
$2$-morphisms by
\begin{gather}\label{eq:elfunctor}
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2.5,0) to (2.5,-2) to (-.5,-2) to (-.5,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (2.5,2) to (2.5,0) to (-.5,0) to (-.5,2) to (0,2);
\fill[myorange, opacity=0.8] (1.5,2) to [out=270, in=0] (1,1.5) to [out=180, in=270] (.5,2) to (1.5,2);
\fill[myyellow, opacity=0.3] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (1.5,2) to [out=270, in=0] (1,1.5) to [out=180, in=270] (.5,2) to (0,2);
\draw[rstrand, directed=.999] (1.5,2) to [out=270, in=0] (1,1.5) to [out=180, in=270] (.5,2);
\draw[bstrand, directed=.999] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2);
\end{tikzpicture}
,
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-1) to (0,0) to (1,1) to (1.5,1) to (1.5,-1) to (0,-1);
\fill[myorange, opacity=0.8] (0,0) to (-1,1) to (1,1) to (0,0);
\fill[mypurple, opacity=0.8] (0,1) to (0,0) to (-1,1) to (-1.5,1) to (-1.5,-1) to (0,-1);
\draw[dstrand, Ymarked=.55] (-1,1) to (0,0);
\draw[dstrand, Ymarked=.55] (1,1) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,-1);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (1,-1) to (1,-.6) to [out=90, in=315] (.55,0) to [out=45, in=270] (1,1) to (1.5,1) to (1.5,-1) to (1,-1);
\fill[myorange, opacity=0.8] (-.5,1) to [out=270, in=135] (0,.3) to [out=45, in=270] (.5,1) to (-.5,1);
\fill[mypurple, opacity=0.8] (-1,-1) to (-1,-.6) to [out=90, in=225] (-.55,0) to [out=135, in=270] (-1,1) to (-1.5,1) to (-1.5,-1) to (-1,-1);
\fill[myblue, opacity=0.3] (1,-1) to (1,-.6) to [out=90, in=315] (.55,0) to [out=235, in=325] (-.55,0) to [out=225, in=90] (-1,-.6) to (-1,-1) to (1,-1);
\fill[myred, opacity=0.3] (.5,1) to [out=270, in=45] (0,.3) to (-.55,0) to [out=135, in=270] (-1,1) to (.5,1);
\fill[myyellow, opacity=0.3] (-.5,1) to [out=270, in=135] (0,.3) to (.55,0) to [out=45, in=270] (1,1) to (-.5,1);
\draw[ystrand, directed=.35] (1,-1) to (1,-.6) to [out=90, in=270] (-.5,1);
\draw[rstrand, rdirected=.35] (-1,-1) to (-1,-.6) to [out=90, in=270] (.5,1);
\draw[bstrand, directed=.55] (-1,1) to [out=270, in=180] (0,-.2) to [out=0, in=270] (1,1);
\end{tikzpicture}
,
\end{aligned}
\;\;
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2.5,0) to (2.5,2) to (-.5,2) to (-.5,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\draw[dstrand, Xmarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-2) to [out=90, in=180] (1,-1) to [out=0, in=90] (2,-2) to (2.5,-2) to (2.5,0) to (-.5,0) to (-.5,-2) to (0,-2);
\fill[myorange, opacity=0.8] (1.5,-2) to [out=90, in=0] (1,-1.5) to [out=180, in=90] (.5,-2) to (1.5,-2);
\fill[myyellow, opacity=0.3] (0,-2) to [out=90, in=180] (1,-1) to [out=0, in=90] (2,-2) to (1.5,-2) to [out=90, in=0] (1,-1.5) to [out=180, in=90] (.5,-2) to (0,-2);
\draw[rstrand, directed=.999] (.5,-2) to [out=90, in=180] (1,-1.5) to [out=0, in=90] (1.5,-2);
\draw[bstrand, directed=.999] (2,-2) to [out=90, in=0] (1,-1) to [out=180, in=90] (0,-2);
\end{tikzpicture}
,
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,1) to (0,0) to (1,-1) to (1.5,-1) to (1.5,1) to (0,1);
\fill[myorange, opacity=0.8] (0,0) to (-1,-1) to (1,-1) to (0,0);
\fill[mypurple, opacity=0.8] (0,1) to (0,0) to (-1,-1) to (-1.5,-1) to (-1.5,1) to (0,1);
\draw[dstrand, Xmarked=.55] (-1,-1) to (0,0);
\draw[dstrand, Xmarked=.55] (1,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,1);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (1,1) to (1,.6) to [out=270, in=45] (.55,0) to [out=315, in=90] (1,-1) to (1.5,-1) to (1.5,1) to (1,1);
\fill[myorange, opacity=0.8] (-.5,-1) to [out=90, in=225] (0,-.3) to [out=315, in=90] (.5,-1) to (-.5,-1);
\fill[mypurple, opacity=0.8] (-1,1) to (-1,.6) to [out=270, in=135] (-.55,0) to [out=225, in=90] (-1,-1) to (-1.5,-1) to (-1.5,1) to (-1,1);
\fill[myblue, opacity=0.3] (1,1) to (1,.6) to [out=270, in=45] (.55,0) to [out=125, in=55] (-.55,0) to [out=135, in=270] (-1,.6) to (-1,1) to (1,1);
\fill[myred, opacity=0.3] (.5,-1) to [out=90, in=315] (0,-.3) to (-.55,0) to [out=225, in=90] (-1,-1) to (.5,-1);
\fill[myyellow, opacity=0.3] (-.5,-1) to [out=90, in=225] (0,-.3) to (.55,0) to [out=315, in=90] (1,-1) to (-.5,-1);
\draw[ystrand, rdirected=.35] (1,1) to (1,.6) to [out=270, in=90] (-.5,-1);
\draw[rstrand, directed=.35] (-1,1) to (-1,.6) to [out=270, in=90] (.5,-1);
\draw[bstrand, rdirected=.55] (-1,-1) to [out=90, in=180] (0,.2) to [out=0, in=90] (1,-1);
\end{tikzpicture}
,
\end{aligned}
\;\;
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (2.5,0) to (2.5,-2) to (-.5,-2) to (-.5,0) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to [out=270, in=180] (1,-1) to [out=0, in=270] (2,0);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (2.5,2) to (2.5,0) to (-.5,0) to (-.5,2) to (0,2);
\fill[mypurple, opacity=0.8] (1.5,2) to [out=270, in=0] (1,1.5) to [out=180, in=270] (.5,2) to (1.5,2);
\fill[myblue, opacity=0.3] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2) to (1.5,2) to [out=270, in=0] (1,1.5) to [out=180, in=270] (.5,2) to (0,2);
\draw[ystrand, directed=.999] (0,2) to [out=270, in=180] (1,1) to [out=0, in=270] (2,2);
\draw[rstrand, directed=.999] (1.5,2) to [out=270, in=0] (1,1.5) to [out=180, in=270] (.5,2);
\end{tikzpicture}
,
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-1) to (0,0) to (1,1) to (1.5,1) to (1.5,-1) to (0,-1);
\fill[mypurple, opacity=0.8] (0,0) to (-1,1) to (1,1) to (0,0);
\fill[myorange, opacity=0.8] (0,1) to (0,0) to (-1,1) to (-1.5,1) to (-1.5,-1) to (0,-1);
\draw[dstrand, Xmarked=.55] (-1,1) to (0,0);
\draw[dstrand, Xmarked=.55] (1,1) to (0,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,-1);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (1,-1) to (1,-.6) to [out=90, in=315] (.55,0) to [out=45, in=270] (1,1) to (1.5,1) to (1.5,-1) to (1,-1);
\fill[mypurple, opacity=0.8] (-.5,1) to [out=270, in=135] (0,.3) to [out=45, in=270] (.5,1) to (-.5,1);
\fill[myorange, opacity=0.8] (-1,-1) to (-1,-.6) to [out=90, in=225] (-.55,0) to [out=135, in=270] (-1,1) to (-1.5,1) to (-1.5,-1) to (-1,-1);
\fill[myyellow, opacity=0.3] (1,-1) to (1,-.6) to [out=90, in=315] (.55,0) to [out=235, in=325] (-.55,0) to [out=225, in=90] (-1,-.6) to (-1,-1) to (1,-1);
\fill[myred, opacity=0.3] (.5,1) to [out=270, in=45] (0,.3) to (-.55,0) to [out=135, in=270] (-1,1) to (.5,1);
\fill[myblue, opacity=0.3] (-.5,1) to [out=270, in=135] (0,.3) to (.55,0) to [out=45, in=270] (1,1) to (-.5,1);
\draw[ystrand, directed=.55] (-1,1) to [out=270, in=180] (0,-.2) to [out=0, in=270] (1,1);
\draw[rstrand, rdirected=.35] (-1,-1) to (-1,-.6) to [out=90, in=270] (.5,1);
\draw[bstrand, directed=.35] (1,-1) to (1,-.6) to [out=90, in=270] (-.5,1);
\end{tikzpicture}
,
\end{aligned}
\;\;
\begin{aligned}
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (2.5,0) to (2.5,2) to (-.5,2) to (-.5,0) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to [out=90, in=180] (1,1) to [out=0, in=90] (2,0) to (0,0);
\draw[dstrand, Ymarked=.55] (2,0) to [out=90, in=0] (1,1) to [out=180, in=90] (0,0);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,-2) to [out=90, in=180] (1,-1) to [out=0, in=90] (2,-2) to (2.5,-2) to (2.5,0) to (-.5,0) to (-.5,-2) to (0,-2);
\fill[mypurple, opacity=0.8] (1.5,-2) to [out=90, in=0] (1,-1.5) to [out=180, in=90] (.5,-2) to (1.5,-2);
\fill[myblue, opacity=0.3] (0,-2) to [out=90, in=180] (1,-1) to [out=0, in=90] (2,-2) to (1.5,-2) to [out=90, in=0] (1,-1.5) to [out=180, in=90] (.5,-2) to (0,-2);
\draw[ystrand, directed=.999] (2,-2) to [out=90, in=0] (1,-1) to [out=180, in=90] (0,-2);
\draw[rstrand, directed=.999] (.5,-2) to [out=90, in=180] (1,-1.5) to [out=0, in=90] (1.5,-2);
\end{tikzpicture}
,
\\
&
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,1) to (0,0) to (1,-1) to (1.5,-1) to (1.5,1) to (0,1);
\fill[mypurple, opacity=0.8] (0,0) to (-1,-1) to (1,-1) to (0,0);
\fill[myorange, opacity=0.8] (0,1) to (0,0) to (-1,-1) to (-1.5,-1) to (-1.5,1) to (0,1);
\draw[dstrand, Ymarked=.55] (-1,-1) to (0,0);
\draw[dstrand, Ymarked=.55] (1,-1) to (0,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,1);
\end{tikzpicture}
\xmapsto{\elfunctor}
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (1,1) to (1,.6) to [out=270, in=45] (.55,0) to [out=315, in=90] (1,-1) to (1.5,-1) to (1.5,1) to (1,1);
\fill[mypurple, opacity=0.8] (-.5,-1) to [out=90, in=225] (0,-.3) to [out=315, in=90] (.5,-1) to (-.5,-1);
\fill[myorange, opacity=0.8] (-1,1) to (-1,.6) to [out=270, in=135] (-.55,0) to [out=225, in=90] (-1,-1) to (-1.5,-1) to (-1.5,1) to (-1,1);
\fill[myyellow, opacity=0.3] (1,1) to (1,.6) to [out=270, in=45] (.55,0) to [out=125, in=55] (-.55,0) to [out=135, in=270] (-1,.6) to (-1,1) to (1,1);
\fill[myred, opacity=0.3] (.5,-1) to [out=90, in=315] (0,-.3) to (-.55,0) to [out=225, in=90] (-1,-1) to (.5,-1);
\fill[myblue, opacity=0.3] (-.5,-1) to [out=90, in=225] (0,-.3) to (.55,0) to [out=315, in=90] (1,-1) to (-.5,-1);
\draw[ystrand, rdirected=.55] (-1,-1) to [out=90, in=180] (0,.2) to [out=0, in=90] (1,-1);
\draw[rstrand, directed=.35] (-1,1) to (-1,.6) to [out=270, in=90] (.5,-1);
\draw[bstrand, rdirected=.35] (1,1) to (1,.6) to [out=270, in=90] (-.5,-1);
\end{tikzpicture}
,
\end{aligned}
\end{gather}
together with similar assignments for the other generators.
\end{definition}
The following lemma recalls \cite[Claim 3.19]{El1}. We sketch its proof for the convenience of the reader and refer
to \cite[Proof of Claim 3.19]{El1} for more details.
\begin{lemma}\label{lemma:el-well-def}
The $2$-functor $\elfunctor$ is well-defined.
\end{lemma}
\begin{proof}
We only need to show that \eqref{eq:basic-webs-rels-isoa}--\eqref{eq:basic-webs-rels-c}
hold in the image of $\elfunctor$.
The isotopies \eqref{eq:basic-webs-rels-isoa}--\eqref{eq:basic-webs-rels-isoc}
are clearly preserved. For \eqref{eq:basic-webs-rels-a}
we have already verified this in \fullref{example:some-relations}, while
\eqref{eq:basic-webs-rels-b} follows from \eqref{eq:rm-third}, \eqref{eq:rm-first}
and \eqref{eq:circle-primary}
together with $\partial_{{\color{myblue}b}}(\frobel{{\color{myorange}o}}{{\color{myred}r},{\color{myyellow}y}})=-\qnumber{2}$.
The relation \eqref{eq:basic-webs-rels-c} is a bit more involved
(but not hard),
and can be proved by using \eqref{eq:square} on the $\elfunctor$-image of the square.
\end{proof}
We say that a $2$-functor from an ungraded $2$-category
(whose $2$-morphisms are all of degree zero, by convention) to a graded $2$-category
is a degree-zero $2$-equivalence, if it is a bijection on objects,
essentially surjective on $1$-morphisms,
faithful on $2$-morphisms, and full onto degree-zero $2$-morphisms.
Using this notion, the quantum Satake correspondence can be formulated as in \cite[Theorem 3.21]{El1}:
\begin{theoremqed}\label{theorem:q-satake}
The $2$-functor $\elfunctor$ is a degree-zero $2$-equivalence.
\end{theoremqed}
\begin{remark}\label{remark:q-satake}
Elias actually proves \fullref{theorem:q-satake} in
much more generality. For us the important
case is over the ring $\C_{\intqpar}=\mathbb{C}[\varstuff{q},\varstuff{q}^{-1}]$, which then
implies that \fullref{theorem:q-satake} holds
over any ground ring we are going to use.
\end{remark}
Let $\cRKLg{m,n}=\cRKLg{m,n}(\varstuff{X}^m\varstuff{Y}^n)$ denote the (unique) projection
of $\varstuff{X}^m\varstuff{Y}^n$ onto the irreducible direct summand $\algstuff{L}_{m,n}$, regarded as a
$2$-morphism in $\sltcatgop$ with rightmost color ${\color{mygreen}g}$.
We call $\cRKLg{m,n}$ the (right-green) $\mathfrak{sl}_{3}$-clasp.
Similarly, we define $\cRKLo{m,n}$, $\cRKLp{m,n}$ and
$\cKLg{m,n}$, $\cKLo{m,n}$, $\cKLp{m,n}$. Note that there is actually a different clasp for each
product of $m$ factors $\varstuff{X}$ and $n$ factors $\varstuff{Y}$, but these clasps are all closely related, as we will see in
\fullref{lemma:choice-does-not-matter}. For now, it suffices to consider only the one for $\varstuff{X}^m\varstuff{Y}^n$.
\begin{remark}\label{remark:clasps-formulas}
The $\mathfrak{sl}_{3}$-clasps have a diagrammatic incarnation, obtained
by coloring the diagrammatic clasps from \cite[Theorem 3.3]{Kim}
(which gives the $\mathfrak{sl}_{3}$-clasps in terms of a recursion),
or (using slightly
different conventions) from \cite[(1.8) and Section 3.2]{El3}.
\end{remark}
The colored $\mathfrak{sl}_{3}$-clasps are $2$-morphisms in
$\aDiag[\qpar]$, but do not belong to $\subcatquo[\infty]$, since
their left- and rightmost colors are always secondary colors. Thus,
we need to `biinduce them up to $\emptyset$' in order to have
their appropriate analogs in $\subcatquo[\infty]$:
\begin{definition}\label{definition:clasps-categorified-quotient}
The colored (right-${\color{dummy}\textbf{u}}$) clasps $\CRKLx{m,n}$ are defined by
\[
\CRKLx{m,n}=
\twomorstuff{id}_{\emptyset{\color{dummy}d}{\color{dummy}\textbf{v}}}
\circ_{h}\cRKLx{m,n}\circ_{h}
\twomorstuff{id}_{{\color{dummy}\textbf{u}}{\color{dummy}c}\emptyset}
\]
with $\circ_{h}$ being the horizontal composition in $\Adiag[\qpar]$.
Here $\cRKLx{m,n}$ has leftmost color ${\color{dummy}\textbf{v}}$, and ${\color{dummy}c},{\color{dummy}d}$
are compatible colors where we prefer ${\color{myblue}b}$ over ${\color{myred}r}$ over ${\color{myyellow}y}$.
We define $\CKLx{m,n}$ similarly.
\end{definition}
The colored clasps are idempotent $2$-morphisms
in $\subcatquo[\infty]$, which depend on the same choices as the colored $\mathfrak{sl}_{3}$-clasps
and, additionally, on the choice of ${\color{dummy}c},{\color{dummy}\textbf{u}}$. Again, this dependence is not essential, as
we will show in \fullref{lemma:choice-does-not-matter}, so we abuse notation.
\begin{example}\label{example:clasps}
Using \cite[Theorem 3.3]{Kim}, one can write down the colored $\mathfrak{sl}_{3}$-clasps
explicitly, e.g. in the case $e+1=2$:
\begin{gather*}
\cRKLg{2,0}=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[myorange, opacity=0.8] (0,0) to (0,2) to (-1,2) to (-1,0) to (0,0);
\fill[mypurple, opacity=0.8] (-1,0) to (-1,2) to (-2,2) to (-2,0) to (-1,0);
\draw[dstrand, Xmarked=.55] (0,0) to (0,2);
\draw[dstrand, Xmarked=.55] (-1,0) to (-1,2);
\end{tikzpicture}
+
\tfrac{1}{\qnumber{2}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (-.5,.75) to (-.5,1.25) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[myorange, opacity=0.8] (-1,2) to (-.5,1.25) to (0,2) to (-1,2);
\fill[myorange, opacity=0.8] (-1,0) to (-.5,.75) to (0,0) to (-1,0);
\fill[mypurple, opacity=0.8] (-1,0) to (-.5,.75) to (-.5,1.25) to (-1,2) to (-2,2) to (-2,0) to (-1,0);
\draw[dstrand, Xmarked=.8] (-.5,1.25) to (-1,2);
\draw[dstrand, Xmarked=.8] (-.5,1.25) to (0,2);
\draw[dstrand, Xmarked=.75] (-.5,1.25) to (-.5,.75);
\draw[dstrand, Xmarked=.6] (-1,0) to (-.5,.75);
\draw[dstrand, Xmarked=.6] (0,0) to (-.5,.75);
\end{tikzpicture}
,\quad
\cRKLg{1,1}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to (0,2) to (-1,2) to (-1,0) to (0,0);
\fill[mygreen, opacity=0.8] (-1,0) to (-1,2) to (-2,2) to (-2,0) to (-1,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,2);
\draw[dstrand, Xmarked=.55] (-1,0) to (-1,2);
\end{tikzpicture}
-
\tfrac{1}{\qnumber{3}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (1,0) to (0,0) to [out=90, in=00] (-.5,.75) to [out=180, in=90] (-1,0) to (-2,0) to (-2,2) to (-1,2) to [out=270, in=180] (-.5,1.25) to [out=0, in=270] (0,2) to (1,2) to (1,0);
\fill[mypurple, opacity=0.8] (0,0) to [out=90, in=00] (-.5,.75) to [out=180, in=90] (-1,0) to (0,0);
\fill[mypurple, opacity=0.8] (0,2) to [out=270, in=00] (-.5,1.25) to [out=180, in=270] (-1,2) to (0,2);
\draw[dstrand, Xmarked=.85] (-1,0) to [out=90, in=180] (-.5,.75) to [out=0, in=90] (0,0);
\draw[dstrand, Xmarked=.85] (0,2) to [out=270, in=0] (-.5,1.25) to [out=180, in=270] (-1,2);
\end{tikzpicture}
,\quad
\cRKLg{0,2}=\!
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[mypurple, opacity=0.8] (0,0) to (0,2) to (-1,2) to (-1,0) to (0,0);
\fill[myorange, opacity=0.8] (-1,0) to (-1,2) to (-2,2) to (-2,0) to (-1,0);
\draw[dstrand, Ymarked=.55] (0,0) to (0,2);
\draw[dstrand, Ymarked=.55] (-1,0) to (-1,2);
\end{tikzpicture}
+
\tfrac{1}{\qnumber{2}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\fill[mygreen, opacity=0.8] (0,0) to (-.5,.75) to (-.5,1.25) to (0,2) to (1,2) to (1,0) to (0,0);
\fill[mypurple, opacity=0.8] (-1,2) to (-.5,1.25) to (0,2) to (-1,2);
\fill[mypurple, opacity=0.8] (-1,0) to (-.5,.75) to (0,0) to (-1,0);
\fill[myorange, opacity=0.8] (-1,0) to (-.5,.75) to (-.5,1.25) to (-1,2) to (-2,2) to (-2,0) to (-1,0);
\draw[dstrand, Xmarked=.75] (-1,2) to (-.5,1.25);
\draw[dstrand, Xmarked=.75] (0,2) to (-.5,1.25);
\draw[dstrand, Xmarked=.75] (-.5,.75) to (-.5,1.25);
\draw[dstrand, Xmarked=.8] (-.5,.75) to (-1,0);
\draw[dstrand, Xmarked=.8] (-.5,.75) to (0,0);
\end{tikzpicture}
\end{gather*}
Using $\elfunctor$ and biinduction, we get for example:
\[
\CRKLg{1,1}
=
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-4.5,0) to (-4.5,3) to (-3.5,3) to (-3.5,0) to (-4.5,0);
\draw[very thin, densely dotted, fill=white] (4.5,0) to (4.5,3) to (3.5,3) to (3.5,0) to (4.5,0);
\fill[myblue, opacity=0.3] (-3.5,0) to (-3.5,3) to (-2.5,3) to (-2.5,0) to (-3.5,0);
\fill[myblue, opacity=0.3] (-1.5,0) to (-1.5,3) to (-.5,3) to (-.5,0) to (-1.5,0);
\fill[myblue, opacity=0.3] (1.5,0) to (1.5,3) to (.5,3) to (.5,0) to (1.5,0);
\fill[myblue, opacity=0.3] (3.5,0) to (3.5,3) to (2.5,3) to (2.5,0) to (3.5,0);
\fill[mygreen, opacity=0.8] (2.5,0) to (2.5,3) to (1.5,3) to (1.5,0) to (2.5,0);
\fill[mygreen, opacity=0.8] (-2.5,0) to (-2.5,3) to (-1.5,3) to (-1.5,0) to (-2.5,0);
\fill[mypurple, opacity=0.8] (-.5,0) to (-.5,3) to (.5,3) to (.5,0) to (-.5,0);
\draw[ystrand, directed=.55] (-2.5,0) to (-2.5,3);
\draw[ystrand, directed=.55] (-1.5,3) to (-1.5,0);
\draw[ystrand, directed=.55] (1.5,0) to (1.5,3);
\draw[ystrand, directed=.55] (2.5,3) to (2.5,0);
\draw[rstrand, directed=.55] (-.5,0) to (-.5,3);
\draw[rstrand, directed=.55] (.5,3) to (.5,0);
\draw[bstrand, directed=.55] (-3.5,0) to (-3.5,3);
\draw[bstrand, directed=.55] (3.5,3) to (3.5,0);
\end{tikzpicture}
-
\tfrac{1}{\qnumber{3}}\,
\begin{tikzpicture}[anchorbase, scale=.4, tinynodes]
\draw[very thin, densely dotted, fill=white] (-4.5,0) to (-4.5,3) to (-3.5,3) to (-3.5,0) to (-4.5,0);
\draw[very thin, densely dotted, fill=white] (4.5,0) to (4.5,3) to (3.5,3) to (3.5,0) to (4.5,0);
\fill[myblue, opacity=0.3] (-3.5,0) to (-3.5,3) to (-2.5,3) to (-2.5,0) to (-3.5,0);
\fill[myblue, opacity=0.3] (1.5,0) to [out=90, in=0] (0,1.25) to [out=180, in=90] (-1.5,0) to (1.5,0);
\fill[myblue, opacity=0.3] (-1.5,3) to [out=270, in=180] (0,1.75) to [out=0, in=270] (1.5,3) to (-1.5,3);
\fill[myblue, opacity=0.3] (3.5,0) to (3.5,3) to (2.5,3) to (2.5,0) to (3.5,0);
\fill[mygreen, opacity=0.8] (1.5,0) to [out=90, in=0] (0,1.25) to [out=180, in=90] (-1.5,0) to (-2.5,0) to (-2.5,3) to (-1.5,3) to [out=270, in=180] (0,1.75) to [out=0, in=270] (1.5,3) to (2.5,3) to (2.5,0) to (1.5,0);
\fill[mypurple, opacity=0.8] (-.5,0) to [out=90, in=180] (0,.5) to [out=0, in=90] (.5,0) to (-.5,0);
\fill[mypurple, opacity=0.8] (.5,3) to [out=270, in=0] (0,2.5) to [out=180, in=270] (-.5,3) to (.5,3);
\draw[ystrand, directed=.55] (-2.5,0) to (-2.5,3);
\draw[ystrand, directed=.75] (1.5,0) to [out=90, in=0] (0,1.25) to [out=180, in=90] (-1.5,0);
\draw[ystrand, directed=.75] (-1.5,3) to [out=270, in=180] (0,1.75) to [out=0, in=270] (1.5,3);
\draw[ystrand, directed=.55] (2.5,3) to (2.5,0);
\draw[rstrand, directed=.75] (-.5,0) to [out=90, in=180] (0,.5) to [out=0, in=90] (.5,0);
\draw[rstrand, directed=.75] (.5,3) to [out=270, in=0] (0,2.5) to [out=180, in=270] (-.5,3);
\draw[bstrand, directed=.55] (-3.5,0) to (-3.5,3);
\draw[bstrand, directed=.55] (3.5,3) to (3.5,0);
\end{tikzpicture}
\]
(The outer ${\color{myblue}b}$-regions come from our
choice in \fullref{definition:clasps-categorified-quotient}.)
\end{example}
The next lemma shows that the colored clasps are independent of the choices that we made in their definition:
\begin{lemma}\label{lemma:choice-does-not-matter}
Let $\CRKLx{m,n}$ be a colored clasps and let $(\CRKLx{m,n})^{\prime}$ be defined similarly,
but with some difference in the involved choices. Then there exists an
invertible $2$-morphisms $\twomorstuff{f}$ in $\subcatquo[\infty]$
such that $\CRKLx{m,n}\circ_{v}\twomorstuff{f}=\twomorstuff{f}\circ_{v}(\CRKLx{m,n})^{\prime}$.
\end{lemma}
\begin{proof}
If the ordering of the factors $\varstuff{X}$ and $\varstuff{Y}$ for $\CRKLx{m,n}$ and $(\CRKLx{m,n})^{\prime}$ differs
by precisely one pair, then \fullref{lemma:thecrossings-2} shows the claim.
If $\CRKLx{m,n}$ and $(\CRKLx{m,n})^{\prime}$ differ by precisely one choice of compatible color
for `biinduction', then \fullref{lemma:clasps-well-defined} shows the claim.
Then the general statement then follows by induction.
\end{proof}
\begin{corollary}\label{corollary:clasps-ideal}
If a two sided $2$-ideal in $\subcatquo[\infty]$ contains
a colored clasp $\CRKLx{m,n}$, then it contains
all colored clasps $(\CRKLx{m,n})^{\prime}$
which differ from $\CRKLx{m,n}$ by some of the choices involved in their definition.
\end{corollary}
\subsubsection{From \texorpdfstring{$\mathfrak{sl}_{3}$}{sl3} to singular bimodules: the root of unity case}\label{subsec:quotient-algebra-categorified}
From now on we work over $\mathbb{C}$ by specializing $\varstuff{q}$ to $\varstuff{\eta}$
which, as usual, is a $2(e+3)^{\mathrm{th}}$ primitive, complex root of unity.
Formally this is done by repeating the above for the $\C_{\intqpar}$-linear $2$-categories
which are scalar extended to
$\C_{\inte}=\mathbb{C}[\varstuff{q},\varstuff{q}^{-1},\qnumber{2}^{-1},\dots,\qnumber{e{+}1}^{-1}]$.
We denote these using
$[e]$ as a subscript, and the specialization at $\varstuff{q}=\varstuff{\eta}$ we denote by
$\underline{\phantom{a}}\otimes_{\C_{\inte}}\mathbb{C}$. We also exclude the case $e=0$, which
is a bit special and can easily be dealt with later on.
First of all, all previous definitions and
results in the generic case are still valid in this case, except \fullref{lemma:webs-and-slt}
(which we do not need in the following) and the definition of
the (various) clasps for $m+n > e+1$.
In particular, \fullref{lemma:el-well-def} and \fullref{theorem:q-satake} still hold
for the specialization at $\varstuff{q}=\varstuff{\eta}$.
\begin{lemma}\label{lemma:slt-soergel-clasps}
The colored clasps are well-defined in $\subcatquo[\inte]$
(seen as a $2$-subcategory of $\adiag[\inte]$)
for $0\leq m+n\leq e+1$, and uniquely determined
up to conjugation by an invertible $2$-morphism.
\end{lemma}
\makeautorefname{lemma}{Lemmas}
\begin{proof}
Decomposing $\varstuff{X}^m\varstuff{Y}^n$ generically, i.e.
for $\algstuff{U}_{\qpar}(\mathfrak{sl}_{3})$, works similarly as in the
root of unity case as long as $m+n\leq e+1$, see e.g. \cite[Section 3]{AP}.
Thus, one can use the specializations of the projectors
from the generic case in the root of unity case.
(Alternatively, using \fullref{remark:clasps-formulas}, one checks
that the coefficients of the colored $\mathfrak{sl}_{3}$-clasps specialize properly.)
Finally, note that \fullref{lemma:el-well-def}
and \ref{lemma:choice-does-not-matter} also hold
for $\varstuff{q}$ being specialized to $\varstuff{\eta}$.
\end{proof}
\makeautorefname{lemma}{Lemma}
Having all the above established, we
can define the $2$-category of
trihedral Soergel bimodules of level $e$:
\begin{definition}\label{definition:categorified-quotient}
Let $\claspideal[{e}]$ be the two-sided $2$-ideal, called vanishing $2$-ideal of level $e$,
in $\subcatquo[\inte]\otimes_{\C_{\inte}}\mathbb{C}$ generated by
\begin{gather*}
\left\{\CRKLx{m,n} \mid m+n=e+1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\}
=
\left\{\CKLx{m,n} \mid m+n=e+1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\},
\end{gather*}
where we write e.g. $\CRKLx{m,n}=\CRKLx{m,n}\otimes_{\C_{\inte}}1$ for simplicity.
We define
\[
\subcatquo[e]=(\subcatquo[\inte]\otimes_{\C_{\inte}}\mathbb{C})/\claspideal[{e}],
\]
which we call the $2$-category of trihedral Soergel bimodules of level $e$.
\end{definition}
\begin{remark}\label{remark:before-after}
Note that we specialize before taking the quotient, as Andersen--Paradowski do in order to define
$\sltcat[e]$ in \cite{AP}, where they take the quotient of the already specialized category $\sltcat[\varstuff{\eta}]$ by the
ideal of so-called negligible modules. (This is explicitly described in e.g.
\cite[Section 3.3]{BK1}.) Similarly, we always specialize first throughout.
\end{remark}
Let $\twocatstuff{H}_{e}$
be the two-sided $2$-ideal in $\twocatstuff{Q}_{\inte}^{\Seset}\otimes_{\C_{\inte}}\mathbb{C}$ generated by
\begin{gather*}
\left\{\cRKLx{m,n} \mid m+n=e+1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\}
=
\left\{\cKLx{m,n} \mid m+n=e+1,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\}.
\end{gather*}
The maximally singular version of $\subcatquo[e]$ is
\[
\Subcatquo[e]=(\Subcatquo[\inte]\otimes_{\C_{\inte}}\mathbb{C})/\elfunctor[\varstuff{\eta}]\left(\twocatstuff{H}_{e}\right).
\]
Here we again specialize $\varstuff{q}$ to $\varstuff{\eta}$ and use the same conventions as before.
We state a non-trivial consequence of the
quantum Satake correspondence from \fullref{theorem:q-satake}
and \fullref{remark:q-satake}.
\begin{lemma}\label{lemma:Satake}
$\twomorstuff{S}_{\inte}$ gives rise to
a degree-zero $2$-equivalence
$\elfunctor[e]\colon\slqmodgop\to\Subcatquo[e]$.
\end{lemma}
\begin{proof}
Because $\twomorstuff{S}_{\inte}$ is a degree-zero $2$-equivalence
before quotienting by any clasps, by \fullref{remark:q-satake}, it sends
indecomposable $1$-morphisms in $\twocatstuff{Q}_{\inte}^{\Seset}$
to indecomposable $1$-morphisms in $\Subcatquo[\inte]$, and
the cell structures of $\twocatstuff{Q}_{\inte}^{\Seset}$ and $\Subcatquo[\inte]$ are
isomorphic under $\twomorstuff{S}_{\inte}$.
Clearly, $\twomorstuff{S}_{\inte}$ specializes to $\elfunctor[\varstuff{\eta}]\colon \Subcatquo[\inte]\otimes_{\C_{\inte}}\mathbb{C}\to
\twocatstuff{Q}_{\inte}^{\Seset}\otimes_{\C_{\inte}}\mathbb{C}$, which descends to the $2$-functor
$\elfunctor[e]\colon\slqmodgop\to\Subcatquo[e]$. Both $\elfunctor[\varstuff{\eta}]$ and $\elfunctor[e]$
are essentially surjective on $1$-morphisms and full onto degree-zero $2$-morphisms.
What is not immediately clear, is that $\elfunctor[e]\colon\slqmodgop\to\Subcatquo[e]$ is
also faithful on $2$-morphisms: since $\subcatquo[\inte]$
has $2$-morphisms of negative degree, the
degree-zero part of $\claspideal[e]$ could a priori be bigger
than $\elfunctor[\varstuff{\eta}]\left(\twocatstuff{H}_{e}\right)$. To show that that is not the case, we
use the following roundabout argument.
From \cite{AP} (cf. \fullref{remark:before-after})
we know that we have an equivalence of $2$-categories
\[
(\twocatstuff{Q}_{\inte}^{\Seset}\otimes_{\C_{\inte}}\mathbb{C})/\twocatstuff{H}_{e}
\cong\slqmodgop,
\]
Hence, the indecomposable $1$-morphisms $\morstuff{F}$ in $\twocatstuff{Q}_{\inte}^{\Seset}\otimes_{\C_{\inte}}\mathbb{C}$ for which
$\twomorstuff{id}_{\morstuff{F}}\in\twocatstuff{H}_{e}$ are
strictly greater than
the ones for which $\twomorstuff{id}_{\morstuff{F}}\not\in\twocatstuff{H}_{e}$, in
the two-sided cell preorder. By the observations in the first paragraph,
the same must hold for the indecomposable
$1$-morphisms in $\subcatquo[\inte]\otimes_{\C_{\inte}}\mathbb{C}$ with respect to the
$2$-ideal $\claspideal[e]$, which
is generated by $\elfunctor[\varstuff{\eta}]\left(\twocatstuff{H}_{e}\right)$. This shows
that $\elfunctor[e]\colon\slqmodgop\to\Subcatquo[e]$
is faithful on $2$-morphisms,
since $\slqmodgop$ is semisimple.
\end{proof}
\begin{proposition}\label{proposition:cat-the-quoalgebra}
The isomorphism from \fullref{proposition:cat-the-algebra}
gives an isomorphism
$\subquo[e]\xrightarrow{\cong}\GGcv{\Kar{\subcatquo[e]}}$
of algebras.
\end{proposition}
\begin{proof}
This follows from the discussion
above: By \fullref{corollary:clasps-ideal} and \fullref{lemma:slt-soergel-clasps} the vanishing $2$-ideal of level $e$
contains all colored clasps of level $e+1$.
By by \fullref{proposition:cat-the-algebra}
these decategorify to the $\RKLx{m,n}$ in the definition
of $\subquo[e]$, while \fullref{lemma:Satake} ensures that
the Grothendieck classes of the remaining $\CRKLx{m,n}$
form a basis of $\GGcv{\Kar{\subcatquo[e]}}$.
\end{proof}
\subsection{Generalizing dihedral Soergel bimodules}\label{subsec:dihedral-SB}
As before, we list certain analogies to the dihedral case.
\begin{dihedral}\label{remark:dihedral-SB1} The Hecke algebra $\hecke(\typeat{1})$ of
\fullref{remark:dihedral-group1} is categorified by Soergel bimodules of affine type $\typea{1}$.
Here the Hecke algebra $\hecke(\typeat{2})$ is categorified by Soergel bimodules of affine type $\typea{2}$.
The difference is that now biinduction of the maximally singular bimodules only gives a proper $2$-subcategory.
\end{dihedral}
\makeautorefname{proposition}{Propositions}
\begin{dihedral}\label{remark:dihedral-SB2}
The Satake $2$-functor from \eqref{eq:elfunctor}
exists in the dihedral case as well, with a bicolored
version of quantum $\mathfrak{sl}_2$-modules
as the source $2$-category. This $2$-category
has a diagrammatic incarnation in terms of
a $2$-colored Temperley--Lieb calculus \cite[Section 4.3]{El2}.
The Soergel bimodules of finite Coxeter type
$\typei$ can then be defined by annihilating the ideal generated by the colored
Jones--Wenzl projectors (i.e. colored $\mathfrak{sl}_2$ clasps) of level $e+1$ in this $2$-colored
Temperley--Lieb calculus. Moreover, while
the colored $\mathfrak{sl}_{3}$-clasps satisfy the recursion
in \fullref{lemma:recursion}, their $\mathfrak{sl}_2$ counterparts satisfy
the Chebyshev recursion from \fullref{remark:dihedral-group2}.
Finally, the analogs of \fullref{proposition:cat-the-algebra}
and \ref{proposition:cat-the-quoalgebra}
hold as well.
\end{dihedral}
\makeautorefname{proposition}{Proposition}
\noindent\textbf{Missing proofs from \texorpdfstring{\fullref{section:funny-algebra}}{\ref{section:funny-algebra}}.}
\begin{proof}[Proof of \fullref{lemma:multiplication}]
Recall that the $2$-category $\adiag[\varstuff{q}]$ categorifies
$\hecke$, i.e.
\begin{gather}\label{eq:EW-cat}
\hecke\xrightarrow{\cong}\GGcv{\Kar{\adiag[\varstuff{q}]}},
\end{gather}
such that the KL elements are sent
to the Grothendieck classes of the indecomposable $1$-morphisms (with a fixed choice of grading),
c.f. \cite{El1}, \cite{EW} or \fullref{remark:cat-affine-a2}. Furthermore,
by \fullref{lemma:quotient-of-affine}, the algebra
$\subquo$ can be embedded into $\hecke$ by sending the colored KL elements in $\subquo$ to
KL elements in $\hecke$. Thus, we can
identify elements of $\subquo$ with
Grothendieck classes in $\GGcv{\Kar{\adiag[\varstuff{q}]}}$:
The element
$\rklx{k,l}=\theta_{{\color{dummy}\textbf{u}}_{k+l}}\cdots\theta_{{\color{dummy}\textbf{u}}_1}\theta_{{\color{dummy}\textbf{u}}_0}\in \hecke$,
with ${\color{dummy}\textbf{u}}_0={\color{dummy}\textbf{u}}$, corresponds to
\begin{gather}\label{eq:KL-element}
[
\emptyset{\color{dummy}c}_{k+l}{\color{dummy}\textbf{u}}_{k+l}{\color{dummy}c}_{k+l}\emptyset
\cdots
\emptyset{\color{dummy}c}_1{\color{dummy}\textbf{u}}_1{\color{dummy}c}_1\emptyset{\color{dummy}c}_0{\color{dummy}\textbf{u}}_0{\color{dummy}c}_0\emptyset
]
\in\GGcv{\Kar{\adiag[\varstuff{q}]}},
\end{gather}
where we can chose any compatible primary colors by
\fullref{lemma:clasps-well-defined}. In fact, with \fullref{lemma:clasps-well-defined} in mind, we will
denote all compatible primary colors simply by ${\color{dummy}c}$ from now on.
Using \fullref{lemma:remove-white}, we see that the element
in \eqref{eq:KL-element} is
equal to
\[
\vnumber{2}^{k+l}\,
[
\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}_{k+l}{\color{dummy}c}
\cdots{\color{dummy}c}{\color{dummy}\textbf{u}}_1 {\color{dummy}c}{\color{dummy}\textbf{u}}_0{\color{dummy}c}\emptyset
]
\in\GGcv{\Kar{\adiag[\varstuff{q}]}}.
\]
Next, we use $\elfunctor$ from \eqref{eq:elfunctor}.
By definition,
$[\elfunctor]$ maps $[\varstuff{X}^k\varstuff{Y}^l]\in\GGcv{\sltcatgop}$ to
\[
[
\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}_{k+l}{\color{dummy}c}
\cdots{\color{dummy}c}{\color{dummy}\textbf{u}}_1 {\color{dummy}c}{\color{dummy}\textbf{u}}_0{\color{dummy}c}\emptyset
]
\in\GGcv{\Kar{\adiag[\varstuff{q}]}},
\]
and to similar expressions with different rightmost color.
By \fullref{remark:sl3-cat-GG}, this implies that $[\elfunctor]$
maps $[\algstuff{L}_{m,n}]$ to
\begin{gather}\label{eq:KL-element2}
{\textstyle\sum_{k,l}}
\vnumber{2}^{-k-l}d^{k,l}_{m,n}
[
\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}_{k+l}{\color{dummy}c}\emptyset
\cdots
\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}_1{\color{dummy}c}\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}_0{\color{dummy}c}\emptyset,
]
\end{gather}
and again to similar expressions with different rightmost color.
By \fullref{theorem:q-satake}, $\elfunctor$ is a degree-zero $2$-equivalence.
In particular, it sends the simple $1$-morphisms to indecomposable $1$-morphisms.
This implies that the element
in \eqref{eq:KL-element2} is the Grothendieck
class of an indecomposable $1$-morphism of $\Adiag[\varstuff{q}]$.
Biinduction preserves indecomposibility, so our element $\RKLx{m,n}$ corresponds to the Grothendieck
class of an indecomposable $1$-morphism in $\GGcv{\adiag[\varstuff{q}]}$.
By the categorification theorem from \eqref{eq:EW-cat},
we see that $\RKLx{m,n}$ corresponds to a
KL basis element in $\hecke$, and $\rklx{k,l}$ to the Grothendieck class of
a Bott--Samelson bimodule.
From the above, we obtain the first
equation in \eqref{eq:multiplication}, since
\[
[
\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}{\color{dummy}c}\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}{\color{dummy}c}\emptyset
]
=\vnumber{2}
[
\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}{\color{dummy}c}{\color{dummy}\textbf{u}}{\color{dummy}c}\emptyset
]
=
\vfrac{3}
[
\emptyset{\color{dummy}c}{\color{dummy}\textbf{u}}{\color{dummy}c}\emptyset
],
\]
by \fullref{lemma:remove-white} and \fullref{example:some-relations}.
Using $[\elfunctor]$ and \eqref{eq:sl3-thingy-a}, we deduce the
second equation in \eqref{eq:multiplication}.
Similarly, one can prove the third
equation in \eqref{eq:multiplication} using \eqref{eq:sl3-thingy-b}.
\end{proof}
\begin{proof}[Proof of \fullref{proposition:two-bases}]
Let ${\color{dummy}\textbf{u}}$ be fixed for now. Recalling
the notation from \fullref{section:sl3-stuff},
by \fullref{lemma:multiplication} and its
proof given above, there is
a $\C_{\vpar}$-linear isomorphism between the scalar extension
$\GGcv{\sltcatgop}$ and
$\C_{\vpar}\left\{\rklx{k,l}\mid (k,l)\in X^+\right\}$, defined by
\begin{gather*}
[\varstuff{X}^k\varstuff{Y}^l]
\mapsto
\vnumber{2}^{-k-l}\,\rklx{k,l}.
\end{gather*}
This shows that the $\rklx{k,l}$ are all linearly independent,
and they are also linearly
independent of $1$, of course.
Since ${\color{dummy}\textbf{u}}$ was arbitrary and there are no relations in $\subquo$ which
allow us to change the rightmost color in a word, it
follows that
\[
\left\{1 \right\}\cup \left\{\rklx{k,l}\mid (k,l)\in X^+,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\}
\]
is a basis of $\subquo$.
Because $d^{m,n}_{m,n}=1$, and $d^{k,l}_{m,n}=0$ if $k+l>m+n$,
the above immediately implies that
\[
\left\{\RKLx{m,n}\mid (m,n)\in X^+,\; {\color{dummy}\textbf{u}}\in\Gset\Oset\Pset\right\}
\]
is also a basis,
since the transformation between the two
sets of elements defined by \eqref{eq:the-expressions}
is triangular with diagonal factors $\vnumber{2}^{-m-n}\neq 0$.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,575
|
# SMART SCHOOLS
BETTER THINKING AND LEARNING FOR EVERY CHILD
David Perkins
The Free Press
New york London Toronto Sydney
Copyright © 1992 by David Perkins
All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the Publisher.
The Free Press
A Division of Simon & Schuster Inc.
1230 Avenue of the Americas
New York, N.Y. 10020
www.SimonandSchuster.com
First Free Press Paperback Edition 1995
Printed in the United States of America
printing number
10
**Library of Congress Cataloging-in-Publication Data**
Perkins, David N.
Smart schools: Better Thinking and Learning for Every Child/
David Perkins.
p. cm.
Includes bibliographical references (p. ) and index.
ISBN 0-02-874018-1
ISBN 13: 978-0-0287-4018-8
eISBN 13: 978-1-4391-0840-6
1. Thought and thinking—Study and teaching—United States.
2. Learning, Psychology of.
3. Comprehension.
I. Title.
LB1590.3.P473 1992
370.15′2—dc20 92-13763
CIP
**To my schoolchildren:
Ted, Alice, and Tom**
## **CONTENTS**
**Acknowledgments**
**1. SMART SCHOOLS**
Using What We Know
Goals: Toward Generative Knowledge
Means: Thoughtful Learning
Precedents: Swings of the Pendulum
Prospects: Putting What We Know to Work
Connections: Some Issues Seen Anew
Mission: Smart Schools
**2. THE ALARM BELLS**
A Shortfall: Fragile Knowledge
A Shortfall: Poor Thinking
A Deep Cause: The Trivial Pursuit Theory
A Deep Cause: The Ability-Counts-Most Theory
A Consequence: Economic Erosion
Defining the Problem
**3. TEACHING AND LEARNING: Theory One and Beyond**
Introducing Theory One
The Devastating Critique Levied by Theory One
Three Ways to Put Theory One to Work
The Bogeyman of Behaviorism
Beyond Theory One
Our Most Important Choice Is What We Try to Teach
**4. CONTENT: Toward a Pedagogy of Understanding**
What Is Understanding?
Understanding and Mental Images
Levels of Understanding
Powerful Representations
Generative Topics
An Example of Teaching for Understanding
**5. CURRICULUM: Creating the Metacurriculum**
The Idea of the Metacurriculum
Levels of Understanding
Languages of Thinking
Intellectual Passions
Integrative Mental Images
Learning to Learn
Teaching for Transfer
An Example of Teaching the Metacurriculum
**6. CLASSROOMS: The Role of Distributed Intelligence**
The Idea of Distributed Intelligence
Distributing Cognition in the Classroom
The Fingertip Effect
Who's Boss When?
An Example of Person-Plus Teaching
**7. MOTIVATION: The Cognitive Economy of Schooling**
The Idea of a Cognitive Economy
The Cool Cognitive Economy of the Typical Classroom
Creating a Hot Cognitive Economy
School Restructuring: A Cognitive Economic Revolution
Teaching to the Wrong Test
Teaching to the Right Test: The Idea of Authentic Assessment
The Cognitive Economy Meets the Money Economy
An Example of Progress Toward a Hot Cognitive Economy
**8. VICTORY GARDENS FOR REVITALIZED EDUCATION**
Better Teaching and Learning in Review
Example 1. Expert Tutoring
Example 2. Biology for Young Inquirers
Example 3. History for Thinkers
Example 4. A Textbook from the Past
Example 5. A Metacourse for Computer Programming
Example 6. Escalante Himself
A Smart School Is Something Rather Special
**9. THE CHALLENGE OF WIDE-SCALE CHANGE**
Facing the Necessities of Scale
Making Change Work
Advancing Thoughtful Professionalism
What We Know Can Make a Difference
**Appendix A CHECKLIST FOR CHANGE**
**Notes**
**References**
**Index**
## **ACKNOWLEDGMENTS**
Some of the ideas expressed in _Smart Schools_ were developed under grants from the MacArthur Foundation for integrative work on the teaching of thinking, from the Spencer Foundation for research toward a pedagogy of understanding, and from the Pew Foundation and the MacArthur Foundation for research and development work on project-based learning and after-school programs. The help of these foundations is very much appreciated. Of course, the ideas expressed here are my own and do not necessarily relect their positions or policies.
Several colleagues were kind enough to comment on an earlier draft of this book. My thanks to Phillip Cousins, Howard Gardner, Peter Kugel, Jack Lochhead, Jerry Murphy, Steven Rhodes, John Thurner, Shari Tishman, and Chris Unger. Their counsel was very useful, sparking a number of important changes. I am particularly grateful to Susan Arellano, my editor at The Free Press; her thoughtful counsel made this a better book in a number of ways.
Special thanks go to Diane Downs, who both found many sources and managed the production of the manuscript, and to Tina Blythe, Joyce Conkling, and Noel White, who provided invaluable help in locating sources, making corrections in the text, and otherwise facilitating the production of this book. Their systematic and assiduous efforts made the preparation of the book much easier than it would otherwise have been.
Finally, _Smart Schools_ also benefits from my close work with a number of individuals in addition to the above over the past several years. They are many, and I will not try to list them here. But I am grateful. The intricate and exciting challenge of good educational practice is nothing to be pondered solo!
## CHAPTER 1
**SMART SCHOOLS**
The Hanging Gardens of Babylon counted as one of the seven wonders of the ancient world, alongside the Colossus of Rhodes, the pyramids of Egypt, and the Temple of Artemis at Ephesus. Word comes down to us of a terraced wonderland of fountains, trees, and flowers, rising up from the banks of the Euphrates. King Nebuchadnezzar II constructed this sumptuous adjunct to the royal palace more than half a millennium before the birth of Christ.
Of these ancient wonders only the pyramids remain. Today great physical constructions play second fiddle to the wonders of everyday life—for instance, the transistor, which packs little boxes with great powers of voice, image, and computation; or, more humble yet, the light bulb. How hard it is to imagine life without light available at the flick of a forefinger!
And another invention: schools. Yes, schools. A wonder, really. A very new thing, if we mean public schools, schools for everyone, schools as part of a massive committed mission to bring to virtually all of a population with its multifarious ambitions, misgivings, talents, and quirks basic knowledge, skills, and insights. Schools are wonders in the same way that light bulbs are—too much a part of everyday life to amaze us, but, from a historical perspective, quite novel and exotic in their ambitions and accomplishments.
Not, it must be said, that schools always seem to function in as wondrous a way as we would like. Not that we are so happy with how schools work and what they achieve. Not that society gives over to schools and teachers the resources and the honors they deserve. But with all that, still a wonder indeed. Gripe how we will about what schools are _not_ doing these days, they are already doing things undreamt of a couple of centuries ago, much less in Nebuchadnezzar's day.
### USING WHAT WE KNOW
Dreams are where the dilemma starts. Although schools already achieve things undreamt of earlier, we have more ambitious dreams today. We want schools to deliver a great deal of knowledge and understanding to a great many people of greatly differing talents with a great range of interests and a great variety of cultural and family backgrounds. Quite a challenge—and why aren't we doing better at it?
Some say, "We don't know enough. We don't know how learning really works. We don't know how teachers really think about their craft. We don't know how to cope with cultural diversity. We don't know how schools can work better as institutions. We just don't know enough."
I think they're wrong. Of course, we want to know and understand more about all those things. But we know enough now to do a much better job of education. We know because we have made an effort to find out. Over the past quarter century, psychologists have come to understand more deeply how learning works and how to motivate learning. Sociologists have studied how classrooms and schools as institutions work, what makes them resistant to change, and how to foster change. Innovations in various educational settings around the world allow us to compare experiences across contexts and cultures. We know a lot about how to educate well. In the later chapters of this book, I'll do my best to prove this.
The problem comes down to this: We are not putting to work what we know. In the school down the street, in the school across the river, students are learning and teachers are teaching in much the same way they did twenty or even fifty years ago. In the age of CDs and VCRs, communications satellites and laptop computers, education remains by and large a traditional craft.
Of course, the educational landscape sparkles with isolated innovative programs. Some individual teachers are ardent experimenters, trying worthwhile things. Some initiatives score important successes here and there. But most are limited. Most do not put to work in any full and rounded way what we know about teaching and learning. We do not have a knowledge gap—we have a monumental _use-of-knowledge_ gap.
To close this gap, we need schools that put to work, day in and day out, what we know about how to educate well. We can call such schools "smart schools"—schools wide awake to the opportunities of better teaching and learning. We can think of smart schools as exhibiting three characteristics:
_Informed_. Administrators, teachers, and indeed students in the smart school know a lot about human thinking and learning and how it works best. And they know a lot about school structure and collaboration and how that works best.
_Energetic_. The smart school requires spirit as much as information. In the smart school, measures are taken to cultivate positive energy in the structure of the school, the style of administration, and the treatment of teachers and students.
_Thoughtful_. Smart schools are thoughtful places, in the double sense of caring and mindful. First of all, people are sensitive to one another's needs and treat others thoughtfully. Second, both the teaching/learning process and school decision-making processes are _thinking centered_. As we shall soon see, putting thinking at the center of all that happens is crucial.
Informed, energetic, and thoughtful—three broad characteristics for the smart school. These characteristics are not revolutionary. They are common sense by and large. But they are not common practice. In most schools, faculty and students are not well informed about how teaching, learning, thinking, collaboration, and other such elements of schooling work best. In all too many schools, energy levels are low; students, teachers, and administrators fight a thousand frustrations. And most schools do not put thinking at the center of the learning process or at the center of working together with one another.
In this book, I want to describe in broad stroke the contemporary science of teaching and learning that can inform teachers, students, and administrators about how learning works best. I want to touch on factors that create positive energy in a school setting. And I want to focus particularly on the role of _thoughtfulness_ in the teaching/learning process, the key to genuine learning that serves students well. My hope is that this book, along with other publications and events, will help communities everywhere to work toward smart schools.
The goals of education are a good place to start.
### GOALS: TOWARD GENERATIVE KNOWLEDGE
What do we want of education? This is the key question for the entire enterprise. Unless we know what we want and pursue it with ingenuity and commitment, we are not very likely to get it.
Of course, in a broad sense, we know all too well what we want. It can be put in a single word: _everything_. In _Popular Education and Its Discontents_ , Lawrence Cremin, late historian of education at Columbia University, especially emphasized how we bedevil education with agendas. We try to solve all our problems by assigning them to educators—not only knowledge but citizenship, moral rectitude, comfortable social relations, a more able work force, and so on.
It is easy to like the sound of all of these goals. Most of us would be happy to see public education working away at them insofar as it can. But we should also wonder whether the educational enterprise has a core.
One reason to worry about a core is that the "everything" agenda for schools is an energy vampire. It drains teachers, students, and administrators. Think how crucial an energetic spirit is to any institution you want to thrive. Nothing drains energy more than having too many things to do and too little time to do any of them anywhere near well. I certainly am not saying that schools should focus very narrowly on reading, 'riting, and 'rithmetic, for example. But I am saying, in common voice with many others these days, that some focus is imperative.
So even though we want everything, what do we want _most?_ Without apology, let me attempt an answer. Here at a minimum is what we want, three general goals that stick close to the narrower endeavor of education. These are goals almost no one would argue with:
Retention of knowledge
Understanding of knowledge
Active use of knowledge
A summary phrase for the goals taken together might be "generative knowledge"—knowledge that does not just sit there but functions richly in people's lives to help them understand and deal with the world.
No futuristic agenda this! These goals are not meant to sound exotic. They do not reach for anything very new. They follow directly from the core function of education, passing knowledge from one generation to the next. Whatever else a school is doing, if a school is not serving these goals well, it hardly deserves the name of school.
Lest these goals sound altogether too narrow, let me emphasize how broadly I mean "knowledge." While the term sounds somewhat circumscribed, the English language seems to offer no perfect word to cover the many kinds of learning. So let it be knowledge, emphasizing that this includes factual knowledge, skills, know-how, reflectiveness, familiarity with puzzlements as well as solutions, good questions to ask as well as good answers to give, and so on. As to its content, think in terms of typical subject matters, if you like—reading, writing, mathematics, science, history, and so on. They will do for the present.
We need to pursue every one of these three goals to achieve generative knowledge—knowledge that serves people well in later academic and nonacademic pursuits, knowledge that empowers the new generation to build even further.
Take, for example, goal number one, retention. Having knowledge for the Friday quiz does learners little good unless they still have it when they need it months or years later. Or take goal number two, understanding. There is little point in having knowledge that is not understood. Of course, not everything has to be understood completely. But, for example, if you do not understand when to use the arithmetic or algebra you know, it cannot do you much good. If you do not understand why history unfolds as it does, you will be ill equipped to grasp current events, vote wisely, or steer your own life with an eye on historical forces.
As to active use, the third goal, there is little gain in simply having knowledge and even understanding it for the quiz if that same knowledge does not get put to work on more worldly occasions: puzzling over a public issue, shopping in the supermarket, deciding for whom to vote, understanding why political turmoil persists at home and abroad, dealing with an on-the-job human-relations problem, and so on.
Retention, understanding, and the active use of knowledge... three goals of education hardly anyone can argue with. Of course, one can have other sets of fundamental hard-to-argue-with goals for education besides these. In his 1982 book _The Paideia Proposal: An Educational Manifesto_ , Mortimer Adler advocates the trio of (1) the acquisition of organized knowledge; (2) development of intellectual skills; (3) enlarged understanding of ideas and values. I like Adler's goals. Retention, understanding, and the active use of knowledge include them, when we remember that knowledge has a broad interpretation that includes skills.
However, I like my terms better, because they describe not only what the learner gets but what the learner is supposed to be able to do with it afterwards. In particular, retention and active use point toward action. Not stopping at acquisition, they declare that the learner can go on to do things. Understanding too points toward action. As we shall see in chapter 4, understanding involves what we will call "understanding performances."
### MEANS: THOUGHTFUL LEARNING
They seem innocuous, the three goals proposed here. They do not ask for any more than what we have always been asking for. They do not sound like much of a wake-up call for schools.
But I will let you in on a secret: These goals by themselves are enough to lead us to an ambitious vision of smart schools. Simple and agreeable though they are, they demand a great deal of schooling. Contemporary educational practice in the United States and in many other settings comes nowhere near achieving reasonable versions of these goals.
Nowadays, students emerge from primary, secondary, and even college education with remarkable gaps in basic background knowledge about the world they live in. A case in point: Most seventeen-year-olds cannot identify the date of the U.S. Civil War within half a century. In addition, students do not understand much of what they are taught. After education that directly treats important and accessible principles of physics, biology, and mathematics, many people persist in fundamental misconceptions about the world around them. And further, people do not use what they know. At home or in business, people fail to muster basics of writing, reading, and relating to others that have been prominent in their educational experiences. Chapter 2 says much more about all this.
The bottom line is that we are not getting the retention, understanding, and active use of knowledge that we want. If what we are doing is not working, what do we do instead? What do these shortfalls argue for?
The research and experience of educators, psychologists, and sociologists over a number of years offer a clear answer, the harvest of what might be called an emerging new science of teaching and learning. It is not a completely original answer. Many thoughtful people from Socrates on have expressed the same spirit. But the contemporary understanding of human thinking and learning has buttressed their insights with an array of careful evidence that makes the conclusion difficult to challenge.
The answer is this: We need _thoughtful learning_. We need schools that are full of thought, schools that focus not just on schooling memories but on schooling minds. We want what policy analyst Rexford Brown in a recent study of schools called "a literacy of thoughtfulness." We need educational settings with thinking-centered learning, where students learn by thinking through what they are learning about.
While the chapters to come will revisit this theme again and again, that in a nutshell is the message of extensive research on the nature of human thinking and learning. The rationale can be boiled down to a single sentence: _Learning is a consequence of thinking_. Retention, understanding, and the active use of knowledge can be brought about only by learning experiences in which learners think about and think with what they are learning.
Notice how this single sentence turns topsy-turvy the conventional pattern of schooling. The conventional pattern says that, first, students acquire knowledge. Only then do they think with and about the knowledge that they have absorbed. But it's just the opposite: Far from thinking coming after knowledge, knowledge comes on the coattails of thinking. As we think about and with the content that we are learning, we truly learn it.
Indeed, this even holds for the simplest kind of learning, straight memorization. Over and over again, studies have demonstrated that we memorize best when we analyze what we are learning, find patterns in it, and relate it to knowledge we already have. In other words, when we think about it. As early as 1888, the renowned American psychologist William James expressed the point eloquently this way:
... the art of remembering is the art of _thinking;_... when we wish to fix a new thing in either our own mind or a pupil's, our conscious effort should not be so much to _impress_ and _retain_ it as to _connect_ it with something else already there. The connecting is the thinking; and if we attend clearly to the connection, the connected thing will certainly be likely to remain within recall. [Italics are James's.]
Therefore, instead of knowledge-centered schools, we need thinking-centered schools. This is no luxury, no utopian vision of an erudite and elitist education. These are hard facts about the way learning works.
### PRECEDENTS: SWINGS OF THE PENDULUM
The idea of informed, energetic schools focused on thoughtful learning is hardly new. Indeed, it has figured centrally in the history of education in the United States. Sometimes it has been seen as a mainstay of the educational process, sometimes as an elitist enterprise, neither possible nor needed for the majority of students. The pendulum swings back and forth.
During the first half of this century, one of the persistent champions of thoughtful learning in the United States was the seminal educational philosopher John Dewey, a founder of the progressive education movement. Dewey had this to say about the essential role of thoughtfulness in schooling:
Of course, intellectual learning includes the amassing and retention of information. But information is an undigested burden unless it is understood... And understanding, comprehension, means that the various parts of the information acquired are grasped in their relations to one another—a result that is attained only when acquisition is accompanied by constant reflection upon the meaning of what is studied.
Dewey and other advocates of progressivism envisioned a child-centered education that took account of children's interests and abilities and built on that foundation. Education, Dewey maintained, should take as its foundation what the child knew and build from there toward intellectual insight into and appreciation of the landmarks of culture and science—the wisdom of Shakespeare, Newton, and others.
But progressivism took an odd turn, one quite contrary to Dewey's picture of it. In the child-centered spirit, others began to see schooling as practical preparation for everyday life, serving students who by and large lacked the intellectual ability to aspire to more. In the mid 1940s, "life adjustment education" became the watchword, and subjects like business English and business arithmetic became the paragons of the educational enterprise. For a while, most folks seemed satisfied with a less ambitious model of education. The pendulum had swung away from Dewey.
Then, in October 1957, Russia preempted American ambitions in space and challenged the image of the United States as the premier technological power with the launching of Sputnik, the first space satellite. Concerns over the intellectual quality of the nation rekindled visions of a more ambitious kind of education.
Through the 1960s and early 1970s, a spirit of innovation held sway, and new curricula, conceived in universities, came into the classrooms to run the reality gauntlet of teachers and students. Those were the days of the controversial "new math," which urged students studying elementary arithmetic to learn the logical foundations of the subject matter—set theory, the distinction between a number and a numeral, number systems with bases other than 10. Those also were the days of _Man: A Course of Study_ , an innovative social studies program developed by Jerome Bruner and his colleagues that asked schoolchildren to open their eyes to a broader view of the human condition. Children learned how evolution worked, compared baboon and human societies, and became acquainted with the ingenious survival strategies and the spiritual dimensions of Netsilik Eskimos. Also in that period, _Project Physics_ was developed, a serious and thoughtful effort to humanize physics by providing a curriculum and materials rich not only in concepts but in the historical, social, and biographical roots of the science.
The great moral of that era was that most of these programs did not fare well in practice. It was not that they did not achieve their instructional aims when well implemented. But committed, thoughtful implementations were few and far between. And there was an energy problem. It was so much easier to do something else, to stick to more conventional texts and aspirations.
Those years were the previous time around for a concerted wide-scale effort to create something like smart schools. But the pendulum swung away from that for a while. In the late 1970s it was back to the basics. Sound foundatiorial skills of reading, 'riting, and 'rithmetic became the educational priority in the face of sorry performance by an alarming percentage of the nation's youth. But bit by bit, the educational community became aware that "back to the basics" did not provide the hoped-for payoffs. The problems highlighted in the previous section and reviewed in more detail in chapters to come began to emerge. Youngsters did not know what it seemed they should, Youngsters did not understand what they were learning. They could not solve problems with the knowledge they had gained.
This inspired the contemporary effort to rethink and reform educational practice, much of it in the general direction of thoughtful learning. The current zeal to restructure schools generally brings with it an emphasis on students' thoughtful engagement with content. Mortimer Adler's _Paideia Proposal_ , mentioned earlier, envisions schools with high academic standards and an emphasis on discussion of and thinking about great works and ideas. Theodore Sizer of Brown University has become the philosophical leader of a number of "essential schools," high schools which reduce the number of subject matters for the sake of more deeply pursuing core subject matters and emphasize the idea of "authentic work," where students engage in genuine intellectual inquiry. The "whole language" movement urges involving students in a rich range of writing and other language-oriented activities across subject matters. New standards for the learning of mathematics proposed by the National Council of Teachers of Mathematics underscore the importance of problem solving and mathematical inquiry.
And so here we are again today, involved in the quest for a thoughtful, energetic kind of education that serves well those three key hard-to-argue-with goals celebrated earlier: retention, understanding, and the active use of knowledge.
### PROSPECTS: PUTTING WHAT WE KNOW TO WORK
This time around in the quest for the smart school, do we have any hope of doing better? After all, during the previous swings of the pendulum, efforts to make education more informed, energetic, and thoughtful drew on some of the most ingenious figures of the era along with ample government support. What makes us think that today, when we are no smarter and government commitment to education is sparser, we can do better?
Knowledge. The answer is knowledge. Because of the last quarter century of research and experience, stimulated in good part by the not-quite-successful initiatives of the late 1960s and early 1970s, we know far more about the quest for effective schools than ever before. A new science of teaching and learning is emerging. We can put that knowledge to work.
I have already said that current reforms do not take full advantage of this knowledge. On the contrary, different movements and programs typically have distinctive historical and philosophical roots. They generally proceed with little awareness of other knowledge resources that might help them advance their missions. Indeed, an aim of this book is to put in one place a number of broad principles reflecting the new understanding of teaching and learning emerging from research and experience, so that anyone can put them to work wherever they seem helpful. So what is this knowledge? What does it say to us about the problems? And how does it point to possible solutions? The following chapters try to put the pieces of the puzzle together, building a better picture of the smart school. A preview is in order.
1. _The Alarm Bells_. Researchers have taken a careful look at just what students are achieving. The shortfalls in retention, understanding, and the active use of knowledge are well documented, and severe consequences for economic development seem likely. All this underlines the need for the smart school.
2. _Teaching and Learning: Theory One and Beyond_. If students are to learn with good retention, understanding, and active use of knowledge, some very basic and well-established principles of teaching and learning have to get much more attention than they typically do, even in innovative settings. Theory One spells these principles out. Beyond Theory One, other instructional methods such as cooperative learning offer further leverage.
3. _Content: Toward a Pedagogy of Understanding_. What is it to understand something? Contemporary psychology is building an understanding of understanding. Even current reforms often do not recognize how much learning for understanding demands in the way of artful instruction. This chapter explores what understanding is and how to build learners' understandings.
4. _Curriculum: Creating the Metacurriculum_. In the past several years, how people think and how they can learn to think better have been major areas of inquiry for psychologists and philosophers. Effective learning turns out to involve much more than just acquiring the facts. Students must not just _know_ the content but _think_ with it. This recommends supplementing the content-oriented curriculum with something missing in most current efforts to restructure schools—a "metacurriculum" that pays attention to higher-order thinking and learning.
5. _Classrooms: The Role of Distributed Intelligence_. Schools tend to treat students as solo learners who do most of the real intellectual work of learning in their heads. But a revisionary view of intelligence recognizes that people inherently think with one another cooperatively and with the help of artifacts from paper and pencil to computers. This calls for a basic reorganization of what usually happens in classrooms.
6. _Motivation: The Cognitive Economy of Schooling_. Many schools are wastelands of undermotivated students and teachers. But to what extent do even innovative school settings give students and teachers good reason to invest themselves? This chapter points up how the gains and costs of classroom life—the "cognitive economy" of classrooms—often inadequately reward students and teachers for serious intellectual investment. It examines how current efforts toward school restructuring and alternative methods of assessment can help to build a cognitive economy supportive of thoughtful teaching and learning.
7. _Victory Gardens for Revitalized Education_. The previous five chapters highlight five key dimensions of educational change—instruction, content, the curriculum, classroom organization, and motivation. But what do texts and programs that score well on these dimensions look like? While examples have been given all the way along, this chapter puts in one place several case studies, viewing them from the perspectives of all five dimensions and toward building a clearer image of schooling minds.
8. _The Challenge of Wide-Scale Change_. While wonderful educational achievements can be seen on a small scale in many schools and school systems, wide-scale innovation remains a daunting challenge. A large part of the challenge rests in helping teachers to develop new knowledge and skills and helping educational institutions to change in fundamental ways that make room for thoughtful teaching and learning. Fortunately, sociologists and educators have learned much about the process of teacher and institutional change in recent years. This knowledge, put to work, promises wide-scale progress toward more effective education.
The nine chapters build a vision of the smart school, the school that, informed about teaching, learning, collaboration, and other keys to effective education, fosters an energetic culture of thoughtful teaching and learning. Taken together, the chapters underscore a central point: A culture of thoughtfulness is not a simple thing. It is not just a matter of attitude or style or skill. It is not just a matter of longer class periods for greater depth or more writing in all subject matters. Like any culture, a school culture of thoughtful teaching and learning is a complex construction, built only with commitment, insight, and knowledge. Because we understand better today what such a culture requires, we are in a better position to create smart schools.
### CONNECTIONS: SOME ISSUES SEEN ANEW
The idea of thoughtful learning does not stand apart from other contemporary themes in education. It overlaps and illuminates a number of the issues that figure in the lively discourse around education these days. For instance:
**Slow Learners.** Traditionally, schools have addressed slow learners with tracking and remedial programs that assume they need to focus almost exclusively on routine basics. In such classrooms, rote learning and drill-and-practice dominate even more than in ordinary classrooms.
It's a mistake. Thoughtful learning is the way learning works best. Thoughtful learning is just as important for slow learners as anyone else, honoring rather than demoralizing slow learners, motivating them more, and helping them to achieve more. Remember the energetic character a smart school needs. Let's face it: Slow learners are typically bored by what schools ask them to do. And no wonder! So thoughtful learning is for everyone, not just the gifted or the regular student.
**At-Risk Students.** "At risk" has become a broad and somewhat vague label for youngsters whose economic and family background forecasts poor performance in school and a high dropout rate. Many such students are slow learners, but not so much because of any lack of raw ability as of attitudes and skills ill-tuned to the academic expectations of school. The fact is that some economic and family backgrounds leave children much less prepared for school than do others.
Thoughtful learning is for at-risk students as much as it is for slow learners. At-risk students need the energy, involvement, and learning-to-learn that comes with thoughtful learning. While today's schools tend to widen cultural gaps rather than bridge them, it doesn't have to be like that. The smart school can create a safe, protected atmosphere and help to build the curiosity, confidence, and skills of at-risk students.
**Assessment.** It's widely recognized among today's educators that conventional multiple-choice, knowledge-oriented testing does not serve the cause of education well. Such testing drives teachers and students toward rote styles of instruction that may help with retention of knowledge but have little hope of building understanding or the active use of knowledge. The smart school requires the new concepts of assessment discussed in chapter 7.
**School Governance.** Traditionally, the principal leads the school much as the captain of a ship commands the crew. Contemporary lessons from the business and school communities alike suggest that a strongly hierarchical, nonparticipatory process of governance misses opportunities. Significant teacher, parent, and indeed student participation in school governance can boost motivation and involvement and harvest everyone's intelligence toward the good of the enterprise. This does not mean that principals should have no authority. Of course they should. It means that the smart school needs to foster a thoughtful involvement not just for students in their classrooms but for the adults committed to the school as well.
**School Choice.** The basic idea of school choice says that parents and students should be able to select the school to attend within a region. A marketplace metaphor figures here: Schools not doing a good job will fail to draw students and, if they can't get their acts together, go out of business. Other, more effective schools will take their place.
School choice is a complex issue, and an unrestricted market economy should be viewed with caution (remember how far from _laissez faire_ economics the real world of business has come). However, the idea of parents and students energetic about school choice, informed about the options, and choosing thoughtfully the kind of school that would serve them best certainly resonates with the idea of the smart school. Moreover, if school choice plans are to succeed, it's essential that parents and students have good choices. The notion of the smart school can help society to create such choices.
**School Restructuring.** Innovators concerned with school restructuring locate the malaise of education in organizational features of the school that drastically lower energy and make thoughtful learning difficult. Dull and ineffective patterns of education stay locked in place by short class periods, too many subject matters, conventional testing, command-style leadership, and so on. Thoughtful teaching and learning cannot take hold and thrive in such settings. Efforts to restructure schools typically emphasize fundamental changes in patterns of governance, class periods, curriculum, and testing in order to liberate and energize teachers and learners to get on better with the business of education. Most definitely then, some degree of restructuring is fundamental to the smart school.
**Preservice and In-Service Teacher Education.** To achieve substantially better education, society must invest seriously in renovated preservice education and expanded in-service education. Parents and school boards are notoriously grudging about in-service time: "The teachers should be teaching our kids!" But such attitudes fail to recognize the rapid pace of development of new ideas about teaching and fail to honor how much teachers can learn from both one another and outside sources. Teachers cannot be informed, energetic, and thoughtful in settings, preservice or in-service, that fail to inform them and shrink from providing them the time and encouragement to build energy and reflect deeply on educational practice. Schools need restructuring not just to foster students' thoughtful learning but teachers'—and administrators'—thoughtful learning as well. Chapters 7 and 9 look at some aspects of this challenge.
### MISSION: SMART SCHOOLS
No book can attempt everything. This is not a how-to-do-it book. Teachers will not find formulas for teaching here (nor would they want them). It is not a technical review of research. Those looking for research summaries will discover that many other sources do a far more detailed job of that. It is not a meticulous blueprint for school change. Parents, principals, and members of school boards will find many useful notions but not a stepwise plan. Nor does this book deal much with the special problems of particular populations—poverty, ethnic differences, drugs. Nor with the organizational dilemmas of parent participation, teacher empowerment, and so on.
Instead, this book is a wake-up call. Whatever can be done about the particular woes of particular populations and the overall organization of schools and schooling, education ultimately depends on what happens in classrooms around subject matters between teachers and learners. That is fundamental. We know a great deal about it today. We need to put to work what we know toward making informed, energetic, and thoughtful schools.
What will you discover here? Most of all, information and ideas that can help to inform and energize schools and foster thoughtful learning. These pages offer an overview of the new science of teaching and learning. Although it cannot be complete, it will, I hope, be provocative and empowering.
I hope that parents will take to heart the risks of a diffuse education that tries to serve all agendas and find a common ground in the key goals of retention, understanding, and the active use of knowledge. I hope that business people will recognize the harm done by a routine, rote education that yields uninformed and disillusioned graduates and lend their ingenuity to furthering informed, energetic, and thoughtful teaching and learning.
I hope that teachers will discover an optimism and direction to combat the energy-draining pressures and frustrations of most educational settings, finding affirmation of many of their insightful practices, as well as new ways of thinking about teaching and learning. I hope that school administrators will come upon useful justifications for innovation in the face of discouraged and wary communities.
I hope that citizens will awake to a new interest in the power of public education and lend their views, voices, and votes to creating smart schools. I hope that politicians will recognize that ineffective education weighs a society down, sapping its potential, and appreciate how crucial a change toward thoughtful learning can be for intellectual and economic vitality.
The time is right. Expounding on the theme of education as a social invention, Jerome Bruner, one of the founding fathers of cognitive psychology and an innovative educator, wrote, "For it is psychology more than any other discipline that has the tools for exploring the limits of man's perfectibility." Thanks to advances in cognitive psychology over the last decades, we have a better, albeit far from complete, understanding of human thinking and learning—its mechanisms, proclivities, and opportunities. Thanks to the vigorous work of scholars studying the school milieu, we have a better understanding of teacher and institutional change. Thanks to diverse advances and innovations in education around the world, we have a better opportunity to compare and draw conclusions.
But the use-of-knowledge gap remains a plain reality. If we can only get those fundamentals into focus and widely appreciated, we can create smart schools in every community. We can make schools even more ingenious inventions than they already are: wonders of the world indeed.
For the sake of flow, citations for ideas and sources mentioned in the text appear in the Notes organized by chapter and section at the end of the book. The full references appear in the References section that follows the Notes.
## CHAPTER 2
**THE ALARM BELLS**
Sometimes a memory catches us by surprise, in the midst of something else entirely, telling us that there are connections we have not sought out and perhaps do not even welcome. So it was when I sat down a few weeks ago to draft the first lines of the essay that unexpectedly turned into this book. I discovered myself thinking of a poem I had not read for many years, a poem that nearly every schoolchild encounters, one of the most doggedly onomatopoeic poems in the English language, Edgar Allan Poe's "The Bells."
So I found a copy of the poem to remind myself what it said. Here are a few of its lines:
Hear the loud alarum bells—
Brazen bells!
What a tale of terror, now their turbulency tells!
In the startled ear of night
How they scream out their affright!
Too much horrified to speak, They can only shriek, shriek,
Out of tune
In the process, I puzzled out what brought "The Bells" to mind. It was, of course, the troubles of education. They seem to be sounding from every direction—the woes of teachers, the unease of parents, the infighting of school boards, the restiveness of students, the discouraging findings of various investigative committees. Truly we hear resounding throughout the land Poe's "alarum bells" concerning the educational enterprise.
Poe's bells reminded me of another image of chaos. In _Popular Education and Its Discontents_ , Lawrence Cremin committed a chapter to what he terms "The Cacophony of Teaching." By this, Cremin alludes specifically to the many helter-skelter ways that we in the United States seek to educate—through the public schools, television, museums, preschool programs, special education, and so on, each with its own goals, philosophies of education, economic structures, and hidden curricula, and so on. _A mot juste_ if ever there was one, "cacophony" (although, Cremin emphasizes, not necessarily an unproductive cacophony) underscores the dilemma of making sense of education in a context of conflicts and crosscurrents.
With these images of turmoil so powerfully asserting themselves, there seems no better course than to listen to the bells, the cacophony, the assault of sound and fury, and try to discern the pattern of "alarum."
For a preview, we hear at least two broad shortfalls in educational achievement: _fragile knowledge_ , which means that students do not remember, understand, or use actively much of what they have supposedly learned; and _poor thinking_ , which means that students do not think very well with what they know. Searching for causes, we can discover at least two very pervasive contributing factors: a Trivial Pursuit theory of learning, which pervades educational practice and says that learning is a matter of accumulating facts and routines; and an Ability-Counts-Most theory of achievement, which says that what a person learns depends mostly on how smart the person is, not on how hard the person tries. Wondering about consequences, we can find at least one of great concern: a kind of economic erosion, where the rich get richer while the poor get poorer, and both economic productivity and the average standard of living fall behind those of many other nations. Research suggests that educational problems are a principal cause!
So let's look at the details.
### A SHORTFALL: FRAGILE KNOWLEDGE
It's at least irritating and to many dismaying that many youngsters do not know bits of information they _ought_ to know. For example, as mentioned earlier, a recent survey disclosed that some two thirds of seventeen-year-old schoolchildren in the United States cannot place the U.S. Civil War to within a half century. Eighty percent do not know what Reconstruction is. Two students in three think that Jim Crow laws actually helped black Americans. Half do not know that Stalin led the Soviet Union during World War II. Almost half do not know that the attack on Pearl Harbor occurred during the period between 1939 and 1943. Three in five do not know about the internment of Japanese Americans. A similar number misdefine the Holocaust. Thirty-six percent date Watergate before 1950, and one in five before 1900. Forty-five percent classify Israel as one of the nations occupied by the Soviet Union after World War II. One in three cannot locate France when given a map of Europe, Two in three cannot pick out Walt Whitman as the poet who wrote "Leaves of Grass."
Missing knowledge, we could call it. Missing from the minds of students who have been exposed to it and might have remembered it. Certainly it is reasonable to expect students to emerge from their education with a fund of basic knowledge that orients them to the world around them and equips them to understand its unfolding events and ideas—what is happening where and when and why.
At the same time, people too often see this missing knowledge as the principal shortfall of education. If only kids remembered the facts and skills they've been taught, everything would be fine!
Unfortunately, it's not that simple. Schooling minds is more than schooling memories. Missing knowledge is too crude a diagnosis of the malady. Research shows that there are many more problems of knowledge than just plain not having it. Three such are _inert knowledge, naive knowledge, and ritual knowledge_.
**Inert Knowledge.** Startlingly often, students have knowledge that they remember when directly quizzed, but do not use otherwise. It doesn't come to mind in more authentically open-ended situations of need, such as writing an essay, pondering the morning's headlines, considering alternative professions, selecting a new stereo, or for that matter, studying another subject. Knowledge of this sort is called inert. As the phrase suggests, inert knowledge is the knowledge equivalent of a couch potato: It's there, but it doesn't move around or do anything.
Conventional instruction—reading textbooks and listening to lectures—tends to produce inert knowledge. For example, cognitive psychologist John Bransford and his colleagues conducted an experiment in which some students read items of information about nutrition, water as a standard of density, solar-powered airplanes, and other matters in the usual textbookish way, with the intent to remember. Other students read the same items of information in the context of thinking about the challenges of a journey through a South American jungle. For instance, the students read about the density of water in the context of how much water the travelers would have to carry.
Later, both groups of students were given the task of planning a desert expedition. The students who had studied the information in the conventional way made hardly any use of it. But the students who had studied the same information in the problem-solving context of the jungle journey made rich and extensive use of the information, pondering the kinds of foods that would sustain people the best, worrying about the weight of water, and so on.
For another example, research that colleagues and I conducted on students' computer programming abilities disclosed an often startling gap between knowledge that high school students could remember and knowledge that they used actively. One student, for instance, was struggling with a problem that required a FOR-NEXT command to solve, one of the most fundamental commands in the BASIC programming language. The student didn't recognize what to do. Had the student forgotten about the FOR-NEXT command altogether? An investigator sitting with the student asked whether a FOR-NEXT would help. Oh yes! The student immediately and effectively used the command to solve the problem.
Notice what this shows. The student retained the knowledge in question and even knew how to use it effectively. But the student _did not think to use it!_ Unusual? Not at all. These students often knew and understood relevant programming commands that they did not think to employ in the midst of writing a program. When reminded of particular commands but not their details, many students slotted the commands into place and solved the programming problems.
The same appears to happen in all subject matters. Students retain knowledge they often cannot use actively for problem solving and other activities.
**Naive Knowledge.** One of the discomforting disclosures of the past two decades has been students' fragile grasp of many key concepts in science and mathematics. Students commonly display naive ideas about things even after considerable instruction.
For instance, youngsters in the first half of elementary school often believe the world is flat. This is reasonable to start with—after all, the world _looks_ flat as you gaze from a height out to the horizon. However, many youngsters, even after receiving some instruction with globes, still believe that the world is flat! Often, they have come to think it is flat in a fancier way, like a hemisphere: rounded on the bottom but flat on top, or like a disk, with a round periphery but flat on top and bottom.
"Well, they're young yet," we could say. "There's no hurry. Few students end up believing that the world is flat in the long run." All this is true as far as it goes. But the same thing happens at much more advanced levels, where students are unlikely ever to get it straight later.
As part of a project directed by astrophysicist Irwin Shapiro of Harvard University, Matthew Schneps and Phillip Sadler organized the making of a short film called _A Private Universe_ that has won some attention in educational circles recently. In the film, graduating students of Harvard University were asked a very basic question about the world around them: Why is it hot in the summer and cold in the winter? All the students had studied this at one time or another. Almost everyone does in high school. But many students revealed a fundamentally mistaken conception, suggesting that summers are hotter because the Earth is closer to the sun in the summer.
This is not the right explanation. This is not the explanation they supposedly learned. Moreover, it does not even make sense in terms of other facts people often know. Most of us can recall hearing that when it's summer in the Northern hemisphere, it's winter in the Southern hemisphere, and vice versa. Well, if the Earth is closer to the sun in the summer, it should be summer in both Northern and Southern hemispheres at once. The "closer to the sun" theory is not only mistaken but does not make sense in terms of other information.
Over the past two decades, researchers have looked for students' naive theories in science and mathematics at all levels of education—elementary, secondary, and college. And they have found such theories in abundance. The point is not that students have naive theories before instruction, but that they still have them _after_ instruction, often immediately after. To be sure, when students are asked to repeat facts or apply formulas, they are very often right. But when they are asked to explain or interpret, students often reveal the old naive theory intact.
While the persistence of naive knowledge has been most studied in mathematics and the sciences, it appears to have equivalents in the humanistic subject matters. In his recent book, _The Unschooled Mind_ , Howard Gardner has pointed out that stereotypes are in effect naive theories that students harbor. We would like to think that the teaching of history and literature do much to alter religious, racial, and ethnic stereotypes. Certainly, in many places today, the emphasis falls on multiculturalism, with religious, racial, and ethnic groups of all sorts represented in what students study. Nonetheless, like students' naive theories in science and mathematics, stereotypes seem to survive and even thrive.
How can this happen? How can students seem to learn something new and yet preserve their naive theories? The answer seems to lie in recognizing yet another kind of knowledge.
**Ritual Knowledge.** Rather than coming to a new full understanding of something like the world's roundness or how other people are very like ourselves, learners often seem to acquire what might be called ritual knowledge. They learn the school game. They learn how you are supposed to talk about the world—you use the word _round_. They learn the routines of problem solving with equations. They learn about important black or Hispanic figures in U.S. history. Unfortunately, these schoolish performances make little connection to their intuitions about the way things are. When asked to explain something or ponder a situation or express a view, they reveal the old naive theories, as much alive as ever.
Sometimes, these rituals can even be articulated point-blank by the learners. Here is a marvelous and rather famous example collected by researchers a number of years ago. A good performer in math had this to say about her strategy:
I know what to do by looking at the examples. If there are only two numbers I subtract. If there are lots of numbers I add. If there are just two numbers and one is smaller than the other it is a hard problem. I divide to see if it comes out even and if it doesn't I multiply.
It is not that these students by and large are resisting what they have been taught, displaying thoughtful or even thoughtless skepticism. It is rather that they have not really gotten straight what they have been taught. So they substitute rituals that, like the one above for solving arithmetic problems, work rather well in the artificial world of the typical classroom.
Meanwhile, their naive theories survive unaltered or only partly altered. So, just as "Sunday Christians" do not connect up their daily moral lives with what happens in the church on Sunday, school learners do not connect up the phenomena around them with what gets said from the pulpit at the front of the classroom.
The Fragile Knowledge Syndrome
The summary lesson here is that the problem of knowledge is much more than a problem of missing knowledge, although that is part of it. To put a name to the overall malady, one might speak of fragile knowledge. Students' knowledge is generally quite fragile in several different and significant ways.
_Missing knowledge._ Sometimes important pieces of knowledge are just plain missing.
_Inert knowledge._ Sometimes present, but inert. So it lets the student pass the quiz but does not help otherwise.
_Naive knowledge._ Sometimes the knowledge takes the form of naive theories and stereotypes, even after considerable instruction designed to provide better theories and combat stereo-types.
_Ritual knowledge._ The knowledge that students acquire often has a ritual character, useful for schoolish tasks but not much else.
Notice how these four problems with knowledge run contrary to the goals of education underscored in the introduction—retention, understanding, and the active use of knowledge. Just plain missing knowledge is, of course, not retained. Naive and ritual knowledge do not constitute true understandings. And inert knowledge does not see active use, even though there for the test.
The problems of missing, inert, naive, and ritual knowledge combine in a learner to display a distinctive cluster of behaviors, the fragile knowledge syndrome. Imagine what this is like. Suppose, for example, that you are looking over the shoulder of Brian, who is tackling some fraction computation problems. For the simplest problems, Brian proceeds nicely. Encountering a mixed number, Brian has no idea what to do with it—a knowledge gap. On another problem, Brian obtains an answer that needs reducing but forgets to reduce it, even though he knows how. On an addition problem, Brian cancels a 3 in the numerator of one term against a 3 in the denominator of the other, mistakenly believing that cancellation works for sums as well as products. However, on a similar problem, he does not happen to try canceling and solves the problem correctly.
In short, you see an odd mix of competence and shortfall. Certainly, Brian and other students know a good deal about what they are doing. Yet the entire performance is fragile, troubled with knowledge gaps, inert knowledge, naive knowledge, and ritual knowledge. In consequence, performance is sometimes correct and sometimes incorrect on very similar problems, and problems with any peculiarity tend to throw the student off.
The fragile knowledge syndrome is not something to worry about only in elementary school because high school and college students manage fine, nor is it a matter of concern only for science and mathematics because the humanities fare better, nor a matter of more versus less "practical" subject studies. Dorothy, in the midst of her college-level course on American poetry of the twentieth century, may forget who wrote "Ars Poetica" (missing knowledge). She may not think to mention T. S. Eliot's notion of the objective correlative for an essay question the professor designed to elicit the idea (inert knowledge). She may persist in the belief that "good is what I like" despite the professor's efforts to build finer discrimination (naive knowledge, a tacit theory of aesthetics). Despite all this, Dorothy may score some points with the professor by dutifully defining and advocating structuralist literary criticism (ritual knowledge).
Not only is the fragile knowledge syndrome all too real, and all too present, but for weaker students it is all too painful. Many lower-ability students continue week after week, month after month, dealing with knowledge that is very fragile for them, full of gaps, confusions, and so on.
Even good students have had some experiences like that. We all have considerable fragile knowledge—missing, inert, naive, ritual. Remember a time when you were studying something that came hard and others were doing better? Remember how confused and disoriented you felt, how hard it was to keep wading along? Now imagine a learner for whom most subject matters are like that on most days. No wonder students get discouraged and drop out. Fragile knowledge hurts!
### A SHORTFALL: POOR THINKING
Gary Larson, the notable and notorious cartoonist, displays admirable sensitivity to one of the most fundamental fears of students. In a telling cartoon by Larson entitled "Hell's Library," we see the licking flames of the inferno surrounding a tall bookcase. And what books! The titles read like this: _StoryProblems, More Story Problems, Even More Story Problems, Big Book of Story Problems_, and so on.
Students (including ourselves at one time or another) are right to be fearful, because investigations show that students have enormous difficulty with story problems in mathematics, far more so than with the numerical operations they have practiced so assiduously. They know more or less how to add, subtract, multiply, and even divide. More advanced students know the manipulative rules of algebra and even the calculus. But often they cannot puzzle out just what the problem is asking for. "Should I add, subtract, multiply, or divide? Should I set up simultaneous questions? Should I integrate by parts?" They know how to do any of these things. But they're not sure what to pick. So they resort to _ad hoc_ strategies. They cannot think with what they know.
Thinking with what you learn is of course one of the goals of education. In fact, it's part of the third major goal of education expounded in the introduction: the active use of knowledge. Now and again, there are opportunities to use knowledge actively without a lot of thought, as when you check a bill in a restaurant to be sure it's totaled correctly. By and large, however, using knowledge actively means thinking with it—to solve problems, make inferences, generate plans, and so on. So what signs do we see that students are learning to do this? Few enough signs in the area of solving story problems in mathematics!
What about other kinds of thinking? One common, thought-demanding task arises in reading, when students are asked to read, interpret, and explain. Many reading tests ask youngsters to make elementary inferences from what they read. If, for example, Senator Fitzmorrison supported the antipornography bill but worked against the gun-control bill, what generalization could you tentatively make about his political leanings?
Unfortunately, students prove to be remarkably poor at reading between the lines and drawing appropriate generalizations and extrapolations from what they read. The National Assessment of Educational Progress offers a discouraging quote about students' standing here:
Students seem satisfied with their initial interpretations of what they have read and seem genuinely puzzled at requests to explain or defend their points of view. As a result, responses to assessment items requiring explanations of criteria, analysis of text or defense of a judgment or point of view were in general disappointing. Few students could provide more than superficial responses to such tasks, and even the "better" responses showed little evidence of well-developed problem-solving or critical-thinking skills.
To turn to another cognitively demanding activity, students are no more astute about what they write. According to research conducted by cognitive psychologists Carl Bereiter and Marlene Scardamalia at the Ontario Institute for Studies in Education, most students write using a tacit "knowledge-telling strategy." If spelled out in so many words, this strategy basically says, "Write down something you know about the topic. Then write down something else you know. Then write down something else. And when you have enough, write down something that sounds like an ending and hand it in."
The original work of Bereiter and Scardamalia concerned precollege students. However, many college professors, hearing a profile of the knowledge-telling strategy, react with a shock of recognition: "Yes—that's what many of my students' papers are like!"
With the knowledge-telling strategy as their mainstay, students generally do not organize their knowledge into thoughtful theses and arguments. Moreover, they are not even terribly good at tapping the knowledge they have—inert knowledge again! Bereiter and Scardamalia report an experiment where before writing, students were asked simply to think of key words that they might use in an essay. The students who did this simple exercise had considerably more to say in their essays than those who didn't. Apparently, students do not necessarily know how to stir the mental pot. So they have less to say than they might, even in the straight knowledge-telling mode.
Even plain old rote memory points up many students' sluggish thinking. As mentioned in the introduction, extensive research shows that if your goal is simply knowledge retention, your best approach is strategic and thought demanding. Students learn facts much better if they organize them, actively relate them to prior knowledge, use visual associations, quiz themselves, elaborate and extrapolate what they are reading or hearing. Many students, unfortunately, opt for a straight rehearsal model of memorizing—read it over and over, say it over and over. While repetition is some aid to memory, it helps much less than strategies that involve more elaborate processing of the information in question.
But perhaps all this sorry thinking is the fallout of a general disaffinity with schooling. Youngsters might think a lot better about something closer to their minds and hearts. Maybe. Rexford Brown, who recently wrote about a number of efforts to foster more thoughtful education in Schools of Thought, is skeptical. He writes of watching a teacher leading a discussion around a music video by Paul Simon, "Boy in the Bubble." The teacher tends to lapse into a didactic style. But the students too seem off the wavelength. Brown reflects as follows:
I realize that I have just been assuming that students would love to talk about a rock video, because they watch them so often; but what this episode tells me is that teenagers are as inexperienced in looking critically at videos as they are in looking critically at texts. They don't know how to look critically at anything. Distancing themselves from an event or an experience, analyzing its parts and their relationships, and elaborating its various meanings for themselves and others—these are not things that many teenagers do naturally, even with events or experiences that mean a great deal to them.
The bottom line? Thinking is in trouble too, not just knowledge. At a recent conference, codirector Lauren Resnick of the University of Pittsburgh Learning Research and Development Center emphasized that so-called "higher order thinking isn't higher order." Higher-order thinking refers to reasoning, argument, problem solving, and so on. Resnick urged that thinking should not be seen as an esoteric add-on to good solid knowledge and routine skills. On the contrary, the most basic and seemingly elementary performances require active strategic thinking. If students do not learn to think with the knowledge they are stockpiling, they might as well not have it.
### A DEEP CAUSE: THE TRIVIAL PURSUIT THEORY
It is a neat intellectual move to look for subtle unities in the fabric of a civilization and an era. The Renaissance, for example, held deep currents that came to the surface in wellsprings of art, science, politics, commerce, and everyday dress.
In that spirit, let me suggest that "trivial pursuit" may be one of the submarine torrents of the American character. First and foremost, it refers to the popular game wherein each player advances by displaying his or her breadth of knowledge in various categories. But underneath the fun, I wonder whether the soaring enthusiasm for this game signals a naive love for the idea of wisdom as knowledge and knowledge as facts and routines. We are speaking here of generations that, now enthusiastic about "Trivial Pursuit," were breast-fed on the golden age of quiz shows, with such extravaganzas as "The $64,000 Question."
Deep currents or not, trivial pursuit makes a metaphor for many features of contemporary education. To this point, I have emphasized shortfalls in educational outcomes—achievements of knowledge and thinking. But what are the causes? The answer is inevitably complex, and the story will continue to unfold throughout this book. But here it is worth highlighting two broad attitudes toward teaching and learning that contribute to the malaise on many levels and in many ways. In fact, remembering the earlier discussion of naive theories, both are naive theories. Naive theories held not just by many students but by many educators as well. Here is the first:
_Learning is a matter of accumulating a large repertoire of facts and routines._
Notice how this contrasts with the fundamental principle stressed in the introduction, learning is a consequence of thinking. That principle votes for a much more active, thoughtful kind of learning than accumulating facts and routines.
What does naive theory #1 say in slightly more expanded form? I can do no better than to quote a few fine lines from the beginning of my colleague Vito Perrone's recent Letter to Teachers. He writes:
There is, it seems, more concern about whether children learn the mechanics of reading and writing than grow to love reading and writing; learn about democratic practice rather than have practice in democracy; hear about knowledge... rather than gain experience in personally constructing knowledge;... see the world narrowly, simple and ordered, rather than broad, complex, and uncertain.
But wait a minute! Who really believes this naive theory about accumulating facts and routines? Do I know anyone who espouses it? Do you? No, probably not. But many educators and other people behave as though they believe it. That's what we mean by a tacit theory. Educators do not argue that education is about accumulating large repertoires of facts and routines. But this is overwhelmingly what happens in classrooms, where, as in other settings, actions speak louder than words.
One measure of the trivial pursuit model of education simply asks how often classroom events depart from it. For example, John Goodlad reports in _A Place Called School_ that only 5 percent of class time is spent on average in discussion. Ernest Boyer mentions in _High School_ an investigation revealing that fewer than 1 percent of teachers' questions to students invite them to respond in a richer way than answering a factual question or displaying a routine procedure.
Similar points can be made about textbooks. The educator-psychologist David Olson and his colleague Janet Astington at the Ontario Institute for Studies in Education have systematically surveyed junior-high science textbooks for their use of what Olson calls "mental state verbs," which make reference to such important elements of thinking as "hypothesize," "explain," and so on. They found that such references occur rarely, demonstrating a systematic squeezing out of this "language of thinking" from textbooks.
The tests that in many ways drive the educational system are another testimony to the deep reality of the trivial pursuit model. By and large, those tests press for fact upon fact, procedure upon procedure, emphasizing multiple-choice responding rather than thoughtful performance on complex, open-ended tasks.
The emphasis on coverage, all too familiar to those who work in schools, also points to the problem. The common teacher concern when trying out an innovation is "But I have to be sure to cover my text." A school administrator I know dramatized how entrenched the coverage philosophy was by relating how teachers she supervised would tell her, "But I have to cover my material." "Who told you that?" she would ask. They could not answer. Indeed, the school system in question explicitly did not make coverage a priority. But the coverage mentality is so much a part of the current culture of teaching that it was assumed. Now coverage is not pointless, of course. On the positive side, teachers who react this way are displaying a laudable dedication to acquainting their students with as much knowledge as possible. However, on the negative side, the result is a constant trading of depth for breadth.
Moreover, the conspiracy for coverage extends well beyond the classroom into the textbook industry. Over the past two decades, science texts have grown fat with superficial and disconnected information about every facet of science imaginable, with virtually no prospect of students' really understanding or retaining much of so vast a compendium. The same thing tends to occur in many other subject matters as special interest groups and scholars and others push for this or that point to be added. Everything, it seems, is important. And again, no one individual or group in this pandemonium of knowledge pushing is acting foolishly. Many, many ideas and perspectives are important in many contexts, and choosing is an intellectually difficult and even politically risky task. But the alternative is the familiar political compromise: By serving all agendas, we serve none of them well. Worse, we serve most of them downright poorly.
But the trivial pursuit model lives on. The case for this trivial pursuit has even had its popular champions, most notably E.D. Hirsch. In his wide-selling _Cultural Literacy_ , Hirsch argued that American education should work assiduously to ensure that students gain a broad base of superficial acquaintance with a large number of concepts from different disciplines. Hirsch went so far as to offer a list of concepts people should be acquainted with, including, for example, Atomic Weight, Cleopatra, Pearl Harbor, Relativity, and The Three Little Pigs.
Hirsch's position should certainly not be seen as a naive trivial pursuit model. But neither, in my view, should it be seen as particularly enlightened. I agree with Hirsch that broad superficial acquaintance with a large number of ideas is one important outcome of a good general education. However, it is an outcome hard to strive for directly. Here's why.
The vision projected by Hirsch, although carefully hedged, is that schools can somehow "go through" the fund of needed superficial knowledge and kids will end up culturally literate. Well, it won't work. Remember the core precept that learning is a consequence of thinking. Without a thoughtful process of learning, "going through" will not even yield well-retained knowledge, much less understood and actively used knowledge.
We can have what Hirsch wants, but not by teaching youngsters point-blank what's on Hirsch's list. To be retained, understood, and used actively, that knowledge needs to be accumulated over many years as a consequence of thinking: the good learning that occurs when students engage school content thoughtfully.
### A DEEP CAUSE: THE ABILITY-COUNTS-MOST THEORY
While the trivial pursuit model of education underlies much of educational malpractice, there is another major cause, nicely pointed up by comparing American and Japanese attitudes toward mathematics education. Japanese achievements in this subject matter as well as others have aroused admiration, envy, and not a little puzzlement as investigators have sought to understand the key ingredients.
One of those ingredients has to do with who or what gets the credit for learning and the blame for failure to learn. Ask a Japanese parent why a child isn't learning mathematics successfully, and you get a very clear answer: The child isn't trying hard enough. Ask an American parent, and as a rule, you get a very different reply: Math is a tough subject, or sometimes: The child isn't up to math.
What is true of mathematics is true of other subjects as well. It marks a contrast not only between American and Japanese attitudes but between American attitudes and those of several other nations with more successful educational systems. Predominant in American culture is an "abilities" theory of success and failure. In fact, it is another one of those naive theories. Let's write it down that way:
_Success in learning depends on ability much more than effort._
If you learn something, it's because of your inherent innate ability to catch on to that something, given modest exposure. If you don't, it's because you lack the ability. For you, the subject is too tough.
In contrast, Japanese and certain other cultures cultivate an "effort" model of success and failure. Assiduous extended effort grabs you the gold ring of learning, and ability gaps are surmounted by increasing effort. Although "The Little Engine That Could" is an American institution among children's stories, it is as though Japanese children have read it more often and taken it to heart more deeply.
Naive theory #2, harbored by many parents, has just as much of a home in the minds of teachers and school administrators. Rexford Brown lists the idea that "most students do not have the intelligence required by a literacy of thoughtfulness" as one of the six most common reservations voiced by teachers and administrators asked about more thinking-centered approaches to teaching.
Of course, who says naive theory #2 is really naive? Maybe the parents and teachers and administrators are right. Maybe it's hard fact. Is it the case that effort is substitutable for ability to a considerable extent? Can students master ideas initially beyond them through protracted, motivated, and appropriately guided effort?
The news from research is good here. Although there are bound to be some limits in trading off ability against effort, many findings from laboratory and classroom research testify to the leverage of effort. In a way, the point is a simple one. Some people take longer to learn certain things. However, if we organize education so that those who need more time have the opportunity and motivation to put in more time, they achieve much more.
So naive theory #2 really is naive. But does it actually do any harm? Well, it seems to in some youngsters. Evidence on this point comes from research conducted by University of Illinois psychologist Carol Dweck and her colleagues. Probing students' tacit theories of learning, they have classified learners along a continuum ranging from "entity learners" to "incremental learners." The latter are more aggressive learners. They believe, akin to the Japanese model, that learning comes by increments; you have to hang in there and persist, winning your way to an understanding. In contrast, entity learners harbor the philosophy that learning something new is taking in an entire entity all at once. You either get it or you don't. Learning is a matter of "catching on" fairly quickly. If you don't catch on, the concept is beyond you for the time being, so why try? Interestingly, sometimes relatively bright learners, in an IQ sense, are also entity learners; they may lack stamina and strategies for dealing with situations when the learning gets tough.
If naive theory #2 does mischief in the minds of some youngsters, what about in the minds of teachers? A startling demonstration here is the classic "Rosenthal effect." In the mid-1960s, researcher Robert Rosenthal conducted a simple experiment in San Francisco. He informed some teachers that specific students showed higher IQ scores than others. In fact, the students identified as high-IQ were chosen at random by Rosenthal. At the end of the year, Rosenthal compared the performance of the students said to be gifted with those not. The supposedly gifted students actually had performed better—and not just by the measure of the teacher's subjective grades but on objective tests.
What had happened? Perhaps the teachers' manner toward the supposedly gifted students had built up their self-confidence. Perhaps they had helped these students in subtle ways. Whatever the case, the teachers' belief in ability translated into better learning for those students. But they were not more able. Rather, more was expected of them!
In summary, one might say that American schools are a virtual empire of ability. The teaching is there to feed those of greater ability all they can take and to herd along the rest. Ability, not effort, is seen as the primary determining factor in how much John or Jane can learn. Tracking sorts students into appropriate ability channels, moving each along at the pace dictated by intrinsic gifts and limitations. Students buy in or drop out at various stages in part according to their estimates of whether they have the ability for the learning to be done.
Of course, any teacher knows that motivation is important and that trying helps. It would be foolish to suggest that effort is not seen as a causal factor in learning. Nonetheless, the tacit American model is ability centered. Ability has priority as a causal influence. While effort helps, it cannot really overcome ability gaps.
In other cultures as well as in laboratory research, this premise has been challenged and has been proven false to a considerable degree. Effort can be the primary explanation for successes and shortfalls of learning, with ability playing second string to explain differences left over after effort has been taken into account. We need an effort-centered model.
### A CONSEQUENCE: ECONOMIC EROSION
A chilling diagnosis of the ills of American education came my way recently via that very traditional conduit of information, the lecture. Marc Tucker is director of the National Center on Education and the Economy. At a recent conference, Tucker reported a systematic series of comparisons done between the educational practices of the United States and several other nations. He relayed a number of findings about the interlocked enterprises of education and economic productivity. The findings suggest why American schools are in trouble and where American society might end up unless something is done.
Economic prosperity and productivity are key elements in the picture. In the United States, the average standard of living has been declining slightly for a number of years. While the top 30 percent of wage earners have gained, the bottom 70 percent have lost even more. In other words, the culture is becoming economically polarized.
Several nations in the world have a higher standard of living than the United States, including Japan, Switzerland, Singapore, Denmark, and West Germany. They have high wages, very little unemployment, and high productivity. An especially interesting comparison looks at the relationship between "direct workers," those that actually assemble the products or provide the services, and "indirect workers," the administrators and support staff. In these countries, the ratio of indirect workers to direct workers is substantially lower. That is, there are fewer indirect workers per direct worker. Somehow, the direct workers are getting more of the work done by themselves!
A look at the organization of work in these countries reveals how this can happen. The direct workers typically do not operate in assembly line fashion. They function in teams and do varied tasks. They assemble. They troubleshoot. They refine. They test. In other words, within their own circle, they take care of many of the problems that otherwise would get spread out inefficiently through a complex hierarchy of administration and specialization. Their wages are higher than direct workers in the United States because they do more of the work, including the more thought-demanding sides of the work.
So how come direct workers are up to these challenges? Education! They are very well educated, both in the sense of basic schooling and the sense of technical preparation for their particular roles.
And how did they get so well educated? Marc Tucker's analysis identifies a number of characteristics of the educational process in these nations that ensure a generally and technically well-educated direct-labor force. Generalizing the pattern over nations, these are some key elements:
**Examination Systems Independent of the Teacher.** There are examination systems that gauge student achievement. Receiving educational credentialing requires passing these examinations. The teacher does not compose and administer the exams but rather is cast in the role of working with students to prepare them for the exams. The examinations are less fact-and-routine-procedure based, more open-ended and thought demanding. They may be unconventional in character; for example, involving project work or portfolios. There are general examinations and examinations related to particular professions.
**Credentials Required for Employment.** Employment is virtually unavailable without appropriate credentialing through the examination systems.
**Safety Nets for Dropouts.** Inevitably, people vary in ability. Many do not pass on the first attempt at an examination. But the effort-centered model rules instead of the ability-centered model. You can try as often as you want. Since some people may not find conventional instruction suitable to their learning styles or other predilections, certain of these nations provide a myriad of alternative forms of education. Agencies provide intensive counseling and keep potential dropouts in the educational loop. The whole philosophy is effort based. "Keep in there. Keep trying in different ways. We'll counsel you and guide you. And you'll get your credentials."
**A Labor Market.** One could imagine that well-credentialed individuals might have trouble actually locating a job. The system would break down at that point. However, these countries also have some form of labor market involving computer systems that keep track of job profiles and nationwide needs. This makes it much easier to match up a potential employee with a potential employer.
The circumstances in the United States contrast in virtually every respect with this picture. There is no nationwide, nor in most states even a statewide, achievement exam for those aiming to be direct workers. Exams are by and large composed and administered by teachers with pass/fail decisions made by the teachers, who are therefore placed in a conflict of interest. Typically, educational credentials are not required for direct workers. Contrary to popular belief, even a high school diploma, meaningless as it often is, has been shown not to be a determining feature in whether people get jobs. Despite our concern with at-risk students, there is no very effective safety net that rescues them and keeps them in the educational loop in some fashion. And there is no labor market.
The bottom line, according to Tucker, is that nations with a more effective system are steadily gaining in productivity and outperforming the U.S. labor force. American productivity is stagnant; it has not increased appreciably in many years. As poorer nations with a more effort-based philosophy catch on to the powerful configuration exhibited by these economic leaders, the United States is likely to find more and more nations above it on the productivity ladder. A trivial pursuit model of knowledge and an ability-centered rather than effort-centered concept of the causes of successful learning are more than mistakes: They exact a deadly cost in declining prosperity.
### DEFINING THE PROBLEM
This entire chapter has been an exercise in one of the most fundamental steps of problem solving—defining the problem. So let's review. When we listen for shortfalls in educational outcomes, the alarm bells of American education sound the woes of fragile knowledge and poor thinking. Fragile knowledge goes far beyond the problem of missing knowledge—the dates-and-places ignorance which so troubles so many people. It includes not only missing knowledge but inert knowledge that does not function actively in thinking, naive knowledge representing entrenched misconceptions, and ritual knowledge reflecting superficial, schoolish performances without authentic understanding. As to thinking with the content that they learn, students generally perform rather poorly in solving word problems, making inferences, explaining concepts, constructing arguments, and writing essays.
When we listen to underlying philosophy and method, we find some broad contributing causes. The alarm bells sound out concerning two naive theories. Naive theory #1 is the tacit trivial pursuit theory of learning that drives the educational enterprise. Getting an education is by and large a matter of accumulating a large bank of fairly specific knowledge and routine skills from which you can make withdrawals to deal with particular situations. Naive theory #2 says that abilities count most in learning, much more than effort. You either get it or you don't.
As to consequences, these educational problems yield a pattern of economic stagnation and decline in the work force, as American industry proves unable to compete efficiently with other nations that have their acts together better.
With all this, it must be acknowledged that in many ways America in its vastness and eclecticism poses particular problems: racial and ethnic diversity, the malaise of the inner city, the (in many ways laudable) lack of the strong centralized educational policies characteristic of many nations. Many articles and books deal exclusively and pointedly with such particular problems.
* * *
KEY IDEAS TOWARD THE SMART SCHOOL
**SHORTFALLS: THE ALARM BELLS**
**Fragile Knowledge.** Missing, inert, naive, and ritual knowledge. The fragile knowledge syndrome.
**Poor Thinking.** Poor handling of story problems in mathematics. Poor inferences from reading. The knowledge-telling strategy when writing. Repetition rather than elaborative memory strategies.
**The Trivial Pursuit Model.** Overwhelming emphasis on factual questions. Little "language of thinking" in the classroom. Short-answer, right-or-wrong test questions. Emphasis on coverage.
**Ability Centered Rather than Effort Centered.** Low achievement blamed on ability, not effort. "Entity learners" rather than "incremental learners."
**Economic Decline.** Need examination systems independent of the teacher, credentials required for employment, safety nets for dropouts, a labor market.
* * *
But the agenda of this book is more general. The remaining chapters address fundamentals of teaching and learning, fundamentals for anyone and everyone. Yes, the United States and many other nations have special problems. But the reality of these special problems does not mean that we can ignore facts and fundamentals—the facts of inadequate achievement in knowledge, thinking, and prosperity and the fundamentally misguided naive theories that tacitly underlie much of current practice.
The straightforward, hard-to-argue-with goals mentioned earlier—retention, understanding, and the active use of knowledge—are being addressed by some nations in systematic and effective ways. They are demanding goals, and to achieve them we need informed, energetic, and thoughtful schools—smart schools. Which, by and large, we do not have. Not here. Not in results. Nor in method. In Poe's words, it's worth paying heed to the "moaning and the groaning of the bells."
## CHAPTER 3
**TEACHING AND
LEARNING
Theory One and Beyond**
We could call it the "savior syndrome." Like other folk in other circumstances, educators seem all the time to be looking for a savior. The savior _du jour_ shifts around quite a bit. Once it was behaviorism, then discovery learning. More recently, " time on task" has been a popular idea: If only we could keep schoolchildren at it in a serious, engaged way for long enough, the kinds of learning that we want would surely occur. A current favorite is cooperative learning, where students work in small collaborative groups to master skills and ideas.
Yes, the "savior syndrome" label smacks of cynicism. It reveals a kind of impatience with the hunger for the quick fix, the _deus ex machina_ that will put things right in the classroom. Education is a complex undertaking. The hope and eagerness with which the savior of the moment is greeted, assaulted, and all too often reduced to a trivialized version becomes tiresome.
But certainly no cynicism should apply to any of the candidate saviors, properly viewed. For example, the group-learning techniques that come with cooperative learning are marvelous things with immense promise for improving educational practice. Although not fashionable today, behaviorism has its place in giving us a clear way to think about motivation in the classroom in terms of how students get rewarded for various behaviors—some desirable and some not so desirable.
So the problem with the syndrome does not lie in the candidates. Rather, the savior syndrome is symptomatic of one of the most misleading premises of educational reform: What we need is a new and better method. If only we had improved ways of inculcating knowledge or inducing youngsters to learn, we would attain the precise arithmetic, the artful writing, the astute reading, and all the other outcomes that we cherish.
I don't think so. There are three reasons why a new and better method is a red herring. Here they are in brief (the rest of the chapter is devoted to elaborating them). 1. We have plenty of sophisticated instructional methods but do not use them, or not very well. 2. Most instruction does not even meet minimal criteria for sound methods, never mind sophisticated ones. Our first urgency is putting into practice reasonably sound methods. 3. Given reasonably sound methods, the most powerful choice we can make concerns not method but curriculum—not how we teach but what we choose to try to teach.
Therefore, educational reform toward smart schools should be driven not by method but by curriculum—riot by more sophisticated visions of how to teach, valuable though they are, but by a broader, more ambitious vision of what we want to teach.
I'll build this argument by offering a conception of sound method, emphasizing how little we see of it, underscoring how much it can accomplish, and exploring briefly some more sophisticated methods. Then I'll return to the key point: Given reasonable method, our most powerful choice becomes what we try to teach.
### INTRODUCING THEORY ONE
A rather good theory of teaching and learning can be stated in a single sentence. The theory is not terribly sophisticated. It does not require elaborate laboratory research to test and justify. But pursuing its implications can take us a long way toward a much improved vision of classroom practice. So simple is this theory, so much a rough-hewn, first-order approximation to the conditions that foster learning, that we will call it Theory One, saving higher numbers for fancier theories.
Theory One says this:
_People learn much of what they have a reasonable opportunity and motivation to learn_.
How could so outrageously bland a statement about teaching possibly imply anything about better classroom practice? Admittedly, Theory One seems entirely too mousey for the job. But this is the Mouse That Roared. To see its power, we need to elaborate somewhat on the implications of the one-sentence version of Theory One. What is "reasonable opportunity and motivation to learn"? Without resorting to any technical knowledge about learning, one might commonsensically put down the following conditions:
_Clear information_. Descriptions and examples of the goals, knowledge needed, and the performances expected.
_Thoughtful practice_. Opportunity for learners to engage actively and reflectively whatever is to be learned—adding numbers, solving word problems, writing essays.
_Informative feedback_. Clear, thorough counsel to learners about their performance, helping them to proceed more effectively.
_Strong intrinsic or extrinsic motivation_. Activities that are amply rewarded, either because they are very interesting and engaging in themselves or because they feed into other achievements that concern the learner.
So there it is, Theory One, a commonsense conception of good teaching practice. Theory One aims simply to establish a baseline. For any performance we want to teach, if we supply clear information about the performance by way of examples and descriptions, offer learners time to practice the performance and think about how they are handling it, provide informative feedback, and work from a platform of strong intrinsic and extrinsic motivation, we are likely to have considerable success with the teaching.
### THE DEVASTATING CRITIQUE LEVIED BY THEORY
ONE
Theory One was a Mouse That Roared, I promised. Here's the roar. Simple though it is, Theory One yields a devastating critique of much educational practice. Consider, for example, explanation. If there is one thing that teachers need to do day in, day out, it is explain new ideas and reexplain old ones. In terms of Theory One, good explanation is first and foremost a matter of clear information. Then how well handled is the enterprise of explanation?
Some insight comes from research on "direct explanation" conducted by educational psychologists Laura Roehler, Gerald Duffy, and their colleagues. These investigators sought to characterize what good explanation demanded. Their analysis emphasized such features as the presentation of conceptually accurate, explicit, meaningful, and sequenced information. Good direct instruction includes information about not only the "what" but also the "how" and the "when" of the matter at hand; for example, not only what a particular technique is but how and when to use it. Good direct instruction involves monitoring students' evolving understandings and their points of confusion and uncertainty in order to clarify them. For example, in teaching the reading strategy of asking yourself questions to read for, a sample of direct explanation might sound something like this:
_Okay, folks, we've talked about how to ask yourself questions. But when? When is it worthwhile? Certainly when you're studying for information and understanding. Because you'll remember and understand better. For instance, when you read a section in your science book, look at the title. Look ahead at the subheadings. Ask yourself, "What do Iwant to understand about this?" Make a short list of questions in your mind_.
_Now let's check if I've been clear enough. Roger, what's another time you think it would be worthwhile asking yourself questions before you read?_
_(Roger says: while reading a short story.)_
_Aha. I see some puzzled faces. Some disagreement maybe. Let's look at the pros and cons of that "when."_
Explanation seems such a mainstay of teaching that one might think that good explanation would be routinely found in classroom practice. However, Roehler and Duffy found nothing of the sort. Their studies disclosed that teachers varied widely in the quality of the explanations offered. Some were very good, but some offered vague explanations. "Whats," "hows," and "whens" were sketchy, teachers did not probe their students to cross-check the students' evolving understandings, and so on.
While this and other research speaks to a less-than-Theory One level of practice in many classrooms, we need not rely on research to construct a critique. Common knowledge about typical practices will suffice. For example, history is usually taught by asking students to learn its "story" as it unfolds at particular times in particular places, for example, the French or the Industrial Revolution. Some typical aims of history instruction include: (1) cultivating students' understanding of important historical events, not just what happened but _why_ it happened; (2) preparing students to understand current events in light of historical precedent and contrast; (3) providing students with an historical background so that they can understand allusions, the context of works of literature, and so on.
By the measure of Theory One—never mind loftier instructional theories—typical history instruction does an extraordinarily poor job of working toward these objectives. Take the first goal, cultivating understanding of why things happen as they do. A classic question students get asked here is "What were the causes of the U.S. Civil War?" And students can answer, at least when the text is fresh in their minds, because contemporary textbooks spell out the causes of the Civil War.
The catch? It's right there in Theory One: students need thoughtful practice of their understanding, but asking students to recite causes from the textbook practices just their memory, not their understanding.
What could be done to engage learners in practicing their understandings instead? For a modest improvement, the teacher might ask, "The text has talked about three contributing causes. Which do you think is most important and why?" This demands that the students reason about what they have learned. For a bigger boost, the teacher might say, "A British textbook lists these three causes of the U.S. Civil War. They sound different from what we have been reading. What do you think are the strong points and weak points of this analysis from a British textbook?" For a more aggressive approach yet, the teacher might say, "Let's set up a debate for the entire class period tomorrow. One group will defend the British textbook's interpretation, another the U.S. textbook's interpretation."
These examples are not put forth as the ideal or only way to do a better job on understanding history. There are many innovative approaches to teaching history. But they do show some simple ways of getting Theory One's thoughtful practice into the picture.
Now let's look at the second goal, enabling students to see current events from an historical perspective. Again, the typical shortfall is lack of thoughtful practice. Normal history instruction does not engage students in making connections to contemporary events. History teachers do not often ask students to think about questions like these: "In the news over the last few days, we saw an attempted coup in Russia. How was this coup like and not like a civil war? Could a coup turn into a civil war? Were there any contributing causes anything like those of the U.S. Civil War?"
As to the third goal, the background function of historical knowledge in keeping with E. D. Hirsch's notion of cultural literacy, conventional history instruction provides information but again no practice in treating historical knowledge as a backdrop. Students do not usually read novels or poems in history class and are not primed to see such works of literature against their historic backdrops. History teachers do not usually model for students what it would mean to read a story or poem alert to this backdrop—what ideas would come to mind, how they would color the literary expression, what new meanings they would suggest. All this, indeed, is seen as the work of the English teacher, if anyone. The history teacher expects that such knowledge will "pop up" on appropriate occasions in English or other classes. But remember from chapter 2 the problem of inert knowledge: Knowledge does not pop up reliably.
To all this a skeptic might say, "Oh, you want to turn history into a current-events class or an English class." Not at all. The point is not that history classes should be dominated by discussions of today's newspapers and novels instead of history. The point is that typically there is no practice at all, not even a little, of performances that are often held to be goals of history instruction. We do not need a very sophisticated conception of teaching and learning to levy criticism on usual practice. It fares poorly by the most elementary measures of Theory One.
However, perhaps history, being one of the most reductively taught subject matters, has been unfairly used as the butt of this critique. What, then, about math? If we mean computation, the score is not so bad. Instruction does provide extensive information, elaborate practice with feedback, and so on. Practice could be more thoughtful, and motivation may be a weak link here. But computational achievements are reasonable if not stellar.
If we have in mind solving story problems, the "story" is different. Performance on story problems is chronically poor. Why? For one reason, schooling provides a good deal of practice—but not, typically, thoughtful practice. It's rare that students are encouraged to reflect on how they attack word problems. For instance, a teacher might build a discussion with students about how to tackle algebra problems around questions like these: "How do you begin? Do you read the problem more than once? Does that help? Does it help if you make a diagram? A table? A mental movie of what's happening in the 'story' of the problem? What helps for you?"
Another problem is clear information input. Here the process of problem solving is at stake. Yet, teachers do not normally model for students the mental processes involved in puzzling out a story problem. Yes, they work problems on the blackboard: "Step 1: You see that there are two key variables here—so let x stand for the distance to Plainsville and y the time of departure." But they do not usually dramatize the thinking process blow by blow. Yet they can, in this style for instance:
So we've looked over the problem. We have the car heading toward Plainsville. We need unknowns. What don't we know? Well, we don't know how far it is to Plainsville. That could be an unknown. Or when the car left. That could be another. The problem does say when the car arrived, so that's known. Now there's gas mileage—that's unknown. But do we need that information? Maybe not. Let's see now. What's our best choice of unknowns here?
As to the understanding of mathematics concepts, much the same argument can be made as for history. Understanding means more than repeating explanations found in the book, and youngsters typically do not see models of that kind of generative thinking, nor are they asked to engage in such thinking themselves. But they can be. For instance:
Now everyone pair up with someone near you. We've talked about common denominators and why they're needed. I'd like each pair to make up a way of explaining to someone two years younger what common denominators are and why they're important. You can use diagrams or anything you want. After you work in pairs for a while, I'll ask some of you to present your explanations.
Again, as in the case of history, repeating ideas from the text does not exercise understanding. To practice their understandings, learners must engage in activities that require reasoning and explanation.
In Appreciation of Teachers
Much of this critique seems to hit hard at teachers. While it's true that classroom practices commonly fall short of the desirable, teachers in general are not to blame, for two reasons.
First of all, many teachers inevitably share the trivial pursuit model endemic to the American culture. That this model is mistaken is not at all obvious. Although history gives us many champions of a more constructive approach to education—Socrates and John Dewey, for example—the weaknesses of a doggedly knowledge-oriented approach have become especially plain only in light of recent research in cognitive science.
Second, many teachers know Theory One intuitively and would like to follow through on it all the time. And some manage to do so part of the time. The trouble is that the realities of the school day make this difficult. Lee Shulman of Stanford University wryly puts the matter this way:
Teaching is impossible. If we simply add together all that is expected of a typical teacher and take note of the circumstances under which those activities are to be carried out, the sum makes greater demands than any individual can possibly fulfill. Yet, teachers teach.
Theodore Sizer, in _Horace's Compromise_ , writes eloquently of an English teacher, Horace Smith, who labors with great art and dedication in behalf of his students' learning. The catch is that Horace, like most teachers, has too much to manage. So he compromises. For instance, he assigns far less writing than he thinks ideal, although more than many teachers do, and offers far less feedback than he would like. Sizer puts Horace's plight this way:
Most jobs in the real world have a gap between what would be nice and what is possible. One adjusts. The tragedy for many high school teachers is that the gap is a chasm, not crossed by reasonable and judicious adjustments. Even after adroit accommodations and devastating compromises—only _five minutes per week_ of attention on the written work of each student and an average of ten minutes of planning for each fifty-odd-minute class—the task is already crushing, in reality a sixty-hour work week.
Many of the instructional ideas in this and later chapters will seem quite useless to teachers in their present settings. And they're right! Because most current educational settings neither labor very hard to build teachers' understandings of new instructional perspectives nor allow teachers the flexibility or freedom from the coverage fetish to pursue more enlightened instruction. This is why improvement in teaching practices has to go hand in hand with some restructuring of the way schools work as organizations (see chapters 7 and 9).
Theory One Revisited
To generalize, Theory One gives a poor grade to a good deal of educational practice. And now it's easier to see how. The first Theory One condition, clear information, should include clear explanations and monitoring of students' understanding of those explanations. But often this is not found. Theory One also says that students need clear information about process—about what the performances should look like blow by blow rather than just about the facts students are to use. However, typical instruction does not provide this process information; for instance, through teachers' modeling by thinking aloud while working problems.
Thoughtful practice should mean practicing the very performances one is seeking to develop. But surprisingly commonly, we do not actually engage students in the target performances but other, substitute performances. For instance, we do not actually ask youngsters to connect their historical knowledge to current events or to read literature with the historical backdrop in mind but rather simply test their knowledge base to check whether the historical data are there. When there is plenty of practice, as with story problems, it often is not very thoughtful—students simply do problem after problem without any encouragement to strategize reflectively on how they approach problems and what works for them.
As to informative feedback, apart from the general problem of being clear, the crowded curriculum and the number of students a teacher must deal with often simply do not allow for very much feedback. And as to motivation, it's plainly the case that many students find their school experiences quite disconnected from their out-of-school lives and professional aspirations.
Curing these problems will require helping teachers toward a richer and more discriminating conception of instructional method, the concern of this chapter. It will require building a new set of priorities in the educational milieu, priorities that give teachers the time and space to pursue a more ambitious agenda for students' learning. Chapter 7, concerning the "cognitive economy" of schooling, and Chapter 9, concerning the prospects of wide-scale educational change, speak to this challenge.
### THREE WAYS TO PUT THEORY ONE TO WORK
Of course, Theory One is not a teaching method. Rather it is a set of principles that any good method should satisfy. That is, any good teaching method incarnates Theory One, fleshes out its principles in ways to suit the particular needs of the learner and the moment.
Good teaching demands different methods for different occasions. Theory One should underlie them all. Here's a case in point: In _The Paideia Proposal_ , Mortimer Adler highlights three different ways of teaching. He calls them didactic instruction, coaching, and Socratic teaching. All three put Theory One to work in different ways. Let's take a look.
Didactic Instruction
By this, Adler means good, clear presentation of information by teachers and texts. The enterprise is most centrally one of explanation: laying out the whats, whys, and wherefores of a topic.
Recent research has clarified some of the components of good explanation. Earlier, I mentioned the work of Roehler, Duffy, and their colleagues on direct explanation. For another source, Gaea Leinhardt has outlined several characteristics of good explanation in instructional practice. By way of example, imagine a teacher teaching the concept of an "ecological niche." Leinhardt's principles and how they might translate into action follow:
_Identification of goals for the students_. (Teacher: "We want to understand the meaning of an ecological niche, so we can use the concept to describe the plants and animals in ecologies and compare ecologies with one another.")
_Monitoring and signaling progress toward the goals_. (Teacher: "Francis, when you said that sharks are predators in the sea, you made a smart connection: We hadn't talked about sea creatures at all. But sure, sharks are predators. What other sea predators can we identify?")
_Giving abundant examples of the concepts treated_. (Teacher: "Let's compare the animals in our local forests with those in Australia, Alaska, and Madagascar.)
_Demonstration, including offering complementary representations, highlighting links among them, and identifying conditions for use and nonuse of the concepts_. (Teacher: "As we watch this film about African animals, we'll stop it and talk about the niches we see. And we'll ask whether different animals are always in different niches or sometimes the same niche and why.")
_Linkage of new concepts to old through identification of familiar, expanded, and new elements_. (Teacher: "Niche is an odd word. Who knows what a niche in the ordinary sense is?")
_Legitimizing a new concept or procedure by means of principles the students already know, cross-checks among representations, and compelling logic_. (Teacher: "Is the niche concept really such a useful way of talking about ecologies? Well, let's explore that. Let's think about other situations where we talk about roles in a system; for instance, the roles of people in a business or a school.")
In intensive research on an adroit teacher's classroom practices, Leinhardt found all these elements at work.
How does Theory One match with Leinhardt's concept of good explanation? Plainly, the elements of didactic instruction spelled out by Leinhardt by and large concern clear information. They are elaborations of what it would take to be very clear with learners about what they are learning. Leinhardt also touches on informative feedback in her mention of monitoring and signaling progress toward the goals; and on motivation in the emphasis on clarifying the conditions for use and nonuse and on legitimizing.
Coaching
The second kind of teaching identified by Adler was coaching. Notice how neatly coaching and didactic teaching work together. Without didactic teaching to present some kind of information base on a new topic, learners have nothing to practice. Given clear information, however, the question arises: How does the teacher's role shift? Coaching offers an answer.
The metaphor with sports is meant quite seriously. In football, gymnastics, hockey, or track, the coach stands back, observes the performances, and provides guidance. The coach applauds strengths, identifies weaknesses, points up principles, offers guiding and often inspiring imagery, and decides what kind of practice to emphasize. This role makes just as much sense in a writing teacher's or mathematics teacher's classroom as it does on the playing field.
Imagine, for example, a teacher coaching high school students writing short stories with good "narrative hooks," those opening sentences that draw a reader in, such as Charles Dickens's famous, "It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness..." from _A Tale of Two Cities_. A teacher's comment to one student might sound something like this:
Charles, I love your first line, "When I got to the cookie jar, there was only one thing left, and it wasn't a cookie." That's terrific. It really establishes a mystery. But then you give it away in the next paragraph. Maybe you should keep the mystery going for a while to keep the reader involved.
Again, how does Theory One fit in to the idea of coaching? Plainly, coaching highlights the two middle concerns of Theory One: thoughtful practice and informative feedback. Assigning practice, encouraging learners to be mindful of what they are doing, and offering feedback are the coach's principal activities. At the same time, the coach needs to strive for clarity, clear information again. Moreover, the coach's relationship with the learners commonly taps powerful mechanisms of motivation.
Socratic Teaching
Adler's third method of instruction was Socratic teaching. Both didactic teaching and coaching are relatively directive, working to inform and shape the performances of the learners. Where, it's fair to ask, are the students engaged in a more open-ended way, supported in their explorations of a topic but not told what to do all the time? Where is there occasion not just to learn answers but to learn the art of inquiry? Socratic teaching is a reply to that question.
In typical Socratic teaching, the teacher poses a conceptual conundrum or snags one from the ongoing conversation. The teacher urges exploration of the issue. What do you think? What positions could one take? What definitions do we need? Views are advanced, approaches proposed. The Socratic teacher acts as kindler and tender of the conversation, helping as the paradoxes vex too much, irritating with counterexamples and potential inconsistencies as premature satisfaction sets in.
Let's imagine a teacher talking with students about "zero."
TEACHER: The "zero" is one of the great inventions of our number system. Let's have a little argument around zero to see if we can understand its importance better. Now pretend I'm a member of the Anti-Zero club. I say we don't need it. It just wastes space. Let me hear some arguments against that. Don't be shy!
STUDENT: Yeah, well, but how do you write down nothing?
TEACHER: Well, if there's nothing, you don't need to write down anything. You don't write down a 1 or 2 or anything.
STUDENT: But see, suppose something comes out to nothing, like you have nothing in your bank account, what do you write down?
TEACHER: You just leave a blank.
ANOTHER STUDENT: Yeah, but how can you tell whether you know you have nothing or you just forget to write down what you have?
TEACHER: That's a good point!
YET ANOTHER STUDENT: Well, but wait a minute. Roman numerals don't have a zero, and they must have done something to show nothing. So it must be okay somehow.
Cognitive scientist Allan Collins has analyzed the key moves of Socratic teaching this way.
Select both positive and negative examples to illustrate all qualities relevant to the issue under consideration.
Vary cases systematically to help focus on specific facts.
Employ counterexamples to question students' conclusions.
Generate hypothetical cases to encourage reasoning about related situations that might not occur naturally.
Use hypothesis identification strategies to force articulation of a particular working hypothesis.
Use hypothesis evaluation strategies to encourage critical evaluation of predictions and hypotheses.
Promote identification of other predictions that might explain the phenomenon in question.
Employ entrapment strategies to lure students into making incorrect predictions and premature formulations.
Foster tracing of consequences to a contradiction to encourage the careful formation of sound and consistent theories.
Encourage the questioning of answers provided by authorities such as teacher and textbook to promote independent thought.
How does Theory One apply? As to clear information, the Socratic teacher normally does not supply piles of information. However, the teacher does police clarity of information supplied by the participants with probing questions and with the aim of encouraging all involved to examine the information critically. (Classically, Socratic interactions are undertaken concerning problems where people already have a fund of experience that they can draw on for information.) The Socratic teacher involves the learners in continuous thoughtful practice as they collaborate and contend to sort out the issue. The teacher provides immediate feedback by way of encouragement and critiques, encouraging others in the conversation to do the same. Finally, the teacher capitalizes on the intrinsic motivation of big, motivating questions that touch us all, such as Plato's "What is justice?" and on the engaging cooperative/competitive texture of lively conversations.
If Theory One is central to these three rather different ways of teaching, what accounts for the contrasts among them? In a word, agenda. Didactic teaching serves a need that arises in instructional contexts, that of expanding learners' repertoire of knowledge. Coaching serves another need: ensuring effective practice. Socratic teaching serves yet others: helping learners to work through concepts for themselves that they might not truly grasp in any other way, as well as giving them a chance to engage in and learn about inquiry. It is only a slight exaggeration to say that, if you combine the conditions laid down by Theory One with each of those agendas, you get the respective methods. In other words, Theory One has different incarnations depending on the instructional agenda of the moment. Three key incarnations of Theory One are didactic instruction, coaching, and Socratic teaching.
### THE BOGEYMAN OF BEHAVIORISM
One of my favorite essays was authored by B. F. Skinner, founder of behavioristic learning theory and progenitor of a range of behavioristic classroom practices. The essay is called "On 'Having' a Poem." Behaviorism proposes that human behavior can be explained as a large set of inborn and learned reflexives—responses to stimuli. One doesn't even need to talk about thoughts or minds. Writing about creativity from a behaviorist's standpoint, Skinner proposed that we must eschew all talk about the poet's mind, a misleading concept lacking any real concrete meaning. Rather, a poet "has" a poem in much the same way that a hen lays an egg: a result of the physical constitution of the poet and the rewards in the environment that have "reinforced" over the years the poet's behavior of "having" other good poems.
Of course, I do not agree at all with Skinner's account of the matter. But I admire his style. The core metaphor—having a poem is like a hen laying an egg—is outrageous enough to press us to rethink our categories and premises.
So in return for such stimulation, it seems worth a brief detour to (a) moderate the barrage of criticism fired at behaviorism for the present ills of education and (b) spell out how behaviorism relates to Theory One.
Behaviorism is often blamed for the current problems of education. And not without some justification. In its heyday, behaviorism was the reigning theory of learning and teaching. It cultivated a kind of excess atomism in which performances were broken down into microperformances—the 30 key subskills of effective reading and such—that students never put back together again in meaningful, thoughtful performances, a behaviorist's Humpty Dumpty. Also, by ignoring human thinking as an invalid "folk theory," behaviorism discouraged some people from interacting with students in ways that made plain the workings of the mind.
However, with that acknowledged, the commonplace accusation that "this is an old-fashioned behaviorist classroom; we need to get beyond that" usually is simply false. Very few classrooms are run in an effective behaviorist manner. Well-crafted instruction in the style of behaviorism involves careful adjustments in the teacher's actions, how students work together, and other matters, so that what is most rewarding for students is pursuing the learning objectives, not side activities. The typical classroom, in contrast, allows students many other rewarding paths, some of them disruptive.
Indeed some of those unproductive paths are rewarding for students exactly because they are disruptive. I remember a high school English teacher of mine, let's call him Mr. Davis, who had the habit of saying "well" a lot. Some students cooked up the idea of forming a pool over how many "wells" Mr. Davis would say in a week. Almost everyone in the class invested. Several students became official talliers.
Mr. Davis found out about the pool quite soon and, to subvert students' efforts to guess the number, tried rather unsuccessfully to stop saying "well." The class that week enjoyed many a giggle at his efforts. The pool became the main interest of several days, although we went through the motions of normal English classes.
Of course, Mr. Davis made the wrong behaviorist move in trying to cut out his "wells." His behavior rewarded us, making the affair more rather than less interesting. He should simply have ignored the whole thing, minimizing the damage. To be sure, this example is mild compared with the harassment teachers sometimes face in inner city schools, which really can't be ignored.
The moral: Good control of the reward structure in teaching is a complex science and art. The typical classroom is as simplistic by the measure of behaviorism as it is by the measure of its successor, cognitive psychology. Moreover, behaviorism straightforwardly encompasses a number of important learning principles. In fact, Theory One is quite consistent with behaviorism. Providing information, arranging practice, offering informative feedback, and establishing motivation are all notions that a good behaviorist would subscribe to. This does not mean that Theory One is a behaviorist theory, because Theory One allows what behaviorism denies: talk about the mind and mental processes, including teacher modeling of mental processes. But it does mean that classroom practices that do not live up to Theory One do not live up to behaviorism.
If we want insight into the roots of educational malpractice, we can find some important hints in the influence of behaviorism on the classroom. But we need to look much further and in other directions—toward the Trivial Pursuit and Ability-Counts-Most theories discussed in the last chapter, for instance, both of which dominant the scene and work against the thoughtful process of teaching and learning demanded by Theory One.
### BEYOND THEORY ONE
Theory One can be seen as a kind of a milepost, marking the first mile toward more sophisticated theories. Theory One is a pretty good theory of instruction. If we conducted education assiduously according to Theory One and its two easiest incarnations, didactic instruction and coaching, we would get considerably better results than we do.
But this is no reason for serene satisfaction. A variety of research on teaching and learning has refined our understanding beyond Theory One. A look at several approaches to teaching and learning illustrates the rich resources available to teachers.
A Constructivist Perspective
Many educators today take a constructivist view of educational practice. Such a view conceives the learner as an active agent, "constructing meanings" in response to the instructional situation. Constructivism, effort centered rather than ability centered, denies the notion that the learner passively absorbs information provided by the teacher or textbook. Rather, even when the task is sheer memorization, the learner plays a very active role, struggling to understand, formulating tentative conceptions, testing those conceptions out on further instances.
What does this mean in practice? Things like this:
One might engage students in puzzling over and experimenting with what makes some things sink in water while others float. With artful coaching, they may recreate the principle of displacement.
One might ask young students just learning arithmetic to invent their own ways of adding and subtracting, without teaching the standard algorithms. This actually works!
One might ask students to build better more empowering conceptions of themselves as writers by keeping a writer's diary. What things work for them? What things don't? When their writing goes well or not, what seem to be the causes?
In other words, a constructivist approach puts students in the driver's seat to a surprising extent, asking them to find their way through large parts of the learning, but of course with teacher coaching.
How does this go beyond Theory One? Theory One is consistent with constructivism but does not lay specific emphasis on the importance of the learner's working through ideas with a good measure of autonomy to achieve understanding. (How much autonomy is controversial, even within a constructivist perspective.)
A Developmental Perspective
The challenges of learning can often be clarified through a developmental perspective that examines the age and sophistication of the learner, one that asks about developmentally appropriate attainments and means of instruction. Prior to the 1980s, a developmental perspective usually meant the theories of Jean Piaget, the famous Swiss developmental psychologist. Piaget held that youngsters pass through several developmental stages, culminating during adolescence in the stage of "formal operations," which enables thinking in a formal, logical way across diverse disciplines. Piaget argued that little could be done to accelerate development through these stages. Moreover, efforts to teach a topic would fail if they demanded patterns of reasoning beyond the learner's stage.
Not everyone was persuaded about the limits to learning suggested by Piaget. Jerome Bruner, writing in 1960, articulated one of the best-known statements about children's potential in education: "We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any state of development."
Research conducted in the 1960s, 1970s, and 1980s confirmed the ingenuity of many of Piaget's questions and methods of inquiry but undermined many of his principal tenets. Contrary to Piaget's belief that stage advance comes at its own pace, a number of teaching experiments have accomplished stage advance by using a variety of instructional methods. Contrary to Piaget's notion that stages have a universal cross-disciplinary character, it appears that the progressive mastery of more sophisticated patterns of reasoning is often discipline specific. And contrary to his precept that certain patterns of logical reasoning are simply not accessible to young children, investigators have found again and again that children can display such patterns of reasoning when the content is familiar, representations are concrete, and supports for short-term memory (for instance, paper and pencil) are available.
What are the implications for the practice of education? Here are some:
"Advanced" ideas, like the control of variables, can figure in simple form in science experiments done in early elementary school. Formerly, such ideas were kept out of elementary science.
Concrete or familiar materials can make very abstract ideas accessible; witness the classic experiments of Jerome Bruner with arrays of blocks representing the factoring of algebraic expressions.
Familiarity counts for a lot and makes abstraction and complexity accessible. For example, one can encourage elementary school youngsters in reasoning about complex causal patterns, such as mutual escalation, with examples drawn from such familiar territory as family squabbles.
Moreover, in several specific areas there are stage-like or other models of children's developing understandings—for example, children's ways of approaching arithmetic problems, their understanding of narrative and metaphor, or their moral reasoning. These schemes, while not as sweeping as the original Piagetian conception, can provide discipline-specific and topic-specific guides to well-tuned instruction.
How does all this go beyond Theory One? By recognizing broad developmental trends and specific developmental patterns in subject matters simply not addressed by Theory One.
Cooperative and Collaborative Learning
There is considerable evidence that children can learn much better in well-configured cooperative groupings than solo. Most any cooperative grouping may help to achieve certain ends—for example, better socialization—but for gains in conventional educational objectives, careful design is needed. Investigators of cooperative learning David Johnson and Roger Johnson of the University of Minnesota at Minneapolis and Robert Slavin of Johns Hopkins University agree that effective cooperative learning calls for joint responsibility of all children for the group performance.
Here is a practical example, the well-known "jigsaw" method:
1. Students form groups of four and divide a given topic into subtopics, each student in a group taking responsibility for teaching the others one subtopic.
2. Say the subtopics are _X, Y, Z_ , and _W_. The students responsible for X all leave their small groups and form a larger group that learns about _X_ from the text, teacher, and other sources. Likewise for _Y, Z_ , and _W_.
3. Then the students go back to their small groups and teach one another their subtopics.
4. After testing, each student receives as a grade the average performance of his or her group. Thus, each is motivated to see that all do well.
Researcher/educators William Damon and Erin Phelps draw a contrast between cooperative learning and peer collaboration. The jigsaw method typifies cooperative learning: Students work in groups on the same task and, within groups, often divide up the task into subtasks. In peer collaboration, pairs or small groups of students work on the same task simultaneously, thinking together as they puzzle over its demands and work through its complexities. The task may be specific to the group. Damon and Phelps argue that peer collaboration offers more "mutuality"—more extensive, intimate, and connected discourse. They suggest that peer collaboration serves better when youngsters face novel and complex concepts.
Cooperative learning and peer collaboration go beyond Theory One in using the dynamics of groups to promote thoughtful learning as youngsters talk and think together and in tapping the intrinsic motivation of social contact to keep students interested in their academic activities.
Intrinsic Motivation
Theory One avers the importance of extrinsic and intrinsic motivation but casts no vote between them. The tradeoffs of motivating students through either intrinsic or extrinsic means have been investigated in a number of studies. Briefly, performances motivated by such extrinsic rewards as grades, lollipops, or dollar bills tend not to persist once the reward structure is dropped. The activity is simply not seen as interesting in itself. In contrast, efforts to cultivate children's intrinsic interest in a rich activity, such as the reading of literature, are more likely to lead to sustained, self-motivated involvement.
This is obvious as far as it goes, but there is a more subtle finding: Strong extrinsic reward tends to undermine intrinsic interest. In other words, if an activity is both interesting in itself _and_ rewarded in extrinsic ways, children's intrinsic interest tends to wane. In one classic experiment, children engaged in an art activity. Two groups received a reward in the form of a certificate while another group did not. Later, the children encountered another opportunity to work with the art materials or do other things. The children who had received the extrinsic reward of the certificate turned to the art materials much _less_ than the unrewarded children. The extrinsic reward had undermined their intrinsic interest.
Moreover, intrinsic interest is related to creativity: People are more likely to perform creatively if driven by strong intrinsic motivation. In another provocative study, Teresa Amabile of Brandeis University asked college and graduate students serious about writing to rank order lists of reasons for their interest. Some writers rank ordered a list of intrinsic reasons ("You enjoy the opportunity for self-expression") while others rank ordered a list of extrinsic reasons ("You enjoy public recognition of your work").
Shortly before and after this activity, all the writers wrote haiku. A panel of judges carefully rated their output for creativity, with high reliability between judges. The poems written before rank ordering the lists showed about equal creativity, but the poems written afterward by the writers who rank ordered the list of extrinsic motivators showed markedly less creativity than the poems from the other writers. Amabile's interpretation: The poets who rank ordered a list of extrinsic motives experienced a temporary drop in intrinsic motivation. This in turn hurt the quality of their poems.
Here are some examples of educational practice that take these points seriously:
A teacher coaches students in developing their own criteria of quality for stories and poems they write.
A teacher involves students in writing story problems in mathematics for one another to heighten the intrinsic interest of the problems.
On some quizzes, a teacher has students grade one another. But the teacher gives strong feedback about what kinds of answers make more and less sense and why.
This perspective on intrinsic motivation goes beyond Theory One in a very straightforward way: Theory One says nothing about interactions between intrinsic and extrinsic motivation, but there are important interactions to worry about when we want to build strong intrinsic motivation, as we often do.
Honoring Multiple Intelligences
Developmental psychologist Howard Gardner has articulated a Theory of Multiple Intelligences, arguing that conventional IQ conceptions of human intelligence are too monolithic. Gardner builds a case for seven different dimensions of human intelligence—seven intelligences—associated with distinctive symbol systems and modes of representation. For example, logical/mathematical intelligence involves ability with the formal notations of mathematics. Linguistic intelligence involves artistry with words on paper or in the air—the artistry of the poet, the novelist, the orator. Musical intelligence demands adroit handling of musical structures, instruments, and notation. Gardner's other four intelligences are spatial (architects, graphic artists), bodily/kinesthetic (sports, dance), interpersonal intelligence (politics, management), and intrapersonal intelligence (self-reflection).
Gardner makes the point that conventional educational practice focuses largely on linguistic and mathematical intelligence. Yet, he urges, the multiple character of human intelligence demands a wider horizon if we are to honor people's varied abilities. This would include finding ways to make music, the visual arts, dance and sports, interpersonal skills, and skills of self-reflection more substantive and salient presences in classrooms and curricula.
What might this mean in practice? Things like this:
Involving students in projects that allow many alternative modes of symbolic expression—visual art, language, music.
Creating group projects that invite students to work in media and symbol systems with which they feel most closely attuned.
Bringing a greater diversity of symbol systems into subject matters; for instance, engaging students in writing essays in mathematics class, or essays about mathematics in English class, or doing cartoons with witty captions in English.
The idea of multiple intelligences goes beyond Theory One in highlighting the diversity of human ability and the consequent need to diversify instructional opportunities. Theory One says nothing directly about this.
Situated Learning
Recently, cognitive psychologists Allan Collins, John Seely Brown, James Greeno, Lauren Resnick, and others have underscored a troubling feature of typical classroom learning: its decontextualized character. What happens in school mathematics, writing, or the study of history, for example, bears little resemblance to what mathematicians, authors, or historians do. Nor does it resemble in-context uses of mathematics, writing, or history by nonprofessionals—in the supermarket, on the tax form, in formulating a personal statement for a job application, in understanding current events.
These scholars have also pointed out that, in authentic contexts, effective learning gets supported in a number of ways absent in the typical classroom. For example, apprentice-like relationships are common. Knowledge and skills figure conspicuously in making progress on tasks that need doing. A social network functions to support performance and sustain relevant learning.
They sum up the circumstances with a well-chosen term, "situated learning." Truly effective learning should be situated in a culture of needs and practices that gives the knowledge and skill being learned context, texture, and motivation. Much can be done to move classroom practices toward situated learning: for example:
Students can learn writing skills by publishing a newsletter for fellow students.
Students can learn principles of flight by experimenting with paper airplanes and other simple gizmos that fly.
Students can learn about statistics by engaging in research relevant to their immediate surroundings; for instance, statistics for the school sports teams.
All this too goes beyond Theory One in that Theory One says nothing specific about the importance of situating learning in contexts with real audiences, needs worth pursuing, and so on.
Beyond, But...
Theory One is only Theory One. That has been the important message of the past few pages. There are many ideas in contemporary education—I've given only a sample—that add more sophistication to the basics of Theory One. But—and this is important—although these other perspectives add to Theory One, one should always keep Theory One in mind. Because in general these other perspectives do not automatically take care of the four core concerns of Theory One.
Take constructivism. Simply allowing youngsters to try to figure out something for themselves by experimentation does not ensure that they're getting informative feedback (because some experiments yield very obscure results) or that they're particularly motivated (because many things to investigate aren't automatically and instantly interesting to many children).
Or take cooperative learning. Simply putting students in small groups does not substitute for their having sources of clear information. The muddled accounts of important concepts in many science texts will still be just as muddled. And while students can give one another _informative feedback_ up to a point, for subtleties the teacher's input may be crucial.
Or take situated learning. Learners will still need _clear information_ about how to tackle tasks. In many situations, they will not get good informative feedback automatically. For instance, the authors of a student newsletter might need to talk to fellow students to get reactions.
The moral should be clear. It's fine to go beyond Theory One, but many efforts to do so in the spirit of constructivism, cooperative learning, peer collaboration, situated learning, or any other approach tend to lose the basics of Theory One in the shuffle. Those basics do not typically take care of themselves. They need teachers' thoughtful attention, whatever the grander plan.
### OUR MOST IMPORTANT CHOICE IS WHAT WE TRY TO TEACH
The challenge of educating youngsters presents innumerable choices. Two kinds of choices routinely faced are those about method—how to teach—and those about content—what to try to teach. Very often, method gets treated as the most critical choice. To be sure, there are concerns about excess content and reductive content, and there are important contemporary initiatives to reform content. However, most of the action rotates around methods—let's try cooperative learning, let's try discovery learning, let's try the Madeline Hunter method, and so on.
The implicit message is that we are fairly satisfied with what we are trying to teach... if only we taught it better, so that youngsters really understood it, thought critically about it, and used it. Hence the "savior syndrome," the continuing quest for the magic method that will finally inculcate in youngsters the knowledge and skill that we cherish.
I believe that this emphasis on the quest for new and better method is a mistake. Not how we teach but what we try to teach is our most important choice. Here's why.
Reason 1: _There is not that much choice to be exercised about basic method_. Any instructional method should incorporate the fundamentals of Theory One. Among basic incarnations of Theory One, such as didactic teaching, coaching, or Socratic teaching, choice is driven by need. What does the teacher or instructional designer need to do?
• Do you need to get across a complex bundle of ideas and information where the learners have little background? Then you had better employ didactic teaching to start with; there is nothing to coach yet, and a Socratic approach is ill suited to convey all that information.
Do you need to ensure thoughtful practice and informative feedback? Then you had better coach. Students will need your help on what kinds of tasks to tackle, how to approach them, how to handle blocks, how to work with one another.
Is there a puzzling concept that youngsters are not likely to grasp without working it through in their own minds in a very active way? Then you had better employ Socratic techniques, because Socratic techniques support students in open-ended inquiry.
But suppose the things to be learned have _all_ these needs (as many things do)? Then weave together didactic, coaching, and Socratic approaches. Again, choice is driven by need.
Of course, there is much more choice to be exercised about fancier methods. Should we use cooperative learning here? What kind of cooperative learning? How can we build in more intrinsic motivation? Should we harness the seven intelligences? These are indeed worthwhile questions to ask and answer in context-appropriate ways. However, even if we never got to those questions and never used fancier methods, Theory One and its basic variations would take us a long way.
Reason 2: _The shortfalls we want to address by newer methods are often more a matter of what we try to teach_. For example, many educators would like students to develop better thinking and learning strategies. But they do nothing to teach such strategies. Many educators would like students to carry ideas from the classroom out into their lives away from school. But they do nothing to help youngsters make such connections. Virtually all educators want students to understand what they are learning, not just acquire rote knowledge and skills. But most educators do not get students to practice their understandings. Instead, students end up practicing remembering.
Notice the pattern: We want better thinking and learning strategies. We want connections to life outside of school. We want understanding. And we want other things. But by and large we do not actually teach those things—not in the sense of providing direct information about them, not in the sense of providing thoughtful practice or informative feedback, not in the sense of making plain those objectives and pursuing them directly with students to harness intrinsic motivation. This is the great paradox of education: To a startling extent, we do not really try to teach what we want students to learn.
* * *
KEY IDEAS TOWARD THE SMART SCHOOL
**TEACHING AND LEARNING:
THEORY ONE AND BEYOND**
**Basic Sound Method**
**Theory One**. Clear information. Thoughtful practice. Informative feedback. Strong intrinsic or extrinsic motivation.
**Three Incarnations of Theory One**. Didactic teaching, coaching, Socratic teaching.
**Further Choices**
**Beyond Theory One**. A constructivist perspective. A developmental perspective. Collaborative learning. Care that extrinsic motivation does not undermine intrinsic motivation. Honoring multiple intelligences. Situated learning.
**Our Most Important Choice Is What We Try to Teach**. Teach to the target performances actually desired. Use didactic, coaching, Socratic, or other methods as demanded by the target performances, meeting Theory One standards.
* * *
In reconsidering what we try to teach, the single most helpful move may be to redescribe educational objectives in terms of performances rather than knowledge possessed. This idea smacks somewhat of the movement a few years back to frame educational goals as "behavioral objectives," long lists of overt, witnessable performances. Perhaps inevitably, such lists commonly trivialized the behaviors they aimed to cultivate. Nothing so destructively dissective is intended here. Rather, when I say "describe objectives in terms of performances," I mean describe in simple broad terms what we want learners to be able to do—explain a concept in one's own words, give fresh examples of it, and so on. The chapters to come offer many illustrations.
When we look at an educational setting and ask, "What performances are students actually asked to do?" the answer is always revealing. Often, it's plain that the students are only asked to display specific knowledge and routine skills. They are not asked to learn strategically, to make connections to life outside of school, to explain or inquire or practice understanding in other ways. Unsurprisingly, people learn to do much of what they actually practice doing and not a lot else.
So newer methods of teaching alone will not help. They are not even the heart of the problem, since we have plenty of solid and plenty of fancy ways to teach already. The heart of the problem is not teaching what we really want students to learn. If we face up to what we want students to learn, we then know a good deal about how to approach it: àla Theory One, provide information capturing the performance in question, provide needed background knowledge, offer thoughtful practice, generate informative feedback, and build motivation.
So let's make two slogans out of it all:
1. _Our most important choice is what we try to teach!_
2. _Our most important craft is solid Theory One teaching!_
What we try to teach will determine more than anything else what students learn, if we teach at least as well as Theory One asks. So the smart school needs to inform and energize. It must give teachers and administrators the time, encouragement, and knowledge resources (1) to think more deeply about what is worth teaching and learning and (2) to advance the craft of Theory One teaching. Both these are central to the smart school.
## CHAPTER 4
**CONTENT
Toward a Pedagogy of Understanding**
Several years ago I made a presentation on youngsters' misconceptions about science and mathematics at a conference. I reviewed a few misconceptions and talked about their causes. Whatever the audience gained from the experience, I learned the most after the closing questions. I had packed up my transparencies and was on the way to another session, when two of the people who had heard the presentation drew me aside.
"We have a small question," one of them said. "Just a point of curiosity really."
"Okay, sure," I said.
"One of the misconceptions you talked about was kids' belief that you can take the square root of a sum. The square root of _a_ square plus _b_ square equals _a_ plus _b_."
√a2 + b2 = a + b
"Right, and that isn't so."
"Yes, we understand that. But our question is, _why_ isn't it so? It looks as though it _ought_ to be so."
The question took me aback. At first I had no idea how to respond. Had they asked me why some mathematical relationship _is_ so, I would have tried to offer a proof or at least a qualitative explanation. But why does this relation _not_ hold? Well, it just doesn't. You don't explain _that_.
Then I had an insight, which I gratefully explained to the two, about what made their question hard and what it said about our very different perspectives on the world of mathematics. Though an educator and cognitive psychologist now, I was trained as a mathematician. I knew from years of experience that every valid mathematical relationship is hard won. Relationships that "look nice," such as the one listed above, often do not hold up. The universe of nice-looking relationships is full of chaff, with the apparatus of mathematical proof winnowing out the grain.
However, my questioners' experience of mathematics would have been quite different. They had no exposure to building mathematical systems. They had learned mostly the received content of mathematics, the many beautiful relationships that do hold up. From this kind of experience, it is very natural to conclude that nice-looking relationships generally work out, to expect validity, and to react with surprise when a nice-looking relationship betrays that expectation.
In short, I learned that my questioners and I had very different understandings of not just the square root but something much broader—the whole enterprise of mathematics. They saw mathematics as a matter of formally validating relationships that looked promising and would probably hold up. I saw mathematics as a matter of drawing from a sea of possible relationships the few valid ones. It was those that needed explanation, not the many invalid ones.
The moral of the story is that understanding is a multilayered thing. It has to do not just with particulars but with our whole mindset about a discipline or subject matter. The story is testimony to the dangers of an overly atomistic take on teaching subject matters, one that does not pay heed to how individual facts and concepts form a larger mosaic that has its own spirit, style, and order. If a pedagogy of understanding means anything, it means understanding the piece in the context of the whole and the whole as the mosaic of its pieces.
Pedagogy is simply an erudite word for the art of teaching. A pedagogy of understanding would be an art of teaching for understanding. Yes, that surely is a large part of what education needs. Remember the "fragile knowledge syndrome" from chapter 2: An abundance of research shows that youngsters generally do not understand very well what they are learning. They suffer from deep-rooted misconceptions and stereotypes. And they are often just plain bewildered by difficult ideas: subjunctive tenses in English, Hamlet's indecision, Archimedes's principle of displacement, why it's hotter in the summer, how slavery could have taken such a strong hold in the South. We all want to teach for understanding, of course, and we often think that we are. But too often, quite apparently, we are not.
The last chapter drew a strong moral: Our most important choice is what we try to teach. This implies that teaching for better understanding is more than just a matter of superior method. It requires teaching something more or something else, choosing differently what we try to teach. To teach for better understanding, we should teach different stuff.
But what sort of stuff? What is understanding made of?
### WHAT IS UNDERSTANDING?
The Role of "Understanding Performances"
Chapter 1 offered three hard-to-disagree-with goals for education: retention, understanding, and the active use of knowledge. Understanding plays a particularly pivotal role in this trio for two reasons. First of all, the kinds of things you might do to understand a concept better are some of the best things you can do to remember it well. Looking for patterns in ideas, finding personal examples, and relating new ideas to prior knowledge, for example, all serve understanding and also lock information into memory. Second, active use of knowledge comes hard without understanding. What can you do with knowledge that you don't understand?
But understanding is a somewhat puzzling objective for education. I have often been put off by goal statements in sample lesson plans or curriculum designs that take the form "Students will understand such-and-such." How, after all, can we tell whether a student has attained this precious state of understanding? It's not something that you can measure with a thermometer, nor very readily with a multiple-choice quiz.
A comparison between _knowing_ and _understanding_ underscores the mysterious character of understanding. Take Newton's laws. These are the cornerstones of classical physics. The first law of Newton says, more or less, that an object keeps going in the same direction at the same speed unless some force diverts it. This was not at all obvious before Newton's insight. After all, we do not very often see objects moving in the way Newton described. In our everyday world there are plenty of forces around to divert objects in motion. Friction slows them down to a stop. Gravity bends the paths of thrown objects into a curve that comes back to earth. So it's far from clear that, left alone, objects keep going at the same speed in the same direction.
If my goal as teacher is that a student _know_ Newton's laws, I can check the student's achievement by asking for a recitation, or perhaps the writing of formulas. I can even insist that the student do some algebraic manipulations to show that the knowledge is not pure rote but at least somewhat operational.
However, suppose my goal is that the learner _understand_ Newton's laws. Then if I have my students recite them, write them out algebraically, and even execute a few manipulations, I still can't tell whether my students understand. My students easily could be showing me "canned" performances with hardly any understanding of what the laws really imply or explain or why they are valid.
The mystery boils down to this: Knowing is a state of possession, and I can easily check whether learners possess the knowledge they are supposed to. But understanding somehow goes beyond possession. The person who understands is capable of "going beyond the information given," in Jerome Bruner's eloquent phrase. To understand understanding, we have to get clearer about that "beyond possession."
Understanding Performances
So let us view understanding not as a state of possession but one of enablement. When we understand something, we not only possess certain information about it but are enabled to do certain things with that knowledge. These things that we can do, that exercise and show understanding, are called "understanding performances."
For example, suppose that someone understands Newton's first law. What kinds of understanding performances might that someone be able to show? Here are several:
_Explanation._ Explain in your own words what it means to go at a constant speed in the same direction and what sorts of forces might divert an object.
_Exemplification._ Give fresh examples of the law at work. For instance, identify what forces divert the paths of objects in sports, in steering cars, in walking.
_Application._ Use the law to explain a phenomenon not yet studied. For instance, what forces might make a curve ball curve?
_Justification._ Offer evidence in defense of the law; formulate an experiment to test it. For instance, to see the law at work, how can you set up a situation as little influenced by friction and gravity as possible?
_Comparison and Contrast._ Note the form of the law and relate it to other laws. What other laws can you think of that say that something stays constant unless such-and-such?
_Contextualization._ Explore the relationship of this law to the larger tapestry of physics; how does it fit into the rest of Newton's laws, for example? Why is it important? What role does it play?
_Generalization._ Does the form of this law disclose any more general principles about physical relationships, principles also manifested in other laws of physics? For instance, do all laws of physics say in one way or another that something stays constant unless such-and-such?
And so on in similar spirit.
Some of these understanding performances are quite modest in their demands; for instance, making up a fresh example of Newton's first law at work. Perhaps a student knows examples about football, and so makes up one about baseball or soccer or playing frisbee. Others are quite challenging: for instance, the last one about generalization. The variety demonstrates some important points about understanding.
First of all, we identify understanding through generative performances, where learners "go beyond the information given." Understanding consists in a state of enablement to display such understanding performances.
Second, different understanding performances demand somewhat different kinds of thinking. To justify Newton's first law is not quite the same as to apply it, although there are parallels in the reasoning.
Third, understanding is not a matter of "either you get it or you don't." It is open ended and a matter of degree. You can understand a little about something (you can display a few understanding performances) or a lot more about something (you can display many varied understanding performances), but you cannot understand everything about something because there are always more extrapolations that you might not have explored and might not be able to make.
This performance perspective on understanding illuminates what a pedagogy of understanding should attempt: to enable students to display a variety of relevant understanding performances surrounding the content that they are learning. It also harks back to the basic principle emphasized in the introduction: Learning is a consequence of thinking. Notice how all these understanding performances demand thinking—to generate explanations, find new examples, generalize, and so on.
Finally, as mentioned earlier, this performance perspective on understanding connects to the previous chapter's moral that our most important choice is what we try to teach. If we want students to understand, we should make the choice of teaching them understanding performances about Newton's first law or anything we want them to understand. We should provide clear information, thoughtful practice, informative feedback, and good motivation, just as Theory One says. But we don't, by and large. We do not very often even engage students in understanding performances like generating explanations, fresh examples, and justifications. Then we wonder why they don't understand!
### UNDERSTANDING AND MENTAL IMAGES
Suppose one day, sitting quietly on your living room sofa, you find yourself in an Eastern mood. Summoning your powers of concentration and contemplation, you levitate into the air. Rising up toward the ceiling, you pass through it.
The question is: Where would you find yourself? Perhaps in a bedroom or a bathroom. Perhaps in an attic. Perhaps in the apartment of the people who live above you. The curious thing about this exercise in imagination is that typically you can say where you would end up, even though the path through the ceiling is one you have never before traveled.
Notice that you've gone beyond the information given. Your trip through the ceiling is an understanding performance that reveals your understanding of the place you leave—an understanding more integrated than just a list of all the different routes you travel in your house.
This bit of mental gymnastics reveals at work one of the most important resources of mind we have—the mental image. Mental images help to explain how the trip through the ceiling works. Over the years, you have built up a mental image of your living space. It is like a map or a three-dimensional model. It shows how the various rooms relate to one another. Therefore, when asked what would happen should you float through the ceiling, you are in a position to answer. You look at the map in your mind—the mental image—chart your course, and read off your destination.
Mental images in the sense I use the phrase here are not limited just to environments or even to the strictly visual. People have mental images of what stories are supposed to be like.
What happens when you tell a child "Goldilocks and the Three Bears," but, seeing it's getting late, you stop at the point where Goldilocks is sleeping in the baby bear's bed, which is "just right." You say, "So that's all for tonight."
"But you didn't finish the story," your child says.
"Oh," you say. "Did I tell you the story before?"
"Nope," your child says. "But it doesn't sound done!"
Stories have a shape to them. They need mystery or challenge and resolution. Children develop a mental image of the shape of stories quite early in life—not a visual image, but a kind of general feel for how stories go. After the child has that image, you can't cop out with Goldilocks snoozing in bed.
Mental Images Enable Understanding Performances
There is an important connection between a pedagogy of understanding and this notion of mental images. Understanding performances might be called the overt side of understanding—what people do when they show understanding. But what about the internal side of understanding? What do people have in their heads when they understand something?
Contemporary cognitive science has a favorite answer: mental images (many psychologists would say "mental models" to mean the same thing). Roughly defined, a mental image is a holistic, highly integrated kind of knowledge. It is any unified, overarching mental representation that helps us work with a topic or subject. For example, our mental images of our homes and neighborhoods help us navigate them (as well as allowing journeys of the imagination through ceilings). Mental images of what stories are like help us understand and make up stories (as well as stop us from palming off nonstories on our children). Other mental images help us understand topics and themes in history, science, or any other subject matter.
How do mental images do this? They give us something to reason with when we attempt understanding performances. Because you have a mental image of your home, you can work with that mental image when I ask you to predict (an understanding performance) where you'd end up if you floated through the ceiling. Because you have a feel for the shape of a story, if I ask you to make one up, your general story image gives you something to work from. Whatever the understanding performance is—explaining, extrapolating, exemplifying—if you have the right mental images they will help you do it.
The mental images talked about so far concern very basic things such as the layout of your home or the shape of a story. But mental images can also concern very abstract and sophisticated matters. Consider, for example, the mental image of the organization of the chemical elements provided by the periodic table. The table itself is, of course, an overt image on paper. But in so far as people at least partially internalize what it says, it becomes a mental image too.
And notice how abstract it is, either on paper or in your head. The periodic table is a map of sorts, but not a map of a physical space. Rather, the spatial relationships on the periodic table denote cyclic patterns in the chemical behavior of the elements, with close neighbors on the chart sharing certain physical properties. The map thereby enables a variety of understanding performances, as people reason from the spatial relationships on the table to reach predictions about the chemical behavior of the elements.
For another kind of mental image, consider personalities. Ponder for a moment the mental images of characters you build up by reading _Othello_. To test their vividness, try this thought experiment. Suppose that Othello's neighbor stops by two-thirds of the way through the play and bears vehement witness to Desdemona's good conduct. Would Othello say, "Well, okay, I guess it was all in my head"? Certainly not! If you have a mental image of Othello (not in the literal sense of what he looks like but in the sense of a feel for his personality), you know immediately and intuitively that Othello would remain uneasy. He is compulsively suspicious of Desdemona's faithfulness. What about Iago? Hearing about the neighbor's testimony, would he leave town, fearful of disclosure? Certainly not! If you have a mental image of Iago's character, you know immediately this would be too meek for Iago. He would try some further treachery to discredit the neighbor and stoke Othello's fears all the more.
For an example even more abstract than the periodic table or a personality, consider my mental image of mathematics that figured in the introduction of this chapter, where a mathematical relationship that "looks nice" also looks suspect. Like any mental image, this one enables understanding performances. Because of it, I approach new mathematical propositions with appropriate expectations—skepticism and a demand for justification. Remember the people who approached me after the lecture on science and mathematics misconceptions, asking why one of the formulas I discussed wasn't true. They showed a more credulous mental image: mathematical relationships that "look nice" look true, too. And there would be consequences for their understanding performances. They would approach a new mathematical proposition with overconfidence in its likely validity and puzzlement if it fell through.
Understanding Performances Build Mental Images
So mental images equip people for understanding performances. And sometimes people get mental images through direct instruction—as when we teach the periodic table.
But the relationship between mental images and understanding performances is not a one-way street, from images to performances. It's a two-way street: Understanding performances build mental images.
For example, we do not normally learn our way about a new neighborhood by memorizing a map. We wander around. We cope with challenges, such as getting to the grocery store or the barber shop. We explain how to find _X_ or _Y_ to our spouse, who explains to us how to find _Z_ or _W_. All these physical understanding performances of getting acquainted with a neighborhood build up a coherent mental image over time.
For another example, where do your children get their mental image of what a story is like? Certainly not from your giving them a formal definition of a story. Instead, from hearing lots of stories, asking questions about them, acting them out, and so on.
Or, for still another example, where did I get my "looks nice means suspect" mental image of mathematical relationships. No one ever told me point-blank as a mathematics student that neat-looking propositions should be viewed skeptically. I learned this by encountering many such propositions, by struggling with their proofs or disproofs, by calibrating my hopes and expectations to the real, albeit abstract, terrain of mathematics. By learning my way around the conceptual neighborhood of mathematics much as people learn their ways around real physical neighborhoods.
In summary, mental images and understanding performances occur in a kind of reciprocal relationship. Helping students acquire mental images by whatever means—including direct instruction—equips them for understanding performances. But also,, involving students in understanding performances—efforts to predict, explain, resolve, exemplify, generalize, and so on—helps them build up mental images. So there is a kind of partnership between mental images and understanding performances. They feed one another. The two are, you could say, the yin and yang of understanding.
Understanding performances and mental images become the interlocking elements that yield a pedagogy of understanding. But how might one play out this conception? If our most important choice is what we try to teach, what kinds of understanding performances should we try to teach? And what kinds of mental images? The sections to come stake out further what to try to teach in a pedagogy of understanding.
### LEVELS OF UNDERSTANDING
_If you can't solve it in ten minutes, you can't solve it at all._
Many students of mathematics believe essentially this. It's their credo for mathematics problem solving. Mathematics educator Alan Schoenfeld at the University of California at Berkeley has written on students' mindsets about mathematics, one of them being this "ten-minute rule." Such an attitude undermines persistence, and although blind persistence is no virtue, intelligent persistence is one of the most powerful resources for learning and problem solving.
Notice an interesting feature of the ten-minute rule: It does not concern any particular piece of mathematical content, not square roots nor the Pythagorean theorem nor the quadratic formula. Instead, it's general. It's an overarching posture toward the mathematical enterprise. In fact, the ten-minute rule is a mental image about mathematics. Although expressed in verbal form, it's essentially a holistic attitude toward the character of mathematical problems. Either they yield quickly or they don't yield at all. Either you get it quickly or you don't. And, as images do, this image influences understanding performances.
Here is another example of a very general mental image about mathematics. Dan Chazen, an investigator of mathematics learning, has found that students of Euclidean geometry have some very odd ideas about the nature of proof. Catch students after successfully doing a proof, and ask them whether they might possibly find an exception. Often they will say, "Oh yeah, if you look hard enough. Maybe an unusual triangle or a quadrilateral where the theorem does not hold up."
This is an odd image of proof to have. A formal deductive proof establishes a theorem always and forever, with no exceptions. But somehow, many students miss this point and end up with a mental image of proofs as simply pretty good evidence that does not completely settle the question. Note again, as in the case of the ten-minute rule, how this attitude toward proof has no link to any specific theorem. It's general.
Why do I bring up these particular examples of mental images? To underscore two points: (1) mental images that students have are often pivotal to their understanding of a subject matter; (2) the mental images are often not part of what is ordinarily called content. They are more general and overarching. Typical content instruction rarely touches on them directly. But teachers, by listening to what students say, watching how they behave, asking general questions, and learning what research shows, can become more aware of these general mental images and can then include in their instruction direct attention to these overarching mental images that sometimes impair and sometimes empower students.
Is there any way of organizing the general images that students harbor? We might say that there are different levels of understanding involved in understanding a subject matter. Learners need to understand things not only about particular concepts but about the whole enterprise of a subject matter, the game of mathematics, of history, of literary criticism. The understanding of particulars sits within the context of these umbrella understandings.
Colleague Rebecca Simmons and I developed a four-tier analysis of levels of understanding. Our levels were:
**Content.** Knowledge and know-how concerning the facts and routine procedures of a subject matter. The relevant performances are not by and large understanding performances but are reproductive: repeating, paraphrasing, executing routine procedures. The mental images are particular and, although important, somewhat parochial: the layout on paper of long division, a synoptic "mental movie" of the Civil War. Conventional education exposes students to a lot of knowledge at this level.
**Problem Solving.** Knowledge and know-how concerning the solution of characteristic textbook problems in the subject matter. The relevant performances are one kind of understanding performance: problem solving in the textbook sense; for instance, solving word problems or diagraming sentences in English. The mental images involve problem-solving attitudes and strategies: The negative ten-minute rule fits here, along with its opposite, which says "you often can get a problem to yield by intelligent perseverance." Here too are familiar problem-solving strategies like dividing a problem into parts. Conventional education provides abundant practice in problem solving, but very little direct instruction in problem-solving-related knowledge!
**Epistemic.** Knowledge and know-how concerning justification and explanation in the subject matter. The relevant understanding performances include generating justifications and explanations; for example, justifying a critical opinion in literature or explaining causes in history. The mental images express the forms of justification and explanation appropriate to the discipline. For instance, from the geometry example earlier, the "pretty good evidence" image of proof versus the "really reliable" image. Apart from Euclidean geometry, conventional education gives very little attention to justification and explanation. In contrast with problem solving, students are generally not even engaged in activities of justification and explanation.
**Inquiry.** Knowledge and know-how concerning the way results are challenged and new knowledge constructed in the subject matter. The relevant performances include advancing new hypotheses (new to oneself at least), challenging assumptions, and so on. The mental images include the spirit of adventuring and a sense of what makes for a "good bet" hypothesis—potentially illuminating and valid. As with the epistemic level, conventional education gives very little attention to the inquiry level.
In summary, there is a great deal of important knowledge and know-how concerning a subject matter that simply does not sit at the content level. Conventional instruction gives the higher levels of understanding hardly any attention. Yet the very spirit and structure of subject matters, fields, and disciplines lie at these levels.
So do important contrasts among the subject matters. In mathematics, evidence consists in deductive proof. Examples will not do. In physics, just the reverse applies: Although you may deduce predictions from a given theory, the ultimate litmus consists in a check with empirical reality. Noting such contrasts and extrapolating their implications for activities within mathematics, physics, and other disciplines are parts of what it is to understand the subject matters individually and collectively.
A pedagogy of understanding therefore demands at least a Theory One treatment of subject matter knowledge at these levels. In particular, to run through Theory One again, students need clear information at all these levels: The instruction should present or otherwise cultivate the development of relevant mental images. Also, students need thoughtful practice with the understanding performances characteristic of the levels in order to sharpen the performances and strengthen the mental images. They need informative feedback to refine their performances. And they need intrinsic and extrinsic motivation, which should consist in good part in awakening learners to the power and perspective afforded by a more bird's-eye view of a subject matter.
The smart school gives teachers the opportunity to think, talk with one another, and learn some more about higher levels of understanding in their subject matters and the encouragement to pay serious attention to higher levels of understanding in their teaching. Such teaching is not terribly technical or demanding. It reaches only a little further than many proactive teachers already do. What might instruction in this spirit look like? Suppose students are studying the well-known sonnet by William Words-worth, "The world is too much with us."
_Content level._ The teacher might go over the lines of the poem, clarifying terms and allusions, and would expect the students to "know" the poem and information about it on a quiz later.
_Problem-solving level._ Interpretation is a typical problem in literature. The teacher might ask students to work up interpretations of key lines, such as "I'd rather be / A Pagan suckled in a creed outworn." What is Wordsworth saying there? What would it mean to be such a pagan, and why did Wordsworth think of that as a worthwhile contrast to people's usual "getting and spending" from line 2? Also, the teacher might suggest strategies that students could use to tackle such problems of interpretation and might coach them along.
_Epistemic level._ The teacher might press for reasons for their interpretations: What's your evidence and argument that this line means what you say? Moreover, the teacher might engage the students in a discussion of what counts as evidence for a literary interpretation and what kinds of evidence one should look for.
_Inquiry level._ So far, all has turned on teacher-generated questions, like the one about the "creed outworn" line. In addition, or instead, the teacher might encourage the students to find their own puzzles in the poem and might talk explicitly with them about what makes a literary puzzle worth pursuing.
Now some readers may be put off by this example, because it does not truly involve students' immersing themselves in the poem and finding their own personal reactions to it. That approach is fine too!—full of understanding performances, in fact. Both, in my view, are important parts of literary study. However, I've chose this approach to show how readily straightforward literary criticism can reach beyond the content level to encompass all the levels of understanding.
### POWERFUL REPRESENTATIONS
How do we represent things to make them understandable? Sometimes in stories. Here's an example:
A grammarian fell into a well one day and had difficulty climbing up the slippery sides. A little later a Sufi chanced by and heard the man's cries for succor. In the casual language of everyday life, the Sufi offered aid. The grammarian replied, "I would certainly appreciate your help. And by the way, you have committed an error in your speech," which the grammarian proceeded to specify. "A good point," acknowledged the Sufi. "I had best go off awhile and try to improve my skills." And so he did, leaving the grammarian at the bottom of the well.
This is a story from a literary and cultural tradition we do not so often encounter, the Islamic tradition of Sufi teaching tales, the same tradition that yielded the widely known parable of the three blind men and the elephant. It is a representation designed to cultivate understanding. Like many such, it has an analogical character. The tale does not so much concern Sufis or grammarians specifically as it concerns academicism, grace, and getting your priorities right. In fact, the story gives us a mental image about such things. If we take it to heart, we may understand some of our own foolishness better.
The Sufi tradition of teaching tales is just one example of the use of succinct stories in building mental images. And all such stories are but one kind of representation among many that can serve the pedagogy of understanding by helping people build mental images.
From Sufis to Physics
Take diagrams. Most diagrams used in science problem solving are specific to a quantitative problem: so much mass at such-and-such a height, or something of the sort. But qualitative diagrams can represent more general situations.
Imagine that you have been studying the mechanics of motion. This problem is posed: A rocket travels through space in free fall, unpowered. The captain, aiming to redirect the course of the rocket, turns it so that it points sideways to its direction of travel and fires the engines. Your task: Draw a diagram showing qualitatively the path the rocket will take.
Such a question can elicit a variety of responses. Some typical ones appear in the following diagram. You can probably find your response among them.
This is just the sort of question that commonly discloses students' misconceptions about the science they have been studying. Note how it involves no numbers and invites no routine calculations. In fact, responses _A_ and _B_ in the diagram are mistaken, whereas _C_ is correct. The problem with responses _A_ and _B_ is that they treat the initial motion of the rocket as though it was somehow eliminated by the action of the newly fired engines. According to the laws of Newton, that initial momentum remains and continues to influence the trajectory of the object. So the right answer is version C, which shows a smooth turn as the momentum in the rightward direction builds up while the original motion remains.
A teacher might use a task like this and a diagram like the one below (perhaps having the students construct the diagram by generalizing from their own responses) to help students understand Newton's principles. The qualitative diagram highlights ideas that never tend to come up in normal quantitative diagrams.
Even better might be diagrams that actually move, to give students something closer to the experience of Newtonian motion. Happily, we have those. Educational psychologists Barbara White and Paul Horwitz developed a computer environment called ThinkerTools that provides students with a Newtonian world to play with. In this environment, friction can be turned off or up. Gravity can be turned off and on and set to different strengths. Dots move about according to user-applied impulses, manifesting Newton's laws moment to moment. Moving objects can leave behind copies of themselves every second to show changes in their direction and velocity. In other ways, as well, the ThinkerTools environment demonstrates Newtonian motion clearly, allows learners to manipulate it, and makes key features of it salient by additional notational devices. In other words, ThinkerTools allows learners to build a better mental image of what Newtonian motion is like. Research done with the ThinkerTools environment has demonstrated that learners attain considerably more understanding of Newtonian motion than through more conventional instruction.
ThinkerTools is one case in point among many. A considerable body of evidence shows that carefully chosen representations can provide learners with mental images that enhance their understanding. Educational psychologist Richard Mayer recently reported an extensive series of experiments in which science concepts were taught both conventionally and accompanied by some sort of conceptual model, typically a visual representation that illustrated in some simple fashion what the concept meant and how it worked. For instance, a lesson on radar included a five-step diagram that showed a radar pulse moving outward from the source, striking an object, and bouncing back, with the total travel time measured to determine the distance. A lesson on the concept of density showed volume by the number of equal-sized boxes and density by the number of equal-mass particles in every box.
Mayer discovered that students' verbatim recall of the concepts taught did not differ much with or without the conceptual models. But when the conceptual models made up part of the lesson, recall of the gist of the message was superior. Moreover, the students showed much better performance on problems that asked them to extrapolate from what they had learned (understanding performances again). Such advantages appeared for weaker students but not for stronger ones, who, it seemed, constructed their own conceptual models. Interestingly, Mayer also discovered that conceptual models presented _after_ a lesson yielded no positive effects. Mayer suggests that conceptual models presented after a lesson about a concept run up against students' already formed ideas and fail to penetrate.
Concrete, Stripped, Constructed Analogs
Colleague Christopher Unger and I generalized some key characteristics from many of the powerful representations that have proved effective in building students' understandings. Quite commonly, powerful representations are what might be called _concrete, stripped, constructed analogs_. There is a lesson to be learned about the use of representations by focusing on what each of these terms means.
_Analogs._ Most such representations provide some kind of analogy with the real phenomenon of interest. For example, pictures of rockets and dots for their paths are not real rockets and trajectories. In the "ThinkerTools" Newtonian-motion computer simulation, dots on a screen are not real objects in motion, but behave like them.
_Constructed._ Most commonly, the analogs are fabricated for the purpose at hand. Analogs based on common knowledge are often misleading. For example, one can characterize the atom as a small solar system, but in several ways the analogy is deceptive. By drawing the diagrams, programming the computer simulations, or telling the stories as we want them to be rather than relying on direct allusion to everyday experience, one can avoid this.
_Stripped._ Most such representations eliminate extraneous clutter to highlight the critical features. For example, the diagram above lacks detail and the "ThinkerTools" environment does not show rockets or flying saucers in motion, but simply dots.
_Concrete._ Most such representations make the phenomenon in question concrete, reducing it to examples, visual images, and so on.
Of course, not all representations that powerfully enhance understanding have to fit this profile. Representations of diverse sorts play important roles in learning and understanding. Still, it was with some amusement that I discovered, months after Unger and I had formulated the four criteria above, that the ancient tradition of Sufi tales fit them just as well as ThinkerTools!
The tale of the grammarian, for example, is an analog for a more general class of situations where fussy correctness threatens to override what's really important. The story makes the idea behind it concrete. The story is plainly constructed, not an actual experience. And finally, the story is stripped: nowhere near as elaborated in terms of character and setting as it would be were it presented primarily as a literary work. ThinkerTools may be a product of twentieth-century technology, but the human ingenuity to create powerful representations is ancient.
All this may sound rather technical, an unwelcome barrier for busy teachers. True! Although I've tried to cast some analytical light on what makes representations powerful, intuitions are worth trusting here. There is no need to worry about a checklist of key features of representations. Teachers—and students—can tap their intuitions. Does a representation look and feel as though it's making things clearer? If not, can you make up an image or analogy that works better? Can you trim down an image or analogy, getting rid of clutter to make the point clearer? What's really important is not technical criteria but the free and imaginative reach for diverse representations to build understandings.
### GENERATIVE TOPICS
We've been discussing what to do with a topic to teach the topic for understanding: engage students in understanding performances, involve higher levels of understanding, use powerful representations.
But what about the choice of topics itself? Might some topics not lend themselves much more to a pedagogy of understanding than others do? To be sure, much can be made of any topic by sufficiently artful teaching. But this does not mean that all topics are created equal. One might speak of "generative topics," topics that particularly invite understanding performances of diverse kinds, topics that make teaching for understanding easy.
Again we are back to the theme of what we choose to try to teach. Many of the topics taught in the conventional treatment of the subject matters do not appear to be very generative. They are not chosen for their outreach, their import, their connectability. A pedagogy of understanding invites reorganizing the curriculum around generative topics that provoke and support a variety of understanding performances.
It is even possible to lay down some standards for a good generative topic. Here are three, stemming from collaborative work of colleagues Howard Gardner, Vito Perrone, and me:
_Centrality_ —the topic should be central to a subject matter or curriculum.
_Accessibility_ —the topic should allow and invite teachers' and students' understanding performances rather than seeming sparse or arcane.
Richness—the topic should encourage a rich play of varied extrapolation and connection making.
But what are some good-bet generative topics? Here is a sample drawn from the collaboration mentioned above:
**Natural Sciences.** Evolution, focusing on the mechanism of natural selection in biology and on its wide applicability to other settings, like pop music, fashion, the evolution of ideas. The origin and fate of the universe, focusing qualitatively on "cosmic" questions, as in Stephen Hawking's _A Brief History of Time_. The periodic table, focusing on the dismaying number of elements identified by early investigators and the challenge of making order out of the chaos. The question "What is real?" in science, pointing up how scientists are forever inventing entities (quarks, atoms, black holes) that we can never straightforwardly see but as evidence accumulates, come to think of as real.
**Social Studies.** Nationalism and internationalism, focusing on the causal role of nationalistic sentiment (often cultivated by leaders for their own purposes), as in Hitler's Germany, in world history, and in the prevailing foreign policy attitudes in America today. Revolution and evolution, asking whether cataclysmic revolutions are necessary or evolutionary mechanisms will serve. Origins of government asking where, when, and why different forms of government emerged. The question "What is real?" in history, pointing up how events can look very different to different participants and interpreters.
**Mathematics.** Zero, focusing on the problems of practical arithmetic that this great invention resolved. Proof, focusing on the different ways of establishing something as "true" and their advantages and disadvantages. Probability and prediction, highlighting the ubiquitous need for simple probabilistic reasoning in everyday life. The question "What is real?" in mathematics, emphasizing that mathematics is an invention and that many mathematical things initially were not considered real (for instance, negative numbers, zero, and even the number one).
**Literature.** Allegory and fable, juxtaposing classic and modern examples and asking whether the form has changed or remains essentially the same. Biography and autobiography, contrasting how these forms reveal and conceal "the true person." Form and the liberation from form, examining what authors have apparently gained from sometimes embracing and sometimes rejecting certain forms (the dramatic unities, the sonnet). The question "What is real?" in literature, exploring the many senses of realism and how we can learn about real life through fiction.
Many of these topics do not sound very much like those typically focused on in the subject matters. Compare them to "mixed numbers" or "factoring" from mathematics; "poetic feet" or "adverbs" from English; "Abraham Lincoln's early years" from history; and so on. To be sure, we have to remember what was acknowledged earlier: Much can be made of any topic. But truly generative topics reach for depth and breadth to a degree that more customary topics do not. They can provide the basis for foundational reorganizing of instruction, as indeed very similar topics and questions have in the Coalition of Essential Schools organized by Theodore Sizer and his colleagues.
This should not be taken as a blanket condemnation of the typical organization of subject matter instruction. Inevitably, there are a number of more specific and localized topics that invite weaving into the fabric of the subject matter. However, as underscored earlier, in general there is far too _much_. The trivial pursuit model has led to huge compilations of bits and pieces.
The smart school wants it just the other way. Working toward informed, energetic, and thoughtful learning, the smart school encourages teachers to think deeply about what they are teaching and why and gives them time and background information to help. In the smart school, there are fewer bits and pieces, and they cluster around more general and pregnant generative topics.
### AN EXAMPLE OF TEACHING FOR UNDERSTANDING
Concretely, what does all this mean for classroom practice? What happens in the truly thoughtful school? Suppose, for example, that youngsters in a class have just read one of the tales from the Sufi tradition alluded to earlier.
* * *
KEY IDEAS TOWARD THE SMART SCHOOL
**CONTENT: A PEDAGOGY OF UNDERSTANDING**
**The Nature of Understanding**
**Understanding Performances.** Explanation, exemplification, application, justification, comparison and contrast, contextualization, generalization, and the like.
**Mental Models.** Breadth, coherence, generativity, accessibility for mental models in the strong sense. Mental models enable understanding performances. Understanding performances build up mental models.
**Levels of Knowledge.** Content, problem solving, epistemic, inquiry.
**Representations for Understanding.** A variety of media and symbol systems. Often concrete, stripped, constructed analogs.
**Generative Topics.** Centrality, accessibility, richness.
* * *
**The Conventional Classroom.** The teacher asks the student for a definition of fable. The teacher then asks the students to recount what happened in the story. And why do we call it a fable? Does it fit the definition? Then the students discuss the point or moral of the story. And that's that.
**The Thoughtful Classroom.** The fable is a locus for a much richer and more complex array of activities:
_Generative topics._ The fable is one example within an ongoing topic concerned with allegory and fable. "Why do we have fables?" the teacher asks. "Do all cultures have fables?" "Where do you think fables come from?" As the youngsters get the spirit of connection making, the teacher encourages them to ask their own questions: "Why do fables stick around for so long?" "Why do fables sort of sound alike?" "Are any of the fables we read really helpful?"
_Mental images._ The teacher uses many classic exemplars of fables, and contrasting examples of nonfables are used to help the students develop a feel for what fables are like. "What about jokes?" the teacher asks. "Are jokes fables? How are they like and not like fables? Can you find a joke that's more like a fable than most jokes?"
"Do jokes ever have 'morals,' like fables do?" one student asks. The class ponders that one. Can anyone think of a joke that has a moral? Two or three jokes are told. One of them does have a moral. The class discusses how jokes and fables are the same and different.
_Understanding performances._ Questions like those above ask the students to engage in understanding performances, explaining, choosing, extrapolating, developing arguments, and so on. Such questions can encourage learners to relate fables to their everyday lives, find cases where the fable would offer sound advice, make up cases where the fable would offer poor advice, create another fable with the same moral, revise the fable to offer a different moral.
While students can pursue some of these enterprises through class discussion, others are too large. They demand group work or work at home to afford time to think things through and craft a product.
_Levels of understanding._ The teacher involves students in exploring how you know what a fable really means. "How do you test your interpretation against the story?" the teacher asks. "Is this different from other ways that we test ideas in literature or in other fields?" "Let's work through an example. We'll all write what we think the moral of this fable is. Then we'll see if we agree. If not, we'll look for evidence. And we'll pay attention to what kinds of evidence we need."
The students give it a try. Of course, they all don't draw the same moral. Thinking on the Sufi story about the grammarian, maybe one child says, "It means that you shouldn't say bad things to people who might help you."
"Okay, can you find some evidence for that in the story?" the teacher asks.
"Well, the grammarian, he says what's wrong with the way the Sufi talked. And then the Sufi walked away."
"Good. How about another moral?" Another student answers. "It's about what's important. Getting out of the well is more important than grammar. It's life or death."
"And what's your evidence?"
"Well, it's sort of the same. The grammarian tells the Sufi his mistakes. But the grammarian isn't thinking about his own situation. He's not thinking about what's important."
After a little more of this, the teacher shifts focus. "Okay, we'll see what other morals there might be in a moment. But lets stand back and look at what we're doing. We're looking for evidence in the story. How do we do that? What do we look for? What's evidence?"
After a pause and a little puzzlement, the students begin to respond. "Well, you sort of read it over and look for things the story says that fit." "You think about what happens in the story." "You can't just make it up. You have to find some words that say the evidence." "Sometimes you can use the same evidence in more than one way." In this sort of exchange, teacher and students begin to deal directly and explicitly with the epistemic level of understanding.
_Powerful representations._ The teacher asks the students to make up unifying metaphors that express the students' mental images of fables. One child suggests, "A fable is like a peach, with a tasty outside but a tough center. That's the moral."
Another says "A fable is like a joke, only not always funny. It's like a joke 'cause it tells the story but you have to get the point yourself."
Of course, all this is simply by way of example. Many different and rich lessons cultivating students' understanding could be built around the same fable. To offer a formula for a pedagogy of understanding would defeat the enterprise at the outset, working against the extrapolative character of understanding performances.
But to disdain formulas is not to eschew guidelines. The notions of understanding performances, mental images, higher levels of understanding, powerful representations, and generative topics offer a broad framework for changes in what we try to teach in the smart school.
## CHAPTER 5 **CURRICULUM
Creating the Metacurriculum**
_What do you do when you don't understand something if you want to understand it better? For example, what would you do if you wanted to understand something like Abraham Lincoln's Gettysburg Address, long division, or a suit of armor, just by thinking about it? What questions would you ask yourself about it?_
This was the question my colleagues Heidi Goodrich, Jill Mirman, Shari Tishman and I concocted for youngsters to probe their ideas about how to approach a problem of understanding. We wondered what their ideas were about the challenge of understanding and what strategies they knew. One thing we quickly discovered was that even fourth and fifth graders had some pretty sophisticated notions. One fourth grader wrote:
First I would ask myself: What is it? Then, why we need it? How does it work or how it happens? If for example I can't understand a word, I read the title and think about what the title means. Then I would read the sentence before it and after it twice. Next I would read that sentence and replace the word with a word that might fit in its place.
Some chose to generate specific questions rather than mention their general strategies.
1. How do you make a Lego? 2. How do we think? 3. How do we move? 4. How did we get the alphabet? 5. How do taste buds taste? 6. How do we hear? 7. How do we read? 8. How do I do this test? 9. What is this test about? 10. How hard is this test?
Apparently, not very hard for this curious youngster. Of course, whereas these are two particularly rich responses in their distinctive ways, there were also sparse responses.
First when my teacher asks me if I understand I would not answer yes. I would only answer _that_ if I knew the answer and if I didn't know the answer I would ask her to explain it more clearly.
And, even more baldly:
I have no idea what goes on in my head when I don't understand something.
However sparsely or richly, all these youngsters, cued by a simple question, proved able to reflect upon their own thinking and learning. Moreover, many of the children revealed quite specific and sophisticated ideas about the thinking and learning process. Their responses illustrate "metacognition"—thinking about thinking (including learning). The liveliness of their answers and the practical utility of many of their strategies speak to the importance of what we could call the metacurriculum.
### THE IDEA OF THE METACURRICULUM
The basic idea of the metacurriculum is a simple one. It says that our usual notions of subject-matter content leave out higherorder knowledge.
But what does higher-order knowledge mean? For a start, let's proceed by way of example:
From the previous section, fourth and fifth graders' ideas about questions to ask themselves to understand something are higher-order knowledge—knowledge about how to get knowledge and understanding.
General problem-solving strategies like "divide a problem into subproblems" are higher-order knowledge—knowledge about how to think well.
Familiarity with ideas like hypothesis and evidence—and with what you do with such ideas, such as make hypotheses and test them by seeking evidence—is higher-order knowledge about thinking.
Knowledge about what evidence is like in different subject matters—formal proof in mathematics, experiment in science, argument from the text and from historical context in literature—is higher-order knowledge about the way the subject matter works.
As these examples suggest, what makes higher-order knowledge higher order is its aboutness. Higher-order knowledge is about how ordinary subject-matter knowledge is organized and about how we think and learn.
People sometimes worry that higher-order knowledge consists only of generalities disconnected from the subject matters. But on the contrary, much of it specifically concerns particular subject matters and is arguably an essential part of understanding a subject matter. Look again at the last bulleted item above. Certainly, understanding how mathematics or science or literature works requires understanding what evidence is like in those disciplines.
Of course, not all higher-order knowledge is knowledge about particular disciplines. Much of it concerns people's knowledge of how they think and learn. This is often called "metacognitive" knowledge, because it is knowledge about how cognition works. The fourth grader with all the questions like "What is it?" and "Why do we need it?" showed considerable metacognitive knowledge about how to build an understanding. My colleague Robert Swartz and I have defined four levels of metacognition: tacit, aware, strategic, and reflective. Tacit learners are unaware of their metacognitive knowledge. Aware learners know about some of the kinds of thinking they do—generating ideas, finding evidence—but are not strategic in their thinking. Strategic learners organize their thinking by using problem solving, decision making, evidence seeking, and other kinds of strategies. Finally, reflective learners not only are strategic about their thinking but reflect on their thinking-in-progress, ponder their strategies, and revise them.
Whether about the subject matter or one's own thinking, all these kinds of higher-order knowledge are parts of the metacurriculum. The idea of the metacurriculum puts more flesh on the basic notion expressed two chapters ago that _our most important choice is what we try to teach_. The normal curriculum deals with conventional content and rarely touches the metacurriculum, another kind of content that addresses the learner and the subject matters from a higher-order perspective.
Motivating the Metacurriculum
Nice, this higher-order perspective. But is it necessary? Do we really need it for the smart school?
We do if we want to accomplish the hard-to-argue-with goals advanced in chapter 1. It was already argued there that conventional educational practice does not achieve what we want for any of these objectives. We do not see the retention, or the understanding, or the active use of knowledge that we would like to see.
But how does the metacurriculum speak to these shortfalls? By dealing directly with all three. In particular, the metacurriculum includes skills of memorization, thus dealing directly with retention. It treats the conceptual organization of the subject matters and thinking, thus dealing directly with understanding. And it includes attention to transfer of learning, thus dealing directly with the active use of knowledge.
Besides this general argument, there is the testimony of various individuals and organizations who have thought hard about the dilemmas of education in recent times. For example, in the widely read report _High School: A Report on Secondary Education in America_ , Ernest L. Boyer, president of the Carnegie Foundation for the Advancement of Teaching, speaks to the need for enhanced literacy in a broad sense. He includes not just better mechanics of reading and writing, but higher-order skills of dealing thoughtfully with texts.
In like spirit, the report of Project 2061, an effort to reconceive the content of science and mathematics instruction, urges attention not just to selected conventional content (Newton's laws, atomic theory) but to several "meta-aspects" of science and mathematics (the nature of scientific inquiry, the evolution of scientific thought, the practice of scientific thinking). Recent recommendations of the National Council of Teachers of Mathematics emphasize the centrality of flexible problem solving to the learning of mathematics.
Building the Metacurriculum
If the metacurriculum is so important, what does it look like? First of all, the metacurriculum is not a separate curriculum, with its own class periods. The metacurriculum should be blended in rather than added on. The metacurriculum is infused into the usual teaching of the subject matters, enriching and amplifying them. Without pretending to exhaustiveness, we can certainly list some key components of this metacurriculum. Here they are in preview, with upcoming sections saying more about each of them:
_Levels of understanding_. As in the previous chapter, kinds of knowledge "above" the level of content knowledge in their abstraction, generality, and leverage (e.g., problem-solving strategies).
_Languages of thinking_. Verbal, written, and graphic languages that assist thinking in and across subject matters.
_Intellectual passions_. Feelings and motives that mobilize the mind toward good thinking and learning.
Integrative mental images. Mental images that tie a subject matter or large parts of it together into a more coherent and meaningful whole.
Learning to learn. Building students' ideas about how to conduct themselves most effectively as learners.
Teaching for transfer. How to teach so that students use in other subject matters and outside of school what they learn in a particular subject matter.
If the metacurriculum reminds us of anything prominent these days on the educational scene, it is thinking skills. For more than a decade, many educators have worked hard to understand the nature of complex cognition and sought ways to teach students how to think better. Indeed, this has been one of my own major areas of research and materials development.
However, the metacurriculum is much more than thinking skills. It is a larger concept in several ways. Whereas thinking skills generally do not focus on the subject matters, the metacurriculum concerns their conceptual organization as well. Whereas thinking skills usually are seen as cross-disciplinary, the metacurriculum emphatically includes discipline-specific skills. Whereas thinking skills by name and nature center on thinking, the metacurriculum includes integrative mental images and teaching for transfer.
Certainly, the inspiration for the metacurriculum comes from contemporary efforts to teach thinking skills. But ambitious as they are, those efforts may not be ambitious enough to mesh well with the larger enterprise of education. By going further, the notion of the metacurriculum may help to make plain how essential to youngsters' learning a higher-order perspective is.
Let us look at some pieces of the metacurriculum in more detail.
### LEVELS OF UNDERSTANDING
The notion of levels of understanding is already familiar from the previous chapter. It means here what it did there: Attention should be paid not just to facts and routines but to the problemsolving, epistemic, and inquiry levels of understanding. Recall that the problem-solving level concerned how to solve typical problems in a discipline. The epistemic level had to do with the nature of evidence and explanation in a discipline. The inquiry level addressed the kinds of questions and explorations characteristic of the discipline.
This is a good place to underscore how very useful such higher-order knowledge can be. Investigations of mathematical problem solving conducted by Alan Schoenfeld have shown substantial gains from instruction in good problem management and the use of problem-solving strategies. For instance, students who learn to monitor their progress on problems, asking themselves "Am I making progress with this approach?" "If not, can I find another approach?" "How can I check my answer?" and so on, make better use of their knowledge of mathematics in the solving of problems.
How about the epistemic and inquiry levels? Researchers Posner, Strike, Hewson, and Gertzog argue that the acquisition of science concepts with genuine understanding depends in part on the "conceptual ecology" within which the particular concept sits, including matters of standards of inquiry, how things are supposed to fit together, what's a puzzle and what isn't, and so on. A number of interesting interventions in science and mathematics learning operate at the epistemic and inquiry levels emphasized by Posner and his colleagues.
For example, educator-software designers Judah Schwartz and Michal Yerushalmy developed a software environment called The Geometric Supposer. The Supposer makes it very easy for students to do geometric constructions on computer. Students can bisect a line segment, drop an altitude, or try out other constructions freely and flexibly. This allows them to poke around with geometry, looking for interesting relationships and formulating conjectures that could become theorems. Of course, a conjecture that passes the test of "coming out right" when worked out several times on the Supposer still needs a proper proof.
Students using the "Supposer" commonly rediscover important classic theorems rather than learning them by rote from the textbook. The moral: The Supposer works at the inquiry level, making Euclidean geometry an inquiry-oriented subject, which it usually is not. Moreover, the Supposer has payoffs at the epistemic level (evidence and explanation), because it emphasizes the confusing distinction between particular constructions where a conjecture holds up and an actual logical proof.
For another example, experiments by John Clement and colleagues on youngsters' understanding of Newton's laws encourage students to use analogy to detect logical incoherences in their own conceptualizations of physics phenomena. For instance, some of their experiments concern force and bending: If you place a book on a table, does the table bend slightly and push back on the book? The answer is crucial to learners' understanding of Newtonian mechanics. Many students say no at first. They agree, however, that if you place a book on a spring, or a very thin board, those certainly bend and push back. So where to draw the line? By imagining thicker and thicker boards, many students realize that a logically simpler picture comes from saying that even thick tables push back and finally, that any push has an opposing push—there is always a reaction force in Newtonian terminology. The moral: Clement's technique works at both inquiry and epistemic levels. As to inquiry, it involves students in reasoning out something for themselves, albeit with the teacher's support. As to the epistemic level, it highlights argument by analogy and the importance of simplicity in explanation.
For a third example, specialists in the teaching of thinking Robert Swartz and Sandra Parks have developed a number of lessons to demonstrate the infusion of thinking strategies into subject matter instruction. One illustrative lesson focuses on decision making in history. The lesson looks closely at Harry Truman's decision to use the atomic bomb to end World War II. Students read a testimonial from Truman about the care with which he took the decision and how troubling he thought it to be. Then the students brainstorm alternative plans, putting themselves in Truman's shoes: What else could have been done? They analyze the consequences of other possible plans, including a land invasion. Along the way, they read original source documents that inform their reasoning. Representatives of the American military comment on likely losses from a land invasion. Japanese generals articulate their steadfast determination to carry the war to the limit.
At the beginning of the exercise, many students view Truman's decision as appalling. By the end, many are less sure. They have learned something about the historical circumstances and the importance of examining options and consequences. The moral: The Truman lesson operates at the inquiry level by projecting students into an active historical role. The lesson also carries an epistemic message about history: the evidential weight of original source materials.
With such examples in mind, problem-solving, epistemic, and inquiry knowledge become obvious essentials for the content of the metacurriculum.
### LANGUAGES OF THINKING
As mentioned earlier, one natural part of the metacurriculum is the teaching of thinking skills, an enterprise that has generated considerable activity and controversy in education over the past two decades. Some thinking skills fall naturally within the levels of understanding just discussed. Others, though, seem less related to the disciplines: skills of decision making, everyday practical problem solving, or communication. Since levels of understanding by no means soaks them all up, they deserve separate discussion.
One problem with the thinking skills movement has been the narrow sound of the term "skills." Indeed, hardly anyone associated with efforts to teach thinking is happy with it. A much broader and more flexible way to understand the enterprise is as cultivating languages of thinking.
The Resources of English
One language of thinking is part of ordinary English. Chapter 2 mentioned a clear example: the investigations at the Ontario Institute for Studies in Education of David Olson, Janet Astington, and Richard Wolfe. They looked at the extent to which certain school textbooks included the ordinary everyday vocabulary of thinking that English provides—terms such as "hypothesize," "believe," "predict," and so on. Disturbingly, their studies revealed that such words rarely appeared in textbooks at all. Authors apparently avoided them on the grounds that students wouldn't understand.
The consequence of this dumbing down of textbooks is dismaying. Students rarely encounter, and so lack familiarity with, a very fundamental vocabulary concerned with the critical and creative exploration of ideas. Therefore, one important part of the metacurriculum has nothing to do with thinking skills in any special sense. It simply says, "Let's get back into place in the schools an important part of our common linguistic heritage."
Arthur Costa, a former president of the Association for Supervision and Curriculum Development and a vigorous campaigner and consultant toward the smart school, directly addresses teachers' use of language in a well-known article called "Do You Speak Cogitare?" By "Cogitare," Costa means ways of using the English language that exercise the vocabulary of thinking and foster thoughtfulness. Costa emphasizes by contrast how teachers can frame their utterances differently to promote thoughtfulness. For example:
Teachers can use the vocabulary of thinking. Instead of saying, "Let's look at these two pictures," a teacher might say, "Let's _compare_ these two pictures." Or instead of "What do you think will happen when...?" "What do you _predict_ will happen when...?"
Teachers can handle discipline in ways that encourage thoughtfulness. Instead of saying, "Be quiet!" a teacher might say, "The noise you're making is disturbing us. Is there a way you can work so that we don't hear you?" Instead of "Sara, get away from Shawn!" a teacher might say, "Sara, can you find another place to do your best work?"
Teachers can provide questions rather than solutions. Instead of saying, "For our field trip, remember to bring spending money, comfortable shoes, and a warm jacket," a teacher might say, "What must we remember to bring with us on our field trip?"
Teachers can press for specificity. When a student says, "Everybody has one," the teacher might say, "Everybody? Who exactly?" Or when a student says, "This cereal is more nutritious," the teacher might say, "More nutritious than what?"
By shifts of locution such as these, Costa shows how teachers can use language artfully to make the classroom a more thoughtful place. Ultimately, as children do in any language environment, they will begin to pick up and internalize the idiom.
The Language of Strategies
Besides the everyday language of thinking ("believe," "predict," and so on), there is the language of thinking strategies. There are numerous efforts to improve particular kinds of thinking—problem solving, decision making, causal reasoning. Usually in such efforts, students are introduced to concepts and strategies for handling better the kind of thinking in question.
Good causal reasoning involves a set of significant terms and concepts—cause, effect, sufficient vs. contributing cause, multiple causes, and so on. There are standards to abide by and cautions about what to avoid: For instance, correlation is not sufficient evidence for causation. An increase in crime rates with the advent of television (correlation between the two events) does not prove that television causes crime. Maybe both were caused by some other event. Maybe it's coincidence. This is an important pitfall to know about: Many students (and not only students!) take correlation as strong evidence of causation; it is not.
Causal reasoning is a good "language" to cultivate in students because there are numerous applications in the curriculum: exploring causes of war, drugs, crime, or on the not-so-grim side, what makes a rocket work, an air conditioner cool, or even a poem speak to its audience with power and eloquence. The concepts, words, and strategies of the language of causation include some everyday ideas—like cause and effect—but also more technical ideas—like contributing cause and correlation. These are not so much a part of our commonplace linguistic heritage, but they are ideas to keep in mind in order to reason well about causes and effects.
There is considerable evidence that some thinking concepts and strategies can be taught to students in a fairly direct way with worthwhile results. For example, several years ago a team of researchers, including myself, wrote and tested a course called Project Intelligence (now distributed in the United States under the name Odyssey). The course was developed at Harvard university and the Cambridge consulting firm of Bolt, Beranek, and Newman under contract with the government of Venezuela.
The course taught a number of concepts and strategies in the areas of classification, decision making, inventive thinking, problem solving, and more. The lessons on decision making introduced a simple but powerful tool for organizing the exploration of a decision. You make a table with brainstormed options down the side and brainstormed criteria across the top. The boxes of the table give you a place to evaluate each option by each criterion. The lessons on inventive thinking highlighted the concept of design. The lessons introduced powerful questions that you can ask about any design—"What are its different purposes?" "How do its features serve those purposes?"—to develop students' appreciation for the ingenuity of ordinary objects like pencils and doorknobs. Then the students learned design strategies to help them invent simple gadgets and tackle other creative problems.
Project Intelligence was tested with a particularly elaborate set of measures. The course showed a major impact on seventhgrade students' cognition, including measures of the particular thinking concepts and strategies taught _and_ measures of general scholastic ability and intelligence. One unfortunate feature of the study was that we did not have a chance to do follow-up work, to see how the students performed six months or a year later. However, the initial results were very encouraging.
A number of positive findings for this and other programs are discussed in _The Teaching of Thinking_ by Raymond Nickerson, David Perkins, and Edward Smith. Accordingly, the teaching of concepts and strategies for reasoning about cause and effect, the soundness of beliefs, decision situations, and the like becomes an important part of the metacurriculum.
Thinking on Paper
"Languages of thinking" inevitably suggests verbal languages. But this is somewhat misleading because some interesting programs and experiments have actually involved visual symbols. Joseph Novak and his colleagues at Cornell University have conducted a number of studies of students' use of "concept mapping," a way of diagramming complex conceptual relationships. Similar techniques are called "webbing" and "mind mapping." The general idea is to create a network of lines connecting words and brief phrases. For example, to diagram the ecology of a pond, you might connect "tadpoles" to "frogs" with a line labeled "grow into." You might connect "frogs" to "flies" with a line that says "eat."
In some cases at least, students find that concept mapping represents an effective means of reviewing and consolidating their understanding of subject matter content. In the same spirit, Beau Jones, Jay McTighe, Sandra Parks, and John H. Clarke are among the investigators and developers who, concerned with effective reading and related performances, have explored pictorial formats for helping students generate and organize ideas. For instance, a neat graphic technique for comparing and contrasting uses two intersecting circles to compare, say, a sonnet by Shakespeare with one by Wordsworth. In the intersection of the two circles, students list features common to both. In the circle for each sonnet but outside of the intersection, the students list features distinctive to one or the other.
Pictorial languages of thinking have an advantage. They "download" onto paper complex patterns of thinking—the whole ecology of a pond or a dozen or more contrasts between sonnets by Shakespeare and Wordsworth. The downloading is important. One problem with pushing students' thinking beyond typical classroom levels is that it introduces an additional cognitive load for students. Another problem is that sessions largely conducted orally afford limited opportunity to look back and reexamine a line of thought. Thinking on paper helps solve these problems. Students do not have to hold so much information in mind at once, and they can look back at what they have written to rethink and revise it.
In this process the powerful resources of two traditional text forms, the essay and the story, should not be overlooked. Both can be potent means of laying out ideas. They do not take the place of resources like concept mapping, which are less formal and more flexible, but they do offer formats for the shaping and refined expression of ideas. Nor should we neglect other forms of writing that afford more flexibility than essays or stories—thought diaries, brainstormed lists and notes, and the like.
The Culture Connection
The notion of languages of thinking has another advantage over talk of skills: its cultural spin. It suggests that education is as much a process of acculturation as of learning particular pieces of knowledge. To achieve thoughtful learning, we need to create a culture of thoughtful learning in the classroom. This is a matter of how teachers talk to students, students to teachers, and students to one another. And talk here is of course a matter not just of the words used but of manner and style and goals.
For example, the "whole language" movement has over the past several years inspired and enabled many teachers to draw their students into classroom cultures of thinking and writing that draw on a number of different text forms. A theory-based perspective on teaching and learning rather than a method or package, whole-language teaching emphasizes how skill in reading and writing develops through authentic involvement in reading and writing activities. In this approach, students do not write mock pieces as exercises for the teacher; they write diaries, stories, advertisements, and arguments that have genuine contemplative and communicative functions—a reflective diary, a contribution to a school newspaper, stories for other students to read and enjoy.
All this reflects a developmental understanding of how language facility evolves. The whole-language perspective emphasizes how instrumental natural language learning is. Toddlers learn their mother tongues because each bit of skill and understanding can help them do something that makes sense and looks worthwhile in context. The same mechanisms can be harnessed in the classroom, where learning activities should not be exercise-like rituals directed toward some vaguely promised goal of mastery but rather should make sense and look worthwhile in a context of communication.
Whether classrooms attain a culture of thoughtfulness has been a direct object of research. Fred Newmann of the Wisconsin Center for Education Research at the University of WisconsinMadison has investigated what might be termed "the thoughtful classroom," examining a number of variables concerning how much a teacher models, expects, and makes time for thoughtfulness in learning. Many of these variables have to do with patterns of language use.
Newmann and his colleagues gauged the extent to which teachers explored explanations and conclusions and encouraged students to offer reasons for claims and to reach for imaginative ideas. They looked for students actively sustaining attention to a topic, engaging in discussion with one another, and raising questions. Newmann discovered that in classrooms with these and similar characteristics, students picked up a thoughtful mindset. They tended to write more elaborated and probing statements on a given topic.
Teachers face challenges when they seek to draw students into a culture of thoughtful learning. For example, working-class students who achieve their way into more difficult studies emphasizing reasoning and imagination may find the problems recognized by Harvard educator Sara Lawrence Lightfoot in _The Good High School:_
To the working-class student who has strived mightily to gain a loftier place, the intellectual play may seem threatening and absurd. With such high stakes, how can he dare to test out alternative propositions? He must search out the right answer. How can he spin out fantasies of adventurous projects? He must take the sure and straight path.
Lightfoot goes on to note how artful teaching can draw in students through staging debates, maintaining a lively pace, and other means. The "play" of which she speaks is, of course, the serious but engaging enterprise of thoughtful learning.
In summary, then, the general area of languages of thinking offers a major body of content for the metacurriculum, including (1) restoration to the classroom of such familiar English thinking terms as belief, hypothesis, evidence; (2) cultivation of concepts and strategies for decision making, problem solving, and related kinds of thinking; (3) introduction of ways of thinking on paper, such as concept mapping and use of traditional text forms, to help manage the problem of cognitive load and afford more opportunities for capturing thoughts and reflecting on them; (4) generally fostering the culture of a thoughtful classroom.
Complicated for teachers? Difficult for at-risk students and slow learners? Yes, if you had to do it all in a semester. But not everything has to be achieved or even attempted at once. Imagine instruction over the years keeping the language of thinking in active use, occasionally introducing a deeper perspective on key kinds of thinking such as causal reasoning or decision making, occasionally acquainting students with concept mapping and other tools for thinking on paper, and always working to keep prior ideas alive and carry them a little further. Time is one of the great resources of public education. Despite the crowded curriculum, and especially since some of the stuff crowded into it is not worth doing, there is ample time to build a smooth ramp up to the truly thoughtful classroom.
### INTELLECTUAL PASSIONS
Culture was mentioned earlier as a matter of language and communication. But culture is also a matter of passions—what is felt about what—about thinking and learning, for instance. We need to make room for the role of affect in schooling, generally, and in thoughtful teaching and learning, specifically. In a short essay on the role of aesthetics in education, Arthur Costa puts it this way:
The addition of aesthetics implies that learners become not only cognitively involved, but also enraptured with the phenomena, principles, and discrepancies they encounter in their environment. In order for the brain to comprehend, the heart must first listen.
However, schools generally give little reason for the heart to listen. In his recent examination of school reform, _In the Name of Excellence_ , Robert Toch writes with concern of the general neglect of the human side of schools. He uses the voices of children to indict a system that leaves them disaffected from education:
"School? It's just a getting out of the house thing. Kids don't come to learn, they just come," said a high school junior from California. A senior from Virginia put it this way: "I'm just doing my time." About efforts to reform schools, Toch warns:
... to date, the widespread disinterestedness among students and the schools' contribution to the problem has received scant attention within the excellence movement. In its eagerness to strengthen the quality of academics, the movement has neglected this crucial _human_ element of the crisis in public education.
Of course, thinking, good thinking, _is_ spirited. Philosophers more than psychologists have underscored this point. John Dewey, who shaped educational theory throughout the first half of this century and helped to found the progressive movement in education, emphasized the importance of cultivating both habits and attitudes of reflective thought. He urged the importance of three attitudes particularly: open-mindedness, wholeheartedness, and responsibility.
Israel Scheffler, noted philosopher of education at Harvard University, writes of the "cognitive emotions," a calculated oxymoron. While emotions are sometimes considered the enemy of good thinking, Scheffler urges that certain emotions—love of truth, commitment to fairness, zest for exploration—serve the agenda of thinking. Indeed, those very phrases demonstrate passionate language about thinking. Teachers who couch what they say in a passionate language of thinking and honor commitment to thinking in their other behavior telegraph to their students a committed culture of thinking.
In the same vein, Richard Paul, a west-coast philosopher and prominent member of the thinking skills movement, speaks of "strong sense" versus "weak sense" critical thinking. Roughly, weak sense critical thinking is the craft of reasoning—formulating sound reasons, combining them into a wellstructured arguments, rebutting counterarguments, and so on. Paul emphasizes that one can become adroit in this craft without an authentic commitment to fairness, without openness to genuinely divergent points of view. Such a commitment involves the will and the passion to maintain open-mindedness about very different perspectives from one's own—not in the empty sense of "anything goes" amiable tolerance but with thoughtful reflection. It is critical thinking in this strong sense, Paul urges, that teachers need to model and encourage in the classroom if students are to divest themselves of prejudice and other forms of narrow thinking.
Also concerned with commitment in thinking, philosopher Robert Ennis urges the importance of "thinking dispositions." The idea of a disposition contrasts with the idea of ability: Whereas the ability to swim refers to know-how, the disposition refers to inclination. You can have the know-how without the inclination or the inclination without the know-how; both are important. Ennis emphasizes that building up thinking abilities counts for little unless teachers also cultivate thinking dispositions. Teachers can emphasize and model appropriate thinking dispositions during lectures and discussions, they can strive to bring in alternative points of view, they can honor divergent perspectives within the classroom. Without such attention, dispositions toward good thinking are not likely to take, whatever technical skills youngsters learn.
Recently, colleagues Eileen Jay, Shari Tishman, and I developed a model of good thinking that makes dispositions its central theme. We propose that seven dispositions make up the essence of what it is to be a good thinker:
1. The disposition to be broad and adventurous.
2. The disposition toward sustained intellectual curiosity.
3. The disposition to clarify and seek understanding.
4. The disposition to be planful and strategic.
5. The disposition to be intellectually careful.
6. The disposition to seek and evaluate reasons.
7. The disposition to be metacognitive.
Our concept of dispositions, a little different from that of Ennis, actually includes abilities within dispositions, so that dispositions become the most central thing, the heart of good thinking.
Classrooms offer ample occasions to cultivate all these dispositions. For instance, in discussions about an essay or a concept in mathematics or science, students have an opportunity to clarify and concretize what was said. In planning a paper or an experiment, students have an opportunity to be planful and strategic. In taking a test or organizing homework time, students have an opportunity to be metacognitive. And these are only a few of many, many occasions. But they are opportunities likely not to be taken unless the teacher encourages such dispositions by naming them, modeling them, creating time for them, helping students see how to pursue them, and rewarding them.
Teachers I know already have an intuitive feel for thinking dispositions. But the low-energy culture of conventional schools and the initial attitudes of many students work against them. In contrast, the high-energy culture of the smart school (see chapter 7) gives teachers the time and encouragement to celebrate and cultivate thinking dispositions.
### INTEGRATIVE MENTAL IMAGES
Central to a pedagogy of understanding, and no less to the metacurriculum, is the idea that the teaching of the subject matters involves much more than teaching bits and pieces of content. Learners need an integrative sense of the subject matter: "How does it all hang together?" They need overarching mental images of its structure, so that they see how its strands interweave to make a whole fabric.
The last chapter emphasized how powerful representations could clarify particular, hard concepts by providing learners with mental images. Here it's worth adding that carefully constructed, powerful representations also can integrate a subject matter.
Steven Schwartz, other colleagues, and I developed supplementary materials and a teachers' guide to provide a higher order approach for students learning computer programming. We called the package a "metacourse." One important feature of these materials was an overall organizing image of the computer as a data factory, with a laborer in the factory that moved around following the commands in the program. The data factory image gave students a general tool for imagining just what the computer did with a program—what a program "meant" to the computer. The intervention proved quite successful in boosting students' programming achievement in comparison with that of control groups.
We have been developing a similar intervention for elementary algebra. For that, we adopted a different overarching metaphor: the "algebra workplace." In the imagery of the algebra workplace, algebra parts hang on the wall above a work table and toward the left—letters, numbers, equal signs, plus, minus, and so on. Hanging above the work table on the right are algebra tools such as commutativity and adding the same thing to both sides of an equation. Doing algebra is treated as a matter of working at the table to build and modify algebra objects, using the parts and the tools.
Of course, powerful as analogical imagery is, it is not the only kind of mental image for integrating a field. Sometimes, wellchosen categories can provide a mental image to do the job.
University of Massachusetts scholar Edwina Rissland, investigating instruction in mathematics, developed a triad of organizing concepts: concepts, examples, and results. The three work together as a team. Take, for instance,, a concept like right triangle. This concept has typical examples: standard diagrams of a right triangle. It also has special-case examples: an isosceles right triangle or the well-known 3-4-5 right triangle, which is 3 units long on one side, 4 units on the other, and 5 units on the hypotenuse. Then there are associated results, most obviously the famous Pythagorean theorem, which says that the sums of the squares of the two sides equal the square of the hypotenuse. Indeed, the 3-4-5 right triangle illustrates this relationship: 9 (3 squared) plus 16 (4 squared) equals 25 (5 squared). Rissland reports that the framework of concepts, examples, and results, used persistently as an organizing scheme for instruction, seems to help learners considerably toward mastery of mathematics.
The previous section mentioned concept maps, a technique developed by Novak and others. These network-like diagrams allow constructing integrative representations of complex disciplines and subject matters. They give teachers and students another resource to use in representing whole subject matters or large parts of them.
In summary, integrative mental images of varied kinds can help students toward a cohesive understanding of particular subject matters and, more broadly, of the interrelations among the subject matters.
### LEARNING TO LEARN
One of the most basic results in the psychology of learning is that humans—and even some animals—do not just learn: They learn to learn. They develop behaviors and concepts serving the endeavor of learning itself. This process begins very early. Youngsters who are just beginning to talk fairly well already have and express ideas about the way memory works. As the quotes at the beginning of this chapter illustrate, fourth graders can have quite sophisticated notions about how to learn.
Unfortunately, the conceptions of learning that students arrive at are not always the best ones. Just as many students have misconceptions about key ideas in physics or mathematics, many students have misconceptions about learning. As mentioned in chapter 2, University of Illinois researcher Carol Dweck and her colleagues have investigated youngsters' theories about the nature of learning and their own learning processes. They distinguish between what they term "entity learners" and "incremental learners." An extreme entity learner believes that "you either get it or you don't." Learning something is a matter of "catching on," and if you don't catch on in a few minutes, you probably will not catch on at all. In contrast, "incremental learners" see learning as more a piecemeal process requiring persistence. Students with an entity attitude toward learning have a theory about the nature of learning that is fundamentally mistaken and counterproductive. The more productive incremental attitude is worth cultivating.
Investigators also have examined students' monitoring of attention—a matter of staying on the task at hand versus drifting off it. Very often, poor learners are poor attention monitors: They have not learned to track their own cognitions very well and do not notice when they drift off task. In contrast, youngsters good at attention monitoring can not only stay on task but track "in the background" what else is going on.
In general, research has shown that in the course of the years, and starting relatively early in life, people develop a number of conceptions about good learning—what strategies are good for reading, understanding, and memorizing, for example. With age, the strategies gradually get more sophisticated, in some learners reaching a high degree of refinement.
For instance, Michelene Chi of the Learning Research and Development Center at the University of Pittsburgh investigated how different students used examples in studying physics. She found that some students had developed the craft of learning well from textbook examples. These students paid careful heed to the logic of examples, working through them step by step and trying to explain to themselves how each step functioned. Other students looked at examples more casually and tried to solve new problems by loose analogy with textbook examples. Chi's research showed that the students who looked at examples carefully also understood and solved new physics problems better.
Other investigations at the Learning Research and Development Center involved students working with microcomputer environments designed to support discovery learning of electrical and economic principles. Important differences among students were found. Some students paid systematic attention to control of variables when they did experiments in the computer environments; others did not. Some kept careful records of their steps, made systematic plans, and tested hypotheses. Understandably, the students who approached the task in a more sophisticated way learned considerably more from the environments.
All this concerns the learning strategies that students spontaneously develop. What happens when learning strategies are taught to students? At least sometimes, substantial advantages can result. In an integrative study of attempts to teach metacognitive reading strategies, the researchers Haller, Child, and Walberg synthesized 20 studies to find an average "effect size" of .71. This means that on the average, these treatments improved students' reading on the measures in question by 70 percent of a standard deviation. An effect size this large is considered very good in instructional interventions. Among the most potent strategies were backward and forward searching in the text to clarify obscure points and self-questioning strategies to monitor progress and regulate one's reading.
Others have taken a more broad-band approach to elevating students' learning abilities. A number of years ago, Benjamin Bloom and Lois Broder conducted an investigation of more and less effective college students and created a pilot program to boost the academic performance of the lower achievers. They painstakingly analyzed differences between better and worse students, finding a number of counterproductive behaviors in the weaker students: impulsive responses to problems on the basis of superficial cues, little effort to understand a problem thoroughly, indifference to gaps in their knowledge, and a general "either you get it or you don't" attitude. They worked with students both individually and in groups and had them think aloud and compare their approach on sample problems to that of model problem solvers who were careful and systematic. The students treated individually, and those in groups that met for at least seven sessions, showed markedly better performance and grades.
Another approach to developing students' academic abilities was developed by Charles Wales and Robert Stager at West Virginia University around 1970. Called "guided design," the approach engages groups of students in working step by step through open-ended problems that use subject matter knowledge. The guidance in guided design comes partly from an organized pattern of problem solving that highlights such steps as identifying the problem, gathering information, and generating and evaluating candidate solutions. The guidance also comes by way of sample resolutions of these steps, given to students after they have made some progress on their own. The sample resolutions are not to be taken as "right answers" but as further fuel for thinking about the problem. In 1970, the approach became the core of a required freshman course for engineering students at West Virginia University. An examination of student performance over the next several years suggests that, because of the guided design component, the students performed better academically and a greater percentage graduated from the program.
None of this means that efforts to get students to learn to learn always works. Indeed, the general consensus in the educational community seems to be that many study skills programs are remarkably ineffective for a variety of reasons. For instance, they often stand separate from the academic mainstream, carry no credit, and are seen as embarrassingly "remedial" by students. Nonetheless, there are enough success stories to make the case that learning to learn can happen in useful ways in educational settings when we make it central and important enough for students to pay serious attention.
### TEACHING FOR TRANSFER
It's a basic premise of education: We don't learn fractions arithmetic to pass the fractions arithmetic quiz. We don't diagram sentences for the sake of diagraming sentences. Ideally at least, the subject matters speak to one another and to life outside the walls of the classroom.
This point engages what has become one of the most important and contentious themes in the psychology of learning—transfer of learning. "Transfer" means learning something in one situation and then applying it in another, significantly different one—for instance, putting the math you learn in school to work in physics class or the supermarket. The dilemma for educators is that often, transfer does not occur. For example science instructors commonly complain of having to reteach mathematics to their students, even though the students seem to be doing well enough in their math classes. Why has their mathematical knowledge not traversed the corridor between the math room and the physics room?
A graphic way to tell the story of transfer as it has played out over the years involves three theories: the Bo Peep theory, the Lost Sheep theory, and the Good Shepherd theory.
The Bo Peep Theory
The Bo Peep theory of transfer is the tacit theory that operates in typical classroom settings. This theory says that useful transfer happens automatically. It takes care of itself. Like Bo Peep's sheep, appropriate knowledge gets drawn to its points of utility: "Leave them alone and they'll come home/wagging their tails behind them."
The trouble with the Bo Peep theory is that an overwhelming body of evidence shows it to be false. All too often, desirable transfer does not occur spontaneously. Youngsters do not think to use their mathematics skills in the supermarket, their social studies knowledge in the workplace, their reading skills acquired in English class in history, and so on.
The classic findings on the question of transfer were established not long after the turn of the century by the notable educational research pioneer E. L. Thorndike. Among his various studies, he investigated whether Latin "trained the mind," as it had been said to do. Comparing matched groups of students who had and hadn't studied Latin, Thorndike found not one whit of difference. In other, more straightforward experiments, Thorndike also found little transfer.
The Lost Sheep Theory
A continuing history of negative findings concerning transfer has fostered what we can call the Lost Sheep theory. The Lost Sheep theory simply says that transfer is a lost cause. People do not on the whole carry knowledge and skills from one context to another. Indeed, some psychologists argue, knowledge and skill may be too context bound by nature to allow much useful transfer. Moreover, when knowledge in context A genuinely and usefully does apply to context B, people commonly fail to see the connection.
Although this claim has been vigorously defended by some, in my view it is simply mistaken. It results from an oversimple conception that does not specify when transfer should be expected and when it should not. Indeed, one reason against general pessimism about transfer is that despite the spate of negative findings, some experiments seeking transfer show positive findings.
Researchers Clements and Gullo investigated transfer of cognitive skills from learning a computer language. They taught the language in an especially mindful way, the teacher working closely with the students to prompt them to ask themselves thoughtful questions about what they were doing and try to answer them. While most investigations of transfer from learning computer programming have been negative, these investigators found enhanced performance on certain tests of flexible thinking.
For another example, consider the Philosophy for Children program developed by philosopher Matthew Lipman and his colleagues. This program involves separate courses for several grades, beginning in middle elementary school. The courses involve the students in reading short novels, especially written for Philosophy for Children, that bring up in a fairly natural way philosophical issues concerning the sureness of our conclusions, what the right thing to do is, and the like. Teachers engage the students in probing discussions of these issues as the students go through the novels. The program in no way directly teaches reading skills, much less mathematics skills. Nonetheless, research seeking spinoff effects found that students who had participated in Philosophy for Children showed better performance in reading and mathematics, as well as in more general tests of reasoning.
For yet a third example, University of Arizona investigator Gavriel Salomon and colleagues engaged students in using a computer-aided reading tool called the Reading Partner. The tool prompted students with questions to ask themselves while reading, such as "What image can I make of what I read?" "What can I predict from the story's title?" "What's a summary of the preceding paragraphs?" and "What are the key sentences here?" The students were strongly encouraged to respond to these cues. Their reading improved substantially. More to the point, a month later the investigators administered a writing task to the students. Those who had worked with the Reading Partner showed better writing performance; they had made fertile generalizations from reading to writing.
So transfer does sometimes happen. But why sometimes yet so often not? More refined models have begun to clarify when to expect transfer. Gavriel Salomon and I presented a theory that distinguishes between two fundamentally different mechanisms of transfer—"low road" and "high road." Low road transfer depends on the reflexive activation of well-practiced patterns. It is automatic and mindless. In contrast, high road transfer depends on effortful, mindful abstraction of principles from one context to apply them in another.
Salomon and I argue that the studies that have failed to find transfer generally did not establish the conditions for either low road or high road transfer. Students had not thoroughly practiced the knowledge and skills in question in diverse contexts to set them up for low road transfer. Nor had they been encouraged in mindful abstraction that would have led to high road transfer. But contrast the studies that did find transfer: Clements and colleagues teaching a computer language emphasized thoughtful self-questioning; the Philosophy for Children program stressed thoughtful analysis of issues; the Reading Partner highlighted questioning oneself about the text read. In other words, all promoted mindful high-order reflection. They established the conditions for high road transfer.
The Good Shepherd Theory
All this amounts to the third, and preferred, theory of transfer—the Good Shepherd theory. The Good Shepherd theory acknowledges that the Bo Peep theory will not do: Transfer does not occur spontaneously nearly as often as we would like. At the same time, the Good Shepherd theory denies the Lost Sheep theory: Contrary to its pessimistic claim, strong transfer is quite possible. The thing is, one cannot expect transfer without "shepherding" it, by setting up the learning conditions that foster transfer.
In an ingenious series of experiments, Ann Brown, of the University of California at Berkeley, investigated whether children would transfer abstract concepts from one context of application to another. In one study, she and colleagues showed that children as young as three could catch on to the parallels between similar problems and solve one problem by analogy with another, provided the children were asked ("shepherded," one might say) to look for similarities. For instance, the children saw the connection between someone helping a boy out of a hole by reaching down with a hoe and someone helping a girl adrift in a boat by reaching out with a fishing pole. In another study, she and colleagues demonstrated that youngsters just as young could learn to look for such connections over a series of problems so that they did not have to be prompted every time.
From these and other studies, Brown concluded that transfer is more likely when (1) the knowledge to be transfered figures in cause/effect relationships; (2) there is emphasis during learning on flexibility and the possibility of multiple application; (3) there is some effort to disembed the principle from the initial learning context. The latter two conditions correspond to Salomon and Perkins's conditions for high road transfer.
Shepherding Transfer
In summary, it seems that students can transfer knowledge and skills from subject matter to subject matter and to a variety of out-of-school contexts, provided that the instruction sets up the conditions for transfer. Regrettably, most instruction proceeds in ways that do not favor transfer. But a number of instructional practices can help.
They fall into two broad categories called bridging and hugging. Bridging means that the teacher helps the students to make connections between what they are studying and other areas—something from another subject matter or their out-of-school lives.
Bridging is not hard to do. It simply means taking time to get students to make outreaching connections. A teacher might ask students studying the U.S. Civil War to explore analogies with topical events in Northern Ireland or the secessionist movement in Canada. When students are studying oscillators in physics class, the teacher might provoke them to find oscillating systems in their everyday lives (dripping faucets, swaying tree branches, backyard swings) and try to identify the energy sources that keep the oscillators going.
Hugging, in contrast, means keeping the instruction close to (hugging) the very target performances one wants to cultivate, so that transfer is less of a problem. This is commonplace in music and drama instruction: One practices the very thing one is to perform. But the principle often gets bypassed in more academic instruction. For example, students may put in practice time on topic sentences largely by selecting topic sentences among multiple choices or identifying topic sentences in paragraphs. Neither one gives intensive practice in actually writing paragraphs with topic sentences.
Once teachers are alert to the routine lack of hugging, it's easy to build well-hugged instruction into the school day. Rather than choosing topic sentences from lists, students might better spend most of their practice time composing paragraphs with an emphasis on good topic sentences. For feedback, they can exchange papers, each trying to identify his or her neighbor's topic sentences. The teacher can adjudicate problems and confusions. This gets kids writing paragraphs, but one can easily go further. More holistic approaches to developing writing skills would recommend full-fledged writing activities with a communicative emphasis, good topic sentences being one among several agendas.
One special kind of hugging is called problem-based learning. In this technique, students learn a body of knowledge by working at problems that require the knowledge, which is not presented in advance but looked up as needed. Research conducted by John Bransford and his colleagues shows that problem-based learning leads to more flexible and generative application of the knowledge later. It's a matter of hugging: Because students learned the knowledge in the context of problem-solving tasks, the knowledge is better organized in their minds for later problem solving.
* * *
**Levels of Knowledge.** Content, problem solving, epistemic, inquiry.
**Languages of Thinking.** Thinking terms in the English language. Thinking strategies. Graphic organizers. Culture of "the thoughtful classroom."
**Intellectual Passions.** Cognitive emotions. Strong-sense critical thinking. Dispositions.
**Integrative Mental Models.** Integrative images. Integrative verbal category systems.
**Learning to Learn.** "Incremental" rather than "entity" learning encouraged. Attention monitoring. Effective learning from examples. Strategic reading, other learning strategies.
**Teaching for Transfer.** Shepherding transfer by bridging and hugging.
* * *
### AN EXAMPLE OF TEACHING THE METACURRICULUM
I hope that the foregoing pages have made something of a case for the metacurriculum. But making a case is not quite the same as painting a picture of what it would really be like. Just as at the end of the last chapter—for a pedagogy of understanding—we can imagine. Indeed, we do not have to imagine very hard, because in many places many teachers are teaching parts of the metacurriculum—teaching languages of thinking, teaching for transfer, employing overarching mental images, cultivating the critical spirit.
Suppose our class is studying the U.S. Constitution, certainly a hallowed topic in the content curriculum. In the conventional course of events, students would probably find themselves reading parts of the Constitution, learning something about the function and importance of particular components, such as the Bill of Rights, and answering fact-oriented questions that displayed their knowledge of what the Constitution said. The metacurriculum would call for something more, something deeper. For example, the class might start by exploring the Constitution using some language of thinking. One language from my own research and materials development work is called "knowledge as design."
Knowledge as design asks learners to analyze things as designs that serve a purpose. Familiar with the approach, a student might ask, "What's the purpose of the preamble?" Puzzling among themselves, the students begin to come up with answers:
"The preamble is a kind of preview, just like it says."
"The preamble says what the purpose of the Constitution is."
"The preamble is kind of inspiring; it says that we all commit ourselves to these ideas as one people."
Knowledge as design has been introduced to the students not just as a thinking strategy but as part of learning to learn. They've come to recognize its set of key questions as a good tool for getting inside a topic.
Okay so far, the teacher thinks. But the teacher wants to push the conversation deeper, urging the students beyond these surface purposes to hidden ones. "Fine," the teacher says. "But you know, I'm really curious about this. It's so well expressed and it says so much." The teacher quite honestly but quite deliberately is displaying an intellectual passion, curiosity. The teacher wants students to see this and value it. "What else is going on here? Can you find any more subtle purposes, ones hard to see at first?"
"Well," one student might answer, warming up to the hunt for something mysterious, "it's a little deceptive, this 'We the People,' because there was a lot of disagreement. And the people really weren't everyone. Like only males could vote, for instance. So it pretends there's a unity that isn't really there."
"Do you think the authors of the Constitution meant to gloss things over there," the teacher says. "Or did they think there was more unity?"
The students disagree, giving the teacher a chance to bring in a higher level of understanding. "Well, let's see now," the teacher says, deliberately sounding skeptical—more intellectual passions. "Can we really tell at all how people might have been thinking 200 years ago? What kind of evidence could we possibly have?" This question provokes a commonsense exploration of how historical interpretations get justified—the epistemic level of understanding.
Concerned with transfer, the teacher wants to connect the discussion with other situations. During the next period, the teacher broadens the compass of the discussion, bringing in other documents that have set nations upon a path—the Declaration of Independence, the Magna Carta. "You know, documents like these really do change the world. How are these the same? How are they different?" the teacher asks. Probes like this are bridging questions. They foster transfer from other topics the students already have studied in the history course.
The teacher wants to bring matters closer to home, for more transfer. "Do we in the class or school or town have any documents like this?" Perhaps a student recalls that the school has a constitution, as some schools do. Perhaps the students have never read it. What are their rights and responsibilities? Maybe they had better find out.
Or perhaps, if there is no constitution, one ought to be drafted. The students might undertake this. What would they like their rights and responsibilities to be? And who else would have to agree? And would they? And why or why not? Such a project would prove an arena for problem solving, decision making, understanding, and a dozen other kinds of thinking.
Suppose students pursued some such project. Reflecting back and going for the big picture, the teacher might ask, "What generalizations can you make about your document, and the U.S. Constitution, and the Magna Carta? What's important that they all have in common? Make a chart, make a diagram." This is an invitation to construct some kind of integrative mental image that captures key features of documents that declare rights and responsibilities. Instead of offering such an image on a platter (which is fine sometimes), the teacher nudges the youngsters to construct their own images.
Of course, as with the example at the end of the last chapter, this is just one cut through the apple of opportunity. If probing the purposes of the preamble sounds too analytical for the students in question, the teacher might ask them to act out different people—housewives, small farmers, slave owners, businessmen—reacting to the preamble. From that, the purposes might be drawn. If finishing a constitution for the school, or even starting one, makes too big a deal of what was supposed to be a short unit, the teacher might stop with a period's worth of discussion and save major projects for another occasion.
Whatever the style, there are ample opportunities to orient instruction toward higher levels of understanding, introduce and exercise languages of thinking, cultivate intellectual passions, seek out integrative mental images, foster learning to learn, and teach for transfer. The smart school makes the most of these opportunities. It informs and energizes teaching by giving teachers time and support to learn about the opportunities and by arranging curriculum, assessment, and scheduling to encourage tapping them.
Opportunities, yes, and necessities as well. Because those three seemingly innocuous goals of education—retention, understanding, and the active use of knowledge—not only invite but demand much more attention to the metacurriculum. We are simply not likely to see much of the three without contributing directly to students' overarching conceptions of the subject matters and to their artful orchestration of their own mental resources.
## CHAPTER 6
**CLASSROOMS
The Role of Distributed Intelligence**
Here is a tale of three notebooks. Alfredo begins the first as a lad fifteen years of age, in his history class, where he and the other students are studying the League of Nations and the United Nations. In this notebook, he enters a variety of information about the two organizations. The history teacher encourages thoughtfulness, and Alfredo, reflectively inclined himself, adds to his notebook a number of ideas about what happened and why and what it meant.
But there is something odd about Alfredo's notebook: What is written there does not count as part of what he has learned. With the final exam coming up in two weeks, Alfredo makes sure that most of what appears in the notebook is also in his head, because the final exam will be closed book, including the essay questions. What is written in the notebook does not count, even though it reflects effortful organization and added reflection. To be sure, side effects of the cognitive effort Alfredo has invested will help him to remember the content, but the notebook itself earns no credit.
Alfredo keeps his second notebook around the epic series of Dungeons and Dragons games that he and several friends maintain. The status of this notebook is quite different. There is no question that the diagrams of dungeons, the notes about crucial hazards, and so on comprise part of what Alfredo has learned. When he doesn't happen to remember, he looks it up. Not only is Alfredo's notebook a resource for him, but the kids are resources for one another. In contrast with the classroom setting, in Dungeons and Dragons the youngsters cooperate as well as compete. They rely upon each other's knowledge and thinking.
Alfredo's third notebook gets started fifteen years later, when Alfredo is a young engineer, part of a technical team designing a new bridge over the Hudson River. Not only is a team of people involved but also a team of physical supports for cognition. Alfredo's notebook, full of ideas and specifications, joins with a computer-aided design system, books full of specifications and regulations, journals with engineering advances, memos from various team members to each other, a physical mock-up of the proposed bridge design, hand calculators, and much more.
Compared with the Dungeons and Dragons game or the engineering profession, the typical classroom begins to look like an odd place. Emphatically and in manifold ways, schools address what might be called the "person-solo." It is the person-solo who should acquire knowledge and skills. It is the person-solo who should work out math problems and write essays. It is the person-solo who should have all knowledge and skill in his or her head rather than tucked away in easily accessible sources.
At least, someone might say, youngsters are encouraged to work out their ideas with pencil and paper; so schools do not ignore the role of physical supports in cognition. Well, sometimes. Thought about more carefully, the pencil and paper tolerated in exams seem to have another purpose: Students are not so much encouraged to think on paper as to use the paper and pencil to display their thinking. The paper and pencil are there not as powerful vehicles for supporting cognition but as a conduit of communication for showing the teacher in-the-head cognitions.
At least, someone else might say, there are open-book exams. Yes, open-book exams are in the right spirit, recognizing that in out-of-school contexts people routinely draw fluently on all sorts of information sources. But they are only a token reflection of this overwhelming trend.
To attach a metaphor to this contrast, we could compare the person-solo with the "person-plus." The solo mode of operation—noncollaborative and without extensive physical and information resources—is an oddity. People normally function in their homes, workplaces, and playplaces in person-plus kinds of ways, with intensive use of physical and information resources, interaction, and interdependency. Plainly, this is no accident. People operate as persons-plus because it is empowering and engaging.
### THE IDEA OF DISTRIBUTED INTELLIGENCE
Lest we take schools too much to task for a unique shortfall, there is at least one other bastion of the person-solo perspective. It is, regrettably, psychological theory and experimentation. The classic question of psychology is "What goes on in the mind," or, from the standpoint of B. F. Skinner's behaviorist psychology (which does not believe in minds), "How does the individual organism react to stimuli?" Just as in the classroom, psychological experiments typically proceed with a minimum of physical and social support for the subject. Psychologists wonder what the subject can and will do without much equipment, and certainly without another person, to help. There are exceptions, but they hardly impugn the reality of the trend.
However, in several quarters a new assessment has been made of this entrenched "personcentric" view of the human organism. Roy Pea of Northwestern University has written recently about what he calls "distributed intelligence." Others, including myself, have picked up this theme. We argue that human cognition at its richest almost always occurs in ways that are physically, socially, and symbolically distributed. People think and remember with the help of all sorts of physical aids, and we commonly construct new physical aids to help ourselves yet more. People think and remember socially, through interaction with other people, sharing information and perspectives and developing ideas. The work of the world gets done in groups! Finally, people sustain thinking through socially shared symbol systems—speech, writing, the technical argot of specialties, diagrams, scientific notations, and so on.
A more modest term than distributed intelligence for this dispersal of intellectual functioning across physical, social, and symbolic supports is "distributed cognition." But there is a point to Pea's more provocative use of the term intelligence. Taken broadly, intelligence refers simply to effective cognitive functioning. And intelligence is at stake here. People can function more intelligently in person-plus than in person-solo kinds of ways.
Defenders of classic notions of intelligence would complain, "But this isn't real intelligence. Real intelligence is in people's heads. Part of what you're talking about lies in the hand calculator or the notebook, not in the person." The rebuttal would be, "But the person-with-calculator-and-notebook is the actual functioning system. The person-plus system is what gets things done in the world. Its intelligence is more to the point than that of the person-solo."
Another contribution to the notion of distributed intelligence or distributed cognition comes from University of Arizona researcher Gavriel Salomon, long-time observer and investigator of the role of technologies in learning, writing with Tamar Globerson and myself. The authors draw a distinction between the effects _with_ and _of_ technology, including computers and television but also such ordinary technologies as paper and pencil. Effects _of_ are the residues left when we are away from the technology. Perhaps, for example, we speak more articulately because we have written so many paragraphs. Effects _with_ a technology are the empowerment that results when we have the technology at hand, actually thinking on paper, writing with word processors, communicating with telecommunications systems, and so on. Both effects _with_ and effects _of_ are part of the person-plus phenomenon, to be sought and cherished.
One might sum up the person-plus perspective in two principles:
1. The surround—the immediate physical, social, and symbolic resources outside of the person—participates in cognition, not just as a source of input and receiver of output but as a vehicle of thought. The surround in a real sense does part of the thinking.
2. The residue left by thinking—what is learned—lingers not just in the mind of the learner but in the arrangement of the surround as well; yet it is just as genuinely learning for all that. The surround in a real sense holds part of the learning.
These precepts imply a very different posture toward Alfredo's school notebook than most classrooms tolerate. The notebook is both an arena of thinking and a container of learning. Alfredo does not just think and put his thoughts down in the notebook. Alfredo thinks with and through the notebook. Alfredo has not just learned what he remembers from his writings in the notebook. Alfredo, the person-plus, functions with his notebook available as a resource. What is in the notebook, whether the person-solo remembers it or not, is part of what the person-plus has learned.
Of course, this does not mean that knowledge in notebooks is always just as good as knowledge in your head. Which is the best place to store up knowledge depends on matters like how often the knowledge is used, how quickly you can get to it when you need it, and so on. But the best place is often not in your head. Often, you can maintain far larger and more accessible and accurate knowledge structures in a notebook or a computer database. What really counts is not where the knowledge is—inside or outside the skull—but what might be called the "access characteristics" of relevant knowledge—what kind of knowledge is represented, how it is represented, how readily it is retrieved, and related matters. Whatever place—in the surround or in the head—gives the person-plus the best access characteristics is the place to use.
### DISTRIBUTING COGNITION IN THE CLASSROOM
Outside of schools and psychological laboratories, person-plus is more the rule than the exception. We operate in close alliance with our physical, social, and symbolic surrounds. What would it mean to nudge classroom practice in that direction? Without trying to be comprehensive, here are some ideas drawn from a variety of innovative practices in education.
Distributing Cognition Physically
The traditional means of distributing cognition in the classroom almost all have to do with input—texts, lectures, posters, films, and so on. Output—what students say and write—is much less varied in format, a matter of problem sets and fill-in-the-blank and essay questions. And this output is usually not seen as a process of thinking something through on paper. Rather, it's a way of exercising and testing students' person-solo, in-the-head thinking.
But rich opportunities exist for changing that. One of the most familiar is the keeping of journals where students write about themes from their subject matters and their own evolving understanding of those themes. Such journals look to be good both for the students' understanding of the subject matter and their own metacognitive development.
John Barell, a figure in current efforts to cultivate thinking in the classroom, has discussed some singularly impressive examples of youngsters' journal reflections on what they are learning in school and what they are not learning. One journal format that Barell developed helps students to articulate and assess both their problem-solving and problem-finding skills. Here, a high school student thinks through a troubling problem she is trying to solve, a problem not unlike the one we struggle with in this book.
I guess I could call myself smart. I mean I can usually get good grades. Sometimes I worry, though, that I'm not equipped to achieve what I want, that I'm just a tape recorder repeating back what I've heard. It scares me... I do my work, but I don't have the motivation. I've done well on Iowa and PSAT tests but they are always multiple choice. I worry that once I'm out of school and people don't keep handing me information with questions and Scantron sheets I'll be lost.
School is kind of unrealistic that way. Kids who do well often just repeat what the teacher has said...
Other modes of journal keeping enable students to track their thinking about a particular assignment as they work through it.
Yet another innovation that has received considerable attention concerns student portfolios. Not just in writing but in science, mathematics, and other subjects, students build portfolios of key products—essays, notes, diagrams, and so on. A portfolio is a selective enterprise: Not everything goes into it, just those items that students think most powerfully reflect their understandings and their expressions of those understandings. The portfolio functions as an object of review and assessment for the teacher, but also as a gauge of progress and occasion of reflection for the learner.
An extension of the idea of portfolios is the "process-folio," developed by the Arts PROPEL project under the direction of Harvard University educators Howard Gardner and Dennis Wolf. In contrast to a portfolio, which presents the best end-products of a student's work, a process-folio is a record of learning that focuses on the student's process of working through a creative activity. Arts PROPEL process-folios are used both for in-class and across-district comparison of student work in the arts. They provide an excellent way to document project work over time and to aid reflection alone or with other individuals.
It's easy to hear teachers saying at this point, "I don't have time to read dozens of journals. I don't have time to review dozens of portfolios." True, if we think of journals and portfolios as traditional student products—yesterday's math assignment—that get handed in for the teacher to go over—each one, every time—in detail. However, in the smart school, we can think in different ways. Although this is not the place for a disquisition on journal or portfolio technique, there are all sorts of tricks to their efficient use. One is that the teacher keep in touch with students' work, but not look at everything from each student all the time. Another is that, part of the time, students function as respondents for one another.
Computer technology has provided a range of new physical vehicles for supporting students' cognitions. The hand calculator is a notable and notorious example. Some years ago, youngsters' use of hand calculators generated considerable furor. This has settled down somewhat. The perspective in many quarters that both person-solo arithmetic and person-with-calculator arithmetic have roles to play: It is simply not an either/or proposition. Nor need the use of hand calculators undermine hand arithmetic proficiency. Moreover, hand calculators afford person-plus learning opportunities that should not be missed. By empowering students to handle large numbers easily, hand calculators allow them to focus on other facets of mathematical understanding.
Heavy-duty computer resources are also making themselves felt. These include word processors, computer programming environments such as Logo, spreadsheets, computer-aided drawing systems, databases—and, of course, special-purpose tutorial environments for cultivating particular skills: sometimes routine, such as arithmetic operations, and sometimes not so routine, such as aspects of higher-order thinking.
For example, working out of the Media Laboratory at the Massachusetts Institute of Technology, Idit Harel involved fourth graders at an inner-city public elementary school in Boston, Massachusetts, in a software-design project concerning fractions arithmetic, one of the all-time killer topics in the elementary curriculum. The students' challenge: to write tutorial programs in the computer language Logo to help third graders understand fractions basics better.
Of course, the real point was for them to understand fractions and programming better. The students designed their programs over a number of weeks, writing before and after each computer session in their Designer's Notebooks. The notebooks served as vehicles for planning and reflection, cultivating metacognition. Harel gathered data to show that this design experience gave the students a much better understanding of fractions and a much better understanding of the Logo computer language than the usual instruction in either. Harel also points up changes in their attitude toward mathematics and in their general reflectiveness.
For another example, desktop publishing can be a vehicle for lively learning activities with multiple payoffs. In Scottsdale, Arizona, at the Kiva Elementary School, students studying ancient Egypt synthesized their knowledge into a _NationalEnquirer_-style, four-page newspaper called _King Tut's Chronicle_. Headlines blared "Cleo in Trouble Once Again?" Readers could check out their horoscopes and the latest price of mummified cloth and pyramid blocks on the stock market. A "Dear Cleopatra" column offered advice, while news on Nile boat races satisfied sports fans. An unusual lesson in history, yes—but also lessons in contemporary media, writing, and cooperation, all rolled into one.
Distributing Cognition Socially
Any educator conscious of the contemporary scene will know what this most often means: cooperative learning. As mentioned in chapter 3, considerable research has shown that techniques of cooperative learning can boost student achievement. Educational psychologists Ann Brown and Annemarie Palincsar, reviewing the research on cooperative learning, emphasize that what beneficial effects occur are not to be attributed merely to the formation of groups. All turns on what happens in the groups—how materials are used, what kinds of interactions are encouraged, and so on.
Socially distributed intelligence inevitably depends on the physical distribution of intelligence. For example, it is often advised that cooperative groups share a common workspace and resources, with one set of source materials for all the group members and one scribe to catch and organize the ideas of the group. If the group develops a chart, there is one draft in front of all, a common focus around which the members interact.
Another interesting facet of cooperative learning concerns specialization. In the simplest of cooperative learning arrangements, everybody tries to get better at the same thing—say, doing algebra manipulation problems. The structure of assessment in the classroom encourages the group members to help one another out. For example, each group member may receive for a grade the average of the test scores of all members tested individually. Thus, it is worth each learner s while to improve the skills of all.
But more elaborate cooperative learning techniques introduce specialization of function. Remember the well-known "jigsaw" method outlined in chapter 3: The materials to be learned are divided into about four parts. Students are organized into home groups of four. Each student leaves his or her home group to join a learning group that studies one part of the material. Then the students return to their home groups; they must teach one another what they learned in the learning groups.
Of course, everyone need not end up knowing the same thing and having the same skills. Although we would like students to master a common core, there is ample room for specialization. Note that in the world of persons-plus outside the walls of schools, specialization is normal: We rely on one another. This immensely practical arrangement has the additional benefit of honoring individual worth.
A broad view of the potentials of working together comes from William Damon of Brown University and Erin Phelps of Radcliffe College. They write of "peer education" as an encompassing category that includes peer tutoring, cooperative learning, and peer collaboration. In peer tutoring, same-age or slightly older students tutor others in their areas of strength. In cooperative learning, students in a class get grouped into learning teams with the same learning goal; often, they divide the work within groups so that each student plays a distinct role, as in the jigsaw method just mentioned. In peer collaboration, pairs or small groups of students work together simultaneously on the same task, which may be individual to the group.
Damon and Phelps highlight two dimensions of importance in understanding the tradeoffs of these different forms of peer learning: equality and mutuality. Equality refers to the equal status of the participants. For instance, peer tutoring offers more equality of status than typical teacher/student relationships, but still maintains some hierarchy: The tutor has a dominant position. Cooperative learning and peer collaboration alike score high on equality.
Mutuality asks to what extent the discourse among learners is extensive, intimate, and connected. Peer collaboration ranks high in mutuality. Peer tutoring varies in mutuality according to the tutor's interactive skills and the learner's openness to the learning experience. Cooperative learning varies in mutuality depending on the extent to which tasks are divided among different students and on competitiveness, a motivator often used in cooperative learning.
The bottom line: Damon and Phelps argue on principle and research evidence that equality and mutuality contribute to good learning in peer learning situations. Therefore, peer collaboration serves well. Since peer tutoring and cooperative learning do not necessarily rate high on equality and mutuality, Damon and Phelps urge making the most use of versions of the two that do.
What do different sorts of peer learning look like? The jigsaw method, just sketched above, is a cooperative learning technique. For a different sort of example, consider "pair problem solving," a peer collaboration method good for developing metacognition and problem-solving abilities.
Pair problem solving was devised by mathematics/science educators Arthur Whimbey and Jack Lochhead and employed and investigated extensively by them. This tactic focuses on problem solving and organizes students into pairs. One student tackles the problem, reporting his or her thoughts. The other has two responsibilities: (1) to understand the first student's thinking, right or wrong (for instance, by asking questions to clarify when the first student has not said enough); (2) not to intervene, even if the first student makes mistakes. (However, if a student is especially prone to minor errors, Whimbey and Lochhead recommend that the listener draw seeming errors to the attention of the problem solver to support getting through the problem.) After the problem solving itself, the students discuss the problem and switch roles for the next problem.
What does this actually sound like? Suppose we have a familiar type of story problem: _If Aaron can rake the lawn in three hours and Boris in four, how long will it take them to do the job together_.
PROBLEM SOLVER: Well, let's see. Certainly they'll do it faster together, 'cause Aaron will do one part and Boris another part.
LISTENER: Okay.
PROBLEM SOLVER: The trick is to figure out how much time they save. Or how much time they take. Let's see... I suppose it would kind of average out.
LISTENER: What do you mean by "average out?"
PROBLEM SOLVER: Well, if Aaron takes three hours and Boris four hours, the two of them together might take three and a half hours, sort of splitting the difference. But I'm not sure.
LISTENER: So let me see if I understand. Aaron is better off working without Boris. Boris actually makes things take longer.
PROBLEM SOLVER: Yeah, I guess so. Wait a minute, that doesn't make sense. 'Cause Boris does part of the lawn so Aaron doesn't have to do it all. So it's not the average. Well let me start over.
Lochhead explains the logic of pair problem solving this way. Metacognitive awareness and reflection are important aspects of effective problem solving. But this is a difficult skill for students to sustain when they are worrying about the problem at the same time. Pair problem solving splits metacognitive awareness into two roles. The problem solver gets practice articulating ongoing thoughts, and the listener gets practice making sense of them and probing for clarity. The need for the two to communicate captures thoughts that otherwise would stream by in the rapids of moment-to-moment cognition. Eventually, students are pressed to combine the two roles, reporter and interpreter, in themselves, internalizing the initially social process of reflection.
Of course, peer learning techniques should not be seen as the only approach to distributing cognition socially. Socratic teaching, discussed in chapter 3, is another pattern of cooperative cognition in groups of medium size. Joint class projects can involve two dozen students, each in a somewhat specialized role. Drama activities distribute roles. And so on, through an abundance of opportunities.
Distributing Cognition Symbolically
In a way, to speak of physical and social distribution of cognition is already to include symbolic distribution, because symbol systems of various kinds—words, diagrams, equations—are the medium of exchange among people. But some direct attention to symbol systems is worthwhile.
One of the prejudices to be broken down in efforts to distribute cognition symbolically concerns the entanglements of particular symbol systems with particular subject matters. Mathematics, for example, is usually seen as the stamping grounds of formal notations. Yet in recent years many mathematics educators have advocated essayistic writing in mathematics: Youngsters discuss their approaches to particular problems, their understandings of key mathematics concepts, how mathematics connects to some slice of life outside of school, such as household budgets or government tactics to stabilize the economy.
Likewise, there is no particular reason why stories in literature always have to be talked and written about. Stories can be diagramed, taxonomized, mimed, turned into plays. Imagine students reading Dickens's _A Christmas Carol_ and then improvising before/after vignettes around the characters. (Bob Cratchit asks for a raise before Scrooge's experience with the ghosts and after.) Three or four pairs of children take a crack at the improvisation. That provides a basis for discussing "character," what it means and why it's important.
The symbolic distribution of intelligence also recalls the emphasis in the last chapter on languages of thinking. One barrier to distributing cognition symbolically in the classroom is the impoverishment of classroom language, the failure to cultivate a common vocabulary about inquiry, explanation, argument, and problem solving. Distributing cognition symbolically calls for a concerted effort to bring languages of thinking into play in the classroom and reawaken them frequently, week by week, hour by hour.
Another important direction for the symbolic distribution of cognition concerns text forms. By and large, schooling relies on the essay, the story, and whatever forms of note taking students fall into spontaneously. But the fact is that neither essay nor story forms are very good for exploring alternatives and organizing patterns, although both can express what one has thought about. More telegraphic and flexible kinds of thinking on paper serve thinking better than these more extended and constrained forms. Remember again from the last chapter the importance of graphic organizers, ways of thinking on paper. Brainstormed lists, concept maps, charts, and two-dimensional tables are among the simple layouts that can be useful as students work to build a conception of something.
In analyzing a short story, for example, students may be better advised to begin with a concept map of it than just to think about it in their heads or start immediately to write about it. In planning an experiment in science, some students may prefer to lay out the steps in a diagram, with branches for contingencies, rather than in a written list. Sometimes key points about a period of historical development might be better captured on sortable three-by-five cards than laid down linearly in a notebook in the order served up by the text. As these points make plain, part of the agenda in distributing cognition symbolically is to foster a wide range of flexible symbolic resources, a kind of broad-band literacy.
Physical, social, and symbolic distribution of intelligence in the classroom—a whole brew of innovations toward the smart school. But where do teachers and administrators gain the know-how to put such ideas into practice? There is no one answer, but also no lack of answers. First of all, it's important to recognize that a good start can be made without any special help. In a school setting that encourages experimentation, any teacher can try some of these ideas. It is not difficult, for example, to ask students to keep a personal journal on something for a few weeks or to call for an essay in math class or a diagram of a short story. As to cooperative and collaborative learning, most teachers have had some introduction to these techniques already.
To be sure, many practices benefit from much more information and advice. Refined use of cooperative learning methods requires techniques for setting up joint responsibility of every student in each group for the other students' learning. Innovations involving computers call for some minimal technical tutorial. But a veritable industry of consultants and published materials stands waiting to supply such needs. And in the smart school, with its commitment to informing and energizing instruction, teachers and administrators have time to experiment and learn.
### THE FINGERTIP EFFECT
Many of the foregoing examples are familiar. If anything is new, it is not the individual ideas but the way of seeing them. They all are part of the mission to redistribute cognition more broadly in the classroom, a person-plus approach to instruction that suggests many ways of reorganizing the learning process.
However, lest all this seem too utopian, there is a fundamental factor that threatens the whole enterprise. I like to call it "the fingertip effect."
The fingertip effect is a belief that many innovators have had in the impact of a new technology or other innovation, such as cooperative learning or peer tutoring. The essence of this belief can be stated in a sentence: _When we put opportunities at learners' fingertips, they take the opportunities_.
For example, the fingertip effect forecasts that when we make word processors available to young writers, they will take the opportunity of making structural revisions of their stories and essays, something very inconvenient with paper and pencil. When we make programming languages available and familiar to learners, they will discern powerful analogies between programming and other areas, carrying over skills from one to another. When we make cooperative groupings part of the classroom, students will seize the opportunity to adopt mutually supportive patterns of thinking and learning. And so on.
In short, belief in the fingertip effect is belief in the immediate opportunism of the human organism. The forecast that follows boils down to this: All we have to do to mediate change is to set up physical or social structures (word processors, cooperative groups) that afford opportunities. Change will then follow naturally, as learners seize those opportunities.
The trouble with the fingertip effect is that it doesn't happen. Not with any reliability. Not in the short term. The impact of word processors on students' writing is a classic case. It is simply not true that students gravitate toward fundamental structural revisions of their texts when given the conveniences of word processors. Rather, they tend to use the word processors for minor, local revisions such as correcting spelling.
Likewise, it is simply not true that organizing youngsters into cooperative groups immediately yields great benefits. Initially, the participants do not know how to work well in groups. Also, certain group structures turn out to foster achievement more effectively than others. While the opportunities to collaborate are created simply by grouping students, _follow-through_ depends on much more than the existence of opportunity.
Once it's recognized that the hoped-for fingertip effect does not routinely occur, reasons why are easy to find. Here are some of them.
**Opportunities Not Recognized.** Students who have had little chance to engage in structural revisions in text do not even recognize its importance. There is nothing in their experience to spur them to seize the opportunity. In contrast, people with ample experience in writing "the hard way," who have coped with the vexations of structural revision by paper and pencil or typewriter, will immediately begin to exploit such resources. In general, the new opportunities afforded by an innovation commonly cannot be recognized by novices.
**Cognitive Burden.** It's also important to realize that the opportunities afforded by innovations often bring with them a bewildering array of new things. A word processor, for example, does not await attentively for you to tell it what to write. There is a lot to learn about what it will do and what keys make it do those things. Cooperative groups bring with them puzzles of decision making, communication, task tracking, and responsibility that youngsters are not used to dealing with. "Here we are in a group. Okay. Who's boss? Is there a boss? How do we decide what to do first?" and so on. Quite apart from whether they are in a position to discern opportunities in principle, students commonly run headlong into considerable confusion and disorientation.
**Motivational Structure.** Just because an opportunity appears does not mean that learners feel motivated to take it. A common problem in cooperative groups is that the ablest member ends up doing the task. The others copy, getting an almost free ride. To be sure, the opportunity for more evenhanded collaboration is there... but why bother? The ablest member will do a better job solo. And the ablest member often likes it that way—it's more quickly done and better done, he or she thinks. Accordingly, advocates of cooperative learning have had to discover how to configure groups and group responsibilities carefully to ensure full participation.
None of this should be surprising. There is no reason why simply providing a resource should yield immediate and profound transformations. But belief in the fingertip effect needs explicit identification because, not uncommonly, innovators have argued for the immediate transformative power of simply putting something into place—computers, television, typewriters, cooperative groups, or whatever. When such initiatives fail, as they commonly do, blame falls on the medium. "Computers can't help, after all."
This is too hasty. The problem is not the new medium but the lack of mediation. The lesson of the fingertip effect is that the discovery of opportunities needs to be guided. Teachers can help students to unearth the opportunities of technical resources, such as computers and calculators. Teachers can arrange fruitful interaction patterns in cooperative groups.
Some innovative software environments provide cuing systems to remind students of opportunities. For example, writing environments developed by Gavriel Salomon, mentioned earlier, and by Collette Daiute of the Harvard Graduate School of Education, prompt students occasionally with reminders about good things to ask themselves. Salomon's Writing Partner from time to time poses questions like this:
Do you want your composition to persuade or to describe?
What kind of an audience are you addressing?
What are some of your main points?
Does this lead me to the conclusion I want to reach?
As these examples make plain, the questions put forward by the Writing Partner are not meant to raise subtle points about composition but quite basic ones—which, however, the learner can easily neglect. Such writing environments go beyond the conventional word processor, which offers opportunity aplenty but little guidance, by stimulating opportunity finding and opportunity taking.
Part of the lure of the fingertip effect seems to lie in the cherished belief that educational transformations should be natural, not forced. The image of simply putting something into place—say, a word processor—and seeing wonderful learning experiences unfold organically is seductive. But innumerable lost hopes argue for a more hard-headed posture toward the fingertip effect. We must not expect new technologies, the grouping of students, and like innovations to do the job by themselves. We must accept the responsibility of mediating students• good use of these person-plus resources.
### WHO•S BOSS WHEN?
If intelligence can be distributed in various ways, a provocative question arises: "Who•s boss when?" Put more formally, people, societies, and even some machine systems have what might be called an "executive function." There are mechanisms that guide the overall activity, confronting decision points and deciding when different tasks should be undertaken. So we can ask: When cognition is distributed, how, specifically, is the executive function distributed?
It is not hard to recognize a number of scenarios. Most often, we think of people as deciding for themselves. Decision making is a person-solo undertaking, whatever other cognitive functions might be distributed. But this is only one possibility. For example, during conventional instruction, the teacher decides what is best to do next. The students carry out the teacher's agenda, their own executive function limited to minor decision points within that agenda. A text or workbook includes a tacit and sometimes explicit set of executive guides: Read the chapter from the beginning; answer the questions at the end of the chapter; fill in the blanks.
In summary, in all sorts of ways, learners (and others) commonly cede executive control to some part of the surround—the text, the worksheet, the teacher. Now all this may seem to be a setup for a revolutionary statement I'm about to make, something along the lines that learners need to be liberated from the autocracy of the surround. But nothing of the sort is intended.
On the contrary, ceding executive function to the surround is one of the most effective cognitive strategies we have. In everyday life, we do it all the time. When you follow a map or a set of directions for assembling a bicycle, you are ceding executive function, and soundly so. The map "knows" more about the landscape than you do. The manufacturer knows more about how to put the bicycle together than you do. To be sure, you retain the right (and the risks) of override. But unless there are reasons to exercise that right, you do well to take advantage of the prefabricated executive.
Further, we as a whole society cede executive function to certain political units—mayors, presidents, governors, and so on. We cede legal judgment to written law, precedent, and a judicial system. In cases of civil conflict, we may cede judgment to a mediator. In cases of management, the wise manager commonly cedes large classes of judgments to subordinates to avoid the overwhelming task of keeping track of everything himself or herself.
With all this in mind, two questions about the executive function seem particularly important:
1. Is there an appropriate executive function anywhere in the person-plus system in question?
2. When learners cede executive function, do they ever get it back?
The first question is important exactly because the answer is commonly no. Indeed, this connects to the problem of the fingertip effect. Why is it that students do not seize the opportunities that various technologies afford? Because, all too often, the technologies are presented as a kind of cognitive sandbox. The invitation is to build what you can in the sandbox. But neither in the student nor in the technology is there any executive function guiding the student to recognize and exercise the opportunities.
And why is it that at first students often have trouble working fruitfully in cooperative groups? Consider their previous experience. Earlier, they did what they or the teacher wanted. But cooperative learning brings new questions of distribution of executive function within the group. Learners need, with appropriate guidance, to feel their way into consensus patterns, tie-breaking tricks, and the like to function well.
The second question—do learners get the executive function back—is important because a good deal of educational practice vests executive function in teacher or materials "temporarily." But the students never recover it.
A prime example is problem selection. Textbooks and teachers do virtually all the problem choosing for students, deciding which problems are worth attention and usually in what order. After a period of practice, the assignments stop. Then we are puzzled when students fail to see the opportunities to apply what they have learned on a final exam with mixed problem types. Here's John or Jane looking at the math final, for example. Now problem number seven... is that a related-rates problem? Do I need to use simultaneous equations? Wasn't there a formula? Poor John and Jane. Before, they always knew what approach to use because the exercises at the end of the chapter always went with the method in the chapter. Now, they don't have that crutch.
Likewise, learners commonly fail to see applications of what they have learned in another subject matter or in everyday life—the problem of transfer of learning. No wonder! They've had no experience identifying problems and thinking how what they already know might connect to those problems. They have never exercised the executive function of deciding what problems to tackle and how.
This is no reason to conclude the opposite: that teachers, texts, and computer tutoring systems should turn over the task of navigating through learning opportunities to the students from the first. Some people do argue for something like this, but I see no evidence that it is particularly effective. If anything, students have no idea what to do with initial near-total freedom, not only because they are not used to the elbow room but because they lack the knowledge base about the topic that would inform wise decisions.
Rather, the main implication is that sooner or later the executive function should return to individual students or groups of students so that they learn how to guide their own thinking and learning. But when exactly?
Suppose that some students are just learning a cooperative learning technique. They most likely will need step-by-step direction. Suppose that they already have employed this technique two or three times with considerable direction. Then the teacher can say, "Remember how we did that? Who can tell me?" A few questions and answers in that spirit will get students off on the right track. Or suppose that the students have grown familiar with the technique. The teacher need only say, "Okay, it's our usual group thing. Let's go!"
In other words, when to pass the executive function to learners has a fairly straightforward answer: Pass along as much as you can as soon as you can. Teachers who know their students can best judge this. It's a matter of the nature of the task and the sophistication of the students and, often, a matter for teachers to experiment with as they learn what their students can handle. But youngsters are really shortchanged when they rarely exercise the executive function at all. This is one of the most significant gaps in conventional education.
### AN EXAMPLE OF PERSON-PLUS TEACHING
We have made our way from Alfredo's three notebooks to an enlarged view of educational practice. Most classroom conduct, like most psychological research, leans toward an emphatically person-solo perspective on cognition, neglecting the many ways in which people tap the resources of the surround (including other people) to support, share, and conduct cognitive processing.
A smart school should be different. Let's take a person-plus approach to thinking and learning. Let's treat the person-plus-surround as one system, scoring as thinking what gets done partly in the surround and treating as learning traces left in certain parts of the surround (such as a notebook). Let's challenge the hegemony of the person-solo posture.
Mounting such a challenge means paying attention to several things. First of all, opportunities must be sought (and in many current innovations are sought) to distribute cognitive functioning more widely, with the help of physical artifacts such as computers, of social configurations such as cooperative groups, and of shared symbol systems such as various languages of thinking. Second, we must be wary of the misleading belief in the fingertip effect—the idea that simply introducing ways of distributing cognition will make things happen. Rather, the taking of opportunities needs to be mediated. Third, the distribution of the executive function—who decides what to do—needs particular attention to ensure that somewhere in a system there is always a good executive function and that it eventually ends up with the learners.
* * *
KEY IDEAS TOWARD THE SMART SCHOOL
**CLASSROOMS: DISTRIBUTED INTELLIGENCE**
**Physical Distribution of Intelligence.** Notes, diaries, portfolios, calculators, computers, and the like.
**Social Distribution of Intelligence.** Learning in groups with common group test. Pair problem solving. Socratic teaching, drama activities.
**Symbolic Distribution of Intelligence.** Essays in mathematics and the sciences. Diagrams, taxonomies in literature. Varied text forms—stories, essays, lists, concept maps, charts, two-dimensional tables.
**The (Unreliable) Fingertip Effect.** Benefits of new physical, social, and symbolic configurations not automatic. Needed: help in recognizing opportunities, managing the cognitive burden. Careful design for motivation.
**Executive Function in Managing Tasks.** Good executive somewhere in the system, not necessarily the student. The student eventually gets the executive function.
* * *
So what would this actually look like? Imagine a class facing a puzzle: How would you arrange a fair race between a dog and an ant? Of course, this puzzle is not all there is to the project these youngsters are undertaking. It is simply a motivating device. The students are investigating the general theme of animal locomotion. Within this theme can be found a number of questions relating to biology, mathematics, and physics. For example:
_What different strategies for locomotion do animals display (for instance, two-leg walking, or four-leg, or six- or eight-, or as many as in millipedes, along with flying insect-style, flying bird-style, and so on)?_
_In what ways are these modes of locomotion adaptive for the life-style of the organism involved? What are their trade-offs (for example, flying has high energy demands but affords speed, escape from ground creatures, and a bird's-eye view)?_
_At what speeds do animals travel, and how are they measured—in absolute terms, relative to the size of the creature, relative to predators' speeds (plenty of mathematics enters here)?_
Of course, merely posing such questions does not demand a person-plus perspective. Person-plus enters during the students' pursuit of the project. They work in small groups—socially distributed cognition. They brainstorm questions to pursue regarding animal locomotion and capture them on paper—physically distributed cognition. They employ symbol-handling techniques introduced by the teacher for supporting exploratory thinking, such as listing, concept mapping, and flowcharts—symbolically distributed cognition.
Skeptical of the fingertip effect, the teacher does not expect that these social, physical, and symbolic resources will automatically help the youngsters progress in their inquiries. The teacher coaches the students to ensure that distributed cognition works well. The teacher helps the groups to divide their work into roles—Alan will handle the stopwatch, Beatrice will handle the frogs, Carlos will handle the ruler. The teacher helps students think about what kinds of graphic organizers would serve them well. Comparing animals' speeds? What about a table of numbers or a bar graph? Classifying different ways that different animals move? How about a concept map? And so on.
As the different groups choose their particular topics of inquiry within the general theme of animal locomotion, distributed cognition broadens and deepens. One group employs field notes and sketches to investigate how ants crawl. Another uses video-tape to examine with stop-action the gaits of humans and dogs. The groups employ the language of mathematics—numbers, formulas, and tabular arrangements—to calculate and compare rates of motion.
Toward the end, the groups collaborate on a large concept map, covering much of the wall of the classroom, that assembles and organizes what they have discovered about animal locomotion. And they may even stage a race between a dog and an ant, with different finish lines chosen to represent each creature's functional context.
Inquiry projects such as this, because of their richness and length, inherently call for a person-plus approach. Otherwise, how is one to divide the work and manage the flood of ideas and information? Moreover, outside of school settings, the activities of life tend to be project-like: many-sided, complex, ongoing, as in pursuing a hobby, advancing a profession, or even simply planning a picnic. Accordingly, this animal locomotion example is something of a pitch for project-based learning as well as for a person-plus perspective. In addition, recalling the last two chapters, it plainly involves understanding performances and the metacurriculum in several ways.
However, if project-based learning seems too complicated an enterprise to mount in a particular classroom, there is ample room for the benefits of a person-plus perspective anyway. Students can work in groups to address the exercises at the end of the chapter. Students can deploy concept mapping and other symbolic devices to synthesize what they have learned from textbook readings. They can maintain portfolios of their best essays or proofs of theorems. They can cooperate on papers. The serious use of distributed intelligence in the classroom comes in many sizes and shapes, some larger and more complex, others that can be started today on a whim, something for virtually everyone.
A person-plus agenda for education acknowledges the broad trajectory of the development of civilization, from one-pebble-per-sheep accounting systems to hieroglyphics to the alphabet and beyond. It's remarkable how vigorously people have recruited into the cognitive enterprise not only other people but the quiescent physical things around us, arranging and transforming them so that they become—to use a lovely phrase coined by Gavriel Salomon—"partners in cognition."
## CHAPTER 7 **MOTIVATION
The Cognitive Economy of Schooling**
"Why are we studying this?"
No teacher likes to hear this question! Such words of skepticism about the educational enterprise are among the least welcome utterances that students voice. When someone asks, "Why are we studying this?" any teacher hears the message all too well. To one student at least, it's not obvious at all. There's no point to the enterprise. So why expect me to buy in?
So what do we do? We bemoan students' blindness. We wonder why they don't recognize the importance of knowledge and skill for their futures. We wonder why they don't tune in to the rich rewards of Shakespeare or algebra. But sometimes students who offer such misgivings may be seeing all too clearly. Here, for example, is a bundle of questions one fourth grader wrote down about fractions:
1. How much is one half of a fraction? 2. Why do we use fractions? 3. Why [ _sic_ ] do fractions have to do with math? 4. Do they have to do with anything else?
Dumb questions? Not at all! Such questions as these do not sound like the obtuse recalcitrance of a student without vision. On the contrary, they are all too pointed. "Why do fractions have to do with math?" suggests that the math curriculum has not made plain how different sides of mathematics relate to one another. "Do they have to do with anything else?" suggests that links have not been made between the mathematics class and the rest of life. With such a persistent lack of connectivity, why indeed are these students studying fractions?
Certainly, we need to work to connect things up more. This is one of the agendas of a pedagogy of understanding, as discussed in chapter 4, and of teaching for transfer, as discussed in chapter 5. But the "Why are we studying this?" question and its kin point up an even larger concern about educational practice: What is it that could and should keep students learning at their best? And let's not forget teachers: What is it that could and should keep them teaching at their best? How, indeed, should we talk about the complex networks of motives that all too often produce low energy and negativity? And how can we see ways toward more vigorous and thriving smart schools with their high positive energy?
Here is a way of talking about all that. We can call it "the cognitive economy."
### THE IDEA OF A COGNITIVE ECONOMY
Consider the economy of a classroom. Not the money economy—how much teachers get paid and textbooks cost. Rather, the metaphorical economy of gains and costs that students encounter.
As in any economy, there are a myriad of gains and costs. The central gain for students is the knowledge and skills that they acquire. But besides this there are a number of other gains as well: intrinsic interest in some of what is taught, a sense of mastery, good grades, credentialing, teacher approval, peer status, and social interaction (not just in the halls, but in cooperative learning and peer collaboration). As in a real economy, many of these gains have value because of their later consequences: credentialing and skills for the sake of a better fate in the job market, for example.
As in a real economy, there are also a number of costs. Time and cognitive effort are the most obvious. But there are others as well: boredom, fear of failure, actual experiences of failure, feelings of isolation, and uncomfortable competition.
Teachers are also players in the economy of gains and costs in the classroom. One central gain for teachers is their students' academic achievement. On the whole, teachers feel very committed to student learning. Another gain is pleasure in the artful exercise of the craft of teaching. Another includes the respect of students, principal, other teachers, parents, and the community. And of course there are salary and professional advancement. But there are costs aplenty. They include time, effort, boredom, lack of appreciation, lack of control, and feeling beleaguered by dozens of agendas.
What kind of an economy is all this? Not a money economy, because money, although part of the picture, does not seem so central as other gains and costs. Let's call it a cognitive economy. After all, schools and classrooms deal with the cognitive achievement of their students more than anything else. Of course, many other noncognitive factors play important roles in schooling. But that's okay. We speak of Iowa as having a corn economy and the Arab countries an oil economy, even though Iowans and Arabs deal with many other things. We do so because those economies rotate around corn and oil, respectively. In just this sense, classrooms and schools have a cognitive economy—matters rotate around the cognitive achievement of students, whatever else goes on.
To clarify the idea of a cognitive economy a little further, it must be acknowledged that the economic metaphor is a loose one. For instance, there is no equivalent of money in the cognitive economy of the classroom; certainly not grades, because they are only one of several kinds of gain, and they do not function as a medium of exchange.
But there is a liberating advantage to the loose fit. In particular, lacking an equivalent of money, the cognitive economy of schools and classrooms does not press us to reduce all costs and benefits to some standard cognitive currency. While the common yardstick of the dollar in Keynesian economics yields mathematical rigor, it also generates unsettling equivalences: How many dollars is health worth? How many dollars is a life worth? In the cognitive economy of schools and classrooms, we can frame costs and benefits more loosely but flexibly in terms of the diverse qualities that count—effort invested, confusions encountered, experiences cherished, understandings gained, skills attained. Most of all, rewards in a cognitive economy need not be selfish in character. Altruism counts as well.
How the Idea of the Cognitive Economy Gives Reason
We usually discuss why students and teachers behave the way they do in terms of motives. And talk of motives is okay as far as it goes. But it fails to capture something crucial and widely neglected about the educational scene: Students and teachers are rational agents.
Talk simply of motivation suggests a picture of students and teachers responding blindly to the pushes and pulls in the school setting, even, in behaviorist terms, the "contingencies of reinforcement." But the economic metaphor pictures all as rational agents, players in the economic game, considering the costs and gains and generally responding rather reasonably. When a student says, "Why are we studying this?" the student has good reason for the question in the context of school as it is. Because the gains the student wants to see coming his or her way are often either not obvious or not there at all.
In _The New Meaning of Educational Change_ , Michael Fullan has recently commented wryly on teachers' rationality and innovation: "Teachers' reasons for rejecting many innovations are every bit as rational as those of the advocates promoting them." Fullan highlights some of the criteria teachers spontaneously use in sizing up an innovation: Does the change address a real need? Will students show an interest? How do I know the change will have the desired impact? Is it clear what I, the teacher, need to do? What about the time, energy, skills I need? What about conflict with other agendas? Often, proposed innovations make little sense from this very rational teacher perspective.
Of course, the reasonableness of people in the cognitive economy should not be overdrawn: People in real economies only behave with limited rationality, as stressed in the classic work of cognitive scientist and economist Herbert Simon of Carnegie Mellon University. Nonetheless, it is terribly important to recognize that students are not generally acting foolishly or blindly when they ask questions like "Why are we studying this?" Teachers are not acting irresponsibly when they hang back from an innovation, saying, "I really don't have the time."
Eleanor Duckworth, an academic colleague at the Harvard Graduate School of Education, writes eloquently of "giving reason to people"—probing to understand their logic when they seem to be struggling obtusely. They almost always have an intelligent cut, albeit sometimes a mistaken one. Giving reason is part of a pedagogy of understanding. And it applies just as much to motives as to concepts. We should also give reason to students who seem to be questioning or even disdaining the educational enterprise and to teachers who respond coolly to yet one more agenda. They are generally behaving quite sensibly.
### THE COOL COGNITIVE ECONOMY OF THE TYPICAL CLASSROOM
Let us give reason to the cognitive economy of the ordinary classroom. We don't have to be happy with it, but at least we can understand how it works.
Remember how smart schools need high positive energy. Basically, this is a matter of the cognitive economy. One might call the cognitive economy of the typical classroom a "cool" rather than a "hot" cognitive economy—one that does not motivate the energy needed for complex cognition of students but runs at an altogether lower level of cognitive demand. Let us probe why it functions in the way it does.
The Costs of Complex Cognition
One reason for the cool, low-energy, cognitive economy is the cost of complex cognition. Of course, reformers usually focus on the gains and feel them to be high. Complex cognition has more intrinsic interest and promises more payoff outside of school and later in life. But consider the cost to learners: Complex cognition demands much more effort. It creates greater risk of failure. It introduces the discomforts of disorientation, as learners struggle to get their heads around difficult ideas. Peer status for complex cognition is certainly mixed; who wants to be known as a "brain"? And very commonly, so far as grades and teacher approval go, complex cognition buys students no more than the simpler path of getting the facts straight and the algorithms right. No wonder, then, that students perfectly reasonably do not automatically gravitate toward complex cognition.
The Lack of Connectivity and Consequences
For complex cognition, or equally, for basic knowledge and skills, one can ask, "What does it connect with? Where are the gains?" As intimated by the fourth grader's questions about fractions, often the curriculum does not make plain the intellectual or practical significance of even the basics, never mind complex cognition. To be sure, there are consequences of lackluster performance for grades and teacher approval (but see below). However, at least some of the consequences for everyday life are doubtful. How does school knowledge prepare the ghetto dweller to cope? In terms of the labor market, as mentioned in chapter 2, research shows that possession of a high school diploma makes virtually no difference in access to jobs for students who are not college bound.
A Sole-Source and Single-Choice Economy
Teachers- _cum_ -texts constitute almost the sole source of information in the typical classroom. And the fare offered by the teacher- _cum_ -text usually is not a menu, but take it or leave it. Plainly, real economies gain much of their vigor from multiple sources for goods and from the flexibility that members of the economy have to seek after their own individual interests. Analogously, students might learn from one another much more than they typically do and might have more choices about what they learn within a subject matter than they typically do. None of this is to deny the importance of a core of knowledge and skills that all students should acquire to some reasonable degree of mastery. Nor is it to argue for a wide range of electives at the expense of a solid core. But the sole-source, single-choice, cognitive economy of the typical classroom seems far more extreme than needed to provide for the common core.
The often abysmal quality of textbooks makes the typical sole source an unfortunate one, with plenty of cognitive cost in boredom and little cognitive gain in insight. Textbook publishers have responded to market pressures and economies of scale by producing textbooks that try to meet a myriad of requirements in different states and the pressures of a hundred special-interest groups. One common tactic is called "mentioning." Publishers say something somewhere in their texts about as many nations, ethnic groups, ideas, issues, and people as they can. Inevitably, it's an exercise in overweening superficiality. Textbooks also use readability formulas to control the difficulty level, formulas that have nothing to do with clarity of exposition and less with engaging style. The result: bad writing in "readable" textbooks.
Teachers' and Schools' Conflict-of-interest Position
Of course, the students are not the only players in this cognitive economy. As a key source of information and guidance, the teacher also holds a unique position as the administrator of grades and advancement. Indeed, the gains and costs from the standpoint of the teacher put teachers in a bit of a bind. On the one hand, research shows that teachers have very high intrinsic motivation. They value youngsters' learning and will work hard to foster it. The gain of seeing youngsters learn spurs them onward in their efforts to teach their best.
However, there are costs of setting high standards, costs associated with the culture of the school and the society outside the school. Teachers get paid not only for teaching but for moving students along through the grades. It is embarrassing for a teacher to hold back many students. Students are discouraged, and parents can be furious. Complaints come to the principal. In many contexts, complaints may also come to the principal if teachers attempt something "highfalutin," peer collaboration or the teaching of thinking or writing without due attention to spelling, for instance.
So teachers feel pressed not to demand too much. Demanding a lot creates a lot of trouble and a lot of cost. So teachers behave entirely reasonably: They do the best they can to encourage good learning but without demanding it. Inevitably, there sets in grade inflation, and more generally, "standards inflation," not a bad analogy to the phenomenon of inflation in real economies.
The same logic applies to whole state educational systems. Over the past few years, efforts to legislate tougher graduation requirements in many settings have generally backfired. Schools have responded not by finding better ways to teach challenging subjects but by padding the menu with watered-down subjects. A few years ago, Florida legislated stiffer high school graduation requirements—three years of science and mathematics, for instance. With what result? Weaker students enrolled on a massive scale in a new, academically negligible "Fundamentals of Biology" course and other "Fundamentals of..." courses acknowledged to require little intellectually of students. "Informal Geometry" calls for no formal proofs, while "Fundamentals of Math II" essentially repeats "Fundamentals of Math I," which is itself a low-demand remedial course. Such courses attract not only the weakest students but a fair number of moderately able students looking for the easy way out.
The Token Investment Strategy
Teachers are beleaguered with agendas. Responsible for teaching the traditional subject matters, they are also asked to impart basic knowledge about health, sex, and the risk of AIDS, imbue youngsters with citizenship values, detect students with special needs or talents and respond appropriately, build students' writing skills whatever the subject matter, foster good thinking, ensure "fun" participation, meet with parents, and so on and so on.
How to manage it all? Very commonly, teachers adopt an adroit strategy: "token investment." They invest a little bit of their resources in everything. In the cognitive economy of typical classrooms and schools, this strategy makes very good sense. 1. It does something for the students about each agenda, even if not as much as one would like. 2. It protects the teacher against the accusation from principals, parents, or students of doing nothing about an agenda, a high-cost accusation.
The token investment strategy should not be seen as selfish and uncommitted. On the contrary, it often constitutes an heroic effort to accomplish as much as possible. The Horace Smith that Theodore Sizer writes about in _Horace's Compromise_ is a case in point. As mentioned in chapter 3, Horace is a conspicuously committed teacher, who asks students to write often, responds to their work, and so on. But to serve all his students and all the agendas, he spreads himself thin. He ends up spending only about five minutes a week on each student's written work and ten minutes planning each class, considerably less effort on any one thing than he thinks appropriate. This is the compromise.
Students, too, are not strangers to the token investment strategy. Student lives are full of demands and desires—some of them academic, many of them not. How to manage the many agendas? Often, token investment. Students quickly learn that it is much better to do a little something on an assignment than nothing, which would trigger a harsh response from the teacher.
Effective though the token investment strategy is in the cool cognitive economy of the ordinary classroom, it of course has at least two flaws from a larger perspective. First of all, students would arguably gain more in the long run by in-depth attention to certain priorities. Second, for many of the agendas, token treatment of many of the agendas arguably benefits students not at all. While the token investment strategy presumes that the small investment in X will achieve a modicum of learning about X, often, without more ample and recurrent treatment, X never takes hold in students' minds at all. In a day or a week, it is gone.
Coverage
While innovators criticize the overemphasis on coverage and teachers generally acknowledge its ills, the impulse to coverage persists. Beyond condemning coverage, we need to recognize the cognitive-economic forces that sustain it.
Coverage can be seen as a special case of the token investment strategy. Students receive a token exposure to a large number of topics in, let us say, science. As in general with the token investment strategy, this puts teachers, texts, and curricula in a highly defensible position. "Students had an opportunity to learn this" can be the response to a parent who complains about a question missed on the final exam or the SAT, to a congressperson who complains that youngsters do not know basic facts about government, to a scientist who complains that students do not know how many planets circle the solar system.
To be sure, there is more to the dominance of coverage than the defense-oriented, token investment strategy. As discussed in chapter 2, a trivial pursuit conception of knowledge and understanding pervades much of education. However, the cognitive economy has a role in sustaining this conception. As happens throughout human affairs, pragmatics reinforce ideologies. Just as in the days of the Crusades, the lure of booty from the East encouraged ideological commitments to a Christian world, so the defensibleness of coverage subtly encourages commitment to a conception of learning as the accumulation of information and skills.
### CREATING A HOT COGNITIVE ECONOMY
We have to do better. Certainly, if we want to attain those three crucial achievements of retention, understanding, and the active use of knowledge, a different cognitive economy is in order—a hot rather than a cool one, a cognitive economy that sustains high-energy, complex cognition in the classroom rather than the low-energy, simplistic cognition of facts and routines.
But how to build a hot cognitive economy? The basic rule is this: An innovation has to make sense in economic terms—cognitive economic terms. An innovation that demands complex cognition of students has to bring with it (a) conspicuous gains and (b) minimally increased costs. After all, the greater cognitive demand is itself a cost that must be offset for the innovation to make sense.
So what might work? What will not work is simply upping cognitive demand—more Shakespeare, calculus sooner, history based on original source materials, no wimpy courses. All that may or may not be a good idea, but by itself it simply increases cognitive complexity, which is a cost, while doing nothing to minimize cost or ensure real and visible gains persuasive to the students.
What can work are the strategies suggested in earlier chapters. They have not been casually chosen, but selected with an eye toward their viability in the cognitive economy of a classroom. A list for review may help bring the prospects into focus:
_Theory One_ , the basic theory of teaching and learning that asks for clear information, thoughtful practice, informative feedback, and strong intrinsic or extrinsic motivation, can reduce the cost of complex cognition by making it more accessible and by reducing students' risk of failure and can increase the gain by helping students to learn more.
_Intrinsic motivation_ in particular can be boosted by giving students more choice about exactly what they work on and more information sources than teacher and text, instead of the sole-source, single-choice economy typical of classrooms (this does not mean an "anything goes" setting).
_Coaching and Socratic teaching_ can reduce the cost of complex cognition in risk and fear by supporting students in the learning process in ways that didactic instruction cannot and can increase the gain by helping students to learn more and by providing a more interesting, interactive style.
_A pedagogy of understanding, with its focus on understanding performances, mental images, and powerful representations_ , can reduce the cost of complex cognition by making difficult ideas more accessible and can increase the gain through greater learning and through the increases in intrinsic motivation that come with understanding what one is learning.
_Higher levels of understanding_ , patterns of problem solving, explanation and justification, and inquiry appropriate to the subject matters can increase the gain by helping students feel more oriented to a subject matter and more empowered to do things with it.
_Generative topics_ , when they become the bread and butter of the curriculum, can increase the gain by making the subject matters more intrinsically interesting and more connected to applications beyond the classroom. Good choice of a limited set of generative topics can help teachers evade the token investment strategy, where they spread themselves thin.
_From the metacurriculum, languages of thinking, integrative mental images, and learning to learn_ can reduce the cost of complex cognition by making difficult topics more accessible and can increase the gain by empowering students as thinkers and learners.
_The intellectual passions_ , modeled and encouraged by teachers, can increase the gain by developing proactive, thoughtful mindsets in students.
_Teaching for transfer_ , with ample attention day by day, can increase the gain by making the payoffs of more demanding studies clearer and preparing learners to transfer school learning to other classes and to applications beyond the classroom.
_The physical distribution of intelligence_ , via writing and other media, can reduce the cost of complex cognition by reducing cognitive load.
_The social distribution of intelligence_ , via peer tutoring, cooperative learning, and peer collaboration, can reduce the cost of complex cognition through group support and the comfort of groups when tasks are hard and can increase the gain through the enjoyments of working with others and the payoffs of learning how to do that well.
_The symbolic distribution of intelligence_ , through drawing a number of different symbolic forms into instruction in the different subject matters (stories, concept maps, diaries, improvisatory drama, picturing) can reduce the cost of complex cognition by making ideas accessible in symbol systems better suited to the topic and/or different individual learners and can increase the gain by equipping all students to work with a diversity of symbol systems.
So what's the catch? If all this worked that well, we would see it in every school district. And there is a catch; in fact, two of them.
**Catch #1.** These strategies for the hot cognitive economy all require up-front investment. Although they can pay for themselves once in place and part of the system, there is a high initial-effort cost, and some dollar cost, in expanded programs of teacher development, rearranging old patterns of administration, getting students used to the new pattern, assembling appropriate materials, and so on.
Often, we do not see a hot cognitive economy because the transition to it is hard. But the smart school has a commitment to making the up-front investments needed for a hot cognitive economy. Moreover, for a somewhat reduced cost, sources such as Michael Fullan's _The New Meaning of Educational Change_ tell us much about how to manage such change processes less painfully, as the next chapter explores.
**Catch #2.** Besides needing to pay the initial cost of change, schools need visions of what to change to. Otherwise, it's easy to get lost. In a way, there is too much to try out. Education abounds with new ways to teach fractions or Dickens, scientific thinking or good citizenship, a vast and confusing menagerie of options. What's required—but found much more rarely than individual, promising innovations—are holistic visions of schooling that provide an overall direction. Many current efforts to restructure schools come with visions that lean in the direction of the smart school. I hope that the idea of the smart school as outlined in this book will bring into clearer focus what such schools could be like.
### SCHOOL RESTRUCTURING: A COGNITIVE ECONOMIC REVOLUTION
A conspicuous feature of the United States educational landscape over the past few years has been efforts to "restructure" schools. As mentioned in chapter 1, behind this initiative lies the recognition that many organizational features of schools—short class periods, too many subject matters, overauthoritative leadership—make more enlightened instructional practices tough to launch and tougher to sustain. School restructuring strives to take down these barriers, liberating teachers, administrators, and students to pursue a brighter path.
One of the best known examples of restructuring high schools is the Coalition of Essential Schools, founded by Theodore Sizer of Brown University. Sizer and his colleagues do not offer a detailed model for what such a school should be like. Rather, Sizer articulates a set of principles, his well-known "nine points." Through these nine points, Sizer urges a focus on only a few academic subjects, with emphasis on adolescents' using their minds well. He calls for simple and universal goals, a limited number of important skills suitable for all students. But students of different abilities and interests may pursue them in different ways. Education must be personalized rather than impersonal, emphasizing individual needs and supportive individual relationships among teachers and students.
Sizer envisions the student-as-worker in contrast with the teacher-as-deliverer-of-instructional-services. Teachers coach students, who engage in meaningful extended inquiry. Graduation requires an "exhibition" by a student, demonstrating his or her grasp of the school's expectations. Through such exhibitions, students must show that they can accomplish worthwhile inquiry or other projects. In the long term, Sizer urges that such a program need only cost 10 percent per student more than a traditional high school.
Notice how Sizer's vision remakes the cognitive economy of the conventional school. With more time for fewer subjects and skills, the cost in frustration of students and teachers reaching for depth becomes much lower. With student-as-worker, both students and teachers get rewarded for nontraditional roles—the student a serious inquirer and the teacher a coach rather than lecturer. The emphasis on exhibitions from students wires student-as-worker to graduation requirements. Sizer's budget estimate calls for more dollar investment, but not that much more, keeping the dollar cost of a hot cognitive economy reasonable.
The Coalition of Essential Schools does not expect each participating school to cast itself in the same mold. The faculty and students at a Coalition school need to find their own way, albeit with input about what others have done. One well-known incarnation of Sizer's principles is the Central Park East secondary school in New York City. Most of the instruction gets organized into two broad subject areas, humanities and mathematics/science, to help knit together particular subjects. Teachers generally manage more than one subject, often teaching in teams. Each student takes responsibility for two hours of pro bono work per week, either out in the community or within the school. Some work in the cafeteria, others in the library, others help deliver food to needy folks in the neighborhood.
A spirit of thoughtfulness prevails. The Central Park East Secondary School has a credo called "The Promise," which swears to develop the students' minds. Students are to learn to ask and answer four key questions in each and every subject matter:
1. From whose viewpoint are we seeing or reading or hearing? From what angle or perspective?
2. How do we know what we know? What's the evidence, and how reliable is it?
3. How are things, events, or people connected to each other? What is the cause and what the effect? How do they "fit" together?
4. So what? Why does it matter? What does it all mean? Who cares?
Central Park East does not rest at the end of a sumptuous suburban lane amidst graceful elms. It sits in the center of a tough urban neighborhood plagued by drugs, poverty, and the usual array of urban ills. So how does it fare? In this vexed setting, attendance is routinely 90 percent and the drop-out rate almost zero. Remarkable! Obviously, the students have decided the payoffs are right: The gains are worth the costs. The hot cognitive economy is alive and well at Central Park East.
Another example of school restructuring is the Comer model, initiated by James P. Comer of Yale Medical School and the Child Study Center at Yale. Comer, a black psychiatrist, developed an acute sensitivity to the dilemmas of disadvantaged students attempting to survive and learn in the alien world of conventional schools. Their plight both resembled and differed from his own as a child. Growing up in poverty, he nevertheless benefited from the support of his mother and adults who were, as he put it, "locked into a conspiracy to see that I grew up a responsible person."
How could schools help disadvantaged children receive the support Comer had enjoyed at home. Comer conceived and put into practice a school structure that reached out to individual students. Its first premise is community involvement. The principal leads a governing council of teachers, counselors, and parents to set directions for the school. Attention goes first where it seems to be most needed—replacing boarded-up windows with glass, making the playground safe. Comer schools encourage parents to be a presence in the classroom as assistants, tutors, or aids. Social events help to bond teachers, parents, and youngsters.
An especially distinctive feature of Comer schools is a "mental health" team consisting of guidance counselors, school psychologists, special-education teachers, nurses, and classroom teachers. The team serves as a safety net for youngsters that might drop out or get expelled in conventional schools. One of their main jobs: case-by-case analysis of any child having difficulty, working over a period of time to help the child thrive.
In his recent book _Smart Schools, Smart Kids_ , Edward Fiske relates the story of a first grader he calls Robert at Columbia Park Elementary School in Landover, Maryland. An authentic cutup, Robert had a repertoire that included sticking pencils in other children's ears and pouring chocolate milk on his cafeteria mashed potatoes. Trouble with Robert was constant.
The guidance counselor and Robert's teacher helped Robert's mother to evolve an explicit plan of rewards and punishments that might lure Robert away from his mischief. Robert's behavior improved some, but not enough.
The mental health team met to consider Robert's difficulties. In the midst of a long interview, Robert told them one thing he wanted—more time with a favorite teacher. It was arranged. They also worked with Robert's mother, guiding her to administer rewards and punishments as she had to but always to make her love clear and unconditional. They set up an information hotline for Robert's mother: When Robert behaved well, someone would call home to say so. Robert would be welcomed home with praise.
Eventually, all these efforts took hold. Robert went half a day, then a day, then three days without trouble. School counselor John Haslinger says:
On the fourth day in a row, I'll never forget it, he came up to me and shouted, "I am going to go for fifty days, then a hundred days." His arm was raised in the air like a champion. And he did it! He became a good kid, a success story. It's very gratifying.
Comer schools illustrate the idea that one key to a hotter cognitive economy among students is a hotter cognitive economy among adults linked to the school. In all too many urban settings, teachers and administrators sleepwalk through their responsibilities, dulled and discouraged by the endless pressures and problems. Comer schools broaden the constituency by emphatically including parents in the thought and work of the school, and they revitalize everyone's interest by asking all to be thoughtful and responsible, not just for the school in a general sense but for each individual child.
There are, of course, many other models for restructuring schools. But restructuring does not come easy. There are endless roadblocks in the form of entrenched attitudes and expectations, limited textbooks, commando-style leaders, and so on. Often, schools find themselves trapped like insects in an ancient amber of outdated regulations and accountability. Perhaps no mechanism figures here more dominantly than conventional assessment, to which we turn.
### TEACHING TO THE WRONG TEST
The inherent conflict-of-interest dilemma of the teacher in the typical classroom has not gone unnoticed. Indeed, there is a familiar solution to it: external testing. By this I mean that some sort of test is declared the official gauge of student performance at a key point in the students' schooling—say, the end of elementary school. The teacher does not make up the test, although the teacher may grade it if this can be done objectively. To advance, students must pass the test.
It's easy to see how this dissolves teachers' conflict of interest. With the external test in place, the teacher no longer is in a position to strike compromises, demanding less "cognitive coin" from the students to ensure that most of them will pass. Students must work to achieve well on the external tests, and teachers must help them to make sure that most do.
Of course, neither students nor teachers always welcome the pressure imposed by an external testing system. Moreover, the installation of such a system can certainly occur in clumsy, high-handed ways. Finally, occasional external testing does not take the place of the much more frequent testing a teacher needs to do to keep in touch with students' progress. Nonetheless, it's important to recognize how some external testing _liberates_ teachers to pursue instruction in closer partnership with their students.
So, by this scheme, both students and teachers think carefully about what the external test demands, and the teacher tries to teach to it. "Aha, teaching to the test," someone will say. "That's a bad thing."
Not necessarily. There is nothing wrong at all with teaching to a test _provided the test tests the outcomes you really want_.
The trouble with the external testing maneuver is that typically the test is a reductive one, reflecting a trivial pursuit conception of education, emphasizing knowledge retention through a multiple-choice or fill-in-the-blank format (with its tendency to cultivate inert knowledge) and the execution of algorithms. When teachers teach to such tests, students generally get better at them but not better at the complex cognition we're looking for.
Test bashing has become one of the most popular sports in the educational arena. The tests are awful, the story goes. The tests drive the system. If the tests were not so awful, that might be acceptable. But how could they be so awful?
Besides designing better tests (see the next section), we have to understand why tests are as awful as they are. It is not accident or obtuseness that makes tests awful, but factors in the cognitive economy. First of all, money is one part of the cognitive economy, and more cognitively demanding, open-ended tests can be more costly to administer and grade.
But there are less obvious and more pernicious factors at work, too. First of all, installing external tests as gatekeepers to credentialing does not really remove the conflict of interest teachers face. It only "promotes" that conflict of interest up the hierarchy of education to the school-system level or the state level. Now it is the school system or state which decides what tests to use and what thresholds count as passing. So now it is they who receive pressure toward straightforward, simple testing from teachers, parents, school boards, and government leaders, who all want to see successful students.
Of course, such pressure does not take the form of a direct request to "be lenient." But a test system that looks esoteric, that does not resonate with popular attitudes about the importance of spelling, that asks fancy questions that parents cannot understand, and that does not yield "reasonable" numbers of passes may be deemed unfair, racist, and ill chosen. Those who established it may get fired or lose elections.
This is not to say that school-system- or state-mandated external tests are a bad idea. Levels of the hierarchy above that of the individual classroom teacher are probably better able to hold the line. So moving the conflict of interest upward from the classroom level serves education well. But we should not deceive ourselves into thinking that the conflict of interest has vanished.
Besides the conflict-of-interest problem, there is another one with external tests. Cognitively demanding tests introduce a complication. When the testing is straightforward—facts and routines—teachers know quite a bit about how to teach to the test. They can enter into fairly effective partnerships with their students to help them toward mastery and over the hurdle of the test. Moreover, texts feed into the process nicely with their heaps of facts and piles of problems at the ends of the chapters.
However, when the test demands complex cognitive performances, teachers do not on the whole know how to teach for it. Nor should they, since neither their preservice educations nor the texts they are using give much information about complex performances like, for example, the understanding performances discussed in chapter 4 or some of the kinds of thinking discussed in chapter 5.
Consequently, installing a test that demands complex cognition without revamped instruction is a recipe for disaster. While the simple, reductive, external test will effectively drive the system, the cognitively demanding test will not, because the system is not able to respond. Widespread failure can result, impugning the competence of those administrators at the school system or state level responsible for putting the test in place.
None of this is meant as an argument for knuckling under and making do with reductive tests. But it is meant as an appeal to recognize that, in the cognitive economy of classrooms, school systems, and state education systems, reductive tests thrive not because people are starkly unenlightened but because they are responding to the cost in risk and cost in resources of cognitively demanding tests.
### TEACHING TO THE RIGHT TEST: THE IDEA OF AUTHENTIC ASSESSMENT
If teaching to the wrong test does so much mischief, what is the right test to teach to? Over the past several years, a vision of and a name for such a test has emerged. The name is authentic assessment.
Well chosen as a term, authentic assessment implies that the test in question tests students by engaging them in examples of the very target performances we really want. An authentic test of writing stories asks students to write stories and gauges their response on the basis of the richness of the stories they write. Authentic assessment of mathematical attainments engages youngsters in working through open-ended problems that require mathematical reasoning and appraises how well students do and in what ways.
Concretely, what do such questions look like? Mathematics is a good area to pick for some examples, since mathematics testing so commonly takes a simple figure-out-an-answer form. One useful source is a booklet called _Assessment Alternatives in Mathematics_ , produced by the Lawrence Hall of Science, Berkeley, California. One problem in gist goes like this:
_A tape recorder is just beginning to play a tape. The tape passes over the head of the tape recorder at a constant speed. The challenge: Draw a qualitative graph showing how the length of the tape on the uptake reel changes with time; and draw another showing how the radius of the tape on the uptake reel changes with time; and yet another showing how the radius of the tape on the feed reel changes with time, with an explanation of why_.
Notice that such a problem calls for no numbers. It is a qualitative problem that gives no opportunity to turn the crank of some routine. Rather, it requires the learner to reason about what would happen and represent it by using graphs and giving a verbal explanation.
Another kind of problem challenges students to compose conventional problems rather than solve them; for example, "Write a word problem that would probably involve multiplying 59 times 12 for its solution." Yet another asks students to make up a lesson plan to teach younger students what multiplication is all about, using whatever materials they want from such resources as blocks, beans, balance scales, tiles, graph paper, calculator, and so on.
To reach elsewhere for examples, one popular category of activities is called "Fermi problems" after the Nobel Prize-winning physicist Enrico Fermi, who offered such problems for fun. A typical Fermi problem is "Estimate the number of pencils in Chicago."
Outrageous, isn't it! The problem is deliberately ill defined. Plainly, no more than an estimate can be derived. That estimate in turn is going to depend on other estimates, such as pencils per home or pencils per person. But after a little thought, the problem does not seem so impossible after all. Population figures can be obtained from almanacs, encouraging use of library resources. Pencils-per-person or pencils-per-home estimates can be obtained from home surveys done by students themselves. Students may seek to refine estimates by taking into account institutions other than homes. Would stationery and other stores that sell pencils add appreciably to the estimate? Would schools in Chicago add appreciably to the estimate?
As these examples suggest, items for authentic assessment have a number of salient characteristics:
They are open-ended rather than one-right-answer problems.
They are not solvable by applying a routine method.
They require substantive understanding of meaning (in the case of mathematics, of arithmetic operations and other mathematical knowledge).
They demand considerably more time than conventional problems; accordingly, an assessment might only involve one or a very few authentic assessment problems.
They call for pulling together a number of different ideas from the subject matter.
They often involve writing as well as formal manipulations such as computation.
They usually have a complex product: an essay, a lesson plan, a problem set for others to solve.
Note that there is a fringe benefit to authentic assessment. Because of all these characteristics, doing an authentic assessment problem tends to be very much a learning as well as a testing experience. Authentic assessment problems stretch the learner even as they create an occasion for a learner to display mastery and understanding. Inherently, they test for, and therefore press for, transfer and understanding, two principal concerns of the metacurriculum. Indeed, in classrooms that emphasize authentic assessment, little distinction appears between assessment and other activities. Youngsters are simply assessed in terms of the rich thinking and learning activities underway. Teaching, learning, and assessment merge into one seamless enterprise.
Besides the kind of problem posed, there are also variations in the form of the product and the time scale. An assessment may be based on a test administered over a few hours, or on a portfolio where students keep what they judge to be their own best problem write-ups, or on notebooks that students keep for an entire semester. Small-group projects can be used, and the projects may endure for a day, a week, or longer. As mentioned earlier, the Coalition of Essential Schools calls for "exhibitions" from students, demonstrating their academic prowess to qualify for graduation. The possibilities are endless.
### THE COGNITIVE ECONOMY MEETS THE MONEY ECONOMY
So why don't we do it everywhere tomorrow? Because there are counterforces. Let us look back to the cognitive economy of the classroom and some of the dilemmas posed by so greatly enlarged a conception of assessment. The gain of authentic assessment is, of course, to create a system where complex cognition carries much greater value. Authentic assessment upgrades performances requiring persistence, understanding, problem-solving abilities, and ready use of resources at the expense of more routine problems. Remember, in the normal classroom, complex cognition costs more in time, effort, and risk but typically is not rewarded more. By attaching greater value to complex cognition, one makes it a more fallible commodity.
* * *
KEY IDEAS TOWARD THE SMART SCHOOL
**MOTIVATION: THE COGNITIVE ECONOMY**
**The Cognitive Economy of the Typical Classroom.** The high cost compared to gains of complex cognition. The lack of connectivity and consequences. A sole-source and single-choice economy. The teacher's conflict-of-interest position. The token investment strategy. Coverage as a pattern of token investment.
**Teaching to the Wrong Test.** Usually the test does not test the performances we really want. More demanding tests drive the system effectively only when teachers know how to teach to them.
**Instruction.** Theory One and beyond. A pedagogy of understanding. The metacurriculum. Distributed intelligence.
**Authentic Assessment.** Open-ended rather than one-right-answer problems. Not solvable by routine methods. Require substantive understanding of meaning. Demand considerably more time than conventional problems. Require pulling together several ideas from the subject matter. Often involve writing as well as computation. Usually have a complex product: essay, lesson plan, problem set for others to solve. Portfolios representing products and process often kept.
* * *
The fly in this smooth ointment is, of course, the cost of authentic assessment within the cognitive economy of the classroom, the school system, and the state. As mentioned already, such assessments are more costly, in time, effort, and money, to compose and grade. And they are more costly in quite a different sense: the risks of a backlash when you demand more of students. And, to guard against that backlash, they are more costly in putting into place instructional innovations that adequately prepare students for higher cognitive complexity. It's an expensive proposition all around.
Thus it is that authentic assessment sounds wonderful on paper but tends not to market well in the actual cognitive economy of classrooms, school systems, and states. What is the solution to this dilemma?
There is no _magic_ solution. We simply have to face up to the problems inherent in a cognitive economy. One initial hindrance but long-term help is that much of the cost is a front-end investment, as noted earlier. Ongoing work to illustrate and systematize approaches to authentic assessment, work such as that distilled in Assessment Alternatives in Mathematics, mentioned above, helps make authentic assessment less costly not only in dollars but in teachers' and students' uncertainties and anxieties. Ongoing work toward better instruction—for instance, via a pedagogy of understanding, the metacurriculum, and better distributed cognition—helps make the kind of instruction that can prepare students for authentic assessment more economical in all the senses of the cognitive economy.
The current enthusiasm for school choice reflects a recognition of economic conundrums. In conventional settings, the local public school is a virtual monopoly. Apart from costly private schools, it is the sole source of educational services and thus largely protected from the consequences of any shortfalls.
School choice says that, under one plan or another, children and their parents get to pick the school they attend within some region. School-choice plans assume that, like consumers everywhere, children and parents want a good deal: in this case, an effective education. Schools that perform poorly will fail to draw students. They will find ways to do better or go out of business.
School choice is a complex and volatile issue. Will children and parents have good information to make choices? Often no, not even if local schools provide information. Will children and parents opt for better education? Not always. The convenience of the school next door commonly outdraws the glitter of the school across town.
And does school choice guarantee a push toward what I have called the smart school? Assuredly not. Many parents and children in disadvantaged settings aspire more to better performance on conventional tests than to a genuinely thoughtful education. To them, the smart school can all too easily look like an upper-middle-class luxury or even a boondoggle. While I have argued that the simple goals of retention, understanding, and the active use of knowledge _require_ the smart school, such lines of reasoning and evidence are hardly known in the ghetto and if known, would be viewed skeptically.
Despite all that, school choice certainly creates a press for better education in general and sometimes for the smart school. Moreover, we have evidence that it does what it is supposed to do: push schools toward more responsive and responsible conduct. For about a decade, the school system in Cambridge, Massachusetts, has followed a limited school-choice plan, where parents and children signal preferences and get their choice when it is consistent with desegregation requirements. Ninety percent of Cambridge children choose public over private schools, up from 70 percent a few years ago. Standardized test scores have risen, whatever their flaws. Racial minorities can no longer be distinguished from the mainstream by lower test scores in the Cambridge system, although social class still makes a difference.
In New York City's District 4, where a school-choice plan has operated since 1982/3, two schools closed for lack of students, and new schools and schools within schools, sharing the same building, have thrived. District 4 now draws over 1,000 students from other areas of New York City. District 4 includes the Central Park East Secondary School, discussed earlier.
So there is motion. The best hope for accelerating the pace and achieving the aim may be the fact that the cognitive economy of classrooms is yoked to the money economy of the nation. As discussed toward the end of chapter 2, a number of countries in the world have made this connection work for them through assessment and credentialing systems that virtually require considerable general and specialized educational achievement in order to enter the job market, even at the blue-collar level. These systems include safety nets to catch those who have trouble. In the United States, as companies cut back on upper management to save money, they must ask employees at the lower end to make more decisions, to respond to greater cognitive demands. Traditionally less-skilled or less-educated employees need to think and learn on the job now, more so than a few years ago, if the money economy is to thrive. The spiraling economic problems in the United States are a powerful motivator in the money economy toward getting our act together in the cognitive economy.
### AN EXAMPLE OF PROGRESS TOWARD A HOT COGNITIVE ECONOMY
How hot is hot? What does a hot cognitive economy look like? And who would be so bold as to try to set one up on a wide scale?
We do not have to grope for examples because the state of Vermont has launched just such an endeavor. Already on a pilot basis and soon statewide, Vermont teachers in the fourth and eighth grades will assess their students in writing and mathematics in a new way.
The program relies on "portfolio assessment." In writing and mathematics, students accumulate portfolios of works during the school year. The works represent a variety of open-ended problems and tasks. They reflect the students' efforts not only in English and mathematics classes but in other subject matters that invite written statement or mathematical analysis. While a portfolio represents a student's work as a whole, students also select for special scrutiny pieces they and their teachers view as their best accomplishments.
A fine picture. But what actually goes into these portfolios? In the case of writing, the pilot initiative developed a profile of important portfolio elements. It's suggested that a portfolio include at least these:
1. Table of contents.
2. A dated "best piece," chosen with the teacher's help.
3. A dated letter from the students to the reviewers, explaining the choice of the "best piece" and the process of its composition.
4. A dated poem, short story, play, or personal narration.
5. A dated personal response to a cultural, media, or sports exhibit or event or to a book, current issue, math problem, or scientific phenomenon.
6. _Fourth grade:_ A dated prose piece from any curriculum area that is not English or Language Arts. _Eighth grade:_ Three dated prose pieces from any curriculum areas that are not English or Language Arts.
For sure, a rich array of products, and students can go well beyond this. As to the assessment itself, teachers evaluate the portfolios on a scale from 1 (not so good) to 4 (outstanding), paying heed to five important characteristics: clear purpose in the piece; coherence of organization; detail used to support the main theme; voice and tone established; and finally, spelling, punctuation, and related matters.
What about the risks of subjectivity? It's worth remembering that assessments of writing are typically subjective once we get beyond spelling and punctuation. If anything, the five criteria make the process more systematic than usual. But the Vermont process takes the question of subjectivity seriously, providing training and, at the end of the year, gathering teachers together with a sample of their students' portfolios to score one another's portfolios, look for discrepancies, detect teachers who seem not as well calibrated with the others, and provide further guidance.
The tale in mathematics is much the same—an emphasis on open-ended problems that may take days to pursue, the building of a portfolio, criteria that look for mathematical insight across a variety of problems and settings, and a four-level rating scheme.
Of course, this program of authentic assessment aims to magnetize the process of teaching and learning in a new direction, toward new poles. All would be for naught if classrooms remain stuck to the all-too-common "What I did this summer" essays and battalions of routine arithmetic problems. But since students have to build portfolios full of their responses to invitingly open-ended tasks, teachers find themselves in a wonderful position: motivated to teach as most have always been wanting to.
So what happens in classrooms? Ann Rainey, an eighth-grade mathematics teacher in Shelburne, asks her students questions like this:
You and a friend read in the newspaper that 7 percent of all Americans eat at McDonald's each day. Your friend says that this is impossible. You know that there are approximately 250,000,000 Americans and approximately 9,000 McDonald's restaurants in the United States. Convince your friend (in writing) that the statistic is possible.
Tough? Yes, if you're a student who's not used to it. You'll have to make estimates. You'll have to plan computations to knit them all together. You'll have to wrap it all up in a written argument that mixes math and essay form. But it's doable. Indeed, it's not so different from the Fermi problem discussed earlier, estimating the number of pencils in Chicago.
And whatever you do with the problem, into your portfolio it goes. Maybe it will be one of your best pieces.
From a state policy stance on assessment to Rainey's particular question about how many Americans eat at McDonald's is quite a stretch. But that is exactly what we need to sustain a hot cognitive economy. Only the most courageous and even outrageous teachers can build hot cognitive economies in classrooms that sit within schools, school systems, and states timid in their aspirations and comfortable with cool cognitive economies. But when up and down the administrative ladder of education the voices sound in chorus, demanding authentic intellectual performance of youngsters, then the smart school with its high positive energies can thrive.
## CHAPTER 8 **VICTORY GARDENS FOR
REVITALIZED EDUCATION**
The first time I ever heard of a victory garden was on public television. One of the popular programs in the Boston area for a number of years was _Crockett's Victory Garden_. For thirty minutes every week, Jim Crockett took the audience out into his garden and demonstrated how ample knowledge along with tender loving care could do remarkable things with a little land.
I also learned why victory gardens were called victory gardens. The concept developed during World War II, when the citizenry was encouraged to help out the war economy during rationing by provisioning themselves insofar as they could. So in odd back lots, small gardens sprang up, lovingly tended, yielding carrots, broccoli, cabbage, lettuce, and other fine vegetables, along with a feeling of unity with the nation's effort.
The spirit and substance of a victory garden can be summed up by this little Victory Garden poem of mine:
With plenty of savvy
And T.L.C.,
You can do a lot
On a small plot.
Which brings us back to the dilemmas of education.
The previous chapters of this book have tried to deal with the "plenty of savvy" mentioned above. They have offered several ways of thinking about better education—ensuring that we teach at least as well as Theory One prescribes, building a pedagogy of understanding, attending to the metacurriculum, arranging better-distributed cognition in the classroom, and cultivating hot cognitive economies, all toward the smart school.
It's fair to wonder how to get these notions off the printed page and put them to work in behalf of better education. One answer is that they can serve as lively tools for evaluating educational settings and designing new ones. This chapter puts the "savvy" in order in three steps: (1) a brief review of the principles behind the smart school; (2) a series of classroom examples that show the principles in action; (3) a discussion of what's special about these principles, how they go not only beyond ordinary instruction but beyond what many innovations attempt.
But there's more to the story. Besides savvy about teaching and learning, a vision of enhanced education needs a way to cope with what we might call the "small plot" problem. There are in fact many successes in education. There are innumerable tales to be told about the wonderful results that occur in one teacher's classroom or sometimes in an entire school. A deservedly popular example is the achievement of Jaime Escalante in developing high school kids' mathematics abilities, celebrated in the film _Stand and Deliver_. These successes teach us a lot.
But they almost always occur on "small plots." They are victory gardens, where unusual savvy and unusual tender loving care have cultivated fine results. How do we think about scaling up, then? What kinds of results can we hope for in most schools for most students—not just meticulous victory gardens but the amber waves of grain we would like to see as our educational crop? That question is the topic of the chapter after this.
### BETTER TEACHING AND LEARNING IN REVIEW
The mainspring of this book is a fundamental principle about learning underscored in the introduction. Simple to state, its message continues through chapter after chapter, rewriting the way we should proceed with matters educational: Learning is a consequence of thinking.
The principle implies a different conception of educational practice than we usually find in the classroom down the street. It calls for educational settings where students learn by way of thinking about and with what they are learning, no matter what the subject matter is—history, mathematics, English, science, geography, you name it. Thoughtful learning in turn requires informed and energetic schools, settings where teachers and administrators know a lot about both learning and working together and have time to learn themselves and where the management style, schedules, and forms of assessment create positive energy in everyone. Thus, the smart school—informed, energetic, and thoughtful.
One might dismiss the smart school as something of a luxury, okay for especially able students and for wealthier communities but nothing so very essential for the larger part of our population. This would be a great mistake. While educational goals make for an endless debate, almost everyone could agree on at least three: retention, understanding, and the active use of knowledge. Because without these three, what students learn in school would not be very useful to them. But given them, the evidence presented in the last seven chapters argues that we need smart schools. We need to put to work the principle that learning is a consequence of thinking. If we do not, we simply will not get the amount or kind of learning that we want.
What we get instead was surveyed in chapter 2. Students acquire fragile knowledge, often inert (not remembered in open-ended situations that invite its use), naive (reflecting stubborn misconceptions and stereotypes), or ritualized (reflecting classroom routines but no real understanding). And students show poor thinking in many ways—inadequate reasoning from texts they read, bafflement in the face of story problems in mathematics, and so on. Behind these shortfalls lie two tacit theories that pervade our education and our culture more broadly. The Trivial Pursuit theory of learning says that learning is a matter of accumulating facts and routines, not a consequence of thinking. The Ability-Counts-Most theory says that ability influences achievement far more than anything else, when in fact effort counts at least as much.
So what do we do? Besides battling these shortfalls and the tacit theories underlying them, we can seek to remake education along five dimensions familiar from the preceding chapters. Here they are:
1. _Theory One_ says that we need instruction that emphasizes clear information, reflective practice, informative feedback, and intrinsic and extrinsic motivation. Teachers can draw on didactic, coaching, and Socratic teaching styles in applying Theory One. And they can explore methods beyond Theory One. Given methods at least as good as Theory One, our most important choice is what we try to teach.
2. _A pedagogy of understanding_ highlights students' engagement in understanding performances (explaining, finding examples, generalizing). Powerful representations can help build up students' mental images and equip them for understanding performances. Instruction needs to give attention to higher levels of understanding, capturing how in different subject matters we solve problems, explain and justify, and inquire. And we need to organize instruction around generative topics that connect in rich ways within the subject matter, to other subject matters, and to life beyond the classroom.
3. _The metacurriculum_ urges us to include attention to higher-order knowledge in several senses: levels of understanding (as above); languages of thinking (terms and concepts like hypothesis and evidence; graphic organizers); intellectual passions; integrative mental images (that knit a large topic or subject matter together); learning to learn; and teaching for transfer.
4. _Distributed intelligence_ calls for a shift from a person-solo to a person-plus organization of classroom activities. Physical resources such as writing materials and computers should support thinking and learning. Thinking and learning should be socially shared as in cooperative learning, peer collaboration, peer tutoring, Socratic interaction, and like relationships. And diverse symbol systems (language, graphic organizers, improvisations) should figure in thinking and learning.
5. _The cognitive economy_ underscores the need to build hot rather than cool cognitive economies, where the extra financial and psychological cost of complex cognition makes sense to students and teachers because of the gains. Dimensions 1-4 above profile a culture of complex cognition and authentic assessment that can help sustain that culture.
Victor Gardens and How They Grow
With these five dimensions toward better education freshly in mind, we can get an even better sense of them by looking at some classroom-sized examples—some small-scale victory gardens, where what we want to see happening is happening.
Victory gardens are to be cherished for many reasons. One reason is that at least on a small plot of the educational turf, something immensely fruitful occurs. Another is that they teach us what is possible. To be sure, what is possible in individual classrooms or even individual schools may not be so on a larger scale, that of those amber waves of grain addressed in the next chapter. Nonetheless, seeing what can happen on a small scale helps us to recalibrate our sense of how much youngsters can learn and how deeply and wisely they can learn it.
### EXAMPLE 1. EXPERT TUTORING
The smallest of small plots, the most intimate of victory gardens, is the individual tutorial relationship—one tutor and one learner. Here is your ultimate teacher/student ratio, profoundly different from the one-to-twenty or one-to-thirty found in most schools. Although not the sort of relationship that one readily replicates in classrooms, the individual tutorial situation teaches us something enormously important about how much better youngsters can do.
Benjamin Bloom of the University of Chicago is the noted formulator of Bloom's taxonomy and the notion of mastery learning. Several years ago, Bloom described "the two sigma problem" in an article called "The Search for Methods of Group Instruction As Effective as One-to-One Tutoring." The Greek letter sigma by convention refers to the standard deviation of a statistical distribution: If we are talking about a bell-shaped curve, how fat is the curve? The two sigma problem Bloom discussed in the article concerned the impact of good tutorial instruction on students. In a variety of cases, good tutoring had advanced a student starting at the _average_ of his or her peers to two standard deviations beyond the average.
How much is that? A lot! For a better sense of the two sigmas, we can talk in percentile terms. An average student falls in the fiftieth percentile: Half the students are scoring worse and half better. Assuming a normal distribution (the bell-shaped curve), a boost two standard deviations beyond the mean puts a learner in the ninety-eighth percentile: 98 percent of students are scoring worse, 2 percent better. This is not only impressive, it is quite amazing. Who would have thought that an average student had that much potential?
Studies of expert tutoring conducted by Stanford psychologist Mark Lepper have helped to illuminate just what it is that expert tutors do to advance learners so much. One very important aspect of their art is letting the learner undertake most of the work. Expert tutors often do not help very much. They hang back, letting the student manage as much as possible. And when things go awry, rather than help directly, they raise questions: "Could you explain that step again?" "How did you get the 7?" "I notice that on this earlier problem you said 8. How was this case different?"
Another interesting characteristic is very little direct praise. Expert tutors commonly emphasize beforehand how tough a problem is, rather than praising the learner's ability afterward. The front-end emphasis on difficulty is an artful move: If the student succeeds with the problem, it's self-rewarding because the student has surmounted an impressive obstacle. If not, well, the problem has been framed as quite difficult—it's only reasonable to look at some more tractable ones before reattempting the tough one.
Applying the Five Dimensions
Our five dimensions of better teaching and learning help to explain why expert tutoring has such an impact. In terms of Theory One, the tutor is in a position to offer clear information, lead the student in reflective practice (the reflection generated by the tutor's probing questions), provide informative feedback (the tutor's questions help the student to generate his or her own feedback), and tap the strong intrinsic motivation of task mastery (hence the tough-problem ploy).
In terms of a pedagogy of understanding, the tutor's emphasis on "How do you get that?" "Why is this different?" and such matters presses for understanding performances; in particular, explanation. This in turn helps build the learner's higher-order reflective knowledge of how to do things—knowledge at the problem solving and epistemic levels in the scheme discussed in chapter 4.
By engaging students in such discourse, tutors encourage a language of thinking, where students give conjectures, reasons, and plans. By pressing students to explain things for themselves, tutors encourage learning to learn. Tutors mediate transfer by bringing in problems different enough to challenge the learner to bridge a gap. In other words, the metacurriculum is well represented.
As to distributed cognition, the tutorial relationship presents an obvious example of cognitive partnership, with the tutor supporting the student as much as needed... and no more! Thus the tutor strives to leave with the student as much executive control of the task as the student can handle. As to a hot cognitive economy, the intimate and artfully sustained relationship between tutor and tutored creates enormous valuing of progress and mastery while supporting the learner's efforts in ways that reduce cognitive cost: the cognitive effort and risk of failure that discourage so many youngsters.
### EXAMPLE 2. BIOLOGY FOR YOUNG INQUIRERS
If the tutorial relationship can be so powerful, it is natural to wonder whether anything as rich can be mounted even at classroom scale. The answer is clearly yes. One interesting case in point is a provocative experimental classroom maintained by University of California at Berkeley researchers Ann Brown and Joseph Campione. Their goal was to bring together a number of innovations into a very mindful experience of learning biology.
A number of contrasts distinguish the experience of youngsters in this classroom from youngsters studying biology in more conventional ways. One is that much of the work occurs by way of cooperative learning. In "research groups," students explore a variety of materials relating to a component of the larger topic being studied by the class. Then they form "learning groups," which consist of one member from each of the research groups. The "expert" on each subtopic assumes responsibility for teaching the other learning group members about it (this is a version of the jigsaw method discussed in chapter 6).
Another contrast is the amount of control students have in determining their curriculum. In the research groups, students make decisions about what is important information and what can be sidelined. They then write their own textbooks—booklets revised and published with the help of computers and used to teach other members of the learning groups about that subtopic.
Another contrast is that great emphasis falls on the why of things: What makes for interdependence in nature? Why do disturbances in one part of the system create problems in another? What causes some animals to survive while others die out?
Researchers that they are, Brown and Campione not only created this educational setting in the midst of an ordinary school but took pains to formally investigate youngsters' response to it. The findings are impressive. In addition to gains on tests of biological knowledge, results showed students' achievements in more global skills. Although reading, writing, and computer use were not taught as such—they were simply very important to the highly interactive conduct of the biology instruction—the participating students showed improvement on standardized measures of all three.
Perhaps more importantly, class discussions revealed that students' thinking processes became increasingly sophisticated and complex. Over time, students began to explain phenomena more precisely and coherently. They used evidence more skillfully to support conclusions, systematically compared and critiqued different viewpoints, and engaged spontaneously in developing and exploring hypothetical situations.
Applying the Five Dimensions
In Theory One terms, this setting makes the students responsible for clear information by putting them in charge of teaching one another, thus also assuring reflective practice as students prepare information for others. How informative feedback comes in is not clear. But the approach to the subject matter and intriguing topics within it as well as the cooperative-learning instruction style ensure plenty of intrinsic motivation.
In terms of a pedagogy of understanding, understanding performances—especially explanation—and good representations figure prominently in the Brown/Campione setting, along with some very generative topics within biology, such as camouflage.
In terms of the metacurriculum, the setting creates enormous emphasis on learning to learn, and the emphasis on communication among the students cultivates languages of thinking. There is considerable attention to transfer of concepts from one context to another.
As to distributed cognition, we have cooperative learning. As to the physical distribution of cognition, computers are used as a medium for students' publishing lessons for one another. As to executive functioning, the whole setting encourages considerable student autonomy. The sum of it all is a rich cognitive economy driven by the students' responsibility for one another's performances, the intrinsic interest of the themes, social connectedness, and related motivators.
### EXAMPLE 3. HISTORY FOR THINKERS
Who fired the shot heard 'round the world—the first shot of the American revolution—on April 19, 1775, in Lexington, Massachusetts? All sources agree that someone fired that shot, but did a British soldier or a rebellious colonist trigger this turning point in history?
The historical evidence leaves the matter in doubt. A number of years ago, Peter Bennett, capitalizing on the uncertainties, developed an instructional unit entitled _What Happened at Lexington Green?_ The unit teaches history and aspects of evidence and argument at the same time. Bennett provided students with original-source testimonials, some American and some British, about the first shot. What do the testimonials suggest? Students are invited to puzzle over the evidence and reach conclusions.
As the lesson unfolds, students learn more about not just the testimonials but the circumstances behind them. One testimonial was offered by a colonist over fifty years after the event. Another, averring that the British fired the first shot, was given by a British soldier—but while in American captivity! Thus simultaneously, students learn something about the first moments of the American Revolution, the dilemmas of historians in interpreting evidence, and the process of evidential reasoning.
Kevin O'Reilly, a high school teacher at the Hamilton-Wenham Regional High School in Massachusetts, has for a number of years taught history in the style and spirit represented by Bennett's materials. His aim is at one and the same time to build in his students a deeper understanding of history and to cultivate their critical-thinking abilities. He avers that it's not enough to teach history in a thoughtful way. History instruction should include point-blank attention to relevant principles of evidence and argument.
A favorite O'Reilly artifice involves five students role-playing a theft. Other students watch. After all return to the classroom, the observers have to ask questions to try to determine what really happened. At first, O'Reilly notes, the questions disclose little. Gradually, some information comes to light. O'Reilly uses this as an occasion to talk about evidence. Evidence only yields information, it's stressed, when the source becomes clear, so that we can assess its reliability. Activities like this help O'Reilly's students recognize the hazards of evidence—how perceptions differ, and how difficult it is moments later, never mind years or centuries later, to nail down what happened.
As the course unfolds, O'Reilly laces his teaching of history with discussions of argument and evidence. He uses the acronym PROP to remind students that good sources prop up the evidence for a conclusion. PROP stands for Primary versus secondary (not an eyewitness) source, Reasons the source may have to distort the evidence, Other witnesses or kinds of evidence that corroborate, and Private versus public statement, the former more reliable because said in confidence. PROP is just one element in a fairly elaborate conceptualization of evidence and argument O'Reilly employs.
O'Reilly has developed his own materials to accompany the teaching of American history in a project he calls Critical Thinking in American History. In a number of ways, these materials challenge students to think through historical episodes and struggle with questions of evidence and argument. For one example, a version of the Lexington Green activity appears. For another, students are challenged by multiple interpretations of the Salem witch trials. What actually happened? Was witchcraft practiced by any of the victims? For yet another, why was the Constitution written and ratified? Students need to examine and assess two very different views, one averring that wealthy landowners shaped the document to ensure that majority rights did not override property rights, another challenging this monolithic motive.
In working with his classes, O'Reilly used essays to gauge the success of the approach. He found that students in his class wrote analyses of historical situations that appeared considerably richer than those written by students from another class.
Applying the Five Dimensions
It is easy to see Theory One characteristics in O'Reilly's approach: plenty of information from multiple sources, ample reflective practice in working through historical events from different perspectives, the intrinsic motivation of O'Reilly's lively style.
There is of course enormous emphasis on a pedagogy of understanding and in particular what was called the epistemic level of understanding: how we know what we know (and do we really know it?). As to the metacurriculum, the language of critical thinking employed in O'Reilly's setting is a straightforward example of languages of thinking. Moreover, the bringing in of immediate experiences and events, such as the staged theft, should cultivate generalization and transfer of ideas.
It's not clear that there are any unusual moves toward distributed cognition. But a hot cognitive economy is apparent: an emphasis on understanding, thinking, working out perspectives, authentic tasks, and so on that engages students in learning as a meaning-making rather than a fact-amassing process.
### EXAMPLE 4. A TEXTBOOK FROM THE PAST
As I write this, I have beside me a high school textbook published by Allyn and Bacon in 1971 called _The People Make a Nation_. Not a surprising name for a history text, but what is found inside would surprise you. It's worth remembering that the 1960s included a number of vigorous efforts to deepen education. This text was one of the outcomes. The classic example would be the famous _Man: A Course of Study_ , developed by psychologist/educator Jerome Bruner and a committed and ingenious group of colleagues. But much has been written about Man: A Course of Study, which also hews a path well away from conventional social studies content. _The People Make a Nation_ is worth a look in part because it cleaves fairly closely to conventional content with section titles such as "Founders and Forefathers," "Government by the People," "Slavery and Segregation," and the like.
But the book's style and method are surprising. I open at random to a page about "Methods of Persuasion." In fact, it is a four-page spread with color photographs illustrating the paraphernalia of political campaigns. We see crowds of people at a rally, campaign buttons, newspaper headlines, baloons with slogans.
We are also faced with a set of questions. The learner is challenged to identify as many methods of persuasion as possible in the illustrations. Then the learner is sent back to an earlier page to list persuasion techniques from an early presidential campaign. The account of the election is not your typical watered-down text; it was adapted from an historical analysis of the Andrew Jackson campaign of 1828 written by Robert Remini.
And that is not the end of it. Students are asked to search magazines for advertisements that use the same persuasion techniques. And the final part of the activity: "Think up some methods of persuasion and apply them to a particular situation."
Impressive! Yes, I may have hit on a particularly lucky page. But throughout this text, emphasis falls on the use of original source materials, artful use of graphics, and challenging learning activities.
Applying the Five Dimensions
Although a textbook is only a part of a complete educational intervention, we see several of our five dimensions clearly at work in _The People Make a Nation_. Front and center appears a pedagogy of understanding, where students are urged to understand the political process through such understanding performances as identifying tactics of persuasion. There is a significant presence of the metacurriculum through building students' critical awareness of persuasion techniques and provoking transfer to contemporary situations (magazine advertisements) and the students' own lives (the request to apply methods of persuasion).
We also find writing, scrapbook, and other activities that download cognition onto paper more so than is typical, a move in the direction of person-plus. The entire book, in the expectations it projects, fuels a more vigorous cognitive economy fostering complex cognition. At the same time, all this is packaged in the conventional wrappings of a textbook designed for mass distribution.
Regrettably, textbooks like this are rarely found in schools today. This copy was given to me by Sandra Parks, an individual who has worked with many school systems on approaches to teaching thinking. Sandra Parks has encouraged teachers to make use of such texts and explains the problems that the teachers encounter. The youngsters have difficulty reading the original source materials in the text. In addition, they have little experience with the critical-thinking skills the text activities require. One has to go slowly. The teachers do not always know how to support the youngsters in their learning process. Neither do they always adhere to the same view of the discipline as the text's. Perhaps in-service was needed here more than the publishers expected. For reasons such as these, texts of this caliber did not succeed in the marketplace.
Indeed, that was the era when the problems of instruction that attempted complex cognition were just being discovered. Today we realize that materials requiring complex cognition demand, from teachers and the materials themselves, much scaffolding of thinking and learning processes. This means that teachers, in turn, need opportunities to learn how to offer such scaffolding. We also realize that thinking and learning processes require explicit articulation—they need to be looked at and talked about as objects, in the spirit of the metacurriculum. Moreover, we understand much more today about the complex challenge of establishing stable, long-term innovations in school settings, as discussed in the following chapter.
Whatever the shortcomings such texts and materials possessed, their weakest point was perhaps their place in the cognitive economy of classrooms, school systems, and states. As the analysis in chapter 7 suggested, they created risks for the teachers that taught with them and the principals that approved them. So much safer to go with less challenging materials! And it was just such materials that wrested the market from them. Today, with a greater feeling of urgency about remaking education, texts in this ambitious style should emerge again and compete in the educational marketplace.
### EXAMPLE 5. A METACOURSE FOR COMPUTER
PROGRAMMING
I would like to describe briefly a project that my colleague Steve Schwartz of the University of Massachusetts, Boston, and I directed as part of the work of the Educational Technology Center at the Harvard Graduate School of Education.
Working with graduate students and participating teachers, we evolved what we termed a "metacourse" for enhancing students' understanding of computer programming and programming abilities. As the name implies, our intervention was a point-blank effort to provide a metacurriculum for programming instruction. In keeping with the idea of the metacurriculum, the aim was not necessarily to displace the usual curriculum of basic content and procedures. It was presumed that the teacher would provide programming instruction in something like his or her usual manner, using a text, class notes, or other means.
Concretely, our metacourse took the form of a dozen or so lessons. These lessons were not to be taught all at once but interspersed with the regular instruction, once every week or two. Additionally, there were guidelines helping the teacher to infuse the ideas in the lessons throughout the normal instruction.
In chapter 5, I mentioned the overarching image of the "data factory"; we used it as a conceptual integrator in this metacourse. The data factory image consists in a conceptual (rather than hardware) map of the computer: a place for programs to be stored, a scratchpad for doing arithmetic, a memory area, a keyboard area for input, a screen area for output, and other features. In the data factory lives a worker called NAB. It's NAB that gets things done. Students are encouraged to tell the story of the execution of a program in terms of actions taken by NAB. For instance, NAB looks at the first instruction in the program storage area. Perhaps it says to place the number 41 in the "bin" of a variable called X in the memory area. So NAB picks up the number and puts it in the bin.
Telling the story this way gives learners a way of envisioning what the computer does. Moreover, we also developed a computer animation in which students actually see NAB running around the data factory, undertaking such jobs.
And what about the name? Well, of course NAB nabs data and manipulates it. But really NAB is an acronym for Not Awfully Bright. Students are told this and why. Many people (research shows) think that computers are intelligent and "understand" what a program is supposed to do. A number of typical programming errors stem from this tacit assumption on the part of students. NAB's name is a way of reminding students that the computer understands nothing. It just does what it is told. Thus, NAB's name is part of the overall enterprise of giving students a strong, clear mental image of the capacities and functioning of the computer.
The metacourse also provided students with several learning and problem-solving strategies targeted on the particular problems of programming. For example, a learning strategy urged the students to pay attention to three key characteristics of a new command: its purpose, or typical applications; its syntax, or how to tell the computer to execute that command; and its action, exactly what happened in the data factory during execution of the command. Thus, purpose, syntax, action become a key phrase that the teachers and students were urged to use throughout the programming course. To keep the data factory imagery and the learning and problem-solving strategies on teachers' and students' minds, large posters went up on the walls of the classrooms, actually showing the data factory and bulleting out the strategies.
Applying the Five Dimensions
In a number of ways, the programming metacourse exhibits features sought in our five dimensions. The thinking and learning strategies offered by the metacourse work against inert knowledge and poor thinking, constituting a metacurriculum for programming. The overarching mental image of the data factory and the purpose/syntax/action learning strategy contribute to a pedagogy of understanding. The posters, as well as the emphasis on envisioning what happens during program execution with use of the data factory posters, help distribute cognition. The entire spirit of the metacourse fosters a cognitive economy that demands more complex cognition.
The materials went through three rounds of testing. The last round was testing at a distance, with very little contact between the development team and the participating teachers and classrooms: only one brief kickoff meeting, an occasional newsletter giving tips during the semester, and a few telephone consultations.
A programming test with several different types of problems was used to compare the classes using the metacourse with control classes. The test disclosed a considerable edge for the metacourse groups: about half a standard deviation. This is, of course, far less than Benjamin Bloom's two standard deviations for expert tutoring. But half a standard deviation is considered a good solid gain for an educational intervention. Also, our metacourse was by design a minimalist intervention; not a whole new course but a few added lessons and practices. Now under development are metacourses for algebra and Euclidean geometry, although formal findings are still some time in the future.
### EXAMPLE 6. ESCALANTE HIMSELF
The introduction to the notion of victory gardens mentioned a classic case: Jaime Escalante's teaching of mathematics and especially advanced placement calculus in Garfield High School in East Los Angeles. This stroll through several victory gardens could hardly be complete without a look at his singular achievement.
When Jaime Escalante arrived at Garfield High School and took the first steps along a path that would lead to national recognition, the setting was anything but conducive. The school's enrollment was largely Latino, more than 95 percent, and came from low-income families with poorly educated parents. The drop-out rate was high. Academic success was rare. It is hard to imagine a less likely arena for building a program that would send nearly as many students to the advanced placement exam in calculus as the famous Bronx High School of Science.
How did Jaime Escalante effect this magic? Perhaps the first thing to be said is that nothing requiring so much dedication could be called magic. Escalante's teaching of mathematics became for him an obsessive twenty-four-hour-a-day endeavor. He would meet with students before school and after school. In the evening, he would visit the parents of students who demanded that they hold jobs that interfered with their studies. A committed and strong-willed individual, Escalante argued, cajoled, and threatened administrators, parents, and, of course, students, all for the sake of inculcating the spirit, understanding, and skill of mathematics.
Could students respond with indifference in the face of his outspoken enthusiasm? Of course, many of them could. Indifference in Garfield High School was a natural state of mind. But Escalante brought more than commitment to the mathematics classroom; he also brought an armamenarium of artful tactics for motivating students. One of the most basic and conspicuous was a motto hung prominently in the classroom:
_Calculus need not be made easy;
it is easy already._
Escalante would not let his students think of calculus as hard.
At the same time, he would not let them take an indolent approach to the subject matter. Early in the term, Escalante made friends with his students, clowning around, devising pet names for the many students whose names he found it difficult to remember. Then the guilt trips began. Completing homework on time was a must. Have you got your homework today? No? Off to the office then. Out of this class. We don't want you here! Students could negotiate their way back in, but only with promises of reform.
Quizzes were frequent, but help was ample. As mentioned earlier, Escalante made himself available before and after school. He would routinely demand the presence of students who needed special attention.
Escalante's skills as a communicator were as peerless as his arts as a motivator. He would frame calculus concepts in several different vivid ways, often using sports metaphors to get the ideas across and make them memorable. Discussing the concept of a limit, he spoke of a pitched baseball approaching the catcher's glove and interacted with youngsters to get them to keep naming the key elements in the metaphor—pitcher, catcher, fastball, curveball. He introduced the concept of absolute value by analogy with the "give and go" in basketball. Miming the give and go with his back to an imaginary basket, he passed to a guard crossing on his right or left. He explained how the absolute value is like that: There are two possibilities. If the absolute value of x is such and such, maybe x is greater than zero. Or maybe x is less than zero.
The solid measure of Escalante's success in mobilizing his students' efforts and intellects is the advanced placement exam administered by the College Board. Grading of the exam occurs on a five-point scale, 1 to 5, and anyone scoring 3 or more has shown a college level of competence in the subject matter. In the most famous incident associated with Escalante, eighteen students from Escalante's AP calculus took the exam in May of 1982. While four passed outright, the scorers raised the concern that fourteen may have copied answers from some common source on one question. An enormous controversy ensued, resolved by giving the fourteen the opportunity to take another version of the exam at the end of August. With only a few days to prepare, twelve students did so, and all passed.
Although this was a very special moment for Escalante and his dedicated students, none of the events of 1982 were a fluke. Escalante and fellow teacher Benjamin Jimenez expanded the program of calculus instruction. In 1986, 84 percent of 93 students passed the exam. In 1987, the figure was 66 percent of 129. Escalante found this disappointing although it was close to the national average of 71 percent. He blamed himself for letting class size grow too large.
Applying the Five Dimensions
Escalante's teaching style is a model of Theory One, with clear, vivid explanations, abundant practice and feedback, and a mountain of motivation. The pedagogy of understanding appears most clearly in Escalante's use of vivid, concrete representations to make calculus concepts immediate for his students and thereby help them build lucid mental images.
As to the metacurriculum, Escalante's approach does not appear to be very "meta." His teaching seems to focus doggedly on the craft of calculus, with little philosophizing about it or efforts to connect it with other subject matters. Some of his language amounts to problem-solving heuristics of a sort, but certainly his approach to the teaching of mathematics does not seem as heuristics-centered as many.
However, as to distributed cognition, Escalante is a master of the use of imagery and social networking. And in the cognitive economy we find Escalante's forte. He establishes in his classrooms a cognitive economy with powerful, multiple rewards for investing effort in the complex cognition demanded by calculus—and sharp rebukes for slackers.
### A SMART SCHOOL IS SOMETHING RATHER SPECIAL
In offering these six classroom-sized examples, it is important to stress that they are unusual, yet not unusual. Percentagewise, they stand out. It is seldom that one finds the energy, the ingenuity, the social pattern, the emphasis on understanding to sustain these victory gardens, precious enclaves of effective teaching and learning that they are.
However, by count rather than percentage, they are not so rare. There are hundreds of such success stories that could be told. Those above have been singled out because they exemplify their kind, not necessarily because they are the best of their kind. They show what can be done when the multiple factors that impact on learning are configured in more potent ways in the local arena of a classroom, text, or set of materials.
But let's think bigger: What is the smart school that this book envisions? How does it go beyond what we occasionally find in ambitious classrooms? How does it go beyond what we occasionally find in a whole school's worth of such classrooms? Is a smart school, in the present sense, any more than a school where teaching and learning proceed in a rather thoughtful way?
A fair question. Here is an answer.
**Extended.** First of all, a smart school is of course a school. A whole school. Or more, a school system. A smart school does not stop at the end of a text or the end of a class. The informed, energetic, and thoughtful culture of the smart school impregnates all subject matters and activities. Ideally, it does not even stop at the end of the school day but filters in subtle ways into the lives of students outside of school.
**Inclusive.** Second, a smart school includes systematically a number of features that both research and educational practice have found to be important to thoughtful learning. I have tried to organize these features into five dimensions: Theory One, a pedagogy of understanding, the metacurriculum, distributed intelligence, and the cognitive economy. The next chapter will add a sixth, concerning effective change.
All six dimensions are important. But many generally worthwhile innovations in education pay no particular heed to one or another. Perhaps they fail to harness the power of distributed intelligence. Or they neglect the metacurriculum. Or they do not do enough to establish a hot cognitive economy.
**Explicit.** The dimensions seek to make explicit and articulate what goes into thoughtful teaching and learning. In the past, many fruitful educational innovations have rested on the good intuitions of their originators. But for wide-scale fundamental change in education, an educational green thumb will not do. We need explicit models of thoughtful learning.
**Elaborated.** Perhaps the easiest way for a generally thoughtful approach to teaching and learning to fall short is through a Swiss-cheese version of the principles laid out here—tasty but full of large holes. For example, in many classrooms students write potentially rich, open-ended essays. This serves well a pedagogy of understanding. But if there is no regular mechanism of informative feedback (not necessarily from the teacher but perhaps from peers), Theory One principles are violated. In many classrooms students reason out problems and debate issues, creating a generally thoughtful classroom. But if there is no explicit metacurriculum, where students have a chance to learn concepts and strategies that support thinking and learning, students will not gain as much as they might through the immersion in thoughtfulness.
In other words, a smart school in the fullest sense is more than a place where teachers and students work in a generally thoughtful way. It requires more than a bundle of tactics that lean toward thoughtfulness—open-ended assignments, peer collaboration, discussion and debate, and the like. These are all steps in the right direction, but they do not necessarily reach the destination. A smart school in the fullest sense is an intricate social mechanism, where multiple factors—in the present rubric, Theory One, a pedagogy of understanding, a metacurriculum, distributed intelligence, a hot cognitive economy, and (in the next chapter) attention to the dynamics of change—all mesh to support informed, energetic, and thoughtful teaching and learning.
To get systematic about all this, Appendix A includes a checklist of questions for each of the six dimensions to help gauge how far a school—or a classroom or text or unit of instruction—has moved. Teachers and school administrators are in the best position to put this checklist to work. They can assess where they stand and ponder directions for innovation. But I am also hoping that parents will look over these questions and ask themselves what kind of an education their children are receiving. I am hoping that school board members will survey the checklist and ponder how education works in the school they serve. I am hoping that educational planners at the state level will pay heed to the checklist and ask how state procedures and policies can encourage the smart school. In the mosaic of principles represented by the six dimensions, we find a demanding vision of what education could be.
## CHAPTER 9
**THE CHALLENGE
OF WIDE-SCALE
CHANGE**
The Escalante syndrome works like this. Media attention gets attracted to a conspicuously successful educational setting; for example, Jaime Escalante's inspired teaching of his calculus students. Celebration follows. Characteristics of the efficacious setting are bruited far and wide. "There," we all say. "See. It's possible. It's great. Let's get on with it." And then comes the last stage of the syndrome: Nothing happens.
Jaime Escalante's admirable achievements can only be applauded, but the Escalante syndrome is another matter. It keeps recurring, such is the hunger for solutions to the educational binds that beset our culture. And it does harm. People gradually learn that none of this seems really to work. Oh, it works for Jaime Escalante, but it does not work for us. So hopelessness begins to set in.
The Escalante syndrome comes from our not thinking carefully enough about our victory gardens and what they really teach us. Enthusiasm for their warm, intimate fermentations of learning leads us to want more of the same everywhere. We do not think hard enough about how special many of those victory garden settings are, how they depend on special people like Jaime Escalante and special conditions like tolerance for experimentation in a given setting.
We have victory gardens, many of them. But what we need is "amber waves of grain." What we need is innovations that work on a wide scale, in the school down the street and the one across the river, in the automotive and human gridlock of the inner city and out on the long reach of the prairie.
The past twenty years have seen an unprecedented amount of research scrutinizing the process of change—how an innovation takes hold or falters, how it persists or withers away. The single greatest moral to be drawn from these thousands of experiences and studies is a profoundly discouraging one: Almost all educational innovations fail in the long term. Even those that get a good start typically fall back into business as usual five years later.
It's easy to blame the peculiarities of the educational malaise in the United States for this history of setbacks. Inner-city poverty and violence sap students' energy and will. Cultural diversity makes it hard for texts and teachers to strike resonant notes across a range of students. Agendas in tension—from fundamentalist backlash to multiculturalism—make a deep and challenging curriculum difficult to sustain.
All this is true and truly part of the problem. At the same time, I remain determined to look to the fundamentals of teaching and learning for understanding the shortfalls of education. While dilemmas of inner-city poverty and violence, cultural diversity, and agendas in tension exacerbate the difficulties of innovation and deserve the ample treatment they receive in pages other than these, in my view they do not constitute the essence of the challenge. That essence is structural in character. It has to do not with the oddities of American circumstance but with principles unconfined by setting or society.
There seem to be three fundamental challenges to far-reaching educational innovation:
_Facing the necessities of scale_. Many innovations workable on a small scale in settings favored in one way or another are simply not suitable for wider-scale use.
_Making change work_. The process of implementing home-grown or "imported" innovations is crucial. Without artful implementation, innovations viable in principle will generally fail.
_Advancing thoughtful professionalism_. Teacher development, both in-service and preservice, is essential if challenging innovations are to be taken in stride.
The moral: For smart schools on a wide scale, we must understand all three of these challenges and pay heed to their nature. Indeed, with Theory One, a pedagogy of understanding, the metacurriculum, distributed intelligence, and the cognitive economy as five crucial dimensions toward schooling minds, effective wide-scale change can be seen as a sixth. If we want every school to be a smart school, we must listen carefully to new understandings of school change.
### FACING THE NECESSITIES OF SCALE
We can cure the Escalante syndrome. We can escape the cycles of discouragement that plague education. We can have powerful innovations not just on the small victory-garden scale but for the amber waves of grain.
But not all innovations. Crucial to the enterprise of educational change is recognizing this reality: Some innovations can work on a wide scale, but many others contain the seeds of their own failure. They have virtually no chance of thriving widely, even though they may have great impact under hothouse conditions.
What are the realities of scale that we must face if we are to see far-reaching change in education? They are part and parcel of the cognitive economy discussed in chapter 7. Many innovations simply ask more in the way of effort and other resources than many teachers and students are willing to pay. Innovations often develop in special circumstances where social, administrative, financial, and other supports make the cognitive economy favorable. But out in the wide world of educational realities, they can't survive.
So here are some survival conditions.
_A wide-scale innovation should not escalate teacher workload_. Many small-scale innovations ask for effort way beyond the call of duty. Jaime Escalante himself, for example, pursued his educational agenda with all the zeal of a missionary. On a wide scale this is simply unrealistic. An innovation cannot add substantially to teachers' already overburdened schedules. It should make things easier, or at least not much harder. Innovations that are conspicuously demanding of time need to come with a plan to lighten other loads. One that dramatically escalates the total time required of teachers will simply fail.
_A wide-scale innovation should allow teachers a creative role_. The opposite sin to making an excessive workload is making hardly any demands at all—providing a package that teachers can execute point by point. The trouble is that, quite understandably, teachers won't do it. They balk. And they should. An effective wide-scale innovation should not be a paint-by-numbers kit. It needs to be respectful of, and take advantage of, teachers' ingenuity in adjusting to their circumstances and making distinctive contributions.
_A wide-scale innovation should avoid extreme demands on teachers' skills and talents_. Many small-scale innovations call for very sophisticated performances from teachers. The expert tutoring discussed in chapter 8 is quite an art. Remember the biology class organized by Ann Brown and Joseph Campione, with its emphasis on cooperative learning and students writing texts for one another? Choreographing such a setting requires consummate skill.
From such victory-garden settings we can learn principles to put to work on a wider scale. But we cannot expect most teachers to undertake them with skill and confidence. At the least, substantial teacher development would be required, far more than normal in-service provides. Moreover, some small-scale innovations call for extraordinary levels of skill in Socratic interaction or other teaching styles that many people will have trouble mastering.
In summary, innovations can certainly ask for a solid level of professional skill, but those that require virtuosic performance from teachers, especially without ample time and support to learn the technique, are doomed.
_A wide-scale innovation should include strong materials support_. In many quarters these days, textbooks, worksheets, posters, and the other physical paraphernalia of teaching get bad press. And not without reason, because they often convey distorted and simple-minded content in boring ways. Moreover, ample experience with innovation shows that materials alone are not the solution.
However, let me suggest that for effective wide-scale change, strong materials support is an essential. First of all, materials do not have to be bad. Materials such as the textbook reviewed in the previous chapter or the metacourse for helping students to understand programming afford stimulation, challenge, and insight. In the second place, teachers cannot always be reinventing every spoke of the wheel of education. Education requires too much time and ingenuity as it is. Third, good materials are enormously supportive. They give a path to follow and milestones along the way. Although the creative teacher will not follow that path blindly, its presence helps keep things oriented.
_A wide-scale innovation should not boost costs a lot_. Obvious though this seems, the fact of the matter is that many educational innovations developed in university or government centers end up, when ready for dissemination, costing far more per student than conventional budget items such as textbooks. Who is going to pay?
On a wide scale, no one. Maybe people will shell out a little more for something special. But they can't dig deeply into pockets that are mostly empty. High per-student costs are anathema if we aim to remake education.
_A wide-scale innovation should fulfill many conventional educational objectives at least as well as conventional instruction_. Consider again some of the innovations discussed in the previous chapter. The metacourse for computer programming improves student performance on programming quizzes. The biology curriculum of Brown and Campione instills plenty of biological knowledge. Yes, these interventions and others seek more thoughtful learning and deeper student understanding. But they serve some more conventional objectives at the same time. It is not difficult to craft an innovation that does at least as well as typical instruction.
The Paradoxes of Scale
Readers will notice some oddities about the needs of scale spelled out above. Oddity number one is that they are by and large obvious. Naturally, innovations that cost a lot more or demand enormous effort will fail. The paradox is that, although obvious, such concerns get short shrift in much educational development work. So the message has to be pounded home again and again.
Oddity number two is that many of these characteristics stand in some tension with one another. For instance, we do not want to add to teachers' workloads, but we want room for teacher creativity, which requires time. We desire innovation, but we want conventional objectives served well, too. All this is paradoxical. Can we really have our cake and eat it, too?
Yes, we can. With clever design of programs and approaches. An instructional approach that asks teachers to invest creative time can come imbedded within a larger framework of school restructuring that frees the needed time. Many instructional approaches that build students' understanding of mathematics or science also enhance performance on conventional problem-solving tests, as well as yield more insightful responses to authentic assessments that probe understanding more deeply. Still and all, it has to be recognized that the demands of scale pull against one another somewhat. Artful balances need to be struck.
Oddity number three is that an innovation is sensitive to any one of these conditions. You might think that an innovation which scores well on half or two thirds of them would do fine. But it's not as easy as that. Individually, these conditions are not niceties but necessities. For instance, almost apart from its other merits, an innovation will not see wide usage if it costs a lot more than normal instruction. It will not see wide usage if it demands a great deal of additional teacher time and effort. It will not see wide usage if it makes extreme cognitive demands on teachers. It only takes one hole to sink the ship.
So simple a point should not need such emphasis. But experience teaches otherwise. Over the years, innumerable innovations, ambitiously crafted for wide-scale impact, have had virtually no chance because of one or another of the flaws listed earlier. Not that everything has to be perfect. But we do well to remember that almost all of a number of conditions have to be met for an innovation to do wide-scale service.
Indeed, one problem with contemporary research and development in education is a dogged victory-garden bias. Almost everyone tries to set up an optimal educational experience next door where everything wonderful happens. Hardly anyone worries about designing for wide impact. This is not a criticism of any individual, because, as emphasized before, we learn a great deal from what can be done in hothouse settings. But it is a criticism of persistent tunnel vision in the research community.
Educators and developers must recognize a simple truth: Wonderful as victory gardens are and as much as we learn from them, not all victory-garden innovations are suitable for wide-scale impact. Attention to the factors that make for wide-scale viability is imperative. Only then can we build toward educational change that works not just next door, where we can administer tender loving care, but down the block, on the next block, and in the next community as well.
### MAKING CHANGE WORK
Educational innovation is one of the gristly challenges of contemporary society. Most innovations do not succeed. If they get beyond the gleam-in-the-eye stage, they generally falter and fail during implementation. If they thrive for a couple of years, they generally fall into disuse when the special circumstances that launched them shift: government funds cease, enthusiastic personnel move away, newer, more trendy priorities emerge.
Yet amidst these worries, something remarkable has happened over the past several years. Educators and scholars concerned with school change have looked hard at the innumerable practical experiments with change carried out in thousands of classrooms and schools nationwide. They have found the fossils of failure in school records and teachers' and administrators' memories. But they have discovered occasional successes. And they have made sturdy progress toward teasing out the conditions for change.
While the previous section dealt with some necessary features of innovations viable on a wide scale, the process of change itself got little attention. Moreover, the story of successful change is understandably complex, certainly beyond the scope of these few pages to treat in detail. Nonetheless a book committed to the smart school can hardly settle for articulating the vision without any words on how to get there.
So how might progress toward a smart school occur? For a comprehensive sourcebook of information on change, one can hardly do better than Michael Fullan's recent _The New Meaning of Educational Change_. Drawing on ideas synthesized there, especially from chapters 5 and 6, let us by way of illustration formulate a particular story of change.
Initiation
Constancia Sanchez, principal of Magellan High School, is on edge. Her vigorous efforts to rally her teachers to improve student performance on SATs and like yardsticks have yielded some incremental gains, but students still score significantly below the state average. And anyway, Constancia, like many educators, finds herself skeptical of the significance of SAT scores, although she is bound by the political milieu to pay some heed to them.
So Constancia Sanchez has pressed her faculty for ideas. "I'm open to your suggestions. Although there's a lot we may not be able to do, I'm ready for some imagination," she says. When two teachers of science, Jeff Orono and Rudy Baker, come to her with a fresh angle and a request for some resources to buy new materials, she pays close attention.
Jeff and Rudy want to build toward a smart school. Familiar with the basic idea of a pedagogy of understanding (perhaps from reading this book), they want to put such a pedagogy in motion in Magellan High's science program. They especially like the idea of understanding performances ("Let's get students _thinking_ science—real inquiry, not canned labs.") and the idea of higher levels of understanding ("Youngsters have got to see and talk about and think about how the game of science is played—how you solve problems, how you justify things—if they're going to understand what science is all about.").
For a framework that highlights understanding, Jeff and Rudy decide to use "knowledge as design," discussed in the example that closes chapter 5. At the heart of knowledge as design lies a key understanding performance: Students think about the purposes of things and how the parts and features of the thing work to fulfill those purposes. ("We can have our students analyze a microscope from top to bottom, for instance—optics, wheels and gears, weight on the bottom for stability. A microscope bundles together half a dozen ideas from physics. And you know, our students can also look at something abstract in terms of parts and purposes, like Newton's law of gravity. Newton made it up for a purpose—explaining why the planets moved as they did. And our students ought to be able to tell us, with a little help, how every part of the law contributes to that purpose.")
Jeff and Rudy also have some ideas about materials they need—some booklets, some lab equipment. Constancia could tap a $2,000 gift from an alumnus to seed a program. Should it be this one? Constancia thinks it over.
"Start small, think big," she says. "It's fine to get started this way. But let's work toward bringing in some other subject matters. Not just science, but some math. Maybe some history. What do you think?"
Jeff and Rudy agree. They will get started themselves and as soon as possible—maybe even from the beginning—marshal involvement from a few others.
* * *
**Clear need** recognized by the principal and the two teachers.
**Strong advocacy** , in this case from Jeff and Rudy.
**Clarity**. Jeff and Rudy have at this point a sufficient sense of the innovation's philosophy and approach. As they get more involved, they will constantly need to work for clarity.
**Practicality**. Jeff and Rudy have done their homework. They see how in broad stroke a pedagogy of understanding can be made to work in their situation. Again, as they and others advance, they will need to think further to sustain a practical vision.
**Resources**. Constancia is ready to supply the modest resources needed.
The initiation stage, like all the stages of change, is peppered with hazards. So far, Constancia, Jeff, Rudy, and the others they draw in have avoided some major pitfalls.
Don't be too simple. Research also suggests that more challenging changes (so long as they are reasonable to attempt) stand a better chance of success than meeker goals. Thus, Constancia ups the ante a bit, suggesting a broader initial effort involving more teachers and subject matters.
Pressure and support. Although pressure has a bad name, the evidence is that successful innovation requires both pressure and support from administrators. Without support, innovation loses momentum. Without some pressure, it tends to sprawl and lose direction. Thus, Constancia presses her faculty for ideas and offers both support and a certain amount of direction. Significant at this early stage, pressure and support will continue to be important throughout the change process.
* * *
Implementation
Jeff Orono and Rudy Baker do their best with a persuasive pitch to a number of other faculty members. Despite approval from Constancia, they run into some discouragement: "I already do pretty much what you're talking about, and I don't have time to rethink it next fall." "I'm already starting this other new thing, and that's all I can handle." "I've decided I really need to help my students build up some basic skills; this is too esoteric for them."
But they secure some interest from Sara Greenbaum in mathematics and, in history, Barbara Finelli and Leland Parks. "Is that enough?" they ask Constancia later.
"Plenty," she says. "Start small, think big! If you try to involve folks who aren't that interested early on, it's just going to bog things down. Give it room to grow."
Over the summer, the five teachers meet a couple of times to plan. They decide to try a number of strategies in their classrooms during the Fall term. They articulate understanding goals for their students and spell out understanding performances their students will engage in to build those understandings. With knowledge as design in mind, they figure out what objects and concepts they want their students to look at as designs. They begin to map out higher levels of understanding in their subject matters. (See chapter 4.)
Also, to help think all this through and begin to draw other teachers in, they decide to spend some of their money for a consultant to do a workshop or two. They contact Rosa Ferris, a local free-lance consultant with a good reputation. But Rosa has a concern about their plan. "I think it's a great thing you're getting into," she says. "And I don't want you to think I'm selling myself. But a couple of workshops isn't going to help much."
Rosa goes on to explain that she finds a series of small-scale orientation meetings with those who will be most involved much more effective. "Even then, don't depend on me," she urges. "You need to develop 'internal consultants,' people within the school system who help others and keep up momentum." Also, Rosa urges, there needs to be a program of peer observation and collaboration.
Is Rosa just selling them a bigger package? "Maybe," history teacher Barbara Finelli says. "But she's right anyway. What will Constancia say?"
Constancia hears their arguments and okays spending a little more for Rosa's regular involvement.
* * *
Implementation, the launch period for an innovative program, typically takes two to three years. During this time, teachers learn about the approach or develop it themselves, most often a mix of the two. They try the approach in practical classroom contexts, discover its problems, and strive to solve them.
Many innovations sputter out during this period. Potential participants prove resistant, plans look impractical, competing agendas appear, support from consultants inside or outside the school system turns out to be weak or misguided, and enthusiasm wanes. But Jeff, Rudy, Sara, Barbara, and Leland have avoided the worst hazards.
**Start small, think big**. Research suggests that often big starts are a mistake. When too many people get involved early on, the initiative drowns in its own social complexity and the marginal commitment of some participants. Thus, it's fine with Constancia that only a few more teachers get involved.
**Continuing sound counsel**. One-shot workshops or consultations almost never have a lasting impact. As teachers actually try to work through innovations in the classroom, they encounter innumerable problems and need counsel along the way. Rosa's suggestion for continuing contact is crucial.
**Internal expertise**. Also apt is Rosa's emphasis on building internal expertise that can in time come to fill her role. However much of Rosa's time the school can buy, it will not in the long run be enough. If the pedagogy of understanding innovation (or any other) is to last, it must be because of strong internal commitments and understanding, not continuing outside help.
* * *
More Implementation
During the fall term, science teachers Jeff Orono and Rudy Baker, who initiated the understanding project in the first place, prove aggressive in trying things out in their classrooms. A rich conversation evolves among them, the other three teachers, and Rosa Ferris.
For the first several weeks of the term, the three teachers Jeff and Rudy recruited find themselves strangely reluctant to take the plunge.
History teacher Leland Parks says, "I already teach this lesson where students play historical roles and improvise what figures might say. They do this Boston Tea Party skit where they're the plotters, arguing about what to try and how to get away with it. Now that would be an 'understanding performance' wouldn't it?"
The rest agree. "So I'll do that again," Leland says, "and maybe emphasize it a little more, give it more time."
This makes sense to everyone. But after a while, it becomes obvious that Leland, his colleague in history Barbara Finelli, and Sara Greenbaum in math are plugging in their best lessons without really attempting much new.
"Frankly," Sara Greenbaum says, "I think that I'm doing a lot of this already. And I don't completely get the idea of 'higher-order understanding.' I mean, I get the general idea. But just what kinds of explanations or examples count as higher order? And how do you know that the students aren't going through the motions?"
"You know," says Leland, "we were talking about that same question a week ago. And the week before. We keep coming back to it and I'm not sure we're getting anywhere."
"Maybe," says Sara, almost reversing her previous stance, "well, maybe we just have to bite the bullet and try out some new things. See how it plays."
They talk this over with Rosa, who not only agrees but adds, "I'm glad you came to this, because it was on my mind too. Sometimes the only way to make sense of a new approach is to get in there and try it and see what happens."
* * *
**Action, then understanding**. Teachers often want to be completely clear about an innovation before trying it. But research and experience suggest that understanding develops gradually as teachers try things, not by talking everything through in advance.
**The hazard of action without understanding**. On the other hand, if the approach highlights straight cultivation of teaching skills without reflection, teachers can develop classroom routines that play well superficially but soon wither for lack of commitment. Fortunately, the five are persistently reflective about what they are trying to do.
**Commitment and ownership evolve**. Sara, the math teacher, is somewhat skeptical in the familiar "I already do this" mold. No doubt she does do much of it. And skepticism is fine so long as participants are interested enough to keep at it. Commitment and ownership evolve gradually. They need not exist from the first, and thinking that they must tends to stall efforts at change.
* * *
And Yet More Implementation
Sara, initially wary of the innovation, becomes in the spring term an ardent champion of it with other teachers, who begin to get cautiously interested. She finds that she has a knack for putting it into words that make sense to teachers from different disciplines. "If you can move in this direction in math, you can do it _anywhere!_ " she says. And she tells a tale from her classroom about eleventh graders who "finally caught on to fractions."
At Sara's urging, the five plan an interest-building workshop with Rosa Ferris for all interested teachers. But the announcement of the workshop several weeks in advance triggers a backlash. It turns out that several teachers of English and social studies have been talking informally about Mortimer Adler's _Paideia Proposal_. "This is really the same idea," they say. "So couldn't we have a Paideia expert come?" It turns out that they made a budget request of Constancia several weeks before and, hard pressed on the budget, Constancia said no.
"We want you to do this workshop," Jeff says to consultant Rosa. "But how can we handle it diplomatically? Any ideas?"
"This is an opportunity, not a problem," says Rosa. "I know a good Paideia person. If your colleagues don't have a connection already, let's suggest him. It's not quite the same thing as the pedagogy of understanding perspective, but it's close enough. Why not invite him to come, and we'll do a joint session. I'll split the fee. Maybe you can form a coalition with the English teachers."
* * *
**Vision building**. A philosophy plus technique isn't enough to make most innovations work. One needs visionaries that can paint pictures of what the school would be like. Sara, unexpectedly even to herself, finds a calling there.
**Power sharing**. When an individual or group attempts to maintain tight control over an innovation, others understandably shy away from it. Here, Jeff and Rudy, who initiated the process, are quite happy to see Sara becoming its most conspicuous spokesperson. And all five are willing to share resources with the Paideia interest group.
Evolutionary planning and problem solving. One can't plan innovations down to a T. All successful innovations have problems that, as they arise, need to be solved through ingenious and committed efforts, with plans revised accordingly. In this spirit, the five are ready to change their workshop plans and explore an alliance with the Paideia group. Rosa projects the right spirit when she says, "This is an opportunity, not a problem."
Continuation
Two years go by. The Pedagogy of Understanding group indeed merges with the Paideia group. By now, some twenty teachers are consistently involved, with several others around the fringes. SAT scores have nudged just above the state average, although the teachers have not been paying a lot of attention to exam cramming.
At the end of that year, some hazards threaten. Jeff and Rudy, who launched the pedagogy of understanding group, and Sara, who became its chief missionary, all leave for other positions. It's sheer chance, but so it happens. Also, the seed money Constancia had used to support the innovation runs out. Will the program falter? Jeff, Rudy, and Sara are all a little embarrassed at leaving and more than a bit worried.
"Don't fuss," says history teacher Leland Parks. "Remember where things are. There are five of us now who do the internal consulting thing. And as to the seed money, we haven't used any of it for a semester. In fact, Constancia warned us to taper off, remember."
Viewed whole, the tale of a few teachers at Magellan High School affords a panorama of fundamental principles of school change and the lessons of several decades' worth of research and practical experience. Change is seen outfitted for a long and trying trek. From the early recognition of clear need, through "start small, think big," on to the idea that action ordinarily precedes understanding, and further still to the people and dollar sensitivity of innovations, we find that lasting change demands a complex art and craft.
* * *
"Continuation" or "institutionalization" is what scholars of school change call the next phase of an innovative program. It's nothing to take for granted. Constancia has kept her eye on the project. She knew something about the likely fate of innovations, even vigorous ones. She felt confident that this one would be with Magellan High for quite a while longer.
**Dollar sensitive**. Successful innovations often falter when special funds disappear. This innovation was never lavishly supported, and Constancia warned its devotees to shrink their expectations. When an innovation does need continuing special resources, its survival often depends on administrators' building them routinely into the budget rather than always treating them as a special request.
**People sensitive**. Successful innovations often end when key people move away. Here, many people have become involved with shared leadership and expertise. So the innovation is not brittle.
Of course, in a way "continuation" or "institutionalization" paints the wrong picture. Both words suggest persistence of a pattern. However, innovations almost inevitably evolve. They turn in unexpected directions, adopt new strategies, discover new grails to seek.
Indeed, more important than any particular innovation is a culture of innovation in a school setting with a cadre of teachers ready to move it forward. And along the way, figuring out what "forward" means.
* * *
Today's teachers, administrators, and university-based developers are coming to a new awareness of the demands of change, and none too soon. The enthusiasm for deeper learning in better schools will not sustain itself against an endless parade of partial measures and faltering forays. We have to make it _work_.
### ADVANCING THOUGHTFUL PROFESSIONALISM
It is time now to turn to the third leg of wide-scale change in education: the role of teachers. The story of Magellan High School can be put to work once more to make a basic point: Smart schools need to be places of thoughtful learning not only for students but for teachers too.
Notice how the teachers at Magellan High face problems of learning with understanding quite akin to those of their students. While their students puzzle over Boyle's law or the Boxer Rebellion, they puzzle over what a pedagogy of understanding means. While their students engage in understanding performances to build understandings, the teachers need to try different teaching approaches—the teachers' understanding performances—to build an understanding of how a pedagogy of understanding translates into practice. While the teachers build students' understanding through mental images, they themselves benefit from Sara's anecdotes, which help the teachers form good mental images of a pedagogy of understanding.
All this has to do with a pedagogy of understanding. But the teachers at Magellan High School benefit from other dimensions of a smart school, too. Take distributed intelligence, for instance. The success of their innovation depends crucially on the socially distributed know-how they establish among many teachers in different disciplines. Or consider learning to learn. Their long-term momentum depends on a kind of learning to learn—learning to manage the process of change.
The inherent thoughtfulness of the teaching process shines through in the Magellan example: Teaching at its best involves active reasoning about the myriad aspects of practice. Stanford University scholar Lee Shulman speaks out directly about the centrality of what he calls pedagogical reasoning to the art and craft of teaching:
As we have come to view teaching, it begins with an act of reason, continues with a process of reasoning, culminates in performances of imparting, eliciting, involving, or enticing, and is then thought about some more until the process can begin again.
Shulman sees teaching as an interweaving of pedagogical reason and action, in which teachers build their understandings of their discipline and the purposes of instruction, develop ways of representing ideas to students, carry out instruction, monitor results, and reflect critically on what they have done, to begin a new cycle.
In sum, a school in the midst of innovation is a setting of fundamental learning for teachers as well as students. And not just routine learning either, the mechanical kind that yields inert, naive, and ritualized knowledge. The same principle applies to both students and teachers: Learning is a consequence of thinking. A smart school, or a school on its way to becoming one, cannot just feature thoughtful learning for students. It has to be an informed and energetic setting for teachers' thoughtful learning, too.
A Home for the Mind
What are the payoffs of such schools? In an essay called "The School as a Home for the Mind," Arthur Costa warns of the costs of a negative school climate:
When... a dismal school climate exists, teachers understandably become depressed. Their vivid imagination, altruism, creativity, and intellectual prowess may soon succumb to the humdrum dailiness of unruly students, irrelevant curriculum, impersonal surroundings, and equally disinterested co-workers.
However, Costa adds, a few sentences later:
When the conditions in which teachers work signal, promote, and facilitate their intellectual growth, they will gradually align their classrooms and instruction to promote students' intellectual growth as well.
A thoughtful climate, or what I have called earlier a culture of thoughtfulness, makes all the difference.
Roland Barth, first director and one of the founders of the Principals' Center at the Harvard Graduate School of Education, sees the plight of teachers in the frenetic character of schools. In _Improving Schools from Within_ , he likens the professional life of a teacher to a "tennis shoe in a laundry dryer."
What is the key to a more orderly and enlightened profession? Of many factors, Barth underscores collegiality. Collegiality means something different from congeniality, Barth emphasizes. It's not just good manners and telling jokes in the teacher's room. Collegiality means working together in a mutually supportive and thoughtful way at the business of education. Barth borrows a four-way characterization of collegiality from Judith Warren Little. In a collegial atmosphere, teachers talk about practice, observe each other, work on curriculum together, and teach each other. The school that serves as a home for teachers' minds is much more likely to become one for students' minds as well.
A collegial climate of mutual learning cannot be plugged in like a new dishwasher, Barth acknowledges. While cautious of categorizing people, he finds it broadly useful to think about teacher development in terms of three kinds of teachers. Group One teachers resist scrutiny and counsel from others and show little tendency to reflect on their own practice. They go through the motions. Group Two teachers—the most numerous, in Barth's view—rethink their own practice according to their classroom experiences. But they do not welcome the eyes, minds, and mouths of outsiders, even outsiders who teach across the hall. Group Three teachers not only pursue self-examination but throw the door open to collegial interaction around their teaching.
All this reveals a path of development toward the smart school: teachers progressing from Group One to Group Two and Group Two to Group Three. Regrettably, Barth notes that school pressures typically trigger just the opposite effect. Critical parents, competitiveness among teachers, rituals of authoritative feedback from principals, and pressures to comply with mandated teaching practices all work to make teachers closet their endeavors more and more, ultimately even from themselves. They settle inward toward the "humdrum dailiness" of which Costa writes.
What can make a difference? Turning to his personal experience as a principal, Barth reports that formal workshops and direct advice had modest impact. What helped more was responsiveness to questions and suggestions from teachers. He learned to listen hard. Did a teacher want a thousand tongue depressors? Perhaps he could do something about it. Would this new angle on fractions arithmetic be worth trying? Why not. And would there be some way he could help? Barth observed that encouraging teachers in their own initiatives rekindled guttering enthusiasm. And the diversity cultivated in different classrooms by teachers of different visions promoted general thoughtfulness: Teachers began to ponder over and talk about what they were variously doing and why.
From such clues as these we recognize once again the risks of a conventional, command-style leadership in educational institutions. It undermines a culture of thoughtfulness at the faculty level and, through ripple effects, at the student level too. Sara Lawrence Lightfoot, professor at the Harvard Graduate School of Education, writes harshly of the plight of teachers not treated with the respect due them:
In the worst schools, teachers are demeaned and infantilized by administrators who view them as custodians, guardians, or uninspired technicians. In less grotesque settings, teachers are left alone with little adult interaction and minimal attention is given to their needs for support, reward, and criticism.
The smart school, in contrast, honors the ingenuity, commitment, and centrality of teachers and provides time, resources, and encouragement toward the expansion and refinement of their craft.
The Asian Example
Roland Barth and many other principals have found some strategies to open the way toward the smart school. But these are more victory gardens—small plots of progress in a generally dusty plain. What would a wide-scale culture of thoughtfulness in teaching be like? Is such a thing possible?
An illustration comes from halfway around the world: the pattern of professional teaching practices in China and Japan. In a recent article, University of Chicago psychologist James Stigler and University of Michigan developmentalist Harold Stevenson offer a bird's-eye view of teachers and teaching in the Asian setting. They underscore how supportive and thoughtful an environment exists in China and Japan for the professional development of teachers and the promotion of potent teaching practices.
To be sure, comparisons with the Asian model have bombarded the American public to the point of irritation. In backlash, critics complain that the United States and other settings are too culturally diverse for the Asian model to work. Maybe. Let us first look over the Asian model and then return to the question of how it might speak to settings marked by great diversity.
**Time to Think.** One of the most notable features of the Asian model is the time teachers have to think about their craft, separately and together. Anyone familiar with the U.S. educational scene recognizes the racehorse pace that teachers must sustain. This was the central point in Theodore Sizer's book _Horace's Compromise_ , touched on more than once in previous pages. Sizer and many others have documented how little time dedicated teachers have to give to lesson planning, paper grading, and other important elements of the teaching profession because they must spend so much time standing at the front of the class or cruising up and down the aisles.
In the Asian context, we find a different and liberating pattern. Teacher/student ratios are much the same as in the United States. But, paradoxically, class sizes are larger. By teaching larger classes, teachers have more time during the school day outside of the classroom for other responsibilities, including the building of their craft. In Beijing, as Stigler and Stevenson report, teachers work in class only three to four hours per day. This does not mean a shorter workday for teachers; on the contrary, they generally spend more hours at the school building than do U.S. teachers. But they invest their time differently.
The Asian pattern invites an immediate objection: those large class sizes. Do not large class sizes lead to poorer learning? That large classes inevitably reduce learning is one of the entrenched myths of education, though challenged by abundant research and experience. Most pertinent to the present context is the marked success of the Asian model and the fact that in such subjects as mathematics it yields students who understand concepts and solve problems far better than U.S. students. More broadly, numerous studies of class size and student learning have shown no effective relationship between the two with class sizes of around twenty or more. In summary, concerns about class size are a poor reason to reject the Asian model. Buying more time to think at the cost of larger class sizes (but with the same teacher/student ratio, which means the same attention per student on assignments and exams) is a good trade that serves teachers and learners well.
**A Shared Culture of the Craft of Teaching.** Asian teachers have time to think. But what do they think about? According to Stigler and Stevenson, much of the time they think about the lessons they teach. They plan lessons, share plans with one another, get critiques, attend workshops, observe other teachers teaching, watch videotapes of teaching practices.
Here, an entrenched attitude says that teachers are born and not made. This is the teacher's version of the Ability-Counts-Most theory criticized in chapter 2. Just as we think that ability counts most for youngsters and thus do not recognize how much effort contributes to performance, so we think that ability counts most for teachers and so do not support a culture of learning for teachers. In contrast, the Asian model is effort centered for teachers as well as students. Through organized time and commitment, teachers can learn to teach much better than native ingenuity alone could allow.
Another entrenched attitude says that we are not copycats: Teachers should develop their own lessons. In contrast, the Asian model urges that teachers have much to learn from one another. They routinely share lesson plans. Teachers strive to emulate others they see as displaying potent practices. A lack of creativity? Not according to Stigler and Stevenson. They say that Asian teachers do not blindly copy but adapt and extend. They make an analogy with different performers of a piece of music: They play the same piece, but they each put their own stamp upon it.
Stigler and Stevenson note that the Asian culture of shared professional development gets sustenance from the very physical structure of educational institutions themselves. Whereas teachers here typically remain isolated in their classrooms, teachers in Asian schools are provided with desks in a large area where they can interact and share their craft. While not actually teaching, teachers work in this common space rather than in their classrooms.
**An Apprenticeship Model of Teacher Development.** Part and parcel of this culture of the craft of teaching is the treatment of novice teachers. The U.S. model is more or less sink or swim. New teachers get assigned classrooms where they must do their best with scant support from older and more experienced colleagues.
The Asian model takes a different approach. It's not assumed that preservice education equips teachers for the trials of managing classrooms and delivering lessons. For at least a year, beginning teachers pair with older ones. Extensive in-service is expected: for beginning teachers in Japan., twenty days per year by law—far more than almost any U.S. teacher enjoys. Master teachers receive leaves for a year to tour other teachers' classrooms, spreading ideas and critiquing lessons.
**But What About Cultural Diversity?** This brings us back to the backlash. Stigler and Stevenson underscore how commonly the complaint is heard: "All well and good for the Japanese or the Chinese with their uniform cultures. But it's not so easy in America, the great melting pot."
Stigler and Stevenson show little sympathy for this argument. They acknowledge that our culture (and, of course, many other cultures around the world) contains great diversity. But most of that diversity, they urge, appears not within individual classrooms but across classrooms and school systems, between, for example, urban and suburban schools. Within a classroom, where the teaching and learning happen, students are no more diverse here than in Asian classrooms.
Stigler and Stevenson conclude—and so do I—that much can be learned from the Asian exemplar. We make a bad tradeoff when we insist on smaller class sizes and keep our teachers busy all day. We would do better to adopt the larger class sizes that allow teachers time to refine their craft. We adopt a mistaken vision of human nature when we treat teachers as born and not made. We would do better to get away from the ability-countsmost model and invest in developing teachers' craft. We serve teachers and students poorly when we toss novice teachers into classrooms and hope they will perform. More networking with experienced teachers seems essential. We undermine the essence of human cultural advance—passing knowledge along—when in the name of creativity we expect teachers always to invent their own lessons instead of passing around well-designed ones. A culture of thoughtful sharing and refinement of lessons would strengthen every teachers' repertoire.
All in all, there is no need to grope about for a vision of a thoughtful culture of teaching. We have examples in Asia. While we cannot clone the Asian model here, some of its basic structural principles would seem to serve a thoughtful culture of teaching in the United States or anywhere else. Schools that are informed and energetic settings of thoughtful learning for teachers as well as students stand within reach.
### WHAT WE KNOW CAN MAKE A DIFFERENCE
The last quarter century has seen an Odyssey of educational experimentation and an Alps of experience and research findings. Comparison with ways of schooling in other countries has only been one element. Our understanding of education has advanced on many fronts, from the details of the learning process in the individual human mind and brain to the broad structural and long-range factors that influence the viability of an innovation on a wide scale.
Some points confirmed by the new science of teaching and learning are hardly news. Figures such as John Dewey, William James, or, for that matter, Aristotle or Plato anticipated them. Those points we know more freshly and firmly and in more detail. Others really do stake out new terrain—for example, emerging conceptions of intelligence, understanding, and the process of wide-scale change.
To revisit ideas from this and previous chapters, let me list some insights toward the smart school that we can tap today.
Research on school and teacher change has alerted us to typical pitfalls and helps us organize processes of change likely to take hold and last.
Comparison with other cultures has provided yardsticks by which to gauge more objectively how well we are doing.
Comparison with other cultures has also revealed wide-scale models of thoughtful instruction and the thoughtful development of professional practice.
Research on the nature of knowledge and understanding has underscored the importance of mental images (or as many psychologists would say, mental models) in building understanding.
Such conceptions of human understanding as the understanding-performance perspective discussed in chapter 4 have begun to inform instruction more widely.
Research and diverse educational experiments on human thinking have shown that instruction can elevate students' abilities to think and learn and have revealed much about the design of such instruction.
Such new conceptions of intelligence as the notion of distributed intelligence explored in chapter 6 have emerged to picture intelligence as a more malleable and accessible commodity than many have heretofore held.
Studies of transfer of learning have reidentified transfer as a serious roadblock to educational impact but also have disclosed how we can teach for transfer effectively.
Techniques of cooperative learning have been widely researched and implemented, providing the know-how for successful classroom use.
Different ways of harnessing social relations for learning—cooperative learning, peer collaboration, peer tutoring, and Socratic interaction, to name a few—have been distinguished, investigated, and put to work more widely and systematically.
Innovators have designed and investigated learning environments where computers or videodisc/computer systems provide an engaging and supportive setting for thoughtful learning activities.
Vigorous exploration of alternative means of assessment (those that get beyond a "facts and algorithms" mentality) has begun to yield useable methods.
This list does not purport to itemize all the recent knowledge gains in education, not even those underscored in this book. But perhaps it serves to make the case that we know a lot more now than the "last time around"—the 1960s and early 1970s—about how to work for smart schools. And today, in the essential schools movement launched by Theodore Sizer, in Mortimer Adler's Paideia schools, and in other initiatives across the American landscape, we see students, teachers, administrators, university people, and others working toward more thoughtful patterns of education and drawing upon this accumulated savvy.
With all these signs of a new spring in educational practice, we should recognize that it's still _early_ spring. Many of the initiatives underway today certainly serve to make schools more thoughtful places. But they do not often enough take full advantage of what we know about thinking and learning. They generally do not have a point-blank metacurriculum. While striving to teach for understanding, they do not usually do so against the backdrop of a model of what understanding is. Very often, caught up in the momentum of innovation and the shine of new ideas, they can neglect fundamental facets of learning underscored in Theory One.
This book, then, points toward quite a special star in the educational firmament. The smart school finds its motivation in the three hard-to-argue-with educational goals of retention, understanding, and active use of knowledge. The smart school makes a commitment to informed, energetic, and thoughtful teaching and learning. The smart school finds its foundation in a rich and evolving set of principles about human thinking and learning. The smart school in its fullest sense will take us beyond even the ingenious educational innovations now to be found. Smart schools were an ambition and an endeavor a quarter-century ago and at other points in history before that. Today they are within reach.
Thinking back to the first pages of chapter 2, I am reminded that Edgar Alan Poe in "The Bells" wrote not only of alarm bells but of other bells with brighter tones, sleigh bells and wedding bells. He never mentioned school bells at all.
I have often wondered why. Perhaps they were too ambiguous in character.
However, Poe does spread out a palette of sounds from which we can choose. Among the options are "the moaning and the groaning of the bells." That is what we are trying to get away from. Another sonority Poe offers I like better: "the tintinnabulation" of the bells. That's what the spirit of a school ought to sound like.
Just as Poe contrived his phrase and his poem to sound the way he wanted, so will we have to make up our schools for the spirit and substance we want them to express. With commitment, effort, and intelligence, we can look forward to an era when schooling will be an upbeat and effective enterprise day in, day out:
Keeping time, time, time,
In a sort of Runic rhyme,
To the tintinnabulation that so musically wells
From the bells, bells, bells, bells,
Bells, bells, bells—
From the jingling and the tinkling of the bells.
##
APPENDIX
**A CHECKLIST FOR CHANGE**
This checklist can help anyone to appraise how far a unit, classroom, curriculum, text, or whole school has moved toward the spirit of the smart school—informed, energetic, and thoughtful. The checklist moves systematically through major features of the five dimensions discussed earlier: Theory One, a pedagogy of understanding, the metacurriculum, distributed intelligence, and the hot cognitive economy. And it adds a sixth, conditions for change. Of course, an innovation need not score well on all these dimensions and subcategories to be worthwhile. Few innovations could reasonably undertake all these agendas at once in a full-blown way.
1. DIMENSION 1. THEORY ONE AND BEYOND
2. Does the instruction offer clear information about topics and processes (for instance, through modeling) that students are to learn?
3. Does the instruction provide for reflective practice, where students practice the very performances they are supposed to achieve and ponder how their learning is going and how they might manage it better?
4. Does the instruction offer informative feedback, helpful to students in improving their performance?
5. Does the instruction use extrinsic and/or intrinsic motivation to ensure students' interest and commitment?
6. Does the instruction take advantage of good didactic teaching when students need information?
7. Does the instruction take advantage of coaching when students are practicing challenging performances?
8. Does the instruction take advantage of Socratic instruction when the students engage in complex inquiry?
9. Does the instruction take advantage of teaching and learning methods beyond Theory One; for example, a constructivist or developmental perspective, cooperative learning, emphasis on intrinsic motivation, honoring multiple intelligences, situated learning?
1. DIMENSION 2. A PEDAGOGY OF UNDERSTANDING
2. Does the instruction engage students in understanding performances as a major part of the learning experience (explaining, finding new examples, generalizing, making analogies)?
3. Does the instruction pay heed to students' existing mental images and try to build mental images that represent well the target concepts?
4. Does the instruction use powerful representations to help create needed mental images?
5. Does the instruction pay direct attention not only to content knowledge but to the problem-solving level of understanding: how problems are solved in the subject matter, including problem-solving strategies?
6. Does the instruction pay direct attention to the epistemic level of understanding: how justification and explanation are handled in the subject matter?
7. Does the instruction pay direct attention to the inquiry level of understanding: what makes good questions and how they are approached in the subject matter?
8. Is the instruction organized around generative topics, which are central to the discipline, accessible to teachers and students, and rich in their ramifications and implications?
1. DIMENSION 3. THE METACURRICULUM
2. Does the instruction pay direct attention to the problem-solving, epistemic, and inquiry levels of understanding? (Same as under Dimension 2, here for completeness.)
3. Does the instruction explicitly use languages of thinking (terms like reason, evidence, hypothesis, strategy; graphic organizers; efforts to cultivate a culture of thinking in the classroom)?
4. Does the instruction model and encourage intellectual passions (intellectual persistence, curiosity, concern with truth and fairness)?
5. Does the instruction employ integrative mental images to knit together large topics or entire subject matters?
6. strategies and reflect upon their own learning processes?
7. Does the instruction involve teaching for transfer, where connections beyond the immediate topic to other topics in the subject matter, other subject matters, and outside of school are explored?
1. DIMENSION 4. DISTRIBUTED INTELLIGENCE
2. Does the instruction take advantage of the physical distribution of intelligence through "thinking on paper" or on computer or other graphic and writing devices?
3. Does the instruction take advantage of the social distribution of intelligence through cooperative learning, tutoring relationships, and other social mechanisms?
4. Does the instruction take advantage of the symbolic distribution of intelligence through varied symbolic vehicles such as concept maps, diagrams, improvisations, and prose?
5. Does the instruction avoid the trap of the "fingertip effect," not assuming that students will simply catch on to the opportunities brought by distributed intelligence but coaching students in good ways to proceed?
6. Does the instruction take care that the executive function (which decides what the task is and how to manage it), if it does not stay with the students, at least returns to them toward the end of the episode of learning so that they have experience with managing their thinking and learning?
1. DIMENSION 5. THE COGNITIVE ECONOMY
2. Does the instruction demand complex cognition (understanding performances, higher levels of understanding, use of languages of thinking) of the students?
3. Does the instruction make plain the gains of the complex cognition through highlighting enjoyment and making connections to other matters in and out of school?
4. Does the instruction minimize the costs of the complex cognition through supporting students in their efforts?
5. Does the instruction make sense to the teacher in terms of cost in effort and other factors?
6. Does the instruction minimize conflict of interest for the teacher through use of external testing at least sometimes?
7. Does the instruction decrease the pressure on teachers toward the token investment strategy through a small number of clear topics and priorities?
8. Does the instruction employ authentic assessment (testing students with open-ended tasks that tap the very performances one wants them to develop) to allow the teacher to teach to the test legitimately and fruitfully?
1. DIMENSION 6. CONDITIONS FOR CHANGE Some Conditions for the Wide-Scale Viability of an Innovation
2. Does not escalate overall teacher workload.
3. Allows teachers a creative role.
4. Avoids extreme demands on teachers' skills and talents.
5. Includes strong materials support.
6. Does not boost costs a lot.
7. Fulfills many conventional educational objects at least as well as conventional instruction.
1. Some Conditions for an Effective Process of Change
2. During initiation (and beyond), an effective process of change:
3. Rests on a clear need discernible by the participants.
4. Enjoys strong advocacy within the institution.
5. Brings clarity of philosophy and approach.
6. Is practical to pursue in the context.
7. Includes the needed financial, human, and other resources.
8. Involves challenge, rather than being very simple to do.
9. Includes both some pressure and good support from administrators.
10. 11. During implementation, an effective process of change:
12. Starts small but thinks big, aiming to include many people and change much.
13. Benefits from regular counsel from outside the institution, counsel that continues for some time.
14. Develops internal experts responsible for orienting new participants, training, and other functions.
15. Moves toward action without expecting everyone to understand everything or buy into everything at first.
16. Avoids unreflective mechanical implementation.
17. Recognizes that commitment and ownership will evolve gradually for many participants.
18. Includes visionaries who paint pictures of what the school could be like.
19. Shares power, avoiding situations where a few try to control the innovation tightly.
20. Recognizes that opportunities and problems will come up along the way and will need to be dealt with as they arise.
21. Moving toward continuation, an effective program of change:
22. Avoids overdependence on external funds, which may sink the program when they disappear.
23. Avoids overdependence on one or two key people, instead distributing expertise over several.
**Some Conditions for Advancing Thoughtful Professionalism**
1. A smart school needs to be an informed and energetic setting of thoughtful learning for teachers and administrators, not just for students. This involves most of the features discussed: emphasis on understanding, attention to thinking (the metacurriculum), distributed intelligence (cooperative and collaborative work), and the rest.
2. Collegiality, including talk about practice, observing one another teach, working on the curriculum together, and teaching one another.
3. An administrative style that is responsive to teachers' ideas and not too directive.
4. Time to think rather than constant teaching.
5. A shared culture of the craft of teaching.
6. Apprenticing of beginning teachers to more experienced ones.
## **NOTES**
**CHAPTER 1 SMART SCHOOLS**
**Goals: Toward Generative Knowledge (pp. 4-6)**
Lawrence Cremin on the multiple agendas of education: Cremin (1990).
Mortimer Adler's _Paideia Proposal:_ Adler (1982).
**Means: Thoughtful Learning (pp. 6-8)**
Students cannot identify the date of the Civil War within a half century: Ravitch and Finn (1987).
Fundamental science misconceptions: See, for example, Clement (1982, 1983); McCloskey (1983); Novak (1987); Perkins and Simmons (1988).
Rexford Brown's study: Brown (1991).
Quote from William James on memory: James (1983), p. 87.
**Precedents: Swings of the Pendulum (pp. 8-11)**
Quote from John Dewey on intellectual learning: Archambault (1964), p. 249.
Progressivism and life adjustment education: Toch (1991), pp. 44-55.
On _Man: A Course of Study:_ Dow (1991).
On _Project Physics:_ Holton, Rutherford, and Watson (1970).
"Back to the basics:" Toch (1991), p. 64.
_Paideia Proposal:_ Adler (1982).
Essential schools: Sizer (1984).
Whole language: Edelsky, Altwerger, and Flores (1991).
New standards for the learning of mathematics: National Council of Teachers of Mathematics (1989).
**Mission: Smart Schools (pp. 16-18)**
Quote from Jerome Bruner on psychology: Bruner (1973a), p. 478.
**CHAPTER 2 THE ALARM BELLS**
Lawrence Cremin on the cacophony of teaching: Cremin (1990).
**A Shortfall: Fragile Knowledge (pp. 21-27)**
Statistics about what students don't know: Ravitch and Finn (1987).
The research of John Bransford and colleagues on inert knowledge: Bransford, Franks, Vye, and Sherwood (1989).
Inert knowledge in computer programming: Perkins and Martin (1986).
Children's belief in a flat earth: Neussbaum (1985).
The film _A Private Universe:_ Schneps (1989).
Misconceptions in science and mathematics in general: for example, Clement (1982, 1983); McCloskey (1983); Novak (1987); Perkins and Simmons (1988).
Howard Gardner's idea of stereotypes: Gardner (1991).
Ritual knowledge—the girl with the clever word-problem strategy: Taba and Elzey (1964).
Fragile knowledge in general: Perkins and Martin (1986).
**A Shortfall: Poor Thinking (pp. 27-31)**
Students' difficulties with story problems in mathematics: see, for instance, Schoenfeld (1985); Nesher (1988); Bebout (1990).
Quote from the National Assessment: National Assessment of Educational Progress (1981).
The knowledge-telling strategy for writing: Bereiter and Scardamalia (1985).
The importance of active thinking in memorization: for example, Baddeley (1982); Higbee (1977).
Quote from Rexford Brown on students' ability to reason about what they are involved with: Brown (1991), p. 187-188.
Lauren Resnick's remark about higher-order thinking: made at the Council of Chief State School Officers Summer Institute, Mystic, Connecticut, July 29-August 1, 1990.
**A Deep Cause: The Trivial Pursuit Theory (pp. 31-34)**
Quote from Vito Perrone on the trivial pursuit character of teaching and learning: Perrone (1991b), p. 2.
Goodlad's information on classroom events: Goodlad (1984).
Boyer's information on classroom events: Boyer (1983).
Lack of the language of thinking in education: Astington and Olson, 1990.
The school administrator's story: this was from Carolee Matsumoto, now at the Educational Development Corporation. Thank you, Carolee.
The idea of cultural literacy, with a list of concepts to be familiar with: Hirsch (1987).
**A Deep Cause: The Ability-Counts-Most Theory (pp. 34-37)**
Japanese versus U.S. attitudes toward effort in learning: White (1987).
Quote from Rexford Brown on the intelligence required by a literacy of thoughtfulness: Brown (1991), p. 240.
The critical role of effort as the key variable in learning is, for instance, well documented in studies of mastery learning: see Bloom (1984). Another set of sources is the voluminous work on "time on task," for instance Denham and Lieberman (1980).
Dweck and colleagues' work on incremental and entity learners: Dweck and Bempechat (1980); Dweck and Licht (1980); Cain and Dweck (1989).
On the Rosenthal effect: Rosenthal and Jacobson (1968).
**A Consequence: Economic Erosion (pp. 37-42)**
Marc Tucker on education and the economy. Tucker (1990).
**CHAPTER 3 TEACHING AND LEARNING:
THEORY ONE AND BEYOND**
**The Devastating Critique Levied by Theory One (pp. 46-53)**
Research in "direct explanation": see Roehler, Duffy, Putnam, Wesselman, Sivan, Rackliffe, Book, Meloth, and Vavrus (1987); Duffy, Roehler, Meloth, and Vavrus (1986).
Other innovative approaches to teaching history: See, for instance, the insightful booklet by Tom Holt (1990).
E. D. Hirsch's cultural literacy (also discussed in chapter 2): Hirsch (1987).
Computational achievements in mathematics are not so bad, but word problems present many difficulties: National Council of Teachers of Mathematics (1989).
On the importance of explicit modeling of thinking processes in mathematics: Schoenfeld (1979, 1980).
Quote from Lee Shulman: Shulman (1983), p. 497.
_Horace's Compromise:_ Sizer (1984), quote from p. 20.
**Three Ways to Put Theory One to Work (pp. 53-58)**
_The Paideia Proposal:_ Adler (1982).
Gaea Leinhardt's research on teaching: Leinhardt (1989).
For a further perspective on coaching, see Collins, Brown, and Newman (1989).
Allan Collins on Socratic teaching: Collins and Gentner (1982); Collins (1988); Collins (1987); Collins and Stevens (1983).
**The Bogeyman of Behaviorism (pp. 58-60)**
"On 'Having' a Poem:" Skinner (1972).
**Beyond Theory One (pp. 60-69)**
Constructivism in education: For a very recent assembly of viewpoints, see Duffy and Jonassen (1991). Also see Liben (1987).
Regarding Piaget, see, for instance, Piaget (1954); Inhelder and Piaget (1958).
Bruner's thesis that any subject can be taught at any age: Bruner (1973c).
Problems with Piaget's theory: See, for example, Brainerd (1983); Case (1984, 1985); Piaget (1972).
Cooperative learning: See Damon (1984); Slavin (1980); Glasser (1986); Johnson, and Johnson, Holvbec-Johnson (1986).
On collaborative versus cooperative learning: Damon and Phelps (1989).
Intrinsic motivation: See Lepper and Green (1978).
Teresa Amabile's experiment with student writers: Amabile (1983), pp. 153-157.
Multiple intelligences: Gardner (1983).
Situated learning: e.g., Brown, Collins, and Duguid (1989).
**CHAPTER 4 CONTENT: TOWARD A PEDAGOGY
OF UNDERSTANDING**
**What Is Understanding? (pp. 75-79)—The Role of Understanding Performances**
"Going beyond the information given:" Bruner (1973b).
The idea of understanding performances: See also Perkins (1988; 1991).
**Understanding and Mental Images (pp. 79-83)**
The idea of mental images or, more commonly in psychological writings, mental models: See, for example, Gentner and Stevens (1983); Johnson-Laird (1983).
Note: Often in psychology, "mental image" is taken to mean a visualization that people create in their "mind's eye." The present use of mental image is broader than that.
**Levels of Understanding (pp. 83-87)**
"If you can't solve it in ten minutes...": Schoenfeld (1985).
Dan Chazen's investigations of geometry learning: Chazen (1989).
Levels of understanding: Perkins and Simmons (1988).
Strategies at the problem-solving level: For example, see Perkins (1990); Polya (1954, 1957).
Epistemic level reasoning: For example, see Perkins (1989); Perkins, Farady, and Bushey (1991); Toulmin (1958).
Inquiry level: For example, see Duckworth (1987); Perkins (1986).
The epistemic and inquiry levels correspond roughly to what Joseph Schwab a number of years ago termed the "syntactic structure" of a discipline: Schwab (1978).
**Powerful Representations (pp. 87-92)**
The Sufi tale: paraphrased from Shah (1970), p. 193.
The rocket trajectory task: McCloskey (1983).
Thinker Tools: White (1984); White and Horwitz (1987).
Richard Mayer's work on conceptual models: Mayer (1989).
On concrete, stripped, constructed analogs: Perkins and Unger (1989).
**CHAPTER 5 CURRICULUM: CREATING THE METACURRICULUM**
**The Idea of the Metacurriculum (pp. 101-104)**
Four levels of metacognition: Swartz and Perkins (1989).
_High School:_ Boyer (1983).
Project 2061: _Science for All Americans_ (1989).
Recommendations of the National Council of Teachers of Mathematics (1989).
**Levels of Understanding (pp. 104-107)**
Good problem management and problem-solving strategies: Schoenfeld (1982); Schoenfeld and Herrmann (1982).
Conceptual ecology: Posner, Strike, Hewson, and Gertzog (1982).
The Geometric Supposer: Schwartz and Yerushalmy (1987).
Does the table push back on the book? Clement (1991).
Lesson on Truman and the atomic bomb: from Swartz and Parks (1992).
**Languages of Thinking (pp. 107-114)**
The everyday language of thinking and its absence from textbooks: Astington and Olson (1990); Olson and Astington (1990).
Speaking "Cogitare": From the article "Do You Speak Cogitare?" in Costa (1991), quotes from pp. 111, 113, and 114.
Project Intelligence and its testing: Herrnstein, Nickerson, Sanchez, and Swets (1986).
_The Teaching of Thinking:_ Nickerson, Perkins, and Smith (1985).
Concept mapping: Novak and Gowin (1984).
Pictorial formats for thinking: Clarke (1990); Jones, Pierce, and Hunter (1988-89); McTighe and Lyman (1988); for Sandra Parks, Black and Black (1990).
Whole language: Edelsky, Altwerger, and Flores (1991).
The thoughtful classroom: Newman (1990a,b).
Quote from Sara Lawrence Lightfoot: Lightfoot (1983), p. 365.
**Intellectual Passions (pp. 114-117)**
Quote from Arthur Costa on aesthetics: Costa (1991), p. 17.
Quotes from _In the Name of Excellence:_ Toch (1991), p. 235.
Dewey's three attitudes of open-mindedness, whole-heartedness, and responsibility: Archambault (1964).
The cognitive emotions: Scheffler (1991).
Strong sense critical thinking: Paul (1990).
Dispositions: Ennis (1986).
A dispositional model of good thinking: Perkins, Jay, and Tishman (in press).
**Integrative Mental Images (pp. 117-119)**
A metacourse for programming: Perkins, Schwartz, and Simmons (1988); Schwartz, Perkins, Estey, Kruidenier, and Simmons (1989).
Rissland's concepts, examples, and results: Rissland (1978).
Concept maps: Novak and Gowin (1984).
**Learning to Learn (pp. 119-122)**
Entity versus incremental learners: Dweck and Bempechat (1980); Dweck and Licht (1980); Cain and Dweck (1989).
Attention monitoring: Miller (1985).
Research on learning from examples: Chi and Bassok (1989).
Learning electrical and economic principles from computer environments: Schauble, Glaser, Raghavan, and Reiner (1991).
Review of metacognitive reading strategy results: Haller, Child, and Walberg (1988).
Helping college students to become better academic performers: Bloom and Broder (1950).
Guided design: Wales and Stager (1977).
Findings on guided design at West Virginia University: Wales (1979).
**Teaching for Transfer (pp. 122-128)**
Early studies of transfer: Thorndike (1923); Thorndike and Woodworth (1901).
Studies of transfer from programming: Clements (1985); Clements and Gullo (1984); Salomon and Perkins (1987).
Findings on Philosophy for Children: Lipman, Sharp, and Oscanyan (1980), appendix B.
The Reading Partner findings: Salomon (1988).
The low road-high road theory of transfer: Salomon and Perkins (1989).
Ann Brown and colleagues' research on transfer: Brown (1989).
Hugging and bridging in teaching for transfer: Fogarty, Perkins, and Barell (1991); Perkins and Salomon (1988).
**An Example of Teaching the Metacurriculum (pp. 128-130)**
Knowledge as design: Perkins (1986).
**CHAPTER 6 CLASSROOMS: THE ROLE OF
DISTRIBUTED INTELLIGENCE**
**The Idea of Distributed Intelligence (pp. 133-135)**
The concept of distributed intelligence: Pea (in press); Perkins (in press); Salomon (in press).
Effects with and of technology: Salomon, Perkins, and Globerson (1991).
**Distributing Cognition in the Classroom (pp. 135-144)**
John Barell's journal-keeping approach and example: Barell (1991), p. 3.
Assessment by portfolios: Valencia (1990); Wiggins (1989); Wolf (1989); Baron (1990).
Processfolios and Arts PROPEL: Zessoules and Gardner (1991); Howard (1990); Wolf (1989); Gardner (1989).
The computer language Logo: Papert (1980).
Idit Harel's design experience around factions: Harel (1991).
_King Tut's Chronicle_ activity: Fiske (1991), pp. 157-8.
Brown and Palinscar's review of cooperative learning: Brown and Palincsar (1989).
On cooperative learning generally: Johnson, Johnson, and Holubec-Johnson (1986); Glasser (1986); Damon (1984); Slavin (1980).
On peer learning (peer tutoring, cooperative learning, and collaborative learning): Damon and Phelps (1989).
On pair problem solving: Whimbey and Lochhead (1982); Lochhead (1985).
**The Fingertip Effect (pp. 144-148)**
On the fingertip effect: Perkins (1985).
Students' response to word processors: Daiute (1985).
Problems with cooperative learning: See the above references on cooperative learning.
Computer support environments for writing: Daiute and Morse (in press); Salomon (1991).
**CHAPTER 7 MOTIVATION: THE COGNITIVE
ECONOMY OF SCHOOLING**
The fourth grader's questions about fractions are drawn from the same study mentioned at the beginning of chapter 5. Thanks again to colleagues Heidi Goodrich, Jill Mirman, and Shari Tishman.
**The Idea of a Cognitive Economy (pp. 156-159)**
Quote on teachers' reasons for rejecting innovations: Fullan (1991), p. 130.
Teachers' spontaneous criteria about worthwhile changes: Fullan (1991), pp. 127-128.
Herbert Simon's notion of limited rationality: Simon (1957).
The concept of "giving reason": Duckworth (1987).
**The Cool Cognitive Economy of the Typical Classroom (pp. 159-164)**
Problems with textbooks: Toch (1991), pp. 225-226.
Watered-down courses in Florida: Toch (1991), p. 104.
_Horace's Compromise:_ Sizer (1984).
**Creating a Hot Cognitive Economy (pp. 164-167)**
_The New Meaning of Educational Change:_ Fullan (1991).
**School Restructuring: A Cognitive Economic Revolution (pp. 167-171)**
Theodore Sizer's "nine points": Sizer (1984), pp. 225-227.
Central Park East Secondary School: Toch (1991), various mentions pp. 260-270.
"The Promise," credo of Central Park East: Quoted from Perrone (1991b), pp. 13-14.
James Comer and Comer Schools: Fiske (1991), 205-220.
"Locked into a conspiracy" quote from Comer: quoted from Fiske (1991), p. 206.
The story of Robert: Fiske (1991), pp. 212-215.
Quote from John Haslinger: Fiske (1991), p. 215.
**Teaching to the Right Test: The Idea of Authentic Assessment (pp. 174-176)**
About authentic assessment: Gifford and O'Connor (1991); Perrone (1991a); Schwartz and Viator (1990).
_Assessment Alternatives in Mathematics:_ Stenmark (1989).
**The Cognitive Economy Meets the Money Economy (pp. 176-180)**
School choice: see Fiske (1991), chapter 7; Toch (1991), pp. 246-263.
Impact of choice in the Cambridge school system: Fiske (1991), pp. 178-179.
Impact of choice in New York District 4: Toch (1991), pp. 256-257.
Countries with a good relationship between money and cognitive economies: Besides the latter part of chapter 2, see Tucker (1990).
**An Example of Progress Toward a Hot Cognitive Economy (pp. 180-182)**
The Vermont authentic assessment program: Fiske (1991), pp. 132-138; Writing Assessment: The Pilot Year (1990).
The contents of a writing portfolio: a direct quote from Vermont Writing Assessment: The Pilot Year (1990).
Ann Rainey's mathematics problem: Fiske (1991), p. 134.
**CHAPTER 8 VICTORY GARDENS FOR
REVITALIZED EDUCATION**
**Example 1. Expert Tutoring (pp. 187-189)**
The two-sigma effect: Bloom (1984).
Research on expert tutoring: Lepper, Aspinwell, Mumme, and Chabay (1990).
**Example 2. Biology for Young Inquirers (pp. 189-191)**
Ann Brown's and Joseph Campione's biology course: Brown and Campione (1990).
**Example 3. History for Thinkers (pp. 191-194)**
Critical thinking and the shot heard 'round the world: Bennett (1970).
Kevin O'Reilly's approach to critical thinking in American history: O'Reilly (1991).
Materials developed by O'Reilly: O'Reilly (1990).
**Example 4. A Textbook from the Past (pp. 194-196)**
_Man: A Course of Study:_ Dow (1991).
_The People Make a Nation:_ Sandler, Rozwenc, and Martin (1971).
**Example 5. A Metacourse for Computer Programming (pp. 196-198)**
A metacourse for programming: Perkins, Schwartz, and Simmons (1988); Schwartz, Perkins, Estey, Kruidenier, and Simmons (1989).
**Example 6. Escalante Himself (pp. 198-201)**
On Jaime Escalante: Matthews (1988).
**CHAPTER 9 THE CHALLENGE OF WIDE-SCALE CHANGE**
**Making Change Work (pp. 210-219)**
_The New Meaning of Educational Change:_ Fullan (1991).
Knowledge as design: Perkins (1986).
For some other perspectives on teaching for understanding, see Gardner (1991); Mayer (1989); Perkins (1991); Perkins and Simmons (1988); Rissland-Michener (1978).
_The Paideia Proposal:_ Adler (1982).
**Advancing Thoughtful Professionalism (pp. 219-227)**
Quote from Lee Shulman: Shulman (1987), p. 13.
Quotes from Arthur Costa: Costa (1991), p. 3.
_Improving Schools from Within:_ Barth (1991).
"Tennis shoe in a laundry dryer": Barth (1991), p. 1.
Collegiality: Barth (1991), chapter 3.
Judith Warren Little on collegiality: Little (1981).
Three kinds of teachers: Barth (1991), chapter 5.
Quote from Sara Lawrence Lightfoot: Lightfoot (1983), p. 334.
Teacher development in Japan: Stigler and Stevenson (1991).
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## **INDEX**
Ability-Counts-Most theory, , , , , , , ,
explained, –37
Active use of knowledge, _see_ Knowledge: active use, retention, understanding of
Adler, Mortimer, , , , , ,
Amabile, Teresa,
Analogs: concrete, stripped, constructed, –92, Aristotle,
Arts PROPEL project,
Asian teaching methods, –27
Assessment,
authentic, –76, –78
_Assessment Alternatives in Mathematics_ (Hall), ,
Astington, Janet, ,
At-risk students, –15, ,
Attention monitoring, ,
Authentic assessment, –76, –78
Aware learners,
Barell, John,
Barth, Roland, –22,
Behaviorism, , , –60,
"The Bells" (Poe), –20,
Bennett, Peter, –92
Bereiter, Carl,
Biology, in victory gardens, –91
Bloom, Benjamin, , ,
Bo Peep theory, –23,
Boyer, Ernest L., ,
Bransford, John, ,
Bridging, , ,
_A Brief History of Time_ (Hawking),
Broder, Lois,
Brown, Ann, , , , , , ,
Brown, John Seely,
Brown, Rexford, , ,
Bruner, Jerome, , , , , ,
Campione, Joseph, , , , ,
Causal reasoning,
Central Park East Secondary School, –69,
Change, –30
checklist for, –35
continuation of, –19
costs of,
implementation of, –17
initiation of, –12
necessities of scale in, –10
teachers' role in, , –27,
Chazen, Dan,
Chi, Michelene,
Child, D. A.,
China, –27
Clarke, John H.,
Class size, ,
Clement, John,
Clements, D. H., ,
Coaching, , , , , , , ,
explained, –55
Coalition of Essential Schools, , –69,
Cognitive economy, , , –82, , –34
authentic assessment in, –76, –78
catches in, –67
cool, –64, ,
example of progress toward, –82
hot, –67, , –82, , , , –94,
money economy and, , , –80
school restructuring and, –71
victory gardens and, , , , , –94, , , ,
Collaborative learning, _see_ Peer collaboration
Collins, Allan, ,
Comer, James P. (school restructuring model of), –71
Complex cognition, –60,
Computer programming
inert knowledge in, –23
integrative mental images in, –18
in victory gardens, –98
Computer technology, distributed intelligence and, –39
Concept mapping, , ,
Constructivism, –62, , ,
Cooperative learning, , , , , , ,
distributed intelligence and, , , , , , , , , , ,
Theory One and, –64, ,
Costa, Arthur, , , , ,
Coverage, –64,
Creativity,
Cremin, Lawrence, ,
Critical thinking, strong vs. weak sense in, –16
_Crockett's Victory Garden_ (television program),
Cultural diversity, –27
Cultural literacy,
_Cultural Literacy_ (Hirsch), –34
Culture connection, –14
Daiute, Collette,
Damon, William, , ,
Denmark,
Desktop publishing, –39
Developmental perspective, –63,
Dewey, John, , , ,
Diagrams, –90
Didactic teaching, , , , , , , ,
explained, –54
Direct explanation, –47, , –83
Direct workers, –38,
Discovery learning,
Distributed intelligence, –13, –54, , , ,
in cognitive economy, , ,
executive function in, –51, ,
fingertip effect and, –48, , , ,
physical, –39, , ,
social, –42, ,
symbolic, –44, ,
in victory gardens, , –87, , , , , ,
Dropouts, ,
Duckworth, Eleanor,
Duffy, Gerald, ,
Dweck, Carol,
Economic erosion, , –40,
Education credentials, ,
English language, resources of, –9
Ennis, Robert,
Entity learners, , , ,
Epistemic knowledge, , , , ,
in metacurriculum, , , , ,
Equality, peer education and, ,
Escalante, Jaime, , –201, –5, ,
Essays, –12
Essential schools, ,
Examinations
as authentic assessment, –76, –78
independent of teacher,
open-book,
teaching to, –74,
Executive function, –51, ,
Exhibitions, ,
Expert tutoring, –89,
Extrinsic motivation, , –65, , ,
Fermi, Enrico (problems invented by), ,
Fingertip effect, –48, , , ,
Fiske, Edward,
Fragile knowledge, –27, , ,
defined,
Fragile knowledge syndrome, –27, ,
Fullan, Michael, , ,
Gardner, Howard, , , ,
Garfield High School, –201
Generative knowledge, –6
Generative performances,
Generative topics, –95, , ,
Geometric Supposer, –6
Gertzog, W. A.,
Globerson, Tamar,
Good explanation, –54
_The Good High School_ (Lightfoot),
Goodlad, John,
Goodrich, Heidi,
Good Shepherd theory, , –26
Graphics, –12
Greeno, James,
Guided design,
Gullo, D. F.,
Hall, Lawrence,
Haller, E. P.,
Hand calculators, –38
Harel, Idit,
Hawking, Stephen,
Hewson, P. W.,
Higher-order thinking, , –2, , , ,
_High School: A Report on Secondary Education in America_ (Boyer), ,
High school diplomas, ,
Hirsch, E. D., –34,
History
Theory One and, –49,
in victory gardens, –94
_Horace's Compromise_ (Sizer), , ,
Horwitz, Paul,
Hugging, –27
Hunter, Madeline,
_Improving Schools from Within_ (Barth),
Incremental learners, , , ,
Indirect workers,
Inert knowledge, –23, , , , , , ,
Inquiry, –86, ,
in metacurriculum, , , ,
Integrative mental images, , –19, , ,
Intellectual passions, , –17, , ,
_In the Name of Excellence_ (Toch), –15
Intrinsic motivation
in cognitive economy, ,
in Theory One, , –66, , , ,
James, William, ,
Japan, , , , , –27
Jay, Eileen,
Jigsaw method, , , ,
Jimenez, Benjamin,
Johnson, David,
Johnson, Roger,
Jones, Beau,
Journals, –37,
Kiva Elementary School,
Knowledge: active use, retention, understanding of, –6, , , , , , ,
in cognitive economy, ,
fragile knowledge syndrome and,
in metacurriculum, , ,
poor thinking and,
Trivial Pursuit theory and,
Knowledge as design, , ,
Knowledge gap, , , –26, ,
Knowledge-telling strategy, ,
Labor market, –40
Language of strategies, –10
Languages of thinking, , –14, , , , , ,
culture connection in, –14
graphics in, –12
language of strategies in, –10
resources of English in, –9
Larson, Gary,
Learning Research and Development Center,
Leinhardt, Gaea, ,
Lepper, Mark,
_Letter to Teachers_ (Perrone), –32
Lightfoot, Sara Lawrence, ,
Lipman, Matthew,
Literacy of thoughtfulness, ,
Literature, generative topics in, –95
Little, Judith Warren,
Lochhead, Jack,
Logo program,
Lost Sheep theory, , –25
McTighe, Jay,
_Man: A Course of Study program_ , ,
Mathematics, , , ,
Ability-Counts-Most theory in,
authentic assessment in, –75
Escalante's work in, , –201, –5, ,
integrative mental images in,
mental images in, –82, –84
new,
poor thinking and, ,
story problems in, , , , ,
Theory One and, –50
Mayer, Richard,
Memorization, , –30,
Mental images, –84, , , , , ; _see also_ Integrative mental images; Powerful representations
change and, ,
defined,
problem solving and,
understanding performances and, –83
victory gardens and, ,
Metacognition, , , ,
Metacurriculum, , –130, ; _see also_ Languages of thinking; Transfer of learning
change and, , , –33
cognitive economy and, –66, ,
example of teaching in, –30
integrative mental images in, , –19, , ,
intellectual fiassions in, , –17, , ,
learning to learn in, , –22, , ,
levels of understanding in, , –7, ,
motivation in, –103
victory gardens and, , , , , , , , , , ,
Mind mapping,
Mirman, Jill,
Missing knowledge, _see_ Knowledge gap
Money economy, , , –80
Motivation; _see also_ Extrinsic motivation; Intrinsic motivation
fingertip effect and, –47
in metacurriculum, –3
in Theory One, , , , –66, , , , , ,
Multiple intelligences, –67, ,
Mutuality, peer education and, ,
Naive knowledge, , –24, , , ,
National Assessment of Educational Progress,
National Council of Teachers of Mathematics, ,
Natural sciences, generative topics in,
Newmann, Fred, –13
New math,
_The New Meaning of Educational Change_ (Fullan), , ,
Nickerson, Raymond,
Novak, Joseph, ,
Odyssey,
Olson, David, ,
"On 'Having' a Poem" (Skinner), –59
Ontario Institute for Studies in Education,
_The Paideia Proposal: An Educational Manifesto_ (Adler), , , ,
Paideia schools,
Pair problem solving, –42,
Palincsar, Annemarie,
Parks, Sandra, , ,
Paul, Richard, ,
Pea, Roy, ,
Pedagogy of understanding, , –98, , , , , , ; _see also_ Epistemic knowledge; Inquiry; Mental images; Powerful representations; Understanding performances
cognitive economy and, , ,
content in, –85, ,
generative topics in, –95, , ,
problem solving in, , ,
teaching implications of, –98
victory gardens and, , , , , , , , , ,
Peer collaboration, , ,
distributed intelligence and, , , ,
Theory One and, –64, ,
Peer education, –42, _see also_ specific types
Peer tutoring, , , , ,
_The People Make a Nation_ (Allyn & Bacon), –96
Perkins, David, ,
Perrone, Vito, –32,
Person-plus systems, , , , , ,
example of, –54
Person-solo systems, , , , , ,
Phelps, Erin, , ,
Philosophy for Children program, ,
Physical distribution of intelligence, –39, , ,
Physics, in pedagogy of understanding, –91
Piaget, Jean, ,
A Place Called School (Goodlad),
Plato,
Poe, Edgar Allan, –20, ,
Poor thinking, –30, , , ,
defined,
_Popular Education and Its Discontents_ (Cremin), ,
Portfolios, , , –81
Posner, G. J.,
Powerful representations, –92, , –98, , , ; _see also_ Mental images
Preservice education,
Principals,
A Private Universe (film), –24
Problem-based learning,
Problem selection, , –50
Problem-solving skills, , ,
distributed intelligence and, –42,
journals and,
in metacurriculum, , , , ,
Process-folios,
Progressivism,
Project 2061,
Project-based learning,
Project Intelligence,
_Project Physics_ program,
Reading, poor thinking and, –29,
Reading Partner, ,
Reflective learners,
Remini, Robert,
Resnick, Lauren, ,
Retention of knowledge, _see_ Knowledge: active use, retention, understanding of
Rissland, Edwina,
Ritual knowledge, , , , , ,
Roehler, Laura, , ,
Rosenthal, Robert,
Russia,
Sadler, Phillip,
Salomon, Gavriel, , , , ,
SAT scores, ,
Savior syndrome, –44,
Scardamalia, M.,
Schemer, Israel,
Schneps, Matthew,
Schoenfeld, Alan, ,
School choice,
School governance,
School restructuring, , –71
Schools of Thought (Brown),
Schwartz, Judah,
Schwartz, Steve,
Shapiro, Irwin,
Shulman, Lee, ,
Simmons, Rebecca,
Simon, Herbert,
Singapore,
Single-choice economy, –61, ,
Situated learning, –69,
Sizer, Theodore, , , , , –68, ,
Skinner, B. F., –59,
Slow learners, ,
Smart Schools, Smart Kids (Fiske),
Smith, Edward,
Social distribution of intelligence, –42, ,
Social studies, generative topics in, –94
Socrates, ,
Socratic teaching, , , , , , , , , ,
described, –58
Sole-source economy, –61, ,
Specialization, –40
Stager, Robert,
_Stand and Deliver_ (film),
Stevenson, Harold, , , ,
Stigler, James, , , ,
Stories, –88, –12,
Story problems in math, , , , ,
Strategic learners,
Strike, K. A.,
Sufi teaching tales, , , ,
Swartz, Robert, ,
Switzerland,
Symbolic distribution of intelligence, –44, ,
Tacit learners,
Teachers,
Asian example of, –27
change and, , , –27,
cognitive economy and, , –62
education of,
examination systems independent of,
metacurriculum and, –30
pedagogy of understanding and, –98
Theory One and, –52
token investment strategy and, –63,
Teaching for transfer, _see_ Transfer of learning
The Teaching of Thinking (Nickerson, Perkins & Smith),
Ten-minute rule, ,
Textbooks, –8,
Trivial Pursuit theory and, –33
in victory gardens, –96
Theory One, , –72, , , , , , –32; _see also_ Coaching; Didactic teaching; Socratic teaching
behaviorism and, –60
constructivism and, –62, , ,
cooperative learning and, –64, ,
critique levied by, –53
developmental perspective and, –63,
feedback in, , , , , , , ,
information in, , , , ,
motivation in, , , , –66, , , , , ,
multiple intelligences and, –67, ,
peer collaboration and, –64, ,
situated learning and, –69,
thoughtful practice in, , , , , , ,
victory gardens and, , , , , , , ,
ThinkerTools, , ,
Thinking dispositions,
Thorndike, E. L.,
Thoughtful classroom, ,
Thoughtful learning, , , –8,
Thoughtful practice, , , , , , ,
Time on task,
Tishman, Shari, ,
Toch, Robert, –15
Token investment strategy, –63,
Transfer of learning, , –27, , , , , ,
Bo Peep theory of, –23,
Good Shepherd theory of, , –26
high road, –25,
Lost Sheep theory of, , –25
low road, –25
Trivial Pursuit theory, , , , –51, , , , –86
explained, –34
Tucker, Marc, , ,
Tutoring, –89,
Two sigma problem, –88
Understanding of knowledge, _see_ Knowledge: active use, retention, understanding of Understanding performances, –79, , , , , , , , ,
application in, ,
comparison and contrast in, ,
contextualization in, ,
defined,
exemplification in, ,
explanation in, , , ,
generalization in, ,
justification in, , ,
mental images and, –83
role of, –76
Unger, Christopher, ,
_The Unschooled Mind_ (Gardner),
Victory gardens, , –203, ,
biology in, –91
computer programming in, –98
Escalante's work in, , –201, –5, ,
expert tutoring in, –89,
history in, –94
textbooks in, –96
Walberg, H. J.,
Wales, Charles,
Webbing,
West Germany,
West Virginia University,
_What Happened at Lexington Green?_ (instructional unit), –92
Whimbey, Arthur,
White, Barbara, –90
Whole language movement, ,
Wolf, Dennis,
Wolfe, Richard,
Word processors, ,
Writing, poor thinking and,
Writing Partner,
Yerushalmy, Michal,
|
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Il trofeo Auld Alliance (in francese Trophée Auld Alliance; in inglese Auld Alliance Trophy) è un premio internazionale di rugby a 15 annualmente in palio tra le squadre nazionali maschili di e in occasione del loro incontro in calendario nel Sei Nazioni.
Istituito nel 2018, vide la sua prima edizione disputarsi a Edimburgo con la vittoria da parte della .
Storia
L'idea di istituire un riconoscimento nacque nel 2017 — a un anno di distanza dalla ricorrenza del centenario della fine della prima guerra mondiale (1918) — da parte di quattro appassionati francesi di rugby, Patrick Caublot, Gilles Mairesse, Christian Raoult e Patrick Sandou che, in occasione dell'incontro del Sei Nazioni contro la allo Stade de France, organizzarono una manifestazione in onore di Eric Millroy, tenente dell'esercito britannico che al momento dell'arruolamento in guerra era capitano della nazionale scozzese e che morì nel 1916 in Francia nella battaglia del bosco di Delville.
Caublot, membro del club rugbistico cittadino di Amiens, nella Somme dove la battaglia ebbe luogo, riuscì a rintracciare gli eredi superstiti di Millroy e a ottenere la loro partecipazione alla celebrazione; tra di essi il pronipote, Douglas Kinloch Anderson, si fece promotore presso la federazione scozzese della stessa iniziativa che Caublot intraprese nei confronti di quella francese, ovvero l'istituzione di un premio permanente da disputarsi in occasione degli incontri del Sei Nazioni tra le selezioni nazionali di e .
Il loro tentativo ebbe successo e le due federazioni si accordarono, di conseguenza, per istituire un premio che, oltre al citato Millroy, onorasse anche la memoria di Marcel Burgun, capitano della nel Cinque Nazioni 1914 e morto in battaglia nel 1915 e, più in generale, tutti i rugbisti di ambo le nazioni morti durante la Grande Guerra, 30 dei quali internazionali per la Scozia e 22 per la Francia.
Il trofeo fu presentato ufficialmente alla stampa allo stadio di Murrayfield, a Edimburgo, il 9 febbraio 2018 dai presidenti delle due federazioni, Bernard Laporte per la e Rob Flockhart per la .
Il suo nome, Auld Alliance Trophy, è un riferimento alla Auld Alliance (scozzese per Vecchia alleanza), intesa stretta nel 1295 tra i regni di Francia e di Scozia in funzione anti-inglese e che Charles de Gaulle definì nel 1942 «la più antica alleanza del mondo».
Il trofeo, una coppa d'argento alta 60 centimetri, è realizzato dagli orafi e argentieri londinesi Thomas Lyte & Co; alla sommità presenta dei fiori incisi, tra i quali quelli di cardo, uno dei simboli nazionali della Scozia.
Nella parte centrale riporta, sia in francese che in inglese (su facce opposte), la dedica «In memoria di Eric Millroy, Marcel Burgun e di tutti i rugbisti francesi e scozzesi caduti nella prima guerra mondiale».
Il basamento è destinato altresì a riportare, incisi anno per anno, i nomi delle squadre che l'hanno vinto.
Due giorni dopo la presentazione alla stampa, domenica 11 febbraio, il trofeo fu presentato al pubblico di Murrayfield prima dell'incontro del Sei Nazioni 2018 tra Scozia e Francia alla presenza dei due discendenti più giovani di Millroy e Burgun, rispettivamente lo scozzese Lachlan Ross e il francese Romain Cabanis, entrambi undicenni.
Il primo vincitore del trofeo fu la Scozia che batté la Francia per 32-26.
In caso di parità il trofeo rimane detenuto presso l'ultimo vincitore.
Palmarès
Note
Voci correlate
Nazionale di rugby a 15 della Francia
Nazionale di rugby a 15 della Scozia
Battaglia del Bosco di Delville
Auld Alliance
Altri progetti
Collegamenti esterni
Auld Alliance
Rugby a 15 in Francia
Rugby a 15 in Scozia
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Q: Pandas convert mixed type column values to numeric where possible I have a dataframe which looks like this:
disp_name measured_value measure_time
temperature 99.3 05/06/2020 13:32:40
pressure 2 05/06/2020 13:32:40
colour orange 05/06/2020 13:32:40
name measure_name 05/06/2020 13:32:40
Currently all the variables in the 'measure_value' column are strings. I want to keep the format of the dataframe as it is but convert all the strings to their correct types, i.e. 99.3 should be a float and 2 should an int. 'orange' and 'measure_name' can remain as str.
The dataframe is quite large and the next step is to loop through with groupby('measure_time'), so I don't want to make separate columns in the table for each variable.
I have tried using pandas to_numeric and astype, but these don't seem to be able to handle mixed type columns.
Is there a way to achieve what I want without writing some complex loop structure which involves copying the whole DataFrame?
A: use object type and convert cells values individually:
# setup:
df = pd.DataFrame({'a':['1','two','3', '4', 'five']})
# convert only values that start with number (enter your own criteria if needed)
df.loc[df.a.str.slice(0,1).isin(list('0123456789')), 'a'] = \
df.loc[df.a.str.slice(0,1).isin(list('0123456789')), 'a'].astype(int)
# test
for i in range (df.shape[0]):
print (df.a[i], type(df.a[i]))
1 <class 'int'>
two <class 'str'>
3 <class 'int'>
4 <class 'int'>
five <class 'str'>
print(df.info())
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 5 entries, 0 to 4
Data columns (total 1 columns):
# Column Non-Null Count Dtype
--- ------ -------------- -----
0 a 5 non-null object
dtypes: object(1)
memory usage: 168.0+ bytes
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{
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Nova Southeastern University's College of Engineering and Computing hosts several events throughout the year to provide prospective students the opportunity to explore our degree program offerings, learn about admissions, financial aid, and more.
Whether you're looking to increase your earning potential or advance your career, these events will provide the opportunity to connect with faculty members, students, alumni, and admission counselors. Events are held at our main campus in Fort Lauderdale-Davie, online, and in various locations.
Learn more about our Master of Science and Doctoral degree programs during one of our interactive virtual sessions. Sign up today!
Thursday, April 11, 2019 at 2:00 p.m.
Can't make any of the above events? Please contact us for a one-on-one session online or on-campus!
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Today's techniques for treating chondral or osteochondral defects commonly include microfracture and transplantation procedures. These techniques typically provide temporary relief, however, concerns remain about limited graft availability, morbidity at the autograft donor site, graft hypertrophy, the need for multiple surgical interventions and excessive cost. These current techniques, though beneficial in temporarily reducing pain, cannot be considered long-term solutions.
DSM is developing technologies to address these deficiencies. Significant preclinical research and development using our proprietary biomaterial technologies has yielded promising designs. Preclinical studies have been conducted with some of the premier academic institutes in the US.
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{"url":"http:\/\/server1.wikisky.org\/starview?object_type=2&object_id=1395&object_name=NGC+5783&locale=HE","text":"WIKISKY.ORG\n\n \u05d1\u05d9\u05ea \u05d4\u05ea\u05d7\u05dc\u00a0\u05de\u05db\u05d0\u05df To Survive in the Universe News@Sky \u05ea\u05de\u05d5\u05e0\u05ea\u00a0\u05d0\u05e1\u05d8\u05e8\u05d5 \u05d4\u05d0\u05d5\u05e1\u05e3 \u05e7\u05d1\u05d5\u05e6\u05ea\u00a0\u05d3\u05d9\u05d5\u05df Blog\u00a0New! \u05e9\u05d0\u05dc\u05d5\u05ea\u00a0\u05e0\u05e4\u05d5\u05e6\u05d5\u05ea \u05e2\u05d9\u05ea\u05d5\u05e0\u05d5\u05ea \u05db\u05e0\u05d9\u05e1\u05d4\n\n# NGC 5783\n\n\u05ea\u05d5\u05db\u05df\n\n### \u05ea\u05de\u05d5\u05e0\u05d5\u05ea\n\n\u05d4\u05d5\u05e1\u05e3 \u05ea\u05de\u05d5\u05e0\u05d4 \u05e9\u05dc\u05da\n\nDSS Images \u00a0 Other Images\n\n### \u05de\u05d0\u05de\u05e8\u05d9\u05dd \u05e7\u05e9\u05d5\u05e8\u05d9\u05dd\n\n Scale Heights of Non-Edge-on Spiral GalaxiesWe present a method of calculating the scale height of non-edge-onspiral galaxies, together with a formula for errors. The method is basedon solving Poisson's equation for a logarithmic disturbance of matterdensity in spiral galaxies. We show that the spiral arms can not extendto inside the forbidden radius'' r0, due to the effect ofthe finite thickness of the disk. The method is tested by re-calculatingthe scale heights of 71 northern spiral galaxies previously calculatedby Ma, Peng & Gu. Our results differ from theirs by less than 9%. Wealso present the scale heights of a further 23 non-edge-on spiralgalaxies. Properties of isolated disk galaxiesWe present a new sample of northern isolated galaxies, which are definedby the physical criterion that they were not affected by other galaxiesin their evolution during the last few Gyr. To find them we used thelogarithmic ratio, f, between inner and tidal forces acting upon thecandidate galaxy by a possible perturber. The analysis of thedistribution of the f-values for the galaxies in the Coma cluster leadus to adopt the criterion f \u2264 -4.5 for isolated galaxies. Thecandidates were chosen from the CfA catalog of galaxies within thevolume defined by cz \u22645000 km s-1, galactic latitudehigher than 40o and declination \u2265-2.5o. Theselection of the sample, based on redshift values (when available),magnitudes and sizes of the candidate galaxies and possible perturberspresent in the same field is discussed. The final list of selectedisolated galaxies includes 203 objects from the initial 1706. The listcontains only truly isolated galaxies in the sense defined, but it is byno means complete, since all the galaxies with possible companions underthe f-criterion but with unknown redshift were discarded. We alsoselected a sample of perturbed galaxies comprised of all the diskgalaxies from the initial list with companions (with known redshift)satisfying f \u2265 -2 and \\Delta(cz) \u2264500 km s-1; a totalof 130 objects. The statistical comparison of both samples showssignificant differences in morphology, sizes, masses, luminosities andcolor indices. Confirming previous results, we found that late spiral,Sc-type galaxies are, in particular, more frequent among isolatedgalaxies, whereas Lenticular galaxies are more abundant among perturbedgalaxies. Isolated systems appear to be smaller, less luminous and bluerthan interacting objects. We also found that bars are twice as frequentamong perturbed galaxies compared to isolated galaxies, in particularfor early Spirals and Lenticulars. The perturbed galaxies have higherLFIR\/LB and Mmol\/LB ratios,but the atomic gas content is similar for the two samples. The analysisof the luminosity-size and mass-luminosity relations shows similartrends for both families, the main difference being the almost totalabsence of big, bright and massive galaxies among the family of isolatedsystems, together with the almost total absence of small, faint and lowmass galaxies among the perturbed systems. All these aspects indicatethat the evolution induced by interactions with neighbors would proceedfrom late, small, faint and low mass Spirals to earlier, bigger, moreluminous and more massive spiral and lenticular galaxies, producing atthe same time a larger fraction of barred galaxies but preserving thesame relations between global parameters. The properties we found forour sample of isolated galaxies appear similar to those of high redshiftgalaxies, suggesting that the present-day isolated galaxies could bequietly evolved, unused building blocks surviving in low densityenvironments.Tables \\ref{t1} and \\ref{t2} are only available in electronic form athttp:\/\/www.edpsciences.org Arcsecond Positions of UGC GalaxiesWe present accurate B1950 and J2000 positions for all confirmed galaxiesin the Uppsala General Catalog (UGC). The positions were measuredvisually from Digitized Sky Survey images with rms uncertainties\u03c3<=[(1.2\")2+(\u03b8\/100)2]1\/2,where \u03b8 is the major-axis diameter. We compared each galaxymeasured with the original UGC description to ensure high reliability.The full position list is available in the electronic version only. A catalogue of spatially resolved kinematics of galaxies: BibliographyWe present a catalogue of galaxies for which spatially resolved data ontheir internal kinematics have been published; there is no a priorirestriction regarding their morphological type. The catalogue lists thereferences to the articles where the data are published, as well as acoded description of these data: observed emission or absorption lines,velocity or velocity dispersion, radial profile or 2D field, positionangle. Tables 1, 2, and 3 are proposed in electronic form only, and areavailable from the CDS, via anonymous ftp to cdsarc.u-strasbg.fr (to130.79.128.5) or via http:\/\/cdsweb.u-strasbg.fr\/Abstract.html Short 21-cm WSRT observations of spiral and irregular galaxies. HI properties.We present the analysis of neutral hydrogen properties of 108 galaxies,based on short 21-cm observations with the Westerbork Synthesis RadioTelescope (WSRT). The results of two HI surveys are analysed toinvestigate the existence of relations between optical and HIproperties, like diameters, hydrogen masses and average surfacedensities. For all galaxies in our sample we find that the HI diameter,defined at a surface density level of 1Msun_\/pc^2^, is largerthan the optical diameter, defined at the 25^th^mag\/arcsec^2^ isophotallevel. The Hi-to-optical-diameter ratio does not depend on morphologicaltype or luminosity. The strongest, physically meaningful, correlationfor the sample of 108 galaxies is the one between logM_HI_ and logD_HI_,with a slope of 2. This implies that the HI surface density averagedover the whole HI disc is constant from galaxy to galaxy, independent ofluminosity or type. The radial HI surface density profiles are studiedusing the technique of principal component analysis. We find that about81% of the variation in the density profiles of galaxies can beexplained by two dimensions. The most dominant component can be relatedto \"scale\" and the second principal component accounts for the variancein the behaviour of the radial profile in the central parts of galaxies(i.e. \"peak or depression\") . The third component accounts for 7% of thevariation and is most likely responsible for bumps and wiggles in theobserved density profiles. Short WSRT HI observations of spiral galaxies.We have obtained short HI observations of 60 late type spiral galaxieswith the Westerbork Synthesis Radio Telescope (WSRT). Several HIproperties are presented, including the radial surface densitydistribution of HI and a position-velocity map. When possible these arecompared to those measured from single-dish observations. We confirmearlier results that there is no serious systematic difference betweenthe WSRT and single-dish observations in total flux and linewidths. The far-infrared properties of the CfA galaxy sample. I - The catalogIRAS flux densities are presented for all galaxies in the Center forAstrophysics magnitude-limited sample (mB not greater than 14.5)detected in the IRAS Faint Source Survey (FSS), a total of 1544galaxies. The detection rate in the FSS is slightly larger than in thePSC for the long-wavelength 60- and 100-micron bands, but improves by afactor of about 3 or more for the short wavelength 12- and 25-micronbands. This optically selected sample consists of galaxies which are, onaverage, much less IR-active than galaxies in IR-selected samples. Itpossesses accurate and complete redshift, morphological, and magnitudeinformation, along with observations at other wavelengths. Galaxy alignmentsLarge areas of the sky around the brightest apparent magnitude galaxieshave been examined. In almost every case where they are not crowded byother right galaxies, clearly marked lines of higher red shift galaxieshave been going through, or originating from, the positions of thesebright apparent magnitude galaxies. It is shown that galaxies of about3000 to 5000 km\/s red shift define narrow filaments of from 10 to 50 degin length. It is found that galaxies of very bright apparent magnitudetend to occur at the center or ends of these alignments. The 20brightest galaxies in apparent magnitude north of delta = 0 deg areinvestigated here. Of the 14 which are uncrowded by nearby brightgalaxies, a total of 13 have well marked-lines and concentrations offainter, higher red shift galaxies. Star formation in spiral galaxies. I - H-alpha observationsThis paper presents large-aperture photometric measurements of H-alpha +forbidden N II emission line strengths in 110 spiral galaxies. Thesegalaxies represent three different samples of objects: (1) low surfacebrightness spiral galaxies; (2) galaxies of extreme arm classifications(flocculent versus grand design); and (3) galaxies with measured (B -H)colors which have been used to study the color-absolute magnituderelation for spiral galaxies. Details of the observations are given, anda comparison is made with previous work. Future papers will use thisdata to study the star-formation rates in the various samples. A deep redshift survey of IRAS galaxies towards the Bootes voidRedshifts were measured for a complete sample of galaxies detected bythe IRAS within 11.5 deg of the center of the void in Bootes discoveredby Kirshner et al (1981). There are 12 IRAS galaxies within the void asdefined by the above authors, seven of which were discovered in thissurvey. One of these has a companion at the same redshift. The resultingdensity of IRAS galaxies in the void is measured to be between 1\/6 and1\/3 of the average density; the uncertainty is dominated by Poissonstatistics. Good agreement is found between the selection function andnumber density derived from the present sample and those derived fromthe all-sky sample of Strauss (1989). The optical spectra of the newlyfound galaxies in the void are typical of IRAS galaxies in the field. The peculiar velocity of the Local Group. II - H I observations of SC galaxiesH I observations of a sample of 163 Sc galaxies have been obtained usingthe Mk IA and Mk II Jodrell Bank radio telescopes. In the presentanalysis, the overall rms error in redshift determination is 5 km\/s andthe rms error in velocity width determination is 10 km\/s. The resultssuggest that Sc galaxies have high internal obscuration and may beoptically thicker in blue light than earlier-type spirals. An orthogonalthree-dimensional classification system based on three uncorrelatedparameters related to linear diameter, quiescent star-formation rate,and embedded starburst-type activity is shown to account for the globalproperties of Sc galaxies with an accuracy close to the limit ofmeasurement error. Uncertainties in 21 centimeter redshifts. I - DataHigh-precision data on the 21-cm redshifts, profile widths, and shapesfor 625 galaxies are presented. Each galaxy is listed in across-identification and morphology table. High-resolution spectra arealso given for each galaxy. Internal redshift consistency is roughly 1km\/s for galaxies for which the S\/N is above 15. No systematic effectshave been found which might influence the observed redshift quantizationat 72.5 km\/s or its submultiples. Arm classifications for spiral galaxiesThe spiral arm classes of 762 galaxies are tabulated; 636 galaxies withlow inclinations and radii larger than 1 arcmin were classified on thebasis of their blue images on the Palomar Observatory Sky Survey (POSS),76 SA galaxies in the group catalog of Geller and Huchra were alsoclassified from the POSS, and 253 galaxies in high-resolution atlaseswere classified from their atlas photographs. This spiral armclassification system was previously shown to correlate with thepresence of density waves, and galaxies with such waves were shown tooccur primarily in the densest galactic groups. The present sampleindicates, in addition, that grand design galaxies (i.e., those whichtend to contain prominent density wave modes) are physically larger thanflocculent galaxies (which do not contain such prominent modes) by afactor of about 1.5. A larger group sample confirms the previous resultthat grand design galaxies are preferentially in dense groups. The extragalactic distance scale derived from 'sosie' galaxies. I - Distances of 167 galaxies which are sosies of 14 nearby galaxiesThe method of 'sosie' galaxies is applied to a large sample of galaxiesextracted from the BGP catalog of H I line data and the Second ReferenceCatalog of Bright Galaxies. The sosies of 14 calibrating galaxies(primary calibrators and galaxies in the nearest groups) are defined asthose having the same parameters, either (1) morphological type T, axisratio R, and maximum rotation velocity VM or (2) T, R, andluminosity index lambdac. Distance moduli directly derivedfrom apparent magnitudes and\/or diameters are provided on the distancescale whose zero point is defined by the adopted distance moduli of thecalibrators. The external mean error (0.4 mag) is competitive with thebest currently available. Morphology of spiral galaxies. I - General propertiesRed Palomar Sky Survey plates are scanned to characterize a completesample of 605 spiral galaxies north of declination -33 deg havinginclination angle less than 56 deg and blue diameter 2-15 arcmin. Theselection of the data and the reduction and parameter-extractionprocedures are explained, and the data and the results of statisticalanalysis are presented in tables and graphs. Findings reported include alow frequency of occurrence for small inclination angles (suggestingdistortion of outer structures), similar distributions of central diskbrightness for types Sa-Sc but not for types Sd-Sm (where mean valuesare smaller), fewer late-type galaxies with large exponential-disk scalelengths, no galaxies with both high central brightness and large scalelength (indicating a limit on angular momentum in galaxy formation), anda correlation between mean surface brightness and absolute magnitude forlater-type galaxies but not for types Sa-Scd. H I line studies of galaxies. IV - Distance moduli of 468 disk galaxiesAbstract image available at:http:\/\/adsabs.harvard.edu\/cgi-bin\/nph-bib_query?1985A&AS...59...43B&db_key=AST A survey of galaxy redshifts. IV - The dataThe complete list of the best available radial velocities for the 2401galaxies in the merged Zwicky-Nilson catalog brighter than 14.5mz and with b (II) above +40 deg or below -30 deg ispresented. Almost 60 percent of the redshifts are from the CfA surveyand are accurate to typically 35 km\/s. Flocculent and grand design spiral structure in field, binary and group galaxiesA 12-division morphological system emphasizing arm continuity, lengthand symmetry has been developed for the classification of all spiralgalaxies according to the regularity of their spiral arm structure. Armclassifications were tabulated for 305 barred and nonbarred spiralgalaxies; of these, 79 are isolated, 52 are binary and 174 are ingroups. Among the isolated SA galaxies, 68 + or - 10% have irregular andfragmented, or 'flocculent', spiral structures. Only 32 + or - 10% havesymmetric spiral arms in the classic grand design pattern. Flocculentspirals are the most common structures of galaxies without companions orbars. Since flocculent galaxies may have bars and companions, and granddesign galaxies may have neither bars nor companions, such perturbationsare neither perfectly effective nor always necessary in the driving ofgrand design patterns.\n\u05d4\u05db\u05e0\u05e1 \u05de\u05d0\u05de\u05e8 \u05d7\u05d3\u05e9\n\n### \u05dc\u05d9\u05e0\u05e7\u05d9\u05dd \u05e7\u05e9\u05d5\u05e8\u05d9\u05dd\n\n\u2022 - \u05dc\u05d0 \u05e0\u05de\u05e6\u05d0\u05d5 \u05dc\u05d9\u05e0\u05e7\u05d9\u05dd -\n\u05d4\u05db\u05e0\u05e1 \u05dc\u05d9\u05e0\u05e7 \u05d7\u05d3\u05e9\n\n### \u05de\u05e9\u05de\u05e9 \u05e9\u05dc \u05d4\u05e7\u05d1\u05d5\u05e6\u05d4 \u05d4\u05d1\u05d0\u05d4\n\n#### \u05ea\u05e6\u05e4\u05d9\u05ea \u05d5\u05de\u05d9\u05d3\u05e2 \u05d0\u05e1\u05d8\u05e8\u05d5\u05de\u05d8\u05e8\u05d9\n\n \u05e7\u05d1\u05d5\u05e6\u05ea-\u05db\u05d5\u05db\u05d1\u05d9\u05dd: \u05e9\u05d5\u05de\u05e8 \u05d4\u05d3\u05d5\u05d1\u05d9\u05dd \u05d4\u05ea\u05e8\u05d5\u05de\u05de\u05d5\u05ea \u05d9\u05de\u05e0\u05d9\u05ea: 14h53m28.40s \u05e1\u05d9\u05e8\u05d5\u05d1: +52\u00c2\u00b004'33.0\" \u05d2\u05d5\u05d3\u05dc \u05d2\u05dc\u05d5\u05d9: 2.692\u2032\u00a0\u00d7\u00a01.622\u2032\n\n\u05e7\u05d8\u05dc\u05d5\u05d2\u05d9\u05dd \u05d5\u05db\u05d9\u05e0\u05d5\u05d9\u05dd:\n \u05e9\u05dd \u05e2\u05e6\u05dd \u05e4\u05e8\u05d8\u05d9 (Edit) NGC 2000.0 NGC 5783 HYPERLEDA-I PGC 53217 \u2192 \u05d4\u05d6\u05de\u05df \u05e2\u05d5\u05d3 \u05e7\u05d8\u05dc\u05d5\u05d2\u05d9\u05dd \u05d5\u05db\u05d9\u05e0\u05d5\u05d9\u05dd \u05de\u05d5\u05d6\u05d9\u05e8","date":"2013-05-23 21:03:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6399843096733093, \"perplexity\": 6521.700661735454}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368703788336\/warc\/CC-MAIN-20130516112948-00014-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
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Bill Kennemer (born 1946) is an American clinical psychologist and Republican politician who represented the 20th district in the Oregon State Senate from 2021 to 2023. Kennemer previously represented Oregon's 39th House district in the Oregon House of Representatives from 2009 to 2019, and Oregon's 12th Senate district from 1987 to 1997.
Early life and education
Kennemer was born in Sacramento, California. He received a Bachelor of Arts degree from Warner Pacific College in 1968 and a PhD from Fuller Graduate School of Psychology in 1975.
Career
Kennemer was a clinical psychologist in private practice for nearly 25 years as well as a professor of psychology. He has also worked as a truck driver and farm hand.
Political career
Kennemer cites his commitment to public service as stemming from an incident in 1952 when, after his family home was destroyed by a fire, the community came together to rebuild it. He was an Oregon State Senator from 1987 to 1996, where he served as Assistant Senate Minority Leader,
and Chair of the Senate Business, Housing and Finance Committee. He was in the BiPartisan Tourism Caucus, and the Fish and Wildlife Caucus, and was a member of the Association of Oregon Counties Legislative Committee.
He was also a member of the Education Commission of the States and the Clackamas County Economic Development Commission.
Upon leaving the Senate, he served as a Clackamas County Commissioner from 1997 to 2008, five times as chair.
In 2008, he narrowly won the closest legislative race in Oregon against first-time candidate Democrat Toby Forsberg for the Oregon House of Representatives seat held by former Minority Leader Wayne Scott. He won four more two-year terms before opting not to seek re-election in 2018.
In 2021, Kennemer was appointed to complete the Senate term of Alan Olsen, who resigned with two years remaining in his term.
Awards
As Clackamas County Commissioner in, he received the Association of Oregon Counties Board of Directors' Outstanding Service Award for 1998. Later, while a state representative, his support and advocacy in animal-related measures saw him labeled as a 2011 "Top Dog" by the Oregon Humane Society.
Memberships and committees
Oregon Trail Foundation, Founding Member
Providence Milwaukie Hospital Foundation Board
North Clackamas Chamber Board of Directors
Warner Pacific College Board of Trustees
Personal life
Having lost his previous wife to cancer, Kennemer is now married to Cherie McGinnis. They share four children and nine grandchildren.
Electoral history
2022
References
External links
Clackamas County Board of Commissioners
Project VoteSmart biography
1946 births
21st-century American politicians
County commissioners in Oregon
Living people
Republican Party members of the Oregon House of Representatives
Republican Party Oregon state senators
Politicians from Oregon City, Oregon
Politicians from Sacramento, California
Warner Pacific University alumni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: Integrating Beyond Compare 3 with base clearcase version 8 on Windows 7 64 bit? Has anyone succeeded in this integration? Previously I integrated fine with Clear Case version 6 on Windows xp.
Now with Clearcase 8 (I am using dynamic views) I have edited the map file as per instructions (and as I did with old clearcase) and restarted clearcase .. but any compare just pulls up the old clearcase diff tool, never BC
A: This is finally working fine. I added beyond compare 3 - and without quotes - and also the full path. Here is an example of one of the lines
text_file_delta compare C:\Program Files (x86)\Beyond Compare 3\BComp.exe
I notice that relative path also works, so maybe it was just having quotes around the path
text_file_delta compare ..\..\..\..\..\Beyond Compare 3\BComp.exe
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,698
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namespace Org.Apache.REEF.Wake.Time.Event
{
/// <summary>
/// Represents the Time at which a component started.
/// </summary>
public class StartTime : Time
{
public StartTime(long timeStamp) : base(timeStamp)
{
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,887
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\section{Introduction}
\subsection{Gromov's surface subgroup question}
The following well-known question is usually attributed to Gromov:
\begin{question}[Gromov]\label{question:Gromov_question}
Let $G$ be a one-ended hyperbolic group. Does $G$ contain the fundamental group of a closed
surface with $\chi<0$?
\end{question}
Hereafter we abbreviate ``fundamental group of a closed surface with $\chi<0$'' to ``surface group'',
so that this question asks whether every one-ended hyperbolic group contains a surface subgroup.
This question is wide open in general, but a positive answer is known in certain special cases,
including:
\begin{enumerate}
\item{Coxeter groups (Gordon--Long--Reid \cite{Gordon_Long_Reid});}
\item{Graphs of free groups with cyclic edge groups and $b_2>0$ (Calegari \cite{Calegari_graph});}
\item{Fundamental groups of hyperbolic $3$-manifolds (Kahn--Markovic \cite{Kahn_Markovic});}
\item{Certain doubles of free groups (Gordon--Wilton, Kim--Wilton, Kim--Oum
\cite{Gordon_Wilton, Kim_Wilton, Kim_Oum});}
\end{enumerate}
(this list is not exhaustive).
The main goal of this paper is to describe how linear programming may be used to settle the
question of the existence of surface subgroups in certain graphs of free groups, either by giving
a powerful computational tool to {\em find} surface subgroups in specific groups, or by reducing
the analysis of this question in infinite families of groups to a finite (tractable) calculation.
There are many reasons why the case of graphs of free groups is critical for Gromov's question, but
we do not go into this here, taking the interest of Gromov's question in this subclass of groups
to be self-evident.
\subsection{Statement of results}
We are able to prove the existence of surface subgroups in the following groups:
\begin{enumerate}
\item{A group $G$ with $b_2>0$ obtained by doubling a free group $F$ along a finite collection
of finitely generated subgroups $F_i$;}
\item{A group $G$ obtained as an HNN extension $F *_\phi$ where $F$ is a free group of fixed rank
and $\phi$ is a {\em random} endomorphism;}
\item{``Sapir's group'' $C=F*_\phi$ for $F=\langle a,b\rangle$ and $\phi:a \to ab,b \to ba$.}
\end{enumerate}
The sense in which this constitutes a significant advance over the results and methods in
\cite{Calegari_graph,Kim_Wilton,Kim_Oum} is that the edge groups
are free groups of {\em arbitrary rank}, whereas in the cited papers the edge groups
were required to be cyclic.
Bullet (1) above is implied by a stronger result about the Gromov norm on $H_2$ of the double of
$F$ along the $F_i$, which we discuss in \S~\ref{subsection:Gromov}.
Bullet (3) is reasonably self-explanatory. A precise statement of bullet (2) is:
\begin{random_folded_theorem}
Let $k\ge 2$ be fixed, and let $F$ be a free group of rank $k$.
Let $\phi$ be a random endomorphism of $F$ of length $n$. Then
the probability that $F*_\phi$ contains an essential surface subgroup is at least $1-O(C^{-n^c})$
for some $C>1$ and $c>0$.
\end{random_folded_theorem}
Here a random endomorphism of length $n$ is one that takes the generators to reduced words of
length $n$ chosen independently and randomly with the uniform distribution. We became interested
in surface subgroups of HNN extensions of free groups after discussions with Mark Sapir, who
conjectured that the subgroup $C$ does not contain a surface subgroup, and thought it was unlikely
that many HNN extensions should contain surface subgroups (other than $\Z^2$ subgroups for
endomorphisms fixing a nontrivial conjugacy class). See also \cite{Crisp_Sageev_Sapir} and
\cite{Sapir}. Therefore it seems safe to say that the
Random $f$-folded Surface Theorem is in many ways very unexpected.
\subsection{Gromov norm}\label{subsection:Gromov}
If $X$ is a $K(\pi,1)$, the Gromov norm of a class $\alpha \in H_2(X;\Q)$, denoted $\|\alpha\|$
is the infimum of $-2\chi(S)/n$ over all closed oriented surfaces $S$ without sphere components,
and all positive integers $n$, so that there is a map $f:S \to X$ with $f_*[S]=n\alpha$. If $G$
is a group, define the Gromov norm on $H_2(G;\Q)$ by identifying this space with $H_2(X;\Q)$
for $X$ a $K(G,1)$. The function $\|\cdot\|$ extends by continuity to $H_2(X;\R)$, where
(despite its name) it defines a pseudo-norm in general.
There is a relative version of Gromov norm for surfaces with boundary, and classes in
$H_2(X,Y)$ for subspaces $Y \subset X$, and when $H_2(X)=0$ this relative Gromov norm is equivalent
(up to a factor of 4) to the {\em stable commutator length} norm, as defined in \cite{Calegari_scl},
Ch.~2 (also see the start of \S~\ref{section:Gromov_norm}).
There are equivalent definitions for pairs $G,\lbrace G_i\rbrace$
where $G$ is a group and $\lbrace G_i \rbrace$ is a family of conjugacy classes of subgroups of $G$.
In \S~\ref{section:traintrack_rationality}
and \S~\ref{section:Gromov_norm} we develop tools to compute stable commutator length in
free groups relative to families of finitely generated subgroups, and show
(Theorem~\ref{theorem:traintrack})
that the unit balls in the norm are finite sided rational polyhedra. By a doubling argument,
we obtain a similar theorem for Gromov norms of groups obtained from free groups by doubling along
a collection of subgroups:
\begin{double_norm_theorem}
Let $F$ be a finitely generated free group, and let $F_i$ be a finite collection of conjugacy
classes of finitely generated subgroups of $F$. Let $G$ be obtained by doubling $F$ along
the $F_i$. Then the unit ball in the Gromov norm on $H_2(G)$ is a finite sided rational
polyhedron, and each rational class is projectively represented by an extremal surface.
\end{double_norm_theorem}
Since extremal surfaces are necessarily $\pi_1$-injective, this shows that a group $G$ as
in the theorem contains a surface subgroup when $H_2(G)$ is nontrivial.
\subsection{Unity of methods}
The Double Norm Theorem and the Random $f$-folded Surface Theorem are logically independent,
and the certificates for $\pi_1$-injectivity of the surface subgroups they promise are
quite different.
However, the surfaces in either case are constructed combinatorially from pieces obtained by
solving a rational linear programming problem; and the nature of the representation of the
surfaces by vectors, and the tools used to set up the linear programming problems, are very similar.
Thus there is a deeper unity of methods underlying the two theorems, beyond the similarity that
both promise surface subgroups in certain graphs of free groups.
\subsection{Acknowledgments}
We would like to thank Sang-Hyun Kim, Tim Susse and Henry Wilton for helpful conversations about the
material in this paper. Danny Calegari was supported by NSF grant DMS 1005246,
and Alden Walker was supported by NSF grant DMS 1203888.
\section{Traintrack Rationality Theorem}\label{section:traintrack_rationality}
\subsection{Graphs and traintracks}
We recall some standard definitions from the theory of graphs, traintracks and immersions,
and their connection to free groups and morphisms between them. See e.g.\/ \cite{Bestvina_Handel}
for background and more details.
We fix a free group $F$ of finite rank and a free generating set for $F$, and realize $F$
as the fundamental group of a rose $R$, identifying the generators of $F$ with the (oriented)
edges of $R$. If $X$ is a graph, an {\em immersion} $X \to R$ is a locally injective
simplicial map taking edges to edges. Every nontrivial conjugacy class in $F$ is represented
by an immersed loop in $R$, unique up to reparameterization of the domain (which is an oriented
circle).
\begin{definition}
Let $T$ be a graph. A {\em turn} is an ordered pair of distinct oriented edges incident to a vertex of $T$,
the first element incoming and the second outgoing. If $e_1$ is the incoming edge and $e_2$ the
outgoing edge, we denote the turn $e_1 \to e_2$.
\end{definition}
Thus, a turn is the same thing as the germ at a vertex of an oriented immersed path in $T$.
\begin{definition}
A {\em traintrack} is a graph $T$ together with a subset of the turns at each vertex which are
called {\em admissible turns}. If $L$ is an oriented 1-manifold, an immersion $L \to T$ is
{\em admissible} if the germ of $L$ is admissible at every vertex of $T$. A {\em traintrack immersion}
is a simplicial map $T \to R$ taking edges to edges, which is locally injective on each
admissible turn.
\end{definition}
Thus if $L \to T$ is admissible, and $T \to R$ is a traintrack immersion, then $L \to R$ is an immersion.
If $X$ is a graph and we fix a simplicial map $X \to R$, we label the oriented edges of $X$ by the
generators of $F$ corresponding to the edges that they map to.
Any oriented 1-manifold mapping $L \to X$ pulls back these labels
so that each component of $L$ is labeled by a cyclic word in $F$. If $T$ is a traintrack
and $T \to R$ is a traintrack immersion and $L \to T$ is admissible, then the labels on the components
of $L$ are cyclically reduced words.
Conversely, suppose we are given a finite set $\Gamma$ of nontrivial conjugacy classes in $F$. We
let $L$ be an oriented simplicial 1-manifold with one component for each element of $\Gamma$, and each
component labeled by the cyclically reduced word representing the given conjugacy class. There is
a unique immersion $L \to R$ compatible with the labels. We say that $L$ is {\em carried}
by a traintrack immersion $T \to R$ if $L \to R$ factors through an admissible map $L \to T$. See
Figure~\ref{figure1}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$b$} at 4 45
\pinlabel {$c$} at 1 9
\pinlabel {$a$} at 33 2
\pinlabel {$b$} at 40 42
\pinlabel {$A$} at 61 46
\pinlabel {$A$} at 51 18
\pinlabel {$c$} at 67 0
\pinlabel {$c$} at 92 2
\pinlabel {$a$} at 103 25
\pinlabel {$b$} at 91 47
\pinlabel {$b$} at 145 37
\pinlabel {$A$} at 174 26
\pinlabel {$b$} at 214 0
\pinlabel {$a$} at 217 47
\pinlabel {$c$} at 262 26
\pinlabel {$c$} at 293 35
\pinlabel {$a$} at 340 5
\pinlabel {$b$} at 410 11
\pinlabel {$c$} at 372 58
\endlabellist
\centering
\includegraphics[scale=0.85]{figure1}
\caption{The pair of loops $L$ maps to the rose $R$, and this
map factors through an admissible map to the traintrack $T$, so $L$ is
carried by $T$.}
\label{figure1}
\end{figure}
\begin{definition}
If $T$ is a traintrack, a {\em weight} $w$ is an assignment of real numbers to the admissible turns
in such a way that for each oriented edge $e$, the sum of numbers associated to turns involving $e$
at one vertex is equal to the sum at the other.
\end{definition}
The space of weights on $T$, denoted $W(T)$, is a real vector space defined over $\Q$. Weights can be non-negative,
integral, and so on. The space of non-negative weights is a convex rational cone $W^+(T)$.
A carrying map $L \to T$ determines a function from admissible turns to non-negative integers, where
the number assigned to a turn is the number of times that $L$ makes such a turn when it passes through
the given vertex. We denote this function $w(L)$.
\begin{lemma}\label{weight_is_degree}
The set of functions $w(L)$ over all carrying maps $L \to T$ is precisely the set of integer
weights in $W^+(T)$.
\end{lemma}
\begin{proof}
Each edge of $L$ contributes $1/2$ to the value of $w(L)$ on the turns at its vertices, so $w(L) \in W^+(T)$.
Conversely, let $w$ be a non-negative integer weight in $W^+(T)$. For each turn $e \to e'$ with weight
$n$, take $n$ disjoint intervals made by gluing the front half of $e$ to the back half of $e'$, and glue
these oriented intervals together (over all turns) compatibly with how they immerse in $T$ to produce $L$.
The defining property of a weight says that this gluing can be done, and $w(L)=w$.
\end{proof}
Note that $w(L)$ does not determine the topology of $L$ (i.e.\/ the number of components).
But it does determine the image of $L$ in $H_1(F)$ under $L \to R$. Thus we obtain
a homomorphism $h:W(T) \to H_1(F)$, defined over $\Q$, so that the image of $[L]$ in $H_1(R)=H_1(F)$
is $h(w(L))$.
\subsection{Fatgraphs and scl}
For an introduction to fatgraphs, see \cite{Penner}.
\begin{definition}
A {\em fatgraph} is a graph $X$ together with a choice of cyclic ordering of the edges incident to each
vertex. A fatgraph admits a canonical {\em fattening} to a compact oriented surface $S(X)$ in such a way
that $X$ sits inside $S(X)$ as a spine to which $S(X)$ deformation retracts. The boundary $\partial S(X)$
is an oriented 1-manifold, which comes with a canonical map $\partial S(X) \to X$ which is the
restriction of the deformation retraction, and is an immersion unless $X$ has 1-valent vertices.
A {\em fatgraph over $F$} is a fatgraph $X$ together with a simplicial map of the underlying graph $X \to R$.
It is {\em reduced} if the composition $\partial S(X) \to X \to R$ is an immersion. See Figure~\ref{figure2}.
\end{definition}
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 13 58
\pinlabel {$A$} at 23 41
\pinlabel {$c$} at 31 13
\pinlabel {$C$} at 36 -4
\pinlabel {$b$} at 72 -4
\pinlabel {$B$} at 78 16
\pinlabel {$A$} at 84 25
\pinlabel {$a$} at 84 43
\pinlabel {$b$} at 121 24
\pinlabel {$B$} at 122 44
\pinlabel {$c$} at 160 5
\pinlabel {$C$} at 161 23
\pinlabel {$b$} at 198 23
\pinlabel {$B$} at 199 44
\pinlabel {$a$} at 161 63
\pinlabel {$A$} at 160 44
\pinlabel {$a$} at 278 11
\pinlabel {$b$} at 340 9
\pinlabel {$c$} at 308 67
\endlabellist
\centering
\includegraphics[scale=1.0]{figure2}
\caption{This fatgraph map $X \to R$ is an immersion, but the fatgraph is
\emph{not} reduced because the boundary is not reduced.}
\label{figure2}
\end{figure}
If $X \to R$ is a fatgraph over $F$ without 1-valent vertices,
and if the underlying map of graphs $X \to R$ is an immersion,
the fatgraph is reduced. The converse is true if $X$ is 3-valent,
but not in general otherwise. All the fatgraphs we consider in this paper will be immersed.
Moreover, throughout \S~\ref{section:traintrack_rationality} they will also be reduced.
However we need to consider unreduced fatgraphs in \S~\ref{subsection:bounded_folding}.
Now, let $f:L \to R$ be an oriented 1-manifold mapping to $R$ by an immersion; equivalently, $L$ and $f$
are determined by the data of a collection $\Gamma$ of nontrivial conjugacy classes in $F$.
\begin{definition}
An {\em admissible surface} for $f:L \to R$ is a compact oriented surface $S$ together with a map
$g:S \to R$ and an oriented covering map $h:\partial S \to L$ so that $f\circ h = g|\partial S$.
\end{definition}
We denote the degree of the covering map $h:\partial S \to L$ by $n(S)$. We say that ad admissible
surface $S$ is {\em efficient} if no component of $S$ is a sphere, and if every component of $S$
is geometrically incompressible; i.e.\/ if there is no essential {\em embedded} loop in $S$ mapping
to a null-homotopic loop in $R$. Any admissible surface can be replaced by an efficient one, by
throwing away sphere components and repeatedly performing compressions. Note that since by hypothesis
every component of $L$ maps to a nontrivial immersed loop in $R$, no component of $S$ is a disk, and
therefore every component of $S$ has non-positive Euler characteristic.
The following proposition is essentially due to Culler \cite{Culler} (see also \cite{Calegari_scl} \S~4.1)
and lets us reduce the study of admissible surfaces to combinatorics:
\begin{proposition}\label{proposition:surface_is_fatgraph}
Every efficient admissible surface for every oriented $f:L \to R$
is homotopic to a surface obtained by fattening a reduced fatgraph over $F$.
\end{proposition}
\begin{definition}
Let $\Gamma$ be a finite collection of conjugacy classes in $F$ whose sum is homologically trivial
(i.e.\/ represents $0$ in $H_1(F)$). The {\em stable commutator length of $\Gamma$}, denoted $\scl(\Gamma)$,
is defined to be the infimum
$$\scl(\Gamma) = \inf_S -\chi(S)/2n(S)$$
over all efficient admissible surfaces $S$ for $L$, where $f:L \to R$ represents $\Gamma$. A surface is
{\em extremal} for $\Gamma$ if equality is achieved.
\end{definition}
The main theorem of \cite{Calegari_rational} says that extremal surfaces exist for any $\Gamma$. For
more background and an introduction to the theory of stable commutator length, see \cite{Calegari_scl}
or \cite{Bavard}.
\subsection{Polygons}
Let $X$ be a reduced fatgraph over $R$ with fattening $S(X)$ and oriented boundary $\partial S(X)$.
There is a decomposition of $S(X)$ into polygons --- canonical up to isotopy --- where all
vertices of each polygon are vertices on $\partial S(X)$,
with one rectangle for each edge of $X$, and one $n$-gon for each
$n$-valent vertex of $X$. Each $n$-gon with $n\ge 3$ may be further decomposed into $n-2$ triangles,
without introducing new vertices; this decomposition is not canonical unless every vertex of $X$ is
at most 3-valent. Thus, we decompose $S(X)$ into two kinds of polygons: rectangles and triangles.
Note that $\chi(X) = \chi(S(X))=-\tau/2$ where $\tau$ is the number of triangles.
See Figure~\ref{figure3}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\endlabellist
\centering
\includegraphics[scale=1.0]{figure3}
\caption{A fatgraph $S(X)$ (left) can be cut into rectangles and polygons
(center), and the polygons can be further cut into triangles (right).}
\label{figure3}
\end{figure}
The edges of the polygons could be {\em boundary edges}, which are edges of $\partial S(X)$, or
{\em internal edges}, which are determined by ordered pairs of vertices of $\partial S(X)$. A polygon is
determined by the cyclic list of its edges; thus, a rectangle has four edges which alternate between
boundary edges and internal edges, while a triangle has three internal edges. Note that
the edge labels on the two boundary edges of a rectangle have inverse labels. Summarizing: a rectangle
piece is determined by the data of a pair of edges of $\partial S(X)$ with inverse labels, while a
triangle is determined by the data of a cyclically ordered list of three vertices of
$\partial S(X)$. In particular, there are finitely many polygon types (at most cubic in the length
of $X$).
Now suppose that $\partial S(X)$ is carried by some immersed traintrack $T \to R$. Each rectangle
determines a pair of edges of $\partial S(X)$ with inverse labels, which are mapped to a pair
of oriented edges of $T$ with inverse labels. At each vertex, $\partial S(X)$ makes some admissible
turn in $T$; we record the information of these admissible turns at the vertices.
Similarly, each triangle determines a cyclically ordered
list of vertices of $\partial S(X)$ which are mapped to a cyclically ordered list of admissible turns
of $T$.
\begin{definition}
Let $T \to R$ be an immersed traintrack. A {\em triangle} over $T$ is a cyclically
ordered list of three admissible turns.
A {\em rectangle} over $T$ is a cyclically ordered list of 4 admissible turns of the form
$e_1 \to e_2$, $e_2 \to e_3$, $e_4 \to e_5$, $e_5 \to e_6$ where $e_2$ and $e_5$
have inverse labels. See Figure~\ref{figure4}.
\end{definition}
\begin{figure}[ht]
\labellist
\small\hair 2pt
\endlabellist
\centering
\includegraphics[scale=1.0]{figure4}
\caption{If a fatgraph boundary $\partial S(X)$ is carried by an
immersed traintrack $T \to R$, then each vertex of each rectangle and triangle
is associated with an admissible turn in $T$ (blue). As we cut $S(X)$ into
rectangles and triangles, we record these admissible turns for each piece;
using this information, we can reassemble the pieces into a fatgraph
carried by $T$. }
\label{figure4}
\end{figure}
A rectangle over $T$ determines two ordered pairs $(e_2\to e_3,e_4\to e_5)$ and $(e_5\to e_6,e_1\to e_2)$
with notation as above; call these pairs the
{\em internal edges} of the rectangle, while the edges $e_2$ and $e_5$ are the {\em boundary edges}.
Similarly, call the three ordered pairs arising
as the boundary of a triangle over $T$ the {\em internal edges} of the triangle.
\begin{definition}
If $T \to R$ is an immersed traintrack, a {\em polygon weight} is an assignment of real numbers to
triangles and rectangles in such a way that for every unordered pair of admissible turns, the
number of times it appears as an internal edge with one ordering is the same as the number of times
it appear as an internal edge with the other ordering.
\end{definition}
The space of polygon weights on $T$, denoted $P(T)$, is a real vector space defined over $\Q$.
The space of non-negative weights is a convex rational cone $P^+(T)$. By the discussion above,
if $X$ is a reduced fatgraph over $R$ with fattening $S(X)$ and oriented boundary $\partial S(X)$
carried by $T$, then after decomposing $S(X)$ into rectangles and triangles,
we obtain a vector $p(X)$ whose coefficients are the number of each kind of polygon over $T$
(note that $p(X)$ depends not just on $X$ but on the decomposition into triangles, although our
notation obscures this).
\begin{lemma}
Let $X$ be a reduced fatgraph over $R$ with fattening $S(X)$ and oriented boundary $\partial S(X)$
carried by $T$. Then $p(X) \in P^+(T)$.
\end{lemma}
\begin{proof}
This is just the observation that the polygons into which $S(X)$ is decomposed are glued together
in pairs along internal edges.
\end{proof}
\begin{lemma}
There is a rational linear map $\partial:P^+(T) \to W^+(T)$ so that if $X$ is a fatgraph with $\partial S(X)$
carried by $T$, then $\partial p(X) = w(\partial S(X))$.
\end{lemma}
\begin{proof}
The map $\partial$ takes each rectangle to the vector consisting of the 4 admissible turns appearing
as vertices, each with weight $1/2$. Define $\partial$ to be zero on triangles, and
extend by linearity. This map has the desired properties.
\end{proof}
Note that $\partial$ takes integer vectors to integer vectors (though we do not use this fact).
\begin{lemma}
There is a rational linear map $-\chi:P^+(T) \to \R$ so that if $X$ is a fatgraph with $\partial S(X)$
carried by $T$, then $-\chi(p(X)) = -\chi(S(X))$.
\end{lemma}
\begin{proof}
Define $-\chi$ to be $1/2$ on every triangle, and $0$ on rectangles.
\end{proof}
Again, $-\chi$ takes integer vectors to integers.
\begin{proposition}\label{proposition:can_be_reduced}
For every non-negative integer weight $p$ in $P^+(T)$ there is some non-negative integer weight
$p'$ with $\partial p = \partial p'$ and $-\chi(p) \ge -\chi(p')$, and such that $p'=p(X)$
for some fatgraph $X \to R$ with $\partial S(X)$ carried by $T$.
\end{proposition}
\begin{proof}
An integral weight $p$ determines a collection of triangles and rectangles where the weight
of each piece determines the number of copies. Polygons can be glued together along the same
internal edge with opposite orderings; by the definition of a weight, this can be done to produce
a surface $S$ without corners. The surface $S$ might
contain some components without rectangles (i.e.\/ consisting entirely of triangles);
throw these pieces away. The surface $S$ might also contain some subsurface made entirely of
triangles with nontrivial topology. Compress these surfaces down to disks, and triangulate
the result without introducing new vertices on the boundary. The result is a new surface which
by construction is of the form $S(X)$ for some fatgraph $X \to R$. The compression did not
affect boundary edges, so $\partial p = \partial p'$. Moreover, compression can only reduce the
number of triangles used, so $-\chi(p) \ge -\chi(p')$. This completes the proof.
\end{proof}
\subsection{Traintrack Rationality Theorem}
For $w \in W^+(T)$ rational and in the kernel of $h:W^+(T) \to H_1(F)$,
we can define $\scl(w)$ to be the infimum of $\scl(\Gamma)/n$ for all
homologically $\Gamma$ represented by an oriented 1-manifold $L$ carried by $T$ with $w(L)=nw$ for some $n$.
The following Traintrack Rationality Theorem is the main theorem of this section.
\begin{theorem}[Traintrack Rationality Theorem]\label{theorem:traintrack}
Let $T$ be a traintrack immersing to $R$, and let $B^+(T)$ denote the kernel of $h:W^+(T) \to H_1(F)$.
The function $\scl$ extends continuously to $B^+(T)$ in a unique way,
where it is convex and piecewise rational linear. For any rational $w\in B^+(T)$ there is some
homologically trivial $\Gamma$ and a fatgraph $X$ over $F$ with $\partial S(X)$ representing $\Gamma$,
in such a way that $\partial S(X)$ is carried by $T$ with $w(\partial S(X))=nw$ and
$\scl(w)=-\chi(S(X))/2n$.
\end{theorem}
In particular, the surface $S(X)$ is extremal for $\partial S(X)$.
\begin{proof}
Define $Q(w)=P^+(w) \cap \partial^{-1}(w)$; this is a convex linear polyhedron, and is rational
if $w$ is rational. Define
$$\scl(w)=\inf_{q \in Q(w)} -\chi(q)/2$$
This is evidently convex and piecewise rational linear on $B^+(T)$. We show that it agrees with
the definition of $\scl(w)$ already given when $w$ is rational, and that there is an extremal
surface obtained from some fatgraph.
The infimum of $-\chi$ on $Q(w)$ is achieved on some nonempty subpolyhedron $E(w)$, which is convex in general,
and rational if $w$ is rational. A nonempty rational polyhedron contains a rational point, and
every rational $p \in E(w)$ can be rescaled to an integer point $np$, which is in $E(nw)$ by linearity
of the maps and $-\chi$; and by Proposition~\ref{proposition:can_be_reduced}, there is some fatgraph $X$ with
$\partial S(X)$ carried by $T$ and with $w(\partial S(X))=nw$ and $-\chi(S(X))=-\chi(np)$.
Conversely, any efficient admissible surface $S$ with $\partial S$ carried by $T$ and
with $w(\partial S)=mw$ for some $m$ can be obtained as $S=S(X)$ for some reduced
fatgraph $X$ over $R$ by Proposition~\ref{proposition:surface_is_fatgraph}. Then
any $p(X)$ satisfies $\partial p(X) = mw$, so $p(X) \in Q(mw)$. But then
$$-\chi(S(X))/2m = -\chi(p(X))/2m \ge -\chi(E(w))/2$$
Thus $\scl(w)=-\chi(E(w))/2$, and the surface constructed from $p$ above was
extremal, as claimed.
\end{proof}
\begin{example}[Verbal traintracks]
Fix a free group $F$ of rank $k$ and a free generating set, and fix a positive integer $\ell$.
Define a traintrack $T_\ell$ whose oriented edges are the set of reduced words in $F$ of length
$\ell-1$ and whose admissible turns are reduced words of length $\ell$, which we think of as
an ordered pair of oriented edges consisting of the prefix and suffix of the given word of length
$\ell-1$.
Let $W_\ell$ denote the weight space, and $W_\ell^+$ the non-negative weights as above. There is
an involution $\epsilon$ on $W_\ell$, which takes $\sigma$ to $-\sigma^{-1}$, where $\sigma^{-1}$
denotes the inverse word to a reduced word $\sigma$. The natural inclusion $W_\ell^+ \to W_\ell$
induces a {\em surjection} $W_\ell^+ \to W_\ell/\epsilon$, and we obtain a rational linear
(pseudo)-norm on $W_\ell/\epsilon$, where the norm $\|[w]\|$ of an equivalence class $[w]$
is the infimum of the $\scl(w)$ over all $w \in W_\ell^+$ mapping to $w$. The linear functions
on $W_\ell/\epsilon$ are precisely real linear combinations of the {\em homogeneous} (big)
{\em counting quasimorphisms of length at most $\ell$} first introduced by Rhemtulla \cite{Rhemtulla}
and studied later by Brooks \cite{Brooks}, Grigorchuk \cite{Grigorchuk} and others. Thus we may
use $W_\ell/\epsilon$ to get an explicit and complete set of linear relations between the
homogeneous counting quasimorphisms supported on words of any bounded length.
For more details, see \cite{Calegari_Walker_sslpv1}, especially \S~4--5.
\end{example}
\section{Gromov Norm of doubles}\label{section:Gromov_norm}
We briefly introduce the Gromov norm on the homology of a space or group, and its relative variants.
\begin{definition}
Let $X$ be a topological space. The Gromov (pseudo)-norm (also called the $L_1$ norm)
of a homology class $\alpha \in H_i(X;\R)$, denoted $\|\alpha\|$,
is the infimum of $\sum |t_i|$ over all real singular $i$-cycles $\sum t_i\sigma_i$ representing
$\alpha$. Similarly define a norm on relative classes $\alpha \in H_i(X,Y;\R)$ for a subspace
$Y \subset X$ from relative $i$-cycles.
\end{definition}
If $G$ is a group, we can define the Gromov norm on $H_*(G)$ by identifying the group
homology with $H_*(K(G,1))$.
\begin{definition}
If $G_i$ is a family of conjugacy classes of subgroups of $G$,
we can build a space $K$ as the mapping cylinder of $\coprod_i K(G_i,1) \to K(G,1)$, and we define
the Gromov norm on $H_*(G,\lbrace G_i\rbrace)$ by identifying group homology with
$H_*(K,\coprod_i K(G_i,1))$.
\end{definition}
In the 2-dimensional case, one has the following geometric interpretation of the Gromov norm:
\begin{proposition}\label{proposition:norm_is_surfaces}
For $\alpha \in H_2(X;\Q)$ there is a formula
$$\|\alpha\|=\inf_S -2\chi(S)/n(S)$$
where the infimum is taken over closed oriented surfaces $S$ without sphere components
for which there are maps $f:S \to X$ with $f_*[S]=n\alpha$ for some $\alpha$.
Similarly, for $\alpha \in H_2(X,Y;\Q)$ the same formula is true, where now the infimum is
taken over compact oriented surfaces $S$ without sphere or disk components for which
there are maps $f:(S,\partial S) \to (X,Y)$ with $f_*[S]=n\alpha$ for some $\alpha$.
\end{proposition}
For more details, see \cite{Gromov_bounded}; for the connection to $\scl$ in the 2-dimensional case,
see \cite{Calegari_scl}.
The following application makes no mention of traintracks in the statement, and is our main
motivation for pursuing this line of reasoning.
\begin{theorem}[Relative Gromov Norm]\label{theorem:relative_Gromov_norm}
Let $F$ be a finitely generated free group, and let $F_i$ be a finite collection of conjugacy classes of finitely
generated subgroups of $F$. Let $H:=H_2(F,\lbrace F_i\rbrace)$ denote relative 2-dimensional
homology. Then the unit ball in the Gromov norm on $H$ is a finite sided rational polyhedron,
and each rational class is projectively represented by an extremal surface with boundary.
\end{theorem}
\begin{proof}
Let $R$ be a rose for $F$, and for each $i$ let $R_i$ be a graph without 1-valent edges
that immerses in $R$ in such a way that the image of $\pi_1(R_i)$ is conjugate to $F_i$.
Such graphs are obtained by Stallings' method of {\em folding} a set of generators for $F_i$;
see \cite{Stallings}. We let $T$ be the traintrack whose underlying graph is the disjoint union
$\cup_i R_i$, and whose admissible turns are exactly the paths in $R_i$ that do not backtrack.
We can build a space $C$ as the mapping cylinder of the immersions $\cup_i R_i \to R$; thus
$C$ retracts to $R$, and contains $\cup_i R_i$ as a subspace. For each component $T_i$ of $T$
there is a rational linear map $h:W^+(T_i) \to H_1(R_i)$, and all together these give a (surjective)
rational linear map
$$h:W^+(T) \to \oplus H_1(R_i) = \oplus H_1(F_i)$$
Note that $\partial:H_2(F,\lbrace F_i\rbrace) \to \oplus H_1(F_i)$ is injective, and
has image equal to the kernel of $\oplus H_1(F_i) \to H_1(F)$, by the long exact sequence,
and $H_2(F)=0$ for a free group $F$.
Any $(S,\partial S) \to (R,\cup_i R_i)$ can be homotoped and compressed until $\partial S \to \cup_i R_i$
is an immersion, which is to say it is carried by $T$. The surface $S$ can be further compressed
until we can write $S=S(X)$ for some fatgraph $X$ over $R$
compatible with $\partial S(X) \to T \to R$. Conversely, any fatgraph $X$ over $R$
with $\partial S(X)$ carried by $T$ represents a class in $H_2(F,\lbrace F_i\rbrace)$.
We can express this in terms of linear algebra as follows.
If, as before, we denote the kernel of $h:W^+(T) \to H_1(F)$ by $B^+(T)$, and factor $h$ as
$$0 \to B^+(T) \to \oplus H_1(F_i) \to H_1(F)$$
then this sequence is exact; i.e.\/ the first map is injective on $B^+(T)$, and its image
is exactly equal to the kernel of $\oplus H_1(F_i) \to H_1(F)$. Note that this is an exact sequence
of $\R^+$-modules, since $B^+(T)$ is merely a cone, and not a vector space. On the other hand,
since all the terms and maps are defined over $\Q$, the sequence is still exact when restricted to
the rational points in each term. Since $\partial:P^+(X) \to B^+(X)$ is surjective, and
$\partial:H_2(F,\lbrace F_i\rbrace) \to \oplus H_1(F_i)$ is injective with image equal to the
kernel of $\oplus H_1(F_i) \to H_1(F)$, we see that we have shown that $h:P^+(T) \to H_2(F,\lbrace F_i\rbrace)$
is surjective, and for any rational $\alpha \in H_2(F,\lbrace F_i\rbrace)$ we have an equality
$$\|\alpha\|=\inf_{p \in h^{-1}(\alpha)} -2\chi(p)$$
Since $h$ is rational linear, since $P^+(T)$ is a convex rational polyhedral cone, and since
$-\chi$ is rational linear on $P^+$, it follows that the unit ball in the Gromov norm is a finite
sided rational polyhedron. Moreover, if $\alpha$ is rational, the infimum is achieved on some rational
$p$, and by Proposition~\ref{proposition:can_be_reduced} any $p$ achieving the minimum is
projectively equivalent to $p(X)$ for some $X$, in which case $S(X)$ is an extremal surface
projectively representing $\alpha$.
\end{proof}
An absolute version of Theorem~\ref{theorem:relative_Gromov_norm} may be obtained by
{\em doubling}.
\begin{definition}\label{def:doubling}
If $G_i$ is a family of conjugacy classes of subgroups of $G$,
we can build a space $DK$ from two copies of the mapping cylinder $K$ of
$\coprod_i K(G_i,1) \to K(G,1)$, identified along $\coprod_i K(G_i,1)$. The
{\em double} of $G$ along the $G_i$ is the fundamental group of $DK$.
\end{definition}
Note that the double is a graph of groups, whose underlying graph has two vertices
(corresponding to the two copies of $G$ in the double) and with one edge between the two
vertices for each $G_i$.
\begin{theorem}[Gromov Norm of Doubles]\label{theorem:double_Gromov_norm}
Let $F$ be a finitely generated free group, and let $F_i$ be a finite collection of conjugacy
classes of finitely generated subgroups of $F$. Let $G$ be obtained by doubling $F$ along
the $F_i$. Then the unit ball in the Gromov norm on $H_2(G)$ is a finite sided rational
polyhedron, and each rational class is projectively represented by an extremal surface.
\end{theorem}
\begin{proof}
This follows formally from Theorem~\ref{theorem:relative_Gromov_norm}. First of all, at the
level of homology there is a natural isomorphism $H_2(F,\lbrace F_i\rbrace) \to H_2(G)$ obtained
by identifying the $F$ factors on both sides of the double. The point is that this map is
surjective, since the $F$ factors have no absolute $H_2$ of their own (apply Mayer-Vietoris).
Any surface representing a relative class in $H_2(F,\lbrace F_i\rbrace)$ may be doubled to
produce a closed surface representing a corresponding class in $H_2(G)$. Conversely, any surface
representing a class in $H_2(G)$ may be split into two subsurfaces on either side of the double,
each representing the same relative class in $H_2(F,\lbrace F_i\rbrace)$. One of these subsurfaces
has $-\chi$ at most half of $-\chi$ of the big surface; doubling that subsurface produces a new
surface representing the same class in $H_2(G)$ with the same or smaller $-\chi$.
It follows that the doubling isomorphism $H_2(F,\lbrace F_i\rbrace) \to H_2(G)$ just multiplies
the norm of a class by $2$, and the double of any extremal surface for a class in
$H_2(F,\lbrace F_i\rbrace)$ is an extremal surface for the corresponding class in $H_2(G)$.
\end{proof}
Since extremal surfaces are $\pi_1$-injective, we obtain the following corollary:
\begin{corollary}[Surface subgroups in doubles]\label{corollary:surface_subgroup_corollary}
Let $F$ be a finitely generated free group, and let $F_i$ be a finite collection of conjugacy
classes of finitely generated subgroups of $F$. Let $G$ be obtained by doubling $F$ along the
$F_i$. If $H_2(G)$ is nontrivial, then $G$ contains a surface subgroup.
\end{corollary}
For example, if $\sum \rank(F_i)>\rank(F)$ then $H_2(G)$ is nontrivial.
\begin{remark}
Theorem~\ref{theorem:double_Gromov_norm} should be compared to the case that $G=\pi_1(M)$
where $M$ is an irreducible 3-manifold. Then $\|\cdot\|$ is equal to twice the
{\em Thurston norm} on $H_2(M)$, whose unit ball Thurston famously proved is a finite-sided
rational polyhedron \cite{Thurston_norm}. There is a crucial difference between the two Theorems:
in a 3-manifold, every integral $\alpha$ is represented by a norm-minimizing embedded surface
$S$, so that $[S] =\alpha$, and therefore $\|\alpha\| \in 4\Z$, whereas for $G$ as in
Theorem~\ref{theorem:double_Gromov_norm}, the denominator of $\|\alpha\|$ can be arbitrary
for $\alpha \in H_2(G;\Z)$. This is true even when $G$ is obtained by doubling a free group
of rank 2 along a cyclic subgroup; see \cite{CW_endomorphism}.
\end{remark}
\section{Random endomorphisms}
\subsection{HNN extensions}
Let $F$ be a finitely generated free group, and let $\phi:F \to F$ be an injective endomorphism. We obtain
an HNN extension $G:=F*_\phi$. Geometrically we can realize $F=\pi_1(R)$ for some rose $R$ as
above, and $\phi$ by a simplicial map $f:R \to R$, and build a mapping torus $K$ which is
a CW 2-complex, with one 2-cell (a square) for each generator of $F$.
There is a natural presentation
$$G:=\langle F, t \; | \; tFt^{-1} = \phi(F)\rangle$$
and a surjection $G \to \Z$ defined by $t \to 1$ and $F \to 0$. Let $\tilde{K}$ denote the
infinite cyclic cover of $K$ associated to the kernel of this surjection; $\tilde{K}$ is
made from $\Z$ copies of $R\times I$, which we denote $K_i$ for $i \in \Z$. Denote the copy of
$R\times 1$ in $K_i$ by $\partial^+ K_i$ and the copy of $R\times 0$ in $K_i$ by $\partial^- K_i$. Then
$\tilde{K}$ is obtained by gluing each $\partial^+ K_i$ to $\partial^- K_{i+1}$ by a map
$f_i$ (which is just $f$ when we identify both domain and range in a natural way with $R$).
For any positive $n$ we denote the union $K_0 \cup_{f_0} K_1 \cup_{f_1}\cdots \cup_{f_{n-1}} K_n$ by
$K_0^n$. Observe that $K_0^n$ deformation retracts to $\partial^+ K_n$, and therefore its fundamental
group is free and isomorphic to $F$.
\subsection{$f$-fatgraphs}
Fix a rose $R$ for $F$ and a simplicial map $f:R \to R$ representing $\phi:F \to F$.
\begin{definition}\label{definition:f_fatgraph}
An $f$-fatgraph $X$ over $R$ ({\em not} assumed to be reduced or without 1-valent vertices)
is a fatgraph $g:X \to R$ together with a decomposition of
$\partial S(X)$ into submanifolds $\partial^-$ and $\partial^+$ (each a union of components)
so that there is an orientation-reversing homeomorphism $f':\partial^- \to \partial^+$
lifting $f$ (i.e.\/ satisfying $gf'=fg$ where by abuse of notation we denote the
composition $\partial S(X) \to X \to R$ by $g$).
\end{definition}
If $X$ is an $f$-fatgraph over $R$, we can replace $g:X \to R$ with a homotopic map
of homotopy equivalent spaces $S(X) \to R\times I$, sending $\partial^-$ to $R\times 0$
and $\partial^+$ to $R\times 1$. By the defining property of an $f$-fatgraph, if we
denote by $S*_f(X)$ the closed oriented surface obtained from $S(X)$ by gluing
$\partial^-$ to $\partial^+$ by $f'$, then the map from $S(X)$ to $K$ factors through
$S*_f(X) \to K$. Thus $f$-fatgraphs induce maps from surface groups to $F*_\phi$.
The converse is the following lemma:
\begin{lemma}
Let $S$ be a closed oriented surface, and $g:S \to K$ a map. Then $S$ and $g$
can be compressed to a surface $g':S' \to K$ which is homotopic to a map of the form
$S*_f(X) \to K$ associated to an $f$-fatgraph $X$ over $R$ with $\partial^-$ immersed in $R$.
\end{lemma}
\begin{proof}
First, throw away sphere components of $S$.
Make $g$ transverse to $R\times 0 \subset K$, so that the preimage is a system of embedded
loops $\Gamma$ in $S$. Inductively eliminate innermost complementary disks by an isotopy.
Furthermore, if some component of $\Gamma$ maps to a homotopically trivial loop in
$R\times 0$, we compress $S$ and $g$ along this loop
If $S_i$ is a component of $S$ that does not meet $\Gamma$ then
$g:S_i \to K$ factors through $S_i \to R\times I$; but any map from a closed oriented surface
to a space homotopic to a graph extends over a handlebody, so $S_i$ can be completely compressed
away. Thus we eventually arrive at $g':S' \to K$ which can be cut open along the remaining
loops $\Gamma'$ to produce a proper map $g'':S'' \to R\times I$, every boundary component of which
maps to an essential loop. Compress $S''$ further if possible. The boundary $\partial S''$
decomposes into $\partial^-$ and $\partial^+$, and the way these sit in $S'$
determines an orientation-reversing homeomorphism $\partial^- \to \partial^+$. We homotope
the map on $\partial^-$ so that it is immersed in $R$, and homotop the map on $\partial^+$
to be equal to its image under $f$. Note that if $f$ is not an immersion, neither is
the map $\partial^+ \to R$ necessarily. But $\partial^+ \to R$ factors through
$\partial^+ \to \partial^{++} \to R$, where the first map folds some intervals into trees, and
the second map is an immersion (this is just Stallings' folding procedure applied to
$\partial^+$, together with the fact that each component maps to an essential loop in $R$).
By Proposition~\ref{proposition:surface_is_fatgraph} there is some reduced fatgraph $X$
with $\partial S(X) = \partial^- \cup \partial^{++}$; adding some trees to $X$ we obtain
a (possibly non-reduced) fatgraph $X'$ with $\partial S(X') = \partial^- \cup \partial^+$,
giving $X$ the structure of an $f$-fatgraph with $S*_f(X') \to K$ homotopic to $S' \to K$.
\end{proof}
This Lemma lets us study surfaces in $K$ (and surface subgroups mapping to $G$) combinatorially. But
actually we are interested in going in the other direction, building $f$-fatgraphs and then
using them to construct surfaces and surface subgroups in $G$.
\subsection{Stacking surfaces and fattening stacks}
If $g:X \to R$ is an immersed fatgraph over $R$ (not necessarily reduced)
then we denote by $f(g):f(X) \to R$
the fatgraph over $R$ with the same underlying topological space as $X$, but with $f(g)=f\circ g$
and $f(X)$ subdivided so that this map takes edges to edges. If $X$ is an $f$-fatgraph, then so
is $f(X)$, and there is a natural orientation-reversing simplicial homeomorphism between
$\partial^+S(X)$ and $\partial^-S(f(X))$. Iterating this procedure, we can build a surface
$$S_n(X):=S(X) \cup S(f(X)) \cup \cdots \cup S(f^n(X))$$
The boundary labels of the $\partial S(f^n(X))$ are words obtained by applying $\phi$
by substitution repeatedly to the generators on the edges of $\partial S(f^i(X))$; i.e.\/ we do
{\em not} perform cancellation if these words are not reduced. See Figure~\ref{figure5}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 11 24
\pinlabel {$a$} at 93 48
\pinlabel {$b$} at 86 52
\pinlabel {$B$} at 69 48
\pinlabel {$a$} at 63 39
\pinlabel {$B$} at 46 32
\pinlabel {$A$} at 0 36
\pinlabel {$b$} at 46 13.5
\pinlabel {$a$} at 65 9
\pinlabel {$b$} at 99 -1
\pinlabel {$b$} at 87 25.5
\pinlabel {$A$} at 75 22.8
\pinlabel {$A$} at 77 28.2
\pinlabel {$A$} at 93 31
\pinlabel {$A$} at 102 30
\pinlabel {$B$} at 101 24
\pinlabel {$B$} at 122 3
\pinlabel {$a$} at 117 42.6
\pinlabel {$b$} at 119 51
\pinlabel {$f(a)$} at 146 22
\pinlabel {$b$} at 174 6
\pinlabel {$b$} at 176 26
\tiny
\pinlabel {$f(A)$} at 157 30
\pinlabel {$f(A)$} at 155 37.5
\pinlabel {$f(A)$} at 173 39
\pinlabel {$f(A)$} at 189 33
\small
\pinlabel {$B$} at 170 16.5
\pinlabel {$B$} at 194 6
\pinlabel {$f(a)$} at 200 41.5
\pinlabel {$f(a)$} at 178 49.5
\pinlabel {$b$} at 171.5 56
\pinlabel {$B$} at 161.5 56
\pinlabel {$f(a)$} at 149 47
\pinlabel {$B$} at 129 57
\endlabellist
\centering
\includegraphics[scale=1.7]{figure5}
\caption{Let $f(a) = AbbaaaaBBABabAbBA$ and $f(b) = b$.
This figure shows a fatgraph $S(X)$ (blue) with boundary $\partial^-S(X) = a$
and $\partial^+S(X) = f(A)$. The fatgraph $S(f(X))$ (red) is glued by
identifying $\partial^+S(X)$ and $\partial^-(S(f(X))$, as shown. Typically,
a failure to be reduced will come from cancellation between $f(a)$ and $f(b)$.
Here we have made $f(a)$ non-reduced for illustrative purposes.}
\label{figure5}
\end{figure}
Each $S(f^i(X))$ deformation retracts to $f^i(X)$, so there is an induced quotient map from $S_n(X)$ to
a graph $X_n$. Now, although each individual $S(f^i(X))$ is homotopy equivalent to $f^i(X)$,
it is {\em not} necessarily true that $S_n(X)$ is homotopy equivalent to $X_n$. However, this
can be guaranteed by imposing a simple condition.
\begin{lemma}\label{lemma:embedding}
Suppose that $\partial^- S(X) \to X$ is an embedding; equivalently, that no vertex of $X$ is in the
image of more than one vertex of $\partial^- S(X)$ under the deformation retraction from
$S(X)$ to $X$. Then $X_n$ admits the structure of a fatgraph in a natural way so that
$S_n(X)=S(X_n)$.
\end{lemma}
\begin{proof}
Each $S(f^i(X))$ deformation retracts to $f^i(X)$, and the tracks (i.e.\/ point preimages)
of this deformation are proper essential arcs which retract to points in the edges of $X$,
and proper essential trees which retract to the vertices of $X$. Glue up the tracks of the
deformation retraction for $S(f^i(X))$ to the tracks in $S(f^{i+1}(X))$ by the identification
of the boundaries; the result is a decomposition of $S_n(X)$ into {\em graphs}, in such a way that
$X_n$ is the quotient space obtained by quotienting each graph to a point. We claim
that each such graph is a tree. Since these trees are disjointly embedded in $S_n(X)$,
we can embed $X_n$ as a spine of $S_n(X)$ in a natural way, giving it the structure of a fatgraph
with $S(X_n)=S_n(X)$.
If $\tau$ is a track in some $S(f^i(X))$, then $\tau$ has at most one boundary point on
$\partial^-$ (by hypothesis). Define an orientation on the edges of $\tau$ in such a way that
the edges all point towards this unique boundary point on $\partial^-$ (if one exists), or towards
the unique point on $f^i(X)$ that $\tau$ deformation retracts to otherwise. See Figure~\ref{figure6}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$\partial^-$} at 28 37
\pinlabel {$\partial^+$} at 26 7
\pinlabel {$\partial^+$} at 57 22
\endlabellist
\centering
\includegraphics[scale=1.5]{figure6}
\caption{The flow points towards $\partial^-$.}
\label{figure6}
\end{figure}
Then each graph
$T$ which is a maximal connected union of tracks in the various $f^i(X)$ gets an orientation on
its edges in such a way that each vertex has {\em at most} one outgoing edge. Thus $T$
can be canonically deformation retracted along oriented edges to a (necessarily unique) minimum,
and $T$ is a tree.
\end{proof}
Now, if $X$ is an $f$-fatgraph, we distinguish, amongst the vertices of $\partial^+$, those
which are in the image of vertices of $\partial^-$ under $f$, and call these {\em $f$-vertices}.
\begin{definition}\label{definition:f_folded}
An $f$-fatgraph $g:X \to R$ is {\em $f$-folded} if it satisfies the following conditions:
\begin{enumerate}
\item{the underlying map of graphs $X \to R$ is an immersion;}
\item{every $f$-vertex in $\partial^+$ maps to a
2-valent vertex of $X$ under the retraction $\partial^+ \to X$;}
\item{no vertex of $X$ is in the image of more than one $f$-vertex in $\partial^+$; and}
\item{the map $\partial^- \to X$ is an embedding.}
\end{enumerate}
\end{definition}
The first condition says that the underlying map of graphs
$X \to R$ is folded in the sense of Stallings.
If $X$ has no 1-valent vertices, this implies that $X$ is reduced, but in general
$\partial S(X)$ will contain consecutive pairs of cancelling letters at 1-valent vertices
of $X$.
\begin{proposition}\label{proposition:f_folded_injective}
Suppose $f:R \to R$ is an immersion, and $X$ is $f$-folded. Then $S*_f(X) \to K$ is $\pi_1$-injective.
\end{proposition}
\begin{proof}
First, since $X\to R$ is an immersion by condition (1), and $f:R \to R$ is an immersion by
hypothesis, it follows that $f^i(X) \to R$ is an immersion for each $i$.
If $S*_f(X) \to K$ is not injective, there is some loop in the kernel. Such a loop lifts
to a loop in the infinite cyclic cover of $S*_f(X)$ which maps to $\tilde{K}$ and is contained
in the preimage of some $K_n$. But this preimage is exactly $S_n(X)$, so it suffices to show
that $S_n(X)$ maps injectively. Condition (4) implies that $S_n(X)$ is homotopy equivalent to
the fatgraph $X_n$, so it suffices to prove that $X_n \to R$ is injective, and to do this it
suffices to show that it is an immersion. But this is a local condition, and is proved by
induction on $n$, since the case $n=0$ is condition (1), and conditions (2) and (3) imply that
each vertex of $X_n$ of valence $>2$ whose restriction to $X_{n-1}$ has valence $2$ is locally
isomorphic to some vertex in $f^n(X)$, which we already saw is immersed in $R$. See Figure~\ref{figure7}.
This completes the proof.
\end{proof}
\begin{figure}[ht]
\labellist
\small\hair 2pt
\endlabellist
\centering
\includegraphics[scale=1.5]{figure7}
\caption{At each $f$-vertex (highlighted), subsequent gluings
attach at most one vertex of valence greater than two. See also
Figure~\ref{figure5}, in which the $f$-vertices are bold.}
\label{figure7}
\end{figure}
\subsection{Bounded folding}\label{subsection:bounded_folding}
For technical reasons, it is important to generalize this proposition and the definition of
$f$-foldedness to the case that $f:R \to R$ is not an immersion, but satisfies a slightly
weaker property, that we call {\em bounded folding}.
If $g:X \to Y$ is a map between graphs taking edges to edges, {\em Stallings folding} shows
how to construct canonically a quotient $\pi:X \to X'$ which is a map between graphs taking edges
to edges, and an immersion $X' \to Y$, so that the composition $X \to X' \to Y$ is $g$.
\begin{definition}
Let $g:X \to Y$ be a map of graphs, and let $X'$ be obtained by folding, so that
$X'$ immerses in $Y$ and there is $\pi:X \to X'$ so that $X \to X' \to Y$ is $g$.
We say that $g$ has {\em bounded folding} if there is a collection of disjoint simplicial
trees $T_i'$ in $X'$ so that each preimage $T_i:=\pi^{-1}(T_i')$ is a connected tree in $X$
containing at most one vertex of valence $>2$,
and $\pi$ is a homeomorphism of $X-\cup_i T_i \to X' - \cup_i T_i'$ and
a proper homotopy equivalence of $T_i \to T_i'$ for each $i$.
Call the union of the $T_i$ the {\em folding region}, and denote it by $\fold(X)$;
the complement of the folding region in $X$ is the {\em immersed region}.
\end{definition}
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 50 30
\pinlabel {$b$} at 63 35
\pinlabel {$A$} at 76 40
\pinlabel {$a$} at 37 31
\pinlabel {$a$} at 24 36.5
\pinlabel {$b$} at 7 43
\pinlabel {$a$} at 36 11
\pinlabel {$a$} at 24 5
\pinlabel {$B$} at 9 -1
\pinlabel {$b$} at 51 13
\pinlabel {$a$} at 65 7
\pinlabel {$B$} at 77 0
\pinlabel {$b$} at 130 35
\pinlabel {$a$} at 144 30
\pinlabel {$b$} at 159 35
\pinlabel {$a$} at 165.5 30
\pinlabel {$b$} at 184 23
\pinlabel {$a$} at 196 16
\pinlabel {$B$} at 210 9
\pinlabel {$B$} at 130 13
\endlabellist
\centering
\includegraphics[scale=1.5]{figure8}
\caption{The gray region, left, indicates all edges involved in folding
(the \emph{folding region}). After folding, the gray region is reduced
to the region at right.}
\label{figure8}
\end{figure}
Note that $\fold(X)$ is precisely the preimage of the set of edges of $X'$ with
more than one preimage. Note also that if $g:X \to Y$ is a map with bounded folding,
then $\pi:X \to X'$ is a homotopy equivalence, so $g$ is $\pi_1$-injective.
Topologically, a map with bounded folding is an immersion outside a small tree neighborhood
of some vertices, and collapses each such neighborhood by a proper homotopy
equivalence to a smaller tree.
Now, the map $f:R \to R$ is not simplicial, since edges of $R$ get generally taken to long
paths in $R$. Let $R_1$ denote a rose with edges labeled by reduced words which are the image of the generators
of $F$ under $\phi:F \to F$ (assume none of these is trivial) and
subdivide edges of $R_1$ so that each edge gets one generator. Then we can factorize $f:R \to R$
as the composition of a {\em homeomorphism} $h_1:R \to R_1$ and a simplicial map $R_1 \to R$.
\begin{definition}
With notation as above, and by abuse of notation, we say that $f:R \to R$
has {\em bounded folding} if $R_1 \to R$ has bounded folding.
\end{definition}
If $f:R \to R$ has bounded folding,
either $R_1 \to R$ is an immersion, or else $\fold(R_1)$ consists of a single tree
with a single vertex of valence $>2$ which corresponds to the vertex of $R$ under
$h^{-1}$. See Figure~\ref{figure9}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 61.5 14
\pinlabel {$b$} at 23 72
\pinlabel {$c$} at 0 9
\pinlabel {$a$} at 135 24
\pinlabel {$b$} at 137 6
\pinlabel {$A$} at 158 3
\pinlabel {$c$} at 166.5 21
\pinlabel {$b$} at 152 37
\pinlabel {$A$} at 136 42
\pinlabel {$a$} at 122 48
\pinlabel {$c$} at 135 57
\pinlabel {$c$} at 132 75
\pinlabel {$B$} at 114 83
\pinlabel {$c$} at 100 71
\pinlabel {$A$} at 102 53
\pinlabel {$A$} at 108 42
\pinlabel {$a$} at 112.5 28.5
\pinlabel {$c$} at 97 34
\pinlabel {$A$} at 83.5 19
\pinlabel {$b$} at 86 3
\pinlabel {$a$} at 102 -2
\pinlabel {$c$} at 117 8
\pinlabel {$A$} at 123.5 17
\endlabellist
\centering
\includegraphics[scale=1.5]{figure9}
\caption{The gray region indicates $\fold(R_1)$ for the endomorphism
$a\mapsto abAcbA$, $b\mapsto accBcAA$, $c \mapsto acAbacA$.}
\label{figure9}
\end{figure}
Let $R_2$ be another rose whose edges are labeled by the {\em unreduced} words,
obtained by applying $\phi$ to each letter of the edge labels of $R_1$, and define $R_n$
similarly by induction. So there are homeomorphisms $h_n:R \to R_n$ and a simplicial map
$R_n \to R$ for which the composition $R \to R$ is $f^n$. By abuse of notation we also write
$f:R_{i-1} \to R_i$ for each $i$. Observe that $\fold(R_n)$ contains $f(\fold(R_{n-1}))$,
and the components of $\fold(R_n)-f(\fold(R_{n-1}))$ are intervals, none
of which contains the image of a vertex of $\fold(R_{n-1})$ (except possibly at an endpoint).
Now, suppose $g:X \to R$ is an $f$-folded $f$-fatgraph over $R$.
We might be able to realize $\partial^- \to R$ by an immersion, but it is unlikely
that $\partial^+ \to R$ can be realized by an immersion if $f:R \to R$ is not an immersion.
\begin{definition}
Let $g:\partial^- \to R$ be an immersion, and let $h:\partial^+ \to R_1$ be obtained by
applying $f$ to both sides of $g$. Define $\Sigma^+$ to be the
preimage $\Sigma^+:=h^{-1}(\fold(R_1))$.
\end{definition}
Note that $\fold(\partial^+)$ is contained in $\Sigma^+$, which is a collection of
intervals (it can't be all of $\partial^+$ because $\partial^- \to R$ is an immersion).
\begin{lemma}\label{lemma:peripheral_tree}
Let $w$ be a nonreduced cyclic word which is nontrivial, and let $V$ be the reduced cyclic word
which is inverse to $w$. Then $w \cup V = \partial S(Y(w))$ for an immersed fatgraph
$g:Y(w) \to R$ which consists of a circle (the embedded image of $V$) with a collection
of rooted trees attached, one for each component of $\fold(w)$.
\end{lemma}
\begin{proof}
This is just the observation that $w$ can be repeatedly Stallings
folded to produce $v$ (the inverse of $V$); if we embed $w$ in the plane, the folds can
all be done to the ``inside'', producing a planar graph $Y(w)$ at the end with inner boundary
$V$ and outer boundary $w$. The embedding in the plane gives $Y(w)$ its fatgraph structure.
\end{proof}
If $w=\partial^+$ and $\Sigma^+$ is as above, each component of $\fold(w)$ is contained in
a component of $\Sigma^+$ and folds up to a tree in $Y(w)$ as
in Lemma~\ref{lemma:peripheral_tree}. The image of the component of $\Sigma^+$ is this
tree together possibly with an interval neighborhood of its root; we call this entire
image a {\em peripheral tree}, and denote the union of these trees by $\Sigma$. See Figure~\ref{figure10}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 98 36
\pinlabel {$b$} at 88 45
\pinlabel {$A$} at 74 48
\pinlabel {$c$} at 60 48
\pinlabel {$b$} at 48 49
\pinlabel {$A$} at 33 49
\pinlabel {$a$} at 17 49
\pinlabel {$b$} at 4 42
\pinlabel {$A$} at -2 31
\pinlabel {$c$} at -1 19
\pinlabel {$b$} at 8 7
\pinlabel {$A$} at 21 2
\pinlabel {$a$} at 35 3
\pinlabel {$c$} at 47.5 3.5
\pinlabel {$c$} at 59 3
\pinlabel {$B$} at 69 3.5
\pinlabel {$c$} at 81 5
\pinlabel {$A$} at 91 9
\pinlabel {$A$} at 99 20
\pinlabel {$a$} at 212 33
\pinlabel {$b$} at 204 38
\pinlabel {$A$} at 185 45
\pinlabel {$c$} at 170 45
\pinlabel {$b$} at 154 45
\pinlabel {$A$} at 149.5 50.5
\pinlabel {$a$} at 138 50
\pinlabel {$b$} at 134 42
\pinlabel {$A$} at 126 34
\pinlabel {$c$} at 127 20
\pinlabel {$b$} at 135 11
\pinlabel {$A$} at 139.5 4
\pinlabel {$a$} at 150 4
\pinlabel {$c$} at 155 8
\pinlabel {$c$} at 164 9
\pinlabel {$B$} at 177 9
\pinlabel {$c$} at 193 9.5
\pinlabel {$A$} at 204 18
\pinlabel {$A$} at 212 22
\endlabellist
\centering
\includegraphics[scale=1.6]{figure10}
\caption{Applying the endomorphism from Figure~\ref{figure9} to the loop $aab$
produces the loop at left, with the folding region in gray. After folding,
the folding region is reduced to a collection of peripheral trees, right.}
\label{figure10}
\end{figure}
\begin{definition}\label{definition:grafting}
Let $w$ be a possibly unreduced nontrivial cyclic word,
and $\Sigma^+$ a collection of embedded intervals
containing $\fold(w)$. Let $Y(w)$ be as in the statement of
Lemma~\ref{lemma:peripheral_tree}, and let $\Sigma$ be the union of peripheral trees in
$Y(w)$.
An inclusion of $Y(w)$ into another immersed fatgraph $X$ is a {\em grafting of $Y(w)$}
if it satisfies the following properties:
\begin{enumerate}
\item{$w$ is a component of $\partial S(X)$;}
\item{all 1-valent vertices of $X$ are in $\Sigma$; and}
\item{every vertex of $\Sigma$ has the same valence in $Y(w)$ as in $X$.}
\end{enumerate}
Let $Y'$ be the fatgraph obtained from $X$ by cutting off the peripheral trees at their roots.
Then we say $X$ is obtained by {\em grafting $Y(w)$ onto $Y'$}.
\end{definition}
\begin{definition}
Suppose $f:R \to R$ has bounded folding, and let $X$ be an $f$-fatgraph $g:X \to R$
immersed in $R$. We say that $g:X \to R$ admits
{\em bounded $f$-folding} if the following is true:
\begin{enumerate}
\item{$X$ is obtained by grafting $Y(\partial^+)$, where as above
$\Sigma^+ \subset \partial^+$ is defined to be $h^{-1}(\fold(R_1))$;}
\item{$g:X \to R$ is $f$-folded in the sense of Definition~\ref{definition:f_folded},
except that it is possible that some $f$-vertices in $\partial^+$ map to a 1-valent
vertex of $X$ on the boundary of a peripheral tree;}
\item{distinct $f$-vertices map to different components of $\Sigma$; and}
\item{the image of $\partial^-$ is disjoint from $\Sigma$.}
\end{enumerate}
\end{definition}
\begin{proposition}\label{proposition:bounded_f_folding_injective}
Suppose $f:R \to R$ has bounded folding, and $g:X \to R$ admits bounded $f$-folding. Then
$S*_f(X) \to K$ is $\pi_1$-injective.
\end{proposition}
\begin{proof}
We can build a surface $S_n(X)$ and a fatgraph $X_n$ as before, where $S_n(X)=S(X_n)$,
since $\partial^- \to X$ is an embedding, and Lemma~\ref{lemma:embedding}.
We claim that $X_n \to R$ has bounded folding,
and is therefore $\pi_1$-injective.
We build $X_n$ from $X$ and $f(X_{n-1})$, by
gluing $\partial^+$ in $X$ to $f(\partial^-)$ in $f(X_{n-1})$. Note that
the inclusion of $X$ in $X_n$ is an {\em embedding}, since $X$ is attached
by identifying $\partial^+$ with $f(\partial^-)$ which {\em embeds} in $f(X_{n-1})$.
We assume by induction that $X_{n-1}$ has bounded folding. Then so does $f(X_{n-1})$, since
$\fold(f(X_{n-1}))-f(\fold(X_{n-1}))$ consists of a union of small intervals, none of which
contains the image of a vertex of $X_{n-1}$ except possibly at the endpoints (this is a general
property of the fact that $f$ has bounded folding and $g$ is an immersion).
We need to check that no two vertices of $X_n$ of valence at least 3 are contained in
the same component of $\fold(X_n)$. The vertices of $X_n$ of valence at least 3 are all
images of a vertex of valence at least 3 either in $f(X_{n-1})$ or in $X$. Moreover,
the vertices of valence at least 3 in $f(X_{n-1})$ are the images of vertices of
valence at least 3 in $X_{n-1}$. By abuse of notation, we
refer to the images of {\em all} the vertices of $X_{n-1}$ in $f(X_{n-1})$ as
$f$-vertices; the ordinary $f$-vertices in $X$ are glued to the $f$-vertices (in the new sense)
of $f(\partial^-)$.
Every $f$-vertex in $f(X_{n-1})$ not in $f(\partial^-)$ is thus separated
from the image of $X$ in $X_n$ by the image of an edge of $X_{n-1}-\partial^-$, and
the endpoints of this edge are necessarily in different components of $\fold(X_n)$. Distinct
$f$-vertices in $f(\partial^-)$ must map to distinct vertices of $X$, and no component of
$\Sigma$ contains the image of more than one of them, by condition (3); thus
components of $\fold(X_n)$ cannot contain more than one such $f$-vertex.
So we just need to check that distinct high-valence vertices of $X$ are not included
into the same component of $\fold(X_n)$. Now, it is not necessarily true that
$\fold(X_n) \cap X$ is equal to $\fold(X)$, but the difference is contained in
$\Sigma$ minus the peripheral trees (i.e.\/ in the intervals of $X$ containing the
roots of the peripheral trees) and by the defining properties of grafting, there are no other
high valence vertices there.
\end{proof}
\begin{remark}
If one is prepared to work with {\em groupoid} generators for $F$ rather than group generators,
this contents of this section are superfluous in most cases of interest. Although most
injective endomorphisms $\phi:F \to F$ are not represented by immersions of some rose $R$,
Reynolds \cite{Reynolds} showed that if $\phi$ is an {\em irreducible endomorphism} which is
{\em not} an automorphism, then there is some graph $R'$ (typically with more than one vertex)
and an isomorphism of $F$ with $\pi_1(R')$, so that $\phi$ is represented by an immersion
$f:R' \to R'$. If one wants to find injective surface subgroups in extensions $F*_\phi$ then
in practice it is much easier to work with $f$-folded $f$-fatgraphs over such an $R'$, than
with boundedly $f$-folded $f$-fatgraphs over a rose $R$.
\end{remark}
\subsection{Random endomorphisms}
\begin{definition}
Fix a free group $F$ and a free generating set. A {\em random endomorphism of length $n$}
is an endomorphism $\phi:F \to F$ which takes each generator to a reduced word of length $n$
sampled randomly and independently from the set of all reduced words of length $n$ with
the uniform distribution.
\end{definition}
We require an elementary lemma from probability:
\begin{lemma}\label{lemma:small_prefix}
Let $\phi:F \to F$ be a random endomorphism of length $n$. There for any positive constant $C$
there is a positive constant $c$
depending only on the rank of $F$ so that with probability $1-O(e^{-n^c})$,
for every two distinct generators or inverses of generators $x$, $y$ the reduced words
$\phi(x)$ and $\phi(y)$ have a common prefix or suffix of length $\le C\log{n}$.
\end{lemma}
\begin{proof}
It suffices to obtain such an estimate for two random words. Generate the
words letter by letter; at each step the chance that there is a mismatch is at least
$(k-1)/k$. The estimate follows.
\end{proof}
It follows that if $\phi:F \to F$ is a random endomorphism, the map $f:R \to R$
has bounded folding, and the diameter of $\fold(R_1)$ in $R_1$ is at most $2C\log{n}$,
with probability $1-O(e^{-n^c})$, where we may choose $C$ as small as we like at the cost of
making $c$ small.
We now come to the main theorem of this section, the Random $f$-folded Surface Theorem:
\begin{theorem}[Random $f$-folded surface]\label{thm:random_f_folded_theorem}
Let $k\ge 2$ be fixed, and let $F$ be a free group of rank $k$. Let $\phi$ be a
random endomorphism of $F$ of length $n$. Then the probability that $F*_\phi$ contains
an essential surface subgroup is at least $1-O(e^{-n^c})$ for some $c>0$.
\end{theorem}
We will prove this theorem by constructing an $f$-fatgraph $X$ for which $g:X \to R$
admits bounded $f$-folding, and then apply
Proposition~\ref{proposition:bounded_f_folding_injective}.
In the sequel we denote generators by smaller case letters $a,b,c$ and so on,
and their inverses by capitals; thus $A:=a^{-1}$, $B:=b^{-1}$ etc.
Let $a,b$ be two generators of $F$, let $\partial^-$ be an oriented circle labeled with the
(reduced) cyclic word $abAB$ and let $\partial^+$ be an oriented circle labeled with the
(possibly unreduced!) word obtained by cyclically concatenating
$\phi(b)$, $\phi(a)$, $\phi(B)$, $\phi(A)$. Note that the label on $\partial^+$ is equal
to the inverse of $\phi(abAB)$ in $F$. We will construct $X$ with $\partial S(X) = \partial^-
\cup \partial^+$ with notation as in Definition~\ref{definition:f_fatgraph}.
We build $X$ as a graph by starting with $\partial^+ \cup \partial^-$ and identifying pairs
of segments with opposite orientations and inverse labels. At each stage,
we obtain a {\em partial fatgraph} (bounding the pairs of edges that have been identified)
and a {\em remainder}. See Figure~\ref{figure11}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 132 74
\pinlabel {$b$} at 111 82
\pinlabel {$b$} at 89 82
\pinlabel {$b$} at 65 82
\pinlabel {$a$} at 44 77
\pinlabel {$a$} at 42 54
\pinlabel {$B$} at 62 45
\pinlabel {$A$} at 85 46
\pinlabel {$B$} at 107 46
\pinlabel {$a$} at 129 51
\pinlabel {$A$} at 179 63
\pinlabel {$A$} at 160 82
\pinlabel {$B$} at 139 64
\pinlabel {$A$} at 158 45
\pinlabel {$a$} at 89 39
\pinlabel {$b$} at 59 28
\pinlabel {$b$} at 40 26
\pinlabel {$b$} at 16 30
\pinlabel {$a$} at -2 20
\pinlabel {$a$} at 15 8
\pinlabel {$B$} at 37 13
\pinlabel {$A$} at 59 9.5
\pinlabel {$B$} at 86 -1.5
\pinlabel {$a$} at 100 18
\pinlabel {$B$} at 84 24.5
\pinlabel {$A$} at 79.4 19.5
\pinlabel {$A$} at 84 14.4
\pinlabel {$A$} at 89 19
\pinlabel {$a$} at 196 41
\pinlabel {$b$} at 171 29
\pinlabel {$b$} at 155 26
\pinlabel {$b$} at 141 26
\pinlabel {$b$} at 124 30
\pinlabel {$a$} at 106 25
\pinlabel {$a$} at 123.5 8.5
\pinlabel {$B$} at 140 13.5
\pinlabel {$B$} at 153 13
\pinlabel {$A$} at 171 11
\pinlabel {$B$} at 191 -2
\pinlabel {$a$} at 207 -2
\pinlabel {$A$} at 212 8
\pinlabel {$A$} at 207 14
\pinlabel {$A$} at 203.5 20
\pinlabel {$B$} at 206 27
\pinlabel {$A$} at 213 34
\endlabellist
\centering
\includegraphics[scale=1.5]{figure11-2}
\caption{To build a fatgraph with desired boundary loops, we can proceed by
gluing small portions of the loops, one at a time. After gluing a small amount of our
loops, we obtain a partial fatgraph and the remainder, which is a
collection of loops with tags.}
\label{figure11}
\end{figure}
When all of $\partial^+ \cup \partial^-$
has been paired (i.e.\/ when the remainder is empty),
the result will be $X$. The proof will take up the next section.
\section{Proof of the Random $f$-folded Surface Theorem}
\subsection{Bounded folding and $f$-vertices}
The loop $\partial^-$ has length 4 and thus 4 vertices. The loop $\partial^+$ has length $4n$,
and has 4 $f$-vertices, which separate it into the segments $\phi(b)$, $\phi(a)$, $\phi(B)$
and $\phi(A)$. The map $\partial^- \to R$ is an immersion already. Also, by
Lemma~\ref{lemma:small_prefix} we have already seen that $\fold(R_1)$ is a tree in $R_1$
containing the vertex, of diameter at most $2C\log{n}$, where $C$ is as small as we like,
with probability $1-O(e^{-n^c})$. Since $h:\partial^+ \to R_1$ (obtained by applying $f$
to both sides of $\partial^- \to R$) is an immersion, it follows that $\Sigma^+:=h^{-1}(\fold(R_1))$
consists of four neighborhoods of the $f$-vertices, each of diameter at most $2C\log{n}$.
Note that $\fold(\partial^+)$ is contained in $\Sigma^+$. We fold $\Sigma$ as much as
possible, obtaining a fatgraph $Y(\partial^+)$ as in Lemma~\ref{lemma:peripheral_tree}
with $\partial^+$ on one side of $S(Y(\partial^+))$ and with (the inverse of)
the reduced representative of this word on the other side. Denote the image of $\Sigma^+$
in this fatgraph by $\Sigma$, and let $\Sigma^-$ be the {\em reduced} words on the other
side of $S(Y(\partial^+))$ (i.e.\/ they are the reduced words obtained from the components
of $\Sigma^+$). Note that $\Sigma^-$ has at most four components, each of length
at most $2C\log{n}$. Notice also that if a component of $\Sigma^+$ can be reduced at all,
it can only be reduced by cancelling a pair of maximal inverse subwords on the sides of
the $f$-vertex, so that the peripheral trees consist of at most a single interval with
the $f$-vertex at the tip.
\begin{lemma}\label{lemma:log_word_present}
For any positive $\epsilon$, if $w$ is a random reduced word in $F$
of length at least $n\epsilon$, then for any $C' < 1/\log(2k-1)$ there is a positive $c$
(depending only on the rank of $F$), so that for any reduced word $v$ of length
$\lfloor C'\log{n} \rfloor$ we can find a copy of $v$ in $w$, with probability $1-O(e^{-n^c})$.
\end{lemma}
For a proof, see e.g.\/ \cite{CW_rigidity}, Prop.~2.3 which gives a precise count of
the number of copies of $v$ in $w$. So if we choose $C \ll 1/\log(2k-1)$ we can find
many disjoint copies of segments in $\partial^+-\Sigma^+$ with labels inverse
to the labels on $\Sigma^-$, and we can pair these segments.
Next, we look for a copy of the word $bbaBAB$ in the remainder, glue the outermost
copies of $b$ and $B$, and glue the resulting $baBA$ loop to $\partial^-$. Note that
$\partial^-$ embeds into the resulting partial fatgraph, and is disjoint from
$\Sigma$ and the $f$-vertices.
The part of the fatgraph we have built so far evidently immerses in $R$.
All that is left of the remainder are reduced cyclic words made from the
segments of $\partial^+$ which are disjoint from
$\Sigma$ and the $f$-vertices. It remains to glue up the remainder so that
the resulting fatgraph is immersed. This is a complicated combinatorial argument with
several steps, and it takes up the remainder of the section.
\subsection{Pseudorandom words}
At this stage of the construction, the remainder consists of a collection of cyclic words made from
the 9 segments in $\partial^+$ disjoint from $\Sigma$ and the $f$-vertices. We can arrange for each
of these segments to be long (i.e.\/ $O(n)$), so that in effect we can think of the remainder
as a finite collection of long reduced cyclic words in $F$ whose sum is homologically trivial.
Note that although each segment making up the cyclic words is (more or less) random, different segments
are not necessarily independent --- some of them are subwords of $\phi(a)$ and
some are subwords of $\phi(A)$, which are inverse. But each individual segment is {\em pseudorandom}
in the following sense.
\begin{definition}
For $T>0$ and $\epsilon>0$, a reduced word $w$ or cyclic word in a free group $F$ of rank $k$
is {\em $(T,\epsilon)$-pseudorandom} if, however
we partition $w$ as
$$w=w' v_1 v_2 \cdots v_\ell w''$$
where $|w'|$ and $|w''|<T$ and where $|v_i|=T$ for each $i$, and for any reduced word
$\sigma$ in $F$ of length $T$, there is an inequality
$$1-\epsilon \le \frac {\text{number of $v_i$ equal to $\sigma$}} {\ell/2k(2k-1)^{T-1}} \le 1+\epsilon$$
\end{definition}
Here the term $2k(2k-1)^{T-1}$ is simply the number of reduced words in $F$ of length $T$, so this
just says that the subwords of $w$ of length $T$ are distributed uniformly, up to multiplicative
error $\epsilon$.
Now, for any fixed $T,\epsilon$, a random reduced word in $F$ of length $n$ will be $(T,\epsilon)$-pseudorandom
for sufficiently big $n$, with probability $1-O(e^{Cn})$ for some $C>0$. This follows from
\cite{CW_rigidity}, Prop.~2.3; in fact, with probability $1-O(e^{n^c})$, one can even let $T$
grow with $n$, at the rate $T=C'\log{n}$ for suitable $C'$ (compare with Lemma~\ref{lemma:log_word_present}).
It follows that for big $n$, each individual subword $\phi(a)$, $\phi(b)$ and their inverses is
$(T,\epsilon)$-pseudorandom with high probability, and so are all their subwords of length $\delta n$
for any fixed positive $\delta$. Thus, after the first stage of the construction, the
remainder consists of a finite collection of cyclic loops, each of which is $(T,\epsilon)$-pseudorandom for any fixed
$T,\epsilon$. Theorem~\ref{thm:random_f_folded_theorem} will therefore be proved if we can show that
any finite collection of $T,\epsilon$-pseudorandom reduced cyclic words whose sum is homologically trivial bounds
a folded fatgraph.
\subsection{Folding off short loops}
We introduce some notation to simplify the discussion in what follows.
In order to distinguish words (with a definite initial letter) from cyclic
words, we delimit a word (in our notation) by adding centered dots on both sides; thus $\cdot w \cdot$
is a word, and $w$ is the corresponding cyclic word.
Furthermore, in the course of our folding, we will obtain segments which are in the boundary of a
partial fatgraph, and it is important to indicate which vertices have valence bigger than 2 in the
fatgraph. We will insist that all vertices of valence $>2$ in the remainder at each stage
will be 3-valent, and use the notation $\perp$ for such a vertex. If it is important to record the
label on the third edge at this vertex, we denote it $\perp^x$ where $x$ is the outgoing letter.
Thus, $\cdot ab\perp^ba\cdot$ denotes a segment in the remainder of length 3 with the label $aba$,
and after the second letter there is a 3-valent vertex in the partial fatgraph, with outgoing edge
label $b$.
So the remainder at this stage consists of a finite collection of cyclic words of the form
$\cdots u_i \perp^{x_i} u_j \perp^{x_j} u_k \cdots$ where each $u_i$ is one of the 9 segments
of $\partial^+$, and $x_i$ is the outgoing letter on the edge of the partial fatgraph built by
the identifications made so far.
Now let $w$ be a $(T,\epsilon)$-pseudorandom word.
We perform the following process. As we read off the
letters of $w$ one by one, we look for a segment $\sigma$ of length 11 of the form $\cdot v_1Pupv_2\cdot$
satisfying the following properties:
\begin{enumerate}
\item{$|v_1|=|v_2|=1$ and $v_1\ne v_2^{-1}$;}
\item{$|p|=|P|=1$ and $P=p^{-1}$; and}
\item{$|u|=7$ and $u$ is {\em cyclically} reduced.}
\end{enumerate}
Then $p$ and $P$ can be glued, producing a new reduced word $w'$ containing $\cdot v_1\perp^P v_2\cdot$
where $\sigma$ was, and a {\em short loop} with the cyclic word $u\perp^p$ on it. We call this
operation {\em folding off a short loop}. The data of short loop is determined by a word $u$ of length
$7$ whose associated cyclic word is cyclically reduced, together with a letter $p$ not equal to
the first letter of $u$ or the inverse of the last letter. This data $(u,p)$ is called the {\em type}
of a short loop. Let $L_k$ denote the number of distinct types of short loops in a free group, so
for example $L_2=4376$.
As we read through a component of the remainder, we fold off short loops at regular intervals whenever
we can, so that the ``stems'' of the loops land at places separated by intervals of even length
(say). To fold off a short loop, we desire the pattern described above, and the segments which
satisfy the pattern are simply a subset of all segments of length $11$.
Because $w$ is $(T,\epsilon)$-pseudorandom and there are finitely many types of segments,
whenever $T$ is large enough, we will find short loops of all kinds,
and they will be nearly equidistributed, as described more fully below.
Thus we obtain in this way a {\em reservoir} of short loops, together with the rest of the
remainder, which is a collection of long cyclic words with many trivalent vertices at the
steps of the short loops, where adjacent trivalent vertices are separated by intervals of even
length (with the possible exception of the nine trivalent vertices associated to the vertices
of the fatgraph produced at the first step). See Figure~\ref{figure12}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$A$} at 156 4
\pinlabel {$b$} at 143 4
\pinlabel {$a$} at 139 6.5
\pinlabel {$a$} at 140 12
\pinlabel {$a$} at 146 20
\pinlabel {$b$} at 144 32
\pinlabel {$a$} at 134 36
\pinlabel {$B$} at 123 31
\pinlabel {$a$} at 121 21
\pinlabel {$B$} at 126 12
\pinlabel {$A$} at 128.5 7
\pinlabel {$b$} at 125 4
\pinlabel {$a$} at 115 4
\pinlabel {$b$} at 111 7
\pinlabel {$a$} at 112 12
\pinlabel {$a$} at 118 19
\pinlabel {$b$} at 116 32
\pinlabel {$a$} at 106 36
\pinlabel {$b$} at 96 32
\pinlabel {$a$} at 93.5 20
\pinlabel {$a$} at 98 13
\pinlabel {$B$} at 101 7
\pinlabel {$a$} at 98 4
\pinlabel {$b$} at 88 4
\pinlabel {$a$} at 83.5 6
\pinlabel {$B$} at 85 12
\pinlabel {$B$} at 90 20
\pinlabel {$A$} at 88 32
\pinlabel {$b$} at 77 36
\pinlabel {$A$} at 67 31
\pinlabel {$b$} at 65 20
\pinlabel {$A$} at 71 12
\pinlabel {$A$} at 73 7
\pinlabel {$b$} at 68 4
\pinlabel {$A$} at 58 4
\pinlabel {$b$} at 46 4
\pinlabel {$a$} at 33 4
\pinlabel {$B$} at 31.5 7
\pinlabel {$B$} at 34 13
\pinlabel {$a$} at 39 21
\pinlabel {$B$} at 37 32
\pinlabel {$B$} at 26 36
\pinlabel {$a$} at 15 30
\pinlabel {$B$} at 13 20
\pinlabel {$a$} at 18.5 13
\pinlabel {$b$} at 21 8
\pinlabel {$b$} at 18 4
\pinlabel {$a$} at 7 4
\endlabellist
\centering
\includegraphics[scale=2.0]{figure12}
\caption{Folding off short loops. A loop can be folded off only when
all of the three properties are satisfied. This ensures the
result is a collection of tagged loops of length exactly $7$.}
\label{figure12}
\end{figure}
Let $T$ be some big odd number. We perform this folding procedure on each successive
subword $v_i$ of a $(T,\epsilon)$-pseudorandom $w$ of length $T$ and obtain a collection of new words
$v_i'$ with trivalent vertices separated by even length intervals,
such that length of $v_i'$ and that of $v_i$ agree mod 9. Since $T$ is odd, we distinguish between
{\em even} $v_i'$, for which the trivalent vertices are an even distance from the initial vertex, and
{\em odd} $v_i'$, for which the trivalent vertices are an odd distance from the initial vertex.
By pseudorandomness, the reservoir
consists of an {\em almost equidistributed} collection of short loops; i.e.\/ the distribution
differs from the uniform distribution by a multiplicative error of $\epsilon$. Furthermore, the
words $v_i'$ themselves are uniformly distributed with multiplicative error $\epsilon$, for each
fixed possible value of $|v_i'|$, and the same is true if one conditions on
the $v_i'$ being even or odd (in the sense above). This is because the distribution of random letters $v_2$ not equal
to $v_1^{-1}$ that follow a subword of the form $v_1Pup$ averaged over all possible $Pup$ is
just the uniform distribution on letters not equal to $v_1^{-1}$.
\subsection{Random pairing of $v_i'$}
The $v_i'$ fall into finitely many families depending on their lengths and {\em parity} --- i.e.\/ whether
they are even or odd in the sense of the previous subsection. Moreover, for each fixed
length and parity, the distribution of words is uniform up to multiplicative error $\epsilon$. Thus, for every
reduced word $\sigma$ of suitable length,
the number of $v_i'$ of any given parity equal to $\sigma$ and the number equal to $\sigma^{-1}$ is very nearly equal.
For each $v_i'$ let $v_i''$ denote the subword of $v_i'$ excluding the first and last letter. We call a pair
$v_i'$ and $v_j'$ {\em compatible} if they satisfy the following conditions:
\begin{enumerate}
\item{the label on $v_i''$ is $\sigma$, and the label on $v_j''$ is $\sigma^{-1}$, for some $\sigma$ reduced;}
\item{the first letter of $v_i'$ is not inverse to the last letter of $v_j'$, and conversely;}
\item{if $|\sigma|$ is even, the parities of $v_i'$ and $v_j'$ are opposite, and if $|\sigma|$ is odd,
the parities agree.}
\end{enumerate}
Let $v_i'$ and $v_j'$ be compatible. Then we can glue $v_i''$ to $v_j''$, and by condition (3)
the trivalent vertices on either side are not identified. Furthermore, because of condition (2) the
new boundary words that result from the gluing are still reduced. See Figure~\ref{figure13}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 158.5 19
\pinlabel {$b$} at 151 15.5
\pinlabel {$b$} at 147 17
\pinlabel {$B$} at 136 18
\pinlabel {$a$} at 135 14.8
\pinlabel {$a$} at 123 14.8
\pinlabel {$B$} at 119.7 18
\pinlabel {$b$} at 108.5 19
\pinlabel {$b$} at 104.9 15
\pinlabel {$A$} at 95 15.5
\pinlabel {$B$} at 91.5 18
\pinlabel {$b$} at 81 19
\pinlabel {$b$} at 78.5 15.2
\pinlabel {$a$} at 68 15.5
\pinlabel {$B$} at 57 15.6
\pinlabel {$B$} at 43 15.6
\pinlabel {$a$} at 39.3 18
\pinlabel {$A$} at 28.2 19
\pinlabel {$b$} at 26.3 15.5
\pinlabel {$a$} at 14 15.5
\pinlabel {$b$} at 6 19
\pinlabel {$b$} at 6 2
\pinlabel {$A$} at 11 5
\pinlabel {$A$} at 15 2
\pinlabel {$a$} at 25.4 1
\pinlabel {$B$} at 28 5
\pinlabel {$b$} at 40 4.8
\pinlabel {$a$} at 44 2
\pinlabel {$A$} at 53.5 2
\pinlabel {$b$} at 56 5
\pinlabel {$A$} at 65.5 5
\pinlabel {$A$} at 69 2
\pinlabel {$a$} at 79 1
\pinlabel {$B$} at 81 5
\pinlabel {$a$} at 92 5
\pinlabel {$B$} at 106 5
\pinlabel {$A$} at 119 5
\pinlabel {$b$} at 124 2
\pinlabel {$B$} at 134 2
\pinlabel {$A$} at 137 5.5
\pinlabel {$B$} at 149 5
\pinlabel {$a$} at 158 2
\endlabellist
\centering
\includegraphics[scale=2.1]{figure13}
\caption{Random pairing of the $v_i'$. The shorter region
which is entirely glued is $v_i''$. Note that the initial and final letters
of the paired words are \emph{not} inverse, so the gluing is limited to
exactly the $v_i''$.}
\label{figure13}
\end{figure}
By pseudorandomness, we can glue all but $\epsilon$ of the total length of the remainder (excluding
the reservoir) in this way, and we are left with some collection $\Gamma$ of cyclic words, where
$|\Gamma|\le \epsilon n$, plus the reservoir.
\subsection{Gluing up $\Gamma$}
The next step of the construction is elementary. We use some relatively small mass of small loops from the
reservoir to pair up with $\Gamma$, so at the end of this step we are left only with loops in the
reservoir. Furthermore, since (for a suitable choice of $\epsilon$) the mass of $\Gamma$ is so small,
even compared to the mass of short loops of any given type, if we can do this construction
while using at most $|\Gamma|$ short loops, the content of the reservoir at the end will still be
almost equidistributed, with multiplicative error some new (but arbitrarily small) constant
$\epsilon'$.
We claim that for any positive $m$ we can glue $m-6$ (or at most $3$ if $m<7$) short loops together
to create a trivalent partial fatgraph with unglued part a loop of length $m$. The cases $m<7$
are elementary, and for $m=7$ one can take a single short loop with no gluing at all. We prove the
general case by induction, by proving the stronger statement that the trivalent fatgraph of length
$m\ge 7$ can be chosen to contain a pair of adjacent segments in its boundary of length $4$ and $3$ each
containing no trivalent vertex in the interior. This is obviously true for $m=7$. Suppose it is true for $m$,
and denote the adjacent segments by $\cdot 4\perp 3\cdot$. We can ``bracket'' this as
$\cdot 3 (1\perp 2)2\cdot$ and bracket another short loop as $4(1\perp 2)$. Gluing the two (bracketed)
segments of length 3, we see obtain a loop of length $m+1$ containing $\cdot 3\perp 4 \perp 2\cdot$,
completing the induction step and proving the claim. See Figure~\ref{figure14}.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\endlabellist
\centering
\includegraphics[scale=1.5]{figure14}
\caption{Attaching a loop of length 7 as shown increases the total
loop length by $1$. By induction, we create a trivalent partial fatgraph
with one unglued loop of any size. We can then label the fatgraph
arbitrarily such that the unglued loop is any desired word.}
\label{figure14}
\end{figure}
If we choose suitable short loop types,
the labels on the resulting partial fatgraph and the edges incident to the trivalent vertices
can be {\em arbitrary}, so we can build a loop that can be used to cancel a loop of $\Gamma$.
\subsection{Linear programming}
Finally, we are left with a reservoir of almost equidistributed short loops. Since our original
collection of words had homologically trivial sum, the same is true for the reservoir. We will show
that any homologically trivial collection of almost equidistributed short loops can be glued up
entirely, thus completing the construction of the $f$-folded fatgraph $X$, and the proof of
Theorem~\ref{thm:random_f_folded_theorem}. In fact, it is easier to show that a {\em multiple}
of any such collection can be glued up; thus the fatgraph we build will be assembled from the
disjoint union of copies of the partial fatgraphs built so far, glued up along the collection of
short loops, and $\partial^\pm$ will consist of the same number of disjoint copies of $[a,b]$ and $\phi([b,a])$.
This is evidently enough to prove the theorem.
The advantage of allowing ourselves to use multiple copies is that we can find a solution to
the gluing problem ``over the rationals''. Formally, let $\mathcal L$ denote the vector space spanned by
the set of types of short loops, let $\mathcal L^+$ denote the cone of vectors with non-negative coefficients,
and let $\mathcal L^+_0$ denote the subcone of homologically trivial vectors. The ``uniform'' vector ${\bf 1}$ is
the vector with all coefficients 1. This is in $\mathcal L^+_0$. Denote by $C$ the non-negative linear span
of the vectors representing collections that can be glued up. Then any rational vector in $C$ can be
``projectively'' glued up. It is easy to see that ${\bf 1}$ is in $C$; we will show that $C$ contains an
open neighborhood of the ray spanned by ${\bf 1}$. Since our collection of short loops is almost
equidistributed, it will be contained in this open neighborhood, and we will be done.
We call a vector {\em feasible} if it is in $C$.
\begin{proposition}\label{prop:rank_2_trivalent}
In the above notation and in a rank $2$ free group,
$C$ contains an open projective neighborhood of the ray spanned by ${\bf 1}$.
That is, there is some $\epsilon>0$ such that $C$ contains an
$\epsilon$ neighborhood of ${\bf 1}$ and thus contains all scalar multiples of
this neighborhood.
\end{proposition}
Proposition~\ref{prop:rank_2_trivalent} is actually proved with a computation.
We defer the proof and show how the arbitrary rank case reduces to rank 2.
First we give some notation necessary for the proof.
We define a function
$\iota$ on tagged loops which takes a loop $v$ to $v^{-1}$, with the tag in the
diametrically opposite position. When the tag switches positions under $\iota$,
there is a choice about what the tag becomes, because there is more than one possible tag
at each location in a word.
Choose tags arbitrarily such that $\iota$ is an involution. Notice that
for any loop $\gamma$, there is an annulus (trivalent fatgraph) with boundary
$\gamma + \iota(\gamma)$. We call this an \emph{$\iota$-annulus}.
\begin{proposition}\label{prop:higher_rank_trivalent}
In the above notation and for any finite rank free group,
$C$ contains an open projective neighborhood of the ray spanned by ${\bf 1}$.
\end{proposition}
\begin{proof}
Suppose we are given any vector $v \in \mathcal L'^+_0$. We must show that there is some
$n$ such that $v + n{\bf 1}$ bounds a trivalent fatgraph. This will prove
the proposition. Therefore, we need to understand when a collection
of loops, plus an arbitrary multiple of ${\bf 1}$, bounds a trivalent fatgraph.
Given a collection of tagged loops $S$ and a single loop $\gamma$,
suppose we can exhibit a trivalent fatgraph $Y$ with boundary
$\gamma + \sum_i \alpha_i$. We say that $\gamma$ is
\emph{fatgraph equivalent} to $\sum_i\iota(\alpha_i)$. Now, if
we can find a trivalent fatgraph with boundary $S + \sum_i \iota(\alpha_i)$,
then the union of this trivalent fatgraph with $Y$ has boundary
$S + \sum_i\iota(\alpha_i) + \alpha_i + \gamma$. In other words,
we have a trivalent fatgraph with boundary $S + \gamma$, plus two $\iota$ pairs.
If we then add all other remaining $\iota$ pairs, we can simply add
$\iota$-annuli to get a trivalent fatgraph with boundary $S + \gamma + {\bf 1}$.
Therefore, for the purpose of finding a trivalent fatgraph with a given boundary $S$
modulo adding some multiple of ${\bf 1}$, if we have some loop $\gamma \in S$,
and we find a collection of loops $\sum_i \alpha_i$ which is fatgraph equivalent to $\gamma$,
then we can throw out $\gamma$ from $S$, replace it with $\sum_i \alpha_i$, and find
a trivalent fatgraph bounding what remains.
To reduce to rank $2$, we'll show that any loop is fatgraph equivalent to a collection
of loops, each of which contains only two generators. After showing this,
we'll explain how to apply Proposition~\ref{prop:rank_2_trivalent} to complete
the proof.
\begin{lemma}\label{lem:equiv_to_rank_2}
Every loop is fatgraph equivalent to a collection of loops,
each containing at most two generators.
\end{lemma}
\begin{proof}
To show that a loop is fatgraph equivalent to another collection of
loops, note that it suffices to show it for \emph{un}tagged loops, provided
there are positions on the fatgraph $Y$ to place tags. This is because
a loop is fatgraph equivalent to itself with a different tag position, for
$\gamma + \gamma'^{-1}$ bounds a trivalent tagged annulus, where $\gamma'^{-1}$
is $\iota(\gamma)$ with the tag in a different position.
Thus, suppose we are given a loop $\gamma$ containing more than $2$ generators.
We partition $\gamma$ into \emph{runs} of a single generator, and because $\gamma$
has at least $3$ generators, we can write, $\gamma = abcx$, where $a$, $b$, and $c$
are runs of distinct generators, and $x$ is the remainder of $\gamma$, which might be
empty. Abusing notation, we will write $a$, $b$, $c$ to denote
a run of any size of the $a$, $b$, $c$ generators.
The trivalent fatgraph shown in Figure~\ref{fig:reduce_to_rank_2}
has boundary of the form $abcx + Bc + bA + CaX$. Note that this
fatgraph $\emph{is}$ trivalent; it cannot fold by the assumption that
$a$, $b$, $c$ are maximal runs of distinct generators. Also, the
edge lengths can be chosen so that all the loops have size $7$.
\begin{figure}[ht]
\labellist
\small\hair 2pt
\pinlabel {$a$} at 34 74
\pinlabel {$b$} at 55 34
\pinlabel {$c$} at 26 11
\pinlabel {$x$} at 10 48
\pinlabel {$B$} at 70 34
\pinlabel {$B$} at 81 60
\pinlabel {$c$} at 82 22
\pinlabel {$A$} at 43 88
\pinlabel {$b$} at 81 76
\pinlabel {$A$} at 72 97
\pinlabel {$X$} at -5 50
\pinlabel {$C$} at 15 -3
\pinlabel {$C$} at 94 11
\pinlabel {$a$} at 100 104
\endlabellist
\centering
\includegraphics[scale=1.0]{reduce_to_rank_2}
\caption{A trivalent fatgraph showing how a loop of the form $abcx$ is fatgraph
equivalent to two loops containing only two generators, plus a loop of the
form $XcA$.}
\label{fig:reduce_to_rank_2}
\end{figure}
Therefore, $abcx$ is fatgraph equivalent to two loops with two
generators, plus one loop of the form $Acx$. The latter loop
has, then, fewer runs, and we can repeat this procedure until $Acx$ contains
only two generators. At that point, we have shown that $\gamma$
is fatgraph equivalent to a collection of loops, each of which contains at
most two generators.
\end{proof}
Now we can complete the proof of Proposition~\ref{prop:higher_rank_trivalent}.
Given a vector $v \in \mathcal L'^+_0$, we may take a sufficient multiple to assume that
$v$ is integral. Let $S$ be the collection of loops represented by $v$.
We've shown that $S$ is fatgraph equivalent to a collection of loops $S'$,
where each loop in $S'$ contains at most two generators. We can write
$S' = \bigcup_{\{x,y\}} S'_{x,y}$, where each $S'_{x,y}$ contains the loops in $S'$
containing the generators $x$ and $y$. There is an ambiguity about where to
put loops which contain a single generator; place them arbitrarily, though we will rearrange
them presently.
Now, each collection $S'_{x,y}$ might not be homologically trivial.
However, after multiplying the original vector $v$ by $7$, we may assume that the
homological defect of each generator in each collection is a multiple of $7$.
Therefore, we can make each collection $S'_{x,y}$ homologically trivial by redistributing
the loops consisting of a single generator (or, if necessary, introducing $\iota$ pairs).
Since each collection $S'_{x,y}$ is homologically trivial, we can apply
Proposition~\ref{prop:rank_2_trivalent} to find a trivalent fatgraph with boundary
$S'_{x,y} + {\bf 1}_{x,y}$, where ${\bf 1}_{x,y}$ denotes the uniform vector
in the set of loops containing generators $x$ and $y$. Taking the union of
these fatgraphs over all $x$, $y$, yields a trivalent fatgraph
which shows that $S' = \bigcup_{\{x,y\}}S'_{x,y}$ is fatgraph equivalent to a
collection of $\iota$ pairs, and therefore fatgraph equivalent to the empty set.
That is, the collection $S$ represented by our original vector $v$ is
fatgraph equivalent to the empty set, which is to say that there is a
trivalent fatgraph with boundary $S + n{\bf 1}$, for $n$ sufficiently large.
This completes the proof.
\end{proof}
We now give the proof in the rank $2$ case, or rather, a description
of the computation which proves it.
\begin{proof}[Proof of Proposition~\ref{prop:rank_2_trivalent}]
We wish to show that the cone $C$ contains an open projective
neighborhood of ${\bf 1}$. To start, we describe some necessary background about
cones and linear programming.
Determining if a point lies in the interior of the cone on
a set of vectors can be phrased as a linear programming problem by using the
following lemma.
\begin{lemma}\label{lem:cone_interior}
Let $x,v_1, \ldots, v_k \in \R^n$. If the $v_i$ span $\R^n$ and there is an expression
$x = \sum_it_iv_i$ with $t_i > 0$ for all $i$, then $x$ lies in the
interior of the cone spanned by the $v_i$.
\end{lemma}
\begin{proof}
To show that $x$ lies in the interior, it suffices to show that
$x$ is not contained in any supporting hyperplane. Thus, let $H$
be any supporting hyperplane for the cone spanned by the $v_i$.
Because the $v_i$ span $\R^n$, there is some $j$ so that $v_j$ is
not in $H$. We have expressed $x = \sum_it_iv_i$ with, in particular, $t_j>0$.
Therefore, if we decompose $\R^n = H \oplus \textnormal{span}\{v_j\}$,
and express $x$ in this decomposition,
we will find that the coefficient of $v_j$ is not zero, so $x$ is
not contained in $H$.
\end{proof}
Using Lemma~\ref{lem:cone_interior}, we observe that if we are given
the $v_i$ as the columns of the matrix $M$, and $x$ is a column vector,
then a feasible point $y$ for the problem $Ay = x$, $y\ge 1$
provides a certificate that $x$ is contained in the interior.
Feasibility testing can be phrased as a linear programming
problem by setting the objective function to zero. We remark that the lower bound
on $y$ is arbitrary; if $x$ is in the interior, the linear program will
succeed for \emph{some} lower bound, but we don't know a priori what it is.
The proof of Proposition~\ref{prop:rank_2_trivalent} therefore reduces
to the following computation.
\begin{enumerate}
\item Find a collection of vectors $V$ in the cone $C$ which together
span the space $\mathcal L_0$.
\item Show that the uniform vector ${\bf 1}$ lies in the cone on $V$.
\end{enumerate}
To find $V$, we simply tried many random vectors in $\mathcal L_0$ and
checked if they were contained in $C$ by checking if they
bounded a trivalent fatgraph.
Both steps (1) and (2) require linear programming:
in order to check that a
vector bounds a trivalent fatgraph, we solve a linear programming problem
derived from the \texttt{scallop} algorithm (\cite{scallop}),
and to show that the uniform vector
lies in the cone on $V$, we solve the linear programming problem
derived from Lemma~\ref{lem:cone_interior}.
In order to make the many linear programming problems in step (1) feasible,
we need to choose \emph{low-density} vectors; that is, vectors with a
small number of nonzeros. Recall there are $4376$ short loops of
length $7$ and rank $2$, and the space of homologically trivial
linear combinations has dimension $4374$.
We found a collection of $9626$ vectors, each with $8$ nonzeros,
which span this $4374$-dimensional space. This required solving
a few tens of thousands of small linear programming problems (i.e. runs of \texttt{scallop}),
which was easily accomplished using the linear programming backend
\texttt{GLPK}\cite{GLPK}.
As we built this collection, we occasionally ran a much larger
linear program to determine if the uniform vector was contained
in the cone (not necessarily in the interior, as that is more difficult to
solve in practice). Once our cone did contain the uniform vector,
we ran one final linear program to verify that it lay in the interior.
Even though these latter linear programs had only a few thousand columns
and a few thousand rows, they proved quite difficult in practice.
Fortunately, the proprietary software package
\texttt{Gurobi}\cite{gurobi}, which offers a free academic license,
was able to solve them in a few minutes. We used \texttt{Sage}\cite{sage}
to facilitate many of the final steps.
\end{proof}
\begin{remark}
It is important to highlight that the initial steps of the proof of the
random $f$-folded surface theorem reduce the problem
of finding a folded surface whose boundary is a given random loop to the
problem of showing that a collection of tagged loops of
a \emph{uniformly bounded} size ($7$) bounds a folded fatgraph, provided
this collection is sufficiently close to uniform. Proposition~\ref{prop:higher_rank_trivalent}
shows that, indeed, a collection of tagged loops of size $7$
sufficiently close to uniform does bound a folded fatgraph. We emphasize
that this linear programming is done \emph{once} to solve this
latter, uniformly bounded, one-time problem.
\end{remark}
This completes the proof of Theorem~\ref{thm:random_f_folded_theorem}.
\subsection{Sapir's group}\label{subsection:Sapir_group}
\begin{definition}
We define {\em Sapir's group} $C$ to be the HNN extension of $F_2:=\langle a,b\rangle$ by the
endomorphism $\phi:a \to ab, b \to ba$.
\end{definition}
In \cite{Sapir}, Problem.~8.1, Sapir posed explicitly the problem of determining whether $C$
contains a closed surface subgroup, and in fact conjectured (in private communication) that
the answer should be negative. This group was also studied by Crisp--Sageev--Sapir and
(independently) Feighn, who sought to find a surface subgroup or show that one did not exist.
Because of the attention this particular question has attracted, we consider it significant
that our techniques are sufficiently powerful to give a positive answer:
\begin{theorem}
Sapir's group $C$ contains a closed surface subgroup of genus 28.
\end{theorem}
\begin{proof}
The theorem is proved by exhibiting an explicit $f$-folded surface. Figure~\ref{sapir_example}
indicates a fatgraph whose fattening has four boundary components, three of which are (conjugates
of) $babaBABA$ and the fourth of which is $\phi^4(babaBABA)^{-3}$. The blue circles mark
the $babaBABA$ components. By taking a 3-fold cover we obtain a fatgraph whose fattening has
six boundary components, three of which are conjugates of $(babaBABA)^3$, and three of which are
conjugates of $\phi^4(babaBABA)^{-3}$. In the HNN extension $F*_\phi$ we can glue these boundary
components in pairs, giving a closed surface $S$ together with a map $\pi_1(S) \to F*_\phi$.
The surface is $f$-folded, and therefore the resulting map of the surface group is injective. To
see this, note that the $babaBABA$ components are disjoint from each other,
the underlying fatgraph is Stallings folded, and the
$f$-vertices (indicated in red) are all 2-valent.
\begin{figure}[htpb]
\centering
\includegraphics[scale=0.3]{sapir_babaBABA}
\caption{A fatgraph bounding $3\cdot babaBABA+\phi^4(babaBABA)^{-3}$}\label{sapir_example}
\end{figure}
\end{proof}
\begin{remark}
In fact, Sapir expressed the opinion that ``most'' ascending
HNN extensions of free groups should not contain surface subgroups, which is contradicted by
the Random $f$-folded Surface Theorem~\ref{thm:random_f_folded_theorem}. On the other hand,
the probabilistic estimates involved in the proof of this theorem are only relevant for
endomorphisms taking generators to very long words, and therefore Sapir's group seems to be
an excellent test case.
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 952
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Bei den VIII. Olympischen Sommerspielen 1924 in Paris fanden drei Wettkämpfe im Segeln statt.
Die Regatten wurden jeweils mit Vorrunde und Finale ausgetragen. Im Finale wurden die Platzierungen der beiden Regatten für das Gesamtergebnis addiert, bei Punktegleichheit entschied die bessere Zeit.
Bilanz
Medaillenspiegel
Medaillengewinner
Ergebnisse
Monotyp 1924
Die Regatten fanden vom 10. bis zum 13. Juli auf der Seine zwischen Meulan und Les Mureaux statt.
6-Meter-Klasse
Die Regatten fanden vom 21. bis zum 26. Juli vor Le Havre statt.
8-Meter-Klasse
Die Regatten fanden vom 21. bis zum 26. Juli vor Le Havre statt.
Literatur
Volker Kluge: Olympische Sommerspiele. Die Chronik I. Athen 1896 – Berlin 1936. Sportverlag Berlin, Berlin 1997, ISBN 3-328-00715-6.
Weblinks
Segeln
1924
Sportveranstaltung in Le Havre
Meulan-en-Yvelines
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
Among many identities, the classical families of orthogonal polynomials satisfy
a three-term recurrence
\begin{equation}\label{eq:3-term}
xp_n(x)=a_np_{n+1}(x)+b_np_n(x)+c_np_{n-1}(x),
\end{equation}
where $a_n,b_n,c_n$ are rational functions of~$n$~\cite{OlverLozierBoisvertClark2010};
a differential-difference equation
\begin{equation}\label{eq:mixed}
\pi(x)p_n'(x)=\alpha_np_{n+1}+\beta_np_n+\gamma_np_{n-1},
\end{equation}
with $\pi(x)$ a polynomial of degree at most~2 and
$\alpha_n,\beta_n,\gamma_n$ rational functions
of~$n$~\cite{Lewanowicz2002};
they also satisfy a second order linear differential equation with
polynomial
coefficients~\cite{Szego1975}.
Furthermore, the classical families of Jacobi, Laguerre, Hermite and Bessel are
\emph{hypergeometric}. They can be expressed
in terms of the functions ${}_2F_1$ and ${}_2F_0$~\cite[\S18.5(iii),\S18.34]
{OlverLozierBoisvertClark2010}, special cases of the generalized hypergeometric
series (see \cref{sec:def_hgm} for definitions).
In this work, we focus on two properties of the classical orthogonal polynomials
that persist for some of the generalized hypergeometric series. The
first one is a
generalization of
the 3-term recurrence~\eqref{eq:3-term}:
\begin{equation}\label{eq:X}
x(A_m(n)p_{n+m}(x)+\dots+A_0(n)p_n(x))=B_\ell(n)p_{n+\ell}
(x)+\dots+B_0(n)p_n(x),
\tag{$\mathcal X$}
\end{equation}
where the $A_i$ and $B_j$ are polynomials in~$n$, but do not depend on~$x$.
In other words, the sequence $(p_n)$ satisfies a linear recurrence with
coefficients that are polynomials of degree
at most~1 in~$x$.
The second relation is a
differential-difference equation
of the form
\begin{equation}\label{eq:D}
\left(C_r(n)p_{n+r}(x)+\dots+C_0(n)p_n(x)\right)'=
D_s(n)p_{n+s}(x)+\dots+D_0(n)p_n(x),\tag{$\mathcal D$}
\end{equation}
where again $C_i$ and $D_j$ are polynomials in~$n$, but do not depend on~$x$.
We did not find any explicit mention of the relation~\eqref{eq:D} for
classical orthogonal polynomials in the literature. It can be seen by
differentiating
the three-term
recurrence \cref{eq:3-term}, which allows one to rewrite $xp_n'$ as a linear combination of shifts of~$p_n$ and $p_n'$ with coefficients that do not depend on~$x$. Then by induction, for all nonnegative integers~$k$, $x^kp_n$ and $x^kp_n'$
can also be written that way.
Another equation of interest is a mixed
difference-differential equation
\begin{equation}\label{eq:M}
\pi(x)p_n'(x)=E_{-s}(n)p_{n-s}(x)+\dots+E_{t}(n)p_{n+t}(x),
\tag{$\mathcal M$}
\end{equation}
with $\pi(x)$ a polynomial in~$x$ and $E_{-s}(n),\dots,E_{t}(n)$
rational functions
of~$n$ that do not depend on~$x$. When the $p_n$ are orthogonal
polynomials, such relations characterize semi-classical polynomials~\cite{Maroni1987,Maroni1991}.
A generalization of the derivation above is given in
\cref{prop:XM-give-XD}. It shows that any solution of both an
equation of type~\eqref{eq:X} and an equation of type~\eqref{eq:M} also
satisfies an equation of type~\eqref{eq:D}.
Actually, \cref{eq:D} is strictly more general, as there are
functions satisfying equations of type~\eqref{eq:X}
and~\eqref{eq:D} that do not satisfy any equation of
type~\eqref{eq:M}. (An example is given in \cref{subsubsec:DmgM}.)
\subsection*{Contribution}
Understanding the extent to which relations~\eqref{eq:X}
and~\eqref{eq:D} exist for
hypergeometric polynomials was our initial motivation for this
study.
We focus on six families of
generalized
hypergeometric series, given in \cref{table:families}. (Basic definitions and
properties are recalled in \cref{sec:def_hgm}.)
We show that they satisfy equations of the
types~\eqref{eq:X} and~\eqref{eq:D} and we give explicit
factorizations of the
linear recurrences that appear in these equations, as products of first
order operators.
\begin{table}[t]
\begin{small}
\begin{align}
{}_{p}F_{m+q}&\!\left(\left.\begin{matrix}
(a_p)\\
\Delta(m;\lambda+n),(b_q)
\end{matrix}\right|x\right),\tag{I}\\
{}_{m+p}F_{q}&\!\left(\left.\begin{matrix}
\Delta(m;1-\lambda-n),(a_p)\\
(b_q)
\end{matrix}\right|x\right),\tag{II}\\
{}_{m+p}F_{m+q}&\!\left(\left.\begin{matrix}
\Delta(m;\mu+n),(a_p)\\
\Delta(m;\lambda+n),(b_q)
\end{matrix}\right|x\right),\tag{III}\\
{}_{m+p}F_{m+q}&\!\left(\left.\begin{matrix}
\Delta(m;1-\lambda-n),(a_p)\\
\Delta(m;1-\mu-n),(b_q)
\end{matrix}\right|x\right),\tag{IV}\\
{}_{2m+p}F_{q}&\!\left(\left.\begin{matrix}
\Delta(m;1-\lambda-n),\Delta(m;\mu+n),(a_p)\\
(b_q)
\end{matrix}\right|x\right),\tag{V}\\
{}_{p}F_{2m+q}&\!\left(\left.\begin{matrix}
(a_p)\\
\Delta(m;\lambda+n),\Delta(m;1-\mu-n),(b_q)
\end{matrix}\right|x\right).\tag{VI}
\end{align}
\end{small}
\caption{Families of generalized hypergeometric functions. There, $m$ denotes a
positive integer; $\lambda\neq\mu$ are two arbitrary, but distinct,
constants; $n$ is a nonnegative integer; $(a_p)$ denotes the sequence
of complex numbers
$a_1,\dots,a_p$ and likewise for~$(b_q)$; $\Delta(m;x)$ is the
sequence $x/m,(x+1)/m,\dots,(x+m-1)/m$.}
\label{table:families}
\end{table}
Family~I generalizes the Bessel functions
\[J_\nu(z)=\frac{\left(\frac z2\right)^\nu}{\Gamma(\nu+1)}{}_0F_1\left
(\left.\begin{matrix}\\ \nu+1\end{matrix}\right|-\frac{z^2}4\right)\]
up to a monomial factor.
Families II and V generalize the extended Laguerre and Jacobi
functions
\begin{align}\label{eq:exLagJac}
&
{}_{p+1}F_{q}\!
\left(
\left.
\begin{matrix}
-n-\lambda,a_1,\dots,a_p\\
b_1,\dots,b_q
\end{matrix}
\right|
x
\right),
&
{}_{p+2}F_{q}\!\left(\left.\begin{matrix}
-n-\lambda,n+\mu,a_1,\dots,a_p\\
b_1,\dots,b_q
\end{matrix}\right|x\right).
\end{align}
The classical Laguerre and Jacobi polynomials are recovered, up to a simple
change of
variable, when
$\lambda=0,p=0,q=1$.
The case $\lambda=0$ of Family~II was used by
Brafman~\cite{Brafman1957} to produce several generating functions.
Other generating functions are known for cases of
families~II, IV, V that correspond to polynomials~\cite[chap.~7]{SrivastavaManocha1984}.
\subsection*{Related work}
The closest relation to our work that we could find in the literature are
explicit
equations of the types~\eqref{eq:D} and~\eqref{eq:X} given by Fields, Luke
and Wimp for the extended Laguerre
and Jacobi functions of \cref{eq:exLagJac}
in their study of Pad\'e approximants of hypergeometric functions~\cite{FieldsLukeWimp1968,Wimp1975,Luke1975}.
In the
case of Jacobi functions, this was later improved by Lewanowicz~
\cite{Lewanowicz1985}.
We show in \cref{sec:examples} how the results of Fields, Luke and Wimp are
recovered as special cases of our main result.
\section{Definitions and Notation}\label{sec:def-and-not}
\subsection{Generalized Hypergeometric Function}\label{sec:def_hgm}
We gather here basic material on \emph{generalized hypergeometric series} (see \cite[chap.16]
{OlverLozierBoisvertClark2010}).
Those are classically
defined as the formal power series
\begin{equation}\label{eq:defhgm}
{}_pF_q\left(\left.\begin{matrix}a_1,\dots,a_p\\ b_1,\dots,b_q
\end{matrix}\right|x\right):=\sum_{k\ge0}\frac{(a_1)_k\dotsm(a_p)_k}{
(b_1)_k\dotsm
(b_q)_k}\frac{x^k}{k!},
\end{equation}
where $(a)_k=a(a+1)\dotsm(a+k-1)$ is the Pochhammer symbol and it is assumed
that none of the $b_i$ is a nonpositive integer, so that the denominators are
nonzero for all~$k$.
In the hypergeometric representations of the classical
orthogonal polynomials, the parameter $a_1$ is $-n$, so that the series
becomes a polynomial of degree~$n$. In general, the power series ${}_pF_q$ is a
polynomial if one of
the~$a_i$ is a negative integer. Except in this situation, it is
divergent in the neighborhood of~0 when $p>q+1$, and convergent
otherwise.
If $U_k$ denotes the coefficient of~$x^k$ in the power
series~\eqref{eq:defhgm}, then it follows from the definition of the
Pochhammer symbol that
\begin{equation}\label{eq:rec_hgm}
(k+1)\prod_{j=1}^q(b_j+k)U_{k+1}=\prod_{j=1}^p(a_j+p)U_k.
\end{equation}
It follows that the
generalized hypergeometric series satisfies a linear differential equation~\cite
[16.8.3]{OlverLozierBoisvertClark2010}
\begin{equation}\label{deq:hgm}
\left(\vartheta(\vartheta+b_{1}-1)\cdots(\vartheta+b_{q}-1)-z(\vartheta+a_{1})%
\cdots(\vartheta+a_{p})\right)w=0,
\end{equation}
where $\vartheta=z\frac{d}{dz}$ and whose order, $\max(p,q+1)$, is minimal.
\subsection{Linear Recurrence Operators}\label{sec:linrecop}
Our results and proofs are more conveniently expressed using linear recurrence
operators. We denote by $S_n$ the \emph{shift operator} that maps a sequence $
(u_n)$ to the sequence $(u_{n+1})$. The linear recurrence operators we consider
are polynomials in~$S_n$ with coefficients that are polynomial or rational
functions in~$n$. Their addition is that of commutative polynomials and
their multiplication follows from the commutation rule $S_na(n)=a(n+1)S_n$, for
any rational function~$a(n)$. These polynomials admit a Euclidean division on
the right, and a Euclidean algorithm that allows for the definition of greatest
common right divisors and least common left multiples (denoted
lclm)~\cite{Ore1933}. They are both defined up to a rational
factor. These
notions are exemplified in the following two results.
\begin{lemma}For distinct $a_1,\dots,a_p$ that do not depend on~$n$,
\begin{multline*}
\operatorname{lclm}\!\left(\frac{n+a_1+1}{n+1}S_n-1,\dots,\frac{n+a_p+1}
{n+1}S_n-1\right)\\
=\left(\frac{n+a_p+p}{n+1}S_n-1\right)\dotsm\left(
\frac{n+a_1+1}{n+1}S_n-1\right)=:Q_p.
\end{multline*}
\end{lemma}
\begin{proof}
Let $M_p$ denote the lclm and $Q_p$ the product.
We first show that the remainder
of the right division of~$Q_p$ by
\[L_b:=S_n-\frac{n+1}{n+b+1}\]
is
\[R_{p,b}(n)=\frac{(a_1-b)\dotsm(a_p-b)}{(n+b+1)\dotsm(n+b+p)}.\]
For $p=0$, the remainder of the division of~$Q_0=1$
by~$L_b$ is~1, which corresponds to the empty product.
Next, by induction, since $Q_p\bmod L_b=R_{p,b}(n)$, it is sufficient to
consider
\begin{align*}
& \left(\frac{n+a_{p+1}+p+1}{n+1}S_n-1\right)R_{p,b}(n)\\
&\quad=
\frac{n+a_{p+1}+p+1}{n+1}R_{p,b}(n+1)S_n-R_{p,b}(n)\\
&\quad=
\frac{n+a_{p+1}+p+1}{n+1}R_{p,b}(n+1)L_b+\left(\frac{n+a_{p+1}+p+1}
{n+b+1}R_{p,b}(n+1)-R_{p,b}(n)\right).
\end{align*}
The remainder of the right division by~$L_b$ factors as
\[R_{p,b}(n)\left(\frac{n+a_{p+1}+p+1}
{n+b+p+1}-1\right)=R_{p,b}(n)\frac{a_{p+1}-b}{n+b+p+1},\]
which concludes the induction.
Taking~$b=a_i$ for $i=1,\dots,p$ makes $R_{p,b}=0$, showing that $M_p$
divides~$Q_p$ for all~$p$.
Another
induction on~$p$ establishes that $M_p=Q_p$. The case
$p=1$ is clear.
If $a_{p+1}$ is distinct
from the other~$a_i$, taking
$b=a_{p+1}$ shows that the remainder~$R_{p+1,a_{p+1}}$ is not~0,
which implies that $\deg M_{p+1}=\deg M_p+1=\deg Q_{p+1}$,
concluding the proof since $M_
{p+1}$ divides $Q_{p+1}$.
\end{proof}
\begin{lemma}\label{lemma:explicit-coeffs}
For arbitrary $a_1,\dots,a_p$ that do
not depend on~$n$, let
$Q_p$ be
the product defined in the previous lemma. Then, the coefficients of its
expansion in powers of~$S_n$ are given by
\[Q_p=\sum_{m=0}^p{\frac{c_m(n)}{(p-m)!}\frac{\prod_{i=1}^p(n+a_i+m)}{\prod_
{i=1}^m
{
(n+i)}}S_n^m},\]
with
\[c_m(n)={}_{p+1}F_p
\!\left(\left.
\begin{matrix}m-p,n+m+a_1+1,\dots,n+m+a_p+1\\
n+m+a_1,\dots,n+m+a_p\end{matrix}\right|1\right).\]
\end{lemma}
\begin{proof}
The proof is by induction. For $p=0$, the empty product $Q_0$ is equal to~1 and
so is the
hypergeometric series, since $0$ is its top parameter.
Next, multiplying the sum for~$Q_p$ by
\[\frac{n+a_{p+1}+p+1}{n+1}S_n-1\] shows that the coefficient of~$S_n^m$
in~$Q_{p+1}$ is
\begin{multline*}
\frac{n+a_{p+1}+p+1}{n+1}\frac{c_{m-1}(n+1)}{(p-m+1)!}\frac{\prod_{i=1}^p
(n+1+a_i+m-1)}{\prod_{i=1}^{m-1}{(n+1+i)}}
-\frac{c_m(n)}{(p-m)!}\frac{\prod_{i=1}^p(n+a_i+m)}{\prod_{i=1}^m
{
(n+i)}}\\
=\frac{\prod_{i=1}^{p+1}{(n+a_i+m)}}{(p+1-m)!\prod_{i=1}^m(n+i)}
\frac{(n+a_{p+1}+p+1)c_{m-1}(n+1)-(p+1-m)c_m(n)}{n+a_{p+1}+m}.
\end{multline*}
The coefficient~$c_{k,m,n,p}$ of $x^k$ in the hypergeometric series
defining~$c_m(n)$ is
\[(m-p)_k\frac{(n+m+a_1+k)\dotsm(n+m+a_p+k)}{(n+m+a_1)\dotsm
(n+m+a_p)},\qquad 0\le k\le m-p\]
and~0 for $k>m-p$. From there,
\begin{align*}
&\frac{(n+a_{p+1}+p+1)c_{k,m-1,n+1,p}-(p+1-m)c_{k,m,n,p}}{n+a_{p+1}+m}\\
&\qquad=\frac{c_{k,m,n,p}(m-p-1)}{n+a_{p+1}+m}\left(\frac{n+a_{p+1}+p+1}
{m-p+k-1}+1\right)\\
&\qquad=c_{k,m,n,p}\frac{m-p-1}{m-p+k-1}\frac{n+m+a_{p+1}+k}{n+m+a_
{p+1}}=c_{k,m,n,p+1}.\qedhere
\end{align*}
\end{proof}
\section{Main result}
\begin{theorem}\label{thm:main}
Let $F_n$ be in any of the six families
in \cref{table:families}, with parameters such that~$F_n$ is a
well-defined hypergeometric series for all~$n\in\mathbb N$.
Then the power series
$xF_n$,
$F_n'$ and $F_n$ are related by the following two recurrence equations
\[\begin{split}
&\left((\epsilon_1\epsilon_2S_n)^{\theta}
\mathcal L_{p}^{q+1-2m\chi\epsilon_1}
\mathcal A\right)(xF_n)=\\
&\qquad\left(
m^{m(\epsilon\epsilon_1-\epsilon_2)}\left(
\frac{(\lambda+n)_m}{
(\mu+n)_m^\epsilon}\right)^{\!\epsilon_1}
(\epsilon_1\epsilon_2S_n)^{m-\theta}
\mathcal L_{q+1}^{p+2m\chi\epsilon_1}
\mathcal F_{0,q+1}\mathcal B\right)
(F_n),\\
&\left(
(\epsilon_1\epsilon_2S_n)^{m-\theta}
\mathcal L_{q}^{p+2m\chi\epsilon_1}
\mathcal B\right)(F_n')=\\
&\qquad
\left(
m^{m(\epsilon_2-\epsilon\epsilon_1)}\left(
\frac{(\lambda+n)_m}{
(\mu+n)_m^\epsilon}\right)^{\!-\epsilon_1}
\!(\epsilon_1\epsilon_2S_n)^{\theta}
\mathcal L_{p}^{q-2m\chi\epsilon_1}
\mathcal A\right)(F_n),
\end{split}\]
where $\epsilon=1$ if $F_n$
depends on~$\mu$ and $\epsilon=0$ otherwise; $\epsilon_1$ and
$\epsilon_2$ in $\{-1,1\}$ are
the signs in front of~$\mu$ and~$\lambda$ in the parameters of the
hypergeometric series
(for convenience, we let
$\epsilon_1=1$ when $\epsilon=0$), and
\begin{gather*}
a(n,k)=\frac{n+\lambda}
{n+\epsilon\mu}\,
\frac{n+\epsilon(\mu+\epsilon_1mk)}
{n+\lambda+\epsilon_2mk},\qquad\alpha(n,k)=\frac1
{n+\lambda+\epsilon_2mk},\\
C_i(n)=
\frac{n+\lambda}{n+\epsilon\mu}
(n+\epsilon\mu-\epsilon\epsilon_1\epsilon_2(\lambda+n+i)),
\quad \mathcal C_i=\frac1{C_i(n)}\left(S_n-\frac{\epsilon\epsilon_1
(n+\lambda)}{\epsilon_2
(n+\epsilon\mu)}\right),
\\
B_i(n,c)=-m\epsilon_2\alpha
(n+i-1,-c)C_{i-1}(n),\\
\mathcal F_{c,i}=\frac1{B_i(n,c)}\left(S_n-\frac{a(n,-c)\alpha
(n+i-1,-c)}
{\alpha(n,-c)}\right),\\
\mathcal B=\mathcal F_{b_q-1,q}\dotsm\mathcal F_
{b_1-1,1},\quad
\mathcal A=\mathcal F_{a_p,p}\dotsm\mathcal F_{a_1,1},
\quad
\mathcal L_i^{j}=\begin{cases}\mathcal C_{j-1}\dotsm\mathcal
C_i&\text{if $i<j$},\\ 1&\text{otherwise},\end{cases}\\
\theta=\begin{cases}m&\quad\text{if
$\epsilon_1=1$,}\\0&\quad\text{otherwise,}\end{cases}
\quad
\chi=\begin{cases}0&\quad\text{if $\epsilon_1=\epsilon_2$,}\\
1&\quad\text{otherwise.}
\end{cases}
\end{gather*}
In the special case when~$\epsilon = 1$ and $\epsilon_1=\epsilon_2$ (families III
and~IV) and furthermore~$\mu-\lambda\in\{0,\dots,\max(p-1,q)\}$, these
formulas do not apply directly as a required~$C_i(n)$ vanishes. An
identity is recovered by truncating the operators, keeping all the
right factors up to the division by this~$C_i(n)$ excluded. In that
situation, both terms of the identity vanish.
\end{theorem}
With extra care, the cases of parameters that make the hypergeometric
series be well defined only for sufficiently large or sufficiently
small~$n$ can be handled as well.
\section{Examples}\label{sec:examples}
\subsection{Legendre Polynomials}
One of the simplest examples, the Legendre polynomials, may help clarify
the notation. These polynomials are given as
\[P_{n}\left(x\right)={{}_{2%
}F_{1}}\left(\left.{-n,n+1\atop1}\right|\frac{1-x}{2}\right).\]
Thus they equal $F_n((1-x)/2)$, where $F_n$ is the special case of
family~(V) with~$p=0$, $m=q=b_1=\lambda=\mu=1$. With these values of the parameters, the theorem gives~$\epsilon=1,\epsilon_1=1,\epsilon_2=-1,\theta=\chi=1$,
\begin{gather*}
a(n,0)=1,\quad\alpha(n,0)=\frac{1}{n+1},\quad
B_1(n,0)=2,\quad
B_2(n,0)=\frac{2n+3}{n+2},
\\
\mathcal B=\mathcal F_{0,1}=\frac1{2}(S_n-1),\quad\mathcal
A=\mathcal L_2^1=\mathcal L_2^2=1,\quad
\mathcal F_{0,2}=\frac{n+2}{2n+3}\left(S_n-\frac{n+1}{n+2}\right),\\
\mathcal L_1^2=\mathcal C_1=\frac{1}{2n+3}(S_n+1).\\
\intertext{The relations given by the theorem are therefore}
\begin{cases}-S_n (xF_n)=\frac{n+2}{2n+3}\left(S_n-\frac{n+1}
{n+2}\right)\frac1
{2}(S_n-1)(F_n), \\
\frac{1}{2(2n+3)}(S_n+1)(S_n-1)(F_n')=-S_n(F_n).
\end{cases}
\end{gather*}
Replacing $x$ by $(1-x)/2$ in the first equation and using the fact
that $F_n'((1-x)/2)=-2P_n'(x)$ in the second one retrieves
relations that can be seen to be equivalent to classical ones~\cite
[18.9.1,18.9.17]{OlverLozierBoisvertClark2010}
\[
\begin{cases}\frac{x-1}2P_{n+1}=\frac{1}{2(2n+3)}((n+2)P_{n+2}-
(2n+3)P_{n+1}+
(n+1)P_n),\\
\frac1{2n+3}(P_{n+2}'-P_n')=P_{n+1}.
\end{cases}
\]
\subsection{Extended Laguerre Polynomials}
These are the special case of family (II) with $m=1$ and $\lambda=1$.
Then $\epsilon=0,\epsilon_1=1,\epsilon_2=-1$ and $\theta=\chi=1$.
We give the explicit factorization of the relations given by
\cref{thm:main}. The first one can be compared with the formula
given by Fields, Luke and Wimp~\cite[Cor.~2.2]{FieldsLukeWimp1968}\footnote{Fields, Luke and Wimp take one of the $a_i$s to be~1, but this does
not impact the results mentioned here.}.
The quantities involved in the theorem are
\[\mathcal C_i=\frac{1}{n+1}S_n,\quad
\mathcal F_{c,i}=\frac{n+i+c}{n+1}S_n-1,\]
whence the formulas
\begin{multline*}
-\left(\frac{S_n}{n+1}\right)^{\max(0,q-p-1)}S_n\prod_
{i=1}^p\left(\frac{n+a_i+i}{n+1}S_n-1\right)(xF_n)=\\
(n+1) \left(\frac{S_n}{n+1}\right)^{\max(0,p+1-q)}\left(\frac{n+q+1}
{n+1}S_n-1\right)\prod_
{i=1}^q\left(\frac{n+b_i+i-1}{n+1}S_n-1\right)(F_n),\\
(n+1)\left(\frac{S_n}{n+1}\right)^{\max(0,p+2-q)}\prod_
{i=1}^q\left(\frac{n+b_i+i-1}{n+1}S_n-1\right)(F_n')=\\
\left(\frac{S_n}{n+1}\right)^{\max(0,q-p-2)}S_n\prod_
{i=1}^p\left(\frac{n+a_i+i}{n+1}S_n-1\right)(F_n),
\end{multline*}
where the products are to be interpreted as follows for
any~$M_i$:
\[\prod_{i=j}^kM_i=\begin{cases}M_k\dotsm M_j,\quad&\text{if $k\ge
j$,}\\ 1&\text{otherwise.}\end{cases}\]
The explicit hypergeometric coefficients given by Fields, Luke and
Wimp come from \cref{lemma:explicit-coeffs}. Note however that
they give a non-homogeneous equation, while our result is a homogeneous one.
\subsection{Special Case}
As an illustration of the special case at the end of the theorem,
consider
\[F_n={}_{2}F_{2}\!\left(\left.\begin{matrix}
\lambda+n+1,a\\
\lambda+n,b
\end{matrix}\right|x\right).\]
The second identity of the theorem does not lead to any division by~0
and gives
\begin{multline*}
\left(-\frac{(n+\lambda+1)(n+\lambda+1-b)}
{n+\lambda}S_n+n+\lambda+2-b\right)(F_n')=\\
\frac{n+\lambda+1}{n+\lambda}S_n\left(-\frac{(n+\lambda+1)(n+\lambda-a)}
{n+\lambda}S_n+n+\lambda+1-a
\right)(F_n).
\end{multline*}
The first one involves $\mathcal C_1$ and $\mathcal F_{0,2}$, both
with a division by $C_1(n)=0$. Stopping before this division gives an
identity where both sides are~0:
\begin{multline*}
\left(S_n-\frac{n+\lambda}{n+\lambda+1}\right)\left(-\frac{(n+\lambda+1)(n+\lambda-a)}
{n+\lambda}S_n+n+\lambda+1-a
\right)(xF_n)=\\
\left(S_n-\frac{n+\lambda}{n+\lambda+1}\right)
\left(-\frac{(n+\lambda+1)(n+\lambda+1-b)}
{n+\lambda}S_n+n+\lambda+2-b\right)(F_n)
=0.\end{multline*}
\section{Proof of the main result}
If~$F_n$ is any of the power series from~I to~VI and $U_{n,k}$ denotes the
coefficient of~$x^k$ in~$F_n$, we compare the actions of certain recurrence operators independent
of~$k$ on the coefficient of~$x^k$ in~$xF_n$ and $F_n$ (for
$\mathcal X$), resp. the coefficient of~$x^{k-1}$ in~$F_n'$ and~$F_n$ (for $\mathcal D$),
i.e.,
\begin{equation}\label{eq:very-simple}
U_{n,k-1} \textrm{ and } U_{n,k},\quad \textrm{ resp. }\quad kU_
{n,k} \textrm{ and } U_{n,k-1}.
\end{equation}
The proof is constructive: iteratively, operators in the shift~$S_n$
\emph{with
coefficients that do not depend on~$k$} are built so that in the
end~$\mathcal X$ and $\mathcal D$ are obtained.
The following two identities are satisfied
by $U_{n,k}$
in all cases:
\begin{equation}
\frac{U_{n+1,k}}{U_{n,k}}
=a(n,k),\qquad
\frac{U_{n,k-1}}{U_{n,k}}=\psi_0(n,k)\frac{(b_1+k-1)\dotsm
(b_q+k-1)k}{(a_1+k-1)\dotsm(a_p+k-1)},\label{eq:shifts}
\end{equation}
with a $\psi_0$ that does not depend on~$(a_p)$ and~$(b_q)$ and with
$a(n,k)$ from the theorem.
Writing~$U_{n+1,k-1}$ as either
$\left.U_{n,k-1}\right|_{n\mapsto n+1}$ or
$\left.U_{n+1,k}\right|_{k\mapsto k-1}$ and using the relations from
\cref{eq:shifts} shows
that $\psi_0$
satisfies
the
relation
\begin{equation}\label{eq:psi}
\psi_0(n+1,k)a(n,k)=\psi_0(n,k)a(n,k-1).
\end{equation}
From there and the initial value~$\psi_0(0,k)$, it follows that
\begin{equation}\label{eq:defpsi_0}
\psi_0(n,k)=(\epsilon_1\epsilon_2)^mm^{m (\epsilon\epsilon_1-\epsilon_2)}
\frac{(n+\lambda+\epsilon_2km-\delta_{\epsilon_2,1}m)_{m}^{\epsilon_2}}{
(n+\mu+\epsilon_1 km-\delta_{\epsilon_1,1}m)_{m}^{\epsilon\epsilon_1}},
\end{equation}
with $\delta$ the Kronecker symbol.
A key property satisfied by these 6~families and that makes our approach work
is that \cref{eq:shifts} implies that for any~$c$
and~$i$,
\begin{equation}\label{eq:key}\frac{a(n,k)}{\alpha(n,k)}-\frac{a
(n,-c)\alpha(n+i-1,-c)}{\alpha
(n,-c)\alpha(n+i-1,k)}=(k+c){B_i(n,c)},\end{equation}
synthesizing a crucial factor~$k+c$, while $B_i(n,c)$ does not depend on~$k$
and therefore neither does $\mathcal F_{c,i}$.
We first assume that $C_i(n)$ does not vanish for $i\in\mathbb N$, and
therefore that neither does $B_i(n,c)$
(the other case is addressed at the end of the proof.)
Introducing
\[
\phi_i(n,k)=\prod_{j=0}^{i-1}\alpha(n+j,k),\qquad
\psi_i(n,k)=\psi_0(n,k)\phi_i(n,k-1),
\]
a direct computation using~\cref{eq:shifts,eq:psi,eq:key} shows that
\begin{equation}\label{eq:fcc+1}
\begin{split}
\mathcal F_{c,i}(\phi_{i-1}(n,k)U_{n,k})&=(k+c)\phi_i
(n,k)U_{n,k},\\
\mathcal F_{c+1,i}(\psi_{i-1}(n,k)U_{n,k})&=(k+c)\psi_i
(n,k)U_{n,k}.
\end{split}
\end{equation}
It follows by induction that
\begin{equation}\label{eq:rec-A-B}
\begin{split}
\mathcal F_{0,q+1}\mathcal B(U_{n,k})&=\phi_{q+1}(n,k)(b_1+k-1)\dotsm(b_q+k-1)kU_
{n,k},\\
\mathcal A(U_{n,k-1})&=\quad\psi_p(n,k)(b_1+k-1)\dotsm(b_q+k-1)kU_{n,k},\\
\mathcal B(kU_{n,k})&=\quad\phi_{q}(n,k)(b_1+k-1)\dotsm(b_q+k-1)kU_{n,k}.
\end{split}
\end{equation}
The first two identities will be used to derive~$\mathcal X$;
$\mathcal D$ will come from the last two.
Note that $\phi_p$ and $\psi_q$ do not depend on the parameters $a_i$
and $b_i$ of the hypergeometric series. These operators thus allow for
reducing the computation to the case when $p=q=0$. The next step is to
increase the smaller of the indices~$q+1$ and~$p$
until a given target is reached. This is obtained with $\mathcal C_i$,
as a direct computation using \cref{eq:shifts,eq:psi} shows that
\begin{equation}\label{eq:Ci}
\mathcal C_i(\phi_i(n,k)U_{n,k})=\phi_{i+1}(n,k)U_
{n,k},\qquad
\mathcal C_i(\psi_i(n,k)U_{n,k})=\psi_{i+1}(n,k)U_{n,k}.
\end{equation}
By induction, it follows that
\begin{equation} \label{eq:Lij}
\begin{split}
\mathcal L_i^j(\phi_i(n,k)U_{n,k})&=\phi_{\max(i,j)}(n,k)U_
{n,k},\\
\mathcal L_i^j(\psi_i(n,k)U_{n,k})&=\psi_{\max(i,j)}(n,k)U_{n,k}.
\end{split}
\end{equation}
Note that since $\mathcal C_i$ does not depend on~$k$, the same
identities hold with factors depending on~$k$ but not~$n$ on both
sides, such as the factors in \cref{eq:rec-A-B}.
The final step is to map a $\psi_i$ to a $\phi_j$ or the converse.
This is obtained by considering $\psi_0(n+m,k)U_{n+m,k}/U_{n,k}$:
using \cref{eq:defpsi_0,eq:shifts} and the definition of the
sequence~$a(n,k)$ gives
\[\psi_0(n+m,k)\frac{U_{n+m,k}}{U_{n,k}}=
(\epsilon_1\epsilon_2)^mm^{m(\epsilon\epsilon_1-\epsilon_2)}
L(k,m,\mu)M(k,m,\mu),\]
with
\begin{align*}
L(k,m,\lambda)&=
\frac{
(n+\lambda)_{m}(n+\lambda+\epsilon_2km+(1-\delta_{\epsilon_2,1})m)_m
^{\epsilon_2}}{(n+\lambda+\epsilon_2mk)_m},\\
M(k,m,\mu)&=\frac{(n+\epsilon(\mu+\epsilon_1km))_m}{{
(n+\epsilon\mu)_m
(n+\mu+\epsilon_1 km+(1-\delta_{\epsilon_1,1})m)_m^
{\epsilon\epsilon_1}}}.
\end{align*}
Both products simplify depending on the values of
$\epsilon,\epsilon_1,\epsilon_2$, giving
\[
L(k,m,\lambda)={(n+\lambda)_m}\times\begin{cases}
1&\text{if $\epsilon_2=1$,}\\
\frac{1}
{(n+\lambda-mk)_{2m}}&\text{otherwise.}
\end{cases}
\]
\[M(k,m,\lambda)=
\frac1{(n+\mu)_m^\epsilon}\times\begin{cases}
1&\text{if $\epsilon_1=1$,}\\
(n+\mu-km)_{2m}&\text{otherwise.}
\end{cases}
\]
Finally, we also have
\[\phi_p(n+m,k-1)=\begin{cases}\phi_p(n,k)&\text{if $\epsilon_2=1$},\\
\frac1{(n+2m+\lambda-mk)_p}&\text{otherwise.}
\end{cases}\]
Combining these identities shows that, when $\epsilon_1=1$,
\begin{align*}
(\epsilon_1\epsilon_2S_n)^m&(\psi_p(n,k)U_{n,k})\\
&
=(\epsilon_1\epsilon_2)^m\psi_p(n+m,k)\frac{U_{n+m,k}}{U_{n,k}}U_
{n,k},\\
&=(\epsilon_1\epsilon_2)^m\psi_0(n+m,k)\frac{U_{n+m,k}}{U_
{n,k}}\phi_p(n+m,k-1)U_{n,k},\\
&=m^{m(\epsilon-\epsilon_2)}L(k,m,\mu)M
(k,m,\mu)\phi_p(n+m,k-1)U_{n,k},\\
&=m^{m(\epsilon-\epsilon_2)}\frac{(n+\lambda)_m}{(n+\mu)_m^\epsilon}
U_{n,k}\times
\begin{cases}\phi_p(n,k)&\text{if $\epsilon_2=1$,}\\
\frac1{(n+\lambda-mk)_{2m}(n+2m+\lambda-mk)_p}&\text{otherwise,}
\end{cases}\\
&=m^{m(\epsilon-\epsilon_2)}\frac{(n+\lambda)_m}{(n+\mu)_m^\epsilon}
\phi_{p+2m\chi}U_{n,k}.
\end{align*}
A similar derivation gives
\[(\epsilon_1\epsilon_2S_n)^m(\phi_q(n,k)U_{n,k})=
m^{m(\epsilon+\epsilon_2)}
\frac{(\lambda+n)_m}{(\mu+n)_m^\epsilon}\psi_{q+2m\chi}
(n,k)U_{n,k}\quad(\epsilon_1=-1).
\]
As a consequence, we obtain that
\[\begin{split}
&\left((\epsilon_1\epsilon_2S_n)^\theta\mathcal L_{p}^
{q+1-2m\chi\epsilon_1}\right)(\psi_p(n,k)U_{n,k})=\\
&\qquad\begin{cases}m^{m(\epsilon-\epsilon_2)}
\frac{(\lambda+n)_m}{
(\mu+n)_m^\epsilon}\phi_{\max(p+2m\chi,q+1)}(n,k)U_{n,k},\quad&
\text{if $\theta=m$,}\\
\psi_{\max(p,q+1+2m\chi)}(n,k)U_{n,k},\quad&\text{otherwise.}
\end{cases}\\
&\left((\epsilon_1\epsilon_2S_n)^{m-\theta}\mathcal L_{q+1}^
{p+2m\chi\epsilon_1}\right)(\phi_{q+1}(n,k)U_{n,k})=\\
&\qquad\begin{cases}
\phi_{\max(p+2m\chi,q+1)}(n,k)U_{n,k},&\quad\text{if $\theta=m$,}\\
m^{m(\epsilon+\epsilon_2)}\frac{(\lambda+n)_m}{
(\mu+n)_m^\epsilon}\psi_{\max(p,q+1+2m\chi)}(n,k)U_
{n,k},&\quad
\text{otherwise.}
\end{cases}
\end{split}\]
Together with \cref{eq:rec-A-B} and \cref{eq:very-simple}, this concludes the proof of
the first
recurrence. The second one is obtained with $q$ in the place of $q+1$.
We conclude with the case when $C_i(n)$ vanishes for some value
of~$i\in\{0,\dots,\max(p-1,q)\}$, that are the values of~$i$ used in
the theorem. From the definition of~$C_i$, this occurs when $\epsilon
= 1$, $\epsilon_1 = \epsilon_2$ and $i_0 := \mu - \lambda \in \{ 0,
\ldots, \max( p-1, q ) \}$. Then
$\chi = 0, C_{i_0} =B_{i_0+1} = 0$ and $C_{i} B_{i+1} \neq 0$ for $i
\neq i_0-1$. We focus on the proof of the first relation between
$xF_n$ and $F_n$ (the proof of the other relation is similar).
There are two cases depending on whether $i_0\le\min(p-1,q)$ or not.
Eqs.~\eqref{eq:fcc+1} hold as long as $i < i_0$. Let $\mathcal G_i:=
B_i \mathcal F_i $.
If $i_0\le\min(p-1,q)$, at index
$i_0$, \cref{eq:fcc+1} can be replaced by:
\begin{align*}
\mathcal G_{c,i_0+1}(\phi_{i_0}(n,k)U_{n,k})&= B_{i_0+1}(n,c) (k+c)\phi_i (n,k)U_{n,k} = 0,\\
\mathcal G_{c+1,i_0+1}(\psi_{i_0}(n,k)U_{n,k})&= B_{i_0+1}(n,c+1)(k+c)\psi_i (n,k)U_{n,k}=0.
\end{align*}
Thus in that case, with the notation $b_
{q+1}:=1$, the first two equations of~\eqref{eq:rec-A-B} become
\begin{align}
\mathcal G_{b_{i_0+1}-1,i_0+1}\mathcal F_{b_{i_0}-1,i_0}\cdots \mathcal F_{b_{1}-1,1} (U_{n,k})&= 0, \notag \\
\mathcal G_{a_{i_0+1},i_0+1}\mathcal F_{a_{i_0},i_0}\cdots \mathcal F_{a_{1},1} (U_{n,k-1})&= 0, \label{eq:Ga}
\end{align}
proving the first relation.
In the other case, \cref{eq:rec-A-B} hold for all
required values
of~$i$. Eqs.~\eqref{eq:Ci} hold for $i < i_0$. Let $\mathcal D_{i} :=
C_i \mathcal C_i$. At index $i_0$, \eqref{eq:Ci} can be replaced by:
\begin{align*}
\mathcal D_{i_0} (\phi_{i_0}(n,k)U_{n,k}) & =C_{i_0} \phi_{i_0+1}(n,k)U_{n,k} = 0,\\
\mathcal D_{i_0} (\psi_{i_0} (n,k)U_{n,k}) & =C_{i_0} \psi_{i_0+1}(n,k)U_{n,k}=0.
\end{align*}
If $p-1 < i_0 \leq q$, \cref{eq:Ga} holds and
\cref{eq:Lij} can be replaced by:
\[
\mathcal D_{i_0} \mathcal C_{i_0-1} \cdots \mathcal C_{p} (\psi_{p} (n,k)U_{n,k}) = 0,
\]
which, combined with the second equation of~\eqref{eq:rec-A-B}, yields
\[
\mathcal D_{i_0} \mathcal C_{i_0-1} \cdots \mathcal C_{p} \mathcal A U_{n,k-1} = 0,
\]
concluding the proof of this case. The case $q < i_0 \leq p - 1$ is
similar.
\section{More Relations?}
\subsection{More hypergeometric families?}
Not all
hypergeometric families satisfy equations like~\eqref{eq:D} or~\eqref{eq:X}. For
instance,
if~$\Phi_n$ is defined by
\[\Phi_n(x)={}_2F_1\left(\left.\begin{matrix}n+1,n+1\\ 1
\end{matrix}\right|x\right)=\sum_{k\ge0}{\frac{((n+1)_k)^2}{k!^2}x^k}=\frac{\sum_{i=0}^n{\binom{n}{i}^2x^i}}{(1-x)^{2n+1}},\]
then computing the behaviour at~1 of both members of an identity of the form
\[\sum_{m=0}^tA_m\Phi_{n-m}(x)=x\sum_{m=0}^sB_m\Phi_{n-m}(x)\]
with at least one of~$A_0$ or $B_0$ being nonzero implies
\[\frac{A_0\binom{2n}{n}}{(1-x)^{2n+1}}+O\!\left(\frac1{(1-x)^{2n-1}}\right)
=\frac{B_0\binom{2n}{n}}{(1-x)^{2n+1}}
-\frac{B_0(n/2+1)\binom{2n}{n}}{(1-x)^{2n}}+O\!\left(\frac1{(1-x)^{2n-1}}\right),
\]
a contradiction.
The key property shared by the families of \cref{table:families} that leads
to the existence of the relations~\eqref{eq:D} and~\eqref{eq:X} is given in
\cref{eq:key}. It could be the case that a variant of this
identity allows for generalizing the approach to other hypergeometric families. We have not been able
to do so.
\subsection{Mixed Difference-Differential Equations}
\subsubsection{\texorpdfstring{\eqref{eq:M}}{(M)} implies
\texorpdfstring{\eqref{eq:D}}{(D)}}
\begin{proposition}\label{prop:XM-give-XD} If a family of power series
$(\phi_n(x))_n$
satisfies
a relation of type~\eqref{eq:X} and a relation of type~\eqref{eq:M},
then it satisfies a relation of type~\eqref{eq:D}.
\end{proposition}
\begin{proof}
We give a simple proof in terms of fractions of Ore
polynomials \cite{Ore1933}. It can be turned into
an effective proof by expressing each of the steps using the
extended Euclidean algorithm.
\Cref{eq:X} amounts to a fraction
\[\mathcal F_\mathcal X=(A_m(n)S_n^{m}+\dots+A_0(n))^{-1}(B_\ell
(n)S_n^\ell+\dots+B_0(n))\]
that maps the sequence~$(\phi_n(x))_n$ to the sequence $(x\phi_n
(x))_n$.
Evaluating the polynomial~$\pi$ at this fraction gives another
fraction~$\mathcal F_\pi=\pi(\mathcal F_\mathcal X)$ that maps~$
(\phi_n(x))_n$ to~$(\pi(x)\phi_n
(x))_n$. Similarly a fraction~$\mathcal F_{\pi'}$ maps~$(\phi_n(x))_n$ to~$(\pi'(x)\phi_n
(x))_n$. Letting $$\mathcal P=E_{-s}(n)S^{-s}+\dots+E_{t}(n)S^t$$
and differentiating $\pi(x)\phi_n(x)$ gives
\[
(\mathcal F_\pi(\phi_n))'=(\pi(x)\phi_n)'=\pi'(x)\phi_n+\pi(x)\phi_n'=
(\mathcal F_{\pi'}+\mathcal P)(\phi_n).
\]
Reducing to the same denominator on both sides yields a
relation
of type~\eqref{eq:D}.
\end{proof}
\subsubsection{\texorpdfstring{\eqref{eq:D}}{(D)}
is more general than \texorpdfstring{\eqref{eq:M}}{(M)}}
\label{subsubsec:DmgM}
The functions
\begin{multline*}
F_n(x)={}_{1}F_{2}\!\left(\left.\begin{matrix}
n+2\\
n,1
\end{matrix}\right|x\right)=\sum_{k\ge0}\frac{(n+k)(n+k+1)x^k}
{n(n+1)\,k!^2}\\
=
\left(1+\frac x{n(n+1)}\right)I_0(2\sqrt{x})+\frac{2n+1}{n(n+1)}
\sqrt{x}I_0'(2\sqrt{x}), \end{multline*}
where $I_0$ is a modified Bessel function, form a special case of
family~III (with $p=\lambda=0, m=q=b_1=1, \mu=2$).
From the
derivative
\[
F_n'(x)=\frac2{n}I_0(2\sqrt{x})+\left(\frac1{
\sqrt{x}}+\frac{\sqrt x}{n(n+1)}\right)I_0'(2
\sqrt{x}),
\]
it follows that the existence of a relation of type~\eqref{eq:M}
for~$F_n$ would imply the existence of a linear differential equation
of order~1 for~$I_0$ with polynomial coefficients. This is impossible,
for instance because~$I_0$ has an infinite number of complex zeros.
By contrast, as a further illustration of the general situation in
\cref{thm:main} with
$\epsilon=\epsilon_1=\epsilon_2=1$, this
function satisfies
\[\frac12S_n\left(\frac{n+2}nS_n-1\right)^2(xF_n)=
(S_n-1)n(n/2+1)(S_n-1)(F_n).\]
This is readily checked: both operators map~$(2n+1)/(n(n+1))$ to~0;
the first one maps~1 to~$1/((n+1)(n+2))$ and~$1/(n(n+1))$ to~0;
the second one maps~1 to~0 and~$1/(n(n+1))$ to~$1/((n+1)(n+2))$.
The other equation provided by the theorem
is
\[-(n/2+1)(S_n-1)(F_n')=\frac{n+2}{2n}S_n\left(
\frac{n+2}nS_n-1\right)(F_n).\]
This can be checked by observing that the left operator maps $
(1,1/(n(n+1)),1/n)$ to $
(0,1/(n(n+1)),(n+2)/(2n(n+1)))$ while the right one maps $(1,1/
(n
(n+1)),(2n+1)/(n(n+1)))$ to $((n+2)/(n(n+1)),0,1/(n(n+1)))$.
\subsubsection{Equations of type~\texorpdfstring{\eqref{eq:M}}{(M)}}
By \cref{thm:main}, families I~to~VI satisfy relations of type \eqref{eq:X}
and \eqref{eq:D}. However, apart from types~I and~II, a relation of
type
\eqref{eq:M}
exists only in special cases, for low numbers of parameters.
We now list such relations. In all cases, the proof reduces
to comparing the coefficients of~$x^k$ on both sides.
\smallskip
\paragraph{\em Families I and II}
These families have only one occurrence of the
parameter~$n$. The value~$\epsilon=0$ leads to
\[a(n,k)=\frac{n+\lambda}{n+\lambda+\epsilon_2mk}\] in the identity
$U_{n+1,k}/U_{n,k}=a(n,k)$. It follows that
\[\frac1{a(n-1,k)}-1=\frac{\epsilon_2mk}{n+\lambda-1}\]
which leads to the existence of a difference-differential equation of
type \cref{eq:M} for all these functions:
\[xF_n'=\frac{n+\lambda-1}{\epsilon_2m}(F_{n-1}-F_n).\]
\smallskip
\paragraph{\em Families III and IV}
For family III, we find
\begin{align*}
F_n&={}_{1}F_{1}\!\left(\left.\begin{matrix}
\mu+n\\
\lambda+n
\end{matrix}\right|x\right):&\quad F_n'&=\frac{n+\mu}{n+\lambda}F_n;\\
F_n&={}_{2}F_{1}\!\left(\left.\begin{matrix}
\mu+n,a\\
\lambda+n
\end{matrix}\right|x\right):&\quad (1-x)F_n'&=\frac{(n+\mu)
(n+\lambda-a)}{n+\lambda}F_{n+1}+(n+\mu)F_n.
\end{align*}
Changing $n$ into~$-n$ in these two relations gives the
analogues
for type IV.
\smallskip
\paragraph{\em Family V}
We obtain
\[F_n={}_{2}F_{0}\!\left(\left.\begin{matrix}
1-\lambda-n,\mu+n\\
-
\end{matrix}\right|x\right):\]
\begin{align*}-x^2F_n'&=
\frac{(n+\mu)(n+\lambda-1)}{(2n+\lambda+\mu)
(2n+\lambda+\mu-1)}F_{n+1}-{2\frac{(n+\mu)(n+\lambda-1)}{(2n+\lambda+\mu)
(2n+\lambda+\mu-2)}F_n}\\
&\qquad
+\frac{(n+\mu)(n+\lambda-1)}{
(2n+\lambda+\mu-1)(2n+\lambda+\mu-2)}F_{n-1};
\end{align*}
\begin{align*}
F_n&={}_{2}F_{1}\!\left(\left.\begin{matrix}
1-\lambda-n,
\mu+n\\
b
\end{matrix}\right|x\right):
\end{align*}
\begin{multline*}x(1-x)F_n'=
\frac{(n+\mu)(n+\lambda-1)(n+\lambda+b-1)}{(2n+\lambda+\mu)
(2n+\lambda+\mu-1)}F_{n+1}\\
-{\frac{(n+\mu)(n+\lambda-1)(\lambda-\mu+2b-2)}{(2n+\lambda+\mu)
(2n+\lambda+\mu-2)}F_n}
-\frac{(n+\mu)(n+\lambda-1)(n+\mu-b)}{
(2n+\lambda+\mu-1)(2n+\lambda+\mu-2)}F_{n-1};
\end{multline*}
\smallskip
\paragraph{\em Family VI}
There, more relations can be found:
\begin{align*}
F_n&={}_{0}F_{2}\!\left(\left.\begin{matrix}
-\\
\lambda+n,1-\mu-n
\end{matrix}\right|x\right):
\end{align*}
\begin{align*}-F_n'&=
\frac{(n+\mu)}{(n+\lambda)(2n+\lambda+\mu)
(2n+\lambda+\mu-1)}F_{n+1}
+{\frac{2}{(2n+\lambda+\mu)
(2n+\lambda+\mu-2)}}F_n\\
&\qquad
+\frac{(n+\lambda-1)}{
(n+\mu-1)(2n+\lambda+\mu-1)(2n+\lambda+\mu-2)}F_{n-1};
\end{align*}
\begin{align*}
F_n&={}_{1}F_{2}\!\left(\left.\begin{matrix}
a_1\\
\lambda+n,1-\mu-n
\end{matrix}\right|x\right):
\end{align*}
\begin{align*}F_n'&=
\frac{(n+\mu)(n+\lambda-a_1)}{(n+\lambda)(2n+\lambda+\mu)
(2n+\lambda+\mu-1)}F_{n+1}
+{\frac{\lambda-\mu-2a_1}{(2n+\lambda+\mu)
(2n+\lambda+\mu-2)}}F_n\\
&\qquad
-\frac{(n+\lambda-1)(n+\mu+a_1-1)}{
(n+\mu-1)(2n+\lambda+\mu-1)(2n+\lambda+\mu-2)}F_{n-1};
\end{align*}
\begin{align*}
F_n&={}_{2}F_{2}\!\left(\left.\begin{matrix}
a_1,a_2\\
\lambda+n,1-\mu-n
\end{matrix}\right|x\right):
\end{align*}
\begin{align*}F_n'&=
-\frac{(n+\mu)(n+\lambda-a_1)(n+\lambda-a_2)}{(n+\lambda)
(2n+\lambda+\mu)
(2n+\lambda+\mu-1)}F_{n+1}\\
&\qquad
+\left({\frac12+\frac{(\lambda-\mu-2a_1)(\lambda-\mu-2a_2)}
{4}\left(\frac1{2n+\lambda+\mu}-\frac1
{2n+\lambda+\mu-2}\right)}\right)F_n\\
&\qquad
-\frac{(n+\lambda-1)(n+\mu+a_1-1)(n+\mu+a_2-1)}{
(n+\mu-1)(2n+\lambda+\mu-1)(2n+\lambda+\mu-2)}F_{n-1};
\end{align*}
\begin{align*}
F_n&={}_{3}F_{2}\!\left(\left.\begin{matrix}
a_1,a_2,a_3\\
\lambda+n,1-\mu-n
\end{matrix}\right|x\right):
\end{align*}
\begin{align*}(1-x)F_n'&=
-\frac{(n+\mu)(n+\lambda-a_1)(n+\lambda-a_2)(n+\lambda-a_3)}{
(n+\lambda)
(2n+\lambda+\mu)
(2n+\lambda+\mu-1)}F_{n+1}\\
&\qquad
+\left(\frac{2(a_1+a_2+a_3)+\mu-\lambda-2}4\right.\\
&\qquad\qquad-\frac{
(\lambda-\mu-2a_1)
(\lambda-\mu-2a_2)(\lambda-\mu-2a_3)}
{8}\times\\
&\qquad\qquad\qquad\qquad\left.\left(\frac1{2n+\lambda+\mu}-\frac1
{2n+\lambda+\mu-2}\right)\right)F_n\\
&\qquad
-\frac{(n+\lambda-1)(n+\mu+a_1-1)(n+\mu+a_2-1)(n+\mu+a_3-1)}{
(n+\mu-1)(2n+\lambda+\mu-1)(2n+\lambda+\mu-2)}F_{n-1}.
\end{align*}
\bibliographystyle{abbrv}
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| null | null |
\section{Introduction}\label{intro}
The moduli space ${\mathcal M}_{g,k}$ of Riemann surfaces of genus $g$ with
$k$ punctures plays an important role in many area of mathematics
and theoretical physics.
In this article we first survey some of our recent works on the geometry of this moduli space.
In the following we assume $g\geq 2$ and $k=0$ to simplify notations.
All the results in this paper work for the general case when
$3g-3+k>0$. We will focus on the K\"ahler metrics on the moduli
and Teichm\"uller spaces, especially the Weil-Petersson metric,
the Ricci, the perturbed Ricci, and the K\"ahler-Einstein metrics.
We will review certain new geometric properties we found and proved for
these metrics, such as the bounded geometry, the goodness and
their naturalness under restriction to boundary divisors. The algebro-geometric corollaries such as the stability of the
logarithmic cotangent bundles and the infinitesimal rigidity of the moduli
spaces will also be briefly discussed. Similar to our previous survey
articles \cite{lsy6, lsy5}, we will briefly describe the basic ideas of our proofs, the details of the proofs
will be published soon, see \cite{lsy3, lsy4}.
After introducing the definition of Weil-Petersson metric in
Section \ref{wpme}, we discuss the fundamental curvature formula of Wolpert
for the Weil-Petersson metric. For the reader's convenience we also briefly give a proof of the
negativity of the Riemannian curvature of the Weil-Petersson
metric. In Section \ref{rcprc} we discuss the Ricci and the perturbed Ricci
metrics and their curvature formulas. In Section \ref{asymp} we describe the
asymptotics of these metrics and their curvatures which are
important for our understanding of their bounded geometry. In
Section \ref{equivalence} we briefly discuss the equivalence of all of the
complete metrics on Teichm\"uller spaces to the Ricci and the
perturbed Ricci metrics, which is a simple corollary of our understanding
of these two new metrics. In Section \ref{goodness} we discuss the goodness of the
Weil-Petersson metric, the Ricci, the perturbed Ricci metric and
the K\"ahler-Einstein metric. To prove the goodness we need much
more
subtle estimates on the connection and the curvatures of these
metrics. Section \ref{natural} contains discussions of the dual Nakano
negativity of the logarithmic tangent bundle of the moduli space
and the naturalness of the Ricci and the perturbed Ricci metrics. In Section \ref{krf} we discuss the K\"ahler-Ricci flow and the K\"ahler-Einstein metric on the moduli space. There
are many interesting corollaries from our understanding of the
geometry of the moduli spaces. In Section \ref{app} we discuss the
stability of the logarithmic cotangent bundle, the $L^2$
cohomology and the infinitesimal rigidity of the moduli spaces as well as the Gauss-Bonnet theorem on the moduli space.
The Teichm\"uller and moduli spaces of polarized Calabi-Yau (CY) manifolds
and Hyper-K\"ahler manifolds are also important in mathematics and high
energy physics. In Section \ref{torelli} we will describe our recent
joint work with A. Todorov on the proof of global Torelli theorem of
the Teichm\"uller space of polarized CY manifolds and Hyper-K\"ahler manifolds. As
applications we will describe the construction of a global
holomorphic flat connection on the Teichm\"uller space of CY manifolds and
the existence of K\"ahler-Einstein metrics on the Hodge completion of such Teichm\"uller
spaces.
\section{The Weil-Petersson Metric and Its Curvature}\label{wpme}
Let ${\mathcal M}_g$ be the moduli space of Riemann surfaces of genus $g$
where $g\geq 2$. It is well known that the ${\mathcal M}_g$ is a complex
orbifold. The Teichm\"uller space ${\mathcal T}_g$, as the space parameterizing marked
Riemann surfaces, is a smooth contractible pseudo-convex domain and
can be embedded into the Euclidean space of the same dimension.
\begin{remark}\label{obd}
Since ${\mathcal M}_g$ is only an orbifold, in the following when we work near
a point $p\in{\mathcal M}_g$ which is an orbifold point, we always work on a
local manifold cover of ${\mathcal M}_g$ around $p$. An alternative way is to
add a level structure on the moduli space so that it becomes smooth
\cite{shue1}. All the following results are still valid. In
particular, when we use the universal curve over the moduli space,
we always mean the universal curve over the local manifold cover.
When we deal with global properties of the moduli space, we can use
the moduli space with a level structure such that it becomes smooth.
We take quotients after we derive the estimates. We can also work on
the Teichm\"uller space which is smooth.
\end{remark}
For any point $p\in{\mathcal M}_g$ we let $X_p$ be the corresponding Riemann
surface. By the Kodaira-Spencer theory we have the identification
\[
T_p^{1,0}{\mathcal M}_g\cong H^1\left ( X_p,T_{X_p}^{1,0}\right ).
\]
It follows from Serre duality that
\[
\Omega_p^{1,0}{\mathcal M}_g\cong H^0\left ( X_p,K_{X_p}^2\right ).
\]
By the Riemann-Roch theorem we know that the dimension
$\dim_{\mathbb C}{\mathcal M}_g=n=3g-3$.
The Weil-Petersson (WP) metric is the first known K\"ahler metric on
${\mathcal M}_g$. Ahlfors showed that the WP metric is K\"ahler and its holomorphic
sectional curvature is bounded above by a negative constant which
only depends on the genus $g$. Royden conjectured that the Ricci
curvature of the WP metric is also bounded above by a negative
constant. This conjecture was proved by Wolpert \cite{wol3}.
Now we briefly describe the WP metric and its curvature formula.
Please see the works \cite{wol4}, \cite{wol5} of Wolpert for
detailed description and various aspects of the WP metric.
Let $\pi:\mathfrak X\to{\mathcal M}_g$ be the universal family over the moduli
space. For any point $s\in{\mathcal M}_g$ we let $X_s=\pi^{-1}(s)$ be the
corresponding smooth Riemann surface. Since the Euler characteristic
$\chi\left ( X_s\right )=2-2g<0$, by the uniformization theorem we know that
each fiber $X_s$ is equipped with a unique K\"ahler-Einstein metric $\lambda$. In
the following we will always use the K\"ahler-Einstein metric $\lambda$ on $X_s$.
Let $z$ be any holomorphic coordinate on $X_s$. We have
\[
\partial_z\partial_{\bar z}\log\lambda=\lambda.
\]
Now we fix a point $s\in{\mathcal M}_g$ and let $(U,s_1,\cdots,s_n)$ be any
holomorphic coordinate chart on ${\mathcal M}_g$ around $s$. In the following
we will denote by $\partial_i$ and $\partial_z$ the local vector fields
$\frac{\partial}{\partial s_i}$ and $\frac{\partial}{\partial z}$ respectively. By the
Kodaira-Spencer theory and the Hodge theory we have the
identification
\[
T_s^{1,0}{\mathcal M}_g\cong \check H^1\left ( X_s,T_{X_s}^{1,0}\right ) \cong
{\mathbb H}^{0,1}\left ( X_s,T_{X_s}^{1,0}\right )
\]
where the right side of the above formula is the space of harmonic
Beltrami differentials. In fact we can explicitly construct the
above identification. We let \[
a_i=a_i(z,s)=-\lambda^{-1}\partial_i\partial_{\bar z}\log\lambda
\]
and let
\[
v_i=\frac{\partial}{\partial s_i}+a_i\frac{\partial}{\partial z}.
\]
The vector field $v_i$ is a smooth vector field on $\pi^{-1}(U)$ and
is called the harmonic lift of $\frac{\partial}{\partial s_i}$. If we let
$B_i=\bar\partial_F v_i\in A^{0,1}\left ( X_s,T_{X_s}^{1,0}\right )$ then $B_i$ is
harmonic and the map $\frac{\partial}{\partial s_i} \mapsto B_i$ is precisely
the Kodaira-Spencer map. Here $\partial_F$ is the operator in the fiber
direction. In local coordinates if we let $B_i=A_i d\bar z\otimes
\partial_z$ then $A_i=\partial_{\bar z} a_i$. Furthermore, it was proved by Schumacher
that if $\eta$ is any relative $(1,1)$-form on $\mathfrak X$ then
\begin{eqnarray}\label{20}
\frac{\partial}{\partial s_i}\int_{X_s}\eta=\int_{X_s} L_{v_i}\eta.
\end{eqnarray}
We note that although $A_i$ is a local smooth function on $X_s$, the
product
\[
A_i\bar A_j=B_i\cdot\bar B_j\in C^\infty(X_s)
\]
is globally defined. We let
\[
f_{i\bar j}=A_i\bar A_j\in C^\infty(X_s).
\]
The Weil-Petersson metric on ${\mathcal M}_g$ is given by
\[
h_{i\bar j}(s)=\int_{X_s}B_i\cdot\bar B_j\ dv= \int_{X_s}f_{i\bar
j}\ dv
\]
where $dv=\frac{\sqrt{-1}}{2}\lambda dz\wedge d\bar z$ is the volume
form on $X_s$ with respect to the K\"ahler-Einstein metric.
Now we describe the curvature formula of the WP metric. We let
$\Box=-\lambda^{-1}\partial_z\partial_{\bar z}$ be the Hodge-Laplacian acting on
$C^\infty(X_s)$. It is clear that the operator $\Box+1$ has no
kernel and thus is invertible. We let
\[
e_{i\bar j}=(\Box+1)^{-1}\left ( f_{i\bar j}\right )\in C^\infty(X_s).
\]
The following curvature formula is due to Wolpert. See \cite{lsy1}
for the detailed proof.
\begin{proposition}\label{wpcurv}
Let $R_{i\bar jk\bar l}$ be the curvature of the WP metric. Then
\begin{eqnarray}\label{30}
R_{i\bar jk\bar l}=-\int_{X_s} \left ( e_{i\bar j} f_{k\bar l}+ e_{i\bar
l} f_{k\bar j}\right ) \ dv.
\end{eqnarray}
\end{proposition}
The curvature of the WP metric has very strong negativity property.
In fact we shall see in Section \ref{natural} that the WP metric is
dual Nakano negative. We collect the negativity property of the WP
metric in the following proposition.
\begin{proposition}\label{negativewp}
The bisectional curvature of the WP metric on the moduli space
${\mathcal M}_g$ is negative. The holomorphic sectional and Ricci curvatures
of the WP metric are bounded above by negative constants.
Furthermore, the Riemannian sectional curvature of the WP metric is
also negative.
\end{proposition}
{\bf Proof.} These results are well known, see \cite{wol3}. Here we
give a short proof of the negativity of the Riemannian sectional
curvature of the WP metric for the reader's convenience. The proof
follows from expressing the Riemannian sectional curvature in term
of complex curvature tensors and using the curvature formula
\eqref{30}.
In general, let $(X^n,g, J)$ be a K\"ahler manifold. For any point $p\in
X$ and two orthonormal real tangent vectors $u,v\in T_p^{\mathbb R} X$, we
let $X=\frac{1}{2}\left ( u-iJu\right )$ and $Y=\frac{1}{2}\left ( v-iJv\right )$ and
we know that $X,Y\in T_p^{1,0}X$. We can choose holomorphic local
coordinate $s=(s_1,\cdots,s_n)$ around $p$ such that
$X=\frac{\partial}{\partial s_1}$. If $v=span_{\mathbb R} \{ u, Ju\}$, since $v$ is
orthogonal to $u$ and its length is $1$, we know $v=\pm Ju$. In this
case we have
\[
R(u,v,u,v)=R(u,Ju,u,Ju)=4R_{1\bar 1 1\bar 1}.
\]
Thus the Riemannian sectional curvature and the holomorphic
sectional curvature have the same sign.
If $v$ is not contained in the real plane spanned by $u$ and $Ju$ we
can choose the coordinate $s$ such that $X=\frac{\partial}{\partial s_1}$ and
$Y=\frac{\partial}{\partial s_2}$. In this case a direct computation shows
that
\begin{eqnarray}\label{rie10}
R(u,v,u,v)=2\left ( R_{1\bar 1 2\bar 2}- Re \left ( R_{1\bar 2 1\bar
2}\right )\rb.
\end{eqnarray}
Now we fix a point $p\in{\mathcal M}_g$ and let $u,v\in T_p^{\mathbb R}{\mathcal M}_g$. Let $X,Y$
be the corresponding $(1,0)$-vectors. Since we know that the
holomorphic sectional curvature of the WP metric is strictly
negative, we assume $v\notin span_{\mathbb R}\{ u,Ju\}$ and thus we can
choose holomorphic local coordinates $s=(s_1,\cdots,s_n)$ around $p$
such that $X=\frac{\partial}{\partial s_1}(p)$ and $Y=\frac{\partial}{\partial s_2}(p)$.
By formulas \eqref{rie10} and \eqref{30} we have
\begin{align}\label{rie40}
\begin{split}
R(u,v,u,v)=& -2\left ( \int_{X_p} \left ( e_{1\bar 1}f_{2\bar 2}+e_{1\bar 2}f_{2\bar 1}-2 Re ( e_{1\bar 2}f_{1\bar 2}) \right ) dv \right )\\
=& -2\left ( \int_{X_p} \left ( e_{1\bar 1}f_{2\bar 2}+e_{1\bar 2}f_{2\bar
1}-e_{1\bar 2}f_{1\bar 2}-e_{2\bar 1}f_{2\bar 1}\right ) dv\right ).
\end{split}
\end{align}
To prove the proposition we only need to show that
\begin{eqnarray}\label{rie50}
\int_{X_p} e_{1\bar 2}f_{2\bar 1}\ dv \geq \int_{X_p} Re \left (
e_{1\bar 2}f_{1\bar 2} \right ) dv
\end{eqnarray}
and
\begin{eqnarray}\label{rie60}
\int_{X_p} e_{1\bar 1}f_{2\bar 2}\ dv\geq \int_{X_p} e_{1\bar
2}f_{2\bar 1}\ dv
\end{eqnarray}
and both equalities cannot hold simultaneously.
To prove inequality \eqref{rie50} we let $\alpha=Re (e_{1\bar 2})$
and $\beta=Im (e_{1\bar 2})$. Then we know
\[
\int_{X_p} e_{1\bar 2}f_{2\bar 1}\ dv=\int_{X_p} \left ( \alpha
(\Box+1)\alpha +\beta (\Box+1)\beta \right ) dv
\]
and
\[
\int_{X_p} Re \left ( e_{1\bar 2}f_{1\bar 2} \right ) dv= \int_{X_p} \left (
\alpha (\Box+1)\alpha -\beta (\Box+1)\beta \right ) dv.
\]
Thus formula \eqref{rie50} reduces to
\[
\int_{X_p} \beta (\Box+1)\beta\ dv\geq 0.
\]
However, we know
\[
\int_{X_p} \beta (\Box+1)\beta\ dv=\int_{X_p} \left ( \Vert
\nabla'\beta\Vert^2+\beta^2 \right ) dv\geq 0
\]
and the equality holds if and only if $\beta=0$. If this is the case
then we know that $e_{1\bar 2}$ is a real value function and
$f_{1\bar 2}$ is real valued too. Since $f_{1\bar 1}=A_1\bar A_1$
and $f_{1\bar 2}=A_1\bar A_2$ and $f_{1\bar 1}$ is real-valued we
know that there is a function $f\in C^\infty (X_p\setminus S, {\mathbb R})$
such that $A_2=f(z)A_1$ on $X_p\setminus S$. Here $S$ is the set of
zeros of $A_1$. Since both $A_1$ and $A_2$ are harmonic, we know
that $\bar\partial^* A_1=\bar\partial^* A_2=0$. These reduce to $\partial_z(\lambda
A_1)=\partial_z(\lambda A_2)=0$ locally. It follows that $\partial_z
f\mid_{X_p\setminus S}=0$. Since $f$ is real-valued we know that $f$
must be a constant. But $A_1$ and $A_2$ are linearly independent
which is a contradiction. So the strict inequality \eqref{rie50}
always holds.
Now we prove formula \eqref{rie60}. Let $G(z,w)$ be the Green's
function of the operator $\Box+1$ and let $T=(\Box+1)^{-1}$. By the
maximum principle we know that $T$ maps positive functions to
positive functions. This implies that the Green's function $G$ is
nonnegative. Since $G(z,w)=G(w,z)$ is symmetric we know that
\begin{align}\label{rie70}
\begin{split}
\int_{X_p} e_{1\bar 1}f_{2\bar 2}\ dv=& \int_{X_p\times X_p}G(z,w) f_{1\bar 1}(w) f_{2\bar 2}(z)\ dv(w)dv(z)\\
=& \frac{1}{2} \int_{X_p\times X_p}G(z,w) \left ( f_{1\bar 1}(w)
f_{2\bar 2}(z)+ f_{1\bar 1}(z) f_{2\bar 2}(w) \right ) dv(w)dv(z).
\end{split}
\end{align}
Similarly we have
\begin{align}\label{rie80}
\begin{split}
\int_{X_p} e_{1\bar 2}f_{2\bar 1}\ dv=& \int_{X_p\times X_p}G(z,w) f_{1\bar 2}(w) f_{2\bar 1}(z)\ dv(w)dv(z)\\
=& \frac{1}{2} \int_{X_p\times X_p}G(z,w) \left ( f_{1\bar 2}(w)
f_{2\bar 1}(z)+ f_{1\bar 2}(z) f_{2\bar 1}(w) \right ) dv(w)dv(z).
\end{split}
\end{align}
Formula \eqref{rie60} follows from the fact that
\begin{align*}
\begin{split}
& f_{1\bar 1}(w) f_{2\bar 2}(z)+ f_{1\bar 1}(z) f_{2\bar 2}(w)-f_{1\bar 2}(w) f_{2\bar 1}(z)- f_{1\bar 2}(z) f_{2\bar 1}(w) \\
= & \left | A_1(z)A_2(w)-A_1(w)A_2(z)\right |^2\geq 0.
\end{split}
\end{align*}
\qed
Although the WP metric has very strong negativity properties, as we
shall see in Section \ref{asymp}, the WP metric is not complete and
its curvatures have no lower bound and this is very restrictive.
\section{The Ricci and Perturbed Ricci Metrics}\label{rcprc}
In \cite{lsy1} and \cite{lsy2} we studied two new K\"ahler metrics: the
Ricci metric $\omega_\tau$ and the perturbed Ricci metric
$\omega_{\tilde\tau}$ on the moduli space ${\mathcal M}_g$. These new K\"ahler
metrics are complete and have bounded geometry and thus have many
important applications. We now describe these new metrics.
Since the Ricci curvature of the WP metric has negative upper bound,
we define the Ricci metric
\[
\omega_\tau=-Ric\left (\omega_{_{WP}}\right ).
\]
We also define the perturbed Ricci metric to be a linear combination
of the Ricci metric and the WP metric
\[
\omega_{\tilde\tau}=\omega_\tau+C\omega_{_{WP}}
\]
where $C$ is a positive constants. In local coordinates we have
$\tau_{i\bar j}=-h^{k\bar l}R_{i\bar jk\bar l}$ and
$\tilde\tau_{i\bar j}=\tau_{i\bar j} +Ch_{i\bar j}$ where $R_{i\bar
jk\bar l}$ is the curvature of the WP metric.
Similar to curvature formula \eqref{30} of the WP metric we can
establish integral formulae for the curvature of the Ricci and
perturbed Ricci metrics. These curvature formulae are crucial in
estimating the asymptotics of these metrics and their curvature. To
establish these formulae, we need to introduce some operators. We
let
\[
P: C^\infty(X_s)\to A^{1,0}\left ( T_{X_s}^{0,1}\right )
\]
be the operator defined by
\[
f\mapsto \partial \left ( \omega_{_{KE}}^{-1}\lrcorner\partial f\right ).
\]
In local coordinate we have $P(f)=\partial_z\left ( \lambda^{-1}\partial_z f\right )
dz\otimes \partial_{\bar z}$. For each $1\leq k\leq n$ we let
\[
\xi_k:C^\infty(X_s)\to C^\infty(X_s)
\]
be the operator defined by
\[
f\mapsto \bar\partial^*\left ( B_k\lrcorner \partial f\right )=-B_k \cdot P(f).
\]
In the local coordinate we have $\xi_k(f)=-\lambda^{-1}\partial_z\left ( A_k
\partial_z f\right )$. Finally for any $1\leq k,l\leq n$ we define the operator
\[
Q_{k\bar l}:C^\infty(X_s)\to C^\infty(X_s)
\]
by
\[
Q_{k\bar l}(f)=\bar P\left ( e_{k\bar l}\right ) P(f)-2f_{k\bar l}\Box
f+\lambda^{-1}\partial_z f_{k\bar l}\partial_{\bar z} f.
\]
These operators are commutators of various classical operators on
$X_s$. See \cite{lsy1} for details. Now we recall the curvature
formulae of the Ricci and perturbed Ricci metrics established in
\cite{lsy1}. For convenience, we introduce the symmetrization
operator.
\begin{definition}
Let $U$ be any quantity which depends on indices $i,k,\alpha,\bar
j,\bar l, \bar\beta$. The symmetrization operator $\sigma_1$ is
defined by taking the summation of all orders of the triple
$(i,k,\alpha)$. That is
\begin{align*}
\begin{split}
\sigma_1(U(i,k,\alpha,\bar j,\bar l, \bar\beta))=& U(i,k,\alpha,\bar
j,\bar l, \bar\beta)+ U(i,\alpha,k,\bar j,\bar l, \bar\beta)+
U(k,i,\alpha,\bar j,\bar l, \bar\beta)\\ + & U(k,\alpha,i,\bar
j,\bar l, \bar\beta)+U(\alpha,i,k,\bar j,\bar l, \bar\beta)+
U(\alpha,k,i,\bar j,\bar l, \bar\beta).
\end{split}
\end{align*}
Similarly, $\sigma_2$ is the symmetrization operator of $\bar j$ and
$\bar \beta$ and $\widetilde{\sigma_1}$ is the symmetrization
operator of $\bar j$, $\bar l$ and $\bar \beta$.
\end{definition}
Now we can state the curvature formulae. We let $T=(\Box+1)^{-1}$ be
the operator in the fiber direction.
\begin{theorem}\label{riccicurv}
Let $s_1,\cdots,s_n$ be local holomorphic coordinates at $s \in
{\mathcal M}_g$ and let $\widetilde{R}_{i\bar j k\bar l}$ be the curvature of
the Ricci metric. Then at $s$, we have
\begin{align}\label{finalcurv}
\begin{split}
\widetilde{R}_{i\bar j k\bar l}=& -h^{\alpha\bar\beta}
\left\{\sigma_1\sigma_2\int_{X_s} \left\{T(\xi_k(e_{i\bar j}))
\bar{\xi}_l(e_{\alpha\bar\beta})+ T(\xi_k(e_{i\bar j}))
\bar{\xi}_\beta(e_{\alpha\bar l})
\right\}\ dv\right\}\\
&-h^{\alpha\bar\beta} \left\{\sigma_1\int_{X_s}Q_{k\bar l}(e_{i\bar
j}) e_{\alpha\bar\beta}\ dv
\right\}\\
&+\tau^{p\bar q}h^{\alpha\bar\beta}h^{\gamma\bar\delta}
\left\{\sigma_1\int_{X_s}\xi_k(e_{i\bar q}) e_{\alpha\bar\beta}\
dv\right\}\left\{ \widetilde\sigma_1\int_{X_s}\bar{\xi}_l(e_{p\bar
j})
e_{\gamma\bar\delta})\ dv\right\}\\
&+\tau_{p\bar j}h^{p\bar q}R_{i\bar q k\bar l}.
\end{split}
\end{align}
\end{theorem}
\begin{theorem}\label{perriccicurv}
Let $\tilde\tau_{i\bar j}=\tau_{i\bar j}+Ch_{i\bar j}$ where $\tau$
and $h$ are the Ricci and WP metrics respectively where $C>0$ is a
constant. Let $P_{i\bar j k\bar l}$ be the curvature of the
perturbed Ricci metric. Then we have
\begin{align}\label{finalpercurv}
\begin{split}
P_{i\bar j k\bar l}=&-h^{\alpha\bar\beta}
\left\{\sigma_1\sigma_2\int_{X_s} \left\{T(\xi_k(e_{i\bar j}))
\bar{\xi}_l(e_{\alpha\bar\beta})+ T (\xi_k(e_{i\bar j}))
\bar{\xi}_\beta(e_{\alpha\bar l})
\right\}\ dv\right\}\\
&-h^{\alpha\bar\beta} \left\{\sigma_1\int_{X_s}Q_{k\bar l}(e_{i\bar
j}) e_{\alpha\bar\beta}\ dv
\right\}\\
&+\widetilde\tau^{p\bar q}h^{\alpha\bar\beta}h^{\gamma\bar\delta}
\left\{\sigma_1\int_{X_s}\xi_k(e_{i\bar q}) e_{\alpha\bar\beta}\
dv\right\}\left\{ \widetilde\sigma_1\int_{X_s}\bar{\xi}_l(e_{p\bar
j})
e_{\gamma\bar\delta})\ dv\right\}\\
&+\tau_{p\bar j}h^{p\bar q}R_{i\bar q k\bar l} +CR_{i\bar j k\bar
l}.
\end{split}
\end{align}
\end{theorem}
In \cite{lsy1} and \cite{lsy2} we proved various properties of these
new metrics. Here we collect the important ones.
\begin{theorem}\label{rcprcpro}
The Ricci and perturbed Ricci metrics are complete K\"ahler metrics on
${\mathcal M}_g$. Furthermore we have
\begin{itemize}
\item These two metrics have bounded curvature.
\item The injectivity radius of the Teichm\"uller space ${\mathcal T}_g$ equipped with any of these two metrics is bounded from below.
\item These metrics have Poincar\'e growth and thus the moduli space has finite volume when equipped with any of these metrics.
\item The perturbed Ricci metric has negatively pinched holomorphic sectional and Ricci curvatures when we choose the constant $C$ to be large enough.
\end{itemize}
\end{theorem}
The Ricci metric is also cohomologous to the K\"ahler-Einstein metric on ${\mathcal M}_g$ in
the sense of currents and hence can be used as the background metric
to estimate the K\"ahler-Einstein metric. We will discuss this in Section
\ref{equivalence}.
\section{Asymptotics}\label{asymp}
Since the moduli space ${\mathcal M}_g$ is noncompact, it is important to
understand the asymptotic behavior of the canonical metrics in order
to study their global properties. We first describe the local
pinching coordinates near the boundary of the moduli space by using
the plumbing construction of Wolpert.
Let $\mathcal M_g$ be the moduli space of Riemann surfaces of genus
$g \geq 2$ and let $\bar{\mathcal M}_g$ be its Deligne-Mumford
compactification \cite{dm1}. Each point $y \in \bar{\mathcal M}_g
\setminus \mathcal M_g$ corresponds to a stable nodal surface $X_y$.
A point $p \in X_y$ is a node if there is a neighborhood of $p$
which is isometric to the germ $\{ (u,v)\mid uv=0,\ |u|,|v|<1 \}
\subset \mathbb{C}^2$.
We first recall the rs-coordinate on a Riemann surface defined by
Wolpert in \cite{wol1}. There are two cases: the puncture case and
the short geodesic case. For the puncture case, we have a nodal
surface $X$ and a node $p\in X$. Let $a,b$ be two punctures which
are glued together to form $p$.
\begin{definition}
A local coordinate chart $(U,u)$ near $a$ is called rs-coordinate if
$u(a)=0$ where $u$ maps $U$ to the punctured disc $0<|u|<c$ with
$c>0$, and the restriction to $U$ of the K\"ahler-Einstein metric on
$X$ can be written as $\frac{1}{2|u|^2(\log |u|)^2} |du|^2$. The
rs-coordinate $(V,v)$ near $b$ is defined in a similar way.
\end{definition}
For the short geodesic case, we have a closed surface $X$, a closed
geodesic $\gamma \subset X$ with length $l <c_\ast$ where $c_\ast$
is the collar constant.
\begin{definition}
A local coordinate chart $(U,z)$ is called rs-coordinate at $\gamma$
if $\gamma \subset U$ where $z$ maps $U$ to the annulus
$c^{-1}|t|^{\frac{1}{2}}<|z| <c|t|^{\frac{1}{2}}$, and the
K\"ahler-Einstein metric on $X$ can be written as
$\frac{1}{2}(\frac{\pi}{\log |t|}\frac{1}{|z|}\csc \frac{\pi\log
|z|}{\log |t|})^2 |dz|^2$.
\end{definition}
By Keen's collar theorem \cite{ke1}, we have the following lemma:
\begin{lemma}\label{gcollar}
Let $X$ be a closed surface and let $\gamma$ be a closed geodesic on
$X$ such that the length $l$ of $\gamma$ satisfies $l <c_\ast$. Then
there is a collar $\Omega$ on $X$ with holomorphic coordinate $z$
defined on $\Omega$ such that
\begin{enumerate}
\item $z$ maps $\Omega$ to the annulus
$\frac{1}{c}e^{-\frac{2\pi^2}{l}}<|z|<c$ for $c>0$; \item the
K\"ahler-Einstein metric on $X$ restricted to $\Omega$ is given by
\begin{eqnarray}\label{precmetric}
(\frac{1}{2}u^2 r^{-2}\csc^2\tau) |dz|^2
\end{eqnarray}
where $u=\frac{l}{2\pi}$, $r=|z|$ and $\tau=u\log r$; \item the
geodesic $\gamma$ is given by the equation $|z|=
e^{-\frac{\pi^2}{l}}$.
\end{enumerate}
We call such a collar $\Omega$ a genuine collar.
\end{lemma}
We notice that the constant $c$ in the above lemma has a lower bound
such that the area of $\Omega$ is bounded from below. Also, the
coordinate $z$ in the above lemma is an rs-coordinate. In the
following, we will keep the notations $u$, $r$ and $\tau$.
Now we describe the local manifold cover of $\bar{\mathcal M}_g$
near the boundary. We take the construction of Wolpert \cite{wol1}.
Let $X_{0,0}$ be a stable nodal surface corresponding to a
codimension $m$ boundary point and let $p_1,\cdots,p_m$ be the nodes
of $X_{0.0}$. The smooth part $X_0=X_{0,0}\setminus \{
p_1,\cdots,p_m \}$ is a union of punctured Riemann surfaces. Fix the
rs-coordinate charts $(U_i,\eta_i)$ and $(V_i,\zeta_i)$ at $p_i$ for
$i=1,\cdots,m$ such that all the $U_i$ and $V_i$ are mutually
disjoint.
Now pick an open set $U_0 \subset X_0$ such that the intersection of
each connected component of $X_0$ and $U_0$ is a nonempty relatively
compact set and the intersection $U_0 \cap (U_i\cup V_i)$ is empty
for all $i$. We pick Beltrami differentials
$\nu_{m+1},\cdots,\nu_{n}$ which are supported in $U_0$ and span the
tangent space at $X_0$ of the deformation space of $X_0$. For
$s=(s_{m+1},\cdots,s_n)$, let $\nu(s)=\sum_{i=m+1}^n s_i\nu_i$. We
assume $|s|=(\sum |s_i|^2)^{\frac{1}{2}}$ small enough such that
$|\nu(s)|<1$. The nodal surface $X_{0,s}$ is obtained by solving the
Beltrami equation $\bar\partial w=\nu(s)\partial w$. Since $\nu(s)$
is supported in $U_0$, $(U_i,\eta_i)$ and $(V_i,\zeta_i)$ are still
holomorphic coordinates for $X_{0,s}$. Note that they are no longer
rs-coordinates. By the theory of Ahlfors and Bers \cite{ab1} and
Wolpert \cite{wol1} we can assume that there are constants
$\delta,c>0$ such that when $|s|<\delta$, $\eta_i$ and $\zeta_i$ are
holomorphic coordinates on $X_{0,s}$ with $0<|\eta_i|<c$ and
$0<|\zeta_i|<c$.
Now we assume $t=(t_1,\cdots,t_m)$ has small norm. We do the
plumbing construction on $X_{0,s}$ to obtain $X_{t,s}$ in the
following way. We remove from $X_{0,s}$ the discs $0<|\eta_i|\leq
\frac{|t_i|}{c}$ and $0<|\zeta_i|\leq \frac{|t_i|}{c}$ for each
$i=1,\cdots,m$, and identify $\frac{|t_i|}{c}<|\eta_i|< c$ with
$\frac{|t_i|}{c}<|\zeta_i|< c$ by the rule $\eta_i \zeta_i=t_i$.
This defines the surface $X_{t,s}$. The tuple
$(t_1,\cdots,t_m,s_{m+1},\cdots,s_n)$ are the local pinching
coordinates for the manifold cover of $\bar{\mathcal M}_g$. We call
the coordinates $\eta_i$ (or $\zeta_i$) the plumbing coordinates on
$X_{t,s}$ and the collar defined by $\frac{|t_i|}{c}<|\eta_i|< c$
the plumbing collar.
\begin{remark}
>From the estimate of Wolpert \cite{wol2}, \cite{wol1} on the length
of short geodesic, we have $u_i=\frac{l_i}{2\pi}\sim
-\frac{\pi}{\log|t_i|}$.
\end{remark}
Let $(t,s)=(t_1,\cdots,t_m,s_{m+1},\cdots,s_n)$ be the pinching
coordinates near $X_{0,0}$. For $|(t,s)|<\delta$, let $\Omega^j_c$
be the $j$-th genuine collar on $X_{t,s}$ which contains a short
geodesic $\gamma_j$ with length $l_j$. Let $u_j=\frac{l_j}{2\pi}$,
$u_0=\sum_{j=1}^m u_j+\sum_{j=m+1}^n |s_j|$, $r_j=|z_j|$ and
$\tau_j=u_j\log r_j$ where $z_j$ is the properly normalized
rs-coordinate on $\Omega^j_c$ such that
\[
\Omega^j_c=\{ z_j\mid c^{-1}e^{-\frac{2\pi^2}{l_j}}<|z_j|<c \}.
\]
>From the above argument, we know that the K\"ahler-Einstein metric
$\lambda$ on $X_{t,s}$, restrict to the collar $\Omega^j_c$, is
given by
\begin{eqnarray}\label{jmetric}
\lambda=\frac{1}{2}u_j^2 r_j^{-2}\csc^2\tau_j .
\end{eqnarray}
For convenience, we let $\Omega_c=\cup_{j=1}^m \Omega^j_c$ and
$R_c=X_{t,s}\setminus \Omega_c$. In the following, we may change the
constant $c$ finitely many times, clearly this will not affect the
estimates.
To estimate the WP, Ricci and perturbed Ricci metrics and their
curvatures, we first need to to find all the harmonic Beltrami
differentials $B_1,\cdots,B_n$ corresponding to the tangent vectors
$\frac{\partial}{\partial t_1},\cdots, \frac{\partial}{\partial
s_n}$. In \cite{ma1}, Masur constructed $3g-3$ regular holomorphic
quadratic differentials $\psi_1,\cdots,\psi_n$ on the plumbing
collars by using the plumbing coordinate $\eta_j$. These quadratic
differentials correspond to the cotangent vectors
$dt_1,\cdots,ds_n$.
However, it is more convenient to estimate the curvature if we use
the rs-coordinate on $X_{t,s}$ since we have the accurate form of
the K\"ahler-Einstein metric $\lambda$ in this coordinate. In
\cite{tr1}, Trapani used the graft metric constructed by Wolpert
\cite{wol1} to estimate the difference between the plumbing
coordinate and rs-coordinate and described the holomorphic quadratic
differentials constructed by Masur in the rs-coordinate. We collect
Trapani's results (Lemma 6.2-6.5, \cite{tr1}) in the following
theorem:
\begin{theorem}\label{imp}
Let $(t,s)$ be the pinching coordinates on $\bar{\mathcal M}_g$ near
$X_{0,0}$ which corresponds to a codimension $m$ boundary point of
$\bar{\mathcal M}_g$. Then there exist constants $M,\delta>0$ and
$1>c>0$ such that if $|(t,s)|<\delta$, then the $j$-th plumbing
collar on $X_{t,s}$ contains the genuine collar $\Omega^j_c$.
Furthermore, one can choose rs-coordinate $z_j$ on the collar
$\Omega_c^j$ such that the holomorphic quadratic differentials
$\psi_1,\cdots,\psi_n$ corresponding to the cotangent vectors
$dt_1,\cdots,ds_n$ have the form $\psi_i=\phi_i(z_j)dz_j^2$ on the
genuine collar $\Omega^j_c$ for $1\leq j \leq m$, where
\begin{enumerate}
\item $\phi_i(z_j)=\frac{1}{z_j^2}(q_i^j(z_j)+\beta_i^j)$ if
$i\geq m+1$; \item
$\phi_i(z_j)=(-\frac{t_j}{\pi})\frac{1}{z_j^2}(q_j(z_j)+\beta_j)$ if
$i=j$; \item $\phi_i(z_j)=(-\frac{t_i}{\pi})
\frac{1}{z_j^2}(q_i^j(z_j)+\beta_i^j)$ if $1\leq i \leq m$ and $i\ne
j$.
\end{enumerate}
Here $\beta_i^j$ and $\beta_j$ are functions of $(t,s)$, $q_i^j$ and
$q_j$ are functions of $(t,s,z_j)$ given by
\[
q_i^j(z_j)=\sum_{k<0}\alpha_{ik}^j(t,s)t_j^{-k}z_j^k
+\sum_{k>0}\alpha_{ik}^j(t,s)z_j^k
\]
and
\[
q_j(z_j)=\sum_{k<0}\alpha_{jk}(t,s)t_j^{-k}z_j^k
+\sum_{k>0}\alpha_{jk}(t,s)z_j^k
\]
such that
\begin{enumerate}
\item $\sum_{k<0}|\alpha_{ik}^j|c^{-k}\leq M$ and
$\sum_{k>0}|\alpha_{ik}^j|c^{k}\leq M$ if $i\ne j$; \item
$\sum_{k<0}|\alpha_{jk}|c^{-k}\leq M$ and
$\sum_{k>0}|\alpha_{jk}|c^{k}\leq M$; \item
$|\beta_i^j|=O(|t_j|^{\frac{1}{2}-\epsilon})$ with
$\epsilon<\frac{1}{2}$ if $i\ne j$; \item $|\beta_j|=(1+O(u_0))$.
\end{enumerate}
\end{theorem}
An immediate consequence is the precise asymptotics of the WP metric
which was computed in \cite{lsy1}. These asymptotic estimates were
also given by Wolpert in \cite{wol6}.
\begin{theorem}\label{wpasymp}
Let $(t,s)$ be the pinching coordinates and let $h$ be the WP
metric. Then
\begin{enumerate}
\item $h^{i\bar i}=2u_i^{-3}|t_i|^2(1+O(u_0))$ and $h_{i\bar i} =\frac{1}{2}\frac{u_i^{3}}{|t_i|^2}(1+O(u_0))$ for $1\leq i\leq m$;
\item $h^{i\bar j}=O(|t_it_j|)$ and $h_{i\bar j}=O\left ( \frac{u_i^3u_j^3}{|t_it_j|}\right )$, if $1\leq i,j \leq m$ and $i\ne j$;
\item $h^{i\bar j}=O(1)$ and $h_{i\bar j}=O(1)$, if $m+1\leq i,j \leq n$;
\item $h^{i\bar j}=O(|t_i|)$ and $h_{i\bar j}=O\left ( \frac{u_i^3}{|t_i|}\right )$ if $i\leq m < j$;
\item $h^{i\bar j}=O(|t_j|)$ and $h_{i\bar j}=O\left ( \frac{u_j^3}{|t_j|}\right )$ if $j\leq m < i$.
\end{enumerate}
\end{theorem}
By using the asymptotics of the WP metric and the fact that
\[
B_i=\lambda^{-1}\sum_{j=1}^n h_{i\bar j}\bar\psi_j
\]
we can derive the expansion of the harmonic Beltrami differentials
corresponding to $\frac{\partial}{\partial t_i}$ and $\frac{\partial}{\partial s_j}$.
\begin{theorem}\label{aj10}
For $c$ small, on the genuine collar $\Omega_c^j$, the coefficient functions
$A_i$ of the harmonic Beltrami differentials have the form:
\begin{enumerate}
\item $A_i=\frac{z_j}{\bar{z_j}}\sin^2\tau_j \left (
\bar{p_i^j(z_j)}+\bar{b_i^j}\right )$ if $i\ne j$;
\item
$A_j=\frac{z_j}{\bar{z_j}}\sin^2\tau_j\left (\bar{p_j(z_j)}+\bar{b_j}\right )$
\end{enumerate}
where
\begin{enumerate}
\item $p_i^j(z_j)=\sum_{k\leq -1}a_{ik}^j\rho_j^{-k}z_j^k
+\sum_{k\geq 1}a_{ik}^jz_j^k$ if $i\ne j$;
\item
$p_j(z_j)=\sum_{k\leq -1}a_{jk}\rho_j^{-k}z_j^k +\sum_{k\geq
1}a_{jk}z_j^k$.
\end{enumerate}
In the above expressions, $\rho_j=e^{-\frac{2\pi^2}{l_j}}$ and the
coefficients satisfy the following conditions:
\begin{enumerate}
\item $\sum_{k\leq -1}|a_{ik}^j|c^{-k}=O\left ( u_j^{-2}\right )$ and
$\sum_{k\geq 1}|a_{ik}^j|c^{k}=O\left ( u_j^{-2}\right )$\\ if $i\geq m+1$;
\item
$\sum_{k\leq -1}|a_{ik}^j|c^{-k}=O\left ( \frac{u_i^3
u_j^{-2}}{|t_i|}\right )$ and $\sum_{k\geq
1}|a_{ik}^j|c^{k}=O\left (\frac{u_i^3u_j^{-2}}{|t_i|}\right )$ \\
if $i\leq m$ and $i\ne j$;
\item $\sum_{k\leq
-1}|a_{jk}|c^{-k}=O\left ( \frac{u_j}{|t_j|}\right )$ and $\sum_{k\geq
1}|a_{jk}|c^{k}=O\left (\frac{u_j}{|t_j|}\right )$;
\item $|b_i^j|=O(u_j)$ if $i\geq m+1$;
\item $|b_i^j|=O\left ( u_j\right ) O\left ( \frac{u_i^3}{|t_i|}\right )$ if $i\leq m$ and $i\ne j$;
\item
$b_j=-\frac{u_j}{\pi\bar{t_j}}(1+O(u_0))$.
\end{enumerate}
\end{theorem}
By a detailed study of the curvature of the WP metric we derived the
precise asymptotics of the Ricci metric in \cite{lsy1}.
\begin{theorem}\label{ricciest}
Let $(t,s)$ be the pinching coordinates. Then we have
\begin{enumerate}
\item $\tau_{i\bar i}=\frac{3}{4\pi^2}\frac{u_i^2}{|t_i|^2}(1+O(u_0))$ and $\tau^{i\bar i}=\frac{4\pi^2}{3}\frac{|t_i|^2}{u_i^2} (1+O(u_0))$, if $i \leq m$;
\item $\tau_{i\bar j}=O\bigg (\frac{u_i^2u_j^2}{|t_it_j|}(u_i+u_j)\bigg )$ and $\tau^{i\bar j}=O(|t_it_j|)$, if $i,j \leq m$ and $i\ne j$;
\item $\tau_{i\bar j}=O\left ( \frac{u_i^2}{|t_i|}\right )$ and $\tau^{i\bar j}=O(|t_i|)$, if $i\leq m$ and $j\geq m+1$;
\item $\tau_{i\bar j}=O\left ( \frac{u_j^2}{|t_j|}\right )$ and $\tau^{i\bar j}=O(|t_j|)$, if $j\leq m$ and $i\geq m+1$;
\item $\tau_{i\bar j}=O(1)$, if $i,j \geq m+1$.
\end{enumerate}
\end{theorem}
In \cite{lsy1} we also derived the asymptotics of the curvature of
the Ricci metric.
\begin{theorem}\label{mainholo}
Let $X_0\in\bar{\mathcal{M}_g}\setminus\mathcal{M}_g$ be a
codimension $m$ point and let $(t_1,\cdots,t_m,s_{m+1},\cdots,s_n)$
be the pinching coordinates at $X_0$ where $t_1,\cdots,t_m$
correspond to the degeneration directions. Then the holomorphic
sectional curvature is negative in the degeneration directions and
is bounded in the non-degeneration directions. More precisely, there
exists $\delta>0$ such that, if $|(t,s)|<\delta$, then
\begin{eqnarray}\label{important100}
\widetilde R_{i\bar ii\bar i}=-
\frac{3u_i^4}{8\pi^4|t_i|^4}(1+O(u_0))
\end{eqnarray}
if $i\leq m$ and
\begin{eqnarray}\label{important200}
\left | \widetilde R_{i\bar ii\bar i}\right |=O(1)
\end{eqnarray}
if $i\geq m+1$. Here $\tilde R$ is the curvature of the Ricci
metric.
Furthermore, on $\mathcal M_g$, the holomorphic sectional curvature,
the bisectional curvature and the Ricci curvature of the Ricci
metric are bounded from above and below.
\end{theorem}
In \cite{lsy3} and \cite{lsy4} we derived more precise estimates of
the curvature of the Ricci and perturbed Ricci metrics which we will
discuss in Section \ref{goodness}.
\section{Canonical Metrics and Equivalence}\label{equivalence}
In addition to the WP, Ricci and perturbed Ricci metrics on the
moduli space, there are several other canonical metrics on ${\mathcal M}_g$.
These include the Teichm\"uller metric, the Kobayashi metric, the
Carath\'eodory metric, the K\"ahler-Einstein metric, the induced Bergman metric,
the McMullen metric and the asymptotic Poincar\'e metric.
Firstly, on any complex manifold there are two famous Finsler
metrics: the Carath\'eodory and Kobayashi metrics. Now we describe
these metrics.
Let $X$ be a complex manifold and of dimension $n$. let $\Delta_R$
be the disk in $\mathbb C$ with radius $R$. Let $\Delta=\Delta_1$
and let $\rho$ be the Poincar\'e metric on $\Delta$. Let $p\in X$ be
a point and let $v\in T_p X$ be a holomorphic tangent vector. Let
$\text{Hol}(X,\Delta_R)$ and $\text{Hol}(\Delta_R,X)$ be the spaces
of holomorphic maps from $X$ to $\Delta_R$ and from $\Delta_R$ to
$X$ respectively. The Carath\'eodory norm of the vector $v$ is
defined to be
\[
\Vert v\Vert_C=\sup_{f\in\text{Hol}(X,\Delta)}\Vert f_\ast
v\Vert_{\Delta,\rho}
\]
and the Kobayashi norm of $v$ is defined to be
\[
\Vert v\Vert_K=\inf_{f\in\text{Hol}(\Delta_R,X),\ f(0)=p,\
f'(0)=v}\frac{2}{R}.
\]
It is well known that the Carath\'eodory metric is bounded from
above by the Kobayashi metric after proper normalization. The first
known metric on the Teichm\"uller space ${\mathcal T}_g$ is the Teichm\"uller metric which is
also an Finsler metric. Royden showed that, on ${\mathcal T}_g$, the Teichm\"uller
metric coincides with the Kobayashi metric. Generalizations and
proofs of Royden's theorem can be found in \cite{masar}.
Now we look at the K\"ahler metrics. Firstly, since the Teichm\"uller space ${\mathcal T}_g$
is a pseudo-convex domain, by the work of Cheng and Yau \cite{cy1}
and the later work of Yau, there exist a unique complete K\"ahler-Einstein metric
on ${\mathcal T}_g$ whose Ricci curvature is $-1$.
There is also a canonical Bergman metric on ${\mathcal T}_g$ which we describe
now. In general, let $X$ be any complex manifold, let $K_X$ be the
canonical bundle of $X$ and let $W$ be the space of $L^2$
holomorphic sections of $K_X$ in the sense that if $\sigma\in W$,
then
\[
\Vert\sigma\Vert_{L^2}^2=\int_X
(\sqrt{-1})^{n^2}\sigma\wedge\bar\sigma<\infty.
\]
The inner product on $W$ is defined to be
\[
(\sigma,\rho)=\int_X (\sqrt{-1})^{n^2}\sigma\wedge\bar\rho
\]
for all $\sigma,\rho\in W$. Let $\sigma_1,\sigma_2,\cdots$ be an
orthonormal basis of $W$. The Bergman kernel form is the
non-negative $(n,n)$-form
\[
B_X=\sum_{j=1}^\infty(\sqrt{-1})^{n^2}\sigma_j\wedge\bar\sigma_j.
\]
With a choice of local coordinates $z_i,\cdots,z_n$, we have
\[
B_X=BE_X(z,\bar z)(\sqrt{-1})^{n^2}dz_1\wedge\cdots\wedge dz_n
\wedge d\bar z_1\wedge\cdots\wedge d\bar z_n
\]
where $BE_X(z,\bar z)$ is called the Bergman kernel function. If the
Bergman kernel $B_X$ is positive, one can define the Bergman metric
\[
B_{i\bar j}=\frac{\partial^2\log BE_X(z,\bar z)}{\partial z_i
\partial \bar z_j}.
\]
The Bergman metric is well-defined and is nondegenerate if the
elements in $W$ separate points and the first jet of $X$.
It is easy to see that both the K\"ahler-Einstein metric and the Bergman metric on
the Teichm\"uller space ${\mathcal T}_g$ are invariant under the action of the mapping
class group and thus descend down to the moduli space.
\begin{remark}\label{bergman}
We note that the induced Bergman metric on ${\mathcal M}_g$ is different from
the Bergman metric on ${\mathcal M}_g$.
\end{remark}
In \cite{mc} McMullen introduced another K\"ahler metric $g_{1/l}$ on
$\mathcal M_g$ which is equivalent to the Teichm\"uller metric. Let
$Log:\ \mathbb R_{+}\to [0,\infty)$ be a smooth function such that
\begin{enumerate}
\item $Log(x)=\log x$ if $x \geq 2$; \item $Log(x)=0$ if $x \leq
1$.
\end{enumerate}
For suitable choices of small constants $\delta,\epsilon>0$, the
K\"ahler form of the McMullen metric $g_{1/l}$ is
\[
\omega_{1/l}=\omega_{WP}-i\delta\sum_{l_{\gamma}(X)<\epsilon}\partial\bar\partial
Log\frac{\epsilon}{l_\gamma}
\]
where the sum is taken over primitive short geodesics $\gamma$ on
$X$.
Finally, since ${\mathcal M}_g$ is quasi-projective, there exists a
non-canonical asymptotic Poincar\'e metric $\omega_{_{P}}$ on
${\mathcal M}_g$. In general, Let $\bar M$ be a compact projective manifold of
dimension $m$. Let $Y\subset \bar M$ be a divisor of normal
crossings and let $M=\bar M\setminus Y$. Cover $\bar M$ by
coordinate charts $U_1,\cdots,U_p,\cdots,U_q$ such that $(\bar
U_{p+1}\cup\cdots\cup\bar U_q)\cap Y=\emptyset$. We also assume
that, for each $1\leq \alpha \leq p$, there is a constant $n_\alpha$
such that $U_\alpha\setminus
Y=(\Delta^\ast)^{n_\alpha}\times\Delta^{m-n_\alpha}$ and on
$U_\alpha$, $Y$ is given by $z_1^\alpha\cdots
z_{n_\alpha}^\alpha=0$. Here $\Delta$ is the disk of radius
$\frac{1}{2}$ and $\Delta^\ast$ is the punctured disk of radius
$\frac{1}{2}$. Let $\{\eta_i\}_{1\leq i\leq q}$ be the partition of
unity subordinate to the cover $\{U_i\}_{1\leq i\leq q}$. Let
$\omega$ be a K\"ahler metric on $\bar M$ and let $C$ be a positive
constant. Then for $C$ large, the K\"ahler form
\[
\omega_{_{P}}=C\omega+\sum_{i=1}^p\sqrt{-1}\partial\bar\partial
\bigg (\eta_i\log\log\frac{1}{\left | z_1^i\cdots z_{n_i}^i\right |}
\bigg )
\]
defines a complete metric on $M$ with finite volume since on each
$U_i$ with $1\leq i\leq p$, $\omega{_{p}}$ is bounded from above and
below by the local Poincar\'e metric on $U_i$. We call this metric
the asymptotic Poincar\'e metric.
In 2004 we proved in \cite{lsy1} that all complete metrics on the
moduli space are equivalent. The proof is based on asymptotic
analysis of these metrics and Yau's Schwarz Lemma. It is an easy
corollaries of our understanding of the Ricci and the perturbed
Ricci metrics. In July 2004 we learned from the announcement of
S.-K. Yeung in Hong Kong University
where he announced he could prove a small and easy part of our results about the equivalences
of some of these metrics by using a bounded pluri-subharmonic function. We received a hard copy of
Yeung's paper in November 2004 where he used a method similar to
ours in \cite{lsy1} to compare the Bergman, the Kobayashi and the
Carath\'eodory metric. It should be interesting to see how one can
use the bounded psh function to derive these equivalences.
We recall that two metrics on ${\mathcal M}_g$ are equivalent if one metric is
bounded from above and below by positive constant multiples of the
other metric.
\begin{theorem}\label{eqall}
On the moduli space ${\mathcal M}_g$ the Ricci metric, the perturbed Ricci
metric, the K\"ahler-Einstein metric, the induced Bergman metric, the McMullen
metric, the asymptotic Poincar\'e metric, the Carath\'eodory metric
and the Teichm\"uller-Kobayashi metric are equivalent.
\end{theorem}
The equivalence of several of these metrics hold in more general
setting. In 2004 we defined the holomorphic homogeneous regular
manifolds in \cite{lsy1} which generalized the idea of Morrey.
\begin{definition}\label{hhrm}
A complex manifold $X$ of dimension $n$ is called holomorphic
homogeneous regular if there are positive constants $r<R$ such that
for each point $p\in X$ there is a holomorphic map $f_p:X\to {\mathbb C}^n$
which satisfies
\begin{enumerate}
\item $f_p(p)=0$;
\item $f_p:X\to f_p(X)$ is a biholomorphism;
\item $B_r\subset f_p(X)\subset B_R$ where $B_r$ and $B_R$ are Euclidean balls with center $0$ in ${\mathbb C}^n$.
\end{enumerate}
\end{definition}
In 2009 Yeung \cite{yeung} used the above definition without
appropriate reference which he called domain with uniform squeezing
property. It follows from the restriction properties of canonical
metrics and Yau's Schwarz Lemma that
\begin{theorem}\label{eqhhrm}
Let $X$ be a holomorphic homogeneous regular manifold. Then the
Kobayashi metric, the Bergman metric and the Carath\'eodory metric
on $X$ are equivalent.
\end{theorem}
\begin{remark}
It follows from the Bers embedding theorem that the Teichm\"uller space of
genus $g$ Riemann surfaces is a holomorphic homogeneous regular
manifold if we choose $r=2$ and $R=6$ in Definition \ref{hhrm}.
\end{remark}
\section{Goodness of Canonical Metrics}\label{goodness}
In his work \cite{mum1}, Mumford defined the goodness condition to
study the currents of Chern forms defined by a singular Hermitian
metric on a holomorphic bundle over a quasi-projective manifold
where he generalized the Hirzebruch's proportionality theorem to
noncompact case. The goodness condition is a growth condition of the
Hermitian metric near the compactification divisor of the base
manifold. The major property of a good metric is that the currents
of its Chern forms define the Chern classes of the bundle. Namely
the Chern-Weil theory works in this noncompact case.
Beyond the case of homogeneous bundles over symmetric spaces
discussed by Mumford in \cite{mum1}, several natural bundles over
moduli spaces of Riemann surfaces give beautiful and useful
examples. In \cite{wol1}, Wolpert showed that the metric induced by
the hyperbolic metric on the relative dualizing sheaf over the
universal curve of moduli space of hyperbolic Riemann surfaces is
good. Later it was shown by Trapani \cite{tr1} that the metric
induced by the WP metric on the determinant line bundle of the
logarithmic cotangent bundle of the Deligne-Mumford moduli space is
good. In both cases, the bundles involved are line bundles in which
cases it is easier to estimate the connection and curvature. Other
than these, very few examples of natural good metrics are known.
The goodness of the WP metric has been a long standing open problem.
In this section we describe our work in \cite{lsy3} which gives a
positive answer to this problem.
We first recall the definition of good metrics and their basic
properties described in \cite{mum1}. Let $\bar X$ be a projective
manifold of complex dimension $\dim_{\mathbb C} \bar X=n$. Let $D\subset \bar
X$ be a divisor of normal crossing and let $X=\bar X\setminus D$ be
a Zariski open manifold. We let $\Delta_r$ be the open disk in ${\mathbb C}$
with radius $r$, let $\Delta=\Delta_1$,
$\Delta_r^*=\Delta_r\setminus\{0\}$ and $\Delta ^*=\Delta
\setminus\{0\}$. For each point $p\in D$ we can find a coordinate
chart $(U,z_1,\cdots,z_n)$ around $p$ in $\bar X$ such that
$U\cong\Delta^n$ and $V=U\cap X\cong \left (
\Delta^*\right )^{m}\times\Delta^{n-m}$. We assume that $U\cap D$ is
defined by the equation $z_1\cdots z_k=0$. We let $U(r)\cong
\Delta_r^n$ for $0<r<1$ and let $V(r)=U(r)\cap X$.
On the chart $V$ of $X$ we can define a local Poincar\'e metric:
\begin{eqnarray}\label{localpo}
\omega_{loc}=\frac{\sqrt{-1}}{2}\sum_{i=1}^k \frac{dz_i\wedge d\bar
z_i}{2|z_i|^2 \left ( \log|z_i|\right )^2}+
\frac{\sqrt{-1}}{2}\sum_{i=k+1}^n dz_i\wedge d\bar z_i.
\end{eqnarray}
Now we cover $D\subset \bar X$ by such coordinate charts
$U_1,\cdots,U_q$ and let $V_i=U_i\cap X$. We choose coordinates
$z_1^i,\cdots,z_n^i$ such that $D\cap U_i$ is given by $z_1^i\cdots
z_{m_i}^i=0$.
A K\"ahler metric $\omega_g$ on $X$ has Poincar\'e growth if for each $1\leq i\leq q$
there are constants $0\leq r_i\leq 1$ and $0\leq c_i<C_i$ such that
$\omega_g\mid_{V_i(r_i)}$ is equivalent to the local Poincar\'e metric
$\omega_{loc}^i$:
\[
c_i\omega_{loc}^i\leq \omega_g\mid_{V_i(r_i)} \leq
C_i\omega_{loc}^i.
\]
In \cite{mum1} Mumford defined differential forms with Poincar\'e growth:
\begin{definition}\label{formpg}
Let $\eta\in A^p(X)$ be a smooth $p$-form. Then $\eta$ has Poincar\'e growth if
for each $1\leq i\leq q$ there exists a constant $c_i>0$ such that
for each point $s\in V_i\left ( \frac{1}{2}\right )$ and tangent vectors
$t_1,\cdots,t_p\in T_s X$ one has
\[
\left | \eta(t_1,\cdots,t_p)\right |^2\leq c_i
\prod_{j=1}^p\omega_{loc}^i (t_j,t_j).
\]
The $p$-form $\eta$ is good if and only if both $\eta$ and $d\eta$
have Poincar\'e growth.
\end{definition}
\begin{remark}
It is easy to see that the above definition does not depend on the
choice of the cover $(U_1,\cdots,U_q)$ but it does depend on the
compactification $\bar X$ of $X$.
\end{remark}
The above definition is local. We now give a global formulation.
\begin{lemma}\label{pglobal}
Let $\omega_g$ be a K\"ahler metric on $X$ with Poincar\'e growth. Then a $p$-form
$\eta\in A^p(X)$ has Poincar\'e growth if and only if $\Vert\eta\Vert_g<\infty$
where $\Vert\eta\Vert_g$ is the $C^0$ norm of $\eta$ with respect to
the metric $g$. Furthermore, the fact that $\eta$ has Poincar\'e growth is
independent of the choice of $g$. It follows that if $\eta_1\in
A^p(X)$ and $\eta_2\in A^q(X)$ have Poincar\'e growth, then $\eta_1\wedge\eta_2$
also has Poincar\'e growth.
\end{lemma}
Now we collect the basic properties of forms with Poincar\'e growth as described
in \cite{mum1}.
\begin{lemma}\label{pgpro}
Let $\eta\in A^p(X)$ be a form with Poincar\'e growth. Then $\eta$ defines a
$p$-current on $\bar X$. Furthermore, if $\eta$ is good then
$d[\eta]=[d\eta]$.
\end{lemma}
Now we consider a holomorphic vector bundle $\bar E$ of rank $r$
over $\bar X$. Let $E=\bar E\mid_X$ and let $h$ be a Hermitian
metric on $E$. According to \cite{mum1} we have
\begin{definition}\label{metricgood}
The Hermitian metric $h$ is good if for any point $x\in D$, assume
$x\in U_i$ for some $i$, and any basis $e_1,\cdots,e_r$ of $\bar
E\mid_{U_i\left ( \frac{1}{2}\right )}$, if we let
$h_{\alpha\bar\beta}=h(e_\alpha,e_\beta)$ then there exist positive
constants $c_i,d_i$ such that
\begin{enumerate}
\item $\left | h_{\alpha\bar\beta}\right |, \left ( \det h\right )^{-1}\leq c_i \left ( \sum_{j=1}^{m_i}\log|z_j|\right )^{2d_i}$;
\item the $1$-forms $\left ( \partial h\cdot h^{-1}\right )_{\alpha\beta}$ are good on $V_i\left ( \frac{1}{2}\right )$.
\end{enumerate}
\end{definition}
\begin{remark}\label{choose1}
A simple computation shows that the goodness of $h$ is independent
of the choice of the cover of $D$. Furthermore, to check whether a
metric $h$ is good or not by using the above definition, we only
need to check the above two conditions for one choice of the basis
$e_1,\cdots,e_r$.
\end{remark}
The most important features of a good metric are
\begin{theorem}\label{megoodpro}
Let $h$ be a Hermitian metric on $E$. Then there is at most one
extension of $E$ to $\bar X$ for which $h$ is good. Furthermore, if
$h$ is a good metric on $E$, then the Chern forms $c_k(E,h)$ are
good and the current $[c_k(E,h)]=c_k(\bar E)\in H^{2k}(\bar X)$.
\end{theorem}
See \cite{mum1} for details. This theorem allows us to compute the
Chern classes by using Chern forms of a singular good metric.
Now we look at a special choice of the bundle $E$. In the following
we let $\bar E=T_{\bar X}(-\log D)$ to be the logarithmic tangent
bundle and let $E=\bar E\mid_X$. Let $U$ be one of the charts $U_i$
described above and assume $D\cap U$ is given by $z_1\cdots z_m=0$.
Let $V=V_i=U_i\cap X$. In this case a local frame of $\bar E$
restricting to $V$ is given by
\[
e_1=z_1\frac{\partial}{\partial z_1},\cdots,e_m=z_m\frac{\partial}{\partial z_m},\
e_{m+1}=\frac{\partial}{\partial z_{m+1}}, \cdots,e_n=\frac{\partial}{\partial z_n}.
\]
Let $g$ be any K\"ahler metric on $X$. It induces a Hermitian metric
$\tilde g$ on $E$. In local coordinate $z=(z_1,\cdots,z_n)$ we have
\begin{eqnarray}\label{induce10}
\tilde g_{i\bar j}=
\begin{cases}
z_i\bar z_j g_{i\bar j}&\ \ \text{if}\ \ \ i,j\leq m\\
z_i g_{i\bar j}&\ \ \text{if}\ \ \ i\leq m<j\\
\bar z_j g_{i\bar j}&\ \ \text{if}\ \ \ j\leq m<i\\
g_{i\bar j}&\ \ \text{if}\ \ \ i,j>m.
\end{cases}
\end{eqnarray}
In the following we denote by $\partial_i$ the partial derivative
$\frac{\partial}{\partial z_i}$. Let
\[
\Gamma_{ik}^p=g^{p\bar q}\partial_i g_{k\bar q}
\]
be the Christoffel symbol of the K\"ahler metric $g$ and let
\[
R_{ik\bar l}^p=g^{p\bar j}R_{i\bar jk\bar l}=g^{p\bar j}\left (
-\partial_k\partial_{\bar l}g_{i\bar j}+g^{s\bar t}\partial_k g_{i\bar t}\partial_{\bar
l}g_{s\bar j}\right )
\]
be the curvature of $g$. We define
\begin{eqnarray}\label{dik}
D_i^k=
\begin{cases}
\frac{z_i}{z_k} & \text{if}\ \ \ i,k\leq m\\
\frac{1}{z_k} & \text{if}\ \ \ k\leq m<i\\
z_i & \text{if}\ \ \ i\leq m<k\\
1 & \text{if}\ \ \ i,k>m
\end{cases}
\end{eqnarray}
and we let
\begin{eqnarray}\label{gammaab}
\Lambda_i=
\begin{cases}
\frac{-1}{|z_i|\log |z_i|} & \text{if}\ \ \ i\leq m\\
1 & \text{if}\ \ \ i>m
\end{cases}.
\end{eqnarray}
Now we give an equivalent local condition of the metric $\tilde g$
on $E$ induced by the K\"ahler metric $g$ to be good. We have
\begin{proposition}\label{logiff}
The metric $\tilde g$ on $E$ induced by $g$ is good on $V\left (
\frac{1}{2}\right )$ if and only if
\begin{align}\label{goodiff}
\begin{split}
& |\tilde g_{i\bar j}|,\ |z_1\cdots z_m|^{-2}\deg(g) \leq c\left ( \sum_{i=1}^m \log|z_i|\right )^{2d} \ \ \text{for some constants}\ c,d>0\\
& \left | D_i^k \Gamma_{ip}^k \right |=O(\Lambda_p)\ \ \text{for all}\ 1\leq i,k,p\leq n\ \text{except}\ i=k=p\\
& \left | \frac{1}{t _i}+ \Gamma_{ii}^i \right |=O(\Lambda_i)\ \ \text{if}\ i\leq m\\
& \left | D_i^k R_{ip\bar q}^k \right |=O(\Lambda_p\Lambda_q).
\end{split}
\end{align}
\end{proposition}
In \cite{lsy3} we showed the goodness of the WP, Ricci and perturbed
Ricci metrics.
\begin{theorem}\label{goodmain}
Let ${\mathcal M}_g$ be the moduli space of genus $g$ Riemann surfaces. We
assume $g\geq 2$. Let $\bar{\mathcal M}_g$ be the Deligne-Mumford
compactification of ${\mathcal M}_g$ and let $D=\bar{\mathcal M}_g\setminus{\mathcal M}_g$ be the
compactification divisor which is a normal crossing divisor. Let
$\bar E=T_{\bar{\mathcal M}_g}(-\log D)$ and let $E=\bar E\mid_{{\mathcal M}_g}$. Let
$\hat h$, $\hat\tau$ and $\hat{\tilde\tau}$ be the metrics on $E$
induced by the WP, Ricci and perturbed Ricci metrics respectively.
Then $\hat h$, $\hat\tau$ and $\hat{\tilde\tau}$ are good in the
sense of Mumford.
\end{theorem}
This theorem is based on very accurate estimates of the connection
and curvature forms of these metrics. One of the difficulties is to
estimate the Gauss-Manin connection of the fiberwise K\"ahler-Einstein metric
where we use the compound graft metric construction of Wolpert
together with maximum principle.
\section{Negativity and Naturalness}\label{natural}
In Section \ref{wpme} we have seen various negative properties of
the WP metric. In fact, we showed in \cite{lsy3} that the WP metric
is dual Nakano negative. This means the complex curvature operator
of the dual metric of the WP metric is positive. We first recall the
precise definition of dual Nakano negativity of a Hermitian metric.
Let $(E,h)$ be a Hermitian holomorphic vector bundle of rank $m$
over a complex manifold $M$ of dimension $n$. Let $e_1,\cdots,e_m$
be a local holomorphic frame of $E$ and let $z_1,\cdots,z_n$ be
local holomorphic coordinates on $M$. The Hermitian metric $h$ has
expression $h_{i\bar j}=h\left ( e_i,e_j\right )$ locally.
The curvature of $E$ is given by
\[
P_{i\bar j\alpha\bar\beta}=-\partial_\alpha\partial_{\bar\beta}h_{i\bar j}+
h^{p\bar q}\partial_\alpha h_{i\bar q}\partial_{\bar\beta}h_{p\bar j}.
\]
\begin{definition}\label{nadef}
The Hermitian vector bundle $(E,h)$ is Nakano positive if the
curvature $P$ defines a Hermitian metric on the bundle $E\otimes
T_M^{1,0}$. Namely, $P_{i\bar
j\alpha\bar\beta}C^{i\alpha}\bar{C^{j\beta}}>0$ for all $m\times n$
nonzero matrices $C$. The bundle $(E,h)$ is Nakano semi-positive if
$P_{i\bar j\alpha\bar\beta}C^{i\alpha}\bar{C^{j\beta}}\geq 0$. The
bundle is dual Nakano (semi-)negative if the dual bundle with dual
metric $(E^*,h^*)$ is Nakano (semi-)positive.
\end{definition}
We have proved the following theorem in \cite{lsy3}
\begin{theorem}\label{nakanomain}
Let ${\mathcal M}_g$ be the moduli space of Riemann surfaces of genus $g$
where $g\geq 2$. Let $h$ be the WP metric on ${\mathcal M}_g$. Then the
holomorphic tangent bundle $T^{1,0}{\mathcal M}_g$ equipped with the WP metric
$h$ is dual Nakano negative.
\end{theorem}
The dual Nakano negativity is the strongest negativity property of
the WP metric.
Now we look at the naturalness of the canonical metrics on the
moduli space. We let ${\mathcal M}_g$ be the moduli space of genus $g$ curves
where $g\geq 2$ and let $\bar{\mathcal M}_g$ be its Deligne-Mumford
compactification. We fix a point $p\in \bar{\mathcal M}_g\setminus{\mathcal M}_g$ of
codimension $m$ and let $X=X_p$ be the corresponding stable nodal
curve. The moduli space ${\mathcal M}(X)$ of the nodal surface $X$ is
naturally embedded into $\bar{\mathcal M}_g$. Furthermore, since each element
$Y$ in ${\mathcal M}(X)$ corresponds to a hyperbolic Riemann surface when we
remove the nodes from $Y$, the complement can be uniformized by the
upper half plane and thus there is a unique complete K\"ahler-Einstein metric on
$Y$ whose Ricci curvature is $-1$. We note that the moduli space
${\mathcal M}(X)$ can be viewed as an irreducible component of the
intersection of $m$ compactification divisors.
By the discussion in Section \ref{wpme} there is a natural WP metric
$\hat h$ on ${\mathcal M}(X)$. The curvature formula \eqref{30} is still valid
for this WP metric and it is easy to see that the Ricci curvature of
the WP metric $\hat h$ is negative. We can take $\hat\tau=-Ric\left (
\omega_{\hat h}\right )$ to be the K\"ahler form of a K\"ahler metric on ${\mathcal M}(X)$.
This is the Ricci metric $\hat\tau$ on ${\mathcal M}(X)$.
In \cite{ma1} Masur showed that the WP metric $h$ on ${\mathcal M}_g$ extends
to $\bar{\mathcal M}_g$ and its restriction to ${\mathcal M}(X)$ via the natural
embedding ${\mathcal M}(X)\hookrightarrow\bar{\mathcal M}_g$ coincides with the WP
metric $\hat h$ on ${\mathcal M}(X)$. This implies the WP metric is natural.
In \cite{wol7} Wolpert showed that the WP Levi-Civita connection
restricted to directions which are almost tangential to the
compactification divisors limits to the lower dimensional WP
Levi-Civita connection. In \cite{lsy3} we proved the naturalness of
the Ricci metric.
\begin{theorem}\label{ricnatural}
The Ricci metric on ${\mathcal M}_g$ extends to $\bar{\mathcal M}_g$ in non-degenerating
directions. Furthermore, the restriction of the extension of $\tau$
to ${\mathcal M}(X)$ coincides with $\hat\tau$, the Ricci metric on ${\mathcal M}(X)$.
\end{theorem}
\section{The K\"ahler-Ricci Flow and K\"ahler-Einstein Metric on the Moduli Space}\label{krf}
The existence of the K\"ahler-Einstein metric on the Teichm\"uller space was based on the
work of Cheng-Yau since the Teichm\"uller space is pseudo-convex. By the
uniqueness we know that the K\"ahler-Einstein metric is invariant under the action
of the mapping class group and thus is also the K\"ahler-Einstein metric on the
moduli space. It follows from the later work of Yau that the K\"ahler-Einstein
metric is complete. However, the detailed properties of the K\"ahler-Einstein
metric remain unknown.
In \cite{lsy2} we proved the strongly bounded geometry property of
the K\"ahler-Einstein metric. We showed
\begin{theorem}\label{strong}
The K\"ahler-Einstein metric on the Teichm\"uller space ${\mathcal T}_g$ has strongly bounded
geometry. Namely, the curvature and its covariant derivatives of the
K\"ahler-Einstein metric are bounded and the injectivity radius of the K\"ahler-Einstein metric
is bounded from below.
\end{theorem}
This theorem was proved in two steps. Firstly, we deform the Ricci
metric via the K\"ahler-Ricci flow
\begin{eqnarray}\label{bdd10}
\begin{cases}
\frac{\partial g_{i\bar j}}{\partial t}=-(R_{i\bar j}+g_{i\bar j})\\
g_{i\bar j}(0)=\tau_{i\bar j}
\end{cases}
\end{eqnarray}
Let $h=g(s)$ be the deformed metric at time $s\ll 1$. By the work of
Shi \cite{shi1} we know that the metric $h$ is equivalent to the
initial metric $\tau$ and is cohomologous to $\tau$ in the sense of
currents. Thus $h$ is complete and has Poincar\'e growth. Furthermore, the
curvature and covariant derivatives of $h$ are bounded.
We then use the metric $h$ as a background metric to derive a priori
estimates for the K\"ahler-Einstein metric by using the Monge-Amper\'e equation
\[
\frac{\det\left ( h_{i\bar j}+u_{i\bar j}\right )}{\det h_{i\bar j}}=e^{u+F}
\]
where $F$ is the Ricci potential of the metric $h$. If we denote by
$g$ the K\"ahler-Einstein metric and let
\[
S=g^{i\bar j}g^{k\bar l}g^{p\bar q}u_{;i\bar qk}u_{;\bar jp\bar l}
\]
and
\[
V=g^{i\bar j}g^{k\bar l}g^{p\bar q}g^{m\bar n} u_{;i\bar qk\bar
n}u_{;\bar jp\bar lm}+ g^{i\bar j}g^{k\bar l}g^{p\bar q}g^{m\bar n}
u_{;i\bar nkp}u_{;\bar jm\bar l\bar q}
\]
to be the third and fourth order quantities respectively. We have
\begin{align*}
\begin{split}
\Delta^{'}\left [(S+\kappa)V\right ]\geq & C_{1}\left
[(S+\kappa)V\right ]^2-C_{2} \left [(S+\kappa)V\right
]^{\frac{3}{2}}-C_{3}
\left [(S+\kappa)V\right ]\\
& -C_{4}\left [(S+\kappa)V\right ]^{\frac{1}{2}}
\end{split}
\end{align*}
where $\Delta'$ is the Laplace operator of the K\"ahler-Einstein metric $g$ and
$C_1>0$.
It follows from the mean value inequality that $S$ is bounded.
Furthermore, by the above estimate and the maximum principle we know
$V$ is bounded. In fact this method works for all higher order
derivatives of $u$ and we deduce that the K\"ahler-Einstein metric has strongly
bounded geometry.
The K\"ahler-Ricci flow and the goodness are closely tied together.
Firstly, since the most important feature of a Mumford good metric
is that the Chern-Weil theory still holds, we say metrics with this
property are intrinsic good. In \cite{lsy4} we showed
\begin{theorem}
Let $\bar X$ be a projective manifold with $\dim_{\mathbb C} \bar X=n$. Let
$D\subset\bar X$ be a divisor with normal crossings, let $X=\bar
X\setminus D$, let $\bar E=T_{\bar X}(-\log D)$ and let $E=\bar
E\mid_X$.
Let $\omega_g$ be a K\"ahler metric on $X$ with bounded curvature and
Poincar\'e growth. Assume $Ric(\omega_g)+\omega_g=\partial\bar\partial f$ where $f$ is a
bounded smooth function. Then
\begin{itemize}
\item There exists a unique K\"ahler-Einstein metric $\omega_{_{KE}}$ on $X$ with
Poincar\'e growth.
\item The curvature and covariant derivatives of curvature of the
K\"ahler-Einstein metric are bounded.
\item If $\omega_g$ is intrinsic good, then $\omega_{_{KE}}$ is
intrinsic good. Furthermore, all metrics along the paths of
continuity and K\"ahler-Ricci flow are intrinsic good.
\end{itemize}
\end{theorem}
\section{Applications}\label{app}
In this last section we briefly look at some geometric applications
of the canonical metrics. The first application of the control of
the K\"ahler-Einstein metric is the stability of the logarithmic cotangent bundle
of the Deligne-Mumford moduli space. In \cite{lsy2} we proved
\begin{theorem}\label{stab}
Let $\bar E=T_{\bar{\mathcal M}_g}^*\left ( \log D\right )$ be the logarithmic
cotangent bundle. Then $c_1(\bar E)$ is positive and $\bar E$ is
slope stable with respect to the polarization $c_1(\bar E)$.
\end{theorem}
An immediate consequence of the intrinsic goodness of the K\"ahler-Einstein metric
is the Chern number inequality. We have
\begin{theorem}
Let $\bar E= T_{\bar{\mathcal M}_g}(-\log D)$ be the logarithmic tangent
bundle of the moduli space. Then
\[
c_1(\bar E)^2 \leq \frac{6g-4}{3g-3}c_2(\bar E).
\]
\end{theorem}
An immediate consequence of the dual Nakano negativity and the
goodness of the WP metric is the positivity of the Chern numbers of
this bundle. We have
\begin{theorem}
The Chern numbers of the logarithmic cotangent bundle
$T_{\bar{\mathcal M}_g}^*(\log D)$ of the moduli spaces of Riemann surfaces
are all positive.
\end{theorem}
The dual Nakano negativity of a Hermitian metric on a bundle over a
compact manifold gives strong vanishing theorems by using Bochner
techniques. However, in our case the base variety ${\mathcal M}_g$ is only
quasi-projective. Thus we can only describe vanishing theorems of
the $L^2$ cohomology. In \cite{sap1}, Saper showed that the $L^2$
cohomology of the moduli space equipped with the Weil-Petersson
metric can be identified with the ordinary cohomology of the
Deligne-Mumford moduli space. Our situation is more subtle since the
natural object to be considered in our case is the tangent bundle
valued $L^2$ cohomology. Parallel to Saper's work, we proved in
\cite{lsy4}
\begin{theorem}\label{iden}
We have the following natural isomorphism
\[
H_{(2)}^* \left ( \left ( {\mathcal M}_g,\omega_\tau\right ),\left (
T_{{\mathcal M}_g},\omega_{_{WP}}\right )\rb \cong H^*\left ( \bar{\mathcal M}_g, T_{\bar{\mathcal M}_g}\left (
-\log D\right )\rb.
\]
\end{theorem}
Now we combine the above result with the dual Nakano negativity of
the WP metric. In \cite{lsy4} we proved the following Nakano-type
vanishing theorem
\begin{theorem}\label{van}
The $L^2$ cohomology groups vanish:
\[
H_{(2)}^{0,q}\left ( \left ( {\mathcal M}_g,\omega_\tau\right ),\left (
T_{{\mathcal M}_g},\omega_{_{WP}}\right )\rb=0
\]
unless $q=3g-3$.
\end{theorem}
As a direct corollary we have
\begin{corollary}\label{rigid}
The pair $\left ( \bar{\mathcal M}_g,D\right )$ is infinitesimally rigid.
\end{corollary}
Another important application of the properties of the Ricci,
perturbed Ricci and K\"ahler-Einstein metrics is the Gauss-Bonnet theorem on the
noncompact moduli space. Together with L. Ji, in \cite{jlsy} we
showed
\begin{theorem}
The Gauss-Bonnet theorem holds on the moduli space equipped with the
Ricci, perturbed Ricci or K\"ahler-Einstein metrics:
\[
\int_{{\mathcal M}_g} c_n(\omega_\tau) = \int_{{\mathcal M}_g} c_n(\omega_{\tilde\tau})
=\int_{{\mathcal M}_g} c_n(\omega_{_{KE}}) =\chi({\mathcal M}_g) =
\frac{B_{2g}}{4g(g-1)}.
\]
Here $\chi({\mathcal M}_g)$ is the orbifold Euler characteristic of ${\mathcal M}_g$ and
$n=3g-3$.
\end{theorem}
The explicit topological computation of the Euler characteristic of
the moduli space is due to Harer-Zagier \cite{hz1}. See also the
work of Penner \cite{penn1}.
As an application of the Mumford goodness of the WP metric and the
Ricci metric we have
\begin{theorem}
\[
\chi(T_{\bar{\mathcal M}_g}(-\log D))= \int_{{\mathcal M}_g} c_n(\omega_\tau)=
\int_{{\mathcal M}_g} c_n(\omega_{_{WP}})= \frac{B_{2g}}{4g(g-1)}
\]
where $n=3g-3$.
\end{theorem}
It is very hard to prove the Gauss-Bonnet theorem for the WP metric
directly since the WP metric is incomplete and its curvature is not
bounded. The proof is based substantially on the Mumford goodness
of the WP metric.
By using the goodness of canonical metrics this theorem also gives
an explicit expression of the top log Chern number of the moduli
space.
\begin{theorem}
\[
\chi(\bar{\mathcal M}_g, T_{\bar{\mathcal M}_g}(-\log D))=\chi({\mathcal M}_g)=
\frac{B_{2g}}{4g(g-1)}.
\]
\end{theorem}
\section{Global Torelli Theorem of the Teichm\"uller Spaces of Polarized Calabi-Yau Manifolds (Joint with Andrey Todorov)}\label{torelli}
The geometry of the Teichm\"uller and moduli spaces of polarized Calabi-Yau
(CY) manifolds are the central objects in geometry and string
theory. One of the most important question in understanding the
geometry of the Teichm\"uller
and moduli space of polarized CY manifolds is the global Torelli problem which
asks whether the variation of
polarized Hodge structures determines the marked polarized Calabi-Yau structure. In the rest of this article, after briefly discussing the deformation theory of CY manifolds and the geometry of period domain, we will describe the global Torelli theorem of the Teichm\"uller spaces of polarized CY manifolds and its proof. See \cite{lsty1} for details.
Let $M$ be a Calabi-Yau manifold of dimension $\dim
_{\mathbb{C}}M=n$. Here we assume $n\geq 3$. Let $L$ be an ample
line bundle over $M$. By definition we assume that the canonical
bundle $K_M$ is trivial. Let $X$ be the underlying real
$2n$-dimensional manifold. We know that there is a nowhere vanishing
holomorphic $(n,0)$-form on $M$ which is unique up to scaling. The
Teichm\"uller space of $(M,L)$ is the connected, simply connected, reduced
and irreducible manifold parameterizing triples $(M,L,(
\gamma_1,\cdots,\gamma_{b_n}))$ where $M$ is a CY manifold, $L$ is
the polarization and $(\gamma_1,\cdots,\gamma_{b_n})$ is a basis of
the middle homology group $H_n(X,{\mathbb{Z}})/tor$. Such triples
are called marked polarized CY manifolds.
\subsection{Deformation Theory of Polarized Calabi-Yau Manifolds}
We first recall the
deformation of complex structures on a given smooth manifold. Let $X$ be a smooth
manifold of dimension $\dim_{\mathbb{R}} X=2n$ and let $%
J_0$ be an integrable complex structure on $X$. We denote by
$M_0=(X,J_0)$ the corresponding complex manifold.
Let $\phi\in A^{0,1}\left (M_0,T_{M_0}^{1,0}\right )$ be a Beltrami
differential. We can view $\phi$ as a map
\begin{equation*}
\phi:\Omega^{1,0}(M_0)\to \Omega^{0,1}(M_0).
\end{equation*}
By using $\phi$ we define a new almost complex structure $J_\phi$ in
the following way. For a point $p\in M_0$ we pick a local
holomorphic coordinate chart $(U, z_1,\cdots,z_n)$ around $p$. Let
\begin{eqnarray} \label{defmain}
\Omega_\phi^{1,0}(p)=\text{span}_{\mathbb{C}}\{ dz_1+\phi(dz_1),
\cdots, dz_n+\phi(dz_n)\}
\end{eqnarray}
and
\begin{equation*}
\Omega_\phi^{0,1}(p)=\text{span}_{\mathbb{C}}\{ d\bar
z_1+\bar\phi(d\bar z_1), \cdots, d\bar z_n+\bar\phi(d\bar z_n)\}
\end{equation*}
be the eigenspaces of $J_\phi$ with respect to the eigenvalue
$\sqrt{-1}$ and $-\sqrt{-1}$ respectively.
The almost complex structure $J_\phi$ is integrable if and only if
\begin{eqnarray} \label{int}
\bar\partial\phi=\frac{1}{2}[\phi,\phi]
\end{eqnarray}
where $\bar\partial$ is the operator on $M_0$.
It was proved in \cite{tod1} and \cite{tian1} that the local
deformation of a polarized CY manifold is unobstructed.
\begin{theorem}
\label{unobs} The universal deformation space of a polarized CY
manifold is smooth.
\end{theorem}
The operation of contracting with $\Omega_0$ plays an important role
in converting bundle valued differential forms into ordinary
differential forms. The following lemma is the key step in the proof
of local Torelli theorem.
\begin{lemma}
\label{iso} Let $(M,L)$ be a polarized CY $n$-fold and let
$\omega_g$ be the unique CY metric in the class $[L]$. We pick a
nowhere vanishing holomorphic $(n,0)$-form $\Omega_0$ such that
\begin{eqnarray} \label{normalization}
\left ( \frac{\sqrt{-1}}{2}\right )^n(-1)^{\frac{n(n-1)}{2}%
}\Omega_0\wedge\bar\Omega_0=\omega_g^n.
\end{eqnarray}
Then the map $\iota:A^{0,1}\left ( M, T_M^{1,0}\right )\to
A^{n-1,1}(M)$ given by $\iota(\phi)=\phi\lrcorner\Omega_0$ is an
isometry with respect to the natural Hermitian inner product on both
spaces induced by $\omega_g$. Furthermore, $\iota$ preserves the
Hodge decomposition.
\end{lemma}
In \cite{tod1} the existence of flat coordinates was established and
the flat coordinates played an important role in string theory
\cite{bcov1}. Here we recall this construction. Let $\mathfrak{X}$
be the universal family
over ${\mathcal{T}}$ and let $\pi$ be the projection map. For each $p\in{%
\mathcal{T}}$ we let $M_p=(X,J_p)$ be the corresponding CY manifold.
In the following we always use the unique CY metric on $M_p$ in the
polarization class $[L]$.
By the Kodaira-Spencer theory and Hodge theory, we have the
following identification
\begin{equation*}
T_p^{1,0}{\mathcal{T}}\cong {\mathbb{H}}^{0,1}\left (
M_p,T_{M_p}^{1,0}\right )
\end{equation*}
where we use ${\mathbb{H}}$ to denote the corresponding space of
harmonic forms. We have the following expansion of the Beltrami
differentials:
\begin{theorem}
\label{flatcoord} Let $\phi_1,\cdots,\phi_N \in
{\mathbb{H}}^{0,1}\left ( M_p,T_{M_p}^{1,0}\right )$ be a basis.
Then there is a unique power series
\begin{eqnarray} \label{10}
\phi(\tau)=\sum_{i=1}^N \tau_i\phi_i +\sum_{|I|\geq 2}\tau^I\phi_I
\end{eqnarray}
which converges for $|\tau|<\varepsilon$. Here $I=(i_1,\cdots,i_N)$
is a multi-index, $\tau^I=\tau_1^{i_1}\cdots\tau_N^{i_N}$ and
$\phi_I\in A^{0,1}\left ( M_p,T_{M_p}^{1,0}\right )$. Furthermore,
if $\Omega$ is a nowhere vanishing holomorphic $(n,0)$-form, then
the family of Beltrami differentials $\phi(\tau)$ satisfy the
following conditions:
\begin{align} \label{charflat}
\begin{split}
&\bar\partial_{M_p}\phi(\tau)=\frac{1}{2}[\phi(\tau),\phi(\tau)] \\
&\bar\partial_{M_p}^*\phi(\tau)=0 \\
&\phi_I\lrcorner\Omega =\partial_{M_p}\psi_I
\end{split}%
\end{align}
for each $|I|\geq 2$ where $\psi_I\in A^{n-2,1}(M_p)$. Furthermore,
by
shrinking $\varepsilon$ we can pick each $\psi_I$ appropriately such that $%
\sum_{|I|\geq 2}\tau^I\psi_I$ converges for $|\tau|<\varepsilon$.
\end{theorem}
The coordinates constructed in the above theorem are just the flat
coordinates described in \cite{bcov1}. They are unique up to affine
transformation and they are also the normal coordinates of the
Weil-Petersson metric at $p$. In fact the first equation in
\eqref{charflat} is the obstruction equation and the second equation
is the Kuranishi gauge which fixes the gauge in the fiber. The last
equation which characterized the flat coordinates around a point in
the Teichm\"uller space is known as the Todorov gauge.
From Theorem \ref{flatcoord} the local Torelli theorem and the
Griffiths transversality follow immediately. However, Theorem
\ref{flatcoord} contains more information.
By using the local deformation theory, in \cite{tod1} Todorov
constructed a canonical local holomorphic section of the line bundle
$H^{n,0}=F^n$ over any flat coordinate chart $U\subset
{\mathcal{T}}$ in the form level. This canonical section plays a
crucial role in the proof of the global Torelli theorem.
We first consider the general construction of holomorphic
$(n,0)$-forms in \cite{tod1}.
\begin{lemma}
\label{constructn0} Let $M_0=(X,J_0)$ be a CY manifold where $J_0$
is the complex structure on $X$. Let $\phi\in A^{0,1}\left (
M_0,T_{M_0}^{1,0}\right )$ be a Beltrami differential on $M_0$ which
define an integrable complex structure $J_\phi$ and let
$M_\phi=(X,J_\phi)$ be the CY manifold whose underlying
differentiable manifold is $X$. Let $\Omega_0$ be a nowhere
vanishing holomorphic $(n,0)$-form on $M_0$ and let
\begin{eqnarray} \label{n0expansion}
\Omega_\phi=\sum_{k=0}^n
\frac{1}{k!}(\wedge^k\phi\lrcorner\Omega_0).
\end{eqnarray}
Then $\Omega_\phi$ is a well-defined smooth $(n,0)$-form on
$M_\phi$. It is holomorphic with
respect to the complex structure $J_\phi$ if and only if $%
\partial(\phi\lrcorner\Omega_0)=0$. Here $\partial$ is the operator with
respect to the complex structure $J_0$.
\end{lemma}
By combining Lemma \ref{constructn0} and Theorem \ref{flatcoord} we
define the canonical family
\begin{eqnarray} \label{can10}
\Omega^c=\Omega^c(\tau)=\sum_{k=0}^n \frac{1}{k!} \left (
\wedge^k\phi(\tau) \lrcorner\Omega_0\right )
\end{eqnarray}
and we have
\begin{corollary}
\label{expcoh} Let $\Omega^c(\tau)$ be a canonical family defined by %
\eqref{can10} where $\phi(\tau)$ is defined as in \eqref{charflat}.
Then we have the expansion
\begin{align}\label{cohexp10}
\begin{split}
[\Omega^c(\tau)]=& [\Omega_0]+\sum_{i=1}^N \tau_i[\phi_i\lrcorner\Omega_0]\\
&+\frac{1}{2}\sum_{i,j}\tau_i\tau_j \left [ {\mathbb{H}}(\phi_i\wedge\phi_j%
\lrcorner\Omega_0) \right ]+\Xi(\tau)
\end{split}
\end{align}
where $\Xi(\tau)\subset \bigoplus_{k=2}^n H^{n-k,k}(M_p)$ and $%
\Xi(\tau)=O(|\tau|^3)$.
\end{corollary}
The most important application of the cohomological expansion %
\eqref{cohexp10} is the invariance of the CY K\"ahler forms. The
theorem plays a central role in the proof of the global Torelli
theorem. This theorem was implicitly proved in \cite{bato1}. Please
see \cite{lsty1} for a simple and self-contained proof.
\begin{theorem}
\label{invarkaform} For each point $p\in{\mathcal{T}}$, let
$\omega_p$ be the K\"ahler form of the unique CY metric on $M_p$ in
the polarization class $[L]$. Then $\omega_p$ is invariant. Namely,
\begin{equation*}
\nabla^{GM}\omega_p=0.
\end{equation*}
Furthermore, since ${\mathcal{T}}$ is simply connected, we know that $%
\omega_p$ is a constant section of the trivial bundle
$A^2(X,{\mathbb{C}})$ over ${\mathcal{T}}$.
\end{theorem}
\subsection{The Teichm\"uller Space of Polarized Calabi-Yau Manifolds}
Now we recall the construction of the universal family of marked
polarized CY manifolds and the Teichm\"uller space. See \cite{ltyz} for
details. Let $M$ be a CY manifold of dimension $\dim_{\mathbb{C}}
M=n\geq 3$. Let $L$ be an ample line bundle over $M$. We call a tuple $%
(M,L,\gamma_1,\cdots,\gamma_{h^n})$ a marked polarized CY manifold
if $M$ is a CY manifold, $L$ is a polarization of $M$ and $\left \{
\gamma_1,\cdots,\gamma_{h^n}\right \}$ is a basis of
$H_n(M,{\mathbb{Z}})/tor $.
\begin{remark}\label{simpli}
To simplify notations we assume in this section that a CY manifold
$M$ of dimension $n$ is simply connected and $h^{k,0}(M)=0$ for
$1\leq k\leq n-1$. All the results in this section hold when these
conditions are removed. This is due to the fact that we fix a
polarization.
\end{remark}
Since the Teichm\"uller space of $M$ with fixed marking and
polarization is constructed via GIT quotient, we need the following
results about group actions.
\begin{theorem}
\label{cons20} Let $(M,L,(\gamma_1,\cdots,\gamma_{b_n}))$ be a
marked polarized CY manifold and let $\pi:\mathfrak{X}\to
\mathcal{K}$ be the Kuranishi family of $M$. We let
$p\in\mathcal{K}$ such that $M=\pi^{-1}(p)$. If $G$ is a group of
holomorphic automorphisms of $M$ which preserve the
polarization $L$ and act trivially on $H_n(M,{\mathbb{Z}})$, then for any $%
q\in\mathcal{K}$ the group $G$ acts on $M_q=\pi^{-1}(q)$ as
holomorphic automorphisms.
\end{theorem}
Now we recall the construction of the Teichm\"uller space. We first
note that there is a constant $m_0>0$ which only depends on $n$ such
that for any polarized CY manifold $(M,L)$ of dimension $n$, the
line bundle $L^m$ is very ample for any $m\geq m_0$. We replace $L$
by $L^{m_0}$ and we still denote it by $L$.
Let $N_m=h^0(M,L^m)$. It follows from the Kodaira embedding theorem
that $M$ is embedded into $\mathbb{P}^{N_m-1}$ by the holomorphic
sections of $L^m$. Let $\mathcal{H}_L$ be the component of the
Hilbert scheme which contains $M$ and parameterizes smooth CY
varieties embedded in $\mathbb{P}^{N_m-1}$ with Hilbert polynomial
\begin{equation*}
P(m)=h^0\left ( M, L^m\right ).
\end{equation*}
We know that $\mathcal{H}_L$ is a smooth quasi-projective variety
and there
exists a universal family $\mathcal{X}_L\to \mathcal{H}_L$ of pairs $%
(M,(\sigma_0,\cdots,\sigma_{N_m}))$ where
$(\sigma_0,\cdots,\sigma_{N_m})$ is a basis of $H^0(M,L^m)$. Let
$\tilde{\mathcal{H}}_L$ be its universal
cover. By Theorem \ref{cons20} we know that the group $PGL\left ( N_m,{%
\mathbb{C}}\right )$ acts on $\tilde{\mathcal{H}}_L$ and the family $\tilde{%
\mathcal{X}}_L\to \tilde{\mathcal{H}}_L$ holomorphically and without
fixed points. Furthermore, it was proved in \cite{ltyz} and
\cite{ps} that the
group $PGL\left ( N_m,{\mathbb{C}}\right )$ also acts properly on $\tilde{%
\mathcal{H}}_L$.
We define the Teichm\"uller space of $M$ with polarization $L$ by
\begin{equation*}
{\mathcal{T}}={\mathcal{T}}_L(M)=\tilde{\mathcal{H}}_L/PGL\left ( N_m,{%
\mathbb{C}}\right ).
\end{equation*}
One of the most important features of the Teichm\"uller space is the
existence of universal family.
\begin{theorem}
\label{universal} There exist a family of marked polarized CY manifolds $\pi:%
\mathcal{U}_L\to{\mathcal{T}}_L(M)$ such that there is a point $p\in{%
\mathcal{T}}_L(M)$ with $M_p$ isomorphic to $M$ as marked polarized
CY manifolds and the family has the following properties:
\begin{enumerate}
\item ${\mathcal{T}}_L(M)$ is a smooth complex manifold of dimension $\dim_{%
\mathbb{C}}{\mathcal{T}}_L(M)=h^{n-1,1}(M)$.
\item For each point $q\in{\mathcal{T}}_L(M)$ there is a natural
identification
\begin{equation*}
T_q^{1,0}{\mathcal{T}}_L(M)\cong H^{0,1}\left (
M_q,T_{M_q}^{1,0}\right )
\end{equation*}
via the Kodaira-Spencer map.
\item Let $\rho:\mathcal{Y}\to \mathcal{C}$ be a family of marked polarized
CY manifold such that there is a point $x\in \mathcal{C}$ whose fiber $%
\rho^{-1}(x)$ is isomorphic to $M_p$ as marked polarized CY
manifolds. Then
there is a unique holomorphic map $f:(\mathcal{Y}\to \mathcal{C})\to (%
\mathcal{U}_L\to {\mathcal{T}}_L(M))$, defined up to biholomorphic
maps on the fibers whose induced maps on $H_n(M,{\mathbb{Z}})$ are
the identity map, such that $f$ maps the fiber $\rho^{-1}(x)$ to the
fiber $M_p$ and the
family $\mathcal{Y}$ is just the pullback of $\mathcal{U}_L$ via the map $f$%
. Furthermore, the map $\tilde f:\mathcal{C}\to{\mathcal{T}}_L(M)$
induced by $f$ is unique.
\end{enumerate}
\end{theorem}
It follows directly that
\begin{proposition}
\label{imp1} The Teichm\"uller space
${\mathcal{T}}={\mathcal{T}}_L(M)$ is a smooth complex manifold and
is simply connected.
\end{proposition}
\label{assumption} In the rest of this paper by the Teichm\"uller space ${%
\mathcal{T}}$ of $(M,L, (\gamma_1,\cdots,\gamma_{b_n}))$ we always
mean the
reduced irreducible component of $\tilde{\mathcal{H}}_L/PGL\left ( N_m,{%
\mathbb{C}}\right )$ with the fixed polarization $L$.
It follows from its construction and Theorem \ref{cons20} that the
universal family $\mathcal U_L$ over the Teichm\"uller space ${\mathcal T}$ is
diffeomorphic to $M_p\times{\mathcal T}$ as a $C^\infty$ family where $p\in{\mathcal T}$
is any point and $M_p$ is the corresponding CY manifold.
\subsection{The Classifying Space of Variation of Polarized Hodge Structures}
Now we recall the construction of the classifying space of variation
of polarized Hodge structures and its basic properties such as the
description of its real and complex tangent spaces and the Hodge
metric. See \cite{schmid1} for details.
In the construction of the Teichm\"uller space ${\mathcal{T}}$ we
fixed a
marking of the background manifold $X$, namely a basis of $H_n(X,{\mathbb{Z}}%
)/tor$. This gives us canonical identifications of the middle
dimensional de Rahm cohomology of different fibers over
${\mathcal{T}}$. Namely for any two distinct points
$p,q\in{\mathcal{T}}$ we have the canonical identification
\begin{equation*}
H^n(M_p)\cong H^n(M_q)\cong H^n(X)
\end{equation*}
where the coefficient ring is ${\mathbb{Q}}$, ${\mathbb{R}}$ or
${\mathbb{C}} $.
Since the polarization $[L]$ is an integeral class, it defines a map
\begin{equation*}
L:H^n(X,{\mathbb{Q}})\to H^{n+2}(X,{\mathbb{Q}})
\end{equation*}
given by $A\mapsto c_1(L)\wedge A$ for any $A\in
H^n(X,{\mathbb{Q}})$. We denote by $H_{pr}^n(X)=\ker(L)$ the
primitive cohomology groups where,
again, the coefficient ring is ${\mathbb{Q}}$, ${\mathbb{R}}$ or ${\mathbb{C}%
}$. For any $p\in{\mathcal{T}}$ we let $H_{pr}^{k,n-k}(M_p)=H^{k,n-k}(M_p)%
\cap H_{pr}^n(M_p,{\mathbb{C}})$ and denote its dimension by
$h^{k,n-k}$. The Poincar\'e bilinear form $Q$ on
$H_{pr}^n(X,{\mathbb{Q}})$ is defined by
\begin{equation*}
Q(u,v)=(-1)^{\frac{n(n-1)}{2}}\int_X u\wedge v
\end{equation*}
for any $d$-closed $n$-forms $u,v$ on $X$. The bilinear form $Q$ is
symmetric if $n$ is even and is skew-symmetric if $n$ is odd.
Furthermore, $Q $ is non-degenerate and can be extended to
$H_{pr}^n(X,{\mathbb{C}})$ bilinearly. For any point
$q\in{\mathcal{T}}$ we have the Hodge decomposition
\begin{eqnarray} \label{cl10}
H_{pr}^n(M_q,{\mathbb{C}})=H_{pr}^{n,0}(M_q,{\mathbb{C}})\oplus\cdots\oplus
H_{pr}^{0,n}(M_q,{\mathbb{C}})
\end{eqnarray}
which satisfies
\begin{eqnarray} \label{cl20}
\dim_{\mathbb{C}} H_{pr}^{k,n-k}(M_q,{\mathbb{C}})=h^{k,n-k}
\end{eqnarray}
and the Hodge-Riemann relations
\begin{eqnarray} \label{cl30}
Q\left ( H_{pr}^{k,n-k}(M_q,{\mathbb{C}}), H_{pr}^{l,n-l}(M_q,{\mathbb{C}}%
)\right )=0\ \ \text{unless}\ \ k+l=n
\end{eqnarray}
and
\begin{eqnarray} \label{cl40}
\left (\sqrt{-1}\right )^{2k-n}Q\left ( v,\bar v\right )>0\ \
\text{for}\ \ v\in H_{pr}^{k,n-k}(M_q,{\mathbb{C}})\setminus\{0\}.
\end{eqnarray}
The above Hodge decomposition of $H_{pr}^n(M_q,{\mathbb{C}})$ can
also be described via the Hodge filtration. Let $f^k=\sum_{i=k}^n
h^{i,n-i}$. We let
\begin{equation*}
F^k=F^k(M_q)=H_{pr}^{n,0}(M_q,{\mathbb{C}})\oplus\cdots\oplus
H_{pr}^{k,n-k}(M_q,{\mathbb{C}})
\end{equation*}
and we have decreasing filtration
\begin{equation*}
H_{pr}^n(M_q,{\mathbb{C}})=F^0(M_q)\supset\cdots\supset F^n(M_q).
\end{equation*}
We know that
\begin{eqnarray} \label{cl45}
\dim_{\mathbb{C}} F^k=f^k,
\end{eqnarray}
\begin{eqnarray} \label{cl46}
H_{pr}(X,{\mathbb{C}})=F^{k}(q)\oplus \bar{F^{n-k+1}(q)}
\end{eqnarray}
and
\begin{eqnarray} \label{cl48}
H_{pr}^{k,n-k}(M_q,{\mathbb{C}})=F^k(M_q)\cap\bar{F^{n-k}(M_q)}.
\end{eqnarray}
In term of the Hodge filtration $F^n\subset\cdots\subset F^0=H_{pr}^n(M_q,{%
\mathbb{C}})$ the Hodge-Riemann relations can be written as
\begin{eqnarray} \label{cl50}
Q\left ( F^k,F^{n-k+1}\right )=0
\end{eqnarray}
and
\begin{eqnarray} \label{cl60}
Q\left ( Cv,\bar v\right )>0 \ \ \text{if}\ \ v\ne 0
\end{eqnarray}
where $C$ is the Weil operator given by $Cv=\left (\sqrt{-1}\right
)^{2k-n}v$ when $v\in H_{pr}^{k,n-k}(M_q,{\mathbb{C}})$. The
classifying space $D$ of variation of polarized Hodge structures
with data \eqref{cl45} is the space of all such Hodge filtrations
\begin{equation*}
D=\left \{ F^n\subset\cdots\subset F^0=H_{pr}^n(X,{\mathbb{C}})\mid %
\eqref{cl45}, \eqref{cl50} \text{ and } \eqref{cl60} \text{ hold}
\right \}.
\end{equation*}
The compact dual $\check D$ of $D$ is
\begin{equation*}
\check D=\left \{ F^n\subset\cdots\subset F^0=H_{pr}^n(X,{\mathbb{C}})\mid %
\eqref{cl45} \text{ and } \eqref{cl50} \text{ hold} \right \}.
\end{equation*}
The classifying space $D\subset \check D$ is an open set. We note
that the
conditions \eqref{cl45}, \eqref{cl50} and \eqref{cl60} imply the identity %
\eqref{cl46}.
An important feature of the variation of polarized Hodge structures
is that both $D$ and $\check D$ can be written as quotients of
semi-simple Lie
groups. Let $H_{\mathbb{R}}=H_{pr}^n(X,{\mathbb{R}})$ and $H_{\mathbb{C}}%
=H_{pr}^n(X,{\mathbb{C}})$. We consider the real and complex
semi-simple Lie groups
\begin{equation*}
G_{\mathbb{R}}=\{\sigma\in GL(H_{\mathbb{R}})\mid Q(\sigma u, \sigma
v)=Q(u,v)\}
\end{equation*}
and
\begin{equation*}
G_{\mathbb{C}}=\{\sigma\in GL(H_{\mathbb{C}})\mid Q(\sigma u, \sigma
v)=Q(u,v)\}.
\end{equation*}
The real group $G_{\mathbb{R}}$ acts on $D$ and the complex group $G_{%
\mathbb{C}}$ acts on $\check D$ where both actions are transitive.
This implies that both $D$ and $\check D$ are smooth. Furthermore,
we can embed the real group into the complex group naturally as real
points.
We now fix a reference point $O=\{F_0^k\}\in D\subset\check D$ and
let $B$ be the isotropy group of $O$ under the action of
$G_{\mathbb{C}}$ on $\check D$. Let $\left\{ H_0^{k,n-k}\right\}$ be
the corresponding Hodge
decomposition where $H_0^{k,n-k}=F_0^k\cap \bar{F_0^{n-k}}$. Let $V=G_{%
\mathbb{R}}\cap B$. Then we have
\begin{eqnarray} \label{cl70}
D=G_{\mathbb{R}}/V \ \ \ \text{and}\ \ \ \check D=G_{\mathbb{C}}/B.
\end{eqnarray}
Following the argument in \cite{schmid1} we let
\begin{equation*}
H_0^{+}=\bigoplus_{i\text{ is even}}H_0^{i,n-i} \ \ \ \ H_0^{-}=\bigoplus_{i%
\text{ is odd}}H_0^{i,n-i}
\end{equation*}
and let $K$ be the isotropy group of $H_0^{+}$ in $G_{\mathbb{R}}$.
We note that $H_0^+$ and $H_0^-$ are defined over ${\mathbb{R}}$ and
are orthogonal with respect to $Q$ when $n$ is even. When $n$ is odd
they are conjugate to each other. Thus $K$ is also the isotropy
group of $H_0^-$. In both cases $K$ is the maximal compact subgroup
of $G_{\mathbb{R}}$ containing $V$. This implies that $\tilde
D=G_{\mathbb{R}}/K$ is a symmetric space of noncompact
type and $D$ is a fibration over $\tilde D$ whose fibers are isomorphic to $%
K/V$.
\begin{remark}
\label{question} In the following we will only consider primitive
cohomology classes and we will drop the mark $``$pr$"$. Furthermore,
Since we only need to use the component of $G_{\mathbb{R}}$
containing the identity, we will denote again by $G_{\mathbb{R}}$
and $K$ the components of the real group and its corresponding
maximal compact subgroup which contain the identity.
\end{remark}
We fix a point $p\in{\mathcal{T}}$ and let $O=\Phi(p)\in
D\subset\check D$.
For $0\leq k\leq n$ we let $H_0^{k,n-k}=H^{k,n-k}(M_p)$. Now we let ${%
\mathfrak{g}}={\mathfrak{g}}_{\mathbb{C}}$ be the Lie algebra of $G_{\mathbb{%
C}}$ and let ${\mathfrak{g}}_0={\mathfrak{g}}_{\mathbb{R}}$ be the
Lie algebra of $G_{\mathbb{R}}$. The real Lie algebra
${\mathfrak{g}}_0$ can also be embedded into ${\mathfrak{g}}$
naturally as real points. The Hodge structure $\left\{
H_0^{k,n-k}\right \}$ induces a weight $0$ Hodge
structure on ${\mathfrak{g}}$. Namely ${\mathfrak{g}}=\bigoplus_p {\mathfrak{%
g}}^{p,-p}$ where
\begin{equation*}
{\mathfrak{g}}^{p,-p}=\left\{ X\in {\mathfrak{g}}\mid X\left
(H_0^{k,n-k}\right )\subset H_0^{k+p,n-k-p}\right \}.
\end{equation*}
Let $B$ be the isotropy group of $O\in\check D$ under the action of $G_{%
\mathbb{C}}$ and let $\mathfrak{b}$ be the Lie algebra of $B$. Then
\begin{equation*}
\mathfrak{b}=\bigoplus_{p\geq 0}{\mathfrak{g}}^{p,-p}.
\end{equation*}
Let $V=B\cap G_{\mathbb{R}}$ be the isotropy group of $O\in D$ under
the action of $G_{\mathbb{R}}$ and let $\mathfrak{v}$ be its Lie
algebra. We have
\begin{equation*}
\mathfrak{v}=\mathfrak{b}\cap {\mathfrak{g}}_0\subset
{\mathfrak{g}}.
\end{equation*}
Now we have
\begin{equation*}
\mathfrak{v}={\mathfrak{g}}_0\cap\mathfrak{b}={\mathfrak{g}}_0\cap\mathfrak{b%
}\cap\bar{\mathfrak{b}}={\mathfrak{g}}_0\cap {\mathfrak{g}}^{0,0}.
\end{equation*}
Let $\theta$ be the Weil operator of the weight $0$ Hodge structure on ${%
\mathfrak{g}}$. Then for any $v\in {\mathfrak{g}}^{p,-p}$ we have $%
\theta(v)=(-1)^pv$. The eigenvalues of $\theta$ are $\pm 1$. Let ${\mathfrak{%
g}}^+$ be the eigenspace of $1$ and let ${\mathfrak{g}}^-$ be the
eigenspace of $-1$. Then we have
\begin{equation*}
{\mathfrak{g}}^+=\bigoplus_{p \text{ even}}{\mathfrak{g}}^{p,-p} \ \
\ \text{ and }\ \ \ {\mathfrak{g}}^-=\bigoplus_{p \text{
odd}}{\mathfrak{g}}^{p,-p}.
\end{equation*}
We note here that, in the above expression, $p$ can be either
positive or negative. Let $\mathfrak{k}$ be the Lie algebra of $K$,
the maximal compact
subgroup of $G_{\mathbb{R}}$ containing $V$. By the work of Schmid \cite%
{schmid1} we know that
\begin{lemma}
\label{cartan} The Lie algebra $\mathfrak{k}$ is given by $\mathfrak{k}={%
\mathfrak{g}}_0\cap {\mathfrak{g}}^+$. Furthermore, if we let $\mathfrak{p}%
_0={\mathfrak{g}}_0\cap {\mathfrak{g}}^-$, then
\begin{equation*}
{\mathfrak{g}}_0=\mathfrak{k}\oplus \mathfrak{p}_0
\end{equation*}
is a Cartan decomposition of ${\mathfrak{g}}_0$. The space
$\mathfrak{p}_0$ is $Ad_V$ invariant.
\end{lemma}
We call such a Cartan decomposition the canonical Cartan
decomposition. Here we recall that if
${\mathfrak{g}}_0={\mathfrak{k}}\oplus {\mathfrak{p}}_0$
is a Cartan decomposition of the real semisimple Lie algebra ${\mathfrak{g}}%
_0$, then we know that ${\mathfrak{k}}$ is a Lie subalgebra, $[{\mathfrak{p}}%
_0,{\mathfrak{p}}_0]\subset {\mathfrak{k}}$ and $[{\mathfrak{p}}_0,{%
\mathfrak{k}}]\subset {\mathfrak{p}}_0$.
By the expression of $\mathfrak{v}$ and $\mathfrak{k}$ we have the
identification
\begin{eqnarray} \label{kv100}
\mathfrak{k}/\mathfrak{v}\cong {\mathfrak{g}}_0\cap \left
(\bigoplus_{p\ne 0, \ p \text{ is even}} {\mathfrak{g}}^{p,-p}\right
)
\end{eqnarray}
and the identification
\begin{eqnarray} \label{killing10}
T_O^{\mathbb{R}} D\cong \mathfrak{k}/\mathfrak{v}\oplus
\mathfrak{p}_0.
\end{eqnarray}
Now we look at the complex structures on $D$. By the above
identification we know that for each element $X\in T_O^{\mathbb{R}}
D$ we have the unique decomposition $X=X_++X_-$ where
$X_+\in\bigoplus_{p>0}{\mathfrak{g}}^{-p,p}$ and $X_-
\in\bigoplus_{p>0}{\mathfrak{g}}^{p,-p}$. We define the complex
structure $J$ on $T_O^{\mathbb{R}} D$ by
\begin{eqnarray} \label{cxstructured}
JX=iX_+-iX_-.
\end{eqnarray}
Now we use left translation by elements in $G_{\mathbb{R}}$ to move
this complex structure to every point in $D$. Namely, for any point
$\alpha\in D$
we pick $g\in G_{\mathbb{R}}$ such that $g(O)=\alpha$. If $X\in T_\alpha^{%
\mathbb{R}} D$, then we define $JX=\left ( l_g\right )_*\circ J
\circ \left ( l_{g^{-1}}\right )_*(X)$.
\begin{lemma}
\label{cxstru} $J$ is an invariant integrable complex structure on
$D$. Furthermore, it coincides with the complex structure on $D$
induced by the inclusion $D\subset\check D=G_{\mathbb{C}}/B$.
\end{lemma}
This lemma is well known. See \cite{grsch1} and \cite{ls1} for
details.
There is a natural metric on $D$ induced by the Killing form. By the
Cartan decomposition ${\mathfrak{g}}_0={\mathfrak{k}}\oplus
{\mathfrak{p}}_0$ we know that the Killing form $\kappa$ on
${\mathfrak{g}}_0$ is positive
definite on $\mathfrak{p}_0$ and is negative definite on $\mathfrak{k}/%
\mathfrak{v}$. By the identification \eqref{killing10}, for real
tangent
vectors $X,Y\in T_O^{\mathbb{R}} D$, if $X=X_1+X_2$ and $Y=Y_1+Y_2$ where $%
X_1,Y_1\in \mathfrak{k}/\mathfrak{v}$ and $X_2,Y_2\in
\mathfrak{p}_0$, we let
\begin{eqnarray} \label{killing20}
\tilde\kappa(X,Y)=-\kappa(X_1,Y_1)+\kappa(X_2,Y_2).
\end{eqnarray}
Then $\tilde\kappa$ is a positive definite symmetric bilinear form on $T_O^{%
\mathbb{R}} D$. Now we use left translation of elements in
$G_{\mathbb{R}}$ to move this metric to the real tangent space of
every point in $D$ and we obtain a Riemannian metric on $D$. This is
the Hodge metric defined by Griffiths and Schmid in \cite{grsch1}.
\subsection{Global Torelli Theorem and Applications}
Now we describe the global Torelli theorem.
\begin{theorem}
\label{main} Let $(M,L)$ be a polarized CY manifold of dimension $n$
and let ${\mathcal{T}}$ be its Teichm\"uller space. Let $D$ be the
classifying space of the variation of Hodge structures according to
the middle cohomology of $M$. Let ${\Phi}:{\mathcal T}\to D$ be the period map
which maps each point $q\in{\mathcal T}$ to the Hodge decomposition of the
middle dimensional primitive cohomology of $M_q$ which is a point in
$D$. Then the period map ${\Phi}:{\mathcal{T}}\to D$ is injective.
\end{theorem}
Let us describe the main idea of proving the global Torelli theorem.
See \cite{lsty1} for details. In fact we have proved a stronger
result. For any distinct points $p,q\in{\mathcal T}$, in \cite{lsty1} we
showed that the lines $F^n(M_p)$ and $F^n(M_q)$ do not coincide.
This means the first Hodge bundle already determines polarized
marked Calabi-Yau structures.
The first main component in the proof of Theorem \ref{main} is
Theorem \ref{invarkaform}, namely the K\"ahler forms $\omega$ of the
polarized CY metrics are invariant. From this we know that all the
complex structures corresponding to all points in ${\mathcal T}$ are tamed by
$\omega$. If we fix a base point $0\in{\mathcal T}$ then for any point
$q\in{\mathcal T}$ such that $q\ne 0$, the complex structure on $M_q$ is
obtained by deforming the complex structure on $M_0$ via a unique
Beltrami differential $\phi(q)\in A^{0,1}\left ( M_0,T_{M_0}^{1,0}\right )$.
Thus we obtained the assigning map
\[
\rho:{\mathcal T}\to A^{0,1}\left ( M_0,T_{M_0}^{1,0}\right )
\]
by letting $\rho(q)=\phi(q)$. The assigning map $\rho$ is
holomorphic. In fact, in \cite{lsty4} we proved that the assigning
map $\rho$ is a holomorphic embedding.
According to the work of Todorov \cite{tod1} and our work
\cite{lsty1} we know that $\Omega_q=\sum_{k=0}^n \frac{1}{k!}\left (
\wedge^k \phi(q)\lrcorner\Omega_0\right )$ is a smooth $(n,0)$-form on
$M_q$ where $\Omega_0$ is a properly normalized holomorphic
$(n,0)$-form on $M_0$. It follows from Theorem \ref{flatcoord} and
Lemma \ref{constructn0} that $\Omega_q$ is a holomorphic
$(n,0)$-form on $M_q$. Since the cohomology classes $[\Omega_0]$ and
$[\Omega_q]$ are generators of the Hodge lines $F^n(M_p)$ and
$F^n(M_q)$, it is enough to show that these two classes are not
proportional. Now we look at the Calabi-Yau equation
\[
c_n\Omega_p\wedge\bar\Omega_p= \omega_p^n
\]
where $p\in{\mathcal T}$ is any point, $\Omega_p$ is a properly normalized
nowhere vanishing holomorphic $(n,0)$-form on $M_p$, $\omega_p$ is
the K\"ahler form of the polarized CY metric on $M_p$ and
$c_n=(-1)^{\frac{n(n-1)}{2}}\left (\frac{\sqrt{-1}}{2}\right )^n$. It follows
from Theorem \ref{invarkaform} and the Calabi-Yau equation that if
$[\Omega_q]=c[\Omega_0]$, then $c=1$ and $\phi(q)=0$ which means
that $M_0$ and $M_q$ are isomorphic. This contradicts the assumption
that $q\ne 0$ as points in ${\mathcal T}$ and the global Torelli theorem
follows.
By using the same method we proved the global Torelli theorem of the
Teichm\"uller space of polarized Hyper-K\"ahler manifolds:
\begin{theorem}
Let $(M,L)$ be a polarized Hyper-K\"ahler manifold and let ${\mathcal T}$ be its
Teichm\"uller space. Let $D$ be the classifying space of variation of
polarized weight $2$ Hodge structures according to the data of
$(M,L)$. Then the period map ${\Phi}:{\mathcal T}\to D$ which maps each point
$q\in{\mathcal T}$ to the Hodge decomposition of the second primitive
cohomology of $M_q$ is injective.
\end{theorem}
In \cite{lsty4} we gave another proof of the global Torelli theorem
of the Teichm\"uller space of polarized Hyper-K\"ahler manifolds by directly
showing that the cohomology expansion of the canonical $(2,0)$-forms
has no quantum correction. This implies that the Hodge completion of
the Teichm\"uller space is biholomorphic to the classifying space via the
Harish-Chandra realization.
Another important property of the Teichm\"uller space is the existence of
holomorphic flat connections. We proved the following theorem in
\cite{lsty4}.
\begin{theorem}
There exists affine structures on the Teichm\"uller space ${\mathcal T}$ of polarized
CY manifolds. The affine structures are given by global holomorphic
flat connections on ${\mathcal T}$.
\end{theorem}
It is not difficult by using elementary Lie algebra arguments to
establish the affine structure on the complement of the Schubert
cycle which is the nilpotent orbit containing the base point. In our
case we need to prove that the image of the Teichm\"uller space under the
period map do not intersect certain component of the Schubert cycle
which is codimension one. We call this the partial global
transversality. The problem when partial global transversality holds
is very important in the study of global behavior of the period map.
The Griffiths transversality is too weak to deal with such global
problems of the period map. In \cite{lsty4} we obtained the partial
global transversality by using Yau's solution of the Calabi
conjecture.
As a corollary we proved the holomorphic embedding theorem of the
Teichm\"uller space of polarized CY manifolds in \cite{lsty4}:
\begin{theorem}
The Teichm\"uller space of polarized CY manifolds can be holomorphically
embedded into the Euclidean space of same dimension. Furthermore,
its Hodge completion is a domain of holomorphy and there exists a
unique K\"ahler-Einstein metric on the completion.
\end{theorem}
The methods that we used in \cite{lsty1} can be used to prove the
global Torelli theorem for a large class of manifolds of general
type. Furthermore, these methods can be used to treat the invariance
of the plurigenera for K\"ahler manifolds where the projective case was
proved by Siu.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,090
|
Q: What does the phrase "shadows dance on the gravel underneath it" mean here? Here is a sentence from a sentence:
Sunlight kisses the yellow leaves of a tree by the lake, as shadows
dance on the gravel underneath it.
I am not sure about the phrase ""shadows dance on the gravel underneath it". Is the gravel underneath the tree or the lake?
A: "Underneath it" refers to the noun-phrase "a tree by the lake". If the gravel were underneath the lake, then nobody would be able to see it, and the sunlight would find it hard to make shadows dance on it.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,261
|
{"url":"https:\/\/ora.ox.ac.uk\/objects\/uuid:28f4eb50-987f-4590-b2d2-07b0b9341609","text":"Journal article\n\n### Biased random walks on a Galton-Watson tree with leaves\n\nAbstract:\n\nWe consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\\gamma= \\gamma(\\beta) \\in (0,1)$, depending on the bias $\\beta$, such that $X_n$ is of order $n^{\\gamma}$. Denoting $\\Delta_n$ the hitting time of level $n$, we prove that $\\Delta_n\/n^{1\/\\gamma}$ is tight. Moreover we show that $\\Delta_n\/n^{1\/\\gamma}$ does not converge in law (at least for large values of $\\beta$). We prove that along the seq...\n\n### Access Document\n\nPublisher copy:\n10.1214\/10-AOP620\n\n### Authors\n\nFribergh, A More by this author\nGantert, N More by this author\nMore by this author\nInstitution:\nUniversity of Oxford\nDepartment:\nOxford, MPLS, Statistics\nJournal:\nAnnals of Probability\nVolume:\n40\nIssue:\n1\nPages:\n280-338\nPublication date:\n2007-11-23\nDOI:\nISSN:\n0091-1798\nURN:\nuuid:28f4eb50-987f-4590-b2d2-07b0b9341609\nSource identifiers:\n204289\nLocal pid:\npubs:204289\nLanguage:\nEnglish\nKeywords:","date":"2020-10-24 09:06:47","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\": 0.9195163249969482, \"perplexity\": 595.0191032338004}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107882103.34\/warc\/CC-MAIN-20201024080855-20201024110855-00645.warc.gz\"}"}
| null | null |
import pytest
from unittest import mock
from operator import attrgetter
from django.core.management import call_command
from osf_tests.factories import (
PreprintProviderFactory,
PreprintFactory,
ProjectFactory,
RegistrationProviderFactory,
RegistrationFactory,
)
def sorted_by_id(things_with_ids):
return sorted(
things_with_ids,
key=attrgetter('id')
)
@pytest.mark.django_db
class TestRecatalogMetadata:
@pytest.fixture
def preprint_provider(self):
return PreprintProviderFactory()
@pytest.fixture
def preprints(self, preprint_provider):
return sorted_by_id([
PreprintFactory(provider=preprint_provider)
for _ in range(7)
])
@pytest.fixture
def registration_provider(self):
return RegistrationProviderFactory()
@pytest.fixture
def registrations(self, registration_provider):
return sorted_by_id([
RegistrationFactory(provider=registration_provider)
for _ in range(7)
])
@pytest.fixture
def projects(self, registrations):
return sorted_by_id([
ProjectFactory()
for _ in range(7)
] + [
registration.registered_from
for registration in registrations
])
@mock.patch('api.share.utils.update_share')
def test_recatalog_metadata(self, mock_update_share, preprint_provider, preprints, registration_provider, registrations, projects):
# test preprints
call_command(
'recatalog_metadata',
'--preprints',
'--providers',
preprint_provider._id,
)
expected_update_share_calls = [
mock.call(preprint)
for preprint in preprints
]
assert mock_update_share.mock_calls == expected_update_share_calls
mock_update_share.reset_mock()
# test registrations
call_command(
'recatalog_metadata',
'--registrations',
'--providers',
registration_provider._id,
)
expected_update_share_calls = [
mock.call(registration)
for registration in registrations
]
assert mock_update_share.mock_calls == expected_update_share_calls
mock_update_share.reset_mock()
# test projects
call_command(
'recatalog_metadata',
'--projects',
'--all-providers',
)
expected_update_share_calls = [
mock.call(project)
for project in projects # already ordered by id
]
assert mock_update_share.mock_calls == expected_update_share_calls
mock_update_share.reset_mock()
# test chunking
call_command(
'recatalog_metadata',
'--registrations',
'--all-providers',
f'--start-id={registrations[1].id}',
'--chunk-size=3',
'--chunk-count=1',
)
expected_update_share_calls = [
mock.call(registration)
for registration in registrations[1:4]
]
assert mock_update_share.mock_calls == expected_update_share_calls
mock_update_share.reset_mock()
# slightly different chunking
expected_update_share_calls = [
mock.call(registration)
for registration in registrations[2:6] # already ordered by id
]
call_command(
'recatalog_metadata',
'--registrations',
'--all-providers',
f'--start-id={registrations[2].id}',
'--chunk-size=2',
'--chunk-count=2',
)
assert mock_update_share.mock_calls == expected_update_share_calls
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,423
|
\section{Introduction}
\label{sec:introduction}
The behavior of spin glasses in a magnetic field is still controversial.
While the infinite-range (mean-field)
Sherrington-Kirkpatrick (SK)
model\cite{sherrington:75}
has a line of transitions at finite field known
as the Almeida-Thouless (AT) line,\cite{almeida:78} it has not been definitely
established
whether an AT line occurs in more realistic models with
short range interactions.
Previous numerical
studies\cite{bhatt:85,ciria:93b,kawashima:96,billoire:03b,marinari:98d,houdayer:99,krzakala:01,takayama:04,young:04}
have yielded conflicting results: some data
support the existence of an AT line in short-range spin glasses while others
claim its absence. Recently,\cite{young:04} a new approach using the
correlation length\cite{cooper:82,palassini:99b,ballesteros:00,martin:02}
at {\em finite
fields} has been applied to the three-dimensional Edwards-Anderson Ising
spin-glass model. The data of Ref.~\onlinecite{young:04}
indicate that, even for small fields,
there is no AT line in
three-dimensional spin glasses.
Here we use the techniques of Ref.~\onlinecite{young:04} to
study the crossover between mean-field models and
short-range spin glasses \textit{continuously}
by using a one-dimensional Ising chain with random power-law
interactions. The model's advantages are twofold: first, a large range of
system sizes can be studied, and second, by tuning the power-law exponent of
the interactions, the universality class of the model can be changed
continuously from the mean-field universality class to the short-range
universality class. We find that
there appears to be \textit{no} AT line
for the range of the power-law exponent corresponding to a non-mean-field
transition in zero field.
By analogy, this suggests
that there is also no AT line for short-range spin glasses, at least below the
upper critical dimension.
The paper is organized as follows. In Sec.~\ref{sec:details}, we introduce in
detail the model, observables, and numerical method used. In
Sec.~\ref{sec:results}, we present our results, and in
Sec.~\ref{sec:conclusions}, we summarize our findings.
\section{Model, Observables, and Numerical Details}
\label{sec:details}
The Hamiltonian of the one-dimensional long-range Ising spin glass
with random power-law interactions\cite{bray:86b,fisher:88} is given by
\begin{equation}
{\cal H} = -\sum_{\langle i,j \rangle} J_{ij} S_i S_j - \sum_i h_i S_i\; ,
\label{eq:hamiltonian}
\end{equation}
where $S_i = \pm 1$ represents Ising spins evenly
distributed on a ring of length $L$ in order to ensure periodic boundary
conditions. The sum is over all spins on the chain and the couplings $J_{ij}$
are given by\cite{katzgraber:03}
\begin{equation}
J_{ij} = c(\sigma)\frac{\epsilon_{ij}}{r_{ij}^\sigma}\; ,
\label{eq:bonds}
\end{equation}
where the $\epsilon_{ij}$ are chosen according to a Gaussian distribution
with zero mean and standard deviation unity,
and $r_{ij} = (L/\pi)\sin[(\pi |i - j|)/L]$ represents the
{\em geometric} distance between the spins on
the ring.\cite{distances}
The power-law exponent $\sigma$ characterizes the
interactions and, hence, determines the universality class of the model.
The constant $c(\sigma)$ in Eq.~(\ref{eq:bonds}) is chosen to give a
mean-field
transition temperature $T_{\rm c}^{\rm MF} = 1$, where
\begin{equation}
\left(T_{\rm c}^{\rm MF}\right)^2 = \sum_{j\ne i} [ J_{ij}^2]_{\rm av} =
c(\sigma)^2 \sum_{j\ne i} \frac{1}{r_{ij}^{2\sigma}} \; .
\label{eq:tcmf}
\end{equation}
Here $[\cdots]_{\rm av}$ denotes an average over disorder.
\begin{table}[!tb]
\caption{
A summary of the behavior for different ranges of $\sigma$ in one space
dimension and at zero field.
IR means infinite
range, i.e., $\sum_{j\ne i} J_{ij}^2$ diverges unless the bonds $J_{ij}$
are scaled by an inverse power of the system size. LR means that the behavior
is dominated by the long-range tail of the interactions, and SR means that
the behavior is that of a short-range system.
\label{tab:ranges}
}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}} l l }
\hline
\hline
$\sigma$ & behavior \\
\hline
$\sigma = 0$ & SK model \\
$0 < \sigma \le 1/2$ & IR \\
$1/2 < \sigma < 2/3$ & LR (mean field with $T_{\rm c} > 0$) \\
$2/3 < \sigma \le 1$ & LR (non-mean field with $T_{\rm c} > 0$) \\
$1 < \sigma \le 2$ & LR ($T_{\rm c} = 0$) \\
$\sigma \ge 2$ & SR ($T_{\rm c} = 0$) \\
\hline
\hline
\end{tabular*}
\end{table}
In Eq.~(\ref{eq:hamiltonian}), the spins couple to site-dependent random
fields $h_i$ chosen from a Gaussian
distribution with zero mean $[h_i]_{\rm av} = 0$ and standard deviation
$[h_i^2]_{\rm av}^{1/2} = H_{\rm R}$. For a symmetric distribution of bonds,
the sign of $h_i$ can be ``gauged away'' so a uniform field is completely
equivalent to a bimodal distribution of fields with $h_i = \pm H_{\rm R}$.
While the AT line is usually studied for the case of a uniform field,
the SK model with Gaussian
random fields (as considered here)
also shows an AT line. For short-range three-dimensional spin glasses it has
been shown in Ref.~\onlinecite{young:04} that results for
Gaussian-distributed random fields agree within error bars with results for a
uniform field. The use of Gaussian-distributed random fields has the advantage
over a uniform external field in that we can apply a useful equilibration
test,\cite{katzgraber:01,katzgraber:03,young:04} see Eq.~(\ref{eq:U}) below.
From Eq.~(\ref{eq:tcmf}), we see that for $\sigma \le 1/2,\
c(\sigma)$ varies with a power of the system, $c(\sigma) \sim
L^{-(1-2\sigma)/2}$, for large $L$. We shall denote systems in this region as
``infinite range'' (IR). The extreme limit of $\sigma = 0$ gives the SK model,
whose solution is the mean-field (MF) theory for spin glasses. For
$\sigma > 1/2,\ c(\sigma)$ tends to a constant as $L \to\infty$. As discussed
in an earlier work (see Ref.~\onlinecite{katzgraber:03} and references
therein),
for $1/2 < \sigma \le 1$, the system has a finite-temperature transition into a
spin-glass phase in a long-range (LR) universality class at zero field.
For $1 < \sigma \le
2$, the system has $T_{\rm c} = 0$
and the critical behavior is also determined by
the LR universality class. For $\sigma > 2$, we have a short-range (SR)
universality class with $T_{\rm c} = 0$.
Finally, we note\cite{kotliar:83} that for
$1/2 < \sigma < 2/3$ the critical behavior is mean-field-like, while for
$2/3 < \sigma \le 1$ it is non-mean field like.
This behavior is summarized in Table \ref{tab:ranges}.
Critical exponents depend continuously on
$\sigma$ in the LR regime,
but are independent of $\sigma$ in the SR region. Here
we focus on the regime $1/2 < \sigma \le 1$ because there
the system exhibits a finite-temperature transition that can be
tuned continuously away from the
mean-field universality limit by changing the exponent $\sigma$.
To determine the existence of an AT line, we compute the two-point correlation
length.\cite{palassini:99b,ballesteros:00,young:04}
We calculate the wave-vector-dependent
spin-glass susceptibility which is defined by
\begin{equation}
\chi_{\rm SG}(\mathbf{k}) = \frac{1}{N} \sum_{i, j} \left[\Big(
\langle S_i S_j\rangle_T - \langle S_i \rangle_T \langle S_j\rangle_T
\Big)^2 \right]_{\rm av}\!\!\!\!\! e^{i\mathbf{k}\cdot(\mathbf{R}_i -
\mathbf{R}_j)} ,
\label{eq:chisg}
\end{equation}
where $\langle \cdots \rangle_T$ denotes a thermal average.
Note that at zero
field $\langle S_i \rangle_T$ can be set to zero.
The correlation length of the finite system is then given by
\begin{equation}
\xi_L = \frac{1}{2 \sin (k_\mathrm{min}/2)}
\left[\frac{\chi_{\rm SG}(0)}{\chi_{\rm SG}({\bf k}_\mathrm{min})} -
1\right]^{1/(2\sigma-1)},
\label{xiL}
\end{equation}
where ${\bf k}_\mathrm{min} = (2\pi/L, 0, 0)$ is the smallest nonzero
wave vector.
The reason for the power $1/(2\sigma-1)$ is that at long wavelengths,
we expect a \textit{modified} Ornstein-Zernicke form\cite{cooper:82,martin:02}
\begin{equation}
\chi_{\rm SG}({\bf k}) \propto \left(v + k^{2\sigma - 1}\right)^{-1}
\end{equation}
for the long-range case, where $v$ is a measure of the deviation from
criticality. It follows that the bulk correlation length $\xi$ diverges
for $v \to 0$ like $v^{-1/(2\sigma-1)}$.
\begin{figure}
\includegraphics[width=\columnwidth]{equil.eps}
\vspace*{-1.0cm}
\caption{(Color online)
Sample equilibration plot for $\sigma = 0.85$, $L = 128$, $H_{\rm R} = 0.1$,
and $T = 0.10$ (the lowest temperature simulated at finite fields).
Data for the average energy $U$, and
$U(q_l, q)$ defined in Eq.~(\ref{eq:U}),
as a function of equilibration time
$t_{\rm eq}$. They approach their common value
from opposite directions and, once they agree, do not change on further
increasing $t_{\rm eq}$.
The inset shows data for the
correlation length divided by system size as a function of equilibration time.
The data are independent of $t_{\rm eq}$ once $U$ and
$U(q_l, q)$ agree.
}
\label{fig:equil}
\end{figure}
\begin{table}[!tb]
\caption{
Parameters of the simulations for $H_{\rm R} = 0.0$. $N_{\rm sa}$
is the number of samples, $N_{\rm sw}$ is the total number of Monte Carlo
sweeps for each of the $2 N_T$ replicas for a single sample,
$T_{\rm min}$ is the lowest temperature simulated, and $N_T$ is the number
of temperatures used in the parallel tempering method for each system size
$L$ and power-law exponent $\sigma$.
\label{tab:simparams0}
}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}} c r r r r r }
\hline
\hline
$\sigma$ & $L$ & $N_{\rm sa}$ & $N_{\rm sw}$ & $T_{\rm min}$ & $N_{T}$ \\
\hline
$0.55$ & $32$ & $5000$ & $10240$ & $0.405$ & $15$ \\
$0.55$ & $64$ & $5000$ & $10240$ & $0.405$ & $15$ \\
$0.55$ & $128$ & $5000$ & $20480$ & $0.405$ & $15$ \\
$0.55$ & $256$ & $5000$ & $102400$ & $0.405$ & $15$ \\
$0.55$ & $512$ & $5000$ & $32768$ & $0.630$ & $11$ \\[2mm]
$0.65$ & $32$ & $5000$ & $10240$ & $0.405$ & $15$ \\
$0.65$ & $64$ & $5000$ & $10240$ & $0.405$ & $15$ \\
$0.65$ & $128$ & $5000$ & $20480$ & $0.405$ & $15$ \\
$0.65$ & $256$ & $5000$ & $102400$ & $0.405$ & $15$ \\
$0.65$ & $512$ & $5000$ & $524288$ & $0.405$ & $15$ \\[2mm]
$0.75$ & $32$ & $5000$ & $10240$ & $0.405$ & $15$ \\
$0.75$ & $64$ & $5000$ & $10240$ & $0.405$ & $15$ \\
$0.75$ & $128$ & $5000$ & $20480$ & $0.405$ & $15$ \\
$0.75$ & $256$ & $5000$ & $102400$ & $0.405$ & $15$ \\
$0.75$ & $512$ & $2500$ & $524288$ & $0.405$ & $15$ \\[2mm]
$0.85$ & $32$ & $5000$ & $10240$ & $0.405$ & $15$ \\
$0.85$ & $64$ & $5000$ & $20480$ & $0.405$ & $15$ \\
$0.85$ & $128$ & $5000$ & $102400$ & $0.405$ & $15$ \\
$0.85$ & $256$ & $5000$ & $204800$ & $0.405$ & $15$ \\
$0.85$ & $512$ & $2500$ & $204800$ & $0.405$ & $15$ \\[2mm]
\hline
\hline
\end{tabular*}
\end{table}
\begin{table}[!tb]
\caption{
Parameters of the simulations for $H_{\rm R} = 0.1$. $N_{\rm sa}$
is the number of samples, $N_{\rm sw}$ is the total number of Monte Carlo
sweeps for each of the $4 N_T$ replicas for a single sample, $T_{\rm min}$ is
the lowest temperature simulated, and $N_T$ is the number of temperatures used
in the parallel tempering method for each system size $L$ and power-law
exponent $\sigma$. \label{tab:simparams}
}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}} c r r r r r }
\hline
\hline
$\sigma$ & $L$ & $N_{\rm sa}$ & $N_{\rm sw}$ & $T_{\rm min}$ & $N_{T}$
\\
\hline
$0.55$ & $32$ & $5000$ & $81920$ & $0.100$ & $26$ \\
$0.55$ & $64$ & $5000$ & $327680$ & $0.100$ & $26$ \\
$0.55$ & $128$ & $5000$ & $1310720$ & $0.100$ & $26$ \\
$0.55$ & $256$ & $2000$ & $1048576$ & $0.405$ & $15$ \\
$0.55$ & $512$ & $2000$ & $65536$ & $0.760$ & $9$ \\[2mm]
$0.65$ & $32$ & $5000$ & $81920$ & $0.100$ & $26$ \\
$0.65$ & $64$ & $5000$ & $327680$ & $0.100$ & $26$ \\
$0.65$ & $128$ & $5000$ & $1310720$ & $0.100$ & $26$ \\
$0.65$ & $256$ & $2000$ & $1048576$ & $0.195$ & $20$ \\
$0.65$ & $512$ & $2000$ & $524288$ & $0.500$ & $13$ \\[2mm]
$0.75$ & $32$ & $5000$ & $81920$ & $0.100$ & $26$ \\
$0.75$ & $64$ & $5000$ & $327680$ & $0.100$ & $26$ \\
$0.75$ & $128$ & $5000$ & $1310720$ & $0.100$ & $26$ \\
$0.75$ & $256$ & $2000$ & $8388608$ & $0.100$ & $26$ \\[2mm]
$0.85$ & $32$ & $5000$ & $81920$ & $0.100$ & $26$ \\
$0.85$ & $64$ & $5000$ & $327680$ & $0.100$ & $26$ \\
$0.85$ & $128$ & $2000$ & $16777216$ & $0.100$ & $26$ \\[2mm]
\hline
\hline
\end{tabular*}
\end{table}
The correlation length divided by the system size $\xi_L/L$ has
the following scaling property:
\begin{equation}
\frac{\xi_L}{L} = \widetilde{X}\left(L^{1/\nu}[T - T_{\rm c}(H_{\rm
R})]\right) \; ,
\label{fss}
\end{equation}
where $\nu$ is the correlation length exponent and $T_{\rm c}(H_{\rm R})$ is
the transition temperature for a field of strength $H_{\rm R}$. This
behavior is similar
to that of the
Binder ratio,\cite{binder:81}
but it shows a clearer signature of the transition as the data are not
restricted to a finite interval.
In order to test equilibration of the Monte Carlo method, we also compute the
link overlap\cite{katzgraber:03} $q_{l}$ given by
\begin{equation}
q_l = \frac{2}{N}\sum_{\langle i,j\rangle}
\frac{[J_{ij}^2]_{\rm av}}{(T_{\rm c}^{MF})^2}
[
\langle S_i^\alpha S_j^\alpha S_i^\beta S_j^\beta \rangle_T
]_{\rm av} \; ,
\label{eq:ql}
\end{equation}
where $T_{\rm c}^{\rm MF}$ is given by Eq.~(\ref{eq:tcmf}) and $\alpha$ and
$\beta$ refer to two replicas of the system with the same disorder.
In addition, we compute the spin overlap $q$ given by
\begin{equation}
q = \frac{1}{N}\sum_{i = 1}^{N}
[ \langle S_i^\alpha S_i^\beta \rangle_T ]_{\rm av} .
\label{eq:q}
\end{equation}
Because both the fields and interactions have a Gaussian distribution,
integrating by parts the expression for the average energy per spin $U$
gives\cite{young:04,katzgraber:03,katzgraber:01}
\begin{equation}
U \equiv U(q_l, q) = -{(T_{\rm c}^{\rm MF})^2 \over 2 T} ( 1 - q_l)
- {H_{\rm R}^2 \over T} (1 - q) \, .
\label{eq:U}
\end{equation}
\begin{figure*}[!tb]
\includegraphics[width=\columnwidth]{xi-h0.00-s0.55.eps}
\includegraphics[width=\columnwidth]{xi-h0.00-s0.65.eps}
\vspace*{-1.2cm}
\includegraphics[width=\columnwidth]{xi-h0.00-s0.75.eps}
\includegraphics[width=\columnwidth]{xi-h0.00-s0.85.eps}
\vspace*{-1.0cm}
\caption{(Color online)
Each figure shows data for $\xi_L/L$ vs $T$ for $H_{\rm R}=0$ for different
system sizes, for a particular value of $\sigma$.
For all values of $\sigma$, the data cross indicating that there is a spin-glass
transition at finite temperature.
}
\label{fig:heq0}
\end{figure*}
As shown in Fig.~\ref{fig:equil},
when starting from a random spin configuration,
$U$ approaches its equilibrium value
from above while $U(q_l, q)$ approaches its equilibrium value from below.
Once $U = U(q_l, q)$, the data do not change by further
increasing the number of
Monte Carlo steps, which shows that the system is
in equilibrium.
It is
also important to ensure that other observables are also
in equilibrium once $U = U(q_l, q)$, and this is
shown in the inset to Fig.~\ref{fig:equil} for the case of the correlation
length.
The simulations are done using the parallel tempering Monte Carlo
method.\cite{hukushima:96,marinari:96} The method is not as efficient in a
field,\cite{moreno:03,billoire:03} but nevertheless it performs considerably
better than simple Monte Carlo.
In order to compute the products of up to four thermal averages in
Eq.~(\ref{eq:chisg}) without bias, we simulate four
copies (replicas) of the system with the same bonds and fields at each
temperature. Simulations are performed at zero field, as well as at
$H_{\rm R} = 0.1$, a field that is considerably smaller than the
$\sigma$-dependent transition temperature $T_{\rm c}$.
Parameters of the
simulations at zero and finite fields are presented in Tables
\ref{tab:simparams0} and \ref{tab:simparams}, respectively.
\section{Results}
\label{sec:results}
We first consider the case of zero field and take
$\sigma = 0.55$, $0.65$, $0.75$, and $0.85$. The values $\sigma = 0.75$ and
$0.85$ are in the non-MF region (see Table \ref{tab:ranges}) while
$\sigma = 0.55$ is in the MF region and, furthermore, is close to the
value ($\sigma = 1/2$) where the system becomes infinite range. The value
$\sigma = 0.65$ is close to the point $\sigma = 2/3$ where the critical
behavior changes from MF to non-MF.
The data are shown in
Fig.~\ref{fig:heq0}.
In all cases, the data cross
at a transition temperature which we determine as $T_{\rm c} = 1.03(3)$
for $\sigma = 0.55$, $0.86(2)$ for $\sigma=0.65$,
$0.69(1)$ for $\sigma = 0.75$, and $0.49(1)$ for $\sigma = 0.85$.
Note that $T_{\rm c}$ decreases continuously with increasing $\sigma$ and is
expected to drop
to zero at $\sigma = 1$.\cite{katzgraber:03} For the SK model ($\sigma = 0$),
one has $T_{\rm c} = 1$, essentially the result we find for $\sigma = 0.55$,
so it is possible that $T_{\rm c}$ has little variation with $\sigma$
for $\sigma \le 0.55$.
Next we consider $H_{\rm R} = 0.10$ and show the data in
Fig.~\ref{fig:hgt0}. The results for $\sigma = 0.75$ and $0.85$, which are
in the non-mean-field regime, show no sign of a transition. However,
the data for $\sigma = 0.55$ do show a signature of a transition at $T_{\rm
c} = 0.96(2) < T_{\rm c}(H_{\rm R} = 0)$. Whether this would persist up to
infinite system sizes is not clear, but it certainly cannot be ruled out.
The results for $\sigma = 0.55$ show that the method used here \textit{is}
capable of detecting an AT line in the presence of a field.
For $\sigma = 0.65$, the data shows a marginal behavior.
Since $\sigma = 0.65$ is close to the value of $2/3$ which
separates MF and non-MF behavior in zero field, this marginal behavior may
indicate that $2/3$ is also the borderline value below which an AT line occurs.
An alternative possibility, which
we cannot rule out, is that an AT line only occurs in the infinite-range
region ($\sigma < 1/2$) but that as $\sigma$ is decreased toward $1/2$,
one needs to study larger system
sizes to see the absence of a transition.
\begin{figure*}[!tb]
\includegraphics[width=\columnwidth]{xi-h0.10-s0.55.eps}
\includegraphics[width=\columnwidth]{xi-h0.10-s0.65.eps}
\vspace*{-1.2cm}
\includegraphics[width=\columnwidth]{xi-h0.10-s0.75.eps}
\includegraphics[width=\columnwidth]{xi-h0.10-s0.85.eps}
\vspace*{-1.0cm}
\caption{(Color online)
Each figure shows results for $\xi_L/L$ vs $T$ for $H_{\rm R}=0.10$ for
different system sizes, for a particular
value of $\sigma$.
For $\sigma = 0.55$, the data cross at $T_{\rm c} = 0.96(2)$
showing that an AT line seems to be present.
For $\sigma = 0.65$, the data show close to marginal behavior.
This may indicate that
$\sigma = 0.65$ is close to the borderline value for having an AT line.
For $\sigma = 0.75$ and $\sigma = 0.85$, the data do not cross for any
temperature down to $T = 0.10$, which is considerably smaller than the zero
field transition temperatures. This indicates that there is no AT line.
Overall the results indicate that, on increasing $\sigma$, the AT line
disappears.
}
\label{fig:hgt0}
\end{figure*}
\section{Conclusions}
\label{sec:conclusions}
We have considered
a one-dimensional spin-glass model with long-range interactions
that allows
the universality class to be changed from
the infinite-range limit to the short-range case by tuning the power-law
exponent $\sigma$ of the interactions. We find that
there does not appear to be an AT line in a field for models with $\sigma$ in
the range where there is
non-mean-field critical behavior at zero field. However, in the
region of $\sigma$ that is not infinite-range but has mean-field critical
behavior
($1/2 < \sigma < 2/3$), there does appear to be an AT line.
These conclusions rely on extrapolating from finite sizes to the thermodynamic
limit. It would be
particularly interesting to know if the conclusion that there is an AT line
for $\sigma = 0.55$ persists in the thermodynamic limit, or whether the AT
line really only occurs in the infinite-range case ($\sigma < 1/2$). It is
possible that as $\sigma$ is decreased, larger sizes or larger values of
$H_{\rm R}$ are needed to probe the asymptotic behavior.
Therefore, it would be desirable
to simulate a range of values of $H_{\rm R}$, especially for $\sigma = 0.55$.
We have some results for $H_{\rm R} = 0.2$ for a relatively small number of
samples and for sizes only up to $L=96$,
which indicate a crossing at a lower temperature than for $H_{\rm R} =
0.1$. However, we are unable to carry out a systematic study of the dependence
on $H_{\rm R}$ because
the results presented above already required considerable computer
time, and the parallel tempering algorithm becomes less efficient at larger
fields.
Making an analogy between the one-dimensional
long-range model for different values of
$\sigma$ and short-range models for different values of space dimension $d$,
we infer that there is no AT line for short-range spin glasses in the
non-mean-field regime, i.e., below the upper critical dimension $d_{\rm u}=6$.
However, there
may be an AT line above the upper critical dimension. Speculations along these
lines have also been made very recently by Moore.\cite{moore:05}
\begin{acknowledgments}
A.~P.~Y.~acknowledges support from the National Science Foundation
under NSF Grant No.~DMR 0337049.
The simulations were performed on the Hreidar and Gonzales clusters
at ETH Z\"urich. We thank M.~A.~Moore for helpful discussions and suggestions.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,860
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Our manifolds and valves can be designed by you. Online valve manifold configurator that allows you to design your own valve/manifold. You can choose your I/O type, select a specific valve for each station and even choose blanking plates for future expansion. After designing your valve/manifold, you can download the CAD model and BOM for your assembly.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,862
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explodes è un singolo promozionale del gruppo musicale Kasabian, pubblicato il 30 aprile 2014 in promozione del loro quinto album 48:13.
La canzone
In un'intervista di NME Sergio Pizzorno parla così del brano:
Tracce
Classifiche
Note
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,023
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from BoostBuild import Tester, List
t = Tester()
t.set_tree("railsys")
t.run_build_system("--v2", subdir="program")
t.cleanup()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 103
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The ISIS Apocalypse
The History, Strategy, and Doomsday Vision of the Islamic State
William McCants
St. Martin's Press
New York
Begin Reading
Table of Contents
About the Author
Copyright Page
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For Mom, who's always asking me simple, difficult questions about the Middle East. As the Arabic saying goes, al-sahl al-mumtani', the easy is elusive.
Introduction
June 2014 was going to be quiet. Government employees in Washington, DC, disperse for the summer to beat the heat, which means those of us who make a living commenting on the government do too. It'd be a good time to play with my kids and get back to my hobby, writing a scriptural history of the Qur'an. Articles about al-Qaeda and the civil war in Syria, my usual beat, could wait. No one cared.
They cared a few days later when the so-called Islamic State group marched across Iraq and conquered its second-largest city, Mosul. Mass executions, enslaved women, and crucifixions soon followed, parading across cable news and Twitter. The danse macabre hasn't stopped since. Black flags rose, and government buildings were painted the same somber shade. The Islamic State's leaders proclaimed the establishment of God's kingdom on earth, called the caliphate. Prophecy was fulfilled, they said, and Judgment Day approached.
The new caliphate was expansive and flush with weapons and cash, reportedly in the billions. At its head was the self-styled caliph Ibrahim al-Baghdadi, an Iraqi religious scholar who took up arms after the United States invaded Iraq in 2003. Councils and governors advised the caliph, whose provinces stretched from Mosul to the outskirts of Aleppo in Syria, the distance from Washington, DC, to Cleveland, Ohio. Baghdadi's followers inside the caliphate numbered in the tens of thousands. Thousands more applauded him in Europe and the Middle East. The group threatened to topple American allies in the Middle East, destabilize world energy markets, foment revolution abroad, and launch attacks against Europe and the United States.
Questions flew. How had the Islamic State conquered so much land? Why was it so brutal? Why would such a murderous group claim to do God's bidding and fulfill prophecy? Did it really have anything to do with Islam, the world's second-largest religion? And what threat did it pose to the international community?
Readers who want more than sound-bite answers to these questions face a daunting challenge. Much of the Islamic State's propaganda is in Arabic and cloaked in medieval theological language that confuses Arabic-speaking Muslims, much less English-speaking non-Muslims. The Islamic State's bureaucracy is layered in secrecy, and few details have emerged about its inner workings. But like any bureaucracy, the Islamic State leaves a paper trail of emails and couriered messages. Some of them have been leaked by dissenters who post private Islamic State memoranda on password-protected discussion forums; other memos were released by the U.S. government, which captured them during raids. Followers and critics of the Islamic State have taken and posted pictures of its fatwas that were never meant to circulate online. Making sense of it all would require a guide proficient in Islamic theology and history, modern jihadism, clandestine bureaucracies, and Arabic.
That's what I am, and I am going to take you on a tour of the Islamic State. We will explore its origins, meet its leaders, boo its fans, and cheer its detractors. You will read its propaganda, study its strategies, eavesdrop on its internal debates, and follow its tweets. Along the way, I will explain its obscure allusions to Islamic history and theology so you can understand the ways the Islamic State uses and abuses Islam. You will be able to appreciate how the Islamic State thinks of itself, how its self-understanding has affected its political fortunes, and what will happen if those fortunes change again.
Like most of what the Islamic State does, its extreme brutality defies the conventional jihadist playbook. We're used to thinking of al-Qaeda's former leader Osama bin Laden as the baddest of the bad, but the Islamic State is worse. Bin Laden tamped down messianic fervor and sought popular Muslim support; the caliphate was a distant dream. In contrast, the Islamic State's members fight and govern by their own version of Machiavelli's dictum, "It is far safer to be feared than loved." They stir messianic fervor rather than suppress it. They want God's kingdom now rather than later. This is not Bin Laden's jihad. In what follows, I will tell you why the Islamic State's jihad is different and why that difference matters.
One
Raising the Black Flag
On a mild August morning in 2014, a passerby noticed a black flag hanging outside a rundown duplex in suburban New Jersey. He could not make out the flag's black-and-white Arabic, but he recognized the design from the news. All summer, American televisions and computer screens had been filled with reports of horrific acts committed by a renegade al-Qaeda group in Syria and Iraq, accompanied by foreboding images of masked jihadists waving the flag. From Morocco to Mindanao, jihadists were fighting under the banner to realize their dark vision of God's rule on earth. Alarmed, the passerby sent a picture of the house and its flag to his friend Marc Leibowitz, a former Israeli paratrooper working as an investment manager in New Jersey, who promptly tweeted the picture and the address with the caption "Scary!" The prospect of a jihadist proudly displaying his colors in America guaranteed the tweet went viral. Leibowitz also informed Homeland Security.
When the police arrived, the flag's owner, Mark Dunaway, had no idea anyone had tweeted a picture of it. Dunaway had converted to Islam a decade ago, he explained, and flew the flag to mark Muslim holidays. "Every Muslim uses that black flag," he said. "You'll find it in any mosque in the world. I am an American citizen and I love my country, but I am also a Muslim and I use that flag to say I'm a Muslim." Still, Dunaway could see why people would be concerned, and he took down the flag. "I understand now that people turn on CNN and see the flag associated with jihad, but that's not the intention of that flag at all. It says 'There is only one god, Allah, and the prophet Muhammad is his messenger.' It's not meant to be a symbol of hate. Islam is all about unity and peace. I am not a part of any group like that, and I'm not anti-American. I love my country, but I am a Muslim."
Doubtless, Dunaway sincerely believed he did not support a militant group by flying the flag, as attested by the police's disinterest in the case and his neighbors' testimonials. Dunaway, like many Muslims and even Middle East experts, did not know the flag was designed by an al-Qaeda offshoot, the Islamic State, after it proclaimed its statehood in 2006. It certainly wasn't in every "mosque in the world" as Dunaway thought. He and others were confused because the Islamic State had used terror and Twitter to advertise its brand and Islamic tradition to obscure its meaning.
Before the Islamic State declared itself the caliphate reborn that summer, it had been ambiguous about the flag's meaning and the cause it represented. Was it the flag of an Islamic state or the flag of the Islamic state, the caliphate that had once ruled land from Spain to Iran and whose prophesied return would herald the end of the world? The Islamic State encouraged the second interpretation but let the global community of jihadists read into the flag and the "state" what they would.
And read into them they did, with many taking up the flag and promoting the Islamic State as the fulfillment of prophecy long before it declared itself as such. The Islamic State's cause proved so compelling among jihadists that in 2014 the organization supplanted its former master, al-Qaeda, to lead the global jihadist movement. The spread of the flag, then, traces the spread of an idea and chronicles a major changing of the guard in the global jihadist movement over the past nine years. It also represents a revolution in how jihadists think about acquiring power and holding onto it.
Although it took nearly a decade to play itself out, the Islamic State was destined to fall out with al-Qaeda from the start. Al-Qaeda leader Osama bin Laden and his deputy Ayman al-Zawahiri wanted to build popular Muslim support before declaring the caliphate. The Islamic State wanted to impose a caliphate regardless of what the masses thought. The dispute that divided parent from child was there from the Islamic State's conception.
Problem Child
In 1999, a hotheaded Jordanian street-tough-turned-jihadist, Abu Mus'ab al-Zarqawi, arrived in Qandahar, Afghanistan, seeking an audience with al-Qaeda's leaders. The young Zarqawi wanted to foment revolution in the Fertile Crescent, the land stretching from the eastern Mediterranean through Iraq. Zarqawi had been to Afghanistan before, just after the defeat of the Soviets in 1989. Too late to fight in the war, he soon returned to Jordan, where he failed as a terrorist and spent time behind bars for his effort. Now out of prison, Zarqawi had come back to Afghanistan to gather money and recruits for his cause.
Al-Qaeda's man in Qandahar, Sayf al-Adl, did not contact Zarqawi immediately. A former special forces colonel in the Egyptian military, Sayf had learned to watch and wait. He had Zarqawi followed.
Sayf's spy reported that Zarqawi frequently argued with other jihadists because of his extreme views on who should count as a good Muslim. Zarqawi especially disliked the Shi'a, one of the two major sects in Islam. Zarqawi, a Sunni, disagreed with the Shi'i doctrine that Muhammad's son-in-law and some of his male descendants were infallible and the only legitimate political and religious leaders of the early Muslim community. He also believed the modern Shi'i state of Iran colluded with the West to oppress Sunnis. When Sayf finally met Zarqawi, he found him a man of few words who sincerely wanted to bring Sunni Islam back to "the reality of human life." But Zarqawi did not have a lot of specific ideas for how to do it.
Sayf relayed his impressions of Zarqawi to his bosses in al-Qaeda. Al-Qaeda's leader, Bin Laden, was the son of a wealthy Saudi building contractor, and his deputy, Ayman al-Zawahiri, was a surgeon who had run an Egyptian terror group before merging part of it with al-Qaeda. Both men would oversee the 9/11 attacks, which were premised on their belief that the American infidel should be killed wantonly. But when it came to Muslims, Bin Laden and Zawahiri were more cautious. They believed Muslim support was crucial for driving the Americans out of the Middle East and establishing Islamic states. It wouldn't do to make enemies on all sides, especially over theological differences. Some have even speculated that Bin Laden's own mother grew up in a small Shi'i sect. Unity of mission rather than unity of mind was what was needed.
Despite their misgivings about Zarqawi's extreme views, Sayf recommended his bosses support the Jordanian hothead because they had so few Palestinian or Jordanian allies. They consented but would not invite Zarqawi to join al-Qaeda; he would have refused anyway. Rather, they coordinated and cooperated with him "in serving our common goals."
Sayf and his companions came up with a plan for Zarqawi to establish a training camp in Afghanistan to attract jihadists from Jordan, Palestine, Syria, Lebanon, and Turkey. Herat was chosen because of its proximity to the Iranian border, where it was easy to move men and materiel across. Over time, Syrians, Jordanians, Palestinians, Lebanese, and Iraqis arrived. Zarqawi also reached out to the Kurdish Ansar al-Islam organization in northern Iraq.
By the beginning of 2001, Zarqawi was no longer a jihadist neophyte in the eyes of Sayf. He had "begun to think and plan strategically for the future." Reading widely about world events and Islamic history, Zarqawi was struck by the figure of Nur al-Din Zengi, the ruthless medieval ruler of a dominion stretching from Aleppo in Syria to Mosul in Iraq who had driven the crusaders from Syria.
Zarqawi undoubtedly admired Nur al-Din's ambition and remorseless efficiency. In one account, Nur al-Din had besieged a crusader citadel in Syria. The crusaders finally capitulated and approached Nur al-Din to discuss terms. "He would not consent to their request," as a medieval Muslim historian euphemistically put it. When crusader reinforcements arrived to lift the siege, they saw the citadel wall "and the dwelling of its inhabitants were entirely in ruins."
"[Zarqawi] was always asking for any book available about Nur al-Din and his protégé Saladin," Sayf recalled, referring to the ruler of Egypt who battled Richard the Lionheart during the Crusades. "I believe that what he read about Nur al-Din and his launch from Mosul in Iraq played a big role in influencing Abu Mus'ab [al-Zarqawi] in his decision to go to Iraq after the fall of the Islamic Emirate in Afghanistan" in 2001.
The "Islamic Emirate of Afghanistan" had been established in 1996 by the Taliban, conservative Sunnis who swept to power in the chaotic aftermath of the Soviet withdrawal from the country in 1989. In medieval Islamic thought, an "emirate" (imara), or government of a region, is subordinate to the "state" (dawla), the empire ruled by the caliph. But in the absence of a caliph, jihadists today sometimes use "state" and "emirate" interchangeably when talking about the government of a country they'd like to create. The Taliban's emirate brought order to Afghanistan by strictly enforcing Islamic law. It also gave shelter to likeminded jihadist groups such as al-Qaeda and Zarqawi's outfit.
After the fall of the Taliban, Zarqawi and Sayf fled Afghanistan for Iran. There they discussed where Zarqawi should go next. After "long study and deliberation," Sayf later wrote, Zarqawi's group decided to relocate to Iraq, where their "appearance" and "dialect" would help them blend in. Zarqawi and Sayf anticipated that the Americans would "invade Iraq sooner or later" to overthrow the regime. "It was necessary for us to play a major role in the confrontation and resistance," Sayf recalled. "This is our historical opportunity . . . to establish the state of Islam, which would play the greatest role in lifting injustice and bringing truth to this world, by God's permission. I was in agreement with brother Abu Mus'ab [al-Zarqawi] in this analysis."
For Sayf and presumably for Zarqawi, the "state of Islam" was the caliphate itself. Sayf used to believe that "the Islamic state of the caliphate" would develop from the Taliban's Islamic emirate in Afghanistan. But the American invasion in 2001 had ended that dream. Iraq was a second chance.
In 2002 and early 2003, Zarqawi set about building his clandestine network in Iraq. When the Americans invaded in March 2003, Zarqawi's cells in Baghdad were ready to greet them. Zarqawi himself arrived in June. By the end of August, his new group, Monotheism and Jihad, had bombed the Jordanian embassy and the United Nations headquarters in Baghdad, as well as the mosque of Imam Ali, one of the holiest shrines of Shi'i Islam. The subsequent departure of the UN mission and rising fury of Iraq's majority Shi'a signaled the beginning of a bloody sectarian civil war.
Zarqawi's group had not pulled off the attack alone. It had help from former security officers in Saddam Hussein's government, casualties of the Bush administration's purge of Saddam party loyalists. They, like other disenfranchised Arab Sunnis, feared the rise of the country's Shi'i population who had lived under the yoke of Saddam and minority Sunni rule for decades. There was a reckoning coming, and Sunni jihadists and nationalists were willing to put aside their ideological differences for the time being to unite against a common foe: the Americans and the majority Shi'a who stood to benefit from the occupation.
Zarqawi's hatred of the Shi'a was all-consuming. To his mind, the Shi'a were not just fifth columnists, selling out the Sunnis to the Americans. They were servants of the Antichrist, who will appear at the end of time to fight against the Muslims. The Americans served the same master.
Zarqawi's hatred of the Shi'a made him lose sight of his long-term political goals. When he applied for membership in al-Qaeda in February 2004, he did not mention an Islamic state or a plan for achieving it. Rather, he explained his strategy for winning over the Sunnis, defeating the transitional government, and driving the infidels from Iraq: Provoke the Shi'a. "If we are able to strike them with one painful blow after another until they enter the battle, we will be able to reshuffle the cards. Then, no value or influence will remain to the Governing Council or even to the Americans, who will enter a second battle with the Shi'a. This is what we want, and, whether they like it or not, many Sunni areas will stand with the mujahidin." (Mujahidin, or "those who fight in a jihad," is how jihadists refer to themselves.)
Zarqawi knew he would be criticized as "hasty and rash," "leading the Muslim community into a battle for which it is not ready, a battle that will be revolting and in which blood will be spilled." So be it. "This is exactly what we want, since right and wrong no longer have any place in our current situation. The Shi'a have destroyed all those balances."
If al-Qaeda's leaders would assent to his strategy, Zarqawi offered to swear allegiance to them, joining his group to theirs: "If you agree with us on it, if you adopt it as a program and path, and if you are convinced of the idea of fighting the sects of apostasy, we will be your readied soldiers, working under your banner, complying with your orders, and indeed swearing fealty to you publicly and in the news media, vexing the infidels and gladdening those who preach the oneness of God."
Al-Qaeda's leaders were wary. Bin Laden and Zawahiri wanted to compel the U.S. military to leave the Middle East and to stop supporting local autocrats. Their strategy was to attack the Americans and stir Muslim resentment against them. Building popular Muslim support for their cause was vital; the caliphate could not be established without it. In contrast, Zarqawi wanted to first overthrow local autocrats and eliminate the "traitorous" Shi'a, whom he believed were collaborating with the Americans to subjugate the Sunnis. His strategy was to ignite a sectarian civil war. Popular support mattered far less to Zarqawi than it did to Bin Laden and Zawahiri. He could will a caliphate into being regardless of what its subjects might say.
Despite their reservations, Bin Laden and Zawahiri accepted Zarqawi's oath of allegiance, joining his Monotheism and Jihad group to their own in October 2004. Al-Qaeda had just mounted a disastrous terror campaign in Saudi Arabia and was desperate for a role in the growing Sunni insurgency in Iraq. Zarqawi may have wanted to tap into al-Qaeda's network of private Gulf funders, operational expertise, and recruitment apparatus. Thus, al-Qaeda in Iraq was born.
Zarqawi was elated. "Our noble brothers in al-Qaeda understand the strategy of the Monotheism and Jihad group in the land of the two rivers, the land of the caliphs," he declared in his pledge to al-Qaeda's leaders, "and their hearts are overjoyed by its method there." "Perhaps," wrote Zarqawi, the group would establish the "caliphate according to the prophetic method." As we will see later, Zarqawi was alluding to an Islamic prophesy of the caliphate's return shortly before the End of Days.
Although Bin Laden and Zawahiri shared Zarqawi's desire to reestablish the caliphate, they advised him to proceed slowly and build popular support. In July 2005, Zawahiri wrote Zarqawi, urging him to establish an Islamic "emirate" only after the jihadists had expelled the United States from Iraq. The jihadists were to then "develop" and "consolidate" the emirate as far as they could inside the Sunni areas of Iraq until it reached "the level of the caliphate." The mission of the jihadists thereafter was to protect the caliphate's domain and expand its borders until the Day of Judgment.
Despite encouraging Zarqawi to establish an emirate after the American withdrawal, Zawahiri warned him not to attempt it before securing the support of the Sunni masses. Al-Qaeda's "two short-term goals" of "removing the Americans and establishing an Islamic emirate or caliphate in Iraq" both required "popular support from the Muslim masses in Iraq and the surrounding countries." "In the absence of this popular support," Zawahiri predicted, "the Islamic mujahid movement would be crushed in the shadows."
Zawahiri counseled Zarqawi to overlook the heresies of Sunni religious scholars, whose support al-Qaeda needed, and to cooperate with Sunni community leaders. Zarqawi should also stop broadcasting hostage beheadings. The beheadings may thrill "zealous young men," Zawahiri chided, but the Muslim masses "will never find them palatable." In general, the jihadists "shouldn't stir questions in the hearts and minds of the people about the benefit of our actions . . . we are in a media battle in a race for the hearts and minds of our [Muslim] community."
Zawahiri even went so far as to question Zarqawi's attacks on Shi'i civilians, the cornerstone of Zarqawi's strategy to provoke a sectarian civil war. "My opinion is that this matter won't be acceptable to the Muslim populace however much you have tried to explain it, and aversion to this will continue." In addition to jeopardizing public support, Zawahiri doubted the morality of these sectarian attacks. "Why kill ordinary Shi'a considering that they are forgiven because of their ignorance? And what loss will befall us if we did not attack the Shi'a?"
Another al-Qaeda leader in Bin Laden's inner circle, Atiyya Abd al-Rahman, was blunter in a December 2005 letter to Zarqawi. Echoing the nineteenth-century Prussian military theorist Carl von Clausewitz, Atiyya reminded Zarqawi that "policy must be dominant over militarism. This is one of the pillars of war that is agreed upon by all nations, whether they are Muslims or unbelievers." Atiyya cautioned that unless the jihadists' "short-term goals and successes" serve their "ultimate goal and highest aims," they would simply exhaust themselves to no effect.
Atiyya reminded Zarqawi of the fate of the Algerian jihadists in the 1990s. After Algerian Islamists had won the first round of voting for parliament in 1991, the military had cancelled the elections. Some Islamists turned to violence and, as the civil war dragged on, a jihadist faction began to murder civilians. Their short-term tactical successes won through brutality blinded them to how much they had alienated the Muslim masses. As Atiyya reminded Zarqawi, "They destroyed themselves with their own hands, with their lack of reason, delusions, their ignoring of people, their alienation of them through oppression, deviance, and severity, coupled with a lack of kindness, sympathy, and friendliness." It was not their enemy that defeated them; "they defeated themselves."
Atiyya knew what he was talking about. In 1993, al-Qaeda had sent the young Libyan to Algeria to liaise with the jihadist groups there. Rather than welcoming Atiyya, the worst of the groups imprisoned him. Atiyya managed to escape months later, but he was haunted by the experience. "I think [he] is still having nightmares about it," recounted someone who knew Atiyya's story.
Like Zawahiri, Atiyya reminded Zarqawi of the long-term objective he was fighting for: the establishment of the caliphate. "My brother," Atiyya wrote, "what use is it for us to delight in some operations and successful strikes when the immediate repercussion is a defeat for us of our call, and a loss of the justice of our cause and its logic in the minds of the masses who make up the people of the Muslim nation?" "You need all of these people," Atiyya observed, if you want "to destroy a power and a state and erect on its rubble the state of Islam."
"What am I commanding you to do?" Atiyya asked rhetorically. "Remedy the deficiency."
Atiyya detailed what he and other al-Qaeda leaders expected of Zarqawi. He should make no major strategic decisions without first consulting Bin Laden and Zawahiri. And he was to win over and work with influential Sunnis in Iraq, even the heterodox. Stop killing them, "no matter what."
Atiyya also counseled Zarqawi to stop insisting Sunni rebels join his organization and leave other jihadist groups: "Whether they come into the organization with us or not . . . they are our brothers."
Zarqawi initially agreed with Zawahiri and Atiyya that expelling the Americans was the priority. "First, we will expel the enemy," he explained in an interview. "Then we will establish the State of Islam." After that, the jihadists would "embark on conquest of Muslim lands to reclaim them," and then set their sights on the infidels.
But by April 2006, Zarqawi had changed his mind. When he announced a consultative council composed of several jihadist groups including al-Qaeda, he described it as the "nucleus for establishing an Islamic state." That state, he proclaimed, would be established in three months.
After the United States killed Zarqawi on June 7, 2006, al-Qaeda in Iraq carried out its leader's dying wish. Rather than wait to establish the Islamic state until after the Americans withdrew and the Sunni masses backed the project, as Bin Laden and Zawahiri wanted, the Islamic State was proclaimed on October 15, 2006. As we will see in the following chapter, the timing of the Islamic State's announcement was based on an apocalyptic schedule. The al-Qaeda front group that made the announcement insisted that Muslims in Iraq pledge allegiance to a certain Abu Umar al-Baghdadi and acknowledge him as "commander of the faithful." No one had ever heard of him, not even other jihadists.
State of Confusion
The Islamic State called its mysterious leader the "commander of the faithful" to encourage jihadists to think of him as the caliph without explicitly saying so. Historically, Muslims reserved that title for the early Islamic caliphs, the spiritual and temporal heads of a vast empire. Abu Umar al-Baghdadi's alleged descent from the Prophet's tribe, which many Muslims consider a prerequisite for the caliph, hinted at his entitlement to the position. Abu Umar even claimed descent from one of the Prophet's grandsons, Husayn, in an attempt to appeal to those who would confine leadership of the Muslim world to descendants from Muhammad's family, just as medieval caliphs had done.
The name of the Islamic State was equally ambiguous. The group had called itself a state rather than an emirate, the latter a more common word used by jihadists to describe small territory ruled by an emir. The word "state" in Arabic, dawla, can either mean a modern nation-state or evoke the memory of medieval caliphates like the Dawla Abbasiyya, which spanned Mesopotamia, the Persian Gulf, and North Africa. The Islamic State played on this ambiguity to encourage its followers to view it as the proto-caliphate, sometimes calling itself "the Islamic State in Iraq" rather than its official name of "the Islamic State of Iraq."
Although most of the Islamic State's members at this time were part of al-Qaeda in Iraq, the Islamic State made no mention of al-Qaeda or of members' preexisting oaths of allegiance to Bin Laden or to Mullah Omar, the head of the Taliban. Only four months earlier, Abu Ayyub al-Masri (aka Abu Hamza al-Muhajir), the head of al-Qaeda in Iraq after Zarqawi and later the founder of the Islamic State, had proclaimed his undying loyalty to Mullah Omar as "commander of the faithful" and to Bin Laden as the head of al-Qaeda. "We . . . are an arrow in your quiver. Shoot us where you wish for we are naught but an obedient soldier."
Al-Qaeda's supporters around the globe were confused. Was the new Islamic State part of al-Qaeda or something different? The jihadist pundit Akram Hijazi complained that al-Qaeda had not released an official statement and there was no sign of official coordination between al-Qaeda and the Islamic State. Bin Laden and Zawahiri had pledged oaths of allegiance to Mullah Omar, as had the head of al-Qaeda in Iraq. Were those oaths dissolved now? Who was in charge? Why, Hijazi wondered, was the hitherto unknown Abu Umar al-Baghdadi not just named a governor under the authority of the commander of the faithful, Mullah Omar?
Another jihadist Internet commentator summarized the confusion rampant in the private discussion boards where jihadists hung out before the advent of Twitter. "How can we pledge allegiance to Abu Umar al-Baghdadi when we may have pledged allegiance to Mullah Omar? What do we do with the pledge of allegiance to Shaykh Osama if we want to pledge allegiance to Shaykh Abu Umar?"
The truth was that Bin Laden and Zawahiri had been caught by surprise. "The decision to announce the State was taken without consulting the leadership of al-Qaeda," American al-Qaeda spokesman Adam Gadahn confided in a private letter. As he saw it, the unauthorized announcement "caused splits in the ranks of the mujahids and their supporters inside and outside Iraq." Zawahiri would later recall that "the general command of [al-Qaeda] and its emir Shaykh Osama bin Laden (God bless him) were not asked for permission, consulted, or even warned just prior to the announcement of the establishment of the Islamic State of Iraq."
Behind the scenes, the Islamic State sought to heal its rift with Bin Laden and Zawahiri. The former head of al-Qaeda in Iraq and the actual leader of the new Islamic State, Abu Ayyub al-Masri, assured his bosses that the "commander of the faithful," Abu Umar al-Baghdadi, had pledged an oath of allegiance to Bin Laden in front of the jihadist brothers in Iraq. They did not announce it publicly "due to some political considerations that they saw in Iraq at that time."
Masri was attempting to preserve the Islamic State's ties to al-Qaeda while encouraging the public to think of it as a separate entity. He wanted the world to view his group as a state and not a terrorist franchise. Ambiguity, again, was critical to the Islamic State's early propaganda.
Masri used the same strategy in his public statements. A month after the Islamic State's founding, Masri hailed it as an important step in the "program of the Islamic caliphate." In the same statement, Masri pledged his allegiance to the shadowy "commander of the faithful," Abu Umar al-Baghdadi, announced the dissolution of al-Qaeda, and reassigned all its fighters to the Islamic State of Iraq. "All of them have pledged allegiance unto death in the path of God," Masri assured the commander of the faithful. "You will only find us listening to what you say and obeying what you command."
In the five succeeding months, Abu Umar al-Baghdadi and the Islamic State hammered home the same point: Al-Qaeda in Iraq was no more. "Al-Qaeda is but one of the groups in the Islamic State," Abu Umar declared in December 2006." "It is more correct to say," instructed the Islamic State's ministry of media, "that the brothers previously in the organization of al-Qaeda in Iraq became part of the 'army of the State,' which includes dozens of battalions and thousands of fighters from the remaining jihadist groups that pledged loyalty to the commander of the faithful after the announcement of the State." On April 19, 2007, an Islamic State spokesman announced that Masri was now the minister of war in the State's first cabinet.
Al-Qaeda's leaders were not only angered that the Islamic State had challenged Bin Laden's authority by not seeking his approval. They were also convinced the Islamic State had declared itself too soon. In the spring of 2007, a senior al-Qaeda leader, Abu al-Walid al-Ansari, asked a string of pointed questions of the group. Why had the state been declared now rather than later? Had its appointment of the commander of the faithful followed Islamic rules? Why had the Islamic State announced that anyone who opposed it was a sinner? Ansari reminded the Islamic State that it needed the broad support of the people it wished to govern and the consent of their leaders if it was going to survive. By declaring itself prematurely, the Islamic State had taken on the burdens of governing and invited foreign intervention, both of which could prove lethal to the nascent enterprise.
Other jihadists were even more pointed in their criticisms. The Kuwaiti scholar Hamid al-Ali argued that an authentic Islamic State should be able to impose its authority over those it governs. The Islamic State did not meet that standard. The Islamic State also failed to meet classical Islamic requirements for establishing an Islamic government, at least according to al-Ali. "It is not recognized in Islam to pledge allegiance to an unknown, concealed leader who has no authority . . . [or] established state" capable of imposing Islamic law, al-Ali wrote. Mullah Omar could declare a state in Afghanistan in the 1990s because he actually ruled it at the time; Abu Umar al-Baghdadi ruled nothing. Declaring a state in Iraq under false pretenses, al-Ali charged, had divided the jihadist movement in Iraq, which should be united under the banner of jihad rather than the banner of a single group.
Al-Qaeda's leaders in Pakistan looked on in shame and dread as its problem child stumbled in its first few months. Atiyya Abd al-Rahman, Bin Laden's chief of staff who had sent the 2005 letter upbraiding Zarqawi, shared his worry with a confidant "about the brothers making political gaffes." "You must have heard Abu Umar's recent sermon," he wrote, referring to the Islamic State's nominal leader, Abu Umar al-Baghdadi. "In my view, it was filled with obvious errors. There were things in it that a commander should never say." The speech gave the impression that they were "extremists and gave life to the notion that they are self-absorbed and too hasty!" Atiyya worried that if "they continue in this way, they will become corrupt and . . . lose the people," allowing the enemy to turn the populace against them. "None of the enemies scare me, I swear, no matter who they are, or how intimidating they may be. . . . But I do worry about our and our brothers' mistakes, bad behavior, and lack of wisdom at times." Reminiscent of the letter he had written Zarqawi, Atiyya confided that he himself had chastised the Islamic State's leaders: "I was a little hard on them."
Despite al-Qaeda's private misgivings, its leaders presented a united front in public and endorsed the establishment of the State. They probably wanted to keep a hand in the Iraq game and avoid further dissension in the ranks. "I want to clarify that there is nothing in Iraq by the name of al-Qaeda," said Zawahiri in a December 2007 question-and-answer session. "Rather, the organization of [al-Qaeda in Iraq] merged, by the grace of God, with other jihadist groups in the Islamic State of Iraq, may God protect it. It is a legitimate emirate established on a legitimate and sound method. It was established through consultation and won the oath of allegiance from most of the mujahids and tribes in Iraq."
Al-Qaeda's affiliate in Iraq may have become part of the Islamic State and the Islamic State may have privately joined al-Qaeda, but the public would not know the nature of the relationship between the two groups for years to come. The Islamic State itself never addressed the question publicly, again relying on ambiguity to imply greater power and autonomy than it possessed. The Islamic State's ambiguous audacity would capture the jihadist imagination and become crucial in its later rise to power. Nothing embodied the propaganda strategy better than the Islamic State's flag.
Making the Black Flag
When the Islamic State first announced itself on October 15, 2006, it had no flag of its own. It was not until January 2007 that al-Qaeda's media distribution arm, al-Fajr, released a picture of the Islamic State's new flag. Anonymous authors affiliated with the Islamic State explained its design.
Unsurprisingly, the Islamic State had turned to the Prophet's example for inspiration, quoting passages from Islamic scripture and historical accounts. "The flag of the Prophet, peace and blessings be upon him, is a black square made of striped wool," according to one account. Another depicts Muhammad "standing on the pulpit preaching" surrounded by fluttering black flags. "On the flag of the Prophet was written 'No god but God, and Muhammad is the Messenger of God.'" The flag even had a name: "the eagle."
Although the authors acknowledged other reports of green, white, and yellow flags, they concluded the Islamic State's flag will be black because most of the reports about Muhammad mention a black flag. "The commander of the faithful [Abu Umar al-Baghdadi] issued his decree, informed by knowledgeable people, that the flag of the Islamic State is black." The authors were equally confident when explaining the banner's text, which is the Muslim profession of faith. "What is written on the flag is what is written on the flag of the Messenger of God, peace and blessings be upon him: 'No god but God, Muhammad is the Messenger of God.'"
The Islamic State's design of the Muslim profession of faith is unique, different from every other attempt to replicate the Prophet's flag: "No god but God" is scrawled in white across the top and "Muhammad is the Messenger of God" is stacked in black inside a white circle. As the authors noted, they took the circle's design from a seal of the Prophet used on letters supposedly written on his behalf and housed in Topkapi Palace in Turkey. The seal's design, the authors argued, comports with historical reports of what the Prophet's seal looked like. Never mind that modern scholars doubt the letters' authenticity. We are meant to believe the Islamic State had inherited the Prophet's seal, just as the early caliphs had.
Why make a flag? In addition to following the Prophet's example, the Islamic State wanted a symbol to rally people to its cause. The State quoted a nineteenth-century Ottoman historian and official, Ahmad Cevdet Pasha, to make the point: "The secret in creating a flag is that it gathers people under a single banner to unify them, meaning that this flag is a sign of the coming together of their words and a proof of the unity of their hearts. They are like a single body and what knits them together is stronger than the bond of blood relatives."
Like all fundamentalist attempts to revive the early days of their faith, the Islamic State's leaders had to choose among contrasting scriptures and histories from their religion's past to paint a portrait of what they aspired to in the present and future. Their choices display the cultural biases and modern sensibilities they try so hard to displace. They selected a stark black for the flag rather than green, yellow, or white; the color suits their Manichean worldview, which permits no gray areas between the binaries of right and wrong, believer and unbeliever. The white scrawl across the top, "No god but God," is deliberately ragged, meant to suggest an era before the precision of Photoshop even though the flag was designed on a computer.
Even the Islamic State's quotation of Ahmad Cevdet Pasha unwittingly betrays modern sensibilities. Influenced by European notions of nationalism yet desiring to hold together the multi-ethnic Ottoman Empire under sovereign Turkish rule, Cevdet Pasha imagined Islam and its symbols to be the glue. The sentiment underlies his utilitarian outlook on religion. "The only thing uniting Arab, Kurd, Albanian and Bosnian is the unity of Islam. Yet the real strength of the Sublime State lies with the Turks." Try as they might to re-create the imagined utopian era of the Prophet, the people who designed the Islamic State's flag were still captives of their age. The quest for authenticity is a very modern pursuit.
As with its flag, the Islamic State unwittingly organized and described itself in modern ways. A jihadist pundit complained that the Islamic State used modern words to describe its bureaucracy: "[Words] in the expressions 'Spokesman on behalf of the Islamic State of Iraq' and 'the Minister of Education' are found in Arab and Islamic history but their form appears closer to the reality of today than any Islamic reality." The same pundit also griped that confining the state to Iraq was too close to modern notions of the state. The original caliphal state had been a large empire with ever-expanding borders, not a state contiguous with any particular nation like the modern nation-state.
Despite its ambiguity, the Islamic State was dropping hints that it aspired to be more than a modern nation-state. Its flag carried the seal of the Prophet, a sign of authority inherited by the caliphs. As we will see, the flag's color also evoked a powerful early caliphate. But there was something more. The Islamic State ended its explanation of its flag's design with a prayer: "We ask God, praised be He, to make this flag the sole flag for all Muslims. We are certain that it will be the flag of the people of Iraq when they go to aid . . . the Mahdi at the holy house of God." The house of God is the Ka'ba in Mecca, the holiest shrine in Islam, and the Mahdi is the Muslim savior who will appear there in the years leading up to the apocalypse. The Islamic State was signaling that its flag was not only the symbol of its government in Iraq and the herald of a future caliphate; it was the harbinger of the final battle at the End of Days.
The Rightly Guided One
Legends of the black flag and the Muslim savior, the Mahdi, first circulated during the reign of the Umayyad dynasty, which ruled the Islamic empire from the ancient city of Damascus in the seventh and eighth centuries AD. The dynasty's founders, the Umayya clan, had seized the caliphate from Muhammad's son-in-law and grandsons, which infuriated many Muslims. The father of the dynasty's founder Mu'awiya had persecuted Muhammad and his early followers before he later converted. The founder's mother had even eaten the liver of Muhammad's uncle.
People unhappy with Umayyad rule and the way they had seized power circulated prophecies of a man of the Prophet's family who would return justice to the world. They called the man the Mahdi, Arabic for "the Rightly Guided One." Many of the prophecies envision the Mahdi appearing in the End of Days to lead the final battles against the infidels. It is the Islamic version of the Christian Battle of Armageddon. The Final Hour and Day of Judgment will soon follow.
To give the prophecies added heft, they were often attributed to Muhammad. "[The Mahdi's] name will be my name, and his father's name my father's name" went one. "He is a man from my family" went another.
Like most Islamic prophecies of the End of Days, those about the Mahdi are not found in the preeminent scripture of Islam, the Qur'an, which Muslims believe preserves God's revelation to Muhammad. Rather, the prophecies are found in voluminous compendia of the words and deeds of the Prophet and his companions, known as ahadith. Because the ahadith were written down decades or even centuries after the Prophet's death, they often reflect later political, social, and theological developments rather than what actually happened. Muslims argue over the veracity of individual ahadith and the contradictions between them the way some Christians debate the reliability of the Gospels and their discrepancies.
End-Time prophecies were an especially inviting target for fabricators. In the internecine wars that tore apart the early Muslim community, each side sought to justify its politics by predicting its inevitable victory and the other side's preordained defeat. What better way to do this than to put the prophecy in the mouth of the Prophet? Prophecies proliferated, reaching into the thousands. When the politics evaporated, the prophetic residue remained. Throughout the centuries, new politics would give the residue new meaning, a phenomenon familiar to readers of the Christian Book of Revelation for nearly two millennia.
Over the years, the prophecies of the Mahdi have inspired many claimants. How could they resist? Whether the claimant was sincere or not, claiming the spiritual and political power of the Mahdi made a potent recruiting pitch. Think Jesus and George Washington rolled into one.
"The common people, the stupid mass, who make claims with respect to the Mahdi," wrote the medieval Muslim historian Ibn Khaldun, "assume that the Mahdi may appear in a variety of circumstances and places." Because the masses are gullible, he observed, leaders wrap themselves in the savior's mantel to mobilize them.
Militant messiahs are not unique to Islam. Whether in the Middle Ages or the modern world, groups that want to overturn the social and political order often use apocalyptic language. The Jewish "messiah" Bar Kokhba led an insurgency against the Romans, which the emperor Hadrian brutally repressed, reportedly killing hundreds of thousands, desecrating holy sites, and banning Jews from Jerusalem. The 100,000 European foreign fighters who flooded into Palestine under the banner of the First Crusade believed they were hastening the End of Days. In modern times, some members of the Israeli settler group Gush Emunim sought to hasten the coming of the messiah by blowing up the Dome of the Rock, one of the holiest sites in Islam. The megalomaniacal Christian "savior" Joseph Kony still hides in the Central African Republic, leading the child soldiers in his Lord's Resistance Army.
The two main sects in Islam, Sunni and Shi'a, each had numerous messianic aspirants in the Middle Ages, some of whom established caliphates. These aspirants often claimed the title of Mahdi. "I am Mahdi of the end of time" proclaimed the Sunni founder of the Almohad caliphate in Spain and North Africa. The Shi'i founder of the Fatimid caliphate in Egypt claimed the same for himself and later for his son. Muslims who opposed them were apostates deserving death, a consequence of defying God's anointed.
The first Mahdi didn't actually claim the title himself. In AD 685, a little over fifty years after Muhammad's death, a man named al-Mukhtar led a rebellion against the Umayyads in Iraq in the name of the Mahdi, whom he identified as a grandson of Muhammad. Al-Mukhtar claimed to be the Mahdi's vizier. In addition to leading a rebellion, he is also remembered for prophesying in verse and parading an Ark of the Covenant around Kufa, Iraq. Many of those who rallied to his cause were non-Arab or Jewish converts to Islam who chafed at being treated like second-class citizens in the Arab-dominated government. "They used to say that only three things interrupt prayer," records an early Spanish-Arab historian: A donkey, a dog, and a non-Arab convert to Islam.
The discontent only reinforced the Umayyads' sense of entitlement and fueled their resentment of the new converts who supported the Prophet's family. Were it not for us, complained an early Umayyad caliph, the entire Muslim world would be subservient to the non-Arabs rallying around the Prophet's family. They have become uppity, he allegedly wrote his governor in Iraq, and need to be put in their place.
Supporters of the Prophet's family loosely aligned themselves in what historians call the Hashimite movement, which believed the Mahdi would be a descendent of Muhammad's great-grandfather Hashim. Many of the movement's supporters were from Iran, where Zoroastrian legends prophesied the coming of a club-wielding savior who would appear at the End of Time followed by sable-clad disciples. Influenced by the prophecies, the Hashimite supporters donned black clothes, flew black flags, and carried around wooden clubs called "infidel-bashers."
Iranians and descendants of the Arab conquerors of Iran felt alienated from the remote Umayyad clan ruling from Damascus. "These lands belonged to our ancient fathers!" protested an Arab rebel commander who had grown up in Iran. The early Arab caliphs had once ruled Iran justly, he recalled, and "helped the oppressed." But the Umayyads had made pious people fear the family of the Prophet, so Iranians and Arabs alike had to rise up against them to restore the rule of Muhammad's progeny.
As the revolutionaries built support for their cause, they circulated prophecies of soldiers fighting under black flags who would come from the East to overthrow the Umayyads. Some were put in the mouth of Muhammad's son-in-law Ali, who allegedly foretold the coming of an army from Khorasan, the "land of the rising sun" that includes parts of modern eastern Iran and most of Afghanistan. "The companions of the black flags that will approach from Khorasan are non-Arabs. They will seize power from the Umayyads and kill them under every rock and star." Other black flag prophecies were attributed to the Prophet himself. "The black banners will come from the East, led by men like mighty camels, with long hair and long beards; their surnames are taken from the names of their hometowns and their first names are from kunyas" or teknonyms in the form of Abu So-and-So. "If you see the black banners coming from Khorasan," instructs another, "go to them immediately even if you must crawl over ice because indeed among them is the caliph, al-Mahdi."
In early Islam, the color black was associated not just with mourning but also with revenge for a wrongful death. The pre-Islamic poet Imru al-Qays donned black when he went out to negotiate with the tribe that had murdered his father. When the Arab pagans defeated Muhammad's army at Uhud, his supporters dyed their clothes black to signal their desire to avenge the loss. According to the historian Ibn Khaldun, the opponents of the Umayyads adopted black as their color to avenge the Umayyads' persecution of the Prophet's family. "Their flags were black as a sign of mourning for the martyrs of their family, the Hashimites, and as a sign of reproach directed against the Umayyads who had killed them."
Black flags were also flown by the Prophet in his war with the infidels. "Do not flee with [the flag] from the infidels and do not fight with it against the Muslims," Muhammad reportedly told one of his generals.
When Muslims raised the black flag against the Umayyad caliphs, the caliphs understood the doubly implied threat: vengeance for the family of the Prophet against the "infidel" Muslim rulers who had usurped them.
The seeds of anti-Umayyad propaganda had been sown by a secret network of revolutionaries. The network was led by a shadowy imam, or spiritual leader, descended from the Prophet's uncle Abbas who hoped to use the Hashimite movement to come to power. His agent, Abu Muslim, conducted the propaganda effort and eventually commanded the armed revolt. Their revolutionary agitation on behalf of the Abbasid family came to a head in the Islamic year 129 (AD 746–747), when the imam sent a black "flag of victory" to Abu Muslim in Iran. The flag arrived with a message: "The time has come." Abu Muslim unfurled the black flag, dubbed "the Shadow," on a lance fourteen cubits high and publicly proclaimed the Abbasid's revolutionary call on June 9, 747, the twenty-fifth day of Ramadan, the Muslim month of fasting. After unfurling a second flag, "the Clouds," Abu Muslim and his companions donned black robes. "As the clouds cover the earth, so would the Abbasid preaching," the people were told. "And as the earth is never without a shadow, so it would never be without an Abbasid caliph to the end of time."
Fighting under the black flag, Abu Muslim's armies swept westward into Syria and Iraq. They overthrew the Umayyad caliph and pledged allegiance to a new one, al-Saffah, a brother of the Abbasid imam, who had been executed. The caliph proclaimed himself the Mahdi of the Muslim community, supposedly filling the world with justice and inaugurating the "blessed revolution," dawla mubaraka, from which the new empire took its name, Dawla Abbasiyya. It's thanks to the Abbasids that dawla came to mean "state."
There are striking parallels between the Abbasid revolution and the Islamic State revolution. They share a name (dawla), symbols and colors, apocalyptic propaganda, clandestine networks, and an insurgency in Syria and Iraq. They also claim the right to rule as the Prophet's descendants. The Abbasids had provided a blueprint for how to overthrow a Muslim ruler, establish a new caliphate, and justify both. Apocalypse, caliphate, and revolution were inseparable, just as they are for the Islamic State.
Apocalyptic messages resonate among many Muslims today because of the political turmoil in the Middle East. In 2012, half of all Muslims in North Africa, the Middle East, and South Asia expected the imminent appearance of the Mahdi. And why wouldn't they, given the revolutions sweeping the Arab world? The signs that herald his coming have only multiplied since. A great sectarian war tears Syria asunder. Iraq is in chaos. The "infidel" West has invaded. The "tribulations" (fitan) are too awful and apparent to brook mundane explanations.
Despite the propaganda value of apocalyptic messages, al-Qaeda's leaders were reluctant to use them. Sure, al-Qaeda held press conferences in "Khorasan," part of which is in Afghanistan, backed by a different version of the black flag. The name of al-Qaeda's magazine, Vanguards of Khorasan, evokes the same prophecies, and its media outlet is called al-Sahab, or "the Clouds," perhaps alluding to the Abbasid flag.
But all these examples merely hint at the apocalypse. Al-Qaeda's leaders rarely referred to Islamic End-Times prophecies in their propaganda and never suggested the Mahdi was just around the corner. As one scholar of modern Islamic apocalypticism observes, "al-Qaida, so far as one can judge from its internal correspondence, was for many years impervious to the apocalyptic temptation."
Bin Laden's and Zawahiri's disdain for apocalypticism reflects their generation and class. Until the Iraq war, apocalypticism was unpopular among modern Sunnis, who looked down on the Shi'a for being obsessed with the Mahdi's return. Sunni books on the apocalypse were commercial failures.
Bin Laden and Zawahiri grew up in elite Sunni families, who sniffed at messianic speculation as unbecoming, a foolish pastime of the masses. Their attitude is captured in an article distributed by an al-Qaeda propaganda outfit that discourages Muslims from speculating on who the Mahdi is or when he will appear: "Many people think that the State of Islam will not be established until the Mahdi appears. They neglect to take action, instead raising their hands in prayer that God will hasten his appearance."
Bin Laden had another, more personal reason for disliking messianism. In 1979, the year he graduated from college in Saudi Arabia, a group of Sunni radicals captured the Grand Mosque in Mecca. The mosque encloses Islam's holiest shrine, the Ka'ba. The radicals were there to consecrate one of their number as the Mahdi. Elite Saudi soldiers ended the weeks-long siege with the help of tear gas and French special forces; the Mahdi lay among the dead. The humiliating defeat became a cautionary tale among jihadists: It is too risky to claim the fulfillment of prophecy and then fail.
Although al-Qaeda's leaders downplayed the apocalypse, some of their followers celebrated it. Group members quoted the black flag prophecies when interrogated by members of the Federal Bureau of Investigation (FBI). "If you see the black banners coming from Khorasan, join the army," recited Abu Jandal, a former Bin Laden bodyguard, to FBI agent Ali Soufan. Soufan remembers another Bin Laden associate, Ali al-Bahlul, citing the prophecy during his interrogation in Guantanamo Bay as evidence that al-Qaeda was fighting the final battles of Armageddon.
Senior al-Qaeda officers also cited Islamic prophecies. "The Islamic armies must gather, rely on God, and support His religion and their brothers in Jerusalem" wrote Fadil Harun, Bin Laden's man in Somalia. The "awaited Mahdi" would then appear and lead "an ideological struggle, which will continue until the [Final] Hour as long as an inch of Muslim land in the Holy Land is under the control of the enemies."
Perhaps the most prolific apocalypticist in the al-Qaeda orbit was Abu Mus'ab al-Suri, a Syrian jihadist who devoted over a hundred pages to the End Times in his massive 2004 tome on terrorist strategy. Although the book enjoyed immense popularity among jihadists for its strategic insights, many looked askance at his taste in prophecies. Suri cited the medieval Book of Tribulations written by Nu'aym bin Hammad, which contains prophecies many Muslims consider spurious. But Suri read broadly in the canonical prophecies too and concluded that jihadists should reorient their fight toward the Fertile Crescent, which is where many prophecies locate the final battles of the Apocalypse, as we will see in chapter 5.
Given the rise of messianism in the Middle East and the historical precedent of the Abbasids, it makes sense that the Islamic State would appeal to prophecy to justify its cause. But did the leaders of the Islamic State actually believe what they were saying, or were they cynically cloaking their ambition in messianic garb? And if they were sincere, how could they reconcile the urgent imperatives of the Apocalypse with the patient care required to run a state?
Two
Mahdi and Mismanagement
One way to measure the performance of a terror CEO is by the bounty his enemies put on his head. By that measure, the man who ran the Islamic State in its first year, Abu Ayyub al-Masri, was an abject failure. In 2007, the United States offered $5 million for information leading to Masri's location. In May 2008, the amount had fallen to $100,000. A variety of reasons could be given for the precipitous fall in Masri's stock: He was a terrible manager, isolated from the battlefield, and facing the most powerful military on earth—that of the United States. But many of Masri's bad decisions that first year of the Islamic State's existence can be traced to something more fundamental: his apocalypticism and dogmatic belief that the Islamic State was an actual state.
Masri was an old hand at terrorism. In 1982, he had joined the Islamic Jihad organization later run by fellow Egyptian Ayman al-Zawahiri. When Zawahiri merged his organization with al-Qaeda, Masri went with him, learning to make bombs at an al-Qaeda training camp in the late 1990s. While in Afghanistan, Masri met Abu Mus'ab al-Zarqawi, who later founded al-Qaeda in Iraq. When the United States invaded Iraq in 2003, Masri was already in Baghdad ready to help Zarqawi set up shop. Zarqawi put him in charge of recruiting suicide bombers and overseeing operations in Iraq's Shi'i south. Masri allegedly built the bombs detonated at the United Nations headquarters and the Jordanian embassy in 2003 and oversaw terror campaigns against the Shi'a. As we saw in the last chapter, the attacks paved the way for Iraq's descent into sectarian civil war.
Masri took over al-Qaeda in Iraq after Zarqawi died in June 2006, only to dissolve the group a few months later when the Islamic State was declared. According to the Islamic State's chief judge, Masri rushed to establish the State because he believed the Mahdi, the Muslim savior, would come within the year. To his thinking, the caliphate needed to be in place to help the Mahdi fight the final battles of the apocalypse. Anticipating the imminent conquest of major Islamic cities as foretold in the prophecies, Masri ordered his men to build pulpits for the Mahdi to ascend in the Prophet's mosque in Medina, the Umayyad Mosque in Damascus, and the Aqsa Mosque in Jerusalem. He also ordered his commanders in the field to conquer the whole of Iraq to prepare for the Mahdi's coming and was convinced they would succeed in three months. The Islamic State's forces fanned out across the country, only to be recalled a week later because they were spread too thin. When those close to Masri criticized him for making strategic decisions on an apocalyptic timetable, Masri retorted, "The Mahdi will come any day."
In his public propaganda, Masri's apocalypticism was more restrained but still on display. When Masri was appointed the Islamic State's minister of war, he proclaimed, "The war is in its early stages . . . and this is the beginning of the battles" that herald the Day of Judgment. "We are the army that shall hand over the flag to the servant of God, the Mahdi," he asserted. "If the first of us is killed, the last of us will deliver it."
Masri technically answered to the Islamic State's "commander of the faithful." The original statement announcing the Islamic State did not give the commander's real name, only his nom de guerre Abu Umar al-Baghdadi, which suggested he hailed from Baghdad, the capital of Islam's greatest caliphate, the Abbasid dynasty. At the time, Masri confided to his chief judge that he was still looking for a suitable candidate for the job. "A man will be found whom we will test for a month. If he is suitable, then we will keep him as the commander of the faithful. If not, we will look for someone else."
The man Masri chose, Hamid al-Zawi, was "a normal person and not a leader" in the organization, according to a former member of the Islamic State. Zawi had been a police officer in the Sunni farming town of Haditha until he was fired in the early 1990s for his ultraconservative religious views. Zawi repaired electronics to make ends meet before the Americans invaded in 2003, when he joined al-Qaeda in Iraq and then the Islamic State. "Everyone was astonished [Masri] gave an oath of allegiance to him," the former member recalled. "What were his qualifications?" The selection of Zawi for the post of commander of the faithful was more like a casting call than anything resembling an Islamic procedure for choosing an emir or leader.
In its propaganda, the Islamic State had equivocated on whether it was just a state or something more. In private, Masri equivocated too. Sometimes he said that the announcement of the Islamic State was merely an announcement. Most of the jihadist "brothers" in Iraq took that to mean the Islamic State was just an extension of the emirate declared by Mullah Omar, to whom they had pledged allegiance, just as Bin Laden had done. But at other times Masri claimed that the Islamic State in Iraq was an actual entity unto itself. Masri even confided to his chief judge that he believed Abu Umar al-Baghdadi to be the caliph, although Masri did not intend to proclaim him as such until after the Americans left Iraq.
Because the leaders of the Islamic State thought of it as an actual state, they were high-handed and brutal in dealing with Sunnis who did not want to pledge allegiance to it. In the beginning of 2007, the Islamic State informed the other Sunni rebel groups that they had "no choice." As the State's spokesman put it, "they have to either join us in forming the Islamic State project in the Sunni areas or hand over their weapons to us before we are forced to act against them forcefully." To make its point, the Islamic State began killing or kidnapping insurgent leaders. The Islamic Army, one of the largest Islamist insurgent groups in Iraq, was a particular target. When the Islamic Army refused to bow to the Islamic State's authority, the State killed thirty of its members. In April 2007, the harassed group wrote an open letter to Bin Laden criticizing his Iraqi franchise. "They threaten some members of the group with death if they do not swear allegiance to al-Qaeda or its other names." The Islamic Army also accused the Islamic State of attacking civilians, "especially easy targets such as the imams of mosques, the prayer callers, and unarmed Sunnis." "They try to kill anyone who criticizes [the Islamic State] or goes against them."
It is a major taboo in Islam to kill a fellow Muslim. But the Islamic State argued that those who defied its rule were apostates or rebels so it could kill them without blame.
Still, Abu Umar al-Baghdadi tried to reconcile with the Islamic Army, doubtless worried it would join forces with the United States and the Iraqi government to fight the Islamic State. "Know that I am prepared to shed my blood to spare yours," he told them. "By God, you will hear from us only what is good, and you will see from us only what is good." But the rift quickly turned into a chasm as Islamic State fighters continued attacking their competitors.
Even when the State's leaders wanted to be conciliatory, they often didn't know when their commanders were behaving badly. Part of the problem was that the Islamic State had allowed too many people to join without vetting them properly. A clandestine terrorist group is usually picky about who joins, worried that it will be penetrated by spies or destroyed from the inside by idiots. Because it saw itself as something more than a terrorist group, the Islamic State had opened wide the doors for membership. Former Saddam loyalists from the military and intelligence services rushed in, as did insurgents who took orders from other countries. Some believed in the cause, but many were corrupt or had divided loyalties.
At times, Masri didn't know what his subordinates were doing because he was "totally isolated" and "almost absent from the details of what goes on in the battlefield," recalled his chief judge. Those close to Masri painted a rosy picture that distorted "his view of things and of reality." The reports from the field were similarly filled with happy talk. They didn't "mention the negatives and the problems as they really are."
Compounding the problem, Masri didn't fire or discipline subordinates who abused their power. When his chief judge told Masri that a senior official had confiscated goods from a Sunni merchant in Anbar Province who had been on good terms with the Islamic State, Masri told the judge to mind his own business or face demotion. According to the judge, going over Masri's head to the commander of the faithful, Abu Umar al-Baghdadi, would have been pointless because Abu Umar didn't "know what [was] going on around him."
Already in 2006, Masri was receiving complaints from other Sunni insurgent groups. "Dear Brother [Masri], the mistakes in the field of battle have come to the point [where we are] fed up" wrote a leader of the militia Ansar al-Sunna. A one-time ally of the Islamic State that had considered joining its ranks, the militia was furious at the arrogant and murderous behavior of the State toward it and other Sunni insurgent groups fighting the Americans in Iraq. "Individuals from your group have kidnapped, tortured, and killed people from our group with their [full] knowledge that they were from the Ansar [al-Sunna] group." Fearful of losing the Islamic State's last important ally, Masri apologized profusely in his January 2007 response: "I pledge to become a faithful servant and a loyal guard to you. I would carry your shoes on my head and kiss them a thousand times."
Whether the Islamic State's heavy-handedness came from the top or from rogue elements in the organization (probably a mix of both), the effect was the same. The group's bullying had eroded its base of support among Iraq's Sunnis, which was crucial for keeping alive its state-building enterprise. Ansar al-Sunna, the Islamic Army, and other militias that had complained to the Islamic State began to talk to the Americans in late 2006 and early 2007 about finding a way to rid themselves of the group.
Sunni support was already tepid after years of al-Qaeda murdering Sunni civilians. In the early days of the Sunni insurgency, Zarqawi had pioneered the strategy of provoking the Shi'a into a fight and had warned against attacking Sunnis. His plan worked initially, as Sunnis sought any port in the sectarian storm. But when al-Qaeda began to try to impose its rule in Sunni areas, the locals soured on the organization. In 2004, the inhabitants of Fallujah and the local insurgents who represented them were among the first to resist al-Qaeda, which controlled much of the city. Among their list of complaints were al-Qaeda's religious requirements that were at odds with local Islamic customs, such as full-body veiling. Beatings and broken bones brought reluctant citizens into line but bred resentment. The same happened a year later in 2005 in the city of Qaim on the border of Syria. Islamic State fighters burned down a beauty parlor and torched stores selling music; men who drank alcohol were lashed. By September 2006, Iraqi prime minister Nuri al-Maliki had received pledges of support from over a dozen Sunni tribal leaders, prompting al-Qaeda to issue a statement threatening their lives.
In October 2006, just days before the establishment of the Islamic State, a masked jihadist going by the name Abu Usama al-Iraqi released a video addressed to Bin Laden. Whether a clever government propagandist or a true jihadist insider, Abu Usama channeled the jihadists' mounting frustration with al-Qaeda in Iraq. "From the moment the commander of that group swore an oath of allegiance to you," Abu Usama said to Bin Laden, "we have been faced with a nightmare."
Abu Usama detailed atrocities so awful they "may cause children's hair to turn white." Al-Qaeda in Iraq had liquidated "religious scholars," followed by "acts of persecution against Sunni Muslims, harming their livelihood." The group "had planted explosives in front of homes, schools and hospitals, or under electric transformers, without taking any account of the consequences for society." Al-Qaeda in Iraq even used mothers and sick children as human shields. The group had utterly forgotten Zawahiri's "counsel that they should serve the population and protect it." Without support of the Sunnis, the jihadists had become "easy prey for the crusader occupation and its helpers, the vicious Shi'a."
Abu Usama also accused al-Qaeda in Iraq of greed and sacrilege. Its fighters had confiscated the salaries of government employees, looted religious endowments, "driven out worshippers . . . killed dozens of imams and preachers," and used the mosques to store weapons.
Abu Usama informed Bin Laden that many of the jihadist factions were biting their tongues, waiting for the al-Qaeda leader to censure his Iraqi affiliate. "The other factions are watching the Sunnis sinking deep into hunger and oppression. If any of them speak out in opposition he is silenced . . . He is kidnapped [and placed] in the trunk of a car, and is subjected to severe torture or killed. What kind of behavior is this? Is this religious behavior?"
Abu Usama recounted several stories of al-Qaeda killing or beating other rebels, including a sixty-five-year-old man in the 1920 Revolution Brigade, a Sunni insurgent group. Tired of the abuse, the rebel group began killing al-Qaeda members after mosque prayers in one city after another, which pushed al-Qaeda to further terrorize "people in order to deter them." Rumors circulated that the Americans and the Shi'a were working with al-Qaeda. Surely no Sunni fighters would do this to other Sunnis. Abu Usama counseled Bin Laden to dissociate himself from the group.
The establishment of the Islamic State did nothing to silence the critics. Its nominal head, Abu Umar al-Baghdadi, released an audio statement on March 13, 2007, to refute the "lies" circulating about the group's intolerance and brutality. He also laid down nineteen tenets the Islamic State's subjects were to abide by. All Shi'a and those who assisted the American occupiers were infidels. Christians and Jews living in the Islamic State had to submit to its authority and pay a protection tax or be killed. Any form of secularism, in which Abu Umar included mainstream political Islam and Arab socialism, would not be tolerated. Shari'a law was to be the law of the land and "anything encouraging deviance" was forbidden, such as satellite television and the uncovered faces of women. In the eyes of Iraq's beleaguered Sunnis, Abu Umar's charm offensive was more offensive than charming.
Abu Umar's directives were of a piece with the Islamic State's political theory. In a January 2007 document, the State argued that Islam requires a government to perform the following functions: impose the fixed punishments specified in Islamic scripture, the so-called hudud; resolve conflicts and quarrels; provide security; and prosecute criminals and sinners. The government should also tend to the economic welfare of its subjects. In its own words, the government should "oversee the provision of foodstuffs and relief materials and organize the selling of oil and gas and the other necessities of life to ease the suffering of the people." Because the Islamic State attempted to fulfill the duties of a state, it believed it should be treated as one.
The Islamic State never really controlled any cities so it couldn't do much to improve local economies. But it could dress up its terrorism in religious garb to subdue the locals and lay claim to statehood. The pillar of this strategy was imposing the hudud: beheading or crucifying bandits who kill people while robbing them; death by stoning for adultery; cutting off a hand or foot for theft; and flogging for fornication, drinking alcohol, and falsely accusing someone of fornication or adultery. (Some also say Islamic scripture prescribes death or lesser penalties for apostasy and rebellion.) There had to be two eyewitnesses for all of the crimes except adultery, which required four. Because the evidentiary standards were so high, especially for adultery, it was hard to get a conviction in premodern times. But because the penalties have to be enforced by a state, ultraconservative Muslims see their implementation as a touchstone for an authentic Islamic government. Imposing the hudud, therefore, could burnish one's ultraconservative credentials and bolster one's claims to be a state. The penalties are also frightening, and some of them, such as the punishment for apostasy, can be used against enemies, which makes them handy for subjugating people.
Unfortunately for the Islamic State, it was too zealous in applying the hudud and therefore scared too many people. As one Sunni explained, "I saw an Al Qaeda man behead an 8-year-old girl with my own eyes. . . . We want American support because we fought the most vicious organization in the world here . . . I would rather work with the Americans than the Iraqi Army. The Americans are not sectarian people." Sunni tribal leaders were furious at the abuse of their fellow tribesmen and annoyed that the Islamic State had horned in on their illicit smuggling activities. Local "Awakening" councils of Sunni tribal militias, so-called because they awoke to the threat of al-Qaeda and then the Islamic State, formed to expel Islamic State fighters from their towns. Their cause was helped immeasurably by the Islamic State falling out with the other Sunni insurgent groups, which were now far less inclined to close ranks with it.
Disgruntled
In November 2007, al-Qaeda's leaders in Afghanistan and Pakistan received word that a certain Abu Sulayman al-Utaybi had arrived and was on his way to see them. Abu Sulayman, a Saudi in his late twenties, had served as the chief judge of the Islamic State in its first year. Although he had publicly defended the group from its detractors, Abu Sulayman was privately worried about its missteps and declining popularity. When he raised his concerns with Abu Ayyub al-Masri, the shadow leader of the Islamic State, Masri brushed him off. Incensed, Abu Sulayman tried to send a letter to al-Qaeda badmouthing his boss. When Masri got wind of the letter Abu Sulayman intended to send, he fired him. The former chief judge packed his bags and set out for Afghanistan and Pakistan to hand deliver his complaints to al-Qaeda's leaders.
Although al-Qaeda's leaders knew Abu Sulayman had been fired, they were surprised when he showed up on their doorstep. Before meeting with him, they wrote Masri. "Why did [Abu Sulayman] leave you and come here?" they asked. "Is it perhaps something bad or some problems? What is your recommendation on the brother? . . . What was the reason that you dismissed him from his work?"
When al-Qaeda's leaders got no answer from Masri, which a messenger blamed on "encrypted files that did not open," they went ahead and met with Abu Sulayman.
Abu Sulayman's allegations against Masri—some of which were already presented in this chapter—were damning. Al-Qaeda in Iraq had declared the Islamic State under false pretenses, he charged. People think many militias and tribes pledged allegiance to the Islamic State; in fact, he claimed, only the heads of a few militias, some of which did not even exist, pledged their allegiance. "Among them were those who had never carried weapons in their entire lives." As for the tribes, they never pledged their support for establishing the Islamic State.
As you'll recall, al-Qaeda had repeatedly warned its Iraqi franchise not to declare a state until it had the support of the Sunni masses and their leaders. Abu Sulayman was alleging they had neither, despite the Islamic State's claims to the contrary. The Islamic State had not just defied its bosses; it had risked undermining the project of restoring the caliphate by turning the enterprise into a joke.
Beyond accusing Masri of declaring the Islamic State under false pretenses, Abu Sulayman claimed his former boss had made poor strategic decisions because he believed the apocalypse was imminent. As Abu Sulayman explained to al-Qaeda's leaders, Masri was unduly influenced by prophecies "about the tribulations [preceding the Day of Judgment], especially regarding the Mahdi."
For Abu Sulayman, Masri's greatest sin was his tolerance of corruption in the Islamic State. Several of the State's deputies were depraved and irreligious, Abu Sulayman alleged. They "corrupt the organization and the State from the inside, intellectually, methodologically and ethically," and they hurt the Islamic State's "public relations." Masri would not stand up to them, Abu Sulayman complained, because he was "weak." Masri himself set a poor example, reissuing old propaganda footage "as though it were new operations."
Not once did Abu Sulayman blame the figurehead of the Islamic State, Abu Umar al-Baghdadi, for failing to lead; he understood that Masri was truly in charge. By Abu Sulayman's estimation, the Islamic State was "approaching an abyss" and would not last much longer if al-Qaeda did not get Masri under control. The Islamic State's leaders were corrupt, apocalypse addled, or just incompetent. They had lied to al-Qaeda about the public's support for their cause, and they actually believed their terror group was the caliphate reborn.
After reading Abu Sulayman's letter and a summary of his complaints, Ayman al-Zawahiri cautioned the other al-Qaeda leaders to "not rush to accept everything [Abu Sulayman] says until we verify it and ask [Masri] and his brothers and see their opinion and response. . . . Maybe [Abu Sulayman] is being unfair or is carrying a grudge or something else."
In a letter to Masri, al-Qaeda's leaders demanded he address Abu Sulayman's allegations point by point. Apocalyptic thinking is "very dangerous," they warned, because it "corrupts policy and leadership" and makes for hasty decisions. Corrupt leaders in the Islamic State should be expelled; otherwise their bad behavior would destroy the organization and "deface al-Qaeda and its method." Ever mindful of Muslim public opinion, al-Qaeda worried its rogue affiliate would irreparably tarnish its brand.
Al-Qaeda's leaders ordered Masri to put them in direct contact with the Islamic State's nominal head, Abu Umar, and with the other "trusted brothers" in the organization. Al-Qaeda's leaders worried they did not have enough points of contact in the Islamic State. They were also concerned that if Masri and Abu Umar died, the line of succession would be unclear. The subtext was that Masri had gathered too much power to himself.
Al-Qaeda's leaders received contact information for Abu Umar al-Baghdadi but no answers to their questions. Frustrated, Zawahiri bypassed Masri and wrote directly to Abu Umar on March 6, 2008, to get answers to Abu Sulayman's accusations. "We want to have a full response from you about them." The answer finally came not from Abu Umar but from Masri, who responded to al-Qaeda's letter and Abu Sulayman's charges by playing for time—or "slow rolling," in American bureaucratese. "A letter reached us containing the accusations of Abu Sulayman and many requests for explanation. Before answering those requests, I want you to tell me: has [al-Qaeda's leadership] read this letter before it was sent? If not, I want you to disclose the text to them because it contains many matters and it is important that we know they agree with all of it."
Exasperated, al-Qaeda's leaders responded on March 10. "As I have explained to you," one of them wrote, "[they] have all seen Abu Sulayman's words and they have asked us to write you in order to inquire about these words, claims, and accusations." When Masri once again failed to respond to their satisfaction, al-Qaeda's leaders summoned him to the woodshed in "Khorasan" (Afghanistan or Pakistan) to answer the charges directly. Abu Sulayman, who had started the row, was killed by U.S. forces in Afghanistan before his former boss could confront him.
What Went Wrong?
We don't know if Masri ever met with al-Qaeda's leaders, but it didn't matter. He was never able to turn the Islamic State around. From a high of 2,500 civilians killed per month by the Islamic State in early 2007, the rate had declined to about 500 a month by the end of 2008. The Sunni tribes, backed by the U.S. surge and Iraqi troops, had constrained the State's movements, and American special operations forces had demolished its leadership. By 2009, jihadists were derisively calling the Islamic State a "paper state." Even Masri's Yemeni wife was giving him a hard time. "Where is the Islamic State of Iraq that you're talking about?" she reproached him in their mud hut south of Tikrit. "We're living in the desert!"
Pessimism also filled the jihadist discussion boards online. "The Painful Truth: Al-Qaeda Is Losing the War in Iraq," wrote one commenter. Another lamented, "The people of Iraq completely betrayed the mujahids and allied with everyone who had turned away from religion, except those whom my Lord had mercy on. . . . The situation of the mujahids has become extremely difficult." Jihadists couldn't understand why a group that had God's support was failing: "The Islamic State of Iraq is still in the right," forum members wondered. "So why are things becoming so difficult for it [and] its enemies joining against it from all sides?"
The jihadist discussion boards were usually friendly territory for the Islamic State and its predecessor al-Qaeda in Iraq, where they had pioneered the distribution of propaganda. They opted for snuff films rather than al-Qaeda's usual pedantry and uploaded multiple links to videos in numerous sizes and formats. Extreme violence attracted eyeballs to the propaganda, and redundant links and decentralized distribution kept it online. The Islamic State would use many of the same techniques a few years later to recruit on Twitter. But at the moment, the private discussion boards had turned against the State and become a water cooler around which discontented jihadists gathered to grouse about the group's failures.
The Islamic State's own members were complaining too. In an analysis captured by U.S. forces, a member of the Islamic State cataloged an array of reasons for the State's failure: The organization did not understand Iraqis well; foreign fighters attracted by Islamic State propaganda did not get along with local members of the group; the Islamic State's commanders did not coordinate well with one another and their large number weakened the chain of command; and, finally, good old-fashioned bureaucratic stovepiping prevented commanders from knowing what one another were doing.
From abroad, al-Qaeda's inner circle watched in horror as the Islamic State dragged the al-Qaeda brand through the dust. Al-Qaeda owned the Islamic State's every excess and failure because outsiders still called the State by its old name: al-Qaeda in Iraq. And al-Qaeda had yet to denounce the group.
In Pakistan, al-Qaeda's American propagandist, Adam Gadahn, seethed. A former death metal fan from California, Gadahn had converted to Islam in his late teens, gravitating toward an ultraconservative version of the religion. Two years after his conversion, Gadahn moved to Pakistan, where he married and soon joined al-Qaeda. No softy, Gadahn had celebrated al-Qaeda's atrocities for years in videos distributed online. But the Islamic State was too extreme even for him. Privately commenting on the reign of Abu Umar al-Baghdadi, Gadahn thought it was absurd that the Islamic State had believed "the authenticity of their fictitious State" when it could not even "defend itself" or others. The Islamic State had demanded a protection tax from Christians, a right reserved only for a real Islamic state, and it treated Muslims who disagreed with it like infidels, "targeting [their] mosques with explosives." Al-Qaeda had not ordered these actions, but its reputation would "be damaged more and more as a result of the acts and statements of this group." Gadahn advised al-Qaeda's leader to "cut its organizational ties" to the Islamic State for such behavior.
Gadahn's sentiments were echoed a world away in Somalia, where the secret al-Qaeda affiliate al-Shabab controlled much of the country. In its midst was Fadil Harun, one of the earliest members of al-Qaeda. Harun had grown up in the Comoros, a former French colony off the coast of Mozambique. Like Gadahn a decade later, Harun embraced an ultraconservative version of Islam in his teens and traveled to Pakistan, where he joined al-Qaeda and learned to make bombs. Harun became one of al-Qaeda's most skilled operatives, helping to plan the bombing of the U.S. embassy in Nairobi and later running al-Qaeda's cell in east Africa.
Harun had known Masri from his time in Afghanistan during the Taliban era. "He was very close to us, as we lived with him in Afghanistan during the second period when the [Taliban's] Islamic Emirate was established." Harun admired Masri's piety but worried when he heard his old friend had taken over leadership of al-Qaeda in Iraq and then the Islamic State. Masri was part of al-Qaeda's Egyptian contingent, which was known to be too strict and too indiscriminate in its killing of Muslim civilians. "We know very well that the group is strict in some controversial issues where there is room for disagreement," Harun recounted in his memoir. "They only stick to their opinion," Harun lamented when his worries were later justified by the Islamic State's excesses. The State should have learned from the "mistakes of others" and not "ignore the people of a country."
Bin Laden too privately bemoaned the Islamic State's brutality, especially its rough treatment of the Sunni tribes. In a 2010 letter, he reflected on early American missteps with the tribes that had given the jihadists an opportunity to win them over: "The enemy entered Iraq without any knowledge of the area or the Iraqi people, who have a strong tribal background; therefore, the Iraqis supported the mujahidin." But when the Islamic State attacked tribal youth in Anbar for signing up with the Awakening militias, their tribes retaliated. The youth were "going to join the security forces for financial reasons," complained Bin Laden, and presented no "imminent threat" to the jihadists. The incident taught the jihadists a hard lesson in not initiating a blood feud with tribal members.
Despite his private misgivings, Bin Laden doggedly refused to renounce the Islamic State. Watching the train wreck from afar, al-Qaeda's leader could only offer the chagrined executive's passive excuse: "Mistakes were made."
Masri's and Abu Umar al-Baghdadi's disastrous reign was ended by American and Iraqi soldiers who killed the men in a joint raid on Masri's mud hut in April 2010. In the following three months, thirty-four more Islamic State leaders would be killed or captured, crippling the organization but also making room for new leadership. The Islamic State announced the appointment of its new commander of the faithful, Abu Bakr al-Baghdadi, in May although it would be two years before he issued a public statement. Like his predecessor, most jihadists had never heard of him.
In the three years since its birth in the autumn of 2006, the Islamic State had managed to humiliate its absentee lords in al-Qaeda and lose every bit of territory it claimed to rule. An authoritarian, arrogant style coupled with mismanagement, apocalyptic zeal, and unfocused brutality against an ever-widening circle of enemies was poorly matched against the strength and resolve of the State's opponents. Although the organization would come back to life again as the last American troops departed in 2011, for the moment its prospects were bleak.
But just as the flag of the Islamic State was trampled underfoot in Iraq, jihadist fanboys and al-Qaeda's own affiliates began to lift it up, keeping the dream of the caliphate alive during a bleak period for all of al-Qaeda's affiliates preceding the chaos of the Arab Spring, which would renew the fortunes of the global jihadist movement.
Three
Bannermen
Friends and family remembered nineteen-year-old Nayif al-Qahtani as "quiet" and "shy," the kind of guy who obeys his mom and doesn't get into trouble. "I never saw any change in his behavior or any inclination toward extremism," remembered Qahtani's brother. The Saudi teen was the last guy you'd expect to stand in front of the emir of al-Qaeda's branch in Yemen, about to pitch a bold idea.
You need an online magazine, Qahtani said, and I'm the man to do it.
The emir of al-Qaeda in Yemen, Nasir al-Wuhayshi, liked the idea. Every terrorist organization worthy of the name needs a magazine to needle its enemies and attract recruits. But Wuhayshi probably had misgivings about the young Saudi's ability to pull it off. Qahtani had no university education, having left Saudi Arabia at age seventeen to join the jihad in Yemen. He also had no skill at designing magazines. Still, Wuhayshi admired Qahtani's pluck, perhaps seeing a little of himself in the teen. When Wuhayshi was around the young man's age, he had left his native Yemen for Afghanistan, serving as Bin Laden's secretary for six years. Modest "with flashes of sarcastic humor," Wuhayshi had endeared himself to the al-Qaeda rank and file.
Wuhayshi blessed Qahtani's proposal in late 2007. Over the next few months, Qahtani worked feverishly to find and format the content. To give the magazine some added heft, the teen even conducted an "Interview with One of the Most Wanted People in [Saudi Arabia]"—which was actually an interview with himself. The magazine debuted in January 2008. Its title, The Echo of Battles, alluded to the final battles of the apocalypse.
Anonymous jihadists online applauded the magazine in their private discussion forums. Most of them lived in the Arab world, so they could appreciate the Arabic text. If anyone disliked the magazine, they either were silenced by the jihadists who administered the forums or kept quiet. Such uniformity was typical, achieved through censorship or self-censorship. Freewheeling jihadist debates online would come later with the advent of Twitter.
There was plenty to criticize in the magazine, especially its visual presentation. The layout was ugly, with pictures floating across a vacuum of white interspersed with blue and black text. Qahtani's inexperience as a graphic designer was showing.
Over the next few months, Qahtani either vastly improved as a graphic designer or he had some help. When the second issue of Echo was released in March 2008, the title dripped with fire and blood, and colorful borders corralled the text. On the masthead and silhouetted on every page was the new logo of al-Qaeda in Yemen: two scimitars crossed over outlines of Yemen and Saudi Arabia. Fluttering between the scimitars was the black flag of the Islamic State. When the leaders of al-Qaeda in Yemen released a video the following year announcing the group's merger with the moribund al-Qaeda in the Arabian Peninsula (AQAP), the Islamic State's flag hung in the background. At the very moment the Islamic State was stumbling in Iraq, another al-Qaeda affiliate had taken up its standard.
Why had al-Qaeda in Yemen adopted the flag of another al-Qaeda affiliate? Why not create its own flag, or use the black flag Bin Laden displayed in his press conferences? After all, al-Qaeda in Yemen answered to Bin Laden, just as the Islamic State did. Perhaps the leaders of al-Qaeda in Yemen just liked the design of the Islamic State's flag and the notoriety attached to it. But more likely they admired what the flag represented: the imminent return of the caliphate. As we will see, soon after al-Qaeda in Yemen adopted the Islamic State's flag, it attempted to establish its own Islamic government—as would several other al-Qaeda affiliates that began displaying the flag around the same time in other countries. By flying the Islamic State's symbol, the al-Qaeda affiliates were embracing what it represented. They were not joining the Islamic State in Iraq but rather endorsing its project. It was not a project Bin Laden liked.
Twelve Thousand
Two and a half years after Qahtani debuted his first issue of The Echo of Battles, AQAP launched its second online magazine, Inspire. Another young man, twenty-four-year-old Samir Khan, was at the helm.
Like Qahtani, Khan was born in Saudi Arabia. But as a child he had moved to New York and later to North Carolina with his Pakistani parents. Despite being raised in the West, Khan ended up in the same place as Qahtani. Entranced by the 9/11 attacks and infuriated by the American invasion of Iraq, Khan started a website devoted to jihadist propaganda, which he edited from his parents' home in North Carolina. Ignoring the pleas of his parents and local Muslim leaders to renounce jihadism, Khan left for Yemen in 2009 to join his hero, Anwar al-Awlaki.
Born in America, Awlaki had returned to his parents' homeland of Yemen during his teens and again in 2004. A skilled preacher, he had one foot in radical Islamist eddies and another in the Muslim mainstream. By the time Khan arrived, Awlaki had fully radicalized, joining AQAP and recruiting disciples online to carry out attacks in Europe and the United States. The propagandist and the blogger began to collaborate on a new way to reach Muslim youth in the West. AQAP's first English magazine, Inspire, was born.
Awlaki's article in the inaugural issue was directed to the "American people and Muslims in the West." For the Muslims in the West, his message was stark: "You either leave or you fight." Those who stay in "infidel" lands should follow the example of Awlaki's acolytes Nidal Hasan and Umar Farouk Abdulmutallab. Hasan was a Palestinian-American soldier in the U.S. military who killed thirteen people at Fort Hood, Texas. Abdulmutallab was a British-educated son of a wealthy Nigerian banker who attempted to blow up an airplane over Detroit.
As for Muslims who leave infidel lands, Awlaki invited them to come to Yemen. The Prophet had prophesied the appearance of an army of "twelve thousand" men who would "come out of Aden-Abyan" in the south to "give victory to Allah and His Messenger," Muhammad. Awlaki believed the fulfillment of the prophecy was fast approaching.
The prophecy was a favorite of Awlaki's comrades in AQAP. The group's Arabic magazine, Echo of Battles, frequently quoted it, especially after AQAP clashed with government forces in Abyan Province in the summer of 2010. In July, an AQAP leader announced, "An army of 12,000 fighters is being prepared in Aden and Abyan. By this army, we will establish an Islamic Caliphate."
Jihadist anticipation of a caliphate in Yemen was rising. AQAP's Echo magazine published an exchange on the subject between a pseudonymous Yemeni jihadist and the world's most influential jihadist scholar, Abu Muhammad al-Maqdisi. The jihadist asked if the fighting in southern Yemen was a fulfillment of the "twelve thousand" prophecy and whether it meant the caliphate would be established in Yemen.
Maqdisi was well equipped to answer. The Palestinian had spent most of his adult life studying Islamic scriptures and jihadist treatises. His knowledge had attracted jihadist protégés, including Abu Mus'ab al-Zarqawi, although the two had fallen out over Zarqawi's excesses in the Iraq war. If anyone knew what the "twelve thousand" prophecy portended for the future of the caliphate in Yemen, it would be Maqdisi.
On the question of the caliphate, Maqdisi responded cautiously:
Some of those who talk about this hadith say that it contains a subtle allusion that the caliphate will happen in Aden Abyan and Yemen will be the capital of the caliphate because the armies will usually start out from the seat of the caliphate. . . . That is not necessarily so; rather perhaps it means Yemen will help the caliphate or help its people and its Muslim army, or recruit the like of this army for repelling the enemy attacking some of the lands of Muslims or for driving away an occupier or apostate.
Not wanting to discourage his questioner, Maqdisi was more enthusiastic in his response to the question about whether AQAP was fulfilling the "twelve thousand" prophecy. Affirming the glorious destiny of the jihadists in Yemen, Maqdisi quoted a second prophecy in which Muhammad instructs faithful Muslims to go to Syria in the End Times. If they can't go to Syria, Muhammad advised, they should go to Yemen. AQAP was fulfilling the prophecy, Maqdisi argued, because the "crusaders" feared the growing strength of the fighters gathering in Yemen, "who fight beneath the banner of monotheism."
Maqdisi closed his essay by recommending that the jihadists in Yemen follow the advice of Abu Mus'ab al-Suri. A Syrian, Suri was the red-haired Carlos the Jackal of the global jihad, turning up in one conflict after another. He's the author of the large book on jihadist strategy and End-Time prophecy we met in chapter 1. Having witnessed many failed revolutions, Suri had developed a keen sense for what not to do.
In an October 1999 essay on how to wage a jihad in Yemen, Suri quoted the same prophecies mentioned by Maqdisi and explained why Yemen was the ideal place for launching a revolution in the Arabian Peninsula. Suri was convinced that the jihadists should be fighting to mobilize popular support against the crusaders and their local allies. It is the "key of jihad." Rather than attacking the government in Yemen, Suri advised the jihadists to view the country as a base in which to gather their forces and launch attacks on the rest of the Arabian Peninsula. If any Arab countries retaliated against the jihadists in Yemen, the Yemeni people would view those countries as lackeys of the crusaders. Suri ended his analysis by quoting the "twelve thousand" prophecy.
Bin Laden broadly agreed with Suri's strategic perspective and discouraged al-Qaeda's affiliate in Yemen, AQAP, from overthrowing the local government or establishing a state. The head of AQAP, Nasir al-Wuhayshi, had written Bin Laden sometime in 2010, "If you want Sana'a, today is the day." (Sana'a is the capital of Yemen.) Bin Laden replied, "We want to establish the Sharia of God in Sana'a only if we are able to preserve [the state we create]." Bin Laden reminded Wuhayshi that no Islamic government founded by jihadists will endure as long as America "continues to possess the ability to topple any state we establish." "We have to remember that the enemy toppled the Taliban and Saddam's regime."
This was a big change from Bin Laden's public position in 2007, when he had defended the Islamic State in Iraq against charges that it had been established prematurely. At that time, he had acknowledged that the United States could "make war on any state and bring down its government." But that was no excuse for not trying. The subsequent dismantlement of the Islamic State had made him more cautious.
Instead of establishing a state in Yemen, Bin Laden preferred to use it as a base for operations against the United States: "I believe that Yemen should remain peaceful and kept as a reserve army for the [Muslim] community. It is well known that one of the requirements for plunging into wars is to have a reserve army and to continue exhausting the enemy in the open fronts until the enemy becomes weak, which would enable us to establish the state of Islam. Therefore, the more we can escalate operations against America, the closer we get to uniting our efforts to establish the state of Islam, God willing." Bin Laden acknowledged that, from a jihadist perspective, Yemen's president is an apostate. But he is an ineffectual one, which gives jihadists room to maneuver. "It is not in our interest to rush in bringing down the regime. In spite of this regime's mismanagement, it is less dangerous to us than the one America wants to exchange it with. . . . The Salafists and the jihadist Salafists were able to take advantage of his regime and target America from Yemen, as some of the mujahidin went to Somalia or traveled to us, which allowed us to assign our brothers to conduct international operations."
If the Yemeni government attacks the jihadists, then they should defend themselves, Bin Laden counseled. But they should not go on the offensive against the government, which would only turn the population against their enterprise.
Note the importance of popular support in Bin Laden's strategic thinking and his focus on driving the Americans from the Middle East. Toppling local Muslim regimes and establishing states would have to wait. Those goals would only alienate the locals because achieving them would require attacking fellow Muslims. And even if the jihadists were successful, the United States still stood in the way.
Bin Laden was angered by news that AQAP members had attacked Yemeni government forces in the summer of 2010, which contradicted his guidance. He couldn't believe AQAP's leaders had disobeyed him, so he blamed the group's youth for carrying out the operations. Perhaps they weren't sufficiently apprised of al-Qaeda's America-first policy, Bin Laden wrote. Such operations were excusable if they were "for the mujahidin's self-defense only."
Just as AQAP should avoid attacks on the local government, Bin Laden advised them to "avoid killing anyone from the tribes." To his mind, the tribes were pivotal to the success of the jihadist state-building enterprise, which would be doomed without their backing. "We must gain the support of the tribes who enjoy strength and influence before building our Muslim state," wrote Bin Laden. Muhammad's predecessors had ignored this maxim at their peril, Bin Laden observed. The Prophet had succeeded because he had been able to co-opt the tribes in Medina in whole or in part.
Bin Laden was right. When Muhammad's own tribe in Mecca turned against him, the Prophet allied with tribal factions to the north in the town of Medina. Using a mix of religious persuasion, the promise of spoils, marriages, intimidation, and violence, Muhammad succeeded in bringing many of the tribes of the Arabian Peninsula over to his side by the end of his life.
Bin Laden's counsel to avoid alienating the tribes was part of his larger strategy to win popular Muslim support. Many of Yemen's citizens are members of tribes, which enjoy more autonomy than in most Arab countries. Bin Laden was also wary of triggering a blood feud, a form of retaliatory justice that tribes sometimes initiate when one of their members is killed.
In a draft letter to AQAP's leader, Bin Laden mused at length on the insurmountable challenges facing any Islamic state established in the Arab world. The state would have to meet the "demands and needs of its people." Because Arab states have made people dependent on them for everything, any Islamic state that does not deliver public goods efficiently would quickly lose support. The inevitable sanctions and aerial bombardment would further hasten the state's demise.
A revolutionary movement today needs more than just military might to topple a government or control a country. While putting aside the external enemy, a movement needs to have the resources in place to meet the needs and demands of the society, as it makes its way to controlling a city or a country. A movement cannot expect, however, a society to live without for a long time, even if that society happens to be a great supporter of that movement. People often change when they see persistence in a shortage of food and medicine, and the last thing they want to see is having their children die for lack of food or medicine.
Failure would be costly, warned Bin Laden. "The impact of losing a state can be devastating, especially if that state is at its infancy. . . . The public often has all sorts of interpretations for the word failure. Nonetheless, the public does not like losers. The public is only interested in the results and it often ignores the details and conditions which led to one's success or failure. If the public stigmatizes a group, the group will likely fail to rally that public for support, be it to build or defend a state."
Behind the scenes, at least one member of al-Qaeda's senior leadership was unhappy with the guidance Bin Laden gave to AQAP. Atiyya Abd al-Rahman, Bin Laden's chief of staff who had chastised Zarqawi and the Islamic State in Iraq for their excesses, worried his boss was too obsessed with attacking the Americans. Bin Laden was ignoring opportunities to establish Islamic governments, which was out of step with what the jihadist rank and file wanted. In their eyes, the whole point of jihad was to establish Islamic governments. "The young men want to go to the 'front' and they want 'operations,'" Atiyya reminded Bin Laden. "Is it proper to say: stop the escalation, we do not want war in Yemen!? I do not support this choice. None of the brothers here, who gave you their opinions, supports it. We definitely see it as a mistake." Giving the Yemeni government a pass and allowing other Islamists to take over, which is what Bin Laden's strategy amounted to, should not be an option.
Despite his misgivings, Atiyya deferred to Bin Laden's wishes. AQAP did not. In early 2011, popular protests broke out in the capital of Yemen as part of the Arab Spring uprisings. Taking advantage of the political crisis, AQAP seized parts of Abyan Province in the South, including one of its major towns, Ja'ar. "As soon as we took control of the areas," confided AQAP's leader, Nasir al-Wuhayshi, "we were advised by the General Command [of al-Qaeda] here not to declare the establishment of an Islamic emirate or state for a number of reasons." The reasons al-Qaeda's General Command gave Wuhayshi echoed Bin Laden's earlier guidance: "We wouldn't be able to treat people on the basis of a state since we would not be able to provide for all their needs, mainly because our state is a state of the downtrodden. Moreover, we fear failure in the event that the world conspires against us. If this were to happen, people may start to despair and believe that jihad is fruitless."
AQAP's leaders "deemed that [al-Qaeda's] advice was wise and decided not to declare a state." But they did declare several "emirates" in the areas they controlled, breaking with the spirit of the advice in the letters from Bin Laden and al-Qaeda's senior leaders. A senior jihadist ideologue ominously warned AQAP: "You have seized territories even though you know from prior experience that you will not be able to protect them or your presence in them." Despite Bin Laden's warning and the Islamic State's failure, AQAP couldn't resist the opportunity to take power.
Although AQAP unilaterally seized territory, it was not inured to the need to burnish its public image. Mindful of the mistakes of the Islamic State in Iraq, AQAP vowed to win "hearts and minds" and garner the support of the tribes in the South, whom AQAP's leaders saw as a greater threat to their enterprise than the United States.
As part of its new public relations strategy, AQAP created a front group in 2011 called Supporters of the Shari'a (Ansar al-Shari'a). We don't know exactly why the group chose that name, but we can guess. Western media was in the habit of shortening al-Qaeda's full name, Qa'idat al-Jihad (Base of Jihad), which annoyed Bin Laden because the word "al-Qaeda," or "base," has nothing to do with Islam. Bin Laden privately acknowledged the al-Qaeda brand was toxic, needlessly bringing down heat on groups that adopted it. He had actually considered changing his group's name to something else, such as "Monotheism and Jihad" or "Restoration of the Caliphate." Doing that, he believed, would force the media and the U.S. government to acknowledge the Islamic nature of the group and reinforce the idea that the West was at war with Islam. The same considerations were likely behind AQAP's choice of Ansar al-Shari'a as the name for its new front group.
The term "Ansar," or "supporters," harkens back to the Ansar of Medina, the people who helped Muhammad and his followers in their time of need when they emigrated from Mecca to escape persecution for their beliefs. The Shari'a is God's guidance for humanity, which includes prohibitions and ethical teachings. The whole point of setting up an Islamic state is to enforce the Shari'a.
Islam was born in a tribal society that lacked formal laws or a state to enforce them. Each tribe abided by its own customs and cult. The founder of Islam, Muhammad, was able to create a state because he appealed to a greater religious identity to unite the fractious tribes. Doing that required him to promulgate laws that could transcend tribal identity and regulate a state. As a consequence, Islam's scriptures cover everything from inheritance to warfare. (Judaism emerged in a similar environment, which accounts for some of the parallels between the two religions.) Although some of Muhammad's legislation is harsh by modern standards, jihadists view it as a ready-made kit for enforcing rule in tribal societies. And Muhammad's life provides a model for how to do it successfully.
AQAP's decision to set up a front group reflected its shift away from terrorism and toward state building. In an April 2011 interview, one of AQAP's senior religious leaders, Adil al-Abab, explained the group's change in focus. At times, Abab sounded like any politician tasked with tending to the mundane aspects of governing: "The largest problem that we face here is the lack of public services such as sewage and water, and we are trying to find solutions." AQAP could govern even more territory, he asserted, were it not for the lack of "administrative staff and financial resources that would make us able to provide services to the people." Abab made a special plea to wealthy Muslims to donate to their cause: "We have here in Ja'ar full plans for projects we want to achieve for the people such as water, sewage and cleaning, and we want to make contracts with investors so as to arrange these affairs. These projects are ready, and the regime has failed in carrying out these services. Now we have toppled the regime here and we are waiting for your investments." Still, Abab reminded his listeners that bread and butter alone would not feed the Muslim soul; they must also submit to the hudud, the fixed punishments mandated by Islamic scripture. But this was also part of AQAP's hearts-and-minds strategy, according to Abab:
Three days ago, here in Ja'ar, the Ansar al-Sharia caught a thief stealing while drunk, and I met him the next day. I asked him whether he was drunk, and he said "yes." I said did you know that drinking alcohol is prohibited, and he also said "yes." I said did you know that your punishment would be whipping, he said: "Yes but cleanse me please." May God reward him. We whipped him forty lashes and that was the first implementation of hudud here. So we catch some people and chase others, we apply hudud as much as we can whenever we have the ability.
In Abab's telling, the drunk was not alone in his desire to be punished in accordance with the Qur'an. "People are very happy with this, and they ask us: 'Where have you been all this time?' They never thought of us as such moral people with these ethics. They always thought that the al-Qaeda organization was evil because of the distortions they have been hearing about us. Whenever we meet people today, they rejoice at our existence and we talk to them about the Sharia and that if it is applied, God will send rain with welfare to the earth."
AQAP developed an online media campaign to disseminate interviews with the happy inhabitants of the emirates they established. "How is it working for you now?" an AQAP interviewer asked a Ja'ar resident after the group turned the power back on. "Wonderfully!" the man replied. A resident in the town of Seehan enthused, "Even the children, look at the children, they are happy! We used to wish for this, our grandfathers used to wish for this."
Of course, there were fewer smiles off camera. Men could no longer wear soccer shorts. Music and dancing were banned. Alleged spies were shot and then crucified, women accused of witchcraft were beheaded, and the hands of thieves were amputated. "They accused me of stealing," said one. "They detained me in a room for five days. They kept beating me hard . . . and tortured me with electric shocks. They would pour some water on my chest and then place a wire on it and I would feel as if I had been thrown hard. After five days, they gave me an injection, and I slept. When I woke up, my hand was not there."
Predictably, AQAP's brutal rule turned the population against it within a year. As one Ja'ar resident put it, "In the beginning when they came here, they were simple people and weak. We were one of those people who were harmed by the government, because the government stole from us, and we were without work. We aligned with them in the beginning. We found out, thank God, before we did anything with them, we found out that they are liars . . . they love blood, and they are terrorists."
AQAP also managed to alienate the local tribes despite wooing them with food and water. In Ja'ar, the tribes were roused in 2011 by the cousin of a man AQAP executed for allegedly spying. By the summer of 2012, southern tribal militias dubbed Popular Committees worked with the Yemeni government and the United States to overrun AQAP positions, ending the group's experiment in state building.
After AQAP lost its territories, Adil al-Abab, the spokesperson for its Ansar al-Shari'a front group, ruminated on its failed effort to govern. Abab acknowledged that "some will wonder what were the gains achieved by the Ansar al-Sharia?" He spun AQAP's failure as a good first attempt, dispelling the criticism that jihadists could not govern. They implemented Islamic law, supplied swift justice and competent security, and provided public goods. Abab agreed that AQAP had not governed perfectly, but how could it when the Yemeni government had bombarded the cities under AQAP's control? Abab did not explicitly blame the tribes' betrayal for AQAP's losses, but he came close: "Know, O shaykhs of the tribes, that there is a project run by national security that aims in the first place to corrupt your sons and employ them as lackeys and agents."
In private, AQAP's leaders were less sanguine. Yes, the jihadists had acquired some valuable experience in governing, and their subjects saw they cared about more than fighting. "But after that, the West and the East gathered against us, and fought us with one hand," lamented AQAP's leader Wuhayshi. "After four months of fighting, we were forced to withdraw." "The whole world was against us."
"The most effective weapon of the enemy," Wuhayshi admitted, were the Popular Committees formed of disgruntled tribesmen and some Islamist opponents of AQAP. "Those are the groups which were gathered by the enemy to become a de facto army of the area." In Iraq, Sunni tribal militias backed by the United States and the central government had ended the Islamic State's attempt to rule in 2007. In Yemen, the same strategy had worked against another al-Qaeda affiliate.
In the end, the reasons for the collapse of the AQAP state were similar to the reasons for the Islamic State's failure in Iraq: fickle tribal allies, resentful subjects, and powerful foreign enemies—all reasons Bin Laden had cited for delaying the establishment of an Islamic state in Yemen.
The prophesy of "twelve thousand" wouldn't be fulfilled any time soon. No matter—the young founders of AQAP's online magazines who cherished the prophesy did not live long enough to be disappointed. Nayif al-Qahtani, Samir Khan, and Anwar al-Awlaki all died in American air strikes before the nascent AQAP state collapsed in 2012.
"Try to Win Them Over through the Conveniences of Life"
Just before AQAP lost its territory in 2012, its leader offered advice on governing to al-Qaeda's branch in North Africa, called al-Qaeda in the Islamic Maghreb (AQIM). Flush with weapons from the storehouses of Libya's fallen president and working with other jihadist groups, AQIM had just invaded and occupied northern Mali. "The places under your control are a model for an Islamic state," Wuhayshi wrote in a private letter to AQIM's leader, Abu Mus'ab Abd al-Wadud. "The world is waiting to see what you'll do next and how you'll manage the affairs of your state." But "your enemies want to see you fail," Wuhayshi cautioned. "They are throwing obstacles in your path to prove to people that the mujahideen are people that are only good for fighting and war, and have nothing to do with running countries and the affairs of society."
Wuhayshi recommended a hearts-and-minds strategy: Win people over by providing public goods and by leniently applying Islamic law. The people are "hard-pressed by their needs and by the hard toil of making a living." So the jihadists should "try to win them over through the conveniences of life and by taking care of their daily needs like food, electricity and water." "What we've observed during our short experience," Wuhayshi related, is that "providing these necessities will have a great effect on people, and will make them sympathize with us and feel that their fate is tied to ours." AQAP had learned it was better to be loved and feared than feared alone.
Wuhayshi advised AQIM to apply Islamic law very gradually because the people in the region were not used to its strictures. "You can't beat people for drinking alcohol when they don't even know the basics of how to pray." Wuhayshi recounted his group's own experience in Yemen: "Only after monotheism took hold of people's hearts did we begin enforcing these punishments. Shari'a rule doesn't mean enforcing punishments, as some people believe, or have been made to believe. We have to correct this misconception for the sake of the people. Try to avoid enforcing Islamic punishments as much as possible, unless you are forced to do so."
Wuhayshi was less upbeat in a letter written a few months later, after the United States and its allies helped local tribesman drive AQAP from its territories. "We advise you not to be dragged into a prolonged war. Hold on to your previous bases in the mountains, forests, and deserts and prepare other refuges for the worst-case scenario. This is what we came to realize after our withdrawal."
Whether or not AQIM's leader Abd al-Wadud ever received the advice, he shared Wuhayshi's outlook. For over twenty years, Abd al-Wadud had fought in jihadist insurgencies against Algeria and other North African governments. Although an admirer of the ruthless Zarqawi and commander of his own brutal terrorist organization, Abd al-Wadud had come to see the wisdom of the light touch. In 2012, he issued a "set of directions and recommendations" to "the brother emirs in the Sahara by which they should operate."
Abd al-Wadud cautioned against monopolizing "the political and military stage." "We should not be at the forefront" of the movement to liberate northern Mali from the government's control. Instead, the jihadists should build a broad coalition and abstain from excommunicating others. "The aim of building these bridges is to make it so that our mujahideen are no longer isolated in society, and to integrate with the different factions, including the big tribes and the main rebel movement and tribal chiefs." Abd al-Wadud reminded the emirs that "the people of jihad serve as the directing and leading vanguard that works to implement this project amid our Islamic nation and among the various sectors of its people." The jihadists simply do not have the "military, financial, and structural capability" to govern the state alone and stand up to the inevitable foreign pressure.
AQIM and its cohorts had mixed success in rallying the support of the diverse tribes of the Sahel, the belt of semi-arid land cutting across Mali that separates North Africa from sub-Saharan Africa. In the years leading up to the fall of Timbuktu in 2012, jihadists were at times able to capitalize on racial and economic tensions to attract tribal recruits. However, internal tribal divisions and preexisting prejudices frustrated their efforts. Racial and ethnic tensions among tribes proved a major problem, as the jihadists treated darker-skinned recruits worse than their light-skinned counterparts. Black fighters were reportedly assigned insultingly menial tasks and were even abandoned in Timbuktu after jihadist forces left town.
The need for local support weighed heavily on Abd al-Wadud's mind. He chastised the jihadists for "the extreme speed with which you applied Shari'a, not taking into consideration the gradual evolution that should be applied in an environment that is ignorant of religion, and a people which hasn't applied Shari'a in centuries." Failure to apply the Shari'a gradually would inevitably "engender hatred toward the mujahideen." Citing examples of what not to do, Abd al-Wadud rebuked the jihadists for severely applying religious punishments and for destroying the shrines of saints venerated by locals.
In the months preceding Abd al-Wadud's letter, jihadists in Timbuktu had harshly punished male and female "adulterers," lashing a young couple for having a child out of wedlock and stoning another young couple to death for alleged premarital sex. They flogged locals who dared to indulge in less egregious but still "un-Islamic" activities such as smoking or walking around without a veil. The jihadists even banned music, integral to the beloved rituals associated with the animist-infused version of Islam commonly practiced in Mali.
Locals were especially incensed when the jihadists razed the shrines of local Muslim saints. "All Muslims know the tomb is a holy place," a local religious scholar complained. "It's not something you attack and destroy. It's anti-Islamic. People in the community are angry."
Visiting the shrines of saints is a major no-no in the ultraconservative Islam practiced by Salafi jihadists, so the shrines become targets when jihadists take over. Destroying them demonstrates the jihadists' religious bona fides to other ultraconservatives and lets the locals know who's boss. That Abd al-Wadud would criticize his fellow jihadists for destroying shrines shows how sensitive he was to popular sentiment. In the same vein, he criticized commanders for going beyond the requirements of Islamic law by preventing "women from going out" and "children from playing." "Your officials need to control themselves."
Despite their fear, locals were not always shy about expressing their discontent: In Gao, a group protested after a young man accused of theft was sentenced to amputation, and a young woman defiantly removed her veil in public, which she had previously worn by choice. Concerned citizens in Goundam tried to stop the public flogging of Sufis, followers of a mystical practice in Islam, only to be driven away by gunfire. When a woman was beaten for not veiling, she dropped her baby, who nearly died. In response, crowds gathered at the local mosques to prevent the jihadists from saying their prayers. When firing in the air didn't disperse them, the jihadists took shelter in the house of a local official.
Abd al-Wadud reminded his emirs to "take into account in our overall picture" the ever-present threat of hostile foreign powers. Invasion was "probable, perhaps certain;" if not, hostile foreign powers would still put in place a complete "economic, political, and military blockade." The invasion or blockade would "either force us to retreat to our rear bases," "provoke the people against us," or "enflame conflict between us and the other armed political movements in the region." Even if the armed groups initially closed ranks, the foreign powers would use "a carrot-and-stick policy with them to incite them against us." To avoid that outcome, the jihadists should allow their local allies to publicly take the lead to give the appearance of a broad-based nationalist movement rather than a global jihadist enterprise that would provoke foreign intervention. "Better for you to be silent and pretend to be a 'domestic' movement that has its own causes and concerns. There is no call for you to show that we have an expansionary, jihadi, al-Qaeda or any other sort of project."
Abd al-Wadud's "directions and recommendations" came too late. AQIM and its affiliates were never able to truly win over the locals, despite a shared a resentment of Mali's central government. When the French sent in fighter jets and special forces in early 2013, the jihadist proto-state in northern Mali collapsed. The jihadists either fled for neighboring Algeria and Libya or turned to guerrilla warfare. Another al-Qaeda experiment in state building had ended in disaster.
Failed State
In a now-familiar pattern, al-Qaeda's franchise in Somalia, the Shabab, also tried and failed to govern the territory it conquered. In 2008, the Shabab controlled a large expanse of land in central and southern Somalia, prompting one of its most powerful leaders, Mukhtar Robow (aka Abu Mansur), to predict the imminent establishment of an "emirate." Jihadists online also speculated that the emirate's establishment would come any day. In the fall of 2009, the Shabab's leader, Ahmed Abdi Godane, publicly called Bin Laden the group's emir, and Shabab fighters began carrying the Islamic State's flag. The Shabab's leaders secretly asked Bin Laden whether he would recognize them publicly as an al-Qaeda affiliate. They also asked if they could declare themselves an emirate.
Bin Laden mulled the two questions for months, despite gentle reminders from his chief of staff that he needed to respond. Bin Laden knew that, to garner popular support, the Shabab had provided basic services to citizens in rural areas outside the reach of Somalia's weak government. The southern countryside where the group operated its training camps and aggressively recruited was a particular area of focus. Justice and humanitarian relief were the orders of the day, which were in line with Bin Laden's thinking. The strategy initially seemed to work, winning over the public and local tribal leaders, which enabled the Shabab to function as the de facto government in southern rural areas.
But like the other al-Qaeda affiliates, the Shabab squandered its support by meting out harsh punishments for violating its austere brand of Islamic law. Muslims who repeatedly failed to pray the required five times per day were imprisoned with no food until they prayed correctly. The Shabab beat women for not fully covering themselves and stoned them to death for alleged extramarital sex, including a thirteen-year-old girl who had been raped by Shabab men. The Shabab also publicly amputated the feet and hands of people accused of theft. There were protests against the Shabab's brutality, but they usually happened outside of areas the Shabab controlled. People living in Shababland were scared.
In addition to imposing harsh punishments, the Shabab put in place social restrictions that infuriated the locals in less conservative parts of the country. It forbade music, prohibited smoking and playing sports, required women to don the hijab, closed cinemas showing international films, and banned the Internet just as Internet use was spreading in Somalia.
Mirroring the poor example of the Islamic State, AQAP, and AQIM, the Shabab also alienated local Somali tribes. When it began in 2006, the Shabab militia allied with the powerful Hawiye clan, one of the largest in Somalia. After the Ethiopian invasion in 2006, the Shabab exploited nationalist hatred of the occupiers to gather members of different, sometimes hostile, tribes under its umbrella to combat the invaders. The unity was temporary; interclan disputes in the group erupted soon after the 2009 Ethiopian withdrawal. Things continued to go downhill: The Shabab pitted clans against one another, capitalizing on the grievances of one clan to garner support and recruits, which would anger the clan's enemies. The Shabab also stabbed allies in the back; for example, in late 2009, it pushed out its erstwhile friend, the clan-based Islamist militia Hizb al-Islam, from a strategic port city and then made nice with local tribes to maintain its hold on power.
The Shabab's behavior bothered Bin Laden because it went against everything he believed was necessary for establishing long-lived states. So when he finally responded to the Shabab in August 2010, he cautioned them against declaring a state even though he tepidly acknowledged that "a functioning state exists on the ground." But the Shabab didn't have enough judges to administer the large territory it controlled, observed Bin Laden. Even the judges it had didn't "show lenience" toward the locals, especially toward those who didn't practice the Shabab's austere version of Islam. Such practices would only "push [the locals] closer to the enemies."
Bin Laden also felt the Shabab was making a mess of Somalia's economy. The Shabab dominated the charcoal trade, cutting down scarce trees to manufacture fuel to send to the Gulf states. It also heavily taxed local businesses. The charcoal monopoly and the onerous taxes not only bred resentment but also stifled the economy. Bin Laden recommended easing the taxes, relying on religious tithes instead. He also encouraged the Shabab to abandon its monopoly, citing the negative effects of state ownership in the Arab world. Rather than chopping down Somalia's scarce trees for short-term economic gain, Bin Laden suggested planting other kinds of trees that would encourage economic growth and stop deforestation. "You don't fail to notice that due to climate change, there's drought in some areas and floods in others," Bin Laden observed. If you didn't know he ran the world's most notorious terrorist organization, you'd think Bin Laden was an officer working for the United States Agency for International Development.
Despite Bin Laden's misgivings about the Shabab, he was willing to allow it to secretly join al-Qaeda. But he wouldn't announce the merger publicly. To do otherwise, he explained, would needlessly cause "the enemies to escalate their anger and mobilize against you."
Privately, Bin Laden continued to worry the Shabab wasn't doing enough to improve the livelihoods of the people in its lands. In a letter Bin Laden wrote to his chief of staff a week before he died, he observed that "improving the livelihood of people is one of the important goals of the Shari'a and the most prominent duty of the emir, so there must be an effort to establish an economic power." Bin Laden also fretted the Shabab was still too quick to apply the harsh hudud punishments. As he rightly observed, the Prophet had established a high standard for proof that made it hard to implement his punishments. In contrast, the Shabab wasn't giving people the "benefit of the doubt when it comes to dealing with crimes and applying Shari'a."
Like other al-Qaeda affiliates that attempted to govern, the Shabab undermined its efforts by antagonizing powerful foreign governments. In July 2011, Shabab operatives detonated three bombs in Kampala, the capital of Uganda, killing at least seventy-six people as they watched the World Cup finale. Uganda had contributed peacekeepers to Somalia, whom the Shabab wanted to leave. But the attack only stiffened the resolve of African nations to rid themselves of the Shabab. In late 2011, Kenyan forces entered southern Somalia, and the United States began drone strikes soon after. Over the course of 2012, Kenyan, African Union, and Somali government troops seized control of most major Shabab-held cities. By January 2015, the African Union claimed its troops had reclaimed "85 percent" of territory previously held by the Shabab, isolating the group in remote areas in the north and south. The Shabab, like its brethren in Mali and Yemen, managed to survive as a guerrilla group, carrying out periodic terrorist strikes on select targets in Somalia, Kenya, and neighboring states. But its proto-state had vanished.
When the Shabab and the other al-Qaeda affiliates launched rebellions and tried to govern, they flouted some of al-Qaeda Central's directives for winning hearts and minds. But some flouted less than others. Consider Bin Laden's and Zawahiri's advice for insurgencies: Cooperate with other Sunni rebel groups; don't kill tribesmen even if they collaborate with the enemy; don't broadcast the execution of prisoners; and avoid attacks on Muslim civilians even if they're "heretics." AQAP came closest to the al-Qaeda ideal. It collaborated with local rebel groups and sought the support of the tribes, but it also killed tribesmen who worked with its enemies. It beheaded opponents and spies but usually didn't broadcast the acts. It bombed two Shi'i religious processions, but such attacks were rare. After it lost its territory, AQAP's leaders would denounce the filming of beheadings and attacks on Shi'i worshippers. In both cases, they drew a distinction between themselves and the Islamic State, which we can put at the other end of the hearts-and-minds spectrum. Think of AQAP as attempting to win hearts and minds and the early Islamic State as trying to cut them out.
The same holds true for governance. Bin Laden had discouraged his affiliates from establishing Islamic governments before they had popular support. But when his affiliates insisted, he counseled them to prioritize the economic well-being of their subjects and apply the hudud punishments leniently. AQAP had tried to provide public services and to manage their local economies, but it was not lenient when applying the hudud punishments, a failure its leader later came to regret. As for the early Islamic State, it never really had a chance to govern. But it did burn down local businesses and harshly apply the hudud. Again, AQAP was closer to the al-Qaeda ideal, and the Islamic State was far from it.
Either way, all the al-Qaeda affiliates failed to create durable governments. The jihadists could interpret the failure as proof that al-Qaeda's leaders were right all along. Had the affiliates hewed more closely to the hearts-and-minds strategy advocated by Bin Laden and Zawahiri, they would have succeeded. AQAP's and AQIM's leaders certainly felt that way when they evaluated their successes and failures. But the jihadists could also interpret the failures as proof that the al-Qaeda affiliates hadn't been brutal enough. As we'll see, that's pretty much what the Islamic State would decide, although it would do better than it had in its first attempt at providing government services and co-opting the tribes.
Although al-Qaeda's leaders, like most outsiders, believed the hearts-and-minds strategy was the right one for creating a durable state, it was impossible for the affiliates to know for sure whether it was the correct policy because of one confounding variable: Global jihadists provoke powerful foreign countries to attack them. Recall that old-school global jihadists like Bin Laden wanted to drive the Western infidels from Muslim lands and win popular Muslim support before trying to set up their own governments. But when al-Qaeda's affiliates went "glocal" by prioritizing state building, they still threatened the West in word or deed and so invited a powerful response. No government, especially after 9/11, wanted to take the chance that the glocal jihadists were just bluffing. So when the foreign governments attacked and defeated them, the jihadists could never say for sure how well they would have done had they been left alone. The coming civil war in Syria would give them a few years to find out what would happen if the international community let them do as they pleased.
Flying the Flag in the Arab Spring
Although al-Qaeda's branches attempted and failed to establish Islamic governments from 2006 to 2012, al-Qaeda's online fanboys around the world continued to pine for them. On jihadist discussion forums, members adopted the flag of the Islamic State in Iraq as their avatars, and forum administrators set up clocks counting the days since the Islamic State in Iraq had announced its establishment. Other al-Qaeda flags were rare, and no one set up a clock to count the number of days since al-Qaeda's establishment. The idea of the Islamic State was spreading virally even as the group was on life support in Iraq and its bannermen suffered setbacks in Yemen, Mali, and Somalia.
Some jihadists were confused when al-Qaeda's affiliates flew the Islamic State's flag, wrongly assuming those affiliates had joined the Islamic State. "The whole world has pledged allegiance to the Islamic State," exclaimed a jihadist forum member in 2010. As evidence, he posted videos of the Islamic State's flag waved by jihadists in Indonesia, Somalia, and Yemen.
The onset of the Arab Spring confused the flag's meaning further. In the chaos following the revolutions, jihadists formed new groups with the word "Ansar" ("Supporters") in their names. The groups shared al-Qaeda's global jihadist ideology but had no formal ties to the organization. Nevertheless, many of their members began flying the black flag of the Islamic State in Iraq, which was still an affiliate of al-Qaeda at the time. In Mali, Ansar al-Din raised the Islamic State's flag over the north in 2012. Leaders of the Tunisian group Ansar al-Shari'a frequently stood in front of the flag spewing vitriol at the government. Supporters of a different Ansar al-Shari'a group in Libya flew the flag whenever they talked to the press.
Individuals also got into the act. Jihadists flooded their online discussion forums with pictures of the Islamic State flag popping up across the Arab world. Between May and November 2011, members of the Shumukh forum snapped photos of it at a protest in Gaza, along the Lebanese–Israeli border, at a gathering in Rabat, at a mosque in Egypt, and elsewhere.
The flag made its most startling appearance in September 11, 2012. Responding to a film produced by a Christian living in California that insulted the Prophet, protestors gathered at U.S. embassies in the Middle East to display their displeasure. Some flew the Islamic State's flag, and one even managed to hoist it above the U.S. embassy in Egypt.
Many of those flying the flag during the protests were undoubtedly jihadists, Ayman al-Zawahiri's brother Muhammad among them. But non-jihadists also waved the flag. Some were unaware of its source, taking it as a simple reproduction of the Prophet's standard. Others knew its source and relished the provocative meaning attached to it. Demonstrators in Cairo who flew the flag told a reporter "they were not Al-Qaeda supporters, but were using it to protest against America and support Islam." A nearby vendor selling the flag corroborated their sentiment: "They are buying it for Islam and to show America they are wrong." In the same spirit, the refrain "Obama, all of us are Osama" was chanted frequently and scrawled on walls of the U.S. embassy in Cairo. The flag had transcended its origins to become a symbol of political protest. But the group that had designed the flag was about to remind the world of its original meaning.
Four
Resurrection and Tribulation
Despite the success to come, the auguries boded ill for Abu Bakr al-Baghdadi when he assumed leadership of the Islamic State in May 2010. American and Iraqi troops had killed his predecessor while he was at home, which meant the Islamic State had been penetrated by its enemies. Many of the group's leaders had met similar fates at American hands. In response, the Islamic State shifted to a strategy of clandestine terrorism to cope with the setbacks but longed to fight in the open again as an insurgent group. An Islamic state is nothing if it has no land.
Baghdadi was an unlikely executive. He had no bureaucratic or military training. And he was young, born Ibrahim Awwad Ibrahim al-Badri on July 1, 1971, to a lower middle-class farming family in Samarra, Iraq. Despite his humble origins, though, Ibrahim had connections. Two of his uncles served in Saddam's security apparatus, and one of his brothers was an officer in Saddam's army. Another brother died in the army when Ibrahim was young, a casualty of the Iran–Iraq war. Ibrahim himself would never serve in the military because of his poor eyesight.
The city Ibrahim grew up in was famed for the golden-domed shrine containing the remains of the tenth Shi'i imam and his son. Although Ibrahim, a Sunni, would become rabidly anti-Shi'a later in life, he claimed descent from the tenth imam. Through him, Ibrahim traced his lineage all the way back to the first imam, Ali, and his father-in-law, the Prophet. The man who would one day wage a war against the Shi'a was steeped in their mythology and claimed to descend from their leaders.
Neighbors, friends, and detractors remember Ibrahim's family for its piety but differ over its brand of Sunni Islam. Today, Ibrahim's Salafi-jihadist followers say the family was Salafi, an ultraconservative form of Sunni Islam like the kind practiced in Saudi Arabia. Others say Ibrahim was raised a Sufi and didn't become a Salafi until college. Sufism is a mystical strain of Sunni Islam despised by the Salafis. Whatever the case, friends and neighbors uniformly describe him as "quiet," "introverted," and deeply devout. Ibrahim's brother says he was a "stern" child who chided his siblings for minor religious infractions. His nickname in the neighborhood was "the believer."
In high school, Ibrahim has a middling student. He was excellent at math, so-so at Arabic, and terrible at English, barely passing the subject in his second try at the national exam in 1991. Because of his average scores, Ibrahim couldn't study law at the University of Baghdad as he had hoped. So he enrolled in the university's College of Islamic Sciences, where he first studied the Shari'a and then switched to Qur'anic studies.
Upon completing his bachelor's degree in 1996, Ibrahim enrolled at Saddam University for Islamic Studies as a graduate student. (It was renamed the Islamic University after the Americans invaded in 2003.) Saddam had founded the university in 1989, and it soon became an integral part of his effort to patronize Islamic studies to offset the growth of ultraconservative Salafism, which he viewed as a threat to his rule. For his master's thesis, Ibrahim edited a medieval book on Qur'anic recitation. It took him more years to graduate than he would have liked because one of his advisors died and another moved to Yemen. In other words, Ibrahim underwent the normal travails of a graduate student.
In the mid-1990s, Ibrahim joined the Muslim Brotherhood, a fraternal order that seeks to establish Islamic governments. Most Muslim Brotherhood branches peacefully pursue their goal, working within the local political system. Its members are intellectually diverse because the group doesn't have a fixed theological creed other than being vaguely Sunni. There are liberal members and conservative members. Ibrahim fell in with the ultraconservative Salafi members of the group in Baghdad.
After finishing his master's degree in 1999, Ibrahim was accepted into the university's doctoral program. Academically, he continued to study his favorite subject, the recitation of the Qur'an. Intellectually, he moved rapidly to the right, embracing revolutionary jihadist Salafism by 2000, three years before the American invasion of Iraq. "My group does not embrace me" he told a leader of the Muslim Brotherhood when he left the organization.
On June 30, 2004, a year after the Americans invaded Iraq, one of Ibrahim's professors filed a "Follow-Up Form for Students of Graduate Studies." "He has not attended my class," he noted. "Arrested."
Ibrahim was sitting in Camp Bucca, a sprawling American detention center in the Shi'i south that held 24,000 inmates. He had been picked up in February 2004 while visiting a friend in Fallujah whom the Americans were hunting. U.S. government records show Ibrahim was held for ten months as a "civilian detainee," which means the Americans had no evidence he was in an insurgent group. His picture attached to the records shows a man with close-cropped hair and a trimmed mustache sporting a long black beard and large, silver-rimmed glasses beneath dark, bushy brows.
Whether or not Ibrahim had joined the insurgency before landing in Bucca, he certainly did afterward. The prison was known as the "Academy" because it brought together so many jihadists and former members of Saddam's military and security services. "We could never have all got together like this in Baghdad, or anywhere else," remembered Abu Ahmed, a prisoner who knew Ibrahim. "It would have been impossibly dangerous. Here, we were not only safe, but we were only a few hundred metres away from the entire al-Qaida leadership."
Abu Ahmed recalled that Ibrahim held "himself apart from the other inmates, who saw him as aloof and opaque." But the prison guards viewed Ibrahim as a leader who was able to calm disputes between factions. Ibrahim befriended former members of Saddam's military and intelligence services, as well as future members of al-Qaeda in Iraq. The men would meet again outside the wire and rise with Ibrahim through the ranks of the Islamic State after its senior leaders were killed or captured.
Ibrahim didn't join al-Qaeda until 2006, when his militia enlisted in al-Qaeda's umbrella organization, Majlis Shura al-Mujahidin. When the Islamic State declared itself later that year, Ibrahim was made the head of all the Shari'a committees in the group's Iraqi "provinces."
Ibrahim was a multitasker. Despite the weight of his new responsibilities, he successfully defended his Ph.D. dissertation in March 2007. The aspiring scholar had edited part of a medieval commentary on an Arabic poem about how to recite the Qur'an. His advisor, a professor in Tikrit, could not come to Ibrahim's dissertation defense in Baghdad because travel was dangerous so he sent along his comments to the committee. "The study the student wrote is good but it contained some errors which I noted on the pages of the thesis." The professor points out typographical and spelling mistakes and gives advice on how to make a critical edition from conflicting manuscripts. It was the mild criticism of a pleased professor. Ibrahim was awarded a grade of "very good" for his efforts.
Meanwhile, Ibrahim was navigating the Islamic State's internal politics. The connections he had made in Camp Bucca served him well, as did his experience at negotiating between the prison factions. The Iraqis in the Islamic State chafed at the power of the foreign Arab faction headed by Abu Ayyub al-Masri, an Egyptian. The Iraqis rallied around Abu Umar al-Baghdadi, the Iraqi who was nominally in charge of the Islamic State. Although he still played second fiddle to Masri, Abu Umar's stature among younger jihadists had grown, perhaps because of the aura of mystery that surrounded him. Ibrahim managed to win over both Abu Umar and Masri. According to one insider account, Ibrahim served as one of Abu Umar's three couriers, which meant he enjoyed the emir's trust. Some of the emir's letters to Bin Laden were supposedly drafted by Ibrahim, and "their journey always started with him." But another insider portrays Ibrahim as a mere pass-through for correspondence; he never knew "the sender and the receiver."
Whatever the case, Ibrahim's discretion and secrecy kept him alive. When the Islamic State's commander in Baghdad was arrested, he named two couriers who carried messages to the State's leaders. The Americans tracked the couriers to the hideout of Abu Umar and Masri, who didn't survive the encounter. The third courier, Ibrahim, lived to die another day.
Upon the death of their leaders, the eleven members of the Islamic State's Shura Council deliberated on a new emir. Bin Laden's chief of staff, Atiyya Abd al-Rahman, wrote them to suggest a procedure for selecting one:
We suggest the noble brethren in the leadership appoint a temporary leadership to manage affairs until the consultation is complete. We believe it is best that they delay—as long as there is not an impediment or a strong preference for . . . hastening an official permanent appointment—until they send us suggested names and a report about each of them (the name, background information, qualifications, etc.) that we can send to Shaykh Osama so he can advise you.
The procedure was not followed by the Islamic State, either because no one saw the letter in time or because its recipients ignored it. Clandestine communication makes it hard to run a militia from afar.
But slowness and the vagaries of clandestine communication can also create opportunities. The Islamic State's Shura Council couldn't meet in conclave for security reasons, so its members had to correspond separately. The head of the Islamic State's military council, a former colonel in Saddam's army named Hajji Bakr, saw a way to turn the situation to his advantage. Hajji Bakr wrote each member individually, saying the others had agreed that Ibrahim should take charge. Ibrahim was one of the youngest candidates considered, but he had a lot going for him. He claimed descent from Muhammad; he was a member of the Shura Council and close to the previous emir; and he had ties to other powerful members. It also mattered that members of Ibrahim's tribe had been early supporters of the Islamic State. His tribal connections could help the group make a comeback. Nine of the eleven Shura Council members voted for Ibrahim, now taking the nom de guerre Abu Bakr al-Baghdadi, commander of the faithful.
Al-Qaeda's leaders heard about Abu Bakr al-Baghdadi's appointment in May after everyone else did. Distrustful of the new leadership, in July Bin Laden asked his chief of staff for information about Baghdadi and his deputies. "Ask several sources among our brothers there, whom you trust, about them so that the matter becomes clear to us." Several days later, Atiyya promised he would do so. But he was apparently unsuccessful because he wrote the Islamic State's Ministry of Media in September: "The shaykhs [in al-Qaeda] ask you for an introductory paper about your shaykhs in the new leadership." Better yet, "perhaps they can write and introduce themselves."
A representative for the Islamic State's Shura Council wrote back on October 9. He claimed the Islamic State had received al-Qaeda's instructions to select a temporary emir after it had already announced Baghdadi's appointment. Nevertheless, the representative affirmed Baghdadi's loyalty to al-Qaeda and his consent to be a temporary leader. If al-Qaeda had a better candidate to lead, the Islamic State would "hear and obey."
When Bin Laden died a few months later, Baghdadi made a public statement assuring the new head of al-Qaeda, Ayman al-Zawahiri, that the men of the Islamic State were "faithful" to him and al-Qaeda. In a private letter on May 23, 2011, the Islamic State asked if Zawahiri wanted Baghdadi to make a more explicit public pledge of allegiance to him. The al-Qaeda chief apparently declined.
While Baghdadi stalled for time with his leaders in al-Qaeda, he consolidated his hold on power in the Islamic State. At his right hand was the man who had helped him take the throne, Hajji Bakr. Those who knew the bald, white-bearded Hajji Bakr described him as the "prince of the shadows" and Baghdadi's "private minister." According to insiders, the first order of business for the prince of shadows was to purge the Islamic State of leaders he suspected of disloyalty; those who didn't leave their posts willingly were killed. He and his boss replaced them with their Iraqi allies, many of whom had served as officers in Saddam's military and intelligence services. Saddam, who had conducted a similar purge when he came to power, would have been pleased.
His throne secure, Baghdadi set about reviving the Islamic State's flagging fortunes.
Blueprint
On the new emir's desk was a plan to turn things around. Between December 2009 and January 2010, Iraqi jihadists had circulated a "Strategic Plan for Reinforcing the Political Position of the Islamic State of Iraq." The document has the look and feel of a DC think tank report, with analysis and recommendations for policy makers. Think pieces and after-action reports are common in the jihadist movement, but it was unusual to see jihadists openly criticize the Islamic State. The criticism was evidence of how far the group had fallen. The tempo of the Islamic State's attacks was nowhere near its height in early 2007, and the group held no land.
The strategy paper blamed the Islamic State's fall on a "dirty war" waged by its American adversaries, who used "awakened" Sunni tribes against it. "When the Islamic State was at the pinnacle of its power and influence, [the Americans] bombed markets, public places, and mosques, and they killed the opponents of the State, so that the mujahids were blamed. On account of things like this, we saw the influence of the Islamic State fade and disappear and the apostate Awakenings spread."
The delusions continued. The authors of the paper spun online videos of Americans trying to disarm improvised explosive devices (IEDs) as videos of Americans planting the devices and inadvertently blowing themselves up. The Islamic State did not provoke the Sunni tribes by oppressing them; rather, the jihadists' enemies cleverly turned tribal leaders against the jihadists. Young men in the tribes supported the Americans only for money and for pride, styling themselves as defenders of their people.
The Islamic State has fallen, the authors acknowledged, but it will return just as the Taliban returned in Afghanistan after its defeat at the hands of the Americans. The American withdrawal from Iraq would be the time to act. "When the Americans withdraw within two years . . . the situation will be strongest politically and militarily for the Islamic plan to prepare to completely seize the reins of control over all Iraq." But the authors recognized that other factions in Iraq were preparing to do the same.
The authors recommended several ways to overcome the other factions and control Iraq. Uniting them behind the "jihadist program" of the Islamic State was at the top of their list. "It is not about names and titles the Muslims would strive for. Aiding [the program] is a victory for the people of Islam and not a victory for a group, or a title, or a name." Merely fighting the other factions without a goal would be "stupid."
Militarily, the authors contended it would be a waste of time to focus on attacking the American forces in Iraq since they were leaving; rather, the jihadists should train their fire on the Iraqi military and police, whom the Americans hoped would continue to pacify the country for them once they left. By targeting them, the jihadists would instill fear in the hearts of potential recruits. They should focus in particular on the very few units that were capable of fighting against the jihadists. Attacking government troops would also force them to abandon their bases in regions of the country where they were weak. That would open up security vacuums and drain the government's resources when it fought to protect its remaining bases. The jihadists could exploit these vacuums by seizing the territory and any equipment or infrastructure that was left behind. "Make them always preoccupied with internal problems," wrote the authors, quoting ancient China's preeminent military strategist Sun Tzu.
Readers might find it odd that religious zealots who hate nonbelievers would quote Sun Tzu. But the practice is common, evidence of a pragmatic streak among some jihadists. In the early 2000s, for example, jihadists celebrated the strategic insight of Abu Ubayd al-Qurashi, an anonymous author who quoted dozens of non-Muslim strategists in his magazine column, "Strategic Studies." Among others, Qurashi cited Robert Taber's history of guerrilla campaigns, War of the Flea; William Lind's writings on fourth-generation warfare; the Prussian military theorist Carl von Clausewitz; and the Communist revolutionary Mao Zedong. The authors of the "Strategic Plan" were carrying on that tradition.
The authors assumed guerrilla tactics would weaken the Iraqi government. But they also believed the jihadists could not establish their own state without co-opting the Sunni tribes. To do so, the authors advised the Islamic State to copy what the United States had done: give money and weapons to Sunni tribal leaders who were angry with the Islamic State. Doing so had reinforced the tribal leaders' authority and bought the temporary allegiance of their young men. The authors admitted that "the idea to recruit the tribes to eliminate the mujahids was a clever, bold idea and will be used by any occupier in the future because they make hard work easy for the occupier, just as they provide protection against the attacks of the mujahids." But the authors were sure the tribes would rather receive money and weapons from fellow Muslims than from foreign occupiers who disrespected their religion or promoted thugs as tribal leaders. The mujahids should follow the American blueprint of creating tribal councils and militias they can work with, but do it better. The mujahids would be more respectful of local religious practice and power structures, and they could finance the endeavor with captured booty. As we will see, the jihadists' respect for religious sensitivities and power structures was sometimes more theory than practice.
Uniting the jihadists behind a single program, intimidating Iraqi security forces, and co-opting the Sunni tribes would not be enough, asserted the authors. The jihadists also needed a "political symbol" or "avatar." Several things go into the making of such a symbol: the greatness of his sacrifices, his high morals, and his evenhandedness. The head of the Islamic State at the time, Abu Umar al-Baghdadi, had achieved symbolic status. But the authors worried that none of his deputies was high profile enough to fill his symbolic shoes if he died (which happened a few months later).
In concluding their think piece, the authors stressed that jihadists have to instill confidence in those whom they rule. They could do this by protecting the people in lands they control and making them prosper, seeing to the needs of local governors and soldiers, selecting good executives and judges, ruling by Islamic law, implementing the hudud punishments stipulated in Islamic scripture, and distributing money from the treasury. Since the international media is biased against the jihadists, they said, the jihadists would need a media strategy to make sure their good works were known. The jihadists should also consider allying themselves with their opponents when they face a common enemy, just as the Prophet allied with the Jews against the pagans when they attacked him in Medina.
The "Strategic Plan" has a lot in common with The Management of Savagery, a book released online by an al-Qaeda franchise in 2004, two years before the Islamic State's founding. The book explains how to take control of territory, establish a nascent state, and develop into the caliphate.
The author of The Management of Savagery went by the nom de guerre Abu Bakr Naji. We do not know for certain who Naji was in real life, but he was probably from North Africa, based on certain turns of phrase he uses and his frame of reference. No one wrote under that name after 2004, so he is either dead or started writing under a different name.
Naji argued that terrorist groups should carry out "vexation operations" against sensitive enemy targets, such as oil pipelines and tourist sites vital to the nation's wealth. To protect these sensitive targets, local governments would pull in their security personnel, which would open up security vacuums or "regions of savagery" in the periphery of the state. Jihadists would then move into the volatile regions, provide basic security and public goods to win over the population, and establish "Shari'a justice." Securing the support of the tribes in the area would be crucial because the lack of tribal support would doom the jihadists' efforts. Bribes can go a long way, but the jihadists should also try to indoctrinate the tribes. The jihadists should work with other Muslims regardless of their theology, provided they did not oppose the jihadists' program. After the jihadists had established a network of these "administrations of savagery," they should coalesce into an Islamic state and then a caliphate.
Naji did not sugarcoat what it takes to conquer and control these lands: ceaseless, uncompromising violence. "Those who study only jihad as it is written on paper," Naji observed, "will never grasp this point well." The youngsters in particular "no longer understand the nature of wars." Veterans of previous jihads know "it is naught but violence, crudeness, terrorism, frightening others, and massacres." Those who are soft are better off "sitting in their homes." "If not, failure will be their lot and they will suffer shock afterwards." Even if jihadists wanted to act mercifully toward their enemies, their enemies will "not be merciful to us if they seize us." "Thus, it behooves us to make them think one thousand times before attacking us." Even the enemy's women and children can be targeted, he said, as long as it deters the enemy from doing the same—a principle that contradicts the Prophet's prohibition against killing women and children.
Despite Naji's insistence that jihadists are constrained by Islamic scripture, the "Islamic" principles he enunciated override the Prophet's strictures on violence. Maximum latitude for maximum violence is the real interpretive framework for Naji and his acolytes. To Naji's mind, anything less would make the jihadists ineffective insurgents.
Throughout his book, Naji was at pains to convince his jihadist readers that they could learn from the writings and example of non-Muslim insurgents, military strategists, and political theorists. One of the very first people he quoted was not the Prophet but Paul Kennedy, the American historian who wrote about the dangers of military overreach: "If America expands the use of its military power and strategically extends more than necessary, this will lead to its downfall." Naji used the quote as the bedrock of his entire strategy for provoking the United States to overextend itself militarily. Universal laws of insurgency, argued Naji, are usually compatible with Shari'a laws.
Naji's book is popular in jihadist circles. In 2008, Saudi authorities arrested seven hundred people whom they accused of plotting attacks inside the kingdom. According to the authorities, those arrested were consulting the book to help them "revive criminal activities in all regions of the Kingdom in an attempt to change the internal security situation into a stage that resembles the situation in other unsettled regions." A friend with contacts in al-Shabab, the Somali branch of al-Qaeda, related to me in 2010 that members of the group used the book as a blueprint to take over Somalia.
Today, seasoned jihadists in the Islamic State study Naji's book in their camps. Online Islamic State fanboys also celebrate the manual, often sharing links on Twitter where followers can download the text. One such tweet from November 2014 with the hashtag "The Islamic State is coming" touted the book as "the first resource for mujahids in managing their areas of influence." Another tweet going back to September 2012 promoted a PDF of the book, saying "There is no better book, I highly recommend downloading it from the net, reading it, and then burning it so no one from the state finds it among your things." Other tweets quote sections of the book. One from February 2014 has a picture of a passage providing advice on dealing with jihadists who deviate from the norms of the movement, above which he tweets "a lesson from Abu Bakr in his book #TheManagementofSavagery."
Blueprint in hand, the Islamic State was prepared for a comeback.
Gaining Ground
When the Islamic State decided to set up shop in Syria, it already had a network in place. Syrian president Bashar al-Assad had funneled hundreds of jihadists into Iraq to fight against the United States. According to the U.S. government, in 2007, 85 to 90 percent of the foreign fighters in Iraq had come through Syria. The Islamic State had received many of those fighters and had maintained its facilitation network in Syria after the end of the Iraq war.
When Syrians began peacefully protesting against their government in 2011, the Assad regime released an unknown number of jihadists from prison. The release was calculated to foster violence among the protestors and give Assad a pretext for a brutal crackdown. It worked. As a Syrian intelligence officer would later reveal, "The regime did not just open the door to the prisons and let these extremists out, it facilitated them in their work, in their creation of armed brigades."
Taking advantage of the volatility and the release of prisoners, al-Qaeda leader Ayman al-Zawahiri ordered the Islamic State to "form a group and send it to [Syria]." The Islamic State's emir, Abu Bakr al-Baghdadi, dispatched one of his senior Syrian operatives, Abu Muhammad al-Jawlani, to oversee the effort.
The new group initially behaved like its parent, wantonly disregarding civilian casualties. In December 23, 2011, the group that would soon call itself Nusra Front (Jabhat al-Nusra) carried out a huge suicide attack in Damascus, killing dozens. The next six months were more of the same, which raised the ire of other Sunnis rebelling against Assad. In August 2012, Nusra had only about two hundred operatives in Syria.
But then things began to change. Nusra expanded in the north, becoming an insurgent group rather than a clandestine terrorist organization. The development made the group more sensitive to the need for popular support. It began avoiding suicide attacks on civilians and collaborated with the other Sunni insurgent groups. When the United States designated Nusra a terrorist organization on December 11, 2012, and accused it of operating under the umbrella of the Islamic State, Syria's Sunni rebels responded: "We are all the Nusra Front."
Nusra's change in strategy may also have been prompted by the growing influence of Syrian jihadist Abu Khalid al-Suri. Abu Khalid, who was reportedly released from a Syrian prison in late 2011, had long ties to al-Qaeda and its new chief, Ayman al-Zawahiri. The two had met when Abu Khalid was in Pakistan in the early 1990s, although Zawahiri had lost touch with Abu Khalid after his capture in Pakistan in May 2005. The Pakistanis turned Abu Khalid over to the Syrians, who transferred him to the Saydnaya military prison in Syria. The prison held many of Syria's political prisoners, including the jihadists Assad would later release. When the Assad regime freed Abu Khalid, Zawahiri quickly reestablished contact. "He was to me and my brothers such a great advisor," Zawahiri would later remark.
Zawahiri and Abu Khalid were of similar mind about how to conduct an insurgency: The insurgents should win the hearts and minds of the locals. The common insurgent creed had been given a jihadist flavor by Abu Khalid's longtime comrade Abu Mus'ab al-Suri, the apocalypticist and strategist we met earlier, whom Zawahiri called the "professor of the mujahideen."
In the early 1980s, Suri had participated in the Syrian Muslim Brotherhood's revolt against President Hafez al-Assad, Bashar al-Assad's father. The revolt ended in the deaths of tens of thousands of Syrians and ruin for the Brotherhood, forcing Suri into exile and a life of itinerant jihadism. He would spend the next two decades thinking and writing about what went wrong in Syria and in the other failed jihads he observed.
Suri was an early booster of the 1990s jihad in Algeria, which began when the Algerian military cancelled parliamentary elections after Islamists succeeded in an early round of voting. Suri watched in dismay from London as the jihadists in Algeria became more extreme, which turned the public against them.
Suri went on to Afghanistan, where he served on the consultative council advising al-Qaeda. Suri and his comrade Abu Khalid were independent thinkers and differed frequently from al-Qaeda's leader, Bin Laden. In a 1999 email, they reprimanded Bin Laden for violating a pledge to the "commander of the faithful," Mullah Omar. The Taliban leader had asked Bin Laden to stop making media appearances and antagonizing the Western powers, a request Bin Laden ignored despite his oath of allegiance to the mullah. "You should apologize for any inconvenience or pressure you have caused," the two Syrians wrote, "and commit to the wishes and orders of the Leader of the Faithful on matters that concern his circumstances here."
For Suri, 9/11 was the ultimate act of disrespect to one's host because it provoked the United States to destroy the Taliban. "The outcome, as I see it, was to put a catastrophic end to the jihadist current, an end to the period which started back in the beginning of the 1960s of the past century and has lasted up until September 11th. The jihadist current entered the tribulations of the current maelstrom which swallowed most of its cadres over the subsequent three years."
Chastened by two decades of failure, Suri published a massive tome online, The Call of the Global Islamic Resistance, explaining where the jihadist movement had gone wrong and how to put it right again. The 1,600-page book came out around the same time Naji published his Management of Savagery. The two books are similar in many ways but whereas Naji emphasized conquering territory and brutally defending it, Suri promoted popular revolt. "The Islamic movement can only establish the Muslim society through a popular jihad."
Suri's hearts-and-minds strategy was promulgated by Abu Khalid among the jihadists in Syria and was soon championed by Nusra's leaders. Tweets written by Sami al-Uraydi, a Nusra religious advisor, showcased the "pearls" he had gleaned from Suri's writings. Many of them are veiled criticisms of the excesses of the Islamic State. Uraydi quoted Suri as saying that "[t]he Algerian experience confirmed to me and others that the greatest failing of the entire jihadist experience without exception" is the lack of religious education among jihadists, which leads to violent excess among the youth. In another tweet, Uraydi listed nineteen admonitions from Suri's writings that all jihadists should heed. Among them is Suri's caution against excommunicating Muslims and targeting neutral people. Both were tenets the Islamic State ignored.
Nusra itself ignored parts of Suri's advice. The strategist doubted jihadists could overthrow Muslim rulers as long as the United States dominated the Middle East. The American-led crusader-Zionist alliance was too powerful to be ignored; it must first be resisted and repulsed from the Muslim countries before Muslims could contemplate establishing a truly Islamic state. According to Suri,
The goal of the call of resistance is resisting the aggression of the Crusader-Zionist campaigns led by America and its Jewish and Crusader allies among the foreign forces of unbelief, and the local forces of apostasy and hypocrisy cooperating with it. The old strategic goal of confronting governments [in the region] has changed. . . . No one should take from this that the goals of the call of resistance is anything other than establishing the rule of the law of God. . . . But we consider this the final goal that will result from the success of the resistance in repelling these campaigns and bringing about the downfall of the greatest power, America.
Nusra, in contrast, believed a popular front against a Muslim ruler in Syria could succeed, perhaps because the United States was not backing the regime. The group recruited heavily among Syria's Sunnis and involved locals in governing the areas they seized. Nusra collaborated with other rebel groups as long as they weren't too closely aligned with the United States (especially after the United States began bombing Nusra in late 2014). After taking a city, Nusra participated in joint "Shari'a committees" with other rebel groups to settle disputes, provide basic services, and distribute food and medicine.
In contrast, the Islamic State sought to dominate rather than collaborate whenever it could. When it had sufficient manpower, the group preferred to govern alone and to terrify locals who questioned its writ. When the Islamic State's soldiers murdered members of other rebel groups, it refused to submit to arbitration, claiming a mere group cannot hold a state accountable. "The State is the State," wrote Islamic State scholar Turki ibn Mubarak al-Bin'ali. "Have you ever heard of the Prophet's state, or the Rightly Guiding Caliphs' state, or the Umayyad state, or the Abbasid state submitting to the judgment of an independent person? . . . We are a state, so how could you compel us to submit to the judgment of an independent court? . . . Don't you know that an independent court means a different state?"
To get a sense of how these differences played out, consider reported incidents of Nusra and the Islamic State abusing Syrian Sunnis or attacking other rebel groups: From August to December 2013, the Islamic State behaved badly 43 times, Nusra, 4. Put pseudoscientifically, that is a "badness factor" of ten.
The Islamic State's rule in the Syrian town of al-Bab is a case in point. When the Islamic State first arrived in late 2013, it fixed roads and cleaned up the town. But as its power grew, the State imposed harsh restrictions on women, forbade smoking, and forced people on the street to pray at the designated times. Torture and kidnapping kept the locals in line, but resentment grew.
Still, not everyone resented the Islamic State's authoritarianism. Some citizens supported the State's diktat, considering it a welcome change from the abuses and arbitrary rule of other rebel groups. The Islamic State might be brutal, but it cracked down on bandits, drove out corrupt rival groups, and made decisions quickly. In contrast, Nusra often collaborated with those rebel groups, which made it hard to govern and to curb abuses. As Plato observes in the Republic, the tyrant "sprouts from a protectorate root."
Falling Out
Given their different strategic orientations, it is little wonder that the Islamic State was soon at odds with its branch in Syria. The State wanted to carve out a domain in the Arab hinterland between Syria and Iraq. Nusra wanted to embed itself in the Syrian opposition and overthrow the Assad regime.
Nusra and the Islamic State also disagreed over the control of resources, especially oil. Selling oil on the black market ensured a steady stream of millions for funding operations and attracting recruits. As the American journalist Theo Padnos gleaned from his time as a Nusra hostage from 2012 to 2014, "The real issue between the Nusra Front and the Islamic State was that their commanders, former friends from Iraq, were unable to agree on how to share the revenue from the oil fields in eastern Syria that the Nusra Front had conquered."
The spat over oil was enmeshed in tribal politics. Nusra had the strong support of conservative tribesmen in the town of Shuhail in Deir Ezzor Province bordering Iraq. But after a senior Nusra member from a rival tribe defected to the Islamic State, the latter was able to capture a Conoco gas plant in the area. The move touched off fighting between the rival tribes, which finally ended with a truce. The Shuhailis kicked out the Islamic State, but it later returned and defeated the tribe, which also meant the defeat of Nusra in eastern Syria.
Jealousy also played a role in the conflict between the Islamic State and Nusra. Baghdadi suspected Nusra's fighters were more loyal to their leader, Abu Muhammad al-Jawlani, than to him, the commander of the faithful. Baghdadi reportedly sent a private letter to Jawlani, telling him to announce that Nusra was part of the Islamic State. Jawlani refused because it "would not be beneficial to the Syrian revolution." Baghdadi's consigliere, Hajji Bakr, dispatched spies to watch Jawlani in Syria. Unhappy with the slow trickle of information, Hajji Bakr and Baghdadi went to Syria themselves in March 2013.
Baghdadi met with Jawlani directly and again argued that the ties between Nusra and the Islamic State should be made public. Jawlani refused once more, contending that Nusra was in Syria at the request of Zawahiri and would not answer to the Islamic State. Jawlani reminded Baghdadi that al-Qaeda had expressly forbidden the Islamic State from "announcing any official presence of al-Qaeda" in Syria because it would destroy Nusra's popular support. Changing tack, the emir praised Jawlani for his efforts. But after Jawlani left, Baghdadi met separately with each Nusra leader, trying to turn them against him. Although Baghdadi was unsuccessful, fans of the antihero Frank Underwood in the television series House of Cards will appreciate the maneuver for what it was: cunning politicking.
Two days later, on April 9, 2013, Baghdadi publicly revealed that Nusra was a branch of the Islamic State and announced it would be absorbed into a new entity, the Islamic State of Iraq and al-Sham (the Levant). In his statement, Baghdadi disclosed that when the conflict began in Syria, he had deputized Jawlani to lead a contingent into the country. Baghdadi instructed him to work with Islamic State cells already active there to establish a branch of the State. The Islamic State gave the group its marching orders, paid its salaries, and supplied it with men. Baghdadi explained that he had delayed announcing his control of Nusra for security reasons and to avoid tarring the new venture with the bad reputation of the Islamic State. But the time for secrecy had come to an end.
Jawlani responded the next day by declaring Nusra's independence from the Islamic State and pledging an oath of allegiance directly to Zawahiri as leader of al-Qaeda.
Baghdadi, still nominally under al-Qaeda's authority, immediately fired off a private message to Zawahiri, threatening bloodshed and fitna, or schism, if Zawahiri didn't rule in his favor:
It has just now reached me that al-Jawlani has released an audio message announcing his direct oath of allegiance to you. This is what was planned for him to protect himself and those with him from the consequences of the mistakes and disasters he committed. This poor servant [Baghdadi] and those brothers with him here in al-Sham believe it is up to our shaykhs in Khorasan [Afghanistan and Pakistan] to announce a clear, unambiguous position in order to bury this conspiracy before it causes blood to flow and we [sic] become the reason for a new calamity for the umma.
We believe that any support for what this traitor has done, even tacitly, will lead to a great fitna, which will thwart the program for which the blood of Muslims has been shed. Delaying the announcement of the correct position will lead to making the current circumstance the reality, splitting the ranks of the Muslims and diminishing the prestige of the group such that there will be no healthy cure afterward except by shedding more blood.
Zawahiri was exasperated. "We have neither been asked for authorization or advice, nor have we been notified of what had occurred between both sides," he wrote in a private letter to both men. "Regrettably, we have heard the news from the media." Like a parent scolding siblings for fighting, Zawahiri blamed both parties. Baghdadi "was wrong when he announced the Islamic State in Iraq and the Levant without asking permission or receiving advice from us and even without notifying us." Jawlani erred "by announcing his rejection of the Islamic State in Iraq and the Levant, and by showing his links to al-Qaeda without having our permission or advice, even without notifying us." The Islamic State was to renounce its claim on Syria and go back to Iraq. Nusra should continue its fight in Syria but as an al-Qaeda affiliate in its own right. Both men would be subject to review after one year by the "general command" of al-Qaeda. Zawahiri appointed his trusted friend, Abu Khalid, to arbitrate between Nusra and the Islamic State should subsequent differences arise.
Despite the scolding, Nusra had gotten what it wanted. It would no longer answer to the Islamic State and was now al-Qaeda's official affiliate in Syria.
Incensed, Baghdadi publicly rejected Zawahiri's ruling. "I have chosen the command of my Lord over the command in the message that contradicts it." Baghdadi explained his decision privately in a letter dated July 29, 2013, referring to the "muhajirs," or foreign fighters, in the Islamic State: "It became clear to us that obeying our emir was an act of disobedience to God and ruinous for the mujahids with us, especially the muhajirs. So we obeyed our Lord and preferred his good-pleasure over the good-pleasure of the emir. . . . You do not say someone sinned because he disobeyed a command from an emir that he believes is ruinous for the mujahids and an act of disobedience to God the exalted."
Baghdadi was alluding to a tradition from the Prophet: "No obedience in sin but rather obedience to what is right." The tradition circulated in the early years of Islam to justify revolt against Muslim rulers.
Days after Baghdadi rebuffed Zawahiri, the Islamic State's spokesman, Abu Muhammad al-Adnani, publicly explained Baghdadi's decision. Zawahiri erred when he tried to partition the Islamic State according to borders drawn by colonialists. The ruling would only embolden secessionists in the Islamic State's ranks. Clearly, Zawahiri was a biased judge who had not properly listened to the State's complaints.
Al-Qaeda had faced unruly affiliates in the past, especially the Islamic State. But the spats had never been public, which would have represented a direct challenge to al-Qaeda's leaders. Doubtless worried about setting a bad precedent, Zawahiri decided to kick the Islamic State out of al-Qaeda. Thus, on February 2, 2014, al-Qaeda publicly renounced any "tie" or "connection" with the group.
The Islamic State's spokesman, Adnani, shot back, arguing that the State had never really been part of al-Qaeda. In Adnani's telling, al-Qaeda in Iraq had dissolved itself when the Islamic State was declared in 2006. "The Mujahidin came out from the restrictiveness of the organizations to the emancipation of the State," said Adnani, giving the impression that the Islamic State had never answered to Bin Laden and al-Qaeda's other leaders. Al-Qaeda could not disown what it no longer possessed.
Partisans of the Islamic State on Twitter repeated the talking points. "The best news I've heard," quipped one, "is that the State has no ties to you all. Let us reply to the deserters and liars that are saying Baghdadi disobeyed Zawahiri." When Nusra spiritual advisor Sami al-Uraydi chided Baghdadi on Twitter for his hubris, an Islamic State fan responded, "Our shaykh Osama himself called for joining up with the State." Others shared YouTube videos presenting proof that supposedly exonerated Baghdadi from the charge of disobedience.
Annoyed, Zawahiri issued a statement on May 2, 2014, to clarify the nature of the historical relationship between al-Qaeda and the Islamic State. Zawahiri admitted that the Islamic State had been established in 2006 without consulting al-Qaeda's leaders. But its founder, Abu Ayyub al-Masri, had sent a letter after the fact justifying the State's establishment and affirming its loyalty to al-Qaeda. Masri related that the first emir of the Islamic State, Abu Umar al-Baghdadi, had pledged his loyalty to Bin Laden in front of other Islamic State leaders, but they did not broadcast the news due to "political considerations." According to Zawahiri, the same considerations kept Abu Umar's successor, Abu Bakr al-Baghdadi, from publicly pledging allegiance to Bin Laden and then to Zawahiri after Bin Laden's death. To prove his point, Zawahiri cited internal al-Qaeda memos, even those the United States had captured and declassified. Many of the memos have been presented in this book.
Flustered by Zawahiri's document dump, the Islamic State spokesman Adnani could no longer insinuate that the State had never been under al-Qaeda's authority. Changing course, he claimed the Islamic State had bent the knee to al-Qaeda's commanders only on matters outside of Iraq. The State obeyed al-Qaeda's orders not to undertake attacks outside the country, even though its rank and file demanded them in Saudi Arabia, Egypt, Libya, Tunisia, and finally Iran, where al-Qaeda's leaders wanted to "protect [the organization's] interests and its supply lines." As for attacks inside Iraq, the Islamic State's spokesman claimed that the organization ignored al-Qaeda's "repeated request" to stop targeting the Shi'a. In his telling, al-Qaeda never issued a direct order to the Islamic State or asked its leaders about the disposition of their forces inside Iraq.
Here the Islamic State's spokesman is engaging in a bit of historical revisionism. It is true that the State often ignored al-Qaeda's orders for operations inside Iraq, but al-Qaeda was neither deferential nor disinterested. In private letters from 2007–2008, al-Qaeda sent directives to the group and asked about the disposition of its forces.
Regardless, the crucial question was whether al-Qaeda had authority over the Islamic State for matters outside Iraq. By admitting that the Islamic State deferred to al-Qaeda for foreign operations, the spokesman had acknowledged al-Qaeda's writ outside Iraq. Clearly, the Islamic State had defied that authority, as Baghdadi himself had said privately.
The man Zawahiri had chosen to heal the rift between the Islamic State and Nusra, Abu Khalid, was disgusted by the Islamic State's duplicity. "The State group is excommunicating everyone who opposes them or differs with them, spilling their blood with impunity as if they own Islam." Such wanton disregard for life and loyalty, he sniped, could only mean the Islamic State was working for an intelligence agency to destroy the Syrian rebellion. In response, the Islamic State threatened to send five suicide bombers to pay Abu Khalid a visit.
Fitna
Baghdadi's disobedience divided the jihadist community. "With great regret," wrote the administrators of the most popular jihadist discussion forum, "the brothers have split into three camps because of these events": Team Islamic State, Team Nusra, and the mushy middle.
Jihadists in the thousands, many of them young, took to their favorite social media platforms to voice their views. Gone were the days when disagreements among jihadists were confined to private discussion boards, silenced by moderators. The rest of the world could watch the melee in real time, keeping score by tabulating retweets and "likes." The biggest split ever in the global jihadist community happened just when people could use new forms of social media to quickly sort themselves into rival camps. The two phenomena are probably related.
From the Olympian heights, jihadist scholars traded blows that reverberated throughout the community. Influential graybeards like Abu Qatada al-Filistini and Abu Muhammad al-Maqdisi sided with Nusra. Abu Qatada blasted the Islamic State for "trespasses in . . . cooperating with their brothers and opposing them" and for behaving as if it were an actual state rather than an insurgent group like the rest. Baghdadi should take his organization back to Iraq, Abu Qatada counseled, just as Zawahiri ordered.
Maqdisi, the don of jihadist scholarship, mocked the Islamic State's claims that its jihadist opponents were part of a Western-backed conspiracy against it. The idea that jihadists should treat Baghdadi as the head of an actual Islamic state, much less a caliphate, was absurd.
Maqdisi was particularly annoyed that some of his protégés had backed the Islamic State. "I hope my anger reaches them," he reportedly said. The criticism stung. One of them lashed out: "O you who have made war on the [Islamic] State with your fatwas," he said, "you have created this state of division [fitna]."
Online fans of both camps also got in on the action, tweeting support for their side and denigrating the other. A prominent Islamic State cheerleader calling himself Shami Witness led the charge for his team on Twitter. "Nusra is over," he tweeted confidently a few weeks after Nusra seceded from the Islamic State. Its "approach of collaboration with and appeasement of non-jihadi grps [sic]" had ruined the organization. "LOL why are all these Nusra fanboys attacking me at once?" he taunted. "That insecure eh?"
The barbs were not just traded online. In early January 2014, Islamic State members in Aleppo killed a popular commander in a powerful Salafi militia, Ahrar al-Sham, after weeks of skirmishes with other Sunni rebels in Syria. The rebels demanded the Islamic State submit to arbitration to settle the conflict, but the State refused because it would "infringe on the right of the Muslim sovereign and his state." Various jihadists tried to mediate between the two groups, but the effort went nowhere due to the Islamic State's intransigence. When other rebel groups attacked the State for similar transgressions, Ahrar al-Sham reluctantly joined the fight.
Organizing the fight for Ahrar al-Sham was Abu Khalid, Zawahiri's representative who commanded the group's forces in Aleppo Province in northern Syria. On January 17, 2014, Abu Khalid issued a statement condemning the Islamic State, saying it had broken all the rules of Islamic warfare, attacked its fellow jihadists, and violated the tenets of insurgency put forward by Abu Mus'ab al-Suri, Zawahiri, and Bin Laden. Its method was as "far as possible from the proper method." The Islamic State's war on its brothers would only help Syrian president Bashar al-Assad, who stood to gain the most from rebels fighting one another instead of him.
In response, the Islamic State decided to make good on its earlier threat to kill Abu Khalid with suicide bombers.
Fulfilling a Promise
The man charged with carrying out the Islamic State's order was Ahmad Lulu, a thirty-five-year-old native of Aleppo who worked for a relief organization, Syria Khayr. Lulu enjoyed close ties with Ahrar al-Sham's members, particularly people in the group's humanitarian aid office. His job took him to the old crusader city of Antakya (Antioch), Turkey, where licit and illicit aid flowed across the border with Syria. There Lulu met Abu Hurayra, an Islamic State fighter convalescing away from the battlefront. The two men struck up a friendship, which deepened when they returned to Syria.
At Abu Hurayra's prompting, Lulu soon joined the Islamic State. "In the organization, they knew I worked in the relief field," Lulu later explained. "They left me to complete my work in this field, in addition to working with them to spy and gather information."
When fighting broke out between Ahrar al-Sham and the Islamic State, Lulu was summoned by a State security official. "The task . . . was to 'purge the leadership of Ahrar al-Sham.'" Because of the difficulty and sensitive nature of the operation, only a few of the Islamic State's senior leaders knew of it, including Baghdadi and his spokesman, Abu Muhammad al-Adnani.
Lulu was introduced to Abu Maryam al-Iraqi, an Islamic State operative who would assist him. Lulu recounted that he and Abu Maryam "agreed on the details" and planned to meet after Friday prayers. Abu Maryam showed up with three men and a van. Lulu stowed the men in a house, after which he and Abu Maryam went to case one of Ahrar al-Sham's offices in the al-Halak district of Aleppo.
On the third day, February 23, 2014, Lulu was present when some of Ahrar al-Sham's senior leaders, including Abu Khalid, showed up for a meeting. Lulu "left and informed Abu Maryam in precise language." Abu Maryam responded: "Do it and don't wait." Lulu secreted two of the three would-be assassins in a nearby village. "Then I went and confirmed that the leadership [of Ahrar al-Sham] was still present." It was. Lulu returned to the two men. "Go!"
Local activists reported that armed men charged into the meeting, firing their weapons until one of the gunmen detonated a bomb. The explosion killed six, Abu Khalid among them.
Soon after Abu Khalid's death, the rebel offensive against the Islamic State stalled. The rebels had reached the limit of their power and stretched themselves too thin. Over the ensuing months, the Islamic State consolidated its hold over eastern Syria, becoming its undisputed master. Its strategy of going it alone to capture and control territory may have alienated everyone, but it had paid off.
The Islamic State now had a vast war chest from captured oil fields, taxes, and loot, and it ruled its territory unchallenged. Its Sunni opponents were weakened, and President Assad had turned a blind eye, happy to let the Islamic State threaten his domestic and foreign enemies as long as it didn't threaten him; 90 percent of all Syrian air assaults against the rebels fell on the Islamic State's competitors.
Thousands of fighters left Nusra and the other rebel groups to join the Islamic State. Some wanted to play for the winning team, some believed it was doing God's work, some saw it as the Sunnis' only hope, some had no choice, and some just wanted to make a little money. Then there were the foreigners who had come to fight in the final battles of the apocalypse and usher in the caliphate reborn. The Islamic State welcomed them all.
Five
Sectarian Apocalypse
The chaos unleashed by the Arab Spring revolts prompted many Arab Muslims to wonder whether the end of the world was nigh. Even when the protests were peaceful, the sudden churn of Arab politics after decades of stagnation fired the apocalyptic imagination. Theories circulated online that Hosni Mubarak, the deposed president of Egypt, had really been the Antichrist prophesied in early Islamic scripture. Others swore they had seen End-Time heroes moving among the demonstrators in Egypt and Tunisia. Half the Arabs polled in 2012 believed the Mahdi, the Muslim savior, would appear any day. The mounting violence in Syria, or al-Sham, the land of the eastern Mediterranean mentioned in Islamic prophecies as the site of the final battles of the apocalypse, made the doomsday interpretation of events hard to resist.
Musa Cerantonio, an Australian convert to Islam, watched the events unfolding with glee. "There is a reason why Syria, or Sham more specifically, is special for the believers," he told a Melbourne audience in 2012. "It holds a very specific, special, and strategic place in the future" of the Muslim community. God had revealed its future to Muhammad, Cerantonio said, so the Muslims could prepare themselves for battle. That future had arrived.
Sunni prophecies attributed to Muhammad laud the region of al-Sham as the gathering place for the final battle against the infidels. "Go to Sham . . . and those who are not able to go to Sham should go to Yemen," says one Sunni prophecy credited to Muhammad. "Al-Sham is the land of gathering" for the Day of Judgment, according to another.
Although Sham encompasses the entire eastern Mediterranean, jihadists often equate it with modern Syria. It helps that many of the important apocalyptic sites mentioned in the prophecies are located around Damascus. "The Muslims' place of assembly on the day of the Great Battle will be in Ghouta near a city called Damascus, one of the best cities in al-Sham," says one. Ghouta is where the Assad regime killed hundreds with sarin gas in 2013. "A group in my community will continue fighting at the gates of Damascus and its environs and at the gates of Jerusalem and its environs," says another prophecy.
Their relevance to the current conflict is heightened by the fact that these prophecies mention the same places that are the scenes of today's battles, which makes them useful for jihadist recruitment pitches. "O youth of Islam!" declared Abu Bakr al-Baghdadi. "Go forth to the blessed land of Sham!" "Come to your State to raise its edifice. Come . . . for the Great Battles are about to transpire." The apocalyptic pitch "always works," confided an Islamic State fighter to a reporter.
Musa Cerantonio often quoted the Sham prophecies to persuade young Muslims to fight in Syria under the banner of the Islamic State. Cerantonio himself claimed to join them in July 2014, tweeting: "I have arrived in the land of . . . Ash-Sham!" Cerantonio was actually in the Philippines, where he had been living since February of that year. He was arrested a week after his tweet because the Australians had cancelled his passport.
The Strangers
Jihadists, especially foreigners who travel to fight in distant lands, call themselves "strangers." They are strange, they claim, because they adhere to the true Islam that most Muslims neglect. They are strange because they have abandoned their countries for foreign lands to fight the final battles against the infidels.
"Islam began as something strange," the Prophet told his companions, "and it will return to being something strange as it first began, so glad tidings to the strangers." "Who are the strangers?" someone asked. "Those who break off from their tribes," the Prophet replied.
For jihadists, leaving their tribes means leaving their homelands and emigrating to fight elsewhere, just as the Prophet's companions, "the emigrants," did. In Arabic, the word for "stranger" is gharib. The plural is ghuraba'. The word can also mean "foreigner," which is apt for the foreign jihadists who volunteer to fight in distant lands. Ghuraba' is often the name of the camps they set up, and it's the title of a popular hymn they chant. When Abu Mus'ab al-Zarqawi left for Afghanistan in the late 1980s, he called himself al-Gharib, "the Stranger."
Most of the prophecies about the strangers are found in a medieval compendium of the Prophet's words and deeds. In a section titled The Book of Tribulations, End-Time prophecies intermingle with descriptions of the strangers, giving them an apocalyptic hue. These prophecies are of a piece with others of a "saved group" of Muslims who will fight the infidels until the Day of Judgment.
Jihadists of all stripes, not just Islamic State followers, have been stirred by the promise of fighting in the final battles preceding the Day of Judgment. "If you think all these mujahideen came from across the world to fight Assad, you're mistaken," said a jihadist fighting in Aleppo. "They are all here as promised by the Prophet. This is the war he promised—it is the Grand Battle." Another fighter in northern Syria believed the same. "We have here mujahideen from Russia, America, the Philippines, China, Germany, Belgium, Sudan, India and Yemen and other places. They are here because this [is] what the Prophet said and promised, the Grand Battle is happening." God "chooses the best of people to come" to Sham, asserted Abu Muthanna, a Yemeni from Britain. "You see where the muhajirin are," he said, using the Arabic term for "emigrants." "This is the biggest evidence that they are upon the haqq," or truth.
Many of the emigrants or strangers have flocked to the Islamic State's banner. A popular gray-bearded Tunisian commander goes by the nom de guerre "Father of the Strangers" (Abu al-Ghuraba'). The Strangers Media Foundation produces propaganda supporting the Islamic State and criticizing its jihadist detractors. A YouTube video titled "Strangers—Islamic State in Iraq and Sham—Pictures from the Land of the Great Battles" depicts fighters from around the world. A Jordanian blogger collects Islamic State propaganda on his website, Strangers of the Lands of Sham. The strangers have found their home in the Islamic State.
Dabiq
The meadow outside the small village of Dabiq north of Aleppo, Syria, is an unlikely setting for one of the final battles of the Islamic apocalypse. Although close to the Turkish border, "Dabiq is not important militarily," observed a leader in the Syrian opposition. And yet the Islamic State fought ferociously to capture the village in the summer of 2014. Its members believed the great battle between infidels and Muslims would take place there as part of the final drama preceding the Day of Judgment.
Abu Mus'ab al-Zarqawi himself had stirred apocalyptic expectations by citing a prophecy about Dabiq. In it, the Prophet predicts the Day of Judgment will come after the Muslims defeat Rome at al-A'maq or Dabiq, two places close to the Syrian border with Turkey:
The Hour will not come until the Romans land at al-A'maq or in Dabiq. An army of the best people on earth at that time will come from Medina against them. When they arrange themselves in ranks, the Romans will say: "Do not stand between us and those [Muslims] who took prisoners from amongst us. Let us fight with them." The Muslims will say: "Nay, by God, how can we withdraw between you and our brothers. They will then fight and a third [part] of the army will run away, whom God will never forgive. A third [part of the army], which will be constituted of excellent martyrs in the eye of God, will be killed and the third who will never be put to trial will win and they will be conquerors of Constantinople.
For Zarqawi, Dabiq was the ultimate destination of the fire that had "been lit here in Iraq" by the "strangers"; they need only persist in their fight. The Islamic State's first commander of the faithful, Abu Umar al-Baghdadi, quoted the Dabiq prophecy too, reminding Muslims what his followers were fighting for.
The Dabiq prophecy probably first circulated in the early eighth century AD when the Umayyads attempted to conquer Constantinople, the seat of the eastern Roman (or Byzantine) Empire. There the Umayyad caliph Sulayman (Arabic for "Solomon"), stirred by a prophecy that someone bearing the name of a prophet would conquer Constantinople (Soloman is considered a prophet in Islam), prepared for an assault on the city in 716. Too sick to lead the campaign himself, Sulayman handed command to his brother, who led the caliph's troops in a failed siege on the "Roman" capital. Sulayman died waiting for the victory that never came.
Despite early mentions of Dabiq in Islamic State propaganda, the group's statements did not really focus on the village until 2014. In April, an Islamic State spokesman mentioned the ill-fated village as one of several places prophesied to fall to the jihadists. "You were promised Baghdad, Damascus, Jerusalem, Mecca, and Medina. You were promised Dabiq, Ghouta, and Rome." In July, the Islamic State released an English-language magazine named Dabiq. The editors, calling themselves the Dabiq team, explained why they adopted the name: "The area will play a historical role in the battles leading up to the conquests of Constantinople, then Rome." But first the Islamic State had to "purify Dabiq" from the "treachery" of the other Sunni rebels who held it and raise the flag of the caliphate over its land.
A few weeks later, Islamic State fighters took the village from Sunni rebels, killing forty and capturing dozens. The State lost twelve of its own men. Setting up snipers and heavy machine guns on the hill overlooking Dabiq, they repelled an attempt by the Free Syrian Army to retake the area. Islamic State supporters were jubilant, tweeting pictures of their flag from the hilltop together with quotes from the prophecy. "Dabiq is the most important village in all of Syria for them . . . especially the foreign fighters," explained one of the Muslim rebels who fought the Islamic State advance. In October, the State released a video of European jihadist fighters sitting on the same hilltop, daring the infidel West to intervene. "We are waiting for you in Dabiq," challenged Abu Abdullah from Britain. "Try, try to come and we will kill every single soldier." Abdul Wadoud from France quoted the Dabiq prophecy and another predicting that the infidel forces will gather under eighty flags. He assured his listeners that the Islamic State will defeat the infidel enemies allied against it because the prophecies say so.
Jihadist tweets about Dabiq spiked again in September, when the United States began to consider military action against the Islamic State in Syria. Supporters of the State counted the number of nations that had signed up for "Rome's" coalition against it. "Thirty states remain to complete the number of eighty flags that will gather in Dabiq and begin the battle" tweeted one. After Turkey's parliament approved military operations against the Islamic State in Syria and Iraq, the jihadist Twittersphere applauded. "Turkey's entry into the war will permit the foreign invasion of northern Syria, meaning from the plain of Dabiq. The battles [of the End Times] have grown near." "In Dabiq the crusade will end" tweeted another.
The last time the Turks invaded Dabiq, things didn't go well for the Arabs. The Turkish Ottoman sultan, Selim I, defeated the slave armies of the Mamluk Sultanate on the plain of Dabiq in 1516, which gave the Ottoman sultan control of the eastern Mediterranean and eventually Egypt and the Hijaz in western Arabia. The conquest inaugurated five hundred years of Ottoman rule over the Arabs and strengthened the Ottoman claim to the title of caliph. Selim I's grandfather, Mehmed II, had conquered Constantinople from the Byzantine Empire in 1453.
The fact that Turkish Muslims, not infidel Romans, control Constantinople, or Istanbul, today and are working with the infidel West against the Islamic State makes the Dabiq prophecy a poor fit for contemporary events. The inevitable defeat of the State at Dabiq, should it ever confront "Rome," would also argue against the prophecy's applicability. But in the apocalyptic imagination, inconvenient facts rarely impede the glorious march to the end of the world.
Islamic State fighters pray to God to "protect the Islamic State and support it until its army fights the crusaders near Dabiq." An essay on Dabiq was written by a woman claiming to be the mother of Adam Karim al-Mejjati, a child killed with his father who worked for al-Qaeda's branch in Saudi Arabia. Speaking of U.S. President Barack Obama, Adam's mother sneered: "The Creator lures the runaway slave and his dismal procession of evil creatures to the meadow of Dabiq in fulfillment of the Beloved's prophecy and as a lesson to all created things." When a masked British member of the Islamic State beheaded a U.S. aid worker in November 2014, he growled, "Here we are, burying the first American Crusader in Dabiq, eagerly waiting for the remainder of your armies to arrive."
Somewhere nearby, the tomb of the Umayyad caliph Sulayman lay in ruin, destroyed by Islamic State members who consider ornate graves idolatrous. The resting place of the man who wanted to fulfill prophecy by conquering Constantinople did not survive the zealotry of his modern heirs.
Christ and Antichrist
"You have seen us on the hills of al-Sham and on Dabiq's plain chopping off the heads that have been carrying the cross for a long time," proclaimed a camouflaged Islamic State narrator standing on Libya's Mediterranean coast in a video released in February 2015. A row of fourteen Egyptian Christians in orange jumpsuits kneeled before him and his black-clad brothers, members of the State's branch in Libya. Now, announced the narrator, the world would watch as they beheaded the Christians as part of a war with Christianity that will last until Jesus descends from heaven.
Anticipating Jesus's descent and executing his followers probably strikes most readers as odd. The Qur'an portrays Jesus as a messenger of God and his followers as those "nearest in love to the believers" (5:82). But the prophecies attributed to Muhammad outside the Qur'an foresee Jesus returning to fight alongside the Muslims against the infidels. As in the Bible, the appearance of Jesus heralds the Last Days. But instead of gathering the faithful up to heaven, he will lead the Muslims in a war against the Jews, who will fight on behalf of the Antichrist, called the Deceiving Messiah. Jesus will "shatter the crucifix, kill the swine, abolish the protection tax, and make wealth to flow until no one needs any more," says one prophecy attributed to Muhammad and quoted by the first emir of the Islamic State.
According to another prophecy imputed to Muhammad, God will send Jesus the Messiah when the Antichrist appears. He will "descend at the white minaret at the east of Damascus," which popular legend locates in the Umayyad Mosque. Afterward, Jesus will defeat the Antichrist. In February 2013, the Islamic State's spokesman proclaimed, "We will not lay down this flag until we present it to Jesus, the son of Maryam, and the last of us fights the Deceiver." "We will remain, by the permission of God, until the arrival of the Hour and the last of us fight the Deceiver," he vowed in another statement.
Islamic prophecies about the Antichrist give hints about his identity. In most accounts, he is a one-eyed Jew of grotesque appearance. He is either tall with a receding hairline or short with curly hair. His arms are hairy, and one hand is longer than the other. With such vivid clues to work with, Islamic State fans let their imaginations run wild. Most recently, the Antichrist is the blind Grand Mufti in Saudi Arabia, a baby born with one eye, or a Zionist.
Others interpret the prophecy figuratively. "The West is the one-eyed Deceiver," tweeted an Islamic State supporter after the January 2015 attack on the Charlie Hebdo newspaper office in Paris. "When we criticize them, we are anti-Semitic. When they insult our sacred things, it is freedom of expression. Bombing us is a fight against extremism. When we respond, we are terrorists."
There is one prophecy about the Antichrist that the Islamic State and its fans have studiously avoided, even though it is in a collection of prophecies they revere: The Antichrist will "appear in the empty area between Sham and Iraq." That, of course, is precisely where the Islamic State is located.
The Sufyani
Despite fighting bitterly against each other in Iraq and Syria, many of the Sunni and Shi'i militants drawn to the battlefield were motivated by a common apocalyptic belief that they fight in the vanguard of the Mahdi. "I was waiting for the day when I will fight in Syria. Thank God he chose me to be one of the Imam's soldiers," confided twenty-four-year-old Abbas, a Shi'i from Iraq who, like other Shi'a, believes the Mahdi is the twelfth imam, or leader descended from Muhammad. "With every passing day we know that we are living the days that the Prophet talked about," asserted Mussab, a Sunni fighting for al-Qaeda's branch in Syria, the Nusra Front.
Readers might puzzle at the incongruity of Muslims killing one another somehow fulfilling a prophecy of Muslims defeating infidels. But the early Islamic apocalyptic prophecies are intrinsically sectarian because they arose from similar sectarian conflicts waged at the time in Iraq and the Levant. As such, they resonate powerfully with today's sectarian civil wars.
Soon after the death of Muhammad in AD 632, civil wars, or fitan ("tribulations"), consumed the nascent Islamic empire as Muhammad's companions battled one another for political supremacy. The contest was framed in religious terms, which was unavoidable given that Muhammad and his immediate successors, the caliphs, wielded both spiritual and temporal authority. Before and after each tribulation, partisans on both sides circulated prophecies in the name of the Prophet to support their champion. With time, the context was forgotten but the prophecies remained. The word used for the upheavals of the apocalyptic Last Days, "tribulations," was the same word used for the early sectarian civil wars.
To understand the sectarian dimension of Islamic apocalyptic prophecies, consider the example of the enigmatic figure known as the Sufyani. According to the prophecies, the Sufyani descends from Abu Sufyan, the leader of Muhammad's tribe in Mecca who persecuted the Prophet and his early followers. Although Abu Sufyan and his family converted to Islam before the death of the Prophet, Abu Sufyan's son Mu'awiya later fought Muhammad's son-in-law Ali for control of the Islamic empire. Mu'awiya eventually became caliph, establishing the Umayyad dynasty that ruled for nearly a century.
As one might expect, the partisans of the losing side, called the Shi'a of Ali (Partisans of Ali and later just the Shi'a), began circulating words of the Prophet prophesying the new dynasty's downfall at the hands of the Mahdi, the "Rightly-Guided One." The Mahdi would be a member of the Prophet's family who would defeat the dynasty's champion, the Sufyani, in the Levant. "When the Sufyani reaches Kufa [a city in Iraq] and kills the supporters of the family of Muhammad, the Mahdi will come," according to one early prophecy.
Many Shi'a today believe the Sufyani's appearance is imminent, but they do not welcome it because of his antagonistic role in the apocalyptic drama. "All of us believe the Sufyani will fight the Imam [Mahdi], and the Prophet will kill him in the land of al-Sham," asserted a former leader of Kata'ib Hizballah, a Shi'i militia in Iraq. "We want to stress that what is happening in al-Sham is the beginning of the tribulation and is the beginning of the appearance of the army of the Sufyani, which is now called the Free Army." The "Free Army" is the Free Syrian Army fighting against Shi'i Iran's ally in Syria, Bashar al-Assad.
In contrast, Sunnis are more ambivalent about the Sufyani. Though most Sunni prophecies about him are negative, a few are positive because some Sunnis do not believe the Sufyani's kin, the Umayyads, were wrong to seize the caliphate from the Prophet's family. The Sufyani reconciles with the Mahdi in some prophecies and either swears allegiance to him or hands over the caliphate to him. In early Islam, Syrians were particularly partial to the Sufyani because his kin, the Umayyads, had ruled the caliphate from Damascus. A descendent of Abu Sufyan was proclaimed the Sufyani by his Syrian followers when he unwisely rebelled in Damascus against scions of the Prophet's family to restore the Umayyad dynasty around 750. As the ninth-century Iranian historian al-Tabari records, forty thousand rebels "proselytized on his behalf and said, 'He is the Sufyani who has been mentioned.'" The government killed him soon after.
Several decades later, another "Sufyani" rebelled in Aleppo. In 811, Umayyad supporters circulated prophecies of the Sufyani's imminent return, hinting that all signs pointed to an octogenarian descendent of the Umayyad dynasty named Abu al-Amaytar. The "Sufyani" managed to capture and hold Damascus for two years before his defeat at the hands of the same descendants of the Prophet's family who had defeated his Sufyani predecessors.
As in the ninth century, the Sufyani is a national hero for some Sunnis today, especially those in Syria. "God willing, all of us will be in the army of the Sufyani, who will appear in [Syria] by the permission of God," prayed Adnan al-Ar'ur, a popular Syrian Salafi cleric and supporter of the rebellion who currently lives in Saudi Arabia.
The sectarian wrangling over the identity of the Sufyani suggests an important difference between Islamic and Christian End-Time prophecies. Although both envision a fight between good and evil, the Islamic prophecies foretell a period of intracommunal fighting before the Day of Judgment. "The sword of vengeance will fall upon hypocrites before being turned against infidels," observes a scholar of Muslim apocalypticism. Syria and Iraq, ground zero for the final apocalypse, are also ground zero for the sectarian conflict that must precede it.
Yellow Flags, Black Flags
Early Islamic prophecies of the End Times resonated with the conflicts in Syria and Iraq not only because of their geographical setting; even the colors of the belligerents' flags—yellow for Hizballah, black for al-Qaeda and the Islamic State—seemed to match. "If the black banners and the yellow banners meet in the center of al-Sham," a companion of the Prophet prophesied, "the bowels of the earth will be better than its surface."
Some Shi'a identified the yellow flag with Hizballah and the black flag with the Sunni jihadists. "When those who carry the yellow flag engage in a conflict with anti-Shi'a elements in Damascus and Iranian forces join them, this is a sign and a prelude to the coming of his holiness [the Mahdi]," stated Ruhollah Hosseinian, a member of Iran's parliament and the former deputy intelligence minister. "We see that [now] the masters of the yellow flag, meaning Hizballah of Lebanon, are engaged with anti-Shi'a groups in Damascus. Perhaps this is the event that promises the appearance of his holiness, and we must prepare ourselves."
Sunni jihadists saw it the same way. "The [prophecy] of the black banners and the yellow banners is playing out in al-Sham," asserted a Sunni tweeter. "The mujahids have the black flags and Hizb al-Shaytan has the yellow flags." Hizb al-Shaytan, or the Party of Satan, is the jihadist put-down of Hizballah, the Party of God.
When Shi'a and Sunnis assign the yellow banners to Shi'i Hizballah and the black banners to al-Qaeda or the Islamic State, they miss two historical ironies: First, the black flags were originally associated with the Shi'a and others who wanted to restore the caliphate to the Prophet's family, as described in an earlier chapter.
Second, the people who first inspired the yellow flag prophecies were quite anti-Shi'a and lived in North Africa, far from Hizballah's home base in Lebanon. The yellow flag prophecies date to the time of the Berbers' revolt against the Umayyad and then Abbasid governors of the Maghrib, or North Africa, in the 740s and 750s. In one instance, Berbers plundered Kairouan (in present-day Tunisia) in 757 and massacred its inhabitants. They were cheered on by the extreme "yellow" (Sufri) Kharijites, an Islamic sect whose founders had excommunicated and assassinated their enemies, including the first Shi'i imam, Ali.
The Umayyads and their successors, the Abbasids, feared the Berbers would sweep farther east, a fear documented in numerous prophecies of yellow flags invading the Levant from North Africa. According to one prophecy, "the Berbers will come forth to the center of Syria, that will be the sign for the emergence of the Mahdi." Another predicted "a tremor in Syria in which a hundred thousand will be killed and which God will make an [act of] mercy for the faithful and a punishment for the infidels. When that will happen, look out for the men with the grey nags and the yellow flags advancing from the Maghrib until they descend in Syria."
One might expect that the recent entry of infidel armies into Iraq and Syria would lessen the internecine tone of the modern-day prophesying and focus attention on the Mahdi's battle with the infidels. But it has only heightened the sectarian apocalyptic fervor as each sect vies to destroy the other for the privilege of destroying the infidels. Little wonder such a heady enactment of the End-Time drama on the original stage where it was first rehearsed has drawn an unprecedented number of Sunni and Shi'i foreign fighters to the theater. In early 2015, the number of Sunni foreign fighters in Syria and Iraq had reached twenty thousand, most of whom had joined the Islamic State. The number on the Shi'i side was comparable. In the sectarian apocalypse, everyone has a role to play in a script written over a thousand years ago. No one wants to miss the show.
Satan's Slaves
Not all early Islamic prophecies map onto the current conflict so easily. In some cases, the Islamic State has gone out of its way to explain its actions as a fulfillment of prophecy.
Consider the Yazidis, a religious minority in Iraq. Many Muslims wrongly accuse them of devil worship because Yazidis believe the devil was a fallen angel who eventually repented. Many Yazidis live around the Sinjar Mountains close to the Syrian border and the territory held by the Islamic State, which besieged the area in August 2014. Before attacking the Yazidis, the State asked its own scholars if they could be enslaved. The scholars answered affirmatively, arguing that Islamic law permitted the enslavement of Yazidi women on the grounds that they are mushriks (polytheists) and not members of any protected religion mentioned in the Qur'an. "This large-scale enslavement of mushrik families is probably the first since the abandonment of this Shari'ah law," glowed the author of an article explaining the decision in the Islamic State's online magazine.
Although the Qur'an sanctions slavery, Muslim countries formally forbade the practice in the nineteenth and twentieth centuries. In 1981, Mauritania became the last Muslim-majority country to abolish slavery, though it continues informally there and in a few other Muslim countries. In the fall of 2014, the Islamic State proudly celebrated the return of the practice to public view and distributed captured Yazidi "virgins" as sex slaves to its members.
The Islamic State not only cheered the revival of slavery as a major step in the return of Islamic law, which the group wants to impose in its totality; it also hailed slavery's renewal as "one of the signs of the Hour," or Day of Judgment. According to a prophecy attributed to Muhammad, the Prophet foretold the Hour would be close when "the slave girl shall give birth to her master." The prophecy's wording is not clear about the return of slavery, but the Islamic State argued its import was obvious: Slavery is prohibited today, so a slave girl giving birth to her master must mean slavery will return.
As further proof that its interpretation of the prophecy was correct, an Islamic State author cited the Dabiq prophecy. According to that prophecy, the Romans will line up against the Muslims near the small town of Dabiq, Syria, and say, "Leave us and those who were enslaved from amongst us so we can fight them."
The Islamic State sold some of its Yazidi slaves in markets. Worried about price deflation, the State issued a decree in October 2014 fixing the prices for the females: 200,000 Iraqi dinars (around US$170) for children between the ages of one and nine. The price dropped $40 for every ten additional years of age. Christian slaves fetched the same price. The slaves were bought by jihadists and human traffickers, who reportedly shipped them to homes and brothels across the Middle East.
Yazidi children are not shielded from the sexual advances of the Islamic State's fighters. "Is it permissible to have intercourse with a female slave who has not reached puberty?" asked a pamphlet distributed by the fighters in mosques throughout Mosul. "Yes, if she can have intercourse. If she cannot have intercourse, then one should enjoy her without intercourse."
Many of the Yazidi women were shipped to the Islamic State's stronghold in Raqqa, Syria, where they were forced to provide sex on demand for the Islamic State's soldiers. The women's jailers were other women, many of them European Muslims in their teens or early twenties. They were members of the Khansa Brigade in Raqqa, where they roam the streets with guns and grenades policing "un-Islamic" behavior. They offer women who flout the strict dress code a choice between flogging and the "biter." "I did not know what a 'biter' was," recalled one woman. "I thought it is a reduced sentence, I was afraid of whipping, so I chose the 'biter.'" The Khansa women pressed a "sharp object that has a lot of teeth" to her breast. "I screamed from pain and I was badly injured. They later took me to the hospital." "I felt then," she recalled, "that my femininity has been destroyed completely."
The Khansa Brigade is named for a pagan poetess who converted to Islam in the time of Muhammad. "You are the greatest poet among those who have breasts," Khansa was once told by the caliph Umar. "I am the greatest poet among those who have testicles too," she replied.
The European women who have joined the brigade share Khansa's pride, but in service of the Islamic State's men, who do the fighting. "The primary role of the woman who has emigrated here is supporting her husband and his jihad" tweeted a woman calling herself Khansa. A twenty-year-old Scottish woman leading the Khansa Brigade, Aqsa Mahmood, encouraged other women to immigrate to the Islamic State and advised them on how to find a husband. Islamic State fangirls on Twitter mooned over the group's hirsute warriors. Marriage brings status; the death of a husband brings more, as well as financial compensation.
Status is not the only reason young women join the Islamic State. They share many of the men's motives: seeking adventure, wanting to be part of something larger than themselves, indignation at the suffering of their coreligionists, a chance to rebuild the caliphate, or a belief that they are living in the Last Days. And they are no less bloodthirsty. "So many beheadings at the same time, Allahu Akbar, this video is beautiful," exclaimed one after watching the Islamic State's men behead eighteen Syrian hostages and the American aid worker and Muslim convert Peter Kassig in November 2014.
The women of the Islamic State gleefully abide by and enforce the group's strictures on women. They stand guard as the State's men rape their captured sex slaves. "Here I can really be free," a teenage Austrian recruit texted. "I can practise my religion." The United Nations worries she may now be dead.
The Twelve Caliphs
In April 2014, the Islamic State's spokesman hinted that the group was about to make a major announcement. "O God! A state of Islam rules by your Book and by the tradition of your Prophet and fights your enemies. So reinforce it, honor it, aid it, and establish it in the land. Make it a caliphate in accordance with the prophetic method."
The last line alludes to a prophesy of the caliphate's return. "Prophethood will be among you as long as God intends, and then God will take it away when He so wills," Muhammad purportedly told his followers. His prophetic office would be followed by an Islamic empire governed by the Prophet's "successors," or caliphs. They would each rule as the spiritual and temporal head of the Muslim community as the Prophet had. After them, "There will be a mordacious monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a tyrannical monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method."
Afterward, the Prophet "fell silent."
Some Sunni Muslims today interpret the Prophet's silence to mean the world will end soon after the reestablishment of the caliphate. Among them are those who believe the real caliphate hasn't existed since the early days of the Islamic empire. Others think the last real caliphate was abolished by the secular Turkish leader Mustafa Kemal Atatürk in 1924 after the defeat and dismemberment of the Ottoman Empire in World War I. Either way, the caliphate is destined to return.
Jihadists frequently cite the prophecy because they are fighting to restore the caliphate. But most jihadists see it as a distant goal that will be reached after the Muslims unite and regain their former glory. In 2013, for example, al-Qaeda's chief Ayman al-Zawahiri counseled jihadists to focus on unifying the Muslim world. Then they could consider "the establishment of the caliphate in accordance with the prophetic method."
The Islamic State was impatient. Why wait for the fractious Muslim world to unite when the State already controlled so much land? Why pine for a caliphate later when they could have a caliphate now? The Islamic State knew there were skeptics, most of all among the jihadists, who would say the time wasn't right or the State wasn't qualified. But prophecy demands to be fulfilled.
To make the case, the Islamic State turned to its ablest apologist, twenty-nine-year-old Turki ibn Mubarak al-Bin'ali. A young firebrand from Bahrain, in his early twenties Bin'ali was already running afoul of the authorities. While studying religious subjects at a college in Dubai, he was arrested for "extremist ideas" in 2005. Returning to Bahrain, he tried teaching at a school but was kicked out for proselytizing his jihadist beliefs. Bin'ali's next gig as an imam at a mosque didn't work out either: He was fired for hanging up a jihadist fatwa. When the Syrian civil war started, Bin'ali traveled to Syria for relief work and to spread his message in missionary camps. In March 2013, he announced that he had joined the Islamic State. Bin'ali quickly became one of its most prominent apologists, reportedly overseeing its powerful Shari'a committee.
Two weeks after the Islamic State hinted that it was going to fulfill Muhammad's prophecy of a "caliphate in accordance with the prophetic method," Bin'ali wrote a treatise on the subject. He had to establish, first, that the caliphate would return; second, that the Islamic State merited the title; and third, that Abu Bakr al-Baghdadi deserved the job of caliph.
To establish that the caliphate would return, Bin'ali quoted another Islamic prophecy that there will be "twelve caliphs" from the Quraysh, the tribe of Muhammad. Of course, there had already been far more than twelve caliphs descended from the Quraysh tribe, so Bin'ali sided with those who interpreted the prophecy as requiring twelve just caliphs. There had already been five, six, or seven, so only a handful more were destined to appear. Bin'ali cited another prophecy indicating that, in the future, the final caliphs would be direct descendants of Muhammad's son-in-law Ali. The last one either would pave the way for the Muslim savior, the Mahdi, or would be the Mahdi himself.
As for whether the Islamic State should be considered the caliphate, Bin'ali contended the State was already the "nucleus of the anticipated rightly-guiding caliphate." The caliphate requires "power, authority, and control of territory," he noted, which the Islamic State possessed. Bin'ali disputed the idea that the caliphate should extend over all Muslim lands before it is declared or that all Muslims had to select the caliph. The first caliph, Abu Bakr, he pointed out, lost almost all of the Muslims' territory only to regain it later. Many Muslims rejected the fourth caliph, Muhammad's son-in-law Ali, in favor of a rival claimant. Ali's son Hasan controlled even less territory when he was briefly proclaimed caliph; his brother Husayn controlled nothing at all. The later Abbasid insurgents who overthrew the Umayyad caliphs and proclaimed their own caliphate were also afraid to publicly declare their allegiance to their imam at first. The overthrown Umayyad caliphs went to Andalusia (modern Spain) and proclaimed a new caliphate. Muhammad himself controlled only a small amount of territory when he established a state, and his followers pledged themselves to him long before that. Even when Muhammad began establishing his state, he had not subdued every faction within it.
Bin'ali's case that Abu Bakr al-Baghdadi should get the job of caliph is found in a separate treatise. As we have seen, Bin'ali reckoned that a "caliphate according to the prophetic method" would be reestablished by a direct descendent of Ali, Muhammad's son-in-law. So Bin'ali traced Baghdadi's lineage to Ali through a brother of the eleventh Shi'i imam, a startling pedigree for the man hell-bent on eradicating the Shi'a. Conveniently for Baghdadi, the brother of the eleventh imam had denied the existence of the twelfth imam, whom the Shi'a believe went into hiding and will return at the end of time as the Mahdi. (Never mind that Baghdadi's forefather instead claimed the title of twelfth imam for himself.)
Bin'ali next went about establishing Baghdadi's religious credentials. In his telling, Baghdadi obtained a bachelor's and a master's degree in Qur'anic studies and a doctorate in Islamic jurisprudence. Baghdadi even published a book on the rules of Qur'anic recitation and worked as an imam in several mosques. Some of that isn't true—Baghdadi's dissertation was on Qur'anic recitation, and it was never published—but close enough. "Thus is combined in Shaykh Abu Bakr what is separated in others," Bin'ali exclaimed. "Knowledge that ends with the Prophet, peace and blessings be upon him, and a lineage that ends with the Prophet, peace and blessings be upon him!"
Having demonstrated Baghdadi's familial and intellectual bona fides, Bin'ali addressed his managerial experience. During the early days of the U.S. invasion of Iraq, Baghdadi was in positions of authority in some jihadist groups and then rose through the ranks of the Islamic State to become the chief judge. When the head of the Islamic State was killed, the State's senior officials appointed Baghdadi their new leader.
Bin'ali acknowledged that other jihadists had raised doubts about Baghdadi's suitability. They say, "How can Shaykh Abu Bakr al-Baghdadi be the emir when all of the people have not pledged allegiance to him?" Bin'ali replied that Islamic law does not require unanimous agreement on an emir. Requiring everyone to agree is the evil stuff of democracy. But then they say, "How can the authority of Shaykh Abu Bakr al-Baghdadi be recognized when some of the regions were conquered by force but the leaders of the community there have not given their oath of allegiance?" Bin'ali responded that Muslim scholars have ruled that people are to obey a Muslim who conquers them. But "How can the pledge to Shaykh Abu Bakr al-Baghdadi be correct when he is unknown?" Bin'ali parried that he is known to the jihadists and, regardless, Islamic law does not require that every person know the emir personally, only their leaders must. Were it a requirement, the Abbasid family that clandestinely gathered oaths of allegiance before establishing Islam's mightiest caliphate would never have come to power. Still, "How can the authority of Shaykh Abu Bakr al-Baghdadi be correct when he does not have full territorial control?" Not even the Prophet was held to such a high standard! Bin'ali replied.
Bin'ali either knew or anticipated that the Islamic State was about to declare the caliphate with Baghdadi at its head. "The authority of his state has spread throughout Sham, which is why the shaykh became the 'Commander of the Faithful in the Islamic State in Iraq and Sham.' We beseech God to hasten the day in which we will see our shaykh seated upon the throne of the caliphate!"
When Bin'ali's teacher and critic of the Islamic State Abu Muhammad al-Maqdisi read the treatise about the twelve caliphs and its title, "An Investigation into Whether Absolute Political Power Is a Prerequisite for the Caliphate," he immediately understood its implication. "They will rename their organization 'the caliphate' imminently," Maqdisi foretold. "All of us hope for the return of the caliphate," he acknowledged. But it must actually exist to merit the name. "Announcing something prematurely sets it back on account of its absence." "What concerns me greatly is . . . whether this caliphate will be a refuge for oppressed people and a haven for every Muslim or will become a sword hanging over the Muslims who oppose it."
The Islamic State had been animated by the apocalyptic abiogenesis of the Iraq war and invigorated by the inrush of foreign fighters to Syria, many of them seeking a role in the End-Time drama. The rough beast born and raised in two sectarian civil wars was about to come of age.
Six
Caliphate Reborn
In early June 2014, a thousand Islamic State fighters converged on Mosul, the second-largest city in Iraq. With them were their Sunni allies—disgruntled tribesmen and former Saddam loyalists. They faced little resistance. The Iraqi troops and police guarding the city fled, having seen videos of what happened to anyone who opposed the black-clad fighters. Days later, the Islamic State raised its flag over the city's government buildings. It now dominated land stretching from Mosul to the outskirts of Aleppo in Syria, roughly the distance between Washington, DC, and Cleveland, Ohio. The land was once ruled by Nur al-Din Zengi, the scourge of the crusaders who so inspired Zarqawi, the founder of al-Qaeda in Iraq.
Flush with victory, the Islamic State's spokesman issued a proclamation three weeks later:
The sun of jihad has risen, and the glad tidings of goodness have shone forth. Triumph looms on the horizon, and the signs of victory have appeared.
Here, the flag of the Islamic State, the flag of monotheism, rises and flutters. Its shade covers land from Aleppo to Diyala. Beneath it the walls of the tyrants have been demolished, their flags have fallen, and their borders have been destroyed. . . .
It is a dream that lives in the depths of every Muslim believer. It is a hope that flutters in the heart of every mujahid monotheist. It is the caliphate. It is the caliphate—the abandoned obligation of the era. . . .
Now the caliphate has returned. We ask God the exalted to make it in accordance with the prophetic method.
With that, the caliphate was supposedly reborn and prophesy was fulfilled. All Muslims had to now bend the knee to Abu Bakr al-Baghdadi, renamed Caliph Ibrahim al-Baghdadi.
There had not been a credible claimant to the office of caliph in the Middle East since the defeat of the last Ottoman sultan in World War I. In early Islam, the caliphs had wielded great political and spiritual authority over all Muslims. Although the caliph's authority waned over time, the office still carried tremendous symbolic import among Sunnis. When the founder of secular Turkey abolished the office after World War I, many conservative Sunnis were aggrieved. In their eyes, the loss of the caliphate represented the end of Muslim political power and the triumph of the West. It is no great exaggeration to say that Sunni political Islam began as an effort to restore the caliphate. Several world congresses were convened in the 1920s and 1930s to name a new caliph, but political discord led to deadlock. No Muslim country wanted to see its rivals get the office.
Of course, individual Sunnis had tried to claim the caliphate for themselves, but they were ridiculed. A Jordanian jihadist in Afghanistan proclaimed himself caliph in the 1990s. As an al-Qaeda operative later commented, the caliph and his followers foolishly "excommunicated everyone who opposed them." Making too many enemies of people with guns, the movement crumbled quickly, and the caliph had to relocate to London. With no land, no government, and no followers, his claim to be caliph was silly.
All jihadists fight to restore the caliphate, but most see it as a distant goal. Either the Western nations are too powerful to allow it or Muslims are too divided to see it through. The caliphate will come one day, but first Muslims must become strong and united.
By aspiring to the caliphate, the Islamic State challenged the conventional wisdom among jihadists and other Sunni Muslims. You don't have to overthrow Muslim countries to make a caliphate, and you don't have to persuade them to declare one, argued the State. Conquer land and declare your own. You don't have to wait until the Muslim masses want the caliphate, and you don't have to beg them to support your caliphal project. Ignore popular opinion and establish a caliphate by force of arms. You don't have to cower in fear of the West or its allies. Defy them and defeat them.
By establishing a government and declaring it a caliphate, the Islamic State threatened to overturn conventional wisdom completely. Unlike any other rebel pretenders to the caliphate in the modern Middle East, the Islamic State had the money, fighters, weapons, and land to make a plausible case that it was the caliphate reborn. It helped that its caliph had more religious training than any political leader in the Muslim world. Most Sunni Muslims may have rejected the Islamic State as a travesty and a sham, but they could not easily dismiss it as a joke when it declared itself a caliphate in 2014. The Islamic State was too powerful.
Days after the June declaration of the caliphate, the new caliph rolled up in a black SUV to the Nuri Mosque in Mosul, named for Zarqawi's hero Nur al-Din Zengi, who had ordered it built in the twelfth century. Baghdadi wore a black robe and turban. His followers speculated he had donned the black either because the Prophet wore a black turban when he conquered Mecca or because Baghdadi wanted to signal his descent from the Prophet. Both could be true. The Abbasid caliphs, whose memory Baghdadi was invoking, had done the same.
"God, blessed and exalted, has bestowed victory and conquest upon your mujahid brethren, and has granted them power after long years of jihad, patience, and fighting the enemies of God," proclaimed Baghdadi from the pulpit. "They rushed to announce the caliphate and appoint a leader. This is a duty incumbent on Muslims, which had been absent for centuries and lost from the face of the earth." Echoing the words of the first caliph and his namesake Abu Bakr, the new caliph embraced his office and exhorted his flock to hold him to account. "I was appointed to rule you but I am not the best among you. If you see me acting truly, then follow me. If you see me acting falsely, then advise and guide me. . . . If I disobey God, then do not obey me."
The evocative dress and humble words were belied by the luxury watch the new caliph sported. And the black clothes didn't help soothe the jangled nerves of Mosul's residents, whose men were threatened with lashes if they didn't attend. "We were frightened by the way he wore his black clothes. We had never seen a Sunni imam completely dressed in black clothes," said one female eyewitness. "We women were on the second floor crying, terrified that they would hear us and hurt us. We waited for an hour after the sermon, where there were armed men guarding the doors."
Despite the fear of those living under the caliphate, jihadist fanboys of the Islamic State cheered its return. The announcement "will enter history" proclaimed a tweet on July 3, 2014. Other Islamic State fans declared the end of Western-defined borders. In a widely retweeted example, an activist posted two maps: The first has penciled-in borders for a "Sunni-stan," "Kurdistan," "Alawi-stan," and "Druze mountain." The other shows these borders obliterated by the caliphate. The author challenged readers: "Compare the West's dream of division to Baghdadi's caliphate." In a response to the tweet, an Islamic State enthusiast exclaimed, "the Islamic State has no borders, and its conquests will continue, with God's permission."
In its first incarnation in 2006, the Islamic State had fired the jihadist imagination with its ambiguous audacity. It called its emir the commander of the faithful, but it wouldn't outright call him the caliph. The Islamic State's name and flag harkened back to the medieval caliphate's name and flag, but it wouldn't declare a caliphate. The group encouraged others to view it as the caliphate but would never say so itself. Until now.
The Islamic State was nearly destroyed in 2008 because it tolerated no challenge to its authority as a state. For al-Qaeda's leaders, especially Bin Laden, the State's downfall was further proof that popular support had to be secured before a state could be established, much less a caliphate.
Despite its setbacks, the Islamic State's symbol and the idea it represented lived on. The flag of the Islamic State, still an al-Qaeda affiliate, was taken up by other al-Qaeda affiliates. Its state-building enterprise was taken up as well, much to the annoyance of Bin Laden. Some of al-Qaeda's affiliates, mindful of the Islamic State's poor precedent, tried to be more lenient toward the people they ruled and more considerate of their subjects' economic welfare. But they were not considerate enough and made too many enemies, suffering defeat as a consequence.
Despite the failure of its sister affiliates, the Islamic State doubled down on its state-building strategy during the Syrian civil war. When its subordinates in the Nusra Front and its leaders in al-Qaeda objected, the Islamic State rejected both. It would persevere no matter the consequences.
Were this Iraq in 2008, the Islamic State would have been defeated again, ratifying the jihadist conventional wisdom. At that time, the Americans simply had too many troops on the ground and the Sunni tribes were too willing to cooperate with the Shi'a-dominated government in Baghdad against a common foe. But in 2014, Iraq's Sunni tribes no longer trusted Baghdad, the Americans were gone, and government troops could no longer pacify the Arab Sunni hinterland in western Iraq.
The restless hinterland extended next door into Syria, but President Assad was more worried about direct threats to his stronghold closer to the Mediterranean coast. The Sunnis on the border of Iraq could be dealt with later. Assad also found it politically expedient to leave the Islamic State alone to attack his domestic enemies, scare the hell out of his citizens, and terrify the foreign countries that opposed him. He calculated that the average Syrian, Arab, and Westerner would prefer him to the greater evil of the Islamic State.
As a consequence of Baghdad's and Damascus's policies, the restive Sunnis between Syria and Iraq were ripe for the ruling by someone who wanted to establish a state and had enough manpower, muscle, and managerial experience to do it.
The Islamic State fit the bill. Its apocalypse-laced recruiting pitch had drawn thousands of foreign fighters, as had its political theology and state-building program. That gave it enough men to subdue the towns it captured, drive off its competitors, and raise revenue. The Islamic State was used to organizing itself as a state, and it was staffed by former officers from Saddam's military and intelligence services who could run it.
The Islamic State had once been an object lesson in what not to do. Its critics, jihadist and non-jihadist alike, had attributed its defeat in 2008 to its brutality, zealotry, and arrogant belief that it was a state. But by 2014, those were the very qualities that made the Islamic State so successful. While other rebel groups worked together to overthrow governments, the State was busy creating its own.
Branding the Caliphate
When the Islamic State's spokesman proclaimed the caliphate, he prayed that God would make it a "caliphate in accordance with the prophetic method" and thus fulfill prophesy. After the proclamation, the Islamic State went about making sure everyone knew God had answered affirmatively.
Across Syria and Iraq, billboards popped up at Islamic State checkpoints. On the right side, the black flag proclaimed "No god but God. Muhammad is the messenger of God." On the left: "The Islamic State: A Caliphate in Accordance with the Prophetic Method." Islamic State soldiers wore patches emblazoned with the latter slogan, and the group's official letterhead included the words. Photos of State members baking bread in Iraq's Diyala Province circulated on Twitter with the tagline. The phrase was the title of the State's very first Da'wa Gathering, a proselytizing festival held in Raqqa in autumn 2014. Pictures of the event posted on Twitter show an outdoor venue filled with rapt attendees—including some wide-eyed boys—listening to preachers.
The phrase also appeared on coins the Islamic State planned to mint. Not only were the coins meant to end the State's reliance on "dollar-linked fiat currencies;" but also to evoke the coins of the early Islamic caliphate, which are mentioned often in medieval histories of the period. Never mind that there were no "Islamic" coins in the time of Muhammad, an uncomfortable fact the Islamic State acknowledged in passing. Muhammad and his companions used Roman and Persian currency, which bore the images of non-Muslim emperors and religious symbols. It was not until the time of the Umayyad caliphs that Muslims began minting their own coins. (Shocking to later Sunni Muslim sensibilities, the Umayyads may have put an image of Muhammad on their coins.) For a group so against religious innovation and so obsessed with creating a state on the model of the Prophet, the Islamic State sometimes ignored the Prophet's example altogether when he fell short of its exacting standards or failed to sanction something it wanted.
In the eyes of Muslims, the juxtaposition of the slogan and the Islamic State's cruelty strained its claims to represent the Prophet's method. A gate in the town of Hawija, Iraq, welcomed visitors to the "Caliphate in Accordance with the Prophetic Method." Beneath it swung eight lifeless Iraqi troops, suspended by their feet. Horrified Muslims circulated the image on Twitter with the bodies and the slogan highlighted in red.
The Islamic State's personnel and sectarianism also made its slogan an object of satire. One flier circulated online read "A Caliphate in Accordance with the Ba'athist Method." The Ba'ath was the socialist political party that dominated Iraq under Saddam Hussein. Many former Ba'athists now ran the Islamic State.
When Shi'i militias recaptured a town from the Islamic State after the caliphate was announced, they scaled the local water tower festooned with the Arabic slogan and added two dots. The slogan now read "The Caliphate of Ali, a Prophetic Method," alluding to the first Shi'i imam.
As any business knows, brand management isn't easy. But the Islamic State didn't go out of its way to dissociate its slogan from its atrocities.
Debating the Caliphate
The jihadist old guard scoffed at the Islamic State's announcement of the caliphate. In their lead were Abu Qatada al-Filistini and Abu Muhammad al-Maqdisi, the two most influential jihadist scholars alive. Both were Palestinian Jordanians who had written some of the seminal texts in jihadist literature. They wanted the caliphate to return through violent revolution but were unimpressed with the Islamic State's claim. In their eyes, the mere trappings of a caliphate do not a caliphate make.
"There are simpletons who have deluded themselves with their announcement of the caliphate and the application of the hudud penalties," sneered Abu Qatada, referring to the fixed punishments mentioned in Islamic scripture. "The rational person remembers where they came from and what they have done. You see they have dedicated themselves to killing Muslims and mujahids." It is a "heinous conspiracy," charged Maqdisi. How else could one explain the establishment of a state that sought to kill its jihadist opponents, destroy its scholars, shun popular support, and soak the concept of the caliphate in blood?
Furious at the criticism, the Islamic State's spokesman warned darkly: "All who try to sever the ranks will have their heads severed."
The mutual recriminations became so complex and heated by August 2014 that one jihadist foe of the Islamic State even compiled a series of talking points, "A Summary for Discussing the Announcement of the Caliphate." To summarize the summary: The Islamic State's caliphate is illegitimate because a majority of Muslim leaders had not endorsed it. Not even a majority of jihadist leaders had endorsed it. The only people who elevated Baghdadi to caliph were senior leaders in the Islamic State itself.
Unsurprisingly given its fallout with the Islamic State, al-Qaeda also condemned the group's declaration of a caliphate. "We call for the return of the rightly-guiding caliphate according to the prophetic method," declared a senior al-Qaeda leader, "not according to the method of deviating, lying, violating treaties, and breaking pledges. A caliphate based on justice and consultation, affinity and concord, not oppression, excommunication, murder of monotheists, and dividing the ranks of the mujahids."
The head of al-Qaeda, Zawahiri, took a different tack. Rather than denounce the caliphate, he began to hint there was already a caliph: Mullah Omar, the head of the Taliban in Afghanistan. Bin Laden and Zawahiri had pledged their allegiance to Mullah Omar as the commander of the faithful. Although the title is usually reserved for the caliph, Zawahiri had previously taken pains to assure his followers that Mullah Omar was only the commander of the faithful in Afghanistan. "As for the commander of the believers across the world," Zawahiri said in 2008, "this is the leader of the caliphal state that we, along with every faithful Muslim, are striving to restore, God willing." Zawahiri and his boss, Bin Laden, also didn't take their oath to Mullah Omar very seriously, having repeatedly defied his demand that they stop talking to the media.
But after Baghdadi was proclaimed caliph, al-Qaeda began to promote Mullah Omar as a countercaliph. In July 2014, al-Qaeda's media wing released an old video of Bin Laden explaining his decision to give his oath of allegiance to Mullah Omar as commander of the faithful. A questioner asked Bin Laden if his oath implied that he considered Mullah Omar to possess "supreme leadership," the prerogative of the caliphs, which Bin Laden affirmed. Later that same month, al-Qaeda released a newsletter that began with a renewal of the oath of allegiance to the "Commander of the Faithful Mullah Muhammad Omar" and "affirm[ed] that al-Qaeda and its branches in all locales are soldiers in his army, acting under his victorious banner."
In September, Zawahiri upped the ante when he announced the establishment of a new al-Qaeda affiliate, "al-Qaeda in the Indian Subcontinent." Zawahiri stressed the new group was, like al-Qaeda, under the authority of the "Islamic Emirate" ruled by the "commander of the faithful," Mullah Omar. He then proceeded to heap praise repeatedly on the "commander of the faithful."
Still, Zawahiri stopped short of proclaiming Mullah Omar the caliph. He did not explain why, but his reasons are not hard to guess. If Mullah Omar was still alive, which was unclear, he might not want the job. Claiming to rule Afghanistan is much more modest than claiming to rule the entire Muslim world, which would alienate potential allies, such as some of the Gulf states. Moreover, many jihadists had criticized Baghdadi and the Islamic State for declaring the caliphate too soon. Al-Qaeda would be subject to the same criticism if it aped the State (and Zawahiri would not even control the caliphate he declared). Better for the time being to walk the ambiguous middle way between forthrightly declaring a countercaliph and having no caliph at all.
Despite al-Qaeda's protestations, the Islamic State's declaration of the caliphate drew recruits from all over the world. "To all my brothers in Tunisia who have Jihad in their hearts," proclaimed a young Tunisian, "the Caliphate has been established . . . it is a great blessing." For its admirers, the caliphate promised a place of honor for Muslim youth who felt shut out by their political systems or alienated from their societies. A popular recruitment video declared the bankruptcy of nation-states by showing Islamic State citizens ripping their passports: "[We are] tearing these passports, these identities, and . . . these borders, and we will live in one Islamic state. We will spread from the West to the East, and no one but great God will rule us. One Islamic nation, in which we are not linked with identities, cards, or passports."
"We have brothers from Iraq, from Cambodia, from Australia, the UK," proclaimed British jihadists fighting for the Islamic State. A future Egyptian "martyr" extolled the diversity of the group as a reason for other Arabs to join: "Why don't you emigrate? Why don't you leave your homes? Why don't you leave your home as the Prophet left his? Entire families are coming from Uzbekistan. . . . We have families coming from Uzbekistan, from Turkey, from everywhere!"
The Islamic State's momentum also drew fighters from rival rebel groups, including al-Qaeda's official Syrian arm, al-Nusra. From the outset, the two groups attracted more foreign fighters than the other rebel factions in Syria, and both had received a large influx of men from the Free Syrian Army who left the flailing rebel conglomerate for the better-funded and better-organized jihadist groups. But after the Islamic State and Nusra parted ways in 2013, a reported 60 percent of Nusra's foreign fighters went to its rival.
The Baghdad of al-Rashid
The Islamic State's claim to have reestablished a caliphate according to the prophetic method has misled at least one observer to conclude that the State only seeks "its inspiration from the first four caliphates" that followed Muhammad. As the argument goes, not even the grandest of the caliphates, the Abbasid, is a model for the Islamic State.
This is not quite right. The Islamic State was no doubt inspired by the Prophet's model of state building, but it was not the only model that galvanized the group. Take a look at the State's propaganda, and you will see that its leaders have sought, from its founding, to restore the glory days of the Abbasid caliphate that ruled its empire from Baghdad. The State especially celebrated the era of Harun al-Rashid of 1,001 Nights fame. "Know that the Baghdad of al-Rashid is the home of the caliphate that our ancestors built" proclaimed an Islamic State spokesman in 2007. "It will not appear by our hands but by our carcasses and skulls. We will once again plant the flag of monotheism, the flag of the Islamic State, in it." "Today, we are in the very home of the caliphate, the Baghdad of al-Rashid," stated the Islamic State's first ruler, the aptly named Abu Umar al-Baghdadi, the same year. Even after the Islamic State established its primary base of operations in Syria's Raqqa, once a second home to Harun al-Rashid, and captured Mosul in Iraq, its spokesman still referred to "the Baghdad of the Caliphate" or "the Baghdad of al-Rashid." The first caliph of the Islamic State, Abu Bakr, used the alias of al-Baghdadi, "of Baghdad," even though he hailed from Samarra.
The Islamic State's plan to revive the Abbasid caliphate in Baghdad suffers from two problems. The first is cultural: The values of Harun al-Rashid and his court are not the values of the Islamic State. One of Harun's favorite court poets, Abu Nuwas, often wrote verses glorifying wine drinking and sex with boys:
A boy of beckoning glances and chaste tongue . . .
Proffers me wine of hope mixed with despair.
The caliph himself was frequently drunk. In one account, he ordered the death of his boon companion (some say lover), Ja'far the Barmakid, after drinking too much wine. When the assassin refused to carry out the caliph's order because he was inebriated, Harun screamed, "Bring me Ja'far's head, motherfucker!"
Music was also a fixture in the Abbasid court. Although many conservative Muslims frowned on musical instruments then as they do today, Harun's nights were filled with the sounds of lutes, trumpets, and drums. For ultraconservatives like those leading the Islamic State, the devil uses such instruments to entice the unwary. Better to listen to the unaccompanied voice of males singing battle hymns.
In contrast to the State's pogrom against the Shi'a, Harun's court usually got on well with them. Harun's advisors, the Barmakids, were Persian converts from Buddhism who were friendly with the Shi'a. Harun was also on good terms with the Shi'a early in his reign, and his son Ma'mun even nominated a Shi'i imam to succeed him as caliph.
Pagan Greek learning and interreligious debates were celebrated in the Baghdad of Harun, also anathema to the Islamic State. Harun and his advisors funded translations of Greek tracts on science and philosophy, preserving texts that would otherwise be lost in medieval Europe. Jews, Christians, Muslims, and irreligious philosophers debated in court and in private salons around Baghdad. As a British scholar put it, "The pious Muslims of Mecca and Madina who came thither were scandalized to find unbelievers invested with the highest offices at Court, and learned men of every religion holding friendly debate as to high questions of ontology and philosophy, in which, by common consent, all appeal to revealed Scripture was forbidden."
The Islamic State's leaders studiously ignored Harun al-Rashid's religious shortcomings, however, celebrating instead his jihads against Christian Byzantium. But here too Harun is out of step with the State's religious sensibilities, which require constant war against the unbelievers. Harun, the Muslim ruler of North Africa and the Middle East, exchanged several diplomatic embassies with Charlemagne, the premier ruler of the Christian West. One of the gifts Harun sent was an elephant named for the first Abbasid caliph, Abu al-Abbas. The elephant, delivered by a Jew on the caliph's behalf, became an exotic fixture in Charlemagne's court. It was just one of many gifts exchanged between the two rulers to cement an alliance against their mutual enemies: the Umayyad caliphate in Spain and the Christian kingdom of Byzantium.
Selective memory could surmount the cultural contradictions posed by the restoration of "the Baghdad of al-Rashid" but not the demographic problem: Baghdad is a majority-Shi'a city whose inhabitants would not give up without a fight. When the Islamic State captured Mosul in June 2014, tens of thousands of armed Shi'i men marched in a show of force through the streets of Baghdad. The men belonged to the Mahdi Army, just one of several Shi'i militias in Iraq. Shi'i foreign fighters streamed into Iraq too, responding to a call to arms in late 2014 by the influential but usually quietist cleric Grand Ayatollah Sistani. In India, up to thirty thousand Shi'i men supposedly pledged to fight against the Islamic State in Iraq.
Even if the Islamic State could overcome Iraq's Shi'i militias, it would still have to contend with Iran. Iranian special forces had funneled weapons and personnel to Iraq's Shi'a fighting against the American occupation. They did it again in Syria to save their ally Bashar al-Assad and once again in Iraq after the Islamic State stormed Mosul. Qasem Soleimani, the commander of Iran's special forces, who could have been a body double for Sean Connery in The Hunt for Red October, made regular appearances among the Shi'i militias battling the Islamic State. As powerful as the State was, it could not possibly take Baghdad at its current strength.
Given that, outsiders might reasonably conclude that the Islamic State would be foolish to attempt to take Baghdad rather than consolidate its gains in the Sunni-majority areas it now holds. But sometimes historical imperatives override strategic ones.
In its early days, the Islamic State was more sober when assessing its chances of restoring the caliphate of Harun al-Rashid. In 2008, the Islamic State's founder, Abu Ayyub al-Masri, chided his fellow jihadists for their great expectations. "Some of us incorrectly believe that the concept of the state that ought to be established and announced is the state of al-Rashid who spoke to a cloud in the sky, ladled gold like water, and dispatched armies that stretched all the way from Baghdad to his enemies."
The proper frame of reference, argued Masri, is the Islamic state when it was first established by the Prophet. The original state was a state of sacrifice and few supporters. It was located in a land with little water and filled with disease. The Prophet's companions suffered from hunger and poverty. His army was ragtag and coped with major military setbacks, as when it was defeated at the battle of Uhud.
In comparison with the first Islamic state, the Islamic State of 2008 wasn't doing too badly, Masri contended. "Has the Islamic State in Iraq met the requirements of a state with regard to territorial control, power, and extent of influence in comparison to the prophetic state after taking into account the different trials besetting them?" Absolutely, Masri concluded, especially given the power of the enemies arrayed against the State today compared to those who challenged the Prophet. As we have seen, this was wishful thinking.
Nevertheless, Masri understood that running a state is nothing like a running a war. Governing makes the jihadists more vulnerable to military attack and risks angering the population. "Every monotheist knows . . . that the change from jihad to the stage of ruling—the rule of God on the earth—and the return of the Islamic caliphate is a dangerous matter."
Governing the Caliphate
Despite being declared in 2006, the Islamic State never had a chance to govern in its first few years. Although its predecessor, al-Qaeda in Iraq, tried to govern the land it controlled, the Islamic State itself had virtually no land of its own when it was declared in 2006. Neither did it monopolize violence, try as it might, or consistently provide services. In other words, it lacked the essential characteristics of any government, modern or otherwise.
In 2014, the Islamic State had its chance. It was the unquestioned authority in many cities between Syria and Iraq. How would it govern? What had it learned from the experience of its predecessors in al-Qaeda who tried and failed to set up their own governments? Recall that Bin Laden had counseled al-Qaeda's affiliates to keep the welfare and consent of their subjects uppermost in their minds. From that dictum flowed the rest of his governing advice: Be lenient in applying Islamic punishments, focus on the economic well-being of your subjects, and seek their advice and approval.
As al-Qaeda in the Arabian Peninsula had done, the Islamic State addressed the economic needs of its subjects or at least wanted to be seen trying. It distributed videos of fighters handing out food and humanitarian relief, introduced price controls and regulations, and attempted to keep the lights on. But the usual subsidies and supplies distributed by the governments of Syria and Iraq were gone, and war made commerce and agriculture difficult. It didn't help that the Islamic State was more focused on fighting than on governing. The local economies in the State suffered as a consequence.
"We used to blame [Iraqi prime minister] Maliki for everything. Now we cry and hope for the return of those days," said a Sunni living under Islamic State rule. "Before, there was some kind of security, some kind of state. It is incomparable to the current situation." As Bin Laden had feared, the subjects of a jihadist government would be quick to turn against their rulers if they failed to deliver the normal meager services people were accustomed to.
Rather than distribute its wealth equitably or invest in infrastructure and jobs, the Islamic State gave money, fuel, and food to people who cooperated with it. Seeking the most bay'a for its buck, the State funneled its largesse to local leaders with many followers, the tribal shaykhs. Tribes that cooperated could handle their own security and gain an advantage over rival tribes. Tribes that didn't cooperate had their children kidnapped and their members dumped in mass graves. Some Sunni tribes wanted to fight back against the Islamic State, but the Shi'i governments in Damascus and Baghdad did not see it as in their interests to empower tribes that might one day fight against them.
One could argue that the Islamic State hadn't done so badly, given the exigencies of war and the rough-and-tumble nature of tribal politics, especially when compared to the poor standards of governance in the region. But in the one area where it had the most freedom of action, the Islamic State had surpassed all others in authoritarian excess: the meting out of the harsh hudud punishments.
Today most Muslim countries refuse to apply the hudud. One of the few is Saudi Arabia, whose brand of ultraconservative Islam is nearly identical to that of the Islamic State. When the State needed textbooks to distribute to schoolchildren in Raqqa, it printed out copies of Saudi state textbooks found online. Unsurprisingly then, most of the Islamic State's hudud penalties are identical to penalties for the same crimes in Saudi Arabia: death for blasphemy, homosexual acts, treason, and murder; death by stoning for adultery; one hundred lashes for sex out of wedlock; amputation of a hand for stealing; amputation of a hand and foot for bandits who steal; and death for bandits who steal and murder.
But there are two ways the Islamic State distinguishes itself from Saudi Arabia, which it believes is ruled by apostates. Firstly, the State carries out its penalties in public whereas Saudi Arabia hides them because of international censure. "A man was brought to the square, blind-folded," an eyewitness recounted. "A member of ISIS read the group's judgment. Two people held the victim tightly while a third man stretched his arm over a large wooden board." A fourth man wearing latex gloves impassively watched, then calmly reached out and cut off the victim's hand. "It took a long time," recounted the eyewitness. "One of the people who was standing next to me vomited and passed out due to the horrific scene." Not only would such scenes terrify those who watched them and cow them into submission; they would also attract the bloody-minded to the cause.
Secondly, the Islamic State goes the extra mile in its penalties. It opts for eighty lashes for drinking and slander rather than leaving it to the judge's discretion, as in Saudi Arabia. Whereas Saudi Arabia prefers to execute people by beheading, the Islamic State does that and more, throwing people off buildings or crucifying them after shooting them in the head. When Muslims raise a hue and cry that its actions aren't Islamic, the Islamic State's jurists cite chapter and verse. It is almost as if the group does extreme things to create opportunities for demonstrating its scholarly dexterity and burnishing its ultraconservative bona fides.
Consider the immolation of the Jordanian Muslim pilot, Moath al-Kasasbeh, captured by the Islamic State and burned alive for his alleged apostasy. In Islamic scripture, the Prophet Muhammad expressly and repeatedly forbade this form of punishment for apostasy because only God could punish apostates with fire. The Islamic State's scholars, however, argued that Muhammad was just being humble and not actually forbidding the punishment. For support, they cited some medieval Muslim jurists who argued the same. Not convinced? How about this: Some Islamic scriptures say that when people gouged out the eyes of Muhammad's followers, he had their eyes burned out in retaliation. Therefore, if the apostate government of Jordan was going to drop bombs on the Islamic State, then the Islamic State could respond with a similar punishment against the Jordanians. Still not convinced? It doesn't matter—the Islamic State's scholars will go on and on, relishing the chance to show why they're more faithful to scripture than their detractors.
To see how the Islamic State differentiates itself from other ultraconservatives, let's take something more mundane, like smoking tobacco. Smoking came to the Middle East at the end of the sixteenth century and quickly became popular among all classes of society regardless of gender. Moralists didn't like it for the same reasons they didn't like coffee, which had also recently become popular: It gave people a buzz, which seemed akin to intoxication, and it encouraged conviviality in public places, which could lead to loose behavior. Because smoking was new, Islamic scripture didn't have anything to say directly about it, which was also a mark against it in conservative eyes. The case for smoking wasn't helped by the fact that European merchants were involved in the tobacco trade, which fed paranoia about a Christian plot to subvert Muslims. Playing to the conservatives or fearful of rabble-rousing in places where tobacco was smoked in public, governments in the region tried several times to ban it. We find similar attempts to ban smoking across Eurasia around the same time for some of the same reasons. But by the eighteenth century, governments in the Middle East had largely given up because smoking was so popular. The last major effort to outlaw smoking in the early modern period was in eighteenth-century Arabia, where fighters in the ultraconservative Wahhabi movement and its tribal allies in the Saud family banned it in the towns they captured.
Today, smoking remains popular in the Middle East, and to my knowledge no country bans it, not even Saudi Arabia. Western expats who move to the Arab world are often shocked when someone lights up in an office building or classroom. But despite its popularity, smoking is still a touchstone for religious conservatives. One's stance on smoking signals one's seriousness about religion, even though Islamic scripture doesn't touch on it directly.
Jihadists who take over a town and want to impose their authority are faced with a dilemma. On one hand, if they ban smoking, they'll demonstrate their ultraconservative religious credentials. After all, not even Saudi Arabia bans smoking anymore. On the other hand, they'll be wildly unpopular with the locals, especially smokers and shopkeepers who sell cigarettes. How jihadists fall on the issue gives you some sense of where their priorities are in the hearts-and-minds debate. Some, like the al-Qaeda affiliate in Somalia, banned smoking and imposed a fine and thirty days of jail for violators. Others, like al-Qaeda's affiliate Jabhat al-Nusra in Syria, didn't. "These rules will be introduced gradually. We will advise people at first," remarked a Nusra commander to a reporter.
By now, you can guess where the Islamic State came down on the issue. Some smokers had to pay fines; others received forty lashes of the whip. Repeat offenders faced jail time, severed fingers, and even death. Not even Islamic State commanders were exempt. The severed head of a State commander in Syria was found with a cigarette dangling from his mouth and a sign that read: "This is not permissible, Sheikh." There are a few exceptions to the rule. Some local Islamic State commanders have lifted the ban on smoking, as happened in the Iraqi town of Hawija. Their reason for reversing the ban is telling: They wanted to shore up flagging popular support. Being so strict was good for impressing puritans, but it wasn't terribly crowd pleasing.
Enduring and Expanding
In 2007, when the Islamic State was beset on all sides in Iraq, its first emir adopted a one-word slogan: "Enduring." The United States and the Iraqi government had thrown everything they had at the State, but they could not destroy it. The slogan seemed silly by 2008 when the Islamic State was nearly defeated, but it took on the aura of prophecy when the State made a comeback after 2010. When the State moved into Syria, a new word was added to the slogan: "Expanding." Not only had the Islamic State survived, it was now on the march.
The Islamic State believes prophecy requires the conquest of every country on earth. "This religion will reach everywhere day and night reach" the Prophet had foretold, so the Islamic State determined to make it a reality. As its magazine proclaimed, "The shade of this blessed flag will expand until it covers all eastern and western extents of the Earth, filling the world with the truth and justice of Islam and putting an end to the falsehood and tyranny." In 2014, the Islamic State set about laying the groundwork for taking over the world, beginning in Muslim-majority countries.
On November 10, 2014, the Islamic State announced it had received oaths of allegiance from jihadists in Egypt's Sinai, Libya, Yemen, Algeria, and Saudi Arabia. "Glad tidings, O Muslims," Baghdadi celebrated. "We give you good news by announcing the expansion of the Islamic State to new lands." Lands claimed in those countries were no longer independent but rather provinces in the new caliphate. In accordance with prophecy, the Saudis would rid the Arabian Peninsula of non-Muslims, the Yemenis would muster for the last battles, and the Egyptians would prepare for the final assault to retake Jerusalem from the Jews. (Algeria and Libya receive less apocalyptic attention because North Africa does not figure prominently in End-Time prophecies.)
Initially, reality belied the bold predictions and celebrations that supposedly greeted the announcement. The so-called Army of the Caliphate in Algeria was a small group that had split from al-Qaeda's franchise in North Africa. An Islamic State "province" in Libya was just one jihadist group among several vying for control of the backwater town of Derna; its second "province" in the North would later control a few buildings in the coastal city of Sirte and briefly capture the small town of Nawfaliya. The Islamic State's main booster in Yemen refused to give his oath of allegiance, and his parent organization, al-Qaeda in the Arabian Peninsula, posted a rebuke of the State on its Twitter page, accusing the group of degrading the legitimacy of the global jihadist cause. The Islamic State's supporters in Saudi Arabia were nobodies. Only one of the groups, the Ansar Bayt al-Maqdis in the Sinai, was actually viable. Even then, the group's faction in the Nile Valley reportedly remained loyal to al-Qaeda.
But over time, the Islamic State "provinces" gained ground in the Arab countries riven by civil war. In Libya, the spiritual leader of Ansar al-Shari'a, the powerful jihadist group that attacked the U.S. consulate in Benghazi, pledged allegiance to Baghdadi; his acolytes will likely soon follow. The Islamic State "province" in Yemen also grew more powerful as the country descended into civil war. In just four days in March 2015, the group killed 137 people in Yemen's capital and another 29 soldiers in the South. That same month, the powerful Boko Haram group in northern Nigeria pledged allegiance to Baghdadi. The group, notorious for kidnapping and enslaving hundreds of Christian girls, controlled territory the size of Belgium. In Somalia, some high-ranking members of the Shabab, an al-Qaeda affiliate, started urging the group to join the Islamic State.
The accession of new "provinces" to the Islamic State is mutually beneficial. The State demonstrates that it is constantly expanding and thus succeeding in its divine mission. It also gets to draw on the human and financial resources of its new provinces, which it can call on to retaliate against its enemies. For the provinces, they get to sign up for the hottest thing going. Obscure jihadist groups all of a sudden gain notoriety for pledging allegiance, and unruly factions in al-Qaeda affiliates can circumvent their bosses. They also get the help of the Islamic State's formidable media apparatus and perhaps a taste of its spoils.
As we have seen, al-Qaeda fretted endlessly about expanding the number of its affiliates. Because Bin Laden and Zawahiri worried so much about popular Muslim support, they were reluctant to sign on groups that might behave badly and tarnish the al-Qaeda brand. The Islamic State didn't care about popular Muslim support, so it signed on affiliates at breakneck speed. Unlike Bin Laden, who counseled patience and care for Muslim lives, Baghdadi will only stoke his princes' worst impulses.
When the Islamic State first broke with al-Qaeda, Zawahiri tried to compete for market share in the global jihadist movement. He hinted at a countercaliph, declared a new affiliate in India, and denied the legitimacy of Baghdadi's caliphate. But he couldn't compete with the Islamic State's political success. To make matters worse, al-Qaeda's affiliates were restless and had long admired the Islamic State's ambition.
In 2015, rumors began circulating that Zawahiri would dissolve al-Qaeda. If that happens, it will be little cause for celebration. The end of al-Qaeda will free its affiliates to join the Islamic State. If they do, they will dramatically augment the State's strength in the Middle East and provide operatives capable of mounting attacks in the West. With such validation of the Islamic State's slogan, "Enduring and Expanding," it's hard to imagine mere arguments to the contrary would do anything to discredit it.
The Absent Mahdi
Absent from the Islamic State's plans for conquest is the Mahdi, the Muslim savior who will appear at the End of Days to lead the Muslims to victory. Whereas Islamic prophecies predict Muslim fighters will travel to Mecca and Medina to join forces with the Mahdi, he merits no mention in Islamic State propaganda about the conquest of Saudi Arabia. The State's magazine Dabiq predicts the flag of the caliphate "will rise over" Mecca and Medina in Saudi Arabia, and Islamic State fanboys anticipate the group will invade Saudi Arabia any day now. But they mention no Mahdi.
The Mahdi's absence in today's Islamic State propaganda is a major contrast with the statements released by the group's founders. Whereas Abu Ayyub al-Masri foretold that the Islamic State's fighters would hand their black flag to the Mahdi, who could be anyone, today's Islamic State proclaims that their fighters will hand it to Jesus, who will be recognized when he descends from heaven.
In one April 2007 statement, the first Islamic State emir, Abu Umar al-Baghdadi, hoped to lead troops from Iraq to "aid the Mahdi clinging to the curtains of the Ka'ba," the cuboid shrine in Mecca toward which Muslims pray. Abu Umar was presumably referring to a prophecy in which tribes around the Ka'ba war with one another, each believing they have the mandate of heaven. Some flee the battle and find a man who has buried his face in the Ka'ba's coverings, weeping. Because of the man's purity, they pledge allegiance to him. He refuses to lead but changes his mind after they threaten to behead him. The man turns out to be the Mahdi.
The prophecy is unusual for its content—in most other prophecies, the Mahdi is not reluctant to lead. It is also unusual for its provenance. The prophecy appears in The Book of Tribulations by Nu'aym bin Hammad. You will recall that many jihadists and other Muslims consider the prophecies to be spurious. But because the book is the richest collection of prophecies of the End Times, some jihadists find it irresistible. That the Islamic State's first emir succumbed to its gravitational pull is a testament to the apocalyptic fervor of the State's founders.
Preference for one collection of prophecies over another does not explain why the Mahdi is absent from today's Islamic State propaganda. The Mahdi appears often in the "authentic" collections of prophecies favored by the State's current leaders, whose rhetoric is no less apocalyptic than that of their forebears. Yet they omit the Mahdi in their propaganda.
Without access to the Islamic State's internal deliberations, we can only guess why. Caliph Baghdadi watched the first incarnation of the Islamic State nearly destroy itself because its leaders made hasty strategic decisions in the belief that the Mahdi would appear any day. Perhaps chastened by the experience, he has made the caliphate the locus of the group's apocalyptic imagination rather than the Mahdi. That does not mean the Mahdi will not appear soon—only a handful of just caliphs need rule before the Mahdi arrives. But for the moment, the caliphate is a greater priority than doomsday. The shift of eschatological emphasis from the person of the Mahdi to the institution of the caliphate buys the group time to govern while sustaining the apocalyptic moment that has so captivated its supporters. It is not an easy balance to strike.
One of the Islamic State's few recent references to the Mahdi is found in its magazine, Dabiq. On the last page of the fifth issue, there is a prophecy attributed to Muhammad: "If there were not left except a day from the world, God would lengthen that day to send forth on it a man from my family whose name matches my name and whose father's name matches my father's name. He will fill the Earth with justice and fairness as it was filled with oppression and tyranny."
The prophecy is given without written commentary, foregrounding a picture of the sons of Islamic State fighters dressed in battle fatigues. We are meant to infer that a Mahdi born of the Islamic State's dark imagination will one day lead their murderous ranks.
Conclusion
Apocalypse Then and Now
The French scholar of Muslim apocalypticism, Jean-Pierre Filiu, has argued that most modern Sunni Muslims viewed apocalyptic thinking with suspicion before the United States invaded Iraq in 2003. It was something the Shi'a or the conspiracy-addled fringe obsessed over, not right-thinking Sunnis. Sure, the Sunni fringe wrote books about the fulfillment of Islamic prophecies. They mixed Muslim apocalyptic villains in with UFOs, the Bermuda triangle, Nostradamus, and the prognostications of evangelical Christians, all to reveal the hidden hand of the international Jew, the Antichrist, who cunningly shaped world events. But the books were commercial duds.
The U.S. invasion of Iraq and the stupendous violence that followed dramatically increased the Sunni public's appetite for apocalyptic explanations of a world turned upside down. A spate of bestsellers put the United States at the center of the End-Times drama, a new "Rome" careering throughout the region in a murderous stampede to prevent violence on its own shores. The main antagonists of the End of Days, the Jews, were now merely supporting actors. Even conservative Sunni clerics who had previously tried to tamp down messianic fervor couldn't help but conclude that "the triple union constituted by the Antichrist, the Jews, and the new Crusaders" had joined forces "to destroy the Muslims."
The Iraq war also changed apocalyptic discourse in the global jihadist movement. The languid apocalypticism of Osama bin Laden and Ayman al-Zawahiri now had to contend with the urgent apocalypticism of Abu Mus'ab al-Zarqawi, the founder of al-Qaeda in Iraq, and his immediate successors. Iraq, the site of a prophesied bloodbath between true Muslims and false, was engulfed in a sectarian civil war. As Zarqawi saw it, the Shi'a had united with the Jews and Christians under the banner of the Antichrist to fight against the Sunnis. The Final Hour must be approaching, to be heralded by the rebirth of the caliphate, the Islamic empire that had disappeared and whose return was prophesied.
Because of the impending Final Hour, Zarqawi's successor, Abu Ayyub al-Masri, had rushed to establish the Islamic State in 2006 and declare a commander of the faithful, the traditional title of the caliph, supreme religious and political ruler of the early Islamic empire. The Islamic State was meant to be a caliphate in all but name. Masri believed the caliphate had to be in place to fight for the Mahdi, the Muslim savior, who would appear any day. The actual person of the caliph was an afterthought, someone practically plucked off the street. The world wouldn't be around long enough for it to matter.
Chastened by his failed predictions that first year, Masri's messianic ardor cooled. At the same time, the person of the Islamic State's first commander of the faithful, Abu Umar al-Baghdadi, became more substantial. Iraqis in the Islamic State were unhappy with the power amassed by Masri, an Egyptian, and the other Arab foreign fighters. They gravitated toward Abu Umar, a fellow Iraqi, which enhanced his standing in the organization. As his stature grew, so did the power of his office.
When Masri and Abu Umar were killed, the new commander of the faithful, Abu Bakr al-Baghadi, brought even greater substance to the office. Also an Iraqi, Baghdadi was supposedly descended from the Prophet and had scholarly credentials his predecessor lacked. Rather than declare Baghdadi the Mahdi, the Islamic State's scholars argued he was one of the prophesied handful of just caliphs who would rule before the end of the world. The immediacy of the Mahdi's return and the apocalypse to follow was attenuated in favor of building the institution of the caliphate. Messiah gave way to management. It was a clever way to prolong the apocalyptic expectations of the Islamic State's followers while focusing them on the immediate task of state building.
Earlier messianic caliphates had done the same. The Islamic State's medieval heroes, the Abbasids, had swept to power on a wave of apocalyptic fervor, and many of the early Abbasid caliphs had adopted apocalyptic titles like Mahdi. But the Final Hour never quite came. The same happened to the Almohads in Spain and North Africa and the Fatimids in Egypt, a Sunni caliphate and a Shi'i caliphate that had both announced a Mahdi only to downplay the apocalypse once in power.
Although the messianic fervor has cooled in the Islamic State's leadership, the group's apocalyptic rhetoric has intensified. References to the End Times fill Islamic State propaganda. It's a big selling point with foreign fighters, who want to travel to the lands where the final battles of the apocalypse will take place. The civil wars raging in those countries today lend credibility to the prophecies.
The Islamic State has stoked the apocalyptic fire. Its fighters died to capture the militarily unimportant town of Dabiq, Syria, because it's mentioned in the prophecies. The Islamic State's English-language magazine is named after the place. European fighters in the State have filmed themselves from the hillside overlooking the town, explaining to other Europeans that they are living in the End Times.
For Bin Laden's generation, the apocalypse wasn't a great recruiting pitch. Governments in the Middle East two decades ago were more stable, and sectarianism was more subdued. It was better to recruit by calling to arms against corruption and tyranny than against the Antichrist. Today, though, the apocalyptic recruiting pitch makes more sense. Titanic upheavals convulse the region in the very places mentioned in the prophecies. Sunnis and Shi'a are at war, both appealing to their own versions of prophecies to justify their politics.
This is not Bin Laden's apocalypse.
Hearts and Minds
The Islamic State has defied conventional thinking about how to conduct a successful insurgency. Most people, al-Qaeda's leaders among them, can't imagine that political success could come from enraging the masses rather than charming them. The Western theory of counterinsurgency so popular over the past decade is predicated on the idea that governments and insurgents compete for the hearts and minds of the locals. Those who win them over win the conflict. Recall what Ayman al-Zawahiri, the leader of al-Qaeda today, wrote to the bloody-minded Zarqawi: "we shouldn't stir questions in the hearts and minds of the people about the benefit of our actions . . . we are in a media battle in a race for the hearts and minds of our [Muslim] community."
But the Islamic State has deliberately provoked the anger of Muslims and non-Muslims alike with its online videos of outrageous and carefully choreographed violence. It showcases the beheading of prisoners—something Zawahiri had expressly warned against—and dumps enemy soldiers in mass graves while the camera is rolling. The State revels in gore and wants everyone to know it. And yet it has been remarkably successful at recruiting fighters, capturing land, subduing its subjects, and creating a state. Why?
Because violence and gore work. We forget that this terrifying approach to state building has an impressive track record. The pagan Mongols used it to great effect in the thirteenth century to conquer land stretching from the Pacific to the Mediterranean. They were far more brutal than the Islamic State, massacring entire towns that refused to surrender in order to discourage anyone else from resisting. The Bible says the ancient Israelites did the same in their conquest of Canaan. More brutal too was the Saud family and its ultraconservative Wahhabi allies, who came to power three times between 1744 and 1926, when the third and last Saudi state was established. A Spanish traveler who saw firsthand the rise of the Wahhabis in their second incarnation describes the scene when their fighters stormed the Shi'i holy city of Karbala in 1801: "The inhabitants made but a feeble resistance; and the conqueror put to the sword all the men and male children of every age. Whilst they executed this horrible butchery, a Wehhabite [sic] doctor cried from the top of a tower, 'Kill, strangle all the infidels who give companions to God.'" A contemporary Wahhabi historian wrote, "We took Karbala and slaughtered and took its people [as slaves], then praise be to Allah, Lord of the Worlds, and we do not apologize for that and say: 'And to the unbelievers: the same treatment.'" It was one massacre of many, and locals thought twice before resisting.
More recently, the Sunni Taliban came to power in the 1990s in Afghanistan by murdering thousands of unarmed civilians, often Shi'a or ethnic minorities. After capturing cities and villages that resisted, they would line up and gun down the locals, including women and children. According to a report on the Taliban's recapture of Bamian in 1999, "Hundreds of men and some women and children were separated from their families, taken away, and executed; all of them were noncombatants. In addition, houses were razed to the ground, and some detainees were used for forced labor."
All these groups were savvy at working with local tribes with whom they had ethnic and religious ties. Some tribes joined for political advantage over their rivals or because they wanted a share in the spoils. Others saw the fight as a religious duty. Still others didn't want to resist for fear of what would follow. But allying with local tribes is quite different from appealing to the general public. Playing nice with a tribal leader whose followers have guns is not the same thing as trying to win over your average citizen to the cause.
This is not to say that the softer approach can't work. America's own revolution is a case in point, especially when compared to the French Revolution a few years later. The American rebels warred with British troops and avoided deliberate attacks on civilians; in contrast, the French revolutionaries executed an ever-widening circle of "enemies of the revolution" to subdue the public in what the revolutionaries enthusiastically called "The Terror," from which we get the term "terrorism."
The point is that extreme brutality is not incompatible with establishing a new state. It may not be the wisest course of action, and it probably won't create a state many people would want to live in. But that doesn't mean it won't work.
Just as most people can't imagine that brutality would be a winning political strategy, they also can't imagine that any religious scripture would justify such a thing. Muslims and non-Muslims are equally baffled as to how anyone could commit atrocities in the name of God. To explain it, some either assume the perpetrators use scripture cynically or that they are ignorant of its nuances.
To be sure, many of the Islamic State's foot soldiers are ignorant of their own scriptures. Islamic scripture is vast, encompassing not only the Qur'an but also the ahadith, the words and deeds attributed to Muhammad by his followers. Collections of ahadith run into the hundreds of volumes, and that's just the Sunni variety. The Shi'a have their own collections, adding more volumes to the pile. Want to find passages justifying peace and concord? They're in there. Want to find passages justifying violence? They're in there too. Medieval Muslim scholars spent their whole careers trying to reconcile the contradictions between them. It's extremely difficult to do, which is why early Muslims called the effort ijtihad, or "hard work." People chuckled at the news of two men buying a copy of Islam for Dummies on their way to join the Islamic State. But having spent two decades studying the intricacies of Islamic scripture, I empathized with their bewilderment.
Although the Islamic State's soldiers might not know Islamic scripture very well, some of its leaders do. The caliph has a Ph.D. in the study of the Qur'an, and his top scholars are conversant in the ahadith and the ways medieval scholars interpreted it. There are many stupid thugs in the Islamic State, but these guys are not among them.
As to whether the Islamic State uses scripture cynically to justify whatever it wants to do, that's harder to know. It's certainly the case that the State's scholars pick and choose scripture to suit their biases and desires. But anyone who reads and acts on scripture does that. The better question to ask is where those biases and desires come from.
First, the Islamic State's biases: The Islamic State's theology and method of engaging with scripture is nearly identical to Wahhabism, the ultraconservative form of Islam found in Saudi Arabia. It's very different from the kind of Islam you find in other parts of the world. In Wahhabism, religious innovation is bad; medieval scholarly authorities are respected but disregarded if need be; outside cultural influences should be expunged; and the definition of a good believer is very narrow. Wahhabi scholars might reach different conclusions from Islamic State scholars, but they start at much the same place.
Second, the Islamic State's desires: The Islamic State's politics differ profoundly from that of most Wahhabis, who view the Saudi kingdom as a legitimate Islamic government. As the State sees things, no Muslim-majority state in the world deserves to call itself Islamic, which is why it set up its own state and declared a caliphate. To achieve that end, the Islamic State had to wage an insurgency, which it justified with scripture.
Still, brutal insurgency doesn't necessarily follow from Islamic scripture. Bin Laden and other jihadists found plenty of scriptural support for waging a hearts-and-minds campaign in the Muslim world. The Islamic State's scholars acknowledge these passages of scripture but look for ways around them or find other passages that fit their views. Thus, the Islamic State's disagreement with al-Qaeda's leadership isn't scriptural; it's strategic. The State doesn't believe a hearts-and-minds strategy is effective, and for the past few years it has been proven right.
This is not Bin Laden's insurgency.
A Government Feared by the People
The Islamic State defied another bit of conventional jihadist wisdom when it declared itself a caliphate without the support of the Sunni masses. Al-Qaeda's leaders had advised patient coalition building before declaring an Islamic state, much less a caliphate. They looked upon the defeat of the Islamic State in 2008 as an object lesson in what not to do. But even when the Islamic State was nearly destroyed, its idea of immediately establishing Islamic governments without popular consent caught fire among the jihadist rank and file. When the Arab Spring uprisings in 2011 created political instability, several al-Qaeda groups leapt at the chance to establish governments even though Bin Laden warned them not to. The Islamic State also tried again and was far more successful the second time around, parting company with al-Qaeda to realize its ambition.
Part of the State's success had to do with its style of governing. There were carrots: It tried to provide public services, such as fixing potholes, running post offices, and distributing food. It even had a campaign to vaccinate its subjects against polio. Several of the Islamic State's former sister affiliates in al-Qaeda had attempted the same.
The State also rewarded tribal allies with a share of the spoils to buy their support. As one Islamic State commander wrote to Baghdadi, "We have focused on bribing tribesmen and encouraging them to support the mujahids." In dealing with the Sunni tribes, the State benefited a great deal from the tribes' anger toward the central governments in Damascus and Baghdad.
Bin Laden would have approved of the Islamic State's carrots, but he would have objected to its sticks. The al-Qaeda leader had counseled his affiliates to be lenient in their application of the hudud, the harsh fixed punishments mentioned in Islamic scripture. To do otherwise would only alienate the locals. The Islamic State ignored this counsel, either out of religious conviction or to convince other ultraconservatives of its conviction. The hands of thieves were severed, adulterers were stoned, bandits were shot and crucified, all in full public view. The Islamic State's harsh punishments subdued the locals as effectively as massacring its enemies had. Religious convictions and political benefit are not always antithetical.
Bin Laden had also advised his affiliates against attacking tribes that didn't want to cooperate. He worried it would start a cycle of blood feuds. And yet the Islamic State killed hundreds of tribal members when their leaders refused to bend the knee. Rather than inaugurating blood feuds, the massacres silenced dissent because the tribesmen could find no support from their governments or from foreign nations.
That brings us to the major reason why the Islamic State was so successful from 2013 to 2014: It was left alone. Whereas other Sunni rebels in Syria tried to overthrow their government, the State focused on making its own government in the Sunni hinterland. It filled its leadership with ex-Ba'athists from Saddam's military and intelligence services who had been fighting an insurgency for a decade and were accustomed to running an authoritarian government. The Islamic State also replenished its foot soldiers with thousands of foreigners, more than any other rebel group attracted, by using a propaganda mix of apocalypticism, puritanism, sectarianism, ultraviolence, and promises of a caliphate. The head of the National Counterterrorism Center in the United States testified in February 2015 that the rate of influx of foreign fighters to Syria "is unprecedented," exceeding "the rate of travelers who went to Afghanistan and Pakistan, Iraq, Yemen or Somalia at any point in the last 20 years." The government in Baghdad couldn't stop the Islamic State and the government in Syria didn't want to, preferring to keep it around as a boogeyman to antagonize President Assad's enemies and scare his subjects. Better to focus the firepower of the Syrian state on other rebel groups that posed an immediate threat to his rule.
Bolstered by a combination of government neglect, careful planning, brutal tactics, and clever recruitment, the Islamic State had the manpower, money, and territory to make a credible claim to be a state. Whether the Islamic State is actually a caliphate is something for Muslims to decide. But judging purely by political criteria, the Islamic State is the only insurgent group in the Middle East to have made a plausible claim to the office since the fall of the Ottoman caliphate in the early twentieth century. And it did it by defying the common wisdom about how to govern properly.
This is not Bin Laden's caliphate.
Reconciling the Contradictions
The Islamic State today is full of contradictions, which make its actions hard to explain. Apocalyptic language suffuses its propaganda, yet the group is careful in its planning and cunning in its execution, qualities that we do not often associate with apocalypticists. The State subscribes to a puritanical religious ideology, yet it is willing to collaborate with the self-interested leaders of local tribes and with former and current members of Saddam's secularist Ba'ath party. The Islamic State believes it is better to be feared than to be loved and deliberately stokes the anger of international onlookers, but the group also tries to provide government services to its subjects.
One explanation is that these are just criminals or cynics who'll say and do whatever it takes so long as they can seize and hold power. Their apocalyptic and puritanical religious rhetoric is designed to appeal to people who are interested in that sort of thing, but the Islamic State's leaders don't really believe it themselves. They'll dress their actions up in prophecy and adopt the trappings of an austere Islamic caliphate but it's not from conviction. They see religious symbols and laws as useful vehicles for realizing their ambitions, which is not an irrational viewpoint, as I've argued above.
But there's another way to reconcile the contradictions: The group is devoted to establishing an ultraconservative Islamic state at all costs, so it modifies its religious and political doctrines when they get in the way of that goal. The early Islamic State nearly destroyed itself because of the messianic beliefs of its founder. So the State modified its apocalyptic doctrine to focus on institution building rather than on the imminent appearance of a savior. The Islamic State's narrow definition of a good Muslim would make it impossible to collaborate with many of its coreligionists, which would hamper the group's ability to win an insurgency. So the State makes allowances for working with others as long as they do not get in the way of its project. The State's brutality risks turning its subjects against it in large numbers, which would make it impossible for a state to function. So the group sees to the economic well-being of its subjects so it can maintain a low level of support and a tax base to fund its project.
In this interpretation, we'd need to modify our own understanding of what political behaviors follow from a group or state described as "apocalyptic," "religious," or "totalitarian." Apocalypticism does not necessarily demand rash and irrational behavior. A severe religious theology is not incompatible with practical considerations. And using the stick to discipline a population is not incongruous with giving it some carrots.
Whatever one makes of the true motivations of the Islamic State's leaders, the political impact is the same: the founding of a brutal government at war with its neighbors. But the Islamic State's expansionary policy puts at risk what its leaders hold most dear: the continued existence of its state. Perhaps the Islamic State will modify its doctrine as it has modified others in the interest of self-preservation. Or perhaps this is the one doctrine it can't let go because it believes it is destined to be a world-encompassing state. If the latter, it would go against the argument that these are mere thugs who want power. Criminal gangs aren't suicidal.
Either way, the world can't afford to wait and find out.
What to Do
From 2012 to 2014, the wait-and-see approach of the international community emboldened the Islamic State and filled its ranks, making it a real threat to vital U.S. interests in the Middle East. The United States could have decided those interests weren't worth defending, but it was not willing to chance the possible consequences of further inaction: refugees in the region numbering in the tens of millions, repeated shocks to world energy prices, and an ever-expanding proving ground for future militants. The United States' allies were divided and lacked the ability to organize and lead an effective counteroffensive on their own. The Islamic State was also expanding to other hot spots, like Afghanistan, Libya, Yemen, and Egypt, which would inevitably poison the politics of those already tumultuous countries and create further instability in a restless Middle East.
Some of the usual methods for dealing with jihadist statelets might have worked early on in Syria and Iraq. But the Islamic State is too entrenched now for quick solutions. Defeating its government is going to take time. Disrupting the Islamic State's finances will be difficult because the group does not rely much on outside funding. Attacking from the air will degrade the Islamic State but will not destroy it. Its soldiers are in urban areas where they are hard to target without killing thousands of civilians.
Some methods will help only at the margins. Stanching the flow of foreign fighters is very difficult, given Syria's porous borders and the excitement the Islamic State generates among Muslim radicals. Reducing the mass appeal of the State is pointless, given that it doesn't have mass appeal and isn't trying to cultivate it. What little appeal the Islamic State has rests on its ability to endure and expand. Take away either of those and you erode its legitimacy. In this case, the ideological fight is an actual fight.
Some methods work well but have a lot of downsides. Arming the Sunni tribes against the Islamic State doesn't guarantee they'll fight against it. They don't trust the Shi'i governments in Damascus and Baghdad and could just as easily decide to support the Islamic State or sit out the fight.
Arming Arab Sunni rebel groups to fight the Islamic State is no guarantee they'll get the job done either. They're focused on fighting their respective governments in Syria and Iraq and reluctant to tangle with a powerful rival. Furthermore, a number of those groups are religiously extreme and won't contribute to building a pluralistic future in either country.
Arming the Kurds is attractive because they're more pluralistic, but they won't be able to do much against the Islamic State in its Sunni Arab stronghold. The Kurds are an ethnic minority in Syria and Iraq, and they won't take and hold Arab territory far from their homeland. Plus, the Kurds might use the weapons to fight for their own independence, which would create further instability.
Building the capacity of the governments in Iraq and Syria to deal with the Islamic State is also fraught with problems. Bolstering the Shi'a-dominated government in Baghdad might further reinforce the country's sectarian politics that alienated its Sunnis in the first place. It's an even more dangerous policy in Syria because of its leader, who has let the Islamic State flourish to make his bloody methods appear less repugnant and to destroy his opposition from the inside.
The last option, sending in a large contingent of American troops, would enflame public opinion at home and abroad, ensuring that the United States will not be able to see its mission through. It would also absolve local governments of making the tough political choices required to end the Sunni disenfranchisement that fuels the insurgency.
Until the Shi'i governments in Syria and Iraq reach an accommodation with their Sunni citizens, the international coalition against the Islamic State can only constrain its growth. The coalition should continue using air power to diminish the State's ability to raise money and wage war. It shouldn't work with President Assad in Syria because he has deliberately fueled the rise of the Islamic State and probably won't stop. The government in Baghdad hasn't deliberately helped the State come to power, but its anti-Sunni policies have contributed to the State's success. So the coalition should give the government in Baghdad all the intelligence and logistical support it needs to fight against the Islamic State, but it should be wary of supplying more heavy weapons than are necessary to prosecute the war. The weapons could end up in the State's hands, as happened when Iraq's soldiers fled Mosul; but perhaps more important, giving the government whatever it wants discourages it from making the hard political compromises with the Sunnis that will sap the Islamic State's base of support.
The international coalition can also support proxies to fight against the Islamic State, but the support must be carefully calibrated so as to avoid creating more long-term political problems. The coalition should provide air cover and intelligence to Sunni tribal militias and rebel groups that fight against the Islamic State, whether Arab or Kurd. If it looks like the groups are in the fight for the long haul, then the coalition should consider arming them with light weaponry if they need it. Working with the Shi'i militias against the Islamic State is unnecessary, given that they already have a powerful state sponsor in Tehran.
If you think all of that sounds a lot like the coalition's current military strategy, you're right. It's not a great plan, but it's the best option at the moment.
I'm confident that the Islamic State's government in Syria and Iraq will crumble. No modern jihadist statelet has provoked international intervention and survived. But the disappearance of a jihadist statelet doesn't mean the disappearance of the jihadists. They will continue to wage insurgencies, taking advantage of the political instability and social unrest that gave rise to their statelets in the first place. The Islamic State stuck around after its defeat in 2008. So did al-Qaeda's affiliates in Mali, Yemen, and Somalia after they tried and failed to create states.
The question is, how will the jihadists evaluate the demise of the Islamic State? Will it prove to them that Bin Laden was right? Or will it prove that the State just needed to double down on its strategy? As I argued in chapter three, there's no obvious answer to the question because foreign powers always end the experiment prematurely. Even if a government established by global jihadists isn't serious about its rhetorical attacks on foreign nations, those nations won't wait long to find out.
The current political conditions in the Arab world all but ensure that some jihadists will follow the Islamic State's playbook, especially the group's growing number of affiliates or "provinces." Large-scale violence heightens the appeal of apocalyptic narratives, particularly in areas mentioned in the prophecies, and it creates the political vacuums in which armed groups can flourish. Of course, the Islamic State copycats can be defeated using some of the same methods the international coalition is using against the State in Iraq and Syria. But the Islamic State has demonstrated that a modern caliphate is possible, that doomsday pronouncements and extreme violence attract bloodthirsty recruits, and that cutting out the hearts and minds of a population can subdue them faster than trying to win them over. This may not be Bin Laden's jihad, but it's a formula future jihadists will find hard to resist.
Appendices
Sunni Islamic Prophecies of the End Times
To give readers a fuller appreciation of the prophecies cited by the Islamic State and celebrated by its followers, I have translated some of them in the following appendices. The prophecies come from early collections of words and deeds attributed to the Prophet Muhammad. Conservative Sunni Muslims consider most or all of the following prophecies to be authentic, which means they have the status of scripture second only to the Qur'an.
Modern Muslim authors have collected the prophecies and arranged them according to theme or chronology. I used the arrangements found in three books: The first two are from books available on the premier jihadist website, Tawhed.ws. For the sake of comparison with a more mainstream Islamist group, the third arrangement is from a book by a well-known Hamas activist. The final appendix is the first half of a treatise about the return of the caliphate written by the Islamic State's main scholar.
Readers should bear in mind the historical background of these prophecies I gave in the preceding pages. Readers should also know that many Muslims don't know these prophecies, don't care about them, or deny their authenticity altogether. If you're familiar with arguments about prophecies in your own religious tradition, you'll appreciate the range of views on the topic in the Muslim community.
Appendix 1
The Final Days
I have translated several of Muhammad's supposed prophecies of the End Times based on the original sources, arranging them in the chronological order found in The Final Days (al-Ayyam al-Akhira), a book written by Adnan Taha that was published in 1997 by a Jordanian press. The jihadist website Tawhed.ws digitally republished its own edition of the book. I have used some of Taha's section headings but relied on the original sources for the Arabic text of the prophecies, which he either cites or reproduces differently with editorial comment. For the passages from the Qur'an, I have modified Arberry's translation.
The Full Emergence of the Religion and the Present Reality
Verily, God knit together the earth for me. I beheld its easts and its wests. Truly, my community will rule everything that was knit together for me. (Sahih Muslim)
This cause [Islam] will reach everywhere night and day reach. God will leave no house of clay or camel skin without this religion by strengthening the strong and abasing the abased, with which God strengthens Islam and abases unbelief. (Sahih Ibn Hibban)
While I was sleeping, the keys of the treasuries of the earth were brought to me and put in my hand. (Sahih Muslim)
"Prophethood is among you as long as God wills it to be. Then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a mordacious monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a tyrannical monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method." Then he [the Prophet] fell silent. (Musnad Ahmad)
A part of my community will continue fighting for the truth until the Hour comes and until the promise of God is fulfilled. God will cause the hearts of the people to deviate and he will bestow their possessions upon the group until the [Final] Hour arises. (Sunan al-Nasa'i)
After me the ground will collapse in the east, in the west, and in the Arab Peninsula. (al-Mu'jam al-Awsat li-l-Tabarani)
Iraq will withhold its dirhams and its measure of qafiz [a measure for grain]. Sham will withhold its dinars and its measure of mudi [a measure for grain]. Egypt will withhold its dinars and its measure of irdab [a measure for grain]. You will return from whence you began. (Sahih Muslim)
The Stage before the Mahdi
The Hour will not come until a man of the Qahtan tribe appears, driving the people with his stick. (Sahih al-Bukhari)
Day and night will not end until a man called al-Jahjah rules. (Sahih Muslim)
The Hour will not come until the Euphrates lays bare a mountain of gold. The people will fight over it, and ninety-nine out of every hundred will be slain. Every man among them will say, 'Perhaps I am the one who will be saved.' (Sahih al-Bukhari)
The Euphrates will uncover a treasure of gold. The one who finds it should take none of it. (Sahih Muslim)
The Appearance of the Mahdi
Three people will fight for your treasure, each of them the son of a caliph, but none of them shall gain it. Then the black banners will come from the east and they will kill you in a manner no people have been killed before. . . . If you see him, pledge him allegiance even if you have to crawl over snow. For verily, he is the caliph of God, the Mahdi. (Sunan Ibn Majah)
At the end of my community will be a caliph who scatters wealth like dust that cannot be counted. (Sahih Muslim)
The earth will neither perish nor end until a man of my house whose name is my name rules the Arabs. (Musnad Ahmad)
The Mahdi will appear at the end of my community. God will send down rain for him and the earth will give forth its plants. He will distribute the wealth equitably . . . and livestock will multiply and the community will flourish. He will live seven or eight . . . pilgrimage seasons. (Mustadrak al-Hakim)
I give you tidings of earthquakes and the Mahdi who will be sent to all people. He will fill the earth with fairness and justice as it was filled with injustice and oppression. The denizens of heaven and earth will be pleased with him. He will distribute wealth equitably. (Musnad Ahmad)
The Mahdi has a broader brow than I have and a nose more curved. He will fill the earth with fairness and justice as it was filled with injustice and oppression. He will rule seven years. (Sunan Abi Dawud)
A people who have no power, great numbers, or weapons will seek refuge in my House, meaning the Ka'ba. An army will be sent against them that the earth will swallow when they reach the desert. (Sahih Muslim)
Strange that a group of people from my community would go to the House with a man from the Quraysh tribe who sought shelter at the House. When they reached the desert, the earth swallowed them. (Sahih Muslim)
Fighting Rome
You will offer Rome a treaty of security. You will attack them because they are enemies behind you. Then you will arrive at a meadow with hills. A man from Rome will come bearing the cross saying 'The Cross has been victorious.' A man among the Muslims will stand against him and kill him. Rome will betray you and there will be great battles, and they will mass against you. They will come against you with eighty flags, and under each flag will be ten thousand. (Sunan Abi Dawud)
The Hour will not come until the Romans would land at al-A'maq or in Dabiq. An army consisting of the best of the people of the earth at that time will come from Medina. When they will arrange themselves in ranks, the Romans will say: Do not stand between us and those [Muslims] who took prisoners from amongst us. Let us fight with them." The Muslims will say: "Nay, by God, how can we withdraw between you and our brothers? They will then fight and a third [part] of the army would run away, whom God will never forgive. A third [part of the army], which would be constituted of excellent martyrs in God's eye, will be killed and the third who will never be put to trial will win and they will be conquerors of Constantinople. (Sahih Muslim)
The Muslims' place of assembly on the day of the Battle will be in al-Ghutah near a city called Damascus, one of the best cities in al-Sham. (Sunan Abi Dawud)
Conquest of Constantinople
You have heard of a city on one side of which is land and on the other the sea. . . . The Hour will not come until seventy thousand of the tribe of Isaac attack it. When they reach it, they will disembark. They will not fight with weapons or shoot arrows. They will say, "There is no God but God, God is most great!" One of its two sides will collapse. . . . They will say a second time, "There is no God but God, God is most great." Then the other side will collapse. They will say a third time, "There is no God but God, God is most great." A breach will open for them and they will enter and gather the spoils, distributing it among themselves. Thereupon they would hear a cry, "Verily the Deceiver has appeared." They will abandon everything and return whence they came. (Sahih Muslim)
Conquest of Rome
When the Messenger of God was asked which of the two cities will be conquered first, Constantinople or Rome, he said, "The city of Heraclius is first, meaning Constantinople." (Musnad Ahmad)
The Deceiver
The cry will reach them that the Deceiver is behind them at their homes. They will drop what is in their hands and turn back, sending ten horsemen in the vanguard. The Prophet said, "I know their names, the names of their fathers, and the colors of their horses. They will be the best horsemen on the face of the earth on that day or among the best of horsemen on the face of the earth that day." (Sahih Muslim)
While I slept, I saw myself circumambulating the Ka'ba. There was a man with brown skin and long straight hair between two men. Water trickled from his head or was poured over his head. I said, 'Who is this?' They said, 'This is the son of Mary.' Then I went and saw a corpulent man of red complexion, frizzy hair, and blind in one eye, which was swollen like a grape. I said, 'Who is this?' They said, 'The Deceiver.' (Sahih al-Bukhari)
The Deceiving Messiah is a short man with bowed legs, frizzy hair, and blind in one eye that neither protrudes nor sinks in its socket. (Sunan Abi Dawud)
Between his eyes is written 'Infidel.' . . . Every Muslim can read it. (Sahih al-Bukhari)
Where He Will Appear
"He is not in the sea of al-Sham or in the sea of Yemen. Rather, he is in the east, he is in the east." Then he pointed with his hand to the east. (Sahih Muslim)
The Deceiver will appear from the land in the east called Khorasan. (Jami' al-Tirmidhi)
The Deceiver will appear among the Jews of Isfahan. Seventy thousand Jews will be with him. (al-Fath al-rabbani li-tartib Musnad Ahmad)
Seventy thousand Jews from Isfahan wearing shawls will follow the Deceiver. (Sahih Muslim)
Verily, he will appear in the empty land between al-Sham and Iraq, wreaking havoc left and right. (Sahih Muslim)
I know what the Deceiver will have with him. He will have two rivers running. The eye perceives one of them to be white and the other flaming fire. If anyone sees that, let him go to the river that appears to be fire, close his eyes, lower his head, and drink from it, for it is cool water. (Sahih Muslim)
I will tell you of the Deceiver what no prophet has told his people. He will be blind in one eye. He will bring the semblance of paradise and hellfire with him. What he calls paradise will be hellfire. (Sunan Ibn Majah)
We said, "O Messenger of God, how long will he remain on earth." He said, "Forty days. One day like a year, one day like a month, one day like a week, and the remaining days like your days." We said, "O Messenger of God, how quickly will he move on earth." He said, "Like the rain driven by the wind. . . . He will command the sky, and it will rain and the earth will bloom. . . . The treasures will be gathered for him. He will call to a man filled with youth and strike him with the sword, cutting him into two pieces at a distance of an archer from his target. He will then call the man, who will approach with gleaming face and laughing." (Sahih Muslim)
The Deceiver will come but he will be forbidden from entering the rugged pass of Medina. He will camp at one of the salt flats around Medina. A man will go out to him on that day who is the best of the people or one of the best. He will say, 'I bear witness that you are the Deceiver of whom the Messenger of God (peace and blessings be upon him) spoke.' The Deceiver will say, 'If I kill this man and bring him back to life, will you doubt my claim?' They will say no. Then the Deceiver will kill the man and bring him back to life. The man will say, 'By God, what I have seen today has convinced me.' The Deceiver wanted to kill him again but he was not given the authority to do so. (Sahih al-Bukhari)
Medina will be shaken with its people three times. No male or female hypocrites will remain for they will all go out to him. The city will be cleansed of impurity just as the bellows purifies the iron of dross. (Sunan Ibn Majah)
Descent of Jesus (peace be upon him)
God will send the Messiah son of Mary who will descend at the white minaret in the eastern side of Damascus wearing two garments dyed with saffron and placing each hand on the wing of two angels. When he lowers his head, drops of water fall, and when he raises his head, beads like pearls trickle down. Every infidel who catches a scent of his breath can only die, and his breath will reach far away. (Sahih Muslim)
A group from my community will continue fighting for the truth and conquering until the Day of Resurrection. . . . Jesus the son of Mary (peace be upon him) will descend. Their leader will say, "Come pray for us." He will say, "No, some of you are leaders over others as a blessing of God for this community." (Sahih Muslim)
Fighting the Jews
The Hour will not come until the Muslims fight the Jews. The Muslims will kill them until the Jews hide behind rocks and trees. The rocks and trees will say, "O Muslim! O servant of God! This is a Jew behind me so come and kill him." Except for the Gharqad, which is a tree of the Jews. (Sahih Muslim)
On that day the Arabs will be few, and most of them will be in Jerusalem. Their leader will be a righteous man. When he steps forward to lead them in morning prayer, Jesus son of Mary will descend upon them. Their leader will step back to allow Jesus to lead the people in prayer. Jesus will put his hand between his shoulder blades and say, 'Go to the front and pray, for the call to prayer was given in your name.' He will lead them in prayer. After he finishes, Jesus will say, 'Open the gate.' So they will open it, and behind it will be the Deceiver with seventy thousand Jews, each with an ornate sword and green cloak. When the Deceiver gazes upon [Jesus], [the Deceiver] will melt like salt melts in water. He will start to run away, and Jesus (upon him be peace) will say, 'I have one blow for you that you cannot escape.' He will overtake him at the eastern gate of Lod and slay him. Then God will defeat the Jews. There will be no thing created by God for the Jews to hide behind without God causing it to speak and say, 'O Muslim servant of God, this is a Jew, come kill him!'—neither rock, nor tree, nor wall, nor beast—except the Gharqad tree. (Sunan Ibn Majah)
The Appearance of Gog and Magog
They will question you about the Man with Two Horns. Say: 'I will recite to you a story of him. We established him in the land, and We gave him a path to everything; and he followed a path until, when he reached the setting of the sun, he found it setting in a muddy spring, and he found nearby a people. We said, "O Man with Two Horns, either you will inflict pain upon them, or you will take towards them a way of kindness." He said, "As for the oppressor, we will inflict pain upon him, then he will return to his Lord Who will inflict horrible pain upon him. But as for him who believes, and does righteousness, he shall receive as recompense the reward most fair, and we shall speak to him, of our command, easiness." Then he followed a path until, when he reached the rising of the sun, he found it rising upon a people for whom We had not appointed any veil to shade them from it. And so, We encompassed in knowledge what was with him. Then he followed a path until, when he reached between the two barriers, he found on this side of them a people scarcely able to understand speech. They said, "O Man with Two Horns, behold, Gog and Magog are doing corruption in the earth; so shall we assign to thee a tribute, against your setting up a barrier between us and between them?" He said, "That wherein my Lord has established me is better; so aid me forcefully, and I will set up a rampart between you and between them. Bring me ingots of iron!" Until, when he had made all level between the two cliffs, he said, "Blow!" Until, when he had made it a fire, he said, "Bring me, that I may pour molten brass on it." So they were unable either to scale it or pierce it. He said, "This is a mercy from my Lord. But when the promise of my Lord comes to pass, He will make it into powder; and my Lord's promise is ever true." Upon that day We will leave them surging on one another, and the Trumpet will be blown, and We will gather them together. (Qur'an 18:83–99)
There is a ban upon any city that We have destroyed; they shall not return until Gog and Magog are loosed, and they swarm down out of every slope, and nigh has drawn the true promise, and behold, the eyes of the unbelievers staring: "Alas for us! We were ignorant of this; nay, we were evildoers." (Qur'an 21:75–77)
You will continue to fight an enemy until Gog and Magog come, broad of face, small of eye, and covered with hair, coming from every direction, their faces as wide as shields and as thick as a hammer. (Musnad Ahmad)
God revealed to Jesus, "I have brought forth my servants whom none will be able to fight. So take my servants to Mt. Sinai." Then God will loose Gog and Magog, who swarm down every slope. The first of them will pass the Sea of Galilee and drink from it. When the last of them pass, they will say, "There was once water here." They will encircle the prophet of God Jesus and his companions until the head of the bull will be better for one of them than a hundred dinars for one of you today. The prophet of God Jesus and his companions will supplicate God, and God will afflict their enemies' necks with myiasis. In the morning they will die as if one body. Then the prophet of God Jesus and his companions will go down the mountain, where they will find no place on earth that is not filled with their stench and smell. The prophet of God Jesus and his companions will supplicate God and God will send birds with necks like Bactrian camels that will take the bodies and dispose of them where God wills. (Sahih Muslim)
By Him in Whose hand is my soul, the son of Mary (peace and blessings be upon him) will descend among you as a fair ruler. He will break the cross, kill the swine, abolish the protection tax, and pour forth so much wealth that no one will accept it. (Sahih Muslim)
Death of the Believers and Destruction of Mecca and Medina
At that time Allah will send a pleasant wind that will even reach their armpits. It will take the life of every believer and every Muslim. Only the wicked will survive, committing adultery like asses until the Last Hour would come to them. No one will remain on earth saying, "God, God." (Sahih Muslim)
God will send a perfumed breeze by which everyone who has in their heart even a mustard grain of faith will die. Those who remain will have no goodness in them, and they will revert to the religion of their forefathers. (Sahih Muslim)
"Medina will be left in the best way that it is until a dog or wolf enters it and urinates on one of the pillars of the mosque or on the minaret." They asked, "Messenger of God! Who will have the fruit at that time?" He said, "Animals seeking prey, birds and wild beasts." (Muwatta' Malik)
The thin-legged man from Abyssinia will destroy the Ka'ba. He will then plunder its finery and remove its cover. It is as if I am seeing him now, bald and bow-legged, beating on it with his plane and pick. (Musnad Ahmad)
Afterward no one ever rebuilt it. (Musnad Ahmad)
The Great Universal Signs
The Hour will not come until the sun rises from the West. When it rises, the people will see it and all believe. But by then, no soul will benefit from its belief. (Sahih al-Bukhari)
When the Word falls on them, We will bring forth for them a beast from the earth who will speak to them: "People had no faith in Our signs." (Qur'an 27:82)
The first sign is the rising of the sun in the West and the appearance of the beast to the people in the forenoon. Whichever happens first, the other is soon to follow. (Sahih Muslim)
At the end, fire will blaze from Yemen, driving the people to their place of assembly. (Sahih Muslim)
"There you will assemble, there you will assemble, there you will assemble . . . riding and walking and on your faces." He pointed with his hand to al-Sham. "There you will assemble." (Musnad Ahmad)
Appendix 2
The Victorious Group
In 1993, jihadist author Abu Basir al-Tartusi wrote a book about the characteristics of the "victorious group" of Muslims prophesied to fight in the final battles leading up to the Day of Judgment. In it, he provides a brief chronology of the End Times based on a few of Muhammad's prophecies. (I have preserved his subject headings.) Note that Tartusi, a strong critic of the Islamic State, places the reestablishment of the caliphate after the fight against the Romans and the Jews.
1—The Conquest of Rome, the Capital of Italy
When the Messenger of God was asked which of the two cities will be conquered first, Constantinople or Rome, he said, 'The city of Heraclius is first, meaning Constantinople.' (Musnad Ahmad)
2—The Conquest of Constantinople a Second Time
The Hour will not come until the Romans would land at al-A'maq or in Dabiq. An army consisting of the best [soldiers] of the people of the earth at that time will come from Medina [to counteract them]. When they will arrange themselves in ranks, the Romans will say: "Do not stand between us and those [Muslims] who took prisoners from amongst us. Let us fight with them." The Muslims will say: "Nay, by God, how can we withdraw between you and our brothers?" They will then fight and a third [part] of the army will run away, whom God will never forgive. A third [part of the army], which will be constituted of excellent martyrs in God's eye, will be killed and the third who will never be put to trial will win and they would be conquerors of Constantinople. (Sahih Muslim)
3—Fighting the Jews and Victory over Them
The Hour will not come until the Muslims fight the Jews. The Muslims will kill them until the Jews hide behind rocks and trees. The rocks and trees will say, 'O Muslim! O servant of God! This is a Jew behind me so come and kill him.' Except for the Gharqad, which is a tree of the Jews. (Sahih Muslim)
4—The Invasion of India
God will safeguard two groups from my community from hellfire: a group that invades India and a group with Jesus the son of Mary. (Musnad Ahmad)
5—Establishing the Rightly Guiding Caliphate in Accordance with the Prophetic Method
"Prophethood is among you as long as God wills it to be. Then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a mordacious monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a tyrannical monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method." Then he [the Prophet] fell silent. (Musnad Ahmad)
6—The Penetration of Islam into Every House in the World
Verily, God knit together the earth for me. I beheld its easts and its wests. Truly, my community will rule everything that was knit together for me. (Sahih Muslim)
This cause will reach everywhere night and day reach. God will leave no house of clay or camel skin without this religion by strengthening the strong and abasing the abased, with which God strengthens Islam and abases unbelief. (Sahih Ibn Hibban)
While I was sleeping, the keys of the treasuries of the earth were brought to me and put in my hand. (Sahih Muslim)
7—The Descent of Jesus (Peace Be Upon Him) to the Earth
There is no prophet between me and him, meaning Jesus. Verily, he will descend. When you see him you will recognize him as a man of medium height, reddish to fair, wearing two light yellow garments. His head will appear to be dripping though it is not wet. He will fight the people for Islam, breaking the cross, killing the swine, and abolishing the poll tax. In his time, God will destroy every religious community except Islam. He will destroy the Deceiving Messiah. He will dwell on earth for forty years. When he dies, the Muslims will pray for him. (Sunan Abi Dawud)
Appendix 3
The Mahdi Is Preceded by an Islamic State
In 2009, Jawad Bahr al-Natsha, a Hamas activist, published a book titled The Mahdi Is Preceded by an Islamic State. Below I have translated his proof texts and section titles.
Section One: Explicit Proof Texts that the Mahdi Is Preceded by an Islamic Caliphate
Differences will occur at the death of a caliph. A man will emerge among the people of Medina, fleeing to Mecca. Some of the people of Mecca will come to him. They will bring him forward against his will and swear allegiance to him between the corner and the maqam. An army will be sent against him from Sham, which will be swallowed up in the desert between Mecca and Medina. When the people see the righteous of Sham and the troops of the people of Iraq coming to him, they will pledge allegiance to him between the corner and the maqam. Then there will arise a man of the Quraysh tribe whose maternal uncles belong to the Kalb tribe. He will send an army from the Kalb tribe against them, which will prevail. Those who do not witness the booty of the Kalb will be disappointed. He will divide the wealth and treat the people according to the practice of their Prophet (peace and blessings be upon him). He will apply himself to establishing Islam on the earth. He will remain seven years then die. The Muslims will pray over him. (Sunan Abi Dawud)
Three people will fight for your treasure, each of them the son of a caliph, but none of them shall gain it. Then the black banners will come from the East and they will kill you in a manner no people have been killed before. . . . If you see him, pledge him allegiance even if you have to crawl over snow. For verily, he is the caliph of God, the Mahdi. (Sahih Ibn Majah)
Section Two: Implicit Proof Texts
"Prophethood is among you as long as God wills it to be. Then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a mordacious monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a tyrannical monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method." Then he [the Prophet] fell silent. (Musnad Ahmad)
The caliphate will be in my community thirty years, then there will be monarchy after that. (Jami' al-Tirmidhi)
No house of clay or camel skin will remain on the face of the earth but that God has made the word of Islam enter it by strengthening the strong and abasing the abased, whether God . . . strengthens them and makes them among its people or abases them and they bow to it. (Musnad Ahmad)
Appendix 4
Twelve Caliphs
Before the Islamic State declared itself the caliphate, Turki al-Bin'ali, its young scholarly apologist and now head of its powerful Shari'a Committee, wrote a short treatise on why the time was right to do so. Below is a translation of the portion on the rightly guiding caliphs who will appear before the Mahdi. The bolded and underlined portions are Bin'ali's doing.
The Messenger of God (peace and blessings be upon him) said:
"Prophethood is among you as long as God wills it to be. Then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a mordacious monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a tyrannical monarchy. It will be among you as long as God intends, and then God will take it away when He so wills. Then there will be a caliphate in accordance with the prophetic method." Then he fell silent.
The learned Ibn Khaldun (may God bless him) said:
It is mentioned in the Sahih that [the Prophet] said, "This cause will abide until the Hour comes or there are twelve caliphs of the Quraysh tribe over you."
The Prophet made known that some of them will be in the beginning of Islam and some will be at its end. [The Prophet] said, "The caliphate after me will be thirty, thirty-one, or thirty-six years." It ends in the caliphate of al-Hasan and the beginning of Mu'awiya's rule. The beginning of Mu'awiya's rule is a caliphate in the original meaning of the word. He is the sixth caliph. As for the seventh caliph, it is Umar b. Abd al-Aziz.
The other five are of the People of the House [members of the Prophet's family] descended from Ali, which is supported by [the Prophet's] statement: "You are the possessor of two epochs," meaning the community has you [Ali] as a caliph in its beginning and your progeny as caliphs at its end.
Imam Ibn Kathir (may God bless him) said after mentioning the hadith [of Safina], "The prophetic caliphate will last thirty years then God will give His kingdom to whomever He wills":
This hadith is an explicit rebuttal of the rejectionists [the Shi'a] who denied the three caliphates and a refutation of the hateful among the Umayyad tribe and their followers in Sham who denied the caliphate of Ali b. Abi Talib. So how does one reconcile this hadith of Safina and the hadith of Jabir b. Samura found in Sahih Muslim: "The religion will abide until the Hour comes or there are twelve caliphs over you, each of them from the Quraysh tribe"?
The answer is that some people say the religion continued until twelve caliphs ruled, then it collapsed after them in the time of the Umayyad tribe. Others say this hadith is a prophecy of twelve just caliphs from the Quraysh tribe, even if they have not yet come to power. Rather, it is consistent with the fall of the caliphate in the thirty years that followed the prophetic period. After that, there were rightly guiding caliphs, including Umar b. Abd al-Aziz b. Marwan b. al-Hakim the Umayyad, may God be pleased with him. More than one of the imams has endorsed his caliphate, his justice, and his being counted among the rightly guiding caliphs. Even Ahmad b. Hanbal (may God be pleased with him) said about him, "The words of any among the generation that came after [Muhammad's Companions] have no standing except those of Umar b. Abd al-Aziz." Also among those they mentioned is al-Mahdi bi-l-Amr Allah, the Abbasid, and the Mahdi prophesied at the end of time by virtue of his being one of the People of the House and his name being Muhammad b. Abd Allah. He is not the person awaited in the cellar of Samarra, who absolutely does not exist even if the ignorant among the rejectionists [the Shi'a] wait for him."
Therefore, the caliphate is a prophetic promise foretold by [the Prophet]. The just caliphs are twelve in number as mentioned in the two Sahihs, even though they are not consecutive. Rather, five of them appeared in the early period. Some say six, some say seven. The last of them will pave the way for the Mahdi with the help of God, the exalted.
With patience and certitude, the community from east to west anticipates the return of the rightly guiding caliphate and hopes to live in the age of the remaining rightly guiding caliphs!
However, few in the expectant community work for the return of that rightly guiding caliphate. Among those few are many who commit deviant acts of unbelief or sin for the sake of a "lofty and noble purpose." So some of them nominate themselves for presidential elections to gradually apply the Shari'a until the caliphate returns, as they claim!
The least of their evils is having political parties and societies calling for the return of the caliphate through leaflets and posters!
A the minority in the small group working for the return of the caliphate are those who have been successful in their method of returning the rightly guiding caliphate, the method of speech and spear. As the Shaykh of Islam Ibn Taymiyya says (may God bless him): "Religion is invigorated with a book that guides and a sword that assists, and your Lord is sufficient as guide and helper." He also said, "Religion was invigorated with sword and scripture." Moreover, "Everyone must strive to be in conformity with the Qur'an and with God's iron [the sword]. He must ask what he has that can be of service to God. When that happens, the world will serve religion."
That small minority arose to establish Islamic emirates here and there—which are sometimes strong and sometimes in retreat in the face of the fierce global campaigns against them!–until God the exalted favored the establishment of the Islamic State in Iraq and Sham to be the nucleus of the anticipated rightly guiding caliphate, by the aid of God, the exalted.
Notes
Please note that some of the links referenced in this work may no longer be active.
Chapter 1
1. Marc Leibowitz (@Marc_Leibowitz), Twitter post, August 12, 2014, <https://twitter.com/Marc_Leibowitz/status/499230249063043075>.
2. Jessica Remo, "Garwood Man Says Home's Flag Represents Islam, Not ISIS, Is Not Anti-American," NJ.com, August 13, 2014, <http://www.nj.com/union/index.ssf/2014/08/garwood_resident_removes_isis_flag.html>; Abby Phillip, "How the Violent Islamic State Extremists Got Their Signature Flag," Washington Post, August 15, 2014, <http://www.washingtonpost.com/blogs/worldviews/wp/2014/08/15/how-the-violent-islamic-state-extremists-got-their-signature-flag/>.
3. Michael Weiss and Hassan Hassan, ISIS: Inside the Army of Terror [Google edition] (New York: Regan Arts, 2015), 18.
4. For Zarqawi's biography, see Bruce Riedel, The Search for Al Qaeda (Washington, DC: Brookings Institution Press, 2008), 85–115; Weiss and Hassan, ISIS, 14–30.
5. Not much is known about Sayf, and he is frequently misidentified. The best synthesis of what we know can be found in Clint Watts, Jacob Shapiro, and Vahid Brown, Al-Qa'ida's (Mis)adventures in the Horn of Africa, Harmony Program, Combating Terrorism Center at West Point, July 2, 2007, 119–129, <https://www.ctc.usma.edu/posts/al-qaidas-misadventures-in-the-horn-of-africa>.
6. The following account is based on Sayf's brief memoir of his time with Zarqawi. Sayf had known of Zarqawi before they met because details of Zarqawi's trial in Jordan had made the rounds in jihadist circles. See Sayf al-Adl, "Tajrubati ma'a Abi Mus'ab al-Zarqawi," Minbar al-Tawhid wa-l-Jihad, May 2005, <http://www.tawhed.ws/dl?i=ttofom6f>.
7. Steve Coll, The Bin Ladens: An Arabian Family in the American Century (New York: Penguin Press, 2009), 75.
8. Adl, "Tajrubati," 3–6. Zarqawi's mentor, Abu Muhammad al-Maqdisi, said Zarqawi refused to join al-Qaeda in Afghanistan because he could not reconcile his extreme views with Bin Laden's. See Cole Bunzel, "From Paper State to Caliphate: The Ideology of the Islamic State," The Brookings Project on U.S. Relations with the Islamic World, Analysis Paper No. 19 (March 2015): 13, <http://www.brookings.edu/~/media/research/files/papers/2015/03/ideology-of-islamic-state-bunzel/the-ideology-of-the-islamic-state.pdf>.
9. Adl, "Tajrubati," 12.
10. Thomas Asbridge, The Crusades: The War for the Holy Land (London: Simon & Schuster, 2012), 1138–1139.
11. S. J. Allen and Emilie Amt, eds., The Crusades: A Reader, 2nd ed., Readings in Medieval Civilizations and Cultures: VIII, series ed., Paul Edward Dutton (Toronto: University of Toronto Press, 2014), 122.
12. Adl, "Tajrubati," 13.
13. For example, Bin Laden uses both terms interchangeably when discussing what to name the Shabab's government in Somalia. See Osama bin Laden, "Letter from Osama bin Laden to Mukhtar Abu al-Zubayr [English translation]," personal correspondence to Abu al-Zubayr (aka Ahmed Abdi Godane), SOCOM-2012-0000005, Harmony Program, Combating Terrorism Center at West Point, August 7, 2010, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000005-Trans.pdf>. Original Arabic version available at <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000005-Orig.pdf>.
14. Ibid., 20.
15. Ibid., 25.
16. Ibid., 18.
17. Craig Whitlock, "Zarqawi Building His Own Terror Network," Washington Post, October 3, 2004, <http://old.post-gazette.com/pg/04277/388966.stm>; "Report of the Select Committee on Intelligence on Postwar Findings About Iraq's WMD Programs and Links to Terrorism and How They Compare with Prewar Assessments," U.S. Senate Select Committee on Intelligence, 109th Congress, 2nd session, September 8, 2006, 109, <http://www.intelligence.senate.gov/phaseiiaccuracy.pdf>.
18. Weiss and Hassan, ISIS, 36–37.
19. Abu Mus'ab al-Zarqawi, "A-yanqus al-din wa-ana hayy," Kalimat mudi'a: al-Kitab al-jami' li-khutab wa-kalimat al-shaykh al-mu'taz bi-dinihi, June 10, 2006, 303, <http://e-prism.org/images/AMZ-Ver1.doc>.
20. Abu Mus'ab al-Zarqawi, "Zarqawi Letter [English translation]," personal correspondence to Osama bin Laden and Ayman al-Zawahiri, February 2004, U.S. Department of State Archive, <http://2001-2009.state.gov/p/nea/rls/31694.htm>. Portions of the original are in "Risala min Abi Mus'ab al-Zarqawi ila al-Shaykh Usama bin Ladin," Kalimat mudi'a, February 15, 2004, 56–73.
21. Ibid., modified translation.
22. Ibid., modified translation.
23. See Thomas Hegghammer, Jihad in Saudi Arabia: Violence and Pan-Islamism since 1979 (New York: Cambridge University Press, 2010); Sami Yousafzai, "Terror Broker," Newsweek, April 10, 2005, <http://www.newsweek.com/terror-broker-116359>.
24. The anonymous Islamic State insider "Abu Ahmad" said Zarqawi hoped al-Qaeda would augment his group's money and give him access to its network of funders in the Gulf; "al-Haqa'iq al-mukhfa hawla dawlat al-Baghdadi," al-Durar al-Shamiyya, April 5, 2014, <http://eldorar.com/node/45368>.
25. Zarqawi, "al-Bay'a li-tanzim al-Qa'ida bi-qiyadat al-Shaykh Usama bin Ladin," Kalimat mudi'a, October 17, 2004, 171. Prior to his oath of allegiance, Zarqawi had stated frequently that his group in Iraq was fighting to establish an Islamic state. See Bunzel, "Ideology," 15–16.
26. Ayman al-Zawahiri, "Zawahiri's Letter to Zarqawi [English translation]," personal correspondence to Abu Mus'ab al-Zarqawi, Harmony Program, Combating Terrorism Center at West Point, July 9, 2005, <https://www.ctc.usma.edu/posts/zawahiris-letter-to-zarqawi-english-translation-2>. Original Arabic version available at <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/10/Zawahiris-Letter-to-Zarqawi-Original.pdf>.
27. Ibid.
28. Ibid., modified translation.
29. Atiyya Abd al-Rahman, "'Atiyah's Letter to Zarqawi [English translation]," personal correspondence to Abu Mus'ab al-Zarqawi, Harmony Program, Combating Terrorism Center at West Point, December 12, 2005, <https://www.ctc.usma.edu/posts/atiyahs-letter-to-zarqawi-english-translation-2>. Original Arabic version available at <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/10/Atiyahs-Letter-to-Zarqawi-Original.pdf>.
30. Ibid.
31. Craig Whitlock and Munir Ladaa, "Atiyah Abd al-Rahman, Liaison to Iraq and Algeria," Washington Post, 2006, <http://www.washingtonpost.com/wp-srv/world/specials/terror/rahman.html>.
32. Abd al-Rahman, "'Atiyah's Letter to Zarqawi."
33. Ibid.
34. Ibid.
35. Ibid.
36. Quoted in "Limadha nujahid? Liqa' Mu'assasat al-Furqan ma'a al-Shaykh Abi Mus'ab al-Zarqawi," Majmu' kalimat qadat Dawlat al-'Iraq al-Islamiyya, July 17, 2010, <http://up1430.com/central-guide/pencil/elit/the_sum/the_sum_3/pages/authority/14/index.php>. The video featuring the original interview appears to have been released posthumously on June 12, 2006. See "Hadiyyat Mu'assasat al-Furqan hiwar ma'a al-Shaykh Abi Mus'ab al-Zarqawi 'rahimahu Allah,'" Shabakat Ana al-Muslim li-l-Hiwar al-Islami, June 12, 2006, <http://www.muslm.org/vb/archive/index.php/t-190782.html>.
37. Zarqawi, "Hadha balagh li-l-nas," Kalimat mudi'a, April 24, 2006, 511.
38. The statement comes from a fuller version of the video that was captured by U.S. forces in May 2006. See Bunzel, "Ideology," 16. Not all of al-Qaeda's leaders urged Zarqawi to be cautious. Sayf al-Adl advised Zarqawi in May 2005 to quickly "announce the state" because "the affairs and circumstances . . . are ripe and favorable." See Adl, "Tajrubati," 22.
39. "'Majlis Shura al-Mujahidin' yu'lin ta'sis imara Islamiyya fi al-'Iraq," Al Arabiya, October 15, 2006, <http://www.alarabiya.net/articles/2006/10/15/28296.html>.
40. The title of this section is taken from my Foreign Affairs article, "State of Confusion," September 10, 2014, <http://www.foreignaffairs.com/articles/141976/william-mccants/state-of-confusion>.
41. On the Islamic State's ambiguous use of prepositions, see Bunzel, "Ideology," 18.
42. In the nine-minute audio recording announcing the founding of the Islamic State, spokesman Muharib al-Juburi does not refer to al-Qaeda's role in the state, although he provides multiple justifications for the state's establishment. See Muharib al-Juburi, "al-I'lan 'an qiyam Dawlat al-'Iraq al-Islamiyya," October 15, 2006, <https://archive.org/details/song-of-terror-main-8>.
43. Abu Ayyub al-Masri (aka Abu Hamza al-Muhajir), "Sayuhzam al-jam' wa-yuwallun al-dubur," June 13, 2006, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
44. Akram Hijazi, "Ta'qiban 'ala i'lan Dawlat al-'Iraq al-Islamiyya," Almoraqeb, October 19, 2006, <http://www.almoraqeb.net/main/articles-action-show-id-40.htm>.
45. "Abu Abi" [online pseudonym], "al-Qawl al-Fasl fi mas'alat al-bay'a: Hal hiya li-Shaykh Usama am al-Mulla 'Umar am Abu 'Umar al-Baghdadi," Shabakat Filistin li-l-Hiwar, October 16, 2006, <https://www.paldf.net/forum/showthread.php?t=87326>.
46. Adam Gadahn, "Letter from Adam Gadahn [English translation]," personal correspondence to unknown recipient, Harmony Program, Combating Terrorism Center at West Point, January 2011, <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/10/Letter-from-Adam-Gadahn-Translation.pdf>. Original Arabic version available at <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/10/Letter-from-Adam-Gadahn-Original.pdf>.
47. See Ayman al-Zawahiri, "Shahada li-haqan dima' al-mujahidin bi-l-Sham," Mu'assasat al-Sahab, May 3, 2014, 5, <https://pietervanostaeyen.wordpress.com/2014/05/03/dr-ayman-az-zawahiri-testimonial-to-preserve-the-blood-of-mujahideen-in-as-sham/>. A senior al-Qaeda operative, Fadil Harun, was equally critical of the timing of the Islamic State's establishment: "The decision to declare a state was improper and its timing was not appropriate. There is no doubt that al-Qaeda nowadays is not the same as it was during the era of Shaykh al-Zarqawi." Harun rightly guessed Bin Laden had not known of the state's declaration prior to the event. "The news is relayed to him vaguely and after everything happens. Then he has no choice but to support it because we all want to have a state in Iraq, led by our brother al-Baghdadi." Rather than seeking Bin Laden's blessing, "the brothers took over some areas in some provinces and governed by the laws of God." Instead of declaring a state, Harun believes "they should have announced Islamic provinces . . . because according to my knowledge, the concept of a state is greater, requires more obligations and preserving it is more difficult." See Fadil Harun, al-Harb 'ala al-Islam: Qissat Fadil Harun, 2 vols., February 26, 2009, 2: 134–135; vol. 1 available at <https://www.ctc.usma.edu/posts/the-war-against-islam-the-story-of-fazul-harun-part-1-original-language-2>; vol. 2 available at <https://www.ctc.usma.edu/posts/the-war-against-islam-the-story-of-fazul-harun-part-2-original-language-2>.
48. Zawahiri, "Shahada," 1–2.
49. Abu Ayyub al-Masri, "In al-hukm illa li-Allah," November 10, 2006, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
50. Abu Umar al-Baghdadi, "Wa-qul ja'a al-haqq wa-zahaqa al-batil," December 22, 2006, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
51. Statement from the Islamic State Ministry of Media, February 13, 2007, <http://forum.ma3ali.net/showthread.php?t=273221>.
52. Maamoun Youssef, "Al-Qaeda Chief Appointed Minister of War," Washington Post, April 19, 2007, <http://www.washingtonpost.com/wp-dyn/content/article/2007/04/19/AR2007041901149.html>.
53. Abu al-Walid al-Ansari, "Risala nasiha li-Abi 'Umar al-Baghdadi min al-Shaykh Abi al-Walid al-Ansari," Nukhbat al-Fikr, October 2014, <https://ia902608.us.archive.org/11/items/Abu.al.Walid.al.Ansari.New/naseha.pdf>. See Bunzel, "Ideology," 22. On Ansari, see Kévin Jackson, "Al-Qaeda's Top Scholar," Jihadica, September 25, 2014, <http://www.jihadica.com/al-qaedas-top-scholar/#more-2457>.
54. Hamid al-Ali, "al-Su'al: Hal man la yubayi'u (Dawlat al-'Iraq al-Islamiyya) 'usa?! Wa-hal huwa wajib al-'asr?!" H-Alali.net, April 4, 2007, <http://www.h-alali.cc/f_open.php?id=1a55240a-3422-102a-9c4c-0010dc91cf69>.
55. Atiyya Abd al-Rahman, "Letter from Hafiz Sultan [English translation]," personal correspondence from Atiyya Abd al-Rahman to Mustafa Ahmad Uthman Abu al-Yazid (aka Sa'id al-Masri), SOCOM-2012-0000011, Harmony Program, Combating Terrorism Center at West Point, March 28, 2007, <https://www.ctc.usma.edu/posts/letter-from-hafiz-sultan-english-translation-2>. Original Arabic version available at <https://www.ctc.usma.edu/posts/letter-from-hafiz-sultan-original-language-2>. In his 2014 "Shahada," Zawahiri identifies the author and recipient of the letter.
56. Abd al-Rahman, "Letter from Hafiz Sultan."
57. See Gadahn, "Letter from Adam Gadahn."
58. Ayman al-Zawahiri, "al-Liqa' al-rabi' li-Mu'assasat al-Sahab ma'a Ayman al-Zawahiri," December 16, 2007, <https://nokbah.com/~w3/?p=110>. Accessed in December 2014 before it was taken down.
59. The video announcement of the Islamic State's founding shows its spokesman speaking in front of a generic black flag with the Muslim profession of faith inscribed on it. See "Islamic State Announcement Subtitles Eng 15/10/2006," YouTube video, August 30, 2014, <https://www.youtube.com/watch?v=s1DsXuDHIk0>. Videos and online jihadist forums from the period immediately following this announcement indicate that the Islamic State was still using the flag of the Mujahidin Shura Council that had preceded it. See "Jihadiraq001" [online pseudonym], "al-Ihtifal bi-i'lan al-Dawla al-Islamiyya fi al-'Iraq," YouTube video, December 1, 2006, <https://www.youtube.com/watch?v=Ls_vFXvXDN8>; "Mufaja'a: Suwwar wa-film li-juyush Dawlat al-'Iraq al-Islamiyya / Allahu akbar wa-li-l-Allah al-hamd," Shabakat Filistin li-l-Hiwar, October 27, 2006, <https://www.paldf.net/forum/showthread.php?t=88987>.
60. "A Religious Essay Explaining the Significance of the Banner in Islam [English translation]," Harmony Program, Combating Terrorism Center at West Point, January 2007, <https://www.ctc.usma.edu/posts/a-religious-essay-explaining-the-significance-of-the-banner-in-islam-english-translation-2>. Original Arabic version available at <https://www.ctc.usma.edu/posts/a-religious-essay-explaining-the-significance-of-the-banner-in-islam-original-language-2>. I have used my own translations. Al-Qaeda's media distributor, Fajr li-l-I'lam, released the essay around January 23, 2007 ("al-'Alam al-jadid li-Dawlat al-'Iraq al-Islamiyya," Shabakat Filistin li-l-Hiwar, January 23, 2007, <https://www.paldf.net/forum/showthread.php?t=106152>).
61. "Religious Essay."
62. Ibid.
63. Abdülmecit Senturk, "Nâmah Al-Saadah (Blessed Letter) Sent to the Malik (Ruler) of Ghassan and Transfer to the Ottoman State of the Copy of Surah Al-Qadr (chapter on Power in the Holy Qur'an) Written by Caliph Ali," Journal of Rotterdam Islamic and Social Sciences 3, 1 (2012): 2–4. In a 1940 article, historian D. M. Dunlop dismissed the authenticity of the letters on the grounds that paleographic analysis conducted at the British Museum proved the letters were written after the age of the Prophet. See Dunlop, "Another 'Prophetic' Letter," Journal of the Royal Asiatic Society of Great Britain and Ireland 1 (January 1940): 54–60.
64. Deirdre Elizabeth Jackson, Marvellous to Behold: Miracles in Medieval Manuscripts (London: British Library, 2007), 66.
65. "Religious Essay."
66. Dominic Lieven, Empire: The Russian Empire and Its Rivals (New Haven, CT: Yale University Press, 2001), 156.
67. Hijazi, "Ta'qiban."
68. "Religious Essay."
69. Nu'aym bin Hammad, Kitab al-fitan (Beirut: Dar al-Fikr li-l-Taba'a, 1992), 226, 228.
70. The Qur'an is filled with vague references to the Day of Judgment and the end of the world but has few specifics. See Todd Lawson, "Paradise in the Quran and the Music of Apocalypse," in Roads to Paradise: Eschatology and Concepts of the Hereafter in Islam, 2 vols., eds. Sebastian Günther and Todd Lawson (Leiden, Netherlands: Brill, 2015).
71. Ibn Khaldun, The Muqaddimah: An Introduction to History, transl. Franz Rosenthal (Princeton, NJ: Princeton University Press, 1967), 196, 198.
72. Cassius Dio, Roman History 69.12.1–14.3. For an online edition see Dio's Rome [Roman History], Volume V, Books 61–76 (AD. 54–211) An Historical Narrative Originally Composed in Greek During the Reigns of Septimius Severus, Geta and Caracalla, Macrinus, Elagabalus and Alexander Severus: And Now Presented in English Form by Herbert Baldwin Foster, book 69, chs. 12–14, Project Gutenberg EBook #10890, <http://www.gutenberg.org/files/10890/10890-h/10890-h.htm>.
73. Jay Rubenstein, Armies of Heaven: The First Crusade and the Quest for Apocalypse (New York: Basic Books, 2011). For an overview of European apocalypticism prior to the Crusades, see James Palmer, The Apocalypse in the Early Middle Ages (Cambridge, UK: Cambridge University Press, 2014).
74. Bruce Hoffman, Inside Terrorism (New York: Columbia University Press, 2006), 98–100.
75. Jeffrey Kaplan, "The Fifth Wave: The New Tribalism," Terrorism and Political Violence 19, no. 4 (2007): 554; Eleanor Beevor, "Why Cults Work: The Power Games of the Islamic State and the Lord's Resistance Army," War on the Rocks, March 18, 2015, <http://warontherocks.com/2015/03/why-cults-work-the-power-games-of-the-islamic-state-and-the-lords-resistance-army/>.
76. Allen Fromherz, The Almohads: The Rise of an Islamic Empire (London: I.B. Tauris, 2012), 135–186.
77. Farhad Daftary, The Isma¯'ı¯lllı¯s: Their History and Doctrines (Cambridge, UK: Cambridge University Press, 1990), 129.
78. Khalid bin Ibrahim al-Dubayan, "al-'Aqida al-'askariyya 'inda Ibn Tumart mu'assis Dawlat al-Muwahhidin," 20–22, https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=0CEUQFjAE&url=https%3A%2F%2Ffaculty.sau.edu.sa%2Ffiledownload%2Fdoc-2-doc-0cb82dbdcda47e2ad7b7aaf69573906e-original.doc&ei=6qXsVKLbC4G_ggTJ14LQDg&usg=AFQjCNFX62q9sDkibk0Owlbcx7gy0zudrA&sig2=UukgssbVL9WkzIPe1NKFZg&bvm=bv.86475890,d.eXY&cad=rja; Paul E. Walker, ed. and transl., Orations of the Fatimid Caliphs: Festival Sermons of the Ismaili Imams (London: I.B. Tauris, 2009), 72–76.
79. Patricia Crone, God's Rule—Government and Islam: Six Centuries of Medieval Islamic Political Thought (New York: Columbia University Press, 2004), 77–78.
80. Bernard Lewis, Race and Slavery in the Middle East: An Historical Inquiry (New York: Oxford University Press, 1990), 38.
81. Sulaym bin Qays al-Hilali, Kitab Sulaym bin Qays al-Hilali, ed. Muhammad Baqir al-Ansari (Qom, Iran: Nashr al-Hadi, 2000), 282. See also Crone, God's Rule, 85.
82. Patricia Crone, The Nativist Prophets of Early Islamic Iran: Rural Revolt and Local Zoroastrianism (New York: Cambridge University Press, 2012), 126.
83. Ibn Jarir al-Tabari, Tarikh al-Tabari: Tarikh al-rusul wa-l-muluk, vol. 7 (Cairo: Dar al-Ma'arif bi-Misr, 1960–1977), 391. See also Crone, Nativist Prophets, 19.
84. Hilali, Kitab Sulaym, 285.
85. Nu'aym bin Hammad, Kitab al-fitan, 118.
86. Ibid., 188.
87. Khalil Athamina, "The Black Banners and the Socio-Political Significance of Flags and Slogans in Medieval Islam," Arabica T. 36, Fasc. 3 (November 1989): 307–326, 313.
88. Ibn Khaldun, Muqaddimah, 50–51.
89. Faruq Umar, Buhuth fi al-tarikh al-'Abbasi (Beirut: Maktabat al-Nahda, 1977), 246.
90. Abu Hanifa Ahmad bin Dawud al-Dinawari, al-Akhbar al-tiwal (Leiden, Netherlands: Brill, 1888), 186.
91. Athamina, "Black Banners," 309.
92. Ibn Jarir al-Tabari, The History of al-Tabari, Vol. XXVII: The 'Abbasid Revolution, transl. John Alden Williams (Albany, NY: State University of New York Press, 1985), 65–66.
93. Crone, God's Rule, 87–88. Originally, the Arabic word dawla had the same ambiguous meaning as our English "revolution," meaning a turn or change.
94. Bernard Lewis, The Language of Political Islam (Chicago: University of Chicago Press, 1988), 35–36.
95. "The World's Muslims: Unity and Diversity," Pew Research Center, August 9, 2012, <http://www.pewforum.org/2012/08/09/the-worlds-muslims-unity-and-diversity-executive-summary/>. For comparison, 41 percent of American Christians surveyed by Pew in 2010 believed Jesus would return in the next forty years. See "Public Sees a Future Full of Promise and Peril," Pew Research Center, June 22, 2010, <http://www.people-press.org/2010/06/22/public-sees-a-future-full-of-promise-and-peril/>.
96. Jean-Pierre Filiu, Apocalypse in Islam, transl. M. B. DeBevoise (Berkeley: University of California Press, 2011), 186.
97. Ibid., 121–140.
98. "Tawfiq 123" online pseudonym], "Nahnu umma lam yukallifuna Allah bi-ma'rifat shakhs al-Mahdi qablu khurujihi," Shabakat Ana al-Muslim li-l-Hiwar al-Islami, April 23, 2007, <http://www.muslm.org/vb/showthread.php?225702>. See also Reuven Paz, "Global Jihad and the United States: Interpretation of the New World Order of Usama Bin Ladin," Project for the Research of Islamist Movements (PRISM), Occasional Papers 1, 1, PRISM Series of Global Jihad 1 (March 2003), [http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB4QFjAA&url=http%3A%2F%2Fwww.e-prism.org%2Fimages%2FPRISM%2520no%25201.doc&ei=aOrPVODlG9eHsQS48YCIAQ&usg=AFQjCNH4N9N2OqBQDD5E5M7l1Fm_726WIg&bvm=bv.85076809,d.cWc.
99. Thomas Hegghammer and Stéphane Lacroix, "Rejectionist Islamism in Saudi Arabia: The Story of Juhayman Al-'Utaybi Revisited," International Journal of Middle East Studies 39, no. 2 (February 2007): 103–122, 112–113. See also their book-length treatment, The Meccan Rebellion: The Story of Juhayman al-'Utaybi Revisited (Bristol, UK: Amal Press: 2011).
100. Ali H. Soufan and Daniel Freedman, The Black Banners: The Inside Story of 9/11 and the War Against al-Qaeda (New York: W. W. Norton, 2011), xvii–xix.
101. Harun, Harb, 2:41.
102. On the book's reception, see Brynjar Lia, Architect of Global Jihad: The Life of Al-Qaida Strategist Abu Mus'ab al-Suri (New York: Oxford University Press, 2009). Zawahiri mentions the book favorably in his al-Tabri'a: Risala fi tabri'at ummat al-qalam wa-l-sayf min manqasa tuhmat al-khawar wa-l-du'f, March 2008, http://www.ek-ls.org/forum/showthread.php?t=127920&highlight=%C3%ED%E3%E4. I accessed the site in December 2014 before it was taken down.
103. Filiu, Apocalypse, 189.
Chapter 2
1. "Rewards for Justice Targets al-Qaida in Iraq's New Leader," U.S. Department of State, July 3, 2006, <http://iipdigital.usembassy.gov/st/english/article/2006/07/20060703141847idybeekcm0.5817835.html#axzz3Vmd3N5Cz>; Mike Mount, "Reward for Wanted Terrorist Drops," CNN, May 13, 2008, <http://edition.cnn.com/2008/WORLD/meast/05/13/pentagon.masri.value/>.
2. Sadiq al-Iraqi, "Zawjat Abu [sic] Ayyub al-Masri: wasalna Baghdad qablu suqut nizam Saddam, wa-zawji ghamid wa-mutashaddid," al-Riyad, April 29, 2010, <http://www.alriyadh.com/520823>; "Rewards for Justice," U.S. Department of State; Dexter Filkins, "U.S. Identifies Successor to Zarqawi," New York Times, June 15, 2006, <http://www.nytimes.com/2006/06/15/world/middleeast/15cnd-iraq.html?_r=0>.
3. Abu Sulayman al-Utaybi, "Risalat al-Shaykh Abi Sulayman al-'Utaybi li-l-qiyada fi Khurasan," personal correspondence to al-Qaeda leadership, Shabakat Ana al-Muslim li-l-Hiwar al-Islami, April 28, 2007, <http://justpaste.it/do3r>. The letter appeared online November 24, 2013.
4. Abu Ayyub al-Masri, "Sayuhzam al-jam' wa-yuwallun al-dubur," June 13, 2006. Abu Umar al-Baghdadi, the nominal head of the Islamic State, also anticipated going to aid the Mahdi in Mecca after consolidating gains in Iraq, then "invading the Jewish state, and retaking Jerusalem." See Abu Umar al-Baghdadi, "Hasad al-sinin bi-Dawlat al-Muwahhidin," April 17, 2007, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbut al-I'lam al-Jihadi, 2010).
5. Utaybi, "Risalat al-Shaykh." Abu Sulayman's allegation that Abu Umar al-Baghdadi was just Masri's puppet echoes an earlier charge made by Khalid al-Mashhadani, the head of the Islamic State's media operations, who was captured by American forces in July 2007. Mashhadani told his interrogators that Baghdadi's statements were read by an actor and that the man himself was just a myth. See Bill Roggio, "Islamic State of Iraq—an al Qaeda Front," Long War Journal, July 18, 2007, <http://www.longwarjournal.org/archives/2007/07/islamic_state_of_ira.php#ixzz3Prlnd6mh>.
6. "Abu Ahmad" [online pseudonym], "al-Haqa'iq al-mukhfa hawla Dawlat al-Baghdadi," al-Durar al-Shamiyya, April 5, 2014, <http://eldorar.com/node/45368>.
7. Abu Usama al-Iraqi, "Muhattat min jihad al-Amir al-Baghdadi," Muntada al-Minbar al-I'lami al-Jihadi, June 9, 2012, https://www.mnbr.info/vb/showthread.php?t=11332&langid=3&styleid=18.
8. Abu Ahmad, "Haqa'iq."
9. Utaybi, "Risalat al-Shaykh."
10. Sudarsan Raghavan, "Sunni Factions Split with al-Qaeda Group," Washington Post, April 14, 2007, <http://www.washingtonpost.com/wp-dyn/content/article/2007/04/13/AR2007041300294.html>.
11. "Islamic Army in Iraq Accuses al-Qa'ida in Iraq of 'Transgressing Islamic Law,'" Open Source Center, April 5, 2007, <https://groups.yahoo.com/neo/groups/alphacity/conversations/topics/1735>. For background, see "Islamic Army in Iraq," Mapping Militant Organizations, Stanford University, July 23, 2014, <http://web.stanford.edu/group/mappingmilitants/cgi-bin/groups/view/5>.
12. Brian Fishman, "Dysfunction and Decline: Lessons Learned from Inside al-Qa'ida in Iraq," Harmony Project, Combating Terrorism Center at West Point, March 16, 2009, 11–13, <https://www.ctc.usma.edu/posts/dysfunction-and-decline-lessons-learned-from-inside-al-qaida-in-iraq>.
13. Abu Ahmad, "al-Haqa'iq." Abu Ahmad, who claims to be a former member of the Islamic State, alleged that Syrian intelligence agents and Sunni insurgents affiliated with the Saudi government had joined.
14. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," January 25, 2008, <http://iraqslogger.powweb.com/downloads/aqi_leadership_letters_sept_08.pdf>. This is one of several letters in the exchange, portions of which Tony Badran translated in Bill Roggio, Daveed Gartenstein-Ross, and Tony Badran, "Intercepted Letters from al-Qaeda Leaders Shed Light on State of Network in Iraq," Foundation for the Defense of Democracy, September 12, 2008, <http://www.defenddemocracy.org/media-hit/intercepted-letters-from-al-qaeda-leaders-shed-light-on-state-of-network-in/>. I have used Badran's translations when I can, making a few tweaks here and there. The remaining translations are mine.
15. Utaybi, "Risalat al-Shaykh."
16. See the correspondence between Ansar al-Sunna and the Islamic State in Fishman, "Dysfunction," 8–9.
17. For Ansar al-Sunna, see Michael R. Gordon and Bernard E. Trainor, The Endgame: The Inside Story of the Struggle for Iraq, from George W. Bush to Barack Obama (New York: Vintage Books, 2013), 263. For the Islamic Army, see Michael Weiss and Hassan Hassan, ISIS: Inside the Army of Terror [Google edition] (New York: Regan Arts, 2015), 79.
18. Abu Mus`ab al-Zarqawi, "Zarqawi Letter [English translation]," personal correspondence to Osama bin Laden and Ayman al-Zawahiri, February 2004, U.S. Department of State Archive, <http://2001-2009.state.gov/p/nea/rls/31694.htm>.
19. Karl Vick, "Insurgent Alliance Is Fraying in Fallujah, Washington Post, October 13, 2004, <http://www.washingtonpost.com/wp-dyn/articles/A28105-2004Oct12.html>.
20. Ellen Knickmeyer and Jonathan Finer, "Insurgents Assert Control over Town near Syrian Border," Washington Post, September 6, 2005, <http://www.washingtonpost.com/wp-dyn/content/article/2005/09/05/AR2005090500313.html>.
21. "Ba'da sharit Abi Usama al-'Iraqi: Hal daqqat sa'at al-firaq bayna al-Qa'ida wa-sunnat al-'Iraq?" Asharq al-Awsat, October 13, 2006, http://archive.aawsat.com/details.asp?article=387171&issueno=10181#.VMUd5ivF-Ds.
22. "Islamist Sheikh Abu Osama Al-'Iraqi Denounces Al-Qaeda in Iraq for Atrocities against Sunnis," Middle East Media Research Institute (MEMRI), Jihad & Terrorism Studies Project, Special Dispatch No. 1340, October 31, 2006, <http://www.memri.org/report/en/0/0/0/0/0/0/1926.htm>. The video was originally released on October 12, 2006. See "Ibn Alfalojah" [online pseudonym], "Risalat al-Shaykh Abu Usama al-'Iraqi li-l-Shaykh Usama bin Ladin rahimahu Allah 'am 2006," Vimeo video, May 2014, <http://vimeo.com/96742098>.
23. Ibid.
24. Ibid.
25. Ibid.
26. Ibid.
27. See Cole Bunzel's translation in "From Paper State to Caliphate: The Ideology of the Islamic State," The Brookings Project on U.S. Relations with the Islamic World, Analysis Paper No. 19 (March 2015): 38, <http://www.brookings.edu/~/media/research/files/papers/2015/03/ideology-of-islamic-state-bunzel/the-ideology-of-the-islamic-state.pdf>.
28. Abu Umar al-Baghdadi, "Qul inni 'ala bayyina min Rabbi," March 13, 2007, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010). In Husaybah in 2005, residents complained that music stores and satellite dishes were banned and their women were being forced to wear full-body veils. When the Islamic Army of Iraq criticized Baghdadi for his draconian laws, the prohibition of satellite dishes and the requirement to cover women's faces were among its complaints. See Ellen Knickmeyer, "Zarqawi Followers Clash with Local Sunnis," Washington Post, May 29, 2005, <http://www.washingtonpost.com/wp-dyn/content/article/2005/05/28/AR2005052800967.html>; "Islamic Army in Iraq Accuses al-Qa'ida," Open Source Center.
29. Uthman bin Abd al-Rahman al-Tamimi, ed., "I'lam al-anam bi-milad Dawlat al-Islam," Minbar al-Tawhid wa-l-Jihad, January 7, 2007. For a summary of the document, see Brian Fishman, "Fourth Generation Governance: Sheikh Tamimi Defends the Islamic State of Iraq," Combating Terrorism Center at West Point, March 23, 2007, <https://www.ctc.usma.edu/posts/fourth-generation-governance-sheikh-tamimi-defends-the-islamic-state-of-iraq>.
30. Tamimi, "I'lam," 36.
31. Ibid., 38.
32. M. Cherif Bassiouni, The Shari'a and Islamic Criminal Justice in Time of War and Peace (New York: Cambridge University Press, 2014), 134–141.
33. Thomas Friedman, "Letter from Baghdad," New York Times, September 5, 2007, <http://www.nytimes.com/2007/09/05/opinion/05Friedman.html?_r=0>.
34. Mohammed M. Hafez, "Al-Qaeda Losing Ground in Iraq," CTC Sentinel 1, no. 1 (December 15, 2007): n.p., <https://www.ctc.usma.edu/posts/al-qaida-losing-ground-in-iraq>.
35. Wasim al-Dandashi, "Qadi Dawlat al-'Iraq al-Islamiyya Sa'udi lam yakmal dirasatihi," Elaph, April 25, 2007, <http://elaph.com/ElaphWeb/Politics/2007/4/229045.htm>.
36. See postscript in Utaybi, "Risalat al-Shaykh."
37. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," November 19, 2007, <http://iraqslogger.powweb.com/downloads/aqi_leadership_letters_sept_08.pdf?PHPSESSID=b155c5eb6418ac653ca2ce675e6fb7f8>. This is another letter in the exchange cited in note 14.
38. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," January 25, 2008. See note 14.
39. In 2013, Abu Sulayman's letter was posted online, so we know what his allegations were. See Utaybi, "Risalat al-Shaykh." Al-Qaeda also summarized his allegations in their correspondence with Masri.
40. See Utaybi, "Risalat al-Shaykh."
41. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," January 25, 2008.
42. Ibid.
43. Utaybi, "Risalat al-Shaykh."
44. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," January 25, 2008. Badran identifies the "paternal uncle" as Sayf al-Adl, but the email of the "paternal uncle" appended to the end of the message is written by Zawahiri.
45. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," January 25, 2008. Masri's predecessor, Abu Mus'ab al-Zarqawi, was equally apocalyptic in his rhetoric but avoided mentioning the Mahdi in his public statements; see David Cook, "Abu Musa'b [sic] al-Suri and Abu Musa'b [sic] al-Zarqawi: The Apocalyptic Theorist and the Apocalyptic Practitioner," unpublished, 14.
46. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," January 25, 2008.
47. Ibid.
48. "Letter from Ayman al-Zawahiri to Abu Umar al-Baghdadi," March 6, 2008. This is another letter in the batch cited in note 14.
49. "Letter from Unknown al-Qaeda Leader to Abu Ayyub al-Masri," March 10, 2008. This letter quotes Masri's earlier letter.
50. Amit Paley, "Al-Qaeda in Iraq Leader May Be in Afghanistan," Washington Post, July 31, 2008, <http://www.washingtonpost.com/wp-dyn/content/article/2008/07/30/AR2008073003239.html>.
51. William McCants, "Death of a Sulayman," Jihadica, May 13, 2008, <http://www.jihadica.com/death-of-a-sulayman/>.
52. Jessica D. Lewis, "Al-Qaeda in Iraq Resurgent: The Breaking the Walls Campaign, Part I," Institute for the Study of War, Middle East Security Report 14 (September 2013): 8, <http://www.understandingwar.org/sites/default/files/AQI-Resurgent-10Sept_0.pdf>.
53. See Bunzel, "Ideology," 22; Sadiq al-Iraqi, "Zawjat Abu [sic] Ayyub."
54. William McCants, "'The Painful Truth: al-Qaeda Is Losing the War in Iraq,'" Jihadica, October 1, 2008, <http://www.jihadica.com/the-painful-truth-al-qaeda-is-losing-the-war-in-iraq/>.
55. William McCants, "Lamenting Loss of Anbar, Apprehensive of Jihad's Future in Iraq," Jihadica, September 5, 2008, <http://www.jihadica.com/lamenting-loss-of-anbar-apprehensive-of-jihads-future-in-iraq/>.
56. William McCants, "Spinning the Failure of the Islamic State of Iraq," Jihadica, August 13, 2008, <http://www.jihadica.com/spinning-the-failure-of-the-islamic-state-of-iraq/>.
57. Daniel Kimmage and Kathleen Ridolfo, "Iraqi Insurgent Media—The War of Images and Ideas: How Sunni Insurgents in Iraq and Their Supporters Worldwide Are Using the Media," RadioFreeEurope/RadioLiberty, RFE/RL Special Report, June 26, 2007, <http://www.rferl.org/content/article/1077316.html>.
58. Fishman, "Dysfunction," 16–20. For the original, see: <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/09/Analysis-of-the-State-of-ISI-Original.pdf>; for the translation, see: <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/09/Analysis-of-the-State-of-ISI-Translation.pdf>.
59. Adam Gadahn, "Letter from Adam Gadahn [English translation]," personal correspondence to unknown recipient, Harmony Program, Combating Terrorism Center at West Point, January 2011, 7–9, <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/10/Letter-from-Adam-Gadahn-Translation.pdf>. Original Arabic version available at <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/10/Letter-from-Adam-Gadahn-Original.pdf>.
60. Clint Watts, Jacob Shapiro, and Vahid Brown, Al-Qa'ida's (Mis)adventures in the Horn of Africa, Harmony Program, Combating Terrorism Center at West Point, July 2, 2007, 89–100, <https://www.ctc.usma.edu/posts/al-qaidas-misadventures-in-the-horn-of-africa>.
61. Fadil Harun, al-Harb 'ala al-Islam: Qissat Fadil Harun, 2 vols., February 26, 2009, 2: 68–69; vol. 1 available at <https://www.ctc.usma.edu/posts/the-war-against-islam-the-story-of-fazul-harun-part-1-original-language-2>; vol. 2 available at <https://www.ctc.usma.edu/posts/the-war-against-islam-the-story-of-fazul-harun-part-2-original-language-2>.
62. Ibid., 2: 134–135.
63. Osama bin Laden, "Letter to Nasir al-Wuhayshi [English translation]," personal correspondence to Nasir al-Wuhayshi (aka Abu Basir), SOCOM-2012-0000016, Harmony Program, Combating Terrorism Center at West Point, 2010, 13–14, <https://www.ctc.usma.edu/posts/letter-to-nasir-al-wuhayshi-english-translation-2>. Original Arabic version available at <https://www.ctc.usma.edu/posts/letter-to-nasir-al-wuhayshi-original-language-2>. The letter was probably drafted by one of Bin Laden's lieutenants. See Nelly Lahoud et al., "Letters from Abbottabad: Bin Ladin Sidelined?" Harmony Program, Combating Terrorism Center at West Point, May 3, 2012, <https://www.ctc.usma.edu/posts/letters-from-abbottabad-bin-ladin-sidelined>. Bin Laden also advised the Taliban and other jihadist groups in Afghanistan and Pakistan to avoid alienating the tribes, citing the Islamic State as an example of what not to do: "As for the tribes that joined or about to join the American Awakenings project, please give them a stern warning that an excessive reaction from their part towards these tribes will only increase the latter's unity and desire to fight against them. It will be useful to clarify the matter by citing the experience of our brothers in Iraq." See Osama bin Laden, "Letter from Osama bin Laden to 'Atiyya Abd al-Rahman [English translation]," personal correspondence, 432-10-CR-019-S-4-RJD, August 7, 2010, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2015/03/432-10-CR-019-S-4-RJD-Translation.pdf>. Original Arabic version available at <http://www.jihadica.com/wp-content/uploads/2015/03/432-10-CR-019-S-4-RJD-Original.pdf>.
64. Osama bin Laden, "Ila ahlina bi-l-'Iraq," audiotape message, Mu'assasat al-Sahab, broadcast by Al Jazeera on October 22, 2007. Arabic transcript of message available at "Sahab" [online pseudonym], "Tafrigh risalat al-Shaykh: Usama bin Ladan—hafizahu Allah," Shabakat Ana al-Muslim li-l-Hiwar al-Islami, October 24, 2007, <http://www.muslm.org/vb/showthread.php?259268>. English transcript by Laura Mansfield available at Andrew Cochran, "Bin Laden Sounds the Call of Defeat in Iraq (updated 10/23 with transcript)," Counterterrorism Blog, October 23, 2007, <http://counterterrorismblog.org/2007/10/bin_laden_sounds_the_call_of_d.php>.
65. "DOD News Briefing with Gen. Odierno from the Pentagon," news transcript, U.S. Department of Defense, June 4, 2010, <http://www.defense.gov/transcripts/transcript.aspx?transcriptid=4632>; Weiss and Hassan, ISIS, 81.
66. "Bayan min Majlis Shura Dawlat al-'Iraq al-Islamiyya," Markaz al-Fajr li-l-I'lam, May 15, 2010, posted by "Murasil al-Fajr" [online pseudonym] to Muntadayat Shabakat Shumukh al-Islam on May 16, 2010, <http://www.jihadica.com/wp-content/uploads/2015/03/Bayan-min-Majlis-Shura-Dawlat-al-Iraq-al-Islamiyya.pdf>.
67. Abu Bakr al-Baghdadi, "Wa-ya'ba Allah illa an yutimm nurahu," Mu'assasat al-Furqan, July 21, 2012, <https://ia601207.us.archive.org/14/items/2b-bkr-bghdd/143393.pdf>. See also Bunzel, "Ideology," 24.
Chapter 3
1. Gregory D. Johnsen, The Last Refuge: Yemen, al-Qaeda, and America's War in Arabia (New York: W. W. Norton, 2012), 220–221.
2. Fahd Al-Riya'i, "'Weak and Misled' Militant Not Al-Qaeda Material," Saudi Gazette, May 29, 2010, http://www.saudigazette.com.sa/index.cfm?method=home.regcon&contentID=2010052973772.
3. Robert Worth, "Is Yemen the Next Afghanistan?" New York Times, July 6, 2010, http://www.nytimes.com/2010/07/11/magazine/11Yemen-t.html?pagewanted=all&_r=0.
4. Nayif Muhammad al-Qahtani, "Liqa' ma'a ahad al-matlubin (Abu Hammam al-Qahtani)," Sada al-malahim 1 (January 2008): 7–9, available at <https://ia801403.us.archive.org/25/items/Sada-almala7em/Sada-almala7em1.pdf>.
5. Sada al-malahim 2 (March 2008), <https://ia801403.us.archive.org/25/items/Sada-almala7em/Sada-almala7em2.pdf>.
6. Robbie Brown and Kim Severson, "2nd American in Strike Waged Qaeda Media War," New York Times, September 30, 2011, <http://www.nytimes.com/2011/10/01/world/middleeast/samir-khan-killed-by-drone-spun-out-of-the-american-middle-class.html?_r=0>.
7. J. M. Berger, "The Myth of Anwar al-Awlaki," Foreign Policy, August 10, 2011, <http://foreignpolicy.com/2011/08/10/the-myth-of-anwar-al-awlaki/>.
8. Anwar al-Awlaki, "Shaykh Anwar's Message to the American People and Muslims in the West," Inspire 1 (Summer 1431/2010): 56–58, available at Jihadology, <https://azelin.files.wordpress.com/2010/06/aqap-inspire-magazine-volume-1-uncorrupted.pdf>.
9. Sada al-malahim 6, 7, 8, 10, 12, 13, 14, and 15.
10. For background on the fighting, see Johnsen, Last Refuge, 266.
11. Xiong Tong, ed., "Al-Qaida Wing Claims to Form 12,000-Strong Army in Southern Yemen," Xinhua, July 30, 2010, <http://news.xinhuanet.com/english2010/world/2010-07/30/c_13422488.htm>.
12. Ten years ago, a team of Arabists and I used citation analysis (counting footnote references) to determine Maqdisi's rank in the jihadist scholarly hierarchy. He was at the very top. See William McCants, ed., "Militant Ideology Atlas: Executive Report," Combating Terrorism Center at West Point, November 2006, <https://www.ctc.usma.edu/wp-content/uploads/2012/04/Atlas-ExecutiveReport.pdf>. For Maqdisi's biography and teachings, see Joas Wagemakers, A Quietist Jihadi: The Ideology and Influence of Abu Muhammad al-Maqdisi (New York: Cambridge University Press, 2012).
13. Online correspondence between "Abu Abd al-Rahman al-Yamani" online pseudonym] and Abu Muhammad al-Maqdisi, Minbar al-Tawhid wa-l-Jihad, n.d., [http://www.tawhed.ws/FAQ/pr?qid=3352&PHPSESSID=f815327074d11ecd2fd01853dd03b4dc. Maqdisi is addressed respectfully by the questioner: "The Brothers in Yemen love Shaykh Abu Muhammad al-Maqdisi and consider him their imam . . . . Perhaps the shaykh would favor the monotheists in Yemen, who are among the youth of the Peninsula, with his advice and guidance."
14. Ibid. AQAP published the question and Maqdisi's answer in issue 15 of its Sada al-malahim (October-November 2010).
15. Maqdisi, response to "Abu Abd al-Rahman al-Yamani."
16. For the full text, see Abu Mus'ab al-Suri, "Mas'uliyyat ahl al-Yaman tujah muqaddasat al-Muslimin wa-tharwatihim," Minbar al-Tawhid wa-l-Jihad, section 3, <http://www.tawhed.ws/r?i=wksgfnyz>.
17. Ibid., section 4. As for tactics, Suri suggests a range of standard terrorist targets in the Arabian Peninsula: embassies, commercial interests, military bases, tourists, and so forth. Suri also countenanced attacks on targets in Yemen. See ibid., appendix.
18. Ibid., conclusion.
19. Osama bin Laden, "Letter to Nasir al-Wuhayshi [English translation]," personal correspondence to Nasir al-Wuhayshi (aka Abu Basir), SOCOM-2012-0000016, Harmony Program, Combating Terrorism Center at West Point, 2010, 13–14, <https://www.ctc.usma.edu/posts/letter-to-nasir-al-wuhayshi-english-translation-2>. Regarding the letter's date, its author mentions AQAP attacks in Ma'rib and Ataq. The first attack in Ma'rib was on June 7, 2010; the first in Ataq was on July 22, 2010. Both attacks were against military personnel. For a timeline of AQAP-related attacks in Yemen from 2010 to 2012, see Cody Curran et al., "AQAP and Suspected AQAP Attacks in Yemen Tracker 2010, 2011, and 2012," AEI Critical Threats, May 21, 2012, <http://www.criticalthreats.org/yemen/aqap-and-suspected-aqap-attacks-yemen-tracker-2010>.
20. Bin Laden, "Letter to Nasir al-Wuhayshi," 1. I have modified the translation.
21. Ibid., 2.
22. Osama bin Laden, "al-Sabil li-ihbat al-mu'amarat," Mu'assasat al-Sahab, December 29, 2007, www.tawhed.ws/dl?i=24041002; Cole Bunzel, "From Paper State to Caliphate: The Ideology of the Islamic State," The Brookings Project on U.S. Relations with the Islamic World, Analysis Paper No. 19 (March 2015): 28, <http://www.brookings.edu/~/media/research/files/papers/2015/03/ideology-of-islamic-state-bunzel/the-ideology-of-the-islamic-state.pdf>.
23. For Bin Laden's change in strategy, see Ryan Evans, "From Iraq to Yemen: al-Qa'ida's Shifting Strategies," CTC Sentinel (October 2010), n.p., <https://www.ctc.usma.edu/v2/wp-content/uploads/2011/05/CTCSentinel-Vol3Iss101.pdf>.
24. Bin Laden, "Letter to Nasir al-Wuhayshi," 2. My translation.
25. Ibid., 3.
26. Ibid., 5.
27. Ibid., 6.
28. Ibid., 4.
29. Osama bin Laden, "Draft Letter from Osama bin Laden to Nasir al-Wuhayshi [English translation]," personal correspondence, SOCOM-2012-0000017, Harmony Program, Combating Terrorism Center at West Point, n.d., 3, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000017-Trans.pdf>. Original Arabic version available at <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000017-Orig.pdf>. This letter seems to be an earlier draft of SOCOM-2012-0000016; for a detailed explanation, see Nelly Lahoud et al., "Letters from Abbottabad: Bin Ladin Sidelined?" 14, footnote 47, Harmony Program, Combating Terrorism Center at West Point, May 3, 2012, <https://www.ctc.usma.edu/posts/letters-from-abbottabad-bin-ladin-sidelined>. To distinguish the letter cited in this footnote from the other letter from Bin Laden to Wuhayshi, this one will hereafter be referred to as Bin Laden, "Draft Letter."
30. Bin Laden, "Draft Letter," 3.
31. F. Gregory Gause, Oil Monarchies: Domestic and Security Challenges in the Arab Gulf States (New York: Council on Foreign Relations Press, 1994), 24–25.
32. Bin Laden, "Letter to Nasir al-Wuhayshi," 13–14.
33. Bin Laden, "Draft Letter," 4–5.
34. Ibid., 4.
35. Ibid., 5.
36. Atiyya Abd al-Rahman, "Letter from 'Atiyya Abd al-Rahman to Osama bin Laden [English translation]," personal correspondence, 422-10-CR-109-S-4-RJD, July 17, 2010, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2015/03/422-10-CR-109-S-4-RJD-Translation.pdf>. Original Arabic version available at <http://www.jihadica.com/wp-content/uploads/2015/03/422-10-CR-109-S-4-RJD-Original.pdf>.
37. Nasir al-Wuhayshi, "Second Letter from Abu Basir to Emir of Al-Qaida in the Islamic Maghrib," personal correspondence to Abu Mus'ab Abd al-Wadud, August 6, 2012, in "Al-Qaida Papers," Associated Press; English translation and original Arabic version both available at Long War Journal, <http://www.longwarjournal.org/images/al-qaida-papers-how-to-run-a-state.pdf>.
38. See Mustafa al-Sharqawi, "Tatbiq al-shari'a fi al-Yaman 'taqrir hawla Ansar al-Shari'a,'" YouTube video, February 24, 2013, <https://www.youtube.com/watch?v=Yy8qv52DzVA>; "Mujahidi al-Iraq" online pseudonym], "al-Qa'ida tu'lin Abyan al-Yamaniyya imara Islamiyya!" Shabakat Ana al-Muslim li-l-Hiwar al-Islami, April 27, 2011, <http://www.muslm.org/vb/archive/index.php/t-434052.html>; and "Tanzim al-Qa'ida yu'lin al-bayan raqam 1 min idha'at Abyan imara Islamiyya'," Barakish.net, March 29, 2011, [http://www.barakish.net/news02.aspx?cat=0&sub=0&id=17420.
39. Letter from Abu Basir al-Tartusi, "Ila al-ikhwa Ansar al-Shari'a fi al-Yaman," www.abubaseer.bizland.com/hadath/Read/hadath%2092.doc. See also Joas Wagemakers, "Al-Qaida Advises the Arab Spring: Yemen," Jihadica, June 5, 2012, <http://www.jihadica.com/al-qaida-advises-the-arab-spring-yemen/>.
40. Robin Simcox, "Ansar al-Sharia and Governance in Southern Yemen," Hudson Institute, December 27, 2012, <http://www.hudson.org/research/9779-ansar-al-sharia-and-governance-in-southern-yemen>.
41. Agence France-Presse, "Osama Bin Laden Believed in Image, and Considered Al Qaeda Name Change to Improve 'Brand,'" Al Arabiya, June 25, 2011, <http://english.alarabiya.net/articles/2011/06/25/154757.html>; Peter Bergen, "Bin Laden: Seized Documents Show Delusional Leader and Micromanager," CNN, May 3, 2012, <http://www.cnn.com/2012/04/30/opinion/bergen-bin-laden-document-trove/>; Osama bin Laden, "Letter from Osama bin Laden to Mukhtar Abu al-Zubayr [English translation]," personal correspondence to Abu al-Zubayr (aka Ahmed Abdi Godane), SOCOM-2012-0000005, Harmony Program, Combating Terrorism Center at West Point, August 7, 2010, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000005-Trans.pdf>. Original Arabic version available at <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000005-Orig.pdf>. See also Aaron Y. Zelin, "Know Your Ansar al-Sharia," Washington Institute for Near East Policy, September 21, 2012, <http://www.washingtoninstitute.org/policy-analysis/view/know-your-ansar-al-sharia>.
42. Abu Zubayr Adil al-Abab, "Online Question and Answer Session with Abu Zubayr Adel al-Abab, Shariah Official for Member of al-Qaeda in the Arabian Peninsula (AQAP) [English translation]," transl. Amany Soliman, International Centre for the Study of Radicalisation and Political Violence, April 18, 2011, available at Jihadology, <http://azelin.files.wordpress.com/2011/04/ghorfah-minbar-al-ane1b9a3c481r-presents-a-new-audio-message-from-al-qc481_idah-in-the-arabian-peninsulas-shaykh-abc5ab-zc5abbayr-adc4abl-bc4abn-abdullah-al-abc481b-en.pdf>.
43. Ibid., 2.
44. Ibid., 3.
45. Ibid., 7.
46. Ibid., 5.
47. Ibid., 6.
48. William McCants, "Al Qaeda Is Doing Nation-Building. Should We Worry?" Foreign Policy, April 30, 2012, <http://foreignpolicy.com/2012/04/30/al-qaeda-is-doing-nation-building-should-we-worry/>.
49. Simcox, "Ansar al-Sharia and Governance in Southern Yemen."
50. Sudarsan Raghavan, "In Yemen, Tribal Militias in a Fierce Battle with al-Qaeda Wing," Washington Post, September 10, 2012, <http://www.washingtonpost.com/world/middle_east/in-yemen-tribal-militias-in-a-fierce-battle-with-al-qaeda-wing/2012/09/10/0cce6f1e-f2b2-11e1-b74c-84ed55e0300b_story.html>.
51. "Conflict in Yemen: Abyan's Darkest Hour," Amnesty International, December 3, 2012, 19, <http://www.amnestyusa.org/sites/default/files/mde_31.010.2012_conflict_in_yemen_-_abyans_darkest_hour.pdf>.
52. Simcox, "Ansar al-Sharia and Governance in Southern Yemen."
53. Ibid.
54. Raghavan, "Tribal Militias."
55. Casey Coombs, "Echoes of Iraq: Yemen's War Against al-Qaeda Takes a Familiar Turn," Time, August 10, 2012, <http://world.time.com/2012/08/10/echoes-of-iraq-yemens-war-against-al-qaeda-takes-a-familiar-turn/>.
56. Abu Zubayr Adil al-Abab, "Gains and Benefits of Ansar Al-Sharia Control of Parts of the Wiyalah's [sic] of Abyan and Shabwa," Ansar al-mujahideen English forum, July 6, 2012, 17, 21–25, 34, 41, in "Al-Qaida Papers," Associated Press; English translation and original Arabic version available at Long War Journal, <http://www.longwarjournal.org/images/al-qaida-papers-how-to-run-a-state.pdf>.
57. Nasir al-Wuhayshi, "First Letter from Abu Basir to Emir of Al-Qaida in the Islamic Maghreb," personal correspondence to Abu Mus'ab Abd al-Wadud, May 21, 2012, in "Al-Qaida Papers," Associated Press; English translation and original Arabic version both available at Long War Journal, <http://www.longwarjournal.org/images/al-qaida-papers-how-to-run-a-state.pdf>.
58. On the Popular Committees, see Nadwa Al-Dawsari, "The Popular Committees of Abyan, Yemen: A Necessary Evil or an Opportunity for Security Reform?" Middle East Institute, March 5, 2014, <http://www.mei.edu/content/popular-committees-abyan-yemen-necessary-evil-or-opportunity-security-reform>.
59. Wuhayshi, "First Letter."
60. Ibid.
61. Ibid.
62. Wuhayshi, "Second Letter."
63. Wuhayshi mentions in the second letter to Abd al-Wadud that he had learned his first letter had not reached the AQIM leader. He appended it to the second letter. See ibid.
64. Boubker Belkadi, "Ruthless Chief, Head of Al-Qaeda's NAfrica [sic] Branch," Middle East Online, December 13, 2007, <http://www.middle-east-online.com/english/?id=23510>.
65. Abu Mus'ab Abd al-Wadud, "Mali-Al-Qaida's Sahara Playbook," personal correspondence to the commanders of al-Qaeda in the Islamic Maghreb (AQIM), Associated Press, Autumn 2012, 1, <http://hosted.ap.org/specials/interactives/_international/_pdfs/al-qaida-manifesto.pdf>. For dating the document, see Rukmini Callimachi, "In Timbuktu, al-Qaida Left Behind a Manifesto," Associated Press, February 14, 2013, <http://bigstory.ap.org/article/timbuktu-al-qaida-left-behind-strategic-plans>.
66. Abd al-Wadud, "Sahara Playbook," 4.
67. Ibid., 10.
68. Ibid., 9.
69. Ibid., 4.
70. Andrew Lebovich, "The Local Face of Jihadism in Northern Mali," CTC Sentinel 6, no. 6 (June 2013): 4–10, 5, <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/06/CTCSentinel-Vol6Iss64.pdf>.
71. Ibid., 6.
72. Abd al-Wadud, "Sahara Playbook," 5.
73. Ibid.
74. "Shari'ah Penalties Anger Mali Muslims," OnIslam, August 5, 2012, <http://www.onislam.net/english/news/africa/458395-shariah-penalties-anger-mali-muslims.html>.
75. "TSG IntelBrief: The Continuing Crisis in Mali," The Soufan Group, May 29, 2012, <http://soufangroup.com/tsg-intelbrief-the-continuing-crisis-in-mali/>.
76. Celeste Hicks, "Mali War Exposes Religious Fault Lines," Guardian, May 3, 2013, <http://www.theguardian.com/world/2013/may/03/mali-war-religious-faultlines>; see also William G. Moseley, "Assaulting Tolerance in Mali," Al Jazeera, July 16, 2012, <http://www.aljazeera.com/indepth/opinion/2012/07/201271594012144369.html>; Brian J. Peterson, "Mali 'Islamization' Tackled: The Other Ansar Dine, Popular Islam, and Religious Tolerance," African Arguments, April 25, 2012, <http://africanarguments.org/2012/04/25/confronting-talibanization-in-mali-the-other-ansar-dine-popular-islam-and-religious-tolerance-brian-j-peterson/>.
77. "Rebels Burn Timbuktu Tomb Listed as U.N. World Heritage Site," CNN, May 7, 2012, <http://edition.cnn.com/2012/05/05/world/africa/mali-heritage-sites/index.html>.
78. Abd al-Wadud, "Sahara Playbook," 5.
79. Idrissa Fall, "A Look inside Northern Mali," Voice of America, July 26, 2012, <http://www.voanews.com/content/a-look-inside-northern-mali/1447183.html>.
80. Adam Nossiter, "Jihadists' Fierce Justice Drives Thousands to Flee Mali," New York Times, July 17, 2012, <http://www.nytimes.com/2012/07/18/world/africa/jidhadists-fierce-justice-drives-thousands-to-flee-mali.html?_r=0>.
81. "Malians Protest at Strict Islamic Justice," Times of Malta, July 14, 2012, <http://www.timesofmalta.com/articles/view/20120714/world/Malians-protest-at-strict-Islamic-justice.428483>.
82. Abd al-Wadud, "Sahara Playbook," 3.
83. Ibid., 9.
84. "Mali: Reform or Relapse," International Crisis Group, Africa Report, no. 210 (January 10, 2014): 9, <http://www.crisisgroup.org/~/media/Files/africa/west-africa/mali/210-mali-reform-or-relapse-english.pdf>.
85. The Shabab had made territorial gains in the south and central areas of Somalia following the Ethiopian invasion that started in 2006. Local discontent with the occupation in part enabled the group's expansion. See Jonathan Masters and Mohammed Aly Sergie, "Al-Shabab," CFR Backgrounders, Council on Foreign Relations, March 13, 2015, <http://www.cfr.org/somalia/al-shabab/p18650>.
86. Nick Grace, "Islamic Emirate of Somalia Imminent as Shabaab Races to Consolidate Power," Long War Journal, September 8, 2008, <http://www.longwarjournal.org/archives/2008/09/islamic_emirate_of_s.php>. One of the Shabab's founders, one-time al-Qaeda member Talha al-Sudani, had entertained the idea of establishing a state before the creation of the Shabab. Bin Laden's representative in Somalia, Fadil Harun, discouraged Sudani from doing so for many of the same reasons Bin Laden would later cite. See Nelly Lahoud, "Beware of Imitators: al-Qa'ida through the Lens of Its Confidential Secretary," Harmony Program, Combating Terrorism Center at West Point, June 4, 2012, <https://www.ctc.usma.edu/posts/beware-of-imitators-al-qaida-through-the-lens-of-its-confidential-secretary>.
87. An example of such discussions (in Arabic) is provided as a link in William McCants, "When Will Somalia's Shabaab Movement Declare an Islamic State?" Jihadica, November 16, 2008, <http://www.jihadica.com/when-will-somalias-shabaab-movement-declare-an-islamic-state/>.
88. In a September 20, 2009, video message ("Labbayka ya Usama"), Godane addressed Bin Laden as the group's "shaykh and emir." The full video seems to have been removed from YouTube, but screenshots and detailed analysis of the video are available at Christopher Anzalone, "Leaps & Bounds: The Rapid Evolution of Harakat al-Shabab al-Mujahideen's Media," Views from the Occident, September 21, 2009, <http://occident.blogspot.com/2009/09/leaps-bounds-rapid-evolution-of-harakat.html>. Around the same time, soldiers affiliated with the Shabab began displaying the Islamic State flag, flying it from their trucks as they zoomed around Mogadishu and other battlefronts. See, for example, "al-Jihad al-Sumali" [online pseudonym], "Madin ka-l-sayf ma'a junud Shabab al-Mujahidin fi al-Sumal," YouTube video, November 17, 2009, <https://www.youtube.com/watch?v=EHPVsSeR77Q>. The Islamic State flag was the primary flag the Shabab used thereafter.
89. According to American Shabab member Omar Hammami, AQAP reached out to the group on behalf of the leadership in Afghanistan, relaying al-Qaeda's desire that the Shabab join al-Qaeda. Prompted by the letter, the Shabab's leaders gathered to deliberate on the "option of joining al-Qaeda and the option of declaring an Islamic state." Most of the group wanted to declare a state and proclaim its merger with al-Qaeda. But the Shabab's senior leader, Godane, rejected both, arguing without explaining himself that the circumstances were not favorable for either. Hammami groused, "They continued procrastinating in announcing [their affiliation with al-Qaeda] or the [establishment of] the state." As Bin Laden's letters make clear, Godane was merely following the lead of the al-Qaeda chief. Omar Hammami's recollections are available in Aaron Y. Zelin, "New Video Message and Two Documents from Omar Hammami [Abu Mansur al-Amriki]: 'The Final Appeal from the Humble Servant???'" Jihadology, January 7, 2013, <http://jihadology.net/2013/01/07/new-video-message-and-two-documents-from-omar-hammami-abu-man%E1%B9%A3ur-al-amriki-the-final-appeal-from-the-humble-servant/>. The leader of AQAP, Nasir al-Wuhayshi, mentioned the precedent set by the Shabab's decision not to declare a state in a 2012 letter. See Wuhayshi, "Second Letter."
90. In a letter dated June 19, 2010, Atiyya Abd al-Rahman reminded Bin Laden to send a letter to the Shabab, who had been "waiting for your answers and your guidance." See Atiyya Abd al-Rahman, "Letter from Atiyya Abd al-Rahman to Osama bin Laden [English translation]," personal correspondence, 420-10-CR-019-S-4-RJD, June 19, 2010, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2015/03/420-10-CR-019-S-4-RJD-Translation.pdf>. Original Arabic version available at <http://www.jihadica.com/wp-content/uploads/2015/03/420-10-CR-019-S-4-RJD-Original.pdf>. Atiyya nudged him again on July 17, 2010: "My dear Sheikh, the brothers in Somalia are waiting for a letter and orders from you. They are waiting for advice and a decision about the issues they brought up. It would be great if you assigned them something that we could send on, they would be happy with that." See Abd al-Rahman, "Letter from 'Atiyya Abd al-Rahman to Osama bin Laden [English translation]," 422-10-CR-109-S-4-RJD.
91. Rob Wise, "Al Shabaab," Homeland Security & Counterterrorism Program and Transnational Threats Project, Center for Strategic and International Studies, AQAM Futures Project Case Study No. 2 (July 2011): 5, <http://csis.org/files/publication/110715_Wise_AlShabaab_AQAM%20Futures%20Case%20Study_WEB.pdf>.
92. "Meeting Somalia's Islamist Insurgents," BBC News, April 28, 2008, <http://news.bbc.co.uk/2/hi/africa/7365047.stm>.
93. "Somali Troops 'Capture Key Port Town' from al-Shabab," BBC News, October 5, 2014, <http://www.bbc.com/news/world-africa-29495801>; Abdulkadir Khalif, "Al-Shabaab Order Woman Stoned to Death for Sex Offence," Africa Review, October 26, 2012, <http://www.africareview.com/News/Al-Shabaab-order-woman-stoned-to-death-for-sex-offence/-/979180/1598708/-/55afb2z/-/index.html>.
94. Reuters, "Rape Victim Stoned to Death in Somalia was 13, U.N. Says," New York Times, November 4, 2008, <http://www.nytimes.com/2008/11/05/world/africa/05somalia.html?_r=0>.
95. Mohamed Ahmed, "Al Shabaab Amputates Hands, Feet in Jowhar," Somalia Report, April 12, 2011, <http://www.somaliareport.com/index.php/post/512/Al_Shabaab_Amputates_Hands_Feet_In_J>.
96. Mohammed Ibrahim and Jeffrey Gettleman, "Somalis Protest Against Shabab in Mogadishu," New York Times, March 29, 2010, <http://www.nytimes.com/2010/03/30/world/africa/30shabab.html?_r=0>.
97. "Somalia: Al-Shabaab—It Will Be a Long War," International Crisis Group, Africa Briefing No. 99 (June 26, 2014), <http://www.crisisgroup.org/~/media/Files/africa/horn-of-africa/somalia/b099-somalia-al-shabaab-it-will-be-a-long-war.pdf>.
98. "Somalia: Information on Al-Shabaab, Including Areas of Control, Recruitment, and Affiliated Groups (2012–Nov. 2013)," Immigration and Refugee Board of Canada, available at Refworld, United Nations High Commissioner for Refugees, November 26, 2013, <http://www.refworld.org/docid/52cea4e34.html>.
99. Hamza Mohamed, "Al-Shabab Bans Internet in Somalia," Al Jazeera, January 9, 2014, <http://www.aljazeera.com/news/africa/2014/01/al-shabab-bans-internet-somalia-20141981213614575.html>.
100. Ahren Schaefer and Andrew Black, "Clan and Conflict in Somalia: Al-Shabaab and the Myth of 'Transcending Clan Politics,'" Terrorism Monitor 9, no. 40 (November 4, 2011): 7–11, <http://www.jamestown.org/single/?tx_ttnews[tt_news]=38628#.VLV0lCc7Rl8>. The Shabab's relationship with Somali tribes was complex, defined by converging but temporary interests rather than shared ideology or mutual trust. Tribes often gave the Shabab nominal support in exchange for access to the group's considerable resources. See Stig Jarle Hansen, "An In-Depth Look at Al-Shabab's Internal Divisions," CTC Sentinel 7, no. 2 (February 2014): 9–12, 10, <https://www.ctc.usma.edu/posts/an-in-depth-look-at-al-shababs-internal-divisions>.
101. Bin Laden, "Letter from Osama bin Laden to Mukhtar Abu al-Zubayr."
102. Bin Laden, "Letter from Osama bin Laden to 'Atiyya Abd al-Rahman," 432-10-CR-019-S-4-RJD, 2-4. Despite Bin Laden's guidance and Godane's ultimate decision not to declare an emirate, Robow and other senior Shabab leaders again attempted to announce one. After a four-day gathering in May 2011, Robow and other senior leaders issued a statement proclaiming they would rebrand the Shabab as an Islamic state. A group of the Shabab's religious scholars also recommended that the Shabab rebrand itself as the "Islamic Emirate." Godane pointedly rejected the statement and the recommendation. See J. D. and Mohamed Odowa, "Al-Shabaab to Change Name to Imaarah Islamiyah," Somalia Report, May 12, 2011, <http://www.somaliareport.com/index.php/post/2212/Al-Shabaab_to_Change_Name_to_Imaarah_Islamiyah>; Abdul Qadir Muhammad Abdullah et al., "Final Statement of the Conference of Islamic State Scholars in Somalia," December 3, 2011, in "Al-Qaida Papers," Associated Press, <http://hosted.ap.org/specials/interactives/_international/_pdfs/al-qaida-papers-state-scholars.pdf>.
103. Geoffrey Kambere, "Financing Al Shabaab: The Vital Port of Kismayo," CTX 2, no. 3 (August 2012): n.p., <https://globalecco.org/financing-al-shabaab-the-vital-port-of-kismayo>.
104. Bin Laden, "Letter from Osama bin Laden to 'Atiyya Abd al-Rahman," 432-10-CR-019-S-4-RJD.
105. Bin Laden, "Letter from Osama bin Laden to Mukhtar Abu al-Zubayr." At least one al-Qaeda leader, possibly Ayman al-Zawahiri, disagreed with Bin Laden. He argued that al-Qaeda should publicly recognize all its affiliates. See "Letter from Unknown to Osama bin Laden [English translation]," personal correspondence from unknown author to Bin Laden, SOCOM-2012-0000006, Harmony Program, Combating Terrorism Center at West Point, December 13, 2010, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000006-Trans.pdf>. Original Arabic available at <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000006-Orig.pdf>.
106. Osama bin Laden, "Letter from Osama bin Laden to 'Atiyya Abd al-Rahman [English translation]," personal correspondence, SOCOM-2012-0000010, Harmony Program, Combating Terrorism Center at West Point, April 26, 2011, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000010-Trans.pdf>. Original Arabic version available at <http://www.jihadica.com/wp-content/uploads/2012/05/SOCOM-2012-0000010-Orig.pdf>.
107. Ibid.
108. Peter Wonacott and Nicholas Bariyo, "Militants Find Symbolic Targets in Uganda," Wall Street Journal, July 13, 2010, <http://www.wsj.com/articles/SB10001424052748704288204575362400675683926>.
109. Charles Kazooba, "Somalia: AU Ministers Agree to 'Take on' Al Shabaab," AllAfrica, July 26, 2010, <http://allafrica.com/stories/201007250021.html>.
110. Aaron Maasho, "African Union Says Military Still Weakening Somalia's al Shabaab," Reuters, January 7, 2015, <http://www.reuters.com/article/2015/01/07/us-somalia-insurgency-idUSKBN0KG18C20150107>.
111. Malkhadir M. Muhumed, "UN Report: Leader's Death Won't End al-Shabab," Al Jazeera, November 6, 2014, <http://www.aljazeera.com/indepth/features/2014/11/un-report-leader-death-won-end-al-shabab-201411613375772891.html>.
112. Gabriel Koehler-Derrick, ed. "A False Foundation? AQAP, Tribes and Ungoverned Spaces in Yemen," Harmony Program, Combating Terrorism Center at West Point, September 2011, 97–140, <https://www.ctc.usma.edu/v2/wp-content/uploads/2012/10/CTC_False_Foundation3.pdf>.
113. Bill Roggio, "AQAP's Ansar al Sharia Executes 3 US 'Spies,'" Long War Journal, February 13, 2012, <http://www.longwarjournal.org/archives/2012/02/aqaps_ansar_al_sharia_executes.php>; Koehler-Derrick, "False Foundation?" 59.
114. Koehler-Derrick, "False Foundation?" 114.
115. Ibid., 60.
116. "Yemen al Qaeda Leader Criticises IS Beheadings as un-Islamic," Reuters, December 8, 2014, <http://uk.reuters.com/article/2014/12/08/uk-yemen-qaeda-beheadings-idUKKBN0JM26N20141208>; Alessandria Masi, "Difference Between Al-Qaeda And ISIS: Senior AQAP Leader Holds 'Press Conference,' Said Beheadings Are 'Big Mistake,'" I. B. Times, December 8, 2014, <http://www.ibtimes.com/difference-between-al-qaeda-isis-senior-aqap-leader-holds-press-conference-said-1742818>; "TSG IntelBrief: Beyond Bombs, Bullets, and Blades: The Killer Narrative," The Soufan Group, December 11, 2014, <http://soufangroup.com/tsg-intelbrief-beyond-bombs-bullets-and-blades-the-killer-narrative/>.
117. When the Islamic State bombed Shi'i mosques in Sana'a, AQAP issued a statement denying its involvement. "We adhere to the directives of al-Shaykh Ayman al-Zawahiri, may God protect him, which require avoiding the targeting of mosques, markets, and places [where people] mingle in order to protect the lives of innocent Muslims and to prioritize the greater good." See "Bayan nafi al-'alaqa bi-tafjirat masajid al-Huthiyyin fi Sana'a," March 20, 2015, <https://azelin.files.wordpress.com/2015/03/al-qc481_idah-in-the-arabian-peninsula-22denying-a-relationship-with-the-bombings-of-the-e1b8a5c5abthc4ab-mosques-in-e1b9a3anac48122.pdf>; Daniel Byman and Jennifer R. Williams, "Will al Qaeda Be the Great Winner of Yemen's Collapse?" Foreign Policy, April 9, 2015, <http://foreignpolicy.com/2015/04/09/will-al-qaeda-be-the-great-winner-of-yemens-collapse/>.
118. Jean-Luc Marret was probably the first person to apply the term to an al-Qaeda affiliate. He used it to describe the ideology and behavior of al-Qaeda's North African affiliate, which called for attacks on the West but focused its own attacks locally. See his "Al-Qaeda in the Islamic Maghreb: A 'Glocal' Organization," Studies in Conflict & Terrorism 31 (2008): 541–552.
119. "al-Islah" [online pseudonym], "al-'Alim kullahu yansur rayat Dawlat al-'Iraq al-Islamiyya (al-Yaman—al-Sumal—Indunisiyya. Idkhal wa-shuf)," Arrawan, June 27, 2010, <http://z3tr.arrawan.com/showthread.php?45965>.
120. Reuters, "Factbox-Ansar Dine—Black Flag over Northern Mali," Faith World, July 3, 2012, <http://blogs.reuters.com/faithworld/2012/07/03/factbox-ansar-dine-black-flag-over-northern-mali/>.
121. Thomas Joscelyn, "From al Qaeda in Italy to Ansar al Sharia Tunisia," Long War Journal, November 21, 2012, <http://www.longwarjournal.org/archives/2012/11/from_al_qaeda_in_ita.php>.
122. "al-Barlaman al-Libi yurahhib bi-qarar tasnif 'Ansar al-Shari'a' ka-jama'a irhabiyya," Ennahar El Djadid, November 22, 2014, <http://www.ennaharonline.com/ar/arabic_news/227646.html>.
123. William McCants, "Black Flag," slideshow, Foreign Policy, November 7, 2011, <http://foreignpolicy.com/slideshow/black-flag/>.
124. Tara Todras-Whitehill, "In Cairo, Protesters Didn't Pledge Allegiance to a Flag," Al-Monitor, September 12, 2012, <http://www.al-monitor.com/pulse/originals/2012/al-monitor/scenes-from-cairos-embassy-prote.html>.
125. William McCants, "The Sources of Salafi Conduct," Foreign Affairs, September 19, 2012, <http://www.foreignaffairs.com/articles/138129/william-mccants/the-sources-of-salafi-conduct?page=show>.
Chapter 4
1. Charles Lister, "Profiling the Islamic State," Brookings Doha Center, Analysis Paper No. 13 (November 2014): 10–11, <http://www.brookings.edu/~/media/Research/Files/Reports/2014/11/profiling%20islamic%20state%20lister/en_web_lister.pdf>.
2. A large number of Baghdadi's personal documents were collected by German news agencies. See Volkmar Kabisch et al., "Auf der Spur des IS-Anführers," ARD, February 18, 2015, <http://www.tagesschau.de/ausland/baghdadi-101.html>. In an email, Volkmar Kabisch related to me that Ibrahim was born in Samarra, spent his very early childhood in the town of al-Jelam (fifteen miles northeast of Samarra), and then moved with his family to the Jibriyya district of Samarra where he grew up. See also Hisham al-Hashimi, "Ashya' min hayat al-Baghdadi," Almada Newspaper, September 6, 2014, <http://almadapaper.net/ar/printnews.aspx?NewsID=471187>; Janine di Giovanni, "Who Is ISIS Leader Abu Bakr al-Baghdadi?" Newsweek, December 8, 2014, <http://www.newsweek.com/2014/12/19/who-isis-leader-abu-bakr-al-baghdadi-290081.html>; and; "al-Kashf 'an watha'iq hawla haqiqat Abu [sic] Bakr al-Baghdadi," All4Syria, February 27, 2015, <http://all4syria.info/Archive/196498>.
3. Giovanni, "Who Is ISIS Leader Abu Bakr al-Baghdadi?"
4. The brother, Shamsi, is in prison. See "al-Kashf 'an watha'iq hawla haqiqat Abu Bakr al-Baghdadi," All4Syria, February 27, 2015, <http://all4syria.info/Archive/196498>.
5. Kabisch et al., "Auf der Spur des IS-Anführers"; Bashir al-Wandi, "al-Baghdadi . . . irhab bi-nakhat al-Ba'th," al-Hiwar al-Mutamaddin, March 11, 2015, http://www.ahewar.org/debat/print.art.asp?t=0&aid=458880&ac=1.
6. Kabisch et al., "Auf der Spur des IS-Anführers."
7. See Turki bin Mubarak al-Bin'ali (aka Abu Hummam Bakr bin Abd al-'Aziz al-Athari), "Madd al-ayadi li-bay'at al-Baghdadi," <https://archive.org/details/baghdadi-001>, and chapter 6 on Baghdadi's caliphate in this volume.
8. Abu Yusuf al-Tunisi, "Li-awwal marra . . . al-Sira al-dhatiyya li-l-Shaykh Abu Bakr al-Baghdadi," July 15, 2013, available at Pieter Van Ostaeyen, "Abu Bakr al-Baghdadi—a Short Biography of the ISIS Sheikh," pietervanostaeyen, July 15, 2013, <https://pietervanostaeyen.wordpress.com/2013/07/15/abu-bakr-al-baghdadi-a-short-biography-of-the-isis-sheikh/>. For an English translation, see "A Biography of Abu Bakr al-Baghdadi," INSITE Blog on Terrorism and Extremism, SITE Intelligence Group, August 12, 2014, <http://news.siteintelgroup.com/blog/index.php/entry/226-the-story-behind-abu-bakr-al-baghdadi>.
9. Hashimi, "Ashya' min hayat al-Baghdadi."
10. Martin Chulov, "ISIS: The Inside Story," Guardian, December 11, 2014, <http://www.theguardian.com/world/2014/dec/11/-sp-isis-the-inside-story>; Giovanni, "Who Is ISIS Leader Abu Bakr al-Baghdadi?"; Hashimi, "Ashya' min hayat al-Baghdadi."
11. Hashimi, "Ashya' min hayat al-Baghdadi."
12. "al-Kashf 'an watha'iq," All4Syria.
13. Loveday Morris, "Is This the High School Report Card of the Head of the Islamic State?" Washington Post, February 19, 2015, <http://www.washingtonpost.com/blogs/worldviews/wp/2015/02/19/is-this-the-high-school-report-card-of-the-head-of-the-islamic-state/>.
14. "Neue Erkenntnisse über IS-chef Baghdadi: Der kurzsichtige Kalif," Spiegel Online, February 19, 2015, <http://www.spiegel.de/politik/ausland/abu-bakr-al-baghdadi-das-leben-des-is-anfuehrers-a-1019227.html>; "al-Kashf 'an watha'iq," All4Syria. Other accounts say Baghdadi got his bachelor's degree at Saddam University for Islamic Studies; for example, Hashimi, "Ashya' min hayat al-Baghdadi."
15. The date on his physical when he matriculated at Saddam University for Islamic Studies is 1996. See Kabisch et al., "Auf der Spur des IS-Anführers."
16. Amatzia Baram, "From Militant Secularism to Islamism: The Iraqi Ba'th Regime 1968–2003," Woodrow Wilson International Center for Scholars, October 2011, 16–17, <http://www.wilsoncenter.org/sites/default/files/From%20Militant%20Secularism%20to%20Islamism.pdf>.
17. The book Baghdadi edited, Ruh al-murid fi sharh al-'iqd al-farid fi nuzum al-tajwid by Muhammad al-Samarqandi (who died in Baghdad around AD 1378), is obscure, which is why Baghdadi chose it. It wouldn't make sense for an aspiring editor of manuscripts to choose a published work. See the online record here: <http://quran-c.com/display/DispBib.aspx?BID=4167>. For background on Samarqandi, see Umar Rida Kahhala, Mu'jam al-mu'allifin: Tarajim musannafi al-kutub al-'Arabiya, vol. 12 (Beirut: Dar Ihya' al-Turath al-'Arabi, n.d.), 4–5.
18. "al-Kashf 'an watha'iq," All4Syria.
19. For a detailed discussion of Baghdadi's intellectual journey, see my biography of him in "The Believer," The Brookings Essay, Brookings Institution (August 2015).
20. Wael Essam, "'al-Baghdadi' kharaja min sijn Bukka akthar tatarrufan wa-kafara bi-l-'Ikwhan' . . . wa-darrasahu 'alim sufi," al-Quds al-Arabi, October 19, 2014, <http://www.alquds.co.uk/?p=237500>.
21. "Video: Weltspiegel Extra: Das Phantom des IS Terrors," Das Erste, February 18, 2015, <http://www.daserste.de/information/talk/anne-will/videosextern/weltspiegel-extra-das-phantom-des-is-terrors-100.html>. Ibrahim would finish his doctorate in 2007, after he was already a senior leader in the Islamic State.
22. Chulov, "Isis: The Inside Story."
23. Michael Weiss and Hassan Hassan, ISIS: Inside the Army of Terror [Google Edition] (New York: Regan Arts, 2015), 116.
24. Hunter Walker, "Here Is the Army's Declassified Iraq Prison File on the Leader of ISIS," Business Insider, February 18, 2015, <http://www.businessinsider.com/abu-bakr-al-baghdadi-declassified-iraq-prison-file-2015-2>.
25. Some accounts say Baghdadi had already helped found an insurgent group, the Jaysh Ahl al-Sunna wa-l-Jama'a (Chulov, "Isis: The Inside Story"; Weiss and Hassan, ISIS, 116). Others say he hadn't yet taken up arms. See Ruth Sherlock, "How a Talented Footballer Became World's Most Wanted Man, Abu Bakr al-Baghdadi," Telegraph, November 11, 2014, <http://www.telegraph.co.uk/news/worldnews/middleeast/iraq/10948846/How-a-talented-footballer-became-worlds-most-wanted-man-Abu-Bakr-al-Baghdadi.html>.
26. "al-Kashf 'an watha'iq," All4Syria.
27. Chulov, "Isis: The Inside Story."
28. Ibid.
29. Tim Arango and Eric Schmitt, "U.S. Actions in Iraq Fueled Rise of a Rebel," New York Times, August 10, 2014, <http://www.nytimes.com/2014/08/11/world/middleeast/us-actions-in-iraq-fueled-rise-of-a-rebel.html?_r=0>; Hashimi, "Ashya' min hayat al-Baghdadi"; Giovanni, "Who Is ISIS Leader Abu Bakr al-Baghdadi"; Weiss and Hassan, ISIS, 121–122.
30. Aaron Y. Zelin, "Abu Bakr al-Baghdadi: Islamic State's Driving Force," BBC News, July 31, 2014, <http://www.bbc.com/news/world-middle-east-28560449>.
31. "al-Kashf 'an watha'iq," All4Syria.
32. The medieval book was al-La'ali' al-farida fi sharh al-qasida, Abu Abd Allah Muhammad bin Hasan al-Fasi's thirteenth-century commentary on al-Shatibi's twelfth-century Hirz al-amani wa-wajhuhu al-tahani fi al-qira'at al-sab'. Baghdadi edited the portion from the introduction to the chapter on two hamzas ("Protokoll der Disputationskommission," a German translation of the Arabic minutes from Baghdadi's dissertation defense kindly provided to me by Volkmar Kabisch.)
33. A screenshot of the Arabic comments is found in "Video: Weltspiegel Extra."
34. "al-Kashf 'an watha'iq," All4Syria. All the copies of Baghdadi's dissertation reportedly have been stolen.
35. Hashimi, "Ashya' min hayat al-Baghdadi."
36. Weiss and Hassan, ISIS, 85–86.
37. McCants, "Believer."
38. Chulov, "Isis: The Inside Story."
39. Abu Ahmad, "al-Haqa'iq al-mukhfa hawla dawlat al-Baghdadi," al-Durar al-Shamiyya, April 5, 2014, <http://eldorar.com/node/45368>.
40. Tunisi, "al-Sira al-dhatiyya."
41. Chulov, "Isis: The Inside Story"; Weiss and Hassan, ISIS, 113.
42. Abu Ahmad, "al-Haqa'iq."
43. Sherlock, "Footballer."
44. Letter from Atiyya Abd al-Rahman to the Islamic State, April 21, 2010, quoted in Zawahiri, "Shahada li-haqan dima' al-mujahidin bi-l-Sham," Mu'assasat al-Sahab, May 3, 2014, 2, <https://pietervanostaeyen.wordpress.com/2014/05/03/dr-ayman-az-zawahiri-testimonial-to-preserve-the-blood-of-mujahideen-in-as-sham/>. In "Shahada," Zawahiri seems to be mistaken when he says Atiyya's letter was sent after Baghdadi's appointment, unless he means Baghdadi had already been selected before the letter arrived. In the same letter of condolence, Atiyya suggested the Islamic State should unite with other jihadist factions and contemplate a "new structure."
45. Abu Ahmad, "al-Haqa'iq."
46. "@wikibaghdady" [online pseudonym], "Asrar dawlat al-Baghdadi," Twitter post, circa December 14, 2013, <https://docs.google.com/document/d/1wEQ0FKosa1LcUB3tofeub1UxaT5A-suROyDExgV9nUY/edit>; Abu Ahmad, "al-Haqa'iq." According to @wikibaghdady, Hajji Bakr's knowledge of Iraq's army and his loyalty to Abu Umar al-Baghdadi and Abu Ayyub al-Masri endeared him to them
47. Abu Ahmad, "al-Haqa'iq."
48. Tunisi, "al-Sira al-dhatiyya"; Chulov, "Isis: The Inside Story."
49. Hashimi, "Ashya' min hayat al-Baghdadi."
50. Tunisi, "al-Sira al-dhatiyya."
51. On June 19, 2010, Atiyya forwarded a letter from the Islamic State to Bin Laden about the "new command taking charge." See Abd al-Rahman, "Letter from 'Atiyya Abd al-Rahman to Osama bin Laden [English translation]," 420-10-CR-019-S-4-RJD.
52. Osama bin Laden, "Letter from Osama bin Laden to 'Atiyya Abd al-Rahman [English translation]," SOCOM-2012-0000019, Harmony Program, Combating Terrorism Center at West Point, July 6, 2010, <http://www.docexdocs.com/ctc/SOCOM-2012-0000019Trans.pdf>. Original Arabic available at <http://www.docexdocs.com/ctc/SOCOM-2012-0000019Orig.pdf>. In his "Shahada," Zawahiri dates the letter July 6, 2010. Bin Laden wanted to know about Abu Bakr al-Baghdadi, his "first deputy," and his minister of war, Abu Sulayman al-Nasir li-Din Allah.
53. On July 17, 2010, Atiyya informed Bin Laden that he would ask for "information about Abu Bakr al-Baghdadi and his deputy," as well as information about Abu Sulayman al-Nasir li-Din Allah, the minister of war. See Abd al-Rahman, "Letter from 'Atiyya Abd al-Rahman to Osama bin Laden," 422-10-CR-109-S-4-RJD. (For a biography of Abu Sulayman, see Thomas Joscelyn, "The Islamic State of Iraq and the Sham's Quiet War Minister," Long War Journal, June 16, 2014, <http://www.longwarjournal.org/archives/2014/06/the_islamic_states_q.php>). Atiyya also wanted to ask members of other jihadist groups, including "al-Ansar," about the Islamic State's new leaders. The Ansar probably refers to Ansar al-Islam, an Iraqi insurgent group with which the Islamic State had butted heads. Bin Laden alludes to the conflict in an August 7, 2010, letter: "As for what the brothers in Iraq mentioned about the contention they have with Ansar Al Islam group, continue your correspondence with them. Advise them to do their best to avoid disagreement and conflict if possible. Advise them to seek the help of the tribal leaders, ulema and the former members of Ansar Al-Sunnah to resolve the issue." (See Bin Laden, "Letter from Osama bin Laden to Atiyya Abd al-Rahman," 432-10-CR-019-S-4-RJD.) Ansar al-Islam would later merge with the Islamic State when it declared its caliphate in 2014.
54. Letter from Atiyya Abd al-Rahman to the Islamic State's Ministry of Media, September 29, 2010, quoted in Zawahiri, "Shahada," 3.
55. Letter from a representative of the Islamic State's Majlis al-Shura to Atiyya Abd al-Rahman, October 9, 2010, quoted in Zawahiri, "Shahada," 3.
56. "Tanzim al-Qa'ida bi-l-'Iraq yata'ahhad bi-ta'yid al-Zawahiri wa-shann hajamat," Reuters, May 9, 2011, <http://ara.reuters.com/article/topNews/idARACAE7480SO20110509>; Zawahiri, "Shahada," 3.
57. Letter from a Representative for Communications with the Islamic State to Atiyya Abd al-Rahman, May 23, 2011, quoted in Zawahiri, "Shahada," 4.
58. Giovanni, "ISIS Leader"; "@wikibaghdady" [online pseudonym], "Asrar dawlat al-Baghdadi," December 15, 2013, <https://docs.google.com/document/d/1wEQ0FKosa1LcUB3tofeub1UxaT5A-suROyDExgV9nUY/edit>. For background on Hajji Bakr and his contribution to the organization of the Islamic State, see Christoph Reuter, "The Terror Strategist: Secret Files Reveal the Structure of the Islamic State," Spiegel Online, April 18, 2015, <http://www.spiegel.de/international/world/islamic-state-files-show-structure-of-islamist-terror-group-a-1029274.html>.
59. The insider who went by the Twitter handle @wikibaghdady tweeted a lot of the Islamic State's dirty laundry in the winter of 2013 (see "@wikibaghdady" [online pseudonym], "Asrar dawlat al-Baghdadi"). A few months later, an "Abu Ahmad" spoke to a news outlet and related similar information (see Abu Ahmad, "al-Haqa'iq"). See also Hashimi, "Ashya' min hayat al-Baghdadi"; Mitchell Prothero, "How 2 Shadowy ISIS Commanders Designed Their Iraq Campaign," McClatchy DC, June 30, 2014, <http://www.mcclatchydc.com/2014/06/30/231952/how-2-shadowy-isis-commanders.html>. Baghdadi has preferred to put Iraqis in charge of the war machine (this includes many former officers in the Iraqi army) and to install foreigners in the functional offices, such as the offices of recruitment and media. See Hisham al-Hashimi, "Haykaliyyat tanzim Da'ish," Kitabat fi al-Mizan, February 24, 2014, <http://www.kitabat.info/subject.php?id=43152>.
60. "Khitta istratijiyya li-ta'ziz al-mawqif al-siyasi li-Dawlat al-'Iraq al-Islamiyya," Mufakkirat al-Fallujah, December 2009 or January 2010, <https://ia802604.us.archive.org/16/items/Dirasa_dawla_03/khouta.pdf>. The document is dated Muharram 1431 (December 18, 2009–January 10, 2010).
61. Ibid., 3.
62. Ibid., 8.
63. Ibid., 4.
64. Ibid., 6.
65. Ibid., 25.
66. Ibid., 34.
67. Ibid., 36–37.
68. Ibid., 36.
69. See Majallat al-Ansar, vols. 1–28 (January 15, 2002–April 3, 2003), <https://www.tawhed.ws/c?i=325>.
70. "Khitta istratijiyya," 38–39.
71. Ibid., 39–40.
72. Ibid., 43–45.
73. Ibid., 50.
74. Ibid., 51.
75. Ibid., 51–52.
76. For my translation of the book, see Abu Bakr Naji, The Management of Savagery: The Most Critical Stage through Which the Umma Will Pass, transl. William McCants, John M. Olin Institute for Strategic Studies at Harvard University, May 23, 2006, <https://azelin.files.wordpress.com/2010/08/abu-bakr-naji-the-management-of-savagery-the-most-critical-stage-through-which-the-umma-will-pass.pdf>.
77. A major jihadist insider, Husayn bin Mahmud, alluded to Naji's death in an article online with the phrase "may God bless him," which is used only for those who have died ("Sahil al-jiyad fi jam' masadir al-jihad," al-Jabha al-I'lamiyya al-Islamiyya al-'Alamiyya, June 25, 2007).
78. Imam al-Sharif, the former head of Egyptian Islamic Jihad, claims the author is Muhammad al-Hukayma, an Egyptian member of the Islamic Group. In a 2014 interview, Sharif stated that Hukayma had a professional background in media and was working in the al-Qaeda media wing in Iran when he wrote The Management of Savagery. A 2008 Asharq Al-Awsat article reported that Hukayma moved from Iran to Pakistan in 2005. Hukayma announced the merger of the Islamic Group with al-Qaeda in a 2006 video in which he appeared next to Zawahiri (most of the Islamic Group rejected the merger). With a $1 million U.S. bounty on his head, Hukayma was killed in a U.S. air strike in Pakistan in 2008. See "Sabil al-Rishad" online pseudonym], "Ba'da haqa'iq al-'alaqa bayna al-Qa'ida wa Iran," YouTube video, February 19, 2014, [https://www.youtube.com/watch?v=RHQLpR2dLts&sns=tw; "Maqtal Abu Jihad al-Masri mas'ul al-i'lam bi-tanzim al-Qa'ida," Asharq Al-Awsat, November 2, 2014, http://classic.aawsat.com/details.asp?issueno=10626&article=493173#.VJBI0Sc7Rl8. For discussion of Hukayma as the author of The Management of Savagery in the Arabic press, see Mustafa Zahran, "Ansar Bayt al-Maqdis. wa 'Da'ishna' al-mashhad al-Masri," Al Jazeera, November 26, 2014, <http://www.aljazeera.net/home/Getpage/6c87b8ad-70ec-47d5-b7c4-3aa56fb899e2/e5785718-db70-4f05-bb3e-7fda7f466b0f>; Tariq al-Shaykh, "al-Hajma al-fikriyya al-qadima," Al Ahram, November 24, 2014, <http://www.ahram.org.eg/NewsQ/341613.aspx>.
79. Naji, Management of Savagery, 19.
80. Ibid., 11.
81. Ibid., 111–112.
82. Ibid., 34.
83. Ibid., 31.
84. Ibid., 98.
85. "The Messenger of God forbade the killing of women and children" (tradition from Sunan Abi Dawud).
86. For the ways the Islamic State circumvents the medieval Islamic scholarly tradition on war, see Sohaira Siddiqui, "Beyond Authenticity: ISIS and the Islamic Legal Tradition," Jadaliyya, February 24, 2015, <http://www.jadaliyya.com/pages/index/20944/beyond-authenticity_isis-and-the-islamic-legal-tra>.
87. Naji, Management of Savagery, 7. This seems to be a paraphrase of the following: "if a state overextends itself strategically—by, say, the conquest of extensive territories or the waging of costly wars—it runs the risk that the potential benefits from external expansion may be outweighed by the great expense of it all" (Paul Kennedy, The Rise and Fall of the Great Powers [New York: Random House, 1987], xvi).
88. Naji, Management of Savagery, 98.
89. William McCants, "Managing Savagery in Saudi Arabia," Jihadica, June 26, 2008, <http://www.jihadica.com/managing-savagery-in-saudi-arabia/>.
90. William McCants, "Al Qaeda Is Doing Nation-Building. Should We Worry?" Foreign Policy, April 30, 2012, <http://foreignpolicy.com/2012/04/30/al-qaeda-is-doing-nation-building-should-we-worry/>.
91. Hassan Hassan, "The Secret World of Isis Training Camps—Ruled by Sacred Texts and the Sword," Guardian, January 24, 2015, <http://www.theguardian.com/world/2015/jan/25/inside-isis-training-camps>. See also Jack Jenkins, "The Book That Really Explains ISIS (Hint: It's Not the Qur'an)," Think Progress, September 10, 2014, <http://thinkprogress.org/world/2014/09/10/3565635/the-book-that-really-explains-isis-hint-its-not-the-quran/>; David Ignatius, "The Manual That Chillingly Foreshadows the Islamic State," Washington Post, September 25, 2014, <http://www.washingtonpost.com/opinions/david-ignatius-the-mein-kampf-of-jihad/2014/09/25/4adbfc1a-44e8-11e4-9a15-137aa0153527_story.html>.
92. "Alaslami" [online pseudonym] (@M_Alaslami1), Twitter post (account since deleted), November 2014, <https://twitter.com/M_Alaslami1/status/536781484250497024>.
93. "al-Muhami Adil al-Turki" [online pseudonym] (@laweradelturki), Twitter post, September 13, 2012, <https://twitter.com/laweradelturki/status/246294423527383040>.
94. "al-Dar'awi" [online pseudonym] (@der3am), Twitter post, February 9, 2014, <https://twitter.com/der3am/status/432414102598606848>.
95. Lister, "Profiling the Islamic State," 12.
96. Brian Fishman and Joseph Felter, "Al-Qa'ida's Foreign Fighters in Iraq: A First Look at the Sinjar Records," Harmony Project, Combating Terrorism Center at West Point, January 2, 2007, <https://www.ctc.usma.edu/posts/al-qaidas-foreign-fighters-in-iraq-a-first-look-at-the-sinjar-records>.
97. Weiss and Hassan, ISIS, 99–111.
98. Ibid., 139.
99. Phil Sands, Justin Vela, and Suha Maayeh, "Assad Regime Set Free Extremists from Prison to Fire Up Trouble during Peaceful Uprising," The National, January 22, 2014, <http://www.thenational.ae/world/syria/assad-regime-set-free-extremists-from-prison-to-fire-up-trouble-during-peaceful-uprising>.
100. Abu Ahmad, "al-Haqa'iq."
101. Lister, "Profiling the Islamic State," 12.
102. Weiss and Hassan, ISIS, 168.
103. Lister, "Profiling the Islamic State," 13.
104. Thomas Joscelyn, "Syrian Rebel Leader Was Bin Laden's Courier, Now Zawahiri's Representative," Long War Journal, December 17, 2013, <http://www.longwarjournal.org/archives/2013/12/aq_courier_rebel_leader_zawahiri.php>.
105. Thomas Joscelyn, "Zawahiri Eulogizes al Qaeda's Slain Syrian Representative," Long War Journal, April 4, 2014, <http://www.longwarjournal.org/archives/2014/04/zawahiri_eulogizes_a.php>. For the date of Abu Khalid's capture, see Abd al-Rahman al-Hajj, "'Abu Khalid al-Suri' min 'al-Tali'a al-Muqatila' ila 'Ahrar al-Sham' maruran bi 'al-Qa'ida," Al Hayat, February 28, 2014, <http://alhayat.com/Articles/794946>.
106. Joscelyn, "Zawahiri Eulogizes al Qaeda's Slain Syrian Representative."
107. See Brynjar Lia, Architect of the Global Jihad: The Life of Al-Qaida Strategist Abu Mus'ab al-Suri (New York: Oxford University Press, 2009).
108. Alan Cullison, "Inside Al-Qaeda's Hard Drive," Atlantic, September 1, 2004, <http://www.theatlantic.com/magazine/archive/2004/09/inside-al-qaeda-s-hard-drive/303428/>.
109. Abu Mus'ab al-Suri, Da'wat al-muqawama al-Islamiyya al-'alamiyya [The Call of the Global Islamic Resistance], published on jihadist websites in December 2004, 750.
110. Ibid., 1518.
111. A September 2014 Al Jazeera article describes Uraydi as Nusra's "Sharia official." Nusra also issued a video recording of a dialogue with Uraydi in 2013 titled "Our Method and Creed," addressing Uraydi as "our shaykh." See "Wusul al-junud al-fijiyyin al-mufraj 'anhum ila al-Qunaytra," Al Jazeera, September 11, 2014, <http://www.aljazeera.net/home/Getpage/f6451603-4dff-4ca1-9c10-122741d17432/b539af9c-294b-494e-b503-2a8cd5cda739>; Ansar Jabhat al-Nusra, "Jabhat al-Nusra muqabila ma'a al-Duktur Sami al-Uraydi (hafazuhu Allah) bi-'unwan (manhajuna wa-'aqidatuna)," YouTube video, al-Manara al-Bayda', October 21, 2013, <https://www.youtube.com/watch?v=pYb8Rh_Kwpo>.
112. D. Sami Oride (@sami_oride), Twitter post, August 13, 2014, <https://twitter.com/sami_oride/status/499554084543029248>.
113. D. Sami Oride (@sami_oride), Twitter post, March 18, 2014, <https://twitter.com/sami_oride/status/445987493125427200/photo/1>.
114. D. Sami Oride (@sami_oride), Twitter post, February 12, 2014, <https://twitter.com/sami_oride/status/433474436339990528/photo/1>.
115. Suri, The Call of the Global Islamic Resistance, 909.
116. Nusra's popular support inside Syria derives in part from public perceptions of its heavily Syrian composition, although exact numbers are not known. There are reports that Nusra recruits Syrian adolescents as soldiers through its free "educational" programs in local schools and by offering protection or salaries. See "Maybe We Live and Maybe We Die: Recruitment and Use of Children by Armed Groups in Syria," Human Rights Watch, June 2014, 25–26, <http://www.hrw.org/sites/default/files/reports/syria0614_crd_ForUpload.pdf>; "The Islamic Front and Jabhat an-Nusra: Assessing the Sustainability and Future Trajectory of the Syrian Opposition's Most Important Alliance through Analysis of Rhetoric and Local Governance Activity," Courage Services, Inc., April 2014, 3, <http://www.courageservices.com/documents/Islamic%20Front%20and%20Jabhat%20an-Nusra_Courage%20Services_Apr2014.pdf>.
117. Charles C. Caris and Samuel Reynolds, "ISIS Governance in Syria," Institute for the Study of War, Middle East Security Report 22 (July 2014), <http://www.understandingwar.org/sites/default/files/ISIS_Governance.pdf>.
118. "The Islamic Front and Jabhat an-Nusra," 8.
119. In cities where the Islamic State lacks manpower, it relies more on locals to manage day-to-day affairs. See Weiss and Hassan, ISIS, 212–213.
120. Maqdisi quotes the line in his statement "Bayan hawla 'al-Dawla al-Islamiyya fi al-'Iraq wa-l-Sham' wa-l-mawqif al-wajib tujahaha," Minbar al-Tawhid wa-l-Jihad, May 26, 2014, 11-12, <https://tawhed.ws/r?i=26051401>. See the partial translation by Cole Bunzel, "From Paper State to Caliphate: The Ideology of the Islamic State," The Brookings Project on U.S. Relations with the Islamic World, Analysis Paper No. 19 (March 2015): 29, <http://www.brookings.edu/~/media/research/files/papers/2015/03/ideology-of-islamic-state-bunzel/the-ideology-of-the-islamic-state.pdf>.
121. During the period, I kept a record of every report I could find in English and Arabic of the Islamic State and Nusra governing, abusing civilians, and fighting with other rebel groups.
122. Weiss and Hassan, ISIS, 203.
123. Ibid., 208–211.
124. Plato, Republic 8:565d. See Plato in Twelve Volumes, Vols. 5 & 6 translated by Paul Shorey (Cambridge: Harvard University Press, 1969), <http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0168%3Abook%3D8%3Apage%3D565>.
125. The strategic differences between the two groups blurred somewhat in late 2014 when Nusra began to seize territory from other Sunni rebel groups in northern Syria. Reflecting this change, Nusra leaders began to publicly praise Naji's work for its advice on governance. "It should be the reference for the leaders of the mujahids administering the regions" tweeted one official. "Despite all our differences with the State group, it is the organization that has most applied what is in the book, after Jabhat al-Nusra." See "Abu al-Jamajem" [online pseudonym] (@AbuJamajem), Twitter post, March 6, 2015, <https://twitter.com/AbuJamajem/status/573714693828800512/photo/1>. My thanks to Sam Heller for the reference.
126. Theo Padnos, "My Captivity," New York Times Magazine, October 29, 2014, <http://www.nytimes.com/2014/10/28/magazine/theo-padnos-american-journalist-on-being-kidnapped-tortured-and-released-in-syria.html?_r=1>.
127. Weiss and Hassan, ISIS, 190.
128. "@wikibaghdady" [online pseudonym], "Asrar dawlat al-Baghdadi," Twitter posts, December 17 and December 18, 2013, <https://docs.google.com/document/d/1wEQ0FKosa1LcUB3tofeub1UxaT5A-suROyDExgV9nUY/edit>.
129. See Zawahiri, "Shahada," 6.
130. Abu Ahmad, "al-Haqa'iq"; @wikibaghdady, "Asrar dawlat al-Baghdadi," December 17 and December 18. Abu Ahmad asserts that @wikibaghdady's account of Baghdadi's meetings with Nusra leaders is generally correct but that he was wrong when he said Jawlani never met with Baghdadi.
131. Abu Bakr al-Baghdadi, "Wa-bashshir al-mu'minin," April 9, 2013, posted online by "tawtheek00jihad00dawlah" [online pseudonym], February 20, 2014, <https://tawtheek00jihad00dawlah.wordpress.com/2014/02/20/%D9%88%D8%A8%D8%B4%D8%B1-%D8%A7%D9%84%D9%85%D8%A4%D9%85%D9%86%D9%8A%D9%86-%D8%A3%D8%A8%D9%88-%D8%A8%D9%83%D8%B1-%D8%A7%D9%84%D8%A8%D8%BA%D8%AF%D8%A7%D8%AF%D9%8A/>. Baghdadi sent several other Islamic State leaders with Jawlani, including the Islamic State's spokesman Abu Muhammad al-Adnani. See Hashimi, "Ashya' min hayat al-Baghdadi."
132. "Jabhat al-Nusra," Mapping Militant Organizations, Stanford University, November 12, 2014, <http://web.stanford.edu/group/mappingmilitants/cgi-bin/groups/view/493#note24>.
133. Abu Muhammad al-Jawlani, "About the Fields of al-Sham," audio message, al-Manara al-Bayda', April 10, 2013, original Arabic audio message and English translation available at Aaron Y. Zelin, "al-Manarah al-Baydah' Foundation for Media Production presents a new audio message from Jabhat al-Nusrah's Abu Muhammad al-Jawlani (al-Golani): 'About the Fields of al-Sham,'" April 19, 2013, Jihadology, <http://jihadology.net/2013/04/10/al-manarah-al-bay%E1%B8%8Da-foundation-for-media-production-presents-a-new-audio-message-from-jabhat-al-nu%E1%B9%A3rahs-abu-mu%E1%B8%A5ammad-al-jawlani-al-golani-about-the-fields-of-al-sham/>.
134. Letter from Abu Bakr al-Baghdadi to Ayman al-Zawahiri, April 10, 2013, quoted in Zawahiri, "Shahada."
135. Ayman al-Zawahiri, "Letter from Ayman al-Zawahiri to Abu Bakr al-Baghdadi and Abu Muhammad al-Jawlani [English translation]," personal correspondence, May 22, 2013, <http://s3.documentcloud.org/documents/710588/translation-of-ayman-al-zawahiris-letter.pdf>.
136. Letter from Abu Bakr al-Baghdadi to one of the "officials of the group," July 29, 2013, quoted in Zawahiri, "Shahada," 4–5. It's unclear if the "group" refers to al-Qaeda or the Islamic State.
137. Tradition from Sahih Bukhari. Adam Gadahn quotes the tradition in a letter to an unknown recipient in late January 2011. See Adam Gadahn, "Letter from Adam Gadahn [English translation]," SOCOM-2012-0000004, Harmony Program, Combating Terrorism Center at West Point, January 2011, <https://www.ctc.usma.edu/posts/letter-from-adam-gadahn-english-translation-2>. Original Arabic version available at <https://www.ctc.usma.edu/posts/letter-from-adam-gadahn-original-language-2>.
138. Abu Muhammad al-Adnani, "Fa-dharhum wa-ma yaftarun," audio message, Mu'assasat al-Furqan, June 19, 2013, audio message and Arabic transcript both available at Pieter Van Ostaeyen, "An internal Jihadi strife—Jabhat an-Nusra and the Islamic State in Iraq and as-Sham," pietervanostaeyen, June 22, 2013, <https://pietervanostaeyen.wordpress.com/2013/06/22/an-internal-jihadi-strife-jabhat-an-nusra-and-the-islamic-state-in-iraq-and-as-sham/>. See also Bunzel, "Ideology," 26.
139. "Bayan bi-sha'ni 'alaqat jama'at Qa'idat al-Jihad bi-jama'at al-Dawla al-Islamiyya fi al-'Iraq wa-l-Sham," Tanzim Qai'dat al-Jihad—al-Qiyada al-'Amma, January 22, 2014, <http://justpaste.it/ea9k>. Al-Qaeda has always had a complicated relationship with its affiliate in Iraq. In his 2011 private letter, Adam Gadahn claims the Islamic State technically answered to al-Qaeda but "operational relations between the leadership of al-Qaeda and the State have been cut off for quite some time." Gadahn was probably referring to operations inside Iraq because the Islamic State continued to take direction from al-Qaeda for external operations, at least in the early years. In a private letter written on November 19, 2007, Bin Laden asks al-Masri for an update on a plot against Halliburton, and repeated this question on January 25, 2008. Six weeks later, a March 6, 2008, letter asks al-Masri to carry out attacks on the Danes for printing cartoons lampooning the Prophet. See "Letter from unknown al-Qaeda leader to Abu Ayyub al-Masri," November 19, 2007, and January 28, 2008. The letters are part of the batch of letters discussed in chapter 2, note 14.
140. Abu Muhammad al-Adnani, "Ma kana hadha manhajuna wa-lan yakun," April 17, 2014, <http://justpaste.it/makan>.
141. "al-Garshi Gwantanimu" [online pseudonym] (@MKST1111), Twitter post (account since deleted), February 3, 2014, <https://twitter.com/MKST1111/status/430293977561329665>.
142. "Muhib al-Dawla Baqiya" [online pseudonym] (@klklklkl1234), Twitter post (account since deleted), February 9, 2014, <https://twitter.com/klklklkl1234/status/432604752359481344>.
143. See, for example, "Abu Abd al-Muhsin" [online pseudonym] (@majed9432), Twitter post, February 16, 2014, <https://twitter.com/majed9432/status/435196345948078080>; "Munasir al-Islam" [online pseudonym] (@gzrawi9), Twitter post (account since deleted), February 3, 2014, <https://twitter.com/gzrawi9/status/430244970210275328>.
144. Zawahiri, "Shahada."
145. "The brethren in the Shura required an oath from the martyred (as we consider him) Shaykh Abu Umar al-Baghdadi that his emir is Shaykh Usama bin Laden (may God bless him) and that the State follows the Qa'idat al-Jihad group." See Zawahiri, "Shahada," 1–2.
146. Abu Muhammad al-Adnani, "'Udhran amir al-Qa'ida," Wakalat al-Anba' al-Islamiyya, May 11, 2014, <http://www.dawaalhaq.com/?p=12828>.
147. The spokesman, Abu Muhammad al-Adnani, was in prison during the first incarnation of the Islamic State so he might be unfamiliar with the correspondence between its leaders and al-Qaeda. Although Adnani is right that the group ignored al-Qaeda's instructions regarding what the State was to target inside Iraq, internal al-Qaeda memos disprove his claim that al-Qaeda never asked about the disposition of its forces inside the country. "Very quickly write us a report on your conditions and your assessment of the situation in the current stage" an al-Qaeda leader wrote al-Masri on January 25, 2008. "How are things and conditions on the ground especially in Diyala, Mosul, and Baghdad?" the leader asked in the same letter. "We would also like to reiterate our request that you write us complete and detailed reports about your current conditions" instructed a member of al-Qaeda's leadership to the Islamic State's leader on March 10, 2008. Al-Qaeda's leaders framed their instructions as "advice" rather than orders, but that doesn't mean they expected the Islamic State to ignore them. In a letter dated March 6, 2008, Zawahiri "advises" the Islamic State to set up a human resources department "to look for hidden capabilities among the brothers who have joined the State and activate them." Zawahiri further advises the State to establish an independent legal council and Shari'a court to resolve disputes for all Muslims whether they belonged to the State or not. See chapter 2, note 14.
148. Walid Ghanim, "Man huwa 'Abu Khalid al-Suri' wa-kayfa qutila??!" All4Syria, February 24, 2014, <http://www.all4syria.info/Archive/132991>.
149. "Snafi al-Nasr" [online pseudonym] (@Snafialnasr), Twitter post, February 23, 2014, <https://twitter.com/Snafialnasr/status/437590454604136448>. See also Ghanim, "Man huwa 'Abu Khalid al-Suri' wa-kayfa qutila??!"; Hajj, "'Abu Khalid al-Suri.'"
150. See Cole Bunzel, "The Islamic State of Disobedience: al-Baghdadi Triumphant," Jihadica, October 5, 2013, <http://www.jihadica.com/the-islamic-state-of-disobedience-al-baghdadis-defiance/>.
151. Abu Qatada, "Risala min al-Shaykh Abi Qatada ila ikhwanihi al-mujahidin," January 20, 2014, <http://twitmail.com/email/620473351/30/>.
152. Abu Muhammad al-Maqdisi, "Allahumma inni abra' ilayka mimma san'a' ha'ula'," Minbar al-Tawhid wa-l-Jihad, January 25, 2014, <http://www.tawhed.ws/r?i=25011401>. See Cole Bunzel, "The Islamic State of Disunity: Jihadism Divided," Jihadica, January 30, 2014, <http://www.jihadica.com/the-islamic-state-of-disunity-jihadism-divided/>.
153. Bunzel, "Jihadism Divided."
154. Ibid. See also Bunzel, "Ideology," 34.
155. "Shami Witness" [online pseudonym] (@ShamiWitness), Twitter post, May 17, 2014, <https://twitter.com/ShamiWitness/status/467790551777959936>.
156. "Shami Witness" [online pseudonym] (@ShamiWitness), Twitter post, July 12, 2014, <https://twitter.com/ShamiWitness/status/487865881276739584>.
157. Bunzel, "Ideology," 28.
158. Statement by Abu Khalid al-Suri, January 17, 2014, <http://www.hanein.info/vb/image/imgcache/2014/01/1941.jpg>.
159. The following account is based on Lulu's videotaped confession: "al-Ard al-Mubarika" [online pseudonym], "I'tirafat munassiq 'amaliyyat ightiyyal al-Shaykh Abi Khalid al-Suri yatluha kalimat li-l-Shaykh Abi 'Abd Allah," YouTube video, April 24, 2014, <https://www.youtube.com/watch?v=Fvh79FdaBbg>. For a summary, see Ahmad al-Uqda, "Ahrar al-Sham takshif 'an huwiyyat qatil Abu Khalid al-Suri," Siraj Press, April 25, 2014, <http://www.sirajpress.com/%D9%85%D9%82%D8%A7%D9%84/%28%D8%A3%D8%AD%D8%B1%D8%A7%D8%B1-%D8%A7%D9%84%D8%B4%D8%A7%D9%85%29-%D8%AA%D9%83%D8%B4%D9%81-%D8%B9%D9%86-%D9%87%D9%88%D9%8A%D8%A9-%D9%82%D8%A7%D8%AA%D9%84-%D8%A3%D8%A8%D9%88-%D8%AE%D8%A7%D9%84%D8%AF-%D8%A7%D9%84%D8%B3%D9%88%D8%B1%D9%8A/1517/>; see also "'Zaman al-Wasl' tanshur nass i'tirafat 'munassiq ightiyyal Abi Khalid al-Suri'," Zaman al-Wasl, n.d., <https://www.zamanalwsl.net/mobile/readNews.php?id=49018>.
160. "Syria Rebel Leader Abu Khaled al-Suri Killed in Aleppo," BBC News, February 24, 2014, <http://www.bbc.com/news/world-middle-east-26318646>.
161. "Syria: Countrywide Conflict Report #4," Syria Conflict Mapping Project, Carter Center, September 11, 2014, 25, <https://www.cartercenter.org/resources/pdfs/peace/conflict_resolution/syria-conflict/NationwideUpdate-Sept-18-2014.pdf>; Weiss and Hassan, ISIS, 184.
162. See Weiss and Hassan for an overview of the different motives that drew recruits to the Islamic State (ISIS, 146–159).
Chapter 5
1. Yasmine Fathi, "Mubarak's Fall Spawns End of Times Prophecies," Ahram Online, September 25, 2011, <http://english.ahram.org.eg/NewsContentPrint/1/0/22476/Egypt/0/Mubaraks-fall-spawns-End-of-Times-prophecies.aspx>; Charles Cameron, "Arab Spring and Apocalyptic Dawn," Zenpundit, October 2, 2011, <http://zenpundit.com/?p=4358>.
2. "The World's Muslims: Unity and Diversity," Pew Research Center, August 9, 2012, <http://www.pewforum.org/2012/08/09/the-worlds-muslims-unity-and-diversity-executive-summary/>.
3. Musa Cerantonio, "Syria Lecture—Melbourne—Talk 1—Musa Cerantonio," YouTube video, December 24, 2012, https://www.youtube.com/watch?v=yrt4AMMA2vg&feature=youtube_gdata_player.
4. Muhammad Nasir al-Din al-Albani, Takhrij ahadith fada'il al-Sham wa-Dimashq (Riyadh: Maktabat al-Ma'arif, 2000), 10.
5. Ibid., 14.
6. Ibid., 38.
7. Ibid., 64.
8. Abu Bakr al-Baghdadi, "Baqiya fi al-'Iraq wa-l-Sham," audio recording, June 2014, <https://archive.org/details/seham_201307>.
9. Nour Malas, "Ancient Prophecies Motivate Islamic State Militants," Wall Street Journal, November 18, 2014, <http://www.wsj.com/articles/ancient-prophecies-motivate-islamic-state-militants-1416357441>.
10. Louise Cheer, "Facebook Page Calls for Release of Australian Hate Preacher Who Claimed to Be Waging Jihad in the Middle East," Daily Mail, July 12, 2012, <http://www.dailymail.co.uk/news/article-2689850/Facebook-page-calls-release-Australian-hate-preacher.html>. For Cerantonio's apocalyptic worldview, see Graeme Wood, "What ISIS Really Wants," The Atlantic, March 2015, <http://www.theatlantic.com/features/archive/2015/02/what-isis-really-wants/384980/>.
11. Sunan Ibn Majah (<http://sunnah.com/ibnmajah/36/63>).
12. On Zarqawi's nickname, see Loretta Napoleoni, Insurgent Iraq: Al Zarqawi and the New Generation (New York: Seven Stories Press, 2005), 42.
13. See the "Kitab al-fitan" chapter of Ibn Majah's Sunan, <http://sunnah.com/ibnmajah/36>. Abu Mus'ab al-Suri devoted a few pages to the topic of the strangers in his discussion of the End-Time prophecies. See David Cook, "Abu Musa'b [sic] al-Suri and Abu Musa'b [sic] al-Zarqawi: The Apocalyptic Theorist and the Apocalyptic Practitioner," unpublished, 14.
14. Mariam Karouny, "Apocalyptic Prophecies Drive Both Sides to Syrian Battle for End of Time," Reuters, April 1, 2014, <http://www.reuters.com/article/2014/04/01/us-syria-crisis-prophecy-insight-idUSBREA3013420140401>.
15. Robert Mackey, "The Case for ISIS, Made in a British Accent," New York Times, June 20, 2014, http://www.nytimes.com/2014/06/21/world/middleeast/the-case-for-isis-made-in-a-british-accent.html?_&_r=0.
16. For information on the Strangers' media operation, see Cole Bunzel, "A Jihadi Civil War of Words: The Ghuraba' Media Foundation and Minbar al-Tawhid wa'l-Jihad," Jihadica, October 21, 2014, <http://www.jihadica.com/a-jihadi-civil-war/>.
17. Abu Umar al-Hurani, "Ghuraba'—al-Dawla al-Islamiyya fi al-'Iraq wa-l-Sham—Suwwar min ard al-malahim," YouTube video (video has since been deleted), October 29, 2013, <https://www.youtube.com/watch?v=i9JPc4R_oDM>.
18. Ghuraba' bilad al-Sham, personal blog, <http://williamhamdan.blogspot.com/>.
19. "'Da'ish' yudashshan futuhat jadida fi 'Marj Dabiq'," Al Hayat, August 15, 2014, <http://alhayat.com/Articles/4133949/>.
20. Abu Mus'ab al-Zarqawi, "Ayna ahl al-muru'at?" Kalimat mudi'a: al-Kitab al-jami' li-khutab wa-kalimat al-shaykh al-mu'taz bi-dinihi, September 11, 2004, 159, <http://e-prism.org/images/AMZ-Ver1.doc>.
21. Tradition from Sahih Muslim, <http://sunnah.com/muslim/54/44>.
22. Ibid. Zarqawi's public statements were suffused with apocalyptic rhetoric. See Cook, "Abu Musa'b al-Suri and Abu Musa'b al-Zarqawi."
23. Abu Umar al-Baghdadi, "Innama al-mu'minun ikhwatun," January 1, 2009, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
24. E. W. Brooks, "The Campaign of 716–718, from Arabic Sources," Journal of Hellenic Studies 19 (1899): 19–31, 20.
25. Ibn Jarir al-Tabari, The History of al-Tabari, Vol. 24: The Empire in Transition, transl. David Powers (New York: State University of New York Press, 1989), 41.
26. Abu Muhammad al-Adnani, "Wa-la yumakkinanna lahum dinahum alladhi irtada lahum" (April 2014).
27. Dabiq 1 (Ramadan 1435): 3–5.
28. "'Da'ish' yudashshan," Al Hayat.
29. "8 qutila li-tanzim 'al-Dawla' fi ishtibakat ma'a al-Jaysh al-Hurr 'inda qaryat Dabiq bi-Halab," Smart News, September 24, 2014, <https://smartnews-agency.com/news/8-%D9%82%D8%AA%D9%84%D9%89-%D9%84%D8%AA%D9%86%D8%B8%D9%8A%D9%85-%D8%A7%D9%84%D8%AF%D9%88%D9%84%D8%A9-%D9%81%D9%8A-%D8%A7%D8%B4%D8%AA%D8%A8%D8%A7%D9%83%D8%A7%D8%AA-%D9%85%D8%B9-%D8%A7%D9%84%D8%AC%D9%8A%D8%B4-%D8%A7%D9%84%D8%AD%D8%B1-%D8%B9%D9%86%D8%AF-%D9%82%D8%B1%D9%8A%D8%A9-%D8%AF%D8%A7%D8%A8%D9%82-%D8%A8%D8%AD%D9%84%D8%A8>.
30. Malas, "Ancient Prophecies."
31. "There will be a truce between you and [the Romans]. They will betray you and march against you under eighty banners, under each banner 12,000 troops." In another version, "They will gather for the battle. At that time, they will come under eighty banners, under each banner 12,000 troops." See "Kitab al-fitan" in the Sunan of Ibn Majah.
32. "Casey" [online pseudonym], "Video: Message from Dabiq, Wait. We Are Also Waiting," WorldAnalysis.net, October 15, 2014, <http://worldanalysis.net/14/2014/10/video-message-dabiq-wait-also-waiting/>.
33. "@Otaibiah_n511" [online pseudonym], Twitter post (account since deleted), <https://twitter.com/Otaibiah_n511/status/510669519354748928>.
34. "@88adee" [online pseudonym], Twitter post (account since deleted), <https://twitter.com/88adee/status/517764254762414080>.
35. "@999Rhs" [online pseudonym], Twitter post (account since deleted), <https://twitter.com/999Rhs/status/517764791398440960>.
36. Dabiq 5 (Muharram 1436): 33.
37. "al-'Abd al-abiq . . . ila Marj Dabiq," Platform Media, September 11, 2014, <https://www.alplatformmedia.com/vb/showthread.php?p=457470>. My thanks to Cole Bunzel for this source; "Why Islamic State Chose Town of Dabiq for Propaganda," BBC News, November 17, 2014, <http://www.bbc.com/news/world-middle-east-30083303>.
38. Michael D. Danti, "Planning for Safeguarding Heritage Sites in Syria," ASOR Syrian Heritage Initiative, Weekly Report 1 (August 11, 2014), <http://www.asor-syrianheritage.org/wp-content/uploads/2015/03/ASOR_CHI_Weekly_Report_01r.pdf>. Over a thousand years earlier, apocalyptic partisans of the Abbasids had desecrated Sulayman's remains. See Eric Schroeder, Muhammad's People: An Anthology of Muslim Civilization (Mineola, NY: Dover Publications, Inc., 2002), 262.
39. "ISIS and Obama's Summit," The Wall Street Journal, February 16, 2015, <http://www.wsj.com/articles/isis-and-obamas-summit-1424132931>.
40. Abu Umar al-Baghdadi, "Risala ila hukkam al-bayt al-abyad . . . wa-sa'ir ahlafihim min ru'asa' al-duwal al-nasraniyya," November 7, 2008, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
41. The "White Minaret" is the name of Nusra's media outlet, founded before the group split from the Islamic State.
42. Abu Muhammad al-Adnani, "Lan yadurrukum illa adhan," audio recording, February 2013, <https://archive.org/details/forKan.001>.
43. Abu Bakr al-Baghdadi, "Baqiya fi al-'Iraq wa-l-Sham" (June 2014), <https://archive.org/details/seham_201307>.
44. David Cook, Studies in Muslim Apocalyptic (Studies in Late Antiquity and Early Islam, No. 21) (Princeton, NJ: Darwin Press, 2002), 96–97.
45. "@dAwLa_KiLaFa1" [online pseudonym], Twitter post (account since deleted), <https://twitter.com/dAwLa__KiLaFa1/status/551438259356983296>.
46. Guy Taylor, "Apocalypse Prophecies Drive Islamic State Strategy, Recruiting Efforts," Washington Times, January 5, 2015, <http://www.washingtontimes.com/news/2015/jan/5/apocalypse-prophecies-drive-islamic-state-strategy/?page=all>.
47. "Batirashvili Badria" [online pseudonym] (@BadriaArvi), Twitter post (account since deleted), October 23, 2014, <https://twitter.com/BadriaArvi/status/525210983489236992>. Islamic State supporters are not the only group to invoke the Antichrist in the Syrian context: Anti-Assad factions have regularly tweeted about Hizballah commander Hassan Nasrallah under the Arabic hashtag #The_Antichrist_of_the_Resistance," referring to his support for the Assad regime.
48. "Abd al-Malak al-Matiri" [online pseudonym] (@superbinilly), Twitter post, January 11, 2015, <https://twitter.com/katfqanoni203/status/554282910891515904>.
49. Tradition from Sahih Muslim. Abu Qatada al-Filistini, a jihadist critic of the Islamic State, quotes the prophecy favorably in his "Qira'at fi al-Nubu'at: al-Masih al-Dajjal." See Abu Qatada al-Filistini, "Qira'at fi al-Nubu'at: al-Masih al-Dajjal," Minbar al-Tawhid wa-l-Jihad, n.d., 16, <https://www.tawhed.ws/r?i=p4qqi3j8>.
50. "Twelver" Shi'a believe there were twelve legitimate leaders or imams after Muhammad died, beginning with the Prophet's son-in-law Ali and running through his descendants. Twelvers believe the last of these imams went into hiding and will reappear as the Mahdi. See Najam Haider, Shı¯'ı¯ Islam: An Introduction (New York: Cambridge University Press, 2014), 145–166.
51. Karouny, "Apocalyptic Prophecies."
52. Nu'aym bin Hammad, Kitab al-fitan (Beirut: Dar al-Fikr li-l-Taba'a, 1992), 190.
53. "kemowen5" [online pseudonym], "Amin 'amm Hizb Allah al-'Iraqi al-Jaysh al-Hurr huwa jaysh al-Sufyani alladhi yuqatil al-Imam al-Mahdi," YouTube video, May 4, 2013, <https://www.youtube.com/watch?v=sZjIF91kog4>.
54. Cook, Studies in Muslim Apocalyptic, 133.
55. Ibid.
56. Tabari, The History of al-Tabari, Vol. 27: The 'Abbasid Revolution, 177–178. See Asad Ahmed, The Religious Elite of the Early Islamic Hijaz: Five Prosopographical Case Studies (Oxford: Prosopographica et Genealogica, 2011), 119.
57. Barbara Roggema, The Legend of Sergius Bahira: Easter Christian Apologetics and Apocalyptic in Response to Islam (Boston: Brill, 2008), 76.
58. Wilferd Madelung, "Abu 'l-'Amaytar the Sufyani," Jerusalem Studies in Arabic and Islam 24 (2000): 327–343, 329.
59. Moshe Gil, A History of Palestine, 634–1099 (New York: Cambridge University Press, 1992), 293–294.
60. "Fidiyu: al-'Ar'ur yad'u wa-yatamanna bi-anna yakun tahtu rayat jaysh al-Sufyani alladhi sayazhur fi akhir al-zaman wa-yuharib jaysh al-Imam al-Mahdi 'alayhi al-salam," Makadait, January 2, 2014, <http://www.makadait.com/?p=15659>.
61. Jean-Pierre Filiu, Apocalypse in Islam, transl. M.B. DeBevoise (Berkeley: University of California Press, 2011), 199.
62. Nu'aym b. Hammad, Kitab al-Fitan (1993), 162.
63. Ruhollah Hosseinian, "Istishmam bu-yi havadith akhir al-zamani," Fars News Agency, 6 Tir 1362 (June 27, 2013), <http://farsnews.com/newstext.php?nn=13920403000166>.
64. "Subhan Allah" [online pseudonym] (@ahmd878), Twitter post, May 31, 2013, <https://twitter.com/ahmd878/status/340489376469504003>.
65. Wilferd Madelung, "The Sufyani between Tradition and History," Studia Islamica 63 (1986): 4–48, 13.
66. Ibid., 22.
67. Ibid., 20.
68. Robert Windrem, "ISIS by the Numbers: Foreign Fighter Total Keeps Growing," NBC News, March 1, 2015, <http://www.nbcnews.com/storyline/isis-terror/isis-numbers-foreign-fighter-total-keeps-growing-n314731>.
69. Barbara Slavin, "Shiite Militias Mixed Blessing in Iraq, Syria," Al-Monitor, February 9, 2015, <http://www.al-monitor.com/pulse/originals/2015/02/shiite-militias-mixed-blessing-iraq-syria.html>.
70. Zack Beauchamp, "Iraq's Yazidis: Who They Are and Why the US Is Bombing ISIS to Save Them," Vox, August 8, 2014, <http://www.vox.com/2014/8/8/5982421/yazidis-yezidis-iraq-crisis-bombing>; Gerard Russell, "The Peacock Angel and the Pythagorean Theorem," Foreign Policy, August 8, 2014, <http://foreignpolicy.com/2014/08/08/the-peacock-angel-and-the-pythagorean-theorem/>.
71. Dabiq 4 (Dhul-Hijjah 1435): 15.
72. John D. Sutter, "Slavery's Last Stronghold," CNN, <http://www.cnn.com/interactive/2012/03/world/mauritania.slaverys.last.stronghold/>.
73. Ruth Sherlock, "Islamic State Commanders 'Using Yazidi Virgins for Sex,'" Telegraph, October 18, 2014, <http://www.telegraph.co.uk/news/worldnews/islamic-state/11171874/Islamic-State-commanders-using-Yazidi-virgins-for-sex.html>.
74. Tradition from Sahih Muslim.
75. Dabiq 4: 14–17.
76. Dabiq 4: 17.
77. "'Da'ish' yuqim 'suq nukhasa' wa-yuhaddid as'ar bi' al-nisa'," Al-Alam, November 8, 2014, <http://www.alalam.ir/news/1647166>.
78. Hadir Mahmud, "Bi-l-fidiyu . . . jara'im Da'ish fi al-'Iraq wa-Suriya. al-nisa' mathalan," El Badil, November 4, 2014, <http://elbadil.com/2014/11/04/%D8%A8%D8%A7%D9%84%D9%81%D9%8A%D8%AF%D9%8A%D9%88-%D8%AC%D8%B1%D8%A7%D8%A6%D9%85-%D8%AF%D8%A7%D8%B9%D8%B4-%D9%81%D9%8A-%D8%A7%D9%84%D8%B9%D8%B1%D8%A7%D9%82-%D9%88%D8%B3%D9%88%D8%B1%D9%8A%D8%A7-%D8%A7/>.
79. "Manshur jadid li 'Da'ish' . . . as'ila wa-ajwiba hawla sabi wa muwaqa'at 'al-nisa' al-kafirat' jinsiyyan," CNN Arabic, December 13, 2014, <http://arabic.cnn.com/middleeast/2014/12/13/isis-justification-female-slaves>.
80. Russell Myers, "British Female Jihadis Running ISIS 'Brothels' Allowing Killers to Rape Kidnapped Yazidi Women," Mirror, September 10, 2014, <http://www.mirror.co.uk/news/uk-news/british-female-jihadis-running-isis-4198165>; Ruth Sherlock, "Islamic State Commanders 'Using Yazidi Virgins for Sex.'"
81. Anna Steele, "Woman Uses Hidden Camera to Expose Life under Islamic State (+video)," Christian Science Monitor, September 25, 2014, <http://www.csmonitor.com/World/Middle-East/2014/0925/Woman-uses-hidden-camera-to-expose-life-under-Islamic-State-video>.
82. Damien Gayle, "All-Female Islamic State Police Squad Tortured New Mother with Spiked Clamp Device Called a 'Biter' after She Was Caught Breastfeeding in Public," Daily Mail, December 31, 2014, <http://www.dailymail.co.uk/news/article-2890911/All-female-Islamic-State-police-squad-tortured-woman-device-called-biter-caught-breastfeeding-public.html>.
83. al-Hasan al-Yusi, Zahr al-akam fi al-amthal wa-l-hikam (Dar al-Thaqafa, 1981), 833 (the passage is available online here: <http://www.madinahnet.com/books40/%D8%B2%D9%87%D8%B1-%D8%A7%D9%84%D8%A3%D9%83%D9%85-%D9%81%D9%8A-%D8%A7%D9%84%D8%A3%D9%85%D8%AB%D8%A7%D9%84-%D9%88%D8%A7%D9%84%D8%AD%D9%83%D9%85-%D8%B5%D9%81%D8%AD%D8%A9-833>).
84. Khaled Abd al-Muhsin, "'Nisa' fi firash Da'ish' riwayat haqiqiyya li-ahdath damiya. Katib yakshif asrar 'jawari al-khalifa al-Baghdadi'," Al Ahram, December 14, 2014, <http://gate.ahram.org.eg/News/572214.aspx>.
85. Shadiya Sarhan, "Akhtar nisa' Da'ish tanshur 'dalil aramil al-jihadiyyin' 'ala Twitir,": 24, January 26, 2015, <http://24.ae/article/133546/.aspx>.
86. Ali Rajab, "Bi-l-suwwar . . . abraz 'danjawanat Da'ish'," Veto, December 10, 2014, <http://www.vetogate.com/1370965>.
87. Hani al-Zahiri, "'Niswan Da'ish' al-shaqrawat," Al Hayat, December 23, 2014,
88. Carolyn Hoyle, Alexandra Bradford, and Ross Frenett, "Becoming Mulan? Female Western Migrants to ISIS," Institute for Strategic Dialogue, January 2015, 10–15, <http://www.strategicdialogue.org/ISDJ2969_Becoming_Mulan_01.15_WEB.PDF>. The tweeter, Umm Ubaydah, was living in the Islamic State when she made her comments online. See Joshi Herrmann, "Internet Brides of Jihad: How Islamic State Is Using Social Media to Lure Young British Women to Syria," London Evening Standard, December 2, 2014, <http://www.standard.co.uk/lifestyle/london-life/internet-brides-of-jihad-how-islamic-state-is-using-social-media-to-lure-young-british-women-to-syria-9846143.html>.
89. Herrmann, "Internet Brides."
90. "UN Terror Expert: One Austrian 'Jihad Poster Girl' Is Dead after Moving to Syria to Join ISIS," NY Daily News, December 18, 2014, <http://www.nydailynews.com/news/world/expert-austrian-teen-girl-dead-joining-isis-article-1.2049826>.
91. Abu Muhammad al-Adnani, "Ma kana hadha manhajuna wa-lan yakun," audio message, Mu'assasat al-Furqan, April 17, 2014, audio message and English translation available at Pieter Van Ostaeyen, "Message by ISIS Shaykh Abu Muhammad al-'Adnani as-Shami," pietervanostaeyen, April 18, 2014, <https://pietervanostaeyen.wordpress.com/2014/04/18/message-by-isis-shaykh-abu-muhammad-al-adnani-as-shami/>.
92. The Arabic word khalifa, Anglicized as "caliph," means successor.
93. See, for example, Muhammad Nasir al-Din al-Albani, Silsilat al-ahadith al-sahiha, vol. 1 (Riyadh: Maktabat al-Ma'arif li-l-Nashar wa-l-Tawzi'), 34–36; Abd Allah bin Wakil, "al-Malik al-'Adud wa-l-Malik al-Jabari," Ahl al-Hadith, June 14, 2003, <http://www.ahlalhdeeth.com/vb/showthread.php?t=155508>.
94. Ayman al-Zawahiri, "Tawjihat 'amma li-l-'aml al-jihadi," Minbar al-Tawhid wa-l-Jihad, September 16, 2013, <https://www.tawhed.ws/r?i=16091301>. Bin Laden's secretary in Africa, Fadil Harun, alludes to the prophecy several times in his autobiography, connecting the event with the End Times and the appearance of the Muslim savior, the Mahdi: "The caliph, who is not yet present, will definitely appear when the prerequisite conditions are fulfilled. He will pave the way for the appearance of the Mahdi (peace be upon him) and the descent of Jesus (peace be upon him). We will unite under him for the benefit of the entire world because Islam is ordained for the whole of mankind and not for Muslims alone" (Harun, al-Harb 'ala al-Islam: Qissat Fadil Harun, 2 vols., February 26, 2009, 2:723).
95. "al-Bin'ali . . . Mufti Da'ish 'al-Damawi'," Bawabat al-Harakat al-Islamiyya, January 27, 2015, <http://www.islamist-movements.com/25818>.
96. Turki ibn Mubarak al-Bin'ali, "al-Qiyafa fi 'adm ishtirat al-tamkin al-kamil li-l-khilafa," April 30, 2014, 1, available at Jihadica, www.jihadica.com/wp-content/uploads/2014/07/%D8%A7%D9%84%D9%82%D9%8A%D8%A7%D9%81%D8%A9-%D9%81%D9%8A-%D8%B9%D8%AF%D9%85-%D8%A7%D8%B4%D8%AA%D8%B1%D8%A7%D8%B7-%D8%A7%D9%84%D8%AA%D9%85%D9%83%D9%8A%D9%86-%D8%A7%D9%84%D9%83%D8%A7%D9%85%D9%84-%D9%84%D9%84%D8%AE%D9%84%D8%A7%D9%81%D8%A9.doc. See Cole Bunzel, "The Caliphate's Scholar-in-Arms," Jihadica, July 9, 2014, <http://www.jihadica.com/the-caliphate%E2%80%99s-scholar-in-arms/>.
97. Bin'ali, "al-Qiyafa," 2.
98. Ibid., 3.
99. Ibid., 3.
100. Ibid., 4.
101. Ibid., 6.
102. Ibid., 7–8.
103. Ibid., 10.
104. Ibid., 14–15.
105. Ibid., 12–14.
106. Bin'ali, "Madd al-ayadi li-bay'at al-Baghdadi," July 21, 2013, <https://archive.org/details/baghdadi-001>. See Joas Wagemakers, "Al-Qaida Advises the Arab Spring: The Case for al-Baghdadi," Jihadica, September 21, 2013, <http://www.jihadica.com/al-qaida-advises-the-arab-spring-the-case-for-al-baghdadi/>.
107. Bin'ali, "Madd al-ayadi," 3. Juhayman al-Utaybi, the man who stormed the Meccan mosque after declaring his companion the Mahdi, believed the Mahdi would be a descendent of Muhammad's grandsons. In his treatise on the End of Days, Utaybi quotes a prophecy by Muhammad: "The Mahdi is from my descendants from the son of Fatima." See Juhayman al-Utaybi, "al-Fitan wa-akhbar al-Mahdi wa-nuzul Isa ('alayhi al-salam) wa-ashrat al-sa'a," Minbar al-Tawhid wa-l-Jihad, 18.
108. Twelver Shi'a call Ja'far "the Liar" because he denied his brother had a son, which would have made Ja'far the twelfth imam instead of the son. See Moojan Momen, An Introduction to Shi'i Islam: The History and Doctrines of Twelver Shi'ism (New Haven, CT: Yale University Press, 1987), 59–60, 161.
109. Bin'ali, "Madd al-ayadi," 4–5.
110. Ibid., 6–7.
111. Ibid., 11–12.
112. Ibid., 12–13.
113. Ibid., 13–15.
114. Ibid., 15–16.
115. Ibid., 6.
116. Abu Muhammad al-Maqdisi, "Hadha ba'da ma 'indi wa-laysa kulluhu," Minbar al-Tawhid wa-l-Jihad (June–July 2014), <http://tawhed.ws/r?i=01071401>.
Chapter 6
1. "The Capture of Mosul: Terror's New Headquarters," Economist, June 14, 2014, <http://www.economist.com/news/leaders/21604160-iraqs-second-city-has-fallen-group-wants-create-state-which-wage-jihad>.
2. Zana Khasraw Gul, "Who Is in Control of Mosul?" openDemocracy, July 7, 2014, <https://www.opendemocracy.net/zana-khasraw-gul/who-is-in-control-of-mosul>.
3. Associated Press, "Fear, Sectarianism behind Iraq Army Collapse," Haaretz, June 13, 2014, <http://www.haaretz.com/news/middle-east/1.598575>; Nico Prucha, "Is This the Most Successful Release of a Jihadist Video Ever?" Jihadica, May 19, 2014, <http://www.jihadica.com/is-this-the-most-successful-release-of-a-jihadist-video-ever/>.
4. Alexander Mikaberidze, Conflict and Conquest in the Islamic World: A Historical Encyclopedia, vol. 1 (Santa Barbara, CA: ABC-CLIO, 2011), 663–663.
5. Aaron Y. Zelin, "ISIS is Dead, Long Live the Islamic State," Foreign Policy, June 30, 2014, <http://foreignpolicy.com/2014/06/30/isis-is-dead-long-live-the-islamic-state/>.
6. Abu Muhammad al-Adnani, "Hadha wa'd Allah," Al-Battar Media Foundation, June 29, 2014, available at Jihadica, www.jihadica.com/wp-content/uploads/2014/07/%D9%87%D8%B0%D8%A7-%D9%88%D8%B9%D8%AF-%D8%A7%D9%84%D9%84%D9%87.pdf. Full video of the speech is available at "Ajil wa-hamm" [online pseudonym], "Kalimat li-l-shaykh al-mujahid Abi Muhammad al-'Adnani—hadha wa'd Allah," Youtube video, June 29, 2014, <https://www.youtube.com/watch?v=d4OZcLlMAOs>.
7. Fadil Harun, al-Harb 'ala al-Islam: Qissat Fadil Harun, 2 vols., February 26, 2009, 1:686; available at <https://www.ctc.usma.edu/posts/the-war-against-islam-the-story-of-fazul-harun-part-1-original-language-2>. See Kévin Jackson, "The Forgotten Caliphate," Jihadica, December 31, 2014, <http://www.jihadica.com/the-forgotten-caliphate/>.
8. Anwar al-Awlaki, "Allah Is Preparing Us for Victory," transcr. Amatullah, ed. Mujahid fe Sabeelillah, 25–26, <https://ia600609.us.archive.org/31/items/AllahIsPreparingUsForVictory-AnwarAlAwlaki/AllahIsPreparingUsForVictory.pdf>.
9. Ayman al-Zawahiri, "Tawjihat 'amma li-l-'aml al-jihadi," Minbar al-Tawhid wa-l-Jihad, September 16, 2013, <https://www.tawhed.ws/r?i=16091301>.
10. Stern and Berger argue that the shift in discourse from weakness to strength was a major change in jihadist rhetoric; see Jessica Stern and J. M. Berger, ISIS: The State of Terror (New York: HarperCollins, 2015), 108–109, 117.
11. "Li-madha zahara 'al-khalifa' al-Da'ishi bi-l-'imama al-suda'?" Al Arabiya, July 6, 2014, <http://www.alarabiya.net/ar/arab-and-world/iraq/2014/07/06/-%D8%AF%D8%A7%D8%B9%D8%B4-%D8%A7%D9%84%D8%A8%D8%BA%D8%AF%D8%A7%D8%AF%D9%8A-%D9%84%D8%A8%D8%B3-%D8%B9%D9%85%D8%A7%D9%85%D8%A9-%D8%B3%D9%88%D8%AF%D8%A7%D8%A1-%D8%A7%D9%82%D8%AA%D9%81%D8%A7%D8%A1%D9%8B-%D8%A8%D8%A7%D9%84%D9%86%D8%A8%D9%8A-%D9%85%D8%AD%D9%85%D8%AF.html>.
12. "Faris al-Dawla" [online pseudonym] (@M_N_B_A), Twitter post (account since deleted), July 5, 2014, <https://twitter.com/M_N_B_A/status/485437010169982976>; "Li-madha zahara 'al-khalifa' al-Da'ishi bi-l-'imama al-suda'?"
13. Abu Bakr al-Baghdadi, Mosul sermon, July 1, 2014, Arabic transcript available at <http://justpaste.it/gtdd>.
14. Ibid.
15. "Was It a Rolex? Caliph's Watch Sparks Guesses," Al Arabiya, July 6, 2014, <http://english.alarabiya.net/en/variety/2014/07/06/Was-it-a-Rolex-Caliph-s-watch-sparks-guesses.html>.
16. "Li-madha zahara 'al-khalifa' al-Da'ishi bi-l-'imama al-suda'?"
17. "al-Nafir ila ard al-Sham" [online pseudonym] (@daaghu345h), Twitter post, July 3, 2014, <https://twitter.com/daaghu345h/status/484719022441648129>.
18. "Muzmajir al-Sham" [online pseudonym] (@saleelalmajd1), Twitter post, July 3, 2014, <https://twitter.com/saleelalmajd1/status/484745347697479680>.
19. "Abu Hadhifa al-Maqdisi" [online pseudonym] (@glaopalo2), Twitter post (account since deleted), July 3, 2014, <https://twitter.com/glaopalo2/status/484746768199192577>.
20. "@salman15n" [online pseudonym], Twitter post (account since deleted), <https://twitter.com/salman15n/status/529306591896809472>.
21. "@khansaa000" [online pseudonym], Twitter post (account since deleted), <https://twitter.com/khansaa000/status/526113636637757440>; "Abu Yusuf al-Zahiby" [online pseudonym] (@prince_zahab), Twitter post, February 8, 2015, <https://twitter.com/prince_zahab/status/564576740836933632>.
22. "Harad al-Mu'minin" [online pseudonym] (@RBG011), Twitter post (account since deleted), November 13, 2014, <https://twitter.com/RBG011/status/533238697848610816>.
23. "Suwwar al-Dawla al-Islamiyya" [online pseudonym] (@isis_pic), Twitter post (account since deleted), January 17, 2015, <https://twitter.com/isis_pic/status/556557099094663169/photo/1>.
24. "Zakariyya al-Salami #Khilafa" [online pseudonym] (@zakjdidi), Twitter post (account since deleted), November 26, 2014, <https://twitter.com/zakjdidi/status/537517046675034112>.
25. Dabiq 6: 59.
26. Dabiq 5: 18.
27. Robert Hoyland, "Writing the Biography of the Prophet Muhammad," History Compass 5 (2007): 581–602.
28. "S'adat al-Wazira" [online pseudonym] (@s3adt_alwazera), Twitter post, March 11, 2015, <https://twitter.com/s3adt_alwazera/status/575637105520480256>; "Belal" [online pseudonym] (@Faglal), Twitter post, March 7, 2015, <https://twitter.com/Faglal/status/574248240834543616>.
29. "#Asifa_al-Hazm" [online pseudonym] (@alhawazni), Twitter post, March 27, 2015, <https://twitter.com/alhawazni/status/581779621370195968>.
30. "Tania" [online pseudonym] (@TANIA_IRAQ), Twitter post, March 21, 2015, <https://twitter.com/TANIA_IRAQ/status/579326527294160896>.
31. Nearly a decade ago, I directed a team of Arabists who counted the citations in jihadist scholarly texts. Maqdisi was cited most, followed by Abu Qatada. See William McCants, "Militant Ideology Atlas: Executive Report," Combating Terrorism Center at West Point, November 1, 2006, <https://www.ctc.usma.edu/posts/militant-ideology-atlas>. In a strange twist, Maqdisi once cited the study and my blog as proof he was the jihadist big dog (see Robert Worth, "Credentials Challenged, Radical Quotes West Point," New York Times, April 29, 2009, <http://www.nytimes.com/2009/04/30/world/middleeast/30jihad.html?_r=0>).
32. Muhammad al-Da'ma, "Abu Qatada: Jama'at al-Baghdadi laysu ikhwanuna," Asharq Al-Awsat, December 7, 2014, <http://aawsat.com/home/article/238521/%D8%A3%D8%A8%D9%88-%D9%82%D8%AA%D8%A7%D8%AF%D8%A9-%D8%AC%D9%85%D8%A7%D8%B9%D8%A9-%D8%A7%D9%84%D8%A8%D8%BA%D8%AF%D8%A7%D8%AF%D9%8A-%D9%84%D9%8A%D8%B3%D9%88%D8%A7-%D8%A5%D8%AE%D9%88%D8%A7%D9%86%D9%86%D8%A7>.
33. Abu Muhammad al-Maqdisi, "Wa-la takunu ka-allati naqadat ghazlaha min ba'da quwwatin ankanthan," Minbar al-Tawhid wa-l-Jihad, July 12, 2014, <https://www.tawhed.ws/r?i=12071401>, English translation available at "Sheikh Abu Muhammad al-Maqdisi on Proclamation of Caliphate," Kavkaz Center, July 20, 2014, <http://www.kavkazcenter.com/eng/content/2014/07/20/19483.shtml>.
34. "Ittisa' mubaya'at Da'ish fi al-mashraq wa-l-maghrab yuthir al-jadal wa-yada' nihayat li 'al-Qa'ida'," Alkarama Press, October 8, 2014, http://www.karamapress.com/arabic/?Action=ShowNews&ID=94643.
35. Muhammad bin Salih al-Muhajir, "al-Khulasa fi munaqashat i'lan al-khilafa," Minbar al-Tawhid wa-l-Jihad, August 2014, <http://www.jihadica.com/wp-content/uploads/2014/10/al-Khulasa-fi-munaqashat-ilan-al-khilafa.doc>. See Cole Bunzel, "A Jihadi Civil War of Words: The Ghuraba' Media Foundation and Minbar al-Tawhid wa'l-Jihad," Jihadica, October 21, 2014, <http://www.jihadica.com/a-jihadi-civil-war/>.
36. Muhajir, "Khulasa," 6–9.
37. Abu Dujana al-Basha, "Hadhihi risalatuna," audio message, Mu'assasat al-Sahab, posted by "Murasil" [online pseudonym] to Shabakat al-Jihad, September 26, 2014, <http://www.shabakataljahad.com/vb/showthread.php?t=40231>. See Kévin Jackson, "Al-Qaeda Revives Its Beef with the Islamic State," Jihadica, October 15, 2014, <http://www.jihadica.com/al-qaeda-revives-its-beef-with-the-islamic-state/>.
38. Cole Bunzel, "Al-Qaeda's Quasi-Caliph: The Recasting of Mullah 'Umar," Jihadica, July 23, 2014, <http://www.jihadica.com/al-qaeda%E2%80%99s-quasi-caliph-the-recasting-of-mullah-%E2%80%98umar/>. See also Cole Bunzel, "From Paper State to Caliphate: The Ideology of the Islamic State," The Brookings Project on U.S. Relations with the Islamic World, Analysis Paper No. 19 (March 2015): 33–34, <http://www.brookings.edu/~/media/research/files/papers/2015/03/ideology-of-islamic-state-bunzel/the-ideology-of-the-islamic-state.pdf>. Similarly, Al-Qaeda's representative in Somalia, Fadil Harun, believed he had pledged allegiance to Mullah Omar as the commander of the faithful but his authority was confined to Afghanistan. See Harun, Harb, 2:90.
39. See Abu Khalid's and Abu Mus'ab al-Suri's letter scolding Bin Laden for defying Mullah Omar's command in Alan Cullison, "Inside Al-Qaeda's Hard Drive," Atlantic, September 1, 2004, <http://www.theatlantic.com/magazine/archive/2004/09/inside-al-qaeda-s-hard-drive/303428/>.
40. Bunzel, "Al-Qaeda's Quasi-Caliph."
41. William McCants, "Zawahiri's Counter-Caliphate," War on the Rocks, September 5, 2014, <http://warontherocks.com/2014/09/zawahiris-counter-caliphate/>.
42. "Tunisi min tanzim Da'ish watawa"ad al-Tunisiyyin bi-fath Tunis dhabh Rashid al-Ghannushi," YouTube video, September 24, 2014, <https://www.youtube.com/watch?v=dLgW_CdX2b0>. YouTube terminated the account for violating its terms of service.
43. Conflict Studies, "ISIS Propaganda Video—Destroying Passports," YouTube video, September 14, 2014, <https://www.youtube.com/watch?v=8hKo9Y2XHkM>. The video has now been made private.
44. "al-Tayf Niyuz" [online pseudonym], "Yamani: tafaja'a bi-ibnihi al-mukhtafi fi sharit li-'Da'ish'," YouTube video, June 21, 2014, <https://www.youtube.com/watch?v=OlFy4USKL1Y>.
45. <https://www.youtube.com/watch?v=dxQ7xElti0k>. YouTube terminated the account for violating its terms of service.
46. Richard Barrett, "Foreign Fighters in Syria," The Soufan Group, June 2014, <http://soufangroup.com/wp-content/uploads/2014/06/TSG-Foreign-Fighters-in-Syria.pdf>.
47. Jamie Dettmer, "ISIS and Al Qaeda Ready to Gang Up on Obama's Rebels," Daily Beast, November 11, 2014, <http://www.thedailybeast.com/articles/2014/11/11/al-qaeda-s-killer-new-alliance-with-isis.html>.
48. Barrett, "Foreign Fighters," 24. Barrett quotes an Ahrar al-Sham interviewee claiming that "60% to 70%" of Nusra's foreign fighters and "30% to 40%" of Ahrar's foreign fighters joined ISIS in 2013. There were also practical reasons why Nusra foreign fighters joined the Islamic State. Nusra and other al-Qaeda recruitment materials are often in Arabic, in contrast to the State's multilingual media machine. Nusra is more selective in who it allows in the group, preferring foreigners who bring military skills and experience to the table, while the Islamic State usually accepts anyone who will hop aboard. See Jaysh Jabhat Jabhat [sic] al-Nusra al-ilaktruni, "al-Liqa' al-sawti al-awwal ma'a al-shaykh al-fatih Abi Muhammad #al-Jawlani," YouTube video, November 4, 2014, <https://www.youtube.com/watch?v=hFpIMi_hmRE>.
49. Edward N. Luttwak, "Caliphate Redivivus? Why a Careful Look at the 7th Century Can Predict How the New Caliphate Will End," Strategika, August 1, 2014, <http://www.hoover.org/research/caliphate-redivivus-why-careful-look-7th-century-can-predict-how-new-caliphate-will-end>.
50. The phrase "al-Rashid" is not a common epithet for Baghdad in medieval books of Islamic history. When the phrase is used, it usually refers to events in the city of al-Rashid's own day. See, for example, al-Yafi'i's Mirat al-jinan wa-'ibrat al-yaqzan, vol. 2 (Beirut: Dar al-Kutub al-'Ilmiyya, 1997), 92. In modern discourse, the phrase "the Baghdad of al-Rashid" is sometimes contrasted with "the Damascus of al-Walid," one of the great Umayyad caliphs who ruled from the city. The Islamist Hakim al-Mutayri, for example, tweeted: "The battle today into which the Baghdad of al-Rashid and the Damascus of al-Walid have plunged represents the entire Muslim community." Hakim al-Mutayri (@DrHAKEM), Twitter post, June 16, 2014, <https://twitter.com/DrHAKEM/status/478457595229769728>. See also William McCants, "Why ISIS Really Wants to Conquer Baghdad," Markaz, Brookings Institution, November 12, 2014, <http://www.brookings.edu/blogs/markaz/posts/2014/11/12-baghdad-of-al-rashid-mccants>.
51. Muharib al-Jaburi, "al-I'lan 'an qiyam Dawlat al-'Iraq al-Islamiyya," January 15, 2007, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
52. Abu Umar al-Baghdadi, "Wa-yamkurun wa-yamkur Allah," September 15, 2007, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
53. Abu Muhammad al-Adnani, "al-Iqtihamat afja'," November 2012; al-Adnani, "Lan yadurrukum illa adha," February 2013; al-Adnani, "al-Ra'id la yakdhib ahlaha," January 2014; al-Adnani, "Ma asabaka min husna fa-min Allah," June 11, 2014.
54. Hugh Kennedy, When Baghdad Ruled the Muslim World: The Rise and Fall of Islam's Greatest Dynasty (Boston: Da Capo Press, 2004), 123, quoting Julia Bray's translation in Julia Ashtiany, Abbasid Belles Lettres: Cambridge History of Arabic Literature, vol. 2 (Cambridge, UK: Cambridge University Press, 1990), 294–5.
55. Ibid., 72.
56. Ibid., 126–127.
57. Hugh Kennedy, When Baghdad Ruled the Muslim World: The Rise and Fall of Islam's Greatest Dynasty (Boston: Da Capo Press, 2004), 74.
58. On the translation of Greek texts into Arabic during Harun's reign, see L. E. Goodman, "The Translation of Greek Materials into Arabic," in M. J. L. Young et al., Religion, Learning and Science in the 'Abbasid Period (Cambridge, UK: Cambridge University Press, 1990), 482–484. For the translation movement in general, see Dimitri Gutas, Greek Thought, Arabic Culture (New York: Routledge, 1998).
59. Edward Granville Browne, A Literary History of Persia, vol. 1 (New York: Charles Scribner's Sons, 1902), 307.
60. Gene Heck, Charlemagne, Muhammad, and the Arab Roots of Capitalism (Berlin: Walter de Gruyter, 2006), 269–271.
61. Joseph F. O'Callaghan, A History of Medieval Spain (Ithaca, NY: Cornell University Press, 1975), 106; Heck, Charlemagne, 172.
62. Colin Freeman, "Iraq crisis: Baghdad's Shia militia in Defiant 50,000-Strong Rally as Isis Make [sic] Further Gains," Telegraph, June 21, 2014, <http://www.telegraph.co.uk/news/worldnews/middleeast/iraq/10916926/Iraq-crisis-Baghdads-Shia-militia-in-defiant-50000-strong-rally-as-Isis-make-further-gains.html>.
63. Janine di Giovanni, "The Militias of Baghdad," Newsweek, November 26, 2014, <http://www.newsweek.com/2014/12/05/militias-baghdad-287142.html>.
64. Phillip Smyth, "The Shiite Jihad in Syria and Its Regional Effects," Washington Institute for Near East Policy, Policy Focus 138 (2015): 43, <http://www.washingtoninstitute.org/uploads/Documents/pubs/PolicyFocus138_Smyth-2.pdf>.
65. Abu Ayyub al-Masri, "al-Dawla al-Nabawiyya," September 19, 2008, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
66. Ibid., 165.
67. Ibid., 166.
68. Ibid., 168.
69. Ibid., 171.
70. Ibid., 175.
71. Ibid., 180.
72. Ibid., 165.
73. Michael Weiss and Hassan Hassan, ISIS: Inside the Army of Terror [Google edition] (New York: Regan Arts, 2015), 215–216; Sarah Birke, "How ISIS Rules," New York Review of Books, February 5, 2015, <http://www.nybooks.com/articles/archives/2015/feb/05/how-isis-rules/>. For a good summary of the Islamic State's ordinances, see Andrew F. Marsh and Mara Revkin, "Caliphate of Law," Foreign Affairs, April 15, 2015, <https://www.foreignaffairs.com/articles/syria/2015-04-15/caliphate-law>.
74. Maggie Fick, "Special Report: For Islamic State, Wheat Season Sows Seeds of Discontent," Reuters, January 20, 2015, <http://www.reuters.com/article/2015/01/20/us-mideast-crisis-planting-specialreport-idUSKBN0KT0W420150120>.
75. For the Islamic State's tribal politics in Syria, see Charles Lister, "Profiling the Islamic State," Brookings Doha Center, Analysis Paper No. 13 (November 2014): 10–11, <http://www.brookings.edu/~/media/Research/Files/Reports/2014/11/profiling%20islamic%20state%20lister/en_web_lister.pdf>. 24.
76. Hassan Hassan, "Isis Exploits Tribal Fault Lines to Control Its Territory," Guardian, October 25, 2014, <http://www.theguardian.com/world/2014/oct/26/isis-exploits-tribal-fault-lines-to-control-its-territory-jihadi>; Weiss and Hassan, ISIS, 195.
77. "Islamic State Abducts Sons of Tribal Leader in Eastern Syria," ARA News, January 2, 2015, <http://aranews.net/2015/01/islamic-state-abducts-sons-tribal-leader-eastern-syria/>.
78. Reuters, "Syria: Mass Grave May Hold Bodies of 230 Members of Anti-ISIS Tribe," New York Times, December 17, 2014, <http://www.nytimes.com/2014/12/18/world/middleeast/syria-mass-grave-may-hold-bodies-of-230-members-of-anti-isis-tribe.html?_r=0>; "Iraqi Tribal Leader Calls for Help After ISIS Massacre," Al Arabiya, October 30, 2014, <http://english.alarabiya.net/en/News/middle-east/2014/10/30/ISIS-kills-220-from-opposing-Iraqi-tribe.html>.
79. Martin Chulov, "Lack of Political Process in Iraq 'Risks Further Gains for ISIS,'" Guardian, January 18, 2015, <http://www.theguardian.com/world/2015/jan/18/lack-politial-process-iraq-renders-us-coalition-bombs-grave-mistake>.
80. The Gulf Institute (@GulfInstitute), Twitter post, October 8, 2013, <https://twitter.com/GulfInstitute/status/387596457596837890>; David D. Kirkpatrick, "ISIS' Harsh Brand of Islam Is Rooted in Austere Saudi Creed," New York Times, September 24, 2014, <http://www.nytimes.com/2014/09/25/world/middleeast/isis-abu-bakr-baghdadi-caliph-wahhabi.html?_r=1>.
81. "'Shari' al-Baghdadi' . . . Qat' ru'us wa-ayadi wa-rajm wa-sulb (suwwar wa-fidiu)," al-Masry al-Youm, December 21, 2014, <http://www.almasryalyoum.com/news/details/607192>; Mary Atkinson and Rori Donaghy, "Crime and Punishment: Islamic State vs Saudi Arabia," Middle East Eye, February 13, 2015, <http://www.middleeasteye.net/news/crime-and-punishment-islamic-state-vs-saudi-arabia-1588245666>.
82. Rule of Terror: Living under ISIS in Syria, Report of the Independent International Commission of Inquiry on the Syrian Arab Republic, United Nations Office of the High Commissioner for Human Rights, November 14, 2014, 6, <http://www.ohchr.org/Documents/HRBodies/HRCouncil/CoISyria/HRC_CRP_ISIS_14Nov2014.pdf>; "'Shari' al-Baghdadi.'"
83. For the Islamic State's use of ultraviolence to attract recruits, see Stern and Berger, ISIS: The State of Terror, 72–73.
84. See, for example, the Jami' of Tirmidhi: "Do not punish with God's punishment," <http://sunnah.com/tirmidhi/17/42>.
85. "Ma hukm tahriq al-kafir bi-l-nar hatta yamut?" Fatwa 60, January 20, 2015, available at Jihadica, <http://www.jihadica.com/wp-content/uploads/2015/02/IS-fatwas-35-38-40-53-55-57-59-62-65-71.pdf>. English translation available at Cole Bunzel, "32 Islamic State Fatwas," Jihadica, March 2, 2015, <http://www.jihadica.com/32-islamic-state-fatwas/>.
86. James Grehan, "Smoking and 'Early Modern' Sociability: The Great Tobacco Debate in the Ottoman Middle East (Seventeenth to Eighteenth Centuries)," American Historical Review 111, no. 5 (December 2006): 1352, 1356–1358.
87. Ibid., 1360, 1373–1374.
88. Ibid., 1361–1362.
89. Ibid., 1363–1364.
90. Ibid., 1367.
91. "Somalia's Al-Shabaab Bans Smoking Cigarettes, Chewing Khat," Bloomberg, May 9, 2011, <http://www.bloomberg.com/news/articles/2011-05-09/somalia-s-al-shabaab-bans-smoking-cigarettes-chewing-khat>.
92. Loveday Morris, "In Syrian Civil War, Emergence of Islamic State of Iraq and Syria Boosts Rival Jabhat al-Nusra," Washington Post, October 28, 2013, <http://www.washingtonpost.com/world/middle_east/in-syrian-civil-war-emergence-of-islamic-state-of-iraq-and-syria-boosts-rival-jabhat-al-nusra/2013/10/25/12250760-3b4b-11e3-b0e7-716179a2c2c7_story.html>; Balint Szlanko, "Jabhat Al Nusra's New Syria," The National, December 15, 2012, <http://www.thenational.ae/news/world/middle-east/jabhat-al-nusras-new-syria>.
93. Morgan Winsor, "ISIS Beheads Cigarette Smokers: Islamic State Deems Smoking 'Slow Suicide' Under Sharia Law," IB Times, February 12, 2015, <http://www.ibtimes.com/isis-beheads-cigarette-smokers-islamic-state-deems-smoking-slow-suicide-under-sharia-1815192>; "'Da'ish' tastathmir fi al-tadkhin bi-tahrimihi," Al-Alam, March 29, 2015, <http://www.alalam.ir/news/1689723>; "Shabaka Kurdiyya: Tatbiq 'Da'ish' li-'uqubat al-tadkhin tawaqqa'a 12 qatilan min 'anasir al-tanzim," Youm7, February 20, 2015, <http://www.youm7.com/story/2015/2/20/%D8%B4%D8%A8%D9%83%D8%A9-%D9%83%D8%B1%D8%AF%D9%8A%D8%A9--%D8%AA%D8%B7%D8%A8%D9%8A%D9%82%D8%AF%D8%A7%D8%B9%D8%B4-%D9%84%D8%B9%D9%82%D9%88%D8%A8%D8%A9-%D8%A7%D9%84%D8%AA%D8%AF%D8%AE%D9%8A%D9%86-%D8%AA%D9%88%D9%82%D8%B9-12-%D9%82%D8%AA%D9%8A%D9%84%D9%8B%D8%A7-%D9%85%D9%86-%D8%B9%D9%86%D8%A7%D8%B5%D8%B1/2075304#.VcT3fEXlZNO>.
94. "ISIS Reverts Smoking Ban in Kirkuk 'to Gain Popularity,'" Al Arabiya, September 22, 2014, <http://english.alarabiya.net/en/variety/2014/09/22/ISIS-reverts-smoking-ban-in-Kirkuk.html>.
95. Abu Umar al-Baghdadi, "Hasad al-sinin bi-Dawlat al-Muwahhidin," April 17, 2007, al-Majmu' li-qadat Dawlat al-'Iraq al-Islamiyya (Nukhbat al-I'lam al-Jihadi, 2010).
96. Dabiq 5: 3, 12–13, 24.
97. Abu Bakr al-Baghdadi, "Even If the Disbelievers Despise Such," English translation available at <http://ia902205.us.archive.org/17/items/bghd_20141113/english.pdf>; original Arabic version available at <http://ia902205.us.archive.org/17/items/bghd_20141113/arabic.pdf>.
98. Dabiq 5: 25–30.
99. Katie Zavadski, "ISIS Now Has a Network of Military Affiliates in 11 Countries Around the World," New York Magazine, November 23, 2014, <http://nymag.com/daily/intelligencer/2014/11/isis-now-has-military-allies-in-11-countries.html>. AQIM had rejected the Islamic State's caliphate soon after it was announced. See Thomas Joscelyn, "AQIM Rejects Islamic State's Caliphate, Reaffirms Allegiance to Zawahiri," Long War Journal, July 14, 2014, <http://www.longwarjournal.org/archives/2014/07/aqim_rejects_islamic.php>.
100. Paul Cruickshank et al., "ISIS Comes to Libya," CNN, November 18, 2014, <http://www.cnn.com/2014/11/18/world/isis-libya/>; Thomas Joscelyn, "Islamic State 'Province' in Libya Claims Capture of Town," Long War Journal, February 15, 2015, <http://www.longwarjournal.org/archives/2015/02/islamic_state_provin_1.php>; Frederick Wehrey and Ala' Alrababa'h, "Rising Out of Chaos: The Islamic State in Libya," Carnegie Endowment for International Peace, March 5, 2015, <http://carnegieendowment.org/syriaincrisis/?fa=59268>.
101. Bunzel, "Ideology," 32.
102. "Al-Qaeda in Yemen Denounces 'Expansionist' ISIS," Al Arabiya, November 22, 2014, <http://english.alarabiya.net/en/News/middle-east/2014/11/22/Al-Qaeda-in-Yemen-denounces-ISIS-.html>.
103. David D. Kirkpatrick, "Militant Group in Egypt Vows Loyalty to ISIS," New York Times, November 10, 2014, <http://www.nytimes.com/2014/11/11/world/middleeast/egyptian-militant-group-pledges-loyalty-to-isis.html?_r=3>.
104. Jack Moore, "Spiritual Leader of Libya's Biggest Jihadi Group Pledges Allegiance to ISIS," Newsweek, April 8, 2015, <http://www.newsweek.com/top-judge-libyas-biggest-jihadi-group-pledges-allegiance-isis-320408>.
105. "ISIS Claims Increasing Stake in Yemen Carnage," CBS News/Associated Press, March 23, 2015, <http://www.cbsnews.com/news/isis-yemen-carnage-houthi-rebels-advance-in-aqap-territory-toward-hadi/>.
106. David Blair, "Boko Haram Is Now a Mini-Islamic State, with Its Own Territory," Telegraph, January 10, 2015, <http://www.telegraph.co.uk/news/worldnews/africaandindianocean/nigeria/11337722/Boko-Haram-is-now-a-mini-Islamic-State-with-its-own-territory.html>.
107. Harun Maruf, "Experts Say al-Shabab-Islamic State Linkup 'Unlikely,'" Voice of America, March 18, 2015, <http://www.voanews.com/content/experts-say-al-shabab-islamic-state-linkup-unlikely/2684247.html>.
108. David D. Kirkpatrick, "ISIS Finds New Frontier in Chaotic Libya," New York Times, March 10, 2015, <http://www.nytimes.com/2015/03/11/world/africa/isis-seizes-opportunity-in-libyas-turmoil.html>.
109. For a terrific discussion of this principal-agent problem, see Jacob Shapiro, The Terrorist's Dilemma: Managing Violent Covert Organizations (Princeton, NJ: Princeton University Press, 2013).
110. Kamil al-Tawil, "al-Zawahiri yattajih ila hall 'al-Qa'ida,'" Al Hayat, April 3, 2015, <http://www.alhayat.com/Articles/8371259/>.
111. Dabiq 5: 3.
112. Rachael Levy, "Could Saudi Arabia Be the Next ISIS Conquest?" Vocativ, June 23, 2014, <http://www.vocativ.com/world/iraq-world/saudi-arabia-next-isis-conquest/>.
113. Abu Ayyub al-Masri, "Sayuhzam al-jam' wa-yuwallun al-dubur," June 13, 2006.
114. Abu Muhammad al-Adnani, "Lan yadurrukum illa adhan," audio recording, February 2013, <https://archive.org/details/forKan.001>.
115. Baghdadi, "Hasad al-sinin."
116. See Nu'aym b. Hammad, Kitab al-fitan (1993), 211; David Cook, Studies in Muslim Apocalyptic (Studies in Late Antiquity and Early Islam, No. 21) (Princeton, NJ: Darwin Press, 2002), 156.
117. On the apocalyptic moment, see Richard Landes, Heaven on Earth: The Varieties of Millennial Experience (New York: Oxford University Press, 2011), 16.
118. Dabiq 5: 40.
Conclusion
1. Jean-Pierre Filiu, Apocalypse in Islam, transl. M. B. DeBevoise (Berkeley: University of California Press, 2011), 121, 140.
2. Ibid., 80–103; David Cook, Contemporary Muslim Apocalyptic Literature (Syracuse, NY: Syracuse University Press, 2005), 59–83.
3. Filiu, Apocalypse, 121–140.
4. Ibid., 131.
5. Ayman al-Zawahiri, "Zawahiri's Letter to Zarqawi [English translation]," personal correspondence to Abu Mus'ab al-Zarqawi, Harmony Program, Combating Terrorism Center at West Point, July 9, 2005, <https://www.ctc.usma.edu/posts/zawahiris-letter-to-zarqawi-english-translation-2>. Original Arabic version available at <https://www.ctc.usma.edu/v2/wp-content/uploads/2013/10/Zawahiris-Letter-to-Zarqawi-Original.pdf>. I have modified the translation.
6. Paul D. Buell, "Massacre," Historical Dictionary of the Mongol World Empire (Lanham, MD: Scarecrow Press, 2003), 190.
7. For example, see the account of Joshua's conquest of Jericho (Joshua 6:17–22). For an overview of the doctrine of the "ban" (herem), or slaughtering one's enemies, see Susan Niditch, War in the Hebrew Bible: A Study in the Ethics of Violence (New York: Oxford University Press, 1993).
8. Ali Bey al-Abbasi, Travels of Ali Bey in Morocco, Tripoli, Cyprus, Egypt, Arabia, Syria, and Turkey: Between the Years 1803 and 1807, 2 vols. (Longmans, 1816), vol. 2, 152.
9. Alastair Crooke, "You Can't Understand ISIS If You Don't Know the History of Wahhabism in Saudi Arabia," Huffington Post, August 27, 2014, <http://www.huffingtonpost.com/alastair-crooke/isis-wahhabism-saudi-arabia_b_5717157.html>.
10. For a list of the massacres and the related literature, see Khaled Abou El Fadl, Reasoning with God: Reclaiming Shari'ah in the Modern Age (Lanham, MD: Rowman and Littlefield, 2014), 456, footnote 82.
11. Frank Clements, Conflict in Afghanistan: A Historical Encyclopedia (Santa Barbara, CA: ABC-CLIO, 2003), 112.
12. See, for example, Madawi al-Rasheed's explanation of why tribes joined the Wahhabi movement in A History of Saudi Arabia (New York: Cambridge University Press, 2005), 13–23.
13. Max Boot, Invisible Armies: An Epic History of Guerilla Warfare from Ancient Times to the Present (New York: Liveright Publishing, 2013), 64–79.
14. Mehdi Hasan, "This Is What Wannabe Jihadists Order on Amazon Before Leaving for Syria," New Republic, August 22, 2014, <http://www.newrepublic.com/article/119182/jihadists-buy-islam-dummies-amazon>.
15. Aaron Zelin, "The Islamic State of Iraq and Syria Has a Consumer Protection Office," Atlantic, June 13, 2014, <http://www.theatlantic.com/international/archive/2014/06/the-isis-guide-to-building-an-islamic-state/372769/>.
16. Letter from Abd al-Hamid Abu Yusuf, a commander in "Northern Baghdad Province," to Abu Bakr al-Baghdadi, n.d. The Arabic handwritten letter was given to me by the German journalist Volkmar Kabisch. The word for "bribing" here is irsha'.
17. Jamie Crawford and Laura Koran, "U.S. Officials: Foreigners Flock to Fight for ISIS," CNN, February 11, 2015, <http://www.cnn.com/2015/02/10/politics/isis-foreign-fighters-combat/>.
18. Yasir al-Shadhili, "Dirasa Amirkiyya: 5 fi al-mi'a faqat min al-Sa'udiyyin man yata'atif ma'a 'Da'ish," Al Hayat, October 17, 2014, <http://alhayat.com/Articles/5095664/%D8%AF%D8%B1%D8%A7%D8%B3%D8%A9-%D8%A3%D9%85%D9%8A%D8%B1%D9%83%D9%8A%D8%A9--5-%D9%81%D9%8A-%D8%A7%D9%84%D9%85%D8%A6%D8%A9-%D9%81%D9%82%D8%B7-%D9%85%D9%86-%D8%A7%D9%84%D8%B3%D8%B9%D9%88%D8%AF%D9%8A%D9%8A%D9%86-%D9%85%D9%86-%D9%8A%D8%AA%D8%B9%D8%A7%D8%B7%D9%81-%D9%85%D8%B9--%D8%AF%D8%A7%D8%B9%D8%B4->.
Appendices
1. I've consulted the translations and texts on Sunnah.com where possible.
Appendix 1
1. Arthur John Arberry, The Koran Interpreted: A Translation (New York: Touchstone, 1955).
2. Ibid., 29 (tradition from Mustadrak al-Hakim).
3. The last sentence alludes to Qur'an 6:158.
Appendix 3
1. Jawad Bahr al-Natsha, al-Mahdi Masbuq bi-Dawla Islamiyya (Al-Khalil/Hebron: Markaz Dirasat al-Mustaqbal, 2009), 59–60, 66, 74, 99.
2. That is, between the corner (rukn) of the Ka'ba and the "place" (maqam) where Abraham stood.
Appendix 4
1. Turki ibn Mubarak al-Bin'ali, "Al-Qiyafa fi 'adm ishtirat al-tamkin al-kamil li-l-khilafa."
2. "The religion will abide until the Hour comes or there are twelve caliphs over you, each of them from the Quraysh tribe" (tradition from Sahih Muslim).
3. "The caliphate in my community will be thirty years, then there will be monarchy after that" (tradition from Jami' al-Tirmidhi).
Index
The index that appeared in the print version of this title does not match the pages in your e-book. Please use the search function on your e-reading device to search for terms of interest. For your reference, the terms that appear in the print index are listed below.
al-Abab, Adil, –9
Abbasids, , –9, , –11, , , , –3, , , n39
Abd al-Rahman, Atiyya, –14, –19, , –8, n44, n51
Abd al-Wadud, Abu Mus'ab, –4
Abdullah, Abu,
Abdulmutallab, Umar Farouk,
Abu Bakr, , –4
Abu Muslim, –7
Abu Nuwas,
Abu Sufyan, . See also Sufyani
al-Adl, Sayf, –10, n5–6, n38, n44
al-Adnani, Abu Muhammad, –4, , n147
Afghanistan, , , , , , , ,
Herat training camp,
and Taliban, , –10, , , , –30,
Africa
Central African Republic,
Ethiopia, , –1n85
Kenya, ,
Nigeria,
Somalia, , , –8, –70, ,
See also North Africa
African Union,
ahadith, ,
Ahmed, Abu, –6
Ahrar al-Sham, –8
al-Qaeda
al-Fajr (media distribution),
and apocalypticism, –9
full name (Qa'idat al-Jihad),
and Islamic State, –7, , –9, –5, –5, –30, –2, , –2
Vanguards of Khorasan (magazine),
See also Bin Laden, Osama; al-Zawahiri, Ayman
al-Qaeda affiliates
al-Qaeda in Iraq, , –17, , –3, –7, ,
al-Qaeda in Yemen, –9, ,
al-Qaeda in the Arabian Peninsula (AQAP), –68, –7n19, –2n89, n117
al-Qaeda in the Indian Subcontinent, –30
al-Qaeda in the Islamic Maghreb (AQIM), –5, –9, , n99
and "glocalization,"
and Islamic State flag, –71
Nusra Front (Jabhat al-Nusra, Syria), –93, –6, , , , , , n116, n125, n48
al-Shabab (Somalia), , –8, , , , n13, –1n85, n86, n88–9, n100, n102
al-Qaeda in the Arabian Peninsula (AQAP)
Ansar al-Shari'a (Supporters of the Shari'a), –9, ,
attacks in Ma'rib and Ataq, –7n19
The Echo of Battles (online magazine), –50
Inspire (online magazine), 49–50
merger with al-Qaeda in Yemen, , –2n89
Aleppo, Syria, , –8, –2, ,
Algeria, –14, , , –7,
Ali (Muhammad's son-in-law), , , , , –17, , , n50
al-Ali, Hamid,
Almohads, ,
al-Amaytar, Abu,
American Revolution,
Ansar al-Din,
Ansar al-Shari'a, –9, ,
Ansar al-Sunna,
Ansar Bayt al-Maqdis,
al-Ansari, Abu al-Walid,
Antichrist, , , –7, –7, –8, n47
apocalyptic prophecies
and Antichrist, –7, –7, –8, n47
appearance of Gog and Magog, –70
conquest of Constantinople, , –4
conquest of Rome, ,
and Dabiq, –5, , , ,
and Damascus, , , –10, , ,
death of the believers, –1
destruction of Mecca and Medina, –1
End of Days, , –4, , , n107
End-Time, , , , , , , , , , , –71, –5, n13
establishing the caliphate in accordance with the prophetic method, , –17, , –9, , –4, , –9
fighting Rome, –6
fighting the Jews, –9,
Final Hour, , , –7,
and flags/banners, , –9, –11
great universal signs,
and Hashimites,
invasion of India,
and Jesus, –6, , –70, –5, n95, n94
and Muhammad, , –6, –101, , , , , , , –71, –5
sectarian dimension of, –11
and al-Sham, –102, , –8, , , , –7, , , –1
and Shi'i Islam, , –8, ,
spread of Islam, –5
and strangers/foreigners, –2
and the Sufyani, –9
and Sunni Islam, , , , –9, , , –7, , –71, –5
twelve caliphs, –18, –81
"twelve thousand," –2,
and Yazidis, –14
See also Mahdi (Muslim savior, "the Rightly-Guided One")
apocalypticism
and Abbasids, ,
and Abu Mus'ab al-Suri, ,
and al-Qaeda, –9
and AQAP, –1
apocalyptic propaganda and rhetoric, , –8, , , n45, n22
and generational change, ,
and Iraq War,
and Islamic State, –2, , , –19, , –40, –3, –7, –5, –8
and Masri, –2, –1
and violence, –12, , –9
and Zarqawi, –3, , n45, n22
Arab Spring, , , –71, ,
Ark of the Covenant,
Army of the Caliphate,
al-Ar'ur, Adnan,
al-Assad, Bashar, –6, , –8, –1, , , , ,
al-Assad, Hafez,
Atatürk, Mustafa Kemal,
al-Awlaki, Anwar, –50
al-Badri, Ibrahim Awwad Ibrahim. See al-Baghdadi, Abu Bakr (nom de guerre)
Baghdad, Iraq, , –2, –7, , –4, , n50
al-Baghdadi, Abu Bakr (renamed Caliph Ibrahim al-Baghdadi, nom de guerre), , , ,
and al-Qaeda, , –6, , 140–2
apocalypticism of, ,
appointment as commander of the faithful, , , –8
and Bin Laden,
black clothing as caliph, –4
as caliph of Islamic State, –18, –4, ,
in Camp Bucca (American detention center), –6
education, –6
family and early years, –4
and Nusra Front, –5
and Zawahiri, –9, –6
al-Baghdadi, Abu Umar (nom de guerre), , , , , , , n4–5
allegiance pledged to Bin Laden, ,
appointment as commander of the faithful, –18
and black flag,
death of, , ,
and Masri, –5, –1
and nineteen tenants of Islamic State, –8
as nominal head of Islamic State, –9, , –1,
and proclamation of Islamic State, –16
status of, –7, ,
al-Bahlul, Ali,
banners. See flags and banners
Bar Kokhba, Simon,
Barmakids,
beheading, , , , , –6, , , ,
Benghazi, Libya,
Berbers, –11
Bin Laden, Osama
and Abu Usama al-Iraqi video, –7
and al-Qaeda affiliates, –6, , –9, ,
allegiance pledged to Mullah Omar, , , ,
and apocalypticism, , –7
family, ,
and Islamic State, , –18, –4, –7, –5, –9, –8, –4, , ,
strategy of, –8, –12, –15, –6, , –2
al-Bin'ali, Turki ibn Mubarak, , –18, –81
Boko Haram,
Book of Tribulations, ,
Cairo, Egypt, –1
caliphate, , –10, –7, –1, , –9
Abbasid, , –9, , –11, , , , –3, , , n39
Almohad, ,
countercaliph, –30,
and debate over popular support, , –12, , , –5, –3
Fatimid, ,
Islamic State as, , –17, , –44, –3,
Mamluk,
and Management of Savagery, –3
Ottoman, , , , ,
prophecy of establishing "in accordance with the prophetic method," , –18, , –7, , , –4, , –9
Rashidun, –4, n50
Umayyad, , –7, , –11, , , , , n50
caliphs, , –6, –9, –17, –33,
Abu Bakr, , –4
Abu Bakr al-Baghdadi as, , , , –18, –4, ,
and Abu Umar al-Baghdadi as, ,
Ali, , , , , –17, , , n50
Hasan, ,
Mehmed II,
Mu'awiya, , ,
and Mullah Omar, –30
and prophecy, , , –8, –81
Rashid, –4, n50
Saffah, ,
Selim I, –6
Sulayman, ,
twelve caliphs prophecy, –18, –81
Umar,
See also Mahdi (Muslim savior, "the Rightly-Guided One")
Call of the Global Islamic Resistance, 87
Camp Bucca (American detention center), –6
Cerantonio, Musa, –100
Charlie Hebdo attacks, –7
Christ. See Jesus
Christianity, , , , –6, 109, , , , –6, n95
Clausewitz, Carl von, ,
Constantinople, , , , 173–4
countercaliph, –30,
Crusades, –9, ,
Dabiq, Syria, –5, , , ,
Damascus, Syria, , , , , , , –10, , ,
dawla, , ,
dawla mubaraka,
Day of Judgment, , , , , –2, , , , n70
Deceiver, see Antichrist
al-Din, Ansar,
Dunaway, Mark, –6
Egypt, –9, , , –1, , , ,
emirate, use of the term,
End of Days, , –4, , , n107
End-Time prophecies, , , , , , , , , , , n13
Ethiopia, , –1n85
Fallujah, Iraq, ,
Fatimids, ,
fatwas, , ,
Fertile Crescent, ,
al-Filistini, Abu Qatada, , , n49
Filiu, Jean-Pierre,
Final Hour, , , –7,
flags and banners
black flag, , , –22, –8, , , –10,
of Islamic State, –6, –22, , –9, , –71, , –11, , –6
in legend and history, , –8, ,
and prophecy, , –9, –11
yellow flag, –1, –11
French Revolution,
Gadahn, Adam, , –4, n139
gharib/ghuraba (stranger, foreigner),
Godane, Ahmed Abdi, , n88–9, n102
Gush Emunim,
Hadrian,
Hajji Bakr (nom de guerre), ,
Hammad, Nu'aym bin, ,
Harun, Fadil, , , n47, n86, n94, n38
Harun al-Rashid, –4, n50
al-Hasan, ,
Hasan, Nidal,
Hashim (Muhammad's great-grandfather),
Hashimites, –6
hearts-and-minds strategy, –8, , –9, –7, , –51
Hijazi, Akram,
Hizb al-Islam,
Hizballah,
Hosseinian, Ruhollah,
hudud (fixed punishments in Islamic scripture), –9, –8, –8, , , ,
Hurayra, Abu,
Husayn (Muhammad's grandson), ,
Hussein, Saddam, , , , –6, –9, , –7, –4
Ibn Khaldun, –4, ,
imara (emirate),
Imru al-Qays,
Iran, –9, –6, , , , –4, , n78
Iran-Iraq War,
Iraq
al-Qaeda in Iraq, , –17, , –3, –7, ,
Anbar Province, , ,
Ba'ath party, , –4
Baghdad, , –2, –7, , –4, , n50
Fallujah, ,
Hawija, ,
Kurds in, –7
Mosul, , , , –4, –4,
Shi'a in, –11, , , , , , , –7
Sunni in, , , –9, , , , –8
and US invasion, , –10, , , , –6
and US withdrawal, ,
and Yazidis, –13
al-Iraqi, Abu Maryam,
al-Iraqi, Abu Usama (nom de guerre), –7
Iraqi Hizballah,
ISIS (Islamic State of Iraq and al-Sham/the Levant). See Islamic State
Islam
hudud (fixed punishments), –9, –8, –8, , , ,
Ka'ba (shrine), , , –3, , , n2
origins of, –7,
Qur'an, , , –6, , , , , , , –71, , n70
Salafism, , , –5, ,
Shari'a law, , , –2, , –4,
Wahhabism, , –51
See also apocalyptic prophesies; caliphate; caliphs; mosques; Muhammad; Shari'a law; Shi'i Islam; Sunni Islam
Islamic Army, –5, n28
Islamic Jihad,
Islamic State
and Ahrar al-Sham, –8
and al-Qaeda, –7, , –9, –5, –5, –30, –2, , –2
and apocalypticism, –2, , , –19, , –40, –3, –7, –5, –8
and Bin Laden, , –18, –4, –7, –5, –9, –8, –4, , ,
branding of, –8
as caliphate, , –17, , –44, –3,
coins of,
contradictions of, –5
Dabiq (magazine), –3
Da'wa Gathering, –7
declaration as caliphate, , , –30
and debate over popular support, , –12, , , –5, –3
and "Enduring and Expanding" slogan, –42
establishment of Islamic State of Iraq and al-Sham (the Levant) (2013), –2
flag of, –6, –22, , –9, , –71, , –11, , –6
and jihadism, , , , , , , –81, –7, , –6, , , –2
and Mahdi, , –4, –7
and Management of Savagery, –4, , n78, –11n87
proclamation of Islamic State of Iraq (2006), , –17, ,
and Shi'a, , , , , –3,
Shura Council, –8
and slavery, –14
and "Strategic Plan for Reinforcing the Political Position of the Islamic State of Iraq," –82
strategy of, –51
and Sunni Islam, –9, , , –82, , –4, , –5, , –2, –7
use of the term, –16
and Yazidis, –13
and young women recruits, –14
See also al-Masri, Abu Ayyub; al-Baghdadi, Abu Bakr; al-Baghdadi, Abu Umar
Istanbul, Turkey, . See also Constantinople
Ja'far the Barmakid, , n108
Jandal, Abu,
al-Jawlani, Abu Muhammad, , –2, n130–1
Jerusalem, , , , , , , , n4
Jesus, –6, , –70, –5, n95, n94
Jews, , , , , , , –6, –9,
jihad, , –6, –12, , –6, , , –7, , ,
jihadism and jihadists
and 9/11,
Algerian,
and apocalyptic prophecies, –3, , –3, ,
global movement, , , , –70, , , –1, ,
global recruitment, –1
hearts-and-minds strategy, –8, , –9, –7, , –51
historical,
and Hizballah,
and Islamic State, , , , , , , –81, –7, , –6, , , –2
and Islamic State flag, –6, –70
and Management of Savagery, –4, , n78, –11n87
as mujahidin,
online forums and social media, , , , n59
and razing of shrines,
and smoking tobacco, –9
and "Strategic Plan for Reinforcing the Political Position of the Islamic State of Iraq," –82
See also al-Qaeda; al-Qaeda affiliates; Bin Laden, Osama
Jordan, –8, , , , –8
Judaism, , . See also Jews
Ka'ba (Islamic shrine), , , –3, , , n2
Kampala, Uganda, July 2010 attacks,
al-Kasasbeh, Moath,
Kassig, Peter,
Kennedy, Paul,
Kenya,
Khan, Samir, ,
Khansa Brigade,
Khorasan, –6, –9, , ,
Kony, Joseph,
Kurds, –8
Last Days, , ,
Lebanon, ,
Leibowitz, Marc,
Libya, , , , , , , –1,
Lind, William,
Lord's Resistance Army,
Lulu, Ahmad, –8, n159
Machiavelli, Niccolò,
Maghrib. See North Africa
Mahdi (Muslim savior, "the Rightly-Guided One"), –9, , n45
and al-Qaeda, –9
Bin Laden on, n94
Bin'ali on, , –1
claimants to title of, –5,
historic origins of, –9
and Islamic State, , –4, 146–7
Masri on, , , ,
Muhammad on, –5, n107
Natsha on (in The Mahdi Is Preceded by an Islamic State),
and popular belief, –4, , –8
and Saffah,
and Shi'a, , –8, , –17
and Sunni Islam, , , –9
al-Utaybi on, n107
Mahdi Army,
Mahmood, Aqsa,
Majlis Shura al-Mujahidin,
Mali, –4, , –70,
al-Maliki, Nuri, ,
Mamluk Sultanate,
Management of Savagery, –4, , n78, –11n87
Mao Zedong,
al-Maqdisi, Abu Muhammad, –1, , , , –4n8, n12–13, n120, –5n31
al-Masri, Abu Ayyub, (aka Abu Hamza al-Muhajir), –7,
apocalypticism of, –2, –1,
and bounty on,
death of, , ,
and founding of Islamic State, –17, –5,
as leader of Islamic State, –17, –5, –42, , –5,
as leader of al-Qaeda in Iraq, –17, –2
and Islamic State of Iraq,
and Twelver Shi'i Islam, n50
and Utaybi, –42
and Zarqawi, –2
Mehmed II,
al-Mejjati, Adam Karim,
messiahs, , . See also Mahdi (Muslim savior, "the Rightly-Guided One")
monotheism, , , –2, , , n13
Monotheism and Jihad, , ,
mosques, , ,
Aqsa Mosque (Jerusalem),
attacks on, , , , , , n117, n107
Grand Mosque (Mecca),
Imam Ali Mosque (Najaf),
Nuri Mosque (Mosul),
Prophet's Mosque (Medina),
Umayyad Mosque (Damascus), ,
Mosul, Iraq, , , , –4, –4,
Mu'awiya, , ,
Mubarak, Hosni,
Muhammad, , , , , –7, , , –7
ahadith, ,
and battle of Uhud, ,
prophecies attributed to, , –6, –101, , , , , , , –71, –5
and punishment for apostasy, –8
and Shi'i doctrine of leadership, ,
mujahidin, use of the term, , , , . See also jihadists
al-Mukhtar,
Muslim Brotherhood,
Muslim Brotherhood of Syria,
Muthanna, Abu, –2
9/11 attacks, , , ,
Naji, Abu Bakr (nom de guerre), –4, , n77, n125
Nigeria,
North Africa, , , , ,
and al-Qaeda in the Islamic Maghreb (AQIM), –5, –9, , n99
Algeria, –14, , , –7,
and Almohads, ,
Egypt, –9, , , –1, , , ,
Libya, , , , , , , –1,
Mali, –4, , –70,
Tunisia, , , , , ,
Nur al-Din Zengi, –9, ,
Obama, Barack, ,
oil, , ,
Omar, Mullah Muhammad, , , , , –30, n38
Ottoman Empire, , , , ,
Padnos, Theo,
Pakistan, , , –4, , , , , –5n63, n78
Palestine, , ,
Pasha, Ahmad Cevdet,
prophecies. See apocalyptic prophecies
Prophet. See Muhammad
al-Qahtani, Nayif, –9,
Qur'an, , , –6, , , –71,
and end of days, –70, , n70
portrayal of Jesus in,
recitation of, –5, ,
and slavery,
Qur'anic studies, ,
al-Qurashi, Abu Ubayd,
Quraysh tribe, , , , –80, n2
Ramadan,
al-Rashid, Harun, –4, n50
Robow, Mukhtar (aka Abu Mansur), , n102
al-Saffah, , . See also Abbasids
Saladin,
Salafism, , , –5, ,
Sana'a, Yemen, , n117
Saydnaya military prison,
Saudi Arabia, , –9, , , –6, –40, , ,
Mecca, , , , , , , , , , ,
Medina, , , –3, , , , , ,
Selim I, –5
September 11, 2011, attacks of, , , ,
al-Shabab (al-Qaeda affiliate in Somalia), , –8, , , , n13, –1n85, n86, n88–9, n100, n102
al-Sham, , –102, , –8, , , , –7, , , –1
Shami Witness (online pseudonym),
Shari'a committees, –16,
Shari'a law, , , –2, , –4,
Shi'a Islam
and apocalyptic prophecies, , –8, ,
and Baghdadi, ,
doctrine of leadership,
and flags,
in Iraq, –11, , , , , , , –7
and Islamic State, , , , , –3,
in Syria, , ,
and Taliban,
Twelver Shi'a Islam, n50, n108
and the U.S.,
and Zarqawi, –8, –11, , ,
Sistani, Grand Ayatollah,
slavery, , –14,
smoking tobacco, , , , –9
social media,
blogs,
Twitter, , , , , , –4, , , , –7, , n59
YouTube, ,
Soleimani, Qasem,
Somalia, , , –8, –70, , . See also al-Shabab (al-Qaeda affiliate in Somalia)
Soufan, Ali,
Spain, , , ,
strangers, –2
Strangers Media Foundation,
"Strategic Plan for Reinforcing the Political Position of the Islamic State of Iraq," –82
Sufism, ,
Sufyani, –9
Sulayman, ,
Sun Tzu,
Sunni Islam
1920 Revolution Brigade,
and al-Qaeda, –9
and apocalyptic prophecies, , , , –9, , , –7, , –71, –5
and Baghdadi, –5
and flags,
in Iraq, , , –9, , , , –8
and Islamic State, –9, , , –82, , –4, , –5, , –2, –7
in Syria, –6, –9, , , –4,
and Zarqawi, –8, –14,
See also Taliban
al-Suri, Abu Khalid, –8
al-Suri, Abu Mus'ab, , –2, –7, , n17, n13
Syria
al-Bab,
Aleppo, , –8, –2, ,
Dabiq, –5, , , ,
Damascus, , , , , , , –10, , ,
Free Syrian Army, , ,
Ghouta, ,
Kurds in, –7
Nusra Front (Jabhat al-Nusra), –93, –6, , , , , , n116, n125, 226n48
Raqqa, , , ,
al-Sham, , –102, , –8, , , , –7, , , –1
Shi'a in, , ,
Sunni in, –6, –9, , , –4,
See also al-Assad, Bashar
Syria Khayr,
al-Tabari,
Taber, Robert,
Taha, Adnan,
Taliban, , , , , –5n63
and Afghanistan, , –10, , , , –30,
See also Omar, Mullah Muhammad
Timbuktu, Mali, –3
Tunisia, , , , ,
Turkey, , , , , , ,
"twelve thousand" prophecy, –2, 60
Uganda, Kampala,
Uhud, battle of, ,
Umayyads, , –7, , –11, , , , , n50
United Nations, bombing of Baghdad U.S. headquarters, ,
United States
9/11 attacks, , , ,
and al-Qaeda, , –12, –15, –9, –2, –3, , –61
American Revolution,
attack on Benghazi consulate,
and Bin Laden, , –5, –3,
Camp Bucca (detention center), –6
and deaths of Masri and Abu Umar al-Baghdadi, , , ,
and death of Utaybi,
and death of Zarqawi, ,
drone strikes,
embassies, , –4
Fort Hood shooting,
and invasion of Iraq, , –10, , , , –6
and Islamic State, –15, , –81, , , –5, , –7
and Israel,
and Nusra Front, –8
al-Uraydi, Sami, –8, , n111
al-Utaybi, Abu Sulayman, –42, n5, n39, n107
Wadoud, Abdul,
Wahhabism, , –51
World War I, ,
al-Wuhayshi, Nasir, –8, , , –61, n63, –2n89
Yazidis, –13
Yemen, –55, , , , , –1, , , , , n13
al-Qaeda in Yemen, –9, ,
al-Zarqawi, Abu Mus'ab, –15, –6,
and Abd al-Rahman, –20,
admiration for Nur al-Din Zengi, –9, ,
and al-Qaeda, –15,
apocalypticism of, –3, , n45, n22
death of, ,
as al-Gharib ("the Stranger"),
and Maqdisi,
and Masri, –2
and Monotheism and Jihad, , ,
and Shi'a Islam, –8, –11, , ,
and Sunni Islam, –8, –14,
al-Zawahiri, Ayman, , –17, , , ,
and Abu Khalid al-Suri,
and apocalypticism, , , , n22
assumes leadership of al-Qaeda,
hearts-and-minds approach of, , –13, , , ,
and Islamic State, , –9, , –7, –30
and Zarqawi, –14,
al-Zawi, Hamid,
Transliteration
I generally followed the guidelines of the International Journal of Middle East Studies, although I've capitalized proper nouns. Outside of book and article titles, I haven't used initial ayn because I find it too distracting. Hamza and ayn are represented by an apostrophe.
Acknowledgments
Readers will decide if this book is any good, but I'm certain it would have been much poorer were it not for the help of my family, friends, and colleagues. Assad Ahmed, Cole Bunzel, Ali Chaudrey, and Shadi Hamid all get gold stars and my eternal gratitude for reading and commenting on the early draft, which really helped me clarify my thoughts. Their enthusiasm and encouragement propelled me toward the finish line. Michael O'Hanlon is also in the gold-star club for reading the entire manuscript and reminding me that I couldn't leave the "so what?" question unanswered. I never showed the manuscript to my Dad, but he kept urging me to answer the equally important question, "Why do they do that?"
Anne Peckham and Elizabeth Pearce not only read and commented on the entire manuscript in detail; they also shouldered many of my office duties to give me time to write it. I couldn't have done it without them.
Daniel Benjamin, Ryan Evans, Stephanie Kaplan, Daniel Kimmage, Robert McKenzie, and Clint Watts—nonalarmists all—read individual chapters and let me know when I was getting too carried away or not carried away enough. I sent my research assistant Kristine Anderson on lots of wild goose chases in the thickets of Twitter and YouTube and she managed to come back with some fantastic cyberfowl. Superstar intern Nouf Al Sadiq helped me decipher some awfully blurry screen shots of some awfully messy Arabic handwriting. And Jennifer Williams went above and beyond the call of duty when she put everything on hold to help me tame my unruly footnotes before deadline.
Dan Byman, Mike Doran, Jeremy Shapiro, and Tamara Wittes all heard various parts of the book in their nascent form and encouraged me to keep at it. Tamara loved the idea of telling the story of the Islamic State through its flag. Dan didn't love it enough but found lots of other things he did love. Jeremy, who usually doesn't love anything, thought I had a few smart things to say, which was just as gratifying. And Mike, who has long been a mentor and a friend, urged me to write something that people would want to actually read. I hope you're all pleased with how things turned out.
I'm grateful to Suzanne Maloney and Khaled Elgindy, who gave me the right compliments at the right moment, when things were tough going. Martin Indyk and Bruce Jones asked hard questions and nudged me to defend my ideas in front of the great scholars at Brookings, which kept me on my toes. And Strobe Talbott's enthusiasm for the project not only encouraged me but also made me hopeful that Brookings will want to keep me around a bit longer.
There are plenty of people who kindly answered my questions or generously shared their findings. I especially want to thank Volkmar Kabisch, who passed along the documents he and his team had gathered in Iraq, and Peter Neumann, who put us in touch and shared his own research. J. M. Berger graciously responded to my queries, Thomas Hegghammer pushed me to be more precise, Sam Helfont helped me better understand Ba'athist religious politics in Iraq, and Jomana Qaddour kept me up to date on the Syrian civil war.
A special thanks to my agent extraordinaire, Bridget Matzie, who not only read and commented on the manuscript but also connected me with the wonderful Karen Wolny and her outstanding team at St. Martin's. Readers can thank Karen for constantly reminding me that people like stories and explanations that don't leave them more confused than when they started.
There's a reason why authors usually thank their immediate family last: every subsequent thank you would dim in comparison. Casey and my three sweet little kids spent a very cold winter in DC tiptoeing around the house so I could write this book. I can never thank you enough for your understanding—well, at least Casey understood. My kids just laughed loudly when I shushed them. Now it's time to take that much-delayed summer vacation!
About the Author
David McCants
William McCants directs the project on U.S. Relations with the Islamic World at the Brookings Institution. He is adjunct faculty at Johns Hopkins University and a former U.S. State Department senior adviser for countering violent extremism. McCants has a Ph.D. in Near Eastern Studies from Princeton University and lives in the Washington, D.C. area.
Thank you for buying this St. Martin's Press ebook.
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Contents
Title Page
Copyright Notice
Dedication
Introduction
1: Raising the Black Flag
2: Mahdi and Mismanagement
3: Bannermen
4: Resurrection and Tribulation
5: Sectarian Apocalypse
6: Caliphate Reborn
Conclusion
Appendices: Sunni Islamic Prophecies of the End Times
Appendix 1: The Final Days
Appendix 2: The Victorious Group
Appendix 3: The Mahdi Is Preceded by an Islamic State
Appendix 4: Twelve Caliphs
Notes
Index
Transliteration
Acknowledgments
About the Author
Copyright Page
THE ISIS APOCALYPSE. Copyright © 2015 by William McCants. All rights reserved. For information, address St. Martin's Press, 175 Fifth Avenue, New York, N.Y. 10010.
www.stmartins.com
The Library of Congress has cataloged the print edition as follows:
McCants, William Faizi, 1975–
The ISIS apocalypse : the history, strategy, and doomsday vision of the Islamic State / William McCants.
pages cm
ISBN 978-1-250-08090-5 (hardback)
1. IS (Organization)—History. 2. Terrorism—Religious aspects—Islam. 3. Terrorism—Middle East. 4. Jihad. 5. Strategy. 6. Islamic Empire. 7. End of the world—Political aspects—Middle East. 8. Political messianism—Middle East. 9. Middle East—Politics and government—21st century. I. Title.
HV6433.I722M35 2015
363.3250956—dc23
2015015281
e-ISBN 978-1-4668-9270-5
Our e-books may be purchased in bulk for promotional, educational, or business use. Please contact your local bookseller or the Macmillan Corporate and Premium Sales Department at (800) 221-7945, extension 5442, or by e-mail at MacmillanSpecialMarkets@macmillan.com.
First edition: September 2015
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{"url":"https:\/\/www.dummies.com\/education\/math\/calculus\/calculate-a-cube-root-using-linear-approximation\/","text":"# Calculate a Cube Root Using Linear Approximation\n\nLinear approximation is not only easy to do, but also very useful! For example, you can use it to approximate a cubed root without using a calculator.\n\nHere\u2019s an example. Can you approximate\n\nLike this: Bingo! 4.125.\n\nWell, okay, there\u2019s a little more to it than that. Take a look at the figure and then follow the steps below to get the full picture.\n\nThe line tangent to the curve at (64, 4) can be used to approximate cube roots or numbers near 64.\n\nTo estimate\n\n1. Find a perfect cube root near\n\nYou notice that\n\nis near a no-brainer,\n\nwhich, of course, is 4. That gives you the point (64, 4) on the graph of\n\n2. Find the slope of\n\n(which is the slope of the tangent line) at x = 64.\n\nThis tells you that \u2014 to approximate cube roots near 64 \u2014 you add (or subtract)\n\nto 4 for each increase (or decrease) of one from 64. For example, the cube root of 65 is about\n\nthe cube root of 66 is about\n\nthe cube root of 67 is about\n\nand the cube root of 63 is about\n\n3. Use the point-slope form to write the equation of the tangent line at (64, 4).\n\nIn the third line of the above equation, you put the 4 in the front of the right side of the equation (instead of at the far right which might seem more natural) for two reasons. First, because doing so makes this equation jibe with the explanation at the end of Step 2 about starting at 4 and going up (or down) from there as you move away from the point of tangency. And second, to make this equation agree with the explanation at the end of Step 4. You\u2019ll see how it all works in a minute.\n\n4. Because this tangent line runs so close to the function\n\nnear x = 64, you can use it to estimate cube roots of numbers near 64, like at x = 70.\n\nBy the way, in your calc text, the simple point-slope form from algebra (first equation line in Step 3) is probably rewritten in highfalutin\u2019 calculus terms \u2014 like this:\n\nDon\u2019t be intimidated by this equation. It\u2019s just your friendly old algebra equation in disguise! Look carefully at it term by term and you\u2019ll see that it\u2019s mathematically identical to the point-slope equation tweaked like this: y = y1 + m(xx1). (The different subscript numbers, 0 and 1, have no significance.)","date":"2019-03-19 23:25:48","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8178708553314209, \"perplexity\": 368.13331231407926}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912202161.73\/warc\/CC-MAIN-20190319224005-20190320010005-00047.warc.gz\"}"}
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Q: Data type mismatch in criteria expression in the INSERT statement When running the program i get an error saying 'Data type mismatch in criteria expression.' and the line cmd.ExecuteNonQuery() is highlighted. In my database the datatype for 'ID' is AutoNumber and the datatype for 'Calories Burned' is decimal and everything else is text. I don't know if it is do with fact that when i input data into the text boxes its classed as a string. but if someone could help i would appreciate it a lot.
Private Sub btnAdd_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles btnAdd.Click
Dim cmd As New OleDb.OleDbCommand
' add data to table '
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cnn.Open()
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cmd.Connection = cnn
If Me.txtID.Tag & "" = "" Then
cmd.CommandText = "INSERT INTO [Training log] ([ID], [Runner Name], [Running Average Speed], [Cyclying Average Speed], [Swimming style] , [Calories Burned]) VALUES ('" & Me.txtID.Text & "' , '" & Me.txtRunnerName.Text & "' , '" & Me.txtRunSpeed.Text & "' , '" & Me.txtCycleSpeed.Text & "', '" & Me.txtSwimStyle.Text & "', '" & Me.txtCaloriesBurned.Text & "')"
cmd.ExecuteNonQuery()
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cmd.CommandText = "UPDATE [Training log] SET ID=" & Me.txtID.Text & ", [Runner Name]='" & Me.txtRunnerName.Text & "', [Running Average Speed]='" & txtRunSpeed.Text & "', [Cyclyin Average Speed]='" & txtCycleSpeed.Text & "', [Swimming style]='" & txtSwimStyle.Text & "', [Calories Burned]='" & txtCaloriesBurned.Text & "' WHERE ID='" & txtRunnerName.Tag & "' "
cmd.ExecuteNonQuery()
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A: You may need to remove the single quotes around your non-string values.
cmd.CommandText = "INSERT INTO [Training log] ([ID], [Runner Name], [Running Average Speed], [Cyclying Average Speed], [Swimming style] , [Calories Burned]) VALUES (" & Me.txtID.Text & " , '" & Me.txtRunnerName.Text & "' , '" & Me.txtRunSpeed.Text & "' , '" & Me.txtCycleSpeed.Text & "', '" & Me.txtSwimStyle.Text & "', " & Me.txtCaloriesBurned.Text & ")"
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\section{Introduction} \label{section1}
Let $T$ be a compact torus with its Lie algebra $\mathfrak{t}$ and lattice $\mathfrak{l} \subset\mathfrak{t}$. For a compact symplectic manifold $(M,\omega)$ equipped with a Hamiltonian-$T$-action, we have a \emph{moment map} $\phi\colon M\rightarrow \mathfrak{t}^\ast$, where $\mathfrak{t}^\ast$ is the dual of $\mathfrak{t}$. Then we have the following equation: \begin{equation*} \iota_{X_\xi}\omega=-d\phi^\xi,\mbox{ }\forall\xi\in\mathfrak{t}\end{equation*} where $X_\xi$ denotes the vector field on $M$ generated by the action and $\phi^\xi \colon M\to\mathbb{R}$ is defined by $\phi^\xi (x)=\langle\phi (x),\xi\rangle$. Here, $\langle .,.\rangle$ is the natural pairing of $\mathfrak{t}^\ast$ and $\mathfrak{t}$. M is called a \emph{compact Hamiltonian-$T$-space}.
$\phi^\xi$ is called \emph{the component} of the moment map $\phi$ corresponding to the chosen element $\xi\in\mathfrak{t}$. Suppose that the component of the moment map is \emph{generic}, that is, $\langle\eta,\xi\rangle\neq 0$ for each weight $\eta\in \mathfrak{l}^\ast \subset\mathfrak{t}^\ast$ in the symplectic representation $T_p M$ for every $p$ in the $T$-fixed point set $M^T$, then $\psi=\phi^\xi \colon M\to\mathbb{R}$ is a Morse function with the critical set $M^T$. Under this situation, the Morse index of $\psi$ at each $p\in M^T$ is even. Let $\lambda(p)$ be half of the index of $\psi$ at $p$. Let $\wedge_p^{-}$ be the product of all the individual weights of this representation.
For each $p\in M^T$, the natural inclusion map $i_p\colon p \to M$ induces a map $i_p^\ast\colon H_T^\ast(M)\to H_T^\ast(p)$ in equivariant cohomology. Let $\alpha\in H_T^\ast(M)$, we use the notation $\alpha(p)$ for the image of $\alpha$ under the map $i_p^\ast$. $i_p^\ast$ is called \emph{the localization at p}.
\begin{definition} [\cite{GoldinTolman}]
Let M be a compact Hamiltonian-$T$-space with the moment map $\phi\colon M\to\mathfrak{t}^\ast$ and let $\psi=\phi^\xi \colon M\to\mathbb{R}$ be a generic component of the moment map for some $
\xi\in\mathfrak{t}$, a cohomology class $\alpha_p \in H_T^{2\lambda(p)}(M;\mathbb{Q})$ is a canonical class at the fixed point $p$ with respect to $\psi$ if
\begin{enumerate}
\item $\alpha_p(p)=\wedge_p^{-}$.
\item $\alpha_p(q)=0$ for all $q\in M^T \backslash\{p\}$ such that $\lambda(q)\leq\lambda(p)$.
\end{enumerate}
\end{definition}
Canonical classes do not always exist, see Example 2.2 in \cite{GoldinTolman}. But if canonical classes exist for all $p\in M^T$, then $\{\alpha_p\}_{p\in M^T}$ form a basis of $H_T^\ast(M)$ as a module over $H_T^\ast(pt)\cong H^\ast(BT)$.
Suppose that a set of canonical classes exists, the \emph{equivariant structure constants} for $H_T^\ast(M)$ are the elements $c_{pq}^r \in H_T^\ast(pt)$ given by the equation \begin{equation}\alpha_p \alpha_q =\sum_{r\in M^T} c_{pq}^r \alpha_r.\end{equation}
In \cite{Tymoczko}, explicit formulas for the equivariant structure constants of $H_T^\ast(\mathbb{CP}^n)$ are computed in terms of the localizations of canonical classes at various fixed points in $(\mathbb{CP}^n)^T$. This paper is concerned with the generalization of these formulas to compact Hamiltonian-$T$-spaces, under the assumption that the set of canonical classes exists.
Given a directed graph with vertex set $V$ and edge set $E\subset V\times V$, a path from a vertex $p$ to a vertex $q$ is a $(k+1)$-tuple $r=(r_0,...,r_k)\in V^{k+1}$ so that $r_0=p,r_k=q$ and $(r_{i-1},r_i)\in E$ for all $1\leq i\leq k$.
\begin{definition} [\cite{GoldinTolman}] \label{directedgraph}
Define an oriented graph with the vertex set $V=M^T$ and the edge set
\begin{equation*}
E=\{(r,r')\in M^T\times M^T\mid\lambda(r')-\lambda(r)=1,\alpha_r(r')\neq 0\}.
\end{equation*}
Let $\sum_p^q$ be the set of paths from $p$ to $q$ in $(V,E)$.
\end{definition}
From now on, we call this graph a \emph{moment graph}. Note that if no path connects $p$ and $q$, i.e. $\sum_p^q$ is an empty set, then $\alpha_p(q)=0$.
The realization of the image of a moment map as a graph has been known for a while, see \cite{GulleminZara} for example. Some useful information about the equivariant cohomology of a Hamiltonian-$T$-space can be extracted by using such a graph.
A formula for $\alpha_p (q)$ in terms of the values of a moment map at the points in $M^T$ and the restriction of canonical classes to points of index exactly two higher was derived in \cite{GoldinTolman}. Based on their idea, more formulas are derived in \cite{Zara}. The goal of this paper is totally different: Formulas for some equivariant structure constants are written in terms of other equivariant structure constants and the restriction of canonical classes to the $T$-fixed point set $M^T$. These are the main results in Section \ref{section2}. The complexity of computations involved in our formulas depends heavily on the structure of the moment graph. In some special cases, if the structure of the moment graph is exceptionally simple, our formulas are greatly simplified. We will look at an example in Section \ref{section3}. Note that we don't make use of any extra assumption on the Hamiltonian-$T$-spaces except the existence of a set of canonical classes.
\section{Main Results} \label{section2}
Let $(M,\omega)$ be a compact Hamiltonian-$T$-space and $\psi=\phi^\xi$ be a generic component of the moment map. Assume that a set of canonical classes $\alpha_p\in H_T^{2\lambda(p)}(M;\mathbb{Q})$ exists for all $p\in M^T$, define an oriented graph $(V,E)$ as in Definition \ref{directedgraph}. In this section we compute the equivariant structure constants $c_{pq}^k$ for any $p,q$ in the vertex set of the moment graph. We do the computations following the values of $\lambda(k)$ in asecending order.
\begin{lemma} \label{lemma1}
\begin{equation*}
c_{pq}^k=0
\end{equation*}for $\lambda(k)<\lambda(p)\leq\lambda(q)$, where $p,q,k\in M^T$.
\end{lemma}
\begin{proof}
We begin by writing the equation
\begin{equation}\label{star}
\alpha_p \alpha_q =\sum_{r\in M^T} c_{pq}^r \alpha_r.
\end{equation}
Without loss of generality, we assume that $\lambda(p)\leq\lambda(q)$. Let $t$ be an element in $M^T$ such that $\lambda(t)$ is the minimum value in the set $S=\{\lambda(x)\mid x\in M^T\}$. Since $\lambda(t)<\lambda(q)$, we have $\alpha_q (t)=0$. Localizing (\ref{star}) at $t$ gives
\begin{equation} \label{equation1}
\sum_{r\in M^T} c_{pq}^r \alpha_r(t)=0.
\end{equation}
Since $\alpha_r (t)=0,\forall r\in M^T\backslash\{t\}$, (\ref{equation1}) implies
\begin{equation*}
c_{pq}^t \alpha_t(t)=0.
\end{equation*}
But $\alpha_t(t)\neq 0$, thus we get
\begin{equation} \label{equationt}
c_{pq}^t=0.
\end{equation}
In the set $S=\{\lambda(x)\mid x\in M^T\}$, pick $u\in M^T\backslash\{q\}$ such that $\lambda(u)\leq\lambda(q)$ and $\lambda(u)$ attains the minimum value in the set $S\backslash\{\lambda(t)\}$, where $t\in M^T$ still satisfies the same property as above that $\lambda(t)=\mbox{min}_{x\in M^T}\lambda(x)$. Then $\alpha_q(u)=0$ and hence localizing (\ref{star}) at $u$ gives
\begin{equation} \label{equation2}
\sum_{r\in M^T} c_{pq}^r \alpha_r(u)=0.
\end{equation}
But we know that $\alpha_r(u)=0$ when $u\neq r$ and $\lambda(u)\leq\lambda(r)$.
Also, $c_{pq}^t=0$ by (\ref{equationt}). Hence (\ref{equation2}) gives
\begin{equation*}
c_{pq}^u \alpha_u(u)=0.
\end{equation*}
Since $\alpha_u(u)\neq 0$, we have
\begin{equation}
c_{pq}^u =0.
\end{equation}
By using the same method inductively on the set of values in $S$ which are smaller than $\lambda(p)$, we conclude that
\begin{equation}
c_{pq}^k =0
\end{equation}
for all $k\in M^T$ such that $\lambda(k)<\lambda(p)\leq\lambda(q)$.
\end{proof}
\begin{lemma} \label{lemma2}
\begin{equation*}
c_{pq}^p=0
\end{equation*}for $\lambda(p)\leq\lambda(q)$, where $p,q\in M^T$.
\end{lemma}
\begin{proof}
Note that $\alpha_q(p)=0$. Localizing (\ref{star}) at $p$ gives
\begin{equation}\label{equationp}
\sum_{r\in M^T} c_{pq}^r \alpha_r(p)=0.
\end{equation}For all $k\in M^T\backslash\{p\}$ such that $\lambda(k)<\lambda(p)$, $c_{pq}^k=0$ by Lemma \ref{lemma1}. For all $k'\in M^T\backslash\{p\}$ such that $\lambda(k')\geq\lambda(p)$, $\alpha_{k'}(p)=0$. Hence by (\ref{equationp}),
\begin{equation*}
c_{pq}^p \alpha_p(p)=0.
\end{equation*}Since $\alpha_p(p)\neq 0$, we have
\begin{equation}
c_{pq}^p=0.
\end{equation}
\end{proof}
\begin{lemma} \label{lemma3}
\begin{equation*}
c_{pq}^k=0
\end{equation*}
for $k\in M^T\backslash\{p,q\}$ such that $\lambda(p)\leq\lambda(k)\leq\lambda(q)$.
\end{lemma}
\begin{proof}
Note that $\alpha_q(k)=0$. Localizing (\ref{star}) at $k$ gives
\begin{equation} \label{equation10}
\sum_{r\in M^T} c_{pq}^r \alpha_r(k)=0.
\end{equation}
If $\lambda(p)=\lambda(k)$, then $c_{pq}^u=0$ for $u\in M^T$ such that $\lambda(u)<\lambda(p)=\lambda(k)$ by Lemma $\ref{lemma1}$. And $\alpha_{k'}(k)=0$ for $k' \in M^T\backslash\{k\}$ such that $\lambda(k')\geq\lambda(k)$. (\ref{equation10}) becomes
\begin{equation}
c_{pq}^k \alpha_k(k)=0.
\end{equation}Since $\alpha_k (k)\neq 0$, we get $c_{pq}^k$=0. By using the same localization method inductively on the set $S'\subset S=\{\lambda(x)\mid x\in M^T\}$ that contains all values between $\lambda(p)$ and $\lambda(q)$, we get the result.
\end{proof}
\begin{lemma} \label{lemma4}
\begin{equation*}
c_{pq}^q=\alpha_p(q)
\end{equation*}for $\lambda(p)\leq\lambda(q)$, where $p,q\in M^T$.
\end{lemma}
\begin{proof}
Localizing (\ref{star}) at $q$ gives
\begin{equation}\label{equation11}
\alpha_p (q)\alpha_q(q)=\sum_{r\in M^T} c_{pq}^r \alpha_r(q).
\end{equation}By Lemma \ref{lemma1}, \ref{lemma2} and \ref{lemma3}, $c_{pq}^k=0$ for all $k\in M^T\backslash\{q\}$ such that $\lambda(k)\leq\lambda(q)$. And $\alpha_{k'}(q)=0$ for all $k'\in M^T$ such that $\lambda(k')>\lambda(q)$. Hence by (\ref{equation11}),
\begin{equation}
\alpha_p(q)\alpha_q(q)=c_{pq}^q \alpha_q(q).
\end{equation}Then we divide both sides by $\alpha_q(q)$, which is non-zero, to get the desired result.
\end{proof}
Next, we will consider the equivariant structure constants $c_{pq}^z$ such that $\lambda(z)=1+\lambda(q)$.
\begin{theorem} \label{theoremz}
By the same notations and assumptions as in Lemma \ref{lemma1},
\begin{equation*}
c_{pq}^z=\frac{\alpha_q(z)}{\alpha_z(z)}(\alpha_p(z)-\alpha_p(q))
\end{equation*}where $\lambda(z)=1+\lambda(q)$.
\end{theorem}
\begin{proof}
Let $z\in M^T$ such that $\lambda(z)=1+\lambda(q)$. Localizing (\ref{star}) at $z$ gives
\begin{eqnarray}
\alpha_p(z)\alpha_q(z)&=&\sum_{r\in M^T} c_{pq}^r \alpha_r(z)\nonumber\\
&=&c_{pq}^q \alpha_q(z)+c_{pq}^z \alpha_z(z).\label{equationz}
\end{eqnarray}The second equality holds because $c_{pq}^k=0$ for $k\in M^T\backslash\{q\}$ such that $\lambda(k)\leq\lambda(q)$ by Lemma \ref{lemma1}, \ref{lemma2} and \ref{lemma3}. Also, $\alpha_{k'}(z)=0$ for all $k'\in M^T\backslash\{z\}$ such that $\lambda(k')\geq\lambda(z)$. By (\ref{equationz}),
\begin{equation}\label{equationz2}
c_{pq}^z=\frac{\alpha_p(z)\alpha_q(z)-c_{pq}^q \alpha_q(z)}{\alpha_z(z)}=\frac{\alpha_p(z)\alpha_q(z)-\alpha_p(q) \alpha_q(z)}{\alpha_z(z)}=\frac{\alpha_q(z)}{\alpha_z(z)}(\alpha_p(z)-\alpha_p(q)).
\end{equation}
\end{proof}
\begin{remark} \label{remark1}
We note that if $(q,z)\notin E$, which means that there is no edge connecting $q$ and $z$ in the moment graph, then $\alpha_q(z)=0$ and hence $c_{pq}^z=0$ by (\ref{equationz2}).
\end{remark}
We will then consider the equivariant structure constants $c_{pq}^y$ such that $\lambda(y)=2+\lambda(q)$.
\begin{definition}
In the directed graph defined in Definition \ref{directedgraph}, define the \emph{negative valency}, $V_p^{-}$, at $p\in V$ by
\begin{equation*}
V_p^{-} =\{v\in V\mid (v,p)\in E\}.
\end{equation*}
Define the positive valency, $V_p^{+}$, at $p\in V$ by
\begin{equation*}
V_p^{+}=\{v\in V\mid (p,v)\in E\}
\end{equation*}
and let $| V_p|$ be the number of elements in $V_p$.
\end{definition}
\begin{definition} \label{shifting}
Let the rank of the torus $T$ be $n$. Let $I\subset\mathbb{Q}[t_0, t_1,..., t_n]$ denote the subring generated by $\alpha_p(q)$ for all $p,q\in M^T$. Define a \emph{shifting operator} $\mathfrak{s}_a^b\colon I\rightarrow I$ by
\begin{equation*}
\mathfrak{s}_a^b(\alpha_p(a))=\alpha_p(b)
\end{equation*}for any $p\in M^T$.
\end{definition}
Note that the definition of $\mathfrak{s}_a^b$ can be extended to the ring of fractions of $I$. Now we are in the right place to state our next result.
\begin{theorem} \label{theoremy}
By the same notations and assumptions as in Lemma \ref{lemma1},
\begin{equation*}
c_{pq}^y=\sum_{i=1}^{|V_y^{-}|} \frac{\alpha_{z_i}(y)}{\alpha_y(y)}(\frac{1}{| V_y^{-}|}\mathfrak{s}_{z_i}^y c_{pq}^{z_i}-c_{pq}^{z_i})
\end{equation*}where $y\in M^T$ such that $\lambda(y)=2+\lambda(q)$ and $z_i$ are the elements in $V_y^-$, for $i=1,2,...,|V_y^-|$.
\end{theorem}
\begin{proof}
Let $y\in M^T$ such that $\lambda(y)=2+\lambda(q)$. Localizing (\ref{star}) at $y$ gives
\begin{equation}\label{equation16}
\alpha_p(y)\alpha_q(y)=\sum_{r\in M^T}c_{pq}^r \alpha_r(y).
\end{equation}Note that $c_{pq}^k=0$ if $\lambda(k)\leq\lambda(q)$ and $k\neq q$. Also, $\alpha_{k'}(y)=0$ if $\lambda(k')\geq\lambda(y)$ and $k'\neq y$. For $z\in M^T$ such that $\lambda(z)=1+\lambda(q)$ but $z\notin V_y^-$, $\alpha_z(y)=0$. Hence, (\ref{equation16}) is simplified as
\begin{eqnarray}
\alpha_p(y)\alpha_q(y)&=& c_{pq}^q \alpha_q(y)+\sum_{z\in V_y^{-}} c_{pq}^z \alpha_z(y)+c_{pq}^y \alpha_y(y)\nonumber\\
&=& \alpha_p(q) \alpha_q(y)+\sum_{z\in V_y^{-}} c_{pq}^z \alpha_z(y)+c_{pq}^y \alpha_y(y). \label{equation17}
\end{eqnarray}By rearranging terms in (\ref{equation17}), we get
\begin{equation} \label{equation18}
c_{pq}^y =\frac{\alpha_p(y)\alpha_q(y)-\alpha_p(q)\alpha_q(y)}{\alpha_y(y)}-\frac{\sum_{z\in V_y^{-}}c_{pq}^z \alpha_z(y)}{\alpha_y(y)}.
\end{equation}Denote the elements in $V_y^-$ by $z_1, z_2,..., z_{|V_y^-|}$. By Theorem \ref{theoremz},
\begin{equation*}
c_{pq}^{z_i}=\frac{\alpha_q(z_i)}{\alpha_{z_i}(z_i)}(\alpha_p(z_i)-\alpha_p(q))
\end{equation*} for all $z_i$ in $V_y^-$. By the shifting operators defined in Definition \ref{shifting}, we have
\begin{equation} \label{equation19}
\mathfrak{s}_{z_i}^y c_{pq}^{z_i}=\frac{\alpha_q(y)}{\alpha_{z_i}(y)}(\alpha_p(y)-\alpha_p(q)) .
\end{equation}By (\ref{equation18}), we have
\begin{equation}
c_{pq}^y =\frac{\alpha_{z_i}(y)}{\alpha_y(y)}\mathfrak{s}_{z_i}^y c_{pq}^{z_i} -\frac{\sum_{i=1}^{| V_y^{-}|} c_{pq}^{z_i} \alpha_{z_i}(y)}{\alpha_y(y)}=\frac{\alpha_{z_i}(y)}{\alpha_y(y)}(\mathfrak{s}_{z_i}^y c_{pq}^{z_i} -\sum_{i=1}^{|V_y^-|} c_{pq}^{z_i} )
\end{equation} for each $z_i\in V_y^-$. Adding all these $| V_y^{-}|$ equations together, and then dividing the sum by $| V_y^{-}|$, we have
\begin{equation} \label{equation21}
c_{pq}^y=\sum_{i=1}^{| V_y^{-}|} \frac{\alpha_{z_i}(y)}{\alpha_y(y)}(\frac{1}{| V_y^{-}|}\mathfrak{s}_{z_i}^y c_{pq}^{z_i}-c_{pq}^{z_i}).
\end{equation}
\end{proof}
\begin{remark} \label{remark2}
If there is no path connecting $q$ and $y$ in $(V,E)$ when $\lambda(y)-\lambda(q)=2$, then $\alpha_q(y)=0$. Under this situation, for $z\in V_y^{-}$, $\alpha_z(y)\neq 0$ but $c_{pq}^z=0$ (see Remark \ref{remark1}) since there does not exist any path connecting $q$ and $z$ in $(V,E)$. Thus, $c_{pq}^z \alpha_z(y)=0$ for all $z\in V_y^-$. By (\ref{equation18}), we can conclude that $c_{pq}^y =0$ if $\sum_q^y$ is an empty set.
\end{remark}
Finally, we will consider the equivariant structure constants $c_{pq}^x$ where $\lambda(x)=3+\lambda(q)$.
\begin{theorem} \label{theoremx}
By the same notations and assumptions as in Lemma \ref{lemma1},
\begin{equation*}
c_{pq}^x=\sum_{y\in V_x^-}\frac{\alpha_y(x)}{\alpha_x(x)}(\frac{1}{|V_x^-|}\mathfrak{s}_y^x c_{pq}^y-c_{pq}^y)+\sum_{z\in M^T\backslash (\{q,x\}\cup V_x^-)}\frac{| V_z^+| -| V_x^-|}{|V_x^-|}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z
\end{equation*}where $x\in M^T$ such that $\lambda(x)=3+\lambda(q)$.
\end{theorem}
\begin{proof}
For $x\in M^T$ such that $\lambda(x)=3+\lambda(q)$, localizing (\ref{star}) at $x$ gives
\begin{equation} \label{equation22}
\alpha_p(x)\alpha_q(x)=\sum_{r\in M^T}c_{pq}^r \alpha_r(x).
\end{equation}Note that $c_{pq}^k=0$ if $\lambda(k)\leq\lambda(q)$ and $k\neq q$. $\alpha_{k'}(x)=0$ if $\lambda(k')\geq\lambda(x)$ and $k' \neq x$. For $y\in M^T$ such that $\lambda(y)=2+\lambda(q)$, the term $c_{pq}^y \alpha_y(x)$ is non-zero only if $y\in V_x^-$. By (\ref{equation22}), we have
\begin{eqnarray}
\alpha_p(x) \alpha_q(x)&=&c_{pq}^q \alpha_q(x)+\sum_{z\in M^T\backslash(\{q,x\}\cup V_x^{-})}c_{pq}^z \alpha_z(x) +\sum_{y\in V_x^{-}} c_{pq}^y \alpha_y(x)+c_{pq}^x \alpha_x(x)\nonumber\\
&=&\alpha_p(q) \alpha_q(x)+\sum_{z\in M^T\backslash(\{q,x\}\cup V_x^{-})}c_{pq}^z \alpha_z(x) +\sum_{y\in V_x^{-}} c_{pq}^y \alpha_y(x)+c_{pq}^x \alpha_x(x). \nonumber
\end{eqnarray}The terms included in the second term on the right hand side can be non-zero only when $z\in V_q^{+}$ and $\sum_z^x$ is a non-empty set. Hence, by rearranging the terms, we get
\begin{equation}\label{equation23}
c_{pq}^x=\frac{\alpha_p(x)\alpha_q(x)-\alpha_p(q)\alpha_q(x)-\sum_{z\in V_q^{+}}c_{pq}^z \alpha_z(x)}{\alpha_x(x)} -\frac{\sum_{y\in V_x^{-}}c_{pq}^y \alpha_y(x)}{\alpha_x(x)}.
\end{equation}For $y\in V_x^{-}$, by (\ref{equation18}),
\begin{equation} \label{equation24}
\frac{\alpha_y(x)}{\alpha_x(x)}\mathfrak{s}_y^x c_{pq}^y=\frac{\alpha_p(x)\alpha_q(x)-\alpha_p(q)\alpha_q(x)-\sum_{z\in V_y^{-}}c_{pq}^z \alpha_z(x)}{\alpha_x(x)}.
\end{equation}The last term in the numerator on the right side of (\ref{equation24}) can be non-zero only when $z\in V_q^{+}$ and $\sum_z^x$ is non-empty. By (\ref{equation23}) and (\ref{equation24}), for each $y\in V_x^{-}$,
\begin{equation} \label{equation25}
c_{pq}^x=\frac{\alpha_y(x)}{\alpha_x(x)} \mathfrak{s}_y^x c_{pq}^y-\sum_{z\in V_q^+\backslash V_y^-}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z - \sum_{y\in V_x^-}\frac{\alpha_y(x)}{\alpha_x(x)}c_{pq}^y.
\end{equation}Before adding up the equations (\ref{equation25}) for each $y\in V_x^-$, let us focus on the second term on the right side of (\ref{equation25}). Since we are only interested in those non-zero terms, we only have to take care of all the terms for those $z\in V_q^+$ when there is at least one path connecting $q$, $z$ and $x$. The simplest case is that $| V_z^+|=1$ for all $z\in V_q^+$. That is, $z$ is only connected to one and only one $y\in V_x^-$. In this case, the sets $V_y^-$ for each $y\in V_x^-$ are all disjoint. It implies that $V_q^+$ is a disjoint union of $V_y^-$ for each $y\in V_x^-$. Then by adding (\ref{equation25}) for all $y\in V_x^-$, we get
\begin{equation} \label{equation26}
|V_x^-| c_{pq}^x=\sum_{y\in V_x^-}\frac{\alpha_y(x)}{\alpha_x(x)}\mathfrak{s}_y^x c_{pq}^y-\sum_{y\in V_x^-} \sum_{z\in V_q^+\backslash V_y^-}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z-| V_x^-| \sum_{y\in V_x^-}\frac{\alpha_y(x)}{\alpha_x(x)}c_{pq}^y.
\end{equation}For the second term on the right side, we have
\begin{eqnarray}
\sum_{y\in V_x^-} \sum_{z\in V_q^+\backslash V_y^-}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z &=&
\sum_{y\in V_x^-}(\sum_{z\in V_q^+}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z-\sum_{z\in V_y^-}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z)\nonumber\\
&=&|V_x^-|\sum_{z\in V_q^+}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z-\sum_{y\in V_x^-}\sum_{z\in V_y^-}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z\nonumber\\
&=&|V_x^-|\sum_{z\in V_q^+}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z-\sum_{z\in V_q^+}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z\nonumber\\
&=& (|V_x^-|-1)\sum_{z\in V_q^+}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z.\label{equation27}
\end{eqnarray}Substitute (\ref{equation27}) into (\ref{equation26}) to get
\begin{equation} \label{equation28}
| V_x^-| c_{pq}^x=\sum_{y\in V_x^-}\frac{\alpha_y(x)}{\alpha_x(x)}\mathfrak{s}_y^x c_{pq}^y+(1-| V_x^-|)\sum_{z\in V_q^+}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z-| V_x^-| \sum_{y\in V_x^-}\frac{\alpha_y(x)}{\alpha_x(x)}c_{pq}^y.
\end{equation}Dividing (\ref{equation28}) by $| V_x^-|$, we get
\begin{equation}
c_{pq}^x=\sum_{y\in V_x^-}\frac{\alpha_y(x)}{\alpha_x(x)}(\frac{1}{|V_x^-|}\mathfrak{s}_y^x c_{pq}^y-c_{pq}^y)+\frac{1 -| V_x^-|}{|V_x^-|}\sum_{z\in M^T\backslash (\{q,x\}\cup V_x^-)}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z.
\end{equation}which is our desired formula (when $|V_z^+|=1$ for all $z\in V_q^+$).
More generally, if $|V_z^+|>1$ for some $z\in V_q^+$, we have to take care of those `excessive edges' coming out of each $z\in V_q^+$. For each of these `excessive edges', we have an extra term $-\alpha_z(x)c_{pq}^z/ \alpha_x(x)$ in (\ref{equation27}). The number of these `excessive edges' for each $z\in V_q^+$ is $| V_z^+|-1$. It means that we have an extra term $-(|V_z^+|-1)\alpha_z(x)c_{pq}^z/ \alpha_x(x)$. Hence (\ref{equation27}) becomes
\begin{eqnarray}
\sum_{y\in V_x^-} \sum_{z\in V_q^+\backslash V_y^-}\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z
&=& \sum_{z\in V_q^+}[(|V_x^-|-1)\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z-(|V_z^+|-1)\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z]\nonumber\\
&=& \sum_{z\in V_q^+}(|V_x^-|-|V_z^+|)\frac{\alpha_z(x)}{\alpha_x(x)}c_{pq}^z.\label{equation30}
\end{eqnarray}Substitute (\ref{equation30}) into (\ref{equation26}) and divide (\ref{equation26}) by $|V_x^-|$ to get the desired formula.
\end{proof}
\section{An example: Complex Projective Space} \label{section3}
A simple example for a compact Hamiltonian-$T$-space is $\mathbb{CP}^n$. The $T$-action is defined by $(t_0,...,t_n).[z_0,...,z_n]=[t_0 z_0,...,t_n z_n]$. The moment polytope is the $n$-simplex. By suitably choosing a generic component of the moment map, we get the Morse function. There are $n+1$ vertices in the moment graph. We label the vertices by $p_0, p_1,..., p_n$ in the ascending order of their indices. $|V_{p_i}^+|=|V_{p_i}^-|=1$ for all $i$ except $i=0$ and $i=n$. By Lemma 3.2 in \cite{Tymoczko}, the classes $\alpha_{p_i}$ defined by $\alpha_{p_i}(p_k)=\prod_{j=0}^{i-1}(t_j-t_k)$ for $i\leq k,i=1,...,n$ can be used as the set of canonical classes for $H_T^\ast(\mathbb{CP}^n)$. Thus, we have $\alpha_{p_{k-1}}(p_k)/ \alpha_{p_k}(p_k)=1/(t_{k-1}-t_k)$. By Theorem \ref{theoremz}, \ref{theoremy} and \ref{theoremx}, we have \begin{equation} \label{equation31}
c_{p_i p_j}^{p_k}=\frac{\mathfrak{s}_{p_{k-1}}^{p_k} c_{p_i p_j}^{p_{k-1}}-c_{p_i p_j}^{p_{k-1}}}{t_{k-1}-t_k}
\end{equation} when $\lambda(p_i)\leq\lambda(p_j)$ and $\lambda(p_k)-\lambda(p_j)=1, 2, 3$.
More generally, for $\lambda(p_k)-\lambda(p_j)>3$, it is straightforward to check that (\ref{equation31}) still holds by the localization method used in the proofs of Theorem \ref{theoremz}, \ref{theoremy} and \ref{theoremx}. Hence we have obtained Theorem 4.1 in \cite{Tymoczko} as a special case of our results.
\begin{remark}
The right side of (\ref{equation31}) is the same as $\partial_{k-1}c_{p_i p_j}^{p_{k-1}}$ where $\partial_{k-1}$ is the \emph{divided difference operator} defined in \cite{Tymoczko}. Divided difference operators are also defined in Kasparov's equivariant $KK$-theory. For the definitions and some interesting applications of divided difference operators in $K$-theory and $KK$-theory, see \cite{Leung1} and \cite{Leung2}.
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,076
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16, April 2015: To be launched on the 15th of April 2015 via Indiegogo, Remote EmoFix is now making selfies better than ever and more enjoyable than it was before under the efficient leadership of Co-Founder Vladimir Malakchi.
The hottest and most anticipated technology of this generation is now available to teach people how to take a selfie in an easier and more fun way. It is in fact an excellent Bluetooth device that will help people in capturing their photos as well as videos conveniently via a selfie remote.
Using EmoFix Bluetooth selfie remote, the user doesn't have to touch the phone anymore and it can also be used in wet locations since it has a water-proof, metal body selfie gadget. The project will be represented via Indiegogo to make life easier.
EmoFix talented team has successfully made this selfie shooter through hard work and skills while keeping in their minds the needs of mobile users around the world. Now, taking a group picture or best selfies is convenient and comfortable with a selfie device.
What so good about the selfie app for android is that it can last up to 2 years and more in terms of battery life, range from 30 feet and more, water resistant and made with durable hardware. The best selfie remote will therefore give people complete freedom in capturing videos and photos without the need to touch their mobile devices.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 135
|
namespace boost {
namespace asio {
namespace detail {
class scheduler_operation;
// Base class for all tasks that may be run by a scheduler.
class scheduler_task
{
public:
// Run the task once until interrupted or events are ready to be dispatched.
virtual void run(long usec, op_queue<scheduler_operation>& ops) = 0;
// Interrupt the task.
virtual void interrupt() = 0;
protected:
// Prevent deletion through this type.
~scheduler_task()
{
}
};
} // namespace detail
} // namespace asio
} // namespace boost
#include <boost/asio/detail/pop_options.hpp>
#endif // BOOST_ASIO_DETAIL_SCHEDULER_TASK_HPP
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,190
|
Q: O meu programa continua a ir abaixo Não sei o porquê disto acontecer mas necessito de ajuda para perceber o porquê de o programa estar a crashar. O meu programa pretende usar o algoritmo de Kruskal para encontrar os caminhos mais leves entre cidades (aeroportos e estradas). Para isto, cria uma grafo não dirigido que liga os vértices com os arcos atribuidos.
Código completo:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
// Uma estrutura para representar um arco pesado no grafo.
struct Edge {
int src, dest, weight;
};
// Uma estrutura para representar um grafo ligado, não dirigido e pesado.
struct Graph {
// V -> Número de vértices (Número de cidades), E -> Número de arcos (Número de estradas + conecções por aeroportos).
int V;
int E;
// O grafo é representado como um array de arcos.
// Visto o grafo ser não dirigido, o arco
// da origem (src) ao destino (dest) é igual
// ao arco de dest a src. Ambos são contados como 1 arco.
struct Edge* edge;
};
// Cria um grafo com V vértices e E arcos.
struct Graph* createGraph(int V, int E)
{
struct Graph* graph;
graph->V = V;
graph->E = E;
graph->edge = (struct Edge*)malloc(E * sizeof(struct Edge));
return graph;
};
// Uma estrutura para representar um subconjunto para as funções "find" e "Union".
struct subset {
int parent;
int rank;
};
// Função que procura pelo nº de vezes que o elemento i aparece.
int find(struct subset subsets[], int i)
{
// Encontra a raíz e torna a raíz o predecessor de i.
if (subsets[i].parent != i)
subsets[i].parent
= find(subsets, subsets[i].parent);
return subsets[i].parent;
}
// Função que une os conjuntos x e y.
void Union(struct subset subsets[], int x, int y)
{
int xroot = find(subsets, x);
int yroot = find(subsets, y);
// Agrega a árvore com rank pequeno sob a raíz da árvore com rank maior (Union by Rank).
if (subsets[xroot].rank < subsets[yroot].rank)
subsets[xroot].parent = yroot;
else if (subsets[xroot].rank > subsets[yroot].rank)
subsets[yroot].parent = xroot;
// Se os ranks forem os mesmos, tornar um deles na raíz e incrementar a sua rank por 1.
else
{
subsets[yroot].parent = xroot;
subsets[xroot].rank++;
}
}
// Compara 2 arcos de acordo com os pesos.
// Usado na função "qsort" para ordenar o array de arcos.
int myComp(const void* a, const void* b)
{
struct Edge* a1 = (struct Edge*)a;
struct Edge* b1 = (struct Edge*)b;
return a1->weight > b1->weight;
}
// Função principal para construir a MST usando o algoritmo de Kruskal.
void KruskalMST(struct Graph* graph)
{
int V = graph->V;
struct Edge
result[V]; // Guarda a MST resultante.
int e = 0; // Variável de índice, usada para o result[].
int i = 0; // Variável de índice, usada para arcos ordenados.
// 1º pass: Ordenar todos os arcos por ordem crescente dos pesos.
// Se não podemos alterar o grafo dado, copiamos o array de arcos original.
qsort(graph->edge, graph->E, sizeof(graph->edge[0]),
myComp);
// Alocar memória para criar V subconjuntos.
struct subset* subsets
= (struct subset*)malloc(V * sizeof(struct subset));
// Criar V subconjuntos com 1 só elemento.
for (int v = 0; v < V; ++v) {
subsets[v].parent = v;
subsets[v].rank = 0;
}
// Número total de arcos possível = V-1.
while (e < V - 1 && i < graph->E) {
// 2º passo: Escolher o arco mais leve.Pick the smallest edge.
// Incrementar o índice para a próxima iteração.
struct Edge next_edge = graph->edge[i++];
int x = find(subsets, next_edge.src);
int y = find(subsets, next_edge.dest);
// Se a inclusão do arco não causa um ciclo, incluí-lo no result [] e,
// incrementar o índice do result[] para o arco seguinte.
if (x != y) {
result[e++] = next_edge;
Union(subsets, x, y);
}
// Senão, descartar o arco.
}
printf("Arcos da MST:\n");
printf("V1 V2 Custo\n");
int minimumCost = 0;
int nRoads = 0;
int nAirports = 0;
for (i = 0; i < e; ++i)
{
printf("%d -- %d == %d\n", result[i].src,
result[i].dest, result[i].weight);
if (result[i].src == 0 || result[i].dest == 0) {
nAirports++;
}
else {
nRoads++;
}
minimumCost += result[i].weight;
}
printf("Minimum Spanning Tree com custo minimo: %d\n",minimumCost);
printf("Numero de aeroportos: %d\n",nAirports);
printf("Numero de estradas: %d",nRoads);
return;
}
int main()
{
int V = 0; // Número de vértices(cidades) no grafo.
int A = 0; // Número de aeroportos.
int e = 0; // Número de estradas no grafo.
int E = 0; //Númeto total de arcos no grafo.
int cidade, aeroporto, cidade1, cidade2, custo = 0;
printf("Introduza o numero de cidades: \n");
scanf("%d", &V);
printf("Introduza o numero de aeroportos: \n");
scanf("%d", &A);
printf("Introduza o numero de estradas: \n");
scanf("%d", &e);
E = A + e;
struct Graph* graph = createGraph(V, E);
for (int i = 0; i < A; i++) {
printf("Introduza o custo do aeroporto: \n");
scanf("%d %d", &cidade, &aeroporto);
graph->edge[i].src = cidade;
graph->edge[i].dest = 0; // vértice "céu"
graph->edge[i].weight = aeroporto;
}
for (int j = A; j < A + E; j++) {
printf("Introduza o custo da estrada: \n");
scanf("%d %d %d", &cidade1, &cidade2, &custo);
graph->edge[j].src = cidade1;
graph->edge[j].dest = cidade2;
graph->edge[j].weight = custo;
}
KruskalMST(graph);
return 0;
}
A: João veja este trecho aqui
struct Graph* createGraph(int V, int E)
{
struct Graph* graph;[[(struct Edge*)malloc(E*sizeof(struct Edge))]aqui também]
graph->V = V;
graph->E = E;
graph->edge = (struct Edge*)malloc(E * sizeof(struct Edge));
return graph;
};
o struct Graph * também é um ponteiro, se você não allocar com malloc aqui no começo também, o porgrama não sabe aonde aramazenar grapho->V e ->E... e qualquer outro valor adiante.
Se usar linux, compile com a flag gcc -g ... e depois da gdb ./file.out para debugar.
b numeor_da_linha (break point)
disp nome_da_variavel - para ver a variavel
n (para seguir para proxima instrucao)
run (para começar)
quit (para sair)
help (manual)
São as instruções mais usadas.
Se estiver em outro sistema e precisar testar rapido tem este site abaixo:
https://www.onlinegdb.com/
lá voce clica na linha, define os break points executa com modo de depuração, da run e vai testando linha a linha. Do lado direito ficam as variaveis, para codigos mais curtos é bem conveniente para testar ai e uma mão na roda.
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{
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The best hotels in Phuket offer their guests a unique ambiance of opulence and beauty in this warm and stunning Tropical Island; with its perfect and relaxing weather combined with the picturesque views of its amazing beaches and a plethora of excellent attractions, Phuket is definitely a paradise that is best nicknamed as the Pearl of the Andaman Sea.
Whether you are staying at a peaceful and tranquil resort or one of the finest and most grandiose accommodations around, Phuket hotels will definitely deliver the best and most memorable stay for everyone who visits the island. Most of their hotels display a warm and tranquil ambiance by exhibiting fresh flower displays, excellent amenities, original artworks and a lot more; there are also cocktail lounges and cafes featured so guests will surely enjoy their stay in these hotels.
Trisara in Sanskrit means "The Garden in the Third Heaven" and it is known as Phuket Island's resort that gives the most special and exceptional experience; Trisara is also known as one of the most favored Phuket hotels around, which is why a large number of tourists stay in this stunning accommodation. The hotel is surrounded by a lush tropical forest and alluring gardens that give off a charming ambiance that one can easy relax in. The hotel also features forty-eight spacious pool villas that have picturesque views of the ocean, as well as private residences with two to six bedrooms that are perfect for a big group visiting the area. Another nice thing about the hotel is that it is only fifteen minutes away from the Phuket Airport.
The Andara Resort and Villas will give you an experience of extreme luxury and grandeur; it can be found amid a rich tropical environment while overlooking the Andaman Sea on Phuket's Central West coast. With its excellent sea views, outstanding resort facilities, as well as the available infinity pools, Andara is definitely the perfect choice of Phuket hotels for an unforgettable and relaxing vacation.
The Shore at Katathani is an assortment of villas along the beach that can be found at the southernmost end of Kata Noi Beach. It is a breathtaking location that was created perfectly for those peace lovers who want to escape the chaos of the city. This is also one of the best hotels in Phuket for those who admire the combination of the timeless beach, its luxury and the tranquility it brings.
The JW Marriott Phuket Resort and Spa is one of the best hotels in Phuket to stay in for an active and exciting holiday, no matter what the weather condition is. It is quite intriguing how the hotel has kept a personal touch on such extensive grounds, yet it still brings about a welcoming surprise and charming mystery to all.
Anantara Mai Khao Phuket Villa is kept hidden in the charming and stunning island getaway which is a luxurious haven that extends itself to a pure environment. Anantara Mai Khao is also known as one of the best hotels in Phuket that places a lot of significance on unparalleled relaxation, utmost comfort and privacy for the guests; it also features ninety-one private Phuket pool villas that are dotted perfectly across a massive stretch of beach.
The Marina Phuket Resort is also one of the favorite hotels in Phuket that rest perfectly on a headland that is located south of Karon; it is also famous for being one of the finest west coast beaches in Phuket, reason why a lot of tourists stay in the resort during their visit. Much of this lovely hotel is hidden amid lush and verdant sceneries combined with a web of wooden pathways that provide easy of access to the bungalows, as well as the sea. The rooms that are available along the seaside exhibit breathtaking views of the vistas of the Andaman Sea and the beach; plus, it also features striking views of the sunset. Also, the available jungle-themed rooms offer excellent views of the forest and little peeks of the poolside area – the place being one of the most appealing poolside areas on Phuket.
This beach resort can be found amid the charming tropical lagoons and the azure-like water of the Andaman Sea. Its property exhibits some of the finest and most chic accommodations that have all been renovated with contemporary interiors with hints of modern Thai themes. The Outrigger Laguna resort is situated in a convenient location on the famous Bangtao beach where timeless white sand and clear, tranquil water combines perfectly with its surroundings, creating an idyllic and perfect tropical setting that most guests love. Like a lot of Phuket hotels, this beach resort has an excellent view of the beachfront which means that you can easily access the beach for numerous activities for the family, friends, and couples too.
The BYD Lofts is famed for being one of the most expansive Phuket hotels which are actually serviced apartments along the heart of the Patong Beach. These grand and colossal apartments were created specifically for living, and not just for sleeping; it features exquisite themes that display a lavish lifestyle that is close to numerous locations such as restaurants, convenience stores, bars, shopping areas, as well as entertainment spots. They also provide stylish and fully-furnished accommodations that are sized just like an apartment but with a hotel-styled ambiance.
COMO Point Yamu is one of the newest luxury hotels in Phuket that can be located right at the peak of the Yamu cave. It overlooks the enchanting Andaman Sea, as well as the dramatic limestone of Phang Nga Bay. The hotel features lovely interiors created by an Italian designer and offers a unique display of modern Thai luxury in combination with the COMO Shambhala wellness spa, and also two other top notch dining spots that serve a variety of authentic Thai and Italian dishes.
Hyatt Regency Phuket Resort is one of the newest additions to the collection of luxury hotels in the island of Phuket. It can be found situated on a hill on the Millionaires Mile which is a sheltered area along the southern tip of Kamala Beach. The beach resort features a contemporary theme plus exquisite views that overlook the Andaman Sea, as well as Phuket's west coast luxury hills.
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Home Magazine 2019 Spring Alumnus of Point Loma (APL) Award Winner: Danielle Cervantes Stephens
Alumnus of Point Loma (APL) Award Winner: Danielle Cervantes Stephens
Journalist, professor, nonprofit writer, student — Danielle Cervantes Stephens (00) has a career that is still evolving. In each stage and place, her intellect, faith, and work ethic have stood out.
In 2006, Cervantes won a Pulitzer for her team's journalism work at the San Diego Union Tribune. In 2007, she added adjunct professor to her resume. By 2011, she had moved from daily journalism work to writing and researching for nonprofits while teaching. Today, in addition to those pursuits, she is working on a master's degree in French literature in preparation for pursuing her Ph.D. in comparative literature.
As an undergrad at PLNU, Cervantes studied literature and political science.
"Literature taught me to understand how people think, and political science taught me how people behave and how to work with people," she said.
Cervantes also minored in women's studies and took many courses in French, giving her a rich and diverse liberal arts background.
"Another super influential thing that happened was in 1999, I went to Israel over Christmas break with Mary Conklin and Bob Smith," she said. "I re-found my faith there, and it just changed me. I had been raised Nazarene, but I learned new things that inspired me."
The summer between her sophomore and junior years at PLNU, Cervantes was barely making rent when she landed a much-needed temporary job at the San Diego Union Tribune. She impressed enough over the next several summers that she was offered a regular position in April 2000. She started out as a researcher and librarian before being promoted to senior reporter specializing in investigations and data journalism.
"Literature taught me to understand how people think, and political science taught me how people behave and how to work with people."
As her writing career unfolded, Cervantes saw her role as more than simply storytelling.
"I am a sixth generation San Diegan, so when I was working on [certain] stories, it was about doing better for the taxpayers," she said.
One example was the story set that won the 2006 Pulitzer Prize for National Affairs Reporting — the story of allegations of bribery and fraud against Rep. Duke Cunningham. Another story where Cervantes felt she was able to serve the public came in 2008 when she and her partner covered the city's rebuilding efforts after the Witch Creek and Guejito wildfires.
"We really dug in after the 2007 fires because the City of San Diego tried to clean that up, and they did a horrible job with the contractors," she said. "We triggered a federal criminal investigation. The city settled and a few million dollars went back to San Diego taxpayers."
In 2007, Cervantes and her colleague were removed from City Hall for "civil disobedience" while they were attempting to view public records. The year 2007 was also when Cervantes married Dave Stephens and when she began teaching as an adjunct professor at PLNU.
As a professor, she strives to impact the lives of her own students the way her PLNU professors influenced her.
"Teaching isn't the right word," she said. "I'm trying to raise this generation of journalists. That's why I call them my 'watchpups' instead of 'watchdogs.' It's this little family I have tried to develop over the years."
"I'm trying to raise this generation of journalists. That's why I call them my 'watchpups' instead of 'watchdogs.' It's this little family I have tried to develop over the years."
In addition to inspiring her students to good work, Cervantes has a desire to help students struggling with mental health issues. Cervantes was diagnosed with bipolar disorder at age 40 and has experienced PTSD and OCD in response to traumas from her past. Her mental illness has been one of the greatest challenges of Cervantes' life. But it means she has great empathy to share with others who are suffering.
"I know I am created the way I am for a reason," she said.
The professors who saw her, cared about her, and supported her during college made a tremendous difference — she goes so far as to say that PLNU saved her life. Now she is committed to helping others.
Whether in the classroom as student or professor, writing, or serving at her church, Cervantes uses her education and faith to love and serve others.
Related Article: APL award winner Jenny Harris' work to bring self-sustaining food sources to people across the world.
"How Are You Single?" And Why That's the Wrong Thing to Ask
In Praise of a Nonelite Education
The Gift of Singleness
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Il Gran premio della giuria: U.S. Dramatic è un premio assegnato dalla giuria del Sundance Film Festival al miglior film statunitense candidato nella sezione competitiva del concorso.
Vincitori
1985 - Blood Simple - Sangue facile, regia di Joel ed Ethan Coen
1986 - La prima volta, regia di Joyce Chopra
1987 - Dimensioni parallele, regia di Gary Walkow; Waiting for the Moon, regia di Jill Godmilow
1988 - Heat and Sunlight, regia di Rob Nilsson
1989 - True Love, regia di Nancy Savoca
1990 - Chameleon Street, regia di Wendell B. Harris Jr.
1991 - Poison, regia di Todd Haynes
1992 - In the Soup - Un mare di guai, regia di Alexandre Rockwell
1993 - Public Access, regia di Bryan Singer; Ruby in Paradise, regia di Victor Nuñez
1994 - What Happened Was..., regia di Tom Noonan
1995 - I fratelli McMullen, regia di Edward Burns
1996 - Fuga dalla scuola media, regia di Todd Solondz
1997 - Sunday, regia di Jonathan Nossiter
1998 - Slam, regia di Marc Levin
1999 - Three Seasons, regia di Tony Bui
2000 - Girlfight, regia di Karyn Kusama; & Conta su di me, regia di Kenneth Lonergan
2001 - The Believer, regia di Henry Bean
2002 - Personal Velocity - Il momento giusto, regia di Rebecca Miller
2003 - American Splendor, regia di Shari Springer Berman e Robert Pulcini
2004 - Primer, regia di Shane Carruth
2005 - Forty Shades of Blue, regia di Ira Sachs
2006 - Non è peccato - La quinceañera, regia di Richard Glatzer e Wash Westmoreland
2007 - Padre Nuestro, regia di Christopher Zalla
2008 - Frozen River - Fiume di ghiaccio, regia di Courtney Hunt
2009 - Precious, regia di Lee Daniels
2010 - Un gelido inverno (Winter's Bone), regia di Debra Granik
2011 - Like Crazy, regia di Drake Doremus
2012 - Re della terra selvaggia (Beasts of the Southern Wild), regia di Benh Zeitlin
2013 - Prossima fermata Fruitvale Station (Fruitvale Station), regia di Ryan Coogler
2014 - Whiplash, regia di Damien Chazelle
2015 - Quel fantastico peggior anno della mia vita (Me & Earl & the Dying Girl), regia di Alfonso Gomez-Rejon
2016 - The Birth of a Nation - Il risveglio di un popolo (The Birth of a Nation), regia di Nate Parker
2017 - I Don't Feel at Home in This World Anymore, regia di Macon Blair
2018 - La diseducazione di Cameron Post, regia di Desiree Akhavan
2019 - Clemency, regia di Chinonye Chukwu
2020 - Minari, regia di Lee Isaac Chung
2021 - CODA, regia di Sian Heder
2022 - Nanny, regia di Nikyatu Jusu
Collegamenti esterni
Premi del Sundance Film Festival
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Big Lots (NYSE:BIG) disclosed a 30% slide in fourth-quarter profits on Friday, but the company's earnings, revenue and share buyback plans impressed Wall Street.
Slumping shares of the close-out retailer surged more than 15% on the stronger-than-expected earnings.
Big Lots said it earned $84.4 million, or $1.45 a share, last quarter, compared with a profit of $120.3 million, or $2.09 a share, a year earlier. Analysts had called for EPS of $1.40.
Revenue fell 6.2% to $1.64 billion, narrowly topping the Street's view of $1.61 billion. U.S. net sales slumped 7.3% to $1.57 billion. Same-store sales declined 3%.
Gross margins dipped to 38.2% from 39.7%, but still exceeded consensus calls for 37.9%.
Big Lots also said its board of directors signed off on a fresh $125 million share buyback program. The plan is set to go into effect on March 11.
Looking ahead, Big Lots projected 2014 non-GAAP EPS of $2.25 to $2.45, compared with the Street's view of $2.40. Same-store sales are seen flat to up 2%.
For the current quarter, the company estimates non-GAAP EPS of 40 cents to 45 cents, compared with 70 cents the year before. Analysts had been modeling for first-quarter EPS of 50 cents. Same-store sales are seen ranging between slightly positive to slightly negative.
Meanwhile, Big Lots announced it has added the title of executive vice president to Timothy Johnson, who has served as chief financial officer since 2012.
Shares of Columbus, Ohio-based Big Lots raced 16.34% higher to $34.03 ahead of Friday's opening bell. The rally positions Big Lots to wipe out its 2014 slump of 9%.
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license: >
Licensed to the Apache Software Foundation (ASF) under one
or more contributor license agreements. See the NOTICE file
distributed with this work for additional information
regarding copyright ownership. The ASF licenses this file
to you under the Apache License, Version 2.0 (the
"License"); you may not use this file except in compliance
with the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing,
software distributed under the License is distributed on an
"AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
KIND, either express or implied. See the License for the
specific language governing permissions and limitations
under the License.
---
# Файл config.xml
Многие аспекты поведения приложения могут контролироваться с помощью глобального конфигурационного файла `config.xml`. Этот платформо-зависимый XML-файл основан на спецификации W3C [Упакованые веб-приложения (Packaged Web Apps)][1] и расширен для указаниях основных функций Cordova API, плагинов и настроек специфичных для платформы.
[1]: http://www.w3.org/TR/widgets/
Для проектов, созданных с использованием Cordova CLI (описанном в разделе Интерфейс командной строки) этот файл можно найти в директории верхнего уровня:
app/config.xml
Обратите внимание, что до версии 3.3.1-0.2.0, файл располагался в `app/www/config.xml`, и что размещение файла в этом месте по-прежнему поддерживается.
При использовании командной строки для построения проекта, версии этого файла копируются без изменений в различные подкаталоги `platforms/`, например:
app/platforms/ios/AppName/config.xml
app/platforms/blackberry10/www/config.xml
app/platforms/android/res/xml/config.xml
Этот раздел описывает параметры глобальной и кросс платформенной конфигурации. В следующих разделах для платформо-зависимых параметров:
* Конфигурация iOS
* Конфигурация Android
* Конфигурации BlackBerry 10
В дополнение к различным конфигурационным параметрам описанным ниже, можно также настроить основной набор изображений для приложения для каждой целевой платформы. Смотрите раздел Иконки и Заставки для дополнительной информации.
## Основные элементы конфигурации
Этот пример показывает значения по умолчанию в файле `config.xml` сформированном командой CLI `create`, описанной в разделе Интерфейс командной строки:
<widget id="com.example.hello" version="0.0.1">
<name>HelloWorld</name>
<description>
A sample Apache Cordova application that responds to the deviceready event.
</description>
<author email="dev@callback.apache.org" href="http://cordova.io">
Apache Cordova Team
</author>
<content src="index.html" />
<access origin="*" />
</widget>
Следующие элементы конфигурации появляются в файле верхнего уровня `config.xml` и поддерживаются на всех существующих платформах Cordova:
* Атрибут `id` элемента `<widget>` указывает идентификатор приложения в обратном формате доменных имен и атрибут `version` его полный номер версии в нотации майор/минор/патч.
* Элемент `<name>` определяет официальное имя приложения, как он отображается на главном экране устройства и в интерфейсе магазина приложений.
* Элементы `<description>` и `<author>` определяют метаданные и контактную информацию, которые могут отображаться в каталоге магазина приложений.
* Необязательный элемент `<content>` определяет стартовую страницу приложения в каталоге верхнего уровня веб ресурсов. Значением по умолчанию является `index.html` , которая обычно находится в каталоге верхнего уровня `www` проекта.
* Элементы `<access>` определяют набор внешних доменов, с которым приложение имеет право взаимодействовать. Значение по умолчанию, показанное выше позволяет осуществлять доступ к любому серверу. Смотрите раздел Руководство по разрешению доступа к доменам для подробностей.
* Элемент `<preference>` задает различные параметры как пару атрибутов `name`/`value`. Имя каждого параметра, указанного в атрибуте `name` указывается без учета регистра. Многие параметры являются уникальными для конкретных платформ, как это указано в начале этой страницы. В следующих разделах подробно описаны настройки, которые применяются к более чем одной платформе.
## Глобальные настройки
Следующие глобальные настройки применяются для всех платформ:
* `Fullscreen` позволяет скрыть строку состояния в верхней части экрана. Значение по умолчанию — `false` . Пример:
<preference name="Fullscreen" value="true" />
* `Orientation` позволяет заблокировать поворот приложение при изменении положения устройства. Возможные значения `default`, `landscape`, или `portrait` . Пример:
<preference name="Orientation" value="landscape" />
**Примечание**: `default` значение означает что разрешены *обе* ориентации, альбомная и портретная. Если вы хотите использовать настройки по умолчанию для каждой платформы (обычно только для портретная ориентация), удалите этот элемент из файла `config.xml`.
## Много-платформенный настройки
Следующие параметры применяются для более чем одной платформы, но не ко всем из них:
* `DisallowOverscroll` (логическое значение, по умолчанию `false` ): Установите в `true` если вы не хотите чтобы интерфейс отображал каких-либо обратной связи, когда пользователи прокручивают за начало или конец содержимого.
<preference name="DisallowOverscroll" value="true"/>
Применяется к Android и iOS. На iOS, совершение перелистывание за границу контента (overscroll) плавно возвращает положение контента назад в исходное положение. На Android это действие производит более тонкий светящийся эффект вдоль верхнего или нижнего края контента.
* `BackgroundColor`: Задайте цвет фона приложения. Поддерживает шестнадцатеричное значение размером 4 байта, где первый байт представляющий альфа-канал и стандартные значения RGB для следующих трех байтов. В этом примере указывается голубой цвет:
<preference name="BackgroundColor" value="0xff0000ff"/>
Применяется к Android и BlackBerry. Переопределяет CSS, который доступен для *всех* платформ, например:`body{background-color:blue}`.
* `HideKeyboardFormAccessoryBar`(логическое значение, по умолчанию `false` ): Установите в `true` чтобы скрыть дополнительную панель инструментов, которая появляется над клавиатурой, помогая пользователям перемещаться из одной формы ввода на другой.
<preference name="HideKeyboardFormAccessoryBar" value="true"/>
Применяется к iOS и BlackBerry.
## Элемент *feature*
При использовании командной строки для построения приложений, вы используете команду `plugin`, чтобы включить API устройства. Это не изменяет файл `config.xml` верхнего уровня, так что элемент `<feature>` не применяется к вашему рабочему процессу. Если вы работаете непосредственно в SDK и используете платформо-зависимый файл `config.xml` в качестве источника, вы используете элемент `<feature>`, чтобы включить API функции устройства и внешних плагинов. Эти элементы обычно присутствуют с разными значениями в платформо-зависимых файлах `config.xml`. К примеру, таким образом можно указать Device API для проектов Android:
<feature name="Device">
<param name="android-package" value="org.apache.cordova.device.Device" />
</feature>
Вот как этот элемент появляется в проектах iOS:
<feature name="Device">
<param name="ios-package" value="CDVDevice" />
</feature>
Смотрите подробную информацию о том, как определить каждую функции в разделе Справочник API. Также смотрите Руководство по разработке плагинов для получения дополнительной информации по плагинам.
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ACP Summit in Port Moresby may result in more partnerships and investments
With the recent conclusion of the ACP Summit held at Port Moresby, 79 leaders from the African, Caribbean and Pacific countries have gathered to discuss lessons learned from each of their respective areas. Possible alliances and collaborative efforts were also topics of discussion for future developments. But the summit held particular interest for developing an "earliest adoption of a United Nations General Assembly Resolution to develop an international legal framework governing human rights and climate change" according to the recently published In Depth News article that came out online.
Source image: news.pngfacts.com
This comes with the leaders also seeking a joint conference between them and the UN, along with international financial entities and the European Commission. This is in line with their longing for an "intensification of South-South and Triangular Cooperation to build the productive capacities of ACP countries," to which, the UN states that they are open to do. Roy Trivedy who is an attendee of the said conference and who is the United Nations Resident Coordinator also confirms its support for the said countries, most especially for Papua New Guinea pushing for sustainable development goals. He says that the Papua New Guinea-based UN country team is in full support of this.
Though focused in the regard of human rights and the effects of climate change, the ACP also discussed issues from different sectors as well that were mentioned in the statement made by Papua New Guinea Prime Minister Peter O'Neill where he says, "the Summit adopted the Port Moresby Declaration that captures the main outcomes of the leaders' discussions, and the Waigani Communiqué that sets the way forward for the future of the ACP Group beyond the expiry of the ACP-EU Cotonou Partnership Agreement in 2020… I believe this Summit has inspired us to take the giant steps in our approach that will shape and transform the ACP Group into a more dynamic force."
Improved Governance to Drive more Investments and Improve Citizens' Lives
While it wasn't directly mentioned, the Waigani Communiqué and the Port Moresby Declaration where both documents and declarations communicate plans to improve the lives of its citizens through efficient and proper governance that will promote trade and investments where it is most needed, particularly for the entrepreneurship, technology, private sector building capacity and technological sectors. This could mean more infrastructure and real estate policies and construction.
There were also discussions for achieving financial stability and self-sufficiency through a long-term endowment fund and venturing into solutions through a collaboration of both the public and private sectors.
The nearing expiration of the ACP-EU Cotonou agreement or partnership is also a growing concern. But members of the ACP countries will remain persistent in trying to enhance and renew this agreement, while at the same time negotiating an improved agreement starting in 2018. The Summit will focus on issues that are of interest to both delegations and what has been gained by both the ACP and EU during this time as part of its negotiation.
The benefits and focus of the financial partnership
But with the approach of the expiration of the ACP-EU Cotonou agreement, there are also more opportunities that come as a form of support that may help alleviate poverty, issues with infrastructure and homelessness which is a big issue right now in Papua New Guinea. In an interview with The Financial, Marcelo Minc who is the ADB Country Director in Papua New Guinea states that it has "been engaged with PNG for 45 years and in that time the partnership has rehabilitated transport infrastructure, improved air safety, provided basic health services to the rural population, extended financial services to the unbanked, and removed barriers to jobs and business creation."
This statement signals ADB's renewed and continued commitment to Papua New Guinea where the cumulative ADB assistance to PNG has increased from $1.27 billion last 2009 to roughly $2.13 billion last year. This is largely due to the different projects that were put in place during the 45 years of financial assistance by ADB which reached $1.04 billion in 2015.
This financial partnership will focus on infrastructure in the provinces through their local governments who play a key role in the implementation of projects. This will allow new opportunities for investment and employment that can possibly assist poor families into a better future. And though the primary focus of this assistance was for transport upgrades and the construction of infrastructural developments, ADB hopes that this may also provide better opportunities for health services, banking services and also a more suitable environment for those who wish to do business in Papua New Guinea. This could mean better conditions for those who are also wishing to go into the property market.
With the continued efforts of the government and partnering countries and institutions around the world, Papua New Guinea is definitely bound to see better years ahead. It's only a matter of good governance and time that will tell how everything will unfold. But this is surely an exciting time for Papua New Guinea.
Subscribe to our Newsletter to receive news and updates about New Developments, Free Land Titles News, Real Estate & Housing Demand trends and other programmes to help PNG home buyers!
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|
\section{Introduction}
According to the Riemann uniformization theorem, there exists
only three conformal types of simply connected Riemannian surfaces,
namely
$$
\begin{array}{c}
S^2\\[6pt]
K=1
\end{array}
\qquad
\qquad
\begin{array}{c}
\mathbb{R}^2\\[6pt]
K=0
\end{array}
\qquad
\qquad
\qquad
\begin{array}{c}
H^2\\[6pt]
K=-1.
\end{array}
$$
In the Lorentz case considered in this paper, the relevant geometry is the
so-called ``fourth'' geometry of Poincar\'e \cite{Poin} as opportunely
mentioned in \cite{KS2}, i.e., the Lorentz geometry of the hyperboloid of
one sheet
$$
\begin{array}{c}
H^{1,1}
\\[6pt]
K=\pm1.
\end{array}
$$
\begin{quote}
``\textsc{La quatri\`eme g\'eom\'etrie.} ---
Parmi ces axiomes implicites, il en est un qui semble m\'eriter
quelque attention, parce qu'en l'abandonnant, on peut construire une
quatri\`eme g\'eom\'etrie aussi coh\'erente que celle d'Euclide, de
Lobatchevsky et de Riemann.
[\dots]
Je ne citerai qu'un de ces th\'eor\`emes et je ne choisirai pas le plus
singulier~: \textit{une droite r\'eelle peut \^etre perpendiculaire \`a
elle-m\^eme}.''
\begin{flushright}
Henri Poincar\'e
La science et l'hypoth\`ese (1902)
\end{flushright}
\end{quote}
Let us, nevertheless, emphasize that a Lorentz uniformization
theorem is still not available, as of today---the problem lying
in the classification of the conformal boundaries \cite{Kul,Wei}.
This study has been triggered by previous work of Kostant and
Sternberg \cite{KS1,KS2} who first pointed out an intriguing relationship
between the Schwarzian derivative of a diffeomorphism of null infinity
$\mathbb{T}$ of the Lorentz hyperboloid $H^{1,1}$ and the transverse Hessian
of the conformal factor associated with this diffeomorphism (viewed as a
conformal transformation of~$H^{1,1}$). We contend that this
correspondence stems from a particular geometric object, namely the
cross-ratio as a four-point function associated with the canonical
projective structure of the projective line.
Such an observation prompted us to further investigate
the relationship between (i) the conformal geometry of the hyperboloid of
one sheet~$H^{1,1}$ and (ii) the Virasoro group, $\mathop{\rm Vir}\nolimits$.
Our contribution has therefore consisted in identifying several conformal
classes of Lorentz metrics on $H^{1,1}\cong\mathbb{T}^2-\Delta$ within the space
of projective structures on $\Delta\cong\mathbb{T}$, i.e., the (regular) dual of
$\mathop{\rm Vect}\nolimits(\bbT)$ \cite{Kiri}. In doing so, we have been able to give an
explicit, yet non standard, realization of the generic coadjoint orbits
\cite{Kiri,KY,Wit,Guieu1,Guieu2} of the Virasoro group in the framework of
$2$-dimensional real conformal geometry. Note that Iglesias
\cite{Igl} has also obtained other realizations of such orbits in quite a
different context.
\medskip
The paper is organized as follows.
\begin{itemize}
\item
Section \ref{TheLorentzHyperboloidOfOneSheet} describes in various ways
the Lorentz cylinder $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta$ and its
associated conformal structure for special projective
structures of null infinity, i.e. the circle $\mathsf{S}$.
\item
In Section \ref{TheSchwarzianDerivative}, we briefly introduce the
Schwarzian $1$-cocycle $\mathbf{S}$ of $\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$, while in Section
\ref{ConformalTransformations}, we recall the Kostant-Sternberg Theorem
\cite{KS2} and the basic notions attached to conformal Lorentz
structures on surfaces.
\item
Our main results are presented in Section
\ref{SymplecticStructuresOnConformalClassesOfMetrics} where special,
infinite-dimensional, conformal classes of metrics $\mathop{\textrm{\rm g}}\nolimits$ on $\mathcal{H}$
are shown to be symplecto\-morphic to coadjoint orbits of the
group~$\mathop{\rm Vir}\nolimits$---central extension of $\mathop{\rm Conf}\nolimits_+(\mathcal{H})\cong\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$. The
$\mathop{\rm Conf}\nolimits_+(\mathcal{H})$-orbit of the flat Lorentz metric on the cylinder
corresponds to a zero central charge orbit, whereas the central charge $c$
of the other generic $\mathop{\rm Vir}\nolimits$-orbits we investigate is related to the
(constant) curvature $K$ of $(\mathcal{H},\mathop{\textrm{\rm g}}\nolimits)$ by $cK=1$. We, likewise, derive
the Bott-Thurston cocycle within the same framework.
\item
Some perspectives are finally drawn in Section \ref{ConclusionAndOutlook}.
It is, in particular, expected that our results allow for generalizations
that would, e.g., relate Kulkarni's Lorentz surfaces and
universal Teichm\"uller space.
\end{itemize}
\bigskip\noindent
\textbf{Acknowledgments:}
It is a pleasure for us to acknowledge enlightening conversations with
V.~Ovsienko and P.~Iglesias during the preparation of this article. We
would also like to thank H.~Heyer and J.~Marion, for the nice organization
of the Colloquium on \textit{Analysis on Infinite-Dimensional Lie Groups
and Algebras} held at CIRM in September 1997; a great many thanks for
their unwavering patience.
\goodbreak
\section{The Lorentz hyperboloid of one sheet}
\label{TheLorentzHyperboloidOfOneSheet}
\subsection{An adjoint orbit in $\mathop{\mathfrak{sl}(2,\bbR)}\nolimits$}
The single sheeted hyperboloid $H^{1,1}_c\hookrightarrow\mathbb{R}^{2,1}$ defined
for
$c\in\mathbb{R}^*_+$ by
\begin{equation}
x^2+y^2-t^2=c
\label{HypEq}
\end{equation}
carries a canonical Lorentz metric\footnote{In the physics literature
$H^{1,1}_1$ is called anti-de Sitter spacetime.} given by the
induced quadratic form
\begin{equation}
\mathop{\textrm{\rm g}}\nolimits_c=dx^2+dy^2-dt^2.
\label{inducedMetric}
\end{equation}
\begin{proposition}[\cite{Kul}, \cite{Wolf}]\label{Killing}
The hyperboloid of one sheet
$
H^{1,1}_c\cong\mathbb{R}\times\mathbb{T}
$
with radius $r=\sqrt{c}\neq0$ is the homogeneous space
$$
H^{1,1}_c=\mathop{\mathrm{SL}(2,\bbR)}\nolimits/\mathop{\mathrm{SO}}\nolimits(1,1)
$$
which is symplectomorphic to the $\mathop{\mathrm{SL}(2,\bbR)}\nolimits$-adjoint orbit of
$$
\pmatrix{r&0\cr0&-r}\in\mathop{\mathfrak{sl}(2,\bbR)}\nolimits.
$$
As a Lorentz manifold, $H^{1,1}_c$ is a space form of constant
curvature\footnote{Since $\mathop{\textrm{\rm g}}\nolimits\longrightarrow-\mathop{\textrm{\rm g}}\nolimits$ yields $K\longrightarrow-K$
and preserves the Lorentz signature $(+,-)$, we will admit $c<0$
in (\ref{curvature}); see \cite{Ghys2}. Recall that $K=\frac{1}{2}R$ where
$R$ is the scalar curvature.}
\begin{equation}
K=\frac{1}{c}
\label{curvature}
\end{equation}
whose group of direct isometries is $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$.
\end{proposition}
\medskip
\begin{remark}\label{spaceOfOrientedGeodesicsOfH2}
{\rm
The unit hyperboloid $H_1^{1,1}$ is also symplectomorphic to the manifold
of oriented geodesics of the Poincar\'e disk $H^2\cong\mathop{\mathrm{SL}(2,\bbR)}\nolimits/\mathop{\mathrm{SO}}\nolimits(2)$.
}
\end{remark}
From now on we will write $H$ as a shorthand notation for $H^{1,1}$
provided no confusion occurs.
\bigskip
\goodbreak
The following expression for the Lorentz metric (\ref{inducedMetric}) on
$H$ will prove useful. In view of (\ref{HypEq}), write
$x = \varrho\sin\theta$,
$y = \varrho\cos\theta$,
$r = \varrho\sin\phi$,
$t = \varrho\cos\phi$
so that the metric (\ref{inducedMetric}) takes the form
$\mathop{\textrm{\rm g}}\nolimits_c=r^2\csc^2\!\phi\,(d\theta^2-d\phi^2)$. Putting now
$\theta_1=\theta+\phi$ and $\theta_2=\theta-\phi$, we obtain
\begin{equation}
\mathop{\textrm{\rm g}}\nolimits_c =
\frac{4c\,d\theta_1d\theta_2}{\left|e^{i\theta_1}-e^{i\theta_2}\right|^2}
\label{KostantSternberg1}
\end{equation}
with (see (\ref{curvature}))
\begin{equation}
c\in\mathbb{R}^*,
\label{cneq0}
\end{equation}
yielding the canonical Killing metric on the
hyperboloid
\begin{equation}
H\cong{}\mathbb{T}\times\mathbb{T}-\Delta
\label{KostantSternberg2}
\end{equation}
globally parametrized by
$\theta_1,\theta_2\in\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$ with
$\theta_1\neq\theta_2$. See, e.g.,~\cite{KS1}.
The transverse null foliations $\theta_1=\mathop\mathrm{const.}\nolimits$ and
$\theta_2=\mathop\mathrm{const.}\nolimits$ correspond to the rulings of the hyperboloid,
and the diagonal $\Delta$ is the conformal boundary \cite{Kul} (or
null infinity \cite{Penr}) of~$H$.
\subsection
The Cayley-Klein model}\label{KleinSection}
The material of this Section has been borrowed from \cite{Cartan3} with
a slight adaptation to our framework.
\begin{definition}
{\rm
An \textit{involution} of $\mathop{\bbR P^1\!}\nolimits$ is an homography $s\in\mathop{\mathrm{PGL}(2,\bbR)}\nolimits$ such that
$s^2=\mathop{\mathrm{id}}\nolimits$ and $s\neq\mathop{\mathrm{id}}\nolimits$. We will denote $\mathcal{I}$ the space of involutions.
}
\end{definition}
In the projective plane $P$ associated to the vector space $\mathop{\mathfrak{sl}(2,\bbR)}\nolimits$, there is
a distinguished conic $C$, defined by the light cone.
\begin{lemma}
The space of involutions is naturally identified with $P-C$.
\end{lemma}
The determinant map $\det:\mathop{\mathrm{GL}(2,\bbR)}\nolimits\longrightarrow\mathbb{R}^*$ descends, after
projectivization, as a map $\delta:\mathop{\mathrm{PGL}(2,\bbR)}\nolimits\longrightarrow\mathbb{Z}/(2\mathbb{Z})$, that
defines the two connected components of the projective group. Then, we can
define $\mathcal{I}_+ =\mathcal{I}\cap \delta^{-1}(1)$ the space of direct involutions and
$\mathcal{I}_- =\mathcal{I}\cap \delta^{-1}(-1)$ the space of anti-involutions.
Let us denote by $D$ the interior of the convex hull of $C$ and by $\mathcal{K}$
the complement of $D\cup C$ in $P$.
\begin{proposition}
The space of direct involutions is naturally isomorphic to the disk $D$
and the space of anti-involutions to $\mathcal{K}$.
\end{proposition}
\begin{remark}
\rm{
Topologically, $\mathcal{K}$ is a M\"obius band.
}
\end{remark}
\begin{proposition}
The $2$-fold covering of orientations for $\mathcal{I}_-$ is $C\times{}C-\Delta$.
The restriction of the projection
$\pi:\mathop{\mathfrak{sl}(2,\bbR)}\nolimits\!-\singleton{0}\longrightarrow{}P$ to the Lorentz hyperboloid
$H$ is a $2$-fold covering on $\mathcal{K}$.
\end{proposition}
There exists an isomorphism $P\cong\RPn{2}$ such that the conic
$C$ is mapped onto the unit circle $\bbT$ in the affine plane
$\singleton{t=1}$, where $x,y,t$ are homogeneous coordinates in $\mathbb{R}^3$.
This isomorphism is given by the map
$$
X =
\pmatrix{a&b\cr{}c&-a}
\longmapsto\frac{1}{2}
\pmatrix{2a\cr{}b+c\cr{}b-c}.
$$
Thus, we verify that the light
cone, whose equation is given by
$\det(X)=0$, is mapped onto the conic of homogeneous equation
$x^2 + y^2 - t^2 = 0$.
In the Klein model, the complement
$
\mathcal{K}=\{z\in\mathbb{C}\,|\,|z|>1\}
$
of the closed unit disk thus represents the projectivized hyperboloid
$P(H)$ in $\mathbb{R} P^2\cong{}P(\mathop{\mathfrak{sl}(2,\bbR)}\nolimits)$. It is the space of geodesics of the
open unit disk, i.e., of the hyperbolic plane in the Klein model. See
Remark~\ref{spaceOfOrientedGeodesicsOfH2}.
\subsection{Projective structures}
In order to gain some insight into the preceding results, let us briefly
recall the notion of projective structure \cite{Cartan1,Cartan2,CF,Wil}. To
that end, we need the
\begin{definition}\label{projStructDef}
{\rm
A \textit{projective structure} $\varpi$ on a $n$-dimensional connected
manifold~$\mathcal{M}$ is given by the following data:
\begin{enumerate}
\item
an immersion
$\Phi:\widetilde{\mathcal{M}}\longrightarrow\mathbb{R}{}P^n$ defined on the universal
covering $\widetilde{\mathcal{M}}$ of~$\mathcal{M}$,
\item
a homomorphism
$T:\pi_1(\mathcal{M})\longrightarrow\mathrm{PSL}(n+1,\mathbb{R})$
\end{enumerate}
such that
\begin{equation}
\forall a \in\pi_1(\mathcal{M})
\quad
\Phi\circ a=T(a)\circ\Phi.
\label{holo}
\end{equation}
One calls $\Phi$ the \textit{developing map} and $T$ the
\textit{holonomy} of the structure.
}
\end{definition}
We denote by $\varpi = [\Phi,T]$ the associated projective structure.
The developing map and the holonomy characterizes the structure up to
conjugation by the projective group, that is:
$$
\forall A \in \mathrm{PGL}(n+1,\mathbb{R})
\quad
[A\circ\Phi,A\cdot{}T\cdot{}A^{-1}] = [\Phi,T].
$$
Such a structure is equivalently given by an atlas of projective charts
$\varphi_i: U_i \subset \mathcal{M} \longrightarrow \mathbb{R}{}P^n$ with transition
diffeomorphisms in $\mathrm{PGL}(n+1,\mathbb{R})$.
In the $1$-dimensional case under study, and, more particularly in the case
of the circle~$\mathsf{S}$, a projective structure $\varpi$ is given by a pair
$(\Phi,M)$ with $\Phi:\mathbb{R}\longrightarrow\mathop{\bbR P^1\!}\nolimits$ an
immersion and $M\in\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$. Condition $\ref{holo}$
then reads
$$
\Phi(\theta+2\pi) = M\cdot \Phi(\theta).
$$
\goodbreak
It is a classic result \cite{Segal2,Guieu2} that the space $\mathcal{P}(\mathsf{S})$ of
all projective structures on~$\mathsf{S}$ is an affine space modeled on the space
$\mathcal{Q}(\mathsf{S})$ of quadratic differentials $q=u(\theta)\,d\theta^2$ of~$\mathsf{S}$.
The projective atlas associated with $q$ is obtained by locally solving the
third order non-linear differential equation
$
q=S(\Phi)
$
where $S$ stands for the Schwarzian derivative (see below).
From now on, we restrict considerations
to either choices of projective structures on $\mathsf{S}$, namely
\begin{enumerate}
\item
the torus $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$ defined by the following developing
map\footnote{We use the notation $[z]=\mathbb{R}{}z$ for all
$z\in\mathbb{C}-\singleton{0}$.} (with trivial holonomy)
\begin{equation}
\Phi(\theta)=[e^{i\theta}]
\qquad
\mbox{or}
\qquad
\Phi(\theta)=2\tan\frac{\theta}{2},
\label{T}
\end{equation}
\item
the projective line $\mathop{\bbR P^1\!}\nolimits$ defined by the developing map
\begin{equation}
\Phi(\theta)=\tan\theta
\qquad
\mbox{or}
\qquad
\Phi(t)= t.
\label{P1}
\end{equation}
\end{enumerate}
\subsection{Lorentzian metric and cross-ratio}
Let us describe, following Ghys \cite{Ghys2}, how the canonical Lorentz
metric (\ref{KostantSternberg1}) on anti-de Sitter space
(\ref{KostantSternberg2}) indeed originates from the cross-ratio
\begin{equation}
(z_1,z_2,z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_1-z_4)(z_2-z_3)}
\label{crossRatio}
\end{equation}
of four points on the projective line \cite{Cartan2}.
Let us fix $(\theta_1,\theta_2)\in\mathbb{T}^2-\Delta$ and consider then a nearby
point $(\theta_3,\theta_4)=(\theta_1+d\theta_1,\theta_2+d\theta_2)$.
Put $z_j=e^{i\theta_j}$ for $j=1,\ldots,4$ and perform a Taylor
expansion of the cross-ratio (\ref{crossRatio}) at
$(\theta_1,\theta_2)$, so that
\goodbreak
\begin{eqnarray*}
(z_1,z_2,z_3,z_4)
&=&
\frac{
e^{i\theta_1}e^{i\theta_2}
\left(1-e^{id\theta_1}\right)\left(1-e^{id\theta_1}\right)
}
{
\left(e^{i\theta_1}-e^{i(\theta_2+d\theta_2)}\right)
\left(e^{i\theta_2}-e^{i(\theta_1+d\theta_1)}\right)
}\\
\\
&=&
\frac{
\left(-id\theta_1\right)\left(-id\theta_2\right)
}
{
\left(e^{i\theta_1}-e^{i\theta_2}\right)
\left(e^{i\theta_2}-e^{i\theta_1}\right)
e^{-i\theta_1}e^{-i\theta_2}
}
+\cdots\\
\\
&=&
\frac{
-d\theta_1 d\theta_2
}
{
\left|e^{i\theta_1}-e^{i\theta_2}\right|^2
}
+\cdots
\end{eqnarray*}
where the ellipsis ``$\cdots$'' stands for ``terms of order $\geq3$''.
One can thus claim that, up to higher order terms, the metric
(\ref{KostantSternberg1}) on the unit hyperboloid $H$
(\ref{KostantSternberg2}) is given by
$
\mathop{\textrm{\rm g}}\nolimits_1
=
-4\left(z_1,z_2,z_1+dz_1,z_2+dz_2\right)+\cdots
$ or, equivalently, by
\begin{equation}
\mathop{\textrm{\rm g}}\nolimits_1
=
-4\,\lim_{\varepsilon\rightarrow0}{
\frac{1}{\varepsilon^2}
\left(z_1,z_2,z_1+\varepsilon{}dz_1,z_2+\varepsilon{}dz_2\right)
}
\label{theMetric}
\end{equation}
which is therefore conspicuously $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$-invariant.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.5]{Klein.EPSF}
\caption{\label{Klein}\textit{The Klein model}}
\end{center}
\end{figure}
\medskip
Resorting to Definition \ref{projStructDef}, we then have the
\begin{theorem}
Consider the hyperboloid $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta$ where the circle~$\mathsf{S}$
has a projective structure defined by
$\Phi\in\mathop{\mathrm{Diff}}\nolimits_\mathrm{loc}(\mathbb{R},\mathop{\bbR P^1\!}\nolimits)$ as in (\ref{T}) or (\ref{P1}).
Then,
$\mathcal{H}$ carries a natural $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$-invariant metric of the form
\begin{equation}
\mathop{\textrm{\rm g}}\nolimits_1 =
(\Phi\times\Phi)^*\frac{4\,dt_1dt_2}{(t_1-t_2)^2}.
\label{generalg1}
\end{equation}
\end{theorem}
\textit{Proof:}
The cross-ratio (\ref{crossRatio}) is $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$-invariant and so is the
Lorentz metric $4dt_1dt_2/(t_1-t_2)^2$ of $\mathop{\bbR P^1\!}\nolimits\times\mathop{\bbR P^1\!}\nolimits-\Delta$ given by
(\ref{theMetric}) with $z_j=t_j$ (see (\ref{P1})). In any cases (\ref{T})
or (\ref{P1}), the metric (\ref{generalg1}) defined on $\mathbb{R}^2-\Gamma$
where $\Gamma=(\Phi\times\Phi)^{-1}(\Delta)$ is automatically
$\pi_1(\mathsf{S})$-invariant thanks to (\ref{holo}). It is invariant, as well,
under the universal covering $\mathop{\mathrm{\widetilde{PSL}}(2,\bbR)}\nolimits$ of $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$. Hence, this metric
descends to $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta=\pi\times\pi(\mathbb{R}^2-\Gamma)$ where
$\pi:\mathbb{R}\to\mathsf{S}$ is the universal covering map. The projected metric
$\mathop{\textrm{\rm g}}\nolimits_1$ is then clearly $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$-invariant.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
Example (\ref{KostantSternberg1})
corresponds to the developing maps (\ref{T}); as for the first developing
map in (\ref{P1}), it leads via (\ref{generalg1}) to the metric of the
Klein model of Section \ref{KleinSection} (see Figure \ref{Klein}).
\section{The Schwarzian derivative}\label{TheSchwarzianDerivative}
\subsection{Osculating homography of a diffeomorphism}
Let $\varphi:\mathop{\bbR P^1\!}\nolimits\longrightarrow\mathop{\bbR P^1\!}\nolimits$ be a diffeomorphism and let $t_0\in\mathop{\bbR P^1\!}\nolimits$. We want
to find the homography $h\in\mathop{\mathrm{PGL}(2,\bbR)}\nolimits$ that best approximates the
diffeo\-morphism~$\varphi$ at this point $t_0$.
\begin{proposition}
This homography $h$ exists and is unique. It is completely defined by the
conditions
\begin{eqnarray*}
\hfil h(t_0) &=& \varphi(t_0),\\
\hfil h'(t_0) &=& \varphi'(t_0),\\
\hfil h''(t_0) &=& \varphi''(t_0).
\end{eqnarray*}
\end{proposition}
The diffeomorphism $h^{-1}\circ\varphi$ has the $2$-jet of
the identity at $t_0$. The difference between $h$ and $\varphi$
starts, hence, at the third order derivative. (See, e.g.,~\cite{Ghys2}.)
\begin{definition}
{\rm
The \textit{Schwarzian derivative} of $\varphi$ at the point $t_0$ is
$$
S(\varphi)(t_0) := \left( h^{-1} \circ \varphi \right) '''(t_0).
$$
}
\end{definition}
The quantity $S(\varphi)(t_0)$ measures how much does the diffeomorphism
$\varphi$ differ from an homography at the point $t_0$. All projective
information about $\varphi$ is encoded into the Schwarzian derivative. If we
identify the real projective line with $\mathbb{R}\cup\singleton{\infty}$ by:
$[x,y] \longmapsto t=y/x$, we obtain the classical formula:
\begin{equation}
S(\varphi) =
\left(
{\varphi'''(t)\over\varphi'(t)}-{3\over2}{\varphi''(t)^2\over\varphi'(t)^2}
\right) dt^2.
\label{TheFormule}
\end{equation}
The graph $\Gamma_\varphi$ of our diffeomorphism is a simple closed curve
on $\mathop{\bbR P^1\!}\nolimits\times\mathop{\bbR P^1\!}\nolimits$.
\begin{definition}
{\rm
The homography $h$ and its graph $\Gamma_h$ are respectively called the
\textit{osculating homography} and the \textit{osculating hyperbola} of
$\varphi$ at $t_0$.
}
\end{definition}
\subsection{The Schwarzian as a projective differential invariant}
\begin{theorem}[\cite{Green}]
The Schwarzian derivative is a third-order complete differential invariant for
the group of diffeomorphisms of the projective line.
\end{theorem}
More precisely, if $\varphi$ and $\psi$ are two diffeomorphisms of $\mathop{\bbR P^1\!}\nolimits$,
then
$$
S(\varphi) = S(\psi)
\qquad
\Leftrightarrow
\qquad
\exists A \in \mathop{\mathrm{PSL}(2,\bbR)}\nolimits,\ \psi = A \circ \varphi.
$$
\begin{theorem}[\cite{Bott,Kiri,Roger,Segal}]
The Schwarzian $S$ given by (\ref{TheFormule}) is a non trivial
$1$-cocycle, i.e.,
$$
S(\varphi\circ\psi) = \psi^*S(\varphi) + S(\psi)
\qquad
\forall\varphi,\psi\in\mathop{\mathrm{Diff}}\nolimits_+(\mathop{\bbR P^1\!}\nolimits),
$$
on the group of orientation-preserving diffeomorphisms of $\mathop{\bbR P^1\!}\nolimits$ with
values in the $\mathop{\mathrm{Diff}}\nolimits_+(\mathop{\bbR P^1\!}\nolimits)$-module of real quadratic differentials
$\mathcal{Q}(\mathop{\bbR P^1\!}\nolimits)$ of of $\mathop{\bbR P^1\!}\nolimits$. Its kernel is $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$.
\end{theorem}
\begin{remark}
{\rm The Schwarzian cocycle (\ref{TheFormule}) is uniquely characterized
(up to a constant factor) by the property of having kernel $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$.
}
\end{remark}
\subsection{Cartan formula of the cross-ratio}
\label{CartanFormulaSection}
A useful means for calculating the
Schwarzian derivative of a smooth map of the projective line is given by
\begin{theorem}[\cite{Cartan2}]\label{CartanThm}
Consider a smooth map $\varphi:\mathbb{R} P^1\longrightarrow\mathbb{R} P^1$ and
four points
$t_1,\ldots,t_4\in\mathbb{R} P^1$ tending to $t\in\mathbb{R} P^1$;
putting
$\tau_j=\varphi(t_j)$ one has
\begin{equation}
\frac{(\tau_1,\tau_2,\tau_3,\tau_4)}{(t_1,t_2,t_3,t_4)}-1
=
\frac{1}{6}
S(\varphi)(t) (t_1-t_2)(t_3-t_4)+\hbox{\rm [higher order terms]}
\label{SchwarzCartan}
\end{equation}
where $S(\varphi)$ denotes the Schwarzian derivative (\ref{TheFormule}) of
$\varphi$.
\end{theorem}
This expression still makes sense for any smooth map of the
circle $\mathsf{S}$ endowed with some projective structure given, for example, by
(\ref{T}) or (\ref{P1}). We, indeed, have the
\begin{definition}
{\rm
Let $\varphi:\mathsf{S}\longrightarrow\mathsf{S}$ be a smooth map identified with one
of its representatives\footnote{Choose any element of
$C^\infty(\mathbb{R})$ that commutes with $\pi_1(\mathsf{S})$.} in
$C^\infty_{\pi_1(\mathsf{S})}(\mathbb{R})$, then the \textit{Schwarzian} of $\varphi$ is
the pull-back of the Schwarzian (\ref{SchwarzCartan}) of the induced map
$\tilde\varphi$ of
$\mathop{\bbR P^1\!}\nolimits$, namely
\begin{equation}
\mathbf{S}(\varphi)=\Phi^*S(\tilde\varphi).
\label{generalSchwarzian}
\end{equation}
}
\end{definition}
We note that (\ref{generalSchwarzian}) yields a well-defined quadratic
differential on $\mathsf{S}$ since one trivially finds $a^*\mathbf{S}(\varphi)=\mathbf{S}(\varphi)$ in
view of $T(a)^*S(\tilde\varphi)=S(\tilde\varphi)$ for all
$a\in\pi_1(\mathsf{S})\cong\mathbb{Z}$.
\begin{proposition}
One has, locally,
\begin{equation}
\mathbf{S}(\varphi)=S(\varphi)+\varphi^*S(\Phi)-S(\Phi).
\label{generalSchwarzianLoc}
\end{equation}
\end{proposition}
\textit{Proof:}
Using $\tilde\varphi\circ\Phi=\Phi\circ\varphi$, one easily finds
$\Phi^*S(\tilde\varphi)(\theta)
=
S(\varphi)(\theta)+S(\Phi)(\varphi(\theta))\,\varphi'(\theta)^2
-S(\Phi)(\theta)$.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\goodbreak
\section{Conformal transformations}\label{ConformalTransformations}
\subsection{Conformal Lorentz structures}
Let us recall some basic definitions and facts about $2$-dimensional
Lorentzian conformal geometry.
\begin{definition}[\cite{Kul}]\label{DefConf}
{\rm
A \textit{conformal Lorentz structure} on a surface~$\Sigma$ is
characterized by a pair of transverse foliations; in other words, it is
given by a splitting
\begin{equation}
T\Sigma=T_1\Sigma\oplus{}T_2\Sigma
\label{splitting}
\end{equation}
into two trivial line bundles (light-cone field).
We call $N_1$ and $N_2$, respective\-ly, the spaces of leaves of the
two foliations of $\Sigma$.
}
\end{definition}
The leaves composing the ``grid'' associated to these foliations are,
locally, given by
$$
N_1:\theta_1=\mathop\mathrm{const.}\nolimits, \qquad N_2:\theta_2=\mathop\mathrm{const.}\nolimits
$$
The conformal structure is characterized by the global inter\-section
properties of the (null) leaves of $N_1$ and $N_2$.
One can associate to the splitting (\ref{splitting}) a class of
metrics on $\Sigma$, locally, of the form
$\mathop{\textrm{\rm g}}\nolimits=F(\theta_1,\theta_2)\,d\theta_1d\theta_2$ where $F$ is some smooth
positive function. If~$\mathop{\textrm{\rm g}}\nolimits$ is any metric with prescribed null cone field
$T_1\Sigma\oplus{}T_2\Sigma$, we denote by
\begin{equation}
\class{\mathop{\textrm{\rm g}}\nolimits}=\{F\cdot\mathop{\textrm{\rm g}}\nolimits|F\in{}C^\infty(\Sigma,\mathbb{R}^+_*)\}
\label{classg1}
\end{equation}
the class of metrics conformally equivalent to $\mathop{\textrm{\rm g}}\nolimits$. Thus, a conformal
Lorentz structure \cite{Wei} on $\Sigma$ is equivalently defined by
$(\Sigma,\class{\mathop{\textrm{\rm g}}\nolimits})$.
\begin{definition}\label{DiffConf}
{\rm
A diffeomorphism $\varphi$ of $(\Sigma,\mathop{\textrm{\rm g}}\nolimits)$ is called
\textit{conformal}---we write $\varphi\in\mathop{\rm Conf}\nolimits(\Sigma,\mathop{\textrm{\rm g}}\nolimits)$---if
\begin{equation}
\varphi^*\!\mathop{\textrm{\rm g}}\nolimits=f_\varphi\cdot{\mathop{\textrm{\rm g}}\nolimits}
\quad
\mbox{for some\ }
f_\varphi\in{}C^\infty(\Sigma,\mathbb{R}^+_*).
\label{defConfDiff}
\end{equation}
The function $f_\varphi$ is called the \textit{conformal factor}
associated with $\varphi$.}
\end{definition}
\begin{remark}
{\rm
Definition \ref{DiffConf} is general and holds in the
Riemannian case. It is, for instance, well known that $\mathop{\rm Conf}\nolimits(H^2)=\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$.
In the Lorentzian case, the conformal
group of $H^{1,1}$ is, however, infinite dimensional; more
precisely, we will see that $\mathop{\rm Conf}\nolimits(H^{1,1})=\mathop{\mathrm{Diff}}\nolimits(\bbT)$.
}
\end{remark}
\subsection{Conformal geometry of the Lorentz hyperboloid}
We have seen (\ref{KostantSternberg2}) that the global intersection
properties of the rulings of the hyperboloid yield (see
Figure~\ref{T2MinusDelta})
\begin{equation}
H=\mathbb{T}\times\mathbb{T}-\Delta
\label{KostantSternberg2bis}
\end{equation}
\begin{figure}[h]
\begin{center}
\begin{picture}(7,7)
\put(0,0){\framebox(7,7)}
\put(0,0){\line(1,1){7}}
\put(0,0){\vector(0,1){3.5}}
\put(0,0){\vector(1,0){3.5}}
\put(0,7){\vector(1,0){3.5}}
\put(7,0){\vector(0,1){3.5}}
\put(6,.3){$\theta_1$}
\put(.3,6){$\theta_2$}
\put(4.5,5.5){$\Delta$}
\end{picture}
\end{center}
\caption{\label{T2MinusDelta}The hyperboloid}
\end{figure}
whose metric (\ref{KostantSternberg1},\ref{theMetric}) is
given by
\begin{equation}
\mathop{\textrm{\rm g}}\nolimits_1 =
\frac{4\,d\theta_1d\theta_2}
{\left|e^{i\theta_1}-e^{i\theta_2}\right|^2}.
\label{g1}
\end{equation}
In view of the previous definitions \ref{DefConf} and \ref{DiffConf}, any
conformal (grid-preserving) diffeomorphism $\varphi$ of a Lorentz surface
$(\Sigma,\mathop{\textrm{\rm g}}\nolimits)$ is, locally, of the form $\varphi_1\times\varphi_2$ where $\varphi_j
\in \mathop{\mathrm{Diff}}\nolimits(N_j)$. A (global) conformal diffeomorphism of~$\Sigma$ must
preserve the two foliations by lines.
In our case, such a transformation of $\mathbb{T}^2-\Delta$ must preserve not
only the meridians and parallels of $\mathbb{T}^2$, but the diagonal $\Delta$ as
well. Therefore, $\varphi_1(\theta) = \varphi_2(\theta)$ for all
$\theta\in\mathbb{T}$, whence the
\begin{proposition}[\cite{KS2}]
There exists a canonical isomorphism
$$
\mathop{\mathrm{Diff}}\nolimits(\Delta)\stackrel{\cong}\longrightarrow\mathop{\rm Conf}\nolimits(H)
$$
given by the diagonal map: $\varphi\longmapsto\varphi\times\varphi$.
\end{proposition}
\goodbreak
Let us recall the
\begin{theorem}[\cite{KS2}]\label{KS.th}
(i) Let $\varphi\in\mathop{\mathrm{Diff}}\nolimits_+(\bbT)\cong\mathop{\rm Conf}\nolimits_+(H)$ be given. Then
$f_\varphi=(\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_1)/\mathop{\textrm{\rm g}}\nolimits_1\longrightarrow1$ as one tends to the conformal
boundary
$\Delta$.
(ii) The conformal factor $f_\varphi$ extends smoothly to
$H\cup\Delta=\mathbb{T}^2$ and has, moreover, $\Delta$ as its critical set.
(iii) One has $\mathop{\mathrm{Hess}}\nolimits(f_\varphi)|\Delta=\frac{1}{3}\widetilde{S}(\varphi)$
where
\begin{equation}
\widetilde{S}(\varphi)
=
S(\varphi) +
\frac{1}{2}\left(\varphi'(\theta)^2-1\right) d\theta^2.
\label{SchwKS}
\end{equation}
(iv) The Schwarzian $\widetilde{S}(\varphi)$ completely determines $f_\varphi$.
\end{theorem}
Our proof proceeds as follows. Comparison with the definition
(\ref{theMetric}) of the metric $\mathop{\textrm{\rm g}}\nolimits_1$ on the hyperboloid $H$ in terms of
the cross-ratio prompts the following computation. Given
any $\varphi\in\mathop{\mathrm{Diff}_+}\nolimits(\mathbb{T})$ viewed as a conformal diffeomorphism
(\ref{defConfDiff}) of $(H,\mathop{\textrm{\rm g}}\nolimits_1)$, apply the Cartan formula
(\ref{SchwarzCartan}) in the case of a diffeomorphism of the circle
$\mathbb{T}$, and get
\begin{eqnarray}
\frac{(\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_1)(\theta_1,\theta_2)}{\mathop{\textrm{\rm g}}\nolimits_1(\theta_1,\theta_2)}-1
&=&
\label{fminusone1}
f_\varphi(\theta_1,\theta_2)-1\\
&=&
\frac{1}{6}
S(\tilde\varphi)(e^{i\theta})
\left(e^{i\theta_1}-e^{i\theta_2}\right)^2+\cdots
\label{SchwarzHat1}
\end{eqnarray}
where $\tilde\varphi(e^{i\theta})=e^{i\varphi(\theta)}$ and
$\theta_j\longrightarrow\theta$ for $j=1,2$.
A tedious calculation using (\ref{SchwarzCartan}) leads to
\begin{lemma}
If $\tilde\varphi\in\mathop{\mathrm{Diff}_+}\nolimits(\mathbb{T})$ is represented by\footnote{
We denote by $\mathop{\mathrm{Diff}}\nolimits_{2\pi\mathbb{Z}}(\mathbb{R})$ the universal covering of
$\mathop{\mathrm{Diff}_+}\nolimits(\bbT)$, i.e., the group of those diffeomorphisms $\varphi$ of
$\mathbb{R}$ such that $\varphi(\theta+2\pi)=\varphi(\theta)+2\pi$.
}
$\varphi\in\mathop{\mathrm{Diff}}\nolimits_{2\pi\mathbb{Z}}(\mathbb{R})$, one has
\begin{equation}
S(\tilde\varphi)(e^{i\theta})
=
-\left(
S(\varphi)(\theta)
+
\frac{1}{2}\left(\varphi'(\theta)^2-1\right)
\right)
e^{-2i\theta}.
\label{SchwarzHat}
\end{equation}
\end{lemma}
From (\ref{fminusone1})--(\ref{SchwarzHat}) one obtains
\begin{equation}
f_\varphi(\theta_1,\theta_2)-1
=
\frac{1}{6}
\left(
S(\varphi)(\theta)
+
\frac{1}{2}\left(\varphi'(\theta)^2-1\right)
\right)
(\theta_1-\theta_2)^2
+\cdots
\label{SchwarzHat2}
\end{equation}
that is, theorem 1 in \cite{KS2}. In particular, the conformal factor
$f_\varphi$ extends to the diagonal $\Delta\subset\mathbb{T}^2$ (its critical set)
and $f_\varphi|\Delta=1$, its transverse Hessian being related to the
modified Schwarzian derivative (see (\ref{SchwarzHat2})) by
$
\mathop{\mathrm{Hess}}\nolimits(f_\varphi)
=
\frac{1}{3}\widetilde{S}(\varphi)
$.
The fourth item of theorem \ref{KS.th} will be a consequence of Theorem
\ref{main}.
\goodbreak
\medskip
We are thus led to the
\begin{theorem}\label{FirstTheorem}
(i) Given any $\varphi\in\mathop{\rm Conf}\nolimits_+(H)$ of $H=\mathbb{T}^2-\Delta$ and
$c\neq0$, the twice-symmetric tensor field $\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_c-\mathop{\textrm{\rm g}}\nolimits_c$ of $H$
extends to null infinity~$\Delta$ and defines a non trivial
$1$-cocycle
\begin{equation}
S_c:\varphi
\longmapsto
\frac{3}{2}
\left(\strut\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_c - \mathop{\textrm{\rm g}}\nolimits_c\right)\!|\Delta
\label{TheCocycle}
\end{equation}
of $\mathop{\mathrm{Diff}_+}\nolimits(\mathbb{T})$ with values in the module $\mathcal{Q}(\mathbb{T})$ of quadratic
dif\-ferentials of the circle, given by the (modified) Schwarzian
derivative (\ref{SchwKS}):
\begin{equation}
S_c=c\,\widetilde{S}.
\label{Schwarzc}
\end{equation}
(ii) There holds
$H^1(\mathop{\mathrm{Diff}_+}\nolimits(\mathbb{T}),\mathcal{Q}(\mathbb{T}))=\mathbb{R}\left[S_1\right]$.
\end{theorem}
\textit{Proof:}
From the formul\ae\ (\ref{SchwarzHat2}) and (\ref{g1}) one immediately
gets
\begin{eqnarray*}
\left(\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_1 - \mathop{\textrm{\rm g}}\nolimits_1\right)|\Delta
&=&
\left({\textstyle\frac{2}{3}}\,\widetilde{S}(\varphi)(\theta)(\theta_1-\theta_2)^2
d\theta_1d\theta_2|e^{i\theta_1}-e^{i\theta_2}|^{-2}+\cdots\right)
\!\Big|\Delta\\[6pt]
&=&
\left({\textstyle\frac{2}{3}}\,\widetilde{S}(\varphi)(\theta)d\theta_1d\theta_2+\cdots\right)
\!\Big|\Delta\\[6pt]
&=&
\frac{2}{3}\,\widetilde{S}(\varphi)(\theta)\,d\theta^2\\[6pt]
&=&
\frac{2}{3}\,\widetilde{S}(\varphi).
\end{eqnarray*}
Then (\ref{Schwarzc}) is clear by (\ref{KostantSternberg1}) and
(\ref{cneq0}).
At last, part (ii) follows immediately from the knowledge that
$H^1(\mathop{\mathrm{Diff}_+}\nolimits(\mathbb{T}),\mathcal{Q}(\mathbb{T}))$ is $1$-dimensional \cite{GF,Fuchs} and
generated by the class of the Schwarzian.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
\begin{remark}
\rm{
The cocycle $\varphi\longmapsto\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_c - \mathop{\textrm{\rm g}}\nolimits_c$ of $\mathop{\rm Conf}\nolimits_+(H)$ with
values in the space of twice-covariant symmetric tensor fields is obviously
trivial. Non triviality of the cocycle (\ref{TheCocycle}) quite remarkably
stems from the ``restriction'' of the latter to null infinity
$\Delta$.\footnote{This observation is due to Valentin Ovsienko.}
}
\end{remark}
\begin{proposition}
The group of direct isometries of the hyperboloid is
\begin{equation}
\mathop{\rm Isom}\nolimits_+(H,\mathop{\textrm{\rm g}}\nolimits_c)=\ker(S_c)\cong\mathop{\mathrm{PSL}(2,\bbR)}\nolimits.
\label{IsomCurved}
\end{equation}
\end{proposition}
\textit{Proof:}
Using (\ref{TheCocycle}), we find that the group
$\mathop{\rm Isom}\nolimits_+(H,\mathop{\textrm{\rm g}}\nolimits_c)\subset\mathop{\mathrm{Diff}_+}\nolimits(\bbT)$ of direct isometries is clearly a
subgroup of $\ker({S_c})\cong\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$. Conversely, for any
$\varphi\in\ker({S_c})$, and thanks to (\ref{SchwarzHat1}), the conformal
factor in (\ref{fminusone1}) is $f_\varphi=1$, i.e.,
$\varphi\in\mathop{\rm Isom}\nolimits_+(H,\mathop{\textrm{\rm g}}\nolimits_c)$.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
Theorem \ref{FirstTheorem} still holds true for the $\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$-invariant
metric (\ref{generalg1}) on $\mathsf{S}\times\mathsf{S}-\Delta$. In fact, a calculation akin to
that of (\ref{fminusone1},\ref{SchwarzHat1}) leads to
\begin{proposition}\label{TheGeneralCocycleTh}
Given any $\varphi\in\mathop{\rm Conf}\nolimits_+(\mathcal{H})$ of $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta$ where $\mathsf{S}$ is
endowed with the projective structure (\ref{T}) or (\ref{P1}), one has
\begin{equation}
\mathbf{S}(\varphi)=
S_1(\varphi)
=
\frac{3}{2}
\left(\strut\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_1 - \mathop{\textrm{\rm g}}\nolimits_1\right)\!|\Delta
\label{TheGeneralCocycle}
\end{equation}
where the metric $\mathop{\textrm{\rm g}}\nolimits_1$ on $\mathcal{H}$ is given by (\ref{generalg1}) and the
universal Schwarzian~$\mathbf{S}$ by
(\ref{generalSchwarzian},\ref{generalSchwarzianLoc}).
\end{proposition}
\goodbreak
\subsection{Conformal geometry of the flat cylinder}
Let us envisage, for a moment, the flat induced
Lorentz metric
\begin{equation}
\mathop{\textrm{\rm g}}\nolimits_0=d\theta_1d\theta_2
\label{g0}
\end{equation}
on the cylinder $H=\mathbb{T}^2-\Delta$. (A non significant constant factor might
be introduced in the definition (\ref{g0}) of $\mathop{\textrm{\rm g}}\nolimits_0$.)
In this special case, the $\mathop{\mathrm{Diff}_+}\nolimits(\bbT)$-cocycle $S_0$ defined, in the
same manner as in (\ref{TheCocycle}), by
\begin{equation}
S_0(\varphi)=\left(\strut\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_0-\mathop{\textrm{\rm g}}\nolimits_0\right)\vert\Delta
\label{S0}
\end{equation}
is, plainly, a coboundary since $\mathop{\textrm{\rm g}}\nolimits_0$ admits a prolongation to $\Delta$.
We, indeed, have
$S_0(\varphi)(\theta)=(\varphi'(\theta)^2-1)\,d\theta^2$. Notice that flatness
of the metric is now related to triviality of the associated cocycle.
\begin{proposition}
The group of direct isometries of the flat cylinder is
\begin{equation}
\mathop{\rm Isom}\nolimits_+(H,\mathop{\textrm{\rm g}}\nolimits_0)=\ker(S_0)\cong\mathbb{T}.
\label{isomzero}
\end{equation}
\end{proposition}
\textit{Proof:}
Solving $\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_0=\mathop{\textrm{\rm g}}\nolimits_0$ and $\varphi'(\theta)>0$ gives
$\varphi(\theta)=\theta+t$ with $t\in\mathbb{T}$, that is $\varphi\in\ker(S_0)$.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\section{Symplectic structure on conformal classes of metrics on
$\mathsf{S}\times\mathsf{S}-\Delta$}
\label{SymplecticStructuresOnConformalClassesOfMetrics}
We analyze, in this section, the structure of the conformal classes of the
previously introduced metrics $\mathop{\textrm{\rm g}}\nolimits_c$ and $\mathop{\textrm{\rm g}}\nolimits_0$ on the ``hyperboloid''
$\mathcal{H}$ and relate them to the generic coadjoint orbits \cite{Kiri} in
the regular dual of the Virasoro group. It should be recalled that the
conformal class of $\mathop{\textrm{\rm g}}\nolimits_1$ has first been identified with the homogeneous
space $\mathop{\mathrm{Diff}_+}\nolimits(\bbT)/\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$ in \cite{KS1}.
\subsection{Homogeneous space $\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})/\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$}
\label{DiffOverPSL2}
\subsubsection{Conformal classes of curved metrics}
Consider first the curved case. If $c\neq0$, denote by $M_c$ the
space of metrics on $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta$ related to $\mathop{\textrm{\rm g}}\nolimits_c=c\,\mathop{\textrm{\rm g}}\nolimits_1$
(\ref{KostantSternberg1}) by a conformal diffeomorphism (see
(\ref{classg1})), \textit{viz.}
$$
M_c=\{\mathop{\textrm{\rm g}}\nolimits\in\class{\mathop{\textrm{\rm g}}\nolimits_1}\,|\,\mathop{\textrm{\rm g}}\nolimits=\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_c,\varphi\in\mathop{\rm Conf}\nolimits_+(\mathcal{H})\}.
$$
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.53]{ConfClasses.EPSF}
\caption{\label{ConfClassesEPSF}
\textit{The conformal classes of metrics on $\mathbb{T}^2-\Delta$}}
\end{center}
\end{figure}
These classes $M_c$ of metrics (see Figure \ref{ConfClassesEPSF}) turn out
to have a symplectic structure of their own.
\begin{theorem}\label{ConfOrbitsTh}
If $c\neq0$, the homogeneous space
\begin{eqnarray*}
M_c
&=&
\mathop{\rm Im}\nolimits(\varphi\longmapsto\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_c)\\
&\cong&
\mathop{\rm Conf}\nolimits_+(\mathcal{H})/\mathop{\rm Isom}\nolimits_+(\mathcal{H},\mathop{\textrm{\rm g}}\nolimits_c)
\end{eqnarray*}
is endowed with (weak) symplectic structure $\omega_c$
which reads
\begin{equation}
\omega_c(\delta_1\!\mathop{\textrm{\rm g}}\nolimits,\delta_2\!\mathop{\textrm{\rm g}}\nolimits)
=
\frac{3}{2}
\int_\Delta{\strut\!i_{\xi_1}L_{\xi_2}\!\mathop{\textrm{\rm g}}\nolimits}
\label{theSymplecticStructure}
\end{equation}
where $\delta_j\!\mathop{\textrm{\rm g}}\nolimits=L_{\xi_j}\!\mathop{\textrm{\rm g}}\nolimits$ with $\xi_j\in\mathop{\rm Vect}\nolimits(\mathsf{S})$.
\end{theorem}
\textit{Proof:}
From (\ref{omega1}) below, $\omega_c$ is, indeed, skew-symmetric in its
arguments. It is, clearly, also closed. We then have
$\delta_2\!\mathop{\textrm{\rm g}}\nolimits_c\in\ker\omega_c$ iff
$\omega_c(\delta_1\!\mathop{\textrm{\rm g}}\nolimits_c,\delta_2\!\mathop{\textrm{\rm g}}\nolimits_c)=0$ for all $\xi_1\in\mathop{\rm Vect}\nolimits(\mathsf{S})$,
i.e., iff $L_{\xi_2}\!\mathop{\textrm{\rm g}}\nolimits_c\!|\Delta=0$, that is iff $\delta_2\!\mathop{\textrm{\rm g}}\nolimits_c=0$
in view of (\ref{TheCocycle}) and (\ref{IsomCurved}).
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
We will prove that $M_c$ is symplectomorphic to a
Kirillov-Segal-Witten $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$-orbit \cite{Kiri,Segal,Wit} for the
affine coadjoint (anti-)action $\mathop{\rm Coad}\nolimits_\Theta$ on $\mathcal{Q}(\mathsf{S})$ defined by
\begin{equation}
\mathop{\rm Coad}\nolimits_\Theta(\varphi)q = \mathop{\rm Coad}\nolimits(\varphi)q+\Theta(\varphi)
\label{CoadS}
\end{equation}
where the $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$-coadjoint (anti-)action reads
\begin{equation}
\mathop{\rm Coad}\nolimits(\varphi)q=\varphi^*q
\label{DefCoad}
\end{equation}
and where $\Theta$, a $1$-cocycle of $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$ with values in
$\mathcal{Q}(\mathsf{S})$, is a particular Souriau cocycle \cite{JMS}.
\goodbreak
\subsubsection{Intermezzo}
This technical section presents the standard $\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$-cocycles
in a guise adapted to any projective structure (\ref{T},\ref{P1}) on the
circle~$\mathsf{S}$.
Consider the line element
$$
\lambda=\Phi^*d\theta
$$
on $\mathsf{S}$ associated with the developing map
$\Phi\in\mathop{\mathrm{Diff}}\nolimits_\mathrm{loc}(\mathbb{R},\mathop{\bbR P^1\!}\nolimits)$. Actually, $\lambda$ is a
$\pi_1(\mathsf{S})$-invariant line-element of $\mathbb{R}$ which therefore descends to
$\mathsf{S}$.
Let $\varphi$ be a representative in
$\mathop{\mathrm{Diff}}\nolimits_{\pi_1(\mathsf{S})}(\mathbb{R})$ of a diffeomorphism of $\mathsf{S}$ and let
$\tilde\varphi=\Phi\circ\varphi\circ\Phi^{-1}$ denote the diffeomorphism it
induces on $\mathop{\bbR P^1\!}\nolimits$.
\begin{proposition}
(i) The \textrm{Euclidean cocycle} $\mathbf{E}(\varphi)=\Phi^*E(\tilde\varphi)$ where
$E(\tilde\varphi)=\log((\tilde\varphi^*d\theta)/d\theta)$ reads
\begin{equation}
\mathbf{E}(\varphi)=\log\left(\frac{\varphi^*\lambda}{\lambda}\right).
\label{Euclide}
\end{equation}
\goodbreak
(ii) The \textrm{affine cocycle} $\mathbf{A}(\varphi)=\Phi^*dE(\tilde\varphi)$ is then
\begin{equation}
\mathbf{A}(\varphi)=d\mathbf{E}(\varphi).
\label{Affine}
\end{equation}
(iii) The \textrm{Schwarzian cocycle}
$\mathbf{S}(\varphi)=\Phi^*S(\tilde\varphi)$ (see
(\ref{generalSchwarzian},\ref{generalSchwarzianLoc})) retains the form
\begin{equation}
\mathbf{S}(\varphi)
=
\lambda\,d\left(\frac{\mathbf{A}(\varphi)}{\lambda}\right)
-\frac{1}{2}\mathbf{A}(\varphi)^2.
\label{NiceSchwarzian}
\end{equation}
\end{proposition}
\textit{Proof:}
We easily prove (iii) by noticing that the Schwarzian
(\ref{TheFormule}) can be written in term of the affine coordinate
$\theta$ of $\mathop{\bbR P^1\!}\nolimits$ as
$$
S(\tilde\varphi)
=
d\theta\,d\left(\frac{\tilde\varphi''(\theta)}{\tilde\varphi'(\theta)}\right)
-
\frac{1}{2}\left(
\frac{\tilde\varphi''(\theta)}{\tilde\varphi'(\theta)}\,d\theta
\right)^2
$$
and the affine cocycle
as $A(\tilde\varphi)=(\tilde\varphi''(\theta)/\tilde\varphi'(\theta))\,d\theta$.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
For example, the $\mathop{\mathrm{Diff}_+}\nolimits(\bbT)$-Schwarzian in angular
coordinate is recovered with
$\Phi$ as in (\ref{T}); one finds
$$
\mathbf{S}(\varphi)(\theta)=\widetilde{S}(\varphi)(\theta),
$$
i.e., the modified Schwarzian derivative (\ref{SchwKS}). See also
\cite{Segal}.
\begin{proposition}\label{TheInfinitesimalCocycle}
The infinitesimal Schwarzian takes either forms
$$
\mathbf{s}(\xi)=s_1(\xi)
$$
for any $\xi\in\mathop{\rm Vect}\nolimits(\mathsf{S})$
with\footnote{Recall that $\mathop{\rm Div}\nolimits\xi=(L_\xi\lambda)/\lambda$.}
\begin{equation}
\mathbf{s}(\xi)
=
\lambda\,d\left(\frac{d\mathop{\rm Div}\nolimits\xi}{\lambda}\right)
\label{gf}
\end{equation}
and
$$
s_1(\xi)
=
{\textstyle\frac{3}{2}}{}\left(L_\xi\mathop{\textrm{\rm g}}\nolimits_1\strut\right)\!|\Delta.
$$
\end{proposition}
\textit{Proof:}
This follows clearly from (\ref{TheGeneralCocycle}) and
(\ref{NiceSchwarzian}).
\hskip 2truemm \vrule height3mm depth0mm width3mm
\begin{remark}
{\rm
In local affine coordinate on $\mathop{\bbR P^1\!}\nolimits$, the infinitesimal Schwarzian
(\ref{gf}) of $\xi=\xi(t)\partial/\partial{t}$ retains the familiar form
$$
\mathbf{s}(\xi)=\xi'''(t)\,dt^2.
$$
}
\end{remark}
\subsubsection{A Virasoro orbit}
With these preparations, let us formulate the
\begin{proposition}\label{dalphaProp}
Endow $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$ with the $1$-form $\alpha$
defined
by
\begin{equation}
\alpha(\delta\varphi)
=
\frac{1}{2}\int_{\mathsf{S}}{\!\mathbf{A}(\varphi)\delta{}\mathbf{E}(\varphi)}
\label{alpha}
\end{equation}
where
$\delta\varphi=\delta(\varphi\circ\psi)$ with
$\delta\psi=\xi\in\mathop{\rm Vect}\nolimits(\mathsf{S})$ at $\psi=\mathop{\mathrm{id}}\nolimits$.
(i) The exterior derivative of $\alpha$ is given, for
$\xi_1,\xi_2\in\mathop{\rm Vect}\nolimits(\mathsf{S})$, by
\begin{equation}
d\alpha(\delta_1\varphi,\delta_2\varphi)
=
\int_{\mathsf{S}}{\!\mathbf{S}(\varphi)([\xi_1,\xi_2])}
+
\underbrace{
\int_{\mathsf{S}}{\!d(\mathop{\rm Div}\nolimits\xi_1)\,\mathop{\rm Div}\nolimits\xi_2}.
}_{\mathop{\bf GF}\nolimits(\xi_1,\xi_2)}
\label{dalpha}
\end{equation}
(ii) If $\sigma$ denotes the canonical symplectic structure of the
$\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$-affine coadjoint orbit $\mathcal{O}$ of the origin with Souriau
cocycle $\mathbf{S}$ (see (\ref{CoadS})), namely if
\begin{eqnarray}
\mathcal{O} &=& \mathop{\rm Im}\nolimits(\mathbf{S})\\
&\cong&\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S}))/\mathop{\mathrm{PSL}(2,\bbR)}\nolimits
\label{O}
\end{eqnarray}
then
\begin{equation}
d\alpha=\mathbf{S}^*\sigma.
\label{KKS}
\end{equation}
\end{proposition}
\textit{Proof:}
Since
$
d\alpha(\delta_1\varphi,\delta_2\varphi)
=
\mathop{{\textstyle{\frac{1}{2}}}}\nolimits\int_{\mathsf{S}}{d(\delta_1\mathbf{E}(\varphi))\delta_2\mathbf{E}(\varphi)}
-
\mathop{{\textstyle{\frac{1}{2}}}}\nolimits\int_{\mathsf{S}}{d(\delta_2\mathbf{E}(\varphi))\delta_1\mathbf{E}(\varphi)}
$
let us first remark that
$$
\delta_j\mathbf{E}(\varphi)=\mathbf{A}(\varphi)(\xi_j)+\mathop{\rm Div}\nolimits\xi_j
$$
with the above notation. If we posit for convenience $a=\mathbf{A}/\lambda$, and
note that
$\lambda(\xi_1)\mathop{\rm Div}\nolimits\xi_2-\lambda(\xi_2)\mathop{\rm Div}\nolimits\xi_1=\lambda([\xi_1,\xi_2])$,
a lengthy calculation then leads to
$$
d\alpha(\delta_1\varphi,\delta_2\varphi)
=
\int_{\mathsf{S}}{(da-\mathop{{\textstyle{\frac{1}{2}}}}\nolimits{}a^2\lambda)\lambda([\xi_1,\xi_2])}
+
\int_{\mathsf{S}}{d(\mathop{\rm Div}\nolimits\xi_1)\mathop{\rm Div}\nolimits\xi_2}.
$$
Whence the sought equation (\ref{dalpha}).
Now, the affine coadjoint orbit of $q_1\in\mathcal{Q}(\mathsf{S})$ given
by the action~(\ref{CoadS}) carries a canonical
symplectic structure $\sigma$ which reads \cite{JMS}:
\begin{equation}
\sigma(\delta_1q,\delta_2q)
=
\langle{}q,[\xi_1,\xi_2]\rangle + f(\xi_1,\xi_2)
\label{Souriau}
\end{equation}
at $q=\mathop{\rm Coad}\nolimits_\Theta(\varphi)q_1$; here $f\in{}Z^2(\mathop{\rm Vect}\nolimits(\mathsf{S}),\mathbb{R})$ is the
derivative of the group-cocycle
$\Theta\in{}Z^1(\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S}),\mathcal{Q}(\mathsf{S}))$ at the identity. The
expression~(\ref{dalpha}) of $d\alpha$ clearly matches that of
$\sigma$ (\ref{Souriau}) with $q_1=0$, $\Theta=\mathbf{S}$ and $f=\mathop{\bf GF}\nolimits$
where the Gelfand-Fuchs cocycle \cite{GF} reads
\begin{equation}
\mathop{\bf GF}\nolimits(\xi_1,\xi_2)
=
-\int_\mathsf{S}{\!\mathbf{s}(\xi_1)(\xi_2)}
\label{GelfandFuchs}
\end{equation}
according to (\ref{gf}).
\hskip 2truemm \vrule height3mm depth0mm width3mm
\goodbreak
Our main result is then given by
\begin{theorem}\label{main}
The map
\begin{equation}
J_c:
g\longmapsto
\frac{3}{2}\left(\strut\mathop{\textrm{\rm g}}\nolimits-\mathop{\textrm{\rm g}}\nolimits_c\right)\!\big|\Delta
\label{Jc}
\end{equation}
establishes a symplectomorphism\footnote{It is the momentum
map of the hamiltonian action of $\mathop{\rm Conf}\nolimits_+(\mathcal{H})$ on
$(M_c,\omega_c)$.}
\begin{equation}
J_c:(M_c,\omega_c)\longrightarrow(\mathcal{O}_c,\sigma_c)
\label{ourSymplecto}
\end{equation}
between the metrics of $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta$ conformally
related to $\mathop{\textrm{\rm g}}\nolimits_c$ and the affine coadjoint orbit $\mathcal{O}_c=c\cdot\mathcal{O}$
(see (\ref{O})) with central charge $c$, the inverse curvature
(\ref{curvature}).
\end{theorem}
\textit{Proof:}
Let us denote by $\mathop{\textrm{\rm g}}\nolimits_c:\mathop{\rm Conf}\nolimits_+(\mathcal{H})\longrightarrow{}M_c$ the orbital map
and let us put $\mathop{\textrm{\rm g}}\nolimits=\mathop{\textrm{\rm g}}\nolimits_c(\varphi)=\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_c$. We find, using
(\ref{theSymplecticStructure}),
\begin{eqnarray*}
\omega_1(\delta_1\!\mathop{\textrm{\rm g}}\nolimits,\delta_2\!\mathop{\textrm{\rm g}}\nolimits)
&=&
{\textstyle\frac{3}{2}}
\int_\Delta{\!i_{\xi_1}L_{\xi_2}(\mathop{\textrm{\rm g}}\nolimits-\mathop{\textrm{\rm g}}\nolimits_1)}
+
{\textstyle\frac{3}{2}}
\int_\Delta{\!i_{\xi_1}L_{\xi_2}(\mathop{\textrm{\rm g}}\nolimits_1)}\\[6pt]
&=&
{\textstyle\frac{3}{2}}
\int_\Delta{\!(\mathop{\textrm{\rm g}}\nolimits-\mathop{\textrm{\rm g}}\nolimits_1)([\xi_1,\xi_2])}
+
{\textstyle\frac{3}{2}}
\int_\Delta{\!i_{\xi_1}L_{\xi_2}(\mathop{\textrm{\rm g}}\nolimits_1)}\\[6pt]
&=&
\int_\Delta{\!S_1(\varphi)([\xi_1,\xi_2])}
-
\int_\Delta{\!s_1(\xi_1)(\xi_2)}\\[6pt]
&=&
\int_\Delta{\!\mathbf{S}(\varphi)([\xi_1,\xi_2])}
-
\int_\Delta{\!\mathbf{s}(\xi_1)(\xi_2)}
\end{eqnarray*}
with the help of Propositions \ref{TheGeneralCocycleTh} and
\ref{TheInfinitesimalCocycle}.
Note that we have taken into account the skew-sym\-metry
of the Gelfand-Fuchs cocycle introduced in (\ref{dalpha}) and
(\ref{GelfandFuchs}). One thus gets
\begin{equation}
\omega_1(\delta_1\!\mathop{\textrm{\rm g}}\nolimits,\delta_2\!\mathop{\textrm{\rm g}}\nolimits)
=
\langle\mathbf{S}(\varphi),[\xi_1,\xi_2]\rangle+\mathop{\bf GF}\nolimits(\xi_1,\xi_2)
\label{omega1}
\end{equation}
and, since $\mathop{\textrm{\rm g}}\nolimits_c=c\,\mathop{\textrm{\rm g}}\nolimits_1$,
$$
\omega_c=c\,\omega_1.
$$
Thanks to (\ref{dalpha}) and (\ref{KKS}), one can claim that
\begin{eqnarray*}
d\alpha
&=&
\mathop{\textrm{\rm g}}\nolimits_1^*\omega_1\\
&=&\mathbf{S}^*\sigma.
\end{eqnarray*}
At last, this clearly entails
$$
\omega_c=J_c^*\sigma_c
$$
where $\sigma_c=c\,\sigma$ is the canonical symplectic structure on
$\mathcal{O}_c$.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\goodbreak
The following diagram summarizes our claim.
$$
\diagram
{
\mathop{\rm Conf}\nolimits_+(\mathcal{H}) & \hfl{\cong}{} & \mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})\cr
\vfl{\mathop{\textrm{\rm g}}\nolimits_c}{} && \vfl{}{c\,\mathbf{S}}\cr
M_c & \hfl{\cong}{J_c} & \mathcal{O}_c\cr
}
$$
\subsection{Homogeneous space $\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})/\bbT$}
\label{DiffOverS1}
Consider then the flat case (\ref{g0}) and introduce the space $M_0$
of metrics (see Figure \ref{ConfClassesEPSF}) on $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta$ related
to $\mathop{\textrm{\rm g}}\nolimits_0$ by a conformal diffeomorphism, \textit{viz.}
$$
M_0=\{\mathop{\textrm{\rm g}}\nolimits\in\class{\mathop{\textrm{\rm g}}\nolimits_1}\,|\,\mathop{\textrm{\rm g}}\nolimits=\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_0,\varphi\in\mathop{\rm Conf}\nolimits_+(\mathcal{H})\}.
$$
\begin{theorem}
The homogeneous space
\begin{eqnarray*}
M_0
&=&
\mathop{\rm Im}\nolimits(\varphi\longmapsto\varphi^*\!\mathop{\textrm{\rm g}}\nolimits_0)\\
&\cong&
\mathop{\rm Conf}\nolimits_+(\mathcal{H})/\mathop{\rm Isom}\nolimits_+(\mathcal{H},\mathop{\textrm{\rm g}}\nolimits_0)
\end{eqnarray*}
is endowed with a (weak) symplectic structure $\omega_0$ which reads
\begin{eqnarray}
\omega_0(\delta_1\!\mathop{\textrm{\rm g}}\nolimits,\delta_2\!\mathop{\textrm{\rm g}}\nolimits)
&=&\label{omegazero}
\int_\Delta{\strut\!i_{\xi_1}L_{\xi_2}\!\mathop{\textrm{\rm g}}\nolimits}\\[6pt]
&=&
\int_\Delta{\strut\!\!\mathop{\textrm{\rm g}}\nolimits([\xi_1,\xi_2])}
\label{omegazeroBis}
\end{eqnarray}
where $\delta_j\!\mathop{\textrm{\rm g}}\nolimits=L_{\xi_j}\!\mathop{\textrm{\rm g}}\nolimits$ with $\xi_j\in\mathop{\rm Vect}\nolimits(\mathsf{S})$.
\end{theorem}
\textit{Proof:}
Since $\mathop{\textrm{\rm g}}\nolimits_0$ can be prolongated to $\Delta$, (\ref{omegazero}) may
be rewritten as (\ref{omegazeroBis}) which is manifestly
skew-symmetric in its arguments. The closed $2$-form $\omega_0$ is weakly
non degenerate as
$\delta_2\!\mathop{\textrm{\rm g}}\nolimits\in\ker\omega_0$ iff $L_{\xi_2}\!\mathop{\textrm{\rm g}}\nolimits\!|\Delta=0$, i.e.
$\delta_2\!\mathop{\textrm{\rm g}}\nolimits=0$ in view of (\ref{S0}) and (\ref{isomzero}).
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
In fact, $M_0$ is symplectomorphic to a $\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$-coadjoint
orbit \cite{Kiri} as shown below.
\goodbreak
Let us consider the following quadratic differential
\begin{equation}
q_0=\mathop{\textrm{\rm g}}\nolimits_0\!|\Delta\in\mathcal{Q}(\mathsf{S})
\label{q0}
\end{equation}
so that the $\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$-coadjoint (anti-)action\footnote{We, indeed,
have $\mathop{\rm Coad}\nolimits(\varphi)(q_0)=(q_0\circ\mathop{\rm Ad}\nolimits)(\varphi)$ for all $\varphi\in\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$.}
$\mathop{\rm Coad}\nolimits$ given by (see (\ref{DefCoad}))
$\mathop{\rm Coad}\nolimits(\varphi):q_0\longmapsto{}q=\varphi^*q_0$, reads according to (\ref{S0}):
\begin{equation}
q=q_0+S_0(\varphi).
\label{Coad}
\end{equation}
\begin{proposition}\label{dalphazeroProp}
Endow $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$ with the $1$-form $\alpha_0$ defined by
$$
\alpha_0(\delta\varphi)
=
-\int_{\mathsf{S}}{\!(\varphi^*q_0)(\xi)}
$$
where, again,
$\delta\varphi=\delta(\varphi\circ\psi)$ with
$\delta\psi=\xi\in\mathop{\rm Vect}\nolimits(\mathsf{S})$ at $\psi=\mathop{\mathrm{id}}\nolimits$.
(i) We have, for any $\xi_1,\xi_2\in\mathop{\rm Vect}\nolimits(\mathsf{S})$,
$$
d\alpha_0(\delta_1\varphi,\delta_2\varphi)
=
\int_{\mathsf{S}}{\!(\varphi^*q_0)([\xi_1,\xi_2])}.
$$
(ii) The $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$-coadjoint orbit through $q_0$ (\ref{q0}) is
\begin{eqnarray}
\mathcal{O}_{q_0}
&=&\mathop{\rm Im}\nolimits(q_0\circ\mathop{\rm Ad}\nolimits)\\
&\cong&
\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})/\bbT
\label{Oq0}
\end{eqnarray}
and is endowed with the symplectic $2$-form $\sigma_0$ such that
$$
d\alpha_0=(q_0\circ\mathop{\rm Ad}\nolimits)^*\sigma_0.
$$
\end{proposition}
\textit{Proof:}
If $\delta_j\varphi$ is associated with $\xi_j\in\mathop{\rm Vect}\nolimits(\mathsf{S})$ at
$\varphi\in\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$, one readily finds $\delta_jq=L_{\xi_j}q$ and
$d\alpha_0(\delta_1\varphi,\delta_2\varphi)
=
-\alpha_0([\delta_1,\delta_2]\varphi)
=
\langle{}q,[\xi_1,\xi_2]\rangle
$
which descends as the canonical symplectic $2$-form $\sigma_0$ of
$\mathcal{O}_{q_0}$, namely
$$
d\alpha_0(\delta_1\varphi,\delta_2\varphi)
=
\sigma_0(\delta_1q,\delta_2q).
$$
We then simply check that $\ker(d\alpha_0)$ is $1$-dimensional and
integrated by $\ker(S_0)\cong\mathbb{T}$ (see (\ref{isomzero}) and (\ref{Coad})).
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
The ``flat'' counterpart of Theorem \ref{main} is now at hand.
\begin{theorem}\label{mainzero}
The map
\begin{equation}
J_0:
\mathop{\textrm{\rm g}}\nolimits\longmapsto\mathop{\textrm{\rm g}}\nolimits\big|\Delta
\end{equation}
establishes a symplectomorphism\footnote{It is the momentum
map of the hamiltonian action of $\mathop{\rm Conf}\nolimits_+(\mathcal{H})$ on
$(M_0,\omega_0)$.}
\begin{equation}
J_0:(M_0,\omega_0)\longrightarrow(\mathcal{O}_{q_0},\sigma_0)
\end{equation}
between the metrics of $\mathcal{H}=\mathsf{S}\times\mathsf{S}-\Delta$ conformally
related to $\mathop{\textrm{\rm g}}\nolimits_0$ and the coadjoint orbit $\mathcal{O}_{q_0}$ (see \ref{Oq0}) with
zero central charge.
\end{theorem}
\textit{Proof:}
Clear.
\hskip 2truemm \vrule height3mm depth0mm width3mm
\subsection{Bott-Thurston cocycle and contactomorphisms}
It is know since the work of Kirillov \cite{Kiri} that the
$\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$-homogeneous spaces we dealt with in Sections
\ref{DiffOverPSL2} and \ref{DiffOverS1} are, in fact, genuine coadjoint
orbits of the Virasoro group, $\mathop{\rm Vir}\nolimits$, i.e., the $(\mathbb{R},+)$-central
extension \cite{TW} of $\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$ that can be recovered as follows in
our setting.
Let us emphasize that the $1$-form $\alpha$ (\ref{alpha}) on $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$
fails to be invariant. So, let us equip
$\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})\times\mathbb{R}$ with the following ``contact'' $1$-form~$\widehat{\alpha}$, \textit{viz.}
\begin{equation}
\widehat{\alpha}(\delta\varphi,\delta{t})=\alpha(\delta\varphi)+\delta{t}.
\label{hatAlpha}
\end{equation}
Now, the $2$-form $d\widehat{\alpha}$ is $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$-invariant and plainly
descends to $M_1$ as~$\omega_1$ (see (\ref{KKS}) and
(\ref{Jc},\ref{ourSymplecto})). We now have the
\begin{proposition}
Lifting $\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})$ into the group of auto\-morphisms of
$(\mathop{\mathrm{Diff}}\nolimits_+(\mathsf{S})\times\mathbb{R},\widehat{\alpha})$ yields the Virasoro group $\mathop{\rm Vir}\nolimits$
with multiplication law
\begin{equation}
(\varphi_1,t_1)\cdot(\varphi_2,t_2)
=
\Big(
\varphi_1\circ\varphi_2,t_1+t_2
\underbrace{
-\frac{1}{2}\int_{\mathsf{S}}{\!\mathbf{E}(\varphi_1\circ\varphi_2)\mathbf{A}(\varphi_2)}
}_{\mathop{\bf BT}\nolimits(\varphi_1,\varphi_2)}
\Big)
\label{BT}
\end{equation}
where $\mathop{\bf BT}\nolimits$ is the Bott-Thurston cocycle \cite{Bott} of
$\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})\cong\mathop{\rm Conf}\nolimits_+(\mathcal{H})$.
\end{proposition}
\textit{Proof:}
Using the cocycle relation
$\mathbf{E}(\varphi\circ\psi)=\psi^*\mathbf{E}(\varphi)+\mathbf{E}(\psi)$---see (\ref{Euclide})---and
(\ref{Affine},\ref{alpha}), one immediately finds
\begin{eqnarray*}
\alpha(\delta(\varphi\circ\psi))
&=&
\mathop{{\textstyle{\frac{1}{2}}}}\nolimits\int_\mathsf{S}{\!\psi^*(\mathbf{A}(\varphi)\delta(\mathbf{E}(\varphi))}
+
\mathop{{\textstyle{\frac{1}{2}}}}\nolimits\int_\mathsf{S}{\!\mathbf{A}(\psi)\delta(\mathbf{E}(\varphi\circ\psi))}\\[6pt]
&=&
\alpha(\delta\varphi)
+
\delta\left[
\mathop{{\textstyle{\frac{1}{2}}}}\nolimits\int_\mathsf{S}{\!\mathbf{E}(\varphi\circ\psi)\mathbf{A}(\psi)}
\right]
\end{eqnarray*}
for all $\varphi,\psi\in\mathop{\mathrm{Diff}_+}\nolimits(\mathsf{S})$. Looking for those maps
$(\varphi,t)\longmapsto(\varphi^\star,t^\star)$ such that
$\varphi^\star=\varphi\circ\psi$ and
$\widehat{\alpha}(\delta\varphi^\star,\delta{t^\star})=\widehat{\alpha}(\delta\varphi,\delta{t})$
leads to $t^\star=t+\mathop{\bf BT}\nolimits(\varphi,\psi)+\mathop\mathrm{const.}\nolimits$, hence, to the group law
(\ref{BT}).
\hskip 2truemm \vrule height3mm depth0mm width3mm
\medskip
The triple $(\mathbf{S},\mathop{\bf GF}\nolimits,\mathop{\bf BT}\nolimits)$ is a special instance of a general
structure that has been coined ``trilogy of the moment'' \cite{Igl}.
\begin{remark}
{\rm
It would be interesting to have a conformal interpretation of
the contact structure $\mathop{\rm Vir}\nolimits/(\ker\widehat{\alpha}\cap\ker{d\widehat{\alpha}})$ above
$(M_1,\omega_1)$.
}
\end{remark}
\goodbreak
\section{Conclusion and outlook}\label{ConclusionAndOutlook}
This work prompts a series of more or less ambitious questions connected
with the striking analogies between conformal geometry of Lorentz surfaces
and projective geometry of conformal infinity that we have just discussed.
It constitutes an introduction to a more detailed paper (in preparation)
where the authors wish to tackle the following problems.
\begin{enumerate}
\item Is it possible to realize any Virasoro coadjoint
orbit\footnote{Other isotropy groups are, e.g., the finite coverings of
$\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$ and $1$-parameter subgroups of the form $\bbT\times\mathbb{Z}_n$; see
\cite{Guieu2}.} as a conformal class of Lorentz metrics on the cylinder?
If this is so, spell out the symplectic forms in terms of the classes of
metrics; also study the relationship between the properties of an
orbit and the dynamics of the null foliations in the associated conformal
class. There exists, in fact, a map sending the space of Virasoro
orbits---modules of projective structures on the circle---to the space of
modules of Lorentzian conformal structures on the cylinder; analyze its
properties. More conceptually, given a conic $C$ in the real
projective plane, what are the links between the space of projective
structures on $C$, the space of Lorentzian structures in the exterior of $C$
and the space of Riemannian metrics in the interior of $C$?
\item The Ghys theorem \cite{Ghys,OT} states that any diffeomorphism of the
projective line has at least four points where its Schwarzian vanishes,
i.e., four points where the contact of the graph of the diffeomorphism
with its osculating hyperbola is greater than the generic one. This result
is a Lorentzian analogue of the so-called four vertices
theorem\footnote{Any closed simple curve in the plane admits at least four
points where its Euclidean curvature is critical.} for closed curves in
the Euclidean plane. In our context, the Ghys theorem would imply the
existence, for any conformal automorphism of the hyperboloid, of some
particular points where this diffeomorphism is closer than usual to an
isometry.
\item The orbit $\mathop{\mathrm{Diff}}\nolimits(\bbT)/\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$ embeds symplectically in the universal
Teichm\"uller space $T(1)=\mathop{\mathrm{QS}}\nolimits(\bbT)/\mathop{\mathrm{PSL}(2,\bbR)}\nolimits$, where $\mathop{\mathrm{QS}}\nolimits(\bbT)$ denotes the
group of quasi-symmetric homeomorphisms of the circle~\cite{NV}. With the
help of the quantum differential calculus of Connes, it is possible to
construct extensions of the three fundamental cocycles $\mathbf{E}$, $\mathbf{A}$ and
$\mathbf{S}$ to the group
$\mathop{\mathrm{QS}}\nolimits(\bbT)$~\cite{NS}. Can one construct a ``quantum analogue'' of the
Lorentzian hyperboloid whose conformal class may be identified with~$T(1)$?
\end{enumerate}
Let us finally mention two other subjects closely connected with
our problem, namely the geometry of the Wess-Zumino-Witten model \cite{FG}
and Douglas' proof of the Plateau problem revisited by Guillemin,
Kostant and Sternberg \cite{GKS}.
|
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| 8,455
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{"url":"https:\/\/www.semanticscholar.org\/paper\/Phenomenology-of-the-top-mass-in-realistic-extended-Appelquist-Evans\/1a59fc04e31ecb06655fff1e648529e2e870d39b","text":"# Phenomenology of the top mass in realistic extended technicolor models\n\n@article{Appelquist1996PhenomenologyOT,\ntitle={Phenomenology of the top mass in realistic extended technicolor models},\nauthor={Thomas Appelquist and Nick Evans and Stephen B. Selipsky},\njournal={Physics Letters B},\nyear={1996},\nvolume={374},\npages={145-151}\n}\n\u2022 Published 22 January 1996\n\u2022 Physics\n\u2022 Physics Letters B\n8 Citations\n\n## Figures from this paper\n\n\u2022 Physics\n\u2022 2005\nWe consider a minimal technicolour theory with two techniflavours in the adjoint representation of an SU(2) technicolour gauge group which has been argued to feature walking dynamics. We show how to\n\u2022 Physics\n\u2022 2007\nWe calculate the Z boson propagator correction, as described by the S parameter, in technicolor theories with extended technicolor interactions included. Our method is to solve the Bethe-Salpeter\n\u2022 Physics\n\u2022 2006\nWe construct extended technicolor (ETC) models that can produce the large splitting between the masses of the $t$ and $b$ quarks without necessarily excessive contributions to the $\\rho$ parameter or\nWe study the role new gauge interactions in extensions of the standard model play in air showers initiated by ultrahigh-energy cosmic rays. Hadron-hadron events remain dominated by quantum\n\u2022 Physics\n\u2022 2010\nWe construct and analyze an ultraviolet extension of a model in which electroweak symmetry breaking is due to both technifermion and top-quark condensates. The model includes dynamical mechanisms for\n\n## References\n\nSHOWING 1-10 OF 41 REFERENCES\n\n\u2022 Physics\nPhysical review. D, Particles and fields\n\u2022 1995\nThe diagonal contribution can naturally explain the LEP result of Rb in some models of the ETC theory and some mechanisms which genera~e negative contribution to the T parameter are needed to explain the the experimental value of R b by the diagonal ETC boson.","date":"2023-02-01 10:36:21","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\": 0.4993171989917755, \"perplexity\": 4652.097493872748}, \"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-2023-06\/segments\/1674764499919.70\/warc\/CC-MAIN-20230201081311-20230201111311-00676.warc.gz\"}"}
| null | null |
Molecules and Radicals
Magnetic properties of cobalt(II) complex with 3-(3, 5-dimethylpyrazol-1-yl)quinoxaline-2-one
Effective magnetic moment of dichloro-bis[3-(3,5-dimethylpyrazol-1-yl)quinoxaline-2-one]cobalt(II) measured using Faraday method is given in this chapter.
Get Access PDF
Magnetic Properties of Paramagnetic Compounds
Book DOI
Chapter DOI
A. Gupta (3)
Editor Affiliation
3 Department of Chemistry, University of Delhi, Delhi, India
R. T. Pardasani (10)
P. Pardasani (20)
10 Department of Chemistry, School of Chemical Sciences and Pharmacy, Central University of Rajasthan, Bandar Sindri, Ajmer, India
20 Department of Chemistry, University of Rajasthan, Jaipur, India
R. T. Pardasani, P. Pardasani (2017) A. Gupta (ed.) SpringerMaterials
Magnetic properties of cobalt(II) complex with 3-(3, 5-dimethylpyrazol-1-yl)quinoxaline-2-one Molecules and Radicals 31D (Magnetic Properties of Paramagnetic Compounds)
10.1007/978-3-662-53971-2_377 (Springer-Verlag Berlin Heidelberg © 2017) Accessed: 20-01-2021
|
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| 6,993
|
/* LICENSE_BEGIN
MIT License
SPDX:MIT
https://spdx.org/licenses/MIT
See LICENSE.txt file in the top level directory.
LICENSE_END */
#include "zed5/core/base/base.h"
#include <chrono>
#include "zed5/core/util/backoff.h"
#include "zed5/core/util/getopt.h"
using namespace org::zed5;
using core::BackoffSequence;
using core::CommandLineParser;
using core::Flag;
using core::ParsedCommandLine;
BackoffSequence<std::chrono::seconds> makeBackoffSequence() {
// Parameters from RFC6762#section-5.2.
using namespace std::literals::chrono_literals;
return BackoffSequence<std::chrono::seconds>::Builder()
.withInitialRetransmissionTime(1s)
.withPerIterationScalingFactor(2U)
.withMaximumRetransmissionTime(1h)
.withJitter()
.build();
}
int main(int argc, char **argv, char **env) {
UNUSED(env);
ParsedCommandLine cmdline = CommandLineParser()
.with(Flag("watch", "true"))
.with(Flag("namw", Flag::Argument::REQUIRED, ".local."))
.parse(argc, argv);
return 0;
}
|
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| 8,275
|
/* -------------------------------------------------------------- */
/* (C)Copyright 2001,2008, */
/* International Business Machines Corporation, */
/* Sony Computer Entertainment, Incorporated, */
/* Toshiba Corporation, */
/* */
/* All Rights Reserved. */
/* */
/* Redistribution and use in source and binary forms, with or */
/* without modification, are permitted provided that the */
/* following conditions are met: */
/* */
/* - Redistributions of source code must retain the above copyright*/
/* notice, this list of conditions and the following disclaimer. */
/* */
/* - Redistributions in binary form must reproduce the above */
/* copyright notice, this list of conditions and the following */
/* disclaimer in the documentation and/or other materials */
/* provided with the distribution. */
/* */
/* - Neither the name of IBM Corporation nor the names of its */
/* contributors may be used to endorse or promote products */
/* derived from this software without specific prior written */
/* permission. */
/* */
/* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND */
/* CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, */
/* INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF */
/* MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE */
/* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR */
/* CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, */
/* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT */
/* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; */
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/* OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, */
/* EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */
/* -------------------------------------------------------------- */
/* PROLOG END TAG zYx */
#ifdef __SPU__
#ifndef _TAND2_H_
#define _TAND2_H_ 1
#include <spu_intrinsics.h>
#include "cos_sin.h"
#include "divd2.h"
/*
* FUNCTION
* vector double _tand2(vector double angle)
*
* DESCRIPTION
* The _tand2 function computes the tangent of a vector of "angle"s
* (expressed in radians) to an accuracy of double precision floating
* point.
*
*/
static __inline vector double _tand2(vector double angle)
{
vec_int4 octant;
vec_ullong2 select;
vec_double2 cos, sin;
vec_double2 num, den;
vec_double2 toggle_sign, answer;
/* Range reduce the input angle x into the range -PI/4 to PI/4
* by performing simple modulus.
*/
MOD_PI_OVER_FOUR(angle, octant);
/* Compute the cosine and sine of the range reduced input.
*/
COMPUTE_COS_SIN(angle, cos, sin);
/* For each SIMD element, select the numerator, denominator, and sign
* correction depending upon the octant of the original angle.
*
* octants angles numerator denominator sign toggle
* ------- ------------ --------- ----------- -----------
* 0 0 to 45 sin cos no
* 1,2 45 to 135 cos sin no,yes
* 3,4 135 to 225 sin cos yes,no
* 5,6 225 to 315 cos sin no,yes
* 7 315 to 360 sin cos yes
*/
octant = spu_shuffle(octant, octant, ((vec_uchar16) {
0,1, 2, 3, 0,1, 2, 3,
8,9,10,11, 8,9,10,11 }));
toggle_sign = spu_and((vec_double2)spu_sl(octant, 30),
(vec_double2) spu_splats(0x8000000000000000ULL));
select = (vec_ullong2)spu_cmpeq(spu_and(octant, 2), 0);
num = spu_sel(cos, sin, select);
den = spu_sel(sin, cos, select);
answer = spu_xor(_divd2(num, den), toggle_sign);
return (answer);
}
#endif /* _TAND2_H_ */
#endif /* __SPU__ */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,889
|
module Sociality
class << self
attr_accessor :config
def configure
yield self.config ||= Config.new
end
end
class Config
attr_accessor :allow_sites
def initialize
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,377
|
SK Telecom is First to Launch LTE-Advanced Service
Wednesday, June 26, 2013 #mobile, Korea, LTE-Advanced, SK Telecom
SK Telecom launched the world's first LTE-Advanced (LTE-A) service through smartphones.
The LTE-A service offers download speeds of up to 150 Mbps, which is two times faster than its regular LTE service, and 10 times faster than its 3G network. LTE-A coverage initially is available in Seoul and central city areas of Gyeongg-do and Chungcheong-do. Expansion to 84 cities nationwide is planned.
At top speed, LTE-A users can download an 800MB movie in just 43 seconds.
The LTE-A implementation leverages several advanced mobile network technologies including Carrier Aggregation (CA) and Coordinated Multi Point (CoMP). SK Telecom plans to implement Enhanced Inter-Cell Interference Coordination (eICIC) in 2014.
CA supports up to 150 Mpbs speed by combining two 10 MHz components carriers to form an effective bandwidth of 20 MHz. The company said further advancements should yield rates up to 300Mbps speed by aggregating two 20MHz component carriers by 2015, and become capable of combining three component carriers by 2016. An uplink capability using carrier aggregation should also be possible by 2016. The current CA standards allow for up to five 20 MHz carriers to be aggregated.
SK Telecom is offering LTE-A service at LTE prices. Customers will not have to pay more for the faster service.
Samsung's Galaxy S4 LTE-A smartphone is the first compatible device It will come in two colors, red (exclusively available at SK Telecom) and blue. The operator has secured an initial supply of 20,000 units of Galaxy S4 LTE-A. SK Telecom expects to have seven LTE-A devices on offer later this year.
SK Telecom' LTE-A phones will feature embedded services like 'Safe Message' and 'Safe Data Backup'. Safe Message is designed to protect users from smishing (SMS phishing) attacks by enabling them to check whether the message is sent from a trusted source; and 'Safe Data Backup' enables users to upload personal data stored in their smartphones to the cloud server to keep data safe from smartphone loss and accidental deletion. The company also plans to mount these features on all its to-be released LTE phones as well.
Some additional network service innovations from SK Telecom:
SK Telecom is launching a group video calling service for up to four users. The service, an upgraded version of the 3G network-based multi-party video conferencing service, will support 12 times better video quality and 2 times clearer audio quality.
SK Telecom's 'Btv mobile,' an IPTV service with 550,000 paid subscribers, will begin providing full HD (1080p resolution) video streaming service, for the first time in the world, from early July. Full HD video streaming requires a speed of 2Mbps or above, which is well supported by the LTE-A network.
The company will also launch 'T Baseball Multiview,' to enable users to watch two different games on one screen in July 2013. T Baseball is a free, real-time professional baseball game broadcast service optimized to the LTE network. Launched in August 2012, the service is currently enjoyed by 1.1 million users.
SK Telecom plans to launch 'T Freemium 2.0,' a free multimedia content package that offers three times more contents – e.g. dramas, TV entertainment shows, music videos, sports game highlights, etc.- than its previous version, 'T Freemium,' in July 2013.
SK Telecom plans to launch a new HD video-based shopping service in August 2013 to make shopping more fun and convenient for customers. Users will be able to seamlessly watch 6 different home shopping channels on one screen.
SK Telecom's online/mobile music portal service MelOn just opened a new service category to allow users to listen to original CD quality music by downloading Free Lossless Audio Codec (FLAC) files.
SK Telecom will hold a large-scale contest named 'LTE-A i.con' to boost the creation and provision of diverse innovative contents and applications optimized for the LTE-A network.
http://www.sktelecom.com/
NTT Comm Launches SDN-based Cloud Migration Service
Wednesday, June 26, 2013 #cloud, #SDN, Japan, NTT
NTT Communications is launching the world's first software-defined networking (SDN)-based cloud migration service.
The NTT Comm On-premises Connection links customer on-premises systems with NTT Com's Enterprise Cloud via the Internet using SDN technology.
NTT Comm said overcoming complexity will be the driving factor when considering migrating an application from on-premise to the cloud as this requires servers and data to be transferred sequentially. Conventionally, this involves many changes to network equipment settings and IP addresses in both servers and clients, which is a huge headache for many customers.
NTT Comm's On-premises Connection Service greatly simplifies the process through incorporation of unique SDN technology, which NTT Com developed in cooperation with VMware.
Some highlights:
By connecting on-premises systems and cloud in the same network segment using SDN-compatible gateway equipment installed in the customer's system, the service substantially reduces the workload normally required to design and configure on-premises networks. Also, once in the cloud environment after migration, customers can continue to use the existing IP addresses of their on-premises systems, a further reduction of workload.
Highly secure, encrypted data can be transferred between on-premises systems and Enterprise Cloud via the Internet at up to 100 Mbps. SDN technology enables customers to make changes promptly and flexibly, such as adjusting bandwidth to transfer large data in off-peak hours. Customers can even use their existing Internet connection lines for cloud migration.
On-demand use helps to minimize the cost of cloud migration because payment for the service, including gateway equipment, is on a per-day basis.
The initial fee of 231,000 yen includes the cost of installing gateway equipment at the customer's site. The daily charge is 11,718 yen and the maximum monthly charge has a ceiling of 234,150 yen. Internet connection from on-premises systems and two fixed global IP addresses for gateway equipment are charged separately.
The service is available in Japan starting this week and will be offered later in other markets worldwide.
http://www.ntt.com/aboutus_e/news/data/20130627.html
ZTE Develops Cloud Radio Scheduling/Coordination Functions for LTE
Wednesday, June 26, 2013 #cloud, #LTE, China, SDN, ZTE
ZTE is showcasing a new Cloud Radio Solution for 4G network optimization at the Mobile Asia Expo conference in Shanghai. China Mobile is already testing ZTE's Cloud Radio Solution to optimize trial TD-LTE networks in Guangzhou.
ZTE's Cloud Radio features newly developed Cloud Scheduling and Cloud Coordination modules for LTE network operators.
ZTE's Cloud Scheduling module utilizes a central scheduler to manage network resources on a real-time basis and achieve unified network scheduling. The Cloud Coordination module enables seamless and borderless coordination for the whole network to improve user experience. Cloud Scheduling realizes coordination at the cell level from a macro perspective, while Cloud Coordination realizes coordination at the user level from a micro perspective. The two-level coordination helps operators build smooth LTE networks.
The company said its Cloud Radio solution will prove most use for mitigating signal interference issues at the cell edge. multiple-layer coordination within bases stations and network users, the approach should lead to a better user experience.
In April 2013, ZTE and China Mobile conducted the Phase I Cloud Radio Solution verification in a project site in Guangzhou. When the Cloud Scheduling function was enabled with the network at more than 70% loading, the data throughput volume of cell edge users increased by 40%. When the Cloud Coordination function was enabled, the throughput volume increased between 20% and 100%, with the improvement most pronounced for cell edge users. The test by ZTE and China Mobile was the first site-level verification of LTE-Advanced coordination technology in the global telecommunications industry.
China Mobile has now expanded the scale of the test in order to collect more performance data. In June 2013, ZTE and China Mobile started the Phase II large-scale Cloud Radio verification in the Tianhe commercial area of Guangzhou, a complex wireless environment with a variety of data transmission requirements.
http://wwwen.zte.com.cn/en/press_center/news/201306/t20130627_401238.html
ZTE Hits 1Gbps with LTE-Advanced Carrier Aggregation
Wednesday, June 26, 2013 Chine, LTE, TD-LTE, ZTE
ZTE has achieved wireless data transmission rates of 1 Gbps in a live demonstration using LTE-Advanced.
In the demonstration, ZTE deployed Carrier Aggregation technology with four carrier frequencies on the F-band(1.9G Band) and D-band (2.6G Band). Carrier Aggregation is a core technology in the LTE-Advanced (LTE-A) standard, combining two or more carrier frequencies sharing the same or different bands into one channel, increasing the peak transmission speeds of TD-LTE cells by multiple times. ZTE used three carrier frequencies in the 2.6 G band, and on carrier in the 1.9 G band to complete the 1Gbps (F+D CA @ 4×4MIMO) demonstration successfully.
Broadcom's Latest StrataXGS Switch Delivers 10GbE Mobile Backhaul
Wednesday, June 26, 2013 #Backhaul, Broadcom, Ethernet, Silicon
Broadcom introduced its latest StrataXGS Ethernet switch chip for the aggregation layer of mobile backhaul networks.
The StrataXGS BCM56450 switch features a traffic manager that provides fine-grained quality of service and offers 10Gbps serial interfaces. The single-chip solution, which is part of Broadcom's switching portfolio for network access and aggregation, integrates many capabilities onto a single device, including high bandwidth switching with a control-plane processor, hierarchical traffic management, packet timing recovery, and advanced operation, administration and maintenance (OAM) functions.
The network aggregation layer could be used for multiple network architecture, including those based on VLANs, MPLS or MPLS-TP.
Advanced OAM functions and hardware-based protection switching enable a highly reliable backhaul network
Fully integrated timing solution with integrated digital phase-locked loop and 1588 v2 and synchronous Ethernet hardware support provides highly accurate time synchronization
Carrier Ethernet packet processor delivers line-rate L2 and VLAN switching, L3 routing, MPLS, MPLS-TP, and provider backbone bridging (PBB) while supporting L2, L3, MPLS, or PBB deployment models
Channelization over Ethernet and hierarchical traffic manager with DDR3-based packet buffer memory provides fine grained quality of service (QoS) to aggregate sub-1G microwave and small cell backhaul networks
On-chip ARM-based CPU and integrated 10G optical PHYs reduce bill-of-materials cost and enable a compact system design
The device, which is currently sampling, is implemented in 40nm. Production is slated for 1H 2014.
http://www.broadcom.com/press/release.php?id=s773805
Oracle's Gatekeeper Provides Carrier Billing, QoS Control for OTT Services
Wednesday, June 26, 2013 Oracle
Oracle released a new version of its Communications Services Gatekeeper featuring pre-integrated solutions to help network operators monetize over-the-top (OTT) content.
Essentially, the Oracle Communications Services Gatekeeper is an open, standards-based service exposure platform. The new release provides analytics and partner management portals that leverage APIs to track and bill for specific OTT services.
The Oracle Communications Services Gatekeeper runs on commercial-off-the-shelf platforms and offers built-in integration to Policy and Charging Rules Function (PCRF), online charging systems and network call control.
"In a competitive communications marketplace, it is vital that CSPs can rapidly deploy new and exciting services that meet customer and partner expectations. The latest release of Oracle Communications Services Gatekeeper provides CSPs with industry-standard technology combined with sophisticated management tools and analytics to securely open up networks to third-party services and applications while also helping to monetize those services and increase customer satisfaction," said Bhaskar Gorti, senior vice president and general manager, Oracle Communications.
http://www.oracle.com/us/corporate/press/1965921
ADVA's 16G Fibre Channel Gains Brocade Gen 5 Qualification
Wednesday, June 26, 2013 ADVA, Brocade, DWDM, Storage
ADVA Optical Networking's 16 Gbps Fibre Channel (FC) card has achieved Brocade qualification. The card delivers 16 Gbps FC over wavelength division multiplexing (WDM) networks, enabling enterprises to transport data across increasingly large distances. It has now completed interoperability testing as part of Brocade's Gen 5 Fibre Channel program and has already been deployed by a number of financial organizations.
"The continued explosion in bandwidth demand, when combined with growing expectations to access data-intensive applications, is driving customer need for the combination of ADVA Optical Networking transport solutions linked by Brocade Gen 5 Fibre Channel technologies," said Jack Rondoni, vice president, Data Center Storage and Solutions at Brocade. "The increased performance of Brocade Gen 5 Fibre Channel technology better enables the functionality of the ADVA FSP 3000 by simplifying network operations and improving network resiliency."
http://www.advaoptical.com/
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,381
|
{"url":"https:\/\/aedifion.gitlab.io\/swop\/flows.html","text":"# Flows\n\nWe distinguish three main flow types:\n\n1. Setpoint flows for single setpoints.\n2. Schedule flows for pre-defined schedules.\n3. Controls App flows for messages exchanged directly between a control application and the edge device.\n\n## Setpoint Flows\u00b6\n\n### Write Setpoint Flow\u00b6\n\nThe setpoint flow defines the type and order of messages for writing a single setpoint.\n\nThe usual flow without any errors proceeds as follows (square brackets mark optional steps).\n\n1. The user posts the parameters for the new setpoint to the API, i.e., the issuer in SWOP.\n2. The issuer validates the request, optionally creates internal state for this operation, then requests the setpoint write by sending a NEWSPT (new setpoint) message to the edge device, the receiver in SWOP.\n3. The edge device receives and validates the NEWSPT message, then writes the setpoint.\n4. If the setpoint must be acked (acknowledge flag), the edge device acknowledges success or error and additional state to the issuer in an ACKSPT message.\n5. The issuer receives the optional ACKSPT and updates its internal state of this setpoint operation accordingly.\n\nIf the receiver does not require and acknowledgement, Steps 4 and 5 are omitted.\n\n#### Handling lost NewSetpoint messages\u00b6\n\nThe receiver may be temporarily unreachable, or, more generally, the NEWSPT message may be lost in transit, e.g., when a non-direct transport such as MQTT is used between issuer and receiver. To handle this case, the issuer may choose to repeat its NEWSPT message after an appropriate timeout. It is, however, left to the application logic whether the write operation should be retried in this way or aborted altogether. Note that this is thus not a message or object field but must be a parameter of the API call (which is outside these protocol specifications).\n\n#### Handling lost AcknowledgeSetpoint messages\u00b6\n\nThe issuer may be temporarily unreachable, or, more generally, the ACKSPT message may be lost in transit, e.g., when a non-direct transport such as MQTT is used between issuer and receiver. The receiver knows its acknowledgement has been lost if it receives a duplicate NEWSPT message from the issuer. It then reacts by sending a copy of the lost ACKSPT to the issuer.\n\n## Schedule Flows\u00b6\n\n### Create Schedule Flow\u00b6\n\nThe Create Schedule Flow defines the type and order of messages for creating a new schedule and initializing it on the edge device.\n\nThe usual flow without any errors proceeds as follows.\n\n1. The user posts the parameters for the new schedule to the API, i.e., the issuer in SWOP.\n2. The issuer validates the request, creates internal state for this operation, then initiates the create schedule flow by sending a NEWSCHD (new schedule) message to the edge device, the receiver in SWOP.\n3. The edge device receives and validates the NEWSCHD message, then initiates the schedule.\n4. The edge device acknowledges success or error and additional state to the issuer in an ACKSCHD message.\n5. The issuer receives the acknowledgement and updates its internal state of this schedule operation accordingly.\n\n### Update Schedule Flow\u00b6\n\nThe Update Schedule Flow defines the type and order of messages for updating an active schedule.\n\nThe usual flow without any errors proceeds as follows.\n\n1. The user posts the reference of the target schedule and the parameters for the update to the issuer.\n2. The issuer validates the request, updates internal state for this operation, then initiates the update schedule flow by sending a UPSCHD (update schedule) message to the receiver.\n3. The edge device receives and validates the UPSCHD message, then updates the schedule.\n4. The edge device acknowledges success or error and additional state to the issuer in an ACKSCHD message.\n5. The issuer receives the acknowledgement and may again update its internal state of this schedule operation accordingly.\n\n### Acknowledge Schedule Flow\u00b6\n\nThe Acknowledge Schedule Flow defines the type and order of messages for acknowledging progress of the execution of a schedule from the receiver towards the issuer of the schedule. It is left to the SWOP implementation to decide which aspects of the execution of a schedule are acknowledged from the receiver to the issuer. We recommend to acknowledge success or failure of the execution of each setpoint defined in the schedule.\n\nThe usual flow without any errors proceeds as follows.\n\n1. An event occurs on the receiver side, e.g., a setpoint is written.\n2. The receiver acknowledges success or error and additional state about the occurred event to the issuer in an ACKSCHD message.\n3. The issuer receives the acknowledgement and may again update its internal state of this schedule operation accordingly.\n\n### Delete Schedule Flow\u00b6\n\nThe Delete Schedule Flow defines the type and order of messages for stopping an active schedule and deleting it on the receiver site. The schedule is marked as stopped on the issuer side but it's state should be kept for auditing purposes.\n\nThe usual flow without any errors proceeds as follows.\n\n1. The user posts the reference of the schedule to delete to the issuer in SWOP.\n2. The issuer validates the request, marks the schedule for deletion in the internal state, then initiates the delete schedule flow by sending a DELSCHD (delete schedule) message to the receiver.\n3. The edge device receives and validates the DELSCHD message, then stops and deletes the schedule.\n4. The edge device acknowledges success or error and additional state to the issuer in an ACKSCHD message.\n5. The issuer receives the acknowledgement and may again update its internal state of this schedule operation accordingly.\n\n## Controls App Flows\u00b6\n\nCurrently, the Controls App Flow is based on an MQTT pipeline hosted by aedifion, i.e., an MQTT broker with pre-defined topics.\n\n### Upsert Controls Flow\u00b6\n\nThe Upsert Controls Flow defines the type and order of messages for creating and updating a controls app and initializing it on the edge device.\n\nThe usual flow without any errors proceeds as follows.\n\n1. The user sends an UPSRTCTRL message via MQTT, i.e., the issuer in SWOP.\n2. The edge device receives and validates the UPSRTCTRL message and stores reset values and further connectivity-related values internally.\n3. The edge device acknowledges success or error to the issuer in an ACKUPSRTCTRL message.\n4. The issuer receives the ACKUPSRTCTRL, updates its internal state of this controls operation accordingly and starts to run a controls app.\n5. Sending of new setpoints as an output of a controls app is then based on the Setpoint Flow.\n\nIf the receiver does not require an acknowledgement, Step 4 is omitted.\n\n### Reset Controls Flow\u00b6\n\nThe Reset Controls Flow defines the type and order of messages for writing reset values of a controls app to the BAS.\n\nThe usual flow without any errors proceeds as follows.\n\n1. The user sends a RESETCTRL message via MQTT, i.e., the issuer in SWOP.\n2. The edge device receives and validates the RESETCTRL message and stores reset values and further connectivity related values internally.\n3. The edge device acknowledges success or error to the issuer in an ACKRESETCTRL message.\n4. The issuer receives the ACKRESETCTRL and updates its internal state of this controls operation accordingly.\n\nIf the receiver does not require an acknowledgement, Step 4 is omitted.\n\n### Delete Controls Flow\u00b6\n\nThe Delete Controls Flow defines the type and order of messages for deleting a controls app.\n\nThe usual flow without any errors proceeds as follows.\n\n1. The user sends a DELCTRL message via MQTT, i.e., the issuer in SWOP.\n2. The edge device receives and validates the DELCTRL message and deletes the corresponding controls app data.\n3. The edge device acknowledges success or error to the issuer in an ACKDELCTRL message.\n4. The issuer receives the ACKDELCTRL and updates its internal state of this controls operation accordingly.\n\nIf the receiver does not require an acknowledgement, Step 4 is omitted.","date":"2022-09-25 17:06:10","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\": 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\": 0.23023459315299988, \"perplexity\": 4300.402165276082}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030334591.19\/warc\/CC-MAIN-20220925162915-20220925192915-00757.warc.gz\"}"}
| null | null |
Roman J. Israel, Esq. is a 2017 American legal drama film written and directed by Dan Gilroy. The film stars Denzel Washington and Colin Farrell and follows the life of a civil rights advocate and defense lawyer (Washington) who finds himself in a tumultuous series of events that lead to a personal crisis and the necessity for extreme action.
The project was announced on August 25, 2016, as Gilroy's next directorial effort titled Inner City, but was renamed on June 22, 2017. Principal photography began in March 2017 and took place in Los Angeles and Santa Clarita, California.
Roman J. Israel, Esq. premiered at the 2017 Toronto International Film Festival on September 9, 2017, and was theatrically released in the United States by Sony Pictures Releasing on November 17, 2017. The film grossed just $13 million against its $22 million budget and received mixed reviews from critics. For his performance, Washington was nominated for the Academy Award, the Golden Globe, and the Screen Actors Guild Award.
Plot
Roman J. Israel is a lawyer earning $500 a week at a small law firm in Los Angeles. In his two-partner office, Israel is responsible for preparing briefs, often focusing on the civil rights of their defendants, while William Jackson, the firm's founder and a well-respected professor, focuses on the courtroom appearances that Israel struggles with. Israel has spent years developing a brief that he believes will bring reform to the unfair use of plea bargaining to induce guilty pleas in the justice system. Though short on social skills, Israel is gifted with a phenomenal memory as well as strong personal convictions, which he has pursued at the expense of family.
Jackson suffers a fatal heart attack. The firm is broke and will close, all to be handled by Jackson's former student, George Pierce. Pierce, who greatly admired Jackson and is impressed by Israel's legal mind, offers a job at his own large firm. Israel initially rejects this offer, believing that Pierce is simply a greedy lawyer. Israel meets Maya during a job interview at a local activist network. The interview does not go well, but Maya asks him to speak at an upcoming meeting organizing a protest. Struggling to find a job elsewhere, Israel reluctantly takes up Pierce's job offer.
Israel is a poor fit at his new firm, clashing with senior partner Jesse Salinas over a joke Salinas makes about battered women. After attempting to interest Pierce in his brief, Israel is disappointed to be assigned to handle clients. One such client is Derrell Ellerbee, a young man arrested for murder, who tells Israel that he is willing to divulge the whereabouts of the actual shooter, Carter "CJ" Johnson, and will testify against him. Israel goes behind Pierce's back to negotiate a plea deal with the district attorney. The prosecutor rejects his offer and hangs up on Israel after he insults her unsympathetic counter-offer. As a consequence, Ellerbee is also denied the protective custody he begged for in prison and is murdered as a snitch.
On the same evening, Israel is berated by Pierce for mishandling of Ellerbee's case, then is mugged by a homeless man he attempted to help. He becomes downcast and cynical, illegally using the information he received from Ellerbee to anonymously collect the $100,000 reward for Johnson's location. Israel indulges in luxuries he had previously eschewed. Pierce apologizes to Israel for berating him earlier and for forcing him out of the shadows, accepting that he thrives working behind the scenes as he did at his old firm. Maya calls Israel to ask him out on a date, where she shares some of her struggles with idealism and thanks Israel for his inspiring her. Pierce invites Israel to a luxury box, where he shares some of his big plans for their future at the firm.
Pierce calls Israel to meet a new client arrested for murder. Before the meeting, Pierce resumes the conversation from the game stating that Israel's dedication to justice has touched him, and that he is reorganizing the firm to take on pro bono cases handled by Israel. Pierce offers to work with Israel on his brief. Israel, still a bit despondent, is unmoved by these developments. They go in to see the client, who turns out to be Johnson. Meeting Israel in jail, Johnson accuses him of divulging privileged communications to collect the reward money, and resolves to torment Israel with the threat of jail time or death. Israel suffers a breakdown in which he becomes a law unto himself becoming both the lawyer and the defendant in one and judges his own actions as unlawful. Renouncing his momentary transgression, Israel goes home and returns the reward money with a note apologizing for taking it in the first place. He reconciles with Maya and Pierce, and tries to motivate them to pursue their inner sense of justice. He tells Pierce that he is turning himself in to the police for his crime. As Israel starts walking to a nearby station, he is shot and killed by one of Johnson's henchmen.
In the aftermath, Maya is seen to be renewed in her activism efforts, while Pierce files Israel's brief, in both their names, intent on continuing his efforts to reform the justice system.
Cast
Denzel Washington as Roman J. Israel
Colin Farrell as George Pierce
Carmen Ejogo as Maya Alston
Shelley Hennig as Olivia Reed
Lynda Gravatt as Vernita Wells
Amanda Warren as Lynn Jackson (niece)
Hugo Armstrong as Fritz Molinar
Sam Gilroy as Connor Novick
Tony Plana as Jesse Salinas
DeRon Horton as Derrell Ellerbee
Amari Cheatom as Carter Johnson
Nazneen Contractor as Assistant D.A. Melina Nassour
Joseph David-Jones as Marcus Jones
Henry G. Sanders as Pastor Jack
Production
On August 25, 2016, it was revealed that Dan Gilroy's next directorial project was Inner City, a legal drama in the vein of The Verdict. Gilroy was then courting Denzel Washington to star. It was reported on September 21, 2016 that Sony Pictures was closing a deal to distribute the film, with principal photography scheduled to begin in March 2017. Gilroy's collaborators on Nightcrawler, cinematographer Robert Elswit and editor John Gilroy, worked with him again on the project. On January 31, 2017, it was reported that Colin Farrell was in talks to join the cast. As of February 28, 2017, Ashton Sanders was in talks to join as well, though he was unable to because of scheduling conflicts. In April 2017, Nazneen Contractor and Joseph David-Jones joined the cast. As of April 21, 2017, Inner City had begun filming in Los Angeles. In June 2017, Carmen Ejogo joined the cast as a civil rights worker. On June 22, 2017, the film was renamed Roman J. Israel, Esq.
Music
James Newton Howard composed the film's music, as he previously worked with Gilroy in Nightcrawler. The score is now released at Sony Classical.
Release
The film had its world premiere at the Toronto International Film Festival on September 10, 2017, before its commercial release on November 17, 2017, initially limited, by Sony Pictures Releasing. Following its festival premiere, the film was re-edited to tighten its pacing, with a dozen minutes (including one whole subplot) being shaved off the final runtime, and a key scene regarding Colin Farrell's character being shifted from the third act to earlier in the film.
Reception
Box office
Roman J. Israel, Esq. grossed $12 million in the United States and Canada, and $1.1 million in other territories, for a worldwide total of $13 million.
Released alongside Justice League, The Star and Wonder on November 17, 2017. The film grossed $61,999 from four theaters in its limited opening weekend, for a per-venue average of $15,500. It then expanded to 1,648 theaters the following Wednesday, alongside the openings of Coco and The Man Who Invented Christmas. It went on to gross at $4.5 million over the three-day weekend (and $6.2 million over the five), finishing 9th at the box office. It fell 57% in its second weekend to $1.9 million.
Critical response
On Rotten Tomatoes, the film has an approval rating of 54% based on 177 reviews, with an average rating of 5.80/10. The website's critical consensus reads, "Intriguing yet heavy-handed, Roman J. Israel, Esq. makes the most of — but never quite lives up to — Denzel Washington's magnetic performance in the title role." On Metacritic, which assigns a normalized rating to reviews, the film has a weighted average score 58 out of 100, based 41 critics, indicating "mixed or average reviews". Audiences polled by CinemaScore gave the film an average grade of "B" on an A+ to F scale.
Writing for Rolling Stone, Peter Travers gave the film 3 out of 4 stars, praising Washington and writing, "In no way is his performance a stunt. Washington digs so deep under the skin of this complex character that we almost breathe with him. It's a great, award-caliber performance in a movie that can barely contain it." Richard Roeper of the Chicago Sun-Times gave the film 2 out of 4 stars. He also highlighted Washington, but criticized the narrative, saying, "Roman J. Israel, Esq. has pockets of intrigue, and writer-director Gilroy and Washington have teamed up to create a promising dramatic character. We just never get full delivery on that promise."
In his review for Empire, Simon Braund summarized the political motives in the film viewed as a legal thriller stating, "It (Roman's idealism) illustrates succinctly how at odds with the modern world Roman Israel is. A brilliant legal mind, trapped in the body of a twitchy social misfit, he has all the hallmarks of a true genius-savant — the interpersonal skills of a yeast cell, dress sense of an Open University lecturer circa 1973 and an unshakeable conviction that justice for the poor and dispossessed is a cause worth fighting for. To this deeply unfashionable end, he's spent decades toiling in the shadows at a tiny law firm, making trouble for The Man while compiling a vast, unwieldy brief he hopes will, one day, set the American legal system on its ear".
Owen Gleiberman of Variety wrote: "It leaves us with a character you won't soon forget, but you wish that the movie were as haunting as he is."
Accolades
References
External links
2010s legal drama films
Cross Creek Pictures films
Columbia Pictures films
Escape Artists films
Topic Studios films
Films directed by Dan Gilroy
Films with screenplays by Dan Gilroy
Films produced by Denzel Washington
Films scored by James Newton Howard
American legal drama films
Films set in Los Angeles
2017 drama films
2010s English-language films
2010s American films
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Sectionsfor Coronary artery disease
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Ed Walsh Trial: Golf Outings, Purchases & Political Dealings Scrutinized
Suffolk County Conservative Party Chairman Ed Walsh's political dealings and purchases he made for golf outings, ferry rides to Connecticut, and outdoor shopping malls took center stage during the second day of his ongoing federal fraud trial.
Prosecutors invested significant time Thursday questioning witnesses on records maintained by golf courses, a credit card company, and various retailers to bolster their accusations that Walsh was either hitting the greens, conducting Suffolk County Conservative Party business, gambling at casinos, or shopping while he was supposed to be working at the Suffolk County Sheriff's Office in Riverhead. Walsh, who was a lieutenant, has since retired.
A day earlier, prosecutors promised to layout a "paper trail" proving Walsh was absent from work.
Walsh's political connections also came into play on Day Two. The jury at U.S. District Court in Central Islip also heard testimony from Jerry Wolkoff, a wealthy real estate developer; Anthony Senft, a Suffolk County District Court Judge; and John Ruocco, a businessman who said he met Walsh through Oheka Castle owner Gary Melius.
Walsh was indicted in January 2015 for allegedly defrauding Suffolk of more than $80,000 in no-show work. He was additionally charged with theft of funds and wire fraud last March. Walsh pleaded not guilty and has denied the charges. His attorneys contend that if Walsh was not at his desk, it was because he was conducting business at the behest of his boss, Suffolk County Sheriff Vincent DeMarco.
Prosecutors filed a lengthy pretrial motion March 8 alleging that several attempts by DeMarco to investigate Walsh were killed by Suffolk County District Attorney Thomas Spota—accusations Spota's office has denied.
The trial continued Thursday with testimony from Christina Ofeldt, the bookkeeper at Hampton Hills Golf Course & Country Club, of which Walsh was a "weekday" member who benefited from a police and armed services discount. Walsh was also provided guest passes, which were used on multiple occasions, Ofeldt testified.
The country club had invoices from more than dozen dates between 2010 and 2013 containing purchases for golf rounds and other items that were posted on Walsh's account, including charges for golf carts, beverages, and a cigar, Ofeldt testified.
The government also produced a guest pass for a person identified as "Rich" and marked with an expiration date. Ofeldt, however, admitted that the club does not strictly enforce expiration dates.
Walsh's attorney Leonard Lato got Ofeldt to testify that there was no year specified on the date the guest pass was used—Aug. 13—therefore, impossible to know when the pair actually played. She also said there was no way of knowing if "Rich" was the person's true name and no way of knowing if Walsh left before completing his round.
The jury also heard testimony from Kevin Kline of the Metropolitan Golf Association, who oversees a program called Golf Handicap Information Network, which maintains golfers' statistics and evaluates scores to produce a handicap. Walsh used the system, Kline testified, and made 23 entries between Aug. 15, 2013 and Oct. 22, 2013 at various courses, including Sebonack Golf Course in Southampton. But on cross examination, Kline admitted he was unsure if the days Walsh golfed were actual work days.
Walsh played Sebonack as a guest at least four times, according to the course's director of golf, Jason McCarty. McCarty testified that Walsh and Wolkoff, who was a member of the course, played a round at 2:45 p.m. on May 10, 2013 and chose to walk the course. The pair also played a round with Senft on Friday, May 31, 2013, he testified, adding that Wolkoff is considered an "extremely fast golfer."
The most electric testimony of the day came from the 79-year-old Wolkoff, who admitted to contributing $20,000 to the Suffolk County Conservative Party between 2014 and 2015.
"I got to know Ed Walsh and I thought he was doing a terrific job," Wolkoff said, testifying that his donations were a result of Walsh's aptitude and not to curry favor with elected officials who may be able to assist him with zoning issues.
Wolkoff is in the midst of one of the largest redevelopment projects on Long Island, which would transform the abandoned Pilgrim State Psychiatric Center property in the Town of Islip into a mixed-use "town square."
Wolkoff said he remembered golfing with Walsh at Sebonack but, "I can't remember all the dates."
Asked whether they discussed business issues, Wolkoff said "I might of." Walsh, he testified, once commented that he may have to leave if he got a call from work, but Wolkoff said he didn't recall Walsh abandoning a round early.
When Wolkoff was asked whether Walsh ever facilitated meetings with elected officials, Wolkoff said "No," but later admitted that Walsh did help the developer meet Mary Kate Mullen, then a candidate for Islip Town Council, at his office. Walsh was also in attendance, Wolkoff testified.
Wolkoff said he was considering supporting Mullen's campaign and wanted to meet with her.
The developer testified that he donated $1,000 to the New York State Conservative Party from 2014 to 2016, a sliver compared to the $20,000 he donated to Suffolk's arm of the party.
Testimony transitioned into claims from prosecutors that Wolkoff reneged on an agreement to meet with the government prior to the trial. Catherine Mirabile, a prosecutor for the Eastern District of New York, then suggested Wolkoff met with Lato, an assertion Wolkoff brushed away by claiming all he said was "Hello."
When it was the defense's turn, Wolkoff said he's been a registered Conservative for more than 50 years but also contributed to Democrats and Republicans.
He also boasted about his pace on the golf course. "I like to move quickly," he said, adding that Walsh is not nearly as quick, which Walsh blames on bad knees.
"[Walsh's] life is his work, his family, and his Conservative Party," Wolkoff said.
Regarding his decision not to meet with prosecutors, he exclaimed: "I have nothing to hide."
The jury also heard from John Ruocco, who was the president and CEO of Interceptor Ignition, which had dealings with Melius, the Oheka Castle owner who survived a botched assassination attempt two years ago.
Ruocco testified that Walsh attended a shareholder's meeting on Feb. 21, 2014 at 11 a.m. in Shirley. He noted that Walsh had no business dealings and didn't speak during the two-hour meeting.
Prosecutors also produced evidence allegedly showing Walsh made reservations to travel to Connecticut by using the Cross Sound Ferry on four occasions between 2011 and 2014. One package also included a bus transfer to Mohegan Sun casino, according to testimony.
The government also claimed Walsh went shopping at three stores—Nike, Reebok and Neiman Marcus—on June 16, 2012, and used his American Express card to pay for a woman's blouse at one store and a "Yankees trainer" double extra-large shirt at another.
The defense got one witness to testify that there's no way of knowing that it was Walsh who actually purchased the blouse.
The day ended with testimony from Senft, who was endorsed by the Town of Islip Conservative Party last year when he campaigned for a county judge seat. Senft had originally planned to run for state senate, but dropped out due to a toxic dumping scandal that enveloped the town. At the time, Senft was the town board's park liaison.
Senft also said he found five dates in his Google calendar in which he attended Conservative Party events with Walsh.
Senft's testimony continues Monday when the trial resumes. The prosecution plans to produce photos from various political events Walsh and Senft attended.
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{"url":"https:\/\/www.rdocumentation.org\/packages\/memisc\/versions\/0.99.27.3\/topics\/Table","text":"# Table\n\n0th\n\nPercentile\n\n##### One-Dimensional Table of Frequences and\/or Percentages\n\nTable is a generic function that produces a table of counts or weighted counts and\/or the corresponding percentages of an atomic vector, factor or \"item.vector\" object. This function is intended for use with Aggregate or genTable. The \"item.vector\" method is the workhorse of codebook.\n\nKeywords\nunivar\n##### Usage\n# S4 method for atomic\nTable(x,weights=NULL,counts=TRUE,percentage=FALSE,\u2026)\n# S4 method for factor\nTable(x,weights=NULL,counts=TRUE,percentage=FALSE,\u2026)\n# S4 method for item.vector\nTable(x,weights=NULL,counts=TRUE,percentage=(style==\"codebook\"),\nstyle=c(\"table\",\"codebook\",\"nolabels\"),\ninclude.missings=(style==\"codebook\"),\nmissing.marker=if(style==\"codebook\") \"M\" else \"*\",\u2026)\n##### Arguments\nx\n\nan atomic vector, factor or \"item.vector\" object\n\ncounts\n\nlogical value, should the table contain counts?\n\npercentage\n\nlogical value, should the table contain percentages? Either the counts or the percentage arguments or both should be TRUE.\n\nstyle\n\ncharacter string, the style of the names or rownames of the table.\n\nweights\n\na numeric vector of weights of the same length as x.\n\ninclude.missings\n\na logical value; should missing values included into the table?\n\nmissing.marker\n\na character string, used to mark missing values in the table (row)names.\n\nother, currently ignored arguments.\n\n##### Value\n\nThe atomic vector and factor methods return either a vector of counts or vector of percentages or a matrix of counts and percentages. The same applies to the \"item.vector\" vector method unless include.missing=TRUE and percentage=TRUE, in which case total percentages and percentages of valid values are given.\n\n##### Aliases\n\u2022 Table\n\u2022 Table,atomic-method\n\u2022 Table,factor-method\n\u2022 Table,item.vector-method\n##### Examples\n# NOT RUN {","date":"2021-02-27 18:52:19","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\": 0.21467822790145874, \"perplexity\": 7175.10874411031}, \"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-2021-10\/segments\/1614178359082.48\/warc\/CC-MAIN-20210227174711-20210227204711-00537.warc.gz\"}"}
| null | null |
\section{Technical Details of Shape Building via Hierarchical Assembly}\label{sec:2ham-append}
In this section we present technical details of the construction from Section~\ref{sec:2ham}.
\subsection{Shape decomposition into blocks}\label{sec:shape-decomp}
We present a relatively naive, greedy algorithm for decomposing $S$ into a set $R$ of rectangular prisms, called {\tt block}s, that can combine to form a structure of shape $S$. (See Figure \ref{fig:2HAM-block-decomp} for an example of a shape and its decomposition into {\tt block}s.) Note that for our shape-replicating construction to work for $S$, it also requires that $S$, once divided into rectangular prisms, is block-diffusable. Our algorithm does not ensure block-diffusability, and in fact, we conjecture that there exist shapes for which this is not possible without arbitrarily scaling the shapes.
\begin{enumerate}
\item Define $S' = S$.
\item Initialize the set of {\tt blocks} $B = \emptyset$.
\item For each voxel $v \in S'$, define the function $P$ so that on input $v$, $P(v)$ returns the largest (by volume) rectangular prism (as the set of coordinates contained within it) containing $v$ within $S'$.\label{step:loop1}
\item Let $p_{max}$ be the largest rectangular prism (by volume) returned by $P$ for any $v \in S'$.
\item Add $p_{max}$ as a {\tt block} to the set of {\tt block}s $B$, and remove the voxels of $p_{max}$ from $S'$. (Note that this may make $S'$ into a disconnected set of points, but that is okay.)
\item If $S' \neq \emptyset$, return to step \ref{step:loop1}.
\end{enumerate}
We now have $B$ as a preliminary set of {\tt block}s, which we will modify as necessary to ensure that each {\tt block} has only one adjacent neighbor to which it will need to bind in each direction.
\begin{enumerate}
\item Define the graph $G$ such that for each $b \in B$, $G$ has a corresponding node, and there is an edge between each pair of nodes of $G$ that correspond to {\tt block}s that are adjacent to each other in $S$. \label{step:loop2}
\item Generate a tree $T$ from graph $G$ by removing edges from each cycle until no cycles remain.
\item For each $b \in B$, if there exist $b',b'' \in B$ where $b \ne b' \ne b'' \ne b$ such that $b$ is adjacent to both $b'$ and $b''$ along the same plane in $S$, and there are edges in $T$ (1) between the nodes representing $b$ and $b'$ and (2) the nodes representing $b$ and $b''$, then split $b$ into two new rectangular prisms, $b_1$ and $b_2$, such that each is adjacent to exactly one of $b'$ and $b''$ (this is always possible since all of $b,b',$ and $b''$ are rectangular prisms). \label{step:split}
\item Remove $b$ from $B$ and add $b_1$ and $b_2$ to $B$.
\item If any {\tt block} was split in step \ref{step:split}, loop back to step \ref{step:loop2}.
\end{enumerate}
The tree $T$ is a graph whose edges connect the nodes representing {\tt block}s which must bind to each other in the final assembly. At this point, each $b \in B$ will have at most 1 adjacent $b' \in B$ on each side to which it must bind, and each $b \in B$ will have at least one other $b' \in B$ to which it must bind. We will refer to any pair of {\tt block}s which must bind to each other as \emph{connected}.
\subsection{Scale factor and interface design}\label{sec:scale-and-interface}
We now describe the size and composition of each boundary, called the \texttt{interface}, between connected {\tt block}s. Each \texttt{interface} will include two specially designated glues, one on each end of the {\tt interface}, and assuming the length of the \texttt{interface} is $n$, an $n-2$ tile wide portion in between those glues which will eventually be mapped to a particular ``geometry'' of bumps and dents (i.e. tiles protruding from a surface, and openings for tiles in a surface). No \texttt{interface} can be shorter than $2$. Also, since each \texttt{interface} must be unique, there is only one valid \texttt{interface} of length $2$, and for each $n > 2$ there will be $2^{(n-2)/2}$ valid interfaces because each bit of the assigned number is represented by two bits in the geometry. For a 0-bit, the pattern $01$ is used, and for a 1-bit the pattern $10$ is used. This ensures that each geometry is compatible only with its complementary geometry (see \cite{GeoTiles} for further examples.) Figure \ref{fig:2HAM-block-interfaces2} shows an example of {\tt interface}s which could be added to the {\tt block}s of the example shape from Figure \ref{fig:2HAM-block-decomp}. Note, however, that for the sake of a more interesting example larger {\tt interface}s are shown than would be assigned by the algorithm presented, which would have created one {\tt interface} of size 2, with only White and Black glues, and two of size 4, one with a ``dent'' then ``bump'' to represent $01$ which maps to 0, and one with a ``bump'' then ``dent'' to represent $10$ which maps to 1.
\begin{enumerate}
\item Define the function $\texttt{RECT}$ such that, for each connected pair $b,b' \in B$, $\texttt{RECT}(b,b')$ returns the rectangle along which $b$ and $b'$ are adjacent in $S$, and the function $\texttt{RECTMAX}(b,b') = \texttt{max}(m,n)$ where $m$ and $n$ are the lengths of the sides of the rectangle returned by $\texttt{RECT}(b,b')$ (i.e. it returns the length of the maximum dimension of the rectangle).
\item Initialize the mapping $\texttt{INTERFACE-LENGTH}$ which maps a connected pair $b$ and $b'$ to an integer such that $\texttt{INTERFACE-LENGTH}(b,b') = 2$. (\texttt{INTERFACE-LENGTH} will eventually specify the length of the \texttt{interface} between {\tt block}s.)
\item Define the function \texttt{COUNT} such that, for each $k > 1$, $\texttt{COUNT}(k)$ is equal to the number of connected pairs $b,b' \in B$ such that $\texttt{INTERFACE-LENGTH}(b,b') = k$. (That is, \texttt{COUNT} returns the number of pairs of {\tt block}s that are currently assigned interfaces of length $k$.)
\item While there exists $k > 1$ such that $\texttt{COUNT}(k) > 2^{(k-2)/2}$:
\begin{enumerate}
\item Select a connected pair $b,b'$ where $\texttt{INTERFACE-LENGTH}(b,b') = k$ and update the mapping $\texttt{INTERFACE-LENGTH}$ so that $\texttt{INTERFACE-LENGTH}(b,b') = k+1$.
\end{enumerate}
\item If there exists a connected pair $b,b' \in B$ such that $\texttt{INTERFACE-LENGTH}(b,b') > \texttt{RECTMAX}(b,b')$, this (simplified) construction requires the shape $S$ to be scaled because there are too many {\tt interface}s of one or more lengths for them all to be unique\footnote{The number of unique {\tt interface}s for any length can easily be increased using methods discussed later.}. Therefore, replace $S$ with $S^2$ (the scaling of $S$ by $2$) and restart the construction from shape decomposition, at the beginning of Section~\ref{sec:shape-decomp}.
\end{enumerate}
At this point, the mapping $\texttt{INTERFACE-LENGTH}$ defines a valid mapping of lengths to each \texttt{interface}. We now assign a valid geometric pattern (i.e. a series of ``bumps'' and ``dents'') to each.
\begin{enumerate}
\item Let $s$ equal the value of the maximum of the width, height, and depth of $S$ (i.e. the length of its greatest dimension).
\item For each integer $1 < i \le s$, let $I_i = \{ (b,b') \mid $ where $b,b' \in B$ are connected and $\texttt{INTERFACE-LENGTH}(b,b') = i\}$. Thus, $I_i$ is the set of connected pairs of {\tt block}s which have {\tt interface}s of length $i$.
\item For each $I_i$ where $|I_i| > 0$, assign an arbitrary, fixed ordering to $I_i$ and for $0 < |I_i| < j$, let $I_{i_j}$ be the $j$th connected pair in $I_i$.
\item For each $I_{i_j}$:
\begin{enumerate}
\item Recall that $i$ is the assigned {\tt interface} length.
\item Assign $j$ as the number assigned to the {\tt interface} (after the number of bits is doubled so that each 0-bit is represented by $01$ and each 1-bit by $10$).
\item Let $(b,b') = I_{i_j}$ and $r = \texttt{RECT}(b,b')$
\item As $r$ is a rectangle, it is 2-dimensional and has only two of width ($x$ dimension), height ($y$ dimension), and depth ($z$ dimension). If its width is $\ge i$, we call $r$ an East-West (EW) rectangle. Else, if its height is $\ge i$, we call $r$ a North-South (NS) rectangle. Otherwise, its depth must be $\ge i$ (by design of the algorithm determining the assigned value of $i$, it will fit in at least one dimension of $r$) and we call $r$ an Up-Down (UD) rectangle.
\item Define $\texttt{RECT-ROW}$ as a function such that on input $b,b' \in B$, $\texttt{RECT-ROW}(b,b')$ returns a single row of coordinates as follows. Rectangle $r$ is either EW, NS, or UD and has one other non-zero dimension ($x$, $y$, or $z$) other than the dimension its type is named for. If that other non-zero dimension is $x$ (resp. $y$, resp. $z$), set direction $d = E$ (resp. $N$, resp. $U$). If $\texttt{RECT}(b,b')$ returns EW (resp. NS, resp. UD) rectangle $r$, $\texttt{RECT-ROW}(b,b')$ returns the row furthest in direction $d$ which runs EW (resp. NS, resp UD) in $r$.
\item Let $r' = \texttt{RECT-ROW}(b,b')$. If $r'$ is an EW (resp. NS, resp. UD) rectangle, we define the \texttt{interface} for $I_{i_j}$ such that the easternmost (resp. northernmost, resp. uppermost) location in $r'$ is assigned the Black glue, the adjacent $i-2$ locations are assigned the $i-2$ bits of the binary representation of the number $j$, in order, with the least significant bit in the easternmost (resp. northernmost, resp. uppermost) location, and the next location is assigned the White glue, making it the westernmost (resp. southernmost, resp. downwardmost) location containing a non-zero amount the {\tt interface} information. The other locations of the row of $r'$ are assigned ``empty'' values. Define the function $\texttt{INTERFACE}(b,b')$ such that it returns this \texttt{interface} definition for the entire row of $r'$ for the {\tt interface} between $b$ and $b'$. (Recall that by our construction, any connected pair can have at most one \texttt{interface}.)
\end{enumerate}
\end{enumerate}
\subsection{Growth of a {\tt block}}\label{sec:block-growth}
Each {\tt block} $b \in B$ making up shape $S$ has at most $6$ {\tt interface}s. Because of this constant bound, and the fact that each {\tt block} is a rectangular prism, it is possible to encode all of the information needed to grow an entire {\tt block} $b$ within a sequence of glues, taken from a set of glues that is constant over any shape $S$, that is no longer than the longest dimension of $b$.\footnote{Later we will also briefly mention ways in which the length can actually be as small as the $\log$ of the longest dimension.} We call each such sequence a {\tt gene}. In this section we show how a {\tt gene} can be encoded and how a {\tt block} can then grow from it.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{2HAM-block-growth.png}
\caption{Schematic representation of the order of block growth (without directions shown for every row). Starting from a \texttt{gene} section, the green surface grows upward in a zig-zag pattern. As each row of the green face completes, one plane can grow perpendicularly to it (the first is shown in blue, with the next two in white). Each of these also grows in a zig-zag pattern away from the green face.}
\label{fig:side-growth}
\end{figure}
Each {\tt block} grows so that one of its $6$ faces grows directly upward off of the {\tt block}'s {\tt gene}. The growth of this plane happens in a zig-zag manner, meaning that the first row grows completely from left to right (zigging), then the second from right to left (zagging), and the pattern continues until the growth terminates. (Shown schematically in green in Figure~\ref{fig:side-growth}.) The zig-zag pattern of growth allows for each row to transmit (and update) information it reads from the row below it (to be discussed shortly).
As each row of the first face completes, a plane growing perpendicular to the first face can begin its growth. (The first such plane is shown in light blue in Figure~\ref{fig:side-growth}, and the next two in white.) Every row of each such plane also grows in a zig-zag manner, which allows information to be transmitted from the green initiating rows throughout each plane.
To control the size of each plane, a pair of binary counters are used. The upward facing glues of the {\tt gene} encode a series of bits (which we will call the \emph{green} bits). As the face grows upward, every other row increments the value of the binary number represented by the bits, and every other row checks to see if all bits are equal to 1. If they are all equal to 1, upward growth terminates. An example can be seen in Figure~\ref{fig:counting}.
\begin{figure}
\centering
\begin{subfigure}{0.46\textwidth}
\includegraphics[width=1.0\textwidth]{basic-counting.png}
\caption{\label{fig:counting}}
\end{subfigure}
\hspace{0.06\textwidth}
\begin{subfigure}{0.46\textwidth}
\includegraphics[width=1.0\textwidth]{basic-pattern-rotation.png}
\caption{\label{fig:pattern-rotation}}
\end{subfigure}
\caption{Examples of basic growth patterns in {\tt blocks} in the 2HAM construction. (a) Example of a binary counter which increments every other row, and checks for all 1s on the others. Note that the least significant bit is on the left. When the ``checking'' row detects all 1s, a red tile marks the end of further upward growth. (b) A basic example of the rotation of a pattern.}
\label{fig:2HAM-block-growth-details}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth]{2HAM-block-3-rotations.png}
\caption{Schematic of how information can be transmitted to three sides of one plane of a {\tt block}, with rotations of alternating rows (as shown in Figure~\ref{fig:pattern-rotation}) rotating information to the left and right sides. \label{fig:3-sides}}
\end{figure}
We will call the bits of the counter which control the length of the perpendicular planes (shown as blue and white in Figure~\ref{fig:side-growth}) the \emph{blue} bits. These bits are also encoded in the upward facing glues of the {\tt gene} (i.e. each glue can encode both a green and a blue bit by making $4$ glues, one for each pair of bit values $00$, $01$, $10$, and $11$). However, as each row of the green face assembles, rather than using the blue bits to count, each row presents the blue bits on both its upward and backward facing glues. This allows them to be propagated up throughout the green face, unchanged, and to control the distance grown by each perpendicular plane, which uses them as the bits for its counter.
With the {\tt gene}'s length implicitly encoding the size of one dimension of the growing {\tt block}, and the green and blue counter bits controlling the sizes of the other two dimensions, the {\tt block} grows into a rectangular prism of the correct dimensions. (Note that growing counters, zig-zag growth, rotating bits, etc. are very standard techniques in tile assembly literature - see \cite{IUSA,j2HAMIU,Versus,SolWin07,RotWin00} for just some examples - and issues like growing sides of odd length, despite the zig-zag pattern, are easily handled with a few extra glues that signal for one additional row to grow.)
Each {\tt block} has a fixed orientation relative to the others when they are attached together to form the shape $S$, and since we (arbitrarily) assign each shape a canonical translation and rotation, each {\tt block} has a canonical orientation which allows us to refer to its sides by the directions they face in that orientation. Throughout, we talk about {\tt block}s in term of this orientation, irrespective of the orientation in which they grow.
This (simplified version of the) construction has each {\tt gene} equal to the length of the longest dimension of the {\tt block} it initiates. This could lead to the first surface to grow being any of at least $4$ sides, so without lack of generality we fix a preferred ordering as: North, East, South, West, Up, Down. Therefore, of the multiple faces which share the longest dimension, that appearing first in the ordering grows ``first'' (i.e. as the green face, as shown in Figure~\ref{fig:side-growth}), and with the side attached to the {\tt gene} being that whose coordinates are the smallest along the direction of upward growth of the first face.
\subsubsection{Growth of {\tt interfaces}}
With the dimensions of each {\tt block} correctly controlled, the next thing to ensure is correct growth of the {\tt block}'s {\tt interface}s. As previously mentioned, there are at most $6$ of these (no more than one per side), and each {\tt interface} consists of two outward facing glues (Black and White) with a possible series of ``bumps'' and ``dents'' between them, geometrically encoding the bits of the number which is uniquely assigned to that {\tt interface}. If the {\tt interface} is on the North, East, or Up side, in the location of each bit $b = 1$ there is a tile which extends from the side as a ``bump'', and in the location of each bit $b = 0$, there is no such bump. If the {\tt interface} is on the South, West, or Down side, in the location of each bit $b = 1$ there is an empty tile location (i.e. a ``dent''), and in the location of each bit $b = 0$, there is no such dent. (See Figure \ref{fig:2HAM-block-interfaces2} for examples of {\tt interfaces} with ``bumps'' and ``dents''.)
The information defining each {\tt interface} can be encoded as a series of glues representing the locations of the Black and White {\tt interface} glues plus each of the bits of the assigned {\tt interface} number, as well as the information about whether the $1$-bits are encoded as ``bumps'' or ``dents'' for the particular surface. Using the same technique as mentioned previously for adding information about an extra bit to the glues extending from the {\tt gene}, we can similarly add the information which defines each of the (up to $6$) {\tt interface}s of a {\tt block}. Therefore, we individually discuss the patterns by which the information specifying each {\tt interface} is propagated into the correct locations, and note that all of that information can be encoded in the outward facing glues of the {\tt gene} and then distributed to the proper locations in the {\tt block} during the growth process previously described. Once we've explained how the information about each {\tt interface} arrives at the correct location, we will discuss the tiles which encode it.
There are $6$ sides, and for each side $2$ orientations which must be considered for the possible {\tt interface} on each side (note that on {\tt block} sides which don't have {\tt interface}s, nothing needs to be done beyond the growth of the side to the correct dimensions as previously described). One orientation we will refer to as ``parallel'' to the {\tt gene}, and the other as ``perpendicular'' (although these terms aren't technically accurate for all cases). The parallel cases are depicted in Figure~\ref{fig:parallel-interfaces}, and the perpendicular cases are depicted in Figure~\ref{fig:perpendicular-interfaces}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\textwidth]{block-interfaces-parallel.png}
\caption{Schematic representation of the patterns by which {\tt interface} information is propagated into the correct positions for ``parallel'' {\tt interface}s. (a) A counter is used to determine the correct height for the {\tt interface} on the green side, (b) Two counters are used to position the {\tt interface} on the yellow side. The first counts to the top of the green side, then the bits of the second and the {\tt interface} are rotated onto the yellow colored plane and the second counts to the proper location for the {\tt interface} on that side. (c) Two counters are used to position the {\tt interface} on the back side of the {\tt block}. The first counts to the correct height, then the bits of the second and the {\tt interface} are rotated onto the blue colored plane and the second counter counts the distance to the back surface. (d) To position the {\tt interface} on the pink side, a counter first counts to the correct height, then the bits are rotated to the pink face during the outward growth of the white plane. Note that the side opposite the pink {\tt interface} is positioned analogously but with an opposite rotation, and the bottom {\tt interface} is positioned similar to the top (yellow) but without the necessity of the first counter.}
\label{fig:parallel-interfaces}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\textwidth]{block-interfaces-perpendicular.png}
\caption{Schematic representation of the patterns by which {\tt interface} information is propagated into the correct positions for ``perpendicular'' {\tt interface}s. (a) For a perpendicular {\tt interface} on the green side, the information is split into two halves (depending on the actual length and position of the {\tt interface}). The left half is rotated upward immediately, and the second half has a counter which first moves it upward to the halfway point, and then it is rotated. Note that any offset from the center can be accommodated by shifting the location of the split and the height of the counter. If the {\tt interface} needs to be completely to the left or right, only one rotation is needed, and no splitting of the information or counting is needed. (b) The positioning of the {\tt interface} on the top is the same as for the green side, but a counter first propagates all information to the top, where it is rotated to the yellow surface. (And the same hold for the bottom surface but without the initial counter.) (c) To position the {\tt interface} on the back surface, the same rotations and counting are used as for the green surface. However, then the information from each row is carried all the way to the back surface following the counter which dictates that distance. (d) To position the {\tt interface} for the pink surface, again the same rotations and counting are used to align the information on the green surface, but then the information of each row is rotated to the pink surface as its plane grows away from the green surface. The surface opposite the pink is handled similarly, but with an opposite rotation.}
\label{fig:perpendicular-interfaces}
\end{figure}
It is important to note that the patterns shown in Figures~\ref{fig:parallel-interfaces} and \ref{fig:perpendicular-interfaces} suffice when each {\tt interface} is anywhere from the minimum allowed size (i.e. $2$) up to the maximum size, which is the full length of the side on which it is located. This is because the construction is designed so that the length of the {\tt gene}, and thus the green side, is the length of the longest dimension of the {\tt block}. Thus, there is room for the information in a longest-possible {\tt interface} to be correctly positioned, and shorter {\tt interfaces} can also be correctly positioned by correctly shifting the locations of information in the {\tt gene} so that the counters and rotations will propagate it correctly. Additionally, Figures~\ref{fig:parallel-interfaces} and \ref{fig:perpendicular-interfaces} depict the cases where each {\tt interface} is in the center of its surface, but any position along each surface can be accommodated by simply adjusting initial information alignment along the {\tt gene}, counter values, and/or the location of splits between rotations and counting.
Recall that the {\tt block}s on either side of an {\tt interface} have complementary geometries, i.e. one has ``bumps'' in the $1$-bit locations and the other has ``dents''. Once the information encoding an {\tt interface} reaches the correct location on the correct surface, the locations assigned the Black and White glues of the {\tt interface} receive tiles which have strength-1 glues of those types exposed on the exterior of the {\tt block} for the {\tt block} with a bump {\tt interface}, and the {\tt block} with the dent {\tt interface} receives tiles which expose the complements of those glues (i.e. Black$^*$ and White$^*$, respectively). Additionally, in $1$-bit positions for a {\tt block} with a bump {\tt interface}, tiles attach which have strength-2 glues exposed, allowing the ``bump'' tiles to attach, and signals ensure that all ``bump'' tiles have attached before the Black tile can attach and enable the {\tt interface} to bind to its counterpart. See Figure \ref{fig:bump-tiles} for details of the signals.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{2HAM-bump-tiles.png}
\caption{Templates for tile types which make up a ``bump'' {\tt interface} that extends in the same direction as the growth direction of the plane. These tile types are used for an {\tt interface} that grows from left to right, so each ``Pre-bump'' tile stalls the growth of the {\tt interface} until all of its associated ``Bump'' tiles have attached, ensuring that all ``bumps'' are in place before the Black tile is in place, since that allows the {\tt block} to attach to a {\tt block} with the complementary half of the {\tt interface}.}
\label{fig:bump-tiles}
\end{figure}
\subsubsection{Formation of ``bumps'' and ``dents'', and detachment of {\tt block}s}
The first portion of this construction which requires signals are the formations of the {\tt interface}s and the detachment of {\tt block}s from the {\tt genome}. (To make the exposition easier to understand, we will wait until Section \ref{sec:block-combo} to discuss a layer of signals which overlay each {\tt block}.) These situations are relatively straightforward to handle with signals, and in this section we provide an overview of the various cases and how they are handled. We also provide depictions of templates for the required signal tile types, which abstract away the fact that a variety of additional information (useful for the further growth of a {\tt block}) may be propagated through the signal tiles using standard glues. This is simply handled by making a set of signal tile types for each template provided, with a unique tile type for each glue type which needs to pass additional information through it. For each of the provided sets of templates for signal tile types, there are tile sets generated for each of the various permutations of such glues, as well as the orientations and locations of {\tt interface}s. However, the number of permutations and thus signal tile types is a constant, irrespective of the shape $S$. Which set is to be used for each {\tt interface} is encoded along with the definition of the {\tt interface} in the corresponding {\tt gene}.
The general scenarios we will address are: (1) the growth of an {\tt interface} at the terminal edge of the plane of a {\tt block}, (2) the growth of an {\tt interface} in the middle of a plane of a {\tt block}, and (3) the detachment of a {\tt block} from the {\tt genome}, both when the attached row does and does not need to also encode an {\tt interface}.
\paragraph{The growth of an {\tt interface} at the edge of the plane of a {\tt block}}\label{sec:end-of-plane}
If the ``dents'' of an {\tt interface} appear in the row (or column) immediately adjacent to the edge of the plane in which the {\tt interface} information is being propagated, the White, Black, and $0$-bit tiles (i.e. those of the {\tt interface} not corresponding to ``dent'' locations) are attached with strength-2 glues to the row (or column) preceding the final row (or column). No tiles are ever placed in the locations of the ``dents'', and therefore no signals are required to allow tiles to detach, this basic scenario is handled without any signals.
The growth of the ``bumps'' of an {\tt interface} requires that all ``bump'' tiles are in place before the Black and White glues are in place and active. Otherwise, the {\tt interface} could be missing ``bumps'' and allow incorrect binding of the {\tt block} with another {\tt block}. The templates for the necessary tiles are shown in Figure~\ref{fig:bump-tiles}. Note that any ``Pre-Bump'' tiles will not activate the glue needed for the next tile of the {\tt interface} to attach until its ``Bump'' tile has attached. Therefore, only if all ``Bump'' tiles have attached throughout the {\tt interface}, will the {\tt interface} be able to grow to the point of the ``Black'' tile attaching. (Note that this assumes growth from the ``White'' side of the {\tt interface} to the ``Black'', and appropriately modified tiles exist for {\tt interface}s growing in the opposite direction.) The same tile types are used whether the {\tt interface} is at the edge of a plane (in the growth direction) or in the middle, except that in the middle of a plane the ``bumps'' must be going either into, or out of, the plane and thus the ``bump'' glue is either on the side of the tile facing into the page, or that facing outward. The directions for the White and Black glues are the same.
\paragraph{The growth of an {\tt interface} in the middle of a plane of a {\tt block}}
If the ``dents'' of an {\tt interface} appear in any row (or column) before the end of the plane, in the direction in which the {\tt interface} information is being propagated (i.e. in the middle of the plane), then tiles will initially be placed in the ``dent'' locations in order to allow information to be propagated through those locations, but signals will eventually cause them to deactivate glues and dissociate. The templates for the tile types of an {\tt interface} if it is positioned in such a location are shown in Figure~\ref{fig:dent-tiles}, and a an example sequence showing a portion of a growing plane and {\tt interface} which places, then eventually loses, tiles in the ``dent'' locations can be seen in Figure~\ref{fig:dent-tiles-sequence}.
The case in which the ``bumps'' of an {\tt interface} appear in the middle of a plane is (nearly) identical to the case in Section \ref{sec:end-of-plane} and was discussed there.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{2HAM-dent-tiles.png}
\caption{Templates for tile types which make up a ``dent'' {\tt interface} that is not at the edge of the plane in which it is growing. These tile types are used for an {\tt interface} that grows from left to right. Signal propagation which initiates tile detachment begins once a tile has attached to the north of a White tile, since the zig-zag growth pattern means that a row must have completed growth to the north of the {\tt interface}, growing right to left. This allows the ``dent'' tiles to propagate any needed information via their northern glues before they dissociate. An example growth sequence can be seen in Figure~\ref{fig:dent-tiles-sequence}.}
\label{fig:dent-tiles}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.8\textwidth]{2HAM-dent-tiles-sequence.png}
\caption{From bottom to top, example sequence of growth around ``dent'' tiles and their dissociation, using tiles of the templates shown in Figure~\ref{fig:dent-tiles}.}
\label{fig:dent-tiles-sequence}
\end{figure}
\paragraph{The detachment of a {\tt block} from the {\tt genome}}
We now discuss the detachment of a {\tt block} from the {\tt genome}, which we break into 3 sub-cases: (1) the attached row does not include an {\tt interface}, (2) the attached row encodes an {\tt interface} with ``dents'', and (3) the attached row encodes an {\tt interface} with ``bumps''.
The templates for the tile types used in Case 1 are shown in Figure \ref{fig:block-detach-tiles}. The completion of the first row, consisting of one Left tile, one Right tile, and perhaps many Mid tiles, causes the ``x''glues to bind the tiles of the row together with strength 2 and to deactivate their glues attached to the {\tt genome} to the south. The stable assembly containing the complete first row can detach at any point during the continued growth of the {\tt block} and it will correctly complete. Also, once complete it will be free to bind with the other {\tt blocks} that have complementary {\tt interface}s.
\begin{figure}[ht]
\centering
\includegraphics[width=0.55\textwidth]{2HAM-gene-detach-tiles.png}
\caption{Templates for the tile types which make up the first row of a {\tt block} and allow the {\tt block} to detach once the first row has completed. It is assumed that the first row of a {\tt block} always grows from left to right. The ``x'' glues ensure that all tiles are bound with strength 2, and thus is it safe for the row to detach from the {\tt genome} as soon as it completes. The rest of the {\tt block} correctly grows from this row.}
\label{fig:block-detach-tiles}
\end{figure}
The templates for the tile types used in Case 2 are shown in Figure \ref{fig:detach-dent-tiles}. In this case, a special tile type is also needed to bind to the north of the White tile to ensure that they are bound with strength 2, since its possible for the White tile to be the leftmost and to have a ``dent'' location next to it, meaning it will end up attached only to the tile to its north. Essentially, the tile types ensure that the row to the north completes before any detachments. Then, signal propagation from left to right causes Dent tiles to dissociate, all tiles to deactivate the glues binding them to the {\tt genome}, and 0 and Black tiles to bind to their northern neighbors with strength 2 to ensure the stability of the assembly.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{2HAM-detach-dent-tiles.png}
\caption{Templates for the tile types which make up the first row of a {\tt block} that allow the {\tt block} to detach once the first row has completed, when that first row also has to create an {\tt interface} with ``dents''. Note that the tiles shown here are for the scenario in which the {\tt interface} extends for the entire length of the row, since this is a slightly more complicated case due to the necessity of $tau$-stability of the White and Black tiles. If the {\tt interface} only occupies a portion of the row, a few trivial edits are made to the template for the necessary tile types.}
\label{fig:detach-dent-tiles}
\end{figure}
The templates for the tile types used in Case 3 are shown in Figure \ref{fig:detach-dent-tiles}. In this case, the glues attaching the row to the {\tt genome} are detached via the propagation of an ``x'' signal which also ensures that tiles are bound with strength 2, and a ``w'' signal ensures that all Bump tiles are attached before the White glue can be activated, making sure the {\tt interface} is correctly completed before it is able to bind to another {\tt block}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{2HAM-detach-bump-tiles.png}
\caption{Tiles which make up the first row of a {\tt block} which allow the {\tt block} to detach once the first row has completed, when that first row also has to create an {\tt interface} with ``bumps''. (Note that the tiles shown here are for the scenario in which the {\tt interface} extends for the entire length of the row. If the {\tt interface} only occupies a portion of the row, a few trivial edits are made to the template for the necessary tile types.) Once the row completes with the attachment of a Black tile, the propagation of the ``x'' signal initiates the deactivation of glues attached to the {\tt genome} and the ``w'' signal ensures that all Bump tiles have attached before the White glue is activated. The signals also ensure the $tau$-stability of the row so it can safely detach.}
\label{fig:detach-bump-tiles}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{2HAM-expanding-side.png}
\caption{A basic example of expansion of a {\tt block} in two directions. Note that growth could instead be only in the vertical direction.}
\label{fig:side-increase}
\end{figure}
\subsection{Combination of {\tt block}s to form the target shape}\label{sec:block-combo}
Once a {\tt block} has detached from its {\tt gene}, it is a freely floating supertile which may or may not require additional tile attachments to complete its own growth. However, only {\tt interface}s that have completed are able to bind with strength $2$ to the complementary {\tt interface}s of other {\tt block}s. Additionally, we now discuss a set of signals that allow for a {\tt block} to determine when all tiles have attached. The growth of each plane in a {\tt block} follows the same zig-zag pattern so that the final tile placed in each plane (other than possibly ``bump'' tiles of {\tt interface}s) falls into a single vertical column. These tiles are augmented with signals such that when the final tile of the bottommost plane attaches, it activates a glue that allows it to bind to the tile above it (whose complementary glue will be activated when it attaches). The tile above it in turn passes this signal upward, with each in the column doing the same until the final tile of the top plane is reached. Once that tile (which is of a special type) is placed, it is guaranteed that all tiles of all planes (other than possibly ``bump'' tiles of {\tt interface}s) have attached since each plane signals its completion in order from bottom to top.
Upon receiving the ``completion'' signal, the final tile of the top plane then sends that signal outward, spreading across all tiles on all $6$ surfaces of the {\tt block}. These ``surface'' tiles are all equipped with signals that allow them to receive and pass on this completion signal (and during the growth of the {\tt block} it is always known which tiles will be on a surface since they are at an edge of their plane of growth). The previous description of the signals which activate the Black and White glues (and their complements) on {\tt interface}s was slightly simplified to omit this final detail: the previously described signals which activate those glues actually activate glues facing neighboring tiles so that only at that point they are able to receive the completion signal. It is the reception of this signal which actually activates the Black and White (and Black* and White*) glues on the {\tt interface}s.
The addition of the extra layer of ``completion'' signals ensures that only a {\tt block} that has received all of the tiles of its body can have active {\tt interface}s. Once an {\tt interface} is active and able to bind to the complementary {\tt interface} of another {\tt block}, the {\tt block} combines to a growing supertile consisting of the {\tt block}s forming an assembly of shape $S$. Furthermore, by the definition of a block-diffusable shape and the fact that $S$ is such a shape, it is always possible for a free {\tt block} to attach as needed in any such growing supertile. Thus, the {\tt block}s will eventually form completed, and terminal, assemblies of shape $S$.
\subsection{Possible enhancements to the hierarchical construction}
As previously mentioned, there are many ways in which this construction could be easily modified to further optimize various aspects. For example, to shrink the length of the {\tt genome}, {\tt gene}s could be compressed so that they are no longer required to be as long as the largest dimension of a {\tt block}. Instead, in cases where {\tt interface}s are shorter than {\tt block} side lengths and appropriately positioned, it is possible to shrink the {\tt gene} encoding a block to as small as $\log$-width. This can be done by incorporating counters that also grow out the width of a {\tt block}. (An overview can be seen in Figure~\ref{fig:side-increase}.) Additional, even asymptotically optimal, compression could be achieved by instead encoding the shortest program which can output the {\tt gene} necessary to grow a {\tt block} and then a ``fuel efficient'' Turing machine \cite{jSignals} can be simulated with signal tiles which grow from the {\tt genome} until that encoding is output, allowing {\tt block} growth to proceed from there.
As another example, the necessity to scale certain shapes could be removed by only slightly increasing tile complexity, i.e. the size of $U$. For example, by adding a constant number $m$ of tile types to also be candidates for the ends of {\tt interfaces} (along with the White and Black tiles), the number of {\tt interface}s of each length (which is the limiting number potentially requiring scaling of a shape) can be increased by a factor of on the order of $m^2$. There are many other such variations that can be used to balance several factors of the construction to optimize trade-offs for desired goals. Also, for many variations on the specific algorithm which is used to determine the encoding of $S$ into the {\tt genome}, no changes are even required to $U$, so the algorithm can be modified to favor particular tradeoffs over others (e.g. scale factor over {\tt genome} length) without any other modifications to the system.
Finally, it is easy to combine this construction with the previous constructions. For instance, tile types could be added to $U$ from the construction in Section \ref{sec:simple-replicator} that also create duplicate copies of $\sigma_S$. Additionally, an actual self-replicating system could be built by including the shape-deconstruction capabilities of the construction in Section \ref{sec:deconstruct}. Let $M$ be a Turing machine that performs the following computation. Given an input string consisting of the turns of a path through $\mathbb{Z}^3$ (i.e. the path encoded in a seed assembly genome of the construction in Section \ref{sec:simple-replicator}), it first computes the points of the shape $S$ generated by that path. It then performs the computations for the hierarchical replicator of this section to compute a valid input {\tt genome} for it. Simulation of an arbitrary Turing machine is straightforward even with static aTAM tiles (e.g. \cite{jSADS,jCCSA,SolWin07}) and can additionally be made ``fuel efficient'' using signal tiles \cite{jSignals}. Therefore, there exists a system which can take as input an assembly as for the construction of Section \ref{sec:deconstruct} and use the components of that construction to deconstruct it into a linear genome. Tiles which simulate $M$ then perform the generation of the input {\tt genome} for the hierarchical replicator, which proceeds to make copies of assemblies of shape $S$. This is a more complicated self-replicator which consumes much more fuel (i.e. the TM computation tiles - but note that using techniques of \cite{jSignals} that amount is greatly reduced, and the junk assemblies can all be guaranteed to be of small, constant size) but once the {\tt genome} is computed once it is infinitely replicated along with copies of the shape.
\section{Shape Building via Hierarchical Assembly}\label{sec:2ham}
In this section we present details of a shape building construction which makes use of hierarchical self-assembly. The main goals of this construction are to (1) provide more compact genomes than the previous constructions, and (2) to more closely mimic the fact that in the replication of biological systems, individual proteins are independently constructed and then they combine with other proteins to form cellular structures. First, we define a class of shapes for which our base construction works, then we formally state our result.
Let a \emph{block-diffusable} shape be a shape $S$ which can be divided into a set of rectangular prism shaped blocks\footnote{A rectangular prism is simply a 3D shape that has 6 faces, all of which are rectangles.} whose union is $S$ (following the algorithm of Section \ref{sec:2ham-append}) such that a connectivity tree $T$ can be constructed through those blocks and if any prism is removed but $T$ remains connected, that prism can be placed arbitrarily far away and move in an obstacle-free path back into its location in $S$.
\begin{theorem}\label{thm:2HAM}
There exists a tile set $U$ such that, for any block-diffusable shape $S$, there exists a scale factor $c \ge 1$ and STAM* system $\mathcal{T_S} = (U,\sigma_{S^c},2)$ such that $S^c$ self-assembles in $\calT_S$ with waste size 1. Furthermore, $|\sigma_S|$ is approximately $O(|S|^{1/3})$.
\end{theorem}
To prove Theorem~\ref{thm:2HAM}, we present the algorithm which computes the encoding of $S$ into seed assembly $\sigma_S$ as well as the value of the scale factor $c$ (which may simply be $1$), and then explain the tiles that make up $U$ so that $\mathcal{T_S}$ will produce components that hierarchically self-assemble to form a terminal assembly of shape $S$.
At a high level, in this construction the seed assembly is the \texttt{genome}, which is a compressed linear encoding of the target shape that is logically divided into separate regions (called \texttt{genes}), and each {\tt gene} independently initiates the growth a (potentially large) portion of the target shape called a \texttt{block}. Once sufficiently grown, each {\tt block} detaches from the \texttt{genome}, completes its growth, and freely diffuses until binding with the other {\tt blocks}, along carefully defined binding surfaces called {\tt interfaces}, to form the target shape.
It is important to note that there are many potential refinements to the construction we present which could serve to further optimize various aspects such as \texttt{genome} length, scale factor, tile complexity, etc., especially for specific categories of target shapes. For ease of understanding, we will present a relatively simple version of the construction, and in several places we will point out where such optimizations and/or tradeoffs could be made.
Throughout this section, $S$ is the target shape of our system. For some shapes, it may be the case that a scale factor is required (and the details of how that is computed are provided in Section~\ref{sec:scale-and-interface}). We will first describe how the shape $S$ can be broken into a set of constituent {\tt block}s, then how the {\tt interface}s between {\tt block}s are designed, then how individual {\tt block}s self-assemble before being freed to autonomously combine into an assembly of shape $S$.
\begin{figure}
\vspace{-10pt}
\centering
\begin{subfigure}{0.4\textwidth}
\includegraphics[width=1.0\textwidth]{2HAM-sample-shape.png}
\caption{\label{fig:3D-structure}}
\end{subfigure}
\hspace{0.06\textwidth}
\begin{subfigure}{0.43\textwidth}
\includegraphics[width=1.0\textwidth]{2HAM-sample-shape-components.png}
\caption{\label{fig:3D-structure-decomp}}
\end{subfigure}
\caption{(a) An example 3D shape $S$. (b) $S$ split into 4 {\tt block}s, each of which can be grown from its own \texttt{gene}. Note that the surfaces which will be adjacent when the {\tt block}s combine will also be assigned {\tt interface}s to ensure correct assembly of $S$.}
\label{fig:2HAM-block-decomp}
\end{figure}
\vspace{-10pt}
\subsection{Decomposition into {\tt block}s}
\vspace{-5pt}
Since $S$ is a shape in $\mathbb{Z}^3$, it is possible to split it into a set of rectangular prisms whose union is $S$. We do so using a simple greedy algorithm which seeks to maximize the size of each rectangular prism, which we call a {\tt block}, and we call the full set of {\tt block}s $B$.
After the application of a greedy algorithm to compute an initial set $B$, we refine it by splitting some of the {\tt block}s as needed to form a binding graph in the form of a tree $T$ such that every {\tt block} is connected to at least one adjacent {\tt block}, but also so that each {\tt block} has no more than one connected neighbor in each direction in $T$. This results in the final set of {\tt block}s that combine to define $S$, can join along the edges defined by $T$, and each {\tt block} has at most $6$ neighbors to which it combines. (Figure \ref{fig:2HAM-block-decomp} shows a simple example.)
\begin{figure}
\centering
\begin{subfigure}{0.45\textwidth}
\includegraphics[width=1.0\textwidth]{2HAM-sample-shape-components-interfaces.png}
\caption{\label{fig:3D-structure-decomp-interfaces}}
\end{subfigure}
\hspace{0.06\textwidth}
\begin{subfigure}{0.39\textwidth}
\includegraphics[width=1.0\textwidth]{2HAM-sample-shape-components-interfaces2.png}
\caption{\label{fig:3D-structure-decomp-interfaces2}}
\end{subfigure}
\caption{(a) The {\tt block}s for the example shape $S$ from Figure \ref{fig:2HAM-block-decomp} with example {\tt interface}s included. (b) View from underneath showing more of the {\tt interface}s between {\tt block}s. Note that the actual {\tt interface}s created by the algorithm would be shorter, but to make the example more interesting their sizes have been increased.}
\label{fig:2HAM-block-interfaces2}
\vspace{-20pt}
\end{figure}
\vspace{-10pt}
\subsection{{\tt Interface} design}
\vspace{-5pt}
The {\tt block}s self-assemble individually, then separate from the {\tt genome} to freely diffuse until they combine together via {\tt interface}s along the surfaces between which there were edges in the binding tree $T$. Each {\tt interface} is assigned a unique length and number. The two {\tt block}s that join along a given {\tt interface} are assigned complementary patterns of ``bumps'' and ``dents'' and a pair of complementary glues on either side of those patterns (to provide the necessary binding strength between the blocks). The number assigned to each {\tt interface} is represented in binary and the {\tt block} on one side of an {\tt interface} has a protruding tile ``bump'' in the location of each $1$ bit but not in locations of $0$ bits, and for the {\tt block} on the other side of the {\tt interface} $1$ bits have single tile ``dents'' where a tile is missing. The length of each {\tt interface} dictates which other {\tt interface}s have glues at the correct spacing to allow binding, and the binary pattern of ``bumps'' and ``dents'' guarantees that only the single, correct complementary half can combine with it.
Depending on the shape $S$ and how it is split into {\tt blocks}, it is possible that there are too many {\tt interface}s of a given length ($> 2^{(n-2)/2}$ for an {\tt interface} of length $n$) to be able to assign a unique number to each. Our algorithm will attempt to assign a unique length and number to an {\tt interface} for all lengths $2$ to $n/2$ ($2$ being the minimum since there must be room for the two glues), but since $n$ is the full length of the surface between a pair of {\tt block}s and each bit of the assigned number is represented by a pair of bits, a greater length can't be encoded in the tiles along it. Therefore, if there are too many {\tt interface}s for a unique assignment, the shape $S$ is scaled upward. This is repeated until there can be unique assignments. (Note that there are many ways in which the algorithm could be optimized to reduce the number of shapes for which scaling is necessary, and/or the amount of scaling, especially for particular categories of shapes.) More technical details can be found in Section \ref{sec:scale-and-interface}, and an example of a few {\tt interface}s can be seen in Figure~\ref{fig:2HAM-block-interfaces2}.
\vspace{-10pt}
\subsection{{\tt Block} growth}
\vspace{-5pt}
The growth of each {\tt block} is initiated by a portion of the linear {\tt genome} called a {\tt gene}, which is merely a line of tiles with glues exposed in one direction that encode all of the information required for the {\tt block} to self-assemble to the correct dimensions and with the necessary {\tt interface}s. The techniques used to encode the information and allow the {\tt block}s to grow are very standard tile assembly techniques involving binary counters, zig-zag growth patterns, and rotation of patterns of information. The information to seed the counters and encode the {\tt interface}s is encoded in the outward facing glues of the {\tt gene} and can be done so with the universal tile set $U$ since only a constant amount of information needs to be encoded in any particular {\tt gene} glue, due to the design of {\tt block}s and the fact that each has at most a single {\tt interface} on each side which is no longer than that side. Signals are used for detecting completed growth of {\tt block}s, controlling growth of {\tt interface}s so ``bump'' {\tt interface}s can't complete before all ``bumps'' are in place, and ``dent'' {\tt interface}s can grow beyond ``dent'' locations and then those tiles can fall out, and also so {\tt block}s can dissociate from {\tt gene}s.
\vspace{-10pt}
\subsection{Overview of the hierarchical construction}
\vspace{-5pt}
Once a {\tt block} is freely diffusing and complete, it can combine along its {\tt interface}s with the {\tt block}s that have complementary {\tt interface}s since, due to the fact that $S$ is a block-diffusable shape, free {\tt block}s can always diffuse into the proper locations to form the complete shape. We've described a tile set $U$ that can be used to (1) form the linear seed assembly $\sigma_S$ , and (2) to self-assemble the {\tt block}s which correctly combine to form the target assembly. The STAM* system $\mathcal{T_S} = (U, \sigma_S, 2)$ will produce an infinite number of copies of terminal assemblies of shape $S$ (properly scaled if necessary). The only fuel (a.k.a. consumed, junk assemblies) will be singleton Dent tiles that attached during {\tt block} growth then detached. Note that this construction can be combined with the previous constructions as well, to create a version of a shape self-replicator. Full technical details of the construction, as well as a discussion of possible enhancements, can be found in Section \ref{sec:2ham-append}.
\section{Technical Details of the Self-Replicator that Generates its own Genome}\label{sec:deconstruct-append}
In this section we present technical details of the construction in Section \ref{sec:deconstruct}. We will begin by describing at a high level what information is encoded in the phenotype and genome for a given shape $S$. We will then describe how a phenotype is disassembled to construct a genome, keeping track of what kind of signals and glues are necessary, and where, throughout the process. We will then show that the disassembly is reversible, or in other words, that from a genome we can reconstruct the phenotype, again keeping track of the necessary glues and signals. We will end by showing that, given a shape $S$, there is an efficient algorithm by which we can describe the $STAM^*$ system in which this self-replication occurs including all necessary tiles that make up the phenotype and genome.
\subsection{The Phenotype}
Given a shape $S$, the phenotype $P$ will be a 2-scaled copy of the shape, so that each cube in $S$ corresponds to a $2\times 2\times 2$ block of tiles in $P$. The shape of the phenotype will therefore be identical to $S$ modulo our small, constant scale-factor. $P$ will be made up of tiles from some fixed $STAM^*$ tile system $\mathcal{T}$ which we will define in more detail later.
Let $H$ be a Hamiltonian path that goes through each tile in $P$ exactly once. We will construct $H$ later, but for now assume that it exists. Each tile in $P$ will contain the following information encoded in its glues and signals.
\begin{itemize}
\item Which immediately adjacent tile locations belong to the phenotype
\item Which immediately adjacent tile locations correspond to the next and previous points in the Hamiltonian path
\item Any glues and signals necessary for allowing the deconstruction and reconstruction process to occur as described below
\end{itemize}
\subsection{Disassembly}
Given a phenotype $P$ with embedded Hamiltonian path $H$, the disassembly process occurs iteratively by the detachment of at most 2 of tiles at at time. The process begins by the attachment of a special genome tile to the start of the Hamiltonian path. In each iteration, depending on the relative structure of the upcoming tiles in the Hamiltonian path, new genome tiles will attach to the existing genome encoding the local structure of $H$ (to be used during the reassembly process) and, using signals from these incoming genome tiles, a fixed number of structural tiles belonging to nearby points in the Hamiltonian path will detach from $P$. A property called the \emph{safe disassembly criterion} will be preserved after each iteration assuring that disassembly can continue as described. This process will continue until we reach the last tile in the Hamiltonian path. Once the final genome tile binds to the existing genome and this final tile, signals will cause these final structural tiles to detach and leave the genome in its final state where it can be used to make linear DNA as described above or replicate that phenotype as described below.
\subsubsection{Relevant Tiles and Directions}
In each iteration of our disassembly procedure, indexed by $i$, we will label a few important directions and tiles which will be useful. Since our tiles in this model are not required to reside in a fixed lattice, we define our cardinal directions $\{N, E, S, W, U, D\}$ arbitrarily so that they are aligned with the faces of some arbitrarily chosen tile in our phenotype. These directions will only be used when referring to tiles bound rigidly to the phenotype so there will be no ambiguity in their use.
The first tile, which we will call the \emph{previous structural tile} and write as $S^\text{prev}_i$, is the structural tile to which the genome is attached at the beginning of iteration $i$. This tile will detach from the rest of the phenotype by the end of iteration $i$. The \emph{next structural tile}, written $S^\text{next}_i$, is the structural tile to which the genome will be attached at the end of iteration $i$. Note that in some cases, this may not be the tile corresponding to the next tile in the Hamiltonian path, since we may detach more than one tile in an iteration.
We will refer to the corresponding attached genome tiles accordingly and write $G^\text{prev}_i$ and $G^\text{next}_i$ respectively.
The first direction, which we will call the \emph{next path direction} and write $D^p_i$, represents the direction from the previous structural tile to the next tile in the Hamiltonian path. Next, we will refer to the direction corresponding to the face of the previous structural tile upon which the previous genome tile is attached as the \emph{genome direction} and write $D^g_i$.
We also define a direction called the \emph{dangling genome direction}, written $D^d_i$, relative to the previous genome tile attached to the previous structural tile. At each iteration of the disassembly process new genome tiles will attach to the existing genome and the phenotype. By the end of in iteration, the previous genome tile will have detached from the structure and the next genome tile will be attached to the next structural tile. The dangling genome direction is defined to be the direction relative to the previous genome tile in which the rest of the genome is attached.
Figure \ref{fig:SDC_directions} illustrates what these directions look like in a particularly simple case.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figures/SDC_directions.png}
\caption{The relevant directions before and after an iteration of the disassembly process. The red arrow represents the next path direction, the blue arrow represents the genome direction, and the magenta arrow indicates the dangling genome direction. In this simple case the directions do not change after an iteration, but this is not always the case.}
\label{fig:SDC_directions}
\end{figure}
\subsubsection{The Safe Disassembly Criterion}
To facilitate in showing that the disassembly process works without error, we define a criterion which is preserved through each iteration of the disassembly process effectively acting as an induction hypothesis. We call this criterion, the \emph{safe disassembly criterion} or \emph{SDC}. The SDC is met exactly when all of the following are met:
\begin{enumerate}
\item There is no phenotype tile in the location location in the direction $D^g_i$ relative to the previous structural tile. This essentially means that there was room for the previous genome tile to attach to the previous structural tile.
\item At the current stage of disassembly, there is a path of empty tile locations that connects the previous tile location to a location outside the bounding box of the phenotype. This condition ensures that if our path digs into the phenotype during disassembly, there is a path by which detached tiles can escape and new genome tiles can enter to attach.
\item The dangling genome direction is not the same as the next path direction. This ensures that the existing genome is not dangling off of the previous genome tile in such a way that it would block the attachment of the next genome tile. This also ensures that our genome will never have to branch, though it may take turns.
\item Both the previous genome tile and some adjacent structural tile are presenting glues which allow for the attachment of another genome tile.
\end{enumerate}
\subsubsection{Disassembly Cases}
In each iteration of disassembly, there will be 6 effective possibilities regarding the local structure of the Hamiltonian path. Each of these possibilities will necessitate a different sequence of tile attachments and detachments for disassembly to occur. These cases are illustrated in figure \ref{fig:DR_case_enum_append} and described as follows.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{figures/DR_case_enum.png}
\caption{All of the essentially different cases that require a unique disassembly procedure. We orient these illustrations so that the previous genome direction is always up for convenience. Also note that we always illustrate the dangling genome direction to the left, but this need not be the case, this is just for making visualization easier. In reality, the dangling genome direction could be in any direction relative to the previous genome, so long as it satisfied the SDC condition that it is not the same as the next path direction. Gray squares represent attached structural tiles, green squares represent a location in which it does not matter if an attached structural tile exists, and empty squares represent locations in which no attached structural tile exists.}
\label{fig:DR_case_enum_append}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figures/DR_all_cases.png}
\caption{A side view of the disassembly process for all 6 cases. Each row is a unique case.}
\label{fig:DR_case_enum}
\end{figure}
\begin{lemma}\label{lem:DR_all_cases}
The 6 cases illustrated in Figure \ref{fig:DR_case_enum_append} are all of the possible cases for a disassembly iteration.
\end{lemma}
First note that the next path direction can either be perpendicular to the previous genome direction or not. If it is, we consider two cases. Either the tile location in the next genome direction relative to the next structural tile in the Hamiltonian path contains an attached structural tile or it doesn't. Case 1 is where it doesn't. If on the other hand it does, call the tile in that location the blocking tile; case 2 occurs when the blocking tile follows the next structural tile in the Hamiltonian path and case 3 occurs when it doesn't.
Supposing that the next path direction is not perpendicular to the previous genome direction, either it's the same direction or the opposite direction. By condition 1 of the SDC, it cannot be the same direction since there can be no structural tile attached in that location so all other cases must have the next path direction opposite the previous genome direction.
Now we define the working direction to be the direction opposite the dangling genome direction. This direction will be the direction in which genome tile attachments will occur during the remaining cases. Ultimately this choice is arbitrary, except that the working direction cannot be the dangling genome direction. Let location $a$ be the tile location in the working direction of the previous structural tile and location $b$ be the tile location in the opposite direction of the next path direction of location $a$. Case 4 is when neither location $a$ nor $b$ contains an attached structural tile, case 5 occurs when only location $a$ has an attached tile, and case 6 occurs otherwise.
Notice that since we defined these cases by dividing the possibility space into pieces where either some condition is or isn't met, this enumeration of cases represents all possibilities, thus proving Lemma \ref{lem:DR_all_cases}.
\subsubsection{The Disassembly Process}
Here we describe the disassembly process in enough detail that anyone familiar with basic tile assembly constructions should be able to derive the full details of the process without much difficulty.
Before any of the iterative disassembly cases can occur, the disassembly process begins with the attachment of the initial genome tile. The structural tile corresponding to the first point in the Hamiltonian path will be presenting a strength 2 glue to which this initial genome tile can attach. At this point in the process, this will be the only tile to which anything can attach with sufficient strength. This attachment activates a signal which turns off all glues in this initial structural tile except those holding it to the initial genome tile and the next structural tile in the Hamiltonian path. Also, now that this first genome tile has attached, the next genome tile can cooperatively attach initiating the disassembly process so that in the first iteration, the initial genome tile acts as the previous genome tile and the structural tile to which it's attached acts as the previous structural tile.
In each following iteration, once complete, what used to be called the next structural tile and next genome tile become the previous structural tile and previous genome tile for the next iteration and any relevant directions in the next iteration are specified relative to these new previous tiles.
Each of the cases as described above makes use of a unique sequence of tile genome attachments and signals; however, much of the logic in each of the cases is the same. We will describe two of the cases in greater detail than the rest, specifically cases 1 and 3, since understanding the details of those cases will make understanding the others much easier. Figure \ref{fig:DR_case_enum} illustrates the high level process of each case. It's important to keep in mind that the entire structure of the Hamiltonian path is encoded in the glues and signals of the phenotype tiles. This means that these cases can occur without issue since, for example, in an iteration where case 3 needs to occur, there will only be the glues and signals for case 3 present on the relevant tiles and none that would allow tiles for say case 5 to attach.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{figures/DR_case1_details.png}
\caption{A side view of some of the relevant glues and signals firing during the simplest disassembly case.}
\label{fig:DR_case_details-append}
\end{figure}
\begin{enumerate}
\item This case is the simplest case and is illustrated in Figure \ref{fig:DR_case_details-append}. First, a genome tile $G$ attaches cooperatively to the previous genome tile and the next structural tile. This attachment causes signals to fire in $G$ that activate 2 glues from the latent state to the on state. The first of these glues is a rigid, strength 2 glue that allows $G$ to bind rigidly and with more strength to the next structural tile. The other glue is a flexible, strength 2 glue that allows the genome to more strongly attach to the previous genome tile. The attachment of these glues activate signals which turn the old glues serving the same purpose into the off state. Additionally, signals are activated in the previous genome tile and the next structural tile disabling the glues in both that held onto the previous structural tile. Signals also deactivate any glues in the next structural tile that are attached to all other structural tiles except for the one following it in the Hamiltonian path.
At this point, there are no glues holding the previous structural tile to the genome nor the phenotype. This structural tile is now free to float away from what's left of the phenotype which is possible since the genome to which it was attached is now only bound with a flexible glue to the next genome tile and, by SDC condition 2, there is a path of empty tile locations along which it can escape.
In addition to all of the signals described previously, signals also activate a glue on the next genome tile which enables the attachment of the genome tile that will initiate the next iteration of the disassembly process.
By definition of case 1, SDC conditions 1 and 2 will be met after this process is done. Additionally, since the dangling genome direction now corresponds to the direction of the detached structural tile, condition 3 must also be satisfied. Condition 4 is also satisfied since glues were activated on the upcoming tile in the path to allow for cooperative binding of a new genome tile.
\item This case is largely similar to case 1 except that the next genome tile attaches to the structural tile following the next structural tile in the Hamiltonian path since the next is being blocked. In this case, it will be necessary for this tile to ``know'' that the next genome tile will attach to it. To accomplish this, all of the necessary glues that allowed the disassembly process to occur in the first case exist on this tile instead of the one immediately following the previous structural tile in the Hamiltonian path.
\item In this case, we have to remove the previous structural tile before we can attach the genome to the next structural tile since it is being blocked. We do this by utilizing what we call \emph{utility genome tiles}. These utility tiles are flat tiles that temporarily affix the genome to another part of the phenotype so that the previous tile can safely detach without the genome also detaching.
At first, this case proceeds similar to case 2 (and is illustrated in Figure \ref{fig:DR_case_enum_append}), but with a utility tile attaching to the blocking structural tile instead of the next genome tile. This attachment activates signals which cause the previous structural tile to detach. Since the tile to which the utility tile attached is not immediately adjacent to the previous structural tile, this is done using a chain of signals (which is a common gadget in STAM systems). The detachment of the previous structural tile allows the next genome tile to cooperatively bind to the previous one and to the next structural tile. This attachment causes signals to deactivate glues holding the utility tile in place allowing it to detach.
\item This case is largely degenerate and doesn't involve detachment of any tiles. Instead, utilizing cooperation, the next genome tile attaches to another face of the previous structural tile which also plays the part of the next structural tile. Depending on the tile or lack thereof in the green tile location from Figure \ref{fig:DR_case_enum_append}, the next iteration will either be case 1, 2, or 3.
\item This case is largely similar to case 3 except that the utility tile attaches in a different location. Once this occurs, instead of a new tile attaching cooperatively to the next tile, which is impossible since the next tile is not adjacent to the previous genome tile, a filler genome tile attaches to glues that are now present after the attachment of the utility genome tile. This filler genome tile acts as a spacer and after signals activate its glues, the next genome tile can attach to it and the next genome tile.
There is one consideration that needs to be made in this case. If the tile location illustrated in blue in case 5 of Figure \ref{fig:DR_case_enum_append} is the tile in the Hamiltonian path immediately following the next structural tile, then condition 3 of the SDC will not be met. This is because the dangling genome direction at the start of the next step will be in the same direction as the next path direction. To handle this, we simply require that two filler genome tiles attach between the utility tile and the next genome tile in this case. Since the structure of the Hamiltonian path is known in advance, this is possible, by requiring a different utility tile attach in the case where two filler tiles would be necessary than if only one was. Now, similar to case 3, the utility tile is free to detach following signals from the attachment of the next genome tile.
\item This case is identical to case 5 except that the utility tile attaches in a different location.
\end{enumerate}
\subsection{Reassembly and Replication}
At each iteration of the disassembly process, tiles attached to the genome encoding which tiles were detached. In some stages multiple tiles were detached, but it shouldn't be hard to see how that could be encoded in a single genome tile. Recall that this genome is a ``kinky'' genome. At this point, we could have defined the disassembly process above so that this genome immediately reconstructs the phenotype, the process for which is defined below; however, the definition of self-replicator requires that we construct arbitrarily many copies of the phenotype. Because of this, we can instead define the genome here so that it has the glues and signals necessary to convert into a linear genome as described in Section \ref{sec:simple-replicator-append}.
We refer to the processes described in Section \ref{sec:details-kinkase}. There we use a gadget called kink-ase to convert a linear sequence of genome tiles into a ``kinky'' one which is capable of constructing a shape. This process is easily reversible using a similar gadget which follows the steps in Figure \ref{fig:linear-to-kin} in reverse. This process converts the kinky genome made during the disassembly of our phenotype into a linear genome which can be replicated arbitrarily using the process described in Section \ref{sec:detail-genome}. For our purposes, it's useful to modify this linear genome duplication process so that our linear genome is duplicated into two copies: one that can be further used for genome duplication and one that can be converted back to kinky form and used to reassemble the phenotype. This simply requires that we specify a second set of the corresponding glues and signals on the genome constructed from the disassembly process. This guarantees that we are generating arbitrarily many copies of the phenotype.
Once we have kinky genomes ready to reconstruct the phenotype, we can begin the reassembly process. This process behaves much like the disassembly process, but with the genome being disassembled and the structure being reassembled. Once a reassembly fuel tile attaches to the special tile at the end of the genome, signals will activate glues allowing a structural tile, identical to the last tile in the Hamiltonian path of the original phenotype, to attach. This initiates the reassembly process and each of the tiles in the Hamiltonian path will attach in reverse order as the genome disassembles from the back. This process is in some ways more straightforward than disassembly because the only tiles that detach are genome tiles and they detach completely. In the assembly process, both structural tiles and genome tiles had to detach and the detachment of genome tiles had to happen in such a way that they were still attached by flexible glues to the rest of the genome.
The following is an outline of the reassembly processes for each of the cases. Figure \ref{fig:DR_case_enum_append} can still be used as a reference but be careful to keep in mind that the process is happening in the opposite direction, initiated by the attachment of what was called the \emph{next} structural tile in the disassembly process. In this section we reverse the terminology so that in each iteration, what were the previous structural and genome tiles are now the next structural and genome tiles and vice-versa. In each iteration of this process, the attachment of the previous structural tile to our genome initiates the sequence of attachments, detachments, and signals that allow the next structural tile to attach and the previous genome tile to detach.
\begin{enumerate}
\item This is the most basic case, the attachment of the previous structural tile to the genome activates glues on the next genome tile. This enables the next structural tile to attach cooperatively which causes signals to deactivate glues so that the previous genome tile detaches.
\item The attachment of the previous structural tile in this iteration activates glues on it which immediately allows the next structural tile to attach. Again this attachment activates signals which turn on glues to allow another tile to attach forming the corner. Finally, the next genome tile can bind to this last structural tile which causes glues to deactivate so that the previous genome tile detaches.
\item The attachment of the structural tile to the genome in the previous iteration activates a glue on the genome tile and adjacent structural tile allowing a utility tile to attach. This causes signals to deactivate glues holding the previous genome tile and activating glues on the structural tile to which it was bound. This allows a new structural tile to attach and then the corresponding genome tile. These attachments create signal paths that deactivate glues on the utility tile and the structural tile to which it was attached, allowing it to fall off.
\item This stage just represents the genome tile turning a corner which causes the old genome tile to detach after signals deactivate its glues. This can only happen after case 1, 2, or 3 similar to the analogous case during disassembly.
\item The attachment of the structural tile activates glues which allow the utility tile to attach. This attachment initiates signals which do 3 things. the signals deactivate glues holding the previous genome to the structural tile, the signals deactivate glues holding the utility tile to the old genome tiles, and the signals activate glues on the next genome tile. The next genome tile can then cooperate with the old structural tile to attach a new structural tile. Note that in this case the filler genome tiles from the disassembly will remain attached to the previous genome tile and they will detach as a short chain.
\item This case is almost identical to the previous case with a slightly different binding location for the utility tile.
\end{enumerate}
Note that in each of the cases described above it's possible to reassemble the phenotype structure using the same tiles that were originally in the seed phenotype. As described here, we require that some of the signals in these reassembled phenotype tiles will be fired to facilitate in the reassembly process; however, with a more careful design it wouldn't be difficult to describe a process which reassembles the phenotype without using any signals on the structural tiles if this was a desired property. Additionally, during cases 5 and 6, pairs of filler tiles will detach depending on the next direction of the path in that iteration. This results in our waste size being 2, but again with a more careful design it would be easy to specify tiles which, say, bind to these waste pairs and break them down into single tiles if having waste size 1 was a desired property.
\subsection{Phenotype Generation Algorithm}
In this section, we describe an efficient algorithm for describing the $STAM^*$ system in which this process runs. Given that we require complex information to be encoded in the glues and signals of our components, particularly in the phenotype since it requires an encoded Hamiltonian path, it might seem like we are ``cheating'' by baking potentially intractable computations in these glues and signals. This however is not the case in the sense that, as we will show, all of the required tiles, glues, signals, paths, etc. (all from a fixed, finite set of types) can be described by a polynomial time algorithm given an arbitrary shape to self-replicate.
The algorithm described consists largely of two parts. First, we will determine a Hamiltonian path through our shape, and second we will use this path to determine which glues need to be placed where on our tiles.
\subsubsection{Generating A Hamiltonian Path} \label{sec:hampath_append}
In general, the problem of finding a Hamiltonian path through a graph is \textbf{NP}-complete and may be impossible for many shapes we may wish to use; however, if we scale our shape by a constant factor of 2, that is replace every voxel location with a $2\times 2\times 2$ block of tiles, then not only is there always a Hamiltonian path, but it can be computed efficiently. The algorithm for generating this Hamiltonian path is described in further detail in \cite{Moteins} and was inspired by \cite{SummersTemp}, but we will describe the procedure at a high level here using terminology that is convenient for our purposes.
\begin{enumerate}
\item Given a shape $S$, we first find a spanning tree $T$ through the graph whose vertices correspond to locations in $S$.
\item We embed this spanning tree in a space scaled by a factor of 2 so that each vertex corresponds to a $2\times 2\times 2$ block of locations.
\item To each $2\times 2\times 2$ block in this space, we assign one of two orientation graphs $G_o^1$ or $G_o^2$. These graphs each form a simple oriented cycle through all points. These graphs are assigned so that they form a checkerboard pattern such that no blocks assigned $G_o^1$ are adjacent to any blocks assigned $G_o^2$ and vice versa. Figure \ref{fig:orient_graph} illustrates what the orientation graphs look like for adjacent blocks.
\item For each edge in the spanning tree $T$, we join the orientation graphs corresponding to the vertices of the edge so that they form a single continuous cycle as illustrated in Figure \ref{fig:orient_graph}. This process is described in more detail in \cite{Moteins}.
\item Once we do this for all edges in our spanning tree, the connected orientation graphs will form a Hamiltonian circuit through the $2\times 2\times 2$ blocks corresponding to the tiles in our shape. This is easy to see by analyzing a few cases corresponding to all possible vertex types in the spanning tree and noting that in none of them does the path ever become disconnected. This is done in \cite{Moteins}.
\end{enumerate}
The resulting Hamiltonian path, which we will call $H$, passes through each tile in the 2-scaled version of our shape and only took a polynomial amount of time to compute since spanning trees can be found efficiently and only contain a polynomial number of edges. Given $H$, we can arbitrarily choose some vertex on the surface of our shape to represent the starting point of our path $H_1$ and label the rest of the path in order with respect to this one so that the next point is labeled $H_2$, then $H_3$, and so on. Additionally, we can also keep track of the location in space relative to some fixed origin to which each point in our path belongs and note that, using common data structures and basic arithmetic, determining the index of points in $H$ given a location can be done efficiently.
\subsubsection{Determining Necessary Information to encode in Glues and Signals}
Recall that each case of the disassembly and reassembly processes sometimes required tiles nearby in space to have glues and signals to facilitate each step of the process. We define the following algorithm which is able to describe these glues and signals, showing that we can efficiently describe the tiles necessary for our construction.
Begin with tile $H_1$ and iterate over the entire Hamiltonian path performing the following operations with the current tile labelled $T_i$ and keeping track of a counter $t$ which starts at 0.
\begin{enumerate}
\item Determine which of the 6 disassembly cases would apply to this particular tile by looking at adjacent tile locations and considering only those tiles not yet flagged with a detachment time.
\item At this point, we know exactly which case $T_i$ will use during the detachment process. Assign any glues and signals necessary to this tile and adjacent tiles.
\item Flag $T_i$ as being detached at time $t$.
\item If $T_i$ used case 2, also mark the tile following $T_i$ as being detached at time $t$ and skip the next tile in the path for the next iteration.
\item increment $t$ and $i$.
\end{enumerate}
Our algorithm now knows which glues and signals are necessary for each tile that will make up the phenotype. We can now iterate over all tiles in the construction and make a set consisting of each unique tile in the phenotype. Additionally, the genome tiles necessary for the process are even simpler to define since there is only a small fixed number needed for each case. This shows that the system in which this process occurs can be described efficiently by an algorithm and that we are not doing an unreasonable amount of pre-computation by including the necessary information in our glues and signals.
\subsubsection{Glues for Converting to Linear DNA}\label{sec:kinky-to-linear-append}
The disassembly process above results in arbitrarily many ``kinky'' genomes which are capable of being used to produce a replica of the original phenotype. In order for this process to be possible however, the kinky genome produced by the disassembly process needs glues and signals to indicate locations that should be ``un-kinked'' and replicated. This is no problem however since the only cases in the disassembly process that could induce a kink in our constructed genome are 1, 2, and 3. The kink induced in the genome in any of these cases solely depends on the dangling genome direction and next path direction. Since there are only a finite number of such cases and since our tileset will have a unique set of genome tiles that attach in each such case, we can easily specify the necessary glues and signals to the corresponding genome tiles. This guarantees that the conversion to linear DNA is possible for any genome constructed by the disassembly process.
\subsection{Correctness of Theorem \ref{thm:deconstructor}}\label{sec:decontructor-correctness}
First, we restate Theorem \ref{thm:deconstructor} for convenience:
\begin{theorem}\label{thm:deconstructor-append}
There exists a universal tile set $T$ such that for every shape $S$, there exists an STAM* system $\mathcal{R} = (T,\sigma_{S^2},2)$ where $\sigma_{S^2}$ has shape $S^2$ and $\mathcal{R}$ is a self-replicator for $\sigma_{S^2}$ with waste size 2.
\end{theorem}
We have shown how, given any shape $S$ as input, we can scale it by factor $2$ to $S^2$ and efficiently find a Hamiltonian path through $S^2$. We can then compute the tile types and signals needed at each location to build a phenotype which can serve as a seed supertile for an STAM* system $\mathcal{R}$ using a universal tile set $T$. At temperature $2$, $\mathcal{R}$ will deconstruct the input supertiles to create kinky genome assemblies. Each kinky genome assembly will then first create a copy of the linear genome, and then either continue to create copies of the linear genome, or initiate the growth of a new copy of the phenotype (which consumes the copy of the kinky genome). The new copies of the phenotype will become terminal assemblies, in the shape of $S^2$. The other terminal assemblies are junk assemblies of size $\le 2$ (during the reassembly process for cases 5 and 6, for certain next path directions, pairs of filler tiles will detach), and the linear genome assemblies are never terminal as each facilitates the growth of infinite new copies. Thus, $\mathcal{R}$ is a self-replicator for $S^2$ and since this works for arbitrary shapes at scale factor $2$, $T$ is a universal tile set for shape self-replication for the class of scale factor 2 shapes.
\section{A Self-Replicator that Generates its own Genome}
\label{sec:deconstruct}
In this section we outline our main result: a system which, given an arbitrary input shape, is capable of disassembling an assembly of that shape block-by-block to build a genome which encodes it. We describe the process by which this disassembly occurs and then show how, from our genome, we can reconstruct the original assembly. Here we describe the construction at a high level. The technical details for this construction can be found in the appendix in Section \ref{sec:deconstruct-append}. We prove the following theorem by implicitly defining the system $\mathcal{R}$, describing the process by which an input assembly is disassembled to form a ``kinky'' genome which is then used to make a copy of a linear genome (which replicates itself) and of the original input assembly.
\begin{theorem}\label{thm:deconstructor}
There exists a universal tile set $T$ such that for every shape $S$, there exists an STAM* system $\mathcal{R} = (T,\sigma_{S^2},2)$ where $\sigma_{S^2}$ has shape $S^2$ and $\mathcal{R}$ is a self-replicator for $\sigma_{S^2}$ with waste size 2.
\end{theorem}
In this construction, there are two main components which here we call the \emph{phenotype} and the \emph{kinky genome}. The phenotype, which is the seed of our STAM* system, is a scale 2 version of our target shape made entirely out of cubic tiles. These tiles are connected to one another so that the assembly is $\tau$-stable at temperature 2. We require the phenotype to be a 2-scaled version of $S$ since the disassembly process requires a Hamiltonian path to pass through each of the tiles in $P$. This path describes the order in which the disassembly process will occur. Generally it is often either impossible or intractable to find a Hamiltonian path through an arbitrarily connected graph; however, using a 2-scaled shape we show that it's always possible efficiently. Additionally, the tiles in the phenotype contain glues and signals that will allow the various attachments and detachments to occur in the disassembly process. The genome is a sequence of flat tiles connected one to the next, whose glues encode the construction of the phenotype. In our system, the genome will be constructed as the phenotype is deconstructed and then will be duplicated or used to make copies of the original phenotype. Throughout this section, we refer to the cubic tiles that make up the phenotype as structural tiles and the flat tiles that make up the genome as genome tiles. Additionally, the tiles used in this construction are part of a finite tile set $T$, making $T$ a universal tile set.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{figures/DR_overview.png}
\caption{During disassembly, the genome will be dangling off of a single structural tile in the phenotype. In each iteration, a new genome tile will attach and the old structural tile will detach along the Hamiltonian path embedded in the phenotype.}
\label{fig:DR_overview}
\end{figure}
\vspace{-10pt}
\subsection{Disassembly}
Given a phenotype $P$ with encoded Hamiltonian path $H$, the disassembly process occurs iteratively by the detachment of at most 2 of tiles at at time. The process begins by the attachment of a special genome tile to the start of the Hamiltonian path. In each iteration, depending on the relative structure of the upcoming tiles in the Hamiltonian path, new genome tiles will attach to the existing genome encoding the local structure of $H$ and, using signals from these newly attached genome tiles, a fixed number of structural tiles belonging to nearby points in the Hamiltonian path will detach from $P$. The order in which these detachments happen follow the path $H$ and they will also cause all but the most recently attached genome tile to detach from the structure causing them to dangle, hanging on to the most recently attached genome tile as illustrated in Figure \ref{fig:DR_case_details}.
To show that the disassembly process happens correctly, we break down each iteration into one of 6 cases based on the tiles nearby the next in the Hamiltonian path. We show that these cases are complete and describe the process of disassembly for each one in Section \ref{sec:deconstruct-append}. Figure \ref{fig:DR_case_details} illustrates the process and many of the important signals necessary for the most basic case. In it, a single genome tile attaches causing the previous one to dangle and the previous structural tile to detach. This new genome tile encodes this detachment so that reassembly can occur later and the process continues from there in the next iteration.
\begin{wrapfigure}{r}{0.5\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{figures/DR_case1_details.png}
\vspace{10pt}
\includegraphics[width=0.5\textwidth]{figures/DR_case_enum.png}
\caption{(top) A side view of some of the relevant glues and signals firing during the simplest disassembly case. (bottom) A side view of the local structure of nearby tiles for all 6 essentially different cases in the disassembly process.}
\label{fig:DR_case_details}
\vspace{-20pt}
\end{wrapfigure}
\vspace{-10pt}
\subsection{Reassembly}
Once the genome is built, we show that the original shape can be reconstructed. This occurs when a special structural tile attaches to the genome. This tile is identical to the last tile in the Hamiltonian path of the original phenotype and initiates the reassembly process. Section \ref{sec:deconstruct-append} contains more details of the reassembly process, but essentially that reassembly occurs very similarly to disassembly in reverse - still using the same 6 cases as above and instead of having a new genome tile attach and the old structural tiles detach, the opposite occurs.
\vspace{-10pt}
\subsection{Generating A Hamiltonian Path} \label{sec:hampath}
\begin{lemma}
Any scale factor 2 shape $S^2$ admits a Hamiltonian path and generating this path given a graph representing $S^2$ can be done in polynomial time.
\end{lemma}
The algorithm for generating this Hamiltonian path is described in detail in \cite{Moteins} and was inspired by \cite{SummersTemp}. At a high level, the process proceeds as follows. First we generate a spanning tree through the shape $S$. We then scale the shape by a factor of two, assigning to each $2\times 2\times 2$ block of tiles one of two orientation graphs as illustrated in Figure \ref{fig:orient_graph}. These orientation graphs make a path through the 8 tiles making up a tile block. For each edge in the spanning tree, we connect the corresponding orientation graphs, combining them to form a single orientation graph. Doing this for all edges will leave us with a Hamiltonian path through $S^2$. In fact, we actually define a Hamiltonian circuit which guarantees that during disassembly, the remaining phenotype will always remain connected.
\begin{figure}
\centering
\subfloat{%
\includegraphics[width=0.4\textwidth]{figures/DR_orient.png}%
\label{fig:DR_orient}%
}%
\hspace{20pt}
\subfloat{%
\includegraphics[width=0.4\textwidth]{figures/DR_orient_join.png}%
\label{fig:DR_orient_join}%
}%
\caption{(left) Each $2\times 2\times 2$ block of space is assigned an orientation graph which will be used to help generate the Hamiltonian path through our shape. Adjacent blocks are assigned opposite orientation graphs, the edges of which will help guide the Hamiltonian path around the shape. (right) Orientation graphs of adjacent blocks are joined to form a continuous path}
\label{fig:orient_graph}
\end{figure}
The resulting Hamiltonian path, which we will call $H$, passes through each tile in the 2-scaled version of our shape and only took a polynomial amount of time to compute since spanning trees can be found efficiently and only contain a polynomial number of edges. Additionally, it should be noted that once we generate a Hamiltonian path, an algorithm can easily iterate over the path simulating which tiles would still be attached during each stage of the disassembly process. This means such an algorithm can also easily determine the glues and signals necessary for each tile in the path by considering the appropriate iteration case.
\section{Introduction}
\label{sec:intro}
\vspace{-5pt}
\subsection{Background and motivation}
\vspace{-5pt}
Research in tile based self-assembly is typically focused on modeling the computational and shape-building capabilities of biological nano-materials whose dynamics are rich enough to allow for interesting algorithmic behavior. Polymers such as DNA, RNA, and poly-peptide chains are of particular interest because of the complex ways in which they can fold and bind with both themselves and others. Even when only taking advantage of a small subset of the dynamics of these materials, with properties like binding and folding generally being restricted to very manageable cases, tile assembly models have been extremely successful in exhibiting vast arrays of interesting behavior \cite{RotWin00,SolWin05,IUSA,OneTile,2HAMIU,jCCSA,SummersTemp,DotKarMasNegativeJournal,BeckerRR06,AGKS05g,Dot09}. Among other things, a typical question in the realm of algorithmic tile assembly asks what the minimal set of requirements is to achieve some desired property. Such questions can range from very concrete, such as ``how many distinct tile types are necessary to construct specific shapes?'', to more abstract such as ``under what conditions is the construction of self-similar fractal-like structures possible?''. Since the molecules inspiring many tile assembly models are used in nature largely for the purpose of self-replication of living organisms, a natural tile assembly question is thus whether or not such behavior is possible to model algorithmically.
In this paper we show that we can define a model of tile assembly in which the complexities of self-replication type behavior can be captured, and provide constructions in which such behavior occurs. We define our model with the intention of it (1) being hopefully physically implementable in the (near) future, and (2) using as few assumptions and constraints as possible. Our constructions therefore provide insight into understanding the basic rules under which the complex dynamics of life, particularly self-replication, may occur
We chose to use the Signal-passing Tile Assembly Model (STAM) as a basis for our model, which we call the STAM*, because (1) there has been success in physically realizing such systems \cite{SignalTilesExperimental} and potential exists for further, more complex, implementations using well-established technologies like DNA origami \cite{RotOrigami05,OrigamiTiles,wei2012complex,OrigamiBox,OrigamiSeed} and DNA strand displacement \cite{Qian1196,WangE12182,Simmel2019,ZhangDavidYu2011DDnu,StrandDispInTiles,bui2018localized}, and (2) the STAM allows for behavior such as cooperative tile attachment as well as detachment of subassemblies
We modify the STAM by bringing it into 3 dimensions and making a few simplifying assumptions, such as allowing multiple tile shapes and tile rotation around flexible glues and removing the restriction that tiles have to remain on a fixed grid
Allowing flexibility of structures and multiple tile shapes provides powerful new dynamics that can mimic several aspects of biological systems and suffice to allow our constructions to model self-replicating behavior
Prior work, theoretical \cite{STAMPatternRep} and experimental \cite{SchulYurWinfEvolution}, has focused on the replication of patterns of bits/letters on 2D surfaces, as well as the replication of 2D shapes in a model using staged assembly \cite{RNaseSODA2010}, or in the STAM \cite{STAMshapes}. However, all of these are fundamentally 2D results and our 3D results, while strictly theoretical, are a superset with constructions capable of replicating all finite 2D and 3D patterns and shapes.
Biological self-replication requires three main categories of components: (1) instructions, (2) building blocks, and (3) molecular machinery to read the instructions and combine building blocks in the manner specified by the instructions. We can see the embodiment of these components as follows: (1) DNA/RNA sequences, (2) amino acids, and (3) RNA polymerase, transfer RNA, and ribosomes, among other things. With our intention to study the simplest systems capable of replication, we started by developing what we envisioned to be the simplest model that would provide the necessary dynamics, the STAM*, and then designed modular systems within the STAM* which each demonstrated one or more important behaviors related to replication. Quite interestingly, and unintentionally, our constructions resulted in components with strong similarities to biological counterparts. As our base encoding of the instructions for a target shape, we make use of a linear assembly which has some functional similarity to DNA. Similar to DNA, this structure also is capable of being replicated to form additional copies of the ``genome''. In our main construction, it is necessary for this linear sequence of instructions to be ``transcribed'' into a new assembly which also encodes the instructions but which is also functionally able to facilitate translation of those instructions into the target shape. Since this sequence is also degraded during the growth of the target structure, it shares some similarity with RNA and its role in replication. Our constructions don't have an analog to the molecular machinery of the ribosome, and can therefore ``bootstrap'' with only singleton copies of tiles from our universal set of tiles in solution. However, to balance the fact that we don't need preexisting machinery, our building blocks are more complicated than amino acids, instead being tiles capable of a constant number of signal operations each (turning glues on or off due to the binding of other glues).
\vspace{-10pt}
\subsection{Our results}
\vspace{-5pt}
Beyond the definition of the STAM* as a new model, we present a series of STAM* constructions. They are designed and presented in a modular fashion, and we discuss the ways in which they can be combined to create various (self-)replicating systems.
\vspace{-10pt}
\subsubsection{Genome-based replicator}
\vspace{-5pt}
We first develop an STAM* tileset which functions as a simple self-replicator (in Section \ref{sec:simple-replicator}) that begins from a seed assembly encoding information about a target structure, a.k.a. a \emph{genome}, and grows arbitrarily many copies of the genome and target structure, a.k.a. the \emph{phenotype}. This tileset is universal for all 3D shapes comprised of $1\times 1 \times 1$ cubes when they are inflated to scale factor 2 (i.e. each $1 \times 1 \times 1$ block in the shape is represented by a cube of $2 \times 2 \times 2$ tiles). This construction requires a genome whose length is proportional to the number of cube tiles in the phenotype; for non-trivial shapes the genome is a constant factor longer in order to follow a Hamiltonian path through an arbitrary 3D shape at scale factor 2. This is compared to the Soloveichik and Winfree universal (2D) constructor \cite{SolWin07} where a ``genome'' is optimally shortened, but the scale factor of blocks is much larger.
The process by which this occurs contains analogs to natural systems. We progress from a genome sequence (acting like DNA), which is translated into a messenger sequence (somewhat analogous to RNA), that is modified and consumed in the production of tertiary structures (analogous to proteins). We have a number of helper structures that fuel both the replication of the genome and the translation of the messenger sequence.
\vspace{-10pt}
\subsubsection{Deconstructive self-replicator}
\vspace{-5pt}
In Section \ref{sec:deconstruct}, we construct an STAM* tileset that can be used in systems in which an arbitrarily shaped seed structure, or phenotype, is disassembled while simultaneously forming a genome that describes its structure. This genome can then be converted into a linear genome (of the form used for the first construction) to be replicated arbitrarily and can be used to construct a copy of the phenotype. We show that this can be done for any 3D shape at scale factor 2 which is sufficient, and in some cases necessary, to allow for a Hamiltonian path to pass through each point in the shape. This Hamiltonian path, among other information necessary for the disassembly and, later, reassembly processes, is encoded in the glues and signals of the tiles making up the phenotype. We then show how, using simple signal tile dynamics, the phenotype can be disassembled tile by tile to create a genome encoding that same information. Additionally, a reverse process exists so that once the genome has been constructed from a phenotype, a very similar process can be used to reconstruct the phenotype while disassembling the genome.
In sticking with the DNA, RNA, protein analogy, this disassembly process doesn't have a particular biological analog; however, this result is important because it shows that we can make our system robust to starting conditions. That is, we can begin the self-replication process at any stage be it from the linear genome, ``kinky genome'' (the messenger sequence from the first construction), or phenotype. Finally, since this construction requires the phenotype to encode information in its glues and signals, we show that this can be computed efficiently using a polynomial time algorithm given the target shape. This not only shows that the STAM* systems can be described efficiently for any target shape via a single universal tile set, but that results from intractable computations aren't built into our phenotype (i.e. we're not ``cheating'' by doing complex pre-computations that couldn't be done efficiently by a typical computationally universal system). Due to space constraints we only include a result about the necessity for deconstruction in a universal replicator in Section \ref{sec:need-to-deconstruct}.
\vspace{-12pt}
\subsubsection{Hierarchical assembly-based replicator}
\vspace{-5pt}
For our final construction, in Section \ref{sec:2ham}, our aims were twofold. First, we wanted to compress the genome so that its total length is much shorter than the number of tiles in the target shape. Second, we wanted to more closely mimic the biological process in which individual proteins are constructed via the molecular machinery, and then they are released to engage in a hierarchical self-assembly process in which proteins combine to form larger structures.
Biological genomes are many orders of magnitude smaller than the organisms which they encode, but for our previous constructions the genomes are essentially equivalent in size to the target structures. Our final construction is presented in a ``simple'' form in which the general scaling approximately results in a genome which is length $n^{\frac{1}{3}}$ for a target structure of size $n$. However, we discuss relatively simple modifications which could, for some target shapes, result in genome sizes of approximately $\log{n}$, and finally we discuss a more complicated extension (which also consumes a large amount of ``fuel'', as opposed to the base constructions which consume almost no fuel) that can achieve asymptotically optimal encoding.
\vspace{-10pt}
\subsubsection{Combinations and permutations of constructions}
Due to length restrictions for this version of the paper, and our desire to present what we found to be the ``simplest'' systems capable of combining to perform self-replication, there are several additions to our results which we only briefly mention. For instance, to make our first construction (in Section \ref{sec:simple-replicator}) into a standalone self-replicator, and one which functions slightly more like biological systems, the input to the system, i.e. the seed assembly, could instead be a copy of the target structure with a genome ``tail'' attached to it. The system could function very similarly to the construction of Section \ref{sec:simple-replicator} but instead of genome replication and structure building being separated, the genome could be replicated and then initiate the growth of a connected messenger structure so that once the target structure is completed, the genome is attached. Thus, the input assembly would be completed replicated, and be a self-replicator more closely mirroring biology where the DNA along with the structure cause the DNA to replicate itself and the structure. Attaching the genome to the structure is a technicality that could satisfy the need to have a single seed assembly type, but clearly it doesn't meaningfully change the behavior.
At the end of Section \ref{sec:2ham} we discuss how that construction could be combined with those from Sections \ref{sec:simple-replicator} and \ref{sec:deconstruct}, as well as further optimized. The main body of the paper contains high-level overviews of the definition of the STAM* as well as of the results. Full technical details for each section can be found in the corresponding section of the technical appendix
\section{The Requirement for Deconstruction}\label{sec:need-to-deconstruct}
\begin{definition}
Given a tile set $T$, a \emph{porous assembly} $\alpha$, over tiles in $T$, is one in which it is possible for unbound tiles of one or more types in $T$ to pass freely through either (1) the body of one or more tiles in $\alpha$, or (2) the gaps between tiles in $\alpha$ (which means between bound glues if the tiles are bound to each other), or (3) a combination of both. Conversely, a \emph{non-porous assembly} is one in which no unbound tiles can pass through any of the tile bodies or gaps between tiles.
\end{definition}
For theoretical results, we tend to consider all tile bodies to be solid, or at least solid enough to prevent the diffusion of other tiles through them. Whether or not an assembly is porous then depends upon factors such as the spacing between tiles, lengths of glues, and spacing of glues. For instance, the seed assemblies for the construction in Section \ref{sec:deconstruct} are non-porous assuming glues are spread evenly along the edges of tiles.
In this section we prove that in the STAM* there cannot be a universal shape self-replicator in systems with non-porous assemblies that does not use (an arbitrary amount of) deconstruction.
\begin{theorem}\label{thm:need-to-deconstruct}
Let $U$ be an STAM* tile set such that for an arbitrary 3D shape $S$, the STAM* system $\mathcal{T} = (U,\sigma_S,\tau)$ with $\dom \sigma = S$, $\mathcal{T}$ is a shape self-replicator for $S$ and $\sigma$ is non-porous. Then, for any $r \in \mathbb{N}$, there exists a shape $S$ such that $\mathcal{T}$ must remove at least $r$ tiles from the seed assembly $\sigma_S$.
\end{theorem}
\begin{proof}
We prove Theorem \ref{thm:need-to-deconstruct} by contradiction. Therefore, assume that $U$ is a tile set in the STAM* capable of shape replicating any shape $S$ and that seed assembly $\sigma_S$ is non-porous. Let $t = |U|$, $g$ be the maximum number of glues on any tile type in $U$, and $s$ be the maximum number of signals on any tile type in $U$. Note that for any position in an assembly over tiles in $U$, there is a maximum number of $\lambda = t(3^g)(3^s)$ possible tile types and tile states (accounting for all possible states of glues and signals).
We define a shape $c$ which is an $n \times n \times n$ cube, for some $n \in \mathbb{N}$ to be defined, with every point on the exterior of the cube included in the shape. For every $xy$ plane (i.e. horizontal plane) in the interior of the cube, the points contained within $c$ follow the pattern shown in Figure \ref{fig:imposs-center}, where the grey locations are all included and a subset of the green locations are included. Note that only one plane has a connection to the exterior, and no other tiles of any plane in the interior are adjacent to a location of the exterior. Define the set $C$ as the set of all such $c$ where there is one for each possible pattern of green locations included and excluded.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{impossible-center.png}
\caption{An example interior $xy$ plane within the cube $c$ of the proof of Theorem \ref{thm:need-to-deconstruct}. The plane in this example has the single connection to the exterior of the cube (dark grey), and all light grey locations are included, along with a subset of the green locations.}
\label{fig:imposs-center}
\end{figure}
In order to ensure that only a single location of a single $xy$ plane in the interior of the cube is adjacent to the exterior (i.e. to leave a gap all around) the number of $xy$ planes with occupied locations is $n-4$. The width of each green row is $n-5$. The number of green rows in each $xy$ plane is $(n-4)/2$. Therefore, the number of green interior positions is $(n-4)(n-5)(n-4)/2$. The number of shapes which include every possible subset of those green positions is $2^{(n-4)(n-5)(n-4)/2}$, and this is the size of the set $C$.
Conversely, the number of unit cube locations on the exterior of each $n \times n \times n$ cube is $6(n-1)^2$.
By our assumption, for every $c \in C$, there exists an STAM* system $\mathcal{T}_c = (U,\sigma_c,\tau)$ such that $\mathcal{T}_c$ shape self-replicates $c$. However, for each such $\sigma_c$, the total number of options for a tile in each exterior location (including states) is $\lambda$, and therefore the total number of unique subassemblies composing the exterior surfaces of the cube is $\lambda^{6(n-1)^2}$. Also, since $s$ is the maximum number of signals on any tile type in $T$, $s!$ represents every possible ordering of completion of signals on the tile with the most signals. We can choose a value of $n$ (for the side lengths of the cubes) such that $(s!)\lambda^{6(n-1)^2+1} < 2^{(n-4)(n-5)(n-4)/2}$, since the exponents of the left and right sides grow on the order of $n^2$ and $n^3$, respectively, and all other terms are constants with respect to $n$. Let $n$ be such a sufficiently large value and then note that by the pigeonhole principle, for two $c_1,c_2 \in C$, the systems $\mathcal{T}_{c_1}$ and $\mathcal{T}_{c_2}$ must have identical subassemblies composing the exteriors of their seed assemblies as well as the single tile attaching each exterior to the interior planes. Additionally, there must be an assembly sequence such that the single tile of each exterior subassembly that is connected to the interior planes must experience the same ordering of completion of signals (since anything that could happen on their exteriors must be the same for both, and there were enough assemblies with the same subassemblies to guarantee the same order of completion of their signals for at least two of them). Since $\sigma_{c_1}$ and $\sigma_{c_2}$ are non-porous, there can be no other factors in $\mathcal{T}_{c_1}$ and $\mathcal{T}_{c_2}$ which influence the growth of assemblies, and so both systems must be able to yield the same terminal assemblies. This contradicts that they shape self-replicate $c_1$ and $c_2$ since these are different shapes. Finally, in order to achieve the arbitrary bound $r$ for required tile removals, we can simply adapt our target shape to be a ``chain'' of $r$ cubes (all of which can be made to be unique) connected by a single-tile-wide path of tiles and otherwise completely separated. The previous argument holds for each of the $r$ cubes, and since none can be replicated without the removal of at least one tile, a lower bound of the removal of at least $r$ tiles is established.
\end{proof}
\section{Technical Details of the STAM* Model}\label{sec:prelims-append}
In this section we provide a more detailed set of definitions for the STAM*.
\begin{enumerate}
\item Let $\Sigma$ be an alphabet, and $C: \Sigma^+ \rightarrow \Sigma^+$ be a function which maps each non-empty string over $\Sigma$ to another non-empty string over $\Sigma^+$, such that for all $s \in \Sigma^+$, $C(C(s)) = s$. For a string $s \in \Sigma^+$, we say that $C(s)$ is \emph{complementary} to $s$, and we also use the notation $s^*$ to denote it.
\item Let $D \subset \Sigma^+$ be a finite set of strings which we call \emph{domains}, and let $D^* = \{ s \mid C(s) \in D\}$ be a finite set of strings we call \emph{complementary domains}, or \emph{co-domains}.
\item A \emph{glue type} is an ordered 4-tuple $(d,n,f,l)$ where $d \in D \cup D^*$ is the domain, $n \in \mathbb{Z}^+$ is the \emph{strength}, $f \in \{\texttt{true},\texttt{false}\}$ is a boolean value denoting whether or not the glue is \emph{flexible}, and $l \ge 0$ is the \emph{length} of the glue. For every pair of glue types $g_1 = (d_1,n_1,f_1,l_1)$ and $g_2 = (d_2,n_2,f_2,l_2)$, if $d_1 = d_2$ or $d_1 = d_2^*$, then $n_1 = n_2$, $f_1 = f_2$, and $l_1 = l_2$ (i.e. all glue types sharing the same domains or having co-domains of each other have the same strength, flexibility, and length).
\item For a glue type $g = (d,n,f,l)$, if $f == \texttt{true}$ we say that $g$ is \emph{flexible}, and if $f == \texttt{false}$ we say that $g$ is \emph{rigid}.
\item A \emph{glue} is an instance of a glue type. A glue may be in exactly one of $3$ states at any given time: $\{\texttt{latent,on,off}\}$. Only a glue which is in the $\texttt{on}$ state can bind to another (complementary) glue.
\item A \emph{tile type} is defined as a tuple $(s,G,S)$ where $s$ is the 3-dimensional shape of the tile, $G$ is a set of triples, with each triple consisting of a glue type, a location (the point on the tile's surface where the glue is located), and initial glue state, and $S$ is the set of signals (to be defined). Each face of a tile type can have a finite number of $0$ or more glues. For ease of reference, each tile type has a canonical orientation so that sides can be referenced with respect to that orientation regardless of the orientation of a particular copy.
\item Given a tile of type $t$, and $g_1$ and $g_2$ as glues on $t$, we define a \emph{signal} as an ordered triple $(g_1,g_2,\delta)$, where $\delta \in \{\texttt{activate,deactivate}\}$, specifying that when glue $g_1$ (the \emph{source} glue) forms a bond (which must be with a copy of the complementary glue type on an adjacent tile) that initiates an action intended to cause glue $g_2$ (the \emph{target} glue) to transition from its current state into \texttt{on} if $s == \texttt{activate}$, or \texttt{off} if $s == \texttt{deactivate}$. Note that a glue may be the source glue of multiple signals, but for each pair of source and target glues there can only be one signal defined (i.e. there cannot be multiple signals defined with the same source and target).
\item A \emph{tile} is an instance of a tile type and is represented as a 4-tuple $(t,\vec{l},R,S,\gamma)$ where $t$ is the tile type, $\vec{l} \in \mathbb{R}^3$ is its location, $R$ is a $3\times 3$ orthonormal rotation matrix describing it's orientation relative to a fixed canonical orientation, $S$ is the set of pairs of glues and their current glue states, and $\gamma$ is the set of \emph{signal states} (to be defined). Although each tile type has a canonical orientation, each tile can be in any orientation in 3-dimensional space as long as no portion of it overlaps with any portion of another tile or a pair of glues bound between two other tiles, and its bound glues allow it (i.e. its bound rigid glues form straight lines perpendicular to the glue surfaces extending the correct lengths for the glues, and its bound flexible glues are of the correct length).
\item Given a tile $t$, its set of signal states $\gamma = \{(s,\mu) \mid s$ is a signal on $t$ and $\mu \in \{\texttt{pre,firing,post}\}$ is the current state of $s\}$. A signal is considered to be in state \texttt{pre} if it has never been activated (a.k.a. \emph{fired}), \texttt{firing} if it has fired but not yet completed, and \texttt{post} if it was activated and its target glue's state was changed\footnote{Note that it is possible for multiple signals to target the same glue and, based on the ordering of activations for the state of a glue to change so that the action of a signal is no longer valid while it is firing. In this case, the signal's state changes to \texttt{post} without changing the glue's state.}. Given a signal $s = (g_1,g_2,\delta)$ whose state is $\mu$, when $g_1$ forms a bond if and only if $\mu = \texttt{pre}$ and glue $g_2$ is in a state which can validly transition according to $\delta$ does the signal \emph{fire}, which causes the its state to be changed to \texttt{firing} in $\gamma$. Note that the only allowable transitions are: (1) $\texttt{latent} \rightarrow \texttt{on}$, (2) $\texttt{on} \rightarrow \texttt{off}$, and (3) $\texttt{latent} \rightarrow \texttt{off}$.
\end{enumerate}
\subsection{Detailed STAM* dynamics}
\begin{enumerate}
\item The binding of a glue causes any signals associated with that glue to change states, i.e. fire (if they haven't already fired due to a prior binding event).
\item A glue and its complementary pair which are bound overlap, causing the distance between their tiles to be the length of the glue (not two times the length).
\item The binding of a single rigid glue or two flexible glues on different surfaces lock a tile in place. Two flexible glues on the same surface prevent ``flipping'' (or ``twisting'') but allow ``hinge-like'' rotation.
\item The assembly process proceeds step by step by nondeterministically selecting one of the following types of moves to execute unless and until none is available. While the following set of choices for a next step are made randomly, no action which is valid can be postponed infinitely long.
\begin{enumerate}
\item Randomly select any pair of supertiles, $\alpha$ and $\beta$, which can bind via a sum of $\ge \tau$ strength bonds if appropriately positioned (and binding only via glues in the $\texttt{on}$ state). Position $\alpha$ and $\beta$ to combine them to form a new supertile by binding a random subset of the glues which can bind between them whose strengths sum to $\ge \tau$. For each bound glue which has a signal associated with it, but that signal is still in the \texttt{pre} state, change the signal's state to \texttt{firing}. Note that rigid glues must form bonds which extend perpendicularly from their surfaces, but flexible glues are free to bend to form bonds.
\item Randomly select any supertile which has a cut in its binding graph $< \tau$ (due to one or more glue deactivations), and split that supertile into two supertiles along that cut. We call this operation a \emph{break}.
\item Randomly select any pair of subassemblies (each of one or more tiles) in the same supertile but bound only by flexible glues so that the subassemblies are free to rotate relative to each other, and perform a valid rotation of one of those subassemblies.
\item Randomly select a supertile and pair of unbound glues within it such that the supertile has a valid configuration in which those glues are able to bind (i.e. they are complementary, both in the $\texttt{on}$ state, and the glues can reach each other), and bind them. For each which has a signal associated with it, but that signal is still in the \texttt{pre} state, change the signal's state to \texttt{firing}.
\item Randomly select a signal whose state is \texttt{firing} from any tile and execute it. This entails, based on the signal's definition, that its target glue is either \texttt{activate}d or \texttt{deactivate}d if that is still a valid transition for that glue, and for the signal's state to change to \texttt{post}, marking it as completed and unable to fire again. The STAM* is based on the STAM and it preserves the design goal of modeling physical mechanisms that implement the signals on tiles but which are arbitrarily slower or faster than the average rates of (super)tile attachments and detachments. Therefore, rather than immediately enacting the actions of signals, each signal is put into a state of \texttt{firing} along with all signals initiated by the glue (since it is technically possible for more than one signal to have been initiated, but not yet enacted, for a particular glue). Any \texttt{firing} signal can be randomly selected from the set, regardless of the order of arrival in the set, and the ordering of either selecting some signal from the set or the combination of two supertiles is also completely arbitrary. This provides fully asynchronous timing between the initiation, or firing, of signals and their execution (i.e. the changing of the state of the target glue), as an arbitrary number of supertile binding (or breaking) events may occur before any signal is executed from the \texttt{firing} set, and vice versa.
\end{enumerate}
\end{enumerate}
The multiple aspects of STAM* tiles and systems give rise to a variety of metrics with which to characterize and measure the complexity of STAM* systems. Following is a list of some such metrics.
\begin{enumerate}
\item Tile complexity: the number of unique tile types
\item Tile shape complexity: the number of unique tile shapes, or the maximum number of surfaces on a tile shape, or the maximum difference in sizes between tile shapes
\item Tile glue complexity: the maximum number of glues on any tile type
\item Seed complexity: the size of the seed assembly (and/or the number of unique seed assemblies.
\item Signal complexity: the maximum number of signals on any tile type
\item Junk complexity: the size of the largest terminal assembly which is not considered the ``target assembly'' (a.k.a. \emph{junk assembly}), or the number of unique types of junk assemblies
\end{enumerate}
\section{Preliminaries}
\label{sec:prelims}
\vspace{-5pt}
In this section we define the notation and models used throughout the paper.
We define a \emph{3D shape} $S \subset \mathcal{Z}^3$ as a connected set of $1 \times 1 \times 1$ cubes (a.k.a. unit cubes) which define an arbitrary polycube, i.e. a shape composed of unit cubes connected face to face where each cube represents a voxel (3-D pixel) of $S$.
For each shape $S$, we assume a canonical translation and rotation of $S$ so that, without loss of generality, we can reference the coordinates of each of its voxels and directions of its surfaces, or faces.
We say a unit cube is \emph{scaled by factor $c$} if it is replaced by a $c \times c \times c$ cube composed of $c^3$ unit cubes. Given an arbitrary 3D shape $S$, we say $S$ is \emph{scaled by factor $c$} if every unit cube of $S$ is scaled by factor $c$ and those scaled cubes are arranged in the shape of $S$. We denote a shape $S$ scaled by factor $c$ as $S^c$.
\vspace{-10pt}
\subsection{Definition of the STAM*}
The 3D Signal-passing Tile Assembly Model* (3D-STAM*, or simply STAM*) is a generalization of the STAM \cite{jSignals,jSignals3D,jSTAM-fractals,SignalsReplication} (that is similar to the model in \cite{JonoskaSignals1,JonoskaSignals2}) in which (1) the natural extension from 2D to 3D is made (i.e. tiles become 3-dimensional shapes rather than 2-dimensional squares), (2) multiple tile shapes are allowed, (3) tiles are allowed to flip and rotate \cite{OneTile,jRTAM}, and (4) glues are allowed to be rigid (as in the aTAM, 2HAM, STAM, etc., meaning that when two adjacent tiles bind to each other via a rigid glue, their relative orientations are fixed by that glue) or \emph{flexible} (as in \cite{FTAM}) so that even after being bound tiles and subassemblies are free rotate with respect to tiles and subassemblies to which they are bound by bending or twisting around a ``joint'' in the glue. (This would be analogous to rigid glues forming as DNA strands combine to form helices with no single-stranded gaps, while flexible glues would have one or more unpaired nucleotides leaving a portion of single-stranded DNA joining the two tiles, which would be flexible and rotatable.) See Figure~\ref{fig:glue-example} for a simple example. These extensions make the STAM* a hybrid model of those in previous studies of hierarchical assembly \cite{AGKS05g,DDFIRSS07,j2HAMIU,2HAM-temp1,j2HAMSim}, 3D tile-based self-assembly \cite{CookFuSch11,OptimalShapes3D,BeckerTimeOpt,DDDIU}, systems allowing various non-square/non-cubic tile types \cite{Polyominoes,Polygons,OneTile,GeoTiles,GeoTilesUCNC2019,KariTriHex}, and systems in which tiles can fold and rearrange \cite{FTAM,FlexibleVsRigid,FlexibleCompModel,JonoskaFlexible}.
Due to space constraints, we now provide a high-level overview of several aspects of the STAM* model, and full definitions can be found in Section \ref{sec:prelims-append} of the Technical Appendix.
The basic components of the model are \emph{tiles}. Tiles bind to each other via \emph{glues}. Each glue has a \emph{glue type} that specifies its domain (which is the string label of the glue), integer strength, \emph{flexibility} (a boolean value with {\tt true} meaning \emph{flexible} and {\tt false} meaning \emph{rigid}), and length (representing the length of the physical glue component). A glue is an instance of a glue type and may be in one of three states at any given time, $\{\texttt{latent,on,off}\}$. A pair of adjacent glues are able to bind to each other if they have complementary domains and are both in the {\tt on} state, and do so with strength equal to their shared strength values (which must be the same for all glues with the same label $l$ or the complementary label $l^*$).
A \emph{tile type} is defined by its 3D shape (and although arbitrary rotation and translation in $\mathbb{R}^3$ are allowed, each is assigned a canonical orientation for reference), its set of glues, and its set of \emph{signals}. Its set of glues specify the types. locations, and initial states of its glues. Each signal in its set of signals is a triple $(g_1,g_2,\delta)$ where $g_1$ and $g_2$ specify the \emph{source} and \emph{target} glues (from the set of the tile type's glues) and $\delta \in \{\texttt{activate,deactivate}\}$. Such a signal denotes that when glue $g_1$ forms a bond, an action is initiated to turn glue $g_2$ either {\tt on} (if $\delta == \texttt{activate}$) or {\tt off} (otherwise). A \emph{tile} is an instance of a tile type represented by its type, location, rotation, set of glue states (i.e. $\texttt{latent,on}$ or $\texttt{off}$ for each), and set of \emph{signal states}. Each signal can be in one of the signal states $\{\texttt{pre,firing,post}\}$. A signal which has never been activated (by its source glue forming a bond) is in the {\tt pre} state. A signal which has activated but whose action has not yet completed is in the {\tt firing} state, and if that action has completed it is in the {\tt post} state. Each signal can ``fire'' only one time, and each glue which is the target of one or more signals is only allowed to make the following state transitions: (1) $\texttt{latent} \rightarrow \texttt{on}$, (2) $\texttt{on} \rightarrow \texttt{off}$, and (3) $\texttt{latent} \rightarrow \texttt{off}$.
We use the terms \emph{assembly} and \emph{supertile}, interchangeably, to refer to the full set of rotations and translations of either a single tile (the base case) or a collection of tiles which are bound together by glues. A supertile is defined by the tiles it contains (which includes their glue and signal states) and the glue bonds between them. A supertile may be flexible (due to the existence of a cut consisting entirely of flexible glues that are co-linear and there being an unobstructed path for one subassembly to rotate relative to the other), and we call each valid positioning of it sets of subassemblies a \emph{configuration} of the supertile. A supertile may also be translated and rotated while in any valid configuration. We call a supertile in a particular configuration, rotation, and translation a \emph{positioned supertile}.
Each supertile induces a \emph{binding graph}, a multigraph whose vertices are tiles, with an edge between two tiles for each glue which is bound between them. The supertile is \emph{$\tau$-stable} if every cut of its binding graph has strength at least $\tau$, where the weight of an edge is the strength of the glue it represents. That is, the supertile is $\tau$-stable if cutting bonds of at least summed strength of $\tau$ is required to separate the supertile into two parts.
For a supertile $\alpha$, we use the notation $|\alpha|$ to represent the number of tiles contained in $\alpha$. The \emph{domain} of a positioned supertile $\alpha$, written $\dom \alpha$, is the union of the points in $\mathbb{R}^3$ contained within the tiles composing $\alpha$. Let $\alpha$ be a positioned supertile. Then, for $\vec{v} \in \mathbb{R}^3$, we define the partial function $\alpha(\vec{v}) = t$ where $t$ is the tile containing $\vec{v}$ if $\vec{v} \in \dom \alpha$, otherwise it is undefined. Given two positioned supertiles, $\alpha$ and $\beta$, we say that they are \emph{equivalent}, and we write $\alpha \approx \beta$, if for all $\vec{v} \in \mathbb{R}^3$ $\alpha(\vec{v})$ and $\beta(\vec{v})$ both either return tiles of the same type, or are undefined. We say they're \emph{equal}, and write $\alpha \equiv \beta$, if for all $\vec{v} \in \mathbb{R}^3$ $\alpha(\vec{v})$ and $\beta(\vec{v})$ either both return tiles of the same type having the same glue and signal states, or are undefined.
\begin{wrapfigure}{r}{0.3\textwidth}
\vspace{-10pt}
\centering
\includegraphics[width=0.3\textwidth]{figures/glue-example.png}
\caption{Example showing flat and cubic tiles, and possible behavior of a flexible glue allowing the blue tile to fold upward, away from the red cubic tile, or down against it. In all constructions, we assume lengths for all flexible glues which make the folding and alignment in this figure possible, and length 0 for rigid glues between cubic and flat tiles (as though one tile's glue strand binds into a cavity).\label{fig:glue-example}}
\vspace{-15pt}
\end{wrapfigure}
An STAM* tile assembly system, or TAS, is defined as $\mathcal{T} = (T,C,\tau)$ where $T$ is a finite set of tile types, $C$ is an initial configuration, and $\tau \in \N$ is the minimum binding threshold (a.k.a. temperature) specifying the minimum binding strength that must exist over the sum of binding glues between two supertiles in order for them to attach to each other. The initial configuration $C = \{(S,n) \mid S$ is a supertile over the tiles in $T$ and $n \in \N \cup \infty$ is the number of copies of $S\}$. Note that for each $s \in S$, each tile $\alpha = (t,\vec{l},S,\gamma) \in s$ has a set of glue states $S$ and signal states $\gamma$. By default, it is assumed that every tile in every supertile of an initial configuration begins with all glues in the initial states for its tile type, and with all signal states as $\texttt{pre}$, unless otherwise specified. The initial configuration $C$ of a system $\mathcal{T}$ is often simply given as a set of supertiles, which are also called \emph{seed} supertiles, and it is assumed that there are infinite counts of each seed supertile as well as of all singleton tile types in $T$. If there is only one seed supertile $\sigma$, we will we often just use $\sigma$ rather than $C$.
\vspace{-10pt}
\subsubsection{Overview of STAM* dynamics}
\vspace{-5pt}
An STAM* system $\mathcal{T} = (T,C,\tau)$ evolves nondeterministically in a series of (a possibly infinite number of) steps. Each step consists of randomly executing one of the following actions: (1) selecting two existing supertiles which have configurations allowing them to combine via a set of neighboring glues in the {\tt on} state whose strengths sum to strength $\ge \tau$ and combining them via a random subset of those glues whose strengths sum to $\ge \tau$ (and changing any signals with those glues as sources to the state {\tt firing} if they are in state {\tt pre}), or (2) randomly select two adjacent unbound glues of a supertile which are able to bind, bind them and change attached signals in state {\tt pre} to {\tt firing}, or (3) randomly select a supertile which has a cut $< \tau$ (due to glue deactivations) and cause it to \emph{break} into 2 supertiles along that cut, or (4) randomly select a signal on some tile of some supertile where that signal is in the {\tt firing} state and change that signal's state to {\tt post}, and as long as its action ({\tt activate} or {\tt deactivate}) is currently valid for the signal's target glue, change the target glue's state appropriately.\footnote{The asynchronous nature of signal firing and execution is intended to model a signalling process which can be arbitrarily slow or fast. Please see Section \ref{sec:prelims-append} for more details.} Although at each step the next choice is random, it must be the case that no possible selection is ever ignored infinitely often.
Given an STAM* TAS $\calT=(T,C,\tau)$, a supertile is \emph{producible}, written as $\alpha \in \prodasm{T}$, if either it is a single tile from $T$, or it is the result of a (possibly infinite) series of combinations of pairs of finite producible assemblies (which have each been positioned so that they do not overlap and can be $\tau$-stably bonded), and/or breaks of producible assemblies.
A supertile $\alpha$ is \emph{terminal}, written as $\alpha \in \termasm{T}$, if (1) for every $\beta \in \prodasm{T}$, $\alpha$ and $\beta$ cannot be $\tau$-stably attached, (2) there is no configuration of $\alpha$ in which a pair of unbound complementary glues in the {\tt on} state are able to bind, and (3) no signals of any tile in $\alpha$ are in the {\tt firing} state.
In this paper, we define a shape as a connected subset of $\mathbb{Z}^3$ to both simplify the definition of a shape and to capture the notion that to build an arbitrary shape out of a set of tiles we will actually approximate it by ``pixelating'' it. Therefore, given a shape $S$, we say that assembly $\alpha$ has shape $S$ if $\alpha$ has only one valid configuration (i.e. it is \emph{rigid}) and there exist (1) a rotation of $\alpha$ and (2) a scaling of $S$, $S'$, such that the rotated $\alpha$ and $S'$ can be translated to overlap where there is a one-to-one and onto correspondence between the tiles of $\alpha$ and cubes of $S'$ (i.e. there is exactly $1$ tile of $\alpha$ in each cube of $S'$, and none outside of $S'$).\footnote{In this paper we only consider completely rigid assemblies for target shapes, since the target shapes are static. We could also target ``reconfigurable shapes, i.e. sets of shapes, but don't do so in this paper. Also, it could be reasonable to allow multiple tiles in each pixel location as long as the correct overall shape is maintained, but we don't require that.}
\begin{definition}
We say a shape $X$ \emph{self-assembles in $\calT$ with waste size $c$}, for $c \in \mathbb{N}$, if there exists terminal assembly $\alpha\in\termasm{T}$ such that $\alpha$ has shape $X$, and for every $\alpha\in\termasm{T}$, either $\alpha$ has shape $X$, or $|\alpha| \le c$. If $c == 1$, we simply say $X$ \emph{self-assembles in} $\calT$.
\end{definition}
\vspace{-5pt}
\begin{definition}
We call an STAM* system $\mathcal{R} = (T,C,\tau)$ a \emph{shape self-replicator for shape $S$} if $C$ consists exactly of infinite copies of each tile from $T$ as well as of a single supertile $\sigma$ of shape $S$, there exists $c \in \mathbb{N}$ such that $S$ self-assembles in $\mathcal{R}$ with waste size $c$, and the count of assemblies of shape $S$ increases infinitely.
\end{definition}
\vspace{-5pt}
\begin{definition}
We call an STAM* system $\mathcal{R} = (T,C,\tau)$ a \emph{self-replicator for $\sigma$ with waste size $c$} if $C$ consists exactly of infinite copies of each tile from $T$ as well as of a single supertile $\sigma$, there exists $c \in \mathbb{N}$ such that for every terminal assembly $\alpha \in \termasm{T}$ either (1) $\alpha \approx \sigma$, or (2) $|\alpha| \le c$, and the count of assemblies $\approx \sigma$ increases infinitely.\footnote{We use $\approx$ rather than $\equiv$ since otherwise either both the seed assemblies and produced assemblies are terminal, meaning nothing can attach to a seed assembly and the system can't evolve, or neither are terminal and it becomes difficult to define the product of a system. However, our construction in Section \ref{sec:deconstruct} can be modified to produce assemblies satisfying either the $\approx$ or $\equiv$ relation with the seed assemblies.} If $c==1$, we simply say $\mathcal{R}$ is a self-replicator for $\sigma$.
\end{definition}
The multiple aspects of STAM* tiles and systems give rise to a variety of metrics with which to characterize and measure the complexity of STAM* systems, beyond metrics seen for models such as the aTAM or even STAM. For a brief discussion, please see Section \ref{sec:prelims-append}.
\vspace{-10pt}
\subsubsection{STAM* conventions used in this paper}
\begin{wrapfigure}{r}{0.3\textwidth}
\vspace{-30px}
\centering
\includegraphics[width=0.29\textwidth]{figures/glue_lengths.png}
\caption{The glue lengths used in our constructions: (1) length $2\epsilon$ rigid bonds between cubic tiles, (2) length 0 rigid bonds between flat and cubic tiles, and (3) length $3\sqrt{2}\;\epsilon/2$ flexible glues between flat tiles.}
\label{fig:glue-lengths}
\vspace{-25px}
\end{wrapfigure}
Although the STAM* is a highly generalized model allowing for variety in tile shapes, glue lengths, etc., throughout this paper all constructions are restricted to the following conventions.
\begin{enumerate}
\item All tile types have one of two shapes (shown in Figure~\ref{fig:glue-example}):
\begin{enumerate}
\item A \emph{cubic} tile is a tile whose shape is a $1 \times 1 \times 1$ cube.
\item A \emph{flat} tile is a tile whose shape is a $1 \times 1 \times \epsilon$ rectangular prism, where $\epsilon < 1$ is a small constant.
\item We call a $1 \times 1$ face of a tile a \emph{full} face, and a $1 \times \epsilon$ face is called a \emph{thin} face.
\end{enumerate}
\item Glue lengths are the following (and are shown in Figure \ref{fig:glue-lengths}):
\begin{enumerate}
\item All rigid glues between cubic tiles, as well as between thin faces of flat tiles, are length $2\epsilon$.
\item All rigid glues between cubic and flat tiles are length $0$. (Note that this could be implemented via the glue strand of one tile extending into the tile body of the other tile in order to bind, thus allowing the tile surfaces to be adjacent without spacing between the faces.)
\item All flexible glues are length $\frac{3}{2}\sqrt{2}\epsilon$. \footnote{These glue lengths were chosen so that (1) rigidly bound cubic tiles could each have a flat tile bound to each of their sides if needed and (2) so that two flat tiles attached to diagonally adjacent rigid tiles could be attached via flexible glue.}
\end{enumerate}
\end{enumerate}
Given that rigidly bound cubic tiles cannot rotate relative to each other, for convenience we often refer to rigidly bound tiles as though they were on a fixed lattice. This is easily done by first choosing a rigidly bound cubic tile as our origin, then using the location $\vec{l}$, orientation matrix $R$, and rigid glue length $g$, put in one-to-one correspondence with each vector $\vec{v}$ in $\mathbb{Z}^3$, the vector $\vec{l} + g R \vec{v}$. Once we define an absolute coordinate system in this way, we refer to the directions in 3-dimensional space as North ($+y$), East ($+x$), South ($-y$), West ($-x$), Up ($+z$), and Down ($-z$), abbreviating them as $N,E,S,W,U,$ and $D$, respectively.
\section{Technical Details of the Genome Based Replicator}
\label{sec:simple-replicator-append}
In this section we present technical details of the construction in Section \ref{sec:simple-replicator}.
\subsection{Tile sets}
\label{sec:tiles}
We provide the enumerated sets of tiles in this section.
\subsubsection{\texorpdfstring{$T_\sigma$}{T sigma}}
As shown in Figure \ref{fig:genome_tiles}, all tiles except for the end tile have the same structure of signals and glues, where the glues are a specific mapping to tiles in $T_\mu$. Glues which bind between $T_\sigma$ and $T_\mu$ have the $\mu$ subscript in the glue description. Glues without the $\mu$ subscript bind between the north and south glues of tiles in $T_\sigma$.
\subsubsection{\texorpdfstring{$T_\mu$}{T mu}}
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\textwidth]{figures/SR-mu.png}
\caption{Tile types (non-kink). Note that the red `d' glues have deactivation signals to all glues on the tile, but are omitted for visual clarity. This turns the messenger tile into a `junk' product.}
\label{fig:start-tile-vox}
\end{figure}
The tiles presented in Figure \ref{fig:start-tile-vox} represent the base tiles which make up a messenger sequence. Any glue which contains an `f' subscript is a flexible glue. The tile denoted $Ki$ is a placeholder for both $Kp$ and $Km$ tiles, where all glues which contain an `$i$' can be replaced with $p$ or $m$, respectively. All of the tiles aside from $T_i, T_f, Kp_f \text{ or } E$ can be a predecessor to a turning tile. This requires additional glues and signals in order to attach to a kink-ase structure. These modifications are shown in Figure \ref{fig:kink-mods}, and we note that these glues and signals overlay on top of the tiles in Figure \ref{fig:start-tile-vox}; glues not used in the turning process are omitted. The tiles to the right indicate the specific glues and signals for the $Kp,\:Km$ tiles. The tiles to the left indicate the specific glues and signals which must be present on the predecessor tiles to $Kp$ or $Km$. We note that $Kp$ and $Km$ can also be modified with the tiles on the left hand side. In the case of either two $Kp$ or $Km$ tiles in a row, it is required to leave the flexible glues $f_f,g_f$ \texttt{on} instead of \texttt{off} when the `p' glue on the east side of a tile is bound.
We note that the modifications require a mapping of a specific glue from $T_\sigma$ to $T_\mu$. This is accomplished by adding an additional `m' or `p' to the glue based upon the modification made. Glue which connect $T_\mu$ and $T_\pi$ have the subscript $\pi$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.35\textwidth]{figures/SR-kinkmod.png}
\caption{Tile modifications for use with kink-ase. Note that the dashed square indicates the face that the `p' glue is attached}
\label{fig:kink-mods}
\end{figure}
\subsubsection{\texorpdfstring{$T_\phi$}{T phi}}
\begin{figure}[ht]
\centering
\includegraphics[width=.7\textwidth]{figures/SR-fuel-tiles.png}
\caption{Fuel tiles. Tiles on left utilized during replication of $\sigma$, tiles on right combine to form kink-ase structure}
\label{fig:fuel-tiles}
\end{figure}
The tiles presented in Figure \ref{fig:fuel-tiles} are those that cause the replication of $\sigma$ and form kink-ase. The kink-ase tiles first combine to form supertiles of size 4 as shown in Figure \ref{fig:linear-to-kin}. These supertiles are then able to perform the designated functions of the kink-ase. Similarly, the tiles $\varphi_1$ and $\varphi_2$ combine to a supertile in before replication of $\sigma$ can begin.
\subsubsection{\texorpdfstring{$T_\pi$}{T pi}}
The tiles $T_\pi$ are the structural blocks which recreate a desired shape given an input genome. Two strength 1 glues of the type `c' bind the final structure between cubic tiles in the Hamiltonian path dictated by $\sigma$.
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\textwidth]{figures/SR-struct-tiles.png}
\caption{Structural tiles which create the assembly $\pi$. Note that the `R' tile has a second $Kp^*_{f\pi}$ glue activated , however is omitted for visual purposes.}
\label{fig:struct-tiles}
\end{figure}
\subsection{Genome replication details}
\label{sec:detail-genome}
The replication process of $\sigma$ begins with the attachment of tiles from the set $T_{\sigma}$ to $\sigma$ due to the two strength-1 glues on the north face of individual tiles comprising $\sigma$. We denote the incomplete copy of $\sigma$ as $\sigma^\prime$. Asynchronously, a fuel tile assembly $\varphi$ comprised of two subtiles $\varphi_1, \varphi_2 \in T_\phi$ binds to the leftmost tile of $\sigma$. Upon the binding of a start tile to the north thin face of the start tile of $\sigma^\prime$, the signal provided by $\varphi$ begins a chain reaction binding to the the active `n' glue on the west thin face of the newly attached tile and the signal propagates through the chain of connected $\sigma^\prime$ tiles. Once the end tile $E_\sigma$ is bound to the remainder of $\sigma^\prime$ by the active `n' glue, it returns a signal through its newly activated west glue to fully connect it to the prior tile and then detach from the genome to the south. This signal cascades back through the remaining tiles of $\sigma^\prime$ until reaching $\varphi$, at which point $\varphi$ deactivates its glues. allowing the newly replicated copy of $\sigma$ to separate and begin the process of replicating itself and translating copies of $\mu$.
\subsection{Details of kink-ase folding}
\label{sec:details-kinkase}
This section describes in detail how $\mu$ is converted to $\mu^\prime$ utilizing the kink-ase structure, and an example is shown in Figure \ref{fig:linear-to-kin}.
\begin{enumerate}
\item[A)] kink-ase attaches to a turning tile and the predecessor which will be re-oriented in $\mu$. Simultaneously, glues are activated on the kink-ase cube structure attached to the turning tile to bind the turning tile face and to the kink-ase cube structure attached to the predecessor tile to enable the folding of the cube structure in step D). Note - glues connecting tiles in $\mu$ may be either rigid or flexible depending upon the Hamiltonian path generated for $\pi$. This does not effect any intermediate steps presented.
\item[B)] The turning tile's rear face binds to the kink-ase due to random movement allowed by the flexible glues which attach the kink-ase to the turning and predecessor tiles, i.e. the flexible bond allows the tile to rotate and randomly assume various relative positions. When it enters the correct configuration, the glues bind to ``lock it in''.
\item[C)] Upon connection of turning tile face to kink-ase cube, a signal deactivates the rigid glue attaching the predecessor tile to the turning tile. A signal activates glues on the exposed face of the kink-ase tile attached to cube and turning tile structure. The flexible connection between the predecessor tile and kink-ase ensures $\mu$ does not split into two pieces.
\item[D)] Kink-ase cube and kink-ase tile with activated glue bind on faces when they rotate into the correct configuration, bringing the turning tile into correct geometry with the predecessor tile. The kink-ase cube face adjacent to the predecessor tile activates its glue, allowing for binding with the face of the two. The flexible glue allows for random movement for the complementary glues to attach and bind. Concurrently, the flexible glue on the turning tile is deactivated and a rigid glue of similar type to the turning tile glue deactivated in step C) is activated.
\item[E)] A rigid glue between the turning tile and predecessor tile binds, leading to re-connection between both prior detached portions of $\mu$. Activation of the final glue leads to the turning tile signaling to kink-ase to detatch from $\mu$.
\item[F)] This structure represents $\mu$ after one turning tile has been resolved. A completion signal is passed through glues attaching the turning tile and predecessor tile. This process continues for all turning tiles serially, working backwards from the termination tile. This is to prevent any interference between structures incurred by multiple adjacent turning tiles.
\end{enumerate}
\subsection{Detailed assembly of \texorpdfstring{$\pi$}{pi}}
Figure \ref{fig:piconstruction} shows the detailed process of a step of the assembly of a structure $\pi$. Additionally, Figure \ref{fig:minustile} demonstrates the action of a `+' or `-' glue binding - we reflect the full face on which a cubic tile is bound.
\begin{figure}
\centering
\begin{subfigure}{0.73\textwidth}
\includegraphics[width=1.0\textwidth]{figures/SR-pi-assembly.png}
\caption{ \label{fig:assembling-pi}}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.2\textwidth}
\includegraphics[width=1.0\textwidth]{figures/SR-minus-tile.png}
\caption{\label{fig:minustile}}
\end{subfigure}
\caption{(a) The process of assembling $\pi$ from $\mu^\prime$. Spaces between flat tiles and cubic tiles are exaggerated for illustrative purposes. The red arrows and squares demonstrate the signals propagating through adjacent tiles to solidify connections between two successive cubic tiles in the Hamiltonian path of a phenotype. At the final step of this figure, The cubic tile associated with the ``++'' tile of $\mu^\prime$ has its $c^*$ glue in the \texttt{on} state (b) Demonstration of cube tile placement utilizing $S^-$ genome tile in place of $S^+$ - this causes the glues on the flat tile to place the cube tile on the opposite full face of $\mu^\prime$}
\label{fig:piconstruction}
\end{figure}
\subsection{Proof of Theorem \ref{thm:universal-constructor}}
\label{sec:univ-cons}
We prove Theorem \ref{thm:universal-constructor} via induction. Our base case is the start flat tile and its associated cube. Our inductive step is the addition of a cube and a direction associated with the next step of the Hamiltonian path within $S^2$. This direction is provided by the successor tile in $\mu^\prime$, and all possible directions are enumerated in Figure \ref{fig:SR-correctness}. At each step, we place a cubic tile in its associated direction based upon the flat tile in $\mu^\prime$. We analyze the possible direction of placement. Since $\mu$ is a translation of $\sigma$, $x^-$ is not included as it is the location of the prior cubic tile. As a note, the directions provided in the proof reflect those indicated in Figure \ref{fig:SR-correctness}, not necessarily the absolute reference of the entire system. Additionally, as our genome $\sigma$ has a Hamiltonian path ending on an exterior face of $S$, we can guarantee that diffusion is possible for a tile at any stage of construction
\begin{itemize}
\item[$x^+$:] This placement and output direction is carried out by the ++ tile type - the cubic tile is placed in the existing direction of travel
\item[$y^+$:] This correlates to the $T_i$ and $T_o$ tile type.
\item[$y^-$:] This case is the most complex; we are changing the direction of travel in a direction which takes us through the tile of $\mu^\prime$. This requires the use of the following 4 tiles: $Kpf,T_f,T_f,T_o$. This could also be completed with a set of 3 tiles $Kp, Km, Km$, however this increases fuel usage per $y^-$ from 1 to 3, and overall tile usage from 8 to 19 when including all the singleton tiles utilized to create the kink-ase structures consumed by the 3 turning tiles.
\item[$z^-$:] A single $Km$ tile carries out this tile placement and path change. Note, the prior flat tile must additionally be modified to carry out the turning action by the kink-ase.
\item[$z^+$:] A single $Kp$ tile carries out this tile placement and path change. Note, the prior tile must additionally be modified to carry out the turning action by the kink-ase.
\end{itemize}
After the addition of a tile, we re-orient the frame of reference to align with that shown in Figure \ref{fig:SR-correctness}.
The last tile in the Hamiltonian path will not have a new direction - this is indicated by the end tile. We have then generated the structure $S^2$ utilizing $R$.
\subsubsection{STAM* Metrics of \texorpdfstring{$R$}{R}}
The STAM* metrics of $R$ follow below:
\begin{itemize}
\item Tile complexity $= 57$
\begin{itemize}
\item $|T_\sigma|=22$
\item $|T_\mu|=22$
\item $|T_\pi|=7$
\item $|T_\phi|=6$
\end{itemize}
\item Tile shape complexity $= 2$
\item Signal complexity $= 7$
\item Seed complexity $= O(n)$; each cube in the phenotype must be placed by a tile, with some requiring multiple (e.g. turns). As described above, for any structure with greater than 2 tiles we end up with the following number of tiles in $\sigma$ based upon the changes in directions which must occur: ``start tile'' $+$ ``end tile'' $+ |z^+| + |z^-|+ 2|y^+|+4|y^-|+|x^+|$.
\end{itemize}
\section{A Genome Based Replicator}
\label{sec:simple-replicator}
We now present our first construction in the STAM*, in which a ``universal'' set of tiles will cause a pre-formed seed assembly encoding a Hamiltonian path through a target structure, which we call the \emph{genome}, to replicate infinitely many copies of itself as well as build infinitely many copies of the target structure at temperature 2. We consider 4 unique structures which are generated/utilized as part of the self-replication process: $\sigma,\mu,\mu^\prime$, and $\pi$. The seed assembly, $\sigma$, is composed of a connected set of flat tiles considered to be the \emph{genome}. Let $\pi$ represent an assembly of the target shape encoded by $\sigma$. $\mu$ is an intermediate ``messenger'' structure directly copied from $\sigma$, which is modified into $\mu^\prime$ to assemble $\pi$. We split $T$ into subsets of tiles, $T = \{ T_{\sigma} \cup T_{\mu} \cup T_{\varphi} \cup T_{\pi}\}$. $T_\sigma$ are the tiles used to replicate the genome, $T_\mu$ are the tiles used to create the messenger structure, $T_\pi$ are the cubic tiles which comprise the phenotype $\pi$, and $T_\phi$ are the set of tiles which combine to make fuel structures used in both the genome replication process and conversion of $\mu$ to $\mu^\prime$.
The tile types which make up this replicator are carefully designed to prevent spurious structures and enforce two key properties for the self-replication process. First, a genome is never consumed during replication, allowing for exponential growth in the number of completed genome copies. Second, the replication process from messenger to phenotype strictly follows $\mu \rightarrow \mu^\prime \rightarrow \pi$; each step in the assembly process occurs only after the prior structure is in its completed form. This prevents unexpected geometric hindrances which could block progression of any further step. Complete details of $T$ are located in Section \ref{sec:tiles}.
\vspace{-10pt}
\subsection{Replication of the genome}
The minimal requirements to generate copies of $\sigma$ in $\mathcal{R}$ are the following: (1) for all individual tile types $s\in \sigma, s \in T_\sigma$, and (2) the last tile is the end tile $E_\sigma$. However, in order for the shape-self replication of $S$ two additional properties must hold: (3) the first tile in $\sigma$ is a start tile in the set $(S^+_\sigma,S^-_\sigma)$, and (4) $\sigma$ encodes a Hamiltonian path which ends on an exterior cubic tile. We define the genome to be `read' from left to right; given requirements (2) and (3), the leftmost tile in a genome is a start tile and the rightmost is an end tile. (4) can be guaranteed by scaling $S$ up to $S^2$ and utilizing the algorithm in Section \ref{sec:hampath}, selecting a cubic tile on the exterior as a start for the Hamiltonian path and then reversing the result. This requirement ensures the possibility of cubic tile diffusion at all stages of assembly.
\begin{figure}[ht]
\centering
\begin{subfigure}{0.46\textwidth}
\includegraphics[height=4cm]{figures/SR_copy-seed-tiles.png}
\caption{\label{fig:genome_tiles}}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.46\textwidth}
\centering
\includegraphics[height=4cm]{figures/SR-copy-while-translating.png}
\caption{\label{fig:SR-copy-while-translating}}
\end{subfigure}
\caption{(a) Initial genome replicator tiles. Note that $\otimes \otimes$ represents a two strength 1 glues which are on the full face of the seed tiles opposite from the reader (b) Illustration of an arbitrary translation process occurring at the same time as genome replication. Red tiles are representative of $\varphi$, gold tiles are representative of $\sigma$, and blue tiles are representative of $\mu$.}
\label{fig:SR-replication}
\vspace{-10px}
\end{figure}
Figure \ref{fig:genome_tiles} is a template for the tile set required for the replication of an arbitrary genome. The process of replicating a genome $\sigma$ into a new copy $\sigma^\prime$ is carried out left to right, initiated by a fuel assembly which is jettisoned after all tiles in $\sigma^\prime$ are connected with strength 2. This allows for the genome to be copied without itself being used up, leading to exponential growth. Full detail is available in Section \ref{sec:detail-genome}.
\vspace{-10pt}
\subsection{Translation of \texorpdfstring{$\sigma$}{sigma} to \texorpdfstring{$\mu$}{mu S}}
\emph{Translation} is defined as the process by which the Hamiltonian path encoded in $\sigma$ is built into a new messenger assembly $\mu$. Since the signals to attach and detach $\mu$ from $\sigma$ are fully contained in the tiles of $T_{\mu}$, translation continues as long as $T_{\mu}$ tiles remain in the system. We note that the translation process can occur at the same time as $\sigma$ is replicating. This causes no unwanted geometric hindrances as demonstrated in Figure \ref{fig:SR-copy-while-translating}.
\vspace{-10pt}
\subsubsection{Placement of \texorpdfstring{$\mu$}{mu } tiles}
Messenger tiles from the set $T_\mu$ attach to $\sigma$ as soon as complementary glues on the back flat face of $\sigma$ are activated after the binding of $\varphi$ to $\sigma^\prime$. The process of building $\mu$ does not require a fuel structure to continue, as the messenger tiles have built-in signals to deactivate the glues on $\mu$ which attach $\mu$ to $\sigma$. This allows for a genome to replicate the messenger structure without itself being consumed in any manner.
Each genome tile contains two active strength-1 glues on its full face which are mapped to a single messenger tile type. The process begins by the front full face of a messenger tile attaching to the back full face of a $\sigma$ tile. The messenger tile then begins the process of activating signals to attach to the messenger tile to its west side. There are two variants of the start tile ($S^+,S^-$), but both variants carry out the same task in translation by initiating the binding of tiles in $\mu$ by being the only messenger tile type which initially activates glues on its east thin face. Once a flat tile in $\mu$ is bound to its eastern neighbor, signals are fired from the eastern glues to deactivate the glue connecting $\mu$ to $\sigma$. This leaves $\mu$ as its own separate assembly\footnote{The structure of $\mu$ is initially identical $\sigma$ after translation, but signals and glues vary.} when every tile has attached to its neighbor(s) with strength 2.
\vspace{-10pt}
\subsubsection{Modification of \texorpdfstring{$\mu$}{mu } to \texorpdfstring{$\mu^\prime$}{mu prime } }
The current shape of $\mu$ is such that it could only replicate a trivial 2D structure; $\mu$ must be modified to follow a Hamiltonian path in 3 dimensions as made possible by a set of \emph{turning} tiles. Additionally, in the current state of $\mu$ no cubic tiles can be placed as all the glues which are complementary to cubic tiles are currently in the \texttt{latent} state. Once a glue of type `p' is bound on the start tile, we then consider $\mu$ to have completed its modification into $\mu^\prime$. The `p' glue on turning tiles can only be bound once they have been turned, and as such the turning tiles present in $\mu^\prime$ must be turned before assembly of $\pi$ begins.
Turning tiles modify the shape of $\mu$ by adding `kinks' into the otherwise linear structure by the use of a fuel-like structure called a \emph{kink-ase}. The kink-ase structure is generated from a set of 2 flat tiles and 2 cube tiles. These tiles must first fully bind to each other before connections can be made to a turning tile. The unique form of kink-ase allows for the orientation of two adjacent tiles to be modified without separating $\mu$, shown in Figure \ref{fig:linear-to-kin}. The turning tiles are physically rotated such that the connection between a turning tile and its predecessor along the west thin edge of the turning tile is broken, and then reattached along either the up or down thin edge of the turning tile. Each turning tile requires the use of a single kink-ase, which turns into a junk assembly. The turning process is described in depth in Section \ref{sec:details-kinkase}.
\begin{figure}[ht]
\centering
\begin{subfigure}{0.46\textwidth}
\centering
\includegraphics[height=5cm]{figures/SR-linear-to-kinky.png}
\caption{\label{fig:linear-to-kin}}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.46\textwidth}
\centering
\includegraphics[height=5cm]{figures/SR-correctness.png}
\caption{ \label{fig:SR-correctness}}
\end{subfigure}
\caption{(a) Conversion of one turning tile. Blue tiles indicate $\mu$, whereas the red indicate the kink-ase. (b) The inductive steps required in the creation of $\pi$ which follows a Hamiltonian path given by a $\sigma$. The arrow going into the flat tile is the direction taken by the Hamiltonian path in the prior tile addition step. The five arrows indicate possible directions for the direction of the Hamiltonian path after the placement of the transparent cubic tile.}
\vspace{-10px}
\end{figure}
\vspace{-10pt}
\subsection{Assembly of \texorpdfstring{$\pi$}{pi}}
At the end of translation, two strength-1 glues complementary to tiles in $T_\pi$ are active on all tiles of $\mu^\prime$. The only cubic tile which starts with two complementary glues \texttt{on} is the start cubic tile. Once this cubic tile is bound to the start tile, a strength-1 glue of type `c' is activated on the cube. This glue allows for the cooperative binding of the next cubic tile in the Hamiltonian path to the superstructure of both $\mu^\prime$ and the first tile of $\pi$.
After this process continues and a cubic tile is bound to both its neighbors (or just one neighbor in the case of the start and end tiles) with strength 2, a `d' glue is activated on the face of the cubic tile bound to $\mu^\prime$. This indicates to the flat tile of $\mu^\prime$ that the cube tile is fully connected to its neighbors with strength 2. To prevent any hindrances to the placement of any cubic tiles in $\pi$, the flat tile jettisons itself from the remaining tiles of $\mu^\prime$ by deactivating all active glues and becoming a junk tile\footnote{Due to the asynchronous nature of signals, there may be instances which the addition of cubic tiles of $\pi$ are temporarily blocked. These will be eventually resolved, allowing assembly to continue.}. This process is repeated, adding cube by cube until the end tile in $\mu^\prime$ is reached. Once the end cube has been added to $\pi$, it has shape $S^2$ and $\mu^\prime$ has been disassembled into junk tiles.
\vspace{-10pt}
\subsection{Analysis of \texorpdfstring{$\mathcal{R}$}{R} and its correctness}
\begin{theorem}\label{thm:universal-constructor}
There exists an STAM* tile set $T$ such that, given an arbitrary shape $S$, there exists STAM* system $\mathcal{R} = (T,\sigma,2)$ and $S^2$ self-assembles in $\mathcal{R}$ with waste size $4$.
\end{theorem}
We provide a high level overview of the correctness proof, further described in \ref{sec:univ-cons}. We demonstrate inductively that the construction process of an assembly $\pi$ correctly generates a structure of shape $S^2$, as shown in Figure \ref{fig:SR-correctness}. The intuition is that at each step in the Hamiltonian path, there exists some combination of flat tiles which can correctly orient the placement of every cubic tile in the Hamiltonian path. This overall set of tiles are encoded in $\sigma$, demonstrating the ability of $\mathcal{R}$ to replicate arbitrarily many copies of $S^2$.
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{
"redpajama_set_name": "RedPajamaArXiv"
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Q: A4 thesis-like CSS: taking citations from the main column and placing them into the other I'm attempting to achieve the following layout based on this PhD thesis:
But is there a pure CSS way to take HTML citations out of one column and placing them in line in the other column?
.Grid {
display: flex;
}
.Grid-cell {
border: 1px solid black;
}
.Grid-cell:nth-of-type(1) {
flex: 1 1 70%;
}
.Grid-cell:nth-of-type(2) {
flex: 1 1 30%;
padding-left: 20px;
}
q {
/* Insert here */
}
<div class="Grid">
<div class="Grid-cell">
<p>Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo. Nemo enim ipsam voluptatem quia
voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi <q cite="http://www.mashable.com/">nesciunt</q>. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
velit, sed quia non numquam eius modi tempora incidunt ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? Quis
autem vel eum iure reprehenderit qui in ea voluptate velit esse quam nihil molestiae consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur?</p>
</div>
<div class="Grid-cell"><!-- Insert here --></div>
</div>
A: I think you might be overthinking this one a bit. You don't need to set it up as a flex layout and you don't need to copy elements from one location to another. All you really need is to break a q element out of flow and display it in a predefined margin. We can do this using only padding-right and position: absolute.
Example here:
.Grid {
padding-right: 30%;
}
.Grid-cell {
border: 1px solid black;
}
q[cite] {
position: absolute;
left: 75%;
max-width: 20%;
}
q[cite]:before {
content: attr(data-index);
}
<div class="Grid">
<div class="Grid-cell">
<p>Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo. Nemo enim ipsam voluptatem quia
voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi<sup>1</sup> <q data-index="1. " cite="http://www.mashable.com/">nesciunt</q>. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
velit, sed quia non numquam eius modi tempora incidunt ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? Quis
autem vel eum iure reprehenderit qui in ea voluptate velit esse quam nihil molestiae consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur?</p>
</div>
</div>
If you don't define a top or bottom property on absolutely positioned elements, they determine where they land based on where they'd appear in document flow. Therefore, by only setting position: absolute and left: 75% we've broken your citation out, kept it in-line with its source, and using the data-index property in conjunction with a content: attr(data-index); even given you the ability to number your citations to match their source inline.
A: Not using flexbox.
But using an internal span and positioning this can be achieved.
Incidentally, I have used a cite tag in place of the q tag as q renders automatic quotes. It may be possible to remove them using CSS but I haven't looked into that.
.Grid {
display: flex;
}
.Grid-cell {
border: 1px solid black;
}
.Grid-cell:nth-of-type(1) {
flex: 1 1 70%;
position: relative;
}
.Grid-cell:nth-of-type(2) {
flex: 1 1 30%;
padding-left: 20px;
}
sup {
font-weight:bold;
color: red;
font-size:.6em;
}
cite span {
position: absolute;
right:0;
margin-right: -8em;
}
<div class="Grid">
<div class="Grid-cell">
<p>Sed ut perspiciatis unde omnis iste natus error sit voluptatem accusantium doloremque laudantium, totam rem aperiam, eaque ipsa quae ab illo inventore veritatis et quasi architecto beatae vitae dicta sunt explicabo. Nemo enim ipsam voluptatem quia
voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi<sup>1</sup><cite cite="http://www.mashable.com/"><span>1. My Citation</span></cite>. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
velit, sed quia non numquam eius modi tempora incidunt ut labore et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur? Quis
autem vel eum iure reprehenderit qui in ea voluptate velit esse quam nihil molestiae consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla pariatur?</p>
</div>
<div class="Grid-cell"><!-- Insert here --></div>
</div>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,822
|
1 – The installation assumes that you are using a standard POE switch or injector to connect to the Raspberry Pi.
Orders will start to ship Monday 28th February. UK domestic orders will ship within 48 hours and be sent first class post. Our shipping department does not operate Saturday, Sunday or statutory holidays.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,167
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Mexico: Texas shooting 'act of terrorism' against Mexicans
by: AMY GUTHRIE, Associated Press
People gather in Juarez, Mexico, Saturday, Aug. 3, 2019, in a vigil for the 3 Mexican nationals who were killed in an El Paso shopping-complex shooting. Twenty people were killed and more than two dozen injured in a shooting Saturday in a busy shopping area in the Texas border town of El Paso, the state's governor said. (AP Photo/Christian Chavez)
MEXICO CITY (AP) — Mexico's government said it considers a shooting at a crowded department store in El Paso, Texas that left eight of its citizens dead an "act of terrorism" against Mexicans and hopes it will lead to changes in U.S. gun laws.
Mexican Foreign Minister Marcelo Ebrard met Monday afternoon with local authorities in El Paso and said Mexico will participate in the investigations and trial there, as well as take legal action against those who sold the gun to the shooter.
"An investigation will be opened for terrorism, because that's what it was," Ebrard said at a press conference. "And the extradition request is not off the table."
Ebrard also met with families of the victims and the injured and promised to speed up the repatriation process for the bodies of the eight Mexican victims.
"We agree that it appears racism and white supremacy are serious problems in the United States," Ebrard said.
President Andrés Manuel López Obrador previously said that Mexico will respect the debate that will unfold in the United States following the Saturday attack that killed a total of 22 people, but he believes the discussion could lead to change north of the border.
"There could be a change to their laws because it is stunning what is happening, unfortunate and very powerful," López Obrador said. "I don't rule out that they could change their constitution and laws. These are new times; you have to always be adjusting the legal framework to the new reality."
Many in Mexico were reeling from revelations that the shooting appeared to have been aimed at Hispanics, and Mexicans in particular.
Just minutes before the rampage, U.S. investigators believe the shooter posted a rambling online manifesto in which he railed against a perceived "invasion" of Hispanics coming into the U.S. He then allegedly targeted a shopping area in El Paso that is about 5 miles (8 kilometers) from the main border checkpoint with Ciudad Juarez, Mexico.
Tens of thousands of Mexicans cross the border legally each day to work and shop in the city of 680,000 full-time residents, and El Paso County is more than 80% Latino, according to the latest census data.
The Mexican victims were identified as Sara Esther Regalado of Ciudad Juarez; Adolfo Cerros Hernández of Aguascalientes; Jorge Calvillo García of Torreon, Coahuila; Elsa Mendoza de la Mora of Yepomera, Chihuahua; Gloria Irma Marquez of Ciudad Juarez; María Eugenia Legarreta of the city of Chihuahua; Ivan Filiberto Manzano of Ciudad Juarez; and Juan de Dios Velázquez Chairez of Zacatecas. Other victims may have also been of Mexican descent.
As the news dominated weekend headlines, some in Mexico said the shooting was the result of the simmering resentment that President Donald Trump had stirred early into his presidential campaign when he called Mexicans coming into the U.S. "rapists" and "criminals." The U.S.-Mexico relationship was only further strained after he took office and vowed to build a border wall and slap tariffs on Mexican imports.
On Sunday, López Obrador chose his words carefully when speaking of the shooting.
"In spite of the pain, the outrage" that Mexicans are feeling, he said, the U.S. is headed toward elections and Mexico doesn't want to interfere in the "internal affairs" of other countries. He also said the events in Texas reaffirmed his conviction that "social problems shouldn't be confronted with the use of force and by inciting hate."
Former President Felipe Calderón said via Twitter that regardless of whether the shooting is a hate crime, Trump "should stop his hate speech. He should stop stigmatizing others."
Amatza Gutiérrez, a student from the Mexican capital, said the idea of a shooter targeting Mexicans because of their ethnicity gives her goose bumps.
"I don't understand why anyone would go to that extreme," the 24-year-old said.
AP writer Christopher Sherman contributed to this report.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,808
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Nicolai Gottlieb Blædel, född den 26 december 1816 i Stubbekøbing, död den 15 juli 1879 i Köpenhamn, var en dansk präst. Han var måg till Poul Dons och farfar till Nic Blædel.
Blædel blev student 1836 och teologie kandidat 1841. Efter några års arbete som lärare blev han 1849 präst vid Almindelig Hospital och Abel Cathrines Stiftelse i Köpenhamn, 1853 kaplan vid Garnisonskirken och 1859 sognepræst där. Blædel var en av sin tids mest betydande predikanter i den danska huvudstaden. Hans allvarliga förkunnelse samlade ovanligt många åhörare och genom sitt oförfärade uppträdande under koleraåret 1853 vann han allas aktning. Han var själen i många praktiska kyrkliga verksamheter och banade väg för den inre missionen i Köpenhamn. Han sökte inta en mellanställning mellan de olika kyrkliga riktningarna, av vilka han dock hade svårast att förlika sig med grundtvigianismen, mot vilken han utgav ett par stridsskrifter (1873). Han utgav vidare tre predikosamlingar och Udvidet Konfirmationsundervisning eller evangelisk-luthersk Kirkelære (1876, 2:a upplagan 1884).
Källor
Präster i Danska folkkyrkan
Danska präster under 1800-talet
Personer från Falster
Födda 1816
Avlidna 1879
Män
Salmonsens
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,410
|
using Lidgren.Network;
namespace LmpCommon.Message.Interface
{
public interface IMessageBase
{
/// <summary>
/// Name of the class
/// </summary>
string ClassName { get; }
/// <summary>
/// Class with the data that it handles
/// </summary>
IMessageData Data { get; }
/// <summary>
/// True if the data version property mismatches
/// </summary>
bool VersionMismatch { get; set; }
/// <summary>
/// Specify how the message should be delivered based on lidgren library.
/// This is important to avoid lag!
/// Unreliable: No guarantees. (Use for unimportant messages like heartbeats)
/// UnreliableSequenced: Late messages will be dropped if newer ones were already received.
/// ReliableUnordered: All packages will arrive, but not necessarily in the same order.
/// ReliableSequenced: All packages will arrive, but late ones will be dropped.
/// This means that we will always receive the latest message eventually, but may miss older ones.
/// ReliableOrdered: All packages will arrive, and they will do so in the same order.
/// Unlike all the other methods, here the library will hold back messages until all previous ones are received,
/// before handing them to us.
/// </summary>
NetDeliveryMethod NetDeliveryMethod { get; }
/// <summary>
/// Public accessor to retrieve the Channel correctly
/// </summary>
/// <returns></returns>
int Channel { get; }
/// <summary>
/// Attaches the data to the message
/// </summary>
void SetData(IMessageData data);
/// <summary>
/// Retrieves a message data from the pool based on the subtype
/// </summary>
IMessageData GetMessageData(ushort subType);
/// <summary>
/// This method retrieves the message as a byte array with it's 8 byte header at the beginning
/// </summary>
/// <param name="lidgrenMsg">Lidgren message to serialize to</param>
/// <returns>Mesage as a byte array with it's header</returns>
void Serialize(NetOutgoingMessage lidgrenMsg);
/// <summary>
/// Call this method to send the message back to the pool
/// </summary>
void Recycle();
/// <summary>
/// Gets the message size in bytes
/// </summary>
int GetMessageSize();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,949
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I'm a contract recruiter - how do I link two different JobScore accounts together?
For now, please contact us if you want to link multiple accounts together. We have to do this manually and typically get it done within one business day. If you already have a JobScore account and want to provision a new account for a new customer, please contact us first before creating the new account rather than creating it first and then contacting us.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,204
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Голы́шино () — деревня в составе Добровского сельсовета Горецкого района Могилёвской области Республики Беларусь. Переименована 22 июня 2012 года, прежнее название — Голышено.
Географическое положение
Население
1999 год — 41 человек
2010 год — 17 человек
См. также
Добровский сельсовет
Горецкий район
Примечания
Ссылки
Населённые пункты Республики Беларусь
Национальное кадастровое агентство Республики Беларусь
Государственный комитет по имуществу Республики Беларусь
Населённые пункты Горецкого района
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,929
|
{"url":"https:\/\/math.stackexchange.com\/questions\/2302505\/redistributing-probability-mass","text":"# Redistributing probability mass\n\nConsider some canonical probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ and a random variable $X:\\Omega\\rightarrow[0,\\infty)$ and let's assume that $X$ is unbounded, but has finite expectation.\n\nThat is, $X\\in\\mathcal{L}^{1}$, and for all $M\\in[0,\\infty)$ we have $\\mathbb{P}(X \\geq M)>0$.\n\nI am wondering, if there is a way to construct a bounded random variable $\\hat{X}$ with\n\n1. $\\mathbb{E}[X] = \\mathbb{E}[\\hat{X}]$\n2. $\\hat{X}(\\Omega)\\subset X(\\Omega)$.\n\nIntuitively, one should redistribute probability mass from the tail of the random variable into some bounded domain. But I expect some assumption on the underlying probability space for this to be possible.\n\nHave you seen sothing similar or can you point me in the right direction? Any comments, ideas and suggestions are highly appreciated. Thank you!\n\nThis is not really an answer to your question, but might lead to the conclusion that such $\\hat X$ cannot always be found. In the sequel $\\mathbb N$ denotes the set of positive integers.\n\nLet $(\\Omega,\\mathcal F,\\mathbb P)=(\\mathbb N,\\wp(\\mathbb N),\\mathbb P)$ where $\\mathbb P(\\{n\\})=p_n$.\n\nLet $X:\\mathbb N\\to[0,\\infty)$ be prescribed by $n\\mapsto n$.\n\nFor $n=\\mathbb N=\\{1,2,\\dots\\}$ let $p_n>0$ with (of course) $\\sum_{n=0}^{\\infty}p_n=1$ and also $\\sum_{n=0}^{\\infty}np_n<\\infty$.\n\nQuestion: does there exist a finite partition $\\{P_1,\\dots, P_m\\}$ of $\\mathbb N$ such that $$\\sum_{k=1}^m\\sum_{n\\in P_k}kp_n=\\sum_{n=0}^{\\infty}np_n\\tag1$$\n\nThe RHS of $(1)$ equals $\\mathbb EX$.\n\nThe LHS of $(1)$ equals the expectation of $\\hat{X}$ determined by $\\hat X^{-1}(\\{k\\})=P_k$.\n\nIf you can find a case of non-existence of such partition then you have found an example where no such $\\hat X$ can be found.\n\nProvided that your $X$ is a continuous variable, I think that the simplest strategy would be to cut not only the tail but also the head of your distribution so that the expectation value is unchanged. Then you can concentrate the probability mass you have taken away exactly on the expectation value. This should satisfy your requests: $\\hat X$ goes to some interval $[\\epsilon,k]$ and is constructed so that $\\mathbb{E}[\\hat{X}] = \\mathbb{E}[X]$.","date":"2020-02-19 01:13:13","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\": 0.9645293354988098, \"perplexity\": 105.35951870408603}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875143963.79\/warc\/CC-MAIN-20200219000604-20200219030604-00055.warc.gz\"}"}
| null | null |
\section{Introduction}
String theory (with its extensions) hopefully will provide the primordial
matter for the Universe in such a way that, after evolving from a probably non
singular state at very high energy, we find a universe with the
characteristics of the one we observe today. However, almost every important
question is still waiting for a solution from String Theory. Meanwhile, it has
been very clear from the very beginning that String Theory could provide us
with a form of matter special enough so as to settle what seems to be the main
question which is the one about the resolution of the initial singularity.
It seems that, to achieve this, we need a kind of reduction of degrees of
freedom at high energy, at least with respect to the contribution coming from
the ultraviolet (high momentum) region. When
only perturbative strings were known, thermal duality showed itself as a
mechanism to truncate ten dimensional string theories (Bosonic,
Heterotic, Type II) at high energy to a kind of effective two dimensional
(massless) field theories with polarizations given by the cosmological
constant of a ten dimensional string theory \cite{atickyotros, miguel}. The
problem was that only for Heterotic Superstrings a finite, but vanishing, high
temperature free energy was found. For other string gases, like the ones made
of Type II Closed Superstrings, thermal duality implies a two dimensional
divergent free energy because the two-dimensional polarization degrees of
freedom are measured by the cosmological constant of the ten dimensional Type-0
Bosonic String. In other cases, like heterotic compactifications, the dual
phase at high energy is non sense since it has positive free energy. Only in
two dimensions, for generic heterotic non supersymmetric strings, we know
of a meaningful high temperature dual phase \cite{miguel}, but we
are already in a two dimensional space-time at low energy (low temperature).
Anyhow, the high temperature dual phase is very peculiar because it has the
cosmological constant of the theory as the coefficient of $\beta^{\,-2}$ and
still might serve as a high temperature candidate to solve the Hagedorn
behavior through a phase transition.
What we are going to show here is that the gas of open superstrings in a big
nine dimensional container presents, at high energy, a two dimensional behavior
that is corrected by the characteristic dominant Hagedorn
comportment to produce
an exotic equation of state of the form $p = \rho/(1 + k \sqrt{\rho})$ that can
finally be approximated by the better known
$p\propto\sqrt{\rho}$ equation of state.
The entropic
fundamental relation is given as the sum of the entropy of a gas of massless
particles in two dimensions plus $\beta_H E$ which is the standard
dominant Hagedorn behavior at high energy for the entropy of any gas made
of strings (open or closed). The two dimensional contribution to the entropy
alone would produce a Zeldovich's fluid (also known as stiff fluid) equation
of state $\rho=p$ that has appeared many times, for example in
\cite{BanksFischler}, as a good candidate to be the primordial cosmological
fluid.
The two dimensional behavior of the system is stressed by the fact that
the number of strings at high energy is the same function of $E$ as in a
massless two dimensional ordinary gas.
\section{The gas of open (super)strings}
It is well known that, perturbatively, the critical behavior of a gas of free
open strings at the Hagedorn temperature is such that the Helmholtz free
energy diverges when approaching $T_H$ from below. In macrocanonical terms,
this means that $T_H$ is a maximum temperature in the sense that any open
string gas in equilibrium that, at vanishing chemical potential, has a
finite internal energy $U(T, V)$ is necessarily kept at a temperature below
$T_H$. On the other hand, in a description at given energy and null chemical
potential, it is very natural to think about a maximum temperature because it
would be a maximum for the temperature as a mathematical function of energy
and volume.
\subsection{The macrocanonical description of the gas of open superstrings}
The black body of free open superstrings in the macrocanonical
ensemble description has a free energy, $-PV=F\left( \beta,V\right)$,
given by (see, e.g., \cite{emar1})
\begin{equation}
F\left( \beta\right) =
-\frac{V}{8 \left( 2\alpha'\right)^5}
\int_0^{+\infty}\,\mathrm{d} t\,t^{-6}\, \left[ \theta_3\left( 0,
\frac{\mathrm{i} \beta^2}{4 \alpha' t}\right) -\theta_4 \left( 0,
\frac{\mathrm{i} \beta^2}{4 \alpha' t}\right) \right]\,
\theta_4 ^{-8}\left( 0, \mathrm{i} t/\pi^2\right)\,\,.
\end{equation}
Let us remind some things to the reader.
The exponential growth of the number of states with the mass is here
expressed thorough the behavior of $\theta_4 ^{-8}\left( 0, \mathrm{i}
t/\pi^2\right)$ that certainly diverges when $t$ approaches zero from the right.
The behavior of the fourth Jacobi theta function is encoded in the relation
$\theta_4\left( 0, \mathrm{i}\,/t\right) = t^{1/2}\,\theta_2\left( 0,
\mathrm{i}\,t\right)$ that can be obtained by Poisson resummation of the series
representing the Jacobi theta function.
From all this, one gets that, by using an ultraviolet cutoff,
it is convenient to write the free energy as
\begin{equation}
\label{separate}
\begin{split}
F\left( \beta\right) \approx & -\frac{V}{8 \left( 2\alpha'\right)^5}
\int_{\epsilon}^{+\infty}\,\mathrm{d} t\,t^{-6}\, \left[ \theta_3\left( 0,
\frac{\mathrm{i} \beta^2}{4 \alpha' t}\right) -\theta_4 \left( 0,
\frac{\mathrm{i} \beta^2}{4 \alpha' t}\right) \right]\,
\theta_4 ^{-8}\left( 0, \mathrm{i} t/\pi^2\right) \\
& -\frac{V}{2^{\,9} \pi^{\,8}\left( 2\alpha'\right)^5}\int_0^{\epsilon}\,\mathrm{d}
t\,t^{-2} \mathrm{e}^{\left( 8\alpha'\pi^3 - \beta^2\pi\right)/\left( 4\alpha't\right)}\,\,
,
\end{split}
\end{equation}
where $\epsilon$ is a dimensionless cutoff that is taken small enough so
that the second term on the right hand side of eq. \eqref{separate} is a
good approximation to the contribution to the free energy coming
from the ultraviolet degrees of freedom\footnote{In a quantum field theory,
finite temperature provides an ultraviolet exponential regulator. In String
Theory at finite temperature as described by the analog model, the Hagedorn
behavior appears as an ultraviolet divergence. In the analog model, the
infrared and ultraviolet regions are defined for every quantum field in
the infinite collection of them as vibrational modes of the string. Open
string T-duality exchanges both regimes, but with different objects: a short
distance between D-branes corresponds to a long distance propagation of open
strings.}. Only the Maxwell-Boltzmann contribution has survived, so we are in
the classical statistics approximation. This cut-off can be seen as separating
the infrared from the ultraviolet degrees of freedom. We are going to be interested
on the contribution to the high temperature free energy that precisely comes
from the ultraviolet degrees of freedom (see \cite{atickyotros}).
From here, it is very easy to read the Hagedorn temperature
for open superstrings (treated without a gauge group contribution)
as $1/\beta_{\scriptscriptstyle{H}}=T_H =1/ (\pi \sqrt{8\alpha'})$
The integral in the second term in eq. \eqref{separate} can be evaluated
to give \cite{97uno}
\begin{equation}
\int_0^{\epsilon}\,\mathrm{d} t\,t^{-2}
\mathrm{e}^{\left( 8\alpha'\pi^3 - \beta^{\,2} \pi\right)/\left( 4\alpha't\right)} =
\frac{\beta_{\scriptscriptstyle{H}}^{\,2}}{2\pi^3\left( \beta^{\,2}-\beta_{\scriptscriptstyle{H}}^{\,2}\right)}\,
\mathrm{e}^{-2\pi^3\left(
\beta^{\,2}-\beta_{\scriptscriptstyle{H}}^{\,2}\right)/(\epsilon\beta_{\scriptscriptstyle{H}}^{\,2})}\,\, ,
\label{around}
\end{equation}
that is valid when $\beta > \beta_{\scriptscriptstyle{H}}$. It is now very clear that the free energy
diverges as $\beta$ approaches $\beta_{\scriptscriptstyle{H}}$ and eq. \eqref{around} tells us
exactly how. For a fixed cut-off $\epsilon$, the contribution to the free
energy coming from the ultraviolet region exponentially falls off when $(\beta
- \beta_{\scriptscriptstyle{H}})$ grows. If we then make the approximation $\left( \beta^2
-\beta_{\scriptscriptstyle{H}}^2 \right) = \left( \beta -\beta_{\scriptscriptstyle{H}}\right)^2 +2\beta_{\scriptscriptstyle{H}}\left( \beta -\beta_{\scriptscriptstyle{H}}\right) \approx
2\beta_{\scriptscriptstyle{H}}\left(\beta -\beta_{\scriptscriptstyle{H}}\right)$, the contribution to $\beta P V = -\beta F\left(
\beta, V\right) = \overline{N}\left( \beta, V\right)$ from the ultraviolet part can be
approximated by\footnote{That $-\beta F\left( \beta, V\right) = \overline{N}\left( \beta,
V\right)$ is the result of implementing $\mu=0$ in the macrocanonical description
when Maxwell-Boltzmann statistics is a good approximation \cite{exten}.}
\begin{equation}
\begin{split}
-\beta F^{\,h}\left(\beta\right) = &\, \frac{V}{2\pi
\beta_{\scriptscriptstyle{H}}^{\,8}}\,\int_0^{+\infty}\,\mathrm{d} E\,\mathrm{e}^{-\beta E}\,\mathrm{e}^{\beta_{\scriptscriptstyle{H}} E}
\theta \left( E - \frac{4 \pi^3}{\epsilon\beta_{\scriptscriptstyle{H}}}\right)\\
= &\,\frac{V}{2\pi \beta_{\scriptscriptstyle{H}}^{\,8}}\,\frac{\mathrm{e}^{-\left(\beta
-\beta_{\scriptscriptstyle{H}}\right)\,4\pi^{\,3}/\left(\epsilon\beta_{\scriptscriptstyle{H}}\right)}}{\beta -\beta_{\scriptscriptstyle{H}}}\,\, .
\end{split}
\label{laplace1}
\end{equation}
This integral representation tells us several things. First of all it is easy
to read the high energy behavior of the single open string density of
states given by (see \cite{97uno, ska})
\begin{equation}
\Omega_1^{\,h}\left( V, E\right) =
\frac{V}{2\pi\beta_{\scriptscriptstyle{H}}^{\,8}}\,\,\,\mathrm{e}^{\beta_{\scriptscriptstyle{H}} E}\,\,\theta\left( E
-\frac{4\pi^3}{\epsilon\beta_{\scriptscriptstyle{H}}}\right) ,
\label{omega1}
\end{equation}
where the energy cutoff has been directly derived from the dimensionless
$\epsilon$ parameter.
It is also very clear that the contribution to the free energy near $\beta_H$ is
dominated by $F^{\,h}$ that grows unbounded as long as the first integral in
\eqref{separate}
gives a finite value at $\beta_H$.
The internal energy $U \left( \beta, V\right)$ can easily be computed as
$U\left( \beta, V\right) = \frac{\partial}{\partial\,\beta} \beta F\left( \beta\right)$.
Near the Hagedorn temperature, we have that
\begin{equation}
U^{\,h}\left( \beta, V\right) \approx \frac{V}{2\pi \beta_{\scriptscriptstyle{H}}^{\,8}}\,
\left[ \frac{1}{\left( \beta -\beta_{\scriptscriptstyle{H}}\right)^2} -
\frac{8\pi^{\,6}}{\epsilon^{\,2}\beta_{\scriptscriptstyle{H}}^{\,2}}\right],
\end{equation}
where we have presented it to show the only divergent term. Let us notice that
$0 < \left( \beta -\beta_{\scriptscriptstyle{H}}\right) < \epsilon\beta_{\scriptscriptstyle{H}}/\left( \sqrt{8}\pi^{\,3}\right)$.
The calculation of fluctuations is an important piece in the study of
equivalence between different ensemble descriptions. Fluctuations
computed for the energy $U$ are given around $\beta_{\scriptscriptstyle{H}}$ by
\begin{equation}
\frac{\left[ T^{\,2}\,C_V(T, V)\right]^{1/2}}{U} \approx 2\,\sqrt{\frac{\pi}{V}}
\beta_{\scriptscriptstyle{H}}^{\,4}\left( \beta -\beta_{\scriptscriptstyle{H}}\right)^{1/2}\,\mathrm{e}^{\left( \beta -\beta_{\scriptscriptstyle{H}}\right)
2\pi^3/\left(\epsilon\beta_{\scriptscriptstyle{H}}\right)} = \sqrt{2} \left(\overline{N}^{\,h}\right)^{\,-1/2}
,
\end{equation}
that certainly vanishes as $\beta$ approaches $\beta_{\scriptscriptstyle{H}}$. In fact, fluctuations are
O$(\overline{N}^{-1/2})$ as in a typical closed isothermal system. This
means that equivalence between the fixed temperature and the fixed energy
descriptions for the system of open strings with $\mu=0$ must occur in the
sense that the states at given energy certainly would correspond to states
with well defined averaged energy at a given temperature.
To compare with the fixed energy (or micro) calculation, it is useful to get
the thermodynamical fundamental relation giving the entropy $S$ as a function
of the canonical averaged energy $U$ and the volume $V$ for
$\beta$ bigger and near $\beta_{\scriptscriptstyle{H}}$. One gets
\begin{equation}
S^{\,h}\left( U, V\right) = 2\sqrt{\frac{U\,V}{2\pi\beta_{\scriptscriptstyle{H}}^{\,8}}}
+ \beta_{\scriptscriptstyle{H}}\,U ,
\label{entropyM}
\end{equation}
where we have assumed that $U \gg 8\,\pi^5\,V/\left( \epsilon^{\,2}\beta_{\scriptscriptstyle{H}}^{\,10}\right)$.
Finally, it is worth to remark that the first term in \eqref{separate}
gives the contribution from the string modes for long propagation times. The
massive modes are exponentially suppressed so the relevant contribution comes
from the massless degrees of freedom \cite{kipfrantz}.
\subsection{The fixed energy description of the gas of open
superstrings}
The big number of open dimensions tells us that
quantum statistical corrections are negligible for the system \cite{exten}.
Then, the label $N$ in $\Omega_N$ certainly refers to the number of strings in
the gas because Maxwell-Boltzmann statistics is applicable even when the
system energy is not very high.
$\Omega_1 \left( E, V\right)$ is the key ingredient to get $\Omega_N \left( E, V\right)$
using Laplace convolutions \cite{97uno}.
The problem here is that such integrals cannot be performed
analytically. However, we are interested in getting the high energy
behavior of $\Omega_N$ (for high N). For open strings, this can be
obtained from the convolution of the high energy part of $\Omega_n$ to get
$\Omega_{2n}^{\,h}$, \footnote{In fact,
$\Omega_{2n}\left( E\right) =\frac{n!^{\,2}}{(2n)!}\,\int_{0}^{E}\,\mathrm{d}\, t \,\,
\Omega_n\left( E - t\right)\,\Omega_n\left( t\right)$ \cite{97uno}.}.
The reason is that the pure exponential
growth of $\Omega_1^{\,h}$ implies that the convolution between the high and
low energy parts of the single density of states gives a negligible
contribution to $\Omega_2^{\,h}$ (this is not the case for closed strings when
windings are absent \cite{exten}).
In Fig. \ref{F1}, a numerical computation of $\omega_2
(E, t) = \Omega_1 \left( E-t\right)\,\Omega_1\left( t\right)$ at $E=5$ ($\alpha'=1$) is
showed together with the straight line which represents $\omega_2^{\,h}$ as
obtained from $\Omega_1^{\,h}\left( E, V\right) \sim \mathrm{e}^{\beta_{\scriptscriptstyle{H}} E}$. In it, we
can observe the two energy cutoffs for the validity of this approximation
that, in general, will be placed, for $\omega_2$, at $\lambda$ and $E-\lambda$;
$\lambda = 4\pi^3/\left(\epsilon \beta_{\scriptscriptstyle{H}}\right)$. The plateau in the
interval $[\lambda,5-\lambda]$ signals the feature that
energy will be distributed with equal probability among open strings of
different length, i.e., different energy. This is the
way the equilibrium state
shows the most probable decay of a highly excited open string
(see \cite{manes}). Looking at this plateau from the point of view
of equipartition and the equivalence of ensembles, one might erroneously guess
that the fixed temperature and preserved energy descriptions would not be
equivalent. Instead, one would expect as the most probable configuration that
each open string
shared half of the total energy. However, the computation of energy
fluctuations tells us that this is not the case.
We are actually showing that the equilibrium got by shearing
the total energy among open strings with different lengths is stable. Indeed
this is a worthy feature of the behavior, around the Hagedorn temperature, of
our system.
\begin{figure}[htp]
\let\picnaturalsize=N
\def3.0in{3.0in}
\defabiss.eps{abiss.eps}
\ifx\nopictures Y\else{\ifx\epsfloaded Y\else\input epsf \fi
\let\epsfloaded=Y
\centerline{\ifx\picnaturalsize N
\epsfxsize 3.0in\fi\epsfbox{abiss.eps}}}\fi
\caption{$\omega_2(5,t)$ computed numerically. The straight line represents
$\omega_2$ as given by using $\Omega_1^h$ only. $\alpha' = 1$.}
\label{F1}
\end{figure}
With this approximation one can get
\begin{equation}
\begin{split}
\Omega_N^{\,h} \left( E, V\right) = &\frac{1}{N!\,\left( N-1\right)!}\left( \frac{V}{2\pi
\beta_{\scriptscriptstyle{H}}^{\,8}}\right)^{\,N}\mathrm{e}^{\beta_{\scriptscriptstyle{H}} E}\left(
E-N\lambda\right)^{\,N-1}\,\theta\left( E-N\lambda\right)\\
\approx &
\frac{1}{N!^{\,2}}\left( \frac{V}{2\pi \beta_{\scriptscriptstyle{H}}^{\,8}}\right)^{\,N}\mathrm{e}^{\beta_{\scriptscriptstyle{H}} E}\left(
E-N\lambda\right)^{\,N}\,\theta\left( E-N\lambda\right),
\end{split}
\end{equation}
where in the second equation we have assumed that $N$ is very big.
This high energy density of states for the gas of $N$ strings holds as long as
$E$, the energy of the gas, is bigger than $N\lambda$.
The vanishing of the chemical potential $\mu$ implies that the high energy
entropy is $S^{\,h}=\mathrm{ln}\sum_{N=0}^{\infty}\, \Omega_N^{\,h}$. In fact,
the sum can be approximated by a single term $\Omega_{N^{\,*}}^{\,h}$ for
$N^{\,*}$ such that $\Omega_{N=N^{\,*}}$ is a maximum of the density of
states as a function of the number of particles. We then compute
$\frac{\partial}{\partial N}\,\mathrm{ln}\Omega_N^{\,h}$ to get that, for $E
\gg \lambda N^{\,*}$,
\begin{equation}
\overline{N}^{\,h}\left( E, V\right)= N^{\,*}= \sqrt{\frac{E\,V}{2\pi\beta_{\scriptscriptstyle{H}}^{\,8}}}.
\label{ene}
\end{equation}
It is a very notorious fact that the number of open superstrings at high
energy depends on energy and volume as in a gas of massless particles in two
space-time dimensions described using Maxwell-Boltzmann
statistics\footnote{For the gas of massless particles in $d > 2$ space-time
dimensions with Bose-Einstein statistics, one finds: $\overline{N}\left( E, V\right) =
\frac{\zeta\left( d-1\right)}{\zeta\left( d\right)}\,N^{\,*}\left( E, V\right)$ where $N^{*}$ is
the number of particles with $\mu=0$ using Maxwell-Boltzmann statistics
\cite{exten}. In two space-time dimensions, one gets the finite result:
$\overline{N}\left( E, L\right) = \frac{\sum_{r=1}^{+\infty}\,\theta\left( ER
-r\right)/r}{\zeta\left( 2\right)}\,N^{\,*}\left( E, L\right)$, where $R$ is the radius of the
compact space ($L=2\pi R$) that, as the energy, is supposed to become big as
one takes the thermodynamic limit. This expression at dimension two
results from computing the free energy in the thermodynamic limit by
converting the sum over discrete momenta to an integral together with the
separation of the zero momentum contribution that represents the one of the
the vacuum state (no particle state) that must not be second
quantized.}.
The entropy is then
\begin{equation}
S^{\,h}\left( E, V\right) \approx 2\,\overline{N}^{\,\,h}\left( E, V\right) + \beta_H\,E = 2
\sqrt{\frac{E\,V}{2\pi\beta_{\scriptscriptstyle{H}}^{\,8}}} + \beta_{\scriptscriptstyle{H}} E, \label{entropym}
\end{equation}
for $E \gg 8\pi^{\,5}\,V/\left(\epsilon^{\,2}\beta_{\scriptscriptstyle{H}}^{\,10}\right)$.
It exactly coincides with $S^{\,h}\left( U, V\right)$ as computed in the canonical
description, as we expected from the computation of the energy fluctuations in
the fixed temperature picture. This expression of the entropy as a function
of energy and volume was found in \cite{9408134}, \cite{9707167} (see also
\cite{9902058}), although no physical interpretation of the term which scales
with $E^{1/2}$ in terms of the
number of strings was given\footnote{The equivalence of ensembles was also not
treated. It would be a mistake to think that it is trivial because the
specific
heat is positive and it would be cynic to say that equivalence has been
implicitly
assumed since
the computation of energy fluctuations is so easy that does not deserve
any mention. After all, we go further because we show here that the equilibrium
got shearing the
total energy with equal probability among a set of open strings with different
lengths, i.e. different energies, is compatible with the stability of the
system. This is more
than
stating that equivalence holds, it is a proof of the thermodynamic stability of
the stringy
matter that comes from the preferred decay mode of any highly excited open
string \cite{manes}.}. It is also worth to mention that this entropy
is an extensive function of energy and volume that can be written in terms
of the number of strings as
\begin{equation}
S^{\,h} = 2\overline{N} + 2\pi\beta_{\scriptscriptstyle{H}}^{\,9}\overline{N}^{\,2}/V
\label{ene}
\end{equation}
This behavior in terms of the number of objects can be compared with a "regular"
system for which entropy is $O(N)$.
In \eqref{entropym} and \eqref{ene}, the first term is very easily identified
as
the entropy for a two dimensional gas of massless particles that can be
written as a
function of the squared root of the energy or, exactly, as twice the average
number of objects\footnote{The massless character of
the particles does not seem to be something special because the high
temperature limit for the free energy of a massive field in $d$ space-time
dimensions is dominated by the contribution to the free energy of a massless
field in $d$ dimensions.}. We also
see the nine dimensional volume divided by
the adequate power of the length scale of our problem which is the squared
root of $\alpha'$ (included in $\beta_H$). This is the way an
effective one
dimensional volume appears. This contribution to the entropy is corrected by the
universal volume independent, and then pressureless, contribution coming from
the Hagedorn behavior that is $O(\overline{N}^{\,2})$. It is important to
notice that the term $\beta_H\,E$
shows up in the entropic fundamental relation for any critical string gas in the
macrocanonical and the microcanonical computations. This term gives, for any
gas of fundamental strings, a divergent contribution to $C_V(E)$ and the
effective two dimensional part finally makes $C_V(E)$ finite and positive for
the gas of open superstrings. The two terms in the entropy can be seen as the
contribution of two different types of degrees of freedom at high energy that
finally make $T_H$ a maximum temperature for the system.
\section{The equation of state}
It is very easy to get the equation of state which relates the density of
energy $\rho$ and the pressure of the gas. We get
\begin{equation}
P = \frac{\rho}{1\,+\,\beta_H\,\sqrt{2\pi\beta_H^{\,8}\,\rho}}\,\, .
\end{equation}
Here, what makes the denominator different from the unity, and so the equation
of state different from Zeldovich's one, is precisely the term $\beta_H\,E$
in the entropy. This is dominant at high energy, because what we mean by high
energy to get \eqref{entropyM} and \eqref{entropym} implies the condition
$\sqrt{2\pi \beta_H^{\,10}\,\rho} \gg 1$.
With this approximation the equation of state at very high energy looks
$$
P \approx \sqrt{\frac{\rho}{2\pi\beta_H^{\,10}}}\,\, .
$$
This fluid is causal because the sound would propagate in it with a speed
given by
$$
v_s^{\,2} = \frac{\partial P}{\partial \rho} \approx \frac{1}{2\,
\sqrt{2\pi\beta_H^{\,10}\rho}}\ll 1.
$$
\section*{Acknowledgments}
This work has been partially supported by the Spanish MEC research project
BFM2003-00313/FISI. The work of M. A. C. is partially supported by a
MEC-FPI fellowship. The work of M. S. is partially supported by a MEC-FPU
fellowship.
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Q: How to get one stream from error stream and input stream when calling a script using JSCH I am calling a script file (.sh) located on remote machine using JSCH. During execution the script outputs both error and success statements. The JSCH code what I have written exhibits two streams, InputStream & Error Stream
How can I get an single input stream that contains error and output ?
Channel channel=session.openChannel("exec");
((ChannelExec)channel).setCommand("/opt/sdp/SnapShot/bin/dumpSubscribers.ksh");;
InputStream in=channel.getInputStream();
InputStream error=((ChannelExec)channel).getErrStream();
channel.connect();
**Script output:**
\[2014-01-23 19:41:01] SnapShot: Start dumping database
szTimgExtension: sdp511 enabled
functionOfferSupport active
Failed to prepare statements: ODBC Error 'S0022', TimesTen Error 2211, ODBC rc -1
[TimesTen][TimesTen 7.0.6.8.0 ODBC Driver][TimesTen]TT2211: Referenced column O.START_SECONDS not found -- file "saCanon.c", lineno 9501, procedure "sbPtTblScanOfColRef()" [Unable to prepare statement: <Statement for getting subscriber_offer data.>.]
Database error: ODBC Error 'S0022', TimesTen Error 2211, ODBC rc -1
[TimesTen][TimesTen 7.0.6.8.0 ODBC Driver][TimesTen]TT2211: Referenced column O.START_SECONDS not found -- file "saCanon.c", lineno 9501, procedure "sbPtTblScanOfColRef()" [Unable to prepare statement: <Statement for getting subscriber_offer data.>.]
[2014-01-23 19:41:01] SnapShot: Result files:
/var/opt/fds/TT/dump//SDP1.DUMP_subscriber.v3.csv
/var/opt/fds/TT/dump//SDP1.DUMP_usage_counter.v3.csv
[2014-01-23 19:41:01] SnapShot: Finished dumping database
A: Initialize an Output stream for both to write to, then instead of getInputStream, use setOutputSteam and setErrStream
OutputStream out = new OutputStream();
channel.setOutputStream(out);
channel.setErrStream(out);
Note that 'out' will be closed when the channel disconnects. To prevent that behavior, add a boolean when setting the output stream:
channel.setErrStream(out, true);
channel.setOutputSteam(out, true);
This may be important if the output stream you are using for the JSCH ChannelExec session is being reused elsewhere in your code.
If you need to read the output stream into an input stream, refer to this question.
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\section{Introduction}
In nature, species do not live in isolation but form large networks of interdependences that are often depicted as a set of nodes (species) connected by links (species interactions). This entangled web of life is increasingly threated by several drivers of global change (Tylianakis et al., 2008, 2010), such as climate warming (e.g., Memmott {\em et al.}, 2007), habitat loss and fragmentation (e.g., Tylianakis {\em et al.}, 2007), and invasive species (e.g., Aizen {\em et al.}, 2008). In this network context, species interactions are a component of biodiversity as important as species themselves (Thompson, 2009). We know we are losing species interactions (Aizen {\em et al.}, 2012), which may drive ecological communities towards tipping points (Lever {\em et al.}, 2014). We need, hence, a network thinking to predict future community-wide scenarios (Tylianakis {\em et al.}, 2008).
Since the 1980s, data on who interacts-with-whom in ecological communities have been compiled, focused mainly on food webs and plant-animal mutualistic networks. Data, however, can be found mainly as appendices or supplementary material in the papers where the authors originally published their work. The Interaction Web DataBase (www.nceas.ucsb.edu/interactionweb/) created in 2003 and hosted by the National Center for Ecological Analysis and Synthesis (NCEAS) at the University of California (USA) is, to our knowledge, the only initiative with the aim of centralizing the available information on ecological networks. However, as far as we know, it is currently outdated. In addition, because it only provides datasheets, searching options are not implemented. This limits the potential use of the database as a working tool to tackle scientific questions across networks. In addition, if we aim to reach stackeholders and policy makers, an easy way of visualizing the network dataset compiled throughout the world is of paramount importance to capture their attention.
\section{Features}
We provide the most comprenhensive dataset of plant-animal mutualistic networks to date. In contrast to antagonistic interactions in which one species obtains a benefit at expenses of the other, in plant-animal mutualistic interactions, like those between a plant and a pollinator or between a plant and a seed disperser, both species obtain mutual benefit. Food webs and other ecological interactions such as host-parasite, host-parasitoid, plant-ant, plant-epiphyte, plant-herbivore, and anemone-fish interactions, will be available soon.
\subsection{Location map}
The user interface is based on Google Maps. Over the map, colored circles indicate where the compiled networks are located. Colors depict the type of ecological interaction. The user can zoom in or out and drag the map.
\subsection{Network datasheet}
Some basic information about the network can be obtained when the mouse pointer is over one of the colored circles. By left clicking on it, a detailed information about the network is dynamically generated: number of species, number of interactions, network connectance, locality, geographical coordinates, original reference, and a unique network identifier. The network identifier is intended to be adopted by the community of researchers as a unique tag to identify a given network across future studies. The network of interactions is also displayed along the species names (when they are identified). Depending on the data compiled by the researchers, a matrix of ones and zeros (presence/absence of the interaction, respectively) or a matrix of natural numbers indicating the number of visits performed by a pollinator species on a plant species, is displayed. A java script graphical representation using the D3js library is available for a quick visualization of the network.
\subsection{Data filtering criteria}
Network selection can be filtered directly from the menu bar by selecting the type of interaction (up to now pollination or seed dispersal), type of data (binary: presence/absence of the interactions, or weighted: number of visits), number of species, and number of interactions. The list of networks selected as resulting from the filtering criteria applied is immediately ready for visualization or download. Searchers across networks by species names can be performed from the list of networks selected, which allows a meta-analysis never accomplished before.
\subsection{Data download}
Network data can be downloaded from the location map and from the filtering criteria. Species names can also be included. The following file formats are available: comma-separated values (.csv), Excel spreadsheet format (.xls), Pajek format (.net) that can be imported in Gephi as well for visualization, and as Java Script Object Notation (.json). Downloading a large dataset could take some time because a zip file containing each network as a single file, a file for the references, and a readme file, is dynamically generated. A log file tracking the history of changes for the downloaded network dataset (if exists) is also included in the zip file.
\section{Implementation}
The Web of Life has been designed and implemented in an open-source relational database management system (MySQL). This allows sophisticated and user-friendly searching across networks. It also provides an easy way of incorporating new network data available in the future. In order to minimize spelling mistakes when introducing new data, we do not provide an online interface for data entry, so that users cannot enter and edit their data directly. Users can access the database through any web browser using a variety of operating systems.
\section{Conclusion}
Biodiversity is much more than a list of species. Interactions among species are the {\em glue} of biodiversity. If we aim at predicting future community-wide scenarios and anticipating planetary critical transitions, we have to consider the entangled web of interactions among species. Here, we introduce The Web of Life, a database for visualizing and downloading data of species interaction networks. This repository allows scientists to do research within and between networks compiled at different places all over the world.
\section{Acknowledgements}
We would like to thank Jordi Bascompte's lab members for their suggestions during the design of the web, as well as the authors of the compiled networks for their invaluable effort during the fieldwork. This web service is supported by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) through an Advanced Grant to Jordi Bascompte (grant agreement 268543). M.A.F. holds a postdoctoral fellowship (JAE-Doc) from the Program ``Junta para la Ampliacion de Estudios'' co-funded by the Fondo Social Europeo (FSE).
\newpage
\section{References}
\noindent Aizen, M. A., Morales, C., and Morales, J. (2008). Invasive mutualists erode native pollination webs. {\em PLoS Biology}, 6:e31.
\noindent Aizen, M. A., Sabatino, M., and Tylianakis, J. M. (2012). Specialization and rarity predict nonrandom loss of interactions from mutualist networks. {\em Science}, 335:1486-1489.
\noindent Lever, J., van Nes, E. H., Scheffer, M., and Bascompte, J. (2014). The sudden collapse of pollinator communities. {\em Ecol. Lett.}, 17:350-359.
\noindent Memmott, J., Craze, P. G., Waser, N. M., and Price, M. V. (2007). Global warming and the disruption of plant-pollinator interactions. {\em Ecol. Lett.}, 10:710-717.
\noindent Thompson, J. N. (2009). The coevolving web of life. {\em Am. Nat.}, 173:125-140.
\noindent Tylianakis, J. M., Tscharntke, T., and Lewis, O. T. (2007). Habitat modification alters the structure of tropical host-parasitoid food webs. {\em Nature}, 445:202-205.
\noindent Tylianakis, J. M., Didham, R. K., Bascompte, J., and Wardle, D. A. (2008). Global change and species interactions in terrestrial ecosystems. {\em Ecol. Lett.}, 11:1351-1363.
\noindent Tylianakis, J. M., Lalibert\'e, E., Nielsen, A., and Bascompte, J. (2010). Conservation of species interaction networks. {\em Biol. Cons.}, 143:2270-2279.
\newpage
\section{Figures}
\begin{figure}[!ht]
\includegraphics[width=1.00\textwidth]{fig.pdf}
\caption{Snapshot of The Web of Life. On top, the menu bar showing the data filtering criteria. Over the map, three pop-up windows are displayed: the network list containing the selected networks, the species list for searching across networks, and an example of a network datasheet showing the presence/absence of mutualistic interactions between plant and animal species (from back to front, respectively).}
\end{figure}
\end{document}
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Johann Wilhelm Jahn, auch: Jan, Jano, Janus (* 9. November 1681 in Raben; † 27. August 1725 in Wittenberg) war ein deutscher lutherischer Theologe und Historiker.
Leben
Geboren wurde er als Sohn des Magisters Johann Jahn (Janus) und seiner Frau Salome, der Schwester des Wittenberger Theologen Johann Georg Neumann geboren. Nachdem ihm sein Vater die ersten pädagogischen Grundsätze der damaligen Zeit vermittelt hatte und ihn auch Latein gelehrt hatte, besuchte er die Schule in Schneeberg, wo er vor allen Dingen von Johann Gottfried Sieber, den späteren Leipziger Professor der Theologie, einen zugeneigten Förderer fand. Unter dessen Führung erfuhr er Kenntnisse über die grundlegenden Fähigkeiten und die Sprachwissenschaften, so dass er 1699 sich an der Universität Wittenberg immatrikulieren konnte.
Zunächst absolvierte er, wie zur damaligen Zeit üblich, ein philosophisches Studium und hatte dabei Georg Friedrich Schröer, Christian Vater, Heinrich Heuchner, Johann Baptist Roeschel, Michael Strauch, Christian Röhrensee und Johann Christoph Wichmannshausen als Lehrer. Besonders aber wurde er von Konrad Samuel Schurzfleisch beeinflusst, der ihn auch den Besuch bei den öffentlichen Vorlesungen der Theologen Johann Deutschmann, Philipp Ludwig Hanneken, Caspar Löschers und Gottlieb Wernsdorf der Ältere nahelegte und in seinem Vetter einen Förderer fand.
Am 31. Oktober 1701 erlangte er die akademische Magisterwürde, nachdem unter dem Johann Wilhelm von Berger mit "de lemmatibus poëticis" disputiert hatte. 1702 verfasste er die Leichenpredigt für seine Tante Sabina Dorothea, geb. Leyser, die Witwe von Franz Heinrich Höltich und Christian Donati. Als er sich mit der Dissertation "de origine" (dt.: "Über den Ursprung des Menschenopfers") habilitierte, bahnte er sich damit 1706 einen Weg zum Adjunkten der philosophischen Fakultät, an der er dreimal als Präsens wirkte. Jedoch zog es ihn mehr zur theologischen Fakultät, und nachdem er 1708 die Abhandlung "de Trinitate Platonismi vere & falso suspecta" verteidigt hatte, fand er Aufnahme bei den Kandidaten der theologischen Fakultät.
Seine von nun an gehaltenen Vorlesungen zu geschichtswissenschaftlichen und philosophischen Themen und die zur hebräischen Sprache wurden wegen seines sauberen Stils nicht nur von der Studentenschaft geschätzt, sondern sie brachten ihm auch 1712 eine außerordentliche Professur der moralischen Wissenschaften ein. Kaum hatte er diese jedoch angetreten, berief man ihn als ordentlichen Professor der Sittenlehre und Beredsamkeit an das Gymnasium "Elisabethanum" in Breslau, welches er 1713 antrat. Nachdem Schurzfleisch nach Weimar gezogen war, kehrte er 1714 nach Wittenberg zurück und übernahm die ordentliche Professur der historischen Wissenschaften.
Dennoch hatte er den Wunsch, andere Länder zu bereisen und die Bibliotheken außerhalb von Wittenberg kennenzulernen. Mit der Erlaubnis des königlichen Hofes trat er 1715 diese Reise an, die ihn über einige deutsche Universitätsstandorte, nach Holland, England, Frankreich und Italien führen sollte, wo er ein intensives Bibliotheksstudium betrieb und mit den Gelehrten seiner Zeit in Verbindung kam.
Zurückgekehrt nach Wittenberg, stellte sich das an den unterschiedlichsten Universitäten erworbene Wissen als ausgesprochen nützlich heraus. Im Universitätsbetrieb und durch seine veröffentlichten Schriften wuchs das Ansehen seiner Person. Nachdem Löscher verstorben war und über die Neubesetzung der theologischen Fakultät nachgedacht wurde, schlug man ihn von Seiten der Universität für den Lehrstuhl eines Professors der Theologie vor, obwohl dies bei den Beratern am sächsischen Hof nicht auf sonderlich viel Gegenliebe stieß. Vielmehr wollte man Jahn am sächsischen Hof in ein höheres Amt einsetzen und behauptete daher, dass er für die Professur ungeeignet sei.
Dennoch setzte sich die Universität mit ihrem Wunsch durch, und so trat er 1719 mit der Rede "de optima ratione interpretandi sacras literas" sein Amt an. Er disputierte noch im selben Jahr am 31. Juli mit "de jure decidendi controversias Theologicas" und erwarb sich damit den akademischen Grad eines Lizentiaten der Theologie. Kurz darauf promovierte er zum Doktor der Theologie und übernahm auch die für die vierte theologische Professur vorgesehene Ephorie der kurfürstlichen Stipendiaten. Als Vertreter lutherisch-orthodoxer Positionen trat er vor allem gegen die Hallenser Pietisten auf.
Jahn wird von seinen Zeitgenossen als umgänglicher und aufrichtiger Mann geschildert, der in seinem Amt stets korrekt und fleißig gewesen ist. Dennoch besaß er eine schwächliche physische Konstitution und starb an einem hohen Fieber, nachdem ihm zwei Tage zuvor Lähmungserscheinungen Verstand und Sprache geraubt hatten, um 4 Uhr in der Früh.
Am 15. Juni 1721 vermählte er sich mit Magdalena Elisabeth Wichmannshausen, der Tochter des Professors für morgenländische Sprachen und Vorstehers der Wittenberger Universitätsbibliothek. Aus dieser Ehe ging der Sohn Johann Christoph (* 6. Februar 1722; † 25. Juli 1725 in Wittenberg) und die Tochter Wilhelmina Elisabeth (* 8. Oktober 1723 in Wittenberg; † 3. Juli 1749 in Wittenberg) hervor. Nach seinem Tode heiratete seine Witwe am 9. September 1727 Franz Woken, Professor der morgenländischen Sprachen, und wurde nach dessen Tod am 18. Februar 1734 abermals Witwe.
Werke (Auswahl)
Theologia Aphoristica, In qua Sententia orthodoxa, recentioribus potissimum adversariis opposita, succinctis Aphorismis perspicue proponitur, & selectis argumentis confirmaturheologia aphoristica. Zimmermann, Wittenberg 1710. (Digitalisat)
Theosophia Orthodoxa Hallensium Theologorum Theosophiae Pseudōnymō Atque Fanaticae A Collega Illorum, qui se pro Antibarbaro Seculi nostri gerit, proditae et assertae opposita. Wittenberg, Gerdes 1712. (Digitalisat)
Specimen Errorum Langianorvm, Oder: Deutliche Vorstellung einiger groben Theolog. Irrthümer, Deren Herr Joach. Lange Prof. Publ. Hal. Völlig und also überführet ist, daß er Gewissens halber dieselben öffentlich zu revociren schuldig. Nebst Kurtzer Abfertigung des II Theils der Mittel-Strasse und Ablehnung der daselbst wider die Theosophiam Orthodoxam ausgestossenen Injurien. Ludwig, Wittenberg 1713. (Digitalisat)
Historia Aerae Christianae, Cui Praemittuntur Schediasma De Veritate Historica Et Oratio De Vero Historiae Usu. Kreusig, Wittenberg 1715. (Digitalisat)
Er verfasste mehrere Dissertationen, siehe hierzu Rafft und Jöcher.
Literatur
Michael Ranfft: Leben und Schriften aller Chursächsischen Gottesgelehrten. Wolfgang Deer, Leipzig 1742.
Walter Friedensburg: Geschichte der Universität Wittenberg. Max Niemeyer, Halle (Saale) 1917.
Christian Gottlieb Jöcher: Allgemeines Gelehrten-Lexikon. 1750, Band 2, Sp. 1835
Johann Samuel Ersch, Johann Gottfried Gruber: Allgemeine Encyclopädie der Wissenschaften und Künste, Sektion 2, Teil 14, S. 299
Rhetoriker
Historiker
Lutherischer Theologe (18. Jahrhundert)
Hochschullehrer (Leucorea)
Deutscher
Geboren 1681
Gestorben 1725
Mann
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\section{Introduction\label{sec:Introduction}}
Recent studies concerning learning processes in neural circuits have
highlighted the role of spike timing and synchrony (e.g.~in sensory
systems \cite{johansson_first_2004,decharms_primary_1996,meister_concerted_1995,wehr_odour_1996}),
leading to a view of the learning devices as a class of time coincidence
detectors of a limited number of spikes (at least under certain circumstances).
These observations are at the root of several fundamental questions
concerning neural coding, the most important one possibly being how
do neurons learn to recognize multiple spatiotemporal patterns.
A very stimulating contribution in this field has been the recent
introduction by Gutig and Sompolinsky \cite{gutig_tempotron:_2006}
of a perceptron-like model neuron which is able to process spatio-temporal
patterns, the so called tempotron. In spite of its simplicity, such
a device is capable of decoding the information contained in the synchrony
of spike patterns through a relatively simple supervised gradient
learning rule. Subsequent work \cite{rubin_theory_2010} has analyzed
by statistical physics techniques the storage capacity (i.e.~the
typical maximum number of distinct input-output associations which
the device can in principle be trained to reproduce, assuming that
the inputs and the expected responses are drawn from some probability
distribution) and the geometry of the space of solutions of the tempotron
for continuous synaptic weights.
In a nutshell, the tempotron is the simplest form of an integrate
and fire (IF) neuron, with $N$ input synapses of strength $J_{i}$,
$i=1,\dots,N$ (also called synaptic weights).
In the tempotron, each input corresponds to $N$ sequences
of spikes, where the set of spiking times is denoted by $\{t_{i}\}$.
The tempotron performs a binary classification of the inputs depending
on whether the membrane potential reaches or not the firing threshold
$\theta$ in the given time interval. The potential at time $t$ is
given by $V\left(t\right)=\sum_{i}J_{i}\sum_{t_{i}<t}v\left(t-t_{i}\right)$, where $v\left(t\right)$
is the temporal kernel of the membrane. A standard choice for the
kernel is the exponential one, namely $v\left(t\right)=v_{0}\left(e^{-t/t_{m}}-e^{-t/t_{s}}\right)$,
where $t_{m}$ and $t_{s}$ are the membrane and synaptic integration
time constants. For this model the precise timings and the number
of output spikes (if greater than one) play no role in the binary
classification, allowing for multiple equivalent output spiking
profiles for positive classifications of a given time interval.
As discussed in \cite{rubin_theory_2010}, a key parameter is the
quantity $K=T/\sqrt{t_{m}t_{s}}$ where $T$ is the duration of the
input pattern. When both $N$ and $K$ are large, and $N\gg K$,
certain time correlations can be neglected and
the analysis simplifies. This has allowed the authors of \cite{rubin_theory_2010}
to estimate the storage capacity of the device for the case of continuous
weights and random i.i.d.\ patterns. It is interesting to observe that
these conditions are not far from being actually realistic
\cite{rubin_theory_2010}. The vanishing of the correlations at different
times is due to the sparse regime under which the device operates, and
it means that the width of the kernel $v\left(t\right)$ is much shorter than
the typical interval between incoming spikes; this in turn means that,
under this regime, only quasi-simultaneous input spikes actually contribute
to the depolarization at any given time, which explains why in~\cite{rubin_theory_2010}
the theoretical and experimental results are closely reproduced
with a simplified model in which time is discretized in $K^\textrm{discrete}=K/8$
bins and the output is simply given by a perceptron rule applied on each bin
(see paragraph~\ref{par:Model} for additional details).
Basic devices like the temportron have the potential virtue of touching
those fundamental questions in neural coding which are preserved in
spite of the simplicity of the device itself. In such a framework,
we have approached the problem from a different angle, namely adopting
a computational scheme which that is not based on a gradient-like
computation but is still fully local and distributed. We study the
simplified (time-discretized) tempotron by both the replica method and
the so called message-passing approach (or cavity method) which allows
us to study analytically the storage capacity and at the same time to
derive simple learning protocols (i.e.~training rules for the
modifications of the synaptic strengths, which the device applies upon
receiving input patterns and being made aware of the expected
response, such that at the end of the training the desired set of
input-output associations is learned) which are efficient and do not
rely on any continuity condition of the synaptic weights. We also show
how to adapt the simplest of these learning protocols to address the
original, continuous-time model. For the sake of simplicity we focus
directly on the case of discrete synapses, although the results could
be extended to the continuous case.
For the cases of single and multilayer perceptrons with firing rate
coding and binary synapses we have shown in previous works \cite{braunstein_learning_2006,baldassi_efficient_2007,baldassi_generalization_2009}
that the message-passing approach is indeed efficient in solving the
learning problem for random patterns and that the computational scheme
can be simplified to the point of providing extremely simple learning
protocols. These past results together with the novel ones on spatio-temporal
coding presented here should be of practical interest for large scale
neuromorphic devices and hopefully for providing novel hints on aspects
of synaptic plasticity.
\paragraph{The model\label{par:Model}}
We studied two tempotron scenarios, one in which synaptic conductances
can take values in $\left\{ -1,+1\right\} $, and one in which they
can take values in $\left\{ 0,1\right\} $, focusing on the former
case in simulations. As in the final paragraphs of~\cite{rubin_theory_2010}, we worked under
the simplifying assumption that input and output spike patterns can
be encoded (via binning) as sparse strings of $0$'s and $1$'s, and
that the relationship between the inputs and the output at any given
time bin is given by a perceptron rule. As mentioned
in~\cite{rubin_theory_2010} and in the Introduction, we expect that
this simplification does not qualitatively alter the overall picture
under the sparse regime considered, since even in the
integrate-and-fire model only quasi-simultaneous input spikes affect
the overall depolarization at any given time, due to the fast membrane
decay constant w.r.t.\ the typical inter-spike interval; indeed, our
numerical results (see section \ref{sub:Continuous}) show that it is even
possible to use a time-discretized learning protocol to address the
original continuous-time classification problem.
We thus consider a classification
device with $N$ binary synapses, $J_{i}$ with $i\in\left\{ 1,\dots,N\right\} $,
which has to learn to classify $M=\alpha N$ input patterns. The input
patterns are $N\times K$ matrices (where $K$ here corresponds to the $K^\textrm{discrete}$
discussed in the Introduction) whose elements are $\xi_{it}^{\mu}\in\left\{ 0,1\right\} $
with $i\in\left\{ 1,\dots,N\right\} $, $t\in\left\{ 1,\dots,K\right\} $
and $\mu\in\left\{ 1,\dots,M\right\} $. For each pattern $\mu$ and
at each time step $t$, the device response is given by $V_{t}^{\mu}=\Theta\left(\sum_{i=1}^{N}J_{i}\xi_{it}^{\mu}-\theta\right)$,
where $\Theta\left(x\right)$ is the Heaviside step function and $\theta$
is a threshold; we call the vector $V^{\mu}=\left\{ V_{t}^{\mu}\right\} _{t\in\left\{ 1,\dots,K\right\} }$
the \emph{internal representation} for the pattern $\mu$. Finally,
a pattern $\mu$ is classified according to $s^{\mu}=1-\prod_{t=1}^{K}\left(1-V_{t}^{\mu}\right)$,
i.e.~the overall output $s^{\mu}$ equals $0$ if the internal representation
is a vector of all $0$'s, or it equals $1$ if at least one element
of $V^{\mu}$ is $1$. Each pattern has a desired output $s_{\textrm{exp}}^{\mu}\in\left\{ 0,1\right\} $
which is to be compared to the actual output $s^{\mu}$: the classification
problem is satisfied when $s^{\mu}=s_{\textrm{exp}}^{\mu}$ for all
$\mu$. For notational simplicity, we also define $\sigma^{\mu}=2s^{\mu}-1$
and $\sigma_{\textrm{exp}}^{\mu}=2s_{\textrm{exp}}^{\mu}-1$ when
we need to convert the outputs so that they take values in $\left\{ -1,+1\right\} $.
We studied the case in which all inputs and expected outputs are i.i.d.\ random
variables. We call $f^{\prime}$ the output frequency (i.e.~$s_{\textrm{exp}}^{\mu}=1$
with probability $f^{\prime}$) and $f$ the input frequency (i.e.~$\xi_{it}^{\mu}=1$
with probability $f$). We will assume in the following that $f=\left(1-\left(1-f^{\prime}\right)^{\frac{1}{K}}\right)$:
this ensures that the probability that a vector $\left\{ \xi_{it}^{\mu}\right\} _{t\in\left\{ 1,\dots,K\right\} }$
is composed of all $0$'s is $\left(1-f^{\prime}\right)$, i.e.~has
the same statistics as the internal representations which satisfy
the input/output associations. Clearly, the model reduces to a standard
perceptron when $K=1$.
We assume that $N\gg1$, $K\gg1$ and $N\gg K$; in this case, $f\simeq-K^{-1}\log\left(1-f^{\prime}\right)$,
which shows that the inputs are sparse at large $K$ with our choice
for $f$. For simplicity, all our theoretical results and simulations
will be presented for the case $f^{\prime}=0.5$, which is the value
that maximizes the capacity.
\section{Theoretical analysis}
\subsection{Replica theory}
We studied the device described above within the replica theory, in
a replica symmetric (RS) setting for the internal representations,
in the limit of large $K$, both for the case in which $J_{i}\in\left\{ -1,+1\right\} $
and $J_{i}\in\left\{ 0,1\right\} $, and estimated the entropy, the
critical capacity, the optimal value for the threshold, and studied
the structure of the space of the solutions and the valid internal
representations. The results are almost identical for both $\pm1$
and $0/1$ cases, so in the following we will only specify the model
when a difference arises. We confirmed the results, where possible,
with the cavity method (see section~\ref{sub:Cavity-method}). All
details of the calculations are provided in the Appendix (section~\ref{sec:Appendix:-replica-calculations});
here we summarize the results.
The zero-temperature entropy of the device is defined as
$S\left(\alpha\right) = \frac{1}{N}\left<\log\left(\mathcal{V}\right)\right>$,
where $\mathcal{V}$ is the number of solutions (valid configurations of $J$'s) to the problem associated
with some choice of the patterns $\xi_{it}^\mu$ and their expected outputs $s_{\textrm{exp}}^\mu$ (see also
eq.~\ref{eq:V}), and $\left<\cdot\right>$ denotes the average over the patterns.
$S\left(\alpha\right)$ can't be negative (since the number of valid configurations is an integer number);
the value of $\alpha$ at which $S\left(\alpha\right)$ vanishes is called the \emph{critical capacity},
and represents the typical number of patterns per synapse which can be correctly classified by the device
(i.e.~stored) when the patterns are extracted according to the random i.i.d. distribution which
we are studying.
The RS replica calculation predicts $S\left(\alpha\right) = \log\left(2\right)\left(1-\alpha\right)$,
which interestingly does not depend on $K$ (provided $K \gg 1$). This function
goes to zero at $\alpha_c = 1$, which coincides with the information theoretic
upper bound, i.e.~the device is able to store one bit of information per synapse.
This is in contrast with other related architectures, e.g.~the multi-layer perceptron.
After this point, the entropy is negative, and therefore the RS solution is no longer valid.
The typical value of the overlap between two different solutions to
the same classification problem, defined as $q=\frac{1}{N}\left<\sum_{i=1}^{N}J_{i}^{a}J_{i}^{b}\right>$
for two solutions $\left\{ J_{i}^{a}\right\} _{i\in\left\{ 1,\dots,N\right\} }$
and $\left\{ J_{i}^{b}\right\} _{i\in\left\{ 1,\dots,N\right\} }$,
is constant for all values of $\alpha$, and as low as possible,
i.e.~$q=0$ in the $\pm1$ case and $q=Q^{2}=\nicefrac{1}{4}$
in the $0/1$ case, where $Q$ is the typical fraction of non-null
synapses (which we found to be $Q=\nicefrac{1}{2}$). In terms of
the structure of the space of the solutions, this means that the clusters
of solutions are isolated (point-like).
We expand the threshold in series of $\sqrt{N}$ and write $\theta=\theta_{0}N+\theta_{1}\sqrt{N}$.
The optimal value of $\theta_{0}$ is $0$ in the $\pm1$ case and
$\frac{f}{2}$ for the $0/1$ case; the value of $\theta_{1}$ does
not affect the capacity, but we can set it so that synaptic values
are unbiased:
\begin{equation}
\theta_{1}=-\sqrt{2f\left(1-f\right)}\textrm{erfc}^{-1}\left(2\sqrt[K]{1-f^{\prime}}\right)\label{eq:opt_theta}
\end{equation}
where $\textrm{erfc}^{-1}$ is the inverse of the complementary error
function.
We found that the valid internal representations follow a binomial
distribution at large $K$, i.e.~that the probability distribution
for the value at each time bin is independent of the others. This fact
is in agreement with the continuous model findings
\cite{rubin_theory_2010}, and it is interesting for two reasons: on
one hand, it confirms that different time bins are uncorrelated in the
sparse limit, which is important in order to achieve efficiency in
applying the cavity method. On the other hand, it means that the
distributions of the input and output spike trains are identical (note
that our choice of $f$ only ensures that the all-zero string occur
with the same probability in input and output, but does not imply that
the non-zero strings have the same distribution of the number of
spikes). In turn, this is a necessary condition for recurrent
networks to be built and work under this regime, which would be an
interesting direction for future research.
We also computed how the internal representations are partitioned,
and found that the rescaled entropy of the dominant internal representations
(i.e.~the logarithm of the number of different internal representations
for a pattern which are associated with the largest portions of the
solution space) is given by $\log\left(2\right)\log\left(\frac{K}{\log\left(2\right)}\right)$.
This means that, as $K$ increases, the number of valid dominant internal
representations increases as $K^{M\,\log2}$, while the number of
synaptic states associated with each of them correspondingly shrinks,
so that the overall entropy remains constant.
\subsection{Cavity method\label{sub:Cavity-method}}
The cavity method has been shown~\cite{mezard_space_1989} to provide an alternative
scheme for deriving the results from replica theory in the case of the
binary perceptron with binary $\pm1$ inputs and binary $\pm1$ synapses.
It has also been used on single instances of the learning problem on such
devices (in which case it is known as the Belief Propagation algorithm,
i.e.~BP) to study the space of the solutions for some particular instance
and for deriving heuristic learning algorithms~\cite{braunstein_learning_2006,baldassi_efficient_2007}.
In those studies, the problem is represented as a factor graph in which
synaptic weights are represented by variable nodes, and input patterns
(and their desired outputs) are represented by factor nodes; messages are
exchanged between the two types of nodes along the edges of the graph,
representing marginal probabilities over the states of the variables;
global thermodinamic quantities such as the entropy can be computed from
the messages provided they satisfy the BP equations (which is typically
achieved by reaching a fixed point in an iterative algorithm). Replica theory
results can then be reproduced numerically with BP by averaging the computed
quantities from a large number of samples of sufficient size.
In the case of the present study, however, in which the input patterns take values in
$\left\{0,1\right\}$, the approach used in \cite{braunstein_learning_2006}
can not be applied directly for the sake of reproducing replica theory results,
not even in the perceptron limit $K=1$. This is due to a violation of the underlying
assumption of the cavity method, known as the clustering property, as will be explained
in greater detail at the end of this section, and as a result the standard BP
equations are approximate, rather than becoming asymptotically
exact in the large $N$ regime. Thus, in order to reproduce
the replica theory results, the standard BP equations must be amended;
however, since the heuristic algorithm described in section~\ref{sub:reinBP} is based on
the standard BP, which is simpler and more computationally efficient, and
proves equally effective to the corrected-BP version, we will first derive
the standard BP equations, and describe how to correct them afterwards.
\paragraph{Standard Belief Propagation algorithm\label{par:Standard-BP}}
As mentioned above, messages on the factor graph represent probabilities
over the variable nodes (i.e.~the synaptic weights), and therefore
can be represented by a single real value: as in~\cite{braunstein_learning_2006},
we use average values for this purpose (also called magnetizations).
The BP equations, written in terms of the cavity magnetizations $m$
and $n$, read:
\begin{eqnarray}
m_{i\to\mu} & = & \tanh\left(\sum_{\nu\ne\mu}\tanh^{-1}\left(n_{\nu\to i}\right)\right)\label{eq:m1}\\
n_{\mu\to i} & \propto & P\left(\sigma_{\textrm{exp}}^{\mu}=1-\prod_{t=1}^{K}\Theta\left(\theta-\sum_{j\ne i}J_{j}\xi_{jt}^{\mu}-\xi_{it}^{\mu}\right)\right)+\label{eq:n1}\\
& & -P\left(\sigma_{\textrm{exp}}^{\mu}=1-\prod_{t=1}^{K}\Theta\left(\theta-\sum_{j\ne i}J_{j}\xi_{jt}^{\mu}+\xi_{it}^{\mu}\right)\right)\nonumber
\end{eqnarray}
where $i,j$ are synapse indices and $\mu,\nu$ are pattern indices.
The second equation is the difference between the probabilities that
the pattern $\mu$ is satisfied when synapse $i$ takes the values
$1$ and $-1$, respectively, assuming all other synaptic values are
distributed according to the cavity magnetizations $m_{j\to\mu}$
(for $j\ne i$). These can be computed from the probability that the
internal representation is all zero given the value of $J_{i}$:
\begin{equation}
B_{\mu\to i}\left(J_{i}\right)=\sum_{\left\{ J_{j}\right\} _{j\ne i}}\prod_{j\ne i}\left(\frac{1}{2}+J_{j}\frac{\mu}{2}\right)\prod_{t=1}^{K}\Theta\left(\theta-\sum_{j\ne i}J_{j}\xi_{jt}^{\mu}-J_{i}\xi_{it}^{\mu}\right)\label{eq:B1}
\end{equation}
With this, and using the shorthand notation $B_{\mu\to i}^{\pm}=B_{\mu\to i}\left(\pm1\right)$,
we can write eq.~\ref{eq:n1} as:
\begin{equation}
n_{\mu\to i}=\frac{B_{\mu\to i}^{+}-B_{\mu\to i}^{-}}{B_{\mu\to i}^{+}+B_{\mu\to i}^{-}}\left(1-\frac{2s_{\textrm{exp}}^{\mu}}{2-B_{\mu\to i}^{+}-B_{\mu\to i}^{-}}\right)\label{eq:n2}
\end{equation}
In order to compute efficiently the function $B$, we use the central
limit theorem, which ensures that for large $N$ we have:
\begin{equation}
B_{\mu\to i}\left(J_{i}\right)=\int_{\mathcal{S}_{\mu\to i}\left(J_{i}\right)}\prod_{t=1}^{K}dy_{t}\ \mathcal{N}\left(\bar{y};\bar{a}_{\mu\to i},\bar{\Sigma}_{\mu\to i}\right)\label{eq:B2}
\end{equation}
where $\mathcal{N}\left(\bar{y};\bar{a},\bar{\Sigma}\right)$ is a
$K$-dimensional multivariate Gaussian with mean $\bar{a}$ and covariance
matrix $\bar{\Sigma}$, whose elements are given by:
\begin{eqnarray}
\left(\bar{a}_{\mu\to i}\right)_{t} & = & \sum_{j\ne i}\xi_{jt}^{\mu}m_{j\to\mu}\label{eq:a1}\\
\left(\bar{\Sigma}_{\mu\to i}\right)_{tt^{\prime}} & = & \sum_{j\ne i}\xi_{jt}^{\mu}\xi_{jt^{\prime}}^{\mu}\left(1-m_{j\to\mu}^{2}\right)\label{eq:Sigma1}
\end{eqnarray}
The region of integration is a product of semi-bounded intervals:
$\mathcal{S}_{\mu\to i}\left(J_{i}\right)=\bigotimes_{t=1}^{K}\left(-\infty,\theta-J_{i}\xi_{it}^{\mu}\right]$.
Computing this integral in general is very expensive, and rapidly
becomes infeasible for large $K$. However, the sparsity of the input
patterns implies that diagonal terms are of order $K^{-1}$, while
off-diagonal terms are of order $K^{-2}$ and can be neglected, simplifying
the computation:
\begin{equation}
B_{\mu\to i}\left(J_{i}\right)=\prod_{t=1}^{K}\frac{1}{2}\textrm{\textrm{erfc}}\left(\frac{1}{\sqrt{2}}\left(\frac{\theta-J_{i}\xi_{it}^{\mu}-\left(\bar{a}_{\mu\to i}\right)_{t}}{\left(\bar{\Sigma}_{\mu\to i}\right)_{tt}}\right)\right)\label{eq:B3}
\end{equation}
Equations~\ref{eq:m1},\ref{eq:n2},\ref{eq:a1},\ref{eq:Sigma1}
and~\ref{eq:B3} form a closed system which allows computations to
be performed effectively, and which can conveniently be modified to derive
a heuristic solver algorithm (see section~\ref{sub:reinBP}). However, as stated at the
beginning of this section, these equations fail to exactly reproduce the replica
theory results.
\paragraph{Corrected Belief Propagation algorithm\label{par:Corrected-BP}}
The reason for the failure of the standard BP equations to provide correct
results (e.g.~when computing the entropy) is that when the inputs are unbalanced,
i.e.~they don't average to $0$ (as is necessarily the case when the values are
in $\left\{ 0,1\right\} $), the clustering property, i.e.~the assumption that that
the messages incoming into variable nodes from different factor nodes are uncorrelated,
is violated. This can be seen by considering (see~\cite{mezard_space_1989}):
\begin{eqnarray*}
c_{\mu\nu\to i} & = & \frac{1}{N}\left(\left\langle \left(\bar{a}_{\mu\to i}\right)_{t}\left(\bar{a}_{\nu\to i}\right)_{t}\right\rangle -\left\langle \left(\bar{a}_{\mu\to i}\right)_{t}\right\rangle \left\langle \left(\bar{a}_{\nu\to i}\right)_{t}\right\rangle \right)\\
& = & \frac{1}{N}\left(\left\langle \left(\sum_{j\ne i}\xi_{jt}^{\mu}J_{j}\right)\left(\sum_{j\ne i}\xi_{jt}^{\nu}J_{j}\right)\right\rangle -\left\langle \sum_{j\ne i}\xi_{jt}^{\mu}J_{j}\right\rangle \left\langle \sum_{j\ne i}\xi_{jt}^{\nu}J_{j}\right\rangle \right)\\
& = & \frac{1}{N}\sum_{j\ne i}\xi_{jt}^{\mu}\xi_{jt}^{\nu}\left(1-m_{j\to\mu}m_{j\to\nu}\right)
\end{eqnarray*}
which is $\mathcal{O}\left(1\right)$ unless the average input $\bar{\xi}$
is zero, in which case it is $\mathcal{O}\left(N^{-\frac{1}{2}}\right)$
and becomes negligible. Only in that case, therefore, standard BP equations
become asymptotically correct for large $N$; in all other circumstances,
they only provide an approximation (numerical experiments show that for our
model they sistematically predict a slightly lower entropy than the correct one).
We also note that, if we define $\xi_{it}^{\mu}=\bar{\xi}+\rho_{it}^{\mu}$,
where $\bar{\xi}=f$ and $\rho_{it}^{\mu}\in\left\{ -f,1-f\right\} $
with average $\bar{\rho}=0$, we can split the depolarization as such:
\begin{equation}
\sum_{i}J_{i}\xi_{it}^{\mu} = f\sum_{i}J_{i}+\sum_{i}J_{i}\rho_{it}^{\mu}\equiv f\sqrt{N}T+\sum_{i}J_{i}\rho_{it}^{\mu}
\end{equation}
where we defined the overall magnetization $T=\frac{1}{\sqrt{N}}\sum_{i}J_{i}$.
It becomes apparent that the depolarization distributions induced
by the different patterns are all correlated via $T$, which is a
global quantity. We can however amend the BP algorithm, and derive
correct marginals and therefore correct global thermodynamic quantities,
by studying a related problem, in which this contribution is removed from the
factor nodes and induced by an external field instead; this suggests
the following modification to the cavity equations: we start by choosing
a value for the magnetization, call it $T^{\prime}$, and we consider the
problem with patterns $\rho$ instead of $\xi$, thereby ensuring
that the clustering property holds, and with an additional external field $F$
applied to each variable node, thus modifying eqs.~\ref{eq:m1}
and~\ref{eq:n1} as such:
\begin{eqnarray}
m_{i\to\mu} & = & \tanh\left(\sum_{\nu\ne\mu}\tanh^{-1}\left(n_{\nu\to i}\right) + F\right)\label{eq:m_corr}\\
n_{\mu\to i} & \propto & P\left(\sigma_{\textrm{exp}}^{\mu}=1-\prod_{t=1}^{K}\Theta\left(\theta-\sum_{j\ne i}J_{j}\rho_{jt}^{\mu}-\rho_{it}^{\mu}\right)\right)+\label{eq:n_corr}\\
& & -P\left(\sigma_{\textrm{exp}}^{\mu}=1-\prod_{t=1}^{K}\Theta\left(\theta-\sum_{j\ne i}J_{j}\rho_{jt}^{\mu}+\rho_{it}^{\mu}\right)\right)\nonumber
\end{eqnarray}
The total magnetization $T$ can be obtained from the cavity marginals as:
\begin{equation}
T = \sum_i\tanh\left(\sum_{\mu}\tanh^{-1}\left(n_{\mu\to i}\right) + F\right)\label{eq:T}
\end{equation}
Therefore, we can ensure that, at the fixed point, $T=T^{\prime}$ by just
adding an extra step to the BP iterative process in which $F$ is modified
at each iteration according to the difference $T-T^\prime$.
After convergence, we compute the entropy $S\left(T^{\prime}\right)$,
and via this define $T^{\star}=\textrm{argmax}_{T^{\prime}}S\left(T^{\prime}\right)$.
Then, the marginals computed for the problem defined by $T^{\star}$
are the same as those to the original problem, and are asymptotically
correct (within the RS assumption), allowing us to compute all the
desired properties on a given instance of the original problem via
this modified cavity method. By averaging over many different samples,
we can recover the results of the replica method, as shown for the
entropy curves in fig.~\ref{fig:entropy}.
\begin{figure}
\includegraphics{entroplot}\caption{\label{fig:entropy}Entropy vs. $\alpha$ as computed by the cavity
method at different values of $K$, compared to the one predicted
by the replica theory for $K\gg1$ and $N\gg K$. Each point shows
the average and standard deviation over $10$ random samples with
$N=1000$.}
\end{figure}
\section{Solving single instances: learning algorithms}
\subsection{Reinforced Belief Propagation\label{sub:reinBP}}
Belief propagation equations, in their message passing form over single
problem instances, have repeatedly proven to provide very effective
heuristics when modified
in order to produce optimal configurations \cite{braunstein_encoding_2007,baldassi_efficient_2007,bailly-bechet_finding_2010}.
Two main ways in which this can be achieved are decimation \cite{mezard_analytic_2002,braunstein_survey_2005}
and reinforcement \cite{braunstein_learning_2006}, which can be seen
as a ``soft decimation'' process. In decimation, cycles are performed
which alternate message passing and fixing (or ``freezing'', or
``decimating'') the most polarized free variables, until all variables
are fixed. In reinforcement, the iterative equations have an additional
term which has the role of a time-dependent external field, and which
is computed from the magnetizations obtained at the preceding time
step, so that the system is driven towards a completely polarized
state. Following \cite{braunstein_learning_2006} the reinforced BP
equations are the same as the normal BP equations with one single
difference for eq.~\ref{eq:m1}, which becomes:
\begin{equation}
m_{i\to\mu}^{\tau+1}=\tanh\left(\gamma\left(\tau\right)\tanh^{-1}\left(m_{i}^{\tau}\right)+\sum_{\nu\ne\mu}\tanh^{-1}\left(n_{\nu\to i}^{\tau+1}\right)\right)\label{eq:m2_reinf}
\end{equation}
where $\tau$ is the iteration step, $m_{i}^{\tau}=\tanh\left(\sum_{\nu}\tanh^{-1}\left(n_{\nu\to i}^{\tau}\right)\right)$
is the total magnetization of variable $i$ at iteration $\tau$,
and $\gamma\left(\tau\right)$ is an iteration-dependent reinforcement
parameter, which we set as $\gamma\left(\tau\right)=\gamma_{0}\tau$.
Note that the additional term is proportional to the iteration step,
and therefore dominates for large $\tau$, ensuring that polarization
towards one single configuration is eventually achieved (although
in practice computational problems will arise in difficult or unsatisfiable
situations, due to the limited precision of the floating point representation).
We implemented both the decimation and the reinforcement schemes,
for both the standard version of BP (for which the marginals are
approximate due to correlations between different messages) and the
corrected version (for which marginals are exact, at the cost of increased
computational complexity and running time). Since we did not find
the corrected version to provide any significant advantage over the
na\"ive version (which is not particularly surprising, considering that
the approximation provided by standard BP is rather good, and that the introduction
of the reinforcement term introduces spurious correlations by itself),
here we will only present results for the latter case.
The value of $\gamma_{0}$ is a parameter of the solver algorithm;
higher values of $\gamma_{0}$ make the algorithm greedier, in that
the messages are polarized more quickly but can get trapped into a
non-zero-energy state, while reducing $\gamma_{0}$ improves the accuracy
of the algorithm at the cost of requiring more iterations. In practical
tests, we found that by using eq.~\ref{eq:n2} we were able to reach
values of $\alpha$ as high as $0.7$, but only at the cost of using
extremely small values of $\gamma_{0}$ (of the order of $10^{-7}$
for $N=1000$ and $K=10$), and therefore of an impractically high
computational time.
However, we found heuristically that, by detecting when an excessively
polarized state was reached, and introducing a ``depolarization event''
triggered by such condition, we could achieve the same results with
much higher values of $\gamma_{0}$, and therefore in a much shorter
computational time. More in detail, we introduced, at each iteration
step, a check to detect cases in which any of the terms in the denominator
if eq.~\ref{eq:n2} goes to $0$, indicating a numerical problem
due to excessively polarized magnetizations in a state of non-zero
energy. Whenever this condition is found, we divide all messages $m_{i\to\mu}^{\tau}$
and total magnetizations $m_{i}^{\tau}$ by a positive factor $b$
(thus depolarizing all the messages), and reset $\gamma\left(\tau\right)$
to $0$. In subsequent iterations, we keep increasing $\gamma\left(\tau\right)$
linearly in steps of $\gamma_{0}$ (until another event is detected).
We obtain good results by setting the factor $b$ to $2$ initially,
and increasing it by one at every invocation of this additional depolarization
rule. Indeed, since $\gamma\left(\tau\right)$ does not increase monotonically
any more in this scheme, this modified algorithm will not be guaranteed
to polarize towards a single state, unless the state itself has zero
energy and therefore represents a solution to the problem.
Fig.~\ref{fig:bprein} shows the performance of this algorithm for
$N=1000$ and $K=10$. Setting $10000$ as the maximum number of iterations,
a critical capacity of almost $0.8$ is achieved.
\begin{figure}
\includegraphics[scale=0.6]{bpreinplot}
\caption{\label{fig:bprein}Number of iterations until solving (green curve)
and solving probability (red curve) for different values of $\alpha$,
for a tempotron device with $N=1000$ and $K=10$, with the reinforced
BP algorithm. The parameter $\gamma$ was set to $0.01$. For each
point, $10$ samples were used. The number of iterations was capped
at $10000$.}
\end{figure}
\subsection{Simplified BP-inspired scheme\label{sub:BPI}}
As for the case of the simple perceptron \cite{braunstein_learning_2006,baldassi_efficient_2007,baldassi_generalization_2009},
it is possible to drastically simplify the reinforced BP equations
(in a purely heuristic way), and obtain an online algorithm which,
when parameters are set to their optimal values, proves to be almost
as effective at learning as reinforced BP itself, while dramatically
reducing computational requirements.
This algorithm requires a hidden state $h_{i}$ to be endowed with
each synapse. This hidden state can only assume odd integer values,
and is capped by a maximum absolute value $h^{\max}$, so that each
synapse has a total of $h^{\max}+1$ hidden states. The hidden state
and the synaptic weight $J_{i}$ are related by the simple expression
$J_{i}=\textrm{sign}\left(h_{i}\right)$. In all simulations, we set
the initial state of the $h_{i}$ states by randomly drawing values from
$\left\{ -1,1\right\} $.
The learning protocol turns out to be as follows: patterns $\xi^{\mu}$
are presented in random order to the device, computing the depolarization
$\Delta_{t}^{\mu}=\left(\sum_{i=1}^{N}J_{i}\xi_{it}^{\mu}-\theta\right)$;
from this, we determine $t^{\star}=\textrm{argmax}_{t}\Delta_{t}^{\mu}$
and compute $\Phi^{\mu}=\sigma_{\exp}^{\mu}\Delta_{t^{\star}}^{\mu}$;
depending on the value of $\Phi^{\mu}$, we choose one of three actions:
\begin{description}
\item [{$\Phi^{\mu}>1$}]: do nothing
\item [{$0<\Phi^{\mu}\le1$}]: with probability $r$, update synapses for
which $\xi_{it^{\star}}^{\mu}=1$ and $J_{i}=\sigma_{\exp}^{\mu}$;
with probability $\left(1-r\right)$ do nothing
\item [{$\Phi^{\mu}\le0$}]: update all synapses for which $\xi_{it^{\star}}^{\mu}=1$
\end{description}
The synaptic update rule is always of this form:
\[
h_{i}\to h_{i}+2\sigma_{\exp}^{\mu}
\]
which implies that synaptic values $J_{i}$ are only updated in the
$\Phi^{\mu}<0$ case, and only if $\xi_{it^{\star}}^{\mu}=1$ and
$h_{i}=-\sigma_{\exp}^{\mu}$. As stated above, we impose $-h^{\max}\le h_{i}\le h^{\max}$,
so that the update rule is not applied when $h_{i}=h^{\max}\sigma_{\exp}^{\mu}$.
The probability $r$ of taking an action in the ``barely correct''
case $0\le\Phi^{\mu}\le1$ is a parameter of the algorithm, just as
$h^{\max}$. We determined empirically the optimal values of $r$
and $h^{\max}$ for different values of $N$ and $K$ by extensively
testing the space of the parameters. Our results, shown in fig.~\ref{fig:simplestats},
indicate that $r=0.4$ works best for all values (we explored the
values of $r$ in steps of $0.05)$, and that the optimal value of
$h^{\max}$ is reasonably well fitted by a function $\lambda\sqrt{\nicefrac{N}{K}}$,
where $\lambda=2.06\pm0.02$. The capacity decreases with $N$, for
fixed $K$, but does not seem to tend to $0$ asymptotically (see
inset in fig.~\ref{fig:simplestats}).
\begin{figure}
\includegraphics[scale=0.6]{simpleoptplot}
\caption{\label{fig:simplestats}Optimal critical capacity (left) and optimal
value of the number of hidden states $h^{\max}+1$ (right) for various
values of $N$ and $K$. Critical capacity is defined as the value
of $\alpha$ for which the probability of successfully solving the
problem in $10000$ iterations or less is $0.5$. Optimal $h^{\max}$
is defined as the value of $h^{\max}$ which yields the highest critical
capacity. The parameter $r$ is set to $0.4$ in all plots shown here,
since that was found to be the optimal value independently of other
parameters. At least $40$ random samples were generated for each
combination of $\left(N,\alpha,K,h^{\max},r\right)$ in order to determine
the success probability and therefore the critical capacity. $\alpha$
was explored in steps of $0.05$. The inset in the left panel shows
the critical capacity as a function of $\nicefrac{1}{N}$ for $K=10$;
the solid black line is a fit by an exponential function. The solid
black lines in the right panel show the fit of $h^{\max}+1$ as a
function of $N$ and $K$ via $\lambda\sqrt{\nicefrac{N}{K}}$.}
\end{figure}
\subsection{Generality of the discrete-time model\label{sub:Continuous}}
As a way to verify that the model and learning protocols which we
studied are relevant in a more biologically realistic setting, we
adapted the simplified BP-inspired scheme described in the previous
section to address the continuous-time classification problem (see
Introduction): for a given an instance of the problem, we discretize
the time in $K$ bins, apply the BP-inspired learning protocol
(slightly modified to use continuous inputs), and test the resulting
synaptic weights assignments on the continuous device (see the
Appendix for details,
section~\ref{sec:Appendix:-time-discretization}). We found that in a
device with $N=1000$ synapses, with time constants $t_m=10ms$ and
$t_s=2.5ms$, tested on $T=500ms$ long input patterns discretized in
$50$ bins, this scheme can achieve a classification error lower than
1\% up to $\alpha=0.4$, demonstatrating that indeed under these
conditions not much relevant information is typically lost in the time
discretization process, and that the proposed time-discretized
learning protocol can be effective even in a continuous-time setting.
\section{Conclusions}
We have presented a theoretical analysis of the computational
performance of the tempotron model with discretized time and discrete
synaptic weights. The results show that the device is able to learn
random spatio-temporal patterns at a learning rate which saturates the
information theoretic bounds.
In addition to this, and possibly of more practical interest, we have
been able to derive some novel learning protocols which are local
and distributed and do not rely on a gradient descent process on the synaptic
weights. These algorithms are based on the message-passing method
and extend previous works on rate-coding networks. Specifically, we
have shown that the message-passing algorithms can store spatio-temporal
patterns at very high loads and that even some extremely simplified
versions are still able to store an extensive number of patterns
efficiently. Furthermore, we showed that the discretized-time
algorithm can even be adapted to effectively address the original,
continuous-time version of the problem. Our approach can be applied to
both discrete and continuous synapses.
Many open problems remain to be studied, starting from how these protocols
can be made even simpler in a biologically plausible modeling context.
Still we believe that at least as far as artificial neural systems
is concerned these results could find direct application in neuromorphic
devices.
\section{Appendix: statistical physics analysis and replica calculations\label{sec:Appendix:-replica-calculations}}
\subsection{Entropy\label{sub:Entropy}}
We will consider the case in which synaptic weights take values in
$\left\{ -1,1\right\} $ first.
The volume of the space of the solutions for a given instantiation
of the patterns can be written as:
\begin{equation}
\mathcal{V}=\sum_{\left\{ J_{i}\right\} _{i}}\sum_{\left\{ \tau_{t}^{\mu}\right\} _{\mu t}}\prod_{\mu}\chi\left(s_{\textrm{exp}}^{\mu},\left\{ \tau_{t}^{\mu}\right\} _{t}\right)\prod_{\mu t}\Theta\left(\tau_{t}^{\mu}\left(\frac{1}{\sqrt{N}}\sum_{i}J_{i}\xi_{it}^{\mu}-\frac{\theta}{\sqrt{N}}\right)\right)\label{eq:V}
\end{equation}
where index $i\in\left\{ 1,\dots,N\right\} $ is used for synapses,
index $t\in\left\{ 1,\dots,K\right\} $ is used for time bins, index
$\mu\in\left\{ 1,\dots,\alpha N\right\} $ is used for patterns, the
auxiliary variables $\tau_{t}^{\mu}\in\left\{ -1,1\right\} $ are
the internal representations (they are equal to $2V_{t}^{\mu}-1$,
see section~\ref{sec:Introduction}), and$\chi\left(s,\left\{ \tau_{t}\right\} _{t}\right)=\Theta\left(s-\left(1-\prod_{t=1}^{K}\left(1-\frac{1}{2}\left(1+\tau_{t}^{\mu}\right)\right)\right)\right)$
is a characteristic function ensuring that the internal representation
$\tau_{t}$ is compatible with the output $s$.
From here on, for simplicity, we will omit the subscript $\textrm{exp}$
from the outputs $s^{\mu}$.
In order to compute the entropy, we need to compute the quenched average
$\left\langle \log\mathcal{V}\right\rangle _{\xi,s}$; we do this
by using the replica trick:\cite{braunstein_learning_2006,baldassi_efficient_2007}
\begin{equation}
\left\langle \log\mathcal{V}\right\rangle _{\xi,s}=\lim_{n\to0}\frac{\left\langle \mathcal{V}^{n}\right\rangle _{\xi,s}-1}{n}
\end{equation}
where we compute $\left\langle \mathcal{V}^{n}\right\rangle _{\xi,s}$
for integer values of $n$, and use the analytic continuation to compute
the limit $n\to0$. The average over the replicated volume is:
\begin{equation}
\left\langle \mathcal{V}^{n}\right\rangle _{\xi,s}=\left\langle \sum_{\left\{ J_{i}^{a}\right\} _{ia}}\sum_{\left\{ \tau_{t}^{\mu a}\right\} _{\mu ta}}\prod_{\mu a}\chi\left(s^{\mu},\left\{ \tau_{t}^{\mu a}\right\} _{t}\right)\prod_{\mu ta}\Theta\left(\tau_{t}^{\mu a}\left(\frac{1}{\sqrt{N}}\sum_{i}J_{i}^{a}\xi_{it}^{\mu}-\frac{\theta}{\sqrt{N}}\right)\right)\right\rangle _{\xi,s}\label{eq:Vol^n}
\end{equation}
We used the index $a\in\left\{ 1,\dots,n\right\} $ to denote the
replica. We can now use the integral representation of the $\Theta$
function, $\Theta\left(y\right)=\int_{-\infty}^{\infty}\frac{dx}{2\pi}\int_{0}^{\infty}d\lambda\, e^{ix\left(\lambda-y\right)}$,
and compute the average over the input patterns, using their independence
and the $N\gg1$ limit (in the following, all integrals are assumed
to be on $\left[-\infty,\infty\right]$ unless otherwise specified)
:
\begin{eqnarray}
& & \left\langle \prod_{a}\Theta\left(\tau_{t}^{\mu a}\left(\frac{1}{\sqrt{N}}\sum_{i}J_{i}^{a}\xi_{it}^{\mu}-\frac{\theta}{\sqrt{N}}\right)\right)\right\rangle _{\xi}=\\
& & \quad=\int\prod_{a}\frac{dx_{t}^{\mu a}}{2\pi}\int_{0}^{\infty}\prod_{a}d\lambda_{t}^{\mu a}\,\prod_{a}\exp\left(ix_{t}^{\mu a}\left(\lambda_{t}^{\mu a}-\tau_{t}^{\mu a}\left(\bar{\xi}\frac{1}{\sqrt{N}}\sum_{i}J_{i}^{a}-\frac{\theta}{\sqrt{N}}\right)\right)\right)\cdot\nonumber \\
& & \quad\qquad\cdot\exp\left(-\frac{v_{\xi}}{2N}\sum_{a,b}\tau_{t}^{\mu a}\tau_{t}^{\mu b}x_{t}^{\mu a}x_{t}^{\mu b}\sum_{i}J_{i}^{a}J_{i}^{b}\right)\nonumber
\end{eqnarray}
where $\bar{\xi}=f$ and $v_{\xi}=f\left(1-f\right)$ are the average
value and the variance of the inputs $\xi_{it}^{\mu}$, respectively.
We then introduce order parameters $q^{ab}=\frac{1}{N}\sum_{i}J_{i}^{a}J_{i}^{b}$
and $T^{a}=-\sqrt{N}\bar{J}+\frac{1}{\sqrt{N}}\sum_{i}J_{i}^{a}$
via Dirac-delta functions, and their conjugates $\hat{q}^{ab}$,$\hat{T}^{a}$
via integral expansion of the deltas, and get:
\begin{eqnarray}
\left\langle \mathcal{V}^{n}\right\rangle _{\xi,s} & = & \int\prod_{a}\frac{dT^{a}d\hat{T^{a}}\sqrt{N}}{2\pi}\int\prod_{a\ge b}\frac{dq^{ab}d\hat{q}^{ab}N}{2\pi}\exp\left(\sqrt{N}\sum_{a}\left(T^{a}+\sqrt{N}\bar{J}\right)\hat{T}^{a}-N\sum_{a\ge b}q^{ab}\hat{q}^{ab}\right)\cdot\\
& & \cdot\left(\sum_{\left\{ J_{i}^{a}\right\} _{ia}}\prod_{i}\exp\left(\sum_{a\ge b}\hat{q}^{ab}J_{i}^{a}J_{i}^{b}-\sum_{a}\hat{T}^{a}J_{i}^{a}\right)\right)\cdot\nonumber \\
& & \cdot\left\langle \sum_{\left\{ \tau_{t}^{\mu a}\right\} _{\mu ta}}\prod_{\mu a}\chi\left(s^{\mu},\left\{ \tau_{t}^{\mu a}\right\} _{t}\right)\right.\cdot\nonumber \\
& & \quad\cdot\prod_{\mu t}\left(\int\prod_{a}\frac{dx_{t}^{\mu a}}{2\pi}\int_{0}^{\infty}\prod_{a}d\lambda_{t}^{\mu a}\,\prod_{a}\exp\left(ix_{t}^{\mu a}\left(\lambda_{t}^{\mu a}-\tau_{t}^{\mu a}\left(\bar{\xi}\left(T^{a}+\sqrt{N}\bar{J}\right)-\frac{\theta}{\sqrt{N}}\right)\right)\right)\right.\cdot\nonumber \\
& & \quad\left.\left.\cdot\exp\left(-\frac{v_{\xi}}{2}\sum_{a,b}\tau_{t}^{\mu a}\tau_{t}^{\mu b}x_{t}^{\mu a}x_{t}^{\mu b}q^{ab}\right)\right)\right\rangle _{s}\nonumber \\
& = & \int\prod_{a}\frac{dT^{a}d\hat{T^{a}}\sqrt{N}}{2\pi}\int\prod_{a\ge b}\frac{dq^{ab}d\hat{q}^{ab}N}{2\pi}\exp\left(\sqrt{N}\sum_{a}\left(T^{a}+\sqrt{N}\bar{J}\right)\hat{T}^{a}-N\sum_{a\ge b}q^{ab}\hat{q}^{ab}\right)\cdot\nonumber \\
& & \cdot\left(\sum_{\left\{ J^{a}\right\} _{a}}\exp\left(\sum_{a\ge b}\hat{q}^{ab}J^{a}J^{b}-\sum_{a}\hat{T}^{a}J^{a}\right)\right)^{N}\cdot\nonumber \\
& & \cdot\left\langle \sum_{\left\{ \tau_{t}^{a}\right\} _{ta}}\prod_{a}\chi\left(s,\left\{ \tau_{t}^{a}\right\} _{t}\right)\right.\cdot\nonumber \\
& & \quad\cdot\prod_{t}\left(\int\prod_{a}\frac{dx_{t}^{a}}{2\pi}\int_{0}^{\infty}\prod_{a}d\lambda_{t}^{a}\,\prod_{a}\exp\left(ix_{t}^{a}\left(\lambda_{t}^{a}-\tau_{t}^{a}\left(\bar{\xi}\left(T^{a}+\sqrt{N}\bar{J}\right)-\frac{\theta}{\sqrt{N}}\right)\right)\right)\right.\cdot\nonumber \\
& & \quad\left.\left.\cdot\exp\left(-\frac{v_{\xi}}{2}\sum_{a,b}\tau_{t}^{a}\tau_{t}^{b}x_{t}^{a}x_{t}^{b}q^{ab}\right)\right)\right\rangle _{s}^{\alpha N}\nonumber
\end{eqnarray}
where in the second step we dropped indices $i$ and $\mu$. We expand
the threshold $\theta$ in series of $\sqrt{N}$:
\begin{equation}
\theta=N\theta_{0}+\sqrt{N}\theta_{1}
\end{equation}
from which we immediately get the relation:
\begin{equation}
\bar{J}=\frac{\theta_{0}}{\bar{\xi}}
\end{equation}
This leaves us with:
\begin{eqnarray}
\left\langle \mathcal{V}^{n}\right\rangle _{\xi,s} & = & \int\prod_{a}\frac{dT^{a}d\hat{T^{a}}\sqrt{N}}{2\pi}\int\prod_{a\ge b}\frac{dq^{ab}d\hat{q}^{ab}N}{2\pi}\cdot\\
& & \cdot\exp\left(\sqrt{N}\sum_{a}T^{a}\hat{T}^{a}+N\bar{J}\sum_{a}\hat{T}^{a}-N\sum_{a\ge b}q^{ab}\hat{q}^{ab}\right)\cdot\nonumber \\
& & \cdot\left(\sum_{\left\{ J^{a}\right\} _{a}}\exp\left(\sum_{a\ge b}\hat{q}^{ab}J^{a}J^{b}-\sum_{a}\hat{T}^{a}J^{a}\right)\right)^{N}\cdot\nonumber \\
& & \cdot\left\langle \sum_{\left\{ \tau_{t}^{a}\right\} _{ta}}\prod_{a}\chi\left(s,\left\{ \tau_{t}^{a}\right\} _{t}\right)\prod_{t}\left(\int\prod_{a}\frac{dx_{t}^{a}}{2\pi}\int_{0}^{\infty}\prod_{a}d\lambda_{t}^{a}\,\prod_{a}\exp\left(ix_{t}^{a}\left(\lambda_{t}^{a}-\tau_{t}^{a}\left(\bar{\xi}T^{a}-\theta_{1}\right)\right)\right)\right.\right.\cdot\nonumber \\
& & \quad\left.\left.\cdot\exp\left(-\frac{v_{\xi}}{2}\sum_{a,b}\tau_{t}^{a}\tau_{t}^{b}x_{t}^{a}x_{t}^{b}q^{ab}\right)\right)\right\rangle _{s}^{\alpha N}\nonumber
\end{eqnarray}
In the $N\gg1$ limit, this integral can be computed by the saddle
point method: we introduce the RS Ansatz for the solution: $T^{a}=T\,\forall a$,
$q^{ab}=q\,\forall a,b:a\ne b$, $q^{aa}=Q\,\forall a$, and analogous
expressions for the conjugate parameters. Therefore:
\begin{eqnarray}
\left\langle \mathcal{V}^{n}\right\rangle _{\xi,s} & = & \exp\left(N\bar{J}\hat{T}+N\frac{n}{2}\hat{q}q-Nn\hat{Q}Q\right)\cdot\\
& & \cdot\left(\sum_{\left\{ J^{a}\right\} _{a}}\exp\left(\frac{\hat{q}}{2}\left(\sum_{a}J^{a}\right)^{2}-\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)\sum_{a}\left(J^{a}\right)^{2}-\hat{T}\sum_{a}J^{a}\right)\right)^{N}\cdot\nonumber \\
& & \cdot\left\langle \sum_{\left\{ \tau_{t}^{a}\right\} _{ta}}\prod_{a}\chi\left(s,\left\{ \tau_{t}^{a}\right\} _{t}\right)\prod_{t}\left(\int\prod_{a}\frac{dx_{t}^{a}}{2\pi}\int_{0}^{\infty}\prod_{a}d\lambda_{t}^{a}\,\prod_{a}\exp\left(ix_{t}^{a}\left(\lambda_{t}^{a}-\tau_{t}^{a}\left(\bar{\xi}T-\theta_{1}\right)\right)\right)\right.\right.\cdot\nonumber \\
& & \quad\left.\left.\cdot\exp\left(-\frac{v_{\xi}}{2}\left(q\left(\sum_{a}\tau_{t}^{a}x_{t}^{a}\right)^{2}+\left(Q-q\right)\sum_{a}\left(x_{t}^{a}\right)^{2}\right)\right)\right)\right\rangle _{s}^{\alpha N}\nonumber \\
& = & \exp\left(N\bar{J}\hat{T}+N\frac{n}{2}\hat{q}q-Nn\hat{Q}Q\right)\cdot\nonumber \\
& & \cdot\left(\int Du\left(\sum_{\left\{ J\right\} }\exp\left(-\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)J^{2}+\left(\sqrt{\hat{q}}u-\hat{T}\right)J\right)\right)^{n}\right)^{N}\cdot\nonumber \\
& & \cdot\left(\int\prod_{t}Du_{t}\left\langle \left(\sum_{\left\{ \tau_{t}\right\} _{t}}\chi\left(s,\left\{ \tau_{t}\right\} _{t}\right)\prod_{t}\left(\int\frac{dx_{t}}{2\pi}\int_{0}^{\infty}d\lambda_{t}\,\exp\left(-\frac{v_{\xi}}{2}\left(Q-q\right)\left(x_{t}\right)^{2}\right)\right.\right.\right.\right.\cdot\nonumber \\
& & \quad\left.\left.\left.\left.\cdot\exp\left(ix_{t}\left(\lambda_{t}-\tau_{t}\left(\bar{\xi}T-\theta_{1}-\sqrt{v_{\xi}q}u_{t}\right)\right)\right)\right)\right)^{n}\right\rangle _{s}\right)^{\alpha N}\nonumber \\
& = & \exp\, Nn\left(\bar{J}\hat{T}+\frac{1}{2}\hat{q}q-\hat{Q}Q\right.+\nonumber \\
& & \qquad+\int Du\,\log\left(\sum_{\left\{ J\right\} }\exp\left(-\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)J^{2}+\left(\sqrt{\hat{q}}u-\hat{T}\right)J\right)\right)\nonumber \\
& & \qquad+\left.\alpha\int\prod_{t}Du_{t}\left\langle \log\left(\sum_{\left\{ \tau_{t}\right\} _{t}}\chi\left(s,\left\{ \tau_{t}\right\} _{t}\right)\prod_{t}H\left(-\tau_{t}\frac{\bar{\xi}T-\theta_{1}-\sqrt{v_{\xi}q}u_{t}}{\sqrt{v_{\xi}\left(Q-q\right)}}\right)\right)\right\rangle _{s}\right)\nonumber
\end{eqnarray}
where in the second step we introduced auxiliary Gaussian integrals
(we use the shorthand notation $Du=du\frac{1}{\sqrt{2}\pi}e^{-\frac{u^{2}}{2}}$
and define $H\left(x\right)=\int_x^\infty\ Dy$),
which allows to drop the replica index $a$, and in the last step
we used the $n\to0$ limit. Finally, we obtain the expression for
the entropy:
\begin{eqnarray}
\mathcal{S}=\frac{1}{N}\left\langle \log\mathcal{V}\right\rangle _{\xi,s} & = & \bar{J}\hat{T}+\frac{1}{2}\hat{q}q-\hat{Q}Q+\mathcal{Z}_{J}\left(\hat{Q},\hat{q},\hat{T}\right)+\mathcal{Z}_{S}\left(Q,q,T\right)\\
\mathcal{Z}_{J}\left(\hat{Q},\hat{q},\hat{T}\right) & = & \int Du\,\log\left(\sum_{\left\{ J\right\} }\exp\left(-\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)J^{2}+\left(\sqrt{\hat{q}}u-\hat{T}\right)J\right)\right)\\
\mathcal{Z}_{S}\left(Q,q,T\right) & = & \alpha\int\prod_{t}Du_{t}\left\langle \log\left(\sum_{\left\{ \tau_{t}\right\} _{t}}\chi\left(s,\left\{ \tau_{t}\right\} _{t}\right)\prod_{t}H\left(-\tau_{t}\,\eta\left(u_{t},Q,q,T\right)\right)\right)\right\rangle \\
\eta\left(u,Q,q,T\right) & = & \frac{\bar{\xi}T-\theta_{1}-\sqrt{v_{\xi}q}u}{\sqrt{v_{\xi}\left(Q-q\right)}}\label{eq:eta}
\end{eqnarray}
The expression for $\mathcal{Z}_{J}$ is the familiar expression for
perceptron models, and it can be written more explicitly for the two
cases $J\in\left\{ -1,1\right\} $ and $J\in\left\{ 0,1\right\} $:
\begin{eqnarray}
\mathcal{Z}_{J}^{\pm}\left(\hat{Q},\hat{q},\hat{T}\right) & = & -\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)+\int Du\,\log\left(2\cosh\left(\sqrt{\hat{q}}u-\hat{T}\right)\right)\\
\mathcal{Z}_{J}^{01}\left(\hat{Q},\hat{q},\hat{T}\right) & = & \int Du\,\log\left(1+\exp\left(-\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)+\sqrt{\hat{q}}u-\hat{T}\right)\right)
\end{eqnarray}
The expression for $\mathcal{Z}_{S}$ can be manipulated further:
\begin{eqnarray}
\mathcal{Z}_{S}\left(Q,q,T\right) & = & \alpha\left(1-f^{\prime}\right)K\int Du\log H\left(\eta\left(u,Q,q,T\right)\right)+\\
& & +\alpha f^{\prime}\int\prod_{t}Du_{t}\log\left(1-\prod_{t}H\left(\eta\left(u_{t},Q,q,T\right)\right)\right)
\end{eqnarray}
In the limit of $K\gg1$, we can use the central limit theorem and
keep only the higher order terms in $K$, and obtain:
\begin{equation}
\mathcal{Z}_{S}\left(Q,q,T\right)=\alpha\left(\left(1-f^{\prime}\right)K\Lambda\left(Q,q,T\right)+f^{\prime}\log\left(1-\exp\left(K\Lambda\left(Q,q,T\right)\right)\right)\right)
\end{equation}
where we defined:
\begin{equation}
\Lambda\left(Q,q,T\right)=\int Du\log H\left(\eta\left(u,Q,q,T\right)\right)\label{eq:lambdadef}
\end{equation}
The saddle point equation for $T$ gives:
\[
0=\frac{\partial\mathcal{Z}_{S}}{\partial T}=\alpha K\left(1-f^{\prime}\frac{1}{1-e^{K\Lambda}}\right)\frac{\partial\Lambda\left(Q,q,T\right)}{\partial T}
\]
which implies:
\begin{equation}
\Lambda\left(Q,q,T\right)=\frac{1}{K}\log\left(1-f^{\prime}\right)\label{eq:Lambda}
\end{equation}
This in turn puts to zero $\hat{q}$ and $\hat{Q}$:
\begin{eqnarray*}
\hat{q} & = & -2\frac{\partial\mathcal{Z}_{S}}{\partial q}=-2\alpha K\left(1-f^{\prime}\frac{1}{1-e^{K\Lambda}}\right)\frac{\partial\Lambda\left(Q,q,T\right)}{\partial q}=0\\
\hat{Q} & = & \frac{\partial\mathcal{Z}_{S}}{\partial Q}=\alpha K\left(1-f^{\prime}\frac{1}{1-e^{K\Lambda}}\right)\frac{\partial\Lambda\left(Q,q,T\right)}{\partial Q}=0
\end{eqnarray*}
Optimizing with respect to $\theta_{0}$, i.e.~imposing $\frac{\partial S}{\partial\bar{J}}=0$,
we also get $\hat{T}=0$.
The remaining equations are different for the cases $\pm1$ and $01$.
For the $\pm1$ case:
\begin{eqnarray}
q & = & -2\frac{\partial\mathcal{Z}_{J}}{\partial\hat{q}}=1-\frac{1}{\sqrt{\hat{q}}}\int Du\, u\,\tanh\left(\sqrt{\hat{q}}u-\hat{T}\right)\\
Q & = & \frac{\partial\mathcal{Z}_{J}}{\partial\hat{Q}}=1\\
\bar{J} & = & -\frac{\partial\mathcal{Z}_{J}}{\partial\hat{T}}=\int Du\,\tanh\left(\sqrt{\hat{q}}u-\hat{T}\right)
\end{eqnarray}
The result $Q=1$ is obvious. From $\hat{T}=0$ and $\hat{q}=0$, and
since $\bar{\xi}\neq0$, we get $q=0$ and $\theta_{0}=0$.
For the $01$ case:\cite{braunstein_learning_2006,baldassi_efficient_2007}
\begin{eqnarray}
q & = & -2\frac{\partial\mathcal{Z}_{J}}{\partial\hat{q}}=\int Du\,\frac{1}{1+\exp\left(\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)-\sqrt{\hat{q}}u+\hat{T}\right)}\left(1-\frac{u}{\sqrt{\hat{q}}}\right)\\
Q & = & \frac{\partial\mathcal{Z}_{J}}{\partial\hat{Q}}=\int Du\,\frac{1}{1+\exp\left(\frac{1}{2}\left(\hat{q}-2\hat{Q}\right)-\sqrt{\hat{q}}u+\hat{T}\right)}\\
\bar{J} & = & -\frac{\partial\mathcal{Z}_{J}}{\partial\hat{T}}=Q
\end{eqnarray}
From $\hat{q}=0$ and $\hat{Q}=0$ these simplify to:
\begin{eqnarray*}
Q & = & \bar{J}=\frac{1}{1+e^{\hat{T}}}=\frac{1}{2}\\
q & = & \frac{1}{\left(1+e^{\hat{T}}\right)^{2}}=Q^{2}=\frac{1}{4}\\
\theta_{0} & = & \frac{f}{2}
\end{eqnarray*}
From $q=Q^{2}$ we see that the cross-overlap is as low as possible,
like in the $\pm1$ case: the physical interpretation is that clusters
of solution are isolated, i.e.~point-like.
The only remaining order parameters are $T$ and $\theta_{1}$, which
are related by eq. \ref{eq:Lambda} and give:
\begin{equation}
T=\frac{1}{f}\left(\theta_{1}+\sqrt{2f\left(1-f\right)}\textrm{erfc}^{-1}\left(2\sqrt[K]{1-f^{\prime}}\right)\right)
\end{equation}
Therefore, in order to have unbiased synapses, i.e.~$T=0$, we may
set $\theta_{1}$ to:
\begin{equation}
\theta_{1}=-\sqrt{2f\left(1-f\right)}\textrm{erfc}^{-1}\left(2\sqrt[K]{1-f^{\prime}}\right)\label{eq:theta_1}
\end{equation}
With our choice for the distribution of the inputs, $f=1-\sqrt[K]{1-f}$,
this formula starts from $0$ at $K=1$, has a maximum for $K=6$
and slowly decreases (as $\sqrt{\frac{\log\left(K\right)}{K}}$) to
$0$ as $K$ diverges; the reason for this behaviour is that there
are two competing tendencies at work as $K$ increases: on one hand,
the increase in the length of the internal representation while $f^{\prime}$
is kept constant requires that more and more individual bins fall
below threshold; on the other hand, the sparsification of the inputs
reduces the fluctuations in the depolarization; this second contribution
dominates for large $K$ and so the threshold goes to $0$, but for
practical purposes (i.e.~for biologically relevant values of $K$)
it does not become negligible.
From the above results, we can determine the entropy:
\begin{eqnarray}
\mathcal{S} & = & \log\left(2\right)-\alpha\left(\left(1-f^{\prime}\right)\log\left(1-f^{\prime}\right)+f^{\prime}\log\left(f^{\prime}\right)\right)
\end{eqnarray}
which goes to $0$ at:
\begin{equation}
\alpha_{c}=\left(1-f^{\prime}\right)\log_{2}\left(1-f^{\prime}\right)+f^{\prime}\log_{2}\left(f^{\prime}\right)
\end{equation}
which coincides with the information theoretic upper bound.
\subsection{Distribution of output spikes}
The probability distribution of the number of output spikes (i.e.~$1$'s
in the internal representation) can be obtained by taking the ratio
between the volume of the solution space in which one pattern is restricted
to produce $Y$ spikes and the total volume:
\begin{eqnarray}
P\left(Y\right) & = & \frac{1}{\mathcal{V}}\left\langle \sum_{\left\{ \tau_{t}^{\mu}\right\} _{\mu l}}\delta_{k}\left(\sum_{t}\left(\frac{1+\tau_{t}^{1}}{2}\right),Y\right)\right.\cdot\\
& & \quad\cdot\left.\sum_{\left\{ J_{i}\right\} _{i}}\prod_{\mu}\chi\left(s^{\mu},\left\{ \tau_{t}^{\mu}\right\} _{t}\right)\prod_{\mu t}\Theta\left(\tau_{t}^{\mu}\left(\frac{1}{\sqrt{N}}\sum_{i}J_{i}\xi_{it}^{\mu}-\frac{\theta}{\sqrt{N}}\right)\right)\right\rangle _{\xi,s}\nonumber
\end{eqnarray}
where $\delta_{k}\left(x,y\right)$ is the Kronecker delta function:
\[
\delta_{k}\left(x,y\right)=\begin{cases}
1 & \textrm{if}\, x=y\\
0 & \textrm{otherwise}
\end{cases}
\]
We can write $\mathcal{V}^{-1}=\lim_{n\to0}\mathcal{V}^{n-1}$, restrict
ourselves to integer $n$ and obtain an expression almost identical
to eq.~\ref{eq:Vol^n}, except for the Kronecker delta affecting
pattern $\mu=1$:
\begin{eqnarray}
P\left(Y\right) & = & \left\langle \sum_{\left\{ \tau_{t}^{\mu a}\right\} _{\mu ta}}\delta_{k}\left(\sum_{t}\left(\frac{1+\tau_{t}^{1a}}{2}\right)-Y\right)\right.\cdot\\
& & \quad\cdot\left.\sum_{\left\{ J_{i}^{a}\right\} _{ia}}\prod_{\mu a}\chi\left(s^{\mu},\left\{ \tau_{t}^{\mu a}\right\} _{t}\right)\prod_{\mu ta}\Theta\left(\tau_{t}^{\mu a}\left(\frac{1}{\sqrt{N}}\sum_{i}J_{i}^{a}\xi_{it}^{\mu}-\frac{\theta}{\sqrt{N}}\right)\right)\right\rangle _{\xi,s}\nonumber
\end{eqnarray}
The computation follows the one for the entropy; the only affected
term is $\mathcal{Z}_{S}$, and only one term survives the limit $n\to0$,
giving:
\begin{equation}
P\left(Y\right)=\delta_{k}\left(Y,0\right)\left(1-f^{\prime}\right)+\left(1-\delta_{k}\left(Y,0\right)\right)f^{\prime}\int\prod_{t}Du_{t}\frac{\binom{K}{Y}\prod_{t=1}^{Y}H^{+}\left(u_{t}\right)\prod_{t=Y+1}^{K}H^{-}\left(u_{t}\right)}{1-\prod_{t=1}^{K}H^{-}\left(u_{t}\right)}
\end{equation}
where we wrote $H^{\pm}\left(u\right)=H\left(\mp\eta\left(u,Q,q,t\right)\right)$
(see eq.~\ref{eq:eta}) for short.
For large $K$ and finite $Y$, this approximates to:
\begin{eqnarray}
P\left(Y\right) & = & \delta_{k}\left(Y,0\right)\left(1-f^{\prime}\right)+\left(1-\delta_{k}\left(Y,0\right)\right)f^{\prime}\binom{K}{Y}\frac{e^{\left(K-Y\right)\Lambda}}{1-e^{K\Lambda}}\left(1-e^{\Lambda}\right)^{Y}\\
& = & \binom{K}{Y}\left(e^{\Lambda}\right)^{K-Y}\left(1-e^{\Lambda}\right)^{Y}\nonumber
\end{eqnarray}
where $\Lambda=\Lambda\left(Q,q,T\right)$ (see eq.~\ref{eq:lambdadef}), and in the second step
we used eq.~\ref{eq:Lambda} from the saddle point solution. The
result is a binomial distribution, in which the probability of producing
a spike is $1-e^{\Lambda}=1-\sqrt[K]{1-f^{\prime}}$, which is our
choice for the input frequency $f$.
\subsection{Structure of the internal representations}
We can study the structure of the space of the internal representations
by following \cite{monasson_weight_1996,cocco_analytical_1996}: we
consider the volume of each internal representation $\mathcal{V}_{\mathcal{T}}$,
where $\mathcal{T}=\left\{ \tau_{t}^{\mu}\right\} _{\mu t}$ is an
internal representation, such that the overall volume can be written
as $\mathcal{V}=\sum_{\mathcal{T}}\mathcal{V}_{\mathcal{T}}$; then
we define:
\begin{equation}
\mathcal{V}\left(r\right)=\sum_{\mathcal{T}}\left(\mathcal{V_{\mathcal{T}}}\right)^{r}
\end{equation}
and study the free energy defined by:
\begin{equation}
g\left(r\right)=-\frac{\left\langle \log\left(\mathcal{V}\left(r\right)\right)\right\rangle }{Nr}
\end{equation}
which, once known, allows to derive the size of the internal representations
from the quantity
\begin{equation}
w\left(r\right)=\frac{\partial}{\partial r}\left(-rg\left(r\right)\right)
\end{equation}
(for $r=1$, $w\left(1\right)=\frac{1}{N}\log\mathcal{V}_{\mathcal{T}}^{\star}$
where $\mathcal{V}_{\mathcal{T}}^{\star}$ is the typical volume of
the dominant internal representations), and their number from the
micro-canonical entropy
\begin{equation}
\mathcal{N}\left(r\right)=-\frac{\partial}{\partial\nicefrac{1}{r}}g\left(r\right)
\end{equation}
(for $r=1$, $\mathcal{N}\left(1\right)$ is the logarithm of the
typical number of internal representation of size $\mathcal{V}_{\mathcal{T}}^{\star}$,
divided by $N$).
The computation is performed by using the replica trick for $r$ integer
and then performing an analytic continuation:
\begin{equation}
\left\langle \mathcal{V}\left(r\right)^{n}\right\rangle =\left\langle \sum_{\left\{ \tau_{t}^{\mu a}\right\} _{\mu ta}}\sum_{\left\{ J_{i}^{a\nu}\right\} _{ia\nu}}\prod_{\mu a}\chi\left(s^{\mu},\left\{ \tau_{t}^{\mu a}\right\} _{t}\right)\prod_{\mu ta\nu}\Theta\left(\tau_{t}^{\mu a}\left(\frac{1}{\sqrt{N}}\sum_{i}J_{i}^{a\nu}\xi_{it}^{\mu}-\frac{\theta}{\sqrt{N}}\right)\right)\right\rangle _{\xi,s}
\end{equation}
where we introduced the new internal representation replica index
$\nu\in\left\{ 1,\dots,r\right\} $. The computation follows the steps
of the entropy computation of section~\ref{sub:Entropy}, but requires
the introduction of order parameters with 2 replica indices; in particular
$q^{a\nu,b\phi}=\frac{1}{N}\sum_{i}J_{i}^{a\nu}J_{i}^{b\phi}$, which
in the RS Anzatz can take 3 values:
\[
q^{a\nu,b\phi}=\begin{cases}
Q & \textrm{if}\, a=b,\nu=\phi\\
q_{1} & \textrm{if}\, a=b,\nu\neq\phi\\
q_{0} & \textrm{if}\, a\neq b
\end{cases}
\]
We obtain, for large $K$:
\begin{eqnarray}
g\left(r\right) & = & -\bar{J}\hat{T}-\frac{1}{2}rq_{0}\hat{q}_{0}+\frac{r-1}{2}q_{1}\hat{q}_{1}+\hat{Q}Q-\frac{1}{r}\mathcal{Z}_{J}+\frac{1}{r}\mathcal{Z}_{S}\\
\mathcal{Z}_{J} & = & \int Du\,\log\left(\int Dz\left(\sum_{\left\{ J\right\} }e^{\frac{1}{2}\left(2\hat{Q}-\hat{q}_{1}\right)J^{2}+\left(\sqrt{\hat{q}_{0}}u+\sqrt{\hat{q}_{1}-\hat{q}_{0}}z-\hat{T}\right)J}\right)^{r}\right)\\
\mathcal{Z}_{S} & = & \alpha\left(\left(1-f^{\prime}\right)K\Lambda\left(Q,q_{1},q_{0},T\right)+f^{\prime}\log\left(e^{K\Phi\left(Q,q_{1},q_{0},T\right)}-e^{K\Lambda\left(Q,q_{1},q_{0},T\right)}\right)\right)\\
\Lambda\left(Q,q_{1},q_{0},T\right) & = & \int Du\,\log\left(\int Dz\, H\left(\eta\left(u,z,Q,q_{1},q_{0},T\right)\right)^{r}\right)\\
\Phi\left(Q,q_{1},q_{0},T\right) & = & \int Du\,\log\left(\int Dz\, H\left(-\eta\left(u,z,Q,q_{1},q_{0},T\right)\right)^{r}+H\left(\eta\left(u,z,Q,q_{1},q_{0},T\right)\right)^{r}\right)\hspace{1em}\\
\eta\left(u,z,Q,q_{1},q_{0},T\right) & = & -\frac{u\sqrt{v_{\xi}q_{0}}-z\sqrt{v_{\xi}\left(q_{1}-q_{0}\right)}+\theta_{1}-T\xi}{\sqrt{v_{\xi}\left(Q-q_{1}\right)}}
\end{eqnarray}
The saddle point equations for $r=1$ give the same results as before,
as expected; in particular, we find $q_{0}=q_{1}=0$ in the $\pm1$
case and $q_{0}=q_{1}=\nicefrac{1}{4}$ in the $01$ case, and $g\left(1\right)=-\mathcal{S}$
as expected. Furthermore, we have:
\begin{equation}
\left.\frac{\partial\mathcal{Z_{S}}}{\partial r}\right|_{r=1}=\alpha K\int Du\int Dz\left(H^{+}\left(u,z\right)\log H^{+}\left(u,z\right)+H^{-}\left(u,z\right)\log H^{-}\left(u,z\right)\right)
\end{equation}
where we used the shorthand notation $H^{\pm}\left(u,z\right)=H\left(\mp\eta\left(u,z,Q,q_{1},q_{0},T\right)\right)$.
From this and from the saddle point equations at $r=1$, in particular
from eq.~\ref{eq:Lambda}, we obtain the weight and the entropy of
the dominant internal representations for $f^{\prime}=\nicefrac{1}{2}$:
\begin{eqnarray}
w\left(1\right) & = & \left.\frac{\partial}{\partial r}\left(-rg\left(r\right)\right)\right|_{r=1}=\log2\left(-1+\alpha+\alpha\log K-\alpha\log\log2\right)\\
\mathcal{N}\left(1\right) & = & \left.-\frac{\partial}{\partial\nicefrac{1}{r}}g\left(r\right)\right|_{r=1}=-\alpha\left(\log\log2-\log K\right)\log2
\end{eqnarray}
From these, we can find the leading terms of the number of different
dominant internal representations:
\begin{equation}
e^{N\mathcal{N}\left(1\right)}=\left(\frac{K}{\log2}\right)^{N\alpha\log2}
\end{equation}
and their volume:
\begin{equation}
e^{-Nw\left(1\right)}=2^{N\left(1-\alpha\right)}\left(\frac{\log2}{K}\right)^{N\alpha\log2}
\end{equation}
\section{Appendix: time discretization\label{sec:Appendix:-time-discretization}}
\subsection{Modified BP-inspired learning scheme for continuous inputs}
The learning protocol presented in section~\ref{sub:BPI} can be easily
generalized to the case in which the input patterns $\xi_{it}^\mu$ are
not binary, but positive and continuous: the only required change is
that the update rules, rather then being applied only to those
synapses for which $\xi_{it^\star}^\mu=1$, are applied to all synapses
with probability $p_i^\mu=\min\left(\xi_{it^\star}^\mu, 1\right)$. Therefore,
the actions taken upon determining $t^\star$ and the value
$\Phi^\mu$ are:
\begin{description}
\item [{$\Phi^{\mu}>1$}]: do nothing
\item [{$0<\Phi^{\mu}\le1$}]: with probability $r$, update synapses for
which $J_{i}=\sigma_{\exp}^{\mu}$, each with probability $p_i^\mu$;
with probability $\left(1-r\right)$ do nothing
\item [{$\Phi^{\mu}\le0$}]: update all synapses, each with probability $p_i^{\mu}$
\end{description}
In order for this generalization to be effective without furher
modifications of the algorithm, it is crucial that a normalization
step is applied to the inputs (see next section).
As an additional generalization, we also introduce a robustness
parameter $\rho$ and re-define
$\Phi^{\mu}=\sigma_{\exp}^{\mu}\Delta_{t^{\star}}^{\mu} - \rho \theta$,
where $\theta$ is the firing threshold: this forces the learning algorithm
to seek solutions in which the depolarization is far from the
threshold. In the numerical experiments described in section~\ref{sub:Continuous}
we used the value $\rho=0.2$, increasing it from $0$ in steps of $0.01$ for $1000$
iterations at each step; the other parameters of the model used in those
tests were $N=1000$, $K=50$, $h_{\max}=25$ and $r=0.3$.
\subsection{Pattern time-discretization}
In this section we describe the time-discretization process mentioned
in section~\ref{sub:Continuous}: we consider a continuous-time model
as described in the Introduction; then, for any given input spike
train, we compute the post-synaptic-potential trace
$R_i^\mu\left(t\right)=\sum_{t_{i}^\mu<t}v\left(t-t_{i}^\mu\right)$,
where $v\left(t\right)$ is the temporal kernel of the membrane. We
divide the time window $T$ in $K$ equal bins, and for each bin $k$ we
compute the input $\xi_{ik}^\mu$ as the fraction of the membrane
kernel in that bin:
\begin{equation}
\xi_{ik}^\mu = \frac{1}{t_m-t_s} \int_{k-\textrm{th}\ \textrm{bin}}dt\ R_i^\mu\left(t\right)
\end{equation}
where we used the fact that $\int_0^\infty dt\ v\left(t\right)=t_m-t_s$
with out choice of $v$.
Note that the resulting $\xi_{ik}^\mu$ can be greater than $1$, but this is rare
under the sparsity regime which we considered.
The time-discretized patterns can then be passed to the discrete algorithm for
deriving a vector of synaptic weights, which in turn can be tested on the
original model. In our numerical experiments, we generated input spike
trains by a Poisson process with a rate chosen as to obtain
the correct value of the input frequency $f$ (see section~\ref{par:Model})
after the discretization in $K$ time bins. When testing the solution,
we used the value of the firing threshold for the continuous unit which
gave the lowest number of errors.
\bibliographystyle{iopart-num}
\providecommand{\newblock}{}
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{
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}
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As we wrap up 2018 and look to the new year, we wanted to share a number of the milestones and news you might have missed this past year. We've gathered a collection highlighting our research and engagement projects, and patient and other stakeholder perspectives on our work, as told through blogs, videos, events, and more. We think this look back will give you a sense of why we're so proud of the work we and our awardees have done this past year and why we're so energized about an even more productive year ahead. We hope you will be, too.
It would be impossible to capture all the high-impact results from our funded studies that were reported on our website and in major medical journals in 2018, so we're highlighting a few major themes.
People with serious mental illnesses are more at risk than the general population for preventable medical conditions, and often don't get the basic care they need to address them. As a result, they have more health problems and die 10 to 20 years earlier. This feature story on four PCORI-funded projects shows how they are using stakeholder input to test ways to help people with serious mental illness get the physical health care they need.
One project, led by James Schuster, MD, MBA, at the University of Pittsburgh, compared two ways to help those with serious mental illness who receive mental health care at behavioral health homes—not a residence, but a team-based, patient-centered form of care that integrates primary and behavioral care—stay on top of their care. For more on his team's promising results, and how they hope to disseminate them, see our Periscope from February below.
Yesterday, Dr. Schuster discussed our research into an innovative health care model that aims to improve the lives of those with serious mental illness. https://t.co/lS5qaxB5jl @PCORI #spreadinnovation
— UPMC Health Plan (@UPMCHealthPlan) February 7, 2018
Telehealth was another area of great interest in 2018 and will continue to be. As the use of digital communication tools expands sharply, there is a growing sense that telehealth can help people better manage their health and improve access to care. But a number of questions remain, including how to reach populations with limited access to technology or who need culturally tailored interventions. PCORI has funded numerous studies—including on hepatitis B infection, childhood hearing loss, and severe mental illness—that look to address those issues, and others.
We highlighted two different stakeholder perspectives from our funded telehealth projects in a breakout session at our 2018 Annual Meeting. This session examined the potential of each study to change practice and what needs to be done to speed telehealth's adoption.
We also hosted a Facebook Live conversation with PCORI-funded researcher Dror Ben-Zeev, PhD, and a stakeholder partner on that project, Mark Ishaug, MA, about how telehealth can help improve outcomes for people living with serious mental illness.
And we continued our work to address the nationwide opioid crisis. In October, we cohosted a briefing—featuring two awardees and Sen. Bill Cassidy, MD (R-LA)—on how to address the epidemic of inappropriate opioid use in the United States. One of the awardees, Beth Darnall, PhD, who is studying how well alternatives to opioids might work in managing pain, also authored an op-ed in The Hill on this topic.
.@BethDarnall @StanfordPain describes her PCORI project about chronic pain self-management. #opioidsbriefing https://t.co/SuUydITd4e
— PCORI (@PCORI) October 2, 2018
Stakeholders' Perspectives in Their Own Words
As you know, PCORI is unique in requiring our awardees to include patients and other healthcare stakeholders in their research, from study design through dissemination of results. We were pleased to highlight their perspectives many times in 2018 on how that work has changed their lives.
In one case, we highlighted a study involving the largest cohort of transgender people in research to date. The research team reported in Annals of Internal Medicine that transgender women who receive estrogen treatment may face a higher risk for stroke and dangerous blood clots than previously thought. In a guest blog post, the project lead and a patient partner in the study framed the context and importance of the findings.
Different PCORI stakeholders discuss their perspectives on the value of patient-centered outcomes research (PCOR).
If you attended our Annual Meeting, we hope you got to hear from our opening keynote speaker (and if you didn't, you can now), Amy Berman, about how she lives a full life with stage IV cancer, prioritizing treatments that align with outcomes that matter most to her, while working full-time at the John A. Hartford Foundation. She shared more of her story in a guest blog.
Finally, in another guest blog, the National Organization of Rare Disorders' Vanessa Boulanger, MSc, explained the importance of patient-centered research for people with rare diseases. We continued our focus on research to improve care for those with rare diseases—conditions that affect fewer than 200,000 people nationally. Up to 30 million Americans have a rare disease and most conditions are poorly understood.
The Power of Engagement in Research
Given our commitment to engaging patients and other stakeholders in everything we do, we were thrilled to be included just weeks ago in an international set of authors of an editorial in The BMJ about the importance of evaluating engagement—both to show best practices for conducting engagement and also to show skeptics why engagement is worthwhile.
We also are proud to have launched our Engagement in Health Literature Explorer, a searchable list of articles related to engagement in health research. Without a standard language for describing engagement and what it involves, articles like these previously had been hard to find.
We posted many other pieces about engagement throughout the year, including a blog with tips for recruiting and engaging patients from traditionally hard-to-reach populations.
Why is patient and stakeholder engagement in research important? pic.twitter.com/LJtfWCetEW
— PCORI (@PCORI) July 9, 2018
PCORI's Impact Reflected in Media
Our stakeholder communities and major scientific journals weren't the only ones paying attention this past year to our work and our funded projects. The media did, too.
We counted more than 800 mentions of PCORI-funded studies and other initiatives in a range of media outlets in 2018, including major national news media like the The New York Times, The Washington Post and The Wall Street Journal; broadcast outlets like NBC and NPR; major digital services like Vox, Morning Consult and STAT; news services like Reuters, The Associated Press and Kaiser Health News; specialty, trade and professional outlets like Medscape, MedPage Today, Modern Healthcare, Scientific American; and Washington-focused policy outlets like POLITICO and The Hill.
We Have Much More to Do
By year's end, we had awarded more than $2.3 billion in comparative clinical effectiveness research and related projects. We continued to focus on stakeholder-guided efforts to disseminate and promote the uptake of the results of our funded studies. We were pleased to see so many of those studies produce results summarized for professional and consumer audiences on our website and reported in leading medical journals. And we look forward to more such milestones—and others—in 2019.
But none of these achievements, and others we've reported to you over time, would be possible without the involvement and guidance the stakeholder communities we serve. So, thanks for your help. We look forward to continuing to work with you in the year ahead.
I am very glad to see wisely
Abdiqani Ainan January 3, 2019 1:38 am
I am very glad to see wisely how PCORI grapples to work for improving health and fitness with its dynamic multi-disciplinary team by generating innovative ways and clinical advances of tackling conundrum maladies and developing current standard care .
Thank you for all the work
Jacqueline Baker January 3, 2019 9:27 am
Thank you for all the work you have done in research this past year for 2018. I would still like to see more research in sickle cell disease. Many patients continue to suffer and sadly pass away from this painful disease. Many community based organizations, of which I'm a part of, continue to work hard so sickle cell disease patients lives won't be compromised by this disease. I'm a parent advocate for the sickle cell community in Queens, Brooklyn, and the Bronx. I have two sons with sickle cell disease.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,522
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Goodsell House is a historic home located at Old Forge in Herkimer County, New York. It was built in 1899 and is a -story, wood-frame vernacular Queen Anne–style house with a gable ell. The main block is over a limestone foundation. Also on the property is a -story carriage house/garage and an ice house. It is operated as a local history museum for the town of Webb.
It was listed on the National Register of Historic Places in 2006.
The Town of Webb Historical Association operates the Goodsell Museum as a museum of local history.
References
External links
Town of Webb Historical Association - official site
Houses on the National Register of Historic Places in New York (state)
History museums in New York (state)
Queen Anne architecture in New York (state)
Houses completed in 1899
Museums in Herkimer County, New York
Houses in Herkimer County, New York
National Register of Historic Places in Herkimer County, New York
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{
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/climateecology.wordpress.com\/tag\/science\/","text":"# Ecological Dynamic Programming Optimization in\u00a0Python\n\nIt\u2019s been awhile since I\u2019ve posted anything. I\u2019ll use the excuse that I\u2019ve been busy, but mostly I just forget. Regardless, I\u2019m learning how to do Dynamic Programming Optimization (DPO), which sounds more complex than it is. The reason for this is that DPO allows us to simulate the behavior of individuals who make decisions based on current patch quality.\u00a0 DPO is an exciting tool that forms the foundation of individual-based models, which allow us to assess community and ecosystem dynamics as emergent properties of individual decisions based on well-grounded, physiological principles (hence my interest).The underlying idea, as I understand it, is that individuals assess their future reproductive output prior to making a decision (I\u2019m interested in optimal foraging, so I\u2019ll put this in the context of foraging). We can model how individuals make decisions over the course of a lifetime, and track each individual, which then allows us to make quantitative statements about populations, communities, or ecosystems.\n\nThis all sounds complicated, and it can be difficult. So it\u2019s easiest to jump right in with an example. Here\u2019s the code, along with a basic explanation of what\u2019s going on, for the very first toy example found in \u201cDynamic State Variable Models in Ecology\u201d by Clark and Mangel. In the book, they give the computer code in TRUEBASIC, but\u2026 in all honesty.. no one I know uses that. I could use R, but we all know my feelings on that. So here\u2019s their toy model programmed and annotated in Python.\n\nBackground\n\nSuppose we\u2019re interested in the behavior of fish and how they choose to distribute themselves among three different patches. Two patches are foraging patches and one is a reproduction patch. The first thing we need to do is make an objective fitness function F. In this example,\u00a0F relates body size to reproductive output. It can be thought of as what the organism believes about its final fitness given a particular body size. Let\u2019s make it increasing, but asymptotic, with body mass:\n\n$F = \\frac{A(x-x_c)}{x-x_c+x_0}$\n\nHere, A is sort of a saturation constant beyond which fitness increases minimally with body size, $x_c$ is the body size at which mortality occurs (0), and $x_0$ is the initial body size. Lets set $x_c=0$, $A=60$ and $x_0=0.25*x_{max}$, where $x_{max}=30$ is the individuals maximum possible body size. You have to set the ceiling on body size otherwise organisms grow without bounds.\n\nOK so that\u2019s what the organisms \u201cbelieves\u201d about its fitness at the end of a season given a specific body mass. The question is:\u00a0How does the organism forage optimally to maximize fitness at the end of the its lifetime?\n\nThe obvious answer would be to simulate a foraging decision at every time step moving forward, and then have it decide again at the new time step, etc. This is computationally expensive, so to circumvent this we work backwards from the end of the season. This saves time because there are far fewer paths to a known destination than there are to an unknown one (essentially).\n\nSo we imagine that, at the final time step $t_f$, the organism\u2019s fitness is given by\u00a0F for any given body mass.\n\nimport numpy as np\nimport matplotlib.pyplot as pet\nimport pandas as pd\n\nn_bodysize = 31 # 31 body sizes, from 0-30\nn_time = 20 # 20 time steps, 1-20\nn_patch = 3, # 3 foraging patches\n\n# make a container for body size (0-30) and time (1-20)\nF = np.zeros((n_bodysize, n_time))\n\n# make the function F\nx_crit = 0\nx_max = 30\nacap = 60\nx_0 = 0.25*x_max\nt_max = 20\n\n# calculate organism fitness at the final time point\nfor x in range(x_crit, x_max+1):\nF[x,-1] = acap*(x-x_crit)\/(x-x_crit+x_0)\n[\/code]\n\nNow that we know fitness at the final time point, we can iterate through backwards (called\u00a0backwards simulation) to decide what the optimal strategy is to achieve each body mass. To determine that, we need to know the fitness in each patch. Let\u2019s start with the two foraging patches, Patch 1 and Patch 2.\u00a0We need to know four things about each patch: (1) the mortality rate in each patch (m), (2) the probability of finding food in each patch (p), (3) the metabolic cost of visiting a patch (a), and (4) the gain in body mass if food is successfully found (y). For these two patches, let:\n\nPatch 1: m=0.01, p=0.2, a=1, y=2\n\nPatch 2: m=0.2, p=0.5, a=1, y=4\n\nRight away we can see that Patch 2 is high-risk, high reward compared to Patch 1. In each patch, we calculate the next body size given that an animal does (x\u2018)\u00a0or does not (x\u201d)\u00a0find food:\n\n$x' = x-a_i+y_i$\n\n$x'' = x-a_i$\n\nThose are simple equations. Body size is the current body size minus the metabolic cost of foraging in the patch and, if successful, the energy gain from foraging. Great. Now we can calculate the expected fitness of each patch as the weighted average of F\u2018 and F\u201d given the probability of finding food, all times the probability of actually surviving.\u00a0For these two patches, we make a fitness function (V):\n\n$V_i = (1 - m_i)*[p_i*F_{t+1}(x') + (1-p_i)*F'_{t+1}(x'')]$\n\nThe reproductive patch is different. There is no foraging that occurs in the reproductive patch. Instead, if the organism is above a critical mass $x_{rep}$, then it devotes all excess energy to reproduction to a limit\u00a0 (c=4). If the organism is below the reproductive threshold and still visits the foraging patch, it simply loses mass (unless it dies).\n\nOK this is all kind of complicated, so let\u2019s step through it. We know what fitness is at the final time step because of\u00a0F.\u00a0So let\u2019s step back one time step. At this penultimate time step, we go through every body mass and calculate fitness for each patch.\u00a0Let\u2019s do an example. If x=15, then we need to know fitness in Patch 1, Patch 2, and Patch 3. For Patches 1 and 2, we need to know the weight gain if successful and the weight gain if unsuccessful.\n\n$x' = max(15-a_1+y_1, x_{max})$\n\n$x' = 15-1+2 = 16$\n\n$x''=min(15-a_1, x_{c})$\n\n$x'' = 15-1 = 14$\n\nThe min and max functions here just make sure our organism doesn\u2019t grow without limit and dies if metabolic cost exceeds body mass.\u00a0So these are now the two potential outcomes of foraging in Patch 1 given x=15. The expected fitness of these two body masses is given as\u00a0F(16)\u00a0and\u00a0F(14). Plug all these values into equation\u00a0V for Patch 1 to get the expected fitness of Patch 1 at a body size of 15. We then take the maximum\u00a0for all three Patches, save whichever Patch corresponds to that\u00a0as the optimal foraging decision, and then save\u00a0as the fitness for body size 15 at that time step. So at body size 15,\u00a0for Patch 1 is \u00a039,\u00a0for Patch 2 is 38, and\u00a0for Patch 3 is 37.6, so the individual will foraging in Patch 1 and now fitness for body size 15 at this time step is 39.\n\nRepeat this procedure for every possible body size, and you\u2019ll get the fitness for every body size at the second to last time step as well as the optimal foraging patch for every body size at that time.\n\nThen, step backwards in time. Repeat this whole procedure, except now the\u00a0value for each body mass doesn\u2019t come from the equation\u00a0F, but comes from the fitness we just calculated for each body mass. So for example, if an organism\u2019s foraging decisions at this time step lead it to a body mass of 15, then\u00a0F is now 39. Again, repeat this for every body mass, and then step back, etc.\n\nHere\u2019s the full Python code for how this is done:\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport pandas as pd\nimport seaborn as sns\n\nn_bodysize = 31 # 31 body sizes, 0 - 30\nn_time = 20 # 20 time steps, 1 - 20\nn_patch = 3 # number of patches\n\n#%% CONATINERS\n# make a container for body size (0-30) and time (1-20)\nF = np.zeros((n_bodysize, n_time))\n# make a container for three patches, each with body size (0-30) and time (1-20)\nVi = np.zeros((n_patch, n_bodysize, n_time))\n# make a container for the optimal decision\nd = np.zeros((n_bodysize, n_time))\n\n# make a container for the mortality rates in each patch\nm = np.zeros(n_patch)\n# make a container for the probability of finding food in each path\np = np.zeros(n_patch)\n# make a container for the metabolic cost\na = np.zeros(n_patch)\n# make a container for food gain in each patch\ny = np.zeros(n_patch)\n# make a container for reproductive output in each patch\nc = np.zeros(n_patch)\n\n#%% CONDITIONS\nx_crit = 0\nx_max = 30\nt_max = 20\nx_rep = 4\n\n#%% PARAMETERS\nm[0] = 0.01; m[1] = 0.05; m[2] = 0.02\np[0] = 0.2; p[1] = 0.5; p[2] = 0\na[0] = 1; a[1] = 1; a[2] = 1\ny[0] = 2; y[1] = 4; y[2] = 0\nc[0] = 0; c[1] = 0; c[2] = 4\n\n#%% END CONDITION\nacap = 60\nx_0 = 0.25*x_max\n\n# Calculate Fitness for every body mass at the final time step\nfor x in range(x_crit, x_max+1):\nF[x,-1] = acap*(x-x_crit)\/(x-x_crit+x_0) # maximum reproductive output for each body size at the final time\n\n#%% SOLVER\nfor t in range(0, t_max-1)[::-1]: # for every time step, working backward in time\u00a0#print t\nfor x in range(x_crit+1, x_max+1): # iterate over every body size\u00a0# print x\nfor patch in range(0, 3): # for every patch\nif patch in [0,1]: # if in a foraging patch\nxp = x-a[patch]+y[patch] # the updated body size\nxp = min(xp, x_max) # constraint on max size\nxpp = int(x-a[patch]) # updated body size if no food\nVi[patch,x,t] = (1 - m[patch])*(p[patch]*F[int(xp),t+1] + (1-p[patch])*F[xpp, t+1]) # Calculate expected fitness for foraging in that patch\nelse:\nif x < x_rep: # in a reproduction patch\nxp = max(x-a[patch], x_crit)\nVi[patch, x, t] = (1-m[patch])*F[int(xp), t+1]\nelse:\nfitness_increment = min(x-x_rep, c[patch]) # resources devoted to reproduction\nxp = max(x-a[patch]-fitness_increment, x_crit) # new body size is body size minus metabolism minus reproduction\nVi[patch, x, t] = fitness_increment + (1-m[patch])*F[int(xp),t+1]\nvmax = max(Vi[0,x,t], Vi[1,x,t])\nvmax = max(vmax, Vi[2,x,t])\nif vmax==Vi[0,x,t]:\nd[x,t] = 1\nelif vmax==Vi[1,x,t]:\nd[x,t] = 2\nelif vmax==Vi[2,x,t]:\nd[x,t] = 3\nF[x,t] = max # the expected fitness at this time step for this body mass\n[\/code]\n\nThis model doesn\u2019t really track individual behavior. What it does is provides an optimal decision for every time and every body mass. So we know what, say, an individual should do if it finds itself small towards the end of its life, or if it finds itself large at the beginning:\n\nT,X = np.meshgrid(range(1, t_max+1), range(x_crit, x_max+1))\ndf = pd.DataFrame({'X': X.ravel(), 'T': T.ravel(), 'd': d.ravel()})\ndf = df.pivot('X', 'T', 'd')\nsns.heatmap(df, vmin=1, vmax=3, annot=True, cbar_kws={'ticks': [1,2,3]})\nplt.gca().invert_yaxis()\nplt.show()\n[\/code]\n\nAnd that\u2019s that!! To be honest, I still haven\u2019t wrapped my head around this fully. I wrote this blog post in part to make myself think harder about what was going on, rather than just regurgitating code from the book.\n\n# My Ideal Python Setup for Statistical\u00a0Computing\n\nI\u2019m moving more and more towards Python only (if I\u2019m not there already). So I\u2019ve spent a good deal of time getting the ideal Python IDE setup going. One of the biggest reasons I was slow to move away from R is that R has the excellent RStudio IDE. Python has Spyder, which is comparable, but seems sluggish compared to RStudio.\u00a0I\u2019ve tried PyCharm, which works well, but I had issues with their interactive interpreter running my STAN models.\n\nA friend pointed me towards SublimeText 3, and I have to say that it\u2019s everything I wanted. The text editor is slick, fast, and has lots of great functions. But more than that, the add-ons are really what make Sublime shine\n\n\u2022 Side Bar Enhancements: This extends the side-bar project organizer, allowing you to add folders and files, delete things, copy paths, etc. A must have.\n\u2022 SublimeREPL: Adds interactive interpreters for an enormous number of languages, both R and Python included. Impossible to work without.\n\u2022 \u00a0Anaconda: An AMAZING package that extends Sublime by offering live Python linting to make sure my code isn\u2019t screwed up, PEP8 formatters for those of you who like such things, and built in documentation and code retrieval, for those times you\u2019ve forgotten how the function works. Another must have.\n\u2022 SublimeGIT: For working with github straight from Sublime. Great if you\u2019re doing any sort of module building.\n\u2022 Origami: A new way to split layouts and organize your screen. Not essential, but helpful\n\u2022 Bracket Highlighter: Helpful for seeing just what set of parentheses I\u2019m working in.\n\nSublime and all of these packages are also incredibly customizable, you can make them work and look however you want. I\u2019ve spent a few days customizing my setup and I think its fairly solid. Here are my preferences:\n\nFor the main Sublime, I modified the scrolling map, turned off autocomplete (which I find annoying but can still access with Ctrl+space, adjusted the carat so I could actually see it, changed the font, and a few other odds and ends.\n\n{\n\"always_show_minimap_viewport\": true,\n\"auto_complete\": false,\n\"bold_folder_labels\": true,\n\"caret_style\": &quot;phase&quot;,\n\"color_scheme\": \"Packages\/Theme - Flatland\/Flatland Dark.tmTheme\",\n\"draw_minimap_border\": true,\n\"font_face\": \"Deja San Mono\",\n\"font_size\": 14,\n\"highlight_line\": true,\n\"highlight_modified_tabs\": true,\n\"ignored_packages\":\n[\n\"Vintage\";\n],\n\"preview_on_click\": false,\n\"spell_check\": true,\n\"wide_caret\": true,\n}\n\nFor Bracket Highlighter, I changed the style of the highlight:\n\n{\n\"high_visibility_enabled_by_default\": true,\n\"high_visibility_style\": \"thin_underline\",\n\"high_visibility_color\": \"__default__\",\n}\n\nFor Side-Bar Enhancements, I\u2019ve modified the \u2018Open With\u2019 options. For Anaconda, I changed a few small things and turned off PEP8 linting, which I hate. I don\u2019t hate linting nor PEP8, but I don\u2019t have much use for PEP8 linting constantly telling me that I put a space somewhere inappropriate.\n\n{\n\"complete_parameters\": true,\n\"complete_all_parameters\": false,\n\"anaconda_linter_mark_style\": \"outline\",\n\"pep8\": false,\n\"anaconda_gutter_theme\": \"basic\",\n\"anaconda_linter_delay\": 0.5,\n}\n\nI also installed the Flatland Theme to make it pretty. Here is the end result, also showing the Anaconda documentation viewer that I find so awesome:\n\nI also now use Sublime for all of my R, knitr, and LaTeX work as well. In all, it\u2019s a pretty phenomenal editor that can do everything I need it to and combines at least four separate applications into one (TextWrangler, Spyder, RStudio, TexShop). Now, some day I\u2019ll be able to afford the $70 to turn off that reminder that I haven\u2019t paid (and$15 for LaTeXing).\n\nUPDATE\n\nI forgot to mention snippets. You can create snippets in Sublime that are shortcuts for longer code. For example, I heavily customize my graphs in the same way every time. Instead of typing all the code, I can now just type tplt followed by a tab and I automatically get:\n\nf, ax = plt.subplots()\nax.plot()\n#ax.set_ylim([ , ])\n#ax.set_xlim([ , ])\nax.set_ylabel(&quot;ylab&quot;)\nax.set_xlabel(&quot;xlab&quot;)\nax.spines['top'].set_visible(False)\nax.spines['right'].set_visible(False)\nax.spines['bottom'].set_position(('outward', 10))\n#ax.spines['bottom'].set_bounds()\nax.spines['left'].set_position(('outward', 10))\n#ax.spines['left'].set_bounds()\nax.yaxis.set_ticks_position('left')\nax.xaxis.set_ticks_position('bottom')\nplt.savefig(,bbox_inches = 'tight')\nplt.show()\n\nGreat if you rewrite the same code many times.\n\n# PyStan: A Second Intermediate Tutorial of Bayesian Analysis in\u00a0Python\n\nI promised a while ago that I\u2019d give a more advanced tutorial of using PySTAN and Python to fit a Bayesian hierarchical model. Well, I\u2019ve been waiting for a while because the paper was in review and then in print. Now, it\u2019s out and I\u2019m super excited! My first pure Python paper, using Python for all data manipulation, analysis, and plotting.\n\nThe question was whether temperature affects herbivory by insects in any predictable way. I gathered as many insect species as I could and fed them whatever they ate at multiple temperatures. Check the article for more detail, but the idea was to fit a curve to all 21 herbivore-plant pairs as well as to estimate the overall effect of temperature. We also suspected (incorrectly as it turns out) that plant nutritional quality might be a good predictor of the shape of these curves, so we included that as a group-level predictor.\n\nAnyway, here\u2019s the code, complete with STAN model, posterior manipulations, and some plotting. First, here\u2019s the actual STAN model. NOTE: a lot of data manipulation and whatnot is missing. The point is not to show that but to describe how to fit a STAN model and work with the output. Anyone who wants the\u00a0full code and data to work with can find it on my website or in Dryad (see the article for a link).\n\nstanMod = \"\"\"\ndata{\nint<lower = 1> N;\nint<lower = 1> J;\nvector[N] Temp;\nvector[N] y;\nint<lower = 1> Curve[N];\nvector[J] pctN;\nvector[J] pctP;\nvector[J] pctH20;\nmatrix[3, 3] R;\n}\n\nparameters{\nvector[3] beta[J];\nreal mu_a;\nreal mu_b;\nreal mu_c;\nreal g_a1;\nreal g_a2;\nreal g_a3;\nreal g_b1;\nreal g_b2;\nreal g_b3;\nreal g_c1;\nreal g_c2;\nreal g_c3;\nreal<lower = 0> sigma[J];\ncov_matrix[3] Tau;\n}\n\ntransformed parameters{\nvector[N] y_hat;\nvector[N] sd_y;\nvector[3] beta_hat[J];\n\/\/ First, get the predicted value as an exponential curve\n\/\/ Also make a dummy variable for SD so it can be vectorized\nfor (n in 1:N){\ny_hat[n] <- exp( beta[Curve[n], 1] + beta[Curve[n], 2]*Temp[n] + beta[Curve[n], 3]*pow(Temp[n], 2) );\nsd_y[n] <- sigma[Curve[n]];\n}\n\/\/ Next, for each group-level coefficient, include the group-level predictors\nfor (j in 1:J){\nbeta_hat[j, 1] <- mu_a + g_a1*pctN[j] + g_a2*pctP[j] + g_a3*pctH20[j];\nbeta_hat[j, 2] <- mu_b + g_b1*pctN[j] + g_b2*pctP[j] + g_b3*pctH20[j];\nbeta_hat[j, 3] <- mu_c + g_c1*pctN[j] + g_c2*pctP[j] + g_c3*pctH20[j];\n}\n}\n\nmodel{\ny ~ normal(y_hat, sd_y);\nfor (j in 1:J){\nbeta[j] ~ multi_normal_prec(beta_hat[j], Tau);\n}\n\/\/ PRIORS\nmu_a ~ normal(0, 1);\nmu_b ~ normal(0, 1);\nmu_c ~ normal(0, 1);\ng_a1 ~ normal(0, 1);\ng_a2 ~ normal(0, 1);\ng_a3 ~ normal(0, 1);\ng_b1 ~ normal(0, 1);\ng_b2 ~ normal(0, 1);\ng_b3 ~ normal(0, 1);\ng_c1 ~ normal(0, 1);\ng_c2 ~ normal(0, 1);\ng_c3 ~ normal(0, 1);\nsigma ~ uniform(0, 100);\nTau ~ wishart(4, R);\n}\n\"\"\"\n\n# fit the model!\nfit = pystan.stan(model_code=stanMod, data=dat,\niter=10000, chains=4, thin = 20)\n\nNot so bad, was it? It\u2019s actually pretty straightforward.\n\nAfter the model has been run, we work with the output. We can check traceplots of various parameters:\n\nfit.plot(['mu_a', 'mu_b', 'mu_c'])\nfit.plot(['g_a1', 'g_a2', 'g_a3'])\nfit.plot(['g_b1', 'g_b2', 'g_b3'])\nfit.plot(['g_c1', 'g_c2', 'g_c3'])\npy.show()\n\nAs a brief example, we can extract the overall coefficients and plot them:\n\nmus = fit.extract(['mu_a', 'mu_b', 'mu_c'])\nmus = pd.DataFrame({'Intercept' : mus['mu_a'], 'Linear' : mus['mu_b'], 'Quadratic' : mus['mu_c']})\n\npy.plot(mus.median(), range(3), 'ko', ms = 10)\npy.hlines(range(3), mus.quantile(0.025), mus.quantile(0.975), 'k')\npy.hlines(range(3), mus.quantile(0.1), mus.quantile(0.9), 'k', linewidth = 3)\npy.axvline(0, linestyle = 'dashed', color = 'k')\npy.xlabel('Median Coefficient Estimate (80 and 95% CI)')\npy.yticks(range(3), ['Intercept', 'Exponential', 'Gaussian'])\npy.ylim([-0.5, 2.5])\npy.title('Overall Coefficients')\npy.gca().invert_yaxis()\npy.show()\n\nThe resulting plot:\n\nWe can also make a prediction line with confidence intervals:\n\n#first, define a prediction function\ndef predFunc(x, v = 1):\nyhat = np.exp( x[0] + x[1]*xPred + v*x[2]*xPred**2 )\nreturn pd.Series({'yhat' : yhat})\n\n# next, define a function to return the quantiles at each predicted value\ndef quantGet(data , q):\nquant = []\nfor i in range(len(xPred)):\nval = []\nfor j in range(len(data)):\nval.append( data[j][i] )\nquant.append( np.percentile(val, q) )\nreturn quant\n\n# make a vector of temperatures to predict (and convert to the real temperature scale)\nxPred = np.linspace(feeding_Final['Temp_Scale'].min(), feeding_Final['Temp_Scale'].max(), 100)\nrealTemp = xPred * feeding_Final['Temperature'].std() + feeding_Final['Temperature'].mean()\n\n# make predictions for every chain (in overall effects)\novPred = mus.apply(predFunc, axis = 1)\n\n# get lower and upper quantiles\novLower = quantGet(ovPred['yhat'], 2.5)\novLower80 = quantGet(ovPred['yhat'], 10)\novUpper80 = quantGet(ovPred['yhat'], 90)\novUpper = quantGet(ovPred['yhat'], 97.5)\n\n# get median predictions\novPred = predFunc(mus.median())\n\nThen, just plot the median (ovPred) and the quantiles against temperature (realTemp). With just a little effort, you can wind up with something that looks pretty good:\n\nI apologize for only posting part of the code, but the full script is really long. This should serve as a pretty good start for anyone looking to use Python as their Bayesian platform of choice. Anyone interested can get the data and full script from my article or website and give it a try! It\u2019s all publicly available.\n\n## Chart of the Month: Declining extreme weather events in\u00a0Miami\n\n### Image\n\nHere\u2019s my chart of the month (made using Python). This is a variant of an older post that I cleaned up. I really like the way this turned out. Simple and elegant.\n\n# Why Not? (An Evolution\u00a0Pictogram)\n\nIF:\n\nCheck out David Dogglehoff over there\u2026\n\nAND:\n\nThat\u2019s right. Broccoli, cauliflower, cabbage, brussel sprouts, and kale are all one disgusting species.\n\nTHEN WHY NOT:\n\nI am aware that a) dogs and the veggies are all one species while the primates here are all different generas and families that diverged millions of years ago (rather than a few thousand), so the differences are much more pronounced and that b) the process of selective breeding (for dogs and veggies) is different than speciation, although it is very similar to sexual selection and I don\u2019t know of any research suggesting that sexual selection is NOT how primates diversified. The common ancestor is an artist\u2019s rendition of Pierolapithecus catalaunicus, which is the suggested common ancestor (or close to it) of humans and great apes.\nWe are to the great apes what the chinese crested\/hairless chihuahua are to dogs\n\nAFTER ALL:\n\nWe\u2019re not so different\u2026\n\nNotes: All images from Wikimedia commons except the one of me and the skeleton comparison (which came from Google images).","date":"2017-03-24 08:10:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 16, \"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\": 0.41383516788482666, \"perplexity\": 3843.8417807500387}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-13\/segments\/1490218187744.59\/warc\/CC-MAIN-20170322212947-00607-ip-10-233-31-227.ec2.internal.warc.gz\"}"}
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Ghost Guide
La Llorona and the Ghosts Who Haunt Hotel Congress and Fox Theatre
| By Amanda Oien
| Photography by Kristen Brockel
A chill running down your spine, goosebumps dancing on your skin, a tight gasp for air. Is it a chilly October night, or is that an eerie haunt floating in the dark?
"Wrinkled skin dripping with fetid water, twisting fingernails covered with black muck, wailing and gnashing of teeth— and unspeakable sin. These images haunt the imaginations of millions of people…" Christopher Rodarte, La Llorona: Ghost Stories of the Southwest.
Christopher Rodarte, born in Albuquerque, New Mexico and currently a teacher at Sam Hughes Elementary, said growing up, he heard a lot of stories about La Llorona. "It was really entrenched in my childhood, especially in the northwest region of Albuquerque," Rodarte recalled. "There's a lot of very dark sections of town without lights and it was definitely something we were well aware of as children growing up."
La Llorona is one of the most famous ghosts in Latino folklore. According to Rodarte, the legend of the Wailing Woman may be the oldest ghost story in southwestern United States, South America, and Mexico.
Spanish for the weeping woman, La Llorona, sometimes referred to as "the ditch witch," is a tale of a woman who, filled with anger, drowns her two children in a river. When she realizes what she's done, she searches after them and drowns herself.
The ghost of La Llorona roams riverbanks, lakes, and abandoned wells searching for children to drown, just like her own.
Rodarte's aunts and uncles claimed to have encountered La Llorona and told him about her story as he grew up. "It was really part of the lore, sort of the DNA, growing up, that she was definitely out there in those ditches and los arroyos—and there were plenty of them out there in Albuquerque just like there are in Tucson."
When Rodarte drives through the Albuquerque pecan orchards at night, he still twists the rearview mirror so he can't look back. "When I'm out there past dark, oh you can forget it."
While working on a screenplay, Rodarte encountered so many La Llorona stories that he felt compelled to put them into a book. The more he talked about it, the more people contributed and told him about their first-hand experiences with the ditch witch. And thus, La Llorona: Ghost Stories of the Southwest was created.
"It's absolutely something that continues to exist throughout the culture and has really spread throughout the world," Rodarte said.
Rest in Peace. Or, Not So Much.
From Prohibition to the 1934 fire, from the Dillinger Gang to thrilling nightlife, Hotel Congress has seen a lot in its 100 years. But the three ghosts that tend to wander the halls during Congress' summer lull are one constant visitors can rely on.
David Slutes, the entertainment director at Hotel Congress, said their most common ghost, who predates the current ownership, is the Little Old Man. He is often seen in the window of room 214 or walking the halls in the seersucker suit.
Another ghost,Vince, a resident who came off the train, started living and working in the Hotel in 1966 and never moved out. David laughed and said, "He's a real curmudgeon."
In his pre-ghost days, Vince would often get into spats with housekeeping. He would visit the Cup Café and leave his bagel plate and butter knife in the desk in his room. David said housekeeping would scold Vince, telling him to bus his own plates: "You work and live here. You're not a guest!" they'd say.
In 2001, Vince passed away. A year later, Vince began leaving his bagel plate and butter knives in his desk in room 220—a room that hadn't been occupied. Oftentimes the room would be locked from the inside.
Before becoming entertainment director, Slutes ran the front desk for several years. One evening, a guest came downstairs and asked, "Did you have anyone that would have come in my room last night?" The guest recalled hearing someone in the room, but once they entered and turned on the lights, no one was there. The guest told David that all the towels had been stuffed into the toilet.
Two weeks later, another guest had the same occurrence. This playful ghost found their fun in room 216.
Hotel Congress' most well-known ghost haunts room 241. A woman came to an untimely end in that room, prompting housekeepers to actively avoid it. David said at one point, housekeeping even switched out the paintings in the room to religious iconography, trying to control the spirit.
For awhile, guests had a good time staying in that room. Visitors even requested the "ghost room." But now, Hotel Congress rents room 241 last.
David recalled one experience with room 241 when he covered the overnight at the front desk:
"I see this guy running, he has his bags, saying 'I gotta get out, I gotta get out right now,'" David said.
"If there is such a thing as paranormal activity, that's the room that has it, for sure," David said.
Happy Hauntings
Just down Congress Street is the crown jewel of Tucson: Fox Theatre. During its opening night in April 1930, Tucson partied like it'd never partied before—even Congress Street was waxed for dancing. After closing in the 1970s and sitting empty for 25 years, the Fox Theatre has been restored and is once again a go-to for movies and performances.
Tamara Mack, the house manager for the Fox, said the Theatre's most famous ghost is a little girl. During the closure in the 1970's, the roof caved in and the theatre became a homeless camp. "We don't know all that happened during those times," Mack said. "There's still some curiosities about that."
The Tucson Ghost Society has captured an apparition on video and has recorded six different Electronic Voice Phenomenons, including footage from the lobby of an orb that drifts out of the wall, rises up into the middle of the room, and moves to the back as it becomes a figure.
"I love to try to dispel things, you know, it's an old building, but that one we can't figure it out," Mack said. People have seen The Little Girl at the stairwell and have heard giggling and hiccuping.
Some visitors see The Panhandler, but when they go to help him, he vanishes, Mack said.
Reconstruction brought a new stage to the Fox, but if you look closely, you'll notice an inverted block over the center of the stage. "The theory is that there was a workman who died while working on the stage, and that they did that to honor him," Mack said.
Fox Theatre brings history to people by embracing happy ghosts, as Mack refers to them, and invites paranormal peepers to the Theatre for free ghost tours during their Halloween movie nights, and an annual lights-out ghost hunt.
"Nothing has ever scared me so that I wouldn't want to be here," Tamara said. "I don't get those kind of vibes. It's just things I can't quite explain."
If you find your stomach aching for food after running from La Llorona along the banks of the Santa Cruz River, or your throat shriveled after wandering Downtown Tucson's haunts, finish out the night with a slice of pizza and a brew at Reilly Craft Pizza and Drink. It's to die for…No, really. Before its artisan pizza and beer garden days, it was Reilly's Funeral Home, built in 1906.
Dillinger Days at Hotel Congress
Which hotel should University of Arizona visitors choose?
Thistle: The Floral Shop of Your Holiday Dreams
Wildcat Weekend
Tucson Neighborhood Guide
Tucson Guide Magazine c/o Madden Media
345 E. Toole Ave., Tucson AZ 85701
Phone (520) 322-0895 or (800) 444-8768
Pick Up a Copy Privacy Policy
View our Digital Edition
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| 3,411
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{"url":"http:\/\/mathonline.wikidot.com\/linear-operators-on-linear-spaces","text":"Linear Operators on Linear Spaces\n\n# Linear Operators on Linear Spaces\n\n Definition: Let $X$ and $Y$ be linear spaces over $\\mathbb{R}$ (or $\\mathbb{C}$). A function $T : X \\to Y$ is said to be a Linear Operator from $X$ to $Y$ if it satisfies the following properties: 1) $T(x + y) = T(x) + T(y)$ for all $x, y \\in X$. (Additivity) 2) $T(\\lambda x) = \\lambda T(x)$ for all $x \\in X$ and for all $\\lambda \\in \\mathbb{R}$ (or $\\mathbb{C}$). (Homogeneity) The set of all linear operators from $X$ to $Y$ is denoted $\\mathcal L (X, Y)$.\n\nIt is common to use the notation \"$Tx$\" instead of \"$T(x)$\" to denote the image of $x$ under $T$.\n\nFor example, let $X = \\mathbb{R}$ and $Y = \\mathbb{R}$ and consider the function $T : \\mathbb{R} \\to \\mathbb{R}$ defined for all $x \\in \\mathbb{R}$ by:\n\n(1)\n\\begin{align} \\quad T(x) = 2x \\end{align}\n\nThen $T$ is a linear operator from $\\mathbb{R}$ to $\\mathbb{R}$ since for all $x, y \\in \\mathbb{R}$ we have that:\n\n(2)\n\\begin{align} \\quad T(x + y) = 2(x + y) = 2x + 2y = T(x) + T(y) \\end{align}\n\nAnd for all $x \\in \\mathbb{R}$ and for all $\\lambda \\in \\mathbb{R}$ we have that:\n\n(3)\n\\begin{align} \\quad T(\\lambda x) = 2(\\lambda x) = \\lambda (2x) = \\lambda T(x) \\end{align}\n Definition: Let $X$ and $Y$ be normed linear spaces over $\\mathbb{R}$ (or $\\mathbb{C})$. A linear operator $T : X \\to Y$ is said to be a Bounded Linear Operator if there exists an $M \\in \\mathbb{R}$, $M \\geq 0$ such that for every $x \\in X$ we have that $\\| T(x) \\| \\leq M \\| x \\|$. The set of all bounded linear operators from $X$ to $Y$ is denoted $\\mathcal B(X, Y)$.\n\nNote that we can only consider bounded linear operators if both $X$ and $Y$ are normed linear spaces.\n\nFrom the example above, we see that with the standard Euclidean norm of the absolute value on $X = Y = \\mathbb{R}$, that $T : \\mathbb{R} \\to \\mathbb{R}$ defined for all $x \\in \\mathbb{R}$ by $T(x) = 2x$ is a bounded linear operator since for all $x \\in \\mathbb{R}$:\n\n(4)\n\\begin{align} \\quad \\| T(x) \\| = |T(x)| = |2x| = 2|x| = 2\\|x \\| \\end{align}\n\nWhere $M = 2$.","date":"2018-12-17 06:52:43","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\": 4, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9980451464653015, \"perplexity\": 76.47157753152436}, \"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\/1544376828448.76\/warc\/CC-MAIN-20181217065106-20181217091106-00584.warc.gz\"}"}
| null | null |
I am Justin Cormack and Justinc on en.
If you would like me to license any of the stuff here on another licence, please contact at my talk page, or here.
I have made a gallery of all my pictures at Category:User:Justinc; lots of these could be improved.
My User page en:User:Justinc has been newly decorated.
I have recently set up an account at Flickr. The photos there are not tagged as free, however in most cases I will happily upload them here under a free license (exceptions being recognizable people who I may need to ask for permission first). The pictures there are in many cases less carefully selected than the ones here as it is a cheap backup facility. (The interface is slicker but I dont like the fact it is not as wiki).
I can read a bit of French and German and bits of other european languages, so try them if your English is not good.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 9,140
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Q: How to use Logstash split with field I have the following data stored in the file data.json:
{"result":[{"number":"1"},{"number":"2"}]}
This is my logstash conf file:
input {
file {
path => "C:/work/apps/ELK/data.json"
start_position => "beginning"
sincedb_path => "/dev/null"
}
}
filter {
split {
field => "result"
}
}
output {
elasticsearch {
hosts => "localhost:9200"
index => "so"
document_type => 1
}
stdout { codec => rubydebug }
}
When I run Logstash with this configuration and the data above it I get only one entry in Elastic with the whole text (see image below).
When I modify the data to add new lines for example:
"result":[
{"number":"1"},
{"number":"2"}]
result is still ignored but this time I do have a split, according to the new lines. I get three messages in database:
I tried the solution proposed here with the original json file but again nothing is parsed (like in the first case). I don't understand why.
A: The split filter doesn't work since the field result does not exist. To create it, you need to parse the json you're reading from the file, which will create the fields.
To parse the json, use either the json codec on your input or the json filter.
FYI, the final configuration used:
input {
file {
path => "C:/work/apps/ELK/data.json"
start_position => "beginning"
sincedb_path => "/dev/null"
}
}
filter {
json {
source => "message"
remove_field => ["message", "host", "path"]
}
split {
field => "result"
add_field => {
"number" => "%{[result][number]}"
}
remove_field => "result"
}
}
output {
elasticsearch {
hosts => "localhost:9200"
index => "so"
document_type => 1
}
stdout { codec => rubydebug }
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 799
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Q: Raspberry: "do-release-upgrade" thinks 'Jammy Jellyfish' is still in development I wish to upgrade my Raspberry to Jammy Jellyfish. It was released yesterday and announced in news.
But on my machine it still shows that's beta release!
# cat /etc/lsb-release
DISTRIB_ID=Ubuntu
DISTRIB_RELEASE=21.10
DISTRIB_CODENAME=impish
DISTRIB_DESCRIPTION="Ubuntu 21.10"
# do-release-upgrade
Checking for a new Ubuntu release
No new release found.
# do-release-upgrade -p -m server
Checking for a new Ubuntu release
No new release found.
# do-release-upgrade -c -m server
Checking for a new Ubuntu release
No new release found.
# do-release-upgrade -c -m server --allow-third-party
Checking for a new Ubuntu release
No new release found.
# date
Sat Apr 23 07:49:28 CEST 2022
# do-release-upgrade -d
Checking for a new Ubuntu release
= Welcome to the Ubuntu 'Jammy Jellyfish' development release =
''This release is still in development.''
Thanks for your interest in this development release of Ubuntu.
The Ubuntu developers are moving very quickly to bring you the
absolute latest and greatest software the Open Source Community has to
offer. This development release brings you a taste of the newest features
for the next version of Ubuntu.
== Testing ==
Please help to test this development snapshot and report problems back to the
developers. For more information about testing Ubuntu, please read:
http://www.ubuntu.com/testing
== Reporting Bugs ==
This development release of Ubuntu contains bugs. If you want to help
out with bugs, the Bug Squad is always looking for help. Please read the
following information about reporting bugs
Is this some raspberry specific bug?
I did
apt update && apt upgrade && apt dist-upgrade && apt autoremove && apt install update-manager-core
A: There are currently some issues with upgrading from 21.10 to 22.04. That is why do-release-upgrade is showing jammy as a development release. If you want to take the risk and go ahead with the installation use the -d switch, but I would recommend waiting until the issues are solved.
The release date is the date when the iso is available for download. Upgrades from non-LTS releases may take a few more days and for LTS to LTS upgrades, Ubuntu releases the upgrade after the first point release (22.04.1) three months after release.
I've seen people reporting upgrading from 21.10 and end up with the system not booting, so it's better to be patient…
A: It takes until after the first point release (22.04.1) for do-release-upgrade to become available for general use. This is usually in August.
Some years, when there have been known problems, the team has held back the availability of an update through do-release-upgrade for a long time, until the issues are ironed out. This can be particularly so if you're on the LTS-to-LTS track.
A: I got the same message 'This release is still in development.' but was also prompted to continue. I said "y"and my machine is currently upgrading.
A: I'm not a Linux expert, however, you should use do-release-upgrade without the -d flag. -d flag is for the development version.
I did the same and it's currently upgrading!
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{
"redpajama_set_name": "RedPajamaStackExchange"
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| 5,026
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\section{Introduction}
A formula, usually called Beilinson's formula --- though independently due to Deligne as well --- describes the motivic cohomology group of a smooth projective variety $X$ over a number field as the group of extensions in a conjectured category of mixed motives. If $M$ is the motive $h^i(X)(m)$ and $N=M^{*}(1)=h^i(X)(n)$ where $n=i+1-m$ then
$$\operatorname{Ext}^1_{{\mathcal {MM}}}(M,\Q(1))=\operatorname{Ext}^1_{{\mathcal {MM}}}(\Q(0),N)=\begin{cases} CH^n_{\hom}(X) \otimes \Q & \text{ if } i+1=2n\\ H^{i+1}_{\M}(X,\Q(n)) & \text{ if } i+1\neq 2n\end{cases}$$
Hence, if one had a way of constructing extensions in the category of mixed motives by some other method, it would provide a way of constructing motivic cycles.
One way of doing so is by considering the group ring of the fundamental group of the algebraic variety $\ZZ[\pi_1(X,P)]$. If $J_P$ is the augmentation ideal --- the kernel of the map from $\ZZ[\pi_1(X,P)] \to \ZZ$ --- then the graded pieces $J_P^a/J_P^b$ with $a<b$ are expected to have a motivic structure. These give rise to natural extensions of motives --- so one could hope that these extensions could be used to construct natural motivic cycles.
Understanding the motivic structure on the fundamental group is complicated. However, the Hodge structure on the fundamental group is well understood \cite{hain}. The regulator of motivic cohomology cycles can be thought of as the realisation of the extension of motives as an extension in the category of mixed Hodge structures. So while we may not be able to construct motivic cycles as extensions of {\em motives} coming from the fundamental group - we can hope to construct their regulators as extensions of {\em mixed Hodge structures} (MHS) coming from the fundamental group.
The aim of this paper is to describe this construction in the case of the motivic cohomology group of the self product of a curve. The first work in this direction was due to Harris and Pulte \cite{pult}, \cite{hain}. They showed that the Abel-Jacobi image of the modified diagonal cycle on the triple product of a pointed curve $(C,P)$, or alternatively the Ceresa cycle in the Jacobian $J(C)$ of the curve, is the same as an extension class coming from $J_P/J_P^3$, where $J_P$ is the augmentation ideal in the group ring of the fundamental group of $C$ based at $P$.
In \cite{colo}, Colombo extended this theorem to show that the regulator of a cycle in the motivic cohomology of a Jacobian of a hyperelliptic curve, discovered by Collino \cite{coll}, can be realised as an extension class coming from $J_P/J_P^4$, where here $J_P^4$ is the augmentation ideal of a related curve.
In this paper we extend Colombo's result to more general curves. If $C$ is a smooth projective curve with a function $f$ with divisor $\operatorname{div}(f)=NQ-NR$ for some points $Q$ and $R$ and some integer $N$ and such that $f(P)=1$ for some other point $P$, there is a motivic cohomology cycle $Z_{QR,P}$ in $H^3_{\M}(C\times C,\ZZ(2))$ discovered by Bloch \cite{blocirvine}. We show that the regulator of this cycle can be expressed in terms of an extensions coming from $J/J^4$. When $C$ is hyperelliptic and $Q$ and $R$ are ramification points of the canonical map to $\CP^1$, this is Colombo's result as the image of $Z_{QR,P}$ under the map from $C \times C$ to $J(C)$ takes this cycle to Collino'c cycle.
A crucial step in Colombo's work is to use the fact that the modified diagonal cycle is {\em torsion} in the Chow group $CH^2_{\hom}(C^3)$ when $C$ is a hyperelliptic curve. This means the extension coming from $J_P/J_P^3$ splits and hence does not depend on the base point $P$. This allows her to consider the extension for $J_P/J_P^4$. In general, that is not true --- in fact the known examples of non-torsion modified diagonal cycles come from the curves we consider. Our main contribution is to use an idea of Rabi \cite{rabi} to show that Colombo's arguments can be extended to work in our case as well. As a result we have a more general situation --- which has some arithmetical applications.
We have the following theorem ({\bf Theorem \eqref{mainthm}}):
\begin{thm} Let $C$ be a smooth projective curve and $Q$ and $R$ be two points such that there is a function $f_{QR}$ with $\operatorname{div}(f_{QR})=NQ-NR$ for some $N$. Let $P$ be a third point on $C$. Normalize $f_{QR}$ so that $f_{QR}(P)=1$. Let $Z_{QR}=Z_{QR,P}$ be the element of the motivic cohomology group $H^3_{\M}(C \times C, \ZZ(2))$ constructed by Bloch. There exists an extension $\epsilon^4_{QR,P}$ in $\operatorname{Ext}_{MHS}(\ZZ(-2),\otimes^2 H^1(C))$ constructed from the mixed Hodge structure on the fundamental groups $\pi_1(C\backslash Q,P)$ and $\pi_1(C \backslash R,P)$ such that
%
$$\epsilon^4_{QR,P}=(2g_C+1) \operatorname{reg}_{\Q}(Z_{QR}) $$
in $\operatorname{Ext}_{MHS}(\ZZ(-2),\otimes ^2 H^1(C ))$.
\end{thm}
In other words our theorem states that the regulator of a natural cycle in the motivic cohomology group of a product of curves, being thought of as an extension class is the same as that as a natural extension of MHS coming from the fundamental group of the curve. In fact, it is an extension of {\em pure} Hodge structures.
Our primary motivation are the conjectures relating regulators of the motivic cycles to special values of $L$-functions. One application we have is to the case of modular curves. Beilinson \cite{beil} constructed a cycle in the group $H^3_{\M}(X_0(N) \times X_0(N), \Q(2))$ and showed that its regulator is related to a special value of the $L$-function. We construct the extension of $MHS$ coming from the fundamental group which corresponds to the regulator of this cycle.
Since the mixed Hodge structure on the fundamental group is related to the iterated integrals we also get an expression for the regulator as an iterated integral. In a subsequent paper we apply this in the case of Fermat curves to get an explicit expression for the regulator in terms of hypergeometric functions --- analogous to the works of Otsubo \cite{otsu1},\cite{otsu2}.
{\em Acknowledgements:} This work constitutes part of the PhD. thesis of the first author. We would like to that Najmuddin Fakhruddin, Noriyuki Otsubo, Satoshi Kondo, Elisabetta Colombo, Jishnu Biswas, Manish Kumar, Ronnie Sebastian and Suresh Nayak for their comments and suggestions. We would also like to thank the Indian Statistical Institute, Bangalore for their support while this work was done.
\section{Iterated Integrals}
Let $\alpha:[0,1] \to X$ be a path and $\omega_1,\omega_2,\dots,\omega_n$ be $1$-forms on $X$. Suppose $\alpha^*(\omega_i)=f_i(t) dt$. The {\em iterated integral of length n} is defined to be
$$\int_{\alpha} \omega_1\omega_2\dots \omega_n:=\int_{0 \leq t_1 \leq t_2\leq \dots \leq t_n \leq 1} f_1(t_i)f_2(t_2)\dots f_n(t_n)dt_n\dots dt_2 dt_1.$$
An iterated integral is said to be a {\em homotopy functional} if it only depends on the homotopy class of the path $\alpha$. A homotopy functional gives a functional on the group ring of the fundamental group or path space.
Iterated integrals can be thought of as integrals on simplices and satisfy the following basic properties --- here we have only stated the results for length two iterated integrals, since that is the only type we will encounter in this paper ---
\begin{lem}[Basic Properties] Let $\omega_1$ and $\omega_2$ be smooth $1$-forms on $C$ and $\alpha$ and $\beta$ piecewise smooth paths on $C$ with $\alpha(1)=\beta(0)$. Then
\begin{enumerate}
\item $\displaystyle{\int_{\alpha \cdot \beta} \omega_1 \omega_2=\int_{\alpha} \omega_1\omega_2 + \int_{\beta} \omega_1 \omega_2 + \int_{\alpha} \omega_1 \int_{\beta} \omega_2}$
\item $\displaystyle{\int_{\alpha} \omega_1 \omega_2 + \int_{\alpha} \omega_2 \omega_1 = \int_{\alpha} \omega_1 \int_{\alpha} \omega_2}$
\item $ \displaystyle{\int_{\alpha} df \omega_1 = \int_{\alpha} f \omega_1 - f(\alpha(0)) \int_{\alpha} \omega_1}$
\item $\displaystyle{\int_{\alpha} \omega_1 df = f(\alpha(1))\int_{\alpha} \omega_1 -\int_{\alpha} f\omega_1}$
\end{enumerate}
\label{basicproperties}
\end{lem}
\begin{proof} This can be found in any article on iterated integrals, for instance Hain's excellent article\cite{hain}.\end{proof}
\section{Motivic Cohomology Cycles, Extensions and Regulators}
\subsection{Motivic Cohomology Cycles}
\label{motiviccycles}
Let $X$ be a smooth projective algebraic surface defined over a field $K$. An element of the motivic cohomology group $H^3_{\M}(X,\ZZ(2))=H^1(X,\KK_2)=CH^2(X,1)$ has the following presentation. It consists of sums
$$Z=\sum_i (C_i,f_i)$$
where $C_i$ are curves on $X$ and $f_i:C_i \longrightarrow \CP^1$ are functions on them subject to the co-cycle condition
$$\sum_i \operatorname{div}(f_i)=0.$$
The relations in this group are given by the tame symbols of functions $\{f,g\}$ in $K_2(K(X))$,
$$\tau(\{f,g\})=\sum_Z (-1)^{\operatorname{ord}_Z(f)\operatorname{ord}_Z(g)}\frac{f^{\operatorname{ord}_Z(g)}}{g^{\operatorname{ord}_Z(f)}}.$$
If $L/K$ is a finite extension, let $X_L=X \otimes_{K} L$. There is a norm map
$$Nm_{L/K}: H^3_{\M}(X_L,\ZZ(2)) \longrightarrow H^3_{\M}(X,\ZZ(2)).$$
In the group $H^3_{\M}(X,\ZZ(2))$ there are certain {\em decomposable} cycles coming from the product
$$H^3_{\M}(X,\ZZ(2))_{dec}=\bigoplus_{L/K\; finite} Nm_{L/K} \left( \operatorname{Im} \left(H^1_{\M}(X_L,\ZZ(1)) \otimes H^2_{\M}(X_L,\ZZ(1)) \longrightarrow H^3_{\M}(X_L,\ZZ(2))\right)\right).$$
The quotient is called the group of {\em indecomposable} cycles ---
$$H^3_{\M}(X,\ZZ(2))_{ind}=H^3_{\M}(X,\ZZ(2))/H^3_{\M}(X,\ZZ(2))_{dec}.$$
In general it is not easy to find non trivial elements in this group. One of the aims of this paper is to show that in certain cases the cycles we construct are indecomposable. One way to do that is by computing its regulator.
\subsection{Regulators}
Let $X$ be a smooth projective algebraic surface. The regulator map of Beilinson is a Chern class map from the motivic cohomology group to the Deligne cohomology group.
$$\operatorname{reg}_{\ZZ}: H^3_{\M}(X, \ZZ(2)) \rightarrow H^3_{\D}(X, \ZZ(2))= \frac {(F^1H^2(X,\C))^*}{H_2(X,\ZZ(1))}.$$
The Deligne cohomology group $H^3_{\D}(X,\ZZ(2))$ is a generalised torus.
The map is defined as follows: Let $Z=\sum_i (C_i,f_i)$ be a cycle in $H^3_{\M} (X,\ZZ(2))$, so $C_i$ are curves on $X$ and $f_i$ functions on them satisfying the cocycle condition. Let $\mu_i:\tilde{C_i} \rightarrow C_i$ be a resolution of singularities. We can think of $f_i$ as a function $f_i:\tilde{C_i} \rightarrow \CP^1$. Let $[0,\infty]$ denote the positive real axis in $\CP^1$ and $\gamma_i=\mu_{i*} (f_i^{-1}([0,\infty]))$. Then $\sum_i \operatorname{div}(f_i)=0$ implies that the $1$-chain $\sum_i \gamma_i$ is closed and in fact exact. Hence
$$\sum_i \gamma_i=\partial(D)$$
for some $2$-chain $D$. For $\omega$ a closed $2$-form whose cohomology class lies in $F^1H^2(X,\C)$,
\begin{equation}
\langle\operatorname{reg}_{\ZZ}(Z),\omega\rangle:=\sum_i \int_{C_i} \log(f_i) \omega + 2\pi i \int_D \omega
\label{regulatorformula}
\end{equation}
For a decomposable element $(C,a)$ the regulator is particularly simple:
$$\langle\operatorname{reg}_{\ZZ}((C,a)),\omega\rangle=\int_{C} \log(a)\omega=\log(a) \int_C \omega$$
\subsection{Extensions}
As stated in the introduction, conjecturally, there is a canonical description of the motivic cohomology group as an extension in the category of mixed motives. In our case, if one has a suitable category of mixed motives over $\Q$, ${\mathcal {MM}}_{\Q}$, one expects for a surface $X$,\cite{scho}
$$H^3_{\M} (X,\ZZ(2))\simeq \operatorname{Ext}_{{\mathcal {MM}}_{\Q}} (\ZZ(-2),h^2(X)). $$
One {\em knows} that the Deligne cohomology can be considered as an extension in the category of integral mixed Hodge structures,
$$H^3_{\D} (X,\ZZ(2))\simeq \operatorname{Ext}_{MHS} (\ZZ(-2),H^2(X)) $$
The regulator map above then has a canonical description as the map induced by the realisation map from the category of mixed motives to the category of mixed Hodge structures,
$$\operatorname{Ext}_{{\mathcal {MM}}_{\Q}} (\ZZ(-2),h^2(X)) \stackrel{\operatorname{reg}_{\ZZ}}{\longrightarrow} \operatorname{Ext}_{MHS}(\ZZ(-2),H^2(X)).$$
One can take a further realisation in the category of Real mixed Hodge structures $\R-MHS$ to get the real regulator map to Deligne cohomology with $\R$-coefficients
$$\operatorname{reg}_{\R}:H^3_{\M}(X,\ZZ(2)) \longrightarrow H^3_{\D}(X,\R(2))=\operatorname{Ext}_{\R-MHS}(\R(-2),H^2(X)) \simeq H^2_B(X,\R(1)) \cap H^{1,1}(X)$$
hence the real regulator of a cycle can be viewed as a current on $(1,1)$-forms.
\subsection{The regulator to Real Deligne cohomology}
If $X$ is defined over $\Q$, one has the action of the {\em Frobenius at infinity}, $F_{\infty}$ on $X(\C)$ induced by complex conjugation. Let $X_{\C}=X \times_{\Q} \C$. For a subring $A \subset \R$, $F_{\infty}$ acts on the Deligne cohomology with coefficients in $A$. We define the {\em real} Deligne cohomology group as
$$H^3_{\D}(X_{/\R},\R(2))=H^3_{\D}(X_{\C}, \R(2))^{F_{\infty}=1}$$
This has a description as an extension ---
$$ H^3_{\D}(X_{/\R},\R(2))=\operatorname{Ext}_{\R-MHS}(\R(-2),H^2(X_{\C},\R))^{F_{\infty}=1}\simeq \frac{H^2_{B}(X_{\C},\R(1))^{F_{\infty}=1}}{F^2 H^2_{DR}(X_{/\R} )}$$
and the realisation gives the regulator map
$$\operatorname{reg}_{\R}: H^3_{\M}(X,\Q(2)) \longrightarrow H^3_{\D}(X_{/\R},\R(2))=\operatorname{Ext}_{\R-MHS}(\R(-2),H^2(X_{\C},\R))^{F_{\infty}}$$
The Real Deligne cohomology has a natural $\Q$-structure induced by the Betti and de Rham cohomologies. The determinant of that $\Q$-structure determines an element $c_{BdR} \in \R^*/\Q^*$. The Beilinson conjectures then assert
\begin{itemize}
\item The image of $\operatorname{reg}_{\R}$ gives another $\Q$ structure. The determinant of that $\Q$ structure is an element $c_{\operatorname{reg}} \in \R^*/\Q^*$.
\item If $L(H^2(X),s)$ is the $L$-function of $H^2(X)$, then the first non-zero term in the Taylor expansion at $s=1$, $L^*(H^2(X),1)$ satisfies
$$L^*(H^2(X),1) \cdot c_{BdR} \sim_{\Q^*} c_{\operatorname{reg}} $$
where $\sim_{\Q^*}$ means that this is up to a non-zero rational number.
\end{itemize}
Needless to say Beilinson conjectures are far more general --- relating special values of motivic $L$-functions coming from motives or varieties of arbitrary dimension to certain higher regulators --- but in the interest of brevity we have only stated things in the case at hand.
\subsection{Extensions of Mixed Hodge Structures and Motives.}
As stated above, the regulators of motivic cohomology cycles give extensions of mixed Hodge structures. The key point of this paper is that, in some cases, one can also obtain extensions of mixed Hodge structures in other ways. For instance, it was shown by Hain \cite{hain} that the group ring of the fundamental group $\ZZ[\pi_1(X,p)]$ of a pointed algebraic variety, as well as the graded quotients $J_P^a/J_P^b$, with $a\leq b$, where $J_P$ is the augmentation ideal, carry mixed Hodge structures. Hence natural exact sequences involving them lead to extensions of mixed Hodge structures.
Our aim is to first construct some natural motivic cohomology cycles in the case when $X=C \times C$. Their regulators will give rise to extensions of mixed Hodge structures. We will show that there are natural extensions of mixed Hodge structures coming from the Hodge structure on $\ZZ[\pi_1(C,P)]$ for some suitable point $P$ which give the {\em same} extensions. In particular, since the constructions can be carried out in at the level of mixed motives, if we had a good category of mixed motives the cycle {\em itself} would be an extension in the conjectured category of mixed motives coming from the fundamental group.
There are various candidates for the category of mixed motives \cite{levi} --- Voevodsky and Huber have candidates for the triangulated category of mixed motives and Nori and Deligne-Jannsen have candidates for the Abelian category itself. Cushman \cite{cush} showed that Nori's motives can be used to get a motivic structure on the group ring of the fundamental group --- so one expects that the same sequences we use would give extensions of mixed motives in Nori's category.
An alternative to Cushman's way of constructing Nori motives for the fundamental group was suggested to us by N. Fakhruddin. Nori's category requires a realisation in terms of relative cohomology groups. In the case of the fundamental groups this is given in the paper of Deligne and Goncharov \cite{dego} Section 3, (Proposition 3.4).
If $X$ is a smooth variety and $x_0$ a distinguished point, they show that the Hodge structure on the graded pieces of the group ring of the fundamental group can be realised as the Hodge structure on the relative cohomology groups of pairs $(X^s, \cup_{i=0}^{s} X_i )$,
where
\begin{itemize}
\item $X^s=X \times \dots \times X$ $s$-times
\item $X_0$ is the sub-variety given by $t_1=x_0$ --- namely $x_0 \times X^{s-1}$
\item $X_i$ is the sub-variety given by $t_i=t_{i+1}$ for $0 < i <s$ --- namely $X^{i-1}\times \Delta \times X^{s-(i+1)}$, where $\Delta$ is the diagonal in $X \times X$ in the $i^{th}$ and $(i+1)^{st}$ places.
\item $X_s$ is given by $t_s=x_0$ --- namely $X^{s-1} \times x_0$.
\end{itemize}
We have
$$H^s(X^s /\cup_{i=0}^{s} X_i,\C) \simeq \operatorname{Hom}(J/J^{s+1},\C)=H^0(\bar{B}_s(X),x_0)$$
For example, when $s=1$ we have
$$H^1(X/ \{x_0\},\C) \simeq H^1(X,\C) \simeq \operatorname{Hom}(J/J^2,\C).$$
Hence the motive underlying the Hodge structure on the group ring of the fundamental group $\ZZ[\pi_1(X,x_0)]$ is the motive associated to the pair $(X^s,\cup_{i=0}^{s} X_i)$. Namely, to this object one can associate a de Rham, \'{e}tale and Betti realization which are isomorphic when the field of coefficients is large enough.
In the special case when $C$ is a modular curve $X_0(N)$, we will show that certain natural elements of the motivic cohomology group constructed by Beilinson can be thought of as extensions coming from the fundamental group. In particular, we construct the extension in the Nori category of mixed motives which corresponds to the Beilinson elements.
Kings \cite{king} showed this in the case of $H^2_{\M}(X_0(N),\ZZ(2))$ for Huber's motives.
\section{A Motivic Cohomology Cycle on $C \times C$.}
In this section we construct a motivic cohomology cycle on $C \times C$ first introduced by Bloch in the case of $X_0(37)$. The cycle is similar, in fact, generalises, the cycle constructed by Collino \cite{coll}. This section generalises the work of Colombo \cite{colo} on constructing the extension corresponding to the Collino cycle and hence many of the arguments are adapted from her paper.
\subsection{The cycle $Z_{QR}$}
Let $C$ be a smooth projective curve defined over a number field $K$. Let $Q$ and $R$ be two points on $C$ such that there is a function $f=f_{QR}$ with divisor
$$\operatorname{div}(f_{QR})=NQ-NR$$
for some $N \in \NN$. To determine the function precisely, we choose a distinct third point $P$ and assume $f_{QR}(P)=1$.
There exists notable examples of curves where such functions can easily be found. For instance, {\em modular curves} with $Q$ and $R$ being cusps, {\em Fermat curves} with the two points being among the `trivial' solutions of Fermat's Last Theorem, namely the points with one of the coordinates being $0$, and {\em hyperelliptic curves} with the two points being Weierstrass points.
Consider the cycle in $C \times C$ given by
$$Z_{QR,P}=(C \times Q, 1/f^Q)+(\Delta_C, f^{\Delta})+(R \times C, 1/f^R)$$
where $\Delta_C$ is the diagonal, $f^Q=f_{QR}\times Q$, $f^R=R\times f_{QR}$ and $f^{\Delta}=f_{QR}$ being considered as a function on $\Delta_C$. Since
$$\operatorname{div}_{C \times Q} (1/f^Q)+\operatorname{div}_{\Delta_C}(f^{\Delta})+\operatorname{div}_{R \times C} (1/f^R)=$$
$$=N(R,Q)-N(Q,Q)+N(Q,Q)-N(R,R)+N(R,R)-N(R,Q)=0$$
the cycle $Z_{QR,P}$ gives an element of $H^3_{\M}(C \times C,\ZZ(2))$. From now on we suppress the $P$ and simply write $Z_{QR}$.
This cycle was first described by Bloch \cite{blocirvine} in his celebrated Irvine lecture notes and later variants of this construction were used by Beilinson and others to verify the Beilinson conjectures in some special cases. If $C$ is a hyper-elliptic curve and $f$ is a function supported on the Weierstrass points, so $\operatorname{div}(f)=2Q-2R$, then under the natural map:
%
$$C \times C \longrightarrow J(C)$$
$$(x,y) \longrightarrow (x-y)$$
%
this cycle maps to the Collino cycle \cite{coll}.
\subsection{ The Regulator of $Z_{QR}$}
Let $Z_{QR}$ be the motivic cohomology cycle in $H^3_{\M}( C \times C, \ZZ(2))$. We now obtain a formula for its regulator. The motivic cohomology group of $C \times C$ is the same as that of $h^2(C \times C)$. From the K\"{u}nneth theorem the motive $h^2(C \times C)$ decomposes
$$h^2(C \times C)=h^2(C) \otimes h^0(C) \oplus h^1(C) \otimes h^1(C) \oplus h^0(C) \otimes h^2(C).$$
We will essentially be concerned with $h^1(C) \otimes h^1(C)$. The regulator is a current on forms in $F^1(H^1(C) \otimes H^1(C))$ --- that is, forms of the type $\phi \otimes \psi$ where $\phi$ and $\psi$ are $1$-forms on $C$ and $\phi$ is of type $(1,0)$.
Recall that $f=f_{QR}$ is a function on $C$ with divisor $NQ-NR$ for some $N$. Let $\omega=\phi \otimes \psi$ and $\gamma=f^{-1}([0,\infty])$ be a path on $C$ from $Q$ to $R$. As $f$ is of degree $N$, $\gamma$ is the union of $N$ paths - each lying on a different sheet with only the points $Q$ and $R$ in common. We will denote by $\gamma^i$, $1\leq i \leq N$. Each $\gamma^i$ is a path from $Q$ to $R$. Let $\gamma_Q$, $\gamma_R$ and $\gamma_{\Delta}$ denote the curve $\gamma$ on $C \times Q$, $R \times C$ and $\Delta_C$ respectively and similarly for the components $\gamma^i$. Then from the co-cycle condition one has
$$ \gamma_{\Delta} \cdot \gamma_R^- \cdot \gamma_Q^-=\partial(D)$$
where $D$ is a $2$-chain on $C \times C$. Here for a path $\alpha$, $\alpha^-$ is the inverse: $\alpha^-(t)=\alpha(1-t)$.
From equation \eqref{regulatorformula} one has
\begin{equation}
\langle\operatorname{reg}_{\ZZ}(Z_{QR}),\omega \rangle=-\int_{ C \times Q } \log(f^Q ) \omega + \int_{\Delta_C} \log(f^{\Delta})\omega - \int_{R \times C} \log(f^R) \omega+ 2\pi i \int_{D} \omega.
\label{regfor1}
\end{equation}
Our aim is to find a more explicit expression for $\operatorname{reg}_{\ZZ}$. For this we need an explicit description of $D$.
\begin{lem} Let
\[
a(s,t)=t \hspace{2cm} {\rm and } \hspace{2cm} b(s,t)=\frac{t(1-s)}{1-s(1-t)} \]
Define $F_i:[0,1] \times [0,1] \longrightarrow C \times C$ by
$$F_i(s,t)=\left(\gamma^i(a(s,t)), \gamma^i(b(s,t))\right)$$
for $ 1 \leq i \leq N$ and let
$$D_i=\operatorname{Im}(F_i).$$
Then
\[
\partial(D_i)= \gamma_{\Delta}^{i} \cdot \gamma_R^{i-} \cdot \gamma_Q^{i-}
\]
In particular, if $D=\cup_{i=1}^{N} D_i$ then
\[
\partial(D)= \gamma_{\Delta} \cdot \gamma_R^- \cdot \gamma_Q^-
\]
\end{lem}
\begin{proof} The oriented boundary of $D_i$ is
$$\partial(D_i)=F_i(0,t) \cup F_i(s,1) \cup F_i(1,(1-t))\cup F_i((1-s),0).$$
Restricting $F_i$ to the boundary shows ---
\begin{itemize}
\item $F_i(0,t)=(\gamma^i(t),\gamma^i(t))=\gamma_{\Delta}^{i}(t)$ -- a path from $(Q,Q)$ to $(R,R)$.
\item $F_i(s,1)=(\gamma^i(1)),\gamma^i(1-s))=(R,\gamma^i(1-s))=\gamma_R^-(s)$ -- a path from $(R,R)$ to $(R,Q)$.
\item $F_i(1,(1-t))=(\gamma^i(1-t),\gamma^i(0))=(\gamma^i(1-t),Q)=\gamma_Q^-(t)$ -- a path from $(R,Q)$ to $(Q,Q)$ .
\item $F_i((1-s),0)=(\gamma^i(0),\gamma^i(0))=(Q,Q)$ -- the constant path at $(Q,Q)$.
\end{itemize}
Hence the boundary of $D_i$ is $\gamma^{i}_{\Delta}\cdot \gamma^{i-}_R \cdot \gamma^{i-}_Q$. If $D$ is the union of the $D_i$
then its boundary is the union of the boundaries of the $D_i$.
\end{proof}
We can compute the second integral as an iterated integral as follows.
\begin{lem} Let $\phi$ and $\psi$ be closed $1$-forms on $C$ with $\phi$ holomorphic and let $D_i$ be a disc as in the above lemma. Then
$$\int_{D_i} \phi \otimes \psi = \int _{\gamma^{i}_{\Delta} \cdot \gamma_R^{i-} \cdot \gamma_Q^{i-} } \phi \psi= \int_{\gamma_{\Delta}^{i}} \phi \psi.$$
\end{lem}
\begin{proof} Any closed $1$-form on a disc is exact, hence $\phi|_{D_i}=d \rho$ for some function $\rho$. From Stokes theorem one has
$$\int_{D_i} \phi \otimes \psi = \int_{D_i} d(\rho \psi) = \int_{\partial(D_i)} \rho \psi = \int_{\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \rho\psi.$$
Using Lemma \ref{basicproperties} (3) and choosing $\rho$ such that $\rho(\gamma^{i}_{\Delta}\cdot \gamma^{i-}_R \cdot \gamma^{i-}_Q (0))=\rho((Q,Q))=0$ we have
$$\int_{\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} d\rho \psi = \int_{\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \rho \psi -\rho(\gamma^{i}_{\Delta}\cdot \gamma^{i-}_R \cdot \gamma^{i-}_Q (0)) \int_{\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \psi = \int_{\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \rho \psi. $$
Hence
$$\int_{D_i} \phi \otimes \psi =\int_{\gamma^i_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} d\rho\psi = \int_{\gamma^i_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \phi\psi. $$
Using Lemma \ref{basicproperties} (1) with $\alpha=\gamma_{\Delta}^{i}$ and $\beta=\gamma_R^{i-}\cdot \gamma_Q^{i-}$ we get
$$\int_{\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \phi\psi = \int_{\gamma_{\Delta}^{i} }\phi \psi + \int_{\gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \phi \psi + \int_{\gamma_{\Delta}^{i}} \phi \int_{\gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \psi. $$
Since $\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q=\partial(D_i)$ is exact, one has
$$\int_{\gamma^{i}_{\Delta} \cdot \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \phi= \int_{\gamma_{\Delta}^i} \phi + \int_{ \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \phi=0.$$
Using Lemma \ref{basicproperties} (2) we have
$$\int_{ \gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \psi \phi = \int_{\gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \phi \psi - \int_{\gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \phi \int_{\gamma_{R}^{i-} \cdot \gamma^{i-}_Q} \psi.$$
Combining this with the above expression gives
$$\int_{D_i} \phi \otimes \psi = \int_{\gamma_{\Delta}^{i}} \phi \psi - \int_{\gamma_R^{i-} \cdot \gamma^{i-}_Q} \psi \phi.$$
However, the integral
$$\int_{\gamma_R^{i-} \cdot \gamma^{i-}_Q} \psi \phi=0$$
as $\gamma_Q^{i-}$ and $\gamma_R^{i-}$ are supported on $C \times Q$ and $R \times C$ hence one of $\phi$ or $\psi$ restricted to them will be $0$. So we are left with
$$\int_{D_i} \phi \otimes \psi = \int_{\gamma_{\Delta}^{i}} \phi \psi.$$
\end{proof}
Combining the above lemma with the earlier expression for the regulator we have the following theorem
\begin{thm} Let $Z_{QR}$ be the motivic cohomology cycle in $H^3_{\M}(C \times C,\ZZ(2))$ and $\phi$ and $\psi$ two $1$-forms on $C$ with $\phi$ holomorphic. Let $\omega=\phi \otimes \psi$. Then
%
$$ \langle \operatorname{reg}_{\ZZ}(Z_{QR}), \omega > =-\int_{ C \times Q } \log(f^Q ) \omega + \int_{\Delta_C} \log(f^{\Delta})\omega - \int_{R \times C} \log(f^R) \omega
+ N\pi i \int_{\gamma_{\Delta}^-} \phi \psi. $$
%
Here $N$ is the degree of the map $f_{QR}:C \rightarrow \CP^1$.
In fact, since $\omega|_{C \times Q}=\omega|_{R \times C}=0$, the expression simplifies to
$$ \langle \operatorname{reg}_{\ZZ}(Z_{QR}), \omega \rangle = \int_{\Delta_C} \log(f^{\Delta})\omega + N\pi i \int_{\gamma_{\Delta}^-} \phi \psi $$
\label{regform}
\end{thm}
\begin{proof} This is immediate from the earlier lemmas. The only thing to be remarked is that the $N$ appears because there are $N$ components $D_i$ and all of them contribute the same integral.
\end{proof}
\section{The Fundamental group and Mixed Hodge Structures}
From this section onwards, $H^1(C)$ will denote the integral cohomology group $H^1(C,\ZZ)$ and $H^1(C)_{\Q}$ will denote $H^1(C,\Q)$.
Let $C$ be a smooth projective curve and $P$, $Q$ and $R$ be three points on $C$. Consider the open curve $C_Q=C \backslash \{Q\}$. Let $\ZZ[\pi_1(C_Q,P)]$ be the group ring of the fundamental group of $C_Q$ based at $P$. Let $J_{Q,P}:=J_{C_Q,P}$ denote the augmentation ideal ---
$$J_{Q,P}:=J_{C_Q,P}=\operatorname{Ker} \{ \ZZ[\pi_1(C_Q,P)] \stackrel{\operatorname{deg}}{\longrightarrow} \ZZ \}$$
Let $H^0({\mathcal B}_r(C_Q;P))$ denote the $F$-vector space, where $F$ is $\R$ or $\C$, of homotopy invariant iterated integrals of length $\leq r$. Chen\cite{chen} showed that
$$H^0({\mathcal B}_r(C_Q;P)) \simeq \operatorname{Hom}_{\ZZ}(\ZZ[\pi_1(C_Q,P)]/J_{C_Q.P}^{r+1},F)$$
under the map
$$ I \longrightarrow I(\gamma)=\int_{\gamma} I $$
Using this Hain \cite{hain} was able to put a natural mixed Hodge structure on the graded pieces $J_{Q,P}/J_{Q,P}^r$.
\subsection{The extension $E^3_{Q,P}$}
One can consider the extensions of mixed Hodge structures
$$E^r_{Q,P}: 0 \longrightarrow (J_{Q,P}/J_{Q,P}^{r-1})^* \longrightarrow (J_{Q,P}/J_{Q,P}^{r})^*\longrightarrow (J_{Q,P}^{r-1}/J_{Q,P}^{r})^*\longrightarrow 0$$
The simplest non-trivial case is when $r=3$. In this case $(J_{Q,P}/J_{Q,P}^2)^* \simeq H^1(C_{Q})\simeq H^1(C)$ and $(J_{Q,P}^2/J_{Q,P}^3)^* \simeq \otimes^2 H^1(C)$ and the exact sequence becomes
$$E^3_{Q,P}: 0 \longrightarrow H^1(C) \longrightarrow (J_{Q,P}/J_{Q,P}^3)^* \longrightarrow \otimes^2 H^1(C) \longrightarrow 0$$
Hence $E^3_{Q,P}$ gives an element in $\operatorname{Ext}_{MHS}(\otimes^2 H^1(C),H^1(C))$. A similar construction with $R$ in the place of $Q$ gives us the extension $E^3_{R,P}$, which also lies in the same $\operatorname{Ext}$ group.
There are a few natural morphisms of mixed Hodge structures which allow us to pull back and push forward the extensions
\begin{itemize}
\item There is a surjection $\cup:\otimes^2 H^1(C) \longrightarrow H^2(C) \stackrel{\int_C}{\simeq} \ZZ(-1)$ coming from the cup product composed with
Poincar\'{e} Duality. Let $K$ be the kernel of this map. The exact sequence of Hodge structures
$$0 \longrightarrow K \longrightarrow \otimes^2 H^1(C) \stackrel{\cup}{\longrightarrow} \ZZ(-1) \longrightarrow 0 $$
splits rationally to give a map $\beta: 2g_C \ZZ(-1) \rightarrow \otimes^2 H^1(C)$ and
$$\otimes^2 H^1(C)=K \oplus 2g_C \ZZ(-1)$$
\item Let $\Omega$ denote the polarisation on $C$. There is an injection obtained by tensoring with $\Omega$
$$J_{\Omega}=\otimes \Omega:H^1(C)(-1) \longrightarrow \otimes^3 H^1(C).$$
\end{itemize}
From the K\"{u}nneth theorem one has
$$\operatorname{Ext}(\otimes^2 H^1(C),H^1(C))=\operatorname{Ext}(K \oplus \ZZ(-1), H^1(C))=\operatorname{Ext}(K,H^1(C)) \times \operatorname{Ext}(\ZZ(-1),H^1(C))$$
It is well known that $\operatorname{Ext}(\ZZ(-1),H^1(C))\simeq \operatorname{Pic}(C)$.
From the work of Hain \cite{hain}, Pulte \cite{pult}, Kaenders \cite{kaen} and Rabi \cite{rabi} one has the following theorem
\begin{thm} The class $E^3_{QP}$ in $\operatorname{Ext}(\otimes^2 H^1(C) ,H^1(C))$ is described as follows --
$$E^3_{Q,P}=(m_P, k_{QP})$$
%
where $m_P \in \operatorname{Ext}(K,H^1(C))$ depends only on $P$ and $k_{QP}$ is given by
$$2 g_C Q - 2P -\kappa_C \in \operatorname{Pic}(C)$$
where $\kappa_C$ is the canonical divisor of $C$ and $g_C$ is the genus of $C$.
\label{moddiag}
\end{thm}
We now return to our situation where $P$, $Q$ and $R$ are three points such that there is a function $f_{QR}$ with $\operatorname{div}(f_{QR})=NQ-NR$. Recall that in the group Ext, addition is given by the Baer sum. We will denote this by $\oplus_B$ (or $\ominus_B$ if we are taking differences). Let $E^3_{QR,P}$ denote the Baer difference $E^3_{Q,P} \ominus_{B} E^3_{R,P}$
\begin{lem} Under the hypothesis that there is a function with divisor $\operatorname{div}(f_{QR})=NQ-NR$ the extension $\frac{N}{2g_C} E^3_{QR,P}$ splits. That is
$$\frac{N}{2g_C} E^3_{QR,P} \simeq H^1(C) \bigoplus \otimes^2 H^1(C)$$
\label{splitting}
\end{lem}
\begin{proof} This follows quite easily from Theorem \ref{moddiag}. Since the class of $E^3_{QP}=(m_P, k_{QP})$ where $m_P$ does not depend on $Q$, the class of the difference is
$$E^3_{QR,P}=E^3_{QP} \ominus_{B} E^3_{RP}=(m_P,k_{QP})-(m_P, k_{RP})=(0, k_{QP}-k_{RP}).$$
The class
$$k_{QP}-k_{RP}=(2g_C Q -2P- \kappa_C )- (2g_C R - 2P-\kappa_C)=2g_C(Q-R)$$
in $\operatorname{Pic}(C)$. By hypothesis, this is torsion. Hence the extension whose class is given by the Baer difference splits when multiplied by $\frac{N}{2g_C}$.
\end{proof}
\begin{rem}
This extension represents the class $Q-R$, at least up to a integral multiple, and is hence the first example of the theme of this paper - namely the Abel-Jacobi image of a null-homologous cycle is described in terms of extensions coming from the fundamental group.
\end{rem}
A consequence of this is that there is a morphism of integral mixed Hodge structures
$$r_{3}: \frac{N}{2g_C} E^3_{QR,P} \longrightarrow H^1(C)$$
given by the projection.
\subsection{ The extensions $E^4_{Q,P}$ and $E^4_{R,P}$}
From the work of Hain, Pulte, Harris and others one knows that the class $m_P$ in $\operatorname{Ext}(K,H^1(C))$ corresponds to the extension of mixed Hodge structures determined by the Ceresa cycle in $J(C)$, or alternately, the modified diagonal cycle in $C^3$. We would like to construct a similar class corresponding to the motivic cohomology cycle $Z_{QR}$. To that end, we now consider, with $C, P,Q$ and $R$ as before, the extension corresponding to $r=4$
$$E_{Q,P}^4: 0 \longrightarrow (J_{Q,P}/J_{Q,P}^3)^* \longrightarrow (J_{Q,P}/J_{Q,RP}^4)^* \longrightarrow (J_{Q,P}^3/J_{Q,P}^4)^* \longrightarrow 0$$
We have that $(J_{Q,P}^3/J_{Q,P}^4)^*\simeq \otimes^3 H^1(C)$ and this does not depend on $P, Q$ or $R$. However, from Theorem \ref{moddiag}, $(J_{Q.P}/J_{Q,P}^3)^*$ depends on $Q$ and $P$ and similarly $(J_{R.P}/J_{R,P}^3)^*$ depends on $R$ and $P$.
When $C$ is hyperelliptic $E^3_{Q,P}\otimes \Q$ and $E^3_{R,P}\otimes \Q$ are split. Hence one gets two classes
$$E^4_{Q,P}, E^4_{R,P} \in \operatorname{Ext}(\otimes^3 H^1(C)_{\Q},\otimes^2 H^1(C)_{\Q} \oplus H^1(C)_{\Q})$$
and one can project to get two classes $e^4_{Q,P}$ and $e^4_{R,P}$ in $\operatorname{Ext}(\otimes^3 H^1(C)_{\Q}, H^1(C)_{\Q})$. Colombo \cite{colo} shows that the class
$$e^4_{QR,P}=e^4_{Q,P} \ominus_B e^4_{R,P} \in \operatorname{Ext}(\otimes^3 H^1(C)_{\Q}, H^1(C)_{\Q})$$
corresponds to the extension determined by the cycle $Z_{QR}$ - after pulling back and pushing forward with some standard maps.
Unfortunately, in general the extensions $E^3_{Q,P}$ and $E^3_{R,P}$ are {\em not} split rationally. They correspond to the instances where the Ceresa cycle is non-torsion. The instances where this is known are precisely the cases we have in mind - modular curves and Fermat curves\cite{harr}. Hence we cannot use this argument immediately. However, since we know from Lemma \ref{splitting} that their difference $E^3_{QR,P}$ is split rationally, we would like to get an extension of the form
$$0 \longrightarrow E^3_{QR,P}\otimes_{\Q} \longrightarrow ``E^4_{QR,P}" \longrightarrow \otimes^3 H^1(C)_{\Q} \longrightarrow 0$$
where $``E^4_{QR,P}"$ is a sort of generalised Baer difference of the two extensions $E^4_{Q,P}$ and $E^4_{R,P}$. We cannot simply consider $E^4_{QR,P}=E^4_{Q,P} \ominus_{B} E^4_{R,P}$ as the two extensions lie in different $\operatorname{Ext}$-groups. So we have to consider a generalisation of Baer sums to not necessarily exact sequences which we came across in a paper of Rabi \cite{rabi}.
\subsection{ The Baer sum and a generalisation}
Recall that if we have two exact sequences of modules
$$\begin{CD}
E_j: 0 @>>> A @>f_j>> B_j @>p_j>> C @>>>0
\end{CD}$$
for $j \in \{1,2\}$, the Baer difference $E_1 \ominus_B E_2$ is constructed as follows. We have
$$\begin{CD}
0 @>>> A \oplus A @>f_1\oplus f_2>> B_1 \oplus B_2 @>p_1\oplus p_2>> C\oplus C @>>> 0
\end{CD}
$$
Let $\psi: B_1\oplus B_2 \longrightarrow C$ be the map
$$\psi(b_1,b_2)=p_1(b_1)-p_2(b_2)$$
and let $H=\operatorname{Ker}(\psi)=\{(b_1,b_2)| \; p_1(b_1)=p_2(b_2)\}$. Let $D$ be the image of $\tilde{f}: A \longrightarrow A \oplus A \longrightarrow H$
$$\tilde{f}(a)=(f_1(a),f_2(a))$$
Let $B=H/D$. The map $f: A \oplus A \longrightarrow B$ given by
$$f(a_1,a_2)=(f_1(a_1),f_2(a_2))$$
factors through $(A \oplus A)/A \simeq A$ and so one has a map $\bar{f}:A \longrightarrow B$,
$$a \longrightarrow (f_1(a),0)=(0,-f_2(a))$$
and a sequence
$$\begin{CD}
0 @>>> A @>\bar{f}>> B@>p_1( or \;p_2) >> C @>>> 0
\end{CD}
$$
The class of $B$ in $\operatorname{Ext}(C,A)$ is the Baer difference $E_1 \ominus_B E_2$. The Baer sum $E_1 \oplus_B E_2$ is the sequence obtained when one of the maps $f_2$ or $p_2$ is replaced by its negative. If the modules and morphisms have additional structure --- for instance, if we are working in the category of mixed Hodge structures --- then the Baer sum is also in the same category. The Baer sum is essentially the push-out over $A$ in the category of modules.
\subsubsection{ A generalisation} Now suppose we have two exact sequences, for $j=\{1,2\},$
$$\begin{CD}
E_j: 0 @>>> B_1^j @>f_j>> B_2^j @>p_j>> B_3 @>>>0
\end{CD}$$
and a diagram of the following type:
\[
\begin{CD}
&&0\\&&@VVV\\
&&A_1\\
&&@VVi_jV\\
0 @>>> B_1^j @>f_j>> B_2^j @>p_j>> B_3 @>>>0 \\
&&@VV\pi_j V \\
&&C_1\\
&&@VVV\\
&&0
\end{CD}
\]
where the vertical and horizontal sequences are exact for all values of $j$. We would like to take the Baer difference of the $E_j$ -- but since they do not lie in the same $\operatorname{Ext}$ group we cannot quite do that. However, using the vertical exact sequence, we can still salvage something out of it.
We have $B_1^j \in \operatorname{Ext}(C_1,A_1)$ hence we can form their Baer difference. Let $\BB_1=B_1^1\ominus_B B_1^2$. Define $\BB_2$ as follows: Let $H_2=\operatorname{Ker}(\psi)$, where $\psi$ is the `difference' map
%
$$ \psi: B_2^1\oplus B_2^2 \longrightarrow B_3 $$
$$ \psi((b_2^1,b_2^2)) = (p_1(b_2^1)-p_2(b_2^2))$$
Let $D_2$ be the image of the map
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$$A_1 \longrightarrow B_1^1 \oplus B_1^2 \longrightarrow H_2$$
$$ a \longrightarrow (f_1(i_1(a)),f_2(i_2(a)))$$
Define $\BB_2=H_2/D_2$. We call this the {\em generalised} Baer difference of $B_2^1$ and $B_2^2$. Observe that this is almost the Baer difference of $B_2^1$ and $B_2^2$ in the sense that if $B_1=B_1^1=B_1^2$, then we could take the difference in $\operatorname{Ext}(B_3,B_1)$. Since that is not the case, we do the best we can - we take the difference of the {\em inexact} sequences
%
$$
\begin{CD}
0 @>>>A_1 @>>> B_2^j @>>>B_3 @>>> 0.
\end{CD}
$$
%
As a result of this one has a sequence
%
$$
\begin{CD}
0 @>>> \BB_1 @>f_1 \oplus f_2 >> \BB_2 @>p_1( or \; p_2)>> B_3 @>>>0
\end{CD}
$$
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However, this sequence is {\em not} exact --- $\operatorname{Ker}(p_1)$ is larger than $(f_1\oplus f_2)(\BB_1)$. The next lemma describes this difference.
\\
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\begin{lem}[Rabi\cite{rabi}] Let $\BF=\BB_2/\BB_1$. Then one has the following diagram, in which the horizontal and vertical sequences are exact.
$$
\begin{CD}
&&&&&&0\\
&&&&&&@VVV\\
&&&&&&C_1\\
&&&&&&@VV\phi V\\
0 @>>> \BB_1 @>f>> \BB_2 @>\eta>> \BF @>>>0 \\
&&&&&&@VV\bar{p}V \\
&&&&&&B_3\\
&&&&&&@VVV\\
&&&&&&0
\end{CD}
$$
\label{rabilemma}
\end{lem}
\begin{proof} The horizontal sequence is exact by construction. To show the vertical sequence is exact we have to first describe be map $\phi$.
It is defined as follows. One has maps $\pi_j:B_1^{j} \longrightarrow C_1$. Consider the map
$$\begin{CD}
\phi:C_1\oplus C_1 @>(\pi_1^{-1}, \pi_2^{-1})>> B_1^1 \oplus B_1^2 @>(f_1,f_2)>>H_2
\end{CD}
$$
$$\phi(c_1,c_2)=(f_1(\pi_1^{-1}(c_1)),f_2(\pi_2^{-1}(c_2)))$$
This is well defined modulo the image of $A_1$, the kernel of $\pi_j$ --- which is $D_2$. Also, it maps to the kernel of $\psi$. Hence gives a map to $\BB_2=H_2/D_2$. Further,
the image of the diagonal $\Delta_{C_1}=\{(c,-c) | c \in C_1\}$ is $0$, so this map factors through $(C_1 \oplus C_1 )/\Delta_{C_1} \simeq C_1$. The pre-image of $\Delta_{C_1}$ is the Baer difference $\BB_1$ --- hence the map $\phi$ maps to $\BF$.
As a result we get a sequence
$$0 \longrightarrow C_1 \stackrel{\phi}{\longrightarrow} \BF \stackrel{\bar{p}}{\longrightarrow} B_3 \longrightarrow 0$$
This map is exact as if $b=(b^1_2,b^2_2)$ is in $\BF$ and $\bar{p}(b)=0$, then $p_1(b^1_2)=p_2(b^2_2)=0$. So $b^1_2$ and $b^2_2$ lie in the image of $B_1^1\oplus B_1^2$ --- say $b_2^1=f_1(b_1^1)$ and $b_2^2=f_2(b_1^2)$. Let $c_i=\pi_1(b_1^1)$ and $c_2=\pi_2(b_1^2)$. Then
$$b=\phi(c_1,c_2)$$
so it lies in the image of $\phi: C_1=(C_1 \oplus C_1)/\Delta_{C_1}$.
\end{proof}
As a result of this lemma, we get an extension class $f_{12} \in \operatorname{Ext}(B_3,C_1)$ corresponding to $\BF$. This measures, in a sense, the obstruction to having a exact sequence involving $\BB_1$, $\BB_2$ and $B_3$.
One can also get extension classes $e_1$ and $e_2$ in $\operatorname{Ext}(B_3,C_1)$ by pushing forward the extensions $E_j$ under the maps $\pi_j$. From the construction of the map $\phi$, one has the following corollary of the above lemma.
\\
\begin{cor} Let $e_1$, $e_2$ and $f_{12}$ be the extensions in $\operatorname{Ext}(B_3,C_1)$ described above. Then
$$f_{12}=e_1 \ominus_B e_2$$
\label{newextensions}
\end{cor}
In the next section we apply these constructions in our particular case to get the extension class we want.
\subsection{The extension $e^4_{QR,P}$}
In this section we construct an extension $e^4_{QR,P}$ in $\operatorname{Ext}(\otimes^2 H^1(C),H^1(C))$ which generalises the element $e^4_{Q,P} \ominus_B e^4_{R,P}$ constructed by Colombo. Recall that we have an exact sequence
$$E^3_{Q,P}: 0 \longrightarrow H^1(C) \longrightarrow (J_{Q,P}/J_{Q,P}^3)^* \longrightarrow \otimes^2 H^1(C) \longrightarrow 0$$
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and a similar one for $E^3_{R,P}$. Also, we have the sequence
%
$$E_{Q,P}^4: 0 \longrightarrow (J_{Q,P}/J_{Q,P}^3)^* \longrightarrow (J_{Q,P}/J_{Q,RP}^4)^* \longrightarrow (J_{Q,P}^3/J_{Q,P}^4)^* \longrightarrow 0$$
and a similar one for $E^4_{R,P}$. This gives us a diagram as in Lemma \ref{rabilemma}, with $B_1^1=(J_{Q,P}/J_{Q,P}^3)^*$, $B_1^2= (J_{R,P}/J_{R,P}^3)^* $, $B_2^1= (J_{Q,P}/J_{Q,P}^4)^* $ and $B_2^2= (J_{R,P}/J_{R,P}^4)^* $
$$
\begin{CD}
&&0\\&&@VVV\\
&&H^1(C)\\
&&@VVi_jV\\
0 @>>> (J_{Q,P}/J_{Q,P}^3)^* @>f_j>> (J_{Q,P}/J_{Q,P}^4)^*@>p_j>> \otimes^3 H^1(C) @>>>0 \\
&&@VV\pi_j V \\
&&\otimes^2 H^1(C)\\
&&@VVV\\
&&0
\end{CD}
$$
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Let $E^4_{QR,P}$ denote the generalised Baer difference as in Lemma \ref{rabilemma} and $\BF_{QR}=E^4_{QR,P}/E^3_{QR,P}$. We get an exact sequence
$$
\begin{CD}
&&&&&&0\\
&&&&&&@VVV\\
&&&&&&\otimes^2 H^1(C)\\
&&&&&&@VV\phi V\\
0 @>>> E^3_{QR,P} @>f>> E^4_{QR,P} @>\eta>> \BF_{QR} @>>>0 \\
&&&&&&@VV\bar{p}V \\
&&&&&&\otimes^3 H^1(C)\\
&&&&&&@VVV\\
&&&&&&0
\end{CD}
$$
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We know from Lemma \ref{splitting} that the extension $\frac{N}{2g_C} E^3_{QR,P}$ splits -- so
%
$$\frac{N}{2g_C} E^3_{QR,P} \simeq \otimes^3 H^1(C) \oplus H^1(C)$$
%
Let $e^{23}_{Q,P}$ and $e^{23}_{R,P}$ denote the extensions in $\operatorname{Ext}( \otimes^3 H^1(C), \otimes^2 H^1(C))$ determined by pushing forward the extensions $E^4_{Q,P}$ and $E^4_{R,P}$ respectively. From Corollary \ref{newextensions} we have that
%
$$\BF_{QR}=e^{23}_{Q,P} \ominus_B e^{23}_{R,P}$$
%
From Rabi \cite{rabi}, Corollary 3.3, the extension $e^{23}_{Q,P}$ is
%
$$e^{23}_{Q,P}=E^3_{Q,P} \otimes H^1(C) \oplus_{B} H^1(C) \otimes E^3_{Q,P}$$
and similarly for $e^{23}_{R,P}$. Hence their difference $\BF_{QR}$ splits rationally as well! Precisely,
%
$$\frac{N}{2g_C} \BF_{QR} \simeq \otimes^3 H^1(C) \oplus \otimes ^2 H^1(C)$$
%
Hence from Lemma \ref{rabilemma} we get
%
$$ \frac{N}{2g_C} E^4_{QR,P} \in \operatorname{Ext}( \otimes^3 H^1(C) \oplus \otimes^2 H^1(C), \otimes^3 H^1(C) \oplus H^1(C))$$
%
From the K\"unneth theorem,
%
$$\operatorname{Ext}( \otimes^3 H^1(C) \oplus \otimes^2 H^1(C), \otimes^3 H^1(C) \oplus H^1(C))= \prod_{ i \in \{2,3\}, j \in\{1,3\}} \operatorname{Ext}(\otimes^i H^1(C),\otimes^j H^1(C))$$
%
Define
%
$$e^4_{QR,P} \in \operatorname{Ext}(\otimes^3 H^1(C), H^1(C))$$
to be the projection on to that component. Note that if $C$ is hyperelliptic, this class $e^4_{QR,P}$ is precisely the class $e^4_{QR,P}=e^4_{Q,P} \ominus_B e^4_{R,P}$ constructed by Colombo.
\subsection{Statement of the main theorem}
Armed with the class $e^4_{QR,P} \in \operatorname{Ext}(\otimes^3 H^1(C), H^1(C))$ we can proceed as in Colombo. We first pull back the class using the map $J_{\Omega}$ to get a class in
$$J_{\Omega}^*(e^4_{QR,P}) \in \operatorname{Ext}(H^1(C)(-1),H^1(C)).$$
Tensoring with $H^1(C)$ we get a class in
%
$$J_{\Omega}^*(e^4_{QR,P}) \otimes H^1(C) \in \operatorname{Ext}( \otimes^2 H^1(C)(-1), \otimes^2 H^1(C)).$$
%
Once again pulling back using the map $\beta:2g_C\ZZ(-1) \rightarrow \otimes^2 H^1(C)$ gives us a class in
$$\epsilon_{QR,P}^4 \in \operatorname{Ext}(\ZZ(-2),\otimes^2 H^1(C)) \subset \operatorname{Ext}(\ZZ(-2),H^2(C \times C))$$
Our main theorem is
\begin{thm} Let $C$ be a smooth projective curve and $Q$ and $R$ be two points such that there is a function $f_{QR}$ with $\operatorname{div}(f_{QR})=NQ-NR$ for some $N$ . Let $P$ be a third point on $C$. Normalize $f_{QR}$ so that $f_{QR}(P)=1$. Let $Z_{QR}=Z_{QR,P}$ be the element of the motivic cohomology group $H^3_{\M}(C \times C, \ZZ(2))$ constructed above. Let $\epsilon^4_{QR,P}$ be the extension in $\operatorname{Ext}_{MHS}(\ZZ(-2),\otimes^2 H^1(C))$ constructed above. Then
%
$$\epsilon^4_{QR,P}=(2g_C+1) \operatorname{reg}_{\Q}(Z_{QR}) $$
in $\operatorname{Ext}_{MHS}(\ZZ(-2),\otimes ^2 H^1(C ))$.
\label{mainthm}
\end{thm}
In other words our theorem states that the regulator of a natural cycle in the motivic cohomology group of a product of curves, being thought of as an extension class is the same as that as a natural extension of MHS coming from the fundamental group of the curve. In fact, it is an extension of {\em pure} Hodge structures.
\begin{rem} The dependence on $P$ is not so serious. If we do not normalise $f_{QR}$ with the condition that $f_{QR}(P)=1$ then one has to add an expression of the form $\log(f_{QR}(P)) \int_{C} \cdot$ to the term --- and this corresponds to adding a {\em decomposable} element of the form $(\Delta_C,\log(f_{QR}(P)))$ to our element $Z_{QR}$.
\end{rem}
\subsection{Carlson's representatives}
The proof of the above theorem will follow by showing that they induce the same current. For that we have to understand the how the extension class induces a current. This comes from understanding the Carlson representative. In the section we once again follow Colombo \cite{colo} and adapt her arguments to our situation.
If $V$ is a MHS all of whose weights are negative, then the {\em Intermediate Jacobian of $V$} is defined to be
$$J(V) = \frac{ V_{\C}}{F^0V_{\C} \oplus V_{\ZZ}}$$
This is a generalised torus - namely a group of the form $\C^a/\ZZ^b \simeq (\C^*)^{b} \times (\C)^{a-b}$ for some $a$ and $b$.
An extension of mixed Hodge structures
$$0 \longrightarrow A \stackrel{\iota}{\longrightarrow} H \stackrel{\pi}{\longrightarrow} B \longrightarrow 0$$
is called {\em separated} if the lowest non-zero weight of $B$ is greater than the largest non-zero weight of $A$. This implies that $\operatorname{Hom}(B,A)$ has negative weights. Carlson \cite{carl} showed that
$$\operatorname{Ext}_{MHS}(B,A) \simeq J(\operatorname{Hom}(B_{\C},A_{\C}))$$
This is defined as follows. As as extension of {\em Abelian groups}, the extension splits. So one has a map $r_{\ZZ}:H \rightarrow A$ which is a retraction --- namely $r_{\ZZ}\circ \iota=id$. Let $s_F$ be a section in $\operatorname{Hom}(B_{\C},H_{\C})$ preserving the Hodge filtration. Then the Carlson representative of an extension is defined to be the class of
$$r_{\ZZ} \circ s_F \in J(\operatorname{Hom}(B_{\C},A_{\C}))$$
\subsection{The Carlson representative of $\epsilon^4_{QR}$}
We now describe the explicitly the Carlson representative of the extension $\epsilon^4_{QR}$ constructed in the previous section. This is done in three steps, first for $e^4_{QR}$ and then for its various pullbacks and push forwards to obtain that for $\epsilon^4{QR}$.
We first describe the Carlson representative of the extension
$$e^4_{QR,P} \in \operatorname{Ext}_{MHS}(\otimes^3 H^1(C),H^1(C)).$$
Let $P,Q,R$ be as above. Fix a set of loops $\alpha_1,\alpha_2,\dots ,\alpha_{2g}$ based at $P$ in $C_{Q,R}=C\backslash\{Q,R\}$ such that they give a symplectic basis for $H^1(C)$ - so the intersection matrix is of the form
$$\begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}.$$
Let $dx_i$ be the dual basis of this basis. We may assume that the $1$-forms $dx_i$ are harmonic. Let
$$c(i)=\begin{cases} 1 &\text{ if } i\leq g_C\\
-1 & \text{ if } i>g_C\end{cases}
$$
and let
$$\sigma(i)=i+c(i)g_C$$
so one has
$$\int_{\alpha_i} dx_j=c(i)\delta_{j\sigma(i)} $$
where $\delta_{ij}$ is the Konecker $\delta$ function.
From the above description, we have that the Carlson representative is given by
$$s_F \circ r_{\ZZ} \circ p_1$$
where $p_1$ is the projection of $\frac{N}{2g_C} E^3_{QR,P} \longrightarrow H^1(C)$.
To describe $s_F$ we need a little more. Let $\ominus_{\tilde{B}}$ be the generalised Baer difference. Let
$$s_F: \otimes^3 H^1(C) \longrightarrow \frac{N}{2g_C}E^4_{QR,P} \simeq \frac{N}{2g_C}\left( (J_{Q,P}/J_{Q,P}^4)^* \ominus_{\tilde{B}} (J_{R,P}/J_{R,P}^4)^* \right)$$
be the section preserving the Hodge filtration given by
$$s_F(dx_i \otimes dx_j \otimes dx_k )=(I^{ijk}_Q,I^{ijk}_R).$$
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Here $I^{ijk}_{\bullet} \in (J_{\bullet,P}/J_{\bullet,P}^4)^*$ for $\bullet \in \{Q,R\}$ are two iterated integrals with
\begin{equation}
I^{ijk}_{\bullet}= \frac{N}{2g_C} \left(\int dx_i dx_j dx_k + dx_i \mu_{jk,\bullet}+\mu_{ij,\bullet}dx_k + \mu_{ijk,\bullet} \right)
\label{iteratedformula}
\end{equation}
where $\mu_{ij,\bullet}$, $\mu_{jk,\bullet}$ and $\mu_{ijk,\bullet}$ are smooth, logarithmic $(1,0)$ forms on $C_{\bullet}$ such that
\begin{itemize}
\item $d\mu_{jk,\bullet}+dx_j \wedge dx_k=0$
\item $d\mu_{ij,\bullet}+dx_i \wedge dx_{j}=0$
\item $dx_i \wedge \mu_{jk,\bullet} - \mu_{ij,\bullet}\wedge dx_k +d\mu_{ijk,\bullet}=0.$
\end{itemize}
To compute the element of $\operatorname{Hom}(\otimes^3 H^1(C)_{\C}, H^1(C)_{\C})$ obtained as the projection under $p_1$, we describe it as an element of $H_1(C)_{\C}^*=\operatorname{Hom}(H_1(C)_{\Q},\C)$. The map from
$$H^1(C) \longrightarrow (H^1(C) \oplus H^1(C))/\Delta_{H^1(C)}$$
is given by
$$x \longrightarrow (x,-x)$$
Further, if $\alpha$ is a smooth loop based at $P$, the class in $H_1(C)$ corresponding to it is $\alpha -1$.
So one has $s_F \circ r_{\ZZ} \circ p_1 \in \operatorname{Hom}(\otimes^3 H^1(C)_{\C},H^1(C)_{\C})$
$$s_F \circ r_{\ZZ} \circ p_1(dx_i\otimes dx_j\otimes dx_k)(\alpha)=\int_{\alpha-1} I_Q - \int_{\alpha-1} I_R$$
$$ =\frac{N}{2g_C} \left(\int_{\alpha-1} dx_i (\mu_{jk,Q}-\mu_{jk,R})+(\mu_{ij,Q} -\mu_{ij,R})dx_k + (\mu_{ijk,Q}-\mu_{ijk,R}) \right)$$
\begin{rem} We can choose the logarithmic forms $\mu_{ij,\bullet}$ and $\mu_{ijk,\bullet}$, for $\bullet \in \{Q,R\}$, satisfying the following
\begin{itemize}
\item $\mu_{ij,\bullet}=-\mu_{ji,\bullet}$.
\item For $|i-j|\neq g_{C}$, $\mu_{ij,\bullet}$ is smooth on $C$, as $dx_i \wedge dx_j=0$. As $H^2(C_{Q,R},\ZZ)=0$ and $\mu_{ij,\bullet}$ is smooth, it is orthogonal to all closed forms, that is, $\mu_{ij,\bullet}\wedge dx_k=0$. If $dx_i$ is harmonic, then $\mu_{ij,\bullet}\wedge dx_k=0$.
\item $\mu_{i\sigma(i),\bullet}$ has a logarithmic singularity at $\bullet$ with residue $c(i)$.
\item $\mu_{ij,Q}-\mu_{ij,R}=0$ if $|i-j| \neq g_C$.
\item $\mu_{i\sigma(i),Q}-\mu_{i\sigma(i),R}=\frac{c(i)}{N}d\log(f)$, where $f=f_{QR}$ is a function such that $\operatorname{div}(f)=NQ-NR$. We can normalise $f_{QR}$ once again by requiring that $f_{QR}(P)=1$.
\label{properties}
\end{itemize}
\end{rem}
In terms of the basis of harmonic forms of $H^1(C)$, $\Omega \in \otimes^2 H^1(C)$ is expressed as
$$\Omega=\sum_{i=1}^{g_C} dx_i \otimes dx_{(i+g_C)} - dx_{(i+g_C)} \otimes dx_i=\sum_{i=1}^{2g_C} c(i)dx_i \otimes dx_{\sigma(i)}$$
and under the map
$$\otimes^2 H^1(C) \stackrel{\cup}{\longrightarrow} H^2(C)\simeq \ZZ(-1)$$
$$ \sum_{i=1}^{2g_C} c(i) dx_i \cup dx_{\sigma(i)} =2g_C\cdot \omega_C$$
where $\omega_C$ is the generator of $H^2(C)\simeq \ZZ(-1)$. With the choices of $\mu_{ij,\bullet}$ and $\mu_{ijk,\bullet}$ as above, we have the following theorem
\\
\begin{thm} Let $G_{QR,P} \in \operatorname{Hom}(H^1(C)(-1)_{\C},H^1(C)_{\C})$ be the Carlson representative corresponding to the extension class $J_{\Omega}^*(e^4_{QR,P})$. It is given by
$$G_{QR,P}(dx_k\otimes \Omega) (\alpha_j)=\int_{\alpha_j} \frac{(2g_C+1)}{2g_C}\log(f)dx_k + \frac{N}{2g_C} \int_{\alpha_j} W(dx_k) $$
in $J(\operatorname{Hom}(H^1(C)_{\Q}(-1),H^1(C)_{\Q}))$, where
$$W(dx_k)=\sum_{i=1}^{2g_C} c(i)( \mu_{k i \sigma(i),Q} -\mu_{ki\sigma(i),R})$$
\end{thm}
\begin{proof} Let $S_F$ denote the map $S_F=s_F\circ J_{\Omega}:H^1(C)(-1) \rightarrow e^4_{QR,P}$. This is given by
$$S_F(dx_k \otimes \Omega)=\sum_{i=1}^{2g_C} c(i) s_F( dx_k \otimes dx_i \otimes dx_{\sigma(i)})$$
From \eqref{iteratedformula} one has
$$S_F(dx_k \otimes \Omega)=\left( \sum_{i=1}^{2g_C} c(i) \int I_Q^{ki\sigma(i) },\sum_{i=1}^{2g_C} c(i) \int I_R^{ki\sigma(i)} \right)$$
Evaluating on a path $\alpha_j$ based at $P$ using the maps described above, this is
$$\sum_{i=1}^{2g_C} \int_{\alpha_j-1} c(i) \left( I_Q^{ki\sigma(i)} - I_R^{k i \sigma(i)} \right)$$
From Remark \ref{properties}, the leading terms and several of the lower order terms cancel out and
$$ \mu_{ki,Q}-\mu_{ki,R}=-c(i) \delta_{k\sigma(i)} d\log(f)/N $$
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and finally
%
$$ \mu_{i\sigma(i),Q)}-\mu_{i\sigma(i),R}=c(i)d\log(f)/N$$
%
so what remains is
%
$$ \frac{N}{2g_C} \left( \sum_{i=1}^{2g_C} \int_{\alpha_j-1} \frac{dx_k}{N}\frac{ df}{f} - \int_{\alpha_j-1} \frac{df}{f}\frac{ dx_k}{N} + \sum_{i=1}^{2g_C} c(i) \int_{\alpha_j-1} \left(\mu_{k i \sigma(i),Q}-\mu_{k i \sigma(i),R} \right) \right)$$
Let
$$W(dx_k)= \sum_{i=1}^{2g_C} c(i) \left(\mu_{k i \sigma(i),Q}-\mu_{k i \sigma(i),R} \right) $$
Recall that $\gamma=f^{-1}([0,\infty])$. On $C\backslash \gamma$, $d\log(f)$ is exact. So if $\alpha_j \cap \gamma=\emptyset$ then we can evaluate the integral using Lemma \ref{basicproperties}(3). If $\alpha_j \cap \gamma \neq \emptyset$, one has to do the computation on a path lifting of $\alpha_j$ on a covering of $C$ where $d\log(f)$ is exact. The difference in the two integrals is given by a multiple of $2\pi i \int_{\alpha_j} dx_k$ -- hence is in $\operatorname{Hom}_{\ZZ}(H^1(C)(-1), H^1(C))$ -- which is $0$ in the intermediate Jacobian.
Hence we have, using Lemma \ref{basicproperties}(3) and the fact that we have chosen $f$ with $f(P)=1$,
$$\int_{\alpha_j-1} \frac{dx_k}{N}\frac{ df}{f} = \frac{1}{N} \int_{\alpha_j-1} \log(f)dx_k$$
and
$$\int_{\alpha_j-1} \frac{df}{f}\frac{dx_k}{N}= -\frac{1}{N} \int_{\alpha_j-1} \log(f)dx_k $$
So the integral is
$$ \frac{2g_C+1}{N} \int_{\alpha_j-1} \log(f)dx_k + \int_{\alpha_j} W(dx_k)$$
Multiplying this by the factor $\frac{N}{2g_C}$ gives us the final result.
\end{proof}
\begin{rem} It is convenient to have the iterated integral expression for the Carlson representative as well so we note it here
$$G_{QR,P}(dx_k\otimes \Omega) (\alpha_j)=\frac{2g_C+1}{2g_C} \int_{\alpha_j-1} \frac{df}{f} dx_k + \frac{N}{2g_C} \int_{\alpha_j-1}W(dx_k) $$
\end{rem}
We have computed the Carlson representative of our class in $\operatorname{Ext}(H^1(C)(-1),H^1(C))$. We now tensor with $H^1(C)$ and pull back using the map $\beta:2g_C\ZZ(-1)\longrightarrow \otimes^2 H^1(C)$. This gives us an element of $\operatorname{Ext}(2g_C \ZZ(-2), \otimes^2 H^1(C))$.
\\
\begin{lem} The Carlson representative of the class in $\operatorname{Ext}(\ZZ(-2),\otimes^2 H^1(C))$ is given by
$$F_{QR,P} = (G_{QR,P} \times Id) \circ \beta$$
in $(\otimes^2 H^1(C))^*$. On an element $\alpha_j \otimes \alpha_k$, since $\Omega=\beta(1)$, it is given by
$$ F_{QR,P}(\Omega)(\alpha_j \otimes \alpha_k)= \int_{\alpha_j} \frac{2g_C+1}{N}\log(f) dx_k + \int_{\alpha_j} W(dx_k) $$
\end{lem}
\begin{proof} Recall that
$$\Omega=\sum_i^{2g_C} c(i)dx_i \otimes dx_{\sigma(i)}$$
From above we have
$$(G_{QR.P} \times Id )(\Omega)(\alpha_j \otimes \alpha_k) = \sum_i^{2g_C} c(i) G_{QR,P}(dx_i \otimes \Omega)(\alpha_j) \cdot Id(dx_{\sigma(i)}) (\alpha_k)$$
From the choice of $\alpha_k$ one has
$$Id(dx_{\sigma(i)}) (\alpha_k)= \int_{\alpha_k} dx_{\sigma(i)} =c(k)\delta_{ki}$$
Hence, in the sum above, precisely one term survives -- when $i=k$, and we have
$$(G_{QR.P} \times Id ) (\Omega)(\alpha_j \otimes \alpha_k)= c(k)^2 G_{QR,P}(dx_k \otimes \Omega)(\alpha_j)$$
Since $c(k)^2=1$, we get
$$F_{QR,P}(\Omega)(\alpha_j \otimes \alpha_k)= G_{QR,P}(dx_k\otimes \Omega)(\alpha_j)=$$
$$= \int_{\alpha_j} \frac{2g_C+1}{2g_C}\log(f) dx_k + \frac{N}{2g_C}\int_{\alpha_j} W(dx_k) $$
\end{proof}
We now recall a lemma due to Colombo which relates integrals over the curve $C- \gamma$ with integrals over paths. This is crucial in relating the two expressions for the regulator.
\\
\begin{lem}[Colombo, Prop. 3.3]\cite{colo} Let $\gamma$ be the curve $f^{-1}([0,\infty])$. Let $\alpha$ be a smooth, simple loop on $C$ transverse to $\gamma$. Let $\phi$, $\psi$ and $\omega$ be three smooth $1$-forms on $C$ such that $\phi$, $\psi$ and $\Theta=(\log(f)\psi+\omega)$ are closed and $\phi$ is the Poincar\'{e} dual of the class of $\alpha$. Then
$$\int_{\alpha} \Theta=\int_{C-\gamma} \phi \wedge \Theta + 2\pi i \int_{\gamma} \phi\psi$$
\end{lem}
\begin{proof}
We first recall the following explicit construction of a differential form $\eta=\eta_{\alpha}$ which is the Poincar\'e dual of $\alpha$ as in \cite{frka}, II, Section 3.3. Let $\Omega$ be a tubular neighbourhood of $\alpha$ obtained by covering $\alpha$ by a finite number of co-ordinate discs. We can assume $\Omega$ is an annulus and $\Omega-\alpha$ is the union of two annuli $\Omega^+ \cup \Omega^-$. Assume $\alpha$ is oriented so that $\Omega^-$ is to the left.
Let $\Omega_0$ and correspondingly $\Omega^+_0$ and $\Omega^-_0$ be sub-annuli of the $\Omega$. We can find a $\CC^{\infty}$-function $F$ on $C-\alpha$ such that
$$F(z)=\begin{cases} 1 & \text { if } z \in \Omega^-_0 \\ 0 & \text{ if } z \in C-\Omega^-\end{cases}$$
Let $\eta=\eta_{\alpha}$ be defined as follows
$$\eta=\begin{cases} dF & \text{ on } \Omega-\alpha \\ 0 & \text{ on } (C-\Omega) \cup \alpha \end{cases}$$
The $\eta$ is a smooth, closed, differential form with compact support in $\Omega$ which is the Poincar\'e dual of $\alpha$.
Now $\phi$ is also a closed form dual to $\alpha$. Hence
$$\phi=\eta+dg$$
with $dg$ exact. So we can break the integral in to two parts --
$$\int_{C-\gamma} \phi \wedge \Theta=\int_{C-\gamma} (\eta+dg) \wedge \Theta= \int_{C-\gamma} \eta \wedge \Theta + \int_{C-\gamma} dg \wedge \Theta.$$
We first tackle the second term. One has
$$dg\wedge \Theta=dg \wedge (\log(f)\psi+\omega)=d(g\log(f) \psi + \omega)$$
which is exact. So one can evaluate that integral by Stokes theorem applied to the Riemann surface with boundary, $C-\gamma$. The boundary $\partial (C-\gamma)$ consists of two copies of $\gamma$, $\gamma_1$ and $\gamma_2$ on which the $\log$ differs by $2\pi i$, so one has
$$\log(\gamma_1(t))-\log(\gamma_2(t))=2\pi i$$
%
Conventionally, one orients the boundary of $\partial(\CP^1-[0,\infty])$, which consists of two copies of the line $[0,\infty]$ in such a manner that the plane is always to the {\em right}. With that orientation and its induced orientation on $\gamma$ one has
%
$$\partial(C-\gamma)=-\gamma_i \cup \gamma_2 $$
%
Hence, by Stokes Theorem,
%
\begin{equation}
\int_{C-\gamma} dg \wedge (\log(f)\psi + \omega)=\int_{-\gamma_1} g(\log(f) \psi + \omega) + \int_{\gamma_2} g(\log(f)\psi+\omega)=-2\pi i \int_{\gamma} g\psi
\label{dg}
\end{equation}
Let $P$ be the base point of $\alpha$. We can choose $g$ such that $g(P)=0$ hence, using Lemma \ref{basicproperties}, the above expression can be rewritten as
$$\int_{\gamma} g\psi = \int_{\gamma} (g-g(P))\psi=\int_{\gamma} dg\psi $$
\vspace{\baselineskip}
We now deal with the other part, namely $\int_C \eta\wedge \Theta$. Recall that $\eta$ is the Poincar\'e dual of $\alpha$. First suppose $\alpha \cap \gamma=\emptyset$. Then we can choose $\Omega$ above such that $\Omega \cap \gamma=\emptyset$. So $\Theta$ is then a closed form well defined one the support of $\eta$. Then by Poincar\'e duality one has
\begin{equation}
\int_{C-\gamma} \eta \wedge \Theta=\int_{\alpha} \Theta=\int_{\alpha} ( \log(f) \psi + \omega )
\label{eta}
\end{equation}
Further, since $\phi=\eta+dg$ and $\eta|_{\gamma}=0$ one has $\phi|_{\gamma}=dg|_{\gamma}$. Hence adding equations \eqref{dg} and \eqref{eta} we get the result.
If $\alpha \cap \gamma \neq \emptyset$ then $\log(f)$ is no longer well defined on $\Omega$. Hence we have to compute the integral on the disjoint union of regions which make up $\Omega'=\Omega - \gamma$. Since $\eta$ is supported on $\Omega$ one has
$$\int_{C-\gamma} \eta \wedge \Theta=\int_{\Omega'} \eta \wedge \Theta$$
%
Since $\eta=dF$ on $\Omega-\alpha$ and $0$ elsewhere, we have
$$\partial (\Omega'-\alpha)=\alpha \cup (-\gamma_1 \cap \Omega^-) \cup (\gamma_2 \cap \Omega^-)$$
So from Stokes theorem, the integral becomes
$$\int_{\Omega'-\alpha} dF \wedge \Theta = \int_{\alpha} \Theta + \int_{\gamma \cap \Omega^-} \Theta.$$
As in Equation \eqref{dg} above,
$$ \int_{\gamma \cap \Omega^-} \Theta= \int_{\gamma \cap \Omega^-} (\log(f)\psi+\omega)=-2\pi i \int_{\gamma \cap \Omega^-} dF\psi $$
Since $\eta=0$ outside $\Omega^-$ and $\eta|_{\gamma}=dF|_{\gamma}$ and $\phi|_{\gamma}=\eta|_{\gamma}+ dg|_{\gamma}$ one gets
$$\int_{\alpha} \Theta=\int_{C-\gamma} \phi \wedge \Theta + 2\pi i \int_{\gamma} \phi\psi$$
\end{proof}
We now apply this in the case of interest to us.
\\
\begin{cor} Choose $\alpha_j$ to be simple closed loops transverse to $\gamma$. Then we have
$$F_{QR,P}(\Omega)(\alpha_j \otimes \alpha_k)=(2g_C+1)c(j) )(\int_{C-\gamma} dx_{\sigma(j)} \wedge \left( \log(f))dx_k-\frac{N}{(2g_C+1)}W(dx_k)\right) $$
$$+ 2\pi i \int_{\gamma} dx_{\sigma(j)} dx_k )$$
\end{cor}
\begin{proof} One has $c(j) dx_{\sigma(j)}$ is dual to $\alpha_j$. Hence we can apply the above lemma with
\begin{itemize}
\item $\phi=c(j)dx_{\sigma(j)}$
\item $\psi=dx_k$
\item $\Theta=\log(f)dx_k-\frac{N}{(2g_c+1)}W(dx_k)$
\end{itemize}
Note that $\Theta$ is closed because
$$d\Theta=d\log(f)\wedge dx_k -\frac{N}{(2g_C+1)}dW(dx_k)$$
and
$$dW(dx_k)= \sum_{i=1}^{2g_C} c(i) d \left(\mu_{k i \sigma(i),Q}-\mu_{k i \sigma(i),R} \right) $$
Recall that
$$d\mu_{ijk,Q}=\mu_{ij,Q}\wedge dx_k -dx_i \wedge \mu_{jk,Q}$$
So the sum becomes
$$dW(dx_k)=\sum_{i=1}^{2g_C} c(i)\left( \left( \mu_{ki,Q} \wedge dx_{\sigma(i)} - dx_k \wedge \mu_{i\sigma(i),Q}\right) -\left( \mu_{ki,R} \wedge dx_{\sigma(i)} - dx_k \wedge \mu_{i\sigma(i),R}\right)\right) $$
$$=\sum_{i=1}^{2g_C} c(i) \left( (\mu_{ki,Q}-\mu_{ki,R}) \wedge dx_{\sigma(i)} - dx_k\wedge (\mu_{i\sigma(i),Q}-\mu_{i\sigma(i),R}) \right)$$
$$=c(\sigma(k))(\mu_{k\sigma(k),Q}-\mu_{k\sigma(k),R}) \wedge dx_k - \sum_{i=1}^{2g_C} c(i) \left(dx_k \wedge \frac{c(i)}{N} d\log(f)\right)$$
$$=c(\sigma(k))(\frac{c(k)}{N} d\log(f)) \wedge dx_k - \sum_{i=1}^{2g_C}c(i) \left( dx_k \wedge \frac{c(i)}{N} d\log(f)\right)$$
$$=\frac{(2g_C+1)}{N} d\log(f) \wedge dx_k$$
Hence it cancels out and we have $d\Theta=0$. Applying the proposition we have
$$\int_{\alpha_j} \log(f)dx_k-\frac{N}{(2g_C+1)}W(dx_k)=$$
$$=\int_{C-\gamma} c(j) dx_{\sigma(j)} \wedge \left( \log(f)dx_k-\frac{N}{(2g_C+1)}W(dx_k)\right) + 2\pi i \int_{\gamma} c(j) dx_{\sigma(j)} dx_k$$
Hence
$$F_{QR,P}(\Omega)(\alpha_j \otimes \alpha_k)=(2g_C+1)c(j) \left(\int_{C-\gamma} dx_{\sigma(j)} \wedge \left( \log(f)dx_k-\frac{N}{(2g_C+1)}W(dx_k)\right) + 2\pi i \int_{\gamma} dx_{\sigma(j)} dx_k \right)$$
\end{proof}
$F_{QR,P}(\Omega)$ determines an element of the intermediate Jacobian of $(\otimes^2 H^1(C))^*$
$$J(\otimes^2 H^1(C)^*) \simeq \frac{F^1(\otimes^2 H^1(C,\C))^*}{(\otimes^2 H^1(C,\ZZ))^*}$$
so to determine $F_{QR,P}(\Omega)$ it suffices to evaluate it on elements of $F^1(\otimes^2 H^1(C,\C))^*$, namely linear combinations of forms of the type $\zeta_i \otimes \alpha_j$ and $\alpha_i \otimes \zeta_j$, where $\{\zeta_i\}_{i=1}^{g}$ is a basis for the Poincare duals of the holomorphic $1$-forms, $H^{1,0}(C)$. Let $dz_j$ denote the dual of $\zeta_j$. We can choose the basis $\{\zeta_i\}$ such that it satisfies
$$\int_{\alpha_i} dz_j=\delta_{ij} \hspace{1in} 1\leq i\leq g$$
where $\{\alpha_i\}$ is the symplectic basis. Since $c(j)dx_{\sigma(j)}$ is dual to $\alpha_j$,
%
$$dz_j=dx_{j+g} + \sum_{i=i}^{g} A_{ji}dx_i \hspace{1in} \text { where } A_{ji}=\int_{\alpha_i} dz_j$$
\begin{prop}[Colombo, \cite{colo}, Prop 3.4] The map $F$ evaluated on elements of the form $\zeta_i \otimes \alpha_j$ is
%
$$F_{QR,P}(\Omega)(\zeta_i \otimes \alpha_j)=(2g_C+1) \left(\int_{C-\gamma} \log(f) dz_i \wedge dx_j + 2\pi i \int_{\gamma} dz_i dx_j \right)$$
In other words
$$dz_i \wedge W(dx_j)=0.$$
\end{prop}
\begin{proof} $dz_i$ and $W(dx_j)$ are both $(1,0)$ forms. Hence their wedge product is a $(2,0)$ form and is therefore $0$.
\end{proof}
In fact, the theorem holds for the other term as well.
\begin{prop} For a suitable choice of $\mu_{ijk,Q}$ and $\mu_{ijk,R}$ one has
$$W(dz_i):=W(dx_{(j+g)}) + \sum_{i=i}^{g} A_{ji}W(dx_i)=0$$
\end{prop}
\begin{proof} This is essentially the same as Colombo \cite{colo} Lemma 3.1.
\end{proof}
Hence we have
%
\begin{thm}
$$F_{QR,P}(\Omega)(\alpha_j \otimes \zeta_i)=(2g_C+1) \left(\int_{C-\gamma} \log(f) dx_j \wedge dz_i+ 2\pi i \int_{\gamma} dx_j dz_i\right)$$
\end{thm}
Comparing this with the regulator term in Theorem \ref{regform} we get
\begin{thm} Let $Z_{QR}$ be the motivic cohomology cycle constructed above and $\epsilon_{QR,P}^4$ the extension in $\operatorname{Ext}_{MHS}(\Q(2),\otimes^2 H^1(C))$. We used $\epsilon^4_{QR,P}$ to denote its Carlson representative as well. Then one has
$$ \langle\epsilon^4_{QR,P},\omega\rangle=(2g_C+1)\langle\operatorname{reg}_{\Q}(Z_{QR}),\omega\rangle$$
\end{thm}
\begin{proof} The Carlson representative is given by $F_{QR,P}$ and the result follows from comparing the two expressions.
\end{proof}
Recall that we have assumed in both cases that $f_{QR}(P)=1$. If we do not do that, then one has a term corresponding to the decomposable element $(\Delta_C,\log(f_{QR}(P)))$ that one has to account for. However, if we work modulo the decomposable cycles we can ignore that term.
We can also consider the regulator to the Real Deligne cohomology $H^3_{\D}(C \times C_{/\R},\R(2))$ which is the same as the group $\operatorname{Ext}_{\R-MHS}(\R(2), \otimes^2 H^1(C))$ in the category of $\R$-mixed Hodge structures with the action of the infinite Frobenius. We can take the realisation of the extension $\epsilon^4_{QR,P}$ in that extension group and one has
\begin{thm} The real regulator is
$$\operatorname{reg}_{\R}(Z_{QR})=(2g_C+1) \epsilon_{QR,P}^4$$
and on a $(1,1)$ form $dz_i \otimes d\bar{z}_j$ it is
$$\langle\operatorname{reg}_{\R}(Z_{QR}), dz_i \otimes d\bar{z}_j\rangle=\int_{\bar{\zeta}_j} \log|f^{\Delta}| dz_i$$
where $\bar{\zeta}$ is the Poincar\'{e} dual of $d\bar{z}_j$.
\end{thm}
\iffalse
\subsection{Degenerations of Algebraic Cycles}
Let $C$ be a curve. There are a few "natural" constructions of special cycles on self-product of $C$ --- the modified diagonal cycle on $C^3$, our cycle $Z_{QR,P}$ on $C^2$ and the elements of $K_2(C)$ constructed from functions on $C$. In this section we show how these cycles and their regulators are related.
\subsubsection{The modified diagonal cycle in $K_0( C \times C \times C) $} The modified diagonal cycle is defined as follows - If $P$ is a point on $C$, then one has the diagonals in $C \times C \times C$ ---
$$\Delta_{123}=(x,x,x)$$
$$ \Delta_{12}=(x,x,P), \Delta_{13}=(x,P,x), \Delta_{23}=(P,x,x)$$
$$\Delta_{1}=(x,P,P), \Delta_2=(x,P,x), \Delta_3=(P,P,x)$$
and the modified diagonal cycle is defined to be
$$Z_P=\Delta_{123}-\Delta_{12}-\Delta_{13}-\Delta_{23}+\Delta_1+\Delta_2+\Delta_3$$
and it turns out that this is homologically trivial --- so is an element of
$$CH^2_{\hom}(C^3)=H^3_{\M}(C^3,\Q(2))=\operatorname{Ext}_{{\mathcal {MM}}}(\Q(-2),h^3(C^3)).$$
In fact it lies in $\operatorname{Ext}_{\M}(\Q(-2), \otimes^3 h^1(C))$. The Abel-Jacobi image of this cycle is given by
$$AJ(Z_P)(\omega_1 \otimes \omega_2 \otimes \omega_3) = \int_{D} \omega_1 \otimes \omega_2 \otimes \omega_3$$
where $\omega_i$ are $1$-forms on $C$ and $D$ is a real $3$-cycle such that $\partial(D)=Z_P$. This has an iterated integral expression
$$AJ(Z_P)(\omega_1 \otimes \omega_2 \otimes \omega_3) = \int_{\gamma_1} \omega_2 \omega_3$$
where $\gamma_1$ is the Poincar\'{e} dual of the form $\omega_1$.
\subsubsection {Our cycle in $K_1( C \times C ) $ } Our cycle $Z_{QR,P}$ lies in $H^3_{\M}(\otimes^2 h^1(C),\Q(2))=\operatorname{Ext}_{{\mathcal {MM}}}(\Q(-2),\otimes^2 h^1(C))$. The regulator is given by
$$\langle\operatorname{reg}_{\Q}(Z_{QR,P},\omega_1 \otimes \omega_2\rangle=\int_{\gamma_1} \frac{df_{QR}}{f_{QR}} \omega_2$$
where $\gamma_1$ is the Poincar\'e dual of $\omega_1$.
\subsubsection{A cycle in $K_2(C)$ }There is another case we can consider. Suppose $S$ and $T$ are two more points on $C$ such that there is a function $g_{ST}$ with $\operatorname{div}(g_{ST})=M(S-T)$. Assume for definiteness $g_{ST}(P)=1$. Then we can consider the element
$$Z_{QR,ST,P}=\{f_{QR},g_{ST}\} \in H^1_{\M}(h^1(C),\Q(2))=\operatorname{Ext}_{{\mathcal {MM}}}(\Q(-2),h^1(C))$$
and its regulator is given by
$$\langle\operatorname{reg}_{\Q}(Z_{QR,ST,P}),\omega_1\rangle=\int_{\gamma_1} \frac{df_{QR}}{f_{QR}} \frac{dg_{ST}}{g_{ST}}$$
\vspace{\baselineskip}
\begin{tabular}{||l|l|l||}
\hline
Motivic Cohomology Group & Cycle & Regulator \\
\hline
$H^4_{\M}(C^3,\Q(2))$ & $Z_P$ & $\displaystyle{\langle AJ(\Delta_P),\omega_1 \otimes \omega_2 \otimes \omega_3 \rangle = \int_{\gamma_1} \omega_2 \omega_3}$ \\
\hline
$H^3_{\M}(C^2,\Q(2))$ & $Z_{QR,P}$ & $\langle \displaystyle{\operatorname{reg}_{\Q}(Z_{QR},\omega_1 \otimes \omega_2\rangle = \int_{\gamma_1} \frac{df_{QR}}{f_{QR}} \omega _2}$ \\
\hline
$H^2_{\M}(C,\Q(2)) $ & $ Z_{QR,ST,P} $ & $ \displaystyle{\langle\operatorname{reg}_{\Q}(\{f_{QR},g_{ST}\}),\omega_1\rangle=\int_{\gamma_1} \frac{df_{QR}}{f_{QR}} \frac{dg_{ST}}{g_{ST}} }$\\
\hline
\end{tabular}
\vspace{\baselineskip}
One would like to view them as a sort of degeneration
$$K^2_0(C\times C \times C) \longrightarrow K_1^2(C \times C) \longrightarrow K_2^2(C) \longrightarrow K_3(\CP^1)$$
We do not quite understand this but remark that in the case when $C$ is a modular curve, connection will $L$-functions comes through a similar observation that the height of the modified diagonal cycle is related to the $L$-function of the Rankin-Triple product of three cusp forms, the regulator of the Beilinson element in $K_1$ is related to the usual Rankin-Selberg convolution $L$-function of two cusp forms, which can the though of as a triple product with two cusp forms and one Eisenstein series, and finally the regulator of the Beilinson element in $K_2$ of the modular curve is related to the Rankin-Selberg convolution $L$-function of a cusp form with an Eisenstein series. Finally, the value of the regulator of an element of $K_3^2$ of a cyclotomic field is also related to the Rankin-Selberg product of three Eisenstein series.
There is one situation where this degeneration is understood. This is the case of the Ceresa cycle in the $H^2_{\M}(J(C),\Q(2))$ where $J(C)$ is the Jacobian of a genus 3 curve $C$. This was described by Collino \cite{coll}. Namely - the genus three curve $C$ degenerates to a nodal genus two curve $D_{QR}$. Under this degeneration, the Ceresa cycle degenerates in to the Collino cycle in $H^3_{\M}(J(\tilde{D}_{QR}),\Q(2))$ where $\tilde{D}_{QR}$ is the normalisation. A further degeneration of the curve into a genus $1$ curves with two nodes $E_{QR,ST}$ causes the Collino cycle to degenerate to the element of $H^2_{\M}(\tilde{E}_{QR,ST},\Q(2))$. Finally, this element can be seen to degenerate to an element of $K_3$ of a number field.
As this paper neared completion we became aware of this paper by J.N.N. Iyer and S. M\"uller-Stach \cite{iymu} where they indeed compute the degeneration of the modified diagonal cycle in the case when $C$ is a genus $3$ curve and show that it degenerates to the higher Chow cycles as expected.
\fi
\subsection{A generalization}
We will apply this calculation to compute the regulator in a slightly more general situation, which is particularly relevant to our applications to modular curves and Fermat curves. Let $C$ be a curve and $S \subset C$ a finite subset of points on $C$ such that any divisor of degree $0$ supported on $S$ is torsion. Examples of such sets include the set of points $\{Q,R\}$ above, cusps on a modular curve and the $3N$ `trivial solutions' on the Fermat curve $F_N: X^N+Y^N=Z^N$.
Suppose $f$ is a function whose divisor is supported on $S$,
$$\operatorname{div}(f)=\sum_{P\in S} a_PP$$
then we can construct a motivic cohomology cycle $Z_{f}$ as follows. Define a {\em simple} function to be a function $f_{QR}$ with
$$\operatorname{div}(f_{QR})=NQ-NR$$
with $Q$ and $R$ in $S$. Using the fact that any divisor of degree $0$ on $S$ is torsion, one can decompose $f$ into a product
$$f^k=\prod f_{QR}$$
%
for some simple functions $f_{QR}$ and natural numbers $k$. This is far from unique. Then one can easily check that the cycle
%
$$Z_f=k(\Delta_C,f) - \left( \sum (Q \times C, f_{QR} \times C) + (C \times R, C \times f_{QR}) \right)$$
%
where $\Delta_C$ is the diagonal, is an element of $H^3_{\M} (C \times C, \Q(2))$. One can see that
%
$$Z_f-\sum_{Q,R} Z_{QR}=0 \in H^3_{\M}(C \times C, \Q(2))$$
%
Hence one has
\begin{thm} Let $Z_f$ be the motivic cohomology class corresponding to a function $f$ as above. Then there is an extension class
%
$$\epsilon^4_f \in \operatorname{Ext}_{MHS} (\Q(-2), \otimes^2 H^1(C))$$
%
which corresponds to the regulator of $Z_f$. This class is given by
%
$$\epsilon^4_f=\sum \epsilon^4_{QR}$$
%
\end{thm}
While this theorem is immediate from the earlier considerations, it will be useful in the next section on modular curves.
\section{ Modular curves and the Beilinson conjectures}
One of the few cases where the Beilinson conjectures are known is that of $H^2_{\M}(X,\Q(2))$ where $X$ is a modular curve. To prove it he first decomposed the motive of the modular curve in to motives of modular forms of weight $2$. The $L$-function also decomposes. He then constructed an element of $H^2_{\M}(X,\Q(2))$ and showed that the regulator of the projection on to the various modular form components of this element was related to the special value of the $L$-function. In \cite{king}, Kings constructs an extension in the category of mixed motives defined by Huber which corresponds to the element constructed by Beilinson, in the sense that the realisation of Kings' extension, which is an extension in the category of $\R$-mixed Hodge structures with an $F_{\infty}$-action, is the same as the regulator to the real Deligne cohomology of Beilinson's element.
The other case which Beilinson proved in his seminal paper \cite{beil}was that of $H^3_{\M}(X \times X,\Q(2))$. In this section we show that here too one can construct an extension in the category of mixed Hodge structures coming from the fundamental group which corresponds to Beilinson's element. As remarked earlier, this can likely be made to work at the level of Nori's mixed motives itself using the description of Deligne-Goncharov.
\subsection{An explicit description of Beilinson's element}
We give an explicit description of Beilinson's element. This is done in \cite{basr}. Recall that a {\em modular unit} is a function on a modular curve with its divisor supported on the cusps. For $N$ a square free integer, let $X_0(N)$ denote the modular curve with level $N$ structure. Since $N$ is square free, the set of cusps is represented by points $P_d=[\frac{1}{d}]$ for $d|N$.
Let $\Delta(z)$ denote the Ramanujan $\Delta$-function, the classical modular form of weight $12$ for $SL_2(\ZZ)$. For $N$ a square free integer let
$$\Delta_N(z):=\prod_{d|N} \Delta\left(\frac {Nz}{d} \right)^{\mu(d)}$$
Since $\sum_{d|N} \mu(d)=0$, this is a modular function, in fact, it is a modular unit. One has \cite{basr},
$$\operatorname{div}(\Delta_N)=\prod_{p|N} (p-1) \cdot \sum_{d|N} \mu(N/d) P_d$$
Define a {\em simple unit} to be a modular unit $f_{PQ}$ such that
$$\operatorname{div}(f_{PQ})=kP-kQ$$
for some cusps $P$ and $Q$. One has, by the Manin-Drinfeld Theorem, that $\Delta_N^{\kappa}$ is a product of simple units. More precisely, we have
\cite{basr}, Theorem 3.1,
\begin{thm} Let
$$N=\prod_{i=0}^{r} p_i \hspace{1in} \kappa=\prod_{i=1}^{r} (p_i+1)$$
Then one has
$$\Delta_N^{\kappa}=\prod_{d | (N/p_0)} F_d$$
where $F_d$ is the simple unit with divisor
$$\operatorname{div}(F_d)= \Lambda_d(P_d-P_{dp_0})$$
where
$$\Lambda_d=(p_0-1)\mu(N/d) \prod_{i=1}^r (p_i^2-1)$$
\end{thm}
From the above discussion we have an element $Z_{\Delta_N}$ in $H^3_{\M}(X_0(N) \times X_0(N), \Q(2))$ and the extension class is given by
$$ \epsilon^4_{\Delta_N}=\sum_{d|N/p_0} \epsilon^4_{F_d}$$
\subsection{The Real Regulator}
Beilinson's conjecture, or theorem, in this case, relates the element $Z_{\Delta_N}$ with the special value of the $L$-function of $H^2(X_0(N) \times X_0(N)$ at $s=1$.
The theorem shows that the regulator evaluated on the $(1,1)$ form $\omega_{f,g}=f(z)\bar{g}(z)dz d\bar{z}$ is related to the $L$-function of the motive $L(M_f \otimes M_g,1)$.
$$\langle\operatorname{reg}_{\R}(Z_{\Delta_N}),\omega_{f,g}\rangle \sim_{\Q^*} L(M_f \otimes M_g,1)$$
Beilinson proves the theorem by relating the two sides using the integral formula for the Rankin-Selberg convolution and the Kronecker Limit formula. We use the above expression to get an iterated integral formula for the regulator.
From the Carlson representative for the extension class we have that
$$\langle\operatorname{reg}_{\R}(Z_{\Delta_N}),\omega_{f.g}\rangle=F_{\Delta_N,P} (\Omega)(\omega_{f,g}^*)$$
Combining this with the earlier calculations we have the following iterated integral representation of the regulator.
$$\langle\operatorname{reg}_{\R}(Z_{\Delta_N}),\omega_{f.g}\rangle=\int_{\gamma_{\bar{g}}} \log|\Delta_N(z) f(z)dz=\int_{\gamma_{\bar{g}}} f(z_1)E_N(z_2)dz_1dz_2 $$
where $\gamma_{\bar{g}}$ is the Poincar\'e dual of $\omega_{\bar{g}}$. This follows from the fact that
$$d\log|\Delta_0(N)|=E_N(z)dz$$
where $E_N(z)$ is a holomorphic {\em Eisenstein Series} of weight $2$ for $\Gamma_0(N)$.
\subsection{Remarks on Degenerations}
Collino shows that his cycle can be viewed as a `degeneration' of the Ceresa cycle. We expect that the Bloch-Beilinson cycle too can be viewed as a suitable degeneration of the modified diagonal cycle. In a recent preprint \cite{iymu}, Iyer and M\"uller-Stach have worked out a special case of this. In a subsequent paper, we hope to show this in general and derive some additional consequences.
A curious special case is the case of modular curves. Here, if one looks at the regulators - the regulator of the modified diagonal cycle can be expressed as an iterated integral of two cusp forms over the dual of a third. As you degenerate, the regulator of the Beilinson cycle is an iterated integral of a cusp form and an Eisenstein series over the dual of a cusp form. Degenerating further, one has that the regulator of some elements of $K_2$ of a modular curve can be expressed as the iterated integral of two Eisenstein series over the dual of a cusp form. Finally, one expects that there should be an expression for the special value of the $\zeta$-function of the field of definition of a cusp corresponding to $K_3$ as an integral of two Eisenstein series over the dual of a third Eisenstein series.
\bibliographystyle{alpha}
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{
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We've received a lot of questions and comments following the announcement about the 2018 red snapper season, so we thought it might be beneficial to provide some further explanation. If you have any questions about the following information, please don't hesitate to reach out to us.
Question 1: Do Council members fish?
The makeup of the Council includes fishermen from the recreational and commercial sectors along with state marine resource agency staff and federal agency representatives. So, Council members do fish!
Here's a list of current voting Council Members serving on the South Atlantic Council. Click on each of their names to learn more about them. They fish – some professionally, some not. But among many of their differences remains the single truth that each of these individuals comes equipped with a high level of experience in fisheries.
Each state (NC, SC, GA, & FL) has three representatives – right now all states except SC has 1 commercial, 1 recreational, and one state agency representative. SC has one vacant seat.
*Note: Three new Council Members will be joining us in mid-August to fill the seats of outgoing Council members Charlie Phillips, Zack Bowen, and Ben Hartig. The bios of Kyle Chistiansen (GA), Spud Woodward (GA), and Art Sapp (FL) will be available on our website when their terms begin on August 11th. Steve Poland is the new state designee from NC and his bio will be posted when available. In addition, Mark Brown resigned his appointment on the Council effective July 24th; a new member will be appointed as soon as possible through a process very similar to the normal Council appointment process.
Question 2: Why does it seem that commercial fishermen are favored in red snapper decisions?
Sometimes folks get bogged down in the number of days or number of fish vs. pounds associated with allocations. And truthfully, that's understandable. It can be difficult to track. But here's how it all shakes out.
The recreational fishery will be tracked in number of fish and commercial will be tracked in pounds of fish.
Commercial fishermen are required to report their catch. Their season may not last until December 31, 2018. The commercial fishery will close when the commercial catch meets or is projected to meet the Commercial Annual Catch Limit or on December 31, 2018.
Additionally, the commercial trip limit is 75 pounds per trip. That doesn't give commercial fishermen much incentive to target them. Rather, it's more likely they'll retain those fish as incidental catch when targeting other species.
Headboats are currently required to report their catch so we do have a better handle on their catches.
The number of private recreational fishermen targeting snapper grouper species is not known. And in the absence of requirements to report catch, this makes it hard to track how many fish are harvested by recreational fishermen during a mini-season and throughout the year. So, we have to rely on Marine Recreational Information Program (MRIP) sampling efforts and state sampling efforts to fill in the blanks about private recreational and charter catches. We won't know how many were caught until all the recreational data from MRIP and state samplers are analyzed. Data from the 2018 mini-season likely won't be ready for review until 2019.
NMFS has a formula to estimate the amount of anticipated fishing effort with the amount of fish available to harvest to determine how many days the season can be open. This is part of the rationale for a mini-season. If we have six days of great weather, it is reasonable that 29,656 fish could be harvested by recreational snapper grouper fishermen (which includes charter boats and headboats) in the South Atlantic (NC, SC, GA, & East FL).
Question 3: Why doesn't the science match up with what fishermen are seeing on the water?
Results from fishery independent studies (those conducted by scientists only), actually support what you're seeing on the water – there are a lot of red snapper out there. The science shows that there has been an exponential increase in the number of red snapper in the South Atlantic. That's good news!
The question isn't whether or not there are a lot of red snapper. The question is whether there is evidence of a strong enough range of different aged fish surviving to enter or stay in the fishery in order to support harvest.
We need to see different ages of fish in a population for a healthy fishery - both big fish and small fish. And since red snapper life history is so unusual (they can begin reproducing at age 2 and live to be 50 years old), understanding the age structure can be challenging. But fishery independent studies, coupled with studies using fishery dependent data (data collected from fishermen - commercial reports, dockside intercepts, surveys, etc.), can give us a better snapshot of what the red snapper population actually looks like.
The observations from scientists and fishermen are on the same page.
To make sure we get the best data for management, please continue to collaborate with data collectors. State samplers will be in the field during the mini-seasons and it is critical that you cooperate and allow them to observe, measure, and sample your catch. Each and every one of us, managers and stakeholders alike, has need for better and more timely data.
This was a lot of information. But we felt it was important to answer some of the concerns and questions we've encountered in the last few weeks. Need more info? Visit our website or give us a call. We're happy to help.
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{
"redpajama_set_name": "RedPajamaC4"
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Q: Alteryx connection to Oracle database I am trying to connect to oracle database through Alteryx. I keep getting the same error message as below:
connection error im003 specified driver could not be found due to
system error 126: the specified module could not be found. (simba
oracle odbc driver, c:\program files\simba oracle odbc
driver\lib\oracleodbc_sb64.dll)
I tried downloading the Simba Oracle ODBC Driver which contains the oracleodbc_sb64.dll file. However I still get the same error.
any help is appreciated.
Thanks
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Die Aale (Anguilla, Anguillidae, von lat. anguilla "Aal", Diminutiv von anguis "Schlange"), auch Süßwasseraale genannt, sind eine Knochenfischgattung und Familie aus der Ordnung der Aalartigen (Anguilliformes). Es sind flussabwärts wandernde Wanderfische, die ihr Erwachsenenleben in Süßgewässern verbringen und zum Laichen ins Meer wandern.
Verbreitung
Aale kommen mit jeweils einer Art in Europa (die im deutschen Sprachraum bekannteste Art, der Europäische Aal, Anguilla anguilla), in Nordamerika östlich der Appalachen und in Japan und dem küstennahen China vor. Auch in den Flüssen des südöstlichen Australien und in Neuseeland lebt jeweils eine Art. Außer diesen fünf in gemäßigten bzw. subtropischen Breiten lebenden Arten gibt es noch weitere 15 Arten, die in den Tropen Süd- und Südostasiens, in Neuguinea, im östlichen Afrika und westlichen Australien vorkommen. Diese Aale besiedeln nur Flüsse, an deren Mündung das Meer direkt in die Tiefe abfällt, niemals Flüsse, die in weiten Flachmeeren münden. Aale fehlen in den kalten Polargebieten, im südlichen Atlantik, im östlichen Pazifik und dem angrenzenden Festland (westliches und zentrales Nordamerika, West-, Nord- und Zentralafrika, Mittel- und Südamerika).
Merkmale
Aale werden einen halben bis zwei Meter lang und haben 100 bis 119 Wirbel, die nur schwach entwickelte Fortsätze haben. Charakteristisch ist ihre langgestreckte, schlangenförmige Gestalt. Der Körper ist walzenförmig und im Querschnitt rund, erst im hinteren Drittel, nach dem Anus, flacht er seitlich ab. Die Seitenlinie auf Kopf und Körper ist vollständig entwickelt. Rücken-, Schwanz- und Afterflosse sind zu einem durchgehenden Flossensaum zusammengewachsen. Allen Aalen fehlen die Bauchflossen, die Brustflossen sind dagegen gut entwickelt.
Flossenformel: Dorsale 245–275, Caudale etwa 10, Anale 205–225, Pectorale 17–20.
Kopf
Der Kopf ist nicht vom Körper abgesetzt, der Übergang nur durch die Lage der halbmondförmigen, schmalen Kiemenöffnungen zu sehen, der Unterkiefer steht leicht vor. Die engen Kiemenöffnungen schließen so gut, dass Aale für längere Zeit außerhalb des Wassers überleben können, ohne dass die Kiemen austrocknen. Das Schädeldach wird vor allem von Parietale und "Squamosum" (Pteroticum) gebildet, der Oberkiefer von Maxillare und Palatinum. Ein Prämaxillare (Zwischenkieferbein) fehlt. Die Branchiostegalstrahlen sind lang und in einem vorn und oben offenen Bogen um die Knochen des Kiemendeckels gekrümmt. Die Branchiostegalhäute sind breit. Größe und die Stellung der Augen sind variabel. Aale haben zwei Paar Nasenöffnungen, ein Paar, das als Ausströmöffnung fungiert, liegt direkt vor den Augen, das andere, die Einströmöffnungen, liegt direkt oberhalb der Oberlippe und hat die Gestalt häutiger Röhrchen.
Die Kiefer und das Pflugscharbein sind von kurzen, spitzen Zähnen besetzt, die in Bändern angeordnet sind. Außerdem sind die Schlundknochen mit noch winzigeren Schlundzähnen besetzt. Die Zunge ist fleischig und zahnlos.
Insgesamt ist der Kopf durch seine festen Knochen und seine starke Muskulatur gut zum Wühlen geeignet. Ernährungsbedingt kann der Kopf auch innerhalb einer Art eine sehr verschiedene Gestalt haben (Spitzkopfaal/Breitkopfaal).
Integument
Die länglichen, bis zu 2 mm langen Cycloidschuppen der Aale sind unterhalb der drüsenreichen Schleimhaut in der Unterhaut (Corium) eingebettet. Die Schuppen liegen nebeneinander und überdecken einander nicht. Schuppen, deren Längsrichtungen parallel zueinander sind, stehen in kleinen Feldern zusammen und werden von anderen Schuppenfeldern abgelöst, in denen die Längsrichtungen der Schuppen senkrecht zu der der vorgehenden Schuppen gestellt sind. Die einzelligen Schleim- und Proteindrüsen produzieren einen sehr schlüpfrigen und zähen Schleim, der den gesamten Aalkörper überzieht.
Innere Organe
Das Herz der Aale liegt unmittelbar hinter den Kiemenöffnungen. Wie für fleischfressende Tiere typisch, ist der Darm kurz. Er hat keine Blinddarmanhänge, der Magen ist nicht scharf zur Speiseröhre abgegrenzt und geht über zwei Pylorusklappen in den Darm über. Die langgestreckte, spindelförmige Schwimmblase nimmt 30 bis 50 % der Bauchhöhle ein und steht vorn durch den Duktus pneumaticus mit der Speiseröhre in Verbindung. Die Gonaden erstrecken sich als lange, schmale Bänder entlang der gesamten Leibeshöhle bis hinter den Anus. Sie liegen dorsal, neben Darm und Schwimmblase. Geschlossene Eileiter fehlen, die Samenleiter münden in die Harnblase.
Fortpflanzung
Alle Aale verbringen ihr Erwachsenenleben im Süßwasser und kehren zur Fortpflanzung ins Meer zurück. Dabei legen einige Arten tausende von Kilometern zurück. Nach dem Verlassen der Süßgewässer fressen sie nicht mehr und sterben nach der Ei- bzw. Spermienabgabe. Der Europäische und der Amerikanische Aal laichen in der Sargassosee südlich der Bermuda-Inseln zwischen 20° und 30° nördlicher Breite und 80° und 50° westlicher Länge, der Japanische Aal im westlichen Nordpazifik südlich von Japan nahe Guam und der australische Kurzflossen-Aal und der Neuseeland-Aal im zentralen Pazifik zwischen dem Bismarck-Archipel und Fidschi. Die bekannten Laichgebiete der südostasiatischen Aalarten liegen küstennah in Tiefen unterhalb von 200 Metern. Aale, die in ihre in tieferem Wasser liegenden Laichgebiete wandern – wahrscheinlich zwischen 400 und 500 Meter –, bekommen eine dunklere Haut und stark vergrößerte Augen.
Der eigentliche Laichvorgang ist von Menschen nicht beobachtet worden. Die Eier sinken nicht zu Boden, sondern schweben mit Hilfe zahlreicher Öltropfen im freien Wasser. Auch die Prä-Leptocephali, das erste Larvenstadium, halten sich mit Öltropfen im Dottersack in der Schwebe. Prä-Leptocephali sind langgestreckt und schlank. Sie wurden in Tiefen von 300 bis 100 Metern gefangen, während die darauf folgenden Leptocephali oder Weidenblattlarven zwischen 50 Metern Tiefe und der Wasseroberfläche angetroffen werden. Die Larvenzeit ist bei den tropischen Aalen wegen der geringen Entfernung der Laichplätze nur kurz und beträgt etwa zwei bis drei Monate. Beim Europäischen Aal, der den längsten Weg vom Laichgebiet hat, ist die Larvenzeit auf drei Jahre verlängert. Leptocephali sind weiden- oder lorbeerblattförmig, völlig durchsichtig und haben einen auffallend kleinen Kopf. Ihr Körper wird von der Chorda dorsalis (eine Wirbelsäule ist noch nicht ausgebildet) in einen dorsalen und einen ventralen Teil geteilt, die beide fast gleich groß sind. Die Zahl der Muskelsegmente entspricht genau der späteren Wirbelzahl. Der Flossensaum bildet sich allmählich von vorn nach hinten. Leptocephali ernähren sich von kleinem Plankton. Sie sind phototaxisch und bewegen sich tagsüber nach unten, während sie nachts zur Oberfläche streben. Erreichen sie ihre Maximalgröße, so beginnt über das Glasaalstadium die Umwandlung zum ausgewachsenen Tier.
Äußere Systematik
Die Aale gehören in die Ordnung der Aalartigen (Anguilliformes) und sind innerhalb dieser Ordnung am nächsten mit den in der Tiefsee lebenden Schnepfenaalen (Nemichthyidae), Sägezahn-Schnepfenaalen (Serrivomeridae) und Pelikanaalartigen (Saccopharyngoidei) verwandt. Die wahrscheinlichen verwandtschaftlichen Verhältnisse gibt folgendes vereinfachte Kladogramm wieder.
Innere Systematik
Gegenwärtig werden 20 Arten als gültig anerkannt.
Europäischer Aal (Anguilla anguilla , 1758)
Kurzflossen-Aal (Anguilla australis , 1841)
Anguilla bengalensis , 1831
Anguilla bicolor , 1844
Borneo-Aal (Anguilla borneensis , 1924)
Celebes-Aal (Anguilla celebesensis , 1856)
Neuseeländischer Langflossenaal (Anguilla dieffenbachii , 1842)
Anguilla interioris , 1938
Japanischer Aal (Anguilla japonica , 1847)
Anguilla luzonensis 2009
Anguilla malgumora , 1856
Indopazifischer Aal (Anguilla marmorata , 1824)
Anguilla megastoma , 1856
Anguilla mossambica , 1852
Anguilla nebulosa , 1844
Anguilla obscura , 1872
Anguilla reinhardtii , 1867
Amerikanischer Aal (Anguilla rostrata , 1821)
Mit Anguilla pachyura aus dem Miozän von Öhningen (Baden-Württemberg) ist auch mindestens eine fossile Art bekannt.
Nutzung
Alle Arten werden für die menschliche Ernährung genutzt und sind bedeutende Speisefische. Ihr Bestand ist gefährdet. Sie werden als Frischfisch, geräuchert oder eingelegt in Dosen verkauft. In Aquakulturen werden gefangene Jungaale (Glasaale) großgezogen.
Inzwischen ist ein lukrativer illegaler Handel mit Glasaalen von Europa nach Asien entstanden.
Quellen
Literatur
Frederich W. Tesch: Der Aal. 3. Auflage. Ulmer, 1999, ISBN 978-3-8001-4563-8.
Joseph S. Nelson: Fishes of the World. John Wiley & Sons, 2006, ISBN 0-471-25031-7.
Kurt Fiedler: Lehrbuch der Speziellen Zoologie. Band II, Teil 2: Fische, Gustav Fischer Verlag, Jena 1991, ISBN 978-3-334-00338-1.
Horst Müller: Die Aale. A. Ziemsen Verlag, Wittenberg 1975, ISBN 3-89432-823-1.
Einzelnachweise
Weblinks
Aalartige
Speisefisch
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{"url":"https:\/\/oneapi-src.github.io\/oneDAL\/daal\/algorithms\/pca\/transform.html","text":"# Principal Components Analysis Transform\u00b6\n\nThe PCA transform algorithm transforms the data set to principal components.\n\n## Details\u00b6\n\nGiven a transformation matrix $$T$$ computed by PCA (eigenvectors in row-major order) and data set $$X$$ as input, the PCA Transform algorithm transforms input data set $$X$$ of size $$n \\times p$$ to the data set $$Y$$ of size $$n \\times p_r$$, $$pr \\leq p$$.\n\n## Batch Processing\u00b6\n\n### Algorithm Input\u00b6\n\nThe PCA Transform algorithm accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.\n\nInput ID\n\nInput\n\ndata\n\nUse when the input data is a normalized or non-normalized data set.\n\nPointer to the $$n \\times p$$ numeric table that contains the input data set. This input can be an object of any class derived from NumericTable.\n\neigenvectors\n\nPrincipal components computed using the PCA algorithm.\n\nPointer to the $$p_r \\times p$$ numeric table $$(p_r \\leq p)$$. You can define it as an object of any class derived from NumericTable, except for PackedSymmetricMatrix, PackedTriangularMatrix, and CSRNumericTable.\n\ndataForTransform\n\nOptional. Pointer to the key value-data collection containing the following data for PCA. The collection contains the following key-value pairs:\n\nmean\n\nmeans\n\nvariance\n\nvariances\n\neigenvalue\n\neigenvalues\n\nNote\n\n\u2022 If you do not provide the collection, the library will not apply the corresponding centering, normalization or whitening operation.\n\n\u2022 If one of the numeric tables in collection is NULL, the corresponding operation will not be applied: centering for means, normalization for variances, whitening for eigenvalues.\n\n\u2022 If mean or variance tables exist, it should be a pointer to the $$1 \\times p$$ numeric table.\n\n\u2022 If eigenvalue table is not NULL, it is the pointer to ($$1 \\times \\text{nColumns}$$) numeric table, where the number of the columns is greater than or equal to nComponents.\n\n### Algorithm Parameters\u00b6\n\nThe PCA Transform algorithm has the following parameters:\n\nParameter\n\nmethod\n\nDefault Value\n\nDescription\n\nalgorithmFPType\n\ndefaultDense or svdDense\n\nfloat\n\nThe floating-point type that the algorithm uses for intermediate computations. Can be float or double.\n\nnComponents\n\ndefaultDense\n\n$$0$$\n\nThe number of principal components $$(p_r \\leq p)$$. If zero, the algorithm will compute the result for $$\\text{nComponents} = p_r$$.\n\n### Algorithm Output\u00b6\n\nThe PCA Transform algorithm calculates the results described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm.\n\nResult ID\n\nResult\n\ntransformedData\n\nPointer to the $$n \\times p_r$$ numeric table that contains data projected to the principal components basis.\n\nNote\n\nBy default, this result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except PackedSymmetricMatrix, PackedTriangularMatrix, and CSRNumericTable.\n\n## Examples\u00b6\n\nBatch Processing:\n\nNote\n\nThere is no support for Java on GPU.\n\nBatch Processing:\n\nBatch Processing:\n\nBatch Processing:","date":"2021-04-14 07:40:30","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\": 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\": 0.4040943682193756, \"perplexity\": 1836.7463405075553}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038077336.28\/warc\/CC-MAIN-20210414064832-20210414094832-00103.warc.gz\"}"}
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\section{Introduction}
Infinitary lambda calculus is a generalisation of lambda calculus that
allows infinite lambda terms and transfinite reductions. This enables
the consideration of ``limits'' of terms under infinite reduction
sequences. For instance, for a term $M \equiv (\lambda m x . m m)
(\lambda m x . m m)$ we have
\[
M \to_\beta \lambda x . M \to_\beta \lambda x . \lambda x . M
\to_\beta \lambda x . \lambda x . \lambda x . M \to_\beta \ldots
\]
Intuitively, the ``value'' of~$M$ is an infinite term~$L$ satisfying
$L \equiv \lambda x . L$, where by~$\equiv$ we denote identity of
terms. In fact, $L$ is the normal form of~$M$ in infinitary
lambda calculus.
In~\cite{EndrullisPolonsky2011,EndrullisHansenHendriksPolonskySilva2018}
it is shown that infinitary reductions may be defined
coinductively. The standard non-coinductive definition makes explicit
mention of ordinals and limits in a certain metric
space~\cite{KennawayKlopSleepVries1997,KennawayVries2003,BarendregtKlop2009}. A
coinductive approach is better suited to formalisation in a
proof-assistant. Indeed, with relatively little effort we have
formalised our results in~Coq (see Section~\ref{sec_formalisation}).
In this paper we show confluence of infinitary lambda calculus, modulo
equivalence of so-called meaningless
terms~\cite{KennawayOostromVries1999}. We also show confluence and
normalisation of infinitary B{\"o}hm reduction over any set of
strongly meaningless terms. All these results have already been
obtained in~\cite{KennawayKlopSleepVries1997,KennawayOostromVries1999}
by a different and more complex proof method.
In a related conference paper~\cite{Czajka2014} we have shown
confluence of infinitary reduction modulo equivalence of root-active
subterms, and confluence of infinitary B{\"o}hm reduction over
root-active terms. The present paper is quite different
from~\cite{Czajka2014}. A new and simpler method is used. The proof
in~\cite{Czajka2014} follows the general strategy
of~\cite{KennawayKlopSleepVries1997}. There first confluence modulo
equivalence of root-active terms is shown, proving confluence of an
auxiliary $\epsilon$-calculus as an intermediate step. Then confluence
of B{\"ohm} reduction is derived from confluence modulo equivalence of
root-active terms. Here we first show that every term has a unique
normal form reachable by a special standard infinitary
$N_\ensuremath{{\mathcal U}}$-reduction. Then we use this result to derive confluence of
B{\"o}hm reduction, and from that confluence modulo equivalence of
meaningless terms. We do not use any $\epsilon$-calculus at all. See
the beginning of Section~\ref{sec_inflam_confluence} for a more
detailed discussion of our proof method.
\subsection{Related work}
Infinitary lambda calculus was introduced
in~\cite{KennawayKlopSleepVries1997,KennawayKlopSleepVries1995b}. Meaningless
terms were defined in~\cite{KennawayOostromVries1999}. The confluence
and normalisation results of this paper were already obtained
in~\cite{KennawayKlopSleepVries1997,KennawayOostromVries1999}, by a
different proof method. See
also~\cite{KennawayVries2003,BarendregtKlop2009,EndrullisHendriksKlop2012}
for an overview of various results in infinitary lambda calculus and
infinitary rewriting.
Joachimski in~\cite{Joachimski2004} gives a coinductive confluence
proof for infinitary lambda calculus, but Joachimski's notion of
reduction does not correspond to the standard notion of a strongly
convergent reduction. Essentially, it allows for infinitely many
parallel contractions in one step, but only finitely many reduction
steps. The coinductive definition of infinitary reductions capturing
strongly convergent reductions was introduced
in~\cite{EndrullisPolonsky2011}. Later~\cite{EndrullisHansenHendriksPolonskySilva2015,EndrullisHansenHendriksPolonskySilva2018}
generalised this to infinitary term rewriting
systems. In~\cite{Czajka2014} using the definition
from~\cite{EndrullisPolonsky2011}, confluence of infinitary
lambda calculus modulo equivalence of root-active subterms was shown
coinductively. The proof in~\cite{Czajka2014} follows the general
strategy
of~\cite{KennawayKlopSleepVries1997,KennawayKlopSleepVries1995b}. The
proof in the present paper bears some similarity to the proof of the
unique normal forms property of orthogonal~iTRSs
in~\cite{KlopVrijer2005}. It is also similar to the coinductive
confluence proof for nearly orthogonal infinitary term rewriting
systems in~\cite{Czajka2015a}, but there the ``standard'' reduction
employed is not unique and need not be normalising.
Confluence and normalisation results in infinitary rewriting and
infinitary lambda calculus have been generalised to the framework of
infinitary combinatory reduction
systems~\cite{KetemaSimonsen2009,KetemaSimonsen2010,KetemaSimonsen2011}.
There are three well-known variants of infinitary lambda calculus:
the~$\Lambda^{111}$, $\Lambda^{001}$ and~$\Lambda^{101}$
calculi~\cite{BarendregtKlop2009,EndrullisHendriksKlop2012,KennawayKlopSleepVries1997,KennawayKlopSleepVries1995b}. The
superscripts $111$, $001$, $101$ indicate the depth measure used:
$abc$ means that we shall add $a$/$b$/$c$ to the depth when going
down/left/right in the tree of the
lambda term~\cite[Definition~6]{KennawayKlopSleepVries1997}. In this
paper we are concerned only with a coinductive presentation of the
$\Lambda^{111}$-calculus.
In the $\Lambda^{001}$-calculus, after addition of appropriate
$\bot$-rules, every finite term has its B{\"o}hm
tree~\cite{KennawayKlopSleepVries1995b} as the normal
form. In~$\Lambda^{111}$ and~$\Lambda^{101}$, the normal forms are,
respectively, Berarducci trees and Levy-Longo
trees~\cite{KennawayKlopSleepVries1997,KennawayKlopSleepVries1995b,Berarducci1996b,Levy1975,Longo1983}. With
the addition of infinite $\eta$- or $\eta!$-reductions it is possible
to also capture, respectively, $\eta$-B{\"ohm} or
$\infty\eta$-B{\"o}hm trees as normal
forms~\cite{SeveriVries2002,SeveriVries2017}.
The addition of $\bot$-rules may be avoided by basing the definition
of infinitary terms on ideal completion. This line of work is pursued
in~\cite{Bahr2010,Bahr2014,Bahr2018}. Confluence of the resulting
calculi is shown, but the proof depends on the confluence results
of~\cite{KennawayKlopSleepVries1997}.
\section{Infinite terms and corecursion}\label{sec_corec}
In this section we define many-sorted infinitary terms. We also
explain and justify guarded corecursion using elementary notions. The
results of this section are well-known.
\begin{defi}\label{def_coterms}
A \emph{many-sorted algebraic signature}
$\Sigma=\ensuremath{\langle}\Sigma_s,\Sigma_c\ensuremath{\rangle}$ consists of a collection of
\emph{sort symbols}~$\Sigma_s=\{s_i\}_{i\in I}$ and a collection of
\emph{constructors} $\Sigma_c=\{c_j\}_{j\in J}$. Each
constructor~$c$ has an associated \emph{type}
$\tau(c)=(s_1,\ldots,s_n;s)$ where $s_1,\ldots,s_n,s\in\Sigma_s$. If
$\tau(c)=(;s)$ then~$c$ is a \emph{constant} of sort~$s$. In what
follows we use $\Sigma,\Sigma'$, etc., for many-sorted algebraic
signatures, $s,s'$, etc., for sort symbols, and $f,g,c,d$, etc., for
constructors.
The set~$\ensuremath{{\mathcal T}}^\infty(\Sigma)$, or just~$\ensuremath{{\mathcal T}}(\Sigma)$, of
\emph{infinitary terms over~$\Sigma$} is the set of all finite and
infinite terms over~$\Sigma$, i.e., all finite and infinite labelled
trees with labels of nodes specified by the constructors of~$\Sigma$
such that the types of labels of nodes agree. More precisely, a
term~$t$ over~$\Sigma$ is a partial function from~$\ensuremath{\mathbb{N}}^*$
to~$\Sigma_c$ satisfying:
\begin{itemize}
\item $t(\epsilon){\downarrow}$, and
\item if $t(p) = c \in \Sigma_c$ with $\tau(c)=(s_1,\ldots,s_n;s)$
then
\begin{itemize}
\item $t(pi)=d \in \Sigma_c$ with
$\tau(d)=(s_1',\ldots,s_{m_i}';s_i)$ for $i < n$,
\item $t(pi){\uparrow}$ for $i \ge n$,
\end{itemize}
\item if $t(p){\uparrow}$ then~$t(pi){\uparrow}$ for every $i\in\ensuremath{\mathbb{N}}$,
\end{itemize}
where $t(p){\uparrow}$ means that~$t(p)$ is undefined, $t(p){\downarrow}$ means
that~$t(p)$ is defined, and~$\epsilon \in \ensuremath{\mathbb{N}}^*$ is the empty
string. We use obvious notations for infinitary terms, e.g.,
$f(g(t,s),c)$ when $c,f,g \in \Sigma_c$ and $t,s \in \ensuremath{{\mathcal T}}(\Sigma)$,
and the types agree. We say that a term~$t$ \emph{is of sort~$s$} if
$t(\epsilon)$ is a constructor of type $(s_1,\ldots,s_n;s)$ for some
$s_1,\ldots,s_n\in\Sigma_s$. By~$\ensuremath{{\mathcal T}}_s(\Sigma)$ we denote the set of
all terms of sort~$s$ from~$\ensuremath{{\mathcal T}}(\Sigma)$.
\end{defi}
\begin{exa}
Let $A$ be a set. Let $\Sigma$ consist of two sorts~$\mathfrak{s}$
and~$\mathfrak{d}$, one constructor~$\mathtt{cons}$ of type $(\mathfrak{d},\mathfrak{s};\mathfrak{s})$ and a
distinct constant $a \in A$ of sort~$\mathfrak{d}$ for each element
of~$A$. Then~$\ensuremath{{\mathcal T}}_\mathfrak{s}(\Sigma)$ is the set of streams over~$A$. We
also write $\ensuremath{{\mathcal T}}_\mathfrak{s}(\Sigma) = A^\omega$ and $\ensuremath{{\mathcal T}}_\mathfrak{d}(\Sigma) =
A$. Instead of $\mathtt{cons}(a,t)$ we usually write $a : t$, and we assume
that~$:$ associates to the right, e.g., $x : y : t$ is $x : (y :
t)$. We also use the notation $x : t$ to denote the application of
the constructor for~$\mathtt{cons}$ to~$x$ and~$t$. We define the functions
$\mathtt{hd} : A^\omega \to A$ and $\mathtt{tl} : A^\omega \to A^\omega$ by
\[
\begin{array}{rcl}
\mathtt{hd}(a : t) &=& a \\
\mathtt{tl}(a : t) &=& t
\end{array}
\]
Specifications of many-sorted signatures may be conveniently given
by coinductively interpreted grammars. For instance, the
set~$A^\omega$ of streams over a set~$A$ could be specified by
writing
\[
A^\omega \Coloneqq \mathtt{cons}(A, A^\omega).
\]
A more interesting example is that of finite and infinite binary trees
with nodes labelled either with~$a$ or~$b$, and leaves labelled with
one of the elements of a set~$V$:
\[
T \Coloneqq V \parallel a(T, T) \parallel b(T, T).
\]
As such specifications are not intended to be formal entities but
only convenient notation for describing sets of infinitary terms, we
will not define them precisely. It is always clear what many-sorted
signature is meant.
\end{exa}
For the sake of brevity we often use $\ensuremath{{\mathcal T}} = \ensuremath{{\mathcal T}}(\Sigma)$ and $\ensuremath{{\mathcal T}}_s =
\ensuremath{{\mathcal T}}_s(\Sigma)$, i.e., we omit the signature~$\Sigma$ when clear from
the context or irrelevant.
\begin{defi}\label{def_guarded_corecursion}
The class of \emph{constructor-guarded} functions is defined
inductively as the class of all functions $h : \ensuremath{{\mathcal T}}_s^m \to \ensuremath{{\mathcal T}}_{s'}$
(for arbitrary $m \in \ensuremath{\mathbb{N}}$, $s,s' \in \Sigma_s$) such that there
are a constructor~$c$ of type $(s_1,\ldots,s_{k};s')$ and functions
$u_i : \ensuremath{{\mathcal T}}_s^m \to \ensuremath{{\mathcal T}}_{s_i}$ ($i=1,\ldots,k$) such that
\[
h(y_1,\ldots,y_m) =
c(u_1(y_1,\ldots,y_m),\ldots,u_k(y_1,\ldots,y_m))
\]
for all $y_1,\ldots,y_m \in \ensuremath{{\mathcal T}}_s$, and for each $i=1,\ldots,k$ one
of the following holds:
\begin{itemize}
\item $u_i$ is constructor-guarded, or
\item $u_i$ is a constant function, or
\item $u_i$ is a projection function, i.e., $s_i = s$ and there is
$1\le j \le m$ with $u_i(y_1,\ldots,y_m) = y_j$ for all
$y_1,\ldots,y_m \in \ensuremath{{\mathcal T}}_s$.
\end{itemize}
Let~$S$ be a set. A function $h : S \times \ensuremath{{\mathcal T}}_s^m \to \ensuremath{{\mathcal T}}_{s'}$ is
constructor-guarded if for every $x \in S$ the function $h_x :
\ensuremath{{\mathcal T}}_s^m \to \ensuremath{{\mathcal T}}_{s'}$ defined by $h_x(y_1,\ldots,y_m) =
h(x,y_1,\ldots,y_m)$ is constructor-guarded. A function $f : S \to
\ensuremath{{\mathcal T}}_s$ is defined by \emph{guarded corecursion} from $h : S \times
\ensuremath{{\mathcal T}}_s^m \to \ensuremath{{\mathcal T}}_s$ and $g_i : S \to S$ ($i=1,\ldots,m$) if~$h$ is
constructor-guarded and~$f$ satisfies
\[
f(x) = h(x, f(g_1(x)), \ldots, f(g_m(x)))
\]
for all $x \in S$.
\end{defi}
The following theorem is folklore in the coalgebra community. We
sketch an elementary proof. In fact, each set of many-sorted
infinitary terms is a final coalgebra of an appropriate
set-functor. Then Theorem~\ref{thm_corecursion} follows from more
general principles. We prefer to avoid coalgebraic terminology, as it
is simply not necessary for the purposes of the present paper. See
e.g.~\cite{JacobsRutten2011,Rutten2000} for a more general coalgebraic
explanation of corecursion.
\begin{thm}\label{thm_corecursion}
For any constructor-guarded function $h : S \times \ensuremath{{\mathcal T}}_s^m \to
\ensuremath{{\mathcal T}}_{s}$ and any $g_i : S \to S$ ($i=1,\ldots,m$), there exists a
unique function $f : S \to \ensuremath{{\mathcal T}}_s$ defined by guarded corecursion
from~$h$ and~$g_1,\ldots,g_m$.
\end{thm}
\begin{proof}
Let $f_0 : S \to \ensuremath{{\mathcal T}}_s$ be an arbitrary function. Define~$f_{n+1}$
for $n \in \ensuremath{\mathbb{N}}$ by $f_{n+1}(x) = h(x, f_n(g_1(x)), \ldots,
f_n(g_m(x)))$. Using the fact that~$h$ is constructor-guarded, one
shows by induction on~$n$ that:
\[
\text{$f_{n+1}(x)(p) = f_n(x)(p)$ for $x \in S$ and $p
\in \ensuremath{\mathbb{N}}^*$ with $|p|<n$}
\tag{$\star$}
\]
where~$|p|$ denotes the length of~$p$. Indeed, the base is
obvious. We show the inductive step. Let $x \in S$. Because~$h$ is
constructor-guarded, we have for instance
\[
f_{n+2}(x) = h(x, f_{n+1}(g_1(x)), \ldots, f_{n+1}(g_m(x))) =
c_1(c_2, c_3(w, f_{n+1}(g_1(x))))
\]
Let $p \in \ensuremath{\mathbb{N}}^*$ with $|p| \le n$. The only interesting case is
when $p=11p'$, i.e., when~$p$ points inside~$f_{n+1}(g_1(x))$. But
then $|p'| < |p| \le n$, so by the inductive hypothesis
$f_{n+1}(g_1(x))(p') = f_n(g_1(x))(p')$. Thus $f_{n+2}(x)(p) =
f_{n+1}(g_1(x))(p') = f_n(g_1(x))(p') = f_{n+1}(x)(p)$.
Now we define $f : S \to \ensuremath{{\mathcal T}}_s$ by
\[
f(x)(p) = f_{|p|+1}(x)(p)
\]
for $x \in S$, $p \in \ensuremath{\mathbb{N}}^*$. Using~$(\star)$ it is routine to
check that~$f(x)$ is a well-defined infinitary term for each $x \in
S$. To show that~$f : S \to \ensuremath{{\mathcal T}}_s$ is defined by guarded corecursion
from~$h$ and~$g_1,\ldots,g_m$, using~$(\star)$ one shows by
induction on the length of~$p \in \ensuremath{\mathbb{N}}^*$ that for any $x \in S$:
\[
f(x)(p) = h(x, f(g_1(x)),\ldots,f(g_m(x)))(p).
\]
To prove that~$f$ is unique it suffices to show that it does not
depend on~$f_0$. For this purpose, using~$(\star)$ one shows by
induction on the length of~$p \in \ensuremath{\mathbb{N}}^*$ that~$f(x)(p)$ does not
depend on~$f_0$ for any $x \in S$.
\end{proof}
We shall often use the above theorem implicitly, just mentioning that
some equations define a function by guarded corecursion.
\begin{exa}\label{ex_corec}
Consider the equation
\[
\mathtt{even}(x : y : t) = x : \mathtt{even}(t)
\]
It may be rewritten as
\[
\mathtt{even}(t) = \mathtt{hd}(t) : \mathtt{even}(\mathtt{tl}(\mathtt{tl}(t)))
\]
So $\mathtt{even} : A^\omega \to A^\omega$ is defined by guarded corecursion
from $h : A^\omega \times A^\omega \to A^\omega$ given by
\[
h(t,t') = \mathtt{hd}(t) : t'
\]
and $g : A^\omega \to A^\omega$ given by
\[
g(t) = \mathtt{tl}(\mathtt{tl}(t))
\]
By Theorem~\ref{thm_corecursion} there is a unique function $\mathtt{even} :
A^\omega \to A^\omega$ satisfying the original equation.
Another example of a function defined by guarded corecursion is
$\mathtt{zip} : A^\omega \times A^\omega \to A^\omega$:
\[
\mathtt{zip}(x : t, s) = x : \mathtt{zip}(s, t)
\]
The following function $\mathtt{merge} : \ensuremath{\mathbb{N}}^\omega \times \ensuremath{\mathbb{N}}^\omega \to
\ensuremath{\mathbb{N}}^\omega$ is also defined by guarded corecursion:
\[
\mathtt{merge}(x : t_1, y : t_2) =
\left\{
\begin{array}{cl}
x : \mathtt{merge}(t_1, y : t_2) & \text{ if } x \le y \\
y : \mathtt{merge}(x : t_1, t_2) & \text{ otherwise }
\end{array}
\right.
\]
\end{exa}
\section{Coinduction}\label{sec_coind}
In this section\footnote{This section is largely based
on~\cite[Section~2]{Czajka2015a}.} we give a brief explanation of
coinduction as it is used in the present paper. Our presentation of
coinductive proofs is similar to
e.g.~\cite{EndrullisPolonsky2011,BezemNakataUustalu2012,NakataUustalu2010,LeroyGrall2009,KozenSilva2017}.
There are many ways in which our coinductive proofs could be
justified. Since we formalised our main results (see
Section~\ref{sec_formalisation}), the proofs may be understood as a
paper presentation of formal Coq proofs. They can also be justified by
appealing to one of a number of established coinduction
principles. With enough patience one could, in principle, reformulate
all proofs to directly employ the usual coinduction principle in set
theory based on the Knaster-Tarski fixpoint
theorem~\cite{Sangiorgi2012}. One could probably also use the
coinduction principle from~\cite{KozenSilva2017}. Finally, one may
justify our proofs by indicating how to interpret them in ordinary set
theory, which is what we do in this section.
The purpose of this section is to explain how to justify our proofs by
reducing coinduction to transfinite induction. The present section
does not provide a formal coinduction proof principle as such, but
only indicates how one could elaborate the proofs so as to eliminate
the use of coinduction. Naturally, such an elaboration would introduce
some tedious details. The point is that all these details are
essentially the same for each coinductive proof. The advantage of
using coinduction is that the details need not be provided each
time. A similar elaboration could be done to directly employ any of a
number of established coinduction principles, but as far as we know
elaborating the proofs in the way explained here requires the least
amount of effort in comparison to reformulating them to directly
employ an established coinduction principle. In fact, we do not wish
to explicitly commit to any single formal proof principle, because we
do not think that choosing a specific principle has an essential
impact on the content of our proofs, except by making it more or less
straightforward to translate the proofs into a form which uses the
principle directly.
A reader not satisfied with the level of rigour of the explanations of
coinduction below is referred to our formalisation (see
Section~\ref{sec_formalisation}). The present section provides one way
in which our proofs can be understood and verified without resorting
to a formalisation. To make the observations of this section
completely precise and general one would need to introduce formal
notions of ``proof'' and ``statement''. In other words, one would need
to formulate a system of logic with a capacity for coinductive
proofs. We do not want to do this here, because this paper is about a
coinductive confluence proof for infinitary lambda calculus, not about
foundations of coinduction. It would require some work, but should not
be too difficult, to create a formal system based on the present
section in which our coinductive proofs could be interpreted
reasonably directly. We defer this to future work. The status of the
present section is that of a ``meta-explanation'', analogous to an
explanation of how, e.g., the informal presentations of inductive
constructions found in the literature may be encoded in ZFC set
theory.
\begin{exa}\label{ex_1}
Let~$T$ be the set of all finite and infinite terms defined
coinductively by
\[
T \Coloneqq V \parallel A(T) \parallel B(T, T)
\]
where~$V$ is a countable set of variables, and~$A$, $B$ are
constructors. By $x,y,\ldots$ we denote variables, and by
$t,s,\ldots$ we denote elements of~$T$. Define a binary
relation~$\to$ on~$T$ coinductively by the following rules.
\[
\infer=[(1)]{x \to x}{} \quad
\infer=[(2)]{A(t) \to A(t')}{t \to t'} \quad
\infer=[(3)]{B(s,t) \to B(s',t')}{s \to s' & t \to t'} \quad
\infer=[(4)]{A(t) \to B(t',t')}{t\to t'}
\]
Formally, the relation~${\to}$ is the greatest fixpoint of a
monotone function
\[
F : \ensuremath{\mathcal{P}}(T \times T) \to \ensuremath{\mathcal{P}}(T \times T)
\]
defined by
\[
F(R) = \left\{ \ensuremath{\langle} t_1, t_2 \ensuremath{\rangle} \mid \exists_{x \in V}(t_1 \equiv
t_2 \equiv x) \lor \exists_{t,t'\in T}(t_1 \equiv A(t) \land t_2
\equiv A(t') \land R(t,t')) \lor \ldots \right\}.
\]
Alternatively, using the Knaster-Tarski fixpoint theorem, the
relation~$\to$ may be characterised as the greatest binary relation
on~$T$ (i.e. the greatest subset of $T\times T$ w.r.t.~set
inclusion) such that ${\to} \subseteq F({\to})$, i.e., such that for
every $t_1,t_2 \in T$ with $t_1 \to t_2$ one of the following holds:
\begin{enumerate}
\item $t_1 \equiv t_2 \equiv x$ for some variable $x \in V$,
\item $t_1 \equiv A(t)$, $t_2 \equiv A(t')$ with $t \to t'$,
\item $t_1 \equiv B(s,t)$, $t_2 \equiv B(s',t')$ with $s \to s'$ and
$t \to t'$,
\item $t_1 \equiv A(t)$, $t_2 \equiv B(t',t')$ with $t \to t'$.
\end{enumerate}
Yet another way to think about~$\to$ is that $t_1 \to t_2$ holds if
and only if there exists a \emph{potentially infinite} derivation
tree of $t_1 \to t_2$ built using the rules~$(1)-(4)$.
The rules~$(1)-(4)$ could also be interpreted inductively to yield
the least fixpoint of~$F$. This is the conventional interpretation,
and it is indicated with a single line in each rule separating
premises from the conclusion. A coinductive interpretation is
indicated with double lines.
The greatest fixpoint~$\to$ of~$F$ may be obtained by transfinitely
iterating~$F$ starting with~$T \times T$. More precisely, define an
ordinal-indexed sequence~$(\to^\gamma)_\gamma$ by:
\begin{itemize}
\item $\to^0 = T \times T$,
\item $\to^{\gamma+1} = F(\to^\gamma)$,
\item $\to^\delta = \bigcap_{\gamma<\delta} \to^\gamma$ for a limit
ordinal~$\delta$.
\end{itemize}
Then there exists an ordinal~$\zeta$ such that ${\to} =
{\to^\zeta}$. The least such ordinal is called the \emph{closure
ordinal}. Note also that ${\to^\gamma} \subseteq {\to^\delta}$ for $\gamma \ge \delta$
(we often use this fact implicitly). See
e.g.~\cite[Chapter~8]{DaveyPriestley2002}. The relation~$\to^\gamma$ is
called the \emph{$\gamma$-approximant}. Note that the $\gamma$-approximants
depend on a particular definition of~$\to$ (i.e.~on the
function~$F$), not solely on the relation~$\to$ itself. We
use~$R^\gamma$ for the $\gamma$-approximant of a relation~$R$.
It is instructive to note that the coinductive rules for~$\to$ may
also be interpreted as giving rules for the $\gamma+1$-approximants,
for any ordinal~$\gamma$.
\[
\infer[(1)]{x \to^{\gamma+1} x}{}\quad
\infer[(2)]{A(t) \to^{\gamma+1} A(t')}{
t \to^\gamma t'}\quad
\infer[(3)]{B(s,t) \to^{\gamma+1} B(s',t')}{
s \to^\gamma s' & t \to^\gamma t'}\quad
\infer[(4)]{A(t) \to^{\gamma+1} B(t',t')}{
t\to^\gamma t'}
\]
Usually, the closure ordinal for the definition of a coinductive
relation is~$\omega$, as is the case with all coinductive
definitions appearing in this paper. In general, however, it is not
difficult to come up with a coinductive definition whose closure
ordinal is greater than~$\omega$. For instance, consider the
relation $R \subseteq \ensuremath{\mathbb{N}} \cup \{\infty\}$ defined coinductively by
the following two rules.
\[
\infer={R(n+1)}{R(n) & n \in \ensuremath{\mathbb{N}}} \quad\quad
\infer={R(\infty)}{\exists n \in \ensuremath{\mathbb{N}} . R(n)}
\]
We have $R = \emptyset$,
$R^n = \{m \in \ensuremath{\mathbb{N}} \mid m \ge n\} \cup \{\infty\}$ for
$n \in \ensuremath{\mathbb{N}}$, $R^\omega = \{\infty\}$, and only
$R^{\omega+1}=\emptyset$. Thus the closure ordinal of this
definition is $\omega+1$.
\end{exa}
In this paper we are interested in proving by coinduction statements
of the form $\psi(R_1,\ldots,R_m)$ where
\[
\psi(X_1,\ldots,X_m) \equiv \forall x_1 \ldots x_n . \varphi(\vec{x})
\to X_1(g_1(\vec{x}),\ldots,g_k(\vec{x})) \land \ldots \land
X_m(g_1(\vec{x}),\ldots,g_k(\vec{x})).
\]
and $R_1,\ldots,R_m$ are coinductive relations on~$T$, i.e, relations
defined as the greatest fixpoints of some monotone functions on the
powerset of an appropriate cartesian product of~$T$, and
$\psi(R_1,\ldots,R_m)$ is~$\psi(X_1,\ldots,X_m)$ with~$R_i$
substituted for~$X_i$. Statements with an existential quantifier may
be reduced to statements of this form by skolemising, as explained in
Example~\ref{ex_skolem} below.
Here $X_1,\ldots,X_m$ are meta-variables for which relations on~$T$
may be substituted. In the statement~$\varphi(\vec{x})$ only
$x_1,\ldots,x_n$ occur free. The meta-variables $X_1,\ldots,X_m$
\emph{are not allowed to occur} in~$\varphi(\vec{x})$. In general, we
abbreviate $x_1,\ldots,x_n$ with~$\vec{x}$. The
symbols~$g_1,\ldots,g_k$ denote some functions of~$\vec{x}$.
To prove~$\psi(R_1,\ldots,R_m)$ it suffices to show by transfinite
induction that $\psi(R_1^\gamma,\ldots,R_m^\gamma)$ holds for each
ordinal~$\gamma \le \zeta$, where~$R_i^\gamma$ is the
$\gamma$-approximant of~$R_i$. It is an easy exercise to check that
because of the special form of~$\psi$ (in particular because~$\varphi$
does not contain~$X_1,\ldots,X_m$) and the fact that each~$R_i^0$ is
the full relation, the base case~$\gamma=0$ and the case of~$\gamma$ a
limit ordinal hold. They hold for \emph{any}~$\psi$ of the above form,
\emph{irrespective} of $\varphi,R_1,\ldots,R_m$. Note
that~$\varphi(\vec{x})$ is the same in
all~$\psi(R_1^\gamma,\ldots,R_m^\gamma)$ for any~$\gamma$, i.e., it
does not refer to the $\gamma$-approximants or the
ordinal~$\gamma$. Hence it remains to show the inductive step
for~$\gamma$ a successor ordinal. It turns out that a coinductive
proof of~$\psi$ may be interpreted as a proof of this inductive step
for a successor ordinal, with the ordinals left implicit and the
phrase ``coinductive hypothesis'' used instead of ``inductive
hypothesis''.
\begin{exa}
On terms from~$T$ (see Example~\ref{ex_1}) we define the operation
of substitution by guarded corecursion.
\[
\begin{array}{rclcrcl}
y[t/x] &=& y \quad\text{ if } x \ne y &\quad&
(A(s))[t/x] &=& A(s[t/x]) \\
x[t/x] &=& t &\quad&
(B(s_1,s_2))[t/x] &=& B(s_1[t/x],s_2[t/x])
\end{array}
\]
We show by coinduction: if $s \to s'$ and $t \to t'$ then $s[t/x]
\to s'[t'/x]$, where~$\to$ is the relation from
Example~\ref{ex_1}. Formally, the statement we show by transfinite
induction on~$\gamma \le \zeta$ is: for $s,s',t,t' \in T$, if $s \to
s'$ and $t \to t'$ then $s[t/x] \to^\gamma s'[t'/x]$. For
illustrative purposes, we indicate the $\gamma$-approximants with
appropriate ordinal superscripts, but it is customary to omit these
superscripts.
Let us proceed with the proof. The proof is by coinduction with case
analysis on $s \to s'$. If $s \equiv s' \equiv y$ with $y \ne x$,
then $s[t/x] \equiv y \equiv s'[t'/x]$. If $s \equiv s' \equiv x$
then $s[t/x] \equiv t \to^{\gamma+1} t' \equiv s'[t'/x]$ (note that
${\to} \equiv {\to^\zeta} \subseteq {\to^{\gamma+1}}$). If $s \equiv
A(s_1)$, $s' \equiv A(s_1')$ and $s_1 \to s_1'$, then $s_1[t/x]
\to^\gamma s_1'[t'/x]$ by the coinductive hypothesis. Thus $s[t/x]
\equiv A(s_1[t/x]) \to^{\gamma+1} A(s_1'[t'/x]) \equiv s'[t'/x]$ by
rule~$(2)$. If $s \equiv B(s_1,s_2)$, $s' \equiv B(s_1',s_2')$ then
the proof is analogous. If $s \equiv A(s_1)$, $s' \equiv
B(s_1',s_1')$ and $s_1 \to s_1'$, then the proof is also
similar. Indeed, by the coinductive hypothesis we have $s_1[t/x]
\to^\gamma s_1'[t'/x]$, so $s[t/x] \equiv A(s_1[t/x]) \to^{\gamma+1}
B(s_1'[t'/x],s_1'[t'/x]) \equiv s'[t'/x]$ by rule~$(4)$.
\end{exa}
With the following example we explain how our proofs of existential
statements should be interpreted.
\begin{exa}\label{ex_skolem}
Let~$T$ and~$\to$ be as in Example~\ref{ex_1}. We want to show: for
all $s,t,t' \in T$, if $s \to t$ and $s \to t'$ then there exists
$s' \in T$ with $t \to s'$ and $t' \to s'$. The idea is to skolemise
this statement. So we need to find a Skolem function $f : T^3 \to T$
which will allow us to prove the Skolem normal form:
\[
\text{if $s \to t$ and $s \to t'$ then $t \to f(s,t,t')$
and $t' \to f(s,t,t')$.}
\tag{$\star$}
\]
The rules for~$\to$ suggest a definition of~$f$:
\[
\begin{array}{rcl}
f(x, x, x) &=& x \\
f(A(s), A(t), A(t')) &=& A(f(s,t,t')) \\
f(A(s),A(t),B(t',t')) &=& B(f(s,t,t'),f(s,t,t')) \\
f(A(s),B(t,t),A(t')) &=& B(f(s,t,t'),f(s,t,t')) \\
f(A(s),B(t,t),B(t',t')) &=& B(f(s,t,t'),f(s,t,t')) \\
f(B(s_1,s_2), B(t_1,t_2), B(t_1',t_2')) &=&
B(f(s_1,t_1,t_1'),f(s_2,t_2,t_2')) \\
f(s, t, t') &=& \text{some fixed term if none of the above matches}
\end{array}
\]
This is a definition by guarded corecursion, so there exists a
unique function $f : T^3 \to T$ satisfying the above equations. The
last case in the above definition of~$f$ corresponds to the case in
Definition~\ref{def_guarded_corecursion} where all~$u_i$ are
constant functions. Note that any fixed term has a fixed constructor
(in the sense of Definition~\ref{def_guarded_corecursion}) at the
root. In the sense of Definition~\ref{def_guarded_corecursion} also
the elements of~$V$ are nullary constructors.
We now proceed with a coinductive proof of~$(\star)$. Assume $s \to
t$ and $s \to t'$. If $s \equiv t \equiv t' \equiv x$ then
$f(s,t,t') \equiv x$, and $x \to x$ by rule~$(1)$. If $s \equiv A(s_1)$,
$t \equiv A(t_1)$ and $t' \equiv A(t_1')$ with $s_1 \to t_1$ and
$s_1 \to t_1'$, then by the coinductive hypothesis $t_1 \to
f(s_1,t_1,t_1')$ and $t_1' \to f(s_1,t_1,t_1')$. We have $f(s,t,t')
\equiv A(f(s_1,t_1,t_1'))$. Hence $t \equiv A(t_1) \to f(s,t,t')$
and $t \equiv A(t_1') \to f(s,t,t')$, by rule~$(2)$. If $s \equiv
B(s_1,s_2)$, $t \equiv B(t_1,t_2)$ and $t' \equiv B(t_1',t_2')$,
with $s_1 \to t_1$, $s_1 \to t_1'$, $s_2 \to t_2$ and $s_2 \to
t_2'$, then by the coinductive hypothesis we have $t_1 \to
f(s_1,t_1,t_1')$, $t_1' \to f(s_1,t_1,t_1')$, $t_2 \to
f(s_2,t_2,t_2')$ and $t_2' \to f(s_2,t_2,t_2')$. Hence $t \equiv
B(t_1,t_2) \to B(f(s_1,t_1,t_1'),f(s_2,t_2,t_2')) \equiv f(s,t,t')$
by rule~$(3)$. Analogously, $t' \to f(s,t,t')$ by rule~$(3)$. Other
cases are similar.
Usually, it is inconvenient to invent the Skolem function
beforehand, because the definition of the Skolem function and the
coinductive proof of the Skolem normal form are typically
interdependent. Therefore, we adopt an informal style of doing a
proof by coinduction of a statement
\[
\begin{array}{rcl}
\psi(R_1,\ldots,R_m) &=& \forall_{x_1, \ldots, x_n \in T}
\,.\, \varphi(\vec{x}) \to \\ &&\quad \exists_{y \in T}
. R_1(g_1(\vec{x}),\ldots,g_k(\vec{x}), y) \land \ldots \land R_m(g_1(\vec{x}),\ldots,g_k(\vec{x}),y)
\end{array}
\]
with an existential quantifier. We intertwine the corecursive
definition of the Skolem function~$f$ with a coinductive proof of
the Skolem normal form
\[
\begin{array}{l}
\forall_{x_1, \ldots, x_n \in T} \,.\, \varphi(\vec{x}) \to
\\ \quad\quad
R_1(g_1(\vec{x}),\ldots,g_k(\vec{x}),f(\vec{x}))
\land \ldots \land R_m(g_1(\vec{x}),\ldots,g_k(\vec{x}),f(\vec{x}))
\end{array}
\]
We proceed as if the coinductive hypothesis
was~$\psi(R_1^\gamma,\ldots,R_m^\gamma)$ (it really is the Skolem
normal form). Each element obtained from the existential quantifier
in the coinductive hypothesis is interpreted as a corecursive
invocation of the Skolem function. When later we exhibit an element
to show the existential subformula
of~$\psi(R_1^{\gamma+1},\ldots,R_m^{\gamma+1})$, we interpret this
as the definition of the Skolem function in the case specified by
the assumptions currently active in the proof. Note that this
exhibited element may (or may not) depend on some elements obtained
from the existential quantifier in the coinductive hypothesis, i.e.,
the definition of the Skolem function may involve corecursive
invocations.
To illustrate our style of doing coinductive proofs of statements
with an existential quantifier, we redo the proof done above. For
illustrative purposes, we indicate the arguments of the Skolem
function, i.e., we write~$s'_{s,t,t'}$ in place
of~$f(s,t,t')$. These subscripts $s,t,t'$ are normally omitted.
We show by coinduction that if $s \to t$ and $s \to t'$ then there
exists $s' \in T$ with $t \to s'$ and $t' \to s'$. Assume $s \to t$
and $s \to t'$. If $s \equiv t \equiv t' \equiv x$ then take
$s'_{x,x,x} \equiv x$. If $s \equiv A(s_1)$, $t \equiv A(t_1)$ and
$t' \equiv A(t_1')$ with $s_1 \to t_1$ and $s_1 \to t_1'$, then by
the coinductive hypothesis we obtain~$s'_{s_1,t_1,t_1'}$ with
$t_1 \to s'_{s_1,t_1,t_1'}$ and $t_1' \to s'_{s_1,t_1,t_1'}$. More
precisely: by corecursively applying the Skolem function to
$s_1,t_1,t_1'$ we obtain~$s'_{s_1,t_1,t_1'}$, and by the coinductive
hypothesis we have $t_1 \to s'_{s_1,t_1,t_1'}$ and
$t_1' \to s'_{s_1,t_1,t_1'}$. Hence
$t \equiv A(t_1) \to A(s'_{s_1,t_1,t_1'})$ and
$t \equiv A(t_1') \to A(s'_{s_1,t_1,t_1'})$, by rule~$(2)$. Thus we
may take $s'_{s,t,t'} \equiv A(s'_{s_1,t_1,t_1'})$. If
$s \equiv B(s_1,s_2)$, $t \equiv B(t_1,t_2)$ and
$t' \equiv B(t_1',t_2')$, with $s_1 \to t_1$, $s_1 \to t_1'$,
$s_2 \to t_2$ and $s_2 \to t_2'$, then by the coinductive hypothesis
we obtain~$s'_{s_1,t_1,t_1'}$ and~$s'_{s_2,t_2,t_2'}$ with
$t_1 \to s'_{s_1,t_1,t_1'}$, $t_1' \to s'_{s_1,t_1,t_1'}$,
$t_2 \to s'_{s_2,t_2,t_2'}$ and $t_2' \to s'_{s_2,t_2,t_2'}$. Hence
$t \equiv B(t_1,t_2) \to B(s'_{s_1,t_1,t_1'},s'_{s_2,t_2,t_2'})$ by
rule~$(3)$. Analogously,
$t' \to B(s'_{s_1,t_1,t_1'},s'_{s_2,t_2,t_2'})$ by rule~$(3)$. Thus
we may take
$s'_{s,t,t'} \equiv B(s'_{s_1,t_1,t_1'},s'_{s_2,t_2,t_2'})$. Other
cases are similar.
It is clear that the above informal proof, when interpreted in the
way outlined before, implicitly defines the Skolem function~$f$. It
should be kept in mind that in every case the definition of the
Skolem function needs to be guarded. We do not explicitly mention
this each time, but verifying this is part of verifying the proof.
\end{exa}
When doing proofs by coinduction the following criteria need to be
kept in mind in order to be able to justify the proofs according to
the above explanations.
\begin{itemize}
\item When we conclude from the coinductive hypothesis that some
relation~$R(t_1,\ldots,t_n)$ holds, this really means that only its
approximant~$R^\gamma(t_1,\ldots,t_n)$ holds. Usually, we need to
infer that the next approximant~$R^{\gamma+1}(s_1,\ldots,s_n)$ holds
(for some other elements~$s_1,\ldots,s_n$) by
using~$R^\gamma(t_1,\ldots,t_n)$ as a premise of an appropriate
rule. But we cannot, e.g., inspect (do case reasoning
on)~$R^\gamma(t_1,\ldots,t_n)$, use it in any lemmas, or otherwise
treat it as~$R(t_1,\ldots,t_n)$.
\item An element~$e$ obtained from an existential quantifier in the
coinductive hypothesis is not really the element itself, but a
corecursive invocation of the implicit Skolem function. Usually, we
need to put it inside some constructor~$c$, e.g.~producing~$c(e)$,
and then exhibit~$c(e)$ in the proof of an existential
statement. Applying at least one constructor to~$e$ is necessary to
ensure guardedness of the implicit Skolem function. But we cannot,
e.g., inspect~$e$, apply some previously defined functions to it, or
otherwise treat it as if it was really given to us.
\item In the proofs of existential statements, the implicit Skolem
function cannot depend on the ordinal~$\gamma$. However, this is the
case as long as we do not violate the first point, because if the
ordinals are never mentioned and we do not inspect the approximants
obtained from the coinductive hypothesis, then there is no way in
which we could possibly introduce a dependency on~$\gamma$.
\end{itemize}
Equality on infinitary terms may be characterised coinductively.
\begin{defi}\label{def_bisimilarity}
Let~$\Sigma$ be a many-sorted algebraic signature, as in
Definition~\ref{def_coterms}. Let $\ensuremath{{\mathcal T}} = \ensuremath{{\mathcal T}}(\Sigma)$. Define
on~$\ensuremath{{\mathcal T}}$ a binary relation~${=}$ of \emph{bisimilarity} by the
coinductive rules
\[
\infer={f(t_1,\ldots,t_n) = f(s_1,\ldots,s_n)}{t_1 = s_1 & \ldots &
t_n = s_n}
\]
for each constructor $f \in \Sigma_c$.
\end{defi}
It is intuitively obvious that on infinitary terms bisimilary is the
same as identity. The following easy (and well-known) proposition
makes this precise.
\begin{prop}\label{prop_bisimilarity}
For $t,s \in \ensuremath{{\mathcal T}}$ we have: $t = s$ iff $t \equiv s$.
\end{prop}
\begin{proof}
Recall that each term is formally a partial function from~$\ensuremath{\mathbb{N}}^*$
to~$\Sigma_c$. We write $t(p) \approx s(p)$ if either both
$t(p),s(p)$ are defined and equal, or both are undefined.
Assume $t = s$. It suffices to show by induction of the length of $p
\in \ensuremath{\mathbb{N}}^*$ that $\pos{t}{p} = \pos{s}{p}$ or $t(p){\uparrow},s(p){\uparrow}$,
where by~$\pos{t}{p}$ we denote the subterm of~$t$ at
position~$p$. For $p = \epsilon$ this is obvious. Assume $p =
p'j$. By the inductive hypothesis, $\pos{t}{p'} = \pos{s}{p'}$ or
$t(p'){\uparrow}, s(p'){\uparrow}$. If $\pos{t}{p'} = \pos{s}{p'}$ then
$\pos{t}{p'} \equiv f(t_0,\ldots,t_n)$ and $\pos{s}{p'} \equiv
f(s_0,\ldots,s_n)$ for some $f \in \Sigma_c$ with $t_i = s_i$ for
$i=0,\ldots,n$. If $0 \le j \le n$ then $\pos{t}{p} \equiv t_j = s_j
= \pos{s}{p}$. Otherwise, if $j > n$ or if $t(p'){\uparrow},s(p'){\uparrow}$, then
$t(p){\uparrow},s(p){\uparrow}$ by the definition of infinitary terms.
For the other direction, we show by coinduction that for any $t \in
\ensuremath{{\mathcal T}}$ we have $t = t$. If $t \in \ensuremath{{\mathcal T}}$ then $t \equiv
f(t_1,\ldots,t_n)$ for some $f \in \Sigma_c$. By the coinductive
hypothesis we obtain $t_i = t_i$ for $i=1,\ldots,n$. Hence $t = t$
by the rule for~$f$.
\end{proof}
For infinitary terms $t,s\in \ensuremath{{\mathcal T}}$, we shall therefore use the
notations $t = s$ and $t \equiv s$ interchangeably, employing
Proposition~\ref{prop_bisimilarity} implicitly. In particular, the
above coinductive characterisation of term equality is used implicitly
in the proof of Lemma~\ref{lem_N_confluent}.
\begin{exa}
Recall the coinductive definitions of~$\mathtt{zip}$ and~$\mathtt{even}$ from
Example~\ref{ex_corec}.
\[
\begin{array}{rcl}
\mathtt{even}(x : y : t) &=& x : \mathtt{even}(t) \\
\mathtt{zip}(x : t, s) &=& x : \mathtt{zip}(s, t)
\end{array}
\]
By coinduction we show
\[
\mathtt{zip}(\mathtt{even}(t),\mathtt{even}(\mathtt{tl}(t))) = t
\]
for any stream $t \in A^\omega$. Let $t \in A^\omega$. Then $t = x :
y : s$ for some $x, y \in A$ and $s \in A^\omega$. We have
\[
\begin{array}{rcl}
\mathtt{zip}(\mathtt{even}(t),\mathtt{even}(\mathtt{tl}(t))) &=& \mathtt{zip}(\mathtt{even}(x : y : s), \mathtt{even}(y
: s)) \\
&=& \mathtt{zip}(x : \mathtt{even}(s), \mathtt{even}(y : s)) \\
&=& x : \mathtt{zip}(\mathtt{even}(y : s), \mathtt{even}(s)) \\
&=& x : y : s \quad\text{ (by~CH) }\\
&=& t
\end{array}
\]
In the equality marked with~(by~CH) we use the coinductive
hypothesis, and implicitly a bisimilarity rule from
Definition~\ref{def_bisimilarity}.
\end{exa}
The above explanation of coinduction is generalised and elaborated in
much more detail in~\cite{Czajka2015}. Also~\cite{KozenSilva2017} may
be helpful as it gives many examples of coinductive proofs written in
a style similar to the one used here. The book~\cite{Sangiorgi2012} is
an elementary introduction to coinduction and bisimulation (but the
proofs there are presented in a different way than here, not referring
to the coinductive hypothesis but explicitly constructing a
backward-closed set). The
chapters~\cite{BertotCasteran2004Chapter13,Chlipala2013Chapter5}
explain coinduction in~Coq from a practical viewpoint. A reader
interested in foundational matters should also
consult~\cite{JacobsRutten2011,Rutten2000} which deal with the
coalgebraic approach to coinduction.
In the rest of this paper we shall freely use coinduction, giving
routine coinductive proofs in as much (or as little) detail as it is
customary with inductive proofs of analogous difficulty.
\section{Definitions and basic properties}\label{sec_inflam_intro}
In this section we define infinitary lambda terms and the various
notions of infinitary reductions.
\newcommand{\mathtt{free}}{\mathtt{free}}
\begin{defi}\label{def_infinitary_lambda_terms}
The set of \emph{infinitary lambda terms} is defined coinductively:
\[
\begin{array}{rcl}
\Lambda^\infty &::=& C \parallel V \parallel \Lambda^\infty\Lambda^\infty \parallel \lambda V . \Lambda^\infty
\end{array}
\]
where~$V$ is an infinite set of \emph{variables} and~$C$ is a set of
\emph{constants} such that $V \cap C = \emptyset$. An \emph{atom} is
a variable or a constant. We use the symbols $x,y,z,\ldots$ for
variables, and $c,c',c_1,\ldots$ for constants, and
$a,a',a_1,\ldots$ for atoms, and $t,s,\ldots$ for terms. By~$\ensuremath{\mathrm{FV}}(t)$
we denote the set of variables occurring free in~$t$. Formally,
$\ensuremath{\mathrm{FV}}(t)$ could be defined using coinduction.
We define substitution by guarded corecursion.
\[
\begin{array}{rcl}
x[t/x] &=& t \\
a[t/x] &=& a \quad\text{if } a \ne x \\
(t_1t_2)[t/x] &=& (t_1[t/x]) (t_2[t/x]) \\
(\lambda y . s)[t/x] &=& \lambda y . s[t/x] \quad\text{if } y
\notin \ensuremath{\mathrm{FV}}(t,x)
\end{array}
\]
\end{defi}
In our formalisation we use a de Bruijn representation of infinitary
lambda terms, defined analogously to the de Bruijn representation of
finite lambda terms~\cite{Bruijn1972}. Hence, infinitary lambda terms
here may be understood as a human-readable presentation of infinitary
lambda terms based on de Bruijn indices. Strictly speaking, also the
definition of substitution above is not completely precise, because it
implicitly treats lambda terms up to renaming of bound variables and
we have not given a precise definition of free variables. The
definition of substitution can be understood as a human-readable
presentation of substitution defined on infinitary lambda terms based
on de Bruijn indices.
Infinitary lambda terms could be precisely defined as the
$\alpha$-equivalence classes of the terms given in
Definition~\ref{def_infinitary_lambda_terms}, with a coinductively
defined $\alpha$-equivalence relation~$=_\alpha$. Such a definition
involves some technical issues. If the set of variables~$V$ is
countable, then it may be impossible to choose a ``fresh'' variable
$x \notin \ensuremath{\mathrm{FV}}(t)$ for a term $t \in \Lambda^\infty$, because~$t$ may
contain all variables free. This presents a difficulty when trying to
precisely define substitution. See
also~\cite{KurzPetrisanSeveriVries2012,KurzPetrisanSeveriVries2013}.
There are two ways of resolving this situation:
\begin{enumerate}
\item assume that~$V$ is uncountable,
\item consider only terms with finitely many free variables.
\end{enumerate}
Assuming that a fresh variable may always be chosen, one may precisely
define substitution and use coinductive techniques to prove: if $t
=_\alpha t'$ and $s =_\alpha s'$ then $s[t/x] =_\alpha s'[t'/x]$. This
implies that substitution lifts to a function on the
$\alpha$-equivalence classes, which is also trivially true for
application and abstraction. Therefore, all functions defined by
guarded corecursion using only the operations of substitution,
application and abstraction lift to functions on $\alpha$-equivalence
classes (provided the same defining equation is used for all terms
within the same $\alpha$-equivalence class). This justifies the use of
Barendregt's variable convention~\cite[2.1.13]{Barendregt1984} (under
the assumption that we may always choose a fresh variable).
Since our formalisation is based on de Bruijn indices, we omit
explicit treatment of $\alpha$-equivalence in this paper.
We also mention that another principled and precise way of dealing
with the renaming of bound variables is to define the set of
infinitary lambda terms as the final coalgebra of an appropriate
functor in the category of nominal
sets~\cite{KurzPetrisanSeveriVries2012,KurzPetrisanSeveriVries2013}.
\begin{defi}
Let $R \subseteq \Lambda^\infty \times \Lambda^\infty$ be a binary
relation on infinitary lambda terms. The \emph{compatible closure}
of~$R$, denoted~$\to_R$, is defined inductively by the following
rules.
\[
\begin{array}{cccc}
\infer{s \to_R t}{\ensuremath{\langle} s, t \ensuremath{\rangle} \in R} &\quad
\infer{s t \to_R s' t}{s \to_R s'} &\quad
\infer{s t \to_R s t'}{t \to_R t'} &\quad
\infer{\lambda x . s \to_R \lambda x . s'}{s \to_R s'}
\end{array}
\]
If $\ensuremath{\langle} t, s\ensuremath{\rangle} \in R$ then~$t$ is an \emph{$R$-redex}. A term $t
\in \Lambda^\infty$ is in \emph{$R$-normal form} if there is no $s
\in \Lambda^\infty$ with $t \to_R s$, or equivalently if it contains
no $R$-redexes. The \emph{parallel closure} of~$R$, usually
denoted~$\ensuremath{\Rightarrow}_R$, is defined coinductively by the following rules.
\[
\begin{array}{cccc}
\infer={s \ensuremath{\Rightarrow}_{R} t}{\ensuremath{\langle} s,t\ensuremath{\rangle} \in R} &~
\infer={a \ensuremath{\Rightarrow}_{R} a}{} &~
\infer={s_1s_2 \ensuremath{\Rightarrow}_{R} t_1t_2}{s_1 \ensuremath{\Rightarrow}_{R} t_1 &
s_2 \ensuremath{\Rightarrow}_{R} t_2} &~
\infer={\lambda x . s \ensuremath{\Rightarrow}_{R} \lambda x . s'}{s
\ensuremath{\Rightarrow}_{R} s'}
\end{array}
\]
Let ${\to} \subseteq \Lambda^\infty \times
\Lambda^\infty$. By~$\ensuremath{\to^*}$ we denote the transitive-reflexive
closure of~$\to$, and by~$\to^\equiv$ the reflexive closure
of~$\to$. The \emph{infinitary closure} of~$\to$, denoted~$\ensuremath{\to^\infty}$,
is defined coinductively by the following rules.
\[
\begin{array}{ccc}
\infer={s \ensuremath{\to^\infty} a}{s \ensuremath{\to^*} a} &\quad
\infer={s \ensuremath{\to^\infty} t_1't_2'}{s \ensuremath{\to^*} t_1t_2 & t_1 \ensuremath{\to^\infty} t_1' & t_2 \ensuremath{\to^\infty} t_2'} &\quad
\infer={s \ensuremath{\to^\infty} \lambda x . r'}{s \ensuremath{\to^*}
\lambda x . r & r \ensuremath{\to^\infty} r'}
\end{array}
\]
Let
$R_\beta = \{ \ensuremath{\langle} (\lambda x . s) t, s[t/x] \ensuremath{\rangle} \mid
t,s\in\Lambda^\infty \}$. The relation~$\to_\beta$ of \emph{one-step
$\beta$-reduction} is defined as the compatible closure
of~$R_\beta$. The relation $\ensuremath{\to^*}_\beta$ of
\emph{$\beta$-reduction} is the transitive-reflexive closure
of~$\to_\beta$. The relation $\ensuremath{\to^\infty}_\beta$ of \emph{infinitary
$\beta$-reduction} is defined as the infinitary closure
of~$\to_\beta$. This gives the same coinductive definition of
infinitary $\beta$-reduction as in~\cite{EndrullisPolonsky2011}.
The relation~$\to_w$ of \emph{one-step weak head reduction} is
defined inductively by the following rules.
\[
\begin{array}{cc}
\infer{(\lambda x . s) t \to_w s[t/x]}{} &\quad
\infer{s t \to_w s' t}{s \to_w s'}
\end{array}
\]
The relations~$\ensuremath{\to^*}_w$, $\to_w^\equiv$ and~$\ensuremath{\to^\infty}_w$ are
defined accordingly. In a term $(\lambda x . s) t t_1 \ldots t_m$
the subterm $(\lambda x . s) t$ is the \emph{weak head
redex}. So~$\to_w$ may contract only the weak head redex.
\end{defi}
\begin{defi}
Let $\bot$ be a distinguished constant. A $\Lambda^\infty$-term~$t$
is in \emph{root normal form} (rnf) if:
\begin{itemize}
\item $t \equiv a$ with $a \not\equiv \bot$, or
\item $t \equiv \lambda x . t'$, or
\item $t \equiv t_1 t_2$ and there is no~$s$ with $t_1 \ensuremath{\to^\infty}_\beta
\lambda x . s$ (equivalently, there is no~$s$ with $t_1
\ensuremath{\to^*}_\beta \lambda x . s$).
\end{itemize}
In other words, a term~$t$ is in rnf if $t \not\equiv \bot$ and~$t$
does not infinitarily $\beta$-reduce to a $\beta$-redex. We say
that~$t$ \emph{has rnf} if $t \ensuremath{\to^\infty}_\beta t'$ for some~$t'$ in
rnf. In particular, $\bot$ has no rnf. A term with no rnf is also
called \emph{root-active}. By~$\ensuremath{{\mathcal R}}$ we denote the set of all
root-active terms.
\end{defi}
\begin{defi}
A set $\ensuremath{{\mathcal U}} \subseteq \Lambda^\infty$ is a set of \emph{meaningless
terms} (see~\cite{KennawayVries2003}) if it satisfies the
following axioms.
\begin{itemize}
\item {\bf Closure:} if $t \in \ensuremath{{\mathcal U}}$ and $t \ensuremath{\to^\infty}_\beta s$ then
$s \in \ensuremath{{\mathcal U}}$.
\item {\bf Substitution:} if $t \in \ensuremath{{\mathcal U}}$ then $t[s/x] \in \ensuremath{{\mathcal U}}$ for
any term~$s$.
\item {\bf Overlap:} if $\lambda x . s \in \ensuremath{{\mathcal U}}$ then $(\lambda x
. s) t \in \ensuremath{{\mathcal U}}$.
\item {\bf Root-activeness:} $\ensuremath{{\mathcal R}} \subseteq \ensuremath{{\mathcal U}}$.
\item {\bf Indiscernibility:} if $t \in \ensuremath{{\mathcal U}}$ and $t \sim_\ensuremath{{\mathcal U}} s$
then $s \in \ensuremath{{\mathcal U}}$, where~$\sim_\ensuremath{{\mathcal U}}$ is the parallel closure
of~$\ensuremath{{\mathcal U}}\times\ensuremath{{\mathcal U}}$.
\end{itemize}
A set~$\ensuremath{{\mathcal U}}$ of meaningless terms is a set of \emph{strongly
meaningless terms} if it additionally satisfies the following
expansion axiom.
\begin{itemize}
\item {\bf Expansion:} if $t \in \ensuremath{{\mathcal U}}$ and $s \ensuremath{\to^\infty}_\beta t$ then
$s \in \ensuremath{{\mathcal U}}$.
\end{itemize}
Let $\ensuremath{{\mathcal U}} \subseteq \Lambda^\infty$. Let $R_{\bot_\ensuremath{{\mathcal U}}} = \{ \ensuremath{\langle} t, \bot \ensuremath{\rangle}
\mid t \in \ensuremath{{\mathcal U}} \text{ and } t \not\equiv \bot \}$. We define the
relation~$\to_{\beta\bot_\ensuremath{{\mathcal U}}}$ of \emph{one-step $\beta\bot_\ensuremath{{\mathcal U}}$-reduction} as
the compatible closure of $R_{\beta\bot_\ensuremath{{\mathcal U}}} = R_\beta \cup R_{\bot_\ensuremath{{\mathcal U}}}$. A
term~$t$ is in \emph{$\beta\bot_\ensuremath{{\mathcal U}}$-normal form} if it is in
$R_{\beta\bot_\ensuremath{{\mathcal U}}}$-normal form. The relation~$\ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}}$ of
\emph{$\beta\bot_\ensuremath{{\mathcal U}}$-reduction} is the transitive-reflexive closure
of~$\to_{\beta\bot_\ensuremath{{\mathcal U}}}$. The relation $\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}}$ of
\emph{infinitary $\beta\bot_\ensuremath{{\mathcal U}}$-reduction}, or \emph{B{\"o}hm reduction}
(over~$\ensuremath{{\mathcal U}}$), is the infinitary closure of~$\to_{\beta\bot_\ensuremath{{\mathcal U}}}$. The
relation $\ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}}$ of \emph{parallel $\bot_\ensuremath{{\mathcal U}}$-reduction} is the
parallel closure of~$R_{\bot_\ensuremath{{\mathcal U}}}$.
\end{defi}
In general, relations on infinitary terms need to be defined
coinductively. However, if the relation depends only on finite initial
parts of the terms then it may often be defined inductively. Because
induction is generally better understood than coinduction, we prefer
to give inductive definitions whenever it is possible to give such a
definition in a natural way, like with the definition of compatible
closure or one-step weak head reduction. This is in contrast to
e.g.~the definition of infinitary reduction~$\ensuremath{\to^\infty}$, which
intuitively may contain infinitely many reduction steps, and thus must
be defined by coinduction.
The idea with the definition of the infinitary closure~$\ensuremath{\to^\infty}$ of a
one-step reduction relation~$\to$ is that the depth at which a redex
is contracted should tend to infinity. This is achieved by
defining~$\ensuremath{\to^\infty}$ in such a way that always after finitely many
reduction steps the subsequent contractions may be performed only
under a constructor. So the depth of the contracted redex always
ultimately increases. The idea for the definition of~$\ensuremath{\to^\infty}$ comes
from~\cite{EndrullisPolonsky2011,EndrullisHansenHendriksPolonskySilva2015,EndrullisHansenHendriksPolonskySilva2018}. For
infinitary $\beta$-reduction~$\ensuremath{\to^\infty}_\beta$ the definition is the
same as in~\cite{EndrullisPolonsky2011}. To each derivation of $t
\ensuremath{\to^\infty} s$ corresponds a strongly convergent reduction sequence of
length at most~$\omega$ obtained by concatenating the finite
$\ensuremath{\to^*}$-reductions in the prefixes. See the proof of
Theorem~\ref{thm_strongly_convergent}.
Our definition of meaningless terms differs
from~\cite{KennawayVries2003} in that it treats terms with the~$\bot$
constant, but it is equivalent to the original definition, in the
following sense. Let~$\Lambda_0^\infty$ be the set of
infinitary-lambda terms without~$\bot$. If~$\ensuremath{{\mathcal U}}$ is a set of
meaningless terms defined as in~\cite{KennawayVries2003}
on~$\Lambda_0^\infty$, then~$\ensuremath{{\mathcal U}}_\bot$ (the set of terms from~$\ensuremath{{\mathcal U}}$
with some subterms in~$\ensuremath{{\mathcal U}}$ replaced by~$\bot$) is a set of
meaningless terms according to our definition. Conversely, if~$\ensuremath{{\mathcal U}}$ is
a set of meaningless terms according to our definition, then $\ensuremath{{\mathcal U}} =
\ensuremath{{\mathcal U}}_\bot'$ where $\ensuremath{{\mathcal U}}' = \ensuremath{{\mathcal U}} \cap \Lambda_0^\infty$ ($\ensuremath{{\mathcal U}}'$ then
satisfies the axioms of~\cite{KennawayVries2003}).
To show confluence of B{\"o}hm reduction over~$\ensuremath{{\mathcal U}}$ we also need the
expansion axiom. The reason is purely technical. In the present
coinductive framework there is no way of talking about infinitary
reductions of arbitrary ordinal length, only about reductions of
length~$\omega$. We need the expansion axiom to show that $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$ implies $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$.
The expansion axiom is necessary for this implication. Let~$\ensuremath{{\mathsf O}}$ be
the \emph{ogre}~\cite{SeveriVries2005} satisfying $\ensuremath{{\mathsf O}} \equiv \lambda
x . \ensuremath{{\mathsf O}}$, i.e., $\ensuremath{{\mathsf O}} \equiv \lambda x_1 . \lambda x_2 . \lambda x_3
\ldots$. A term~$t$ is \emph{head-active}~\cite{SeveriVries2005} if $t
\equiv \lambda x_1 \ldots x_n . r t_1 \ldots t_m$ with $r \in \ensuremath{{\mathcal R}}$ and
$n,m\ge 0$. Define $\ensuremath{{\mathcal H}} = \{ t \in \Lambda^\infty \mid t
\ensuremath{\to^*}_\beta t' \text{ with } t' \text{ head-active} \}$, $\ensuremath{{\mathcal O}} = \{
t \in \Lambda^\infty \mid t \ensuremath{\to^*}_\beta \ensuremath{{\mathsf O}} \}$ and $\ensuremath{{\mathcal U}} = \ensuremath{{\mathcal H}} \cup
\ensuremath{{\mathcal O}}$. One can show that~$\ensuremath{{\mathcal U}}$ is a set of meaningless terms (see the
appendix). Consider $\Omega_\ensuremath{{\mathsf O}} = (\lambda x y . x x) (\lambda x y . x
x)$. We have $\Omega_\ensuremath{{\mathsf O}} \ensuremath{\to^\infty}_\beta \ensuremath{{\mathsf O}} \in \ensuremath{{\mathcal O}}$. But $\Omega_\ensuremath{{\mathsf O}}
\notin \ensuremath{{\mathcal U}}$, so~$\ensuremath{{\mathcal U}}$ does not satisfy the expansion axiom. Now,
$\Omega_\ensuremath{{\mathsf O}} \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} \ensuremath{{\mathsf O}} \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} \bot$, but
$\Omega_\ensuremath{{\mathsf O}} \not\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} \bot$, because no finite
$\beta$-reduct of~$\Omega_\ensuremath{{\mathsf O}}$ is in~$\ensuremath{{\mathcal U}}$.
The expansion axiom could probably be weakened slightly, but the
present formulation is simple and it already appeared in the
literature~\cite{SeveriVries2011a,SeveriVries2005,KennawaySeveriSleepVries2005}. Sets
of meaningless terms which do not satisfy the expansion axiom tend to
be artificial. A notion of a set of strongly meaningless terms
equivalent to ours appears in~\cite{SeveriVries2005}. In the presence
of the expansion axiom, the indiscernibility axiom may be
weakened~\cite{SeveriVries2011a,SeveriVries2005}.
In the setup
of~\cite{EndrullisHansenHendriksPolonskySilva2015,EndrullisHansenHendriksPolonskySilva2018}
it is possible to talk about reductions of arbitrary ordinal length,
but we have not investigated the possibility of adapting the framework
of~\cite{EndrullisHansenHendriksPolonskySilva2015,EndrullisHansenHendriksPolonskySilva2018}
to the needs of the present paper.
The axioms of a set~$\ensuremath{{\mathcal U}}$ of meaningless terms are sufficient for
confluence and normalisation of B{\"o}hm reduction
over~$\ensuremath{{\mathcal U}}$. However, they are not necessary. The
paper~\cite{SeveriVries2011} gives axioms necessary and sufficient for
confluence and normalisation.
The following two simple lemmas will often be used implicitly.
\begin{lem}
Let $\ensuremath{\to^\infty}$ be the infinitary and~$\ensuremath{\to^*}$ the
transitive-reflexive closure of~$\to$. Then the following conditions
hold for all $t,s,s' \in \Lambda^\infty$:
\begin{enumerate}
\item $t \ensuremath{\to^\infty} t$,
\item if $t \ensuremath{\to^*} s \ensuremath{\to^\infty} s'$ then $t \ensuremath{\to^\infty} s'$,
\item if $t \ensuremath{\to^*} s$ then $t \ensuremath{\to^\infty} s$.
\end{enumerate}
\end{lem}
\begin{proof}
The first point follows by coinduction. The second point follows by
case analysis on $s \ensuremath{\to^\infty} s'$. The last point follows from the
previous two.
The proof of the first point is straightforward, but to illustrate
the coinductive technique we give this proof in detail. A reader not
familiar with coinduction is invited to study this proof and insert
the implicit ordinals as in Section~\ref{sec_coind}.
Let $t \in \Lambda^\infty$. There are three cases. If $t \equiv a$
then $a \ensuremath{\to^*} a$, so $t \ensuremath{\to^\infty} t$ by the definition
of~$\ensuremath{\to^\infty}$. If $t \equiv t_1t_2$ then $t_1 \ensuremath{\to^\infty} t_1$ and $t_2
\ensuremath{\to^\infty} t_2$ by the coinductive hypothesis. Since also $t \ensuremath{\to^*}
t_1t_2$, we conclude $t \ensuremath{\to^\infty} t$. If $t \equiv \lambda x . t'$
then $t' \ensuremath{\to^\infty} t'$ by the coinductive hypothesis. Since also $t
\ensuremath{\to^*} \lambda x . t'$, we conclude $t \ensuremath{\to^\infty} t$.
\end{proof}
\begin{lem}
If $R \subseteq S \subseteq \Lambda^\infty \times \Lambda^\infty$
then ${\ensuremath{\to^\infty}_R} \subseteq {\ensuremath{\to^\infty}_S}$.
\end{lem}
\begin{proof}
By coinduction.
\end{proof}
The next three lemmas have essentially been shown
in~\cite[Lemma~4.3--4.5]{EndrullisPolonsky2011}.
\begin{lem}\label{lem_beta_subst}
If $s \ensuremath{\to^\infty}_\beta s'$ and $t \ensuremath{\to^\infty}_\beta t'$ then
$s[t/x] \ensuremath{\to^\infty}_\beta s'[t'/x]$.
\end{lem}
\begin{proof}
By coinduction, with case analysis on $s \ensuremath{\to^\infty}_\beta s'$, using
that $t_1 \ensuremath{\to^*}_\beta t_2$ implies $t_1[t/x] \ensuremath{\to^*}_\beta
t_2[t/x]$.
\end{proof}
\begin{lem}\label{lem_beta_beta_fin_append}
If $t_1 \ensuremath{\to^\infty}_\beta t_2 \to_\beta t_3$ then $t_1
\ensuremath{\to^\infty}_\beta t_3$.
\end{lem}
\begin{proof}
Induction on $t_2 \to_\beta t_3$, using Lemma~\ref{lem_beta_subst}.
\end{proof}
\begin{lem}\label{lem_beta_append}
If $t_1 \ensuremath{\to^\infty}_\beta t_2 \ensuremath{\to^\infty}_\beta t_3$ then $t_1
\ensuremath{\to^\infty}_\beta t_3$.
\end{lem}
\begin{proof}
By coinduction, with case analysis on $t_2 \ensuremath{\to^\infty}_\beta t_3$, using
Lemma~\ref{lem_beta_beta_fin_append}.
\end{proof}
\begin{lem}\label{lem_rnf_fwd}
If $t$ is in rnf and $t \ensuremath{\to^\infty}_\beta s$ then $s$ is in rnf.
\end{lem}
\begin{proof}
Suppose~$s$ is not in rnf, i.e., $s \equiv \bot$ or $s \equiv
s_1s_2$ with $s_1 \ensuremath{\to^*}_\beta \lambda x . u$. If $s \equiv \bot$
then $t \ensuremath{\to^*}_\beta \bot$, and thus either $t \equiv \bot$ or it
$\beta$-reduces to a redex. So~$t$ is not in rnf. If $s \equiv
s_1s_2$ with $s_1 \ensuremath{\to^\infty}_\beta \lambda x . u'$, then $t
\ensuremath{\to^*}_\beta t_1t_2$ with $t_i \ensuremath{\to^\infty}_\beta s_i$. By
Lemma~\ref{lem_beta_beta_fin_append} we have $t_1 \ensuremath{\to^\infty}_\beta
\lambda x . u$. Thus~$t$ reduces to a redex $(\lambda x . u)
t_2$. Hence~$t$ is not in rnf.
\end{proof}
\section{Confluence and normalisation of B{\"o}hm
reductions}\label{sec_inflam_confluence}
In this section we use coinductive techniques to prove confluence and
normalisation of B{\"o}hm reduction over an arbitrary set of strongly
meaningless terms~$\ensuremath{{\mathcal U}}$. The infinitary lambda calculus we are
concerned with, including the $\bot_\ensuremath{{\mathcal U}}$-reductions to~$\bot$, shall
be called the $\lambda_{\beta\bot_\ensuremath{{\mathcal U}}}^\infty$-calculus.
More precisely, our aim is to prove the following theorems.
\medskip
{ \renewcommand{\thethm}{\ref{thm_bohm_cr}}
\begin{thm}[Confluence of the $\lambda_{\beta\bot_\ensuremath{{\mathcal U}}}^\infty$-calculus]~
If $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_1$ and $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_2$ then
there exists~$t_3$ such that $t_1 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$ and $t_2
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$.
\end{thm}
\addtocounter{thm}{-1}
}
{ \renewcommand{\thethm}{\ref{thm_bohm_norm}}
\begin{thm}[Normalisation of the $\lambda_{\beta\bot_\ensuremath{{\mathcal U}}}^\infty$-calculus]~
For every $t \in \Lambda^\infty$ there exists a unique $s \in
\Lambda^\infty$ in $\beta\bot_\ensuremath{{\mathcal U}}$-normal form such that $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$.
\end{thm}
\addtocounter{thm}{-1}
}
In what follows we assume that~$\ensuremath{{\mathcal U}}$ is an arbitrary fixed set of
strongly meaningless terms, unless specified otherwise. Actually,
almost all lemmas are valid for~$\ensuremath{{\mathcal U}}$ being a set of meaningless
terms, without the expansion axiom. Unless explicitly mentioned before
the statement of a lemma, the proofs do not use the expansion
axiom. To show confluence modulo~$\sim_\ensuremath{{\mathcal U}}$
(Theorem~\ref{thm_cr_modulo}), it suffices that~$\ensuremath{{\mathcal U}}$ is a set of
meaningless terms. Confluence and normalisation of the
$\lambda_{\beta\bot_\ensuremath{{\mathcal U}}}$-calculus (Theorem~\ref{thm_bohm_cr} and
Theorem~\ref{thm_bohm_norm}), however, require the expansion
axiom. But this is only because in the present coinductive framework
we are not able to talk about infinite reductions of arbitrary ordinal
length. Essentially, we need the expansion axiom to compress the
B{\"o}hm reductions to length~$\omega$.
The idea of the proof is to show that for every term there exists a
certain standard infinitary $\beta\bot_\ensuremath{{\mathcal U}}$-reduction to normal
form. This reduction is called an infinitary $N_\ensuremath{{\mathcal U}}$-reduction
(Definition~\ref{def_leadsto} and Lemma~\ref{lem_N_normalising}). We
show that the normal forms obtained through infinitary
$N_\ensuremath{{\mathcal U}}$-reductions are unique (Lemma~\ref{lem_N_confluent}). Then we
prove that prepending infinitary $\beta\bot_\ensuremath{{\mathcal U}}$-reduction to an
$N_\ensuremath{{\mathcal U}}$-reduction results in an $N_\ensuremath{{\mathcal U}}$-reduction
(Theorem~\ref{thm_N_prepend}). Since an $N_\ensuremath{{\mathcal U}}$-reduction is an
infinitary $\beta\bot_\ensuremath{{\mathcal U}}$-reduction of a special form
(Lemma~\ref{lem_N_to_bohm}), these results immediately imply
confluence (Theorem~\ref{thm_bohm_cr}) and normalisation
(Theorem~\ref{thm_bohm_norm}) of infinitary
$\beta\bot_\ensuremath{{\mathcal U}}$-reduction. Hence, in essence we derive confluence from
a strengthened normalisation result. See Figure~\ref{fig_cr}.
\begin{figure}[ht]
\centerline{
\xymatrix{
{t_1} \ar@{~>}[d]_>>{N_\ensuremath{{\mathcal U}}} & {t} \ar@{->}[l]_>>{\infty}^>>{\beta\bot_\ensuremath{{\mathcal U}}}
\ar@{~>}[dl]^>>{N_\ensuremath{{\mathcal U}}} \ar@{->}[r]^>>{\infty}_>>{\beta\bot_\ensuremath{{\mathcal U}}}
\ar@{~>}[dr]_>>{N_\ensuremath{{\mathcal U}}} & {t_2} \ar@{~>}[d]^>>{N_\ensuremath{{\mathcal U}}} \\
{t_1'} \ar@{}[rr]|{{\displaystyle\equiv}} & & {t_2'}
}
}
\caption{Confluence of infinitary B{\"o}hm reduction.}\label{fig_cr}
\end{figure}
In our proof we use a standardisation result for infinitary
$\beta$-reductions from~\cite{EndrullisPolonsky2011}
(Theorem~\ref{thm_polonsky}). In particular, this theorem is needed to
show uniqueness of canonical root normal forms
(Definition~\ref{def_crnf}). Theorem~\ref{thm_N_prepend} depends on
this. Even when counting in the results
of~\cite{EndrullisPolonsky2011} only referenced here, our confluence
proof may be considered simpler than previous proofs of related
results. In particular, it is much easier for formalise.
We also show that the set of root-active terms is strongly
meaningless. Together with the previous theorems this implies
confluence and normalisation of the
$\lambda_{\beta\bot_\ensuremath{{\mathcal R}}}^\infty$-calculus. Confluence of the
$\lambda_{\beta\bot_\ensuremath{{\mathcal R}}}^\infty$-calculus in turn implies confluence
of~$\ensuremath{\to^\infty}_\beta$ modulo equivalence of meaningless terms. The
following theorem does not require the expansion axiom.
\medskip
{ \renewcommand{\thethm}{\ref{thm_cr_modulo}}
\begin{thm}[Confluence modulo equivalence of meaningless terms]~
If $t \sim_\ensuremath{{\mathcal U}} t'$, $t \ensuremath{\to^\infty}_{\beta} s$ and $t' \ensuremath{\to^\infty}_{\beta}
s'$ then there exist~$r,r'$ such that $r \sim_\ensuremath{{\mathcal U}} r'$, $s
\ensuremath{\to^\infty}_{\beta} r$ and $s' \ensuremath{\to^\infty}_{\beta} r'$.
\end{thm}
\addtocounter{thm}{-1}
}
Note that our overall proof strategy is different
from~\cite{Czajka2014,KennawayVries2003,KennawayKlopSleepVries1997}. We
first derive a strengthened normalisation result for B{\"o}hm
reduction, from this we derive confluence of B{\"ohm} reduction, then
we show that root-active terms are strongly meaningless thus
specialising the confluence result, and only using that we show
confluence modulo equivalence of meaningless
terms. In~\cite{Czajka2014,KennawayVries2003,KennawayKlopSleepVries1997}
first confluence modulo equivalence of meaningless terms is shown, and
from that confluence of B{\"o}hm reduction is derived. Of course, some
intermediate lemmas we prove have analogons
in~\cite{Czajka2014,KennawayVries2003,KennawayKlopSleepVries1997}, but
we believe the general proof strategy to be fundamentally different.
\subsection{Properties of~$\sim_\ensuremath{{\mathcal U}}$}
In this subsection~$\ensuremath{{\mathcal U}}$ is an arbitrary fixed set of meaningless
terms, and~$\sim_\ensuremath{{\mathcal U}}$ is the parallel closure of~$\ensuremath{{\mathcal U}} \times \ensuremath{{\mathcal U}}$. The
expansion axiom is not used in this subsection.
\begin{lem}\label{lem_sim_subst_2}
If $t \sim_\ensuremath{{\mathcal U}} t'$ and $s \sim_\ensuremath{{\mathcal U}} s'$ then $t[s/x] \sim_\ensuremath{{\mathcal U}}
t'[s'/x]$.
\end{lem}
\begin{proof}
By coinduction, using the substitution axiom.
\end{proof}
\begin{lem}\label{lem_sim_fin_beta_2}
If $t \to_{\beta} s$ and $t \sim_{\ensuremath{{\mathcal U}}} t'$ then there is~$s'$ with
$t' \to_{\beta}^\equiv s'$ and $s \sim_{\ensuremath{{\mathcal U}}} s'$.
\end{lem}
\begin{proof}
Induction on $t \to_\beta s$. If the case $t,t' \in \ensuremath{{\mathcal U}}$ in the
definition of $t \sim_\ensuremath{{\mathcal U}} t'$ holds then $s \in \ensuremath{{\mathcal U}}$ by the closure
axiom, so $t' \sim_\ensuremath{{\mathcal U}} s$ and we may take $s' \equiv t'$. Thus
assume otherwise. Then all cases follow directly from the inductive
hypothesis, except when~$t$ is the contracted $\beta$-redex. Then $t
\equiv (\lambda x . t_1) t_2$ and $s \equiv t_1[t_2/x]$. First
assume $t \in \ensuremath{{\mathcal U}}$. Then also $t' \in \ensuremath{{\mathcal U}}$ by the indiscernibility
axiom (note this does not imply that the first case in the
definition of $t \sim_\ensuremath{{\mathcal U}} t'$ holds). Also $s \in \ensuremath{{\mathcal U}}$ by the
closure axiom, so $t' \sim_\ensuremath{{\mathcal U}} s$ and we may take $s' \equiv t'$. So
assume $t \notin \ensuremath{{\mathcal U}}$. Then $\lambda x . t_1 \notin \ensuremath{{\mathcal U}}$ by the
overlap axiom. Hence $t' \equiv (\lambda x . t_1') t_2'$ with $t_i
\sim_\ensuremath{{\mathcal U}} t_i'$. Thus $t_1[t_2/x] \sim_\ensuremath{{\mathcal U}} t_1'[t_2'/x]$ by
Lemma~\ref{lem_sim_subst_2}. So we may take $s' \equiv
t_1'[t_2'/x]$.
\end{proof}
\begin{lem}\label{lem_sim_beta_2}
If $t \ensuremath{\to^\infty}_{\beta} s$ and $t \sim_{\ensuremath{{\mathcal U}}} t'$ then there is~$s'$
with $t' \ensuremath{\to^\infty}_{\beta} s'$ and $s \sim_{\ensuremath{{\mathcal U}}} s'$.
\end{lem}
\begin{proof}
By coinduction. If $s \equiv a$ then $t \ensuremath{\to^*}_\beta s$ and the
claim follows from Lemma~\ref{lem_sim_fin_beta_2}. If $s \equiv
s_1s_2$ then $t \ensuremath{\to^*}_\beta t_1t_2$ with $t_i \ensuremath{\to^\infty}_\beta
s_i$. By Lemma~\ref{lem_sim_fin_beta_2} there is~$u$ with $t_1t_2
\sim_\ensuremath{{\mathcal U}} u$ and $t' \ensuremath{\to^*}_\beta u$. If $t_1t_2, u \in \ensuremath{{\mathcal U}}$ then
$s \in \ensuremath{{\mathcal U}}$ by the closure axiom, and thus we may take $s' \equiv
u$. Otherwise $u \equiv u_1u_2$ with $t_i \sim_\ensuremath{{\mathcal U}} u_i$. By the
coinductive hypothesis we obtain $s_1',s_2'$ with $u_i \ensuremath{\to^\infty}_\beta
s_i'$ and $s_i \sim_\ensuremath{{\mathcal U}} s_i'$. Take $s' \equiv s_1's_2'$. Then $t'
\ensuremath{\to^\infty}_\beta s'$ and $s \sim_\ensuremath{{\mathcal U}} s'$. If $s \equiv \lambda x . s'$
then the argument is analogous to the previous case.
\end{proof}
\begin{lem}\label{lem_sim_trans}
If $t \sim_\ensuremath{{\mathcal U}} s$ and $s \sim_\ensuremath{{\mathcal U}} u$ then $t \sim_\ensuremath{{\mathcal U}} s$.
\end{lem}
\begin{proof}
By coinduction, using the indiscernibility axiom.
\end{proof}
\begin{lem}\label{lem_sim_to_bot}
If $t \sim_\ensuremath{{\mathcal U}} s$ then there is~$r$ with $t \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} r$ and $s
\ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} r$.
\end{lem}
\begin{proof}
By coinduction.
\end{proof}
\subsection{Properties of parallel $\bot_\ensuremath{{\mathcal U}}$-reduction}
Recall that~$\ensuremath{{\mathcal U}}$ is an arbitrary fixed set of strongly meaningless
terms. The expansion axiom is not used in this subsection except for
Corollary~\ref{cor_back_active}, Lemma~\ref{lem_omega_merge},
Corollary~\ref{cor_bohm_beta_append} and
Lemma~\ref{lem_par_bot_preserves_rnf_rev}.
\begin{lem}\label{lem_subst_omega_1}
If $s \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s'$ and $t \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t'$ then $s[t/x]
\ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s'[t'/x]$.
\end{lem}
\begin{proof}
Coinduction with case analysis on $s \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s'$, using the
substitution axiom.
\end{proof}
\begin{lem}\label{lem_omega_to_bohm}
If $t \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$ then $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$.
\end{lem}
\begin{proof}
By coinduction.
\end{proof}
\begin{lem}\label{lem_omega_indisc}
If $t \in \ensuremath{{\mathcal U}}$ and $t \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$ or $s \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t$ then
$s \in \ensuremath{{\mathcal U}}$.
\end{lem}
\begin{proof}
Using the root-activeness axiom and that $\bot$ is root-active, show
by coinduction that $t \sim_\ensuremath{{\mathcal U}} s$. Then use the indiscernibility
axiom.
\end{proof}
\begin{lem}\label{lem_omega_1_collapse}
If $t_1 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_2 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_3$ then $t_1 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}}
t_3$.
\end{lem}
\begin{proof}
Coinduction with case analysis on $t_2 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_3$, using
Lemma~\ref{lem_omega_indisc}.
\end{proof}
\begin{lem}\label{lem_omega_1_postpone}
If $t_1 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_2 \to_\beta t_3$ then there exists~$t_1'$
such that $t_1 \to_\beta t_1' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_3$.
\end{lem}
\begin{proof}
Induction on $t_2 \to_\beta t_3$. The only interesting case is when
$t_2 \equiv (\lambda x . s_1) s_2$ and $t_3 \equiv s_1[s_2/x]$. Then
$t_1 \equiv (\lambda x . u_1) u_2$ with $u_i \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_i$. By
Lemma~\ref{lem_subst_omega_1}, $u_1[u_2/x] \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}}
s_1[s_2/x]$. Thus take $t_1' \equiv u_1[u_2/x]$.
\end{proof}
\begin{lem}\label{lem_fin_bohm_decompose}
If $s \ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}} t$ then there exists~$r$ such that $s
\ensuremath{\to^*}_\beta r \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t$.
\end{lem}
\begin{proof}
Induction on the length of $s \ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}} t$, using
Lemma~\ref{lem_omega_1_postpone} and Lemma~\ref{lem_omega_1_collapse}.
\end{proof}
\begin{cor}\label{cor_omega_fin_postpone}
If $t_1 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_2 \ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$ then there
is~$s$ with $t_1 \ensuremath{\to^*}_\beta s \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_3$.
\end{cor}
\begin{proof}
Follows from
Lemmas~\ref{lem_fin_bohm_decompose},~\ref{lem_omega_1_postpone},~\ref{lem_omega_1_collapse}.
\end{proof}
\begin{lem}\label{lem_omega_1_beta_to_beta}
If $t_1 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_2 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$ then $t_1
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$.
\end{lem}
\begin{proof}
By coinduction. There are three cases.
\begin{itemize}
\item $t_3 \equiv a$. Then $t_1 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_2
\ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}} a$. By
Corollary~\ref{cor_omega_fin_postpone} there is~$s$ with $t_1
\ensuremath{\to^*}_\beta s \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} a$. By
Lemma~\ref{lem_omega_to_bohm} we have $s \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}}
a$. Thus $t_1 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} a$.
\item $t_3 \equiv s_1s_2$. Then $t_1 \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_2
\ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}} s_1's_2'$ with $s_i'
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_i$. By
Corollary~\ref{cor_omega_fin_postpone} there is~$u$ with $t_1
\ensuremath{\to^*}_\beta u \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_1's_2'$. Then $u \equiv u_1u_2$
with $u_i \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_i' \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_i$. By the
coinductive hypothesis $u_i \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_i$. Thus $t_1
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_1s_2 \equiv t_3$.
\item $t_3 \equiv \lambda x . r$. The argument is analogous to the
previous case.\qedhere
\end{itemize}
\end{proof}
The following lemma is an analogon
of~\cite[Lemma~12.9.22]{KennawayVries2003}.
\begin{lem}[Postponement of parallel
$\bot_\ensuremath{{\mathcal U}}$-reduction]\label{lem_bohm_decompose}~
If $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$ then there exists~$r$ such that
$t \ensuremath{\to^\infty}_\beta r \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$.
\end{lem}
\begin{proof}
By coinduction with case analysis on $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$,
using
Lemmas~\ref{lem_fin_bohm_decompose},~\ref{lem_omega_1_beta_to_beta}.
Since this is the first of our coinductive proofs involving an
implicit Skolem function (see Example~\ref{ex_skolem}), we give it
in detail. The reader is invited to extract from this proof an
explicit corecursive definition of the Skolem function.
Assume $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$. There are three cases.
\begin{itemize}
\item $s \equiv a$ and $t \ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}} a$. Then the
claim follows from Lemma~\ref{lem_fin_bohm_decompose}.
\item $s \equiv s_1 s_2$ and $t \ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}} t_1 t_2$
and $t_i \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_i$. By
Lemma~\ref{lem_fin_bohm_decompose} there is $t'$ with $t
\ensuremath{\to^*}_\beta t' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_1t_2$. Because $t_1t_2
\not\equiv \bot$, we must have $t' \equiv t_1't_2'$ with $t_i'
\ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t_i$. By Lemma~\ref{lem_omega_1_beta_to_beta} we
have $t_i' \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_i$. By the coinductive
hypothesis we obtain $s_1',s_2'$ such that $t_i' \ensuremath{\to^\infty}_\beta
s_i' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_i$. Hence $t \ensuremath{\to^\infty}_\beta s_1's_2'
\ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_1s_2 \equiv s$.
\item $s \equiv \lambda x . s'$ and $t \ensuremath{\to^*}_{\beta\bot_\ensuremath{{\mathcal U}}}
\lambda x . t'$ and $t' \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s'$. By
Lemma~\ref{lem_fin_bohm_decompose} there is~$u$ with $t
\ensuremath{\to^*}_\beta u \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} \lambda x . t'$. Then $u \equiv
\lambda x . u'$ with $u' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} t'$. By
Lemma~\ref{lem_omega_1_beta_to_beta} we have $u'
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s'$. By the coinductive hypothesis we
obtain~$w$ such that $u' \ensuremath{\to^\infty}_\beta w \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s'$. Hence
$t \ensuremath{\to^\infty}_\beta \lambda x . w \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} \lambda x . s'
\equiv s$.\qedhere
\end{itemize}
\end{proof}
\begin{cor}\label{cor_bohm_preserves_U}
If $t \in \ensuremath{{\mathcal U}}$ and $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$ then $s \in \ensuremath{{\mathcal U}}$.
\end{cor}
\begin{proof}
Follows from Lemma~\ref{lem_bohm_decompose}, the closure axiom and
Lemma~\ref{lem_omega_indisc}.
\end{proof}
The following depend on the expansion axiom.
\begin{cor}\label{cor_back_active}
If $s \in \ensuremath{{\mathcal U}}$ and $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$ then $t \in \ensuremath{{\mathcal U}}$.
\end{cor}
\begin{proof}
Follows from Lemma~\ref{lem_bohm_decompose},
Lemma~\ref{lem_omega_indisc} and the expansion axiom.
\end{proof}
\begin{lem}\label{lem_omega_merge}
If $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$ then $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$.
\end{lem}
\begin{proof}
By coinduction, analysing $t' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$. All cases follow
directly from the coinductive hypothesis, except when $s \equiv
\bot$ and~$t' \in \ensuremath{{\mathcal U}}$. But then~$t \in \ensuremath{{\mathcal U}}$ by
Corollary~\ref{cor_back_active}, so $t \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$, and thus $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$ by Lemma~\ref{lem_omega_to_bohm}.
\end{proof}
\begin{cor}\label{cor_bohm_beta_append}
If $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s \ensuremath{\to^*}_{\beta} r$ then $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} r$.
\end{cor}
\begin{proof}
By Lemma~\ref{lem_bohm_decompose} we have $t \ensuremath{\to^\infty}_\beta t'
\ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s \ensuremath{\to^*}_{\beta} r$. By
Lemma~\ref{lem_omega_1_postpone} there is~$s'$ with $t'
\ensuremath{\to^*}_\beta s' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} r$. By
Lemma~\ref{lem_beta_beta_fin_append} we have $t \ensuremath{\to^\infty}_\beta s'$,
and thus $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s'$. By
Lemma~\ref{lem_omega_merge} we finally obtain $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} r$.
\end{proof}
\begin{lem}\label{lem_par_bot_preserves_rnf_rev}
If $t \notin U$ and $t \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$ and $s$ is in rnf, then~$t$
is in rnf.
\end{lem}
\begin{proof}
We consider possible forms of~$s$.
\begin{itemize}
\item $s \equiv a$ with $a \not\equiv \bot$. Then~$t \equiv a$ and~$t$ is
in rnf.
\item $s \equiv \lambda x . s'$. Then $t \equiv \lambda x . t'$ with
$t' \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s'$, so~$t$ is in rnf.
\item $s \equiv s_1 s_2$ and there is no~$r$ with
$s_1 \ensuremath{\to^\infty}_\beta \lambda x . r$. Then $t \equiv t_1 t_2$ with
$t_i \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_i$. Then also $t_1 \sim_\ensuremath{{\mathcal U}} s_1$. Suppose
$t_1 \ensuremath{\to^\infty}_\beta \lambda x . r$. By Lemma~\ref{lem_sim_beta_2}
there is~$r'$ with $s_1 \ensuremath{\to^\infty}_\beta r' \sim_\ensuremath{{\mathcal U}} \lambda x
. r$. There are two cases.
\begin{itemize}
\item $r', \lambda x . r \in \ensuremath{{\mathcal U}}$. Then
$(\lambda x . r) t_2 \in \ensuremath{{\mathcal U}}$ by the overlap axiom, and thus
$t \in \ensuremath{{\mathcal U}}$ by the expansion axiom. Contradiction.
\item $r' \equiv \lambda x . r''$ with $r \sim_\ensuremath{{\mathcal U}} r''$. But then
$s_1 \ensuremath{\to^\infty}_\beta \lambda x . r''$. Contradiction.\qedhere
\end{itemize}
\end{itemize}
\end{proof}
\subsection{Weak head reduction}
\begin{thm}[Endrullis,
Polonsky~\cite{EndrullisPolonsky2011}]\label{thm_polonsky}~
$t \ensuremath{\to^\infty}_{\beta} s$ iff $t \ensuremath{\to^\infty}_{w} s$.
\end{thm}
Strictly speaking, in~\cite{EndrullisPolonsky2011} the above theorem
is shown for a different set of infinitary lambda terms which do not
contain constants. However, it is clear that for the purposes
of~\cite{EndrullisPolonsky2011} constants may be treated as variables
not occuring bound, and thus the proof of the above theorem may be
used in our setting. We omit the proof of this theorem here, but we
included the proof in our formalisation.
\begin{lem}\label{lem_w_fin_commute}
If $t \ensuremath{\to^\infty}_w t_1$ and $t \to_w t_2$ then there is~$t_3$ with $t_2
\ensuremath{\to^\infty}_w t_3$ and $t_1 \to_w^\equiv t_3$.
\end{lem}
\begin{proof}
If the weak head redex in~$t$ is contracted in $t \ensuremath{\to^\infty}_w t_1$
then $t \to_w t_2 \ensuremath{\to^\infty}_w t_1$ and we may take $t_3 \equiv
t_1$. Otherwise $t \equiv (\lambda x . s) u u_1 \ldots u_m$, $t_2
\equiv s[u/x] u_1 \ldots u_m$ and $t_1 \equiv (\lambda x . s') u'
u_1' \ldots u_m'$ with $s \ensuremath{\to^\infty}_w s'$, $u \ensuremath{\to^\infty}_w u'$ and $u_i
\ensuremath{\to^\infty}_w u_i'$ for $i=1,\ldots,m$. By Theorem~\ref{thm_polonsky}
and Lemma~\ref{lem_beta_subst} we obtain $s[u/x] \ensuremath{\to^\infty}_w
s'[u'/x]$. Take $t_3 \equiv s'[u'/x] u_1' \ldots u_m'$. Then $t_2
\ensuremath{\to^\infty}_w t_3$ and $t_1 \to_w t_3$.
\end{proof}
\begin{lem}\label{lem_beta_to_rnf}
If $t \ensuremath{\to^\infty}_\beta s$ with~$s$ in rnf, then there is~$s'$ in rnf
with $t \ensuremath{\to^*}_w s' \ensuremath{\to^\infty}_w s$.
\end{lem}
\begin{proof}
By Theorem~\ref{thm_polonsky} we have $t \ensuremath{\to^\infty}_w s$. Because~$s$
is in rnf, there are three cases.
\begin{itemize}
\item $s \equiv a$ with $a \not\equiv \bot$. Then $t \ensuremath{\to^*}_w s$
and we may take $s' \equiv s$.
\item $s \equiv \lambda x . s_0$. Then $t \ensuremath{\to^*}_w \lambda x
. t_0$ with $t_0 \ensuremath{\to^\infty}_w s_0$. So take $s' \equiv \lambda x
. t_0$.
\item $s \equiv s_1 s_2$ and there is no~$r$ with $s_1 \ensuremath{\to^\infty}_\beta
\lambda x . r$. Then $t \ensuremath{\to^*}_w t_1 t_2$ with $t_i \ensuremath{\to^\infty}_w
s_i$. Suppose $t_1 \ensuremath{\to^\infty}_\beta \lambda x . u$. Then $t_1
\ensuremath{\to^*}_w \lambda x . u'$ for some~$u'$, by
Theorem~\ref{thm_polonsky}. By Lemma~\ref{lem_w_fin_commute} there
is~$r$ with $\lambda x . u' \ensuremath{\to^\infty}_w r$ and $s_1 \ensuremath{\to^*}_w
r$. But then $r \equiv \lambda x . r'$, so~$s_1$ reduces to an
abstraction. Contradiction. Hence~$t_1t_2$ is in rnf, so we may
take $s' \equiv t_1t_2$.\qedhere
\end{itemize}
\end{proof}
\begin{lem}\label{lem_w_unique}
If $t \ensuremath{\to^*}_w s_1$, $t \ensuremath{\to^*}_w s_2$ and these reductions have
the same length, then $s_1 \equiv s_2$.
\end{lem}
\begin{proof}
By induction on the length of the reduction, using the fact that
weak head redexes are unique if they exist.
\end{proof}
\newcommand{\mathrm{crnf}}{\mathrm{crnf}}
\begin{defi}\label{def_crnf}
The \emph{canonical root normal form} (crnf) of a term~$t$ is an
rnf~$s$ such that $t \ensuremath{\to^*}_w s$ and this reduction is shortest
among all finitary weak head reductions of~$t$ to root normal form.
\end{defi}
It follows from Lemma~\ref{lem_beta_to_rnf} and
Lemma~\ref{lem_w_unique} that if~$t$ has a rnf then it has a unique
crnf. We shall denote this crnf by~$\mathrm{crnf}(t)$.
\begin{lem}\label{lem_beta_to_crnf}
If $t \ensuremath{\to^\infty}_\beta s$ with~$s$ in rnf, then $t \ensuremath{\to^*}_w \mathrm{crnf}(t)
\ensuremath{\to^\infty}_w s$.
\end{lem}
\begin{proof}
Follows from Lemma~\ref{lem_beta_to_rnf} and
Lemma~\ref{lem_w_unique}.
\end{proof}
\subsection{Infinitary $N_\ensuremath{{\mathcal U}}$-reduction}\label{sec_inf_n_red}
In the $\lambda_{\beta\bot_\ensuremath{{\mathcal U}}}^\infty$-calculus every term has a
unique normal form. This normal form may be obtained through an
infinitary $N_\ensuremath{{\mathcal U}}$-reduction, defined below.
\begin{defi}\label{def_leadsto}
The relation~$\leadsto_{N_\ensuremath{{\mathcal U}}}$ is defined coinductively.
\[
\begin{array}{c}
\infer={t \leadsto_{N_\ensuremath{{\mathcal U}}} a}{t \notin \ensuremath{{\mathcal U}} & \mathrm{crnf}(t) \equiv a} \quad\quad\,
\infer={t \leadsto_{N_\ensuremath{{\mathcal U}}} s_1s_2}{t \notin \ensuremath{{\mathcal U}} & \mathrm{crnf}(t) \equiv t_1t_2 & t_1
\leadsto_{N_\ensuremath{{\mathcal U}}} s_1 & t_2 \leadsto_{N_\ensuremath{{\mathcal U}}} s_2} \\ \\
\infer={t \leadsto_{N_\ensuremath{{\mathcal U}}} \lambda x . s}{t \notin \ensuremath{{\mathcal U}} & \mathrm{crnf}(t) \equiv \lambda x . t'
& t' \leadsto_{N_\ensuremath{{\mathcal U}}} s} \quad\quad\,
\infer={t \leadsto_{N_\ensuremath{{\mathcal U}}} \bot}{t \in \ensuremath{{\mathcal U}}}
\end{array}
\]
\end{defi}
Note that because $\ensuremath{{\mathcal R}} \subseteq \ensuremath{{\mathcal U}}$, every term $t \notin \ensuremath{{\mathcal U}}$ has a
rnf, so~$\mathrm{crnf}(t)$ is defined for $t \notin \ensuremath{{\mathcal U}}$. Also note that
$\leadsto_{N_\ensuremath{{\mathcal U}}}$ is not closed under contexts, e.g.,
$t \leadsto_{N_\ensuremath{{\mathcal U}}} t'$ does \emph{not} imply
$t s \leadsto_{N_\ensuremath{{\mathcal U}}} t' s$.
The infinitary $N_\ensuremath{{\mathcal U}}$-reduction~$\leadsto_{N_\ensuremath{{\mathcal U}}}$ reduces a term to
its normal form --- its B{\"o}hm-like tree. It is a ``standard''
reduction with a specifically regular structure, which allows us to
prove Theorem~\ref{thm_N_prepend}: if
$t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t' \leadsto_{N_\ensuremath{{\mathcal U}}} s$ then
$t \leadsto_{N_\ensuremath{{\mathcal U}}} s$. This property allows us to derive confluence
from the fact that every term has a unique normal form reachable by an
infinitary $N_\ensuremath{{\mathcal U}}$-reduction. See Figure~\ref{fig_cr}. It is crucial
here that canonical root normal forms are unique, and that
Lemma~\ref{lem_beta_to_crnf} holds. This depends on
Theorem~\ref{thm_polonsky} --- the standardisation result shown by
Endrullis and Polonsky.
\begin{lem}\label{lem_N_to_bohm}
If $t \leadsto_{N_\ensuremath{{\mathcal U}}} s$ then $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$.
\end{lem}
\begin{proof}
By coinduction.
\end{proof}
\begin{lem}\label{lem_N_normalising}
For every term $t \in \Lambda^\infty$ there is~$s$ with $t
\leadsto_{N_\ensuremath{{\mathcal U}}} s$.
\end{lem}
\begin{proof}
By coinduction. If~$t \in \ensuremath{{\mathcal U}}$ then $t \leadsto_{N_\ensuremath{{\mathcal U}}} \bot$ and we
may take $s \equiv \bot$. Otherwise there are three cases depending
on the form of~$\mathrm{crnf}(t)$.
\begin{itemize}
\item $\mathrm{crnf}(t) \equiv a$. Then $t \leadsto_{N_\ensuremath{{\mathcal U}}} a$ by the first
rule, so we may take $s \equiv a$.
\item $\mathrm{crnf}(t) \equiv t_1t_2$. Then by the coinductive hypothesis
we obtain $s_1,s_2$ with $t_i \leadsto_{N_\ensuremath{{\mathcal U}}} s_i$. Take
$s \equiv s_1s_2$. Then $t \leadsto_{N_\ensuremath{{\mathcal U}}} s$.
\item $\mathrm{crnf}(t) \equiv \lambda x . t'$. Analogous to the previous
case.\qedhere
\end{itemize}
\end{proof}
\begin{lem}\label{lem_N_confluent}
If $t \leadsto_{N_\ensuremath{{\mathcal U}}} s_1$ and $t \leadsto_{N_\ensuremath{{\mathcal U}}} s_2$ then
$s_1 \equiv s_2$.
\end{lem}
\begin{proof}
By coinduction. If $s_1 \equiv \bot$ then~$t \in \ensuremath{{\mathcal U}}$, so we must
also have $s_2 \equiv \bot$. Otherwise there are three cases,
depending on the form of~$\mathrm{crnf}(t)$. Suppose
$\mathrm{crnf}(t) \equiv t_1t_2$, other cases being similar. Then
$s_1 \equiv u_1u_2$ with $t_i \leadsto_{N_\ensuremath{{\mathcal U}}} u_i$ and
$s_2 \equiv w_1w_2$ with $t_i \leadsto_{N_\ensuremath{{\mathcal U}}} w_i$. By the
coinductive hypothesis $u_i \equiv w_i$. Thus
$s_1 \equiv u_1u_2 \equiv w_1w_2 \equiv s_2$.
\end{proof}
The next two lemmas and the theorem depend on the expansion axiom.
\begin{lem}\label{lem_N_nf}
If $t \leadsto_{N_\ensuremath{{\mathcal U}}} s$ then $s$ is in $\beta\bot_\ensuremath{{\mathcal U}}$-normal
form.
\end{lem}
\begin{proof}
Suppose~$s$ contains a $\beta\bot_\ensuremath{{\mathcal U}}$-redex. Without loss of
generality, assume the redex is at the root. First assume that~$s$
is a $\bot_\ensuremath{{\mathcal U}}$-redex, i.e., $s \in \ensuremath{{\mathcal U}}$ and $s \not\equiv
\bot$. Using Lemma~\ref{lem_N_to_bohm} we conclude
$t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$. Then $t \in \ensuremath{{\mathcal U}}$ by
Corollary~\ref{cor_back_active}. Thus $s \equiv \bot$, giving a
contradiction. So assume~$s$ is a $\beta$-redex, i.e.,
$s \equiv (\lambda x . s_1) s_2$. But by inspecting the definition
of $t \leadsto_{N_\ensuremath{{\mathcal U}}} s$ one sees that this is only possible
when~$\mathrm{crnf}(t)$ is a $\beta$-redex. Contradiction.
\end{proof}
\begin{lem}\label{lem_omega_rnf}
Suppose $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$ and $t,s$ are in rnf.
\begin{itemize}
\item If $s \equiv a$ then $t \equiv s$.
\item If $s \equiv s_1s_2$ then $t \equiv t_1t_2$ with $t_i
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_i$.
\item If $s \equiv \lambda x . s'$ then $t \equiv \lambda x . t'$
with $t' \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s'$.\qedhere
\end{itemize}
\end{lem}
\begin{proof}
First note that by Lemma~\ref{lem_bohm_decompose} there is~$r$
with $t \ensuremath{\to^\infty}_\beta r \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s$.
\begin{itemize}
\item If $s \equiv a$ then $a \not\equiv \bot$ and $r \equiv a$, and thus
$t \ensuremath{\to^*}_\beta a$. But because~$t$ is in rnf it does not reduce
to a $\beta$-redex, so in fact $t \equiv a$.
\item If $s \equiv s_1s_2$ then $r \equiv r_1r_2$ with
$r_i \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_i$. Thus $t \ensuremath{\to^*}_\beta t_1't_2'$ where
$t_i' \ensuremath{\to^\infty}_\beta r_i$. Because~$t$ is in rnf, we must in fact
have $t \equiv t_1t_2$ with $t_i \ensuremath{\to^*}_\beta t_i'$. Then
$t_i \ensuremath{\to^\infty}_\beta r_i \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} s_i$, so
$t_i \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s_i$ by Lemma~\ref{lem_omega_merge}.
\item The case $s \equiv \lambda x . s'$ is analogous to the
previous one.\qedhere
\end{itemize}
\end{proof}
\begin{thm}\label{thm_N_prepend}
If $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t' \leadsto_{N_\ensuremath{{\mathcal U}}} s$ then
$t \leadsto_{N_\ensuremath{{\mathcal U}}} s$.
\end{thm}
\begin{proof}
By coinduction. If $s \equiv \bot$ then~$t' \in \ensuremath{{\mathcal U}}$. By
Corollary~\ref{cor_back_active} also~$t \in \ensuremath{{\mathcal U}}$. Hence
$t \leadsto_{N_\ensuremath{{\mathcal U}}} \bot \equiv s$. If $s \not\equiv\bot$ then
$t' \notin \ensuremath{{\mathcal U}}$ and $t' \ensuremath{\to^*}_w \mathrm{crnf}(t')$. By
Corollary~\ref{cor_bohm_beta_append} we have
$t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} \mathrm{crnf}(t')$. By
Lemma~\ref{lem_bohm_decompose} there is~$r$ with
$t \ensuremath{\to^\infty}_\beta r \ensuremath{\Rightarrow}_{\bot_\ensuremath{{\mathcal U}}} \mathrm{crnf}(t')$. We have $t \notin \ensuremath{{\mathcal U}}$
by Corollary~\ref{cor_bohm_preserves_U}. Then~$r$ is in rnf by
Lemma~\ref{lem_par_bot_preserves_rnf_rev}. Hence
$t \ensuremath{\to^*}_w \mathrm{crnf}(t) \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} \mathrm{crnf}(t')$ by
Lemma~\ref{lem_beta_to_crnf} and Lemma~\ref{lem_omega_merge}. Now
there are three cases depending on the form of~$\mathrm{crnf}(t')$.
\begin{itemize}
\item $\mathrm{crnf}(t') \equiv a$. Then $s \equiv a$, and $\mathrm{crnf}(t) \equiv
a$ by Lemma~\ref{lem_omega_rnf}. Thus $t \leadsto_{N_\ensuremath{{\mathcal U}}} a \equiv s$.
\item $\mathrm{crnf}(t') \equiv t_1't_2'$. Then $s \equiv s_1s_2$ with
$t_i' \leadsto_{N_\ensuremath{{\mathcal U}}} s_i$. By Lemma~\ref{lem_omega_rnf} we have
$\mathrm{crnf}(t) \equiv t_1t_2$ with $t_i \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}}
t_i'$. By the coinductive hypothesis $t_i \leadsto_{N_\ensuremath{{\mathcal U}}}
s_i$. Hence $t \leadsto_{N_\ensuremath{{\mathcal U}}} s_1s_2 \equiv s$.
\item The case $\mathrm{crnf}(t') \equiv \lambda x . u$ is analogous to the
previous one.\qedhere
\end{itemize}
\end{proof}
\subsection{Confluence and normalisation}
Recall that~$\ensuremath{{\mathcal U}}$ is an arbitrary fixed set of strongly meaningless
terms.
\begin{thm}[Confluence of the $\lambda_{\beta\bot_\ensuremath{{\mathcal U}}}^\infty$-calculus]\label{thm_bohm_cr}~
If $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_1$ and
$t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_2$ then there exists~$t_3$ such that
$t_1 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$ and
$t_2 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$.
\end{thm}
\begin{proof}
By Lemma~\ref{lem_N_normalising} there are~$t_1',t_2'$ with
$t_i \leadsto_{N_\ensuremath{{\mathcal U}}} t_i'$ for $i=1,2$. By
Theorem~\ref{thm_N_prepend} we have $t \leadsto_{N_\ensuremath{{\mathcal U}}} t_i'$ for
$i=1,2$. By Lemma~\ref{lem_N_confluent} we have $t_1' \equiv
t_2'$. Take $t_3 \equiv t_1' \equiv t_2'$. We have
$t_i \leadsto_{N_\ensuremath{{\mathcal U}}} t_3$ for $i=1,2$, so
$t_1 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$ and
$t_2 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t_3$ by Lemma~\ref{lem_N_to_bohm}.
\end{proof}
\begin{thm}[Normalisation of the
$\lambda_{\beta\bot_\ensuremath{{\mathcal U}}}^\infty$-calculus]\label{thm_bohm_norm}~
For every $t \in \Lambda^\infty$ there exists a unique
$s \in \Lambda^\infty$ in $\beta\bot_\ensuremath{{\mathcal U}}$-normal form such that
$t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$.
\end{thm}
\begin{proof}
By Lemma~\ref{lem_N_normalising} there is~$s$ with
$t \leadsto_{N_\ensuremath{{\mathcal U}}} s$. By Lemma~\ref{lem_N_nf}, $s$ is in
$\beta\bot_\ensuremath{{\mathcal U}}$-normal form. By Lemma~\ref{lem_N_to_bohm} we have
$t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} s$. The uniqueness of~$s$ follows from
Theorem~\ref{thm_bohm_cr}.
\end{proof}
\subsection{Root-active terms are strongly meaningless}
\begin{defi}\label{def_succ}
We define the relation~$\succ_x$ coinductively
\[
\infer={t \succ_x xu_1\ldots u_n}{u_1,\ldots,u_n \in \Lambda^\infty}\quad\quad
\infer={a \succ_x a}{}\quad\quad
\infer={ts \succ_x t's'}{t \succ_x t' & s \succ_x s'}\quad\quad
\infer={\lambda y . t \succ_x \lambda y . t'}{t \succ_x t' & x \ne y}
\]
In other words, $s \succ_x s'$ iff~$s'$ may be obtained from~$s$ by
changing some arbitrary subterms in~$s$ into some terms having the
form $xu_1\ldots u_n$.
\end{defi}
\begin{lem}\label{lem_subst_succ}
If $t \succ_x t'$, $s \succ_x s'$ and $x \ne y$ then $t[s/y] \succ_x
t'[s'/y]$.
\end{lem}
\begin{proof}
By coinduction, analysing $t \succ_x t'$.
\end{proof}
\begin{lem}\label{lem_beta_succ}
If $t \succ_x s$ and $t \to_\beta t'$ then there is~$s'$ with $t'
\succ_x s'$ and $s \to_\beta^\equiv s'$.
\end{lem}
\begin{proof}
Induction on~$t \to_\beta t'$. The interesting case is when $t
\equiv (\lambda y . t_1) t_2$, $t' \equiv t_1[t_2/y]$, $s \equiv
s_1s_2$, $\lambda y . t_1 \succ_x s_1$ and $t_2 \succ_x s_2$. If
$s_1 \equiv x u_1 \ldots u_m$ then $t' \succ_x x u_1 \ldots u_m s_2$
and we may take $s' \equiv s$. Otherwise $s_1 \equiv \lambda y
. s_1'$ with $t_1 \succ_x s_1'$ (by the variable convention $x \ne
y$). Then $t' \equiv t_1[t_2/y] \succ_x s_1'[s_2/y]$ by
Lemma~\ref{lem_subst_succ}. We may thus take $s' \equiv
s_1'[s_2/y]$.
\end{proof}
\begin{lem}\label{lem_beta_succ_rev}
If $t \succ_x s$ and $s \to_\beta s'$ then there is~$t'$ with $t'
\succ_x s'$ and $t \to_\beta^\equiv t'$.
\end{lem}
\begin{proof}
Induction on~$s \to_\beta s'$, using Lemma~\ref{lem_subst_succ} for
the redex case.
\end{proof}
\begin{lem}\label{lem_succ_rnf}
If $t \succ_x t'$ and~$t$ is in rnf, then so is~$t'$.
\end{lem}
\begin{proof}
Assume~$t'$ is not in rnf. Then $t' \equiv \bot$ or $t' \equiv
t_1't_2'$ with~$t_1'$ reducing to an abstraction. If $t' \equiv
\bot$ then $t \equiv \bot$, so assume $t' \equiv t_1't_2'$ and $t_1'
\ensuremath{\to^*}_\beta \lambda y . u'$ with $x \ne y$. Then $t \equiv
t_1t_2$ with $t_i \succ_x t_i'$. By Lemma~\ref{lem_beta_succ_rev}
there is~$u$ with $t_1 \ensuremath{\to^*}_\beta \lambda y . u$ and $u \succ_x
u'$. But this implies that $t \equiv t_1t_2$ is not in
rnf. Contradiction.
\end{proof}
\begin{lem}\label{lem_subst_active}
If $t_1, t_2 \in \Lambda^\infty$ and~$t_1$ has no rnf, then neither
does $t_1[t_2/x]$.
\end{lem}
\begin{proof}
Assume $t_1[t_2/x]$ has a rnf. Then $t_1[t_2/x] \ensuremath{\to^*}_\beta s$
for some~$s$ in rnf, by Lemma~\ref{lem_beta_to_rnf}. By the variable
convention $t_1[t_2/x] \succ_x t_1$. Hence by
Lemma~\ref{lem_beta_succ} there is~$s'$ such that $t_1
\ensuremath{\to^*}_\beta s'$ and $s \succ_x s'$. Since~$s$ is in rnf, so
is~$s'$, by Lemma~\ref{lem_succ_rnf}. Thus~$t_1$ has a
rnf. Contradiction.
\end{proof}
\begin{lem}\label{lem_sim_subst}
If $t \sim_\ensuremath{{\mathcal R}} t'$ and $s \sim_\ensuremath{{\mathcal R}} s'$ then $t[s/x] \sim_\ensuremath{{\mathcal R}}
t'[s'/x]$.
\end{lem}
\begin{proof}
By coinduction, using Lemma~\ref{lem_subst_active}.
\end{proof}
\begin{lem}\label{lem_beta_sim}
If $t \to_\beta t'$ and $t \sim_\ensuremath{{\mathcal R}} s$ then there is~$s'$ with $s
\to_\beta^\equiv s'$ and $t' \sim_\ensuremath{{\mathcal R}} s'$.
\end{lem}
\begin{proof}
Induction on $t \to_\beta t'$. There are two interesting cases.
\begin{itemize}
\item $t, s \in \ensuremath{{\mathcal R}}$, i.e., they have no rnf. Then also $t' \in
\ensuremath{{\mathcal R}}$, so we may take $s' \equiv s$.
\item $t \equiv (\lambda x . t_1) t_2$, $t' \equiv t_1[t_2/x]$, $s
\equiv (\lambda x . s_1) s_2$ and $t_i \sim_\ensuremath{{\mathcal R}} s_i$. Then $t'
\sim_\ensuremath{{\mathcal R}} s_1[s_2/x]$ by Lemma~\ref{lem_sim_subst}. Hence we may
take $s' \equiv s_1[s_2/x]$.\qedhere
\end{itemize}
\end{proof}
\begin{lem}\label{lem_sim_rnf}
If $t$ is in rnf and $t \sim_\ensuremath{{\mathcal R}} s$, then so is~$s$.
\end{lem}
\begin{proof}
Because~$t$ is in rnf, there are three cases.
\begin{itemize}
\item $t \equiv a$ with $a \not\equiv \bot$. Then $s \equiv t$, so
it is in rnf.
\item $t \equiv \lambda x . t'$. Then $s \equiv \lambda x . s'$,
so~$s$ is in rnf.
\item $t \equiv t_1t_2$ and~$t_1$ does not $\beta$-reduce to an
abstraction. Then $s \equiv s_1s_2$ with $t_i \sim_\ensuremath{{\mathcal R}}
s_i$. Assume $s_1 \ensuremath{\to^*}_\beta \lambda x . s'$. Then by
Lemma~\ref{lem_beta_sim} there is~$t'$ with $t_1 \ensuremath{\to^*}_\beta t'
\sim_\ensuremath{{\mathcal R}} \lambda x . s'$. But then~$t'$ must be an
abstraction. Contradiction.\qedhere
\end{itemize}
\end{proof}
\begin{cor}\label{cor_sim_rnf}
If $t$ has a rnf and $t \sim_\ensuremath{{\mathcal R}} s$, then so does~$s$.
\end{cor}
\begin{proof}
Follows from Lemma~\ref{lem_beta_sim} and Lemma~\ref{lem_sim_rnf}.
\end{proof}
\begin{lem}\label{lem_infbeta_rnf}
If $t \ensuremath{\to^\infty}_\beta s$ and~$t$ has a rnf, then so does~$s$.
\end{lem}
\begin{proof}
Suppose~$t$ has a rnf. Then by Lemma~\ref{lem_beta_to_rnf} there
is~$t'$ in rnf with $t \ensuremath{\to^*}_w t'$. By Theorem~\ref{thm_polonsky}
and Lemma~\ref{lem_w_fin_commute} there is~$r$ with $s \ensuremath{\to^*}_w r$
and $t' \ensuremath{\to^\infty}_\beta r$. Since~$t'$ is in rnf, by
Lemma~\ref{lem_rnf_fwd} so is~$r$. Hence~$s$ has a rnf~$r$.
\end{proof}
\begin{thm}\label{thm_root_active_meaningless}
The set~$\ensuremath{{\mathcal R}}$ of root-active terms is a set of strongly meaningless
terms.
\end{thm}
\begin{proof}
We check the axioms. The root-activeness axiom is obvious. The
closure axiom follows from Lemma~\ref{lem_beta_append}. The
substitution axiom follows from Lemma~\ref{lem_subst_active}. The
overlap axiom follows from the fact that lambda abstractions are in
rnf. The indiscernibility axiom follows from
Corollary~\ref{cor_sim_rnf}. The expansion axiom follows from
Lemma~\ref{lem_infbeta_rnf}.
\end{proof}
\begin{cor}[Confluence of the $\lambda_{\beta\bot_\ensuremath{{\mathcal R}}}^\infty$-calculus]\label{cor_bohm_ra_cr}~
If $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal R}}} t_1$ and $t \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal R}}} t_2$ then
there exists~$t_3$ such that $t_1 \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal R}}} t_3$ and $t_2
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal R}}} t_3$.
\end{cor}
\begin{cor}[Normalisation of the
$\lambda_{\beta\bot_\ensuremath{{\mathcal R}}}^\infty$-calculus]\label{cor_bohm_ra_norm}~
For every $t \in \Lambda^\infty$ there exists a unique $s \in
\Lambda^\infty$ in $\beta\bot_\ensuremath{{\mathcal R}}$-normal form such that $t
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal R}}} s$.
\end{cor}
\subsection{Confluence modulo equivalence of meaningless terms}
From confluence of the $\lambda_{\beta\bot_\ensuremath{{\mathcal R}}}^\infty$-calculus we
may derive confluence of infinitary $\beta$-reduction~$\ensuremath{\to^\infty}_\beta$
modulo equivalence of meaningless terms. The expansion axiom in not
needed for the proof of the following theorem.
\begin{thm}[Confluence modulo equivalence of meaningless
terms]\label{thm_cr_modulo}~
If $t \sim_\ensuremath{{\mathcal U}} t'$, $t \ensuremath{\to^\infty}_{\beta} s$ and $t' \ensuremath{\to^\infty}_{\beta}
s'$ then there exist~$r,r'$ such that $r \sim_\ensuremath{{\mathcal U}} r'$, $s
\ensuremath{\to^\infty}_{\beta} r$ and $s' \ensuremath{\to^\infty}_{\beta} r'$.
\end{thm}
\begin{proof}
By Lemma~\ref{lem_sim_beta_2} and Lemma~\ref{lem_sim_trans} it
suffices to consider the case $t \equiv t'$. By
Corollary~\ref{cor_bohm_ra_cr} there is~$u$ with $s
\ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal R}}} u$ and $s' \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal R}}} u$. By
Theorem~\ref{thm_root_active_meaningless} and
Lemma~\ref{lem_bohm_decompose} there are $r,r'$ with $s
\ensuremath{\to^\infty}_\beta r \sim_{\ensuremath{{\mathcal R}}} u$ and $s' \ensuremath{\to^\infty}_\beta r' \sim_\ensuremath{{\mathcal R}}
u$. Because $\ensuremath{{\mathcal R}} \subseteq \ensuremath{{\mathcal U}}$, by Lemma~\ref{lem_sim_trans} we
obtain $r \sim_\ensuremath{{\mathcal U}} r'$.
\end{proof}
\section{Strongly convergent reductions}\label{sec_inflam_strong_convergence}
In this section we prove that the existence of coinductive infinitary
reductions is equivalent to the existence of strongly convergent
reductions, under certain assumptions. As a corollary, this also
yields $\omega$-compression of strongly convergent reductions, under
certain assumptions. The equivalence proof is virtually the same as
in~\cite{EndrullisPolonsky2011}. The notion of strongly convergent
reductions is the standard notion of infinitary reductions used in
non-coinductive treatments of infinitary lambda calculus.
\newcommand{\ensuremath{\to^{2\infty}}}{\ensuremath{\to^{2\infty}}}
\newcommand{\ensuremath{\to^{\infty*}}}{\ensuremath{\to^{\infty*}}}
\begin{defi}
On the set of infinitary lambda terms we define a metric~$d$ by
\[
d(t,s) = \inf\{2^{-n} \mid t^{\upharpoonright n} \equiv
s^{\upharpoonright{n}} \}
\]
where $r^{\upharpoonright n}$ for $r \in \Lambda^\infty$ is defined
as the infinitary lambda term obtained by replacing all subterms
of~$r$ at depth~$n$ by~$\bot$. This defines a metric topology on the
set of infinitary lambda terms. Let
$R \subseteq \Lambda^\infty \times \Lambda^\infty$ and let~$\zeta$
be an ordinal. A map $f : \{\gamma \le \zeta\} \to \Lambda^\infty$
together with reduction steps
$\sigma_\gamma : f(\gamma) \to_R f(\gamma+1)$ for $\gamma < \zeta$ is a
\emph{strongly convergent $R$-reduction sequence of length~$\zeta$
from~$f(0)$ to~$f(\zeta)$} if the following conditions hold:
\begin{enumerate}
\item if $\delta \le \zeta$ is a limit ordinal then~$f(\delta)$ is
the limit in the metric topology on infinite terms of the
ordinal-indexed sequence $(f(\gamma))_{\gamma<\delta}$,
\item if $\delta \le \zeta$ is a limit ordinal then for every $d
\in \ensuremath{\mathbb{N}}$ there exists $\gamma < \delta$ such that for all~$\gamma'$
with $\gamma \le \gamma' < \delta$ the redex contracted in the
step~$\sigma_{\gamma'}$ occurs at depth greater than~$d$.
\end{enumerate}
We write $s \xrightarrow{S,\zeta}_R t$ if~$S$ is a strongly convergent
$R$-reduction sequence of length~$\zeta$ from~$s$ to~$t$.
A relation ${\to} \subseteq \Lambda^\infty \times \Lambda^\infty$ is
\emph{appendable} if $t_1 \ensuremath{\to^\infty} t_2 \to t_3$ implies $t_1 \ensuremath{\to^\infty}
t_3$. We define~$\ensuremath{\to^{2\infty}}$ as the infinitary closure of~$\ensuremath{\to^\infty}$. We
write~$\ensuremath{\to^{\infty*}}$ for the transitive-reflexive closure of~$\ensuremath{\to^\infty}$.
\end{defi}
\begin{lem}\label{lem_appendable_append}
If $\to$ is appendable then $t_1 \ensuremath{\to^\infty} t_2 \ensuremath{\to^\infty} t_3$ implies $t_1
\ensuremath{\to^\infty} t_3$.
\end{lem}
\begin{proof}
By coinduction. This has essentially been shown
in~\cite[Lemma~4.5]{EndrullisPolonsky2011}.
\end{proof}
\begin{lem}\label{lem_two_to_one}
If $\to$ is appendable then $s \ensuremath{\to^{2\infty}} t$ implies $s \ensuremath{\to^\infty} t$.
\end{lem}
\begin{proof}
By coinduction. There are three cases.
\begin{itemize}
\item $t \equiv a$. Then $s \ensuremath{\to^{\infty*}} a$, so $s \ensuremath{\to^\infty} a$ by
Lemma~\ref{lem_appendable_append}.
\item $t \equiv t_1 t_2$. Then there are $t_1',t_2'$ with $s
\ensuremath{\to^{\infty*}} t_1't_2'$ and $t_i' \ensuremath{\to^{2\infty}} t_i$. By
Lemma~\ref{lem_appendable_append} we have $s \ensuremath{\to^\infty} t_1't_2'$, so
there are $u_1,u_2$ with $s \ensuremath{\to^*} u_1u_2$ and $u_i \ensuremath{\to^\infty}
t_i'$. Then $u_i \ensuremath{\to^{2\infty}} t_i$. By the coinductive hypothesis $u_i
\ensuremath{\to^\infty} t_i$. Hence $s \ensuremath{\to^\infty} t_1t_2 \equiv t$.
\item $t \equiv \lambda x . r$. Then by
Lemma~\ref{lem_appendable_append} there is~$s'$ with $s \ensuremath{\to^\infty}
\lambda x . s'$ and $s' \ensuremath{\to^{2\infty}} r$. So there is~$s_0$ with $s
\ensuremath{\to^*} \lambda x . s_0$ and $s_0 \ensuremath{\to^\infty} s'$. Then also $s_0
\ensuremath{\to^{2\infty}} r$. By the coinductive hypothesis $s_0 \ensuremath{\to^\infty} r$. Thus $s
\ensuremath{\to^\infty} \lambda x . r \equiv t$.\qedhere
\end{itemize}
\end{proof}
\begin{thm}\label{thm_strongly_convergent}
For every $R \subseteq \Lambda^\infty \times \Lambda^\infty$ such
that~$\to_R$ is appendable, and for all $s,t \in \Lambda^\infty$, we
have the equivalence: $s \ensuremath{\to^\infty}_R t$ iff there exists a strongly
convergent $R$-reduction sequence from~$s$ to~$t$. Moreover, if $s
\ensuremath{\to^\infty}_R t$ then the sequence may be chosen to have length at
most~$\omega$.
\end{thm}
\begin{proof}
The proof is a straightforward generalisation of the proof of
Theorem~3 in~\cite{EndrullisPolonsky2011}.
Suppose that $s \ensuremath{\to^\infty}_R t$. By traversing the infinite derivation
tree of $s \ensuremath{\to^\infty}_R t$ and accumulating the finite prefixes by
concatenation, we obtain a reduction sequence of length at
most~$\omega$ which satisfies the depth requirement by construction.
For the other direction, by induction on~$\zeta$ we show that if $s
\xrightarrow{S,\zeta}_R t$ then $s \ensuremath{\to^{2\infty}}_R t$, which suffices for $s
\ensuremath{\to^\infty}_R t$ by Lemma~\ref{lem_two_to_one}. There are three cases.
\begin{itemize}
\item $\zeta = 0$. If $s \xrightarrow{S,0}_R t$ then $s \equiv t$, so $s
\ensuremath{\to^{2\infty}}_R t$.
\item $\zeta = \gamma+1$. If $s \xrightarrow{S,\gamma+1}_R t$ then $s
\xrightarrow{S',\gamma}_R s' \to_R t$. Hence $s \ensuremath{\to^{2\infty}}_R s'$ by the
inductive hypothesis. Then $s \ensuremath{\to^\infty}_R s' \to_R t$ by
Lemma~\ref{lem_two_to_one}. So $s \ensuremath{\to^\infty}_R t$ because~$\to_R$ is
appendable.
\item $\zeta$ is a limit ordinal. By coinduction we show that if $s
\xrightarrow{S,\zeta}_R t$ then $s \ensuremath{\to^{2\infty}}_R t$. By the depth condition
there is $\gamma<\zeta$ such that for every $\delta \ge \gamma$ the
redex contracted in~$S$ at~$\delta$ occurs at depth greater than
zero. Let~$t_\gamma$ be the term at index~$\gamma$ in~$S$. Then by
the inductive hypothesis we have $s \ensuremath{\to^{2\infty}}_R t_\gamma$, and thus $s
\ensuremath{\to^\infty}_R t_\gamma$ by Lemma~\ref{lem_two_to_one}. There are three
cases.
\begin{itemize}
\item $t_\gamma \equiv a$. This is impossible because then there
can be no reduction of~$t_\gamma$ at depth greater than zero.
\item $t_\gamma \equiv \lambda x . r$. Then $t \equiv \lambda x
. u$ and $r \xrightarrow{S',\delta}_R u$ with $\delta \le \zeta$. Hence
$r \ensuremath{\to^{2\infty}}_R u$ by the coinductive hypothesis if $\delta=\zeta$,
or by the inductive hypothesis if $\delta < \zeta$. Since $s
\ensuremath{\to^\infty}_R \lambda x . r$ we obtain $s \ensuremath{\to^{2\infty}}_R \lambda x . u
\equiv t$.
\item $t_\gamma \equiv t_1t_2$. Then $t\equiv u_1u_2$ and the tail
of the reduction~$S$ past~$\gamma$ may be split into two parts:
$t_i \xrightarrow{S_i,\delta_i}_R u_i$ with $\delta_i \le \zeta$ for
$i=0,1$. Then $t_i \ensuremath{\to^{2\infty}}_R u_i$ by the inductive and/or the
coinductive hypothesis. Since $s \ensuremath{\to^\infty}_R t_1t_2$ we obtain $s
\ensuremath{\to^{2\infty}}_R u_1u_2 \equiv t$.\qedhere
\end{itemize}
\end{itemize}
\end{proof}
\begin{cor}[$\omega$-compression]
If~$\to_R$ is appendable and there exists a strongly convergent
$R$-reduction sequence from~$s$ to~$t$ then there exists such a
sequence of length at most~$\omega$.
\end{cor}
\begin{cor}
Let~$\ensuremath{{\mathcal U}}$ be a set of strongly meaningless terms.
\begin{itemize}
\item $s \ensuremath{\to^\infty}_{\beta\bot_\ensuremath{{\mathcal U}}} t$ iff there exists a strongly
convergent $\beta\bot_\ensuremath{{\mathcal U}}$-reduction sequence from~$s$ to~$t$.
\item $s \ensuremath{\to^\infty}_{\beta} t$ iff there exists a strongly convergent
$\beta$-reduction sequence from~$s$ to~$t$.
\end{itemize}
\end{cor}
\begin{proof}
By Theorem~\ref{thm_strongly_convergent} it suffices to show
that~$\to_{\beta\bot_\ensuremath{{\mathcal U}}}$ and~$\to_\beta$ are
appendable. For~$\to_{\beta\bot_\ensuremath{{\mathcal U}}}$ this follows from
Lemma~\ref{lem_omega_merge} and
Corollary~\ref{cor_bohm_beta_append}. For~$\to_\beta$ this follows
from Lemma~\ref{lem_beta_beta_fin_append}.
\end{proof}
\section{The formalisation}\label{sec_formalisation}
The results of this paper have been formalised in the Coq proof
assistant. The formalisation is available at:
\begin{center}
\url{https://github.com/lukaszcz/infinitary-confluence}
\end{center}
The formalisation contains all results of
Section~\ref{sec_inflam_confluence}. We did not formalise the proof
from Section~\ref{sec_inflam_strong_convergence} of the equivalence
between the coinductive definition of the infinitary reduction
relation and the standard notion of strongly convergent reductions.
In our formalisation we use a representation of infinitary
lambda terms with de Bruijn indices, and we do not allow constants
except~$\bot$. Hence, the results about $\alpha$-conversion alluded to
in Section~\ref{sec_inflam_intro} are not formalised either. Because
the formalisation is based on de Bruijn indices, many tedious lifting
lemmas need to be proved. These lemmas are present only in the
formalisation, but not in the paper.
In general, the formalisation follows closely the text of the
paper. Each lemma from Section~\ref{sec_inflam_confluence} has a
corresponding statement in the formalisation (annotated with the lemma
number from the paper). There are, however, some subtleties, described
below.
One difficulty with a Coq formalisation of our results is that in~Coq
the coinductively defined equality (bisimilarity)~$=$ on infinite
terms (see Definition~\ref{def_bisimilarity}) is not identical with
Coq's definitional equality~$\equiv$. In the paper we use~$\equiv$
and~$=$ interchangeably, following
Proposition~\ref{prop_bisimilarity}. In the formalisation we needed to
formulate our definitions ``modulo'' bisimilarity. For instance, the
inductive definition of the transitive-reflexive closure~$R^*$ of a
relation~$R$ on infinite terms is as follows.
\begin{enumerate}
\item If $t_1 = t_2$ then $R^* t_1 t_2$ (where~$=$ denotes
bisimilarity coinductively defined like in
Definition~\ref{def_bisimilarity}).
\item If $R t_1 t_2$ and $R^* t_2 t_3$ then $R^* t_1 t_3$.
\end{enumerate}
Changing the first point to
\begin{enumerate}
\item $R^* t t$ for any term~$t$
\end{enumerate}
would not work with our formalisation. Similarly, the formal
definition of the compatible closure of a relation~$R$ follows the
inductive rules
\[
\begin{array}{cccc}
\infer{s \to_R t}{\ensuremath{\langle} s, t \ensuremath{\rangle} \in R} &\quad
\infer{s t \to_R s' t'}{s \to_R s' & t = t'} &\quad
\infer{s t \to_R s' t'}{t \to_R t' & s = s'} &\quad
\infer{\lambda x . s \to_R \lambda x . s'}{s \to_R s'}
\end{array}
\]
where~$=$ denotes the coinductively defined bisimilarity relation.
Another limitation of Coq is that it is not possible to directly prove
by coinduction statements of the form
$\forall \vec{x} . \varphi(\vec{x}) \to R_1(\vec{x}) \land
R_2(\vec{x})$, i.e., statements with a conjuction of two coinductive
predicates. Instead, we show
$\forall \vec{x} . \varphi(\vec{x}) \to R_1(\vec{x})$ and
$\forall \vec{x} . \varphi(\vec{x}) \to R_2(\vec{x})$ separately. In
all our coinductive proofs we use the coinductive hypothesis in a way
that makes this separation possible.
The formalisation assumes the following axioms.
\begin{enumerate}
\item The constructive indefinite description axiom:
\[
\forall A : \mathrm{Type} . \forall P : A \to \mathrm{Prop} . (\exists x : A . P
x) \to \{x : A \mid P x\}.
\]
This axiom states that if there exists an object~$x$ of type~$A$
satisfying the predicate~$P$, then it is possible to choose a fixed
such object. This is not provable in the standard logic of Coq. We
need this assumption to be able to define the implicit functions in
some coinductive proofs which show the existence of an infinite
object, when the form of this object depends on which case in the
definition of some (co)inductive predicate holds. More precisely,
the indefinite description axiom is needed in the proof of
Lemma~\ref{lem_bohm_decompose}, in the definition of canonical
root normal forms (Definition~\ref{def_crnf}), and in the proofs of
Lemma~\ref{lem_sim_beta_2}, Lemma~\ref{lem_sim_to_bot} and
Lemma~\ref{lem_N_normalising}.
\item Excluded middle for the property of being in root normal form:
for every term~$t$, either~$t$ is in root normal form or not.
\item Excluded middle for the property of having a root normal form:
for every term~$t$, either~$t$ has a root normal form or not.
\item Excluded middle for the property of belonging to a set of
strongly meaningless terms: for any set of strongly meaningless
terms~$\ensuremath{{\mathcal U}}$ and any term~$t$, either $t \in \ensuremath{{\mathcal U}}$ or $t \notin \ensuremath{{\mathcal U}}$.
\end{enumerate}
Note that the last axiom does not constructively imply the third. We
define being root-active as not having a root normal form. In fact, we
need the third axiom to show that if a term does not belong to a set
of meaningless terms then it has a root normal form.
The first axiom could probably be avoided by making the reduction
relations $\mathrm{Set}$-valued instead of $\mathrm{Prop}$-valued. We do not use the
impredicativity of~$\mathrm{Prop}$. The reason why we chose to define the
relations as $\mathrm{Prop}$-valued is that certain automated proof search
tactics work better with $\mathrm{Prop}$-valued relations, which makes the
formalisation easier to carry out.
Because the $\bot_\ensuremath{{\mathcal U}}$-reduction rule, for any set of meaningless
terms~$\ensuremath{{\mathcal U}}$, requires an oracle to check whether it is applicable, the
present setup is inherently classical. It is an interesting research
question to devise a constructive theory of meaningless terms.
Aside of the axioms (1)--(4), everything else from
Section~\ref{sec_inflam_confluence} is formalised in the constructive
logic of Coq, including the proof of Theorem~\ref{thm_polonsky} only
cited in this paper. Our formalisation of Theorem~\ref{thm_polonsky}
closely follows~\cite{EndrullisPolonsky2011}.
In our formalisation we extensively used the CoqHammer
tool~\cite{CzajkaKaliszyk2018}.
\section{Conclusions}
We presented new and formal coinductive proofs of the following
results.
\begin{enumerate}
\item Confluence and normalisation of B{\"o}hm reduction over any set
of strongly meaningless terms.
\item Confluence and normalisation of B{\"o}hm reduction over
root-active terms, by showing that root-active terms are strongly
meaningless.
\item Confluence of infinitary $\beta$-reduction modulo any set of
meaningless terms (expansion axiom not needed).
\end{enumerate}
We formalised these results in Coq. Our formalisation uses a
definition of infinitary lambda terms based on de Bruijn
indices. Strictly speaking, the precise relation of this definition to
other definitions of infinitary lambda terms in the literature has not
been established. We leave this for future work. The issue of the
equivalence of various definitions of infinitary lambda terms is not
necessarily
trivial~\cite{KurzPetrisanSeveriVries2012,KurzPetrisanSeveriVries2013}.
By a straightforward generalisation of a result
in~\cite{EndrullisPolonsky2011} we also proved equivalence, in the
sense of existence, of the coinductive definitions of infinitary
rewriting relations with the standard definitions based on strong
convergence. However, we did not formalise this result.
In Section~\ref{sec_coind} we explained how to elaborate our
coinductive proofs by reducing them to proofs by transfinite induction
and thus eliminating coinduction. This provides one way to understand
and verify our proofs without resorting to a formalisation. After
properly understanding the observations of Section~\ref{sec_coind} it
should be ``clear'' that coinduction may in principle be eliminated in
the described manner. We use the word ``clear'' in the same sense that
it is ``clear'' that the more sophisticated inductive constructions
commonly used in the literature can be formalised in ZFC set
theory. Of course, this notion of ``clear'' may always be debated. The
only way to make this completely precise is to create a formal system
based on Section~\ref{sec_coind} in which our proofs could be
interpreted reasonably directly. We do not consider the observations
of Section~\ref{sec_coind} to be novel or particularly
insightful. However, distilling them into a formal system could
perhaps arise some interest. This is left for future work.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,286
|
Q: Does there exist a function $f(x)$, which is "parallel" to $e^x$ and has a finite "norm"? Does there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
\lim_{M \rightarrow +\infty}\frac{\int_{-M}^{M}f(x)e^xdx}{\Big(\int_{-M}^{M}(f(x))^2dx\int_{-M}^{M}(e^x)^2dx\Big)^{1/2}}=1
$$
but also the limit
$$
\lim_{M \rightarrow +\infty} \int_{-M}^{M} (f(x))^2 dx
$$
exists and finite?
I suppose the answer is no, but I don't know how to prove it.
Motivation. I think of the first limit as of some generalization of the cosine between two vectors (as it is somehow similar to a dot product of two vectors divided by their norms). So the first limit says that functions $f(x)$ and $e^x$ are parallel in some sense (although not in the common sense as there is a limit before the whole fraction).
The second limit says that that we are looking for a function $f(x)$ with a finite $L_2$-norm (again, not exactly the norm, but maybe its principal value).
Functions $f(x) = ae^x$ with $a \in \mathbb{R}_+$ satisfy the first condition (as they are parallel to $e^x$), but do not satisfy the second (as their "norm" is infinite). I wonder if there exists such $f(x)$ which satisfies both.
A: As I found out, the answer is no, there is no such function.
Without loss of generality we can assume that
$$
\lim_{M \rightarrow +\infty} \int_{-M}^{M} (f(x))^2 dx = 2,
$$
which also leads to
$$
\lim_{M \rightarrow +\infty} f(M)= \lim_{M \rightarrow +\infty} f(-M) = 0.
$$
Suppose now that there is a function $f$ such that
$$
\lim_{M \rightarrow +\infty}\frac{\int_{-M}^{M}f(x)e^xdx}{\Big(\int_{-M}^{M}(f(x))^2dx\int_{-M}^{M}(e^x)^2dx\Big)^{1/2}}=\lim_{M \rightarrow +\infty}\frac{\int_{-M}^{M}f(x)e^xdx}{\Big(2\int_{-M}^{M}e^{2x}dx\Big)^{1/2}}=
$$
$$
=\lim_{M \rightarrow +\infty}\frac{\int_{-M}^{M}f(x)e^xdx}{\Big(2\cdot \frac{1}{2}(e^{2M}-\underbrace{e^{-2M}}_{\rightarrow 0})\Big)^{1/2}}=\lim_{M \rightarrow +\infty}\frac{\int_{-M}^{M}f(x)e^xdx}{e^M}=1.
$$
This limit equals to $1$ as the denominator tends to $+\infty$, which means that the numerator also tends to $+\infty$:
$$
\lim_{M \rightarrow +\infty}\int_{-M}^{M}f(x)e^xdx=+\infty.
$$
Now it seems that all the conditions needed to apply L'Hôpital's rule are met, and we can conclude
$$
\lim_{M \rightarrow +\infty}\frac{\int_{-M}^{M}f(x)e^xdx}{e^M}=1 \quad \Rightarrow \quad \lim_{M \rightarrow +\infty}\frac{\Big(\int_{-M}^{M}f(x)e^xdx\Big)'_M}{\big(e^M\big)'_M} \color{blue}{=1}.
$$
But
$$
\lim_{M \rightarrow +\infty}\frac{\Big(\int_{-M}^{M}f(x)e^xdx\Big)'_M}{\big(e^M\big)'_M} = \lim_{M \rightarrow +\infty}\frac{f(M)e^M-\overbrace{f(-M)e^{-M}}^{\rightarrow 0}}{e^M} =
$$
$$
=\lim_{M \rightarrow +\infty}\frac{f(M)e^M}{e^M}=\lim_{M \rightarrow +\infty}f(M) \color{red}{=0},
$$
and we have reached a contradiction.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,884
|
shopt -s nocasematch
set -u # nounset
set -e # errexit
set -E # errtrap
set -o pipefail
#
# Gets a bash shell for a container
#
function help {
echo " "
echo "usage: ${0}"
echo " --container-name [OPTIONAL] The Docker container name. Default: metron-bro-plugin-kafka_zeek_1"
echo " -h/--help Usage information."
echo " "
echo " "
}
CONTAINER_NAME=metron-bro-plugin-kafka_zeek_1
# handle command line options
for i in "$@"; do
case $i in
#
# CONTAINER_NAME
#
# --container-name
#
--container-name=*)
CONTAINER_NAME="${i#*=}"
shift # past argument=value
;;
#
# -h/--help
#
-h | --help)
help
exit 0
shift # past argument with no value
;;
#
# Unknown option
#
*)
UNKNOWN_OPTION="${i#*=}"
echo "Error: unknown option: $UNKNOWN_OPTION"
help
;;
esac
done
echo "Running bash on "
echo "CONTAINER_NAME = $CONTAINER_NAME"
echo "==================================================="
docker exec -i -t "${CONTAINER_NAME}" bash
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,108
|
Mark Allan Segal (born 1951), Mark Segal and his Famous TV Zaps is a social activist and author. He participated in the Stonewall riots and was one of the original founders of the Gay Liberation Front where he created its Gay Youth program. He was the founder and former president of the National Gay Newspaper Guild and purchased the Philadelphia Gay News. He has won numerous journalism awards for his column "Mark my Works," including best column by The National Newspaper Association, Suburban Newspaper Association and The Society of Professional Journalist.
Gay rights activism
Segal was a participant at Stonewall in 1969 and help found the Gay Liberation Front that same year. He was also a member of The Christopher Street Gay Liberation Day committee, which organized the first Gay Pride parade in 1970.
In 1972, after being thrown out of dance competition for dancing with a male lover, Segal crashed the evening news broadcast of WPVI-TV, an act that became known as a "zap" and that he helped popularize. He repeated the action during many other television broadcasts. On 11 December 1973, Segal interrupted Walter Cronkite's broadcast of the CBS Evening News when he ran in front of the camera and held up a yellow sign saying "Gays Protest CBS Prejudice."
In 1975, he went on a hunger strike on behalf of the passage of a law to guarantee equal rights for homosexuals. In 1976, he founded the Philadelphia Gay News, a lesbian, gay, bisexual and transgender (LGBT) newspaper in the Philadelphia area. The publication was inspired by activist Frank Kameny, whom Segal first met in 1970. In 1988, Segal had a televised debate with a Philadelphia city councilman, Francis Rafferty, about Gay Pride Month.
Segal partnered with the Obama Administration to create and build the nation's first official "LGBT Friendly" Senior Affordable housing apartment building. The 19.8 million dollar project known as The John C. Anderson Apartments opened in 2013.
On May 17, 2018, Segal donated 16 cubic feet of personal papers and artifacts to the Smithsonian Institution in Washington DC.
Book
Segal is the author of the book "And Then I Danced: Traveling the Road to LGBT Equality" a memoir of his life and experience as a gay rights activist. The book was named "Best Book" by the National Lesbian Gay Journalist Association in 2015.
Personal life
Segal is Jewish and originally from Mount Airy, Philadelphia. He attended school at Germantown High School and Temple University. His friends include several prominent gay activists like Barbara Gittings, Frank Kameny, Harry Hay and Troy Perry.
On July 5, 2014, Segal married his partner of 10 years, Jason Villemez. At the time, Villemez was 29 and Segal was 63. The ceremony was officiated by Judge Dan Anders, Philadelphia's first openly gay judge.
References
"Pursuits: Q&A: Mark Segal". Philadelphia Magazine, 2009.
External links
Profile of Mark Segal at National Lesbian and Gay Journalists Association
Profile of Mark Segal at Philly.com
Philadelphia Gay News website
Mark Segal's official Facebook page
American LGBT rights activists
Living people
American male journalists
American LGBT journalists
American gay writers
Writers from Philadelphia
American publishers (people)
LGBT Jews
Temple University alumni
1951 births
LGBT people from Pennsylvania
Gay Liberation Front members
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,727
|
And a more positive debt that is linked to investment by the government and that will ultimately have a much higher yield. This has a positive net benefit.
We talk of budget or fiscal deficit when the government spends more per year than its income, so it has to borrow to cover the difference. When the government runs a surplus, it is able to repay part of its debt.
But this difference consists of a heterogeneous aggregate of all expenses incurred. For this reason, the European Commission issued on January 13th a so-called Clarification[i] concerning the type of investments that they will disregard when assessing a country's deficit.
The current situation in Greece is a textbook example of unsustained debt.
Despite a budget surplus before interest payments Greek public debt keeps increasing in percentage of the GDP. This is because to obtain this budget surplus, the Government reduced expenses. This, in turn, contracted the GDP further.
The problem is compounded by the difference between long-term debt and short-term debt. To cover payments of their long-term debt, governments often have to take renewable, short-term loans, which come with even higher interest rates, which in turn increase government debt.
In Greece's case this has led to what we call a "Self-fulfilling Crisis" – when there is a belief amongst lenders that some loans will not be paid back, they will only willing to lend to the government over the short term and at punishingly high interest rates. The increased cost of interest payments makes the unsustainability of public debt a kind of self-fulfilling prophecy.
And even as international agents like the IMF of the European Commission step in to say that they are going to help to ensure loans will be repaid, it can be very difficult to change that overarching belief. In economics, this is what we call inertia: the impact is not immediate because agents do not react instantly to new information. Instead they hold on to their old ideas for a certain amount of time so that the effectiveness of the policy intervention is reduced.
Several methods have been used in the past to manage debt. What steps could Greece take?
Should part of Greece's debt be redeemed?
This is what happened for Germany in 1953 for example through a twist with exchange rates. But today, many of the stronger European economies feel an example needs to be made. If an exception is made for Greece what message will this send to the other European economies struggling with debt? This is the concept of moral hazard.
Should Greece leave the euro area and devaluate its currency?
A way for a State to reduce a debt denominated in its own currency is to devaluate, so the value of repayments to foreign creditors is reduced. In the case of Greece, whose debt is currently denominated in euros, a successful devaluation would require that its European partner agree to convert the debt in some form of Greek euros prior to the exit. It is exactly the same as asking them to forego their loans.
Should Europe pool all their debt together and offer one interest rate?
This is likely the best option, but one that doesn't appeal much to countries who currently enjoy low interest rates or are close to fiscal surpluses. However, Europe will eventually have to find a solution to control Greek debt. This means that all will have to lose a bit so that Greek does not collapse and everyone loses a lot.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,577
|
Rowley Turton up for two national awards
We're pleased to announce that Rowley Turton have been named finalists in two categories at this year's national Retirement Planner Awards.
The two categories are 'Best Individual Pensions Advice Firm of the Year Award - Midlands' and 'Best Estate Planning Adviser of the Year'.
This is the third time running that we have made the finals of these awards, having won the Best Individual Pensions Advice Firm of the Year Award - Midlands' award the two previous years.
As a tax-efficient way to save for retirement, pensions offer a route to long term financial security. To highlight the importance of retirement planning in the UK, the 2019 Retirement Planner Awards celebrate how providers and advisers are rising to the challenge of pension provision, particularly in light of the 2015 pension freedoms and the increasing number of pension scams.
The finalists were selected by an independent panel of expert judges and included consideration of appropriate case studies.
Rowley Turton director Scott Gallacher explained: -
"Life is not a rehearsal, and professional independent financial advice is essential to helping people achieve and maintain their desired lifestyle throughout a 30-year retirement. Consequently, we're delighted to have made the finals of the Retirement Planner awards."
Martin Stanley, a Chartered Financial Planner with Rowley Turton said: -
"The Retirement Planner Awards are a great way of highlighting the value of retirement planning."
The winners of the Retirement Planner Awards will be announced during a lunchtime ceremony on Friday 14th June at the London Marriott Hotel Grosvenor Square.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,863
|
namespace views {
namespace {
// The opacity of the highlight when it is not visible.
constexpr float kHiddenOpacity = 0.0f;
} // namespace
std::string ToString(InkDropHighlight::AnimationType animation_type) {
switch (animation_type) {
case InkDropHighlight::AnimationType::kFadeIn:
return std::string("FADE_IN");
case InkDropHighlight::AnimationType::kFadeOut:
return std::string("FADE_OUT");
}
}
InkDropHighlight::InkDropHighlight(
const gfx::PointF& center_point,
std::unique_ptr<BasePaintedLayerDelegate> layer_delegate)
: center_point_(center_point),
layer_delegate_(std::move(layer_delegate)),
layer_(std::make_unique<ui::Layer>()) {
const gfx::RectF painted_bounds = layer_delegate_->GetPaintedBounds();
size_ = painted_bounds.size();
layer_->SetBounds(gfx::ToEnclosingRect(painted_bounds));
layer_->SetFillsBoundsOpaquely(false);
layer_->set_delegate(layer_delegate_.get());
layer_->SetVisible(false);
layer_->SetMasksToBounds(false);
layer_->SetName("InkDropHighlight:layer");
}
InkDropHighlight::InkDropHighlight(const gfx::SizeF& size,
int corner_radius,
const gfx::PointF& center_point,
SkColor color)
: InkDropHighlight(
center_point,
std::make_unique<RoundedRectangleLayerDelegate>(color,
size,
corner_radius)) {
layer_->SetOpacity(visible_opacity_);
}
InkDropHighlight::InkDropHighlight(const gfx::Size& size,
int corner_radius,
const gfx::PointF& center_point,
SkColor color)
: InkDropHighlight(gfx::SizeF(size), corner_radius, center_point, color) {}
InkDropHighlight::InkDropHighlight(const gfx::SizeF& size, SkColor base_color)
: size_(size), layer_(std::make_unique<ui::Layer>(ui::LAYER_SOLID_COLOR)) {
layer_->SetColor(base_color);
layer_->SetBounds(gfx::Rect(gfx::ToRoundedSize(size)));
layer_->SetVisible(false);
layer_->SetMasksToBounds(false);
layer_->SetOpacity(visible_opacity_);
layer_->SetName("InkDropHighlight:solid_color_layer");
}
InkDropHighlight::~InkDropHighlight() {
// Explicitly aborting all the animations ensures all callbacks are invoked
// while this instance still exists.
layer_->GetAnimator()->AbortAllAnimations();
}
bool InkDropHighlight::IsFadingInOrVisible() const {
return last_animation_initiated_was_fade_in_;
}
void InkDropHighlight::FadeIn(const base::TimeDelta& duration) {
layer_->SetOpacity(kHiddenOpacity);
layer_->SetVisible(true);
AnimateFade(AnimationType::kFadeIn, duration);
}
void InkDropHighlight::FadeOut(const base::TimeDelta& duration) {
AnimateFade(AnimationType::kFadeOut, duration);
}
test::InkDropHighlightTestApi* InkDropHighlight::GetTestApi() {
return nullptr;
}
void InkDropHighlight::AnimateFade(AnimationType animation_type,
const base::TimeDelta& duration) {
const base::TimeDelta effective_duration =
gfx::Animation::ShouldRenderRichAnimation() ? duration
: base::TimeDelta();
last_animation_initiated_was_fade_in_ =
animation_type == AnimationType::kFadeIn;
layer_->SetTransform(CalculateTransform());
// The |animation_observer| will be destroyed when the
// AnimationStartedCallback() returns true.
ui::CallbackLayerAnimationObserver* animation_observer =
new ui::CallbackLayerAnimationObserver(
base::BindRepeating(&InkDropHighlight::AnimationStartedCallback,
base::Unretained(this), animation_type),
base::BindRepeating(&InkDropHighlight::AnimationEndedCallback,
base::Unretained(this), animation_type));
ui::LayerAnimator* animator = layer_->GetAnimator();
ui::ScopedLayerAnimationSettings animation(animator);
animation.SetTweenType(gfx::Tween::EASE_IN_OUT);
animation.SetPreemptionStrategy(
ui::LayerAnimator::IMMEDIATELY_ANIMATE_TO_NEW_TARGET);
std::unique_ptr<ui::LayerAnimationElement> opacity_element =
ui::LayerAnimationElement::CreateOpacityElement(
animation_type == AnimationType::kFadeIn ? visible_opacity_
: kHiddenOpacity,
effective_duration);
ui::LayerAnimationSequence* opacity_sequence =
new ui::LayerAnimationSequence(std::move(opacity_element));
opacity_sequence->AddObserver(animation_observer);
animator->StartAnimation(opacity_sequence);
animation_observer->SetActive();
}
gfx::Transform InkDropHighlight::CalculateTransform() const {
gfx::Transform transform;
// No transform needed for a solid color layer.
if (!layer_delegate_)
return transform;
transform.Translate(center_point_.x(), center_point_.y());
gfx::Vector2dF layer_offset = layer_delegate_->GetCenteringOffset();
transform.Translate(-layer_offset.x(), -layer_offset.y());
// Add subpixel correction to the transform.
transform.ConcatTransform(
GetTransformSubpixelCorrection(transform, layer_->device_scale_factor()));
return transform;
}
void InkDropHighlight::AnimationStartedCallback(
AnimationType animation_type,
const ui::CallbackLayerAnimationObserver& observer) {
if (observer_)
observer_->AnimationStarted(animation_type);
}
bool InkDropHighlight::AnimationEndedCallback(
AnimationType animation_type,
const ui::CallbackLayerAnimationObserver& observer) {
// AnimationEndedCallback() may be invoked when this is being destroyed and
// |layer_| may be null.
if (animation_type == AnimationType::kFadeOut && layer_)
layer_->SetVisible(false);
if (observer_) {
observer_->AnimationEnded(animation_type,
observer.aborted_count()
? InkDropAnimationEndedReason::PRE_EMPTED
: InkDropAnimationEndedReason::SUCCESS);
}
return true;
}
} // namespace views
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,733
|
\section{Introduction}
In order to motivate the problem dealt in this paper, we have considered the
results of an experiment carried out by Doll and Pygott (1952) to assess the
factors influencing the rate of healing of gastric ulcers. Two treatments
groups were compared. Patients in group 2 were treated in bed in hospital for
four weeks. For the first two weeks they were given a moderate strict orthodox
diet and for the last two weeks a more liberal one. They were then reexamined
radiographically, discharged, recommended to continue on a convalescent diet
and advised return to work as soon as they felt fit enough. Patients in group
1 were discharged immediately. They were treated from the outset in the way
that group 2 patients were treated after their month's stay in hospital. In
Table \ref{tttt1}, we present the results showed by Doll and Pygott (1952,
Table IV) for three months after starting the treatments. This article
proposes new families of test-statistics when we are interested in studying
the possibility that the ulcer treatment (Treatment $2$) is better than the
control (Treatment $1$).%
\begin{table}[htbp] \tabcolsep2.8pt \centering
$%
\begin{tabular}
[c]{lcccc}\hline
& Larger & $<\frac{1}{3}$ Healed & $\geq\frac{2}{3}$ Healed & Healed\\\hline
Treatment $1$ & 11 & 8 & 8 & 5\\
Treatment $2$ & 6 & 4 & 10 & 12\\\hline
\end{tabular}
\ \ \ \ \ \ \ \ \ \ \ $\caption{Change in size of ulcer crater.\label{tttt1}}%
\end{table}%
Let $Y$ denote the ordinal response variable and $X$ denote an ordinal
explanatory variable with two categories. The variable $Y$ takes the values
$1$, $2$, $3$ and $4$, which represent different levels of healing, from less
to much capacity to heal the ulcer. The variable $X$ takes the values $1$ and
$2$ according as the treatment group, $1$ is control and $2$ is the treatment
group by itself. We shall initially focus on making statistical inference on
the theoretical probabilities displayed in Table \ref{ttt2}.%
\begin{table}[htbp] \tabcolsep2.8pt \centering
$%
\begin{tabular}
[c]{lcccc}\hline
& Larger & $<\frac{1}{3}$ Healed & $\geq\frac{2}{3}$ Healed & Healed\\\hline
Treatment $1$ & $\Pr(Y=1|X=1)$ & $\Pr(Y=2|X=1)$ & $\Pr(Y=3|X=1)$ &
$\Pr(Y=4|X=1)$\\
Treatment $2$ & $\Pr(Y=1|X=2)$ & $\Pr(Y=2|X=2)$ & $\Pr(Y=3|X=2)$ &
$\Pr(Y=4|X=2)$\\\hline
\end{tabular}
\ \ \ \ \ \ \ \ \ \ \ \ $%
\caption{Theoretical conditional probabilities.\label{ttt2}}%
\end{table}%
There are several ways of formulating the statement \textquotedblleft the
treatment is better than the control\textquotedblright. Initially, we shall
consider that Treatment $2$ is at least as good as Treatment $1$ if the ratio
$\frac{\Pr(Y=j|X=2)}{\Pr(Y=j|X=1)}$ increases as the response category, $j$,
increases, i.e.%
\begin{equation}
\tfrac{\Pr(Y=j|X=2)}{\Pr(Y=j|X=1)}\leq\tfrac{\Pr(Y=j+1|X=2)}{\Pr
(Y=j+1|X=1)}\qquad\text{for every }j\text{,} \label{eq1}%
\end{equation}
and Treatment 2 is better than the Treatment 1 if (\ref{eq1}) holds with at
least one strict inequality.
If we assume that Treatment 2 is at least as good as Treatment 1, i.e.,
(\ref{eq1}) holds, is there any evidence to support the claim that treatment
$2$ is better? In such a case null and alternative hypotheses may be%
\begin{subequations}
\begin{align}
& H_{0}:\;\tfrac{\Pr(Y=j|X=2)}{\Pr(Y=j|X=1)}=\tfrac{\Pr(Y=j+1|X=2)}%
{\Pr(Y=j+1|X=1)}\quad\text{for every }j\text{,}\label{eq2}\\
& H_{1}:\;\tfrac{\Pr(Y=j|X=2)}{\Pr(Y=j|X=1)}\leq\tfrac{\Pr(Y=j+1|X=2)}%
{\Pr(Y=j+1|X=1)}\quad\text{for every }j\quad\text{and}\quad\tfrac
{\Pr(Y=j|X=2)}{\Pr(Y=j|X=1)}<\tfrac{\Pr(Y=j+1|X=2)}{\Pr(Y=j+1|X=1)}%
\quad\text{for at least one }j\text{.} \label{eq3}%
\end{align}
The null hypothesis means that both treatments are equally effective, while
the alternative hypothesis means that Treatment 2 is more effective than
Treatment 1. Note that if we multiply on the left and right hand side of
(\ref{eq2}) and (\ref{eq3}) by $\left( \tfrac{\Pr(Y=j|X=2)}{\Pr
(Y=j|X=1)}\right) ^{-1}$ we obtain
\end{subequations}
\begin{subequations}
\begin{align}
& H_{0}:\;\vartheta_{j}=1\quad\text{for every }j\in\{1,...,J-1\}\text{,}%
\label{eq2b}\\
& H_{1}:\;\vartheta_{j}\geq1\quad\text{for every }j\in\{1,...,J-1\}\quad
\text{and}\quad\vartheta_{j}>1\quad\text{for at least one }j\in
\{1,...,J-1\}\text{,} \label{eq3b}%
\end{align}
where $J$ is the number of ordered categories for response variable $Y$,%
\end{subequations}
\begin{equation}
\vartheta_{j}=\dfrac{\pi_{1j}\pi_{2,j+1}}{\pi_{2j}\pi_{1,j+1}},\quad\forall
j\in\{1,...,J-1\}, \label{2}%
\end{equation}
are \textquotedblleft local odds ratios\textquotedblright\ associated with
response category $j$, and%
\begin{equation}
\pi_{ij}=\Pr(Y=j|X=i). \label{eq5}%
\end{equation}
In case of considering the opposite inequalities given in (\ref{eq3}) or
(\ref{eq3b}), the easiest way to carry out the test is to exchange the
observation of the two rows in the contingency table (in the example,
Treatment $2$ in the first row and Treatment $1$ in the second row). In this
way, the mathematical background is not changed but the interpretation of the
aim is changed. In the example however, there is no sense in considering that
the control ($1$) is better than the treatment ($2$), if the experiment is
carried out with humans and it is assumed that the treatment will not harm
these patients.
The non-parametric statistical inference associated with the likelihood ratio
ordering for two multinomial samples was introduced for the first time in
Dykstra et al. (1995) using the likelihood ratio test-statistic. In the
literature related to different types of orderings, in general there is not
very clear what is the most appropriate ordering to compare two treatments
according to a categorized ordinal variable. In the case of having two
independent multinomial samples, the likelihood ratio ordering is the most
restricted ordering type; for example, if the likelihood ratio ordering holds,
then the simple stochastic ordering also holds. Dardanoni and Forcina (1998)
proposed a new method for making statistical inference associated with
different types of orderings. For unifying and comparing different types of
orderings, they reparametrize the initial model. Different ordering types can
be considered to be nested models and the likelihood ratio ordering is the
most parsimonious one. The advantage of nested models is that the most
restricted models tend to be more powerful for the alternatives that belong to
the most restricted alternatives. In this setting, our proposal in this paper
is to introduce new test-statistics that provide substantially better power
for testing (\ref{eq2}) against (\ref{eq3}).
The structure of the paper is as follows. In Section \ref{Sec:LM}, we have
considered the likelihood ratio order associated with a non-parametric model,
as in Dardanoni and Forcina (1998), but the specification of the model through
a saturated loglinear model is substantially different. Section \ref{Sec:PD}
presents the phi-divergence test-statistics as extension of the likelihood
ratio and chi-square test-statistics.\ The applied methodology in Section
\ref{sec:Main results} for proving the asymptotic distribution of the
phi-divergence test-statistics, based on loglinear modeling, has been
developed by following a completely new and meaningful method even for the
likelihood ratio test. A numerical example is given in Section
\ref{sec:Numerical example}. The aim of Section \ref{sec:Simulation Study} is
to study through simulation the behaviour of the phi-divergence
test-statistics for small and moderate simple sizes. Finally, we present an
Appendix in which we establish the part of the proofs of the results not shown
in Section \ref{sec:Main results}.
\section{Loglinear modeling\label{Sec:LM}}
We display the whole distribution of $\pi_{ij}$, given in (\ref{eq5}), in a
rectangular table having $2$ rows for the categories of $X$ and $J$ columns
for the categories of $Y$ (for the initial example, Table \ref{ttt2}) and we
denote the $2\times J$ matrix $\boldsymbol{\Pi}=(\boldsymbol{\pi}%
_{1},\boldsymbol{\pi}_{2})^{T}$, with two rows of probability vectors,
$\boldsymbol{\pi}_{i}=(\pi_{i1},...,\pi_{iJ})^{T}$, $i=1,2$. We consider two
independent random samples $\boldsymbol{N}_{i}=(N_{i1},...,N_{iJ})^{T}%
\sim\mathcal{M}(n_{i},\boldsymbol{\pi}_{i})$, $i=1,2$, where sizes $n_{i}$ are
prefixed and $\boldsymbol{\pi}_{i}>\boldsymbol{0}_{J}$, that is the
probability distribution of r.v. $\boldsymbol{N}=(\boldsymbol{N}_{1}%
^{T},\boldsymbol{N}_{2}^{T})^{T}$ is product-multinomial. Let%
\begin{equation}
p_{ij}=\Pr(X=i,Y=j), \label{eq5b}%
\end{equation}
be the joint probability distribution. Since $\Pr(X=i,Y=j)=\Pr(Y=j|X=i)\Pr
(X=i)$, i.e. $p_{ij}=\pi_{ij}\frac{n_{i}}{n}$, $i=1,2$, where $n=n_{1}+n_{2}$,
we can express (\ref{2}) also in terms of the joint probabilities%
\begin{equation}
\vartheta_{j}=\dfrac{p_{1j}p_{2,j+1}}{p_{2j}p_{1,j+1}},\quad\forall
j\in\{1,...,J-1\}. \label{2b}%
\end{equation}
Let $\boldsymbol{P}=(\mathbf{p}_{1},\mathbf{p}_{2})^{T}$, with $\mathbf{p}%
_{i}=(p_{i1},...,p_{iJ})^{T}$, $i=1,2$, be the $2\times J$ probability matrix
and
\begin{equation}
\boldsymbol{p}=\mathrm{vec}(\boldsymbol{P}^{T})=(\mathbf{p}_{1}^{T}%
,\mathbf{p}_{2}^{T})^{T} \label{0}%
\end{equation}
a probability vector obtained by stacking the columns of $\boldsymbol{P}^{T}%
$\ (i.e., the rows of matrix $\boldsymbol{P}$). Note that the components of
$\boldsymbol{P}$ are ordered in lexicographical order in $\boldsymbol{p}$. The
likelihood function of $\boldsymbol{N}$ is $\mathcal{L}(\boldsymbol{N}%
;\boldsymbol{p})=k%
{\textstyle\prod\nolimits_{j=1}^{J}}
p_{1j}^{N_{1j}}p_{2j}^{N_{2j}}$, where $k$ is a constant which does not depend
on $\boldsymbol{p}$ and the kernel of the loglikelihood function%
\begin{equation}
\ell(\boldsymbol{N};\boldsymbol{p})=%
{\displaystyle\sum\limits_{j=1}^{J}}
(N_{1j}\log p_{1j}+N_{2j}\log p_{2j}). \label{0b}%
\end{equation}
In matrix notation, we are interested in testing
\begin{equation}
H_{0}:\boldsymbol{\vartheta}=\boldsymbol{1}_{J-1}\text{ versus }%
H_{1}:\boldsymbol{\vartheta}\gneqq\boldsymbol{1}_{J-1}\text{,} \label{4}%
\end{equation}
where $\boldsymbol{1}_{a}$\ is the $a$-vector of $1$-s, $\boldsymbol{\vartheta
}=(\vartheta_{1},...,\vartheta_{J-1})^{T}$. Note that (\ref{4}) involves $J-1$
non-linear constraints on $\boldsymbol{p}$, defined by (\ref{0}). In this
article the hypothesis testing problem is formulated making a
reparametrization of $\boldsymbol{p}$ using the saturated loglinear model, so
that some linear restrictions are considered with respect to the new
parameters. This fact is important and interesting.
Focussed on $\boldsymbol{p}$, the saturated loglinear model with canonical
parametrization is defined by
\begin{equation}
\log p_{ij}=u+u_{1(i)}+\theta_{2(j)}+\theta_{12(ij)}, \label{3}%
\end{equation}
with the identifiabilty restrictions%
\begin{equation}
u_{1(2)}=0,\quad\theta_{2(J)}=0,\quad\theta_{12(1J)}=0,\quad\theta
_{12(2j)}=0,\quad j=1,...,J. \label{ident}%
\end{equation}
It is important to clarify that we have used the identifiability constraints
(\ref{ident}) in order to make easier the calculations and this model
formulation for making statistical inference with inequality restrictions with
local odds-ratios has been given in this paper for the first time. Similar
conditions have been used for instance in Lang (1996, examples of Section 7)
and Silvapulle and Sen (2005, exercise 6.25 in page 345). Let
$\boldsymbol{\theta}_{12}=(\theta_{12(11)},...,\theta_{12(1,J-1)})^{T}$,
$\boldsymbol{\theta}_{2}=(\theta_{2(1)},...,\theta_{2(J-1)})^{T}$ denote
subvectors of the unknown parameters $\boldsymbol{\theta}=(\boldsymbol{\theta
}_{2}^{T},\boldsymbol{\theta}_{12}^{T})^{T}$. The components of
$\boldsymbol{u}=(u,u_{1(1)})^{T}$ are redundant parameters since the term $u$
can be expressed in function of $\boldsymbol{\theta}$ using the fact that $%
{\textstyle\sum\nolimits_{j=1}^{J}}
p_{2j}=\frac{n_{2}}{n}$, i.e.%
\begin{equation}
u=u(\boldsymbol{\theta})=\log n_{2}-\log n-\log\left( 1+%
{\displaystyle\sum\limits_{j=1}^{J-1}}
\exp\{\theta_{2(j)}\}\right) , \label{u}%
\end{equation}
and $u_{1(1)}$ taking into account that $%
{\textstyle\sum\nolimits_{j=1}^{J}}
p_{1j}=\frac{n_{1}}{n}$, i.e.%
\begin{equation}
u_{1(1)}=u_{1(1)}(\boldsymbol{\theta})=\log\frac{n_{1}}{n_{2}}+\log\frac{1+%
{\textstyle\sum\nolimits_{j=1}^{J-1}}
\exp\{\theta_{2(j)}\}}{1+%
{\displaystyle\sum\limits_{j=1}^{J-1}}
\exp\{\theta_{2(j)}+\theta_{12(1j)}\}}. \label{u1}%
\end{equation}
In matrix notation (\ref{3}) is given by%
\begin{equation}
\log\boldsymbol{p}(\boldsymbol{\theta})=\boldsymbol{W}_{0}\boldsymbol{u}%
+\boldsymbol{W\theta}, \label{loglin}%
\end{equation}
where $\boldsymbol{p}(\boldsymbol{\theta})$ is $\boldsymbol{p}$\ such that the
components are defined by (\ref{3}),
\[
\boldsymbol{W}_{0}=%
\begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}
\otimes\boldsymbol{1}_{J}%
\]
is a $2J\times2$ matrix with $\boldsymbol{1}_{a}$\ being the $a$-vector of
ones, $\boldsymbol{0}_{a}$\ the $a$-vector of zeros, $\otimes$ the Kronecker
product; $\boldsymbol{W}$ the full rank design matrix of size $2J\times
2(J-1)$, such that%
\begin{equation}
\boldsymbol{W}=%
\begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}
\otimes%
\begin{pmatrix}
\boldsymbol{I}_{J-1}\\
\boldsymbol{0}_{J-1}^{T}%
\end{pmatrix}
, \label{W}%
\end{equation}
with $\boldsymbol{I}_{a}$\ being the identity matrix of order $a$,
$\boldsymbol{0}_{a\times b}$\ the matrix of size $a\times b$ with zeros. The
condition (\ref{eq1}) can be expressed by the linear constraint
\begin{equation}
\theta_{12(1j)}-\theta_{12(2j)}-\theta_{12(1,j+1)}+\theta_{12(2,j+1)}%
\geq0,\text{ }\forall j\in\{1,...,J-1\}, \label{6}%
\end{equation}
since%
\[
\log\vartheta_{j}=\log p_{1j}-\log p_{2j}-\log p_{1,j+1}+\log p_{2,j+1}%
=\theta_{12(1j)}-\theta_{12(2j)}-\theta_{12(1,j+1)}+\theta_{12(2,j+1)}.
\]
Condition (\ref{6}) in matrix notation is given by $\boldsymbol{R\theta}%
\geq\boldsymbol{0}_{J-1}$, with $\boldsymbol{R}=\boldsymbol{e}_{2}^{T}%
\otimes\boldsymbol{G}_{J-1}=(\boldsymbol{0}_{(J-1)\times(J-1)},\boldsymbol{G}%
_{J-1})$, $\boldsymbol{e}_{a}$ is the $a$-th unit vector and $\boldsymbol{G}%
_{h}$ is a $h\times h$\ matrix with $1$-s in the main diagonal and $-1$-s in
the upper superdiagonal. Observe that the restrictions can be expressed also
as $\boldsymbol{G}_{J-1}\boldsymbol{\theta}_{12}\geq\boldsymbol{0}_{J-1}$, and
$\theta_{1(1)}$\ are $\boldsymbol{\theta}_{2}$\ are nuisance parameters
because they do not take part actively in the restrictions.
The kernel of the likelihood function with the new parametrization is obtained
replacing $\boldsymbol{p}$\ by $\boldsymbol{p}(\boldsymbol{\theta})$ in
(\ref{0b}), i.e.
\[
\ell(\boldsymbol{N};\boldsymbol{\theta})=\boldsymbol{N}^{T}\log\boldsymbol{p}%
(\boldsymbol{\theta})=\boldsymbol{N}^{T}(\boldsymbol{W}_{0}\boldsymbol{u}%
+\boldsymbol{W\theta})=nu(\boldsymbol{\theta})+n_{1}u_{1(1)}%
(\boldsymbol{\theta})+\boldsymbol{N}^{T}\boldsymbol{W\theta}.
\]
Hypotheses (\ref{4}) can be now formulated as%
\begin{equation}
H_{0}:\boldsymbol{R\theta}=\boldsymbol{0}_{J-1}\text{ versus }H_{1}%
:\boldsymbol{R\theta}\geq\boldsymbol{0}_{J-1}\text{ and }\boldsymbol{R\theta
}\neq\boldsymbol{0}_{J-1}\text{.} \label{4b}%
\end{equation}
Under $H_{0}$, the parameter space is $\Theta_{0}=\left\{ \boldsymbol{\theta
}\in%
\mathbb{R}
^{2(J-1)}:\boldsymbol{R\theta}=\boldsymbol{0}_{J-1}\right\} $ and the maximum
likelihood estimator (MLE) of $\boldsymbol{\theta}$ in $\Theta_{0}$ is
$\widehat{\boldsymbol{\theta}}=\arg\max_{\boldsymbol{\theta\in}\Theta_{0}}%
\ell(\boldsymbol{N};\boldsymbol{\theta})$. The overall parameter space is
$\Theta=\left\{ \boldsymbol{\theta}\in%
\mathbb{R}
^{^{2(J-1)}}:\boldsymbol{R\theta}\geq\boldsymbol{0}_{J-1}\right\} $ and the
MLE of $\boldsymbol{\theta}$ in $\Theta$ is $\widetilde{\boldsymbol{\theta}%
}=\arg\max_{\boldsymbol{\theta\in}\Theta}\ell(\boldsymbol{N}%
;\boldsymbol{\theta})$. It is worthwhile to mention that the probability
vectors for both parametric spaces, $\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}})$ and $\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}})$\ can be obtained by following the
invariance property of the MLEs first estimating $\boldsymbol{\theta}$\ and
later plugging it into $\boldsymbol{p}(\boldsymbol{\theta})$, however
$\boldsymbol{p}(\widehat{\boldsymbol{\theta}})$ has an explicit expression,%
\begin{equation}
p_{ij}(\widehat{\boldsymbol{\theta}})=\frac{n_{i}(N_{1j}+N_{2j})}{n^{2}},
\label{ind}%
\end{equation}
where $n_{i}=%
{\textstyle\sum_{j=1}^{J}}
N_{ij}$\ (see Christensen (1997), Section 2.3, for more details).
\section{Phi-divergence test-statistics\label{Sec:PD}}
The likelihood ratio statistic for testing (\ref{4}), equivalent to one given
by Dykstra et al. (1995) but adapted for loglinear modeling, is%
\begin{equation}
G^{2}=2(\ell(\boldsymbol{N};\widetilde{\boldsymbol{\theta}})-\ell
(\boldsymbol{N};\widehat{\boldsymbol{\theta}}))=2n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\overline{p}_{ij}\log\frac{p_{ij}(\widetilde{\boldsymbol{\theta}})}%
{p_{ij}(\widehat{\boldsymbol{\theta}})}, \label{LRT}%
\end{equation}
where $\overline{p}_{ij}=N_{ij}/n$, $i=1,2$, $j=1,...,J$. Taking into account
the identifiability constraints (\ref{ident}) and\ $\widehat{u}%
=u(\widehat{\boldsymbol{\theta}})$, $\widetilde{u}%
=u(\widetilde{\boldsymbol{\theta}})$, $\widehat{u}_{1(1)}=u_{1(1)}%
(\widehat{\boldsymbol{\theta}})$, $\widetilde{u}_{1(1)}=u_{1(1)}%
(\widetilde{\boldsymbol{\theta}})$\ (see formulas (\ref{u})-(\ref{u1})),
(\ref{LRT}) can also be expressed as%
\[
G^{2}=2n(\widetilde{u}-\widehat{u})+2n_{1}(\widetilde{u}_{1(1)}-\widehat{u}%
_{1(1)})+2\boldsymbol{N}^{T}\boldsymbol{W}%
(\boldsymbol{\widetilde{\boldsymbol{\theta}}-}\widehat{\boldsymbol{\theta}}).
\]
The chi-square statistic for testing (\ref{4}) is%
\begin{equation}
X^{2}=n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\frac{(p_{ij}(\widehat{\boldsymbol{\theta}})-p_{ij}%
(\widetilde{\boldsymbol{\theta}}))^{2}}{p_{ij}(\widehat{\boldsymbol{\theta}}%
)}. \label{CS}%
\end{equation}
The Kullback-Leibler divergence measure between two $2J$-dimensional
probability vectors $\boldsymbol{p}$ and $\boldsymbol{q}$ is defined as%
\[
d_{Kull}(\boldsymbol{p},\boldsymbol{q})=%
{\displaystyle\sum\limits_{i=1}^{2}}
\sum_{j=1}^{J}p_{ij}\log\frac{p_{ij}}{q_{ij}}%
\]
and the Pearson divergence measure
\[
d_{Pearson}(\boldsymbol{p},\boldsymbol{q})=\frac{1}{2}%
{\displaystyle\sum\limits_{i=1}^{2}}
\sum_{j=1}^{J}\frac{\left( p_{ij}-q_{ij}\right) ^{2}}{q_{ij}}.
\]
It is not difficult to check that
\begin{equation}
G^{2}=2n(d_{Kull}(\overline{\boldsymbol{p}},\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}}))-d_{Kull}(\overline{\boldsymbol{p}%
},\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}))) \label{N1}%
\end{equation}
and
\begin{equation}
X^{2}=2nd_{Pearson}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}%
}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}})), \label{N2}%
\end{equation}
being $\overline{\boldsymbol{p}}=\boldsymbol{N}/n=(\overline{p}_{11}%
,...,\overline{p}_{1J},\overline{p}_{21},....,\overline{p}_{2J})^{T}$ the
vector of relative frequencies.
More general than the Kullback-Leibler divergence and Pearson divergence
measures are $\phi$-divergence measures, defined as
\[
d_{\phi}(\boldsymbol{p},\boldsymbol{q})=%
{\displaystyle\sum\limits_{i=1}^{2}}
\sum_{j=1}^{J}q_{ij}\phi\left( \frac{p_{ij}}{q_{ij}}\right) ,
\]
where $\phi:%
\mathbb{R}
_{+}\longrightarrow%
\mathbb{R}
$ is a convex function such that%
\[
\phi(1)=\phi^{\prime}(1)=0\text{, }\phi^{\prime\prime}(1)>0\text{, }%
0\phi(\tfrac{0}{0})=0\text{, }0\phi(\tfrac{p}{0})=p\lim_{u\rightarrow\infty
}\tfrac{\phi(u)}{u}\text{, for }p\neq0.
\]
From a statistical point of view, the first asymptotic statistical results
based on divergence measures in multinomial populations were obtained in
Zografos et al. (1990). For more details about $\phi$-divergence measures see
Pardo (2006) and Cressie and Pardo (2002).
Apart from the likelihood ratio statistic (\ref{LRT}) and the chi-square
(\ref{CS}) statistic, we shall consider two new families of test-statistics
based on $\phi$-divergence measures. The first new family is obtained by
replacing in (\ref{N1}) the Kullback divergence measure by a $\phi$-divergence
measure,%
\begin{equation}
T_{\phi}(\overline{\boldsymbol{p}},\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}%
}))=\frac{2n}{\phi^{\prime\prime}(1)}(d_{\phi}(\overline{\boldsymbol{p}%
},\boldsymbol{p}(\widehat{\boldsymbol{\theta}}))-d_{\phi}(\overline
{\boldsymbol{p}},\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}))). \label{5a}%
\end{equation}
The second new family is obtained by replacing in (\ref{N2}) the Pearson
divergence measure by a $\phi$-divergence measure,%
\begin{equation}
S_{\phi}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}}))=\frac{2n}{\phi^{\prime\prime}(1)}d_{\phi
}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}})). \label{5b}%
\end{equation}
If we consider $\phi(x)=x\log x-x+1$ in (\ref{5a}), we get $G^{2}$, and if we
consider $\phi(x)=\tfrac{1}{2}(x-1)^{2}$ in (\ref{5a}), we get $X^{2}$.
Test-statistics based on $\phi$-divergence measures have been used in the
framework of loglinear models for some authors, see Cressie and Pardo (2000,
2002, 2003), Mart\'{\i}n and Pardo (2006, 2008b, 2011).
\section{Asymptotic results\label{sec:Main results}}
As starting point, we shall establish the observed Fisher information matrix
associated with $\boldsymbol{\theta}$, $\mathcal{I}_{F}^{(n_{1},n_{2}%
)}(\boldsymbol{\theta})$, for a loglinear model with product-multinomial
sampling as%
\begin{equation}
\mathcal{I}_{F}^{(n_{1},n_{2})}(\boldsymbol{\theta})=\frac{1}{n}%
\boldsymbol{W}^{T}%
\begin{pmatrix}
n_{1}(\boldsymbol{D}_{\boldsymbol{\pi}_{1}(\boldsymbol{\theta})}%
-\boldsymbol{\pi}_{1}(\boldsymbol{\theta})\boldsymbol{\pi}_{1}^{T}%
(\boldsymbol{\theta})) & \boldsymbol{0}_{J\times J}\\
\boldsymbol{0}_{J\times J} & n_{2}(\boldsymbol{D}_{\boldsymbol{\pi}%
_{2}(\boldsymbol{\theta})}-\boldsymbol{\pi}_{2}(\boldsymbol{\theta
})\boldsymbol{\pi}_{2}^{T}(\boldsymbol{\theta}))
\end{pmatrix}
\boldsymbol{W}, \label{FIM1}%
\end{equation}
where $\boldsymbol{D}_{\boldsymbol{a}}$\ is the diagonal matrix of vector
$\boldsymbol{a}$. To proof (\ref{FIM1}), we take into account that the overall
observed Fisher information matrix for product multinomial sampling is the
weighted observed Fisher information matrix associated with each multinomial
sample, $\mathcal{I}_{F,i}^{(n_{1},n_{2})}(\boldsymbol{\theta})$, $i=1,2$,
i.e.%
\begin{align*}
\mathcal{I}_{F}^{(n_{1},n_{2})}(\boldsymbol{\theta}) & =\frac{n_{1}}%
{n}\mathcal{I}_{F,1}^{(n_{1},n_{2})}(\boldsymbol{\theta})+\frac{n_{2}}%
{n}\mathcal{I}_{F,2}^{(n_{1},n_{2})}(\boldsymbol{\theta}),\\
\mathcal{I}_{F,i}^{(n_{1},n_{2})}(\boldsymbol{\theta}) & =\boldsymbol{W}%
_{i}^{T}(\boldsymbol{D}_{\boldsymbol{\pi}_{i}(\boldsymbol{\theta}%
)}-\boldsymbol{\pi}_{i}(\boldsymbol{\theta})\boldsymbol{\pi}_{i}%
^{T}(\boldsymbol{\theta}))\boldsymbol{W}_{i},\quad i=1,2,
\end{align*}
such that $\boldsymbol{W}^{T}=(\boldsymbol{W}_{1}^{T},\boldsymbol{W}_{2}^{T}%
)$, $\log\boldsymbol{p}_{1}(\boldsymbol{\theta})=u\boldsymbol{1}_{J}%
+u_{1(1)}\boldsymbol{1}_{J}+\boldsymbol{W}_{1}\boldsymbol{\theta}$ and
$\log\boldsymbol{p}_{2}(\boldsymbol{\theta})=u\boldsymbol{1}_{J}%
+\boldsymbol{W}_{2}\boldsymbol{\theta}$.
When $\boldsymbol{\theta}\in\Theta_{0}$, we shall denote $\boldsymbol{\theta
}_{0}$\ to be the true value of the unknown parameter under $H_{0}$, and in
such a case it holds $\boldsymbol{\pi}_{1}(\boldsymbol{\theta}_{0}%
)=\boldsymbol{\pi}_{2}(\boldsymbol{\theta}_{0})=\boldsymbol{\pi}%
(\boldsymbol{\theta}_{0})=(\pi_{1}(\boldsymbol{\theta}_{0}),...,\pi
_{J}(\boldsymbol{\theta}_{0}))^{T}$, where $\boldsymbol{\pi}_{i}%
(\boldsymbol{\theta}_{0})$ is defined as the probability vector with the terms
given in (\ref{eq5}) and related to the loglinear model through
$\boldsymbol{p}_{i}(\boldsymbol{\theta}_{0})=\frac{n_{i}}{n}\boldsymbol{\pi
}_{i}(\boldsymbol{\theta}_{0})$, $i=1,2$. Notice that $\boldsymbol{\pi}%
_{i}(\boldsymbol{\theta}_{0})$ is fixed as $n_{1},n_{2}\rightarrow\infty$ and
we shall assume that%
\[
\nu_{i}=\lim_{n_{i}\rightarrow\infty}\frac{n_{i}}{n},\quad i=1,2,
\]
is fixed but unknown, i.e. $\lim_{n_{i}\rightarrow\infty}\boldsymbol{p}%
_{i}(\boldsymbol{\theta})=\nu_{i}\boldsymbol{\pi}_{i}(\boldsymbol{\theta}%
_{0})$, $i=1,2$. We shall also denote%
\[
\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})=(\pi_{1}(\boldsymbol{\theta
}_{0}),...,\pi_{J-1}(\boldsymbol{\theta}_{0}))^{T},\quad i=1,2.
\]
the $(J-1)$-dimensional vector obtained removing from $\boldsymbol{\pi
}(\boldsymbol{\theta}_{0})$\ the last element. Focussing on the parameter
structure $\boldsymbol{\theta}=(\boldsymbol{\theta}_{12}^{T}%
,\boldsymbol{\theta}_{2}^{T})^{T}$, with $\boldsymbol{\theta}_{12}%
=(\theta_{12(11)},...,\theta_{12(1,J-1)})^{T}$, $\boldsymbol{\theta}%
_{2}=(\theta_{2(1)},...,\theta_{2(J-1)})^{T}$ and the specific structure of
$\boldsymbol{W}$, see (\ref{W}), we shall establish asymptotically the
specific shape of (\ref{FIM1}), a fundamental result for the posterior theorems.
\begin{theorem}
The asymptotic Fisher information matrix of $\boldsymbol{\theta}$,
$\mathcal{I}_{F}(\boldsymbol{\theta})=\lim_{n_{1},n_{2}\rightarrow\infty
}\mathcal{I}_{F}^{(n_{1},n_{2})}(\boldsymbol{\theta})$ when
$\boldsymbol{\theta}\in\Theta_{0}$ is given by%
\begin{equation}
\mathcal{I}_{F}(\boldsymbol{\theta}_{0})=%
\begin{pmatrix}
\boldsymbol{D}_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})}%
-\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{\ast
T}(\boldsymbol{\theta}_{0}) & \nu_{1}\left( \boldsymbol{D}_{\boldsymbol{\pi
}^{\ast}(\boldsymbol{\theta}_{0})}-\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta
}_{0})\boldsymbol{\pi}^{\ast T}(\boldsymbol{\theta}_{0})\right) \\
\nu_{1}\left( \boldsymbol{D}_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}%
_{0})}-\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{\ast
T}(\boldsymbol{\theta}_{0})\right) & \nu_{1}\left( \boldsymbol{D}%
_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})}-\boldsymbol{\pi}^{\ast
}(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{\ast T}(\boldsymbol{\theta}%
_{0})\right)
\end{pmatrix}
. \label{FIM2}%
\end{equation}
\end{theorem}
\begin{proof}
Replacing $\boldsymbol{\theta}$ by $\boldsymbol{\theta}_{0}$ and the explicit
expression of $\boldsymbol{W}$\ in the general expression of the finite sample
size Fisher information matrix for two independent multinomial samples,
(\ref{FIM1}), we obtain through the property of the Kronecker product given in
(1.22) of Harville (2008, page 341) that%
\begin{align*}
\mathcal{I}_{F}^{(n_{1},n_{2})}(\boldsymbol{\theta}_{0}) & =\left(
\begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}
\otimes%
\begin{pmatrix}
\boldsymbol{I}_{J-1}\\
\boldsymbol{0}_{J-1}^{T}%
\end{pmatrix}
^{T}\right) \left( diag\{\tfrac{n_{i}}{n}\}_{i=1}^{2}\otimes(\boldsymbol{D}%
_{\boldsymbol{\pi}(\boldsymbol{\theta}_{0})}-\boldsymbol{\pi}%
(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{T}(\boldsymbol{\theta}%
_{0}))\right) \left(
\begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}
\otimes%
\begin{pmatrix}
\boldsymbol{I}_{J-1}\\
\boldsymbol{0}_{J-1}^{T}%
\end{pmatrix}
\right) \\
& =\left(
\begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}
diag\{\tfrac{n_{i}}{n}\}_{i=1}^{2}%
\begin{pmatrix}
1 & 1\\
1 & 0
\end{pmatrix}
\right) \otimes\left(
\begin{pmatrix}
\boldsymbol{I}_{J-1}\\
\boldsymbol{0}_{J-1}^{T}%
\end{pmatrix}
^{T}(\boldsymbol{D}_{\boldsymbol{\pi}(\boldsymbol{\theta}_{0})}%
-\boldsymbol{\pi}(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{T}%
(\boldsymbol{\theta}_{0}))%
\begin{pmatrix}
\boldsymbol{I}_{J-1}\\
\boldsymbol{0}_{J-1}^{T}%
\end{pmatrix}
\right) \\
& =%
\begin{pmatrix}
1 & \tfrac{n_{1}}{n}\\
\tfrac{n_{1}}{n} & \tfrac{n_{1}}{n}%
\end{pmatrix}
\otimes\left( \boldsymbol{D}_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}%
_{0})}-\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{\ast
T}(\boldsymbol{\theta}_{0})\right) ,
\end{align*}
and then%
\begin{equation}
\mathcal{I}_{F}(\boldsymbol{\theta}_{0})=%
\begin{pmatrix}
1 & \nu_{1}\\
\nu_{1} & \nu_{1}%
\end{pmatrix}
\otimes\left( \boldsymbol{D}_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}%
_{0})}-\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{\ast
T}(\boldsymbol{\theta}_{0})\right) . \label{iF}%
\end{equation}
\end{proof}
The following theorem establishes that the asymptotic distribution of the
families of test statistics (\ref{5a}) and (\ref{5b}) corresponds to a
$J$-dimensional chi-bar squared random variable, a mixture of $J$ chi-squared
distributions. Let $E=\{1,...,J-1\}$ be the whole set of all row-indices of
matrix $\boldsymbol{R}$, $\mathcal{F}(E)$ the family of all possible subsets
of $E$, and $\boldsymbol{R}(S\mathbf{)}$ is a submatrix of $\boldsymbol{R}%
$\ with row-\'{\i}ndices belonging to $S\in\mathcal{F}(E)$. We must not forget
that $\boldsymbol{R}=(\boldsymbol{0}_{(J-1)\times(J-1)},\boldsymbol{G}_{J-1})$
and therefore $\boldsymbol{R}(S)=(\boldsymbol{0}_{card(S)\times(J-1)}%
,\boldsymbol{G}_{J-1}(S))$.
We denote by $\boldsymbol{H}(\boldsymbol{\theta})$\ the following
$(J-1)\times(J-1)$ tridiagonal matrix%
\begin{equation}
\boldsymbol{H}(\boldsymbol{\theta})=\frac{1}{\nu_{1}\nu_{2}}%
\begin{pmatrix}
\frac{\pi_{1}(\boldsymbol{\theta})+\pi_{2}(\boldsymbol{\theta})}{\pi
_{1}(\boldsymbol{\theta})\pi_{2}(\boldsymbol{\theta})} & -\frac{1}{\pi
_{2}(\boldsymbol{\theta})} & & & \\
-\frac{1}{\pi_{1}(\boldsymbol{\theta})} & \frac{\pi_{2}(\boldsymbol{\theta
})+\pi_{3}(\boldsymbol{\theta})}{\pi_{2}(\boldsymbol{\theta})\pi
_{3}(\boldsymbol{\theta})} & -\frac{1}{\pi_{3}(\boldsymbol{\theta})} & & \\
& -\frac{1}{\pi_{3}(\boldsymbol{\theta})} & \frac{\pi_{3}(\boldsymbol{\theta
})+\pi_{4}(\boldsymbol{\theta})}{\pi_{3}(\boldsymbol{\theta})\pi
_{4}(\boldsymbol{\theta})} & \ddots & \\
& & \ddots & \ddots & -\frac{1}{\pi_{J-1}(\boldsymbol{\theta})}\\
& & & -\frac{1}{\pi_{J-1}(\boldsymbol{\theta})} & \frac{\pi_{J-1}%
(\boldsymbol{\theta})+\pi_{J}(\boldsymbol{\theta})}{\pi_{J-1}%
(\boldsymbol{\theta})\pi_{J}(\boldsymbol{\theta})}%
\end{pmatrix}
, \label{H}%
\end{equation}
and by $\boldsymbol{H}(S_{1},S_{2},\boldsymbol{\theta})$ the submatrix of
$\boldsymbol{H}(\boldsymbol{\theta})$\ obtained by deleting from it the
row-indices contained in the set $S_{1}$ and column-indices contained in the
set $S_{2}$.
\begin{theorem}
\label{Th1}Under $H_{0}$, the asymptotic distribution of $S_{\phi
}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}}))$\ and $T_{\phi}(\overline{\boldsymbol{p}%
},\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}}))$\ is%
\[
\lim_{n_{1},n_{2}\rightarrow\infty}\Pr\left( S_{\phi}(\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}%
}))\leq x\right) =\lim_{n_{1},n_{2}\rightarrow\infty}\Pr\left( T_{\phi
}(\overline{\boldsymbol{p}},\boldsymbol{p}(\widetilde{\boldsymbol{\theta}%
}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}}))\leq x\right) =\sum
_{j=0}^{J-1}w_{j}(\boldsymbol{\theta}_{0})\Pr\left( \chi_{(J-1)-j}^{2}\leq
x\right)
\]
where $\chi_{0}^{2}=0$ a.s. and $\{w_{j}(\boldsymbol{\theta}_{0}%
)\}_{j=0}^{J-1}$ is the set of weights such that $\sum_{j=0}^{J-1}%
w_{j}(\boldsymbol{\theta}_{0})=1$ and%
\begin{equation}
w_{j}(\boldsymbol{\theta}_{0})=\sum_{S\in\mathcal{F}(E),\mathrm{card}(S)=j}%
\Pr\left( \boldsymbol{Z}_{1}(S)\geq\boldsymbol{0}_{j}\right) \Pr\left(
\boldsymbol{Z}_{2}(S)\geq\boldsymbol{0}_{(J-1)-j}\right) , \label{eqw}%
\end{equation}
where%
\begin{align*}
\boldsymbol{Z}_{1}(S) & \sim\mathcal{N}\left( \boldsymbol{0}_{\mathrm{card}%
(S)},\boldsymbol{H}^{-1}(S,S,\boldsymbol{\theta}_{0})\right) ,\\
\boldsymbol{Z}_{2}(S) & \sim\mathcal{N}\left( \boldsymbol{0}%
_{(J-1)-\mathrm{card}(S)},\boldsymbol{H}(S^{C},S^{C},\boldsymbol{\theta}%
_{0})-\boldsymbol{H}(S^{C},S,\boldsymbol{\theta}_{0})\boldsymbol{H}%
^{-1}(S,S,\boldsymbol{\theta}_{0})\boldsymbol{H}^{T}(S^{C}%
,S,\boldsymbol{\theta}_{0})\right) ,
\end{align*}
$S^{C}=E-S$ and $\mathrm{card}(S)$ denotes the cardinal of the set $S$.
\end{theorem}
\begin{proof}
By following similar arguments of Mart\'{\i}n and Balakrishnan we obtain
$\boldsymbol{H}(S,S,\boldsymbol{\theta}_{0})=\boldsymbol{R}(S\mathbf{)}%
\mathcal{I}_{F}^{-1}(\boldsymbol{\theta}_{0})\boldsymbol{R}^{T}(S\mathbf{)}$
(see Appendix \ref{ProofTh1ContrA}, for the details). In particular,
$\boldsymbol{H}(\boldsymbol{\theta}_{0})=\boldsymbol{H}(S,S,\boldsymbol{\theta
}_{0})$ with $S=E$, i.e.%
\begin{align*}
\boldsymbol{H}(\boldsymbol{\theta}_{0}) & =\boldsymbol{R}(E\mathbf{)}%
\mathcal{I}_{F}^{-1}(\boldsymbol{\theta}_{0})\boldsymbol{R}^{T}(E\mathbf{)}\\
& \mathbf{=}(\boldsymbol{0}_{(J-1)\times(J-1)},\boldsymbol{G}_{J-1}%
)\mathcal{I}_{F}^{-1}(\boldsymbol{\theta}_{0})(\boldsymbol{0}_{(J-1)\times
(J-1)},\boldsymbol{G}_{J-1})^{T},
\end{align*}
where $\mathcal{I}_{F}(\boldsymbol{\theta}_{0})$ is (\ref{iF}). By following
the properties of the inverse of the Kronecker product for calculating the
inverse of (\ref{iF}),%
\begin{align*}
\mathcal{I}_{F}^{-1}(\boldsymbol{\theta}_{0}) & =%
\begin{pmatrix}
1 & \nu_{1}\\
\nu_{1} & \nu_{1}%
\end{pmatrix}
^{-1}\otimes\left( \boldsymbol{D}_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta
}_{0})}-\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})\boldsymbol{\pi}^{\ast
T}(\boldsymbol{\theta})\right) ^{-1}\\
& =%
\begin{pmatrix}
\frac{1}{\nu_{2}} & -\frac{1}{\nu_{2}}\\
-\frac{1}{\nu_{2}} & \frac{1}{\nu_{1}\nu_{2}}%
\end{pmatrix}
\otimes\left( \boldsymbol{D}_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}%
_{0})}^{-1}+\frac{1}{\pi_{J}(\boldsymbol{\theta}_{0})}\boldsymbol{1}%
_{J-1}\boldsymbol{1}_{J-1}^{T}\right) ,
\end{align*}
and replacing it in the previous expression of $\boldsymbol{H}%
(\boldsymbol{\theta}_{0})$,%
\begin{align*}
\boldsymbol{H}(\boldsymbol{\theta}_{0}) & =\frac{1}{\nu_{1}\nu_{2}%
}\boldsymbol{G}_{J-1}\left( \boldsymbol{D}_{\boldsymbol{\pi}^{\ast
}(\boldsymbol{\theta}_{0})}^{-1}+\frac{1}{\pi_{J}(\boldsymbol{\theta}_{0}%
)}\boldsymbol{1}_{J-1}\boldsymbol{1}_{J-1}^{T}\right) \boldsymbol{G}%
_{J-1}^{T}\\
& =\frac{1}{\nu_{1}\nu_{2}}\left( \boldsymbol{G}_{J-1}\boldsymbol{D}%
_{\boldsymbol{\pi}^{\ast}(\boldsymbol{\theta}_{0})}^{-1}\boldsymbol{G}%
_{J-1}^{T}+\frac{1}{\pi_{J}(\boldsymbol{\theta}_{0})}\boldsymbol{e}%
_{J-1}\boldsymbol{e}_{J-1}^{T}\right) ,
\end{align*}
which is equal to (\ref{H}).\medskip
\end{proof}
Even though there is an equality in (\ref{4b}), $\boldsymbol{\theta}$ is not a
fixed vector under the null hypothesis since such an equality is effective
only for $\boldsymbol{\theta}_{12}$,\ and thus $\boldsymbol{\theta}_{2}$ is a
vector of nuisance parameters. This means that we have a composite null
hypothesis which requires estimation of $\boldsymbol{\theta}\in\Theta_{0}$,
through $\widehat{\boldsymbol{\theta}}$ and we cannot use directly the results
based on Theorem \ref{Th1}. The tests performed replacing the parameter
$\boldsymbol{\theta}_{0}$\ of the asymptotic distribution by
$\widehat{\boldsymbol{\theta}}$ are called \textquotedblleft local
tests\textquotedblright\ (see Dardanoni and Forcina (1998)) and they are
usually considered to be good approximations of the theoretical tests.
In relation to the weights, $\{w_{j}(\boldsymbol{\theta}_{0})\}_{j=1,...,J}$,
there are explicit expressions when $J\in\{2,3,4\}$ based on the matrix given
in (\ref{H}) and formulas (3.24), (3.25) and (3.26) in Silvapulle and Sen
(2005, page 80). When $J=2$, $w_{0}(\boldsymbol{\theta}_{0})=w_{1}%
(\boldsymbol{\theta}_{0})=\frac{1}{2}$. When $J=3$, the estimators of the
weights are%
\begin{equation}
\left\{
\begin{array}
[c]{l}%
w_{0}(\widehat{\boldsymbol{\theta}})=\tfrac{1}{2}-w_{2}%
(\widehat{\boldsymbol{\theta}}),\\
w_{1}(\widehat{\boldsymbol{\theta}})=\frac{1}{2},\\
w_{2}(\widehat{\boldsymbol{\theta}})=\tfrac{1}{2\pi}\arccos\widehat{\rho}%
_{12},
\end{array}
\right. \label{weightsJ=3}%
\end{equation}
where%
\begin{equation}
\widehat{\rho}_{ij}=\tfrac{\widehat{\sigma}_{ij}}{\sqrt{\widehat{\sigma}%
_{ii}\widehat{\sigma}_{jj}}}=-\sqrt{\frac{(N_{1i}+N_{2i})(N_{1,j+1}%
+N_{2,j+1})}{(N_{1i}+N_{2i}+N_{1j}+N_{2j})(N_{1j}+N_{2j}+N_{1,j+1}+N_{2j+1})}%
}, \label{cor}%
\end{equation}
is the correlation associated with the $i$-th and $j$-th variable of a central
random variable with variance-covariance matrix
\[
\boldsymbol{H}(\widehat{\boldsymbol{\theta}})=\frac{1}{\widehat{\nu}%
_{1}\widehat{\nu}_{2}}%
\begin{pmatrix}
\frac{\pi_{1}(\widehat{\boldsymbol{\theta}})+\pi_{2}%
(\widehat{\boldsymbol{\theta}})}{\pi_{1}(\widehat{\boldsymbol{\theta}})\pi
_{2}(\widehat{\boldsymbol{\theta}})} & -\frac{1}{\pi_{2}%
(\widehat{\boldsymbol{\theta}})}\\
-\frac{1}{\pi_{2}(\widehat{\boldsymbol{\theta}})} & \frac{\pi_{2}%
(\widehat{\boldsymbol{\theta}})+\pi_{3}(\widehat{\boldsymbol{\theta}})}%
{\pi_{2}(\widehat{\boldsymbol{\theta}})\pi_{3}(\widehat{\boldsymbol{\theta}})}%
\end{pmatrix}
,
\]
where $\pi_{j}(\widehat{\boldsymbol{\theta}})=\frac{N_{1j}+N_{2j}}{n}$. When
$J=4$,%
\begin{equation}
\left\{
\begin{array}
[c]{l}%
w_{0}(\widehat{\boldsymbol{\theta}})=\tfrac{1}{4\pi}\left( 2\pi
-\arccos\widehat{\rho}_{12}-\arccos\widehat{\rho}_{13}-\arccos\widehat{\rho
}_{23}\right) ,\\
w_{1}(\widehat{\boldsymbol{\theta}})=\tfrac{1}{4\pi}\left( 3\pi
-\arccos\widehat{\rho}_{12\cdot3}-\arccos\widehat{\rho}_{13\cdot2}%
-\arccos\widehat{\rho}_{23\cdot1}\right) ,\\
w_{2}(\widehat{\boldsymbol{\theta}})=\tfrac{1}{2}-w_{0}%
(\widehat{\boldsymbol{\theta}}),\\
w_{3}(\widehat{\boldsymbol{\theta}})=\tfrac{1}{2}-w_{1}%
(\widehat{\boldsymbol{\theta}}),
\end{array}
\right. \label{weightsJ=4}%
\end{equation}
which depend on the estimation of the marginal (\ref{cor}) and conditional
correlations%
\[
\widehat{\rho}_{ij\cdot k}=\tfrac{\widehat{\rho}_{ij}-\widehat{\rho}%
_{ik}\widehat{\rho}_{kj}}{\sqrt{(1-\widehat{\rho}_{ik}^{2})(1-\widehat{\rho
}_{kj}^{2})}},
\]
associated with the $i$-th and $j$-th variable, given a value of the $k$-th
variable, of a central random variable with variance-covariance matrix%
\[
\boldsymbol{H}(\widehat{\boldsymbol{\theta}})=\frac{1}{\widehat{\nu}%
_{1}\widehat{\nu}_{2}}%
\begin{pmatrix}
\frac{\pi_{1}(\widehat{\boldsymbol{\theta}})+\pi_{2}%
(\widehat{\boldsymbol{\theta}})}{\pi_{1}(\widehat{\boldsymbol{\theta}})\pi
_{2}(\widehat{\boldsymbol{\theta}})} & -\frac{1}{\pi_{2}%
(\widehat{\boldsymbol{\theta}})} & 0\\
-\frac{1}{\pi_{2}(\widehat{\boldsymbol{\theta}})} & \frac{\pi_{2}%
(\widehat{\boldsymbol{\theta}})+\pi_{3}(\widehat{\boldsymbol{\theta}})}%
{\pi_{2}(\widehat{\boldsymbol{\theta}})\pi_{3}(\widehat{\boldsymbol{\theta}})}
& -\frac{1}{\pi_{3}(\widehat{\boldsymbol{\theta}})}\\
0 & -\frac{1}{\pi_{3}(\widehat{\boldsymbol{\theta}})} & \frac{\pi
_{3}(\widehat{\boldsymbol{\theta}})+\pi_{4}(\widehat{\boldsymbol{\theta}}%
)}{\pi_{3}(\widehat{\boldsymbol{\theta}})\pi_{4}(\widehat{\boldsymbol{\theta}%
})}%
\end{pmatrix}
.
\]
It is interesting to point out that the factor related to the sample size in
each multinomial sample, $\frac{1}{\widehat{\nu}_{1}\widehat{\nu}_{2}}$, have
no effect in the expression of estimator for the weights of the chi-bar
squared distribution These formulas will be considered in the forthcoming
sections. It is worthwhile to mention that the normal orthant probabilities
for the weights given in (\ref{eqw}), can also be computed for any value of
$J$ using the \texttt{mvtnorm} R package (see
\hyperref{http://CRAN.R-project.org/package=mvtnorm}{}{}%
{http://CRAN.R-project.org/package=mvtnorm}%
, for details).
\section{Numerical example\label{sec:Numerical example}}
In this section the data set of the introduction (Table \ref{tttt1}), where
$J=4$, is analyzed. The sample, a realization of $\boldsymbol{N}$, is
summarized in the following vector%
\[
\boldsymbol{n}=(n_{11},n_{12},n_{13},n_{14},n_{21},n_{22},n_{23},n_{24}%
)^{T}=(11,8,8,5,6,4,10,12)^{T}.
\]
The order restricted MLE under likelihood ratio order, obtained through the
\texttt{E04UCF} subroutine of\ \texttt{NAG} Fortran library (%
\hyperref{http://www.nag.co.uk/numeric/fl/FLdescription.asp}{}{}%
{http://www.nag.co.uk/numeric/fl/FLdescription.asp}%
), is%
\[
\widetilde{\boldsymbol{\theta}}%
=(-0.7164,-1.0647,-0.1823,1.5173,1.5173,0.6523)^{T}.
\]
The estimation of the probability vectors of interest is%
\begin{align*}
\overline{\boldsymbol{p}} &
=(0.1719,0.1250,0.1250,0.0781,0.0938,0.0625,0.1563,0.1875)^{T},\\
\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}) &
=(0.1740,0.1228,0.1250,0.0781,0.0916,0.0647,0.1563,0.1875)^{T},\\
\boldsymbol{p}(\widehat{\boldsymbol{\theta}}) &
=(0.1328,0.0938,0.1406,0.1328,0.1328,0.0938,0.1406,0.1328)^{T},
\end{align*}
and the estimation of the weights, based on (\ref{weightsJ=4}), are%
\[
w_{0}(\widehat{\boldsymbol{\theta}})=0.0381,\quad w_{1}%
(\widehat{\boldsymbol{\theta}})=0.2420,\quad w_{2}(\widehat{\boldsymbol{\theta
}})=0.461\,8,\quad w_{3}(\widehat{\boldsymbol{\theta}})=0.2580.
\]
In order to solve analytically the example we shall consider a particular
function $\phi$ in (\ref{5a}) and (\ref{5b}). Taking%
\[
\phi_{\lambda}(x)=\frac{x^{\lambda+1}-x-\lambda(x-1)}{\lambda(\lambda+1)},
\]
we get the \textquotedblleft the power divergence family\textquotedblright%
\[
d_{\phi_{\lambda}}(\boldsymbol{p},\boldsymbol{q})=\frac{1}{\lambda(\lambda
+1)}\left(
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\tfrac{p_{ij}^{\lambda+1}}{q_{ij}^{\lambda}(\widehat{\boldsymbol{\theta}}%
)}-1\right)
\]
in such a way that for each $\lambda\in%
\mathbb{R}
-\{-1,0\}$\ a different divergence measure is obtained, and thus%
\begin{align}
T_{\lambda} & =T_{\phi_{\lambda}}(\overline{\boldsymbol{p}},\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}%
}))=\frac{2n}{\lambda(\lambda+1)}\left(
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\frac{\overline{p}_{ij}^{\lambda+1}}{p_{ij}^{\lambda}%
(\widehat{\boldsymbol{\theta}})}-%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\frac{\overline{p}_{ij}^{\lambda+1}}{p_{ij}^{\lambda}%
(\widetilde{\boldsymbol{\theta}})}\right) ,\label{PD1}\\
S_{\lambda} & =S_{\phi_{\lambda}}(\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}%
}))=\frac{2n}{\lambda(\lambda+1)}\left(
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\frac{p_{ij}^{\lambda+1}(\widetilde{\boldsymbol{\theta}})}{p_{ij}^{\lambda
}(\widehat{\boldsymbol{\theta}})}-1\right) . \label{PD2}%
\end{align}
It is also possible to cover the real line for $\lambda$, by defining
\[
d_{\phi_{\lambda}}(\boldsymbol{p},\boldsymbol{q})=\lim_{\ell\rightarrow
\lambda}d_{\phi_{\ell}}(\boldsymbol{p},\boldsymbol{q}),\quad\lambda
\in\{-1,0\},
\]
and by considering $T_{\lambda}=\lim_{\lambda\rightarrow\ell}T_{\ell}$,
$S_{\lambda}=\lim_{\lambda\rightarrow\ell}S_{\ell}$, for $\lambda\in\{0,-1\}$,
i.e.
\begin{align}
T_{0} & =T_{\phi_{0}}(\overline{\boldsymbol{p}},\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}%
}))=G^{2}=2n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\overline{p}_{ij}\log\frac{p_{ij}(\widetilde{\boldsymbol{\theta}})}%
{p_{ij}(\widehat{\boldsymbol{\theta}})},\label{PD3}\\
T_{-1} & =T_{\phi_{-1}}(\overline{\boldsymbol{p}},\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}%
}))=2n\left(
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
p_{ij}(\widehat{\boldsymbol{\theta}})\log\frac{p_{ij}%
(\widehat{\boldsymbol{\theta}})}{\overline{p}_{ij}}-%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
p_{ij}(\widetilde{\boldsymbol{\theta}})\log\frac{p_{ij}%
(\widetilde{\boldsymbol{\theta}})}{\overline{p}_{ij}}\right) \label{PD4}%
\end{align}
and
\begin{align}
S_{0} & =S_{\phi_{0}}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}%
}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}}))=2nd_{Kull}(\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}%
}))=2n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
p_{ij}(\widetilde{\boldsymbol{\theta}})\log\frac{p_{ij}%
(\widetilde{\boldsymbol{\theta}})}{p_{ij}(\widehat{\boldsymbol{\theta}}%
)},\label{PD5}\\
S_{-1} & =S_{\phi_{-1}}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}%
}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}}))=2nd_{Kull}(\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}}),\boldsymbol{p}(\widetilde{\boldsymbol{\theta}%
}))=2n%
{\displaystyle\sum\limits_{j=1}^{J}}
p_{ij}(\widehat{\boldsymbol{\theta}})\log\frac{p_{ij}%
(\widehat{\boldsymbol{\theta}})}{p_{ij}(\widetilde{\boldsymbol{\theta}})}.
\label{PD6}%
\end{align}
It is well known that $d_{\phi_{0}}(\boldsymbol{p},\boldsymbol{q}%
)=d_{Kull}(\boldsymbol{p},\boldsymbol{q})$ and $d_{\phi_{1}}(\boldsymbol{p}%
,\boldsymbol{q})=d_{Pearson}(\boldsymbol{p},\boldsymbol{q})$, which is very
interesting since $G^{2}$ and $X^{2}$ are members of the power divergence
based test-statistics. It is also worthwhile to mention that $d_{\phi_{-1}%
}(\boldsymbol{p},\boldsymbol{q})=d_{Kull}(\boldsymbol{q},\boldsymbol{p})$.
In Table \ref{t1}, the power divergence based test-statistics for some values
of $\lambda$ in $\Lambda=\{-1.5,-1,-\frac{1}{2},0,\frac{2}{3},1,1.5,2,3\}$,
and their corresponding asymptotic $p$-values are shown. In all of them it is
concluded, with a significance level equal to $0.05$, that an equal effect of
both treatments is rejected and hence the treatment is more effective than the
control to heal the ulcer.
\begin{table}[htbp] \tabcolsep2.8pt \centering
$%
\begin{tabular}
[c]{cccccccccc}\hline\hline
test-statistic & $\lambda=-1.5$ & $\lambda=-1$ & $\lambda=-\frac{1}{2}$ &
$\lambda=0$ & $\lambda=\frac{2}{3}$ & $\lambda=1$ & $\lambda=1.5$ &
$\lambda=2$ & $\lambda=3$\\\hline
\multicolumn{1}{l}{$\overset{}{T_{\lambda}}$} & 6.5323 & 6.3215 & 6.1562 &
\textbf{6.0323} & 5.9261 & 5.8965 & 5.8803 & 5.8965 & 6.0244\\
$p$\textrm{-}$\mathrm{value}(T_{\lambda})$ & 0.0175 & 0.0194 & 0.0211 &
\textbf{0.0225} & 0.0238 & 0.0241 & 0.0243 & 0.0241 & 0.0226\\
\multicolumn{1}{l}{$S_{\lambda}$} & 6.5277 & 6.3189 & 6.1551 & 6.0323 &
5.9270 & \textbf{5.8977} & 5.8815 & 5.8977 & 6.0244\\
$p$\textrm{-}$\mathrm{value}(S_{\lambda})$ & 0.0175 & 0.0195 & 0.0212 &
0.0225 & 0.0238 & \textbf{0.0241} & 0.0243 & 0.0241 & 0.0226\\\hline\hline
\end{tabular}
\ \ \ \ \ \ \ \ \ \ \ \ $%
\caption{Power divergence based test-statistics and asymptotic p-values for the data given Table \ref{tttt1}.\label{t1}}%
\end{table}%
\bigskip
The $p$-values given in Table \ref{t1} were obtained by the following
algorithm:\newline Let $T\in\{{T_{\lambda},S_{\lambda}}\}_{\lambda\in\Lambda}$
be the test-statistic associated with (\ref{4}). In the following steps the
corresponding asymptotic $p$-value, based on the asymptotic distribution of
Theorem \ref{Th1}, is calculated once it is suppose we have $\{w_{j}%
(\widehat{\boldsymbol{\theta}})\}_{j=0}^{J-1}$:
\noindent\texttt{STEP 1: Using }$\boldsymbol{n}$\texttt{\ calculate
}$\boldsymbol{p}(\widehat{\boldsymbol{\theta}})$\texttt{\ taking into account
(\ref{ind}).}\newline\texttt{STEP 2: Using }$\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}})$\texttt{\ calculate value }$t$\texttt{ of
test-statistic }$T$\texttt{ using the corresponding expression in
(\ref{PD1})-(\ref{PD6}).}\newline\texttt{STEP 3: If }$T\leq0$ \texttt{then
compute }$p$\textrm{-}$\mathrm{value}(T):=1$ \texttt{and STOP, otherwise
compute }$p$\textrm{-}$\mathrm{value}(T):=0$.\newline\texttt{STEP 4: }For
$j=0,...,J-2$\texttt{, do }$p$\textrm{-}$\mathrm{value}(T):=p$\textrm{-}%
$\mathrm{value}(T)+w_{j}(\widehat{\boldsymbol{\theta}})\Pr\left(
\chi_{(J-1)-j}^{2}>t\right) $.\texttt{\newline\hspace*{1.6cm}E.g., the NAG
Fortran library subroutine G01ECF can be useful.}
Recently, Shan and Ma (2014) have studied a similar problem as (2a)-(2b), but
considering different alternative hypotheses, since they consider odds ratios
based on cumulative probabilities. Focussed on probabilities rather than
cumulative probabilities, we are going to include the asymptotic version of
their test-statistic in our numerical study as well as later, in the
simulation study: the two sample Wilcoxon test-statistic for discrete data
(ties), also known as Wilcoxon mid-rank test-statistic. Metha et al. (1984)
proposed such a test-statistic for solving exactly the same alternative
hypothesis studied in this paper either as a permutation or as asymptotic
test. Our null and alternative hypotheses are a particular case of their
hypotheses, taking in their Section 4 $\phi^{\ast}=1$. The expression of the
Wilcoxon mid-rank test-statistic is%
\begin{equation}
W=%
{\textstyle\sum\limits_{j=1}^{J}}
r_{j}n_{1j}, \label{wilc}%
\end{equation}
where $r_{1}=(n_{\bullet1}+2)/2$ and $r_{j}=%
{\textstyle\sum\nolimits_{\ell=1}^{j-1}}
n_{\bullet\ell}+\left. \left( n_{\bullet j}+1\right) \right/ 2$,
$j=2,...,J$, $n_{\bullet j}=n_{1j}+n_{2j}$, and the corresponding asymptotic
distribution is normal with mean $\mu_{W}=\left. n_{1}\left( n+1\right)
\right/ 2$ and variance
\[
\sigma_{W}^{2}=n_{1}n_{2}\frac{n+1-\frac{1}{n(n-1)}%
{\textstyle\sum\nolimits_{j=1}^{J}}
(n_{\bullet j}^{3}-n_{\bullet j})}{12}.
\]
The Wilcoxon mid-rank test-statistic for the data of Table \ref{tttt1} is
$W=875$\ and with the corresponding $p$-value, $0.01094$, the same conclusion
is obtained, i.e. rejecting the hypothesis of equal effect of both treatments
with $5\%$ significance level.
\section{Simulation study\label{sec:Simulation Study}}
\subsection{2x2 table: one sided in comparison with the two sided test}
In this section we illustrate in what sense the likelihood ratio test given in
(\ref{LRT}),%
\begin{equation}
G^{2}=2n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
\overline{p}_{ij}\log\frac{p_{ij}(\widetilde{\boldsymbol{\theta}})}%
{p_{ij}(\widehat{\boldsymbol{\theta}})}=2%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
n_{ij}\log\frac{\pi_{ij}(\widetilde{\boldsymbol{\theta}})}{\pi_{ij}%
(\widehat{\boldsymbol{\theta}})}, \label{G 1}%
\end{equation}
is different from the one for the non order restricted alternative hypothesis
(two sided test, in $2\times2$ tables)%
\begin{equation}
\bar{G}^{2}=2n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
\overline{p}_{ij}\log\frac{\overline{p}_{ij}}{p_{ij}%
(\widehat{\boldsymbol{\theta}})}=2%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}/n_{i}}{\pi_{ij}(\widehat{\boldsymbol{\theta}})}=2%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}/n_{i}}{n_{\bullet j}/n}. \label{G bar}%
\end{equation}
For simplicity the case of $J=2$ is taken into account, where the (simple
null) one sided test%
\begin{equation}
H_{0}:\;\vartheta_{1}=1\text{,\qquad vs.}\qquad H_{1}:\;\vartheta
_{1}>1\text{,} \label{tt1}%
\end{equation}
with $\vartheta_{1}=\pi_{11}\pi_{22}/\pi_{21}\pi_{12}=\pi_{11}(1-\pi_{21}%
)/\pi_{21}(1-\pi_{11})$, or
\begin{subequations}
\[
H_{0}:\;\pi_{11}=\pi_{21}\text{,\qquad vs.}\qquad H_{1}:\;\pi_{11}>\pi
_{21}\text{,}%
\]
is tested with (\ref{G 1}), and on the other hand the two sided test
\end{subequations}
\begin{equation}
H_{0}:\;\vartheta_{1}=1\text{,\qquad vs.}\qquad H_{1}:\;\vartheta_{1}%
\neq1\text{,} \label{tt2b}%
\end{equation}
or
\begin{subequations}
\[
H_{0}:\;\pi_{11}=\pi_{21}\text{,\qquad vs.}\qquad H_{1}:\;\pi_{11}\neq\pi
_{21}\text{,}%
\]
is carried out with (\ref{G bar}). The same procedure would be possible to
perform for any $\phi$-divergence based test considered in this paper. We also
consider the mid-rank Wilcoxon test for both version of the alternative
hypothesis. To clarify the parameter space in both tests, we shall rewrite
(\ref{tt1}) and (\ref{tt2b}) as follows
\end{subequations}
\begin{subequations}
\[
H_{0}:\;\vartheta_{1}\in\Psi_{0}\text{,\qquad vs.}\qquad H_{1}:\;\vartheta
_{1}\in\Psi_{1}\text{,}%
\]
where $\Psi_{0}=\{1\}$, $\Psi_{1}=(1,+\infty)$,
\end{subequations}
\begin{subequations}
\[
H_{0}:\;\vartheta_{1}\in\Psi_{0}\text{,\qquad vs.}\qquad H_{1}^{\prime
}:\;\vartheta_{1}\in\Psi_{1}^{\prime}\text{,}%
\]
where $\Psi_{1}^{\prime}=(-\infty,1)\cup(1,+\infty)$. The parameter spaces for
(\ref{tt1}) and (\ref{tt2b})\ are $\Psi=\Psi_{0}\cup\Psi_{1}=[1,+\infty)$ and
$\Psi^{\prime}=\Psi_{0}\cup\Psi_{1}^{\prime}=%
\mathbb{R}
$, respectively. The same hypotheses in term of probabilities are given by
\end{subequations}
\begin{subequations}
\[
H_{0}:\;(\pi_{11},\pi_{21})\in\Lambda_{0}\text{,\qquad vs.}\qquad H_{1}%
:\;(\pi_{11},\pi_{21})\in\Lambda_{1}\text{,}%
\]
where $\Lambda_{0}=\left\{ (\pi_{11},\pi_{21})\in(0,1)\times(0,1):\pi
_{11}=\pi_{21}\right\} $, $\Lambda_{1}=\left\{ (\pi_{11},\pi_{21}%
)\in(0,1)\times(0,1):\pi_{11}>\pi_{21}\right\} $,
\end{subequations}
\begin{subequations}
\[
H_{0}:\;(\pi_{11},\pi_{21})\in\Lambda_{0}\text{,\qquad vs.}\qquad H_{1}%
:\;(\pi_{11},\pi_{21})\in\Lambda_{1}^{\prime}\text{,}%
\]
where $\Lambda_{1}^{\prime}=\left\{ (\pi_{11},\pi_{21})\in(0,1)\times
(0,1):\pi_{11}\neq\pi_{21}\right\} $. The corresponding parameter spaces in
term of probabilities are given by
\end{subequations}
\begin{align*}
\Lambda & =\Lambda_{0}\cup\Lambda_{1}=\left\{ (\pi_{11},\pi_{21}%
)\in(0,1)\times(0,1):\pi_{11}\geq\pi_{21}\right\} ,\\
\Lambda^{\prime} & =\Lambda_{0}\cup\Lambda_{1}^{\prime}=(0,1)\times(0,1).
\end{align*}
The likelihood ratio test-statistics for (\ref{tt1}) and (\ref{tt2b}) are
different since in the numerator of (\ref{G 1}), $\pi_{ij}%
(\widetilde{\boldsymbol{\theta}})$, is obtained maximizing the likelihood
function in $\Lambda$, while the numerator of (\ref{G bar}), $n_{ij}/n$, is
maximized in $\Lambda^{\prime}$.\ Even though both estimators are different,
in practice they require a similar computation:\newline$\bullet$ If $\bar{\pi
}_{11}=\frac{n_{11}}{n_{1}}>\bar{\pi}_{21}=\frac{n_{21}}{n_{2}}$, then
$\pi_{11}(\widetilde{\boldsymbol{\theta}})=\frac{n_{11}}{n_{1}}>\pi
_{21}(\widetilde{\boldsymbol{\theta}})=\frac{n_{21}}{n_{2}}$ and $G^{2}%
=\bar{G}^{2}=2%
{\textstyle\sum\nolimits_{i=1}^{2}}
{\textstyle\sum\nolimits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}/n_{i}}{n_{\bullet j}/n}$;\newline$\bullet$ If $\bar
{\pi}_{11}=\frac{n_{11}}{n_{1}}\leq\bar{\pi}_{21}=\frac{n_{21}}{n_{2}}$, then
$\pi_{11}(\widetilde{\boldsymbol{\theta}})=\pi_{11}%
(\widehat{\boldsymbol{\theta}})=\frac{n_{\bullet1}}{n}\leq\pi_{21}%
(\widetilde{\boldsymbol{\theta}})=\pi_{21}(\widehat{\boldsymbol{\theta}%
})=\frac{n_{\bullet1}}{n}$ and $G^{2}=0$.\newline Hence, taking into account
the asymptotic distributions, i.e. $\frac{1}{2}\chi_{0}^{2}+\frac{1}{2}%
\chi_{1}^{2}$ for (\ref{tt1}) and $\chi_{1}^{2}$\ for (\ref{tt2b}), we obtain%
\[
p\mathrm{-}value(G^{2})=\left\{
\begin{array}
[c]{ll}%
\frac{1}{2}\Pr\left( \chi_{1}^{2}>2%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}/n_{i}}{n_{\bullet j}/n}\right) , & \text{if }%
\frac{n_{11}}{n_{1}}>\frac{n_{21}}{n_{2}},\\
1, & \text{if }\frac{n_{11}}{n_{1}}\leq\frac{n_{21}}{n_{2}},
\end{array}
\right.
\]
and
\[
p\mathrm{-}value(\bar{G}^{2})=\Pr\left( \chi_{1}^{2}>2%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}}{n_{\bullet j}}\right) .
\]
A third test is the composite null one sided test%
\begin{align}
H_{0} & :\;\vartheta_{1}\leq1\text{,\quad(}\vartheta_{1}\in\Psi_{0}^{\prime
}\text{)\qquad vs.}\qquad H_{1}:\;\vartheta_{1}>1\text{,}\quad\text{(}%
\vartheta_{1}\in\Psi_{1}\text{)}\label{tt3}\\
H_{0} & :\;\pi_{11}\leq\pi_{21}\text{,}\quad\text{(}(\pi_{11},\pi_{21}%
)\in\Lambda_{0}^{\prime}\text{)\qquad vs.}\qquad H_{1}:\;\pi_{11}>\pi
_{21}\text{,}\quad\text{(}(\pi_{11},\pi_{21})\in\Lambda_{1}\text{),}\nonumber
\end{align}
with $\Psi_{0}^{\prime}=(-\infty,1]$ and $\Lambda_{0}^{\prime}=\left\{
(\pi_{11},\pi_{21})\in(0,1)\times(0,1):\pi_{11}\leq\pi_{21}\right\} $. For
the corresponding test-statistic,%
\begin{equation}
\widetilde{G}^{2}=2n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
\overline{p}_{ij}\log\frac{\overline{p}_{ij}}{p_{ij}%
(\widetilde{\boldsymbol{\theta}})}=2%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}/n_{i}}{\pi_{ij}(\widetilde{\boldsymbol{\theta}})}:
\label{G tilde}%
\end{equation}
$\bullet$ If $\bar{\pi}_{11}=\frac{n_{11}}{n_{1}}\geq\bar{\pi}_{21}%
=\frac{n_{21}}{n_{2}}$, then $\pi_{11}(\widetilde{\boldsymbol{\theta}}%
)=\frac{n_{\bullet1}}{n}\geq\pi_{12}(\widetilde{\boldsymbol{\theta}}%
)=\frac{n_{\bullet1}}{n}$ and $\widetilde{G}^{2}=2%
{\textstyle\sum\nolimits_{i=1}^{2}}
{\textstyle\sum\nolimits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}/n_{i}}{n_{\bullet j}/n}$;\newline$\bullet$ If $\bar
{\pi}_{11}=\frac{n_{11}}{n_{1}}<\bar{\pi}_{21}=\frac{n_{21}}{n_{2}}$, then
$\pi_{11}(\widetilde{\boldsymbol{\theta}})=\frac{n_{11}}{n_{1}}<\pi
_{21}(\widetilde{\boldsymbol{\theta}})=\frac{n_{21}}{n_{2}}$ and
$\widetilde{G}^{2}=0$.\newline Hence, both one sided test-statistics, the
composite null one, $\widetilde{G}^{2}$, and the simple null one, $G^{2}$, are
almost equal and
\[
p\mathrm{-}value(\widetilde{G}^{2})=\left\{
\begin{array}
[c]{ll}%
\frac{1}{2}\Pr\left( \chi_{1}^{2}>2%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{2}}
n_{ij}\log\frac{n_{ij}/n_{i}}{n_{\bullet j}/n}\right) , & \text{if }%
\frac{n_{11}}{n_{1}}\geq\frac{n_{21}}{n_{2}},\\
1, & \text{if }\frac{n_{11}}{n_{1}}<\frac{n_{21}}{n_{2}}.
\end{array}
\right.
\]
The mid-rank $W$ test-statistic for (\ref{tt1}) and (\ref{tt2b}) is the same,
(\ref{wilc}), as well as the distribution under the null, but%
\[
p\mathrm{-}value(W)=\Pr\left( \mathcal{N}(0,1)<-\frac{\left( r_{1}%
n_{11}+r_{2}n_{12}\right) -\left. n_{1}\left( n+1\right) \right/ 2}%
{\sqrt{n_{1}n_{2}\frac{n+1-\frac{1}{n(n-1)}\left[ \left( (n_{\bullet1}%
^{3}-n_{\bullet1})\right) +\left( (n_{\bullet2}^{3}-n_{\bullet2})\right)
\right] }{12}}}\right)
\]
for (\ref{tt1}) and
\[
p\mathrm{-}value(W)=2\Pr\left( \mathcal{N}(0,1)>\frac{\left\vert \left(
r_{1}n_{11}+r_{2}n_{12}\right) -\left. n_{1}\left( n+1\right) \right/
2\right\vert }{\sqrt{n_{1}n_{2}\frac{n+1-\frac{1}{n(n-1)}\left[ \left(
(n_{\bullet1}^{3}-n_{\bullet1})\right) +\left( (n_{\bullet2}^{3}%
-n_{\bullet2})\right) \right] }{12}}}\right)
\]
\ for (\ref{tt2b}).%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{c}%
{\includegraphics[
height=2.8764in,
width=5.6273in
]%
{T1.pdf}%
}
\\%
{\includegraphics[
height=2.8764in,
width=5.6273in
]%
{T2.pdf}%
}
\\%
{\includegraphics[
height=2.8764in,
width=5.6273in
]%
{W.pdf}%
}
\end{tabular}
\caption{Histograms of $G^{2}$, $\bar{G}^{2}$ and $W$ with $n_1=40$, $n_2=20$ and $\pi _{i1}=0.35$, $i=1,2$. \label{figHH}}%
\end{figure}%
The following short simulation study considers $R=100,000$\ realizations,
$n_{i1}^{(h)}$, $i=1,2$, $h=1,...,R$, of%
\[
N_{i1}\overset{ind}{\sim}\mathcal{B}in(n_{i},\pi_{i1}),\qquad i=1,2,
\]
with $\pi_{11}=\pi_{21}=0.35$ and $n_{1}=40$ and $n_{2}=20$. In Figure
\ref{figHH} a histogram of $G^{2}$, $\bar{G}^{2}$ and $W$\ is shown where the
shape of the density function of each can be recognized. In Table \ref{ttHH},
the simulated significance levels ($\widehat{\alpha}$) and powers
($\widehat{\beta}$) are calculated as the proportion of statistics with
$p$-values smaller than the nominal level $\alpha=0.05$. The test-statistic
based on the Hellinger distance $S_{-1/2}$, given in (\ref{hel}), is also
included. From this simulation study it is concluded that the $G^{2}$
likelihood ratio test-statistic and the $W$\ Wilcoxon mid-rank test for
$2\times2$ contingency tables, are specific procedures for the one sided test
(\ref{tt1}) since the parameter spaces are different, but are strongly related
with the two sided test (\ref{tt2b}) in the way of calculating the value of
the test-statistic and the corresponding $p$-value. It is remarkable that the
simulated significance level for the one-sided $W$\ Wilcoxon mid-rank test for
$2\times2$ contingency tables exhibits a slightly better approximation of the
nominal level\ in comparison with the likelihood ratio test $G^{2}$ for the
one sided test (\ref{tt1}), and the likelihood ratio test $G^{2}$ slightly
better than the test-statistic based on the Hellinger distance $S_{-1/2}$. The
powers of the test-statistics are calculated for $\pi_{11}=0.45>\pi_{21}%
=0.35$. The test-statistic based on the Hellinger distance $S_{-1/2}$ has the
greatest power and the $W$\ Wilcoxon mid-rank test the smallest power for the
one sided test (\ref{tt1}). In Section \ref{Sim} a more extensive simulation
study is considered with a criterion to select the best test-statistic within
a broader class of power divergence based test-statistics. Finally, the two
sided test-statistics, $\bar{G}^{2}$ and $W$, exhibit a worse power than the
one sided test-statistics. This behaviour was obviously expected, since being
$\Psi\subset\Psi^{\prime}$ or equivalently $\Lambda\subset\Lambda^{\prime}$,
the one sided tests have always a better power than the two sided tests.%
\begin{table}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cccccccccccc}\hline
& & $S_{-1/2}$ (one sided) & & $G^{2}$ (one sided) & & $\bar{G}^{2}$ (two
sided) & & one sided $W$ & & two sided $W$ & \\\hline
$\widehat{\alpha}$ & & $0.0567$ & & $0.0559$ & & $0.0533$ & & $0.0495$ &
& $0.0489$ & \\
$\widehat{\beta}$ & & $0.2027$ & & $0.2025$ & & $0.1186$ & & $0.1865$ & &
$0.1149$ & \\\hline
\end{tabular}
\caption{Simulated significance levels ($\pi _{11}=\pi _{21}=0.35$), $\widehat{\alpha
}$, and powers ($\pi _{11}=0.45$, $\pi _{21}=0.35$), $\widehat{\beta }$, for
$S_{-1/2}$, $G^{2}$, $\bar{G}^{2}$ and $W$ test-statistics with $n_{1}=40>n_{2}=20$.\label{ttHH}}%
\end{table}%
\subsection{Power divergence test-statistics: simulated size and
powers\label{Sim}}
In this Section the performance of the power divergence test statistics
(\ref{PD1})-(\ref{PD6}) is studied in terms of the simulated exact size and
simulated power of the test, based on small and moderate sample sizes. A
simulation experiment with seven scenarios is designed in Table \ref{tt},
taking into account the sample sizes of the two independent samples. The pairs
of scenarios (A,G), (B,F) and (C,E) should have very similar exact
significance levels, since the sample sizes of the two samples are symmetrical
(the ratio of one sample is the inverse of the other one). With respect to the
choice of $\lambda$, the parameters for the power divergence test statistics,
the interest is focused on the interval $[-1.5,3]$. Note that the
test-statistics applied in the numerical example are covered as particular cases.%
\begin{table}[htbp] \centering
$%
\begin{tabular}
[c]{cccccccc}\hline
scenarios & sc. A & sc. B & sc. C & sc. D & sc. E & sc. F & sc. G\\\hline
$n_{1}$ & $20$ & $20$ & $20$ & $20$ & $16$ & $10$ & $4$\\
$n_{2}$ & $4$ & $10$ & $16$ & $20$ & $20$ & $20$ & $20$\\\hline
ratio & $5$ & $2$ & $1.25$ & $1$ & $0.8$ & $0.5$ & $0.2$\\\hline
\end{tabular}
\ \ \ \ \ \ \ \ \ $%
\caption{Scenarios, based on sample sizes, for the simulation stydy in a contingency table $2\times 3$.\label{tt}}
\end{table}%
\bigskip
The algorithm described in Section \ref{sec:Numerical example} is taken into
account to calculate the $p$-value of each test-statistic ${T\in\{T_{\lambda
},S_{\lambda}\}}_{\lambda\in\lbrack-1.5,3]}$, with a sample $\boldsymbol{N}$,
and this is repeated independently $R=25\,000$ times. The simulated exact
power was computed as%
\[
{\widehat{\beta}}_{T}={\widehat{\beta}}_{T}(\delta)=\frac{\text{number of
replications of }T\,\text{for which the }p\text{-value is less than }\alpha
}{R},
\]
for the probability vectors%
\begin{align*}
\boldsymbol{\pi}_{i}(\boldsymbol{\theta}(\delta)) & =(\pi_{i1}%
(\boldsymbol{\theta}(\delta)),\pi_{i2}(\boldsymbol{\theta}(\delta)),\pi
_{i3}(\boldsymbol{\theta}(\delta)))^{T}\\
\pi_{ij}(\boldsymbol{\theta}(\delta)) & =\frac{1}{3}\frac{1+i(j-1)\delta
}{1+i\delta},\quad i=1,2,\quad j=1,2,3,
\end{align*}
for $\delta\in\Xi=\{0.1,0.5,1.0,1.5\}$. The simulated exact size was computed
as%
\[
{\widehat{\alpha}}_{T}=\frac{\text{number of replications of }T\,\text{for
which the }p\text{-value is less than }\alpha}{R},
\]
for the probability vectors
\begin{align*}
\boldsymbol{\pi}_{i}(\boldsymbol{\theta}_{0}) & =(\pi_{i1}%
(\boldsymbol{\theta}_{0}),\pi_{i2}(\boldsymbol{\theta}_{0}),\pi_{i3}%
(\boldsymbol{\theta}_{0}))^{T}\\
\pi_{ij}(\boldsymbol{\theta}_{0}) & =\frac{1}{3},\quad i=1,2,\quad j=1,2,3,
\end{align*}
which corresponds to the case of $\delta=0$ for $\boldsymbol{\pi}%
_{i}(\boldsymbol{\theta}(\delta))$.
In Table \ref{tt2} the local odds ratios,%
\[
\vartheta_{j}=\vartheta_{j}(\delta)=\frac{1+(j-1)\delta}{1+2(j-1)\delta}%
\frac{1+2j\delta}{1+j\delta},
\]
$j=1,2$, are shown for $\delta\in\{0\}\cup\Xi$. Notice that in
$\boldsymbol{\vartheta}=\boldsymbol{\vartheta}(\delta)=(\vartheta_{1}%
(\delta),\vartheta_{2}(\delta))^{T}$\ some of the components are further from
$\boldsymbol{\vartheta}(0)=\boldsymbol{1}_{2}$ (null hypothesis), as the value
of $\delta>0$ is further from $0$. This means that a greater value of the
estimation of the power function might be obtained, as $\delta>0$ is greater.
This claim is supported by the fact that some values of the components of
$\boldsymbol{\vartheta}=\boldsymbol{\vartheta}(\delta)$ decrease as $\delta>0$
increases but more slowly than the others increase. In addition, for a fixed
value of $\delta>0$, it is expected a greater value of $\widehat{\beta}%
_{T}(\delta)$, as $n$ is greater (the worst powers in Scenario A and the best
powers in Scenario D). We have also added in Table \ref{tt2}\ the last three
rows for two reasons, first, to show that for any fixed value of $\delta$,
$\pi_{2j}(\boldsymbol{\theta}(\delta))/\pi_{1j}(\boldsymbol{\theta}(\delta))$
is non-decreasing as $j$, the ordinal category, increases and second, to
clarify the meaning of the two asterisks contained in the table. It is clear
that for a big value of $\delta$, $\pi_{i1}(\boldsymbol{\theta}(\delta))>0$
goes to zero on the right for $i=1,2$, but in the practice, due to the empty
cells in the contingency table, the estimator of the ratio $\pi_{21}%
(\boldsymbol{\theta}(\delta))/\pi_{11}(\boldsymbol{\theta}(\delta))$ becomes
$1$ rather than $\frac{1}{2}$ (and $\vartheta_{1}(\delta)$ becomes $1$). This
was our experience when we used values of $\delta$ bigger than $1.5$, i.e. the
power becomes quite little in the practice.%
\begin{table}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{ccccccccccccc}\hline
& & $\delta=0$ & & $\delta=0.1$ & & $\delta=0.5$ & & $\delta=1$ & &
$\delta=1.5$ & & $\delta=\infty$\\\hline
$\vartheta_{1}=\vartheta_{1}(\delta)$ & & $1.000$ & & $1.091$ & & $1.333$ &
& $1.500$ & & $1.600$ & & $2.00^{\ast}$\\
$\vartheta_{2}=\vartheta_{2}(\delta)$ & & $1.000$ & & $1.069$ & & $1.125$ &
& $1.111$ & & $1.094$ & & $1.00$\\\hline
$\pi_{21}(\boldsymbol{\theta}(\delta))/\pi_{11}(\boldsymbol{\theta}(\delta))$
& & $0.33/0.33$ & & $0.28/0.30$ & & $0.17/0.22$ & & $0.11/0.17$ & &
$0.08/0.13$ & & $0.50^{\ast}$\\
$\pi_{22}(\boldsymbol{\theta}(\delta))/\pi_{12}(\boldsymbol{\theta}(\delta))$
& & $0.33/0.33$ & & $0.33/0.33$ & & $0.33/0.33$ & & $0.33/0.33$ & &
$0.33/0.33$ & & $1.00$\\\hline
\end{tabular}
\caption{Theoretical local odd ratios for the Monte Carlo study.\label{tt2}}%
\end{table}%
Once a nominal size $\alpha=0.05$ is established, Table \ref{alfas} summarizes
the simulated exact sizes in all the scenarios for the test-statistic
${T\in\{T_{\lambda},S_{\lambda},W\}}_{\lambda\in\Lambda}$, with $\Lambda
=\{-1.5,-1,-\frac{1}{2},0,\frac{2}{3},1,1.5,2,3\}$. We have plotted $3\times2$
graphs in Figures \ref{fig2}-\ref{fig7} and we refer them as plots in three
rows. In the first row of Figures \ref{fig1}-\ref{fig7} we can see on the left
the exact power in all the scenarios for the test-statistic $\{{T_{\lambda
},W\}}_{\lambda\in\lbrack-1.5,3]}$ and on the right for the test-statistic
$\{{S_{\lambda},W\}}_{\lambda\in\lbrack-1.5,3]}$. In order to make a
comparison of exact powers, we cannot directly proceed without considering the
exact sizes. For this reason we are going to give a procedure based on two
steps, for scenarios B-G.
\noindent\textit{Step 1}: We are going to check for all the power divergence
based test-statistics the criterion given by Dale (1986), i.e.,
\begin{equation}
|\,\text{logit}(1-{\widehat{\alpha}}_{T})-\text{logit}(1-\alpha)\,|\leq e
\label{con1}%
\end{equation}
with $\mathrm{logit}\left( p\right) =\log\left( \frac{p}{1-p}\right) $. We
only consider the values of $\lambda$\ such that ${\widehat{\alpha}}_{T}%
$\ satisfies (\ref{con1}) with $e=0.35$, then we shall only consider the
test-statistics such that ${\widehat{\alpha}}_{T}\in\left[
0.0357,0.0695\right] $, in all the scenarios. This criterion has been
considered for some authors, see for instance Cressie et al. (2003) and
Mart\'{\i}n and Pardo (2012). The cases satisfying the criterion are marked in
bold in Table \ref{alfas}, and comprise those values in the abscissa of the
plot between the dashed band (the dashed line in the middle represents the
nominal size), and we can conclude that we must not consider in our study
${T\in\{T_{\lambda},S_{\lambda},W\}}_{\lambda\in\lbrack-1.5,-0.4)}$.
\noindent\textit{Step 2}: We compare all the test statistics obtained in Step
1 with the classical likelihood ratio test ($G^{2}=T_{0}$) as well as the
classical Pearson test statistic ($X^{2}=S_{1}$). To do so, we have calculated
the relative local efficiencies%
\[
\widehat{\rho}_{T}=\widehat{\rho}_{T}(\delta)=\frac{({\widehat{\beta}}%
_{T}(\delta)-{\widehat{\alpha}}_{T})-({\widehat{\beta}}_{T_{0}}(\delta
)-{\widehat{\alpha}}_{T_{0}})}{{\widehat{\beta}}_{T_{0}}(\delta
)-{\widehat{\alpha}}_{T_{0}}},\qquad\widehat{\rho}_{T}^{\ast}=\widehat{\rho
}_{T}^{\ast}(\delta)=\frac{({\widehat{\beta}}_{T}(\delta)-{\widehat{\alpha}%
}_{T})-({\widehat{\beta}}_{S_{1}}(\delta)-{\widehat{\alpha}}_{S_{1}}%
)}{{\widehat{\beta}}_{S_{1}}(\delta)-{\widehat{\alpha}}_{S_{1}}}.
\]
In Figures \ref{fig2}-\ref{fig7} the powers and the relative local
efficiencies are summarized. The second rows of the figures represent
$\widehat{\rho}_{T}$, while in the third row is plotted $\widehat{\rho}%
_{T}^{\ast}$, on the left it is considered ${T\in}\{{T_{\lambda},W\}}%
_{\lambda\in\lbrack-1.5,3]}$ and ${T\in}\{{S_{\lambda},W\}}_{\lambda\in
\lbrack-1.5,3]}$ on the right. In Figure \ref{fig1} we show only one row since
it represents the atypical case in which the exact powers are less that the
exact significance level for the values of $\lambda$ satisfying the Dale's
criterion and so, it does not make sense to compare the powers.%
\begin{table}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{l}%
$%
\begin{tabular}
[c]{ccccccccccc}\hline\hline
sc & ${\widehat{\alpha}}_{T_{-1.5}}$ & ${\widehat{\alpha}}_{T_{-1}}$ &
${\widehat{\alpha}}_{T_{-1/2}}$ & ${\widehat{\alpha}}_{T_{0}}$ &
${\widehat{\alpha}}_{T_{2/3}}$ & ${\widehat{\alpha}}_{T_{1}}$ &
${\widehat{\alpha}}_{T_{1.5}}$ & ${\widehat{\alpha}}_{T_{2}}$ &
${\widehat{\alpha}}_{T_{3}}$ & ${\widehat{\alpha}}_{W}$\\\hline
$A$ & 0.0013 & 0.0359 & 0.1725 & 0.0745 & \textbf{0.0468} & \textbf{0.0460} &
\textbf{0.0517} & \textbf{0.0586} & 0.0949 & \textbf{0.0509}\\
$B$ & \textbf{0.0670} & \textbf{0.0612} & \textbf{0.0664} & \textbf{0.0597} &
\textbf{0.0541} & \textbf{0.0503} & \textbf{0.0511} & \textbf{0.0536} &
\textbf{0.0619} & \textbf{0.0509}\\
$C$ & 0.0747 & \textbf{0.0686} & \textbf{0.0608} & \textbf{0.0537} &
\textbf{0.0494} & \textbf{0.0485} & \textbf{0.0478} & \textbf{0.0492} &
\textbf{0.0573} & \textbf{0.0485}\\
$D$ & \textbf{0.0688} & \textbf{0.0653} & \textbf{0.0631} & \textbf{0.0577} &
\textbf{0.0538} & \textbf{0.0528} & \textbf{0.0522} & \textbf{0.0530} &
\textbf{0.0572} & \textbf{0.0495}\\
$E$ & 0.0751 & \textbf{0.0691} & \textbf{0.0610} & \textbf{0.0548} &
\textbf{0.0511} & \textbf{0.0502} & \textbf{0.0494} & \textbf{0.0509} &
\textbf{0.0591} & \textbf{0.0512}\\
$F$ & \textbf{0.0665} & \textbf{0.0614} & \textbf{0.0681} & \textbf{0.0616} &
\textbf{0.0554} & \textbf{0.0518} & \textbf{0.0517} & \textbf{0.0539} &
\textbf{0.0615} & \textbf{0.0506}\\
$G$ & 0.0013 & 0.0363 & 0.1802 & 0.0775 & \textbf{0.0477} & \textbf{0.0466} &
\textbf{0.0526} & \textbf{0.0602} & 0.0965 & \textbf{0.0541}\\\hline\hline
\end{tabular}
\ \ $\\
$%
\begin{tabular}
[c]{ccccccccccc}\hline\hline
sc & ${\widehat{\alpha}}_{S_{-1.5}}$ & ${\widehat{\alpha}}_{S_{-1}}$ &
${\widehat{\alpha}}_{S_{-1/2}}$ & ${\widehat{\alpha}}_{S_{0}}$ &
${\widehat{\alpha}}_{S_{2/3}}$ & ${\widehat{\alpha}}_{S_{1}}$ &
${\widehat{\alpha}}_{S_{1.5}}$ & ${\widehat{\alpha}}_{S_{2}}$ &
${\widehat{\alpha}}_{S_{3}}$ & ${\widehat{\alpha}}_{W}$\\\hline
$A$ & 0.2106 & 0.2055 & 0.1572 & 0.0745 & \textbf{0.0429} & \textbf{0.0430} &
\textbf{0.0499} & \textbf{0.0507} & 0.0752 & \textbf{0.0509}\\
$B$ & 0.0799 & 0.0762 & \textbf{0.0638} & \textbf{0.0596} & \textbf{0.0543} &
\textbf{0.0497} & \textbf{0.0509} & \textbf{0.0524} & \textbf{0.0584} &
\textbf{0.0509}\\
$C$ & 0.0729 & \textbf{0.0676} & \textbf{0.0581} & \textbf{0.0537} &
\textbf{0.0505} & \textbf{0.0492} & \textbf{0.0491} & \textbf{0.0501} &
\textbf{0.0583} & \textbf{0.0485}\\
$D$ & \textbf{0.0675} & \textbf{0.0656} & \textbf{0.0620} & \textbf{0.0577} &
\textbf{0.0552} & \textbf{0.0543} & \textbf{0.0541} & \textbf{0.0543} &
\textbf{0.0577} & \textbf{0.0495}\\
$E$ & 0.0745 & \textbf{0.0683} & \textbf{0.0584} & \textbf{0.0547} &
\textbf{0.0518} & \textbf{0.0507} & \textbf{0.0504} & \textbf{0.0515} &
\textbf{0.0598} & \textbf{0.0512}\\
$F$ & 0.0814 & 0.0780 & \textbf{0.0656} & \textbf{0.0616} & \textbf{0.0551} &
\textbf{0.0509} & \textbf{0.0516} & \textbf{0.0528} & \textbf{0.0572} &
\textbf{0.0506}\\
$G$ & 0.2170 & 0.2123 & 0.1653 & 0.0775 & \textbf{0.0446} & \textbf{0.0450} &
\textbf{0.0510} & \textbf{0.0516} & 0.0782 & \textbf{0.0541}\\\hline\hline
\end{tabular}
\ \ \ $%
\end{tabular}
\caption{${\widehat{\alpha}}_{T}$, for ${T\in\{T_{\lambda},S_{\lambda},W\}}_{\lambda\in\Lambda}$ in scenarios of Table \ref{tt}. \label{alfas}}%
\end{table}%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cc}%
${T_{\lambda}}$ & ${S_{\lambda}}$\\%
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscA_Potencia_T.pdf}%
}
&
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscA_Potencia_S.pdf}%
}
\end{tabular}
\caption{Powers for $T_{\lambda}$, $S_{\lambda}$ and $W$ in scenario A. \label{fig1}}%
\end{figure}%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cc}%
${T_{\lambda}}$ & ${S_{\lambda}}$\\%
{\includegraphics[
height=2.4561in,
width=3.1202in
]%
{EscB_Potencia_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.1202in
]%
{EscB_Potencia_S.pdf}%
}
\\%
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscB_Eficiencias_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscB_Eficiencias_S.pdf}%
}
\\%
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscB_Eficiencias_asterisco_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscB_Eficiencias_asterisco_S.pdf}%
}
\end{tabular}
\caption{Power and relative local efficiencies for $T_{\lambda}$, $S_{\lambda}$ and $W$ in scenario B. \label{fig2}}%
\end{figure}%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cc}%
${T_{\lambda}}$ & ${S_{\lambda}}$\\%
{\includegraphics[
height=2.4561in,
width=3.1202in
]%
{EscC_Potencia_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.1202in
]%
{EscC_Potencia_S.pdf}%
}
\\%
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscC_Eficiencias_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscC_Eficiencias_S.pdf}%
}
\\%
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscC_Eficiencias_asterisco_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscC_Eficiencias_asterisco_S.pdf}%
}
\end{tabular}
\caption{Power and relative local efficiencies for $T_{\lambda}$, $S_{\lambda}$ and $W$ in scenario C. \label{fig3}}%
\end{figure}%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cc}%
${T_{\lambda}}$ & ${S_{\lambda}}$\\%
{\includegraphics[
height=2.4561in,
width=3.1202in
]%
{EscD_Potencia_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.1202in
]%
{EscD_Potencia_S.pdf}%
}
\\%
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscD_Eficiencias_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscD_Eficiencias_S.pdf}%
}
\\%
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscD_Eficiencias_asterisco_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscD_Eficiencias_asterisco_S.pdf}%
}
\end{tabular}
\caption{Power and relative local efficiencies for $T_{\lambda}$, $S_{\lambda}$ and $W$ in scenario D. \label{fig4}}%
\end{figure}%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cc}%
${T_{\lambda}}$ & ${S_{\lambda}}$\\%
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscE_Potencia_T.pdf}%
}
&
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscE_Potencia_S.pdf}%
}
\\%
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscE_Eficiencias_T.pdf}%
}
&
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscE_Eficiencias_S.pdf}%
}
\\%
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscE_Eficiencias_asterisco_T.pdf}%
}
&
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscE_Eficiencias_asterisco_S.pdf}%
}
\end{tabular}
\caption{Power and relative local efficiencies for $T_{\lambda}$, $S_{\lambda}$ and $W$ in scenario E. \label{fig5}}%
\end{figure}%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cc}%
${T_{\lambda}}$ & ${S_{\lambda}}$\\%
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscF_Potencia_T.pdf}%
}
&
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscF_Potencia_S.pdf}%
}
\\%
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscF_Eficiencias_T.pdf}%
}
&
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscF_Eficiencias_S.pdf}%
}
\\%
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscF_Eficiencias_asterisco_T.pdf}%
}
&
{\includegraphics[
height=2.463in,
width=3.3667in
]%
{EscF_Eficiencias_asterisco_S.pdf}%
}
\end{tabular}
\caption{Power and relative local efficiencies for $T_{\lambda}$, $S_{\lambda}$ and $W$ in scenario F. \label{fig6}}%
\end{figure}%
\begin{figure}[htbp] \tabcolsep2.8pt \centering
\begin{tabular}
[c]{cc}%
${T_{\lambda}}$ & ${S_{\lambda}}$\\%
{\includegraphics[
height=2.4561in,
width=3.1202in
]%
{EscG_Potencia_T.pdf}%
}
&
{\includegraphics[
height=2.4552in,
width=3.1211in
]%
{EscG_Potencia_S.pdf}%
}
\\%
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscG_Eficiencias_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscG_Eficiencias_S.pdf}%
}
\\%
{\includegraphics[
height=2.4552in,
width=3.0701in
]%
{EscG_Eficiencias_asterisco_T.pdf}%
}
&
{\includegraphics[
height=2.4561in,
width=3.0701in
]%
{EscG_Eficiencias_asterisco_S.pdf}%
}
\end{tabular}
\caption{Power and relative local efficiencies for $T_{\lambda}$, $S_{\lambda}$ and $W$ in scenario G. \label{fig7}}%
\end{figure}%
\pagebreak
The plots are interpreted as follows:\medskip\newline\textbf{a)} In all the
scenarios a similar pattern is observed when plotting the exact power,
${\widehat{\beta}}_{T}$, for $\lambda\in\lbrack-1,3]$ since a U shaped curve
is obtained. This means that the exact power is higher in the corners of the
interval in comparison with the classical likelihood ratio test ($G^{2}=T_{0}%
$) as well as the classical Pearson test statistic ($X^{2}=S_{1}$), contained
in the middle.\medskip\newline\textbf{b)} If we pay attention on the local
efficiencies with respect to $G^{2}$ and $X^{2}$, $\widehat{\rho}_{T}$ and
$\widehat{\rho}_{T}^{\ast}$, to find positive values of them we need to
consider $\lambda\in\lbrack-1,0)$ or $\lambda\in(1,3]$ and thus it confirms
what was said in a). On the other hand, comparing the left hand
($T={T_{\lambda}}$) side of $\widehat{\rho}_{T}$ with the right side
($T={S_{\lambda}}$) and doing the same for $\widehat{\rho}_{T}^{\ast}$, a
slightly higher values of the local efficiencies of ${S_{\lambda}}$ are seen
in comparison with ${T_{\lambda}}$. For this reason we consider that
${\{S_{\lambda}\}}_{\lambda\in\lbrack-1,0)}$ have a better performance than
the classical test-statistics, $G^{2}$ and $X^{2}$ in scenarios B-E and
${\{S_{\lambda}\}}_{\lambda\in(1,3]}$ have a better performance than the
classical test-statistics, $G^{2}$ and $X^{2}$ in scenarios F-G. The Wilcoxon
test-statistic has in all the scenarios worse performance with respect to the
best classical asymptotic statistic, $G^{2}$ for scenarios B-E and $X^{2}$ for
scenarios F-G.\medskip\newline\textbf{c)} What is not so common in comparison
with usual models of categorical data is to find small size sample sizes with
so good performance in exact size as it happens in the case of the likelihood
ratio order. Moreover, the best test-statistic are not very common to be
selected as those with better performance than the classical ones.\newpage
\section{Concluding remark}
The likelihood ratio ordering is a useful technique for comparing treatments
in clinical trials, for this reason it is vitally important to provide
test-statistics to improve the classical ones. Having considered an asymptotic
distribution for two order restricted treatments, the weights needed to manage
the associated asymptotic chi-bar distribution are calculated in a simple way
and the useful matrix for that, $\boldsymbol{H}(\widehat{\boldsymbol{\theta}%
})$, has an easy interpretation in terms of log-linear modeling. The
simulation study highlights the good performance of the all the proposed tests
in relation to the exact size and the comparison is made in terms of the
power. For small and moderate sample sizes there are better choices than the
likelihood ratio test and the Wilcoxon test-statistics inside the family of
$\phi$-divergences. We think that this is a specific characteristic of the
likelihood ordering, and this is the reason of having obtained as the best
test-statistics a set of values of $\lambda\in\lbrack-1,0)\cup(1,3]$ not very
common in the literature of phi-divergence test-statistics. As exception,
notice that%
\begin{align}
S_{-1/2} & =S_{d_{\phi_{-1/2}}}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta
}}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}}))=8n\left( 1-%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
p_{ij}^{\frac{1}{2}}(\widetilde{\boldsymbol{\theta}})p_{ij}^{\frac{1}{2}%
}(\widehat{\boldsymbol{\theta}})\right) \label{hel}\\
& =4n%
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\left( p_{ij}^{\frac{1}{2}}(\widetilde{\boldsymbol{\theta}})-p_{ij}^{\frac
{1}{2}}(\widehat{\boldsymbol{\theta}})\right) ^{2}\nonumber\\
& =4n\mathrm{Hel}^{2}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}%
}),\boldsymbol{p}(\widehat{\boldsymbol{\theta}})),\nonumber
\end{align}
where%
\[
\mathrm{Hel}(\boldsymbol{p}(\widetilde{\boldsymbol{\theta}}),\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}}))=\left(
{\displaystyle\sum\limits_{i=1}^{2}}
{\displaystyle\sum\limits_{j=1}^{J}}
\left( p_{ij}^{\frac{1}{2}}(\widetilde{\boldsymbol{\theta}})-p_{ij}^{\frac
{1}{2}}(\widehat{\boldsymbol{\theta}})\right) ^{2}\right) ^{\frac{1}{2}},
\]
is the Hellinger distance between the probability vectors $\boldsymbol{p}%
(\widetilde{\boldsymbol{\theta}})$\ and $\boldsymbol{p}%
(\widehat{\boldsymbol{\theta}})$. Therefore, one of the test-statistic we are
proposing in this paper is a function of the well-known Hellinger distance,
which has been used in many different statistical problems. We think that the
reason why this happens is related to the robust properties of such a
test-statistic, since when dealing with the likelihood ratio ordering, under
the alternative hypothesis, on the left side of the contingency table empty
cells tend to appear. In particular, the theoretical probability in the first
cell for the second treatment, $\pi_{21}$, is the smallest one and this
circumstance does influence in the results obtained for skew sample sample
sizes in both treatments.
\begin{acknowledgement}
The authors acknowledge the referee. We modified and improved the manuscript
according to comments and questions pointed by the referee.
\end{acknowledgement}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,955
|
Le White Ensign (appelé aussi Saint George's Ensign ; Pavillon de Saint-Georges en français) est le pavillon de la marine militaire britannique, appelée la Royal Navy. Il est composé d'une croix de Saint-Georges rouge sur fond blanc avec l'Union Jack dans le canton supérieur à l'attache. Ce pavillon se bat sur les bateaux de la Royal Navy et aux bases navales opérées par celle-ci. Le Royal Yacht Squadron et les vaisseaux accompagnant le roi l'arborent également.
Sur la terre, pour représenter les marins tués dans les guerres, un White Ensign est toujours présent sur le Cénotaphe à Londres. Le pavillon peut être aussi hissé sur l'église St Martin-in-the-Fields dans la même ville, parce qu'elle est l'église de l'Amirauté. L'Arche de l'Amirauté est normalement décoré avec les White Ensigns aux occasions cérémonielles, comme la fête de Saint-Georges ou l'anniversaire de la Bataille de Trafalgar.
Histoire
Avant la réorganisation de la Royal Navy, en 1864, le White Ensign était le pavillon d'une des trois escadres de la Royal Navy.
Cela changea en 1864, quand un Décret du Conseil attribua le Red Ensign à la marine marchande, le Blue Ensign aux navires de service public ou commandés par un officier réserviste de la Royal Navy, et le White Ensign à la Navy.
Les autres White Ensign
De même que le Royaume-Uni, plusieurs autres nations généralement membres du Commonwealth utilisent le White Ensign en remplaçant l'Union Jack du canton par leur propre drapeau et en supprimant parfois la croix de Saint-Georges.
Notes et références
Notes
Références
Drapeau du Royaume-Uni
Royal Navy
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,320
|
This is a Community-supported project.
If you are interested in becoming a maintainer of this project, please contact us at integrations@bitpay.com. Developers at BitPay will attempt to work along the new maintainers to ensure the project remains viable for the foreseeable future.
# Description
Bitcoin payment plugin for XCart using the bitpay.com service.
## Quick Start Guide
To get up and running with our plugin quickly, see the GUIDE here: https://github.com/bitpay/xcart-plugin/blob/master/GUIDE.md
## Support
**BitPay Support:**
* [BitPay Labs](https://labs.bitpay.com/c/plugins/xcart)
* Post a question in our discussion forums
* [GitHub Issues](https://github.com/bitpay/xcart-plugin/issues)
* Open an issue if you are having issues with this plugin
* [Support](https://support.bitpay.com)
* BitPay merchant support documentation
**XCart Support:**
* [Homepage](http://www.x-cart.com/)
* [SupportForums](http://www.x-cart.com/help.html)
## Troubleshooting
0. Sometimes a download can become corrupted for various reasons. However, you can verify that the release package you downloaded is correct by checking the md5 checksum "fingerprint" of your download against the md5 checksum value shown on the Releases page. Even the smallest change in the downloaded release package will cause a different value to be shown!
* If you are using Windows, you can download a checksum verifier tool and instructions directly from Microsoft here: http://www.microsoft.com/en-us/download/details.aspx?id=11533
* If you are using Linux or OS X, you already have the software installed on your system.
* On Linux systems use the md5sum program. For example:
* md5sum filename
* On OS X use the md5 program. For example:
* md5 filename
1. Ensure a valid SSL certificate is installed on your server. Also ensure your root CA cert is updated. If your CA cert is not current, you will see curl SSL verification errors.
2. Verify that your web server is not blocking POSTs from servers it may not recognize. Double check this on your firewall as well, if one is being used.
3. Check the version of this plugin against the official plugin repository to ensure you are using the latest version. Your issue might have been addressed in a newer version! See the [Releases](https://github.com/bitpay/xcart-plugin/releases/latest) page for the latest.
4. If all else fails, contact us using one of the methods described in the Support section above.
**TIP**: When contacting support it will help us is you provide:
* XCart and BitPay Plugin Version
* Other plugins you have installed
* Some plugins do not play nice
* Configuration settings for the plugin (Most merchants take screen grabs)
* Any log files that will help
* Web server error logs
* Screen grabs of error message if applicable.
## Contribute
Would you like to help with this project? Great! You don't have to be a developer, either. If you've found a bug or have an idea for an improvement, please open an [issue](https://github.com/bitpay/xcart-plugin/issues) and tell us about it.
If you *are* a developer wanting contribute an enhancement, bugfix or other patch to this project, please fork this repository and submit a pull request detailing your changes. We review all PRs!
This open source project is released under the [MIT license](http://opensource.org/licenses/MIT) which means if you would like to use this project's code in your own project you are free to do so. Speaking of, if you have used our code in a cool new project we would like to hear about it! Please send us an email or post a new thread on [BitPay Labs](https://labs.bitpay.com).
## License
Please refer to the [LICENSE](https://github.com/bitpay/xcart-plugin/blob/master/LICENSE) file that came with this project.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 265
|
Nathan Schneider
University of Colorado Boulder, United States
Arild Bergh
Norwegian Defence Research Establishment, Norway
Uta Kohl
University of Southampton, United Kingdom
Methods and Anticipated Results
Front. Hum. Dyn., 26 April 2021
Sec. Social Networks
Volume 3 - 2021 | https://doi.org/10.3389/fhumd.2021.629285
Experimenting With Online Governance
Ofer Tchernichovski1*, Seth Frey2, Nori Jacoby3 and Dalton Conley4
1Department of Psychology, Hunter College, The City University of New York, New York, NY, United States
2Department of Communication, University of California, Davis, Davis, CA, United States
3Research Group Computational Auditory Perception, Max Planck Institute for Empirical Aesthetics, Frankfurt, Germany
4Princeton and National Bureau of Economic Research, Department of Sociology and Office of Population Research, Princeton University, Princeton, NJ, United States
To solve the problems they face, online communities adopt comprehensive governance methods including committees, boards, juries, and even more complex institutional logics. Helping these kinds of communities succeed will require categorizing best practices and creating toolboxes that fit the needs of specific communities. Beyond such applied uses, there is also a potential for an institutional logic itself to evolve, taking advantage of feedback provided by the fast pace and large ecosystem of online communication. Here, we outline an experimental strategy aiming at guiding and facilitating such an evolution. We first review the advantages of studying collective action using recent technologies for efficiently orchestrating massive online experiments. Research in this vein includes attempts to understand how behavior spreads, how cooperation evolves, and how the wisdom of the crowd can be improved. We then present the potential usefulness of developing virtual-world experiments with governance for improving the utility of social feedback. Such experiments can be used for improving community rating systems and monitoring (dashboard) systems. Finally, we present a framework for constructing large-scale experiments entirely in virtual worlds, aimed at capturing the complexity of governance dynamics, to empirically test outcomes of manipulating institutional logic.
Many online communities enjoy little voice regarding the governance of their fora, which are typically controlled by appointed administrators, along with rules and bylaws set by the platform owner—usually a corporation aiming at increasing traffic, or a "benevolent dictator" (Schneider, 2019). However, there is now a trend toward replacing such centralized approaches with more democratic governance methods. Toolkits such as Modular Politics (Schneider et al., 2020), PolicyKit (Zhang et al., 2020a), and others (Bojanowski et al., 2017; Matias and Mou, 2018; Jhaver et al., 2019) now provide online communities with self-governance tools that can be tailored to fit the needs and values of specific communities. These include online voting, juries, petition, elected boards, and even more complex institutional logics (e.g., see https://communityrule.info/templates/).
The relative ease through which online governance tools can be implemented and experimented with, across a variety of platforms, has important implications. Here, we focus on how institutional logic might evolve in online communities and how such an evolution can be guided by experimentation. We present an experimental approach for improving social feedback and for empirically optimizing institutional logic in controlled virtual worlds. We share methods for conducting such virtual worlds experiments, which could bridge between basic research and higher stakes community lead experiments in platform governance (Matias and Mou, 2018).
From an ecological perspective, online communities are ideal settings for the rapid evolution of governance: they are often young and growing, information spreads rapidly, and there is sometimes strong competition among platforms. Moreover, governance tools may rapidly drift from the roles intended by their authors, as unintended consequences of a small change cascade through a population. Such a dynamic was seen, for example, in the "Reddit Blackout," in which community leaders creatively leveraged a simple mechanism for making channels private to affect a protest in which they disabled large portions of the site (one of the top 10 on the Internet), an action which led shortly thereafter to the resignation of the platform's CEO. Such shifts in governance may affect the survival (fitness) of communities, and lead to novel forms of governance via "natural" selection. But, of course, "mutations" in governance are rarely random as in the case of biological systems. Rather, they are directed, adaptive modifications of institutional logic aimed at coping with emerging needs and challenges. Indeed, in this way, they more resemble Lamarckian cultural evolution (Boyd and Richerson, 1988), proposing that an organism (here an organization) can control its evolution based on its experience. Lamarckian and Darwinian evolution often complement each other and there is a large body of work about cultural evolution in humans and other animals (Boyd and Richerson, 2005; Akcay et al., 2013).
Several experimental approaches could potentially guide and facilitate the evolution of online governance. To give an explicit example, we will focus on two fictive cases that may not capture the diverse needs of online communities, but include governance features that should be relevant to many communities. The first case is of an NGO that provides online services to a community (Figure 1A). Services are managed by a board that sets policies and resources for each service. Governance tools include a feedback system and a specialized committee to facilitate satisfaction with services. Note that here community feedback could be as simple as client rating scores for events of service provision. The second case represents a more open-ended challenge: a community that set a petition system to continuously guide governance (Figure 1B). To promote equality of influence, they set an annual petitions quota on each member. They set two committees and a board to process the stream of petitions: petitions are first sorted into clusters by one committee. A second committee then evaluates costs and feasibilities for each cluster. Finally, the board sets policies, which are guided by the petition system. Note that here community feedback is in the form of verbalizing views, grievances and needs, speaking and being heard, i.e., democratic discussion.
Figure 1. Governance structure for (A) NGO providing two types of online services to a community, (B) community governance via a petition system guiding policies and actions. Solid arrows represent primary processes and dotted arrows secondary (slower) processes.
In both cases (Figures 1A,B), the governance structure is relatively simple and yet, outcomes may depend on complex interactions between several variables. For example, in the petition system, the community needs to set a petitions quota, frequencies for committee meetings, thresholds for propagating and filtering petitions in each stage, etc. We outline three approaches for guiding the development of tools for helping such communities in improving their governance: the first is to improve the design of online governance by utilizing knowledge from basic collective action research (section Optimizing collective action). The second approach is to improve the utility of social feedback via experimental manipulations of simulated feedback systems in virtual worlds. Such experiments can be used for improving community rating systems and monitoring (dashboard) systems (section Virtual worlds experiments with social feedback). The third approach aims at capturing some of the complexity of governance dynamics, and empirically optimize institutional logic, by constructing large-scale experiments entirely in controlled virtual worlds (section Methods for exploring governance space with FSG).
We begin with a brief presentation of existing infrastructure that is available for conducting experiments with online governance (see section Materials and Equipment). Then, in the first Methods section, we review, from a translational perspective, recent research on improving collective action. We focus on technologies for efficiently running massive online experiments for studying collective action, which have revolutionized the social sciences (Salganik and Watts, 2009). In particular, new experimental systems (Hartshorne et al., 2019; Almaatouq et al., 2020; Dallinger dallinger.readthedocs.io) now allow the running of online experiments with thousands of participants, while providing for control of not only the content, but also of the network topology of social interactions (Centola, 2019). Via these methods, much progress has been made in understanding how behavior spreads in social networks (Mason et al., 2008), how cooperation evolves over time (Mao et al., 2017), how the wisdom of the crowd can be improved (Hertwig, 2012; Mannes et al., 2014; Prelec et al., 2017), and how cultural innovation propagates (Balietti et al., 2016). However, there is a large gap between our basic knowledge and our ability to translate it into the specifics of institutional logic. We conclude this section by suggesting practical approaches for implementing some of this knowledge for guiding social influences in ratings and petition systems, which could potentially benefit online communities.
In the second Methods section, we present a framework for experimenting with social feedback in modern online governance. We first discuss the challenge of obtaining high-quality information in online feedback systems. We next introduce the approach of designing virtual worlds experiments that simulate feedback systems in a setting that approximates ecologically valid environments. We then present methods for experimenting with improving information quality and engagement. This includes evaluating efficiencies of different strategies in providing greater opportunity for social learning. Finally, we discuss challenges in attempting continuous tuning of social feedback. In particular, we suggest that virtual worlds experiments can be used for testing risks and benefits of coupling between rating systems and marginal investments in service provision. This Methods section may be particularly relevant to the study of online communities that rely on continuous feedback from members.
In the third Methods section we imagine a conceptual meta-framework for conducting large scale virtual worlds experiments aiming at capturing simple governance structure, and then attempt to improve cooperation and crowd wisdom. We propose an approach for designing such virtual world experiments, which simulate governance structures as in the cases presented in Figure 1. We then suggest that an approach some of us recently developed for exploring high dimensional perceptual space using machine learning (Harrison et al., 2020) could be potentially implemented for efficiently exploring the parameter space of governance in virtual world experiments. If successful, such experiments could be useful for tuning and guiding adaptive modifications to the institutional logic of online governance.
The first part of the section Methods is purely conceptual. The second and third parts are based on a framework for conducting virtual world experiments, which we call Ferry Services Game (FSG). We developed this virtual world specifically for experimenting with governance. It is programmed in Unity (https://unity.com/). We made the code freely available for non-profit at https://github.com/oferon/FerryGame. An online demo of a basic FSG game designing options is available at http://u311.org/FerryServiceGame/. Note, however, that using our methods would also require infrastructure for connecting front-end virtual world engines and back-end engines to control the network of social interactions. It is beyond the scope of this article to describe the existing infrastructure in a manner that can guide experimentalists in choosing the proper tools for experimenting with online governance beyond the core methods presented here. Instead, we provide brief guidelines for dealing with three challenges that are unique to governance experiments: proper recruitment, proper compensation, and designing an appropriate experimental infrastructure.
Recruiting participants for massive online experiments can be done based on capturing their interest without paying them, for example, through traditional media (e.g., Müllensiefen et al., 2014), social media, or websites directed to attract public interest (Hartshorne et al., 2019; Zooniverse.org). Theoretically, it is possible to recruit hundred thousands participants in this way (Awad et al., 2018; Hartshorne et al., 2019), but such experiments are difficult to replicate and need to be adapted and gamified to attract and sustain public interest. An alternative approach is to pay participants through an online recruiting service that mediates between workers who would like to participate in experiments and experimenters who want to recruit participants. Currently, the main services that provide these capabilities are Amazon Mechanical Turk (or "MTurk"; Horton et al., 2011; Mason and Suri, 2012) and Prolific (Palan and Schitter, 2018). The inclusion of monetary compensation allows for performing long series of experiments as the experimenter can motivate repeated participation by providing monetary compensation proportional to the participation. Proper compensation design is important because compensation provides motivation that can potentially bias the simulation. We recommend structuring compensation as an object in the virtual world, to provide incentive that is an integral part of the experimental design. For example, while recruiting workers in MTurk, we set only a small fraction of the compensation in the HIT advertisement. The vast majority of the compensation is a bonus that is designed to provide an incentive that is part of the design of the virtual world and its governance (See an example in Figure 3).
In general, the demographic diversity of workers in recruitment services is fairly high with respect to age, gender, and income (Ross et al., 2010). Recently, tools were introduced in order to account for demographic biases in online experiments. For example, cloudresearch.com has been tracking and publishing fluctuations and biases MTurk workers demographics over time. In addition, researchers can use a wide range of questionnaires and pre-screening tasks to reduce biases in recruited populations (Harrison et al., 2020). However, social experiments with governance might be particularly sensitive to biases in participants' motivations to engage, which is likely to differ across recruitment approaches. For example, one may suspect MTurk participants of gaming the experimental system to save time, but we don't know if such factors are sensitive to recruitment methods. The design of the experiments can be made such that saving time would not be beneficial to the participants (for example by giving extra bonus for good performance). In addition, it is therefore advisable, whenever possible, to attempt replication over multiple recruitment methods. For example, one may compliment MTurk experiments with small scale validation experiments in the lab.
The third serious challenge is setting an appropriate experimental infrastructure for experimenting with governance: creating a complex online experiment requires infrastructure that can simplify the design process. The infrastructure allows the experimenter to focus on the unique aspects of each experiment (for example the interaction of participants with experimental logic) and provide a built-in solution for the other aspects of the experiment including (1) Managing the interaction with recruiting services (such as MTurk) to control the recruiting process, (2) compensating participants automatically, potentially allowing for differentiated compensation based on performance, (3) providing database service to record single participant data and synchronize information shared between participants, (4) orchestrating web servers to run the experiment, (5) managing real-time interaction between participants, (6) providing tools and dashboards to monitor the progress and health of the experiment, and (7) providing tools to simulate real participants with bots.
Several platforms focus on the single-participant user experience (jspsych: de Leeuw, 2015; Qualtrics; labaadvance: Finger et al., 2017). While it is possible to design experiments with multiple participants with these platforms, the platforms themselves provide only a few basic resources and abstractions for combining multiple participants. The state of the art in creating complex experiments are "virtual labs" (Psiturk: Gureckis et al., 2016; Dallinger; Empirica: Almaatouq et al., 2020; WEXTOR: Reips and Neuhaus, 2002; LIONESS; oTree: Chen et al., 2016; Breadboard: McKnight and Christakis, 2016; NodeGame: Balietti, 2017; TurkServer: Mao et al., 2017). These are "experiment engines" that provide infrastructure to design complex experiments (as mentioned above) with multiple participants. Because they provide useful abstractions, experimenters can use these platforms to implement complex designs. Finally, there are several platforms for game development (e.g., Unity, Blender, and Unreal), which allow in-browser (WebGL) deployment of multiplayer games.
The implementation presented here is of in-browser WebGL via Unity. A specific implementation of this system for testing a virtual rating system was recently published (Tchernichovski et al., 2019). Beyond this, we acknowledge that, despite the wealth of existing tools, there is not yet a consensus regarding the best ways to connect such virtual environments to the experimental infrastructures we reviewed above. It is also possible that different tools will be applicable for different projects, as different platforms deviate in their complexity and in the technical skills they demand of experimenters. Some platforms require little prior programming experience [e.g., LIONESS (Giamattei et al., 2020), Breadboard (McKnight and Christakis, 2016), WEXTOR (Reips and Neuhaus, 2002)]. Other platforms require significantly more experience: oTree (Chen et al., 2016), nodeGame (Balietti, 2017), Dallinger (https://github.com/Dallinger/Dallinger/), and TurkServer (Mao et al., 2017), empirica (Almaatouq et al., 2020).
Optimizing Collective Action
Modern governance is, to a large extent, a social system for facilitating and coordinating collective action (Bodin, 2017). In democratic societies, collective action stems from collective decisions and requires some voluntary cooperation. Good collective decisions should reinforce cooperation over time, and, vice versa, a high level of cooperation can potentially improve collective decisions. The challenge is, therefore, to both improve the quality of collective decisions and maximize cooperation. We begin with a synthesis across these two topics, which are typically studied separately.
Optimization of Crowd Wisdom
Due to a phenomenon called crowd wisdom, the quality of a collective's decisions can be higher than those of the individuals composing it (Galton, 1907). Galton observed that aggregating evaluations across individuals gave a more accurate estimate than that of the median evaluation across participants. Under what conditions is the crowd wiser than the individuals who compose it? There is a large body of literature about improving crowd wisdom (Hertwig, 2012; Becker et al., 2017; Prelec et al., 2017). In general, the understood model of crowd wisdom suggests that it depends upon a large, unbiased group of independent judges. In this manner, averaging over the responses reduces the noise of an individual's response, therefore making the crowd decision more accurate. However, recent experiments reveal that this is not always the case (Lorenz et al., 2011): Exposing subjects to evaluations given by their peers can often improve collective estimates, by allowing people who are less confident in their estimates to change their mind (Jayles et al., 2017).
In more practical situations that require deliberation and collective decisions, social communication can either improve or undermine crowd wisdom depending on the structure of the social network (Centola, 2010; Becker et al., 2017). Consider, for example, two online communities. In one of them the social communication network is highly centralized, with some popular people serving as "communication hubs" in a small world network (Watts, 1999). In the other community, communications are more distributed and decentralized. Counterintuitively, Becker et al. (2017) found that social influences improved the wisdom of the crowd in the decentralized social network. In other words, pooling biased estimates may be superior to unbiased estimates if the biases themselves are broadly distributed. In contrast, they found that in social networks where there was a high degree of centrality, social influence has the opposite effect: it decreases crowd wisdom.
Note that the structure of a social network may have two distinct influences on crowd wisdom (Figure 2B): First, the structure of the social network may influence signaling quality (the accuracy of individual evaluations) via direct social influences (Jayles et al., 2017). Second, the structure of the social network may also determine how information is "filtered" while propagating through it, potentially affecting crowd wisdom via an implicit (and often obscured) process of data aggregation (Becker et al., 2017). For example, depending on network topology, minority opinions might be filtered out or amplified from a debate in different rates (Li et al., 2013).
Figure 2. Interactions between cooperation and crowd wisdom in collective action. (A) High cooperation and crowd wisdom can potentially facilitate each other. (B) Crowd wisdom (Galton, 1907) depends on signal quality and also on how data are aggregated and weighted. (C) Cooperation is adversely affected by free-ridership (Marwell and Ames, 1979) and defection (Mao et al., 2017).
In sum, crowd wisdom studies indicate that accuracy of collative estimates should improve with sample size and (depending on network topology) with a balanced social influence. Although crowd wisdom and cooperation are typically studied separately, in governance systems that are based on voluntary feedback, the two may interact: both sample size and topologies of social influences are outcomes of cooperation (i.e., of the motivation to participate). If improving crowd wisdom can affect the quality of collective decisions, then crowd wisdom might also, in turn, affect cooperation (Figure 1A).
Tradeoff Between Communication Efficiency and Crowd Wisdom
Many social media platforms aim at maximizing communication efficiency, e.g., by nudging users to add connections and "friends." However, it appears that efficiency and innovations of collective action depend on network structure in a complex manner (Mason et al., 2008). These researchers showed that expansive (wide-ranging) networks increase exploration, which is important in finding optimal solutions for complex problems, whereas highly connected (small world) networks may allow faster convergence but not necessarily on the optimal solution. Other studies further suggested that a focus on efficiency may engender a hidden cost with respect to crowd wisdom: One might imagine that informationally efficient collaboration networks should increase the ability of a group (say a task force) to find an innovative solution to a complex problem. However, networks that are efficient for gathering information are not necessarily efficient for performance (Kearns et al., 2006). For example, in an experiment, Brackbill and Centola (2020) gave groups of data scientists a task to find better solutions for complex statistical modeling problems. Participants were randomly assigned to either an efficient or an inefficient communication network. In both groups, subjects were exposed to the same "load" of information, the only difference was in the network efficiency of propagating information. Interestingly, groups in the efficient networks underperformed, while those assigned to inefficient communication networks reached highly efficient solutions. This result led Brackbill and Centola (2020) to suggest that there exists "a tradeoff between the network structures that promote a solution's rapid diffusion throughout a group and the network structures that promote the discovery of innovative solutions." These results call for translational experiments with governance logic—e.g., regarding the efficiency of communication networks between committees and boards.
Optimization of Cooperation
Early studies outlined central government as a necessary tool for sustaining public goods in order to prevent a "tragedy of the commons" (Hardin, 1968). Three problematic behaviors, in particular, have been studied (Figure 2C): one is free ridership, where people take advantage of public goods but do not contribute enough resources to sustain them (Marwell and Ames, 1979). The second is the "prisoner's dilemma" defection strategy, where a greedy (and myopic) tendency of actors to maximize their immediate gains may erode cooperation over time. And the third is represented by "second-order social dilemmas," in which monitoring, punishment, and other governance activities for preventing free-riders and defection themselves become vulnerable to free-riding or defection (Okada, 2008), because they are also costly and generate positive externalities. Later group and network studies, pioneered by Elinor Ostrom (see e.g., Janssen et al., 2010) and others (Ahn et al., 2009) suggested that subjects can and do creatively evade free-riding and defection in real-world situations that resemble second-order social dilemmas. Ostrom found that people can overcome these three types of behavior to organize locally and make sustainable arrangements for self-governing common pool resources, without any need for external government interventions. People are often willing to engage in altruistic (and costly) punishment of "free riders" (Ostrom, 1990). Consequently, over time, sanctioning institutions tend to be more competitive than institutions that do not punish free riders and defectors (Gürerk et al., 2006). In the same vein, several studies showed that, in a repeated game, reputation can play a major role in sustaining reciprocity (Axelrod and Hamilton, 1981).
Recently, however, massive online experiments showed that cooperation can persist via network level mechanisms even in the absence of punishment or reputational effects (Suri and Watts, 2011; Mao et al., 2017). Mao et al. (2017) performed an online experiment to study the long-term dynamics of cooperation. Participants played a prisoner's dilemma game repeatedly—that is, hundreds of times over several weeks. Although the Nash equilibrium corresponds to zero cooperation in the absence of mechanisms for punishing defectors, about 40% of participants were irrationally resilient cooperators: they would not be the first to defect, despite knowing that they were about to lose money. Interestingly, long-term social learning in the remaining (more rational) players slowly promoted an equilibrium of cooperation, with dynamics unfolding over long time-scales that could have never been revealed in a typical lab study. Further, willingness to directly reciprocate cooperative behavior has been shown to persist even in highly competitive settings, with NBA players continuing to reciprocate assists despite strong individual incentives not to Willer et al. (2012). Using a similar approach, Melamed et al. (2018) showed further that even in the complete absence of reputational memory, simply allowing network dynamics to evolve gives rise to clustering of participants that shield cooperators from defectors. Further, resilience of cooperation was also observed in a more complex online experiment: Melamed et al. (2020) showed that different modes of reciprocity are fairly independent from each other, such that the collapse of one form (such as direct reciprocity) does not necessarily affect the others (indirect and generalized reciprocity).
Translating From Basic Research to Governance "Wisdom"
Although discoveries such as those presented above seem relevant for designing efficient governance systems, it may be challenging to translate them into the specifics of institutional logic. Based on the studies discussed above we propose two practical approaches for improving crowd wisdom and cooperation.
First is the introduction of social influences into crowdsourced feedback and petition systems: The statistical gold standard for designing crowdsourced information systems has been to obtain independent and unbiased evaluations, in order to minimize sampling endogeneity (Heckman, 1979). But as noted above, in governance systems that rely on voluntary cooperation (Figure 1), outcomes may depend on dynamic interactions between cooperation and crowd wisdom. In other words, minimizing social influences may bear the cost of compromising crowd wisdom. More importantly, minimizing social influences may interfere with efforts to perpetuate cooperation (Figure 2A). Sacrificing the independence of evaluations in order to promote a virtuous feedback loop between crowd wisdom and cooperation can therefore make sense. For example, in a field study some of us reported a positive outcome of increasing social influences in a rating system (Tchernichovski et al., 2017). We found that exposing service clients to trends in rating scores—just prior to rating—was associated with a persistent improvement in satisfaction with services over time. It was also associated with the community sustaining a high feedback rate over years. In addition, introducing social influences may promote self-organization of social information. For example, in a petition system, presenting users with similar petitions while filling out the petition form, may prompt users to amend and endorse existing petitions instead of creating redundant new petitions. Finally, as suggested by the Becker et al. (2017) study, setting a decentralized process for deliberating petitions could be advantageous.
Note, however, that the injecting of social influences into early stages of crowdsourced information systems may open gateways for groups of activists with bad intentions to manipulate and distort information. There are well-established mechanisms for online communities to deal with individual bad actors, but much more challenging are disruptions by collective actions of online "mobs" (Trice and Potts, 2018).
Second is the methods of data aggregation and presentation: Typical governance challenges are very different than those presented in most crowd wisdom experiments. In such experiments people are typically asked to make a quantitative estimate where there is a known ground truth, which is rarely the case in governance. However, crowd wisdom might be more relevant to governance at the low level of aggregating feedback from the community. Online community members often share ratings and "likes" to posts and services, generating a constant stream of quantitative data. It is rarely possible to optimize crowd wisdom in aggregates of such subjective evaluations with respect to a ground truth (which is often intractable). However, it is often possible and practical to improve crowd wisdom in order to promote the early detection of trends (Tchernichovski et al., 2017). That is, crowd wisdom can be defined as the speed and accuracy of detecting a change (Tchernichovski et al., 2019). The early detection of trends can be particularly useful in online communities, where social feedback is continuous. Say, for example, that an online community changed a policy regarding publishing posts in their archive. Detecting negative user feedback early could allow a speedy correction before too many users have "voted" by exit (Schneider, 2019). In the next section we will introduce experimental methods of optimizing crowd wisdom in feedback systems, including challenges in sustaining high quality of signaling (Figure 2B), which often suffers from sampling endogeneity and from poor information quality (Stocker, 2006; Moe and Schweidel, 2011; Ho et al., 2014).
In sum, improving crowd wisdom and cooperation should be considered while designing online governance. We highlighted two aspects where this might be particularly relevant: one is in the design of social influences during submission of petitions or evaluations, and the second is in the design of data aggregation in evaluations and ratings. We will return to these issues below while presenting frameworks for virtual worlds experiments with governance.
Virtual Worlds Experiments With Social Feedback
We first briefly review the utilities and limitations of correlational and experimental studies in online community governances. We then introduce virtual world experiments and present methods for experimenting with social feedback.
Correlational Studies in Online Communities
Observational studies on the governance systems of online communities revealed important dynamic relations between evolving governance structure and outcomes. For instance, Tan and Zargham (2021) recently published the GovBase database: a crowdsourced summary of tools used for online governance. The database allows for exploration of how different governance structures that have been adopted by online communities may correlate with (and perhaps predict) the survival of those communities over time. Several studies explored the evolution of governance in online communities. For example, Frey and Sumner (2019) studied governance rules in 5,000 online communities of video game players and described how they evolve over time. They found that the structure (number and scope) of governance rules given population size can, to a certain extent, predict growth of the group of core members. Because users can be enculturated to prefer a new form of governance, and conversely, can exert selective pressure on expressed governance forms by opting out of communities they don't like, it is possible for social feedback loops to drive the evolution of both governance forms and preferences. To this end, a follow-up study by Zhong and Frey (2020) finds evidence that, although influence is evident in both potential directions, the selective pressure mechanism is much stronger than the cultural evolution of preferences. Observational studies have illuminated several questions about the formation of self-directed governance systems. For example, how strong a force is emergent centralization? Shaw and Hill (2014) documented the unexpected emergence of oligarchy in the radically egalitarian domain of "wiki" knowledge bases. They observe that a small administrator class becomes increasingly distinguishable as wikis grow, and that the goals and standards of that class diverge from those of regular volunteers. How do different modalities of positive and negative interactions between agents aggregate to produce social outcomes? Szell et al. (2010) analyze the multidimensional relationships between hundreds of thousands of players in an online game to illuminate the game's emergent system of power dynamics. With a multiplex network approach, they demonstrate alliance and conflict dynamics playing out in a coordinated manner over several types of game action, including trade, conflict, and friendship relations. There are, however, strong limitations to such observational studies: causality is difficult to assert, and the space of scenarios that can be explored is constrained by the current state of the world. With an experimental approach to the governance of online communities, one can test cause and effect directly and explore governance outcomes in social scenarios that cannot be easily observed otherwise.
Experimenting in Social Media Governance
The challenge we would like to focus on hereafter is how to better connect the basic research of optimizing collective actions with observations of online governance (section Correlational studies in online communities) in order to test practical solutions. This would require translational experiments. Experimenting directly with social media governance may raise ethical concerns, but when done properly it can be highly valuable (Lazer et al., 2020). Concerns about early experiments and ethically questionable data-mining have strongly constrained subsequent experiments (Kramer et al., 2014; Lazer et al., 2020). That said, among the most promising recent approaches is the Matias CivilServant System (Matias and Mou, 2018) for community-led experiments in platform governance. An example implementation of CivilServant is a recent experiment that was done in a Reddit community with 13 million subscribers. An announcement about a new community rule was randomly assigned to some of the new members in an online science discussion forum (Matias, 2019). Presenting the announcement informed new participants about the community's rules: no jokes, no abusive content, and so on. The question was if this message would influence newcomers' choices to contribute content, and if it would affect harassment levels. Results showed that the intervention reduced cases of harassment without decreasing participation in the forum. We hope that such an experimental approach would become increasingly useful in optimizing cooperation, hence promoting a virtuous loop between cooperation and high-quality collective decisions, as discussed earlier (Figure 2). One way of achieving this is by complementing community-led experiments with virtual world experiments, where risks and attendant ethical concerns can be minimized. In addition, virtual world experiments provide the possibility to perform much more substantial control experiments and test more radical modification to the services including the possibility of "collapse" of the community, something that should be avoided in a community-led experiments.
A Framework for Virtual Worlds Experiments With Governance
As discussed above, online experiments in crowd wisdom and cooperation have contributed to a deeper understanding of collective behavior in humans. The key to this progress is in technological innovations that now allow for the testing of thousands of participants over hundreds of iterations while controlling the topologies of social communication networks. But given the highly stylized nature of experimental exchange systems, even such large-scale experiments have limited bearing on complex, real-life governance. We now turn to the challenge of conducting online experiments in virtual worlds, with the aim of studying governance in simulated, complex environments. There are two motivations to conduct such experiments: First, the governance systems of the virtual world can permit adventurous experiments in the varieties of social life with low enough stakes that they do not risk any meaningful form of social collapse. Second, placing people in situations that allow them to tweak or elaborate their setting makes it possible to study the evolution of complex institutions from simple ones. This means that we need not only to conduct long-term experiments, but also to place people in situations that mirror the real-world complexities of pooling and sharing multiple resources. Former studies utilize virtual worlds for conducting simple experimental interventions. For example, introducing bots into popular areas in the multiplayer game Second Life allowed researchers to examine how the "age" and "gender" of bots' avatars affected the avatar's chances of obtaining help from other players (Zhang et al., 2020b). They can also provide and validate governance tools for virtual worlds, such as research providing governance tools on the integrated game chat platform Discord (Zhang et al., 2020a), or community monitoring tools on the video game Minecraft (Müller et al., 2015). However, to our knowledge, this article is the first to present a framework for virtual world experiments that are designed specifically for studying governance.
We developed a method for experimenting with governance in a virtual world, which we call Ferry Services Game (FSG). FSG is an in-browser WebGL game (see Material and equipment). Participants (e.g., MTurk workers) manipulate their avatar in a 3D world to collect coins, which are redeemed for money at the end of the game (as a compensation bonus). The 3D world is composed of a long chain of islands (Figure 3A). Participants must use simulated ferry services to visit each island, but some ferries are fast and others are slow, and players are motivated to have faster ferries since time spent on ferries competes with time spent earning revenue (Figures 3B,D). Through several mechanisms, players may gain and contribute information about ferry characteristics. After a ferry ride the player may be prompted to rate the service. Ratings may be pooled and shared with other players via a dashboard. The experimenter sets the distributions of service speeds and delays, and then evaluates the ferry rating scores against the ground truth of that distribution. Therefore, as in many real-world situations, rating information can help players select (or collectively own) better services.
Figure 3. Ferry Services Game. (A) A 3D virtual environment composed of islands with coins scattered in different locations. (B) Collecting coins in each island. (C) Ferry services are used to cross between islands. Dashboard represents ferry rating scores. (D) Ferry services differ in speed and delays. (E) At the end of each ride player might be prompted to rate ferry performance.
Several game parameters are adjustable: Rating the ferry may be either voluntary or obligatory. Players may be given no incentive to rate the ferry (which is often the case in real world rating systems). Alternatively, rating the ferry can be made a social process. For example, in Figure 3C we present a design where players can choose between two ferry services in each island. Here, sharing rating scores with other players would benefit all players by allowing them to quickly pool their knowledge about which ferry service is better. However, this public goods game incentivizes free ridership (Marwell and Ames, 1979). In a different design, only members of a "club" may share rating information, as in a common pool resource game. Here, club governance rules may be either randomly assigned, or set by the players. In this manner, layers of governance can be superimposed on the game, as we will show in the next section. In this section, we focus on low-level experimentation with optimizing social feedback.
Optimization of Feedback Systems With FSG
Many online communities use feedback systems that provide quantitative evaluations such as "likes," and rating scores. Feedback systems have evolved rapidly over the last two decades. They turned from an industry standard mechanism designed for obtaining information from—and retaining—clients, into a rich ecosystem of independent crowdsourced rating platforms, which now guide many everyday decisions.
FSG experiments can be designed for optimizing feedback systems at three levels. First is the optimization of the rating device: Most systems use rating devices that implement Likert scales, e.g., a 1–5 "star" ratings, where the rating device is a trivial "click and submit" radio group. Pooled star ratings can then be made visible to both creators and consumers of content, or in e-commerce, to service providers and potential clients (Figure 4A). There are many concerns about the quality (and honesty) of such rating scores (Luca, 2016). A recent FSG experiment found that even without any conflicts of interest or incentives to cheat, pooled rating scores could explain only about 14% of the variance in the speed of ferries (Tchernichovski et al., 2019). However, pooled rating accuracy was about twice as high when ratings were submitted via a device that imposed time costs of a few seconds on reporting extreme scores. Such an improvement in feedback information quality could be useful in the case of the NGO providing online services presented earlier (Figure 1A): With better rating accuracy the evaluation committee should be able to detect a change in satisfaction with its service outcome much faster. This could have two practical advantages: First, services can respond to the change faster. Second, the sooner a change can be detected the easier it is to identify its cause. In other words, reducing latency in detecting a change is likely to improve reinforcement learning in a multi-agent scenario (Sutton and Barto, 2018).
Figure 4. From rating systems to governance evolution. (A) A FSG experiment with a typical online rating system, where both service provider (e.g., restaurant) and client's behaviors are guided by feedback ratings. (B) Replacing rating scores with a dashboard presenting trends to promote social learning in FSG experiment. (C) A framework for ongoing experimentation with dashboard design.
These results demonstrate the utility of using FSG for testing different designs of rating devices. The FSG game template we shared (https://github.com/oferon/FerryGame) can be used for testing such designs. FSG is coded in the gaming platform Unity, where building arbitrarily complex 3D rating devices (game objects) with physics can be easily implemented. The game can be then compiled as is into either a desktop application for lab experiments, or into a WebGL for online experiments via a browser, or into a phone app, for long-term experiments.
A second manner in which FSG experiments can be used for improving feedback systems is via experimentation with distributed feedback monitoring methods. Performance measurement systems with dashboards are widely used in many industries (Bititci et al., 2000) in order to monitor and coordinate performance benchmarks and optimize institutional learning. In principle, similar dashboard systems can be developed to optimize social learning in a non-commercial setting of online communities. Such dashboards may be particularly useful for online communities that offer services that are used repeatedly by a pool of members. In such cases, it might make sense to present dashboards with trends (Tchernichovski et al., 2017) rather than mean rating scores. FSG experiments can be designed for testing the utility of presenting such trends (Figure 4B). FSG experiments can test, for example, if the presentation of trends in rating scores of ferry services improves social learning, prompting players to adjust their strategies more efficiently. Note that even in a small community that runs a simple operation, social learning may show complex temporal dynamics: the detection of a trend gives both clients and service providers an opportunity for social learning in real time: service providers can "experimentally" adjust their behavior/product in real time in response to trends, while consumers can make choices with more up-to-date information. The challenge is how to tune the parameters of the trend presentation such that each side can learn most effectively from the other? Returning to the case of an NGO providing online services (Figure 1A): should trends in satisfaction with each service type be presented over days, weeks, or months? Presenting short term trends should minimize delay and improve social learning. However, presenting long term trends may promote stability and improve the clarity of social signals. For the NGO, manipulating such parameters can be risky, but FSG experiments can be designed for testing such dashboard calibrations while keeping the temporal dynamics of service provision fixed. For example, one may assign participants into ferry rider and ferry driver groups, and allow them to learn from each other only via a (service provider) dashboard presenting trends. Simulating the dynamics of social learning via feedback in different conditions could then guide the NGO on how to design and tune performance measurement systems in their non-market setting (a monosponistic environment).
Third, FSG experiments can be designed for testing the utility and risks of implementing machine learning for continuous optimization of rating system features. Such features may include, for example, the physics of a rating device (Tchernichovski et al., 2019), or the temporal resolution of a dashboard presenting trends (Tchernichovski et al., 2017), or both. About a decade ago, much of the software industry adopted continuous deployment, such that software is continually released and experimented with (Kevic et al., 2017). Such commercial systems for automated experimentation with platform design are often based on a closed-loop feedback system (Figure 4C). For example, the "style" of a banner in a web page includes many features, such as screen coordinates, size, colors, and fonts, which can be manipulated while monitoring changes in client behavior. Here, the typical feedback is not rating scores, but changes in traffic, clicking on ads, and so on. Using standard machine learning approaches (Mattos et al., 2017; Gauci et al., 2018), such systems can continuously "nudge" the design, with the aim of maximizing outcomes desired by the platform owner. Matias and Mou (2018) suggested that a similar testing approach might be useful in the implementation of online governance policies (Matias, 2019).
One may treat a subset of governance logic in a community as a set of features that can be continuously optimized as in the typical case of an ad banner. This could include gain and delay parameters (how often a certain committee should meet), or setting the threshold of consensus required for votes to enable a petition to pass, etc. In this spirit, we imagine FSG experiments combining continuous rating feedback (which can be seen as continuous voting) with continuous exploration of governance logic (Figure 4C). We emphasize that trying such an approach in real communities could be dangerous and possibly unethical, and that virtual worlds experiments should be regarded as sandboxes where thresholds for tipping points can be safely established. An example of such an FSG experiment would be implementing a machine learning algorithm to find the temporal resolution of a dashboard that maximizes both service usage and satisfaction with service outcomes over time. There are several risks in running such experiments in an online community. For example, people may (wrongly) perceive that information is being manipulated by bad actors, or the algorithm may become unstable, reducing public confidence.
Finally, an experimenter could allow feedback from dashboards to directly guide marginal investments in simulated services via continuous experimentation. The role of such FSG experiments could be to serve as a playground for exploring utility, but even more so, for discovering and then reducing risks prior to deployment in real-world online communities.
Methods for Exploring Governance Space With FSG
The core of the FSG methods we have presented so far is the calibration of feedback systems that are attached to specific activities. For example, in a virtual world where participants collect coins (real money) in islands and ride ferries to get to these islands, the ferry services can be either public goods, common pool resources, or private companies. Either way, participants should care about those services, and may share information about their satisfaction via a dashboard. At this point there is already a need for governance: should information be available to everyone (public good) or only to members (common pool resource)? How to advertise it? How to display the information in an optimal manner? For example, a dashboard could show trends of satisfaction with ferry services before riding a ferry (Tchernichovski et al., 2017, 2019). The experiential challenge is to allow such a dashboard to directly nudge policies and governance logic. For this purpose, we suggest a generic framework for attempting long term FSG experiments that simulate a virtual city, where participants are engaged in a variety of activities, providing other services, instituting taxes and tolls, and forming simple governance institutions (Figure 5).
Figure 5. Ferry Services Game experiment with governance. (A) FSG design simulating governance of pooled resources (services) via feedback in a community (no- market, as in Figure 1A). Groups of participants work in providing simulated services (e.g., driving ferries or minting coins) and another group represents community members who use these services (riding ferries and collecting coins). Feedback from participants' satisfaction and service usage is aggregated into dashboard presenting trends. These trends then feed into governance logic. (B) FSG design simulating a petition governance (as in Figure 1B).
Here, the experimenter needs to design an ecosystem with several groups of players. For example, in Figure 5A, one group of players represents community members who use two common pool resources: coins and ferries. They pay tolls for those services, and rate them. The measure of performance in this group is net coin earning, and the level of cooperation is the feedback rate. The second and third groups of players are service providers: ferry drivers and coin minters. Each of these groups plays a public goods game: in each round these players decide how much of their income (from services and tolls) to invest in these services. The more they invest, the faster the ferries move and the more coins can be collected in each island. If properly designed, such a system may stabilize on different levels of cooperation: the community may decide to pay more or fewer tolls based on their satisfaction with the services. The service providers may decide to invest more or less of their toll income in the quality of these services.
Note that the FSG design presented in Figure 5A is a simulation of the community presented in Figure 1A, including two rating systems, two dashboards displaying trends, a committee and a board. Once participants are recruited to play on a regular basis, communication channels can be used for implementing governance structure including committees, boards and voting. The experimenter could impose community rules or, alternatively, allow participants to negotiate them. We may see some communities where cooperation collapses and others where cooperation persists, in much the manner that online communities can be observed to develop rich and varied governance systems for overcoming their online governance challenges (Frey and Sumner, 2019). In sum, there is a potential value in developing such virtual worlds experiments to facilitate the Lamarckian evolution of governance.
Finally, there are two inherent weaknesses of the approaches we presented for the FSG experiments for exploring governance space, which are worth consideration. The first is a potential experimental failure due to the complexity of the design. How can FSG experiments succeed in exploring a complex governance space? This challenge is somewhat similar to that of experimenting in exploring human perceptual space in the field of cognitive neuroscience. One may ask, for example, what set of acoustic features add up to an abstract percept such as the sound of a violin. Until recently, exploring the space of such high-dimensional acoustic features was not experimentally feasible. Recently, however, combining machine learning with human judgement was shown to be successful in efficiently identifying such perceptual categories (Harrison et al., 2020). In such experiments, participants are presented with a slider, which they manipulate to approximate a category (e.g., determine which sound resembles a violin). Although the participant repeatedly manipulates the same slider, in each round the slider represents a different acoustic feature. The algorithm pools these evaluations across participants in order to explore the perception of an arbitrarily complex space of acoustic features in an efficient manner. At least conceptually, a similar approach could be implemented in FSG experiments. Here, instead of presenting participants with a slider for manipulating values of acoustic features, they can be presented with sliders representing their preferences in governance space. For example, a slider can be presented to cast a preference for the ferry toll, or to vote for a governance rule about feedback quota, etc. The point is, even if this governance space includes several parameters, it may still be possible to efficiently explore it by implementing modern machine learning methods as in Harrison et al. (2020). The utility of such an approach is in detecting and characterizing stable states in the space of governance features.
A second limitation is that a significant aspect of online governance consists of a democratic debate, where participants are verbalizing and communicating views, grievances and needs. Experimenting with feedback systems and exploring governance space using FSG, where governance is based primarily on simple quantitative measures, may fail to capture or acknowledge the minority view and its legitimacy or even nuances in the views of the majority. Although we cannot yet offer specific solutions, we hope that future development of FSG game simulating a petition governance (as in Figure 5B) can allow experimenting with the tradeoff between openness and control. The challenge in such experiments is how to channel activism away from the wild (and mostly futile) social dynamics of echo chambers and internet-storms into petition systems where participant influence is balanced, petitions can evolve, and where one can quantify the extent to which feedback and deliberation can become more constructive in a controlled environment.
We began by reviewing studies of crowd wisdom and studies of cooperation. We claim that with respect to democratic governance, crowd wisdom and cooperation may interact and influence collective action. We then showed that whereas early studies focused on the role of punishment and reputation in sustaining cooperation, more recent studies revealed the importance of the structure (topology) of communication networks in sustaining cooperation. The communication network influences the success of collective action via the manner through which diffusion of knowledge and influence affect crowd wisdom and cooperation. We argued that, on one hand, such studies of network topology are highly relevant to the problem of optimizing governance, but on the other hand translational studies have remained rare. We then presented a framework for virtual worlds experiments, Ferry Service Game (FSG), and presented experimental approaches aiming at overcoming the challenges of conducting translational research in online governance.
From an experimentalist perspective, online communities are wonderful playgrounds for studying the evolution of governance logic. We reviewed several studies that reveal the complex relations between adoption of governance rules and their statistical outcomes. We then discussed how virtual worlds experiments with FSG can complement community-led experiments, making the argument that some of the most exciting directions to explore are too risky and perhaps unethical to experiment with in online communities.
We concluded by suggesting how user feedback systems that provide continuous streams of rating information could potentially be leveraged for continuous optimization of governance logic. We reviewed the idea of using such feedback systems in a manner that generalizes standard commercial design of continuous experimentation with design—which we propose can be extended to governance operation and logic. We propose that large-scale virtual worlds experiments can address this problem. Finally, we presented a method for running FSG experiments with governance. We hope that sharing out methods will encourage adventurous studies in complex virtual social environments aiming at exploring governance beyond the current boundaries of existing governance models. Such experimentation with new types of social contracts could potentially guide our social evolution in new and exciting directions.
OT wrote the initial version. SF and DC edited and expanded the main frame of the MS. NJ wrote section Discussion. All authors worked together editing the manuscript.
This work was supported in part by NSF GCR 2020751 Jumpstarting Successful Open-Source Software Projects With Evidence-Based Rules and Structures.
The handling editor declared a shared research group with the author SF at time of review.
The authors wish to thank Nathan Schneider and the Metagovernance Seminar.
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Keywords: online governance, crowd wisdom, cooperation, costly signaling, collective action, virtual worlds
Citation: Tchernichovski O, Frey S, Jacoby N and Conley D (2021) Experimenting With Online Governance. Front. Hum. Dyn. 3:629285. doi: 10.3389/fhumd.2021.629285
Received: 14 November 2020; Accepted: 23 March 2021;
Published: 26 April 2021.
Nathan Schneider, University of Colorado Boulder, United States
Arild Bergh, Norwegian Defence Research Establishment, Norway
Uta Kohl, University of Southampton, United Kingdom
Copyright © 2021 Tchernichovski, Frey, Jacoby and Conley. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ofer Tchernichovski, otcherni@hunter.cuny.edu
Peer Governance in Online Communities
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test : Test Permission
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<resource class="app.tested.models.testing.User"/>
<permission value="ManageUser" desc="Manage User"/>
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<tag name="User" desc="用户"/>
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{"url":"https:\/\/phys.libretexts.org\/TextBooks_and_TextMaps\/Classical_Mechanics\/Book%3A_Classical_Mechanics_(Tatum)\/19%3A_The_Cycloid\/19.02%3A_Tangent_to_the_Cycloid","text":"$$\\require{cancel}$$\n\n# 19.2: Tangent to the Cycloid\n\nThe slope of the tangent to the cycloid at P is $$dy\/dx$$, which is equal to $$dy\/d\\theta$$, and these can be obtained from Equations 19.1.1\u00a0and 19.1.2.\n\nExercise $$\\PageIndex{1}$$\n\nShow that the slope of the tangent at P is tan $$\\theta$$. That is to say, the tangent at P makes an angle $$\\theta$$ with the horizontal.\n\nHaving done that, now consider the following:\n\nLet A be the lowest point of the circle. The angle $$\\psi$$ that AP makes with the horizontal is given by $$\\tan \\psi = \\frac{y}{x - 2 a \\theta }$$\n\nExercise $$\\PageIndex{2}$$\n\nShow that $$\\psi = \\theta$$ . Therefore the line AP is the tangent to the cycloid at P; or the tangent at P is the line AP.","date":"2018-06-21 21:54:20","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\": 0.8950459361076355, \"perplexity\": 212.28814476454104}, \"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\/1529267864300.98\/warc\/CC-MAIN-20180621211603-20180621231603-00057.warc.gz\"}"}
| null | null |
Q: Optimise primary key and index settings for multiple link tables I have this tables stated as below.
*
*orders is the main table that will store all the details such as the customerID, merchantID, which riderID it was assign. The order details like total, subTotal, tax and also customer details like address latitude, longitude. It also saves every detail like when the order was accepted, ready, delivered.
CREATE TABLE `orders` (
`orderID` int UNSIGNED NOT NULL,
`orderIDHash` varchar(10) NOT NULL,
`customerID` mediumint UNSIGNED NOT NULL,
`merchantID` smallint UNSIGNED NOT NULL,
`branchID` smallint UNSIGNED NOT NULL,
`riderID` smallint UNSIGNED NOT NULL,
`orderDateTime` timestamp NOT NULL,
`subTotal` decimal(10,0) NOT NULL,
`tax` decimal(10,0) NOT NULL,
`total` decimal(10,0) NOT NULL,
`quantity` tinyint NOT NULL,
`customerAddress` varchar(150) NOT NULL,
`customerLatitude` float NOT NULL,
`customerLongitude` float NOT NULL,
`paymentID` varchar(30) NOT NULL,
`paymentType` tinyint NOT NULL,
`instructions` varchar(100) NOT NULL,
`orderType` tinyint NOT NULL,
`neworderNotify` tinyint NOT NULL,
`orderAcceptedDateTime` timestamp NOT NULL,
`merchantUserAcceptedID` smallint NOT NULL DEFAULT '0',
`orderAcceptedNotify` tinyint NOT NULL DEFAULT '0',
`orderReadyDateTime` timestamp NOT NULL,
`merchantUserReadyID` smallint NOT NULL DEFAULT '0',
`orderReadyNotify` tinyint NOT NULL DEFAULT '0',
`orderInDeliveryDateTime` timestamp NOT NULL,
`orderInDeliveryNotify` tinyint NOT NULL DEFAULT '0',
`orderPickerOrDeliveryDateTime` timestamp NOT NULL,
`orderCancelDateTime` timestamp NOT NULL,
`orderStatus` tinyint NOT NULL DEFAULT '0'
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_0900_ai_ci;
ALTER TABLE `orders`
ADD PRIMARY KEY (`orderID`),
ADD UNIQUE KEY `orderIDHash` (`orderIDHash`),
ADD KEY `customerID` (`customerID`),
ADD KEY `merchantID` (`merchantID`),
ADD KEY `riderID` (`riderID`),
ADD KEY `branchID` (`branchID`),
ADD KEY `orderStatus` (`orderStatus`),
ADD KEY `orderAcceptedNotify` (`orderAcceptedNotify`),
ADD KEY `orderReadyNotify` (`orderReadyNotify`),
ADD KEY `orderInDeliveryNotify` (`orderInDeliveryNotify`);
ALTER TABLE `orders`
MODIFY `orderID` int UNSIGNED NOT NULL AUTO_INCREMENT;
COMMIT;
*The second table is the ordersLine. The table basically store all the items related to one particular order. Thus its related to the order table above.
CREATE TABLE `orderLine` (
`orderID` int NOT NULL,
`itemID` mediumint NOT NULL,
`itemQuantity` tinyint NOT NULL,
`itemPrice` decimal(10,0) NOT NULL
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_0900_ai_ci;
ALTER TABLE `orderLine`
ADD PRIMARY KEY (`orderID`,`itemID`);
COMMIT;
*CustomerMessageLog is basically for example is the order table above we have this field orderAcceptedNotify which will change its value say default is 0 then to 1 meaning order is accepted by merchant and to 2 when its already notified to the user. Thus we store the message in this log so when user log he can see all the message. There can be different messageID for different type of messages.
CREATE TABLE `customerMessageLog` (
`orderID` int NOT NULL,
`customerID` mediumint NOT NULL,
`messageID` tinyint NOT NULL,
`messageDateTime` timestamp NOT NULL
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_0900_ai_ci;
ALTER TABLE `customerMessageLog`
ADD PRIMARY KEY (`orderID`,`customerID`,`messageID`);
COMMIT;
*This table is to store all the notified message to the rider.
CREATE TABLE `riderMessageLog` (
`orderID` int NOT NULL,
`riderID` smallint NOT NULL,
`messageID` tinyint NOT NULL,
`messageDateTime` timestamp NOT NULL
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_0900_ai_ci;
--
ALTER TABLE `riderMessageLog`
ADD PRIMARY KEY (`orderID`,`riderID`,`messageID`);
COMMIT;
*This table is to store all the notified message to the merchant.
CREATE TABLE `merchantMessageLog` (
`orderID` int NOT NULL,
`merchantID` smallint NOT NULL,
`branchID` smallint NOT NULL,
`messageID` tinyint NOT NULL,
`messageDateTime` timestamp NOT NULL
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_0900_ai_ci;
ALTER TABLE `merchantMessageLog`
ADD PRIMARY KEY (`orderID`,`merchantID`,`branchID`,`messageID`);
COMMIT;
*This is the table which will store all the orderStatus and also messageID in most of the tables above refer to this table via join.
CREATE TABLE `statusMessage` (
`statusID` tinyint UNSIGNED NOT NULL,
`message` varchar(20) CHARACTER SET utf8mb4 COLLATE utf8mb4_0900_ai_ci NOT NULL
) ENGINE=InnoDB DEFAULT CHARSET=utf8mb4 COLLATE=utf8mb4_0900_ai_ci;
--
-- Dumping data for table `statusMessage`
--
INSERT INTO `statusMessage` (`statusID`, `message`) VALUES
(0, 'Pending'),
(1, 'Payment Completed'),
(2, 'Accepted'),
(3, 'Completed'),
(4, 'Self Picked Up'),
(5, 'Assigned Rider'),
(6, 'In Delivery'),
(7, 'Delivered');
ALTER TABLE `statusMessage`
ADD PRIMARY KEY (`statusID`);
So now the challenge is that I have set the orderID to be DDMMYYHHMMSS which I get from the user phone before order is submitted for payment. Why I did this is cause for the payment gateway I will need to orderID.On the other hand I could also set it as auto increment and can also insert all the order and orderLine before payment but if the payment fails then that is another issue how to handled the inserted order and orderLine. So which is most effective method. Also I would like to hash the primary when its shown to user?
A: customerLongitude float(10,8) chops off about half the world. Simply use FLOAT.
itemPrice float -- Use DECIMAL for monetary values.
Unless a "customer" rarely makes more than one "order", have the customer details in a separate table.
Ditto for "merchant"?
A "customer" makes an "order"; the "order" includes "items" (your "order line"). The point here is that the "items" do not need a link for "customer_id".
Think about what "entities" you have -- customers, merchants, orders, items within an order.
Then think about how they are related. And decide whether they are 1:1, 1:many, or many:many.
*
*1 customer makes many orders
*1 order has many items
Those are handled by having customer_id as a column in orders and order_id as a column in items.
1:1 relations should (usually) be avoided.
many:many requires an extra table with two columns -- namely the ids linking to the two Entities.
That discussion will move you closer to deciding on PRIMARY KEYs. But keep in mind that a PRIMARY KEY is, by definition (in MySQL), a column (or combination of columns) that minimally identifies the rows. Some of the PKs you have suggested do not follow those rules.
Before getting into the rest of the INDEXes, sketch out the desired SELECTs. They will drive what indexes (in addition to the PKs) that you will need.
HashID
For database convenience, you have an AUTO_INCREMENT. For a minor amount of obfuscation, you have added a HashID as some function of the auto_inc. I assume you will set the hash when you create the row.
UNIQUE(HashID) is certainly a way to assure that the row is addressable either way, and you can map between the ids. Having it in the same table (as opposed to an extra table) is probably fine.
I say "minor" because in cryptology the least secure part is the algorithm. You could beef it up with a secret "salt".
If there is some risk of dup hashes, then that can be easily worked around -- DELETE the row, and INSERT a new row to get a new auto_inc id and hash. An occasional loss of an id is not a big deal.
|
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<?php
declare(strict_types=1);
namespace Phpcq\Runner\Output;
use Phpcq\PluginApi\Version10\Output\OutputInterface;
use Symfony\Component\Console\Output\OutputInterface as SymfonyOutputInterface;
final class SymfonyOutput implements OutputInterface
{
/** @var SymfonyOutputInterface */
private $output;
/**
* SymfonyOutput constructor.
*
* @param SymfonyOutputInterface $output
*/
public function __construct(SymfonyOutputInterface $output)
{
$this->output = $output;
}
/** @SuppressWarnings(PHPMD.UnusedFormalParameter) */
public function write(
string $message,
int $verbosity = self::VERBOSITY_NORMAL,
int $channel = self::CHANNEL_STDOUT
): void {
$this->output->write($message, false, $verbosity);
}
/** @SuppressWarnings(PHPMD.UnusedFormalParameter) */
public function writeln(
string $message,
int $verbosity = self::VERBOSITY_NORMAL,
int $channel = self::CHANNEL_STDOUT
): void {
$this->output->writeln($message, $verbosity);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 5,371
|
\section*{Abstract}
We consider a mechanism design problem for energy procurement, when there is one buyer and one seller, and the buyer is the mechanism designer.
The seller can generate energy from conventional (deterministic) and renewable (random) plants, and has multi-dimensional private information which determines her production cost. The objective is to maximize the buyer's utility under the constraint that the seller voluntarily participates in the energy procurement process. We show that the optimal mechanism is a menu of contracts (nonlinear pricing) that the buyer offers to the seller, and the seller chooses one based on her private information.
\vspace{5pt}
Keywords: integration of renewable energy, optimal contract, energy procurement, multi-dimensional private information, mechanism design.
\section{Acknowledgment}
The authors would like to thank Galina Schwartz for her helpful comments on the paper. This work was supported in part by NSF Grant CNS-1238962.
\section{Conclusion}
We proposed a model for energy procurement that captures the current approach for integration of modern renewable and conventional energy resources and takes into account the fact that renewable energy producers behave strategically and may have different production technologies. We analyzed an arbitration between a strategic energy buyer and a strategic energy seller who has the ability to generate energy from a conventional plant and a modern renewable plant and has private information about her production technology and cost. We showed that the optimal contract/mechanism for energy procurement is a nonlinear pricing scheme.
The originally proposed mechanism guarantees interim voluntary participation of the seller. By modifying the seller's payment so as to be weather-dependent we achieved ex-post voluntary participation of the seller. We also presented an alternative payment to the seller that divides the risk (due to the uncertainty in the weather) between the buyer and the seller.
\section{Discussion}
The optimal mechanism/contract for the energy procurement problem formulated in this paper is a nonlinear pricing scheme. The nonlinearity is due to three factors. First, the buyer's utility function $\mathcal{V}(q)$ is not linear in the quantity $q$. Second, for each type of the seller, the cost of production is a nonlinear function of the amount of produced energy. Third, the seller has private information about her technology and cost (seller's type).
The uncertainty about the production from the non-conventional plant makes the total expected production cost function nonlinear even with a fixed marginal cost of production for conventional and renewable plants. The buyer has to pay information rent (monetary incentive) to the seller to incentivize her to reveal her true type. Therefore, the payment the buyer makes to the seller includes the cost of production the seller incurs plus the information rent, which varies with the seller's type; the better the seller's type, the higher is the information rent.
The optimal contract/mechanism discovered in this paper can be implemented indirectly (without reporting the seller's private information) as follows: the buyer offers the seller a menu of contracts (nonlinear pricing scheme); the seller chooses one of these contracts based on her type, and there is no need for an implausible and unnecessary information exchange stage between the seller and the buyer.
The multi-dimensionality of the seller's private information could be due to the different types of energy generators that she owns, or because of a complex cost function with more than one parameter (even with one type of energy generator). The solution approach presented in this paper can be used to solve problems with multi-dimensional private information with similar structure.
The optimal contract defined by Theorem 1 also induces some incentives for investment in infrastructure and technology development. From Lemma 1, the seller with the higher type has a higher utility. Therefore, there is an incentive for the seller to improve her technology and decrease her cost of generation.
It is well-known that in the presence of private information and strategic behavior, in general, there exists no mechanism/contract that is (1) individually rational, (2) incentive compatible, and (3) efficient (Pareto-optimal generally) \cite{rosenthal}. In the optimal contract/mechanism given by Theorem 1, the allocation for the seller's different types is not ex-post efficient (Pareto-optimal) except for the seller's worst type who gets zero utility.
\section{Example}
Consider a seller with a conventional plant and a wind turbine.
The wind turbine's output power curve $g(w)$ is as in figure 1.
We assume that for wind speeds between $v_{ci}$ and $v_r$ the energy generation is given by $\gamma w^3$, where $\gamma$ captures the technology and size of the turbine. Energy generation remains constant for wind speeds between $v_r$ and $v_{co}$, and is zero otherwise.
\vspace{-5pt}
\begin{figure}[h!]
\centering
\includegraphics[width=0.30\textwidth, height=0.10\textheight]{Ex1-1}
\caption{Example - the wind turbine generation curve}
\label{curve}
\end{figure}
We assume that there is a fixed marginal operational cost $\theta_w$ for the wind turbine and a fixed marginal operational cost $\theta_c$ for the conventional plant.
There is a no-production cost $c_0$ that captures the start-up cost for both plants and the capital cost for the seller. Therefore, the seller's type $x\hspace{-2pt}=\hspace{-2pt}(c_0,\theta_w,\theta_c,v_{ci},v_{r},\hspace{-1pt}v_{co},\hspace{-1pt}\gamma)$ is 7-dimensional and her total cost is given by
\begin{eqnarray}
C(q,w,\hspace{-1pt}x)\hspace{-2pt}=\hspace{-2pt}c_0\hspace{-2pt}+\hspace{-2pt}\theta_w\hspace{-2pt}\min\hspace{-2pt}\left\{\hspace{-1pt}q,g(w)\hspace{-1pt}\right\}\hspace{-3pt}+\hspace{-1pt}\theta_c\hspace{-2pt}\max\hspace{-2pt}\left\{\hspace{-1pt}q\hspace{-2pt}-\hspace{-2pt}g(w),\hspace{-1pt}0\hspace{-1pt}\right\}\hspace{-3pt},\hspace{-4pt}
\end{eqnarray}
where $g(w)$ is as in figure \ref{curve}. The wind profile is a class $k\hspace{-2pt}=\hspace{-2pt}3$ Weibull distribution with average speed $5m/s$.
We only consider 6 types for the seller here:\vspace{-6pt}
\begin{eqnarray}
&a&\hspace{-10pt}=\hspace{-2pt}(c_0\hspace{-4pt}=\hspace{-4pt}4,\theta_w\hspace{-4pt}=\hspace{-4pt}0.2,\theta_c\hspace{-4pt}=\hspace{-4pt}1.2,v_{ci}\hspace{-4pt}=\hspace{-4pt}3,v_{r}\hspace{-4pt}=\hspace{-4pt}13,v_{co}\hspace{-4pt}=\hspace{-4pt}20,\gamma\hspace{-4pt}=\hspace{-4pt}1),\nonumber\\
&b&\hspace{-10pt}=\hspace{-2pt}(c_0\hspace{-4pt}=\hspace{-4pt}4,\theta_w\hspace{-4pt}=\hspace{-4pt}0.2,\theta_c\hspace{-4pt}=\hspace{-4pt}1.2,v_{ci}\hspace{-4pt}=\hspace{-4pt}3,v_{r}\hspace{-4pt}=\hspace{-4pt}13,v_{co}\hspace{-4pt}=\hspace{-4pt}20,\gamma\hspace{-4pt}=\hspace{-4pt}2),\nonumber\\
&c&\hspace{-10pt}=\hspace{-2pt}(c_0\hspace{-4pt}=\hspace{-4pt}5,\theta_w\hspace{-4pt}=\hspace{-4pt}0.1,\theta_c\hspace{-4pt}=\hspace{-4pt}1.2,v_{ci}\hspace{-4pt}=\hspace{-4pt}3,v_{r}\hspace{-4pt}=\hspace{-4pt}13,v_{co}\hspace{-4pt}=\hspace{-4pt}20,\gamma\hspace{-4pt}=\hspace{-4pt}1),\nonumber\\ &d&\hspace{-10pt}=\hspace{-2pt}(c_0\hspace{-4pt}=\hspace{-4pt}5,\theta_w\hspace{-4pt}=\hspace{-4pt}0.2,\theta_c\hspace{-4pt}=\hspace{-4pt}1.0,v_{ci}\hspace{-4pt}=\hspace{-4pt}1,v_{r}\hspace{-4pt}=\hspace{-4pt}17,v_{co}\hspace{-4pt}=\hspace{-4pt}28,\gamma\hspace{-4pt}=\hspace{-4pt}2),\nonumber\\ &e&\hspace{-10pt}=\hspace{-2pt}(c_0\hspace{-4pt}=\hspace{-4pt}6,\theta_w\hspace{-4pt}=\hspace{-4pt}0.1,\theta_c\hspace{-4pt}=\hspace{-4pt}1.0,v_{ci}\hspace{-4pt}=\hspace{-4pt}1,v_{r}\hspace{-4pt}=\hspace{-4pt}17,v_{co}\hspace{-4pt}=\hspace{-4pt}28,\gamma\hspace{-4pt}=\hspace{-4pt}1),\nonumber\\ &f&\hspace{-10pt}=\hspace{-2pt}(c_0\hspace{-4pt}=\hspace{-4pt}6,\theta_w\hspace{-4pt}=\hspace{-4pt}0.1,\theta_c\hspace{-4pt}=\hspace{-4pt}1.0,v_{ci}\hspace{-4pt}=\hspace{-4pt}1,v_{r}\hspace{-4pt}=\hspace{-4pt}13,v_{co}\hspace{-4pt}=\hspace{-4pt}28,\gamma\hspace{-4pt}=\hspace{-4pt}2),\nonumber
\end{eqnarray} where the cost unit is $\$1000$ and the energy unit is $MWh$, and there is no worst type. The optimal contract from Theorem 1 is depicted in Figure \ref{example2}. It is a nonlinear pricing scheme. The marginal price varies between $0.33$ and $0.45$ $\$/KWh$. The variation in the marginal price is of the same order as the variation in the expected marginal production cost across the seller's different types.
\begin{figure}[h!]
\centering
\includegraphics[width=0.51\textwidth]{Ex1-2}
\caption{Example - the optimal pricing scheme}
\vspace{-5pt}
\label{example2}
\end{figure}
Since there is no complete order among the different seller's types, we can not compare the seller's types based on the expected revenue or amount of production prior to the design of the mechanism.
However, wherever we have a partial order and can rank a subset of types, we can utilize Lemmas 1 and 3 to predict a ranking a priori. For instance, the seller with type $(b)$ is better than the one with type $(a)$, and we can say a priori that the former has a higher production and revenue. However, we cannot order types $(b)$ and $(c)$.
For the setup of our example, according to the optimal contract, type $(c)$ gets a higher expected revenue than type $(b)$ but produces a lower amount of energy than type $(b)$.
\section{Further Considerations}
\subsection{Commitment and Ex-post Voluntary Participation}
The voluntary participation constraint imposed in problem (P1) is interim. That is, the expected profit with respect to the weather for each type of the seller must be non-negative. By assumption (A9), once the seller agrees on the contract (this agreement takes place before the realization of the weather) she is fully committed to following the agreement, even if the realized profit is negative (because of the realization of the weather)\footnote{Since the seller's reserve utility is zero by not participating (outside option), we can always think of the seller walks away from the agreement for these negative profit realizations and not follow the mechanism rules.}.
To ensure that the seller's realized profit is non-negative for all weather realizations, we impose an ex-post voluntary participation constraint. We replace the interim VP constraint (7) by
\vspace{-1pt}
\begin{eqnarray}
&\hspace{-20pt}\text{Ex-post VP:}&\hspace{-5pt}t(m^*)-C(q,w,x)\geq 0,\forall w,\;\forall x\hspace{-2pt}\in\hspace{-2pt}\chi.
\label{P1'-VP}
\end{eqnarray}
\vspace{-2pt}
To obtain an ex-post voluntary participation constraint, we modify the payment function of the mechanism given by Theorem 1 as follows:
\vspace{-1pt}
\begin{eqnarray}
\tilde{t}(q,w)\hspace{-2pt}=\hspace{-2pt}t(q)-t(q(\underline{x}))+C(q(\underline{x}),w,\underline{x}).
\end{eqnarray}
\vspace{-1pt}
We have $\mathbb{E}_W\hspace{-3pt}\left\{\tilde{t}(q,w)\hspace{-1pt}\right\}\hspace{-3pt}=\hspace{-3pt}t(q)$, and therefore, the seller always chooses the same quantity $q$ under the modified payment function $\tilde{t}(\cdot)$ as under the original payment function $t(q)$ given by (\ref{optt-gen}). Note that for the seller's worst type, we have $\tilde{t}(q(\underline{x}),\hspace{-1pt}w)\hspace{-2pt}=\hspace{-2pt}C(q(\underline{x}),\hspace{-1pt}w,\underline{x})$, and therefore, the total utility of the seller's worst type is zero for all realizations of $W$. Since all other types of the seller are better off than the worst type under $\tilde{t}(\cdot)$ (all types have the choice to produce the same quantity as the worst type)
, the ex-post VP constraint is satisfied for all of the seller's types.
\vspace{-5pt}
\subsection{Risk Allocation}
In the optimal mechanism/contract menu presented by Theorem 1, the buyer faces no uncertainty, and he is guaranteed to receive quantity $q(x)$, and all the risk associated with the realization of the weather is taken by the seller. We wish to modify the mechanism to reallocate the above-mentioned risk between the buyer and the seller. To do so, we modify the payment function so that the risk is reallocated between the buyer and the seller. Consider the following modified payment function with $\alpha\hspace{-2pt}\in\hspace{-2pt}[0,1]$,
\vspace{-1pt}
\begin{eqnarray}
&\hat{t}(x,w)=&t(q(x))+\alpha\left[C(q(x),w,x)\right.\nonumber\\&&\left.-\mathbb{E}_W\left\{C(q(x),w,x)\right\}\right]\label{t-risk}.
\end{eqnarray}
From (\ref{t-risk}) it follows that $\mathbb{E}_W\hspace{-3pt}
\left\{\hat{t}(x,w)\right\} \hspace{-3pt}=\hspace{-2pt}t(q(x))$. Therefore, the strategic behavior of the seller does not change and the seller chooses the same quantity under the modified payment function $\hat{t}(\cdot)$ as under the original payment function $t(q)$ given by (\ref{optt-gen}). Note that for $\alpha=0$ we have the same payment as $t(q)$. For $\alpha=1$, the seller is completely insured against any risk and all the risk is taken by the buyer. The parameter $\alpha$ determines the allocation of the risk between the buyer and the seller; the buyer undertakes $\alpha$ and the seller undertakes $(1-\alpha)$ share of the risk.
\section{Introduction}
The intermittent nature of modern renewable energy resources makes the integration of modern renewable energy generation into the current designed infrastructure for conventional energy generation a challenging problem. Electricity generation from modern renewable energy resources is not predetermined and cannot be treated as conventional generation. Energy generation from wind, solar, and other modern renewable energy resources depends on the weather and is stochastic. This feature results in many technical issues in reliability and stability in generation and transmission, as well as market structure.
Currently, modern renewable energy generators that participate in the regular real-time power market, that is originally designed for conventional generators, are treated as negative loads on the grid, and receive subsidy for each kWh they deliver \cite{east,west}. Due to the increasing share of wind power in energy markets, independent system operators (ISOs) gradually require wind power plants to participate in the day-ahead market, to commit to a fixed amount of generation, and to pay a penalty for each $KWh$ they fail to provide \cite{fink11}.
Along with the current approach, the proposed structure of the smart grid creates additional opportunities to integrate conventional and modern renewable energy resources via flexible loads connected to the grid.
The challenges that arise in the market aspect of the integration of renewable and conventional energy resources motivate the problem formulated and studied in this paper. We follow the current approach for integration of conventional and modern renewable energy resources. This approach requires modern renewable generators to behave as firm power plants that participate in the day-ahead market (in general, a forward market) and commit to producing a predetermined amount of electricity in advance. We want to incorporate new features into our problem formulation, which are mainly motivated by the following observations: (1) Currently, modern renewable generators are paid at a fixed rate, receive subsidy, and do not take any risk, therefore, no strategic behavior is considered. However, as the share of modern renewable generation increases and supportive programs for modern renewable energy decreases, the market becomes more competitive and participants behave strategically. (2) The cost of energy generation varies from 60 to 250 $(\$/MWh)$ due to the different technologies that are available at wind power plants \cite{windcost}. These technologies are the plants' private information. Because modern renewable generators have private production cost and behave strategically, the current market structure, where renewable generators are paid at a fixed rate, receive subsidies, and all the produced electricity is guaranteed to be procured at the real-time market with no risk, is inefficient.
In this paper we consider a model for integration of conventional and modern renewable energy sources with the following features: (F1) Energy market participants are strategic. (F2) The energy seller has the ability to produce both modern renewable and conventional energy. (F3) Energy producers may have different technologies for renewable and conventional energy generation which are their own private information.
By considering a seller with generation capabilities from both conventional and modern renewable energy resources, we postulate a possible future scenario in which the integration of modern renewable and conventional generation is partly done by sellers and the ISO is not fully responsible for the integration. Our model captures the current approach to the integration of conventional and modern renewable energy resources, that considers sellers with only modern renewable generation capability, if we set the cost of conventional generation to be fixed and equal to the penalty rate for each $KWh$ producers fail to provide.
\vspace{-8pt}
\subsection{Literature Review}
Research on resource allocation and resource management with uncertainty in energy markets has addressed two types of problems:
(1) Those which follow the smart grid approach and focus on the demand side with strategic (\text{e.g.} \cite{fahrioglu}) or non-strategic agents (\textit{e.g.} \cite{lina}).
(2) Those where the focus is on the supply side and energy providers face uncertainties in their production.
The problem formulated in this paper belongs to the second class where energy providers are strategic and have private information about their own cost and technology. Research on this class of problems has appeared in \cite{chao},\cite{randomselling},\cite{bitar}, and \cite{jain}. The work in \cite{chao} considers a given uncertain demand and investigates a multi-dimensional auction mechanism for the forward reserved market assuming that the participants have no market power and the equilibrium price is not affected by each individual participant's behavior.
The idea of selling uncertain (random) power to consumers is investigated in \cite{interruppower} and \cite{randomselling}. The problems studied in \cite{bitar} and \cite{jain} consider modern renewable generation and are the most closely related works to the problem we consider here. The work in \cite{bitar} considers a modern renewable generator with stochastic generation and determines the optimal bidding policy for it in the day-ahead market. The authors of \cite{bitar} assume that price is given and fixed, so there is no role for a buyer, and there is a penalty for production deficiency and over production. The work in \cite{jain} considers a problem where an ISO wants to procure energy from modern renewable generators with the assumption that generation from renewable energy resources is free and each generator has private information about the probability density function of its generation; a VCG-based mechanism is proposed for the optimal energy procurement. The proposed VCG-based mechanism is suboptimal for the problem formulated in \cite{jain} when the probability distribution for the generation cannot be parameterized by only a one-dimensional variable (see \cite{krishna}, chapter 14).
From the economics point of view, the problem we formulate in this paper belongs to the class of screening problems. In economics, the one-dimensional screening problem has been well-studied with both linear and nonlinear utility functions \cite{tilman}. However, the extension to the multi-dimensional screening problem is not straightforward and there is no general solution to it. The authors in \cite{multiscreening} study a general framework for a static multi-dimensional screening problem with linear utility. They discuss two general approaches, the parametric-utility approach and the demand-profile approach. The methodology we use to solve the problem formulated in this paper is similar to the demand-profile approach. We consider a multi-dimensional screening problem with nonlinear utilities. The presence of nonlinearities results in additional complications which are not present in \cite{multiscreening} where the utilities are linear\footnote{When a problem is linear, expectation of any random variable can be replaced by its expected value and reduce the problem to a deterministic one.}.
\vspace{-7pt}
\subsection{Contribution}
The contribution of this paper is two-fold. First, we introduce a model that captures key features of the current approach for the integration of modern renewable energy sources into the grid, and postulate a possible broader role for sellers in the future. We consider a strategic seller that has the capability to produce energy from modern renewable and conventional resources. To our knowledge, this is the first model that considers simultaneously both types of electricity generation. As we discuss below, considering both modern renewable and conventional generation with a general production cost results in a multi-dimensional mechanism design problem which is conceptually different from the one-dimensional screening problem that arises when there is only one type of energy resource with simple production cost. The model proposed in this paper captures the scenarios investigated in \cite{jain} and \cite{bitar} as special cases; in \cite{bitar} and \cite{jain} the seller owns only modern renewable generators with free generation and is penalized for production mismatch.
Second, we determine the optimal mechanism for energy procurement from a strategic seller with multi-dimensional private information satisfying both interim and ex-post voluntary participation of the seller. Because of the random nature of renewable energy generation, interim voluntary participation of the seller does not necessarily imply ex-post voluntary participation. To our knowledge, our results present the first optimal mechanism for a strategic seller with conventional and renewable generation, and multi-dimensional private information which also guarantees the seller's ex-post voluntary participation. We show that the current linear pricing for modern renewable energy generation is not efficient and that the optimal pricing method is a nonlinear scheme.
\subsection{Organization}
The paper is organized as follows: In Section 2, we introduce the model and formulate the energy procurement problem. In section 3, we present an outline of our approach and the key ideas toward the solution of the problem formulated in section 2, and state the main result on the optimal mechanism for energy procurement. We illustrate the result by an example in section 4. We discuss the nature of our results in section 5. We extend our results to the energy procurement problem without full commitment for the seller, and propose an optimal contract with arbitrary risk allocation between the buyer and the seller in section 6. We conclude in section 7.
\section{Model Specification and Problem Formulation}
\subsection{Model Specification}
A buyer wishes to make an agreement to buy energy from an energy seller\footnote{From now on, we refer to the buyer as ``he'' and to the seller as ``she''.}.
The seller has the ability to produce energy from conventional (deterministic) generators or from renewable (random) generators that she owns.
Let $q$ be the amount of energy the buyer buys, and $t$ be his payment to the seller. We proceed to formulate the energy procurement problem by making the following assumptions.
\vspace{5pt}
\textbf{(A1)}
The buyer is risk-neutral and his total utility is given by $\mathcal{V}(q)-t$, where $\mathcal{V}(q)$ is the utility that he gets from receiving $q$ amount of energy, and $\mathcal{V}(0)=0$. $\mathcal{V}(\cdot)$ is the buyer's private information\footnote{We assume that the buyer either has an elastic demand, or needs to meet some fixed demand and has an outside option to buy energy if he cannot buy it from the seller (which is the result of his interaction with other players in the market).}.
\vspace{5pt}
\textbf{(A2)} The seller's production cost is given by $C(q,w,x)$, where $x\hspace{-2pt}\in\hspace{-2pt}\chi\hspace{-2pt}\subseteq\hspace{-2pt}\mathbb{R}^n$ is the seller's type (technology and cost) and $w$ denotes the realization of a random variable $W$, \textit{e.g.} weather. $C(q,w,x)$ is convex and increasing in $q$. The start-up cost $C(0,w,x)$ does not depend on the weather $w$ and is given by $x_1$, \textit{i.e.} $C(0,w,x)\hspace{-2pt}=\hspace{-2pt}C(0,x)\hspace{-2pt}=\hspace{-2pt}x_1$.
\vspace{5pt}
\textbf{(A3)} The probability distribution function of $W$, \textit{i.e.} weather forecast, is common knowledge between the buyer and the seller and is given by $F_{W}(w)$.
\textbf{(A4)}
The seller is risk-neutral and her utility is given by her total expected revenue
\begin{equation}
\mathbb{E}_W\left\{t-C(q,W,x)\right\}.
\end{equation}
\vspace{5pt}
\textbf{(A5)}
Define $c(q,x)\hspace{-2pt}=\hspace{-2pt}\frac{\partial \mathbb{E}_W\left\{C(q,W,x)\right\}}{\partial q}$ as the expected marginal cost for the seller's type $x$. Without loss of generality, there exists $m$, $1\hspace{-2pt}<\hspace{-2pt} \hspace{-2pt}m\hspace{-2pt}\leq n$, such that $c(q,x)$ is increasing in $x_i$ for $1\hspace{-2pt}\leq\hspace{-2pt} i\hspace{-2pt}\leq\hspace{-2pt} m$, and decreasing in $x_i$ for $m\hspace{-2pt}<\hspace{-2pt}i\hspace{-2pt}\leq\hspace{-2pt} n$. Moreover, there is an $\underline{x}\hspace{-2pt}\in\hspace{-2pt}\chi$ (the seller's worst type) such that $\underline{x}_i\hspace{-2pt}\leq\hspace{-2pt} x_i$ and $\underline{x_j}\hspace{-2pt}\geq\hspace{-2pt} x_j$ for all $x\hspace{-2pt}\in\hspace{-2pt}\chi$, $1\hspace{-1pt}\leq \hspace{-2pt}i\hspace{-2pt}\leq\hspace{-2pt} m$ and $m\hspace{-2pt}<\hspace{-2pt}j\hspace{-2pt}\leq \hspace{-2pt}n$.
\vspace{5pt}
\textbf{(A6)} The seller's type $x$ is her own private information, the set $\chi$ is common knowledge, and there is a prior probability distribution $f_X$ over $\chi$ which is common knowledge between the buyer and the seller.
\vspace{5pt}
\textbf{(A7)}
Both the buyer and the seller are strategic and perfectly rational, and this is common knowledge.
\vspace{5pt}
\textbf{(A8)}
The buyer has all the bargaining power; thus, he can design the mechanism/set of rules that determines the agreement for energy procurement quantity $q$, and payment $t$\footnote{The buyer is either an ISO or a representative agent for aggregate demand. In the first case, it is realistic to assume that the ISO has all the bargaining power since he is the designer and the regulator of the energy market. In the later case, usually there is no competition on the demand side, but different sellers compete to win a contract with the demand side. Therefore, it is reasonable to assume that in a non-cooperative setting, the demand side has all the bargaining power.}.
\vspace{5pt}
\textbf{(A9)}
After the buyer announces the mechanism for energy procurement and the seller accepts it, both the buyer and the seller are fully committed to following the rules of the mechanism.
As a consequence of assumption (A8) on the buyer's bargaining power and the fact that the seller's utility does not directly depend on the buyer's private information, the solution of the problem formulated in this paper does not depend on whether the buyer's utility $\mathcal{V}(\cdot)$ is private information or common knowledge\footnote{This becomes more clear by looking at the main result given by Theorem 1.}.
Note that in the one-dimensional screening problem, the cost of production induces a complete order among the seller's types, which is crucial to the solution to the optimal mechanism design problem. However, in multi-dimensional screening problems, the expected cost of production induces, in general, only a partial order among the seller's types.
\begin{mydef}
We say the seller's type $x$ is better (resp. worse) than the seller's type $\hat{x}$ if $\mathbb{E}_W\left\{C(q,W,x)\right\}\hspace{-2pt}\leq\hspace{-2pt} \mathbb{E}_W\left\{C(q,W,\hat{x})\right\}$ for all $q\hspace{-2pt}\geq\hspace{-2pt}0$ (resp. $\mathbb{E}_W\left\{C(q,W,x)\right\}\hspace{-2pt}\geq\hspace{-2pt} \mathbb{E}_W\left\{C(q,W,\hat{x})\right\}$) with strict inequality for some $q$.
\end{mydef}
From (A5), the seller's type $x$ is better than the seller's type $\hat{x}$ if and only if $x_i\hspace{-2pt}\leq \hspace{-2pt}\hat{x}_i$ for $1\hspace{-2pt}\leq\hspace{-2pt} i\hspace{-2pt}\leq\hspace{-2pt} m$, and $x_i\hspace{-2pt}\geq\hspace{-2pt} \hat{x}_i$ for $m\hspace{-2pt}<\hspace{-2pt} i\hspace{-2pt}\leq\hspace{-2pt} n$ with strict inequality for some $i$. The following example illustrates assumption (A2)-(A5) and Definition 1.
\textbf{A simple case.} Consider a seller with a wind turbine and a gas generator. The generation from the wind turbine is free and given by $\gamma w^3$, where $\gamma$ is the turbine's technology and $w$ is the realized weather. The gas generator has a fixed marginal cost $\theta_c$. We assume that there is a fixed cost $c_0$ which includes the start-up cost for both plants and the capital cost for the seller. Therefore, the seller's type has $n=3$ dimensions. The generation cost for the seller is given by
\begin{eqnarray}
C(q,w,x)=c_0+\theta_c\max\left\{q-\gamma w^3,0\right\}
\end{eqnarray}
The seller's type $x=(c_0,\theta_c,\gamma)$ is better than the seller's type $\hat{x}=(\hat{c}_0,\hat{\theta}_c,\hat{\gamma})$ if and only if $c_0\leq\hat{c}_0$, $\theta_c\leq \hat{\theta}_c$, and $\gamma\geq \hat{\gamma}$ with one of the above inequalities being strict.
\vspace{-3pt}
\subsection{Problem Formulation}
Let $(\mathcal{M},h)$ be the mechanism/game form (see \cite{mascolell}, Ch. 23) for energy procurement designed by the buyer. In this game form, $\mathcal{M}$ describes the message/strategy space and $h$ determines the outcome function; $h:\mathcal{M}\rightarrow\mathbf{R}_+\times\mathbf{R}$ is such that for every message/action $m\in\mathcal{M}$ it specifies the amount $q$ of procured energy and the payment $t$ made to the seller, \text{i.e.} $h(m)=(q(m),t(m))=(q,t)$.
The objective is to determine a mechanism $(\mathcal{M},h)$ so as to
\begin{equation}
\underset{\left(\mathcal{M},(q,t)\right)}{\textnormal{maximize}}\quad \mathbb{E}_{X,W}\left\{\mathcal{V}(q)-t\right\}
\label{P1}
\end{equation}
under assumptions (A1)-(A9) and the constraint that the seller is willing to voluntarily participate in the energy procurement process. The willingness of the seller to voluntarily participate in the mechanism for energy procurement is called voluntary participation (VP) (or individual rationality) and is written as
\begin{eqnarray}
\text{VP:}\quad t(m^*)-\mathbb{E}_W\left\{C(q(m^*),W,x)\right\}\geq0,\quad\forall x\in \chi
\end{eqnarray}
where $m^*\in\mathcal{M}$ is a Bayesian Nash equilibrium (BNE) of the game induced by the mechanism $(\mathcal{M},h)$. That is, at equilibrium the seller has a non-negative payoff.
We call the above problem \textbf{(P1)}.
\section{Outline of the Approach \& Results}
We prove that the optimal energy procurement mechanism is a pricing scheme that the buyer offers to the seller and the seller chooses a production quantity based on her type. We characterize the optimal energy procurement mechanism by the following theorem, which reduces the original functional maximization problem (P1) to a set of equivalent point-wise maximization problems.
\begin{theorem}
The optimal mechanism $(q,t)$ for the buyer is a menu of contracts (nonlinear pricing) given by
\begin{eqnarray}
\hspace{-20pt}&p(q)=&\textnormal{arg}\max_{\hat{p}}\left\{P\left[x\in\chi|
{\hat{p}}\geq c(q,x)\right]\left(\mathcal{V}'(q)-\hat{p}\right)\right\}\label{opt-p}\hspace{-3pt},\\
\hspace{-20pt}&t(q)=&\int_{0}^q{p(l)dl}+C(0,\underline{x}),\label{optt-gen}\\
\hspace{-20pt}&q(x)=&\textnormal{arg}\max_{l\in\mathbf{R}_+}\mathbb{E}\left\{t\left(l\right)-\mathbb{E}_W\left\{C(l,x,w)\right\}\right\}
\end{eqnarray}
where $\mathcal{V}'(q):=\frac{d\mathcal{V}(q)}{dq}$.
\end{theorem}
The proof of theorem 1 proceeds is several steps. Below we present these steps and the key ideas behind each step. The detailed proof of all results (theorems and lemmas) appearing below can be found in the appendix.
\vspace{5pt}
\textbf{Step 1.} By invoking the revelation principle \cite{dasgupta-revelation}, we restrict attention, without loss of optimality, to direct revelation mechanisms that are incentive compatible (defined below) and individually rational.
\begin{mydef}
A direct revelation mechanism is defined by functions $q:\chi\rightarrow\mathbf{R}_+$ and $t:\chi\rightarrow\mathbf{R}_+$,
and works as follows:
First, the buyer announces functions $q$ and $t$. Second, the seller declares some $x'\in\chi$ as her report for her technology and cost. Third, the seller is paid $t(x')$ to deliver $q(x')$ amount of energy.
\end{mydef}
The seller is strategic and may lie and misreport her private information $x$, \textit{i.e.} we do not necessarily have $x'=x$ unless it is to the seller's interest to report truthfully. We call incentive compatibility (IC) the requirement for truthful reporting, and define the following problem that is equivalent to (P1).
\vspace{5pt}
\textbf{Problem P2:} Determine functions $q:\chi\rightarrow\mathbf{R}_+$ and $t:\chi\rightarrow\mathbf{R}_+$ so as to
\vspace{-5pt}
\begin{eqnarray}
&&{\underset{(q,t)}{\textnormal{maximize}}}\quad \mathbb{E}_{x,W}\left\{\mathcal{V}(q(x))-t(x)\right\}\\
&&\textit{subject to}\nonumber\\&&\hspace{-25pt}IC: x\hspace{-2pt}=\hspace{-2pt}\textnormal{arg}\max_{x'}\mathbb{E}_W\hspace{-2pt}\left[t(x')\hspace{-1pt}-\hspace{-1pt}C(q(x'),w,x)\right],\forall x\hspace{-2pt}\in\hspace{-2pt}\chi\\
&&\hspace{-25pt}VP: \mathbb{E}_W\left[t(x)-C(q(x),W,x)\right]\geq0,\;\forall x\hspace{-2pt}\in\hspace{-2pt}\chi.
\end{eqnarray}
\vspace{5pt}
\textbf{Step 2.} We utilize the partial order among the seller's different types to rank the seller's utility for her different types, and reduce the VP constraint for all the seller's types to the VP constraint only for the seller's worst type.
\begin{lemma}
For a given mechanism $(q,t)$, a better type of the seller gets a higher utility. That is, let $U(x):=\mathbb{E}_W\left\{t(q(x))-C(q(x),W,x)\right\}$ denote the
expected profit of the seller with type $x$. Then,
\begin{enumerate}
\item $\frac{\partial U}{\partial x_i}\leq 0, 1\leq i\leq m$,
\item $\frac{\partial U}{\partial x_i}\geq 0, m< i\leq n$.
\end{enumerate}
\label{ut-order}
\end{lemma}
A direct consequence of Lemma \ref{ut-order}, is that the seller's worst type $\underline{x}$ receives the minimum utility among all the seller's types.
\begin{corollary}
The voluntary participation constraint is only binding for the worst type $\underline{x}$, that is the general VP constraint (7) can be reduced to
\begin{eqnarray}
U(\underline{x}):=\mathbb{E}\left\{t(q(\underline{x}))-C(q(\underline{x}),w,\underline{x})\right\}=0.
\end{eqnarray}
\end{corollary}
\vspace{5pt}
\textbf{Step 3.} We show, via Lemma 2 below, that without loss of optimality, we can restrict attention to a set of functions $t(\cdot)$ that depend only on the amount of energy $q$. That is, the optimal mechanism is a pricing scheme.
\begin{lemma}
For any pair of functions $(q,t)$ that satisfies the IC constraint, we can rewrite $t(x')$ as $t\left(q(x')\right)$.\label{pricing}
\end{lemma}
\textbf{Step 4.} As a consequence of Lemma \ref{pricing}, we determine an optimal mechanism sequentially. First, we determine the optimal payment function $t(\cdot)$, then the optimal energy procurement function $q(\cdot)$. For any function $t(\cdot)$, we determine, for each type $x$ of the seller, the optimal quantity $q^*(x)$ that she wishes to produce as
\vspace{-15pt}
\begin{eqnarray}
&&q^*(x)=\text{arg}\max_{l}\mathbb{E}_W\left\{t\left(l\right)- C(l,w,x)\right\}\label{agent-q}.
\end{eqnarray}
\vspace{-15pt}
Incentive compatibility then requires that the seller must tell the truth to achieve this optimal value, and cannot do better by lying, \emph{i.e.} $q(x)=q^*(x)$ for all $x\in\chi$. For any function $t(\cdot)$, this last equality can be taken as the definition for the associated function $q(\cdot)$. Thus, we eliminate the IC constraint by defining $q(\cdot):=q^*(\cdot)$ and reduce the problem of designing the optimal direct revelation mechanism $(q,t)$ to an equivalent problem where we determine only the optimal payment function $t(\cdot)$ subject to the voluntary participation constraint for the worst type.
\vspace{5pt}
\textbf{Step 5.} To solve this new equivalent problem, we write the buyer's expected utility as the integration of his marginal expected utility, and express the marginal expected utility in terms of the marginal price $p(q):=\frac{\partial t(q)}{\partial q}$ and the minimum payment $t(0)$ (which along with $p(\cdot)$ uniquely determines the payment function $t(\cdot)$). Specifically, in the appendix we show that
\begin{eqnarray}
&\hspace{-90pt}\mathbb{E}_{X}\hspace{-2pt}\left[\mathcal{V}(q^*(X))\right]\hspace{-2pt}-\hspace{-2pt}\mathbb{E}_{X}\hspace{-2pt}\left[t(q^*(X))\right]\hspace{-3pt}=\nonumber\\&\hspace{15pt}\int_{0}^{\infty}\hspace{-3pt}{P\left(x\in\chi | q^*(x)\hspace{-2pt}\geq\hspace{-2pt} l\right)\mathcal{V}'(l)dl}\nonumber\\&\hspace{40pt}-t(0)-\hspace{-3pt}\int_{0}^{\infty}\hspace{-3pt}{P\left(x\hspace{-1pt}\in\hspace{-1pt}\chi | q^*(x)\hspace{-2pt}\geq\hspace{-2pt} l\right)p(l)dl}\label{buyer-max1},
\hspace{-1pt}
\end{eqnarray}
where $\mathcal{V}'\hspace{-1pt}(q)\hspace{-3pt}:= \hspace{-3pt}\frac{d \mathcal{V}(q)}{dq}$.
That is, the buyer's total expected utility is obtained by integrating his marginal utility at quantity $l$, times the probability that the seller's production exceeds $l$, and subtracting the minimum payment $t(0)$.
We show in the appendix that the seller's optimal decision $q^*(x)$ depends only on the marginal price $p(q)$. Thus,
we can write the probability associated with the seller's decision as
\begin{eqnarray}
P\left(x\in\chi | q^*(x)\geq l\right)=P\left[x\in\chi|
p(l)\geq c(l,x)\right]\label{seller-p}.
\end{eqnarray}
That is, the seller is willing to produce the marginal quantity at $l$ if the resulting expected marginal profit is positive, \textit{i.e.} marginal price $p(l)$ exceeds marginal expected cost of generation $c(l,x)$\footnote{This relies on quasi-concavity of the seller's optimal decision program. This is a standard assumption in the literature, e.g. see \cite{multiscreening} and \cite{nonlinear}. Basically, it gives the seller the freedom to decide for each marginal unit of production independently. Therefore, the continuity of the resulting generation quantity must be checked posterior to the design of the optimal contract for each type of the seller.}. Using (\ref{buyer-max1}) and (\ref{seller-p}), we define the following problem that is equivalent to (P2) and is in terms of the marginal price $p(q)$ and the minimum payment $t(0)$.
\vspace{5pt}
\textbf{Problem P3:}
\begin{eqnarray}
&\hspace{-40pt}{\underset{p(\cdot),t(0)}{\textnormal{max}}}&\hspace{-30pt}
\int_{0}^{\infty}\hspace{-9pt}{P\hspace{-2pt}\left[x\hspace{-2pt}\in\hspace{-2pt}\chi|
p(l)\hspace{-2pt}\geq\hspace{-2pt} c(l,x)\right]\hspace{-2pt}\left(\mathcal{V}'(l)\hspace{-2pt}-\hspace{-2pt}p(l)\right)\hspace{-1pt}dl}\hspace{-2pt}-\hspace{-2pt}t(0)\\
&\hspace{-10pt}\textit{subject to}&\nonumber\\&\hspace{-46pt} \text{VP:}&\hspace{-37pt}\mathbb{E}_W\hspace{-2pt}\left\{t(0)\hspace{-1pt}+\hspace{-2pt}\int_0^{q(\underline{x})}\hspace{-10pt}p(r)dr-C(q^*(\underline{x}),w,\underline{x})\right\}\hspace{-4pt}\geq\hspace{-2pt}0.\label{p4-vp}
\end{eqnarray}
\textbf{Step 6.} We provide a ranking for the seller's optimal decision $q^*(x)$ based on the partial order among the seller's types.
\begin{lemma}
For a given mechanism specified by $(t(\cdot),q(\cdot))$, a better type of the seller produces more. That is, the optimal quantity $q^*(x)$ that the seller with true type $x$ wishes to produce satisfies the following properties:
\begin{description}
\item[a)] $\frac{\partial q^*(x)}{\partial x_i}\leq 0,1\leq i\leq m$,
\item[b)] $\frac{\partial q^*(x)}{\partial x_i}\geq 0, m<x\leq n$.
\end{description}
\end{lemma}
In the appendix we show that a consequence of corollary 1 and Lemma 3 is the following result.
\begin{corollary}
The VP constraint is satisfied if $t(0)\hspace{-3pt}=\hspace{-3pt}C(0,\hspace{-1pt}\underline{x})$ and the lowest seller's type payment is equal to her expected production cost, \textit{i.e.} $t(q(\underline{x}))\hspace{-2pt}=\hspace{-2pt}\mathbb{E}_W\hspace{-2pt}\left\{C(q(\underline{x}),W,\underline{x})\right\}$.
\end{corollary}
Based on corollary 2, we define a problem (P4) that is equivalent to (P3) and is only in terms of the marginal price $p(l)$ and the constraint that the payment the seller's lowest type receives is equal to her cost of production.
\vspace{5pt}
\textbf{Problem (P4)}
\begin{eqnarray}
&\hspace{-50pt}{\underset{p(\cdot)}{\textnormal{max}}}&\hspace{-30pt} \int_{0}^{\infty}{\hspace{-5pt}P\left[x\in\chi|
p(l)\geq c(l,x)\right]\left(\mathcal{V}'(l)-p(l)\right)dl}\\
&\hspace{-15pt}\text{subject to}&\nonumber\\
&&\hspace{-50pt}\text{VP:}\hspace{4pt}
C(0,\underline{x})+\int_0^{q(\underline{x})}\hspace{-5pt}{p(l)dl}=\mathbb{E}_W\left[C(q(\underline{x}),W,x)\right]\label{P4-VP}.
\label{p5-con}
\end{eqnarray}
\textbf{Step 7.} We consider a relaxed version of (P4) without the VP constraint (\ref{P4-VP}). The unconstrained problem can be solved point-wise at each quantity $l$ to determine the optimal $p(l)$ as
\vspace{-3pt}
\begin{eqnarray}
\hspace{-20pt}&p(l)=&\hspace{-5pt}\textnormal{arg}\max_{\hat{p}}\left\{P\left[x\hspace{-1pt}\in\hspace{-1pt}\chi|
{\hat{p}}\geq c(q,x)\right]\hspace{-1pt}\left(\mathcal{V}'(q)-\hat{p}\right)\hspace{-1pt}\right\}\label{margp-gen}\hspace{-3pt}
\end{eqnarray}
\vspace{-1pt}
which is the same as (\ref{opt-p}). We show in the appendix that the solution to the unconstrained problem automatically satisfies the VP constraint (\ref{P4-VP}), therefore, the solution to the unconstrained problem determines the optimal marginal price $p(\cdot)$ for the original problem.
Using $p(\cdot)$ and the fact that $t(0)=C(0,\underline{x})$, we determine the optimal function $t(\cdot)$. Using $t(\cdot)$ we find the seller's best response function $q^*(\cdot)$. By incentive compatibility $q^*(\cdot)=q(\cdot)$, and this completely determines the optimal direct revelation mechanism $(q(\cdot),t(\cdot))$ described by Theorem 1.
In essence, Theorem 1 states that at each quantity $l$, the optimal marginal price $p(l)$ is chosen so as to maximize the expected total marginal utility at $l$, which is given by the total marginal utility $\left(\mathcal{V}'(l)-p(l)\right)$ times the probability that the seller generates at least $l$.
\vspace{5pt}
\begin{remark}
In a setup with startup cost for the seller, it might not be optimal for the buyer to require all the seller's types to voluntarily participate in the energy procurement process, since the minimum payment $t(0)$ depends on the production cost of the seller's worst type. In such cases, it might be optimal for the buyer to exclude some ``less efficient'' types of the seller from the contract, select an admissible set of seller's types, and then design the optimal contract for this admissible set of the seller's types\footnote{To find the optimal admissible set, the optimal contract can be computed for different potential admissible sets. Then, the resulting utilities can be compared to find the best admissible set.}. Note that, this is not the case for setups without startup cost. In such setups, if it is not optimal for some type $x$ to be included in the optimal contract, it is equivalent to have $q(x)=0$ for the optimal contract that considers all types of the seller.
\end{remark}
\begin{remark}
In problem (P1), we assume that there exists a seller's worst type which has the highest cost at any quantity among all the seller's types, and we reduce the VP constraint for all the seller's type to only the VP constraint for this worst type. As a result, we pin down the optimal payment function by setting $t(0)=C(0,\underline{x})$ to ensure the voluntary participation of the worst type, which consequently implies the voluntary participation for all the seller's types. In absence of the assumption on the existence of the seller's worst type, the argument used to reduce the VP constraint is not valid anymore and we cannot pin down the payment function and specify $t(0)$ a priori. Assuming that all types of the seller participate in the contract, their decision on the optimal quantity $q^*$ only depends on the marginal price $p(q)$, and therefore, the optimal marginal price $p(q)$ given by (\ref{margp-gen}) is still valid without the assumption on the existence of the worst type. To pin down the payment function $t(\cdot)$, we find the minimum payment $t(0)$ a posteriori so that all types of the seller voluntarily participate. That is,
\vspace{-3pt}
\begin{eqnarray}
t(0)=\max_{x\in\chi}\left[\mathbb{E}_W\left\{C(q(x),w,x)\right\}-\int_0^{q(x)}p(q)dq\right]
\end{eqnarray}
\vspace{-1pt}
where the optimal decision of type $x$ is given by
\vspace{-1pt}
\begin{eqnarray}
q(x)=\textnormal{arg}\max_q \left[\int_0^q p(q)-\mathbb{E}_W\left\{C(q,w,x)\right\}\right].
\end{eqnarray}
\end{remark}
\section*{\LARGE{Appendix}}
\vspace{10pt}
\section*{Details and Proofs of the Results}
\vspace{5pt}
Consider the following problem (P1) formulated in the paper.
\textbf{Problem (P1):}
\begin{eqnarray}
&\hspace{-25pt}{\underset{(\mathcal{M},(q,t))}{\textnormal{maximize}}}& \mathbb{E}_{X,W}\left\{\mathcal{V}(q)-t\right\}
\label{P1}\\\nonumber\vspace{-6pt}\\
&\textit{subject to}\nonumber&\\
&&\hspace{-65pt}\text{VP:}\; t(m^*)\hspace{-2pt}-\hspace{-2pt}\mathbb{E}_W\hspace{-3pt}\left\{C(q(m^*),W,x)\right\}\hspace{-2pt}\geq\hspace{-2pt}0,\forall x\hspace{-2pt}\in \hspace{-3pt}\chi.
\end{eqnarray}
where $\left\{\mathcal{M},(q,t).q:\mathcal{M}\hspace{-1pt}\rightarrow\hspace{-1pt} \mathbb{R}_+,t:\mathcal{M}\hspace{-1pt}\rightarrow\hspace{-1pt} \mathbb{R}_+\right\}$ denotes the mechanism to be designed, and the notation in (1) and (2) is the same as in the paper.
The main result in the paper is given by Theorem 1 stated below.
\begin{theorem}
The optimal mechanism $(q,t)$ for the buyer is a menu of contracts (nonlinear pricing) given by
\begin{eqnarray}
\hspace{-20pt}&p(q)=&\textnormal{arg}\max_{\hat{p}}\left\{P\left[x\in\chi|
{\hat{p}}\geq c(q,x)\right]\left(\mathcal{V}'(q)-\hat{p}\right)\right\}\label{margp-gen}\hspace{-3pt},\\
\hspace{-20pt}&t(q)=&\int_{0}^q{p(l)dl}+C(0,\underline{x}),\label{optt-gen}\\
\hspace{-20pt}&q(x)=&\textnormal{arg}\max_{l\in\mathbf{R}_+}\mathbb{E}\left\{t\left(l\right)-\mathbb{E}_W\left\{C(l,x,w)\right\}\right\}.
\end{eqnarray}
\end{theorem}
\vspace{5pt}
In this note, we provide all the details of the proof of Theorem 1 that were left out of the presentation in the paper due to lack of space. We follow the same steps as in the paper.
\begin{proof}
We proceed to solve (P1) and prove theorem 1 by the following steps.
\vspace{5pt}
\textbf{Step 1.} By invoking the revelation principle \cite{dasgupta-revelation}, we restrict attention, without loss of optimality, to direct revelation mechanisms that are incentive compatible (defined below) and individually rational (individual rationality is equivalent to voluntary participation).
\begin{mydef}
A direct revelation mechanism is defined by functions $q:\chi\rightarrow\mathbf{R}_+$ and $t:\chi\rightarrow\mathbf{R}_+$.
and works as follows:
\begin{itemize}
\item First, the buyer announces functions $q$ and $t$.
\item Second, the seller declares some $x'\in\chi$ as her report for her type.
\item Third, the seller is paid $t(x')$ to deliver $q(x')$ amount of energy.
\end{itemize}
\end{mydef}
Note that the seller is strategic and may lie and misreport her private information $x$, \textit{i.e.} we do not necessarily have $x'=x$.
\vspace{5pt}
\textbf{The revelation principle:}
For any BNE $\tilde{m}^*$ of the game induced by an arbitrary mechanism $(\tilde{\mathcal{M}},(\tilde{q},\tilde{t}))$, there exists an equivalent direct revelation mechanism $\left(\chi,(q,t)\right)$, in which truthful reporting is a BNE of the game induced by $\left(\chi,(q,t)\right)$, and the players' allocation and payment associated with the truth-telling equilibrium are identical to those associated with the BNE $\tilde{m}^*$ of the original mechanism $(\tilde{\mathcal{M}},(\tilde{q},\tilde{t}))$.
\vspace{5pt}
In essence, by invoking the revelation principle we eliminate the problem of finding the optimal message space $\mathcal{M}$ by restricting attention to direct revelation mechanisms ($\mathcal{M}\hspace{-4pt}:=\hspace{-4pt}\chi$), and impose a new set of incentive compatibility (IC) constraints. As a result, we can solve the following problem (P2) to find an optimal direct revelation mechanism.
\vspace{5pt}
\textbf{Problem P2:} Determine functions $q:\chi\rightarrow\mathbf{R}_+$ and $t:\chi\rightarrow\mathbf{R}_+$ so as to
\begin{eqnarray}
&&\hspace{-25pt}{\underset{(q,t)}{\textnormal{maximize}}}\quad \mathbb{E}_{X,W}\left\{\mathcal{V}(q(X))-t(X)\right\}\\\nonumber\\
&&\hspace{-15pt}\textit{subject to:}\nonumber\\&&\hspace{-25pt}IC\hspace{-3pt}: x\hspace{-2pt}=\hspace{-2pt}\textnormal{arg}\max_{x'}\mathbb{E}_W\hspace{-2pt}\left[t(x')\hspace{-2pt}-\hspace{-2pt}C(q(x'),W,x)\right],\forall x\hspace{-2pt}\in\hspace{-2pt}\chi,\\
&&\hspace{-25pt}VP\hspace{-3pt}: \mathbb{E}_W\left[t(x)-C(q(x),W,x)\right]\geq0,\;\forall x\in\chi.
\end{eqnarray}
\vspace{5pt}
\textbf{Step 2.} We utilize the partial order among the seller's different types to order the seller's resulting utility for her different types and reduce the VP constraint for all of the seller's types to the VP constraint only for the seller's worst type.
\begin{lemma}
For a given incentive compatible mechanism $(q,t)$, a better type of the seller gets a higher utility. That is, let
$U(x):=\mathbb{E}_W\left\{t(q(x))-C(q(x),W,x)\right\}$ denote the
expected profit of the seller with type $x$. Then,
\begin{enumerate}
\item $\frac{\partial U}{\partial x_i}\leq 0, 1\leq i\leq m$,
\item $\frac{\partial U}{\partial x_i}\geq 0, m< i\leq n$.
\end{enumerate}
\label{ut-order}
\end{lemma}
\begin{proof}[Proof of lemma 1]
The given mechanism $(q,t)$ is incentive compatible, so we can rewrite $U(x)$ as
\begin{eqnarray}
U(x)=\max_{x'}{\mathbb{E}_W\left\{t(q(x'))-C(q(x'),W,x)\right\}}\label{Lemma1-U}
\end{eqnarray}
By applying the envelope theorem \cite{envelope} on (\ref{Lemma1-U}), we get
\begin{eqnarray}
\frac{\partial U}{\partial x_i}=-\left.\frac{\partial \mathbb{E}_W\left\{C(q(x'),W,x)\right\}}{x_i}\right|_{x'=x}.
\end{eqnarray}
The above equation along with assumption (A5) on the marginal expected cost, gives
\begin{eqnarray}
&&\frac{\partial U}{\partial x_i}\leq 0,1\leq i \leq m\\
&&\frac{\partial U}{\partial x_i}\geq 0,m< i \leq n
\end{eqnarray}
\end{proof}
A direct consequence of Lemma 1 is the following.
\begin{corollary}
The voluntary participation constraint, is satisfied if and only if it is satisfied for the worst type $\underline{x}$, that is the general VP constraint (8) can be reduced to
\begin{eqnarray}
U(\underline{x})=\mathbb{E}_W\left\{t(q(\underline{x}))-C(q(\underline{x}),W,\underline{x})\right\}\geq0.
\end{eqnarray}
\end{corollary}
\textbf{Step 3.} We show, via lemma 2 below, that without loss of optimality, we can restrict attention to a set of functions $t(\cdot)$ that depend only on the amount of delivered energy $q$. That is, the optimal mechanism is a pricing scheme.
\begin{lemma}
For any given pair of functions $(q,t)$ that satisfies the IC constraint, we can rewrite $t(x')$ as $t\left(q(x')\right)$.\label{pricing}
\end{lemma}
\begin{proof}[Proof of lemma 2]
The proof is by contradiction. Assume that there exist $x,x'\in\chi$ such that $q(x)=q(x')$ but $t(x')>t(x)$. Then a seller with type $x$ is always better off by reporting $x'$ instead of her true type $x$, which contradicts the IC constraint.
\end{proof}
\vspace{5pt}
\textbf{Step 4.}
As a consequence of lemma \ref{pricing}, we determine an optimal mechanism sequentially. First, we determine the optimal payment function $t(\cdot)$, then the optimal energy procurement function $q(\cdot)$.
For any given pair of functions $(q,t)$, a seller with true type $x$ solves the following maximization problem to find her optimal report $x'^*$,
\begin{eqnarray}
x'^*=\textnormal{arg}\max_{x'}\mathbb{E}_W\left\{t\left(q(x')\right)-C(q(x'),W,x)\right\}.
\label{agent-gen}
\end{eqnarray}
Considering the solution of (\ref{agent-gen}) for every type $x\hspace{-2pt}\in\hspace{-2pt}\chi$, we can form a function $q^*:\chi\longrightarrow \mathbf{R}_+$ defined by
\begin{eqnarray}
q^*(x):=\textnormal{arg}\max_{l}\mathbb{E}_W\left\{t\left(l\right)- C(l,W,x)\right\}\label{agent-q}.
\end{eqnarray}
For a given direct revelation mechanism, the function $q^*(\cdot)$ defines the optimal quantity of energy procured from each type of the seller. Incentive compatibility then requires that the seller must tell the truth to achieve this optimal value, and cannot do better by lying, \emph{i.e.}
\begin{eqnarray}
q(x)=q^*(x)\quad \forall x\in\chi.
\label{agent-rep}
\end{eqnarray}
The function $q^*(\cdot)$ is induced only by $t(\cdot)$ through (\ref{agent-q}), and therefore, (\ref{agent-rep}) can be taken as the definition for the associated function $q(\cdot)$ that along with $t(\cdot)$ satisfies the set of IC constraints. Thus, we can eliminate the IC constraint by defining $q(\cdot):=q^*(\cdot)$ and reduce the problem of designing the optimal direct revelation mechanism $(q,t)$ to an equivalent problem (P2') where we determine only the optimal payment function $t(\cdot)$ subject to the voluntary participation constraint for the worst type.
\vspace{5pt}
\textbf{Problem P2':} Determine function $t:\mathbf{R}_+\rightarrow\mathbf{R}_+$ so as to
\begin{eqnarray}
&&\hspace{-10pt}{\underset{t(\cdot)}{\textnormal{maximize}}}
\quad \mathbb{E}_{X}\left\{\mathcal{V}(q^*(X))-t(q^*(X))\right\}\label{buyer-obj-p2}\\
&&\hspace{-0pt}\text{subject to}\nonumber\\
&&\hspace{-10pt}\text{VP:}\; \mathbb{E}_W\hspace{-2pt}\left[t(q(\underline{x}))\hspace{-2pt}-\hspace{-2pt}C(q^*(\underline{x}),W,\underline{x})\right]\hspace{-2pt}\geq\hspace{-2pt}0,
\label{ind-3}
\end{eqnarray}
where $q^*$ is given by (\ref{agent-q}).
\vspace{5pt}
\textbf{Step 5.} To solve problem (P2'), we show that the optimal decision of the seller for amount of power $q^*$ depends only on the marginal price $p(q):=\frac{\partial t(q)}{\partial q}$ and express the buyer's expected utility in term of the marginal price.
Consider the buyer's objective (\ref{buyer-obj-p2}). For any function $t(\cdot)$, we can determine from (\ref{agent-q}) the cumulative distribution function for $q^*$, called $F_{q^*}$. Consequently, we can rewrite the buyer's objective as
\begin{eqnarray}
&&\mathbb{E}_{q^*}\left[\mathcal{V}(q^*)-t(q^*)\right]=\int_{0}^{\infty}{\left(\mathcal{V}(l)-t(l)\right)dF_{q^*}(l)}\nonumber\\
&&=\left.\left(F_{q^*}(l)-1\right)\left(\mathcal{V}(l)-t(l)\right)\right|_{0}^{\infty}\nonumber\\&&+
\int_{0}^{\infty}{\left(1-F_{q^*}(l)\right)\frac{d\left(\mathcal{V}(l)-t(l)\right)}{d l}dl}.
\label{maxobj-1}
\end{eqnarray}
We have
\begin{eqnarray}
\left.\left(F_{q^*}(l)-1\right)\left(\mathcal{V}(l)-t(l)\right)\right|_{0}^{\infty}=-t(0)
\label{t0}
\end{eqnarray}
because $\mathcal{V}(0)=0$ by assumption, and $\left(F_{q^*}(\infty)-1\right)=0$.
Because of (\ref{t0}), we can rewrite (\ref{maxobj-1}) as
\begin{eqnarray}
&\mathbb{E}_{q^*}\left[\mathcal{V}(q^*)-t(q^*)\right]&\hspace{-5pt}=\hspace{-3pt}\int_{0}^{\infty}\hspace{-3pt}{P\left(q^*\geq l\right)\left(\mathcal{V}'(l)-p(l)\right)dl}
\nonumber\\&&-t(0)
\label{buyer-max1}
\end{eqnarray}
where $\mathcal{V}'(l)=\frac{d \mathcal{V}(l)}{d l}$.
\vspace{5pt}
We can rewrite $P\left(q^*\geq l\right)$ as
\begin{eqnarray}
&&\hspace{-20pt}P\hspace{-1pt}\left(q^*\hspace{-2pt}\geq\hspace{-2pt} l\right)\hspace{-3pt}=\hspace{-3pt}P[x\hspace{-2pt}\in\hspace{-2pt}\chi|\text{arg}\hspace{-1pt}\max_l\mathbb{E}_W\hspace{-3pt}\left\{t(q)\hspace{-2pt}-\hspace{-2pt}C(q(x),\hspace{-1pt}W,\hspace{-1pt}x)\right\}\hspace{-3pt}\geq\hspace{-2pt} l].\nonumber\\&&
\label{q-demand1}
\end{eqnarray}
We implicitly assume that the seller's problem given by (\ref{agent-q}) is continuous and quasi-concave\footnote{This is a standard assumption in literature, e.g. see \cite{multiscreening} and \cite{nonlinear}. Basically, it can be seen as a situation where the seller can decide for each marginal unit of production independently. Therefore, in general, there is no guarantee that the seller's independent decisions about each marginal unit of production results in a continuous and plausible total production quantity $q$. Therefore, the continuity of the result must be checked posteriori for each type of the seller.}, so that from the first order optimality condition for (\ref{agent-q}) we obtain
\begin{eqnarray}
p(q^*(x))=\left.\frac{\partial \mathbb{E}_W\left\{C(l,W,x)\right\}}{\partial l}\right|_{q^*(x)}.
\label{foc}
\end{eqnarray}
Therefore, each type of the seller wishes to produce more than quantity $l$ if and only if the marginal price $p(q)$ that she is paid at $l$ is higher than the expected marginal cost of production $c(l,x)$ that she incurs at $l$. Consequently, combining (\ref{q-demand1}) and (\ref{foc}) we obtain
\begin{eqnarray}
P\left(q^*\geq l\right)=P\left[x\hspace{-2pt}\in\hspace{-2pt}\chi|
p(l)\geq\frac{\partial \mathbb{E}_W\left\{C(l,W,x)\right\}}{\partial l}\right]
\label{q-demand2}.
\end{eqnarray}
Substituting (\ref{q-demand2}) in (\ref{buyer-max1}), we obtain the following alternative expression for the buyer's objective
\begin{eqnarray}
&\hspace{-8pt}\mathbb{E}_{q^*}\left[\mathcal{V}(q^*)\hspace{-2pt}-\hspace{-2pt}t(q^*)\right]\hspace{-2pt}=&\hspace{-12pt}
\int_{0}^{\infty}\hspace{-6pt}{P\hspace{-2pt}\left[x\hspace{-2pt}\in\hspace{-2pt}\chi|
p(l)\hspace{-2pt}\geq\hspace{-2pt}\frac{\partial \mathbb{E}_W\hspace{-3pt}\left\{C(l,\hspace{-1pt}W,\hspace{-1pt}x)\right\}}{\partial l}\right]}\nonumber\\&&\hspace{10pt}\left(\mathcal{V}'(l)\hspace{-2pt}-\hspace{-2pt}p(l)\right)dl\hspace{-3pt}-t(0).
\label{buyer-max2}
\end{eqnarray}
The buyer seeks to determine the marginal price $p(\cdot)$ and $t(0)$ to maximize the right hand side of (\ref{buyer-max2}), subject to the VP constraint for the seller's worst type.
Based on this consideration, we define the following problem (P3) that is equivalent to problem (P2').
\vspace{10pt}
\textbf{Problem P3:}
\begin{eqnarray}
&&\hspace{-20pt}{\underset{p(\cdot),t(0)}{\textnormal{maximize}}}\quad\hspace{-8pt}
-\hspace{-1pt}t(0)\hspace{-2pt}+\hspace{-4pt}\int_{0}^{\infty}\hspace{-7pt}{P\hspace{-1pt}\left[x\hspace{-2pt}\in\hspace{-2pt}\chi|
p(l)\hspace{-2pt}\geq\hspace{-2pt} \frac{\partial \mathbb{E}_W\hspace{-3pt}\left\{C(l,w,x)\right\}}{\partial l}\right]}\nonumber\\&&\hspace{80pt}\left(\mathcal{V}'(l)-p(l)\right)dl\\
&&\textit{subject to}\quad\nonumber\\&& \hspace{-19pt}\text{VP:}\;\max_{l}{\hspace{-1pt}t(0)\hspace{-2pt}+\hspace{-4pt}\int_0^{q^*\hspace{-1pt}(\underline{x})}\hspace{-11pt}p(r)dr\hspace{-2pt}-\hspace{-1pt}\mathbb{E}_W\hspace{-3pt}\left\{C(q^*\hspace{-1pt}(\underline{x}),W,\underline{x})\hspace{-1pt}\right\}}\hspace{-4pt}\geq\hspace{-2pt}0.
\end{eqnarray}
\textbf{Step 6.} We provide a ranking for the seller's optimal decision $q^*(x)$ based on the partial order among the seller's types via the following lemma.
\begin{lemma}
For a given mechanism specified by $(t(\cdot),q(\cdot))$, a better type of the seller produces more. That is, the optimal quantity $q^*(x)$ that the seller with true type $x$ wishes to produce satisfies the following properties:
\begin{description}
\item[a)] $\frac{\partial q^*(x)}{\partial x_i}\leq 0,1\leq i\leq m$,
\item[b)] $\frac{\partial q^*(x)}{\partial x_i}\geq 0, m<x\leq n$.
\end{description}
\end{lemma}
\begin{proof}[Proof of lemma 3]
Let $x,x'\in\chi$, where $x$ is a better type than $x'$. From IC for seller's type $x$ we have
\begin{eqnarray}
&t(q(x))\hspace{-2pt}-\hspace{-2pt}\mathbb{E}_W\hspace{-3pt}\left\{C(q(x),W,x)\right\}\hspace{-2pt}\nonumber\\&\geq\nonumber\\&\hspace{-2pt} t(q(x'))\hspace{-2pt}-\hspace{-2pt}\mathbb{E}_W\hspace{-3pt}\left\{C(q(x'),W,x)\right\}\label{lemma3-1}
\end{eqnarray}
Similarly from IC for seller's type $x'$ we have
\begin{eqnarray}
&t(q(x'))-\mathbb{E}_W\left\{C(q(x'),W,x')\right\}\nonumber\\&\geq\nonumber\\& t(q(x))-\mathbb{E}_W\left\{C(q(x),W,x')\right\}\label{lemma3-2}
\end{eqnarray}
Subtracting (\ref{lemma3-2}) from (\ref{lemma3-1}), we get
\begin{eqnarray}
&\mathbb{E}_W\left\{C(q(x),W,x')\right\}-\mathbb{E}_W\left\{C(q(x'),W,x')\right\}\nonumber\\&\geq\nonumber\\ &\mathbb{E}_W\left\{C(q(x),W,x)\right\}-\mathbb{E}_W\left\{C(q(x'),W,x)\right\}\label{lemma3-3}
\end{eqnarray}
By assumption (A4), $\frac{d\mathbb{E}_W\hspace{-2pt}\left\{C(q,W,x)\right\}}{dq}\hspace{-2pt}\leq\hspace{-2pt} \frac{d\mathbb{E}_W\hspace{-2pt}\left\{C(q,W,x')\right\}}{dq}$ if $x$ is a better type than $x'$.
Therefore, (\ref{lemma3-3}) holds if and only if
\begin{eqnarray}
q(x)\geq q(x').
\end{eqnarray}
\end{proof}
The following result is a consequence of corollary 1 and lemma 3.
\begin{corollary}
The VP constraint is satisfied if $t(0)=C(0,\underline{x})$ and the lowest seller's type is paid exactly equal to her expected production cost, \textit{i.e.} $t(q(\underline{x}))=\mathbb{E}_W\left\{C(q(\underline{x}),W,\underline{x})\right\}$.
\end{corollary}
\begin{proof}[Proof of corollary 2]
Because of corollary 1, the VP constraint implies
\begin{eqnarray}
U(\underline{x})=t(q(\underline{x}))-\mathbb{E}_W\left[C(q^*(\underline{x}),W,x)\right] = 0,
\end{eqnarray}
which is equivalent to
\begin{eqnarray}
t(0)+\int_0^{q^*(\underline{x})}{p(l)dl}=\mathbb{E}_W\left[C(q^*(\underline{x}),W,x)\right].
\label{VP-red}
\end{eqnarray}
Furthermore, from Lemma 3 it follows that if the worst type wishes to produce more than $q^*(\underline{x})$, then all types produce more than $q^*(\underline{x})$. Therefore,
\begin{eqnarray}
P\left[x\in\chi|
p(l)\geq c(l,x)\right]=1,\;\; \text{for}\; l\leq q^*(\underline{x}).
\label{pforlmin}
\end{eqnarray}
Using (\ref{pforlmin}), we can rewrite the objective function of problem (P3) as,
\begin{eqnarray}
&\hspace{-20pt}&-\left(t(0)+
\int_0^{q^*(\underline{x})}{p(l)dl}\right)+\int_0^{q^*(\underline{x})}{\mathcal{V}'(l)dl}
\nonumber\\&\hspace{-20pt}&+
\int_{q^*(\underline{x})}^{\infty}{P\left[x\in\chi|
p(l)\geq c(l,x)\right]\left(\mathcal{V}'(l)-p(l)\right)dl}\hspace{-1pt}.
\label{obj-red}
\end{eqnarray}
The term $t(0)+\int_0^{q^*(\underline{x})}{p(l)dl}$ appears in both the objective (\ref{obj-red}) and the VP constraint (\ref{VP-red}). Therefore, without loss of optimality, we can assume $t(0)=C(0,\underline{x})$, and set $t(q(\underline{x}))=\mathbb{E}_W\left\{C(q(\underline{x}),W,\underline{x})\right\}$.
\end{proof}
Using corollary 2, we define a problem (P4) that is equivalent to (P3) and is only in terms of the marginal price $p(\cdot)$ and the constraint that the payment the seller's lowest type receives is equal to her cost of production.
\vspace{5pt}
\textbf{Problem (P4)}
\begin{eqnarray}
&\hspace{-50pt}{\underset{p(\cdot)}{\textnormal{max}}}&\hspace{-15pt} \int_{0}^{\infty}{\hspace{-5pt}P\left[x\in\chi|
p(l)\geq c(l,x)\right]\left(\mathcal{V}'(l)-p(l)\right)dl}\\
&&\hspace{-25pt}\text{subject to}\nonumber\\&&\hspace{-30pt}\text{VP:}\;
C(0,\underline{x})+\int_0^{q^*(\underline{x})}{\hspace{-4pt}p(l)dl}\hspace{-1pt}=\hspace{-1pt}\mathbb{E}_W\left[C(q^*(\underline{x}),w,x)\right]\label{P4-VP}\hspace{-1pt}.
\label{p5-con}
\end{eqnarray}
\textbf{Step 7.} We consider a relaxed version of (P4) without the VP constraint (\ref{P4-VP}). The unconstrained problem (P4) can be solved by maximizing the integrand $P\left[x\in\chi|
p(l)\geq c(l,x)\right]\left(\mathcal{V}'(l)-p(l)\right)$ point-wise at each quantity $l$.
The solution of the point-wise maximization problem for the optimal marginal price $p(\cdot)$ is given by,
\begin{eqnarray}
\hspace{-20pt}&p(l)\hspace{-2pt}=&\hspace{-8pt}arg\max_{\hat{p}}\left\{P\left[x\in\chi|
{\hat{p}}\geq c(q,x)\right]\left(\mathcal{V}'(q)-\hat{p}\right)\right\}\label{margp-gen}\hspace{-3pt}.
\end{eqnarray}
Using (\ref{pforlmin}), along with the the fact that the worst type has the highest expected marginal cost, we can simplify (\ref{margp-gen}) for $l\leq q^*(\underline{x})$,
\begin{equation}
p(l)= c(l,\underline{x}),\;\;\text{for}\;l\leq q^*(\underline{x}).
\end{equation}
That is, for $l\leq q^*(\underline{x})$, the minimum marginal price $p(l)$ that ensures all the seller's type are willing to produce more than $q^*(\underline{x})$ is equal to the marginal expected cost for the seller's worst type $c(l,\underline{x})$.
Therefore, the solution to the unconstrained version of problem (P4) satisfies condition (\ref{p5-con}) of problem (P4), and therefore, (\ref{margp-gen}) (which is the same as (3)) is also the optimal solution of problem (P4).
From Corollary 2 and (\ref{margp-gen}), the optimal payment function (nonlinear pricing) is given by,
\begin{eqnarray}
t(q)=\int_{0}^q{p(l)dl}+C(0,\underline{x})
\end{eqnarray}
which is the same as (4).
From (\ref{agent-q}) we determine the optimal energy procurement function,
\begin{eqnarray}
q(x)=\text{arg}\max_{l}\mathbb{E}\left\{t\left(l\right)-C(l,w,x)\right\}
\end{eqnarray}
which us the as (5).
The specification of $t(\cdot)$ and $q(\cdot)$ completes the proof of theorem 1 and the solution to problem (P1).
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,205
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E. CAPE - BIG 5 - Shamwari, Kwandwe, Kwantu, Lalibela, Pumba
African Pride Pumba Private Game Reserve
African Pride Pumba Private Game Reserve - E. CAPE - BIG 5 - Shamwari, Kwandwe, Kwantu, Lalibela, Pumba, THE EASTERN CAPE
Highlands, Makana Municipality, Port Elizabeth, Paterson, THE EASTERN CAPE South Africa (Lodge) Show map
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With a stay at African Pride Pumba Private Game Reserve in Grahamstown, you'll be in a provincial park and minutes from African Pride Pumba Private Game Reserve. This 5-star lodge is within the region of South African Institute for Aquatic Biodiversity and Cathedral of St. Michael and St. George.
Make yourself at home in one of the 13 air-conditioned rooms featuring fireplaces and private plunge pools. Rooms have private balconies or patios where you can take in river and mountain views. Conveniences include direct-dial phones, as well as safes and refrigerators.
Rec, Spa, Premium Amenities
Take time to pamper yourself with a visit to the full-service spa. If you're looking for recreational opportunities, you'll find an outdoor pool and a fitness facility. Additional features include complimentary wireless Internet access, gift shops/newsstands, and a fireplace in the lobby.
Featured amenities include a business center, audiovisual equipment, and multilingual staff. Event facilities at this lodge consist of banquet facilities and a meeting/conference room. A roundtrip airport shuttle is provided for a surcharge (available on request), and free parking is available onsite.
Coffee Tea Maker;Safe;Exterior Room Entrance;Fitness Facility;Business Center;Air Conditioning;Room Service
African Pride Pumba Private Game Reserve Highlands, Makana Municipality, Port Elizabeth, Paterson (Lodge)
Msenge Chalet
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See FAQs for more details.
Number of Rooms/Cottages/Apartments
1 2 3 4 5 6 7 8 9 10 Children
Guest Daytime Contact Number (Eg. +61 2 22222222 or 02 2222 2222)
Guest Evening Contact Number (Eg. +61 2 22222222 or 02 2222 2222)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Children
Exterior Room Entrance
The onsite spa has massage/treatment rooms. Services include manicures and pedicures. The spa is equipped with a sauna. A variety of treatment therapies are provided, including aromatherapy.
Balcony/patio with garden views
Relax - Private plunge pool and fireplace
Food & Drink - Refrigerator, coffee/tea maker, and room service
Practical - Safe
Access via exterior corridors, Air conditioning, Balcony or patio, Coffee/tea maker, Daily housekeeping, Fireplace, In-room climate control (air conditioning), In-room safe, Non-Smoking, Number of bathrooms - 1, Private bathroom, Private plunge pool, Refrigerator, Room service
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The preferred airport for African Pride Pumba Private Game Reserve is Port Elizabeth (PLZ) - 99.1 km / 61.5 mi.
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Included in the published rate is ${iA} adults {{if iC}}and ${iC} children{{/if}}. {{if eA}}Of the extra guests ${eA} may be an adult.{{/if}} {{if eCA}} Extra adults are charged at AU$${eCA} per adult. {{/if}} {{if eCC}} Extra children {{if cAge}} (${cAge} and under) {{/if}} are charged at AU$${eCC} per child. {{/if}} {{if nC}}Please note that this room does not accommodate children{{/if}}
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 5,384
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import {elementByTag, waitForBodyPromise} from '../../../../src/dom';
import {isExperimentOn} from '../../../../src/experiments';
import {dev} from '../../../../src/log';
const ELIGIBLE_TAGS = [
'amp-img',
'amp-anim',
'amp-ad',
'amp-dailymotion',
'amp-jwplayer',
'amp-kaltura-player',
'amp-o2-player',
'amp-pinterest',
'amp-reach-player',
'amp-vimeo',
'amp-vine',
'amp-youtube',
'amp-video',
'amp-twitter',
'amp-facebook',
'amp-instagram',
];
const ELIGIBLE_TAP_TAGS = {
'amp-img': true,
'amp-anim': true,
};
const DEFAULT_VIEWER_ID = 'amp-lightbox-viewer';
const VIEWER_TAG = 'amp-lightbox-viewer';
/**
* Finds elements in the document that meet our heuristics for automatically
* becoming lightboxable and adds `lightbox` attribute to them.
* It may also install a tap handler on elements that meet our heuristics
* to automatically open in lightbox on tap.
* @param {!../../../../src/service/ampdoc-impl.AmpDoc} ampdoc
* @return {!Promise}
*/
export function autoDiscoverLightboxables(ampdoc) {
// Extra safety check, manager should not call this if experiments are off
dev().assert(isExperimentOn(ampdoc.win, 'amp-lightbox-viewer'));
dev().assert(isExperimentOn(ampdoc.win, 'amp-lightbox-viewer-auto'));
return maybeInstallLightboxViewer(ampdoc).then(viewerId => {
const tagsQuery = ELIGIBLE_TAGS.join(',');
const matches = ampdoc.getRootNode().querySelectorAll(tagsQuery);
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const element = matches[i];
if (element.hasAttribute('lightbox') || !meetsHeuristics(element)) {
continue;
}
element.setAttribute('lightbox', '');
// TODO(aghassemi): This is best to do via default action. E.g. we can add
// a tap listener via Action service and invoke lightbox if conditions are
// met.
if (meetsHeuristicsForTap(element)) {
element.setAttribute('on', 'tap:' + viewerId + '.activate');
}
}
});
}
/**
* Decides whether an element meets the heuristics to become lightboxable.
* @param {!Element} element
* @return {!boolean}
*/
function meetsHeuristics(element) {
dev().assert(element);
// TODO(aghassemi): This will become complicated soon, create a pluggable
// system for this.
if (element.getLayoutBox) {
const layoutBox = element.getLayoutBox();
if (layoutBox.left < 0 ||
layoutBox.width < 50 ||
layoutBox.height < 50
) {
return false;
}
}
return true;
}
/**
* Decides whether an already lightboxable element should automatically get
* a tap handler to open in the lightbox.
* @param {!Element} element
* @return {!boolean}
*/
function meetsHeuristicsForTap(element) {
dev().assert(element);
dev().assert(element.hasAttribute('lightbox'));
if (!ELIGIBLE_TAP_TAGS[element.tagName.toLowerCase()]) {
return false;
}
if (element.hasAttribute('on')) {
return false;
}
return true;
}
/**
* Tries to find an existing amp-lightbox-viewer, if there is none, it adds a
* default one.
* @param {!../../../../src/service/ampdoc-impl.AmpDoc} ampdoc
* @return {!Promise<string>} Returns the id of the amp-lightbox-viewer.
*/
function maybeInstallLightboxViewer(ampdoc) {
// TODO(aghassemi): Use the upcoming ampdoc.waitForBody
return waitForBodyPromise(/** @type {!Document} */ (
ampdoc.getRootNode())).then(() => {
const existingViewer = elementByTag(ampdoc.getRootNode(), VIEWER_TAG);
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if (!existingViewer.id) {
existingViewer.id = DEFAULT_VIEWER_ID;
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const viewer = ampdoc.getRootNode().createElement(VIEWER_TAG);
viewer.setAttribute('layout', 'nodisplay');
viewer.setAttribute('id', DEFAULT_VIEWER_ID);
ampdoc.getRootNode().body.appendChild(viewer);
return viewer.id;
});
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 5,304
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{"url":"http:\/\/www.global-sci.org\/intro\/article_detail\/aamm\/13188.html","text":"Volume 11, Issue 4\nA Treecode Algorithm for 3D Stokeslets and Stresslets\n10.4208\/aamm.OA-2018-0187\n\nAdv. Appl. Math. Mech., 11 (2019), pp. 737-756.\n\nPreview Full PDF BiBTex 4 638\n\u2022 Abstract\n\nThe Stokeslet and stresslet kernels are commonly used in boundary element simulations and singularity methods for slow viscous flow. Evaluating the velocity induced by a collection of Stokeslets and stresslets by direct summation requires $\\mathcal{O}(N^2)$ operations, where $N$ is the system size. The present work develops a treecode algorithm for 3D Stokeslets and stresslets that reduces the cost to $\\mathcal{O}(N\\log N)$. The particles are divided into a hierarchy of clusters and\u00a0 well-separated particle-cluster interactions are computed by a far-field Cartesian Taylor approximation. The terms in the approximation are contracted to promote efficient computation. Serial and parallel results display the performance of the treecode for several test cases. In particular the method has relatively simple structure and low memory usage and this enhances parallel efficiency for large systems.\n\n\u2022 History\n\nPublished online: 2019-06","date":"2020-01-26 15:41:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.46448493003845215, \"perplexity\": 1294.4738243586796}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-05\/segments\/1579251689924.62\/warc\/CC-MAIN-20200126135207-20200126165207-00041.warc.gz\"}"}
| null | null |
Ceratosoma amoenum est une espèce de nudibranches de la famille des Chromodorididae et du genre Ceratosoma.
Répartition
Cette espèce se rencontre en Australie et en Nouvelle-Zélande
Habitat
Ceratosoma amoenum peut être observé entre 13 et de profondeur.
Description
Ceratosoma amoenum peut mesurer de l'ordre de 5 à de long.
Le corps est allongé et oblong, arrondi à l'avant et en pointe en arrière avec le dos légèrement convexe. Le manteau est plus petit que le pied. Il est assez lisse et régulier, d'une couleur rose lilas pâle ou violacé. Il présente une rangée centrale de grandes taches oblongues orange vif ou de nombreuses petites taches orange parmi lesquelles se trouvent quelques points rouge et, à l'occasion, quelques taches, latérales proches de la marge, de couleur crème pâle ou blanc jaunâtre. La marge du manteau est blanche. Le pied est plus pâle que le manteau avec les côtés et les extrémités présentant une double rangée irrégulière de taches orange vif arrondies, considérablement plus longues que celles du manteau. Le dessous du pied est de couleur chair pâle. Les spicules du manteau manquent apparemment.
Les rhinophores, de couleur rouge profond avec des bandes blanches, sont clavés et complètement rétractables dans des gaines légèrement surélevées. La partie supérieure est arquée vers l'arrière, stratifiée de 24 à 25 lamelles. Les branchies sont de couleur magenta vif avec des bandes blanches. Les tentacules oraux sont libres, petits et coniques.
L'odontophore comporte environ 65 rangées de dents avec une petite dent centrale et de 60 à 70 dents latérales de chaque côté.
Éthologie
Alimentation
Ceratosoma amoenum se nourrit d'éponges du genre Dysidea et du corail Drifa gaboensis.
Ceratosoma amoenum, espèce carnivore, contient de l'allolaurinterol (), un métabolite algaire que l'on rencontre par exemple chez l'algue rouge Hymenea variolosa, qui pourrait provenir de la prédation des œufs d'espèces du genre Aplysia.
Reproduction
Cette espèce dépose ses œufs, d'un diamètre de , en spirales de quatre tours.
Publication originale
Cheeseman, T. F. 1886. On a new species of Chromodoris. Transactions & Proceedings of the New Zealand Institute, 18: 137.(BHL)
Taxonomie
Cette espèce a été nommée par le zoologiste néo-zélandais Thomas Frederic Cheeseman en 1886 sous le protonyme Chromodoris amoena et transférée dans le genre Ceratosoma.
Notes et références
Note
Références
Liens externes
Chromodorididae
|
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"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
The Hall effect in magnetic conductors has two contributions: a contribution known as the ordinary Hall effect (OHE) associated with Lorentz force and linked to the presence of a magnetic field $\textbf{B}$, and a contribution known as the anomalous Hall effect (AHE) \cite{AHE} linked to magnetization $\textbf{M}$. The AHE has been variously attributed to spin sensitive scattering, which makes its study relevant to spintronics and to topological features of the conduction bands, which have attracted considerable interest in the context of the quantum Hall effect and topological insulators.
The AHE resistivity, $\rho_{xy}^{AHE}$, is phenomenologically described as $\rho_{xy}^{AHE}=R_s\mu_0M_\bot$, where $M_{\bot}$ is the component of the magnetization perpendicular to the film and $R_s$ is called the AHE coefficient. Extrinsic models relate $R_s$ to the longitudinal resistivity, $\rho_{xx}$, and predict $R_{s}=a_{ss}\rho_{xx}+a_{sj}\rho_{xx}^{2}$, where the linear term in $\rho_{xx}$ is attributed to skew-scattering \cite{RsRho1}, and the quadratic term is attributed to the side-jump mechanism \cite{RsRho2}.
Manganites \cite{review_manganite} known for their colossal magnetoresistance \cite{CMR} are particularly intriguing systems for studying AHE due to strong electron correlations and the existence of multiple competing ground states. Matl et al. \cite{matl} showed a linear relation between the AHE coefficient and the longitudinal resistivity. On the other hand, Fukumura et al. \cite{bad metal scaling experiment} showed a scaling behavior of the AHE conductivity $\sigma_{xy} \sim \sigma_{xx}^{1.6-1.8}$ in the hopping conductivity regime of several conductors, including manganites.
Several mechanisms have been suggested to explain the AHE in manganites: a mechanism arising from double exchange quantal phases combined with spin-orbit interaction, which predicts scaling with the reduced magnetization \cite{Lyanda-Geller}, and a real space Berry phase mechanism \cite{ye_skyrmion}, which attributes the AHE to the spatial variation of magnetization induced by skyrmions.
The AHE in manganites has previously been studied with magnetization perpendicular to the film plane. Here, we study the dependence of the AHE in thin films of \lsmo\ (LSMO) on the angle $\theta$ between the magnetic field and the normal to the film plane.
If the Hall effect is determined by the perpendicular components of the magnetic field and the magnetization, a trivial $\cos\theta$ dependence is expected. However, we find that the transverse resistivity ($\rho_{xy}$) follows $\rho_{xy}=a\cos\theta+b\cos3\theta$. We show that the $\cos3\theta$ term is not solely due to the angular dependence of the longitudinal resistivity or the magnetization; therefore, the surprising term is likely a manifestation of an intrinsic transport property that has not been identified so far.
\section{Experiment}
The samples in this study are epitaxial thin films of \lsmo\ grown on single crystal SrTiO$_3$(001) substrates using off-axis RF magnetron sputtering. Growth was carried out at $660^\circ$C in a process gas of 20$\%$ O$_2$ and 80$\%$ Ar at a pressure of 150 mTorr. After growth, the samples were cooled to room temperature at a rate of $10^\circ$C per minute in 1 bar of O$_2$. Film thickness was controlled by deposition time, which was calibrated using ex \emph{situ} x-ray diffraction and x-ray reflectivity measurements.
A 40 nm thick calibration film grown with the same conditions was found to be under tensile strain, with a reduced out-of-plane lattice parameter of 0.385 nm and an in-plane lattice parameter of 0.390 nm. The rocking curve taken around the 002 Bragg reflection had a full width at half maximum of $0.05^\circ$
The films are patterned to allow transverse and longitudinal resistivity measurements, which are performed with a PPMS-9 system (Quantum Design). The magnetic characterization of the films is performed using an MPMS-XL SQUID magnetometer (Quantum Design).
\section{Results and Discussion}
\begin{figure}
\includegraphics[scale=0.45]{25RMvsT_trans.eps}
\caption{(a) Resistivity ($\rho$) as a function of temperature. Inset: Magnetoresistance, $\frac{R_0-R_H}{R_0}*100$, at $\mu_0$H=8 T, as a function of temperature. (b) Reduced magnetization as a function of temperature. (c) $\rho_{xy}$ as a function of magnetic field at different temperatures.}
\label{Fig:MRvsT}
\end{figure}
Figure~\ref{Fig:MRvsT} shows characterization measurements of our films. Figure~\ref{Fig:MRvsT}(a) shows resistivity and magnetoresistance measurements, and Figure~\ref{Fig:MRvsT}(b) shows a field-cooled magnetization measurements with $\mu_0H$=0.05 T. The 22.7 nm thick sample shows bulk-like behavior with a Curie temperature of 300 K and a longitudinal resistivity of 200 $\mu\Omega$-cm at 10 K, pointing to nearly homogeneous electronic and magnetic properties \cite{lsmo_bulk}. Magnetic hysteresis measurements were carried out at 10 K on a 50 nm \lsmo\ film with the field applied perpendicular to the film surface and show the out-of-plane direction to be a hard axis with a saturation magnetization of 3.3 $\mu$B/Mn at fields greater than 2 T. As shown in Figure~\ref{Fig:MRvsT}(b), the resistivity and Curie temperature depend on film thickness, in agreement with earlier works \cite{thickness}.
Figure~\ref{Fig:MRvsT}(c) shows the magnetic field dependence of the Hall effect resistivity, $\rho_{xy}$, at different temperatures. The two slopes are related to AHE which dominates the change at low fields the OHE which dominates the change at high fields.
\begin{figure}
\includegraphics[scale=0.45]{23LONGandTrans_125and175.eps}
\caption{$\rho_{xx}$ (top) and $\rho_{xy}$ (bottom) as a function of the external field angle, $\theta$, at T=125 K (left), and T=175 K (right) for different fields. The lines are fits to Eqs. \ref{Eq:AMR} and \ref{Eq:AHE}, respectively.}
\label{Fig:AHE}
\end{figure}
\begin{figure}
\includegraphics[scale=0.5]{26AMR.eps}
\caption{(top) $\Delta\rho/\rho$ as function of the temperature for different magnetic fields. Inset: Sketch of the relative orientations of the current density J, magnetic field H, magnetization M, and the crystallographic axes. The angle between H and the film normal, $\theta$, is rotating in the (010) plane. (bottom) The ratio between the AMR amplitude, $\Delta\rho/\rho$, at H=4 and H=9 T, as a function of temperature.}
\label{Fig:AMR}
\end{figure}
Figure~\ref{Fig:AHE} shows $\rho_{xx}$ and $\rho_{xy}$ at ${\rm T=125 \ K}$ and at ${\rm T=175 \ K}$ as a function of the angle $\theta$ between an applied magnetic field ($H$) and the normal to the film (film thickness is 22.7 nm). The current path is along [001], and the field is rotated in the (010) plane (see inset of Figure~\ref{Fig:AMR}). The data are shown for $\mu_0 H$ between ${\rm 4 \ T}$ and ${\rm 9 \ T}$, for which the magnetization is saturated and parallel to the applied field. We note that the HE does not follow the expected trivial $\cos\theta$ dependence.
The angular dependence of $\rho_{xx}$ is attributed to the anisotropic magnetoresistance (AMR) as follows: $\rho_{xx}=\rho_1+ \rho_2 \cos2\phi$, where $\phi$ is the angle between the current path and the magnetization \cite{AMR}. In the notation we use here:
\begin{equation}
\rho_{xx}=\rho_0+\Delta \rho \cos2\alpha,
\label{Eq:AMR}
\end{equation}
where $\alpha$ is the angle between the magnetization and the film normal.
The good fit in Figure~\ref{Fig:AHE} (top) indicates that the magnetization approximately follows the external magnetic field direction; however, as we will show next, the magnetization has small deviations from the external field direction due to magnetic anisotropy. Figure~\ref{Fig:AMR} (top) shows the AMR amplitude, $\Delta\rho/\rho$, as a function of the temperature for different magnetic fields, and Figure~\ref{Fig:AMR} (bottom) presents the ratio between the AMR measured in two different fields, ${\mu_0 \rm H=4 \ T}$ and ${\mu_0 \rm H=9 \ T}$. We note that below $\sim 200$ K the AMR is practically field independent for $\mu_0 \rm H \geq 4 \ T$, while at higher temperatures the AMR decreases with increasing field in the same range. A decrease in AMR with increase magnetization was observed before and attributed to increased magnetic uniformity \cite{nonmonotonicAMR}.
As noted above, two effects contribute to $\rho_{xy}$: the OHE and the AHE. Commonly, the OHE is proportional to the perpendicular component of the magnetic field, $B_\bot$, and the AHE is proportional to the perpendicular component of the magnetization, $M_\bot$, yielding:
\begin{equation}
\rho_{xy}=R_0B_\bot+R_s\mu_0M_\bot
\label{Eq:HE}
\end{equation}
where $R_0$ and $R_s$ are the OHE and the AHE coefficients, respectively. If $\textbf{M}$ follows the direction of the applied magnetic field and is constant in magnitude, we expect $\rho_{xy}\propto \cos \theta$. The data presented in Fig.~\ref{Fig:AHE}(bottom) clearly deviates from this expectation, whereas a good fit is found with:
\begin{equation}
\rho_{xy}=a\cos \theta + b\cos 3\theta.
\label{Eq:AHE}
\end{equation}
We fit our data with Eq. \ref{Eq:AHE} in a wide range of temperatures ($5-300$ K) using high magnetic fields ($\mu_{0}H$ between 4 to 9 T). The temperature and field dependence of $a$ and $b$ are shown in Figure~\ref{Fig:parm} for a film thickness of 22.7 nm. A similar behaviour was observed for films with other thicknesses (7 nm and 15.3 nm). We note that at T$\sim$ 150 K the parameter $a$ approaches zero, therefore the contribution of the $\cos 3 \theta$ is more visible.
\begin{figure}
\includegraphics[scale=0.5]{23AandB.eps}
\caption{The fitting parameters from Eq. \ref{Eq:AHE}, (a) a and (b) b, as function of temperature for different magnetic field. Inset: the field dependence of $a$ at T=175 K. The line is a linear fit. (c) The extrapolated $a_0$ (right) and $a_1$ (right) as function of temperature.}
\label{Fig:parm}
\end{figure}
The parameter $a$ exhibits a linear dependence on the magnetic field, and we denote its slope as $a_1$; the inset of Figure~\ref{Fig:parm}(a) shows this dependence at ${\rm T=175 \ K}$. We extrapolate $a$ to zero and mark it as $a_0$. Figure~\ref{Fig:parm}(c) shows the temperature dependence of the extracted $a_0$ (right) and $a_1$ (left). At low temperatures, where $\textbf{M}$ is close to saturation, we may associate the high-field slope, $a_1$, with the OHE, and $a_0$ with the AHE contribution. This assumption clearly breaks close to $T_c$ where field-induced changes in $\textbf{M}$ may affect the high-field slope of $a$.
The low temperature limit of $a_1$ corresponds to a carrier density of $\sim 1.6\times10^{22}$ carriers per cm$^3$ (i.e., $0.9$ holes per Mn site), larger than the nominal doping level ($\sim 0.2$ holes per Mn site). Similar deviations have been reported before and attributed to the inapplicability of the one-band model \cite{OHE_LSMO,OHE_LCMO}.
Figure~\ref{Fig:XYvsXX} shows $a_0$ as a function of the zero-field $\rho_{xx}$ for films of different thicknesses. Although the resistivity changes as a function of the thickness, $a_0$ seems to scale with $\rho_{xx}$, consistent with previous reports \cite{matl}. In addition, we note that $a_0\propto\rho_{xx}^{\gamma}$, where $\gamma$ is in the range of 1-1.2.
\begin{figure}
\includegraphics[scale=0.4]{23XYvsXX.eps}
\caption{The extracted AHE resistivity, $a_0$, as a function of the zero-field longitudinal resistivity.}
\label{Fig:XYvsXX}
\end{figure}
We turn now to discuss possible sources of the surprising $b$ term.
The OHE would have a contribution with a $\cos 3 \theta$ dependence if in addition to the common term, $R_0 B_\bot$, there would also be a term $R_0^* B_\bot^3$ allowed by symmetry. However, the absolute value of such a term is expected to increase with increasing field, contrary to our observations (see Figure~\ref{Fig:parm}(b)).
Assuming that $b$ is related to the AHE, we consider possible effects of the angular dependence of $\rho_{xx}$ and $M$.
Commonly, the AHE coefficient is described as a function of the longitudinal resistivity; i.e., $R_s=R_s(\rho_{xx})$ \cite{RsRho1,RsRho2,RsRho3,RsRhoNoam}; therefore, one would expect changes in $\rho_{xx}$ to induce changes in the AHE. According to Eq. \ref{Eq:AMR}, $\rho_{xx}$ follows $\cos2\alpha$, which may yield a $\cos3\alpha$ term in the AHE. In other words:
\begin{equation}
\begin{array}{ll}
\rho_{xy}^{AHE}&=R_s(\rho)\mu_0 M_{\bot}\\&=[R_s(\rho_0)+\frac{dR_s}{d\rho}\Delta\rho\cos2\alpha]\mu_0|M|\cos\alpha.
\end{array}
\label{Eq:RsRho}
\end{equation}
However, while the change in $\rho_{xx}$, noted as $\Delta\rho/\rho$, is insensitive to the magnetic field in the low temperature regime (see Figure~\ref{Fig:AMR}), $b$ changes with magnetic field (see Figure~\ref{Fig:parm}(b)). Moreover, Figure~\ref{Fig:Ratio} (top) shows the ratio between the measured $b$ and the calculated $b$ according to Eq. \ref{Eq:RsRho} as a function of temperature. As can be seen, the ratio is larger than $10$ in the low temperature regime.
\begin{figure}
\includegraphics[scale=0.5]{23ratio.eps}
\caption{(top) The ratio between the measured $b$ to the calculated value of $b$ according to Eq. \ref{Eq:RsRho}, and (bottom) the difference between the measured and calculated $c_2$, according to Eq. \ref{Eq:RsM} as a function of temperature.}
\label{Fig:Ratio}
\end{figure}
\begin{figure}
\includegraphics[scale=0.4]{23anisotropy.eps}
\caption{(top) $\frac{K_{anis}}{2M_s}$ and (bottom) the ratio between the calculated and measured $K_{anis}$ as a function of temperature. Inset: The angular dependence of the longitudinal resistivity at $\mu_0$H=1 T and T=5 K. The solid line is fit to Eq. \ref{Eq:AMR}, where $\alpha$ is calculated assuming $\frac{K_{anis}}{2M_s}=$0.8 T.}
\label{Fig:Anisotropy}
\end{figure}
Another possible source for $b$ is the angular dependent variation in the magnetization, both in its direction and its magnitude, due to an easy plane anisotropy. Although the magnetization is almost field independent for the magnetic fields that we use, an easy plane anisotropy may cause a small change in the magnetization direction, i.e., the AHE will follow $R_s M \cos(\theta +\Delta\theta) \approx R_s M (\cos\theta -\sin\theta\sin\Delta\theta)$. Considering an easy plane anisotropy and a Zeeman term hamiltonian ($\mathcal{H}=-K_{anis}\sin^2\alpha-M\cdot H\cos(\alpha-\theta)$, where $K_{anis}$ is the anisotropy constant), we obtain that for $H \gg K_{anis}/2M_s$ the deviation from the field direction takes the form $\sin\Delta\theta \approx \frac{K_{anis}}{2M_sH} \cos\theta\sin\theta$. Thus the AHE is given by:
\begin{equation}
\rho_{xy}^{AHE} \approx R_s M \cos\theta (1-\frac{K_{anis}}{2M_sH}\sin^2\theta).
\label{Eq:RsM}
\end{equation}
We extract $\frac{K_{anis}}{2M_s}$ (shown in Figure~\ref{Fig:Anisotropy} (top)) by fitting the longitudinal resistivity to Eq. \ref{Eq:AMR} with $\alpha$ calculated using the easy plane Hamiltonian with $\frac{K_{anis}}{2M_s}$ as a free parameter (as illustrated in the inset of Figure~\ref{Fig:Anisotropy}).
We fit our measurements with $\rho_{xy}^{AHE}=c_1\cos\theta-c_2\cos\theta\sin^2\theta$ and subtract the expected coefficient $c_2$ based on Eq. \ref{Eq:RsM}, substituting the extracted anisotropy constant.
Figure~\ref{Fig:Ratio} (bottom) shows the difference between the measured and the calculated $c_2$. Figure~\ref{Fig:Anisotropy} (bottom) presents the ratio between the anisotropy constant that which should be assumed in order to explain $c_2$ with this scenario and the measured anisotropy constant as a function of temperature. As can be seen, this scenario yields a good description of $c_2$ below 50 K; nevertheless, this source predicts a field dependent anisotropy constant that is significantly higher than the measured one.
In addition to its effect on the magnetization direction, the magnetic anisotropy may affect the magnitude of the magnetization. According to Eq. \ref{Eq:HE}, the AHE is given by $R_s \mu_0 M_\bot$. Since the magnetization is close to saturation and weakly dependent on $H$, we may approximate its field dependence by $M \approx M(H) + \frac{dM}{dH}\Delta H$. Considering the anisotropy effective field $\frac{K_{anis}}{2M_s}$, we obtain that the total field applied in the direction of the magnetization takes the form: $H^*=H-\frac{K_{anis}}{2M_s}\cos\theta$. Thus, the AHE is given by:
\begin{equation*}
\begin{array}{ll}
R_s \mu_0 M_\bot (\theta)& = R_s \mu_0 [M - \frac{dM}{dH}\frac{K}{2M_s}\cos^2\theta]\cos\theta\\
&=R_s \mu_0 [(M-\frac{dM}{dH}\frac{K}{2M_s})+\frac{dM}{dH}\frac{K}{2M_s}\sin^2\theta]\cos\theta.
\end{array}
\end{equation*}
Therefore this source yields a positive contribution to $c_2$, which cannot explain the measured negative $c_2$.
As we have ruled out trivial sources for the $b\cos 3\theta$ term, it appears that other sources should be considered.
We note that as the temperature increases, $|b|$ increases; and as the magnetic field increases, $|b|$ decreases (see Figure~\ref{Fig:parm}(b)). This behavior may suggest that $|b|$ decreases when $\textbf{M}$ approaches saturation either by increasing the field or by decreasing the temperature. It has been shown that spatial variations in the magnetization may yield a contribution to the AHE in manganites \cite{ye_skyrmion}; however, a $\cos{3\theta}$ dependence is not expected in this model. We point out that structural and magnetic symmetries were previously identified as sources for a more complicated angular dependence of the AMR and PHE in epitaxial films of manganites \cite{AMR_manganites}; however, we do not see a simple way to correlate such effects with our observations.
In conclusion, we find that the AHE in \lsmo\ films cannot be described by the simple relation to the perpendicular component of the magnetization. A more careful treatment that takes into account the lattice and magnetic anisotropies is required.
\section{Acknowledgment}
L.K. acknowledge support by the Israel Science Foundation founded by the Israel Academy of Science and Humanities. Work at Yale supported by NSF MRSEC DMR 1119826 and ONR.
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Robert Buchanan (1785–1873) was a Scottish minister and Professor of Logic and Rhetoric at the University of Glasgow, known as a dramatist and poet.
Life
Buchanan was a cadet of the Clan Buchanan, and a native of Callander, where he was born in 1785. He specially distinguished himself in the philosophy classes. After completing his divinity course at the University of Glasgow, he was in 1812 licensed as a preacher of the Church of Scotland by the presbytery of Haddington, and in 1813 was presented to the parish of Peebles.
In 1824 Buchanan was appointed assistant and successor to George Jardine in the chair of Logic and Rhetoric at Glasgow, becoming sole professor in 1827. As a philosopher he was influenced by his teacher James Mylne, and was wary of the philosophy of commonsense. Following Jardine, and with the support of Mylne's successor William Fleming, he resisted attempts to bring Glasgow's courses more in line with those taught in England.
In 1864, Buchanan retired to Ardfillayne, Dunoon. He died on 2 March 1873 and is buried in Dunoon Cemetery.
Legacy
In commemoration of Buchanan's services, the Buchanan prizes were instituted in 1866, for students of the logic, moral philosophy, and English literature classes of the University of Glasgow. By his will he bequeathed £10,000 for the founding of Buchanan bursaries, for the arts classes of the university.
Works
Buchanan was the author of:
Fragments of the Table Round, 1860;
Vow of Glentreuil, and other Poems, 1862;
Wallace, a Tragedy, 1856;
Tragic Dramas from Scottish History, 1868, containing: The British Brothers; Gaston Phœbus; Edinburga; and the tragedies of Wallace and King James the First.
Anonymous, Canute's Birthday in Ireland, a Drama in Five Acts, in 1868.
Buchanan's tragedy Wallace was performed twice for a charitable object at the Prince's Theatre, Glasgow, in March 1862, the major characters being played by students.
Notes
Attribution
1785 births
1873 deaths
19th-century Ministers of the Church of Scotland
Scottish dramatists and playwrights
People from Stirling (council area)
Alumni of the University of Glasgow
Academics of the University of Glasgow
19th-century Scottish poets
19th-century Scottish dramatists and playwrights
Scottish logicians
Scottish philosophers
19th-century British philosophers
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