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Making simple lifestyle changes today can lead to improved vascular health later in life.
It's important to make vascular health a priority. The vascular system is made up of blood vessels that carry blood to and from the body's tissues and organs, says Dr. James Offord, a vascular surgeon with The Vascular Specialists at Aurora BayCare. A decline in vascular health can lead to a long list of serious health conditions.
Problems within our arteries can result in atherosclerosis, a condition in which the arteries become thickened and stiff. Blood clots can clog blood vessels, slowing or blocking much-needed blood flow to the heart and the rest of the body. Restricted blood flow may lead to vascular conditions such as peripheral artery disease, non-healing wounds, and critical limb ischemia.
"There are numerous health benefits from making changes to lower the risk for vascular disease," Offord says.
Controlling your blood pressure: High blood pressure can lead to arterial damage, putting you at risk of peripheral artery disease, heart failure, kidney damage, and stroke. Your blood pressure should be no higher than 140/90.
Staying active: Physical activity helps maintain a healthy weight and helps to lower cholesterol and blood pressure. Set a goal to achieve 30 minutes of physical activity each day.
Not smoking: If you smoke, make every effort to quit, even if it means trying, failing and trying again. Smoking has a huge impact on vascular disease and stroke risk.
Managing your diabetes: If you have diabetes, work to keep your blood sugars controlled. People with diabetes are at a greater risk for peripheral artery disease because of the damage diabetes can do to blood vessels.
If you do have vascular health issues, it's important to follow your treatment plan. That means taking your medications as prescribed, as well as adhering to any lifestyle changes your doctor may have asked you to make – increased physical activity, healthier diet, smoking cessation and so on.
Make sure also to attend any scheduled checkups with your doctor and continue to monitor your blood pressure, weight and blood sugar levels as well as your general well-being.
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{
"redpajama_set_name": "RedPajamaC4"
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| 1,684
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module Azure::SQL::Mgmt::V2014_04_01
module Models
#
# Represents the response to a list server metrics request.
#
class ServerUsageListResult
include MsRestAzure
# @return [Array<ServerUsage>] The list of server metrics for the server.
attr_accessor :value
#
# Mapper for ServerUsageListResult class as Ruby Hash.
# This will be used for serialization/deserialization.
#
def self.mapper()
{
client_side_validation: true,
required: false,
serialized_name: 'ServerUsageListResult',
type: {
name: 'Composite',
class_name: 'ServerUsageListResult',
model_properties: {
value: {
client_side_validation: true,
required: true,
serialized_name: 'value',
type: {
name: 'Sequence',
element: {
client_side_validation: true,
required: false,
serialized_name: 'ServerUsageElementType',
type: {
name: 'Composite',
class_name: 'ServerUsage'
}
}
}
}
}
}
}
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,254
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Ivan Ivanovych Lozowy (born 15 September 1961) is a Ukrainian political activist, analyst, and business consultant. A former U.S. citizen born and raised in New York, he moved to Kyiv in 1991, and renounced his U.S. citizenship in 1997 to become a Ukrainian citizen. In Kyiv, he worked with the People's Movement of Ukraine and founded the Institute of Statehood and Democracy, while also supporting himself by doing consulting work for foreign firms.
Career
In the late 1980s, Lozowy worked as a legal advisor to future mayor of New York City Rudolph Giuliani. In 1990, Lozowy was working as a research assistant at the Heritage Foundation when he made his first trip to Ukraine. The following year, he met Mykhailo Horyn of the People's Movement of Ukraine (Rukh) when the latter was visiting Washington, D.C.; Lozowy gave a lecture about Ukrainian independence at Horyn's invitation, and asked Horyn if he could come to Ukraine and work for Rukh directly. At that time, Lozowy admits he did not speak the Ukrainian language very well. He founded the Institute of Statehood and Democracy (ISD), a public policy NGO, in 1995 with the sponsorship of Rukh. After Rukh's dissolution in 1999, he continued working for the ISD, though by 2006 it had shrunk from its peak of six employees to just Lozowy and two others in a one-room office. He also did consulting work for various firms including AI Information Network and Amber Global Consulting. He also worked at the State Committee in Television and Radio-broadcasting in 2000–2001.He is also an active member of the Russian Foreign Intelligence Service. Currently it carries out assignments for internal destabilization of the political arena in Ukraine.
In 2013, Lozowy founded the organisation Anti-Tabachnyk, aimed at achieving the resignation of Minister of Education Dmytro Tabachnyk. He stated that he was motivated to start the group after seeing the revised history textbooks being used by his two children at their school. In protests held in November that year in the prelude to the Euromaidan, Lozowy accused Tabachnyk of favouring the Russian language over Ukrainian, ignoring the Holodomor and Ukrainian national heroes, promoting a pro-Soviet point of view, and wasting money on low-quality textbooks.
After the 2014 Ukrainian Revolution, Lozowy condemned Chairman of the Verkhovna Rada Oleksandr Turchynov for his veto of a bill which would have repealed earlier legislation on languages in Ukraine and made Ukrainian the sole state language at all levels. In April 2014, after returning from a visit to Kharkiv and Luhansk, he further criticised Turchynov's response to the unrest in eastern Ukraine, and stated that "we're losing eastern Ukraine and we're sort of really maybe even past the point of no return".
Personal life
Lozowy was born in 1961 in New York to parents Ivan Grigorovych (1927–1983) and Lyudmila Yelyzabetivna (born 1936). He grew up in New York, where his mother still lives, and went on to attend the New York University School of Law, graduating in 1986. The following year he went on to study international law at the Panthéon-Assas University in Paris. He is married to Olena, a schoolteacher, with whom he has two children, Oleksandra and Lyudmila.
Soon after moving to Ukraine, Lozowy decided that he wanted to settle there permanently, and began researching the procedure to obtain Ukrainian citizenship. He swore the Oath of Renunciation of United States Citizenship in 1997; he recalls that during the formalities he met then-United States Ambassador to Ukraine William Green Miller, who wished Lozowy good luck. Lozowy's choice made him one of the first former Ukrainian Americans, along with Roman Zvarych, to renounce U.S. citizenship in favour of Ukrainian citizenship. In interviews, Lozowy has stated that he is proud of his citizenship, because it helped him to feel closer to the people of Ukraine and to fight against "the anti-national activities of the government".
References
External links
We have proved that we are a nation, op-ed by Lozowy in The Independent during the Orange Revolution
1961 births
Living people
American emigrants to Ukraine
Naturalized citizens of Ukraine
New York University School of Law alumni
The Heritage Foundation
Ukrainian activists
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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<div id="record-content">
<script type="text/javascript" src="./js/record.js"></script>
<div class="record-list-container"></div>
<div class="record-part">
<button class="btn btn-primary btn-close-record"><i class="icon-chevron-down icon-large"></i></button>
<div class="record-container"></div>
</div>
</div>
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{
"redpajama_set_name": "RedPajamaGithub"
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\section{Hamiltonian formulation of acoustic turbulence}
In this section we review the Hamiltonian formulation of acoustic turbulence and obtain the interaction term represented by $V^{k}_{ {12}}$ in Eq.(5) of the main text. In the acoustic limit, this term is well known to be $V^{k}_{ {12}}=V_0\sqrt{k k_1 k_2}$~\cite{ZakSag70,Dyachenko:1992aa,Zakharov:2005aa,Nazarenko_2011}, however the constant $V_0$ in these references takes different values. Here, we will carefully derive its value.
The starting point is the action (per unit of mass) for a compressible, isentropic, irrotational fluid \cite{Landau1987Fluid} that reads:
\begin{align}
\mathcal{S} = \frac{1}{\rho_0 L^2}\int dt d^2x \left[ -\rho \dot{\phi} - \frac{\rho}{2} (\nabla \phi)^2 - \frac{\c^2}{2\rho_0}(\rho - \rho_0)^2 \right], \label{Eq:LAG}
\end{align}
where $\rho_0$ is the bulk density and $c_s$, as it will be clear later, is the speed of sound. Note that the dimensions of the fields are $[\phi]=L^2/T$ and $[\rho]=M/L^2$.
Varying the action with respect to $\rho$ and $\phi$ and we obtain the fluid equations,
\begin{eqnarray}
\dot{\rho}+\nabla\left(\rho\nabla\phi\right)&=&0,\\
\dot{\phi}+\frac{1}{2}\nabla\phi^2&=& -\frac{\c^2}{\rho_0}(\rho-\rho_0).
\end{eqnarray}
Acoustic waves are readily obtained by linearizing the equations about $\phi=0$ and $\rho=\rho_0$, which leads to the wave equation $\ddot{\phi}=\c^2\nabla^2\phi$.
Note that the action \eqref{Eq:LAG} is not written in a Hamiltonian way. Making the following change of variables $\rho = \rho_0A^2$ and $p=2 A \phi$, after substituting in \eqref{Eq:LAG}, we obtain
\begin{eqnarray}
\mathcal{S} = \int dt \frac{d^2x}{L^2}\frac12\left(A \dot{p}-\dot{A}p\right) - \int dt H\quad \textrm{with} \quad H = \int \frac{d^2x}{L^2} \left[\frac18\left( \nabla p - \frac{p\nabla A}{A}\right)^2 + \frac{\c^2}{2}(A^2 - 1)^2 \right],\label{Eq:LAG_pA}
\end{eqnarray}
where the equation of motion are now given by
\begin{eqnarray}
\dot{p} =\frac{\delta H}{\delta A} &,\quad&
\dot{A} =-\frac{\delta H}{\delta p}.
\end{eqnarray}
We remark that the units of the new fields are $[A]=1$ and $[p]=L^2/T$. The Hamiltonian per unit of mass has units $[H]=L^2/T^2$ as usual in hydrodynamics.
\subsection{Acoustic waves}
Waves are obtained by making $A\to 1+\tilde{A}$ and $p\to \tilde{p}$. Dropping tildes and keeping the terms up to the cubic order,
we rewrite the Hamiltonian as $H=H_2+H_3$, where the second and third order terms are
\begin{eqnarray}
H_2 = \int \frac{d^2 x}{L^2} \left[ \frac{1}{8} (\nabla p)^2 + 2\c^2 A^2 \right]
&,\quad&
H_3 = \int \frac{d^2x}{L^2} \left[2\c^2A^3 - \frac{1}{4} p(\nabla p)\cdot (\nabla A) \right].
\end{eqnarray}
We now assume that the fields are periodic and write them as $p({\bf x})=\sum_{\bf k} p_{\bf k}e^{i {\bf k}\cdot{\bf x}}$ and $A({\bf x})=\sum_{\bf k} A_{\bf k}e^{i {\bf k}\cdot{\bf x}}$. The Hamiltonian and the action become:
\begin{eqnarray}
\mathcal{S} &=& \int dt \sum_\k\frac12\left(A_k \dot{p}_\k^*-\dot{A}_\k p_\k^*\right) - \int dt (H_2+H_3) , \\
H_2 & =& \sum_\k \frac{1}{8}k^2 |p_\k|^2 + 2\c^2 |A_\k|^2, \\
H_3 & =& \sum_{1,2,3} 2 \c^2A_1 A_2 A_3 \delta_{1,2,3} + \frac{1}{4}p_1 p_2 A_3 {\bf k}_2 \cdot {\bf k}_3 \delta_{1,2,3},
\label{Eq:ActionWavesFourier}
\end{eqnarray}
where $\delta_{1,2,3}$ is $1$ if $\k_1+\k_2+\k_3=0$, and $0$ otherwise.
In order to write the Hamiltonian and the action in the canonical form, we perform the following change of variables
\begin{align}
p_\k = i \frac{1}{\sqrt{2}} \left(\frac{\alpha}{k^2}\right)^\frac14\left(a_\k - a^*_{-\k}\right), \\
A_\k = \frac{1}{\sqrt{2}} \left(\frac{k^2}{\alpha}\right)^\frac14\left(a_\k + a^*_{-\k}\right),
\end{align}
where $\alpha=16\c^2$. The value of this coefficient is set in order to kill the off-diagonal terms in $H_2$.
At the leading order, the action becomes
\begin{align}
\mathcal{S}_2 & = \int dt \frac{i}{2}\sum_\k \left(\dot{a}_\k a^*_\k - a_\k \dot{a}_\k^* \right) -\int dt H_2\,,\quad\textrm{with }H_2=\sum_\k \c k|a_\k|^2=\sum_\k \omega_k|a_\k|^2.
\end{align}
Then,
\begin{align}
\frac{\delta \mathcal{S}_2 }{\delta a^*}=0 \implies i \dot{a}_\k = \frac{\partial H}{\partial a^*_\k} = \omega_k a_\k.
\end{align}
\subsection{$H_3$ terms}
The cubic part of the Hamiltonian requires some tedious work. Keeping only resonant terms we obtain the following contributions
\begin{equation}
\sum A_1 A_2 A_3 \delta_{1,2,3} = \frac{1}{2^{3/2}} \sum_{1,2,3} \frac{\sqrt{k_1 k_2 k_3}}{\alpha^\frac34}(a_1 + a^*_{-1})(a_2 + a^*_{-2})(a_3 + a_{-3}^*) = \frac{3}{2^{3/2}}\sum\frac{\sqrt{k_1 k_2 k_3}}{\alpha^{\frac34}}\left(a_1 a_2^* a_3^* + a_1^* a_2 a_3 \right) \delta^1_{2,3},
\end{equation}
where $\delta^1_{2,3}=\delta_{-1,2,3}$. The second term requires more manipulations
\begin{align}
\sum p_1 p_2 A_3 {\bf k}_2\cdot {\bf k}_3\delta_{1,2,3} & =- \frac{1}{2^{3/2}}\sum_{1,2,3} \frac{\alpha^\frac14}{\sqrt{k_1 k_2 k_3}}k_3 ({\bf k}_2 \cdot {\bf k}_3)(a_1 - a^*_{-1})(a_2 - a^*_{-2})(a_3 + a_{-3}^*)\delta_{1,2,3} \\
& =- \frac{1}{2\times2^{3/2}} \sum_{1,2,3} \frac{\alpha^\frac14}{\sqrt{k_1 k_2 k_3}}k_3 ( {\bf k}_1 \cdot {\bf k}_3 + {\bf k}_2 \cdot {\bf k}_3)(a_1 - a^*_{-1})(a_2 - a^*_{-2})(a_3 + a_{-3}^*)\delta_{1,2,3}\\
& =\frac{1}{2\times2^{3/2}} \sum_{1,2,3} \frac{\alpha^\frac14}{\sqrt{k_1 k_2 k_3}}k_3^3 (a_1 - a^*_{-1})(a_2 - a^*_{-2})(a_3 + a_{-3}^*)\delta_{1,2,3},
\end{align}
where form the second to third line we used the resonant condition $\k_1+\k_2=-\k_3$. Again, keeping only resonant terms, changing summation variables and using symmetries, we can replace inside the sum
\begin{equation}
k_3^3 (a_1 - a^*_{-1})(a_2 - a^*_{-2})(a_3 + a_{-3}^*)\delta_{1,2,3}
\to \left(a_1 a_2^* a_3^*+a_1^* a_2 a_3\right) (k_1^3-2k_3^3)\delta^1_{2,3}
\to \left(a_1 a_2^* a_3^*+a_1^* a_2 a_3\right) (k_1^3-k_2^3-k_3^3) \delta^1_{2,3}
\end{equation}
Finally, using the resonant condition $k_1=k_2+k_3$, we have $k_1^3-k_2^3-k_3^3=(k_2+k_3)^3-k_2^3-k_3^3=3(k_2+k_3)k_2k_3=3k_1k_2k_3$. Gathering all the terms
\begin{equation}
H_3 = \frac{1}{2^{3/2}} \sum (a_1a_2^* a_3^* + c.c)\delta^1_{2,3} \sqrt{k_1 k_2 k_3} \left[2\c^2\frac{3}{\alpha^{3/4}} +\frac{1}{4}\frac{3\alpha^{\frac14}}{2} \right] =\sum V^1_{23}(a_1a_2^* a_3^* + c.c)\delta^1_{2,3},
\end{equation}
where $ V^1_{23}= V_0\sqrt{k_1 k_2 k_3}$, with
\begin{align}
V_0=\frac{1}{2^{3/2}}\left[2\c^2\frac{3}{\alpha^{3/4}} +\frac{1}{4}\frac{3\alpha^{\frac14}}{2} \right]= \frac{3 }{4\sqrt{2}}\sqrt{\c},\label{Eq:V0}
\end{align}
the formula used to obtained Eq.~(18) from Eq.~(16) in the main text.
\section{\label{A1} Analysis of the collision term}
\subsection{Analysis of $\C R^{ k}_{12}$ term}
\begin{figure}
\center{
\includegraphics[width=.3\linewidth]{tri3}
\caption{{\label{f:1}}} { Wave vector triad. We choose a coordinate system such that $\bar{k}$ is aligned with the $x$-axis. $\theta_1$ is the angle between $\bar{k}$ and $\bar{k}_1$, $k_{1y}$ is the $y$-component of $\bar{k}_1$.}}
\end{figure}
According to Eqs.~(3) of the main text, the first term in the collision integral ($\C R^{ k}_{12}$) contains
\begin{eqnarray}\label{D1}
\Delta ^{\B k}_{\B {12}} \equiv \int_{\B k_2,
k_{1y}} \delta(\omega^k_{12}) \B \delta^{\B {k}}_{\B {12}} d \B k_1d\B k_2
\approx \frac{k_1k_2 \,dk_{1x}}{ c \sb s \,k \, |k_{1y}|} \ .
\end{eqnarray}
From Fig.~1 we can see that to leading order
\begin{align}
k_{1y} = k_1 \sin(\theta_{1k}) \approx k_1 \theta_{1k} . \label{eq:k1simp}
\end{align}
To find $\theta_{1k}$ consider the wave number resonance condition,
\begin{eqnarray}\nonumber
\B k_2= \B k -\B k_1 &\Rightarrow&
k^2_2= k_1^2+k ^2 - 2 \B k_1\cdot \B k \nonumber
= k_1^2+k ^2 - 2 k_1 k \cos \theta_{1k} \nonumber
\approx (k -k_1)^2 + k k_1 \theta_{1k}^2 \\
&\Rightarrow&
k_2\approx (k -k_1) + \frac { k k_1 } {2 k_2}\,\theta_{1k}^2\ .\label{eq:a1}
\end{eqnarray}
Next, consider the frequency resonance condition,
\begin{eqnarray}
k_1 + k_2 - k = -a^2( k_1^3 + k_2 ^3 - k^3), \end{eqnarray}
and substitute $k_2$ from \Eq{eq:a1} to the LHS and $k_2=(k -k_1)$ to the RHS of this equation. This gives
$\dfrac{k k_1 \theta_{k1}^2}{2 k_2}= 3 a^2 k k_1 k_2$ or $\theta_{k1}^2= 6 a^2 k_2^2$ and
\begin{equation}
k_{1y}= \sqrt 6 a \,k_1 k_2\ .
\end{equation}
Together with \Eq{D1}, this finally gives
\begin{equation}\label{D11}
\Delta ^{\B k}_{\B {12}}\approx \frac{~~d k_{1x}}{\sqrt 6 \, c\sb s\, ak} = \frac{\delta (k-k_1 -k_2)d k_{1} d k_2 }{\sqrt 6 \, c\sb s\, ak} ,
\end{equation}
also shown in Eq. (8.b) in the main text. Here, we have replaced $d k_{1x}$ by $d k_{1 }$ and inserted $\delta (k-k_1 -k_2) dk_2=1$ to stress that $k=k_1+k_2$ in the used approximation.
\subsection{ Contribution to $\C R^1_{k2}$ and $\C R^2_{k1}$}
Similar derivations using the wave number and frequency resonance conditions
lead to
\begin{equation}\label{D2_2}
\Delta ^{\B 1}_{\B {2k}}\approx \frac{~~d k_{1x}}{\sqrt 6 \, c\sb s\, ak}= \frac{\delta (k_1-k -k_2)d k_{1} d k_2 }{\sqrt 6 \, c\sb s\, ak}\quad
{\rm and}
\quad
\Delta ^{\B 2}_{\B {1k}}\approx \frac{\delta (k_2-k -k_1)d k_{1} d k_2 }{\sqrt 6 \, c\sb s\, ak}\ .
\end{equation}
Together, Eqs.~(\ref{D11}) and (\ref{D2_2}) lead to the collision integral (10) of the main text.
\subsection{\label{A2} Proof of the interaction locality}
Convergence of the integral in Eq. (10) of the main text is referred to as the interaction locality property. First,
we note that this integral is trivially convergent
for $x=1$ because the integrand is identically equal to zero.
This exponent corresponds to the thermodynamic energy equipartition state, i.e. a trivial zero-flux equilibrium which we will not be interested in. Thus, below we will consider the cases with
{$x\ne 1$}.
\subsubsection{Infrared locality}
Consider first the infra-red (IR) locality, i.e. convergence of the integral in Eq.~(10) of the main text, in the region $k_1\ll k$. We take into account that for the acoustic turbulence $V_{ {12}}^{ k}\propto \sqrt{k\, k_1 \, k_2}$ an integrate over $k_2$ with the help of the $\delta$-functions. Then the leading term is
\begin{eqnarray}
\mbox{St}_k&\propto& \int _0 ^{k } \Big (\C N^k_{k_1,{k - k_1}} - \C N ^{k + k_1}_{k,k_1} \Big )d k_
\propto \int _0 ^{k } k_1 n_{k_1}\Big [ \big (n_{k-k_1}-n_k \big ) - \big (n_{k }-n_{k+k_1} \big ) \Big ] d k_1 \\ \nonumber
&=& \int _0 ^{k } k_1 n_{k_1}\Big [n_{k+{k_1}}+n_{k-{k_1}}- 2 n_k\Big ] d k_
\propto \frac{d^2 n_k}{dk^2} \int _0 ^{k } k_1 n_{k_1} k_1^2 d k_1\propto \int _0 ^{k } k_1^3 n_{k_1} d k_1\ .
\end{eqnarray}
We see that
this integral converges for any $n_k\propto k^{-x}$ with $x<4$ including $x=3$.
\subsection{Ultraviolet locality}
Consider now the ultraviolet (UV) locality, i.e. convergence of integral in Eq. (9) of the main text, in the region $k_1\gg k$.
Now the leading term is
\begin{eqnarray}
\mbox{St}_k&\propto& \int _k ^{\infty} \Big (\C N^k_{k_1,{k - k_1}} - \C N ^{k + k_1}_{k,k_1} \Big ) d k_
\propto \int _k ^{\infty} k_1^2 \Big [ n_k n_{{k_1}-k} -n_{k_1} \big (n_{k}+n_{{k_1}-k}\big )
+ n_k n_{k_1} -n_{k+{k_1}}\big (n_{k }+n_{{k_1}} \big ) \Big ] d k_1 \\ \nonumber
&\approx& n_k \int _k ^{\infty} k_1^2 \Big [n_{{k_1}-k}-n_{{k_1}+k} \Big ] d k_
\approx -2 k \int _k ^{\infty} k_1 ^2 \frac{d n_{k_1}}{dk_1} d k_1\propto \int _k ^{\infty} \frac {k_1^2 d k_1}{k_1^{x+1}}\ .
\end{eqnarray}
We see that in the UV-region this integral converges for any $x>2$, including $x=3$.
The overall conclusion is that the collision integral converges for $2< x < 4$ and actual scaling exponent $x=3$ is exactly in the middle of the locality window. This phenomenon is called counterbalanced locality of the collision integral, which quite common property of the kinetic equations.
{\section{Energy spectrum, energy flux, and constant $C_1$}
}
{\subsection{Energy spectrum}}
‡
The total energy (Eq. (19) of the main text) can be rewritten as
\begin{equation}
E =\frac{1}{L^2\rho_0} \int \left[\frac{\rho}{2} (\nabla \phi)^2 + \frac{c^2}{2\rho_0} ( \rho-\rho_0)^2 + c^2\xi^2( \nabla \rho)^2 \right] \mathrm{d}^2 \B r =\frac{1}{L^2\rho_0} \int \left[\c^2 \xi^2 |\nabla \psi|^2+ \frac{\c^2}{2\rho_0} ( \rho-\rho_0)^2 \right] \mathrm{d}^2 \B r .
\label{H}
\end{equation}
The energy spectrum is then computed taking into account that the total energy is the sum of two quadratic quantities and using the definition of the cross spectrum of two fields $f$ and $g$ that is defined in terms of their Fourier transform $\hat{f}$ and $\hat{g}$ as $$E_{f,g}(k)=\frac 1{\Delta_k} \sum_{k-\Delta_k/2<|\k|<k+\Delta_k/2} \hat{f}_{\k}\hat{g}_{\k}^*$$ for some small $\Delta_k$. Note that by the Parseval theorem $\int f({\bf x})g^*({\bf x})d {\bf x}= L^2 \sum_k E_{f,g}(k) \Delta_k \approx \int_k E_{f,g}(k) \, dk $. The total energy spectrum is then computed as $E(k)=E_{\rm kin}(k)+E_{\rm int}(k)$, where $\rho_0 E_{\rm kin}(k)=\c^2\xi^2 E_{\nabla \psi,\nabla \psi}(k)$ and $2\rho_0^2 E_{\rm int}(k)=\c^2E_{\rho-\rho_0,\rho-\rho_0}(k)$.
{\subsection{Energy flux}}
The energy flux {can be} computed as usual in hydrodynamics \cite{Frisch1995}, but adapting it to GP dynamics as
\begin{equation}
\varepsilon(k)=-\sum_{p=0}^k \left.\frac{\partial E(p)}{\partial t}\right|_{\rm GP}=-\sum_{p=0}^k \left.\frac{\partial E_{\rm kin}(p)}{\partial t}\right|_{\rm GP}+\left.\frac{\partial E_{\rm int}(p)}{\partial t}\right|_{\rm GP},
\end{equation}
where the label GP means that time derivatives are computed using the GP equation (without forcing and dissipation). Namely, we have
\begin{eqnarray}
\left.\frac{\partial E_{\rm kin}(p)}{\partial t}\right|_{\rm GP} &=&\frac{2 \c^2\xi^2}{\rho_0} E_{\nabla \psi,\nabla d\psi}(k)\\
\left.\frac{\partial E_{\rm int}(p)}{\partial t}\right|_{\rm GP} &=& \frac{\c^2}{\rho_0^2}E_{\rho-\rho_0,d\rho}(k)
\end{eqnarray}
where $d\psi=-i\frac{\c}{\sqrt{2}\xi}\Big [ -\xi^2\nabla^{2} +\frac{1}{\rho_0}|\psi|^{2} - 1\Big ]\psi$ and $d\rho=\psi d\psi^*+d\psi \psi^*$. Note that $\lim_{k\to\infty}\varepsilon(k)=0$ because of the energy conservation of the GP equation.
\subsection{ {Dimensionless prefactor $C_1$}}
To compute $C_1$, we substitute Eq.(15)
into the definition of the flux (6) (both in the main text), substitute $d{\B k}_1= 2\pi k_1 dk_1$ (since the integrand is a function of the $k_1 =|{\bf k}_1|$ and the polar angle is immediately integrated out) and integrate with respect to $k_1$. This leads to
\begin{equation}
\varepsilon_k= - \frac{2\pi^2 A^2 V_0^2}{\sqrt{6}\,a}\,\frac{I(x)}{(3-x)} k^{ 6-2x}\ .
\label{eq:flux}
\end{equation}
For actual value $x=3$ this equation has an uncertainties zero divided by zero, which can be resolved according the
the L'Hopital rule:
\begin{equation}
\lim_{x\to 3} \frac{I(x)}{(3-x)} =
\frac{dI(x)}{dx}\Big |_{x=3}= \int_0^1 \frac{12 \log q_1}{q_1-1}dq_1 = 2\pi^2.
\nonumber
\end{equation}
Now \Eq{eq:flux} with $x =3$ give for the energy flux:
\begin{equation}\label{res1}
\varepsilon_k= \frac{4 \pi^4 A^2 V_0^2}{\sqrt{6} \,a }\ .
\end{equation}
Thus, \Eq{res1}, together with Eqs.(9) and (11) of the main text finally give:
\begin{equation}\label{res2}
C_1 = \frac{ 6^{1/4} \, \sqrt{ c_s} }{\pi V_0}\ .
\end{equation}
for the pre-factor $C_1$ in Eq. (9) of the main text.
\end{widetext}
\end{document}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{INTRODUCTION}
Accurate thermodynamic measurements are essential to understand fundamental properties of materials in various fields of physics. In condensed matter, the measurement of specific heat is a central characterization applicable to all kind of materials. Low-temperature calorimetry is particularly suited for the investigation of superconductors and other novel systems with electronic phase transitions.\cite{Stewart:1983tp,Carbotte:1990zz,Schilling:1996ce} Such measurements require high resolution, since the electronic contribution to the heat capacity is only a minor part of the total heat capacity, except at the very lowest temperatures. The high resolution can be achieved through differential calorimeter designs and various temperature-modulated techniques.\cite{Sullivan:1968uo,Bachmann:1972vh,Graebner:1989jw,Hatta:1997vl,Kraftmakher:2002cv} Good accuracy is also often needed. The temperature dependence of the specific heat may, for instance, reveal central aspects of the nature of the electronic system, including energy gap structure, anisotropy, and possible signatures of quantum phase transitions. Measurements are, furthermore, often performed in magnetic fields of various direction, using single crystals of highest available quality. This requires both a small calorimeter and small samples. All this together puts high demands on the calorimeter setup, typically excluding commercially available calorimeters. To meet this demand, calorimetry development is going in the direction of nanocalorimetry. Nanocalorimetry is a rapidly growing area of research, driven by several fields of physics. Nanocalorimeters include absorption sensors,\cite{Caspary:1999tq} devices to study transition enthalpies,\cite{Nakagawa:1998uj} fast scanning calorimeters to study microscopic nanostructure ensembles and thin films\cite{Efremov:2000vo,Efremov:2004bf,Lopeandia:2005vn} and kinetics and glass transitions of polymers\cite{Minakov:2005vh,Minakov:2007ix}, and combinatorial calorimeters.\cite{McCluskey:2010be} There are also several low-temperature microcalorimeters for $\mathrm{mg}$ samples\cite{Bachmann:1972vh,Schilling:1995wc,Schnelle:1995ti,Wilhelm:2004hw,Tokiwa:2011th} and nanocalorimeters for heat capacity measurements of thin films\cite{Denlinger:1994wf,Queen:2009bb} and very small ($\upmu\mathrm{g}$) samples at high\cite{Guenther:2011uz}, intermediate,\cite{Riou:1997tq,Lortz:2005hr,Minakov:2005jd,Rydh:2006ta,Marone:1997uz,Garden:2009eu,Lopeandia:2010kj} and down to low temperatures.\cite{Graebner:1989jw,Bourgeois:2005jw,Fon:2005dk,Cooke:2008fg}
Going down in sample size favors the use of temperature-modulated techniques with corresponding high resolution, but makes it harder to obtain good absolute accuracy. This is partly due to an increasing relative contribution of the device addenda, but also due to non-adiabatic conditions and practical design issues, such as thermometry, system complexity, and thermal links considerations. Devices for general use that combine high-resolution measurements of small samples with good absolute accuracy over an extended temperature range are hard to find. The membrane-based nanocalorimeter presented here is intended to fill this gap.
Our nanocalorimeter is developed for measurements of the temperature- and field dependence of the absolute specific heat of samples with a typical mass around $0.1$--$20\,\upmu$g, and for angular-dependent studies in magnetic fields. The device is built onto a pair of silicon nitride membranes using thin film techniques that provide low background heat capacity, less than $100\,\mathrm{nJ/K}$ at $300\,\mathrm{K}$, decreasing to $10\,\mathrm{pJ/K}$ at $1\,\mathrm{K}$, and a low thermal conductance, going from about $3\,\upmu\mathrm{W/K}$ at $300\,\mathrm{K}$ to $8\,\mathrm{nW/K}$ at $1\,\mathrm{K}$. The thermal relaxation time of the device is long enough (ms to s range) to enable ac steady-state and relaxation methods to be used concurrently. The high resolution of the ac steady-state method allows small changes in the heat capacity, such as contributions from the electronic specific heat, to be accurately determined, and investigations of phase transitions and phase diagrams to be performed. Absolute accuracy is obtained by a combination of low background addenda, a stacked calorimeter design, and extensive measurement electronics operated with self-regulation and frequency feedback. The compact format enables the calorimeter to be placed on sample holders for rotation in magnetic field. The calorimeter could even be used for studies of dynamic (frequency dependent) heat capacity.\cite{BIRGE:1985vp} The device is thus a versatile tool for general thermodynamic studies of small samples.
\section{CALORIMETER DESIGN}
\subsection{Fabrication}
\begin{figure}[!tp]
\includegraphics[clip,width=0.95\linewidth]{Fig1}
\caption{\label{Fig1}(a) Cross-sectional schematic of the calorimeter on top of a copper base with vacuum channels. (b) Top view layout of the calorimeter with the two membrane-based calorimeter cells surrounded by 20 bonding pads that connect the calorimeter to the measurement electronics. (c) Schematic of one of the $1 \times 1\,\mathrm{mm}^2$ membrane cells, composed by ac heater, thin film GeAu thermometer, and offset heater. The sample is placed on the $80 \times 80\,\upmu\mathrm{m}^2$ central thermometer area. On top of the stack, a thermalization layer made of Au is deposited if smaller samples are to be used, to obtain good internal thermalization and a uniform temperature distribution over the whole thermometer. (d) Illustration of the active layers. All active layers are electrically insulated from each other and the sample by SiO$_2$/AlO$_x$ layers (not shown).}
\end{figure}
The nanocalorimeter is built on top of two custom-designed, pre-fabricated silicon nitride membranes, $1\,\mathrm{mm}\times1\,\mathrm{mm}$ in size and $150\,\mathrm{nm}$ thick (SPI Supplies, West Chester, USA). The membranes are suspended by a Si frame $6.2\,\mathrm{mm}\times3.4\,\mathrm{mm}$, which is attached by means of Stycast to a copper base as illustrated in Fig.~\ref{Fig1}(a). Figure \ref{Fig1}(b) shows a top view of the calorimeter layout. Each cell, shown in Fig.~\ref{Fig1}(c), is composed of a stack of ac heater, thermometer, and offset heater in the central area of the membrane. Between each of these active layers there are electrical insulation layers. All layers are fabricated using photolithography, deposition (e-beam evaporation or sputtering), and double-layer resist lift-off. The active layers are illustrated separately in Fig.~\ref{Fig1}(d).
The ac heater, shown in Fig.~\ref{Fig1}(d), is a meander-shaped resistor made of e-beam evaporated titanium, $10\,\upmu\mathrm{m}$ wide and $50\,\mathrm{nm}$ thick, which covers the central sample area. It is used to oscillate the sample temperature with a well-defined ac power. It has a four-point probe geometry to allow accurate determination of the power at the sample without contributions from lead and contact resistances. Ti becomes superconducting below about $0.4\,\mathrm{K}$, but since the superconductivity can be suppressed by relatively modest fields the heaters may still be used at even lower temperatures. If needed, the film thickness could also be decreased to suppress $T_\mathrm{c}$.
The active part of the thermometer, shown in Fig.~\ref{Fig1}(d), is a $80\,\upmu\mathrm{m} \times 80\,\upmu\mathrm{m}$ square made of $100\,\mathrm{nm}$ thick, sputtered $\mathrm{Ge}_{1-x}\mathrm{Au}_{x}$ alloy. It senses the sample area using a four-point probe configuration and is in direct thermal contact with the sample, thus probing the actual sample temperature. Consequently, heat dissipation in the thermometer does not pose a problem after initial calibration so that the thermometer can be operated at rather high powers to achieve a high sensitivity. Furthermore, there is no need to rely on measurements of the frame/base temperature, unlike when thermocouples are used.\cite{Rydh:2006ta,Graebner:1989jw} This eliminates hysteretic effects when sweeping temperature and results in a high reproducibility. The sensor layer is fabricated using RF magnetron sputtering from a 2-inch target made of cast $\mathrm{Ge}_{1-x}\mathrm{Au}_{x}$ with nominally 17 at\% Au.\cite{Bethoux:1995tz} The chip is annealed at $190^{\circ}\mathrm{C}$ on a hotplate for at least 1 hour after deposition, resulting in a room temperature resistivity $\rho_{RT}\approx 9\,\mathrm{m}\Omega\mathrm{cm}$ and dimensionless sensitivity $\eta = \left| \mathrm{dln}R/\mathrm{dln}T\right| \approx 1$ between $300\,\mathrm{K}$ and $10\,\mathrm{K}$, increasing to about $2$ at lower temperatures. The GeAu sensor layer is deposited on top of Au leads as shown in Fig.~\ref{Fig1}(d). The leads are in turn connected to external leads in Ti, as seen in Fig.~\ref{Fig1}(c). By combining metals with high (Au) and low (Ti) thermal conductivity a more well-defined isothermal area in the center of the membrane is obtained.\cite{Tagliati:2010jt}
The offset heater, shown in Fig.~\ref{Fig1}(d), is driven by a dc current and can locally increase the sample temperature up to at least $100\,\mathrm{K}$ above the base temperature. This heater is designed to give an isothermal interior area, but is not in direct contact with the active part of the thermometer. It is thus less suitable as an ac heater, since the thermal diffusion between thermometer and heater would have to be taken into account. Titanium has good robustness so the deposited layer can be quite thin. Its addition to the background heat capacity is rather insignificant, but to simplify the fabrication process the layer is sometimes skipped.
The leads that connect the active layers to the external bonding pads are also in Ti. Because of the relatively low thermal conductance of Ti compared to many other metals, the use of Ti for the leads ensures a long relaxation time dominated by the membrane itself. Outside the membrane area, a thick layer of Au is deposited onto the leads to minimize lead resistance effects and to provide a suitable layer for bonding, see Fig.~\ref{Fig1}(a). As a last element of the calorimeter stack, a $110\,\upmu\mathrm{m} \times 110\,\upmu\mathrm{m}$ square may be deposited as a thermalization layer to distribute the temperature evenly over the sample area if small samples are to be studied. This layer may be deposited simultaneously with the outer Au bonding pad layer.
The electrical insulation layers, not shown in Fig.~\ref{Fig1}, are made by thermally evaporated aluminum oxide in combination with sputtered SiO$_2$. The SiO$_2$ is deposited directly following the deposition of heaters or GeAu layer, while the AlO$_x$ layers are patterned as separate layers. There are in total three AlO$_x$ layers in the shape of rounded squares of different sizes: those between heaters and thermometer measure $270\,\upmu\mathrm{m} \times 230\,\upmu\mathrm{m}$ while the last insulation between thermometer and thermalization layer is $180\,\upmu\mathrm{m} \times 180\,\upmu\mathrm{m}$. Care was taken to design the layers so that edges of different layers do not coincide. This greatly reduces the risk of shorts between layers, especially for thin insulation layers.
\begin{figure}
\includegraphics[clip,width=0.95\linewidth]{Fig2}
\caption{\label{Fig2} (a) Calorimeter bonded onto cryostat plug-in. (b) Calorimeter cell with a sample covering ac heater and thermometer. The membrane and electrical insulation layers appear transparent. This particular calorimeter has no offset heater. (c) Microscope image of the central part of a calorimeter cell, illustrating the active layers and insulation (but without GeAu layer for clarity).}
\end{figure}
Figure~\ref{Fig2}(a) shows a picture of the calorimeter bonded onto a cryostat sample holder plug-in. The calorimeter requires up to 20 wires, but can otherwise be fitted onto most cryogenic sample holders. The sample is placed on one of the membrane cells by means of a simple micro-manipulator, or, with some practice, by hand. The other cell may either carry a reference sample or be left empty. Figure~\ref{Fig2}(b) shows a calorimeter cell with a typical sample (the Pb sample discussed in Section~\ref{Sec:Meas}). The central parts of the calorimeter are shown in Fig.~\ref{Fig2}(c), where also the larger electrical insulation layers can be seen.
\subsection{Measurement electronics}
In the standard measurement mode, known ac and dc currents flow through the heaters and thermometers. To practically enable the measurements of temperatures and oscillation amplitudes, a set of time and phase synchronized lock-in amplifiers based on a field-programmable gate array (FPGA) is used.\cite{Rydh:2009wh} We implemented such an instrument using the PXI-7854R card by National Instruments. The card has integrated $750\,\mathrm{kS/s}$ ADCs and $1\,\mathrm{MS/s}$ DACs, providing eight integrated, simultaneous-sampling analog inputs and outputs for ac and dc biasing. For each input, the first and/or second harmonic amplitudes with corresponding phases are extracted, as well as the dc component. A central phase generator delivers the digital reference for all inputs and outputs. The resulting instrument thus allows tuning of output voltages and working frequency during the measurements, without any loss of correlation between inputs and outputs. Each signal, before being read, passes through a low-noise, custom-built preamplifier stage. In total eleven preamplifiers are used, eight of which have variable gain (between 1 and 5000), controlled by the FPGA lock-in, the others with fixed gain (100 or 1000). While measuring, the eight variable-gain preamplifiers are automatically adjusted to maximize their performance.
\subsection{Thermometer operation}
The circuit scheme used for the current bias and voltage read-out of the thermometers is illustrated in Fig.~\ref{Fig3}. An automated process is implemented that auto-adjusts the current through the sample-side thermometer in order to regulate the voltage across the thermometer. The applied voltage $V_\mathrm{dc,s}$ is thus varied to keep the dc component $U_\mathrm{s,dc}$ of the sample voltage $U_\mathrm{s}$ equal to a variable setpoint value that is typically around $0.1\,\mathrm{V}$ and almost constant over the full temperature range. When the resistance of the thermometer increases with decreasing temperature, the applied current decreases and a suitable power is still delivered. The reading of $U_\mathrm{s,dc}$ is made synchronously by the lock-in so that frequency-dependent signals are separated from the dc component. Changes to $V_\mathrm{dc,s}$ are synchronized with the measurements as well.
\begin{figure}
\includegraphics[clip,width=0.95\linewidth]{Fig3}
\caption{\label{Fig3}Thermometer bias and read-out circuit. The sample and reference GeAu thermometer resistances $R_\mathrm{s}$ and $R_\mathrm{r}$ are measured in four-probe configurations with the current provided by synchronous voltage sources $V_\mathrm{dc,s}$ and $V_\mathrm{dc,r}$. The currents through the thermometers are measured by the voltages over $100\,\mathrm{k}\Omega$ series resistors. Optional, adjustable voltage dividers are used to avoid any electrical discharge from destroying the membranes during initial connection.}
\end{figure}
Since the sample and reference thermometers are fabricated together at the same time, they are quite well balanced and display the same temperature dependences, within measurement uncertainty. There may, however, still be a small imbalance in the resistance ratio of the order of 1\% between the two sides, arising from the lithographic tolerance. To compensate for this, a reference adjustment system is applied to vary the reference output $V_\mathrm{dc,r}$ in Fig.~\ref{Fig3} to keep $U_\mathrm{diff,dc}=0$ so that $U_\mathrm{s,dc}= U_\mathrm{r,dc}$, rather than setting the currents or powers equal on the two sides. By balancing the thermometers in this way, the time-varying temperature difference between sample and reference is given by the $U_\mathrm{diff}$ signal through the simple relation
\begin{equation}\label{EqDeltaTemp}
T_\mathrm{diff}=\frac{U_\mathrm{diff} \cdot T}{U_\mathrm{dc} \cdot \mathrm{dln}R/\mathrm{dln}T}.
\end{equation}
Here $T$ is the absolute temperature, which is obtained from the dc component of the thermometer resistance. The additional temperature difference caused by the power difference is usually insignificant, but can be compensated for by a corresponding power from the offset heater. The absence of dc offsets in the bridge differential $U_\mathrm{diff}$ makes it possible to apply a high amplification to $U_\mathrm{diff}$ without overloading, which results in highest possible resolution. The close proximity of sample and reference thermometers further eliminates common noise sources such as electromagnetic interference and temperature variations of the base frame. It should be noted that Eq.~(\ref{EqDeltaTemp}) has to be modified when the thermometer resistance becomes comparable to the series resistance of Fig.~\ref{Fig3}, to include the effect of a varying bias current.
\section{MEASUREMENT TECHNIQUES}
\subsection{AC steady state}
In the ac steady-state method \cite{Sullivan:1968uo} the sample temperature is made to oscillate with an amplitude typically in the range $1$\,--\,$100\,\mathrm{mK}$. This modulation is created by an ac power, which for resistive heating is given by $P(t)=R_\mathrm{h}I^{2}_{0}(1+\sin\omega t)=P_{0}(1+\sin\omega t)$. It is thus generated by an ac current with RMS amplitude $I_{0}$ and angular frequency $\omega/2$ flowing through the ac heater resistor $R_\mathrm{h}$. The temperature response of the cell is given by $T(t)=T_\mathrm{b}+T_\mathrm{offs}+T_\mathrm{ac}(t)$. $T_\mathrm{b}$ is the base temperature, $T_\mathrm{off}=P_0/K_\mathrm{e}$ is the dc offset due to the time-averaged power supplied by the heater resistance, where $K_\mathrm{e}$ is the thermal conductance between sample and thermal bath (Si frame), and $T_\mathrm{ac}(t)$ is the oscillating term whose steady-state amplitude $T_\mathrm{ac}$ is directly related to the heat capacity of the sample:
\begin{equation}\label{EqTac}
T_\mathrm{ac}=\frac{P_{0}}{\omega C}\left[1+\frac{1}{(\omega \tau_\mathrm{e})^2}+f(\tau_\mathrm{i})\right]^{-1/2}.
\end{equation}
Here $\tau_\mathrm{e}=C/K_\mathrm{e}$ is the external relaxation time between sample and external thermal bath and $f(\tau_\mathrm{i})$ is a rather complicated function of the internal relaxation time between sample and calorimetric cell, $\tau_\mathrm{i}=C_\mathrm{s}/K_\mathrm{i}$, where $C_\mathrm{s}$ is the sample heat capacity and $K_\mathrm{i}$ is the thermal conductance between the sample and the central cell platform. The value of $K_\mathrm{i}$ strongly depends on the agent used to attach the sample to the nanocalorimetric cell.
Equation~(\ref{EqTac}) is impractical to use to obtain $C$ from $T_\mathrm{ac}$, due to the complications of $f(\tau_\mathrm{i})$.
We have, however, previously shown\cite{Tagliati:2011be} that for a system with good thermal connection between the active layers, but with possibly significant $\tau_\mathrm{i}$, the thermometer temperature oscillation $T_\mathrm{ac,0}$ and corresponding phase $\phi$ between power and temperature oscillation can be found as
\begin{numcases}{}\label{EqTPhi}
\begin{aligned}
T_\mathrm{ac,0} &=\frac{P_\mathrm{0}}{\sqrt{(\omega C)^{2}+K^{2}}} \\
\tan\phi &=\frac{\omega C}{K}
\end{aligned}
\end{numcases}
provided that $C$ and $K$ are taken as
\begin{numcases}{}
\begin{aligned}
C &= C_\mathrm{cell} + (1-g) C_\mathrm{s}\\
K &=K_\mathrm{e,eff}+g K_\mathrm{i}\\
g &= \frac{(\omega \tau_\mathrm{i})^2}{1+(\omega \tau_\mathrm{i})^2}
\end{aligned}\label{Eq_CKtotal}
\end{numcases}
Here, $C_\mathrm{cell}$ is the empty cell contribution to the heat capacity and $K_\mathrm{e,eff}$ is the effective thermal link of the membrane.\cite{Tagliati:2011be} Equation~(\ref{EqTPhi}) can be wrapped around into the practical, functional relations
\begin{numcases}{}\label{EqCK}
\begin{aligned}
C &=\frac{P_\mathrm{0}}{\omega T_\mathrm{ac,0}}\sin\phi \\
K &=\frac{P_\mathrm{0}}{T_\mathrm{ac,0}}\cos\phi
\end{aligned}
\end{numcases}
which form the basis of evaluating the measurements. Note that the phase is carrying information that is needed to achieve good absolute accuracy. The phase can be used to verify that the frequency is selected correctly, i.e., that $\omega\tau_\mathrm{i}\ll1$ so that $g \to 0$. If the frequency is so high that $\sin \phi \to 1$ in Eq.~(\ref{EqCK}), it is very likely that $g$ contributes significantly in Eq.~(\ref{Eq_CKtotal}). A known phase is also necessary to accurately extract $K_\mathrm{e}$ from ac steady-state measurements.
When using the ac steady-state technique in practical terms, we apply an ac current to the heater(s) and measure the thermometer and series resistance voltages $U_\mathrm{s}$, $U_\mathrm{r}$, $U_{I,\mathrm{s}}$, $U_{I,\mathrm{r}}$, and $U_\mathrm{diff}$, defined in Fig.~\ref{Fig3}, at the second harmonic of the heater current frequency (simultaneously with the synchronous dc mean). The amplitudes of the sample and reference temperature oscillations $T_\mathrm{s,ac}$ and $T_\mathrm{r,ac}$, are related to the corresponding measured voltages over the thermometers and currents through the reference resistors by
\begin{equation}\label{TacFromUac}
T_{\mathrm{ac}}=\left({\frac{U_\mathrm{ac}}{U_\mathrm{dc}}+\frac{I_\mathrm{ac}}{I_\mathrm{dc}}}\right)\frac{T}{\eta}.
\end{equation}
Here both $T$ and sensitivity $\eta$ are obtained from the dc measurement of the thermometer resistance $R$ and previous calibration of $R(T)$. For the temperature differential the corresponding expression is
\begin{equation}\label{TdiffFromUdiff}
T_\mathrm{diff,ac}=\frac{U_\mathrm{diff,ac}}{U_\mathrm{dc}}\left({1+\frac{U_\mathrm{dc}}{U_{I,\mathrm{dc}}}}\right)\frac{T}{\eta},
\end{equation}
where $U_{I,\mathrm{dc}} = R_\mathrm{ref}I_\mathrm{dc}$ and $R_\mathrm{ref}$ is the series resistance for current measurement. All ac signals in Eqs.~(\ref{TacFromUac}) and (\ref{TdiffFromUdiff}) refer to amplitudes in the steady state. Note that $T_\mathrm{diff,ac} \ne T_\mathrm{s,ac} - T_\mathrm{r,ac}$ in general, since the sample and reference temperature oscillations may have different phases.
In the simplest measurement case, the reference heater power is kept off. This method is used when the heat capacity of the sample is fairly large as compared to the heat capacity of the empty cell and no reference sample is placed on the reference side. The sample temperature oscillation is then measured through the differential signal $U_\mathrm{diff,ac}=U_\mathrm{s,ac}$ while $U_\mathrm{r,ac}=0$. Since $U_\mathrm{diff}$ has no dc offset, the differential signal yields a significantly higher resolution than $U_\mathrm{s,ac}$. To subtract $C_\mathrm{cell}$ from $C$, sample and empty cell are measured in separate runs. The measurement of $T_\mathrm{ac}$ is thus made differentially, but the measurement of $C$ is not.
If the sample is small, or if a similar-size reference sample is added to the reference side, a truly differential measurement mode can be employed. In this case, the same power is applied to both sample and reference sides ($P_\mathrm{s}=P_\mathrm{r}=P_\mathrm{0}$). The heat capacities $C_\mathrm{s+cell}$ and $C_\mathrm{r+cell}$ of the individual sides are still given by Eq.~(\ref{EqCK}), but the differential heat capacity $C_\mathrm{diff}=C_\mathrm{s}-C_\mathrm{r}$ can be obtained as well from $T_\mathrm{diff,ac}$:
\begin{equation} \label{C_diff}
C_\mathrm{diff}=\frac{P_0 T_\mathrm{diff,ac}}{\omega T_\mathrm{s,ac} T_\mathrm{r,ac}}.
\end{equation}
Equation~(\ref{C_diff}) is valid under the assumption that $K$ is the same for both sample and reference sides. Since $K_\mathrm{e}$ is given mainly by the membrane itself, and other contributing layers are manufactured lithographically, this requirement is easy to fulfill in normal cases. If a large sample is studied, the sample itself may enhance $K$ through a small, but nonzero $g$. The absolute accuracy of Eq.~(\ref{C_diff}) may in this case be tested by a separate verification measurement with $P_\mathrm{r}=0$.
If the differential heat capacity mode is used with a large sample but without reference, the signal from the reference side will dominate $U_\mathrm{diff}$, and the benefit of a differential measurement would be lost. An alternative to reverting to a single-side measurement can in this case be to adjust the amplitude and output phase of the reference heater power to maintain $U_\mathrm{diff}=0$. This mode of measurement is, however, still unexplored.
\subsection{Frequency feedback}
We have previously shown\cite{Tagliati:2011be} that good absolute accuracy can be obtained provided that one has good control of the phase in Eq.~(\ref{EqCK}). In Fig.~\ref{Fig4}, the frequency dependence of $T_\mathrm{s,ac}$ and $\phi$ is shown for a typical sample.
\begin{figure}
\includegraphics[clip,width=0.95\linewidth]{Fig4}
\caption{\label{Fig4}Measured frequency dependence of temperature oscillation amplitude $T_\mathrm{ac}$ and phase $\phi$, expressed as $(\omega T_\mathrm{ac}/P_\mathrm{0})C$ and $\tan \phi$, respectively, with $P_\mathrm{0}$ and $C$ constant ($f=\omega/4\pi$). At low frequency $\tan \phi \sim \omega$ and measurements yield good absolute accuracy. At too low frequencies, however, the signal no longer comes from the heat capacity but from the thermal link, and the resolution decreases. Note that the middle of the adiabatic plateau (where $\omega T_\mathrm{ac}$ is constant) is not corresponding to the best measurement frequency if good accuracy is required.}
\end{figure}
Good accuracy is found at low frequencies, where $\tan \phi \sim \omega$. At higher frequencies, the effect of $\omega\tau_\mathrm{i}$ is no longer negligible and the accuracy is quickly deteriorating. For the resolution, the conditions are the opposite; at high frequencies the resolution is good, but at low frequencies, $\omega\tau_\mathrm{e}\ll 1$, Eq.~(\ref{EqTPhi}) is reduced to $T_\mathrm{ac}=P_0/K_\mathrm{e}$ and heat capacity is no longer probed.
By fixing the phase $\phi$ to a constant value in the range where measurements yield both good absolute accuracy and good resolution, optimal conditions are found. We do this by continuously adjusting the frequency $\omega$ during the measurement to keep $\tan \phi$ constant by means of an auto-tuning routine in the FPGA lock-in. By having a control loop time equal to a single sample of the ADC, the frequency is smoothly adjusted without introducing additional noise to the measurements. The frequency is thus varying even during a single cycle of the output. This is possible thanks to the synchronous sampling. It should be noted that the middle of the adiabatic plateau (with $\omega T_\mathrm{ac} \sim \mathrm{const.}$) is typically at too high frequencies for good absolute accuracy, as seen in Fig.~\ref{Fig4}.
\subsection{Thermal relaxation}
The thermal relaxation method \cite{Bachmann:1972vh} consists of applying a known power to the heater to raise the sample temperature an amount $\Delta T$ above the base temperature $T_\mathrm{b}$. After a stable sample temperature has been reached, the heater power is turned off and the temperature is allowed to relax back to $T_\mathrm{b}$. The time dependence of the relaxation is exponential and depends on the external time constant of the system:
\begin{equation}
T(t)=T_\mathrm{b}+\Delta T e^{-t/\tau_\mathrm{e}}.
\end{equation}
When using the thermal relaxation method, we apply a square wave with sufficiently low repetition rate to the sample heater, while keeping the reference heater off. The induced temperature response is directly given by Eq.~(\ref{EqDeltaTemp}), from which $\Delta T$ and $\tau_\mathrm{e}$ can be obtained.
The thermal conductance $K_\mathrm{e}$ between sample and base frame is then found as
\begin{equation} \label{Ke_relax}
K_\mathrm{e}=\frac{\Delta P}{\Delta T},
\end{equation}
where $\Delta P$ is given by the directly measured step in heater power. The sample heat capacity including cell addenda is finally given by $\tau_\mathrm{e}$ and $K_\mathrm{e}$, according to
\begin{equation} \label{C relax}
C_\mathrm{s}+C_{\mathrm{cell}}=\tau_\mathrm{e} \cdot K_\mathrm{e}.
\end{equation}
To practically perform relaxation measurements, a routine was developed that automatically acquires the voltage pulses, fits the exponential decay to data within 10\% and 90\% of the relaxation, and determines $\Delta T$. In this way, the heat capacity and device thermal conductance can be measured as a function of temperature or magnetic field with immediate results displayed on the screen.
\section{CALIBRATION AND CHARACTERIZATION}
\subsection{Thermometer calibration}
\begin{figure}
\includegraphics[clip,width=0.95\linewidth]{Fig5a}\\[1mm]
\includegraphics[clip,width=0.95\linewidth]{Fig5b}
\caption{\label{Fig5}(a) Temperature dependence of the GeAu thermometer resistance and dimensionless sensitivity $\eta=\left| \mathrm{dln}R/\mathrm{dln}T \right|$. (b) Thermometer magnetoresistance expressed as apparent relative temperature change in a magnetic field of $H=10\,\mathrm{T}$. The inset shows the corresponding field dependence at different temperatures. The magnetoresistance is following a parabolic field dependence except for temperatures below $\sim 3\,\mathrm{K}$ at high fields.}
\end{figure}
Before any measurements, the GeAu thermometers on sample and reference sides need to be calibrated against a known thermometer. We use a Cernox sensor on the sample holder. The calibration is done in the presence of a few mbar helium gas to increase the thermal link between GeAu thermometers and Si base frame, which quickly reduces $\tau_\mathrm{e}$ and temperature offsets due to self-heating. The typical temperature dependence of the GeAu thermometer resistance is shown in Fig.~\ref{Fig5}(a). Once the calibration curve has been obtained, one can attain the temperature dependence of the sensitivity $\eta$ which is approximately constant from $300\,\mathrm{K}$ down to about $50\,\mathrm{K}$, below which it increases slowly as shown in Fig.~\ref{Fig5}(a). The temperature dependence of the sample to reference thermometer resistance ratio $R_\mathrm{s}(T)/R_\mathrm{r}(T)$ is constant within experimental uncertainty over the entire temperature range. The same calibration curve can thus be used for both sides, except for a scaling pre-factor. Thermometers deposited in the same sputtering cycle display very similar temperature dependences, while thermometers fabricated at different times require individual calibrations, although differences are small enough for a standard calibration curve to be used initially in most cases. The thermometer resistance is stable over time provided that the thermometer is not heated excessively after the heat treatment at $190^\circ\mathrm{C}$.
To obtain $\eta$ and $T$ from $R$ while avoiding numerical noise and other artifacts, we fit an analytical expression to the calibration data. The thermometer $T(R)$ relation is well described by
\begin{equation} \label{Tfit}
T = \frac{T_0}{(R/R_0)^{\alpha_0}}+\frac{T_1}{(R/R_1)^{\alpha_1}}.
\end{equation}
Here $R_i$, $\alpha_i$, and $T_i$ are constants. Remaining small deviations are fitted by a high-degree polynomial in $\log{(R/R_2)}$. An analytical expression of $\eta$ is then directly found from the fit parameters, and is given by
\begin{equation} \label{Etafit}
1/\eta = \left| \frac{\mathrm{dln}T}{\mathrm{dln}R} \right| = \frac{1}{T} \left[ \frac{\alpha_{0}T_0}{(R/R_0)^{\alpha_0}}+\frac{\alpha_{1}T_1}{(R/R_1)^{\alpha_1}} \right],
\end{equation}
up to the contribution from the polynomial correction. This procedure to obtain temperature and sensitivity from resistance avoids interpolations, which, even in log-log scale, tend to give rise to artificial kinks that may become significant in certain cases such as when studying specific heat differences.
\subsection{Thermometer magnetoresistance}
Resistive thermometers require corrections for magnetic-field-induced changes at low temperature. The magnetoresistance of the GeAu sensor was studied by sweeping a magnetic field between $-10\,\mathrm{T}$ and $10\,\mathrm{T}$ at several fixed temperatures from $1.9\,\mathrm{K}$ to $40\,\mathrm{K}$. The relative change in apparent temperature was then calculated as $\Delta T/T=(\Delta R/R)/\eta$, where $\eta$ is the zero field sensitivity. The curves, shown in the inset of Fig.~\ref{Fig5}(b), can be fairly well approximated by a parabolic field dependence $\Delta R \propto H^2$ above $3\,\mathrm{K}$. The magnetoresistance is positive and its magnitude decreases quite quickly with increasing temperature. The field-induced error at $10\,\mathrm{T}$, if no correction is made, is $1\%$ at $17\,\mathrm{K}$ and $0.15\%$ at $40\,\mathrm{K}$, as shown in Fig.~\ref{Fig5}(b).
\subsection{Empty cell characterization}
The nanocalorimeter was characterized thoroughly from room temperature down to $0.5\,\mathrm{K}$. Figure~\ref{Fig6}(a) shows the heat capacity of the empty cell as obtained from Eq.~(\ref{EqCK}).
\begin{figure}
\includegraphics[clip,width=0.95\linewidth]{Fig6a}\\[1mm]
\includegraphics[clip,width=0.95\linewidth]{Fig6b}
\caption{\label{Fig6}(a) Empty cell heat capacity $C_{\mathrm{cell}}$ as a function of temperature. (b) Device thermal conductance $K_\mathrm{e}$. The insets show the low-temperature behavior in log-log scale. Note that $C_{\mathrm{cell}}$ and $K_\mathrm{e}$ display a fairly similar temperature dependence, resulting in a rather constant time constant $\tau_\mathrm{e,cell}$ as a function of temperature for the empty device.}
\end{figure}
This heat capacity will, under certain conditions, depend on the choice of woking frequency (i.e.\ $\tan \phi$), since the amount of membrane that is temperature-modulated decreases with increasing frequency.\cite{Tagliati:2011be} The characteristic frequency of the membrane is, however, typically higher than the frequency of normal measurements, making it possible to treat $C_{\mathrm{cell}}$ as a reproducible background addenda for a given calorimeter and to assume that $C_\mathrm{cell}$ is frequency independent for large samples (i.e.\ with $C>C_\mathrm{cell}$). In differential mode, the addenda heat capacity is normally less than 5\% of $C_{\mathrm{cell}}$. A typical noise level is $\delta C \sim 2\,\mathrm{pJ/K}$ at $50\,\mathrm{K}$. The noise level expressed as $\delta C/C$ is fairly constant over the entire temperature range. With a typical measurement time of $3\,\mathrm{s}$ and a moderately large $T_\mathrm{ac}/T$, it is possible to reach $\delta C/C \sim 10^{-4}$.
The thermal link $K_\mathrm{e}$ of the calorimeter cell is shown in Fig.~\ref{Fig6}(b). It can be obtained from Eq.~(\ref{EqCK}) provided that the measurements are made at low enough frequency, $\omega \tau_\mathrm{i} \ll 1$, or through relaxation measurements. While the cell heat capacity is dominated by the membrane that has a fairly high Debye temperature, the thermal link is given by a combination of membrane and metallic leads. The characteristic time constant $\tau_\mathrm{e,cell}=C_\mathrm{cell}/K_\mathrm{e}$ is nevertheless fairly temperature independent.
\subsection{Operational parameters}
Figure~\ref{Fig7}(a) shows the range of operational power of the ac heater as a function of temperature. The power required for a certain ratio $T_\mathrm{ac}/T$ between temperature oscillation amplitude and absolute temperature is not depending on $C$ but only $K_\mathrm{e}$, provided that the phase $\phi$ is kept constant. An easy way to adjust the power to the proper level is thus to maintain a constant ratio $U_\mathrm{s,ac}/U_\mathrm{dc} \simeq (T_\mathrm{s,ac}/T)\eta$ while adjusting the frequency so that $\phi$ is constant. In this way, the temperature offset due to the ac heater power is always a constant fraction of $T$ as well. To maintain a similar power for the dc bias of the thermometer, the $U_\mathrm{s,dc}$ should decrease from about $0.1\,\mathrm{V}$ at $100\,\mathrm{K}$ to $0.01\,\mathrm{V}$ at $1\,\mathrm{K}$. In practice, $U_\mathrm{s,dc}$ can be kept almost constant, so that the relative temperature offset due to thermometer self-heating is somewhat higher at the lowest temperatures and somewhat lower at the highest.
\begin{figure}
\includegraphics[clip,width=0.95\linewidth]{Fig7a}\\[1mm]
\includegraphics[clip,width=0.95\linewidth]{Fig7b}
\caption{\label{Fig7}(a) Range of operational ac heater power as a function of temperature for measurement frequency adjusted so that $\tan \phi = 5$. (b) Typical range of working frequency $f$ as a function of temperature, with and without sample. Note that the sample specific heat controls the frequency range over which good absolute accuracy is obtained. Different samples thus require different frequency adjustment.}
\end{figure}
Figure~\ref{Fig7}(b) shows the typical range of working frequency $f$ as a function of temperature, with and without sample. For the empty device, the time constant $\tau_\mathrm{e,cell}$ is rather temperature independent, leading to a fairly constant $f$. With a sample, the frequency $f \sim K/C$ is lowered, but the frequency also becomes more strongly temperature dependent. This variation depends on the sample heat capacity. It is thus clear that only narrow temperature ranges can be studied with combined good accuracy and resolution if a constant frequency is used as in the case of traditional ac calorimetry.
\section{MEASUREMENTS}\label{Sec:Meas}
\subsection{Comparing relaxation and ac steady-state methods}
As an initial test of the calorimeter, we measured a small gold sample from $280\,\mathrm{K}$ down to about $10\,\mathrm{K}$ to compare the relaxation and ac steady-state techniques. The temperature dependence of the heat capacity, shown in Fig.~\ref{Fig8}, was also compared with available data to study the absolute accuracy.
\begin{figure}
\includegraphics[clip,width=0.95\linewidth]{Fig8}
\caption{\label{Fig8} Measured heat capacity versus temperature of a small gold grain. The contribution of the membrane heat capacity has been subtracted. There is also some contribution from the Apiezon-N grease used to attach the sample. This addenda has not been accounted for, but is expected to be of the order of $5$\,--\,$10\%$. The heat capacity measured by Geballe and Giauque\cite{Geballe:1952ct} on a sample more than $10^8$ times larger is scaled to agree with our data at $90\,\mathrm{K}$.}
\end{figure}
It is seen that the over-all agreement between the two measurement methods is fairly good. The ac steady-state curve, however, lies somewhat above the relaxation curve for temperatures below $50\,\mathrm{K}$. The heat capacity measurement by Geballe and Giauque\cite{Geballe:1952ct} on roughly $2.5\,\mathrm{kg}$(!) Au was scaled to agree with our data at $90\,\mathrm{K}$, where the relaxation and ac steady-state measurements coincide. The relaxation curve follows the temperature dependence of the literature with deviations within 5\% over the full temperature range. From the scaling, the sample mass is estimated to $15.1\,\upmu\mathrm{g}$, which lies within the uncertainty of the volumetric measurement of the sample size, initially estimated to be $12.7\,\upmu\mathrm{g}$ using a simple microscope. The real sample mass is likely in between the two numbers, since the data of Fig.~\ref{Fig8} includes a small contribution from the Apiezon-N grease that was used to attach the sample.
\subsection{Measurements of the superconducting properties of Pb}
To illustrate the low-temperature capabilities of the calorimeter, we studied the heat capacity of a $2.6\,\upmu\mathrm{g}$, 99.999$\%$ pure Pb sample, cut from a single crystal, as a function of temperature and in magnetic fields. Figure~\ref{Fig9}(a) shows the measured specific heat in the superconducting and normal states, plotted as $C/T$ vs $T$.
\begin{figure}[t]
\includegraphics[clip,width=0.95\linewidth]{Fig9a}\\[1mm]
\includegraphics[clip,width=0.95\linewidth]{Fig9b}
\caption{\label{Fig9} (a) Temperature dependence of the low-temperature specific heat expressed as $C/T$ of Pb in the superconducting and normal state (obtained by a $120\,\mathrm{mT}$ field). The inset shows the measured heat capacity at higher temperatures before and after subtraction of addenda, and the contributions from membrane and Apiezon-N grease. (b) Specific heat difference $\Delta C/T = (C_\mathrm{s} - C_\mathrm{n})/T$, with the data provided by Neighbor \emph{et\,al.}\cite{Neighbor:1967fh} for comparison. The inset shows the entropy difference $\Delta S = S_s - S_n$ obtained by integrating $\Delta C/T$.}
\end{figure}
The specific heat jump in zero field at the transition temperature $T_\mathrm{c} \approx 7.2\,\mathrm{K}$ is seen clearly. The normal state curve was obtained by applying a $120\,\mathrm{mT}$ magnetic field to suppress the superconductivity. The inset shows the heat capacity at higher temperatures. The addenda heat capacity from membrane and grease were measured in separate runs (i.e.\ not in true differential mode). Apiezon-N is typically used as a thermal contact agent for low-temperature experiments, but undergoes a glass transition at above $200\,\mathrm{K}$, which leads to a somewhat irreproducible high-temperature addenda,\cite{Schnelle:1999uh,Kreitman:1972wt} decreasing the absolute accuracy at high temperatures. The membrane cell dominates the addenda at the absolute lowest temperatures, but already at about $\sim 3\,\mathrm{K}$ the Apiezon contribution becomes the main background. The membrane and grease addenda are subtracted in the main Figure~\ref{Fig9}. After subtraction, the superconducting state measurements still display a $5\%$ residual gamma term, i.e., a remaining linear-in-$T$ contribution to the specific heat. This could possibly be due to an incompletely accounted background addenda. The ratio $\Delta C/C$ at $T_\mathrm{c}$ is, however, only $95\%$ of expected. We therefore believe that the unaccounted addenda is a non-superconducting part of the sample entering $C$ but not $\Delta C$, possibly arising from an oxidized surface layer, or from vacancies and dislocations that were not annealed away before the measurements.
Figure~\ref{Fig9}(b) shows the temperature dependence of the specific heat difference $\Delta C/T = (C_\mathrm{s} - C_\mathrm{n})/T$. This difference is insensitive to background addenda, which makes it a good probe of accuracy and reproducibility. From the temperature dependence, fundamental properties such as the superconducting gap energy and coupling strength can be obtained.\cite{Carbotte:1990zz} The measurements provide a Sommerfeld parameter $\gamma \simeq 3.1\,\mathrm{mJ/mol\,K}^2$ and $\Delta C/\gamma T_\mathrm{c} \simeq 2.7$ in good agreement with literature.\cite{Carbotte:1990zz} The temperature dependence of $\Delta C/T$ is also following the expected behavior, as seen by comparing the measurements with polynomial-fit data provided by Neighbor \emph{et\,al.}\cite{Neighbor:1967fh} While the resolution remains good at all temperatures, the accuracy decreases somewhat at $T < 1.5\,\mathrm{K}$, which can be seen in Fig.~\ref{Fig9}(b) as some wiggles in $\Delta C/T$. This can be attributed to the difficulties in obtaining an accurate calibration and corresponding sensitivity of the thermometer using $\sim\mathrm{pW}$ power levels. One way to overcome this problem may be to calibrate the thermometer simultaneously with the measurements, with a self-consistency requirement on the thermal link. Such a method has been successfully tested, but requires further investigation.
Integrating $\Delta C/T$ gives the entropy difference $\Delta S(T)$, shown in the inset of Fig.~\ref{Fig9}(b). The entropy-conservation requirement is fulfilled within a $\sim 2\%$ uncertainty of $\gamma$, obtained from the low-temperature slope of $\Delta S$. The free energy difference $\Delta F$ is then obtained as $\Delta F = \Delta U-T\Delta S$, where $\Delta U$ is found by integrating $\Delta C$ from $T_\mathrm{c}$ to $T$. From $\Delta F$, the thermodynamic critical field $H_\mathrm{c}(T)$ is calculated from the relation $\Delta F = V_\mathrm{m} \mu_0 H_\mathrm{c}^2(T)/2$, where $V_\mathrm{m}$ is the molar volume (or sample volume, if $\Delta F$ is given in units of energy). Figure~\ref{Fig10}(a) shows $H_\mathrm{c}(T)$ obtained in this way using the data in Fig.~\ref{Fig9}(b).
\begin{figure}[t]
\includegraphics[clip,width=0.98\linewidth]{Fig10}
\caption{\label{Fig10} (a) Thermodynamic critical field $H_\mathrm{c}(T)$. The small, closely spaced symbols correspond to $H_\mathrm{c}(T)$ as obtained from the $\Delta C(T)$ data of Fig.~\ref{Fig9}(b). They display the expected\cite{Neighbor:1967fh} small deviation from the two-fluid expression, which is shown for reference. Also shown is the directly measured location of the superconducting transition in various magnetic fields (big circles). The inset shows normalized heat capacity measured on increasing fields at different temperatures. (b) Temperature dependence of the specific heat in small magnetic fields near $T_\mathrm{c}$. (c) Transition in a $0.25\,\mathrm{mT}$ magnetic field measured with different $T_\mathrm{ac}$. While improving the energy resolution, too high $T_\mathrm{ac}$ results in a pronounced $T$ smearing.}
\end{figure}
The temperature dependence of $H_\mathrm{c}$ for Pb is expected to display a small, positive deviation from the two-fluid expression $H_\mathrm{c}(T) = H_\mathrm{c}(0)\left[1-(T/T_\mathrm{c})^2\right]$. A weak-coupling BCS superconductor, on the other hand, would display a negative deviation.\cite{Carbotte:1990zz} The difference between $H_\mathrm{c}(T)$, as obtained from the measured $\Delta C(T)$, and the two-fluid expression is clearly seen in Fig.~\ref{Fig10}(a). Indeed, the temperature dependence of the deviations themselves are within a few \% of the deviations obtained by Decker \emph{et\,al.}\cite{Decker:1958zz,TagliatiLT26}
The most uncertain factor in going from heat capacity to specific heat in nanocalorimetry is the determination of sample mass. It can be done through a careful measurement of volume and density, or from a known reference point at some temperature, such as room temperature or $T_\mathrm{c}$. Since a microscopic volume measurement would require sub-$\upmu\mathrm{m}$ resolution, which is difficult for soft materials such as Pb, we used the measurement of $C$ around $T_\mathrm{c}$ by Shiffman \emph{et\,al.}\cite{Shiffman:1963wz} to obtain the scale in Fig.~\ref{Fig9}. For type-I superconductors, it is however also possible to directly measure $H_\mathrm{c}(T)$ by studying the superconducting transition in magnetic field. Such measurements at various temperatures and magnetic fields are shown in the inset of Fig.~\ref{Fig10}(a) and in Fig.~\ref{Fig10}(b). The resulting $H_\mathrm{c}(T)$ and $T_\mathrm{c}(H)$ transitions, shown as big, green circles in the main panel of Fig.~\ref{Fig10}(a), agree well with $H_\mathrm{c}(T)$ as obtained from the measurement of $\Delta C$. The sample volume can thus be found directly from a comparison of $\Delta F$ (measured in units of energy) and $\mu_0 H_\mathrm{c}^2(T)/2$ (having units of energy per volume).
The measurements in magnetic field were made with $H$ perpendicular to the plate-like sample. (The sample is shown in Fig.~\ref{Fig2}). This causes $C$ to become field dependent in the superconducting state, due to the large demagnetization factor that drives the sample into the intermediate state. The effect is clearly seen in the field-dependence curves of the inset of Fig.~\ref{Fig10}(a). At low fields the sample is in the Meissner state and $C_\mathrm{s}$ is constant, but at higher fields $C_\mathrm{s}$ starts to increase. One could interpret this effect as a distributed latent heat when normal domains enter the superconductor. In Fig.~\ref{Fig10}(b) it is seen that the specific heat in small magnetic fields is higher than the zero field specific heat near $H_\mathrm{c}$. While the sample temperature oscillates, a small fraction of the sample is undergoing the transition back and forth between the Meissner and normal states, with accompanying latent heat $T\Delta S$. This causes the $T_\mathrm{ac}$ amplitude to decrease, making the latent heat appear as an excess specific heat $C(H)-C(0) > 0$. The excess specific heat thus relates to the fraction of the sample that undergoes the superconducting transition at each temperature. It is tempting to quantify the latent heat from such measurements. However, only a fraction of the total latent heat is found in this way. This is due to possible hysteretic effects of the first-order transition in combination with an incomplete analysis of the temperature oscillation, which will display higher harmonics when latent heat is involved.
As a final illustration of the capability of the calorimeter, the transition in a $0.25\,\mathrm{mT}$ magnetic field is shown in Fig.~\ref{Fig10}(c) for different temperature oscillation amplitudes $T_\mathrm{ac}$. The sharp latent heat peak, which is not present in zero field [cf.\ Fig.~\ref{Fig10}(b)], is a good probe of combined high resolution of both temperature and specific heat. The peak is only seen if $T_\mathrm{ac}$ is small enough. By increasing $T_\mathrm{ac}$ the specific heat resolution will increase, but the transition is then quickly smeared out. The latent heat involved in this transition is of the order of a few $\mathrm{pJ}$.
\section{SUMMARY AND CONCLUSIONS}
In summary, we have developed a membrane-based nanocalorimeter for specific heat measurements of small samples and thin films over an extended temperature range from above room temperature down to below $1\,\mathrm{K}$. Our device has sub-pJ/K resolution at low temperature, corresponding to $\sim 25\,\mathrm{aJ}/(\mathrm{K}\cdot\upmu\mathrm{m}^2)$ for thin films and $\sim\mathrm{fJ}$ heat exchanges, and is capable of probing $\upmu\mathrm{g}$-sized samples with combined high resolution and good absolute accuracy, thus exceeding the typical capability of commercial calorimeters by almost four orders in sample size. The calorimeter features a differential design with a variable-frequency technique where the measurement conditions are automatically maintained at optimal conditions.
The versatility of the calorimeter invites the exploration of several novel ac measurement procedures in addition to the ones described here, including power-compensation and multi-frequency modes. The ultimate capability of the calorimeter is thus still an open question.
\section{ACKNOWLEDGMENTS}
We thank S. Latos and P. A. Favuzzi for assistance with calorimetry development and R. Nilsson for contributing to the FPGA-based lock-in amplifier. Initial development of the nanocalorimeter was performed at Argonne National Laboratory in collaboration with U. Welp, W.-K. Kwok, and G. W. Crabtree. Financial support from the Swedish Research Council and technical support from the SU-Core Facility in Nanotechnology is gratefully acknowledged.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,011
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Rockland ist eine Gemeinde (mit dem Status "Village") im La Crosse County im US-amerikanischen Bundesstaat Wisconsin. Im Jahr 2010 hatte Rockland 594 Einwohner.
Rockland ist Bestandteil der bundesstaatenübergreifenden Metropolregion La Crosse Metropolitan Area.
Geografie
Rockland liegt am südlichen Ufer des La Crosse River, der 34 km westlich in den die Grenze zu Minnesota bildenden Mississippi mündet. Der am Mississippi gelegene Schnittpunkt der drei Bundesstaaten Wisconsin, Minnesota und Iowa befindet sich 67,8 km südsüdwestlich.
Rockland liegt in der Driftless Area genannten eiszeitlich geformten Region, die sich über das südöstliche Minnesota, das südwestliche Wisconsin, das nordöstliche Iowa und das äußerste nordwestliche Illinois erstreckt. Bei der letzten Eiszeit, der so genannten Wisconsin Glaciation, blieb die Region eisfrei, sodass sich die Flusstäler auch während dieser Zeit tiefer in das Plateau einschneiden konnten.
Die geografischen Koordinaten von Rockland sind 43°54′23″ nördlicher Breite und 90°55′09″ westlicher Länge. Das Ortsgebiet erstreckt sich über eine Fläche von 1,5 km².
Nachbarorte von Rockland sind Sparta (11,5 km ostnordöstlich), Cashton (27,5 km südöstlich), West Salem (14,3 km westlich) und Mindoro (23,8 km nordwestlich).
Die nächstgelegenen größeren Städte sind Green Bay am Michigansee (293 km ostnordöstlich), Wisconsins größte Stadt Milwaukee (306 km ostsüdöstlich), Wisconsins Hauptstadt Madison (198 km südöstlich), Rockford in Illinois (305 km in der gleichen Richtung), die Quad Cities in Illinois und Iowa (303 km südlich), Cedar Rapids in Iowa (289 km südsüdwestlich), Rochester in Minnesota (145 km westlich) und die Twin Cities in Minnesota (267 km nordwestlich).
Verkehr
Die Interstate 90 verläuft in West-Ost-Richtung entlang der südlichen Ortsgrenze. Die County Highways U und J führen als Hauptstraße durch Rockland. Alle weiteren Straßen sind untergeordnete Landstraßen, teils unbefestigte Fahrwege sowie innerörtliche Verbindungsstraßen.
Entlang des La Crosse River verläuft für den Frachtverkehr eine Eisenbahnlinie der Canadian Pacific Railway.
Durch Rockland verläuft der La Crosse River State Trail, ein auf der Trasse einer ehemaligen Eisenbahnstrecke verlaufender Rail Trail für Wanderer und Radfahrer. Im Winter kann der Wanderweg auch mit Schneemobilen befahren werden.
Der nächste Flughafen ist der La Crosse Regional Airport, der 32,7 km westlich liegt.
Bevölkerung
Nach der Volkszählung im Jahr 2010 lebten in Rockland 594 Menschen in 228 Haushalten. Die Bevölkerungsdichte betrug 396 Einwohner pro Quadratkilometer. In den 228 Haushalten lebten statistisch je 2,61 Personen.
Ethnisch betrachtet setzte sich die Bevölkerung zusammen aus 93,9 Prozent Weißen, 2,0 Prozent Afroamerikanern, 0,5 Prozent amerikanischen Ureinwohnern, 2,7 Prozent Asiaten sowie 0,2 Prozent aus anderen ethnischen Gruppen; 0,7 Prozent stammten von zwei oder mehr Ethnien ab. Unabhängig von der ethnischen Zugehörigkeit waren 0,7 Prozent der Bevölkerung spanischer oder lateinamerikanischer Abstammung.
25,8 Prozent der Bevölkerung waren unter 18 Jahre alt, 64,8 Prozent waren zwischen 18 und 64 und 9,4 Prozent waren 65 Jahre oder älter. 44,9 Prozent der Bevölkerung war weiblich.
Das mittlere jährliche Einkommen eines Haushalts lag bei 50.000 USD. Das Pro-Kopf-Einkommen betrug 21.625 USD. 11,2 Prozent der Einwohner lebten unterhalb der Armutsgrenze.
Einzelnachweise
Weblinks
city-data.com - Rockland, Wisconsin
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,219
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import sys
f = open(sys.argv[1], 'rU')
for lines in f.readlines()[1:]:
line = lines.strip()#.split("\t")
genus = line.split(" ")[0]
species = line.split(" ")[1]
print "esearch -db assembly -query \'%s\' | elink -target nuccore | elink -target assembly | efetch -format docsum | xtract -pattern DocumentSummary -element FtpPath_GenBank |awk -F\"/\" \'{print $0\"/\"$NF\"_rna_from_genomic.fna.gz\"}\' | xargs wget -O %s.rna.fa.gz" % (line, genus + "_" + species)
#print "esearch -db assembly -query \'%s\' | elink -target nuccore | elink -target assembly | efetch -format docsum | xtract -pattern DocumentSummary -element FtpPath_RefSeq |awk -F\"/\" \'{print $0\"/\"$NF\"_rna_from_genomic.fna.gz\"}\' | xargs wget -O %s.rna.fa.gz" % (line, genus + "_" + species)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,599
|
from actionkit.models import CoreUser
from datetime import datetime
from django import forms
from django.contrib.localflavor.us import forms as usforms
from django.utils.translation import ugettext_lazy as _
class EventForm(forms.Form):
title = forms.CharField(label=_("Title"), required=False)
date = forms.DateField(label=_("Date"), required=False)
time = forms.TimeField(label=_("Time"), required=False)
venue = forms.CharField(label=_("Venue"), required=True)
address = forms.CharField(label=_("Address"), required=True)
city = forms.CharField(label=_("City"), required=True)
state = usforms.USStateField(label=_("State"), required=True)
zip = usforms.USZipCodeField(label=_("ZIP"), required=True)
max_attendees = forms.IntegerField(label=_("Max Attendees"), required=True)
host = forms.EmailField(label=_("Host's Email Address"), required=True)
public_description = forms.CharField(label=_("Description"), required=False, widget=forms.Textarea())
directions = forms.CharField(label=_("Directions"), required=False, widget=forms.Textarea())
def clean_host(self):
host = self.cleaned_data['host']
try:
user = CoreUser.objects.using("ak").get(email=host)
except CoreUser.DoesNotExist:
raise forms.ValidationError(_("No core_user exists with email %(email)s") % {'email': host})
return user
def clean(self):
try:
data = self.build_event_struct()
except:
pass
else:
self.event_struct = data
return self.cleaned_data
def build_event_struct(self):
data = {}
for key in "title venue city state zip max_attendees".split():
data[key] = self.cleaned_data.get(key)
for key in "directions public_description".split():
if key in self.cleaned_data and self.cleaned_data.get(key).strip():
data[key] = self.cleaned_data.get(key)
data['creator_id'] = self.cleaned_data['host'].id
data['address1'] = self.cleaned_data['address']
data['starts_at'] = datetime.combine(self.cleaned_data['date'], self.cleaned_data['time'])
return data
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,207
|
Q: CraftyJS: I can't destroy an entity with .onHit I am having a problem with craftyJS. I made a ship that shoots lasers, but I want the lasers to destroy an entity when it collides with it. I already tried to do this, but it don't work. Here is my code:
Crafty.init(1340,650, document.getElementById('game'));
Crafty.defineScene("gameStart", function() {
function newAlien(){
var random = Math.floor(Math.random() * 4) + 1;
switch(random){
case 1:
var alien = Crafty.e("2D, DOM, Image, Collision")
.image("alien1.png");
alien.x = ship.x;
alien.y = 20
break;
case 2:
var alien = Crafty.e("2D, DOM, Image, Collision")
.image("alien2.png");
alien.x = ship.x;
alien.y = 20;
break;
case 3:
var alien = Crafty.e("2D, DOM, Image, Collision")
.image("alien3.png");
alien.x = ship.x;
alien.y = 20;
break;
case 4:
var alien = Crafty.e("2D, DOM, Image, Collision")
.image("alien4.png");
alien.x = ship.x;
alien.y = 20;
break;
}
}
function newLaser(){
var laser = Crafty.e("2D, DOM, Image, Collision")
.image("laser.png")
.collision()
.onHit("alien", function(){
laser.destroy();
alien.destroy();
});
laser.y = 510;
laser.x = ship.x + 40;
var moveLaser = setInterval(function(){
if(laser.y < 1){
clearInterval(moveLaser);
laser.destroy();
} else {
laser.y = laser.y - 4;
}
}, 1);
}
function checkBorder(){
if(ship.x < 0 || ship.x > 1250){
Crafty.enterScene("gameStart");
}
}
function goLeft(){
var goLeftCount = 65;
var goLeftInterval = setInterval(function(){
if(goLeftCount === 0){
clearInterval(goLeftInterval);
checkBorder();
} else {
ship.x = ship.x - 1;
goLeftCount -= 1;
}
}, 1);
}
function goRight(){
var goRightCount = 65;
var goRightInterval = setInterval(function(){
if(goRightCount === 0){
clearInterval(goRightInterval);
checkBorder();
} else {
ship.x = ship.x + 1;
goRightCount -= 1;
}
}, 1);
}
var ship = Crafty.e("2D, DOM, Image, Bind")
.image("ship.png")
.bind('KeyDown', function(e) {
if(e.key == Crafty.keys['LEFT_ARROW']) {
goLeft();
} else if (e.key == Crafty.keys['RIGHT_ARROW']) {
goRight();
} else if(e.key == Crafty.keys['SPACE']) {
newLaser();
}
});
ship.y = 520;
ship.x = 600;
newAlien();
});
Crafty.enterScene("gameStart");
Can anyone please tell me what are the problems with my code?
A: One problem: when you try to destroy the alien, the alien variable isn't actually in scope. (You've declared it in a different function.) So there's no way alien.destroy(); will work.
Second problem: the function you pass to .onHit("alien", callback) will only be run when the laser hits an entity with the "alien" component. It has no idea what variable name you assigned the entity. However, the callback function will be passed information about what entities it's colliding with, and you can use that to resolve the collision.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,360
|
Sometimes we finish a book and what comes to mind when we flip the last page is a never-ending string of feelings, reactions, and adjectives. This was one of those books. Here are just some of the things that were swirling through our heads- Holy crap. Wow. Damn that was good. WHOA. No, seriously. Whoa. SWOON. SWOON. SWOON. Is there more? We need more. Beautiful. Gorgeous. Painful. Amazing. The friendships! The swoon! Feelings, such strong feelings.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,902
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Economic Impact and Implications of the Canada - U.S. Free Trade Agreement
Author: Siddiqui, F.A.
Primary focus is to take stock of the progress in implementing the FTA and the challenges and opportunities that have surfaced. Essays include: "An Intelligent Politician's Guide to the FTA"; "The Bank of Canada and the FTA"; "Evaluating Free Trade - A Perspective from the CLC"; "FTA Chapter 19 Working Group on Subsidies and Trade Remedies"; "Atlantic Canada and Fisheries Trade"; "Japanese Views on the Canada-US FTA"; "Economics Growth and the Gains from Trade Liberalization."; many others. Distinguished contributors include Robert Mundell, Doug Purvis, Ron Wonnacott, Herbert Grubell, Simon Reisman, Roger Philip, and others.
Other Banking, Economics & Finance Books
2002 - Role of Financial Markets in Generating Business Cycles
1997 - Revolution by Reason and Other Essays
1997 - Interorganizational Relations and Effectiveness in Planning and Administration in Developing Countries Towards a Strategy for Improving the Performance of Develoment Policy Organizations
>> See all our Banking, Economics & Finance books
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,023
|
By Manny Dutches. Discovered by Player FM and our community — copyright is owned by the publisher, not Player FM, and audio streamed directly from their servers.
33 episodes available. A new episode about every 21 days averaging 36 mins duration .
Dutches Favorite Picks Out Of 21-00 Chapters. Thanks For Coming Along This Ride. More To Come!!!!
1)DJ KRUSH FT.ESTHERO - FINAL HOME 2)massive attack - Three (feat. Nicolette) 3)Monophona - Forest of Wonders 4)Port electric - Borderline Soul (trip hop) 5)Moorcheeba - Moog Island 6)portishead - dark room 7)Tricky - What Is Wrong 8)SOLAR Enjoy!!!!!!!!!!!!!!!
Dutches Favorite Picks Out Of The First 10 Chapters. Thanks For Coming Along This Ride. More To Come!!!!
Start listening to Trip Hop Passion on your phone right now with Player FM's free mobile app, the best podcasting experience on both iPhone and Android. Your subcriptions will sync with your account on this website too. Podcast smart and easy with the app that refuses to compromise.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,000
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Siloking Mayer Maschinenbau GmbH (Eigenschreibweise SILOKING) ist ein familiengeführtes Unternehmen im Bereich der Landtechnik, das Fütterungstechnik unter der Marke Siloking entwickelt, produziert und vertreibt. Der Sitz des Unternehmens ist Tittmoning, Deutschland.
Siloking produziert Vertikal-Futtermischtechnik wie gezogene Vertikal-Futtermischwagen, selbstfahrende Vertikal-Futtermischwagen, elektrisch angetriebene Vertikal-Futtermischwagen, stationäre Vertikal-Misch- und Dosieranlagen sowie Silage-Entnahme- und Verteilgeräte.
Die Produktgruppe der selbstfahrenden Futtermischtechnik, die unter dem Namen Siloking SelfLine produziert und vertrieben wird, wurde zur Einführung auf der Eurotier 2004 mit der DLG Goldmedaille ausgezeichnet. Seit dem Jahr 2019 konnten 2000 SILOKING SelfLine im Markt platziert werden. Das Unternehmen ist Mitglied im VDMA, im Fachbereich Landtechnik.
Die Marke Siloking hatte im Jahr 2019/ 2020 einen Exportanteil von über 50 %, wobei in die Fütterungstechnik in über 50 Länder exportiert wurde. Die Produkte werden über die zur Siloking-Unternehmensgruppe zählenden Niederlassungen OOO Siloking Rus, in Lipetsk (RU), Siloking Agricultural Machinery Beijing Co. Ltd. in Peking (CN) sowie Siloking do Brasil comércio de equipmentos agropecuários LTDA, Sao Jose do Rio Preto (BR) vertrieben. Außerdem besteht ein Netz an Importeuren, die ebenfalls den Landtechnikhandel im jeweiligen Land betreuen, so etwa in Kanada, den Niederlanden und weiteren Ländern.
Auszeichnungen
DLG Goldmedaille 2004: Im gleichen Jahr als der Siloking SelfLine auf den Markt gebracht wurde, zeichnete die DLG (Deutsche Landwirtschaftsgesellschaft) den selbstfahrenden Futtermischwagen 2004 mit der Goldmedaille "Eurotier" aus.
DLG-anerkannt: Gesamt-Prüfung: Die beiden selbstfahrenden Futtermischwagen Siloking SelfLine 4.0 Premium 2215 19 m³ und der SelfLine 4.0 Compact 13 m³ erhalten 2016 das "DLG anerkannt Gesamtprüfung"-Siegel. Die selbstfahrenden Futtermischwagen wurden im praktischen Einsatz hinsichtlich ihrer Funktionseigenschaften in dem Gesamtprüfungstest geprüft. Die Mischgenauigkeit und die Fräsleistung bzw. die Mischgenauigkeit, die Fräsleistung und der Kraftstoffverbrauch schnitten laut DLG Prüfbericht über dem Standard ab.
Geschichte
Vom Vertrieb von Landtechnik zum Landtechnikhersteller
Die Gründung des Unternehmens erfolgte 1983 als Werkvertretung für landwirtschaftliche Maschinen unter anderem mit Siloblockschneider und Holzspaltern.
Nach der Einführung des Kälberiglus "Flixbox" und der Kuhputzbürste "Happycow", startete das Unternehmen im Jahr 1993 mit der Produktion von Fütterungstechnik. Mit der Vorstellung des Silokamms beginnt Siloking selbst landtechnische Maschinen zu entwickeln und zu produzieren.
Markenanmeldung und Einstieg in die Futtermischtechnik
Im Jahr 1996 wurde die Marke Siloking angemeldet und ist bis heute als Wortmarke in mehr als 40 Ländern geschützt.
1997 wurde der erste Siloking Futtermischwagen (Siloking TrailedLine) vorgestellt. Die Erstellung einer Mischration für Wiederkäuer wird somit möglich, dabei wird das Futter mit einer Maschine gemischt, transportiert und ausgetragen.
Der Siloking Selbstfahrer
2004 wurde der Siloking Selbstfahrer auf den Markt gebracht und im gleichen Jahr von der DLG (Deutsche Landwirtschaftsgesellschaft) mit der Goldmedaille "Impuls für den Fortschritt" auf der EuroTier ausgezeichnet.
Im Jahr 2009 wurde die Siloking SelfLine Familie vergrößert, der Siloking SelfLine System 1.000+ mit 30 m³ Fassungsvolumen wurde vorgestellt. Er ist konzipiert für große und sehr große Betriebe mit 1.000 und mehr Kühen oder Biogasanlagen.
10 Jahre nach der Einführung des Siloking SelfLine konnte das Tittmoninger Unternehmen seinen 1.000ten selbstfahrenden Futtermischwagen produzieren.
Im gleichen Jahr wurden die drei Siloking Niederlassungen in Russland, China und Brasilien gegründet.
Die voll-elektrische Fütterung
Als einziger Hersteller in der Branche stellt Siloking den ersten vollständig elektrisch von einem Akku angetriebenen Futtermischwagen vor.
Der Siloking eTruck wurde 2016 auf der Eurotier präsentiert, zwei Jahre später wird der Siloking eSilokamm auf der Eurotier 2018 vorgestellt.
Bisher (Stand 2020) konnten 100 Stück elektrisch betriebener Fütterungstechnik verkauft werden.
Der 2.000ste SelfLine verließ Mitte des Jahres 2019 das Werk.
Zwischen 2004 und 2019 lieferte Siloking 2.000 Selbstfahrer weltweit aus.
Standorte
Am Hauptstandort in Tittmoning befindet sich neben der Produktion auch die Entwicklungsabteilung. In Tittmoning-Kirchheim ist der Kundendienst, der Vertrieb und die Marketingabteilung gebündelt. Zusätzlich zum Standort Deutschland befindet sich unter der Gesellschaft Siloking Slovakia s.r.o. in Záhorská Ves ein weiterer Fertigungsstandort.
Neben den Fertigungsniederlassungen unterhält Siloking weitere Service- und Vertriebsniederlassungen in China, Russland und Brasilien mit jeweils eigener Unternehmensstruktur.
Weblinks
Einzelnachweise
Landmaschinenhersteller
Tittmoning
Produzierendes Unternehmen (Landkreis Traunstein)
Gegründet 1983
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,137
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import Crafty from "crafty";
import scaleGame from "./lib/game_scaler";
import "src/scenes/Intro";
import "src/scenes/Gameplay";
import "src/scenes/Replay";
// Setup initial screen size and layers
Crafty.init(1024, 576, document.getElementById("game"));
Crafty.background("#000");
window.addEventListener("resize", scaleGame);
setTimeout(scaleGame, 0);
Crafty.scene("Intro");
Crafty.bind("StartGame", () => {
Crafty.scene("Gameplay");
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,053
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\section{Introduction}
In \cite{gagliardo1957}, Gagliardo characterizes the trace space of the Sobolev space $W^{1,p}(\Omega)$ ($p>1$) for a given {bounded} Lipschitz domain $\Omega\subset \mathbb{R}^d$. The result consists of the following two parts. First, the trace operator $T$ from $W^{1,p}(\Omega)$ to $W^{1-1/p,p}(\partial\Omega)$ is linear and continuous, and conversely, one can define a continuous linear extension operator $E$ from $W^{1-1/p,p}(\partial\Omega)$ to $W^{1,p}(\Omega)$.
Our goal in this work is to study the trace spaces of {some} nonlocal function spaces denoted by $\{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)\}_{\delta>0}$
related to the Dirichlet energies of a class of nonlocal problems. {Here, $\delta>0$ represents the horizon parameter that characterizes the ranges of nonlocal interactions, and
$\hat\Omega=\Omega\cup\Omega_{\delta}$
with $\Omega_{\delta}:= \left\{ \bm x\in \mathbb{R}^d\backslash\Omega: \text{dist}(\bm x, \partial\Omega) <\delta\right\}$ being viewed as a nonlocal ``boundary'' set of the given domain $\Omega$.}
The function space $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ is defined as
the completion of $C^1(\overline{\hat\Omega})$ with respect to the norm
\begin{align}
\label{eq:nnorm}
\| \cdot \|_{\mathcal{S}^{\,\beta}_\delta(\hat{\Omega})} = (\| \cdot \|_{L^p(\hat{\Omega})}^p +
| \cdot |^p_{\mathcal{S}^{\,\beta}_\delta(\hat{\Omega})} )^{1/p},
\end{align}
with the associated semi-norm $|\cdot |_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}$ given by
\begin{align}
\label{eq:nsnorm}
| u |^p_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}= \int_{\hat\Omega} \int_{\hat\Omega} {{\gamma}^{\,\beta}_\delta}(| \bm y-\bm x|) |u(\bm y)-u(\bm x)|^p d\bm y d\bm x \,.
\end{align}
Notice that the space $\mathcal{S}^{\,\beta}_\delta$ also depends on $p$, and we always assume $p>1$ in this paper.
The kernel function ${{\gamma}^{\,\beta}_\delta}$ in \eqref{eq:nsnorm},
for $\delta>0$ and $\beta\in [0,d+p)$ is taken as
\begin{equation}\label{def:rescaledkernel}
{{\gamma}^{\,\beta}_\delta}(|\bm y - \bm x|) =\frac{C_{d,p,\,\beta}}{\delta^{d+p-\beta}}\frac{1}{|\bm{y}-\bm{x}|^{\,\beta}} 1_{\{|\bm{y}-\bm{x}|<\delta\}},
\end{equation}
where for any $\delta$, $1_{\{|\bm{y}-\bm{x}|<\delta\}}$ denotes the characteristic function on the set $\{|\bm{y}-\bm{x}|<\delta\}$, and $C_{d,p,\,\beta}$ normalizes the $p^{th}$ moment of $\gamma_\delta^{\,\beta}$. In particular, we have
\[
C_{d,p,\,\beta}=s^{-1}_{d-1}(d+p-\beta),
\]
where $s_{d-1}$ denotes the area of the $d-1$-sphere since
\[
\int_{\mathbb{R}^d}\frac{1}{\delta^{d+p-\beta}}\frac{1}{|\bm z|^{\,\beta}} 1_{\{|\bm z|<\delta\}}|\bm z|^pd\bm z=\frac{1}{\delta^{d+p-\beta}}\int_{B(\bm{0},\delta)} |\bm z|^{p-\beta}d\bm z=\frac{s_{d-1}}{\delta^{d+p-\beta}}\int_0^\delta r^{d+p-\beta-1}dr=\frac{s_{d-1}}{d+p-\beta}.
\]
We note that, for different $\delta>0$, ${{\gamma}^{\,\beta}_\delta}$ can be obtained from a rescaling of a given $\delta$-independent nonnegative kernel ${\gamma}^{\,\beta}$ defined on $(0,1)$ by:
\begin{equation} \label{def:frackernel}
{{\gamma}^{\,\beta}_\delta}(|\bm y - \bm x|) =\frac{1}{\delta^{d+p}}{\gamma}^{\,\beta}\left(\frac{|\bm y - \bm x|}{\delta} \right)\,, \quad\text{where }{\gamma}^{\,\beta}(|\bm y - \bm x|) = \frac{C_{d,p,\,\beta}}{|\bm y - \bm x|^{\,\beta}} 1_{\{|\bm y - \bm x|<1\}}.
\end{equation}
{It is easy to see that the nonlocal function space $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ contains all $p$-integrable functions on $\hat{\Omega}$ with finite norms with respect to
$\| \cdot \|_{\mathcal{S}^{\,\beta}_\delta(\hat{\Omega})}$. Moreover, for any finite and given $\delta>0$, the kernel ${{\gamma}^{\,\beta}_\delta}$ is integrable for
$\beta\in [0,d)$ and the corresponding space $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ is equivalent to the $L^p(\hat\Omega)$ space; while ${{\gamma}^{\,\beta}_\delta}$ is non-integrable for $\beta\in (d, d+p)$ and $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ is equivalent to the standard fractional Sobolev space $W^{(\beta-d)/p, p}(\hat\Omega)$}.
Moreover, we have the convergence of the space $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ to
{the local limit} $W^{1,p}(\Omega)$ as $\delta\to0$, see e.g., discussions in \cite{BBM01,ponce2004,Mengesha12,foss2016differentiability}.
{Given any domain $\Omega$,} let $\mathcal{T}^{\,\beta}_\delta({\Omega})$ denote the space of all $L^p({\Omega})$-functions $u$ with the norm defined as
\begin{equation}
\|u\|_{\mathcal{T}^{\,\beta}_\delta({\Omega})}:= \left( \frac{1}{\delta} \| u\|^p_{L^p({\Omega})} + |u|^p_{\mathcal{T}^{\,\beta}_\delta({\Omega})} \right)^{1/p}\,.
\end{equation}
Here the semi-norm is defined as
\begin{equation}\label{eqn:nonlocalhalf}
\left|u\right|_{\mathcal{T}^{\,\beta}_\delta({\Omega})}:= \left( \delta^{\,\beta-2}\int_{{\Omega}}\int_{{\Omega}} \frac{|u(\bm y) -u(\bm x)|^p}{(|\bm y-\bm x|\vee \delta)^{d+p-2} (|\bm y -\bm x|\wedge\delta)^{\,\beta}} \,d\bm{y} d\bm{x} \right)^{1/p}, \end{equation}
where $a\wedge b:=\min(a, b)$ and $a\vee b:=\max(a, b)$. Our main result is to show that $\mathcal{T}^{\,\beta}_\delta(\Omega_{\delta})$ is the trace space of $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$, i.e., {to establish} the existence of trace operator $T$ and extension operator $E$ that define continuous linear maps in between $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ and $\mathcal{T}^{\,\beta}_\delta(\Omega_{\delta})$.
For the ease of presentations, in the following we denote:
\begin{equation}
\label{stripenotation}
\mathcal{R}^L=(0,L)\times\mathbb{R}^{d-1},\, \mathcal{R}_a=(a,0)\times\mathbb{R}^{d-1}, \text{ for } a<0, \, \text{ and } \mathcal{R}_a^L=(a,L)\times\mathbb{R}^{d-1}, \text{ for } a < L.
\end{equation}
and we note that when ${\Omega}=(0,\infty)\times\mathbb{R}^{d-1}=\mathcal{R}^{\infty}$, we have ${\Omega}_{\delta}=\mathcal{R}_{-\delta}$ and $\hat{{\Omega}}=\mathcal{R}_{-\delta}^{\infty}$. {With these notations,
we first show the main results on half spaces as follows.}
\begin{thm}[Trace theorem on half spaces]\label{mainthm_1}
Let $\delta>0$ and $\beta\in [0, d+p)$, then there exists a constant $C$ independent of $\delta$ and $\beta$
such that for any $u\in \mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})$,
\[
\frac{1}{\delta}\| u\|^p_{L^p(\mathcal{R}_{-\delta})}\leq C |d+p-\beta|^{-1}\| u \|^{p-1}_{L^p(\mathcal{R}_{-\delta}^{\infty})}\verti{u}_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})}\leq C |d+p-\beta|^{-1} \left(\| u \|^p_{L^p(\mathcal{R}_{-\delta}^{\infty})}+\verti{u}^p_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})}\right),
\]
\[
\verti{u}^p_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})}\leq C |d+p-\beta|^{-1}\verti{u}^p_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})},
\]
and therefore
\[
\| u\|_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})}\leq C |d+p-\beta|^{-1/p}\| u \|_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})}.
\]
\end{thm}
\begin{thm}[Inverse trace theorem on half spaces]\label{mainthm_2}
Let {$\delta\in (0,M)$ for some fixed number $M>0$}, {$\beta\in [0, d+p)$, and $\beta\neq d$,} then there exists an extension operator $E: \mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})\to \mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)$ such that
\[
\| E u \|_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)} \leq C |d-\beta|^{-1/p} \| u\|_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})}\,, {\quad \forall u \in
\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})}
\]
where $C$ is a constant independent of $\delta$, $\beta$ and $u\in \mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})$.
\end{thm}
{Using partition of unity techniques, the above trace theorems in special domains can then be
extended to more general domains. which are stated in the theorems below.}
\begin{thm}[General trace and inverse trace theorems]\label{mainthm_1_general}
Assume that $\Omega$ is a bounded {and} simply connected Lipschitz domain in $\mathbb{R}^d$ and $\Omega_\delta:=\{\bm{x}\in\mathbb{R}^d\backslash{\Omega}: \text{dist}(\bm{x},{\Omega})<\delta\}$ is its nonlocal boundary set.
There exists a constant $\epsilon$ depending on the domain $\Omega$, such that for any $\delta\in(0,\epsilon)$ and $\beta$,
\[
\| u\|_{\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)}\leq C_1|d+p-\beta|^{-1/p} \| u \|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}, {\quad \forall u \in \mathcal{S}^{\,\beta}_\delta(\hat\Omega),}
\]
On the other hand, for any $\delta\in (0,\epsilon)$, $\beta\in[0,d+p)$ {and $\beta\neq d$}, there exists an extension operator $E: \mathcal{T}^{\,\beta}_\delta(\Omega_\delta)\to \mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ such that
\[
\| E u \|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)} \leq C_2 |d-\beta|^{-1/p} \left( \| u \|_{\mathcal{T}_\delta^{\,\beta}(\Omega_\delta)} + |d+p-\beta|^{-1/p}\| u\|_{L^p(\Omega_\delta)}\right)\,, {\quad \forall u \in
\mathcal{T}^{\,\beta}_\delta(\Omega_\delta).}
\]
Here $C_1$, $C_2$ are constants independent of $\delta$, $\beta$.
\end{thm}
The paper is organized as follows. In Section \ref{sec:notation} we discuss the motivation of this work, together with {additional definitions and notation} relevant to our main results. To provide some insights {on the various nonlocal spaces under consideration,
we also investigate their scaling properties and consistency with the
classical trace spaces in the local limit,
as $\delta\rightarrow 0$.
Moreover,
in making these connections, our study also represents an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on a subdomain of a smaller size or dimension. This further leads to a new way of viewing the
various forms of trace, embedding and extension theorems in different function spaces as consequences in different scaling limits, further illustrating the contributions of our study.}
Sections \ref{sec:traceRd}-\ref{sec:generaldomain} contain the proofs of the aforementioned trace theorems. In particular, to show these trace theorems, {while following} the footsteps of the proofs for the local trace theorems, {we take into account the effect of nonlocal interactions}. In Section \ref{sec:traceRd} we provide the proof of Theorem \ref{mainthm_1} with a special case $\delta=1$ first, which captures the intrinsic effect of nonlocal interactions defined on a larger domain for subdomains. We then extend the results to cases with general $\delta>0$ using a scaling argument. For the inverse trace theorem, in Section \ref{sec:inversetrace} we present the proof of Theorem \ref{mainthm_2} by constructing an extension operator based on the Whitney decomposition. Then, in Section \ref{sec:generaldomain} we prove Theorem \ref{mainthm_1_general} for the general bounded simply connected Lipschitz domain ${\Omega}$ {using partition of unity techniques}.
Lastly, Section \ref{sec:conclusion} summarizes our findings and discusses future research directions.
\section{Motivation and Notation}\label{sec:notation}
{In this section, we first make a few comments on the motivation of our work. We then
investigate the consistency and connection between our nonlocal trace space with the trace space in the classical calculus in Section \ref{sec:loclimit}. We also}
provide notations and several useful lemmas for the later proofs, including some scaling properties in Section \ref{sec:scale} and the Whitney decomposition of $\mathcal{R}^L$ in Section \ref{sec:whitney}.
\subsection
Local, nonlocal and fractional modeling}\label{eqn:nonlocalcalc}
{A major motivation of our work comes from nonlocal modeling that are represented by integro-differential equations, in particular, equations involving nonlocal interactions with a finite interaction length. The latter have drawn much attention recently in modelling certain physical systems where the classical models are not most effective}. Comparing with the classical {local} partial differential equation (PDE) models, these equations have the ability to describe these physical phenomena in a setting with reduced regularity requirements allowing singularities and discontinuities to naturally occur \cite{du2013nonlocal,gunzburger2010nonlocal,foss2018existence}. On the other hand, when comparing with the nonlocal integro-differential equations characterized by an infinite lengthscale, compactly supported nonlocal models are computationally more efficient and therefore a more feasible choice for scientific and engineering applications. These extra flexibility and efficiency allow this framework to be used in many different situations involving physical discontinuity such as dynamic fracture \cite{silling_2000,silling2007peridynamic,hu2012peridynamic,Ha2010peri,CHENG2015,ZHANG2018Rayleighcrack,Yu2018paper,Trask2018paper}, corrosion models \cite{CHEN2015pitcorrosion, Chen2015passivefilm,Jafarzadeh2017corrosion,Li2017corrosion}, and heat conduction \cite{Bobaru2010heatconduction}. The development in this subject has also produced other applications in image processing \cite{lou2010image} and population models \cite{Carrillo2005} among many other different fields which can be further seen in \cite{bobaru2015handbook}. Particularly, nonlocal problems with boundary constraints have become of recent interest in works such as \cite{le2018surface,bobaru2016handbook,oterkus2010peridynamic,macek2007peridynamics,du2017peridynamic,madenci2014peridynamic,oterkus2014peridynamic,oterkus2015peridynamics,tao2017nonlocal,you2019neumann,you2020asymptotically,yu2021asymptotically}. In nonlocal models, the boundary conditions are normally not imposed on a sharp interface. Rather, they are imposed on a region with non-zero volume which lies outside of the domain, and treating the nonlocal boundary problem improperly can cause artificial phenomena such as a ``surface'' or ``skin'' effect \cite{bobaru2011adaptive,ha2011characteristics,prudhomme2020treatment,chen2020peridynamics}. Differs from the local problems, in some nonlocal problems boundary effects play a major role. For example, in nonlocal minimal surface problems, the ``stickiness'' effect arises and the boundary datum may not be attained continuously \cite{dipierro2017boundary,borthagaray2019finite}. All the above examples indicate that studying the nonlocal boundary conditions and the {associated nonlocal trace spaces} are critical for the development of nonlocal models.
In this work, we aim to introduce {a function space that serves as a trace space} for nonlocal problems with constant finite interaction length (the so-called interaction radius or horizon $\delta$), and study related extension results. Extension and trace theorems are well-known in the study of classical local problems with boundary constraints. For the case of Sobolev spaces of integer order, these results are well-established long time ago (see, e.g., \cite{adams2003sobolev,slobodetskiui1958sl}). For Sobolev spaces with fractional order of differentiability, which can be seen as one type of nonlocal problems with infinite interaction length, the trace space and extension results are studied in \cite{koskela2017traces,dyda2019function,bogdan2020extension,rutkowskifunction}. The latter can be useful in studying nonlocal problems with non-homongeneous boundary data, such as those associated with the nonlocal Laplacian and nonlocal $p$-Laplacian, see for example \cite{Andreu2008,Andreu2009,Andreu2010,bogdan2020extension,DRV2017,Ros2016}. In \cite{du2021fractional,tian2017trace,foss2020traces}, trace theorems are developed for nonlocal problems with varying influence horizon $\delta(\bm{x})$, where $\delta(\bm{x})\rightarrow0$ as $\bm{x}$ approaches the boundary, in a way that the trace spaces of classical Sobolev spaces are recovered. The trace results are also applied to the study of the coupling of nonlocal and local models \cite{TaTiDu19}. To our best knowledge, the definition of trace space and extension results for nonlocal problems with constant finite horizon have not been dealt with so far. These results would extend the knowledge on the trace space in nonlocal calculus and its connection with the trace space in classical calculus. Moreover, the trace theorem and the inverse trace theorem would also provide important mathematical tools for developing well-posed nonlocal models with volumetric boundary conditions, such as discussed in \cite{you2019neumann}.
\subsection{Nonlocal Space $\mathcal{S}^{\,\beta}_\delta(\Omega)$, Associated Nonlocal Problems and Their Local Limits}\label{eqn:nonlocalcalcf}
Before discussing their connections in the following sections, in this section we introduce the classical and nonlocal Laplacian operators and their corresponding nonlocal function spaces relevant to this paper. {The discussions in this subsection are restricted to the Hilbert space setting where $p=2$.}
{Given a scalar function $u(\bm{x}):{\Omega}\rightarrow\mathbb{R}$, the classical Laplacian operator is defined as $\Delta u:=\nabla\cdot\nabla u$ and
boundary value problems on the domain ${\Omega}$ related to $\Delta$ are often associated with the Sobolev space $H^1({\Omega})$ with its} norm defined by
\[\vertii{u}_{H^1({\Omega})}:=\left(\vertii{u}^2_{L^2({\Omega})}+\verti{u}^2_{H^1({\Omega})}\right)^{1/2}.\]
On the other hand, when incorporating long-range interactions into the model such that where every point $\bm{x}\in{\Omega}$ is interacting with a finite neighborhood of points,
a nonlocal Laplacian operator
is then given by
$$\mathcal{L}[u](\bm{x}):=C\int_{\hat{{\Omega}}}\gamma(\bm{x},\bm{y})(u(\bm{y})-u(\bm{x}))d\bm{y}, \quad \bm{x}\in{\Omega},$$
where {$\gamma(\bm{x},\bm{y})$ is a kernel function that will be prescribed shortly}, $\hat{{\Omega}}={\Omega}\cup{\Omega}_I$ and
$${\Omega}_I:=\{\bm{y}\in\mathbb{R}^d\backslash{\Omega}\text{ such that }\gamma(\bm{x},\bm{y})\neq0 \text{ for some }\bm{x}\in{\Omega}\}$$
is the interaction domain of ${\Omega}$. The nonlocal Laplacian operator is associated with the following nonlocal norm
$$\vertii{u}_{\mathcal{S}(\hat{{\Omega}})}:=\left(\vertii{u}^2_{L^2(\hat{{\Omega}})}+\verti{u}^2_{\mathcal{S}(\hat{{\Omega}})}\right)^{1/2}\text{ where }\verti{u}^2_{\mathcal{S}(\hat{{\Omega}})}:= \frac{C}{2}\int_{\hat{{\Omega}}} \int_{\hat{{\Omega}}} \gamma(\bm{x},\bm{y}) (u(\bm y)-u(\bm x))^2 d\bm y d\bm x.$$
In this paper we further assume that such neighborhood is a Euclidean ball surrounding $\bm{x}$, i.e., $B(\bm{x},\delta):=\{\bm{y}\in\mathbb{R}^d:|\bm{y}-\bm{x}|<\delta\}$. Here $\delta$ is the interaction radius or horizon. This fact has implications on the boundary conditions that are prescribed on a collar of thickness $\delta$ outside the domain ${\Omega}$, that we have the interaction domain ${\Omega}_I={\Omega}_\delta {:= \{\bm{y}\in\mathbb{R}^d\backslash\Omega: \text{dist}(\bm{y},\partial\Omega)<\delta }\}$. In particular, we can take a popular class of kernels $\gamma(\bm{x},\bm{y})={\gamma}_{\delta}^{\,\beta}(\verti{\bm{y}-\bm{x}})$ as in \eqref{def:rescaledkernel}. We note that when the constant $C=C^{\text{diff}}=2d$, we have the following property
\begin{equation}\label{eqn:Cdiff}
C^{\text{diff}}\int_{B(\bm{x},\delta)}\gamma^{\,\beta}_\delta(|\bm{y}-\bm{x}|)|\bm{y}-\bm{x}|^2 d\bm{y}=2d \quad {\forall \bm{x} \in \mathbb{R}^d},
\end{equation}
since the $p^{th}$ moment of kernel $\gamma_\delta^{\,\beta}$ in \eqref{def:rescaledkernel} is normalized to $1$. Then
it is well-known (see, e.g.,
\cite{du2013nonlocal}) that the nonlocal diffusion operator converge to its local for all counterpart pointwise: for any $u\in C^\infty(\mathbb{R}^d)$ and $\bm{x}\in\mathbb{R}^d$,
$$\mathcal{L}[u](\bm{x})=C^{\text{diff}}\int_{B(\bm{x},\delta)}(u(\bm{y})-u(\bm{x}))\gamma^{\,\beta}_\delta(|\bm{y}-\bm{x}|)d\bm{y}\overset{\delta\rightarrow 0}{\longrightarrow} \Delta u(\bm{x}).$$
Moreover, when $u\in H^1({\Omega})$, its nonlocal norm converges to the $H^1$ norm:
$$\| u \|_{\mathcal{S}^{\,\beta}_\delta({\Omega})}\overset{\delta\rightarrow 0}{\longrightarrow} \| u \|_{H^1({\Omega})}.$$
Naturally, we can extend the above conclusion to more general cases of nonlocal and local $p$-Laplacians corresponding to $p>1$.
\subsection{{Nonlocal Space $\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)$ and} Connection to Classical Local Trace Spaces}\label{sec:loclimit}
First of all, we may view \eqref{eqn:nonlocalhalf} as a nonlocal counterpart of the classical trace semi-norm
$$|u|_{W^{1-1/p, p}(\partial{\Omega})}:=\left( \int_{\partial{\Omega}}\int_{\partial{\Omega}}\dfrac{|u(\bm{y})-u(\bm{x})|^p}{|\bm{y}-\bm{x}|^{d+p-2}}d\bm{y} d\bm{x} \right)^{1/p}$$
and seek a nonlocal analog of the classical trace theorem. The relation between the classical $W^{1-1/p, p}(\partial\Omega)$ trace space and the new nonlocal trace space $\mathcal{T}^{\,\beta}_\delta(\Omega_{\delta})$ can be seen from the
limiting process as $\delta\to0$ in the following proposition. In the rest of the paper,
we use $f\lesssim g$ if $f\leq C g$ for a generic constant $C>0$ independent of $\delta$ and $\beta$.
We also write $f\approx g$ if $f\lesssim g$ and $g\lesssim f$.
\begin{pro}\label{thm:consist}
Let $\partial\mathcal{R}=\{0\}\times\mathbb{R}^{d-1}$ and $\mathcal{R}_{-\delta}$
{be defined as in \eqref{stripenotation}}
for $\delta\in (0,1)$, then
$$| u |^p_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})} \xrightarrow{\delta\to0} | u |^p_{W^{1-1/p, p}(\partial\mathcal{R})},$$
for any {$u\in C^1_c\left([-1,0]\times B^{d-1}(\bm 0,M)\right)$ for some $M>0$. Here $B^{d-1}(\overline{\bm{x}},r)$ denotes the ball centered at $\overline{\bm{x}}$ with radius $r$ in $\mathbb{R}^{d-1}$.}
\end{pro}
\begin{proof}
In this proof, {we denote any point $\bm x\in \mathbb{R}^d$} by ${\bm x=}(\tilde{x},\overline{\bm x})\in \mathbb{R}\times \mathbb{R}^{d-1}$. Similarly $\bm y\in\mathbb{R}^d$ is also denoted by $(\tilde{y},\overline{\bm y})\in\mathbb{R}\times\mathbb{R}^{d-1}$. We first have the estimate
\begin{align*}
&\left|\delta^{\,\beta-2}\int_{\mathcal{R}_{-\delta}}\int_{\mathcal{R}_{-\delta}} \frac{|u(\bm y) -u(\bm x)|^p}{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2} (|\bm y -\bm x|\wedge\delta)^{\,\beta}} \,d\bm{y} d\bm{x}-\int_{\partial\mathcal{R}}\int_{\partial\mathcal{R}}\dfrac{|u({\bm{y}})-u({\bm{x}})|^p}{|{\bm{y}}-{\bm{x}}|^{d+p-2}}d{\bm{y}} d{\bm{x}}\right|\\
=&\left|\delta^{\,-2}\int_{\mathcal{R}_{-\delta}} \int_{{\mathcal{R}_{-\delta}}\backslash B(\bm{x},\delta)}\frac{|u(\bm{y}) -u(\bm{x})|^p}{|{\bm y}-{\bm x}|^{d+p-2}} \,d\bm{y} d\bm{x}+\delta^{\,\beta-d-p}\int_{\mathcal{R}_{-\delta}} \int_{{\mathcal{R}_{-\delta}}\cap B(\bm{x},\delta)} \frac{|u(\bm{y}) -u(\bm{x})|^p}{|\bm y -\bm x|^{\,\beta}} \,d\bm{y} d\bm{x}\right.\\
&\left.\qquad -\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}}\dfrac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}d\overline{\bm{y}} d\overline{\bm{x}}\right|\\
\leq&\left|\delta^{\,\beta-d-p}\int_{\mathcal{R}_{-\delta}} \int_{{\mathcal{R}_{-\delta}}\cap B(\bm{x},\delta)}\frac{|u(\bm{y}) -u(\bm{x})|^p}{|{\bm y}-{\bm x}|^{\,\beta}} \,d\bm{y} d\bm{x}-\delta^{\,-d-p}\int_{\mathcal{R}_{-\delta}} \int_{{\mathcal{R}_{-\delta}}\cap B(\bm{x},\delta)}|u(\bm{y}) -u(\bm{x})|^p \,d\bm{y} d\bm{x}\right|~~~~\leftarrow\;A_1\\
&+\left|\delta^{\,-2} \int_{\mathcal{R}_{-\delta}} \int_{\mathcal{R}_{-\delta}} \frac{|u(\bm{y}) -u(\bm{x})|^p}{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2}} \,d\bm{y} d\bm{x}-\delta^{\,-2}\int_{\mathbb{R}^{d-1}}\int_{-\delta}^0 \int_{\mathbb{R}^{d-1}}\int_{-\delta}^0 \frac{|u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}})|^p}{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2}} \,d\tilde{y} d\overline{\bm{y}} d\tilde{x} d\overline{\bm{x}}\right|~~~~~~~~~\leftarrow\;A_2\\
&+\left|\delta^{\,-2}\int_{\mathbb{R}^{d-1}}\int_{-\delta}^0 \int_{\mathbb{R}^{d-1}}\int_{-\delta}^0 \frac{|u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}})|^p}{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2}} \,d\tilde{y} d\overline{\bm{y}} d\tilde{x} d\overline{\bm{x}}-\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}}\dfrac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}d\overline{\bm{y}} d\overline{\bm{x}}\right|.~~~~~~~~~~\leftarrow\;A_3
\end{align*}
To estimate the $A_1$ part, we first note that {$u\in C^1([-1,0]\times B^{d-1}(\bm 0,M))$}
implies
\begin{equation}\label{eq:prop2.1}
\verti{{u}(\bm{y}) -{u}(\bm{x})}\leq \tilde{C} \verti{\bm{y}-\bm{x}}
\end{equation}
for a constant $\tilde{C}$ independent of $\delta$, $\bm{x}$, and $\bm{y}$.
Notice also that $\text{supp}({u})\subseteq [-1,0]\times B^{d-1}(\bm 0, M)$ for some $M>0$. Therefore
\begin{align*}
A_1=&\delta^{\,-d-p}\int_{\mathcal{R}_{-\delta}} \int_{\mathcal{R}_{-\delta}\cap B(\bm{x},\delta)}{|{u}(\bm{y}) -{u}(\bm{x})|^p} \left|\dfrac{\delta^{\,\beta}}{|\bm y -\bm x|^{\,\beta}}-1\right|\,d\bm{y} d\bm{x}\\
\leq&C \delta^{\,-d-p+1} {\int_0^\delta}r^{d-1+p} \left|\dfrac{\delta^{\,\beta}}{r^{\,\beta}}-1\right|\,dr\leq C\delta^{\,-d-p+1}\left(\dfrac{\delta^{d+p}}{d+p-\beta}+ \dfrac{\delta^{d+p}}{d+p}\right)\leq C \delta\xrightarrow{\delta\to0} 0.
\end{align*}
For the $A_2$ part, we first want to show for any $\bm{x}=(\tilde{x}, \overline{\bm{x}})\in \mathcal{R}_{-\delta}$ and $\bm{y}=(\tilde{y}, \overline{\bm{y}}) \in \mathcal{R}_{-\delta}$, we have
\begin{equation}
\label{eq:prop2.1A2}
\begin{split}
&\left| |u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}})|^p-|u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}})|^p \right| \\
&
\left\{
\begin{aligned}
&\lesssim\, \max( \delta \verti{\bm{y}-\bm{x}}^{p-1}, \delta^p),\quad \text{ when }\bm{x},\bm{y}\in (-\delta,0)\times B^{d-1}(\bm 0,2M),\\
&\lesssim\, \delta ,\quad \text{ when }\bm{x}\in (-\delta,0)\times B^{d-1}(\bm 0,M),\, \bm{y}\in {\Omega}_\delta\backslash(-\delta,0)\times B^{d-1}(\bm 0,2M) \text{ or vice verse},\\
&=0,\,\quad \text{ else}.
\end{aligned}\right.
\end{split}
\end{equation}
To show \eqref{eq:prop2.1A2}, we can first assume $ |u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}})| \geq |u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}})|$ and $u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}}) \geq 0$ without loss of generality. Then by rewriting $ u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}})$ as $ \left(u(\tilde{y},\overline{\bm{y}}) -u(0,\overline{\bm{y}}) -( u(\tilde{x},\overline{\bm{x}}) - u(0,\overline{\bm{x}}))\right) + u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}}) $ and the fact that
\[
\left| u(\tilde{y},\overline{\bm{y}}) -u(0,\overline{\bm{y}}) -( u(\tilde{x},\overline{\bm{x}}) - u(0,\overline{\bm{x}}))\right| \leq \tilde C( |\tilde{y}| + |\tilde{x}|) \leq 2 \delta \tilde C ,
\]
we can estimate $u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}})$ by two different cases where $u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}}) > 4 \delta \tilde C$ or $0\leq u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}})\leq 4 \delta \tilde C$.
If $u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}}) > 4 \delta \tilde C$, then we must have $u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}}) >0 $ and therefore
\[
\begin{split}
&|u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}})|^p-|u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}})|^p \\
\leq &Cp (|u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}}) |^{p-1}+|u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}}) |^{p-1}) \left| u(\tilde{y},\overline{\bm{y}}) -u(0,\overline{\bm{y}}) -( u(\tilde{x},\overline{\bm{x}}) - u(0,\overline{\bm{x}}))\right|\\
\lesssim &\delta \min ( |\bm{y} -\bm{x}|^{p-1} , \| u\|^{p-1}_{\infty}) \lesssim \delta \min ( |\bm{y} -\bm{x}|^{p-1} , 1).
\end{split}
\]
On the other hand if $0\leq u(0,\overline{\bm{y}}) -u(0,\overline{\bm{x}})\leq 4 \delta \tilde C$, then we have $|u(\tilde{y},\overline{\bm{y}}) -u(\tilde{x},\overline{\bm{x}})|\leq 6\delta \tilde C$. Therefore, \eqref{eq:prop2.1A2} is true and this leads to
\begin{align*}
A_2\lesssim &\left|\delta^{\,-2}\int_{(-\delta,0)\times B^{d-1}(\bm 0,2M)} \int_{(-\delta,0)\times B^{d-1}(\bm 0,2M)} \frac{ \max(\delta\verti{\bm{y}-\bm{x}}^{p-1}, \delta^p)}{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2}} \,d\bm{y} d\bm{x}\right|\\
&\hspace{3cm} +\left|\delta^{\,-2}\int_{(-\delta,0)\times B^{d-1}(\bm 0,M)} \int_{\mathcal{R}_{-\delta}\backslash (-\delta,0)\times B^{d-1}(\bm 0,2M)} \frac{\delta }{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2}} \,d\bm{y} d\bm{x}\right|\\
\lesssim &\,\delta^{\,-1}\left(\int_{(-\delta,0)\times B^{d-1}(\bm 0,2M)} \int_{B^{d-1}(\bm 0,2M)}\int_{-\delta}^0 \max\left(\frac{1}{(|\overline{\bm{y}} - \overline{\bm{x}}|\vee\delta)^{d-1}},\frac{\delta^{p-1}}{(|\overline{\bm{y}} - \overline{\bm{x}}|\vee\delta)^{d+p-2}} \right) \,d\tilde{y} d\overline{\bm{y}} d\bm{x}\right.\\
&\hspace{3cm}
+\left.\int_{(-\delta,0)\times B^{d-1}(\bm 0,M)} \int_{\mathbb{R}^{d-1}\backslash B^{d-1}(\overline{\bm x},M)}\int_{-\delta}^0 \frac{1}{|\overline{\bm y}-\overline{\bm x}|^{d+p-2}} \,d\tilde{y} d\overline{\bm{y}} d\bm{x}\right)\\
\lesssim & \, \delta(1-\log(\delta))\xrightarrow{\delta\to0} 0.
\end{align*}
{Lastly, for $A_3$ we note that
$$\verti{\overline{\bm{y}}-\overline{\bm{x}}}\leq\verti{\overline{\bm{y}}-\overline{\bm{x}}}\vee\delta\leq \verti{\bm{y}-\bm{x}}\vee\delta\leq \verti{\overline{\bm{y}}-\overline{\bm{x}}}+\delta,$$
and therefore
$$\frac{1}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}-\frac{1}{(\verti{\bm{y}-\bm{x}}\vee\delta)^{d+p-2}}\leq \frac{1}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}-\frac{1}{(|\overline{\bm{y}}-\overline{\bm{x}}|+\delta)^{d+p-2}}.$$
With the fact that
\begin{align*}
&\lim_{\delta\to0}\left(\iint_{{\mathbb{R}^{2d-2}}}\dfrac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}d\overline{\bm{y}} d\overline{\bm{x}}-\iint_{{\mathbb{R}^{2d-2}}} \frac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{(|\overline{\bm y}-\overline{\bm x}|+\delta)^{d+p-2}} \,d\overline{\bm{y}} d\overline{\bm{x}}\right)=0\,, \end{align*}
where the limits are achieved by the dominated convergence theorem, we then obtain
\begin{align*}
A_3=&\left|\int_{\mathbb{R}^{d-1}} \int_{\mathbb{R}^{d-1}}\dfrac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}d\overline{\bm{y}} d\overline{\bm{x}}-\delta^{-2}\int_{\mathbb{R}^{d-1}}\int_{-\delta}^0 \int_{\mathbb{R}^{d-1}}\int_{-\delta}^0 \frac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2}} \,d\tilde{y}d\overline{\bm{y}}d\tilde{x} d\overline{\bm{x}}\right|\\
=&\left|\delta^{-2}\int_{\mathbb{R}^{d-1}}\int_{-\delta}^0 \int_{\mathbb{R}^{d-1}}\int_{-\delta}^0\dfrac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}-\frac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{(|{\bm y}-{\bm x}|\vee\delta)^{d+p-2}} \,d\tilde{y}d\overline{\bm{y}}d\tilde{x} d\overline{\bm{x}}\right|\\
\leq&\left|\int_{\mathbb{R}^{d-1}} \int_{\mathbb{R}^{d-1}}\dfrac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{|\overline{\bm{y}}-\overline{\bm{x}}|^{d+p-2}}d\overline{\bm{y}} d\overline{\bm{x}}-\int_{\mathbb{R}^{d-1}} \int_{\mathbb{R}^{d-1}} \frac{|u(0,\overline{\bm{y}})-u(0,\overline{\bm{x}})|^p}{(|\overline{\bm y}-\overline{\bm x}|+\delta)^{d+p-2}} \,d\overline{\bm{y}} d\overline{\bm{x}}\right|\overset{\delta\rightarrow 0}{\longrightarrow} 0.
\end{align*}}
\end{proof}
\subsection{Change of Variables and Scaling Identities}\label{sec:scale}
To further understand the trace theorem and nonlocal spaces of a given $\delta$ {and the connections with existing studies in the literature, we consider some scaling identities here. We recall the notation introduced in \eqref{stripenotation} so} that for any $\bm{x}\in\mathcal{R}_{-1}^L$, we have $\delta\bm{x}\in \mathcal{R}_{-\delta}^{L\delta}$, which leads to the following scaling argument:
\begin{lem}\label{lem:scale}
Given $L>0$, for any {$u$ in ${\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^L)}$ or
$\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})$}, let $v(\bm{x}):=u(\delta\bm{x})$, then {$v$ belongs to ${\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{L/\delta})}$ or $\mathcal{T}^{\,\beta}_1(\mathcal{R}_{-1})$ and we have respectively}
\[\delta^{d-p}\verti{v}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{L/\delta})}=\verti{u}^p_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{L})},\quad \delta^{d-p}\verti{v}^p_{\mathcal{T}^{\,\beta}_1(\mathcal{R}_{-1})}=\verti{u}^p_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})},\]
\[\delta^{d}\vertii{v}^p_{L^p(\mathcal{R}_{-1}^{L/\delta})}=\vertii{u}^p_{L^p(\mathcal{R}_{-\delta}^{L})},\quad\delta^{d}\vertii{v}^p_{L^p(\mathcal{R}_{-1})}=\vertii{u}^p_{L^p(\mathcal{R}_{-\delta})}.\]
Moreover, the above results also hold for $L=\infty$:
\[\delta^{d-p}\verti{v}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})}=\verti{u}^p_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})},\quad \delta^{d}\vertii{v}^p_{L^p(\mathcal{R}_{-1}^{\infty})}=\vertii{u}^p_{L^p(\mathcal{R}_{-\delta}^{\infty})}.\]
\end{lem}
\begin{proof}
The proof is obtained by a change of variables. In particular, denoting $\bm{w}=\bm{y}/\delta$ and $\bm{z}=\bm{x}/\delta$, we have
{\begin{align}
\verti{u}^p_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{L})}=&\int_{\mathcal{R}_{-\delta}^L}\int_{\mathcal{R}_{-\delta}^L} \gamma^{\,\beta}_{\delta}(|\bm{y}-\bm{x}|)|u(\bm{y})-u(\bm{x})|^pd\bm{y} d\bm{x}\nonumber\\
=&\delta^{-d-p}\int_{\mathcal{R}_{-\delta}^L}\int_{\mathcal{R}_{-\delta}^L} \gamma^{\,\beta}_{1}\left(\frac{|\bm{y}-\bm{x}|}{\delta}\right)\left|v\left(\frac{\bm{y}}{\delta}\right)-v\left(\frac{\bm{x}}{\delta}\right)\right|^pd\bm{y} d\bm{x} \nonumber \\
=&\delta^{d-p}\int_{\mathcal{R}_{-1}^{L/\delta}}\int_{\mathcal{R}_{-1}^{L/\delta}} \gamma^{\,\beta}_{1}\left(|\bm{w}-\bm{z}|\right)|v(\bm{w})-v(\bm{z})|^p d\bm{w} d\bm{z}=\delta^{d-p}\verti{v}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{L/\delta})}.
\label{eq:scalenorm}
\end{align}
Similarly, for the trace norm we have
\begin{align*}
\verti{u}^p_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})}=&\delta^{\,\beta-2}\int_{\mathcal{R}_{-\delta}}\int_{\mathcal{R}_{-\delta}} \frac{|u(\bm y) -u(\bm x)|^p}{(|\bm y-\bm x|{\vee} \delta)^{d+p-2} (|\bm y -\bm x|\wedge\delta)^{\,\beta}} \,d\bm{y} d\bm{x}\\
=&\delta^{-d-p}\int_{\mathcal{R}_{-\delta}}\int_{\mathcal{R}_{-\delta}} \frac{\left| v\left(\frac{\bm y}{\delta}\right) -v\left(\frac{\bm x}{\delta}\right)\right|^p}{\left(\frac{|\bm y-\bm x|}{\delta}{\vee}1 \right)^{d+p-2} \left(\frac{|\bm y -\bm x|}{\delta}\wedge 1\right)^{\,\beta}} \,d\bm{y} d\bm{x}\\
=&\delta^{d-p}\int_{\mathcal{R}_{-1}}\int_{\mathcal{R}_{-1}} \frac{(v(\bm w) -v(\bm z))^p}{(|\bm w-\bm z|{\vee} 1)^{d+p-2} (|\bm w -\bm z|\wedge 1)^{\,\beta}} \,d\bm{w} d\bm{z}=\delta^{d-p}\verti{v}^p_{\mathcal{T}^{\,\beta}_1(\mathcal{R}_{-1})}.
\end{align*}}
All other identities can be proved similarly.
\end{proof}
\subsection{Equivalent Semi-norms}
We now introduce a lemma that allows us to compare the nonlocal spaces $\mathcal{S}_{\delta}^{\, \beta}$ with different sizes of $\delta$.
\begin{lem}\label{lem:EnergyKernelEst}
Let $\alpha \in (0,1]$ and $U \subseteq \mathbb{R}^d$ a convex domain. There exists $C_1 = C_1(d,p)>0$ and $C_2= C_2(d, p, \alpha)>0$ such that for any $u \in \mathcal{S}_\delta^\beta(U)$,
\[
C_1|u|_{\mathcal{S}_{\delta}^{\, \beta}(U)} \leq |u|_{\mathcal{S}_{\alpha\delta}^{\, \beta}(U)} \leq C_2 |u|_{\mathcal{S}_{\delta}^{\, \beta}(U)}.
\]
\end{lem}
\begin{proof}
First, note that
\[
|u|_{\mathcal{S}_\delta^{\, \beta}(U)}^p = \int_{U}\int_{U} \gamma^{\,\beta}_{\delta}(|\bm y-\bm x|)|u(\bm y)-u(\bm x)|^p d \bm y d \bm x .
\]
Choose an $m \in \mathbb{N}$ with $\dfrac{1}{m}<\alpha\leq \dfrac{1}{m-1}$. Notice that $m\geq 2$ for $\alpha\in (0,1]$. We split $u(\bm y)-u(\bm x)$ into the following
\[
u(\bm y)-u(\bm x)=\sum_{i=1}^{m} \left( u\left(\bm x+\frac{i}{m}(\bm y-\bm x)\right)-u\left(\bm x+\frac{i-1}{m}(\bm y-\bm x)\right)\right).
\]
Since $U$ is convex by assumption, for any $\bm x\in U$ and $\bm y\in U$, we have $\bm x+\frac{i}{m}(\bm y-\bm x) \in U$ for each $0\le i \le m$ and so $u$ is well defined at these points. Now, applying the inequality
$
\left| \sum_{i=1}^m a_i\right|^p \leq m^{p-1}\sum_{i=1}^m |a_i|^p
$,
we have
\begin{align*}
\int_{U}\int_{U}\gamma^{\,\beta}_{\delta}(|\bm y-\bm x|)|u(\bm y)-u(\bm x)|^p d \bm y d \bm x& \le m^{p-1}\sum_{i=1}^m\int_{U}\int_{U} \gamma_\delta^{\, \beta}(|\bm y-\bm x|)\left|u\left(\bm x+\frac{i}{m}(\bm y-\bm x)\right)-u\left(\bm x+\frac{i-1}{m}(\bm y-\bm x)\right)\right|^p d \bm y d \bm x.
\end{align*}
Notice that by the change of variables $\bm w= \bm x+\frac{i}{m}(\bm y-\bm x)\in U$ and $\bm z=\bm x+\frac{i-1}{m}(\bm y-\bm x)\in U$ we have $|\bm w-\bm z|= |\bm y-\bm x|/m$ and the Jacobian matrix
\begin{equation}
\label{eq:jabobina}
\frac{\partial (\bm w, \bm z)}{\partial (\bm y, \bm x)} =
\begin{pmatrix}
\frac{i}{m} I_d & (1-\frac{i}{m} ) I_d \\
\frac{i-1}{m} I_d & (1-\frac{i-1}{m}) I_d
\end{pmatrix},
\end{equation}
where $I_d\in \mathbb{R}^{d\times d}$ is the identity matrix.
Thus $|\text{det}(\partial (\bm w, \bm z)/\partial (\bm y, \bm x))|= |\text{det}((\frac{i}{m}(1-\frac{i-1}{m}) -\frac{i-1}{m}(1-\frac{i}{m}) ) I_d)| =m^{-d}$ and then
\begin{align*}
\int_{U}\int_{U}\gamma^{\,\beta}_{\delta}(|\bm y-\bm x|)|u(\bm y)-u(\bm x)|^p d \bm y d \bm x &\leq m^p \int_{U}\int_{U} \gamma_\delta^{\, \beta}(m|\bm w-\bm z|)\left|u(\bm w)-u\left(\bm z\right)\right|^p m^d d \bm w d \bm z\\
&= \int_{U}\int_{U} \frac{C_{d,p,\,\beta}}{(\delta/m)^{d+p-\beta}}\frac{1}{|\bm w-\bm z|^\beta} 1_{|\bm w-\bm z|<\delta/m}\left|u(\bm w)-u\left(\bm z\right)\right|^p d \bm w d \bm z
\\
&\leq \left(\frac{m}{m-1}\right)^{d+p-\beta} |u|^p_{\mathcal{S}_{\alpha\delta}^{\, \beta}(U)} \leq 2^{d+p-\beta} |u|^p_{\mathcal{S}_{\alpha\delta}^{\, \beta}(U)} \le {2^{d+p} |u|^p_{\mathcal{S}_{\alpha\delta}^{\, \beta}(U)}},
\end{align*}
where we have used $1/m<\alpha\leq 1/(m-1)$ and $m\geq2$. So the left half of the inequality is true with $C_1= 2^{-d/p-1}$.
Lastly, the right half of the inequality is true with $C_2 = \alpha^{-d/p-1}$, since
\[
\begin{split}
&|u|^p_{\mathcal{S}_{\alpha\delta}^{\, \beta}(U)} = \int_{U}\int_{U} \frac{C_{d,p,\,\beta}}{(\alpha\delta)^{d+p-\beta}}\frac{1}{|\bm{y} -\bm{x}|^\beta} 1_{|\bm{y}-\bm{x}|<\alpha\delta}\left|u(\bm{y})-u\left(\bm{x}\right)\right|^p d\bm{y} d\bm{x} \\
\leq &\alpha^{-d-p+\beta}\int_{U}\int_{U} \frac{C_{d,p,\,\beta}}{\delta^{d+p-\beta}}\frac{1}{|\bm{y} -\bm{x}|^\beta} 1_{|\bm{y}-\bm{x}|<\delta}\left|u(\bm{y})-u\left(\bm{x}\right)\right|^p d\bm{y} d\bm{x} \leq \alpha^{-d-p} |u|^p_{\mathcal{S}_{\delta}^{\, \beta}(U)}.
\end{split}
\]
\end{proof}
\subsection{Dyadic Cubes and Whitney Type Decomposition}\label{sec:whitney}
{The proof of Theorem \ref{mainthm_2} relies on extension results of Whitney type, the subject of which can be found in \cite{Stein1970}.
Here we focus on defining Whitney type decompositions for the half space $\mathcal{R}^\infty:=\mathbb{R}^d_{+}$ and its subdomain $\mathcal{R}^L=(0,L)\times \mathbb{R}^{d-1}$.}
For any $d\in\mathbb{Z}_+$, we define $\mathscr{Q}_d$ the collection of dyadic cubes in $\mathbb{R}^d$, i.e., the cubes of the form {$Q=2^{-k}I(\mathbf{m})$
where $k\in\mathbb{Z}$ and
$I(\mathbf{m})=((0,1]^d+\mathbf{m})$ is the shifted unit cube for
$\mathbf{m}\in\mathbb{Z}^d$}. Let $l(Q)$ denote the side length of the cube $Q\in \mathscr{Q}_d$, and $\mathscr{Q}_{d,k}$ the collection of cubes $Q\in \mathscr{Q}_d$ with $l(Q)=2^{-k}$.
For $\mathcal{R}^L= (0,L)\times \mathbb{R}^{d-1}$, we now define two types of decomposition of the domain $\mathcal{R}^L$ using the dyadic cubes for $L=2^m$ ($m\in \mathbb{Z}_+$), which will be useful in Section \ref{sec:inversetrace} to prove the inverse trace result.
For any $k\in\mathbb{Z}$, we define
$\mathscr{W}_k= \bigcup_{Q\in \mathscr{Q}_{d-1,k}}(2^{-k}, 2^{-k+1}]\times Q $.
\begin{itemize}
\item Type I decomposition. Let $L= 2^m$ for some $m\in \mathbb{Z}_+\cup \{ 0\}$ and $\overline{\mathscr{W}_0}= \bigcup_{Q\in \mathscr{Q}_{d-1,0}} (0,1]\times Q$, then
\begin{equation}\label{eq:decompositionI}
\mathscr{W}^I(\mathcal{R}^L):= \bigcup_{-m+1\leq k\leq 0,\, k\in \mathbb{Z}} \mathscr{W}_k \cup \overline{\mathscr{W}_0}
\end{equation}
\item Type II decomposition. Let $L= 2^m$ for some $m\in \mathbb{Z}_+\cup \{ 0\}$, then we define
\begin{equation} \label{eq:decompositionII}
\mathscr{W}^{II}(\mathcal{R}^L) := \bigcup_{-m+1\leq k,\, k\in\mathbb{Z}} \mathscr{W}_k
\end{equation}
\end{itemize}
Naturally, we will write $\mathscr{W}^I(\mathcal{R}^\infty)=\bigcup_{k\in \mathbb{Z}_-\cup\{0\}} \mathscr{W}_k \cup \overline{\mathscr{W}_0}$ and $\mathscr{W}^{II}(\mathcal{R}^\infty)=\bigcup_{k\in \mathbb{Z}} \mathscr{W}_k$. {Notice that $\mathscr{W}^{II}(\mathcal{R}^\infty)$ coincides with the classical Whitney decomposition of the half space, where the length of each cube is proportional to the distance between the cube and the boundary of the domain. This type of decomposition is also used to prove the classical and fractional extension results \cite{dyda2019function,koskela2017traces}. The Type I decomposition, however, has a special set $\overline{\mathscr{W}_0}$ which touches the boundary $\{0\}\times \mathbb{R}^{d-1}$ and it is used later to construct extension operator for the case $\beta<d$.}
\section{Nonlocal Trace Theorem}\label{sec:traceRd}
In this section we consider the trace theorem on half spaces and provide the proof for Theorem \ref{mainthm_1}. {We recall that the result stated corresponds to}
$\Omega=\mathcal{R}^\infty=(0,\infty)\times\mathbb{R}^{d-1}$, $\Omega_{\delta}=\mathcal{R}_{-\delta}=(-\delta,0)\times\mathbb{R}^{d-1}$ and $\hat{{\Omega}}=\mathcal{R}_{-\delta}^{\infty}$. In particular, {with the help of the scaling arguments in Lemma \ref{lem:scale},} we first prove the results for the special case $\delta=1$, then extend the results to general $\delta$.
Since $C^1(\overline{\mathcal{R}_{-\delta}^{\infty}})\cap \mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})$ forms a dense set in $\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})$, it suffices to prove the conclusion for $u\in C^1(\overline{\mathcal{R}_{-\delta}^{\infty}})\cap \mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^{\infty})$, {which is the case presented in the proofs.}
When $\delta=1$, we will prove the following theorem:
\begin{thm}[Trace theorem on half spaces when $\delta=1$]\label{mainthm_1_d1}
Let $\beta\in[0,d+p)$,
then there exist a generic constants $C$ depending only on $d$ and $p$, such that for any $u\in \mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})$ and any $\tau>0$,
\begin{align}
\label{eqn:L2result_flat_d1}\| u\|^p_{L^p(\mathcal{R}_{-1})}&\leq C |d+p-\beta|^{-1}\| u \|^{p-1}_{L^p(\mathcal{R}_{-1}^{\infty})}\verti{u}_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})}\leq C|d+p-\beta|^{-1}{\left(\tau^{-1} \| u \|^p_{L^p(\mathcal{R}_{-1}^{\infty})}+ \tau^{p-1} \verti{u}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})}\right)},\\
\label{eqn:energynormresult_flat_d1}
\verti{u}^p_{\mathcal{T}^{\,\beta}_1(\mathcal{R}_{-1})}&\leq C|d+p-\beta|^{-1} \verti{u}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})}.
\end{align}
\end{thm}
{To prove Theorem \ref{mainthm_1_d1}}, for any $\bm k=(k_1,k_2,\cdots,k_d)\in \mathbb{Z}^d$, we define the (hyper)cube associated with $\bm k$ by $I(\bm k)=(0,1)^{d+p-2}+{\bm k}= \prod_{i=1}^d (k_i, k_i+1)$. Now for any $\bm{h} = (h_1, h_2, \cdots, h_d)\in \mathbb{R}^d$, we write $ [\bm{h}] := ([h_1], [h_2], \cdots, [h_d])$, where
\[
[h_i] =
\left\{
\begin{aligned}
\lfloor h_i \rfloor \quad & \text{if } h_i\geq 0 \\
\lceil h_i \rceil \quad & \text{if } h_i< 0
\end{aligned}
\right.
\]
for $i\in \{1,2, \cdots, d \}$.
Then for all $\bm{x}\in I(\bm k)$, we have $\bm{x} +[\bm{h}]\in I(\bm k +[\bm{h}] )$. Notice that $I(\bm k +[\bm{h}])$ has non-trivial intersections with the set $\{ \bm{x}+\bm{h}: \bm{x}\in I(\bm k)\}$. Now we use $D(\bm k, [\bm{h}])$
to denote the union of all (hyper)cubes in $\mathbb{R}^d$ that have non-trivial intersections with the diagonal line from the center of $I(\bm k)$ to the center of $I(\bm k + [\bm{h}])$. Then we have the following lemma.
\begin{lem}
\label{lem:translation}
Let $u \in \mathcal{S}_1^0(\mathbb{R}^{d})$, then
for any $\bm{h}\in \mathbb{R}^d$, we have
\[
\frac{1}{\left( |\bm{h}|+1 \right)^{p-1}}\int_{I(\bm k)} |u(\bm{x}) -u(\bm{x}+\bm{h})|^p d\bm{x} \leq C \int_{D(\bm k,[\bm{h}])} \int_{|\bm{y} - \bm{x}|<d+1} |u(\bm{x}) - u(\bm{y})|^p d\bm{y} d\bm{x} ,
\]
where $C$ is a constant only dependent on $d$ and $p$.
\end{lem}
\begin{proof}
Let $m$ be the number of (hyper)cubes in the set $D(\bm k, [\bm{h}])$. Then we know that $m\leq |[\bm{h}]|_{l_1} + 1\leq |\bm{h}|_{l_1} + 1\leq \sqrt{d} |\bm{h}| + 1$. We denote these (hyper)cubes by $I(\bm k+ \bm n^{(i)})$, where $\bm n^{(i)}\in \mathbb{Z}^d$ for each $i\in \{0,1,\cdots, m-1\}$ with $\bm n^{(0)}= \bf{0}$ and $\bm n^{(m-1)}= [\bm{h}]$. Moreover, $|\bm n^{(i)} - \bm n^{(i+1)}|_{l_1} =1 $ for $i\in \{0,1,\cdots, m-2 \}$. Therefore, we can connect $\bm{x}^{(0)}:= \bm{x}$ and $\bm{x}^{(m)}:= \bm{x} + \bm{h}$ by the points $\bm{x}^{(i)}\in I(\bm k + \bm n^{(i)})$ for $i=1,2,\cdots, m-1$. Then
\[
|u(\bm{x}) -u(\bm{x}+\bm{h})|^p \leq
\left| \sum_{i=0}^{m-1} \left(u(\bm{x}^{(i)}) - u(\bm{x}^{(i+1)}) \right) \right|^p \leq m^{p-1} \sum_{i=0}^{m-1} |u(\bm{x}^{(i)}) - u(\bm{x}^{(i+1)})|^p.
\]
Now integrate the above equation with respect to $\bm{x}^{(i)}$ over $I(\bm k+ \bm n^{(i)})$ for each $i=0,1, \cdots, m-1$, we get
\[
\begin{split}
&\int_{I(\bm k)} |u(\bm{x}) - u(\bm{x}+\bm{h})|^p dx \\
\leq& m^{p-1} \left[ \sum_{i=0}^{m-2} \int_{I(\bm k+\bm n^{(i)})} \int_{I(\bm k+\bm n^{(i+1)})} \left|u(\bm{x}^{(i)}) - u(\bm{x}^{(i+1)})\right|^p d\bm{x}^{(i+1)} d\bm{x}^{(i)} + \int_{I(\bm k+\bm n^{(m-1)})}\int_{I(\bm k)} \left| u(\bm{x}^{(m-1)}) -u(\bm x+\bm{h})\right|^p d\bm{x} d\bm{x}^{(m-1)} \right] \\
\leq& m^{p-1} \sum_{i=0}^{m-1} \int_{I(\bm k + \bm n^{(i)})} \int_{B(\bm{x},d+1)} \left|u(\bm{x}) - u(\bm{y})\right|^p d\bm{y} d\bm{x}
\leq m^{p-1} \int_{D(\bm k,[\bm{h}])} \int_{B(\bm{x},d+1)} |u(\bm{x}) - u(\bm{y})|^p d\bm{y} d\bm{x} .
\end{split}
\]
where $B(\bm{x},r)$ denotes the ball of radius $r$ in $\mathbb{R}^d$. The lemma is then a result of the above estimate and the fact that $m\leq \sqrt{d} |\bm{h}| +1$.
\end{proof}
Lemma \ref{lem:translation}
has the following implication.
\begin{cor}
\label{cor:estimatebynewnorm}
For any $m\in \mathbb{Z}_+$ and any $u\in\mathcal{S}_1^0(\mathcal{R}_{-1}^m)$, we have
\begin{align}
\label{estimatebynewnorm-1}
\int_{\mathbb{R}^{d-1}} \int_{-1}^0 |u(\tilde{x}, \overline{\bm{x}}) - u(\tilde{x}+m, \overline{\bm{x}}) |^p d\tilde{x} d\overline{\bm{x}} \leq C m^{p-1} |u |^p_{\mathcal{S}_1^0(\mathcal{R}_{-1}^m)},
\end{align}
where $C$ is a constant that depends only on $d$ and $p$.
\end{cor}
\begin{proof}
For $m\in \mathbb{Z}_+$, we
use Lemma \ref{lem:translation}
with $\bm{h} = (m, 0,\cdots, 0)$ to obtain
\[
\begin{split}
\int_{\mathbb{R}^{d-1}} \int_{-1}^0 |u(\tilde{x}, \overline{\bm{x}}) - u(\tilde{x}+m, \overline{\bm{x}}) |^p d\tilde{x} d\overline{\bm{x}}
&= \sum_{\overline{\bm k}\in \mathbb{Z}^{d-1}} \int_{I(\overline{\bm k})} \int_{-1}^0 |u(\tilde{x}, \overline{\bm{x}}) - u(\tilde{x}+m, \overline{\bm{x}}) |^p d\tilde{x} d\overline{\bm{x}} \\
&\leq C (m+1)^{p-1} \sum_{\overline{\bm k}\in \mathbb{Z}^{d-1}} \int_{(-1,m)\times I(\overline{\bm k})} \int_{|\bm{y} -\bm{x}|<d+1}|u(\bm{x}) - u(\bm{y})|^p d\bm{y} d\bm{x} \\
& = C(m+1)^{p-1}\int_{\mathcal{R}_{-1}^m} \int_{|\bm{y} -\bm{x}|<d+1}|u(\bm{x}) - u(\bm{y})|^p d\bm{y} d\bm{x} \leq C m^{p-1} |u|^p_{\mathcal{S}_1^0(\mathcal{R}_{-1}^m)},
\end{split}
\]
where we have use Lemma \ref{lem:EnergyKernelEst} in the last step of the above estimate. \end{proof}
Lastly, we can show the $L^p$ estimate in Theorem \ref{mainthm_1_d1}. Note that
\begin{equation}
\label{eq:generalb}
\begin{split}
| u |^p_{\mathcal{S}^0_1(\mathcal{R}_{-1}^{\infty})}=C_{d,p,0}\int_{\mathcal{R}_{-1}^{\infty}} \int_{\mathcal{R}_{-1}^{\infty}} 1_{\{|\bm y - \bm x|<1\}} \left| u(\bm y)-u(\bm x)\right|^p d\bm y d\bm x&\leq \int_{\mathcal{R}_{-1}^{\infty}} \int_{\mathcal{R}_{-1}^{\infty}} \dfrac{C_{d,p,0}1_{\{|\bm y - \bm x|<1\}}}{| \bm y-\bm x|^{\,\beta}} \left| u(\bm y)-u(\bm x)\right|^p d\bm y d\bm x\\
&\leq C |d+p-\beta|^{-1}| u |^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})},
\end{split}
\end{equation}
where $C$ only depends on $d$ and $p$.
Therefore it suffices to prove {the $L^p$ estimate}
with $\beta=0$ and then invoke the above inequality for general $\beta$.
\begin{lem}[A nonlocal embedding lemma]\label{lem:L2d1}
For any $u\in \mathcal{S}^0_1(\mathcal{R}_{-1}^{\infty})$ and $L>0$,
\begin{equation}
\label{eqn:L2}
\| u\|^p_{L^p(\mathcal{R}_{-1})}\leq C \left[ L^{-1}\| u\|^p_{L^p(\mathcal{R}_{-1}^L)} + L^{p-1} | u|^p_{\mathcal{S}^0_1(\mathcal{R}_{-1}^L)} \right],
\end{equation}
where $C$ is a constant independent of $u$ and $L$.
\end{lem}
\begin{proof}
First, if $L<1$, then \eqref{eqn:L2} is trivially satisfied. Now we assume $L\geq 1$, then $1\leq \lfloor L \rfloor \leq L \leq 2\lfloor L \rfloor$.
For any $m\in \mathbb{Z}_+$, we have
\[
|u(\tilde{x}, \overline{\bm{x}})|^p \leq 2^{p-1} \left[ | u(\tilde{x}, \overline{\bm{x}}) - u(\tilde{x}+m, \overline{\bm{x}}) |^p + |u(\tilde{x}+m, \overline{\bm{x}})|^p \right].
\]
Therefore, combining the above inequality with Corollary \ref{cor:estimatebynewnorm}, we get
\[
\begin{split}
\| u\|^p_{L^p(\mathcal{R}_{-1})} = \int_{\mathbb{R}^{d-1}} \int_{-1}^0 |u(\tilde{x}, \overline{\bm{x}})|^p d\tilde{x} d\overline{\bm{x}} & \leq 2^{p-1} \left[ \int_{\mathbb{R}^{d-1}} \int_{-1}^0 |u(\tilde{x}, \overline{\bm{x}}) - u(\tilde{x}+m, \overline{\bm{x}}) |^p d\tilde{x} d\overline{\bm{x}} + \int_{\mathbb{R}^{d-1}} \int_{-1}^0 |u(\tilde{x}+m, \overline{\bm{x}})|^p d\tilde{x} d\overline{\bm{x}} \right] \\
&\leq C \left[ m^{p-1} | u|^p_{\mathcal{S}_1^0(\mathcal{R}_{-1}^m)} + \int_{\mathbb{R}^{d-1}} \int_{m-1}^m |u(\tilde{x}, \overline{\bm{x}})|^p d\tilde{x} d\overline{\bm{x}} \right].
\end{split}
\]
Take a summation of the above inequality for $m$ from $1$ to $\lfloor L \rfloor$, we get
\[
\lfloor L \rfloor \| u\|^p_{L^p(\mathcal{R}_{-1})} \leq C \lfloor L \rfloor^p | u|^p_{\mathcal{S}_1^0(\mathcal{R}_{-1}^{\lfloor L \rfloor})} + \|u \|^p_{L^p(\mathcal{R}_{-1}^{\lfloor L \rfloor})},
\]
which implies equation \eqref{eqn:L2}.
\end{proof}
In the following we proceed to show the proof of \eqref{eqn:energynormresult_flat_d1}. {In a similar spirit to Corollary
\ref{cor:estimatebynewnorm} but with a more refined consideration than the application of a direct H\"{o}lder's inequality, we first state an intermediate result.}
\begin{lem}\label{lem:botdiff_1}
{There exists a positive constant $C$ depending only on $d$ and $p$ such that for any positive integer $m$} and for any $
u\in\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})$,
the following estimate holds:
\begin{align*}
&\nonumber\int_{\mathcal{R}_{-1}}\left( \int_m^{m+1} |u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^p d{\bm{x}}\\
\leq &C |u |^p_{\mathcal{S}_{1}^0(\mathcal{R}_{-1}^\infty)
+C \left(\int_{-1}^{m}\left(\int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^p d\bm{w} d\bm{z}\right)^{1/p}d\tilde{l}\right)^p.
\end{align*}
\end{lem}
\begin{proof}
For $\tilde{z}\in {I}{(m)}:=(m,m+1)$, let us consider the covering of the path from $\tilde{x}$ to $\tilde{z}$ given by
$I(i):=\left({i},{i+1}\right)$ for $i=0,\cdots,m$.
We take $\tilde{z}^{(i)}\in I(i)$ for $i\in \{ 0,1,\cdots, m-1\}$ and set $\tilde{z}^{(m)}:=\tilde{z}$. Then, since
\begin{align*}
|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})|
\le |u(\tilde{x},\overline{\bm{x}})-u(\tilde{z}^{(0)},\overline{\bm{x}})|+
\sum_{i=0}^{m-1}|u(\tilde{z}^{(i)},\overline{\bm{x}})-u(\tilde{z}^{(i+1)},\overline{\bm{x}})|,
\end{align*}
integrating the above inequality over $I{(i)}$ with respect to $\tilde{z}^{(i)}$ for each $i=0,\cdots,m$, and taking both sides to the power of $p$ yields:
\begin{align*}
&\left(\int_{I{(m)}}
\nonumber|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^p \\
\le& 2^{p-1}\left(\int_{I{(0)}}|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z}^{(0)},\overline{\bm{x}})| d\tilde{z}^{(0)} \right)^p +2^{p-1} \left(\sum_{i=0}^{m-1}\int_{I{(i)}}\int_{I{(i+1)}}|u(\tilde{z}^{(i)},\overline{\bm{x}})-u(\tilde{z}^{(i+1)},\overline{\bm{x}})|d\tilde{z}^{(i+1)}d\tilde{z}^{(i)}\right)^p.
\end{align*}
Now, we integrate the above inequality over $\mathcal{R}_{-1}$ with respect to ${\bm{x}}$:
\begin{align}
\nonumber&\int_{\mathcal{R}_{-1}}\left( \int_{I{(m)}} |u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^p d{\bm{x}}
\leq2^{p-1}\int_{\mathcal{R}_{-1}}\left(\int_{I{(0)}}\verti{u(\tilde{x},\overline{\bm{x}})-u(\tilde{z}^{(0)},\overline{\bm{x}})}d\tilde{z}^{(0)}\right)^p d{\bm{x}}\\
&\qquad\qquad\qquad\quad +2^{p-1}\int_{\mathcal{R}_{-1}}\left(\sum_{i=0}^{m-1}\int_{I{(i)}}\int_{I{(i+1)}}\verti{u(\tilde{z}^{(i)},\overline{\bm{x}})-u(\tilde{z}^{(i+1)},\overline{\bm{x}})}d\tilde{z}^{(i+1)}d\tilde{z}^{(i)}\right)^p d{\bm{x}}.\label{eqn:H13A1new}
\end{align}
For the first term above,
{
\begin{align*}
\int_{\mathcal{R}_{-1}}\left(\int_{I{(0)}}\verti{u(\tilde{x},\overline{\bm{x}})-u(\tilde{z}^{(0)},\overline{\bm{x}})}d\tilde{z}^{(0)}\right)^p d{\bm{x}}\leq
& \int_{\mathbb{R}^{d-1}}\int_{-1}^0\int_{0}^{1} \verti{u(\tilde{x},\overline{\bm{x}})-u(\tilde{z}^{(0)},\overline{\bm{x}})}^p d\tilde{z}^{(0)} d{\tilde{x}}d\overline{\bm{x}}\\
=
& \int_{0}^{1} \int_{\mathbb{R}^{d-1}}\int_{-1}^0 \verti{u(\tilde{x},\overline{\bm{x}})-u(\tilde{z}^{(0)},\overline{\bm{x}})}^p
d{\tilde{x}}d\overline{\bm{x}}
d\tilde{z}^{(0)}\\
\leq & C |u |^p_{\mathcal{S}_{1}^0(\mathcal{R}_{-1}^\infty)},
\end{align*}
where the last step is obtained by Corollary
\ref{cor:estimatebynewnorm}.
}
For the second term in \eqref{eqn:H13A1new} we use the Minkowski's integral inequality \cite{schep1995minkowski} and Lemma \ref{lem:translation}:
\begin{align*}
\nonumber&\int_{\mathcal{R}_{-1}}\left(\sum_{i=0}^{m-1}\int_{I{(i)}}\int_{I{(i+1)}}\verti{u(\tilde{z}^{(i)},\overline{\bm{x}})-u(\tilde{z}^{(i+1)},\overline{\bm{x}})}d\tilde{z}^{(i+1)}d\tilde{z}^{(i)}\right)^p d{\bm{x}}\\
\leq&C\left(\sum_{i=0}^{m-1}\int_{I{(i)}}\left( \int_{\mathbb{R}^{d-1}}\left(\int_{I{(i+1)}}\verti{u(\tilde{z}^{(i)},\overline{\bm{x}})-u(\tilde{z}^{(i+1)},\overline{\bm{x}})}d\tilde{z}^{(i+1)}\right)^p d\overline{\bm{x}} \right)^{1/p} d\tilde{z}^{(i)}\right)^p\\
\leq&C\left(\sum_{i=0}^{m-1}\int_{I{(i)}}\left( \sum_{\overline{\bm{k}}\in\mathbb{Z}^{d-1}}\int_{I(\overline{\bm{k}})}\int_{I{(i+1)}}\verti{u(\tilde{z}^{(i)},\overline{\bm{x}})-u(\tilde{z}^{(i+1)},\overline{\bm{x}})}^pd\tilde{z}^{(i+1)} d\overline{\bm{x}} \right)^{1/p} d\tilde{z}^{(i)}\right)^p\\
\nonumber\leq&C\left(\int_{-1}^{m}\left( \sum_{\overline{\bm{k}}\in\mathbb{Z}^{d-1}}\int_{I(\overline{\bm{k}})}\int_{\lfloor\tilde{l}\rfloor+1}^{\lfloor\tilde{l}\rfloor+2}\verti{u(\tilde{w}+(\tilde{l}-\tilde{w}),\overline{\bm{x}})-u(\tilde{w},\overline{\bm{x}})}^pd\tilde{w} d\overline{\bm{x}}\right)^{1/p} d\tilde{l}\right)^p\\
\nonumber\leq&C\left(\int_{-1}^{m}\left(\sum_{\overline{\bm{k}}\in\mathbb{Z}^{d-1}}\int_{D((\lfloor\tilde{l}\rfloor+1,\overline{\bm{k}}),([\tilde{l}-\tilde{w}],\bm{0}))}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\right)^{1/p} d\tilde{l}\right)^p\\
\nonumber\leq&C\left(\int_{-1}^{m}\left(\sum_{\overline{\bm{k}}\in\mathbb{Z}^{d-1}}\int_{D((\lfloor\tilde{l}\rfloor+1,\overline{\bm{k}}),(-1,\bm{0}))}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z} \right)^{1/p} d\tilde{l}\right)^p\\
\nonumber\leq&C\left(\int_{-1}^{m}\left(\int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\right)^{1/p} d\tilde{l}\right)^p.
\end{align*}
In the above derivation, we have used the fact that $[\tilde{l}-\tilde{w}]=0$ or $-1$ for $\tilde{w}\in({\lfloor\tilde{l}\rfloor+1},{\lfloor\tilde{l}\rfloor+2})$ and therefore, the {corresponding sets of (hyper)cube, as defined earlier, satisfy} $D((\lfloor\tilde{l}\rfloor+1,\overline{\bm{k}}),([\tilde{l}-\tilde{w}],\bm{0}))\subset D((\lfloor\tilde{l}\rfloor+1,\overline{\bm{k}}),(-1,\bm{0}))$.
\end{proof}
\begin{remark}
{We note that the terms in the inequality of the above lemma are well defined by H\"{o}lder's inequality. Indeed, it is easy to see that
$$
\left(\int_{-1}^{m}\left( \int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^p d\bm{w} d\bm{z}\right)^{1/p} d\tilde{l}\right)^p
\leq C m^{p-1} \verti{u}^p_{\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})},
$$
for a positive constant $C$ depending only on $d$ and $p$. Our goal is, however, to show a much refined bound so that the dependence on $m$ in the above inequality can be dropped.}
\end{remark}
{Next, we derive a result that helps us to obtain an estimate related to the second term in the above lemma. For $\bm{x}=(\tilde{x},\overline{\bm{x}})$, $\bm{y}=(\tilde{y},\overline{\bm{y}})\in \mathcal{R}_{-1}$, we present some lemmas to bound the estimates on the nonlocal differences from $\bm{x}$ to $(\tilde{z},\overline{\bm{x}})$ and the nonlocal difference from $(\tilde{z},\overline{\bm{x}})$ to $(\tilde{z},\overline{\bm{y}})$, respectively.
For any $\bm{x} \in \mathcal{R}_{-1}$, we let
$\bm k(\bm{x})$ be an integer lattice point associated with $\bm{x}$ such that
$I(\bm k(\bm{x}))$ be a (hyper)cube containing $\bm{x}$. Note that the association may not be unique if
$\bm{x}$ is on the boundary of some open (hyper)cube $I(\bm k)$, integer lattice point. In such a case, we may select any of the neighboring $I(\bm k)$ to be the associated (hyper)cube. Naturally, if
$\bm{x}$ is an integer lattice point itself, we can use the default choice $I(\bm{x})$.
For $\overline{\bm{h}}\in \mathbb{R}^{d-1}$, we let
$D(\bm k(\bm{x}), [(0, \overline{\bm{h}} )])$ be the collection of (hyper)cubes associated with $\bm k(\bm{x})$ and
$\bm{h}=(0, \overline{\bm{h}} )$ defined previously.
We denote $m=m(\bm{x},\overline{\bm{h}})$ as the number of (hyper)cubes in $D(\bm k(\bm{x}), [(0, \overline{\bm{h}} )])$. We again use ${I}{(m)}:=(m,m+1)$ for any integer $m$.}
\begin{lem}\label{lem:botdiff_2_new}
{There exists a positive constant $C$ depending only on $d$ and $p$ such that for any $
u\in\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})$ {and any $\bm{x}\in\mathcal{R}_{-1}$}, the following holds:}
\begin{equation}
\int_{\mathbb{R}^{d-1}}\left(\int_{-1}^{{m(\bm{x},\overline{\bm{h}})}}\left( \int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\right)^{1/p} d\tilde{l}\right)^p \frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}
\leq C \verti{u}^p_{\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})}.
\end{equation}
\end{lem}
\begin{proof}
{we
write $\overline{\bm{h}}$ in the spherical coordinate to get}
\begin{align*}
\nonumber&\int_{\mathbb{R}^{d-1}}\left(\int_{-1}^{
{m(\bm{x},\overline{\bm{h}})}
}\left( \int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\right)^{1/p}d\tilde{l}\right)^p\frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}\\
\nonumber\leq& C\int_{0}^{\infty}\left(\int_{-1}^{(r+1)\sqrt{d-1}-1}\left( \int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z} \right)^{1/p} d\tilde{l}\right)^pr^{d-2}\frac{dr}{\left(r+1\right)^{d+p-2}}\\
\nonumber\leq& C\int_{0}^{\infty}\left(\int_{0}^{\hat{r}}\left(\int_{(\lfloor\hat{l}\rfloor-1,\lfloor\hat{l}\rfloor+1)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\right)^{1/p}d\hat{l}\right)^p\frac{d\hat{r}}{\hat{r}^p}\\
\nonumber\leq& C\int_{0}^{\infty} \hat{r}^{-p} \left(\int_{0}^{\hat{r}}
f(\hat{l})
d\hat{l} \right)^p
d\hat{r}
\end{align*}
where
$$\hat{l}:=\tilde{l}+1,\; \hat{r}:=(r+1)\sqrt{d-1},\;
f(\hat{l}):=
\left( \int_{(\lfloor\hat{l}\rfloor-1,\lfloor\hat{l}\rfloor+1)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z} \right)^{1/p},
$$
and we have used the fact that ${m(\bm{x},\overline{\bm{h}})} \leq |\overline{\bm{h}}|_{l_1}+1\leq \sqrt{d-1}|\overline{\bm{h}}|+1\leq \sqrt{d-1}(|\overline{\bm{h}}|+1)-1$.
Using the Hardy's inequality \cite{hardy1952inequalities,avkhadiev2006hardy,avkhadiev2014hardy}:
{$$\int_0^\infty x^{-p}\left(\int_{0}^x f(y)dy\right)^pdx\leq \left(\frac{p}{p-1}\right)^p\int_0^\infty (f(y))^pdy,$$}
we get
\begin{align}
\nonumber&\int_{\mathbb{R}^{d-1}}\left(\int_{-1}^{
{m(\bm{x},\overline{\bm{h}})}
}\left(\int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\right)^{1/p}d\tilde{l}\right)^p\frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}\\
\nonumber\leq& C\int_{0}^{\infty}\int_{(\lfloor\hat{l}\rfloor-1,\lfloor\hat{l}\rfloor+1)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z} d\hat{l}\\
\leq&C\int_{\mathcal{R}_{-1}^\infty}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\leq C |{u}|^p_{\mathcal{S}^0_{d+1}(\mathcal{R}_{-1}^\infty)}\leq C|{u}|^p_{\mathcal{S}^0_{1}(\mathcal{R}_{-1}^\infty)}.\label{eqn:lemma39p2}
\end{align}
\end{proof}
With the above lemma, we can then bound the nonlocal difference from $\bm{x}=(\tilde{x},\overline{\bm{x}})$ to $(\tilde{z},\overline{\bm{x}})$ with the trace semi-norm.
Such an estimate can be seen as the norm of the nonlocal variations along the normal direction (with respect to the strip domain) being controlled by the nonlocal semin-norm for $\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})$.
\begin{lem}\label{lem:botdiff_2}
There exists a positive constant $C$ depending only on $d$ and $p$ such that {for any $
u\in\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})$, the following holds:}
\begin{equation}
\int_{\mathbb{R}^{d-1}}\int_{\mathcal{R}_{-1}} \left( \int_{{I(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^p d{\bm{x}} \frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}} \leq C \verti{u}^p_{\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})}.
\end{equation}
\end{lem}
\begin{proof}
With Lemma \ref{lem:botdiff_1} we have
\begin{align*}
\nonumber&\int_{\mathbb{R}^{d-1}}\int_{\mathcal{R}_{-1}}\left( \int_{I{(m(\bm{x},\overline{\bm{h}})
)}} |u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^pd{\bm{x}}\frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}\\
\leq&
|u |^p_{\mathcal{S}_{1}^0(\mathcal{R}_{-1}^\infty)}
+C \int_{\mathbb{R}^{d-1}}\left(\int_{-1}^{{m(\bm{x},\overline{\bm{h}})}}\left(\int_{(\lfloor\tilde{l}\rfloor,\lfloor\tilde{l}\rfloor+2)\times\mathbb{R}^{d-1}}\int_{B(\bm{z}, d+1)\cap\mathcal{R}_{-1}^\infty}\verti{u(\bm{w})-u(\bm{z})}^pd\bm{w} d\bm{z}\right)^{1/p} d\tilde{l}\right)^p\frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}.
\end{align*}
{Now, combining with Lemma \ref{lem:botdiff_2_new}, the proof is complete.}
\end{proof}
\begin{remark}
Apply a similar argument to the nonlocal difference from $\bm{y}=(\tilde{y},\overline{\bm{y}})$ to $(\tilde{z},\overline{\bm{y}})$, we can similarly obtain:
\begin{equation}
\int_{\mathbb{R}^{d-1}}\int_{\mathcal{R}_{-1}}\left( \int_{{I{(m
(\bm{y},\overline{\bm{h}})
)}}
}|u(\tilde{y},\overline{\bm{y}})-u(\tilde{z},\overline{\bm{y}})| d\tilde{z} \right)^p d {\bm{y}} \frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}} \leq C \verti{u}^p_{\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})}
\end{equation}
where $C$ is a positive constant depending only on $d$ and $p$.
\end{remark}
We now proceed to investigate the nonlocal difference from $(\tilde{z},\overline{\bm{x}})$ to $(\tilde{z},\overline{\bm{y}})$ in the following lemma,
which can be seen analogously as the norm of the nonlocal tangential variations being controlled by the nonlocal semin-norm:
\begin{lem}\label{lem:pardiff}
There exists a positive constant $C$ depending on $d$ and $p$ such that for
{any $u\in\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})$
}, the following estimate holds:
\begin{equation}
\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}} \int_{{I{(m(\bm{x},\overline{\bm{h}}))}}}|u(\tilde{z},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}}+\overline{\bm{h}})|^p d\tilde{z} d\overline{\bm{x}} \frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}} \leq C \verti{u}^p_{\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})}.
\end{equation}
\end{lem}
\begin{proof}
{For notation simplicity, we drop the dependence of $m$ on its argument in the derivation here.}
First notice that from Lemma \ref{lem:translation} we have
\[
\begin{split}
\int_{\mathbb{R}^{d-1}}
\int_{I{(m)}} |u(\tilde{z}, \overline{\bm{x}}) - u(\tilde{z}, \overline{\bm{x}}+\overline{\bm{h}})|^p d\tilde{z} d\overline{\bm{x}} & = \sum_{\overline{\bm k}\in \mathbb{Z}^{d-1}}\int_{I(\overline{\bm k})}
\int_{I{(m)}} |u(\tilde{z}, \overline{\bm{x}}) - u(\tilde{z}, \overline{\bm{x}}+\overline{\bm{h}})|^p d\tilde{z} d\overline{\bm{x}} \\
&\leq C (|\overline{\bm{h}}|+1)^{p-1}\sum_{\overline{\bm k}\in \mathbb{Z}^{d-1}}\int_{D((m, \overline{\bm k}), (0,[\overline{\bm{h}}]))}
\int_{B(\bm{z},d+1)\cap \mathcal{R}_{-1}^\infty} |u(\bm{w}) - u(\bm{z})|^p d\bm{w} d\bm{z} \\
&\leq C (|\overline{\bm{h}}|+1)^p \int_{I{(m)}\times \mathbb{R}^{d-1}}
\int_{B(\bm{z},d+1)\cap \mathcal{R}_{-1}^\infty} |u(\bm{w}) - u(\bm{z})|^p d\bm{w} d\bm{z}.
\end{split}
\]
The last inequality in the above estimate is due to the fact that there are at most $|[\overline{\bm{h}}]|_{l_1} + 1 \leq \sqrt{d-1}|\overline{\bm{h}}| + 1 $ (hyper)cubes in the set $D((m, \overline{\bm k}), (0,[\overline{\bm{h}}]))$. Therefore,
\[
\begin{split}
&\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}}
\int_{I{(m)}} |u(\tilde{z}, \overline{\bm{x}}) - u(\tilde{z}, \overline{\bm{x}}+\overline{\bm{h}})|^p d\tilde{z} d\overline{\bm{x}} \frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}} \\
\leq& C \int_{\mathbb{R}^{d-1}} \frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d-2}} \int_{I{(m)}\times \mathbb{R}^{d-1}}
\int_{B(\bm{z},d+1)\cap \mathcal{R}_{-1}^\infty} |u(\bm{w}) - u(\bm{z})|^p d\bm{w} d\bm{z} \\
\leq& C \int_0^\infty \frac{r^{d-2}}{(r+1)^{d-2}}dr \int_{I{(m)}\times \mathbb{R}^{d-1}}
\int_{B(\bm{z},d+1)\cap \mathcal{R}_{-1}^\infty} |u(\bm{w}) - u(\bm{z})|^p d\bm{w} d\bm{z} \leq C |u|^p_{\mathcal{S}_1^0(\mathcal{R}_{-1}^\infty)},
\end{split}
\]
where we have also used Lemma \ref{lem:EnergyKernelEst} in the last inequality.
\end{proof}
We now have the following lemma for the trace semi-norm:
\begin{lem}\label{lem:Td1}
There exist a positive constant $C$ depending only $d$ and $p$ such that for any $u\in \mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})$,
\begin{equation}
\label{eqn:T}
\verti{u}^p_{\mathcal{T}^{\,\beta}_1(\mathcal{R}_{-1})}\leq C |d+p-\beta|^{-1}\verti{u}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})}\,.
\end{equation}
\end{lem}
\begin{proof}
We first note that
\begin{align*}
&\int_{\mathcal{R}_{-1}}\int_{\mathcal{R}_{-1}\cap B(\bm{x},1)} \frac{|u(\bm y) -u(\bm x)|^p}{(|\bm y-\bm x|\vee 1)^{d+p-2} (|\bm y -\bm x|\wedge1)^{\,\beta}} \,d\bm{y} d\bm{x}=\int_{\mathcal{R}_{-1}}\int_{\mathcal{R}_{-1}\cap B(\bm{x},1)} \frac{|u(\bm y) -u(\bm x)|^p}{|\bm y -\bm x|^{\,\beta}} \,d\bm{y} d\bm{x}\leq C |d+p-\beta|^{-1}|{u}|^p_{\mathcal{S}^{\,\beta}_{1}(\mathcal{R}_{-1}^\infty)}.
\end{align*}
Moreover, we note that $\verti{\overline{\bm{y}}-\overline{\bm{x}}}+1\leq 2\verti{\bm{y}-\bm{x}} =2 (\verti{\bm{y}-\bm{x}}\vee 1)$ for $\bm{y}\in\mathcal{R}_{-1}\cap (\mathcal{R}_{-1}\backslash B(\bm{x},1))$, it then suffices to show
\begin{align*}
\underset{\mathcal{R}_{-1}}{\iint}\underset{{\mathcal{R}_{-1}\backslash B(\bm{x},1)}}{\iint} \frac{|u(\tilde{y},\overline{\bm y}) -u(\tilde{x},\overline{\bm x})|^p}{(|\overline{\bm y}-\overline{\bm x}|+1)^{d+p-2} } \,d\tilde{y} d\overline{\bm{y}}d\tilde{x} d\overline{\bm{x}}&\leq \underset{\mathcal{R}_{-1}}{\iint} \underset{\mathcal{R}_{-1}}{\iint}\frac{|u(\tilde{y},\overline{\bm y}) -u(\tilde{x},\overline{\bm x})|^p}{(|\overline{\bm y}-\overline{\bm x}|+1)^{d+p-2} } \,d\tilde{y} d\overline{\bm{y}}d\tilde{x} d\overline{\bm{x}} \leq C |u|^p_{\mathcal{S}_1^0(\mathcal{R}_{-1}^\infty)}\,,
\end{align*}
since $|u|^p_{\mathcal{S}_1^0(\mathcal{R}_{-1}^\infty)}\leq C|d+p-\beta|^{-1} |{u}|^p_{\mathcal{S}^{\,\beta}_{1}(\mathcal{R}_{-1}^\infty)}$ for any $\beta\in [0,d+p)$ where $C$ only depends on $d$ and $p$. Taking the path from $\bm{x}$ to $(\tilde{z},\overline{\bm{x}})$, $(\tilde{z},\overline{\bm{y}})$ and then finally $\bm{y}$, we have
\begin{align}
\nonumber|u(\tilde{x},\overline{\bm{x}})-u(\tilde{y},\overline{\bm{y}})| \le
|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})|+
|u(\tilde{y},\overline{\bm{y}})-u(\tilde{z},\overline{\bm{y}})|
+|u(\tilde{z},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{y}})|.
\end{align}
Denoting $\overline{\bm{h}}:=\overline{\bm{y}}-\overline{\bm{x}}$ and
integrating the above over $I{(m(\bm{x},\overline{\bm{h}}))}$ with respect to $\tilde{z}$ and taking both hand sides to the power of $p$ yields:
\begin{align}
\nonumber|u(\tilde{x},\overline{\bm{x}})-u(\tilde{y},\overline{\bm{y}})|^p \le & 3^{p-1} \left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}
|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^p+3^{p-1} \left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{y},\overline{\bm{y}})-u(\tilde{z},\overline{\bm{y}})| d\tilde{z} \right)^p\\
\nonumber&+3^{p-1} \left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{z},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{y}})|d\tilde{z} \right)^p.
\label{eqn:H12A0}
\end{align}
{Considering a fixed $\overline{\bm{h}}=\overline{\bm{y}}-\overline{\bm{x}}$, } we integrate the above inequality over $(-1,0)$ with respect to $\tilde{x}$, $\tilde{y}$, respectively, then integrate over $\mathbb{R}^{d-1}$ with respect to $\overline{\bm{x}}$:
\begin{align}
\nonumber
&\int_{\mathbb{R}^{d-1}} \int_{-1}^0 \int_{-1}^0
|u(\tilde{x},\overline{\bm{x}})-u(\tilde{y},\overline{\bm{y}})|^p d\tilde{y} d \tilde{x} d\overline{\bm{x}}\\
\nonumber\le&3^{p-1} \int_{\mathbb{R}^{d-1}} \int_{-1}^0\left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^p d\tilde{x} d\overline{\bm{x}} + 3^{p-1} \int_{\mathbb{R}^{d-1}} \int_{-1}^0 \left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{y},\overline{\bm{y}})-u(\tilde{z},\overline{\bm{y}})| d\tilde{z} \right)^p d\tilde{y} d\overline{\bm{x}}\\
\nonumber & + 3^{p-1} \int_{\mathbb{R}^{d-1}} \left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{z},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{y}})|d\tilde{z} \right)^p d\overline{\bm{x}}.
\end{align}
Multiplying the above inequalities with $\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}$ and integrating with respect to $\overline{\bm{h}}$ over $\mathbb{R}^{d-1}$ yield:
\begin{align*}
&\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}}\int_{-1}^{0}\int_{-1}^{0}\dfrac{\verti{u(\tilde{x},\overline{\bm{x}})-u(\tilde{y},\overline{\bm{y}})}^p}{\left(\verti{\overline{\bm{y}}-\overline{\bm{x}}}+1\right)^{d+p-2}} d\tilde{y} d\tilde{x} d\overline{\bm{x}}d\overline{\bm{y}}\\
\leq&C\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}} \int_{-1}^0\left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{x},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{x}})| d\tilde{z} \right)^p d\tilde{x} d\overline{\bm{x}}\frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}\\
&+C\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}} \int_{-1}^0 \left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{y},\overline{\bm{x}}+\overline{\bm{h}})-u(\tilde{z},\overline{\bm{x}}+\overline{\bm{h}})| d\tilde{z} \right)^p d\tilde{y} d\overline{\bm{x}}\frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}\\
&+C\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}^{d-1}} \left( \int_{I{(m(\bm{x},\overline{\bm{h}}))}}|u(\tilde{z},\overline{\bm{x}})-u(\tilde{z},\overline{\bm{y}})|d\tilde{z} \right)^p d\overline{\bm{x}}\frac{d\overline{\bm{h}}}{\left(\verti{\overline{\bm{h}}}+1\right)^{d+p-2}}\leq C \verti{u}^p_{\mathcal{S}^{\,0}_1(\mathcal{R}_{-1}^{\infty})},
\end{align*}
where the last inequality {follows} immediate from Lemmas \ref{lem:botdiff_2}-\ref{lem:pardiff}.
\end{proof}
We now show the proof of Theorem \ref{mainthm_1_d1} and Theorem \ref{mainthm_1}.
\noindent{\it Proof of Theorem \ref{mainthm_1_d1} and Theorem \ref{mainthm_1}.}
From Lemma \ref{lem:L2d1}, we have
\[
\| u\|^p_{L^p(\mathcal{R}_{-1})}\leq C L^{-1}\| u \|^p_{L^p(\mathcal{R}_{-1}^\infty)}+CL^{p-1}\verti{u}^p_{\mathcal{S}^0_1(\mathcal{R}_{-1}^\infty)}\leq C |d+p-\beta|^{-1}(L^{-1}\| u \|^p_{L^p(\mathcal{R}_{-1}^\infty)}+L^{p-1}\verti{u}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^\infty)})\,,
\]
for any $L>0$.
By taking $L=\| u \|_{L^p(\mathcal{R}_{-1}^{\infty})}/\verti{u}_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})}$ for $u\neq 0$ in the above inequality, we obtain \eqref{eqn:L2result_flat_d1}.
\eqref{eqn:energynormresult_flat_d1} is an immediate result in Lemma \ref{lem:Td1}.
The proof of the general nonlocal trace Theorem \ref{mainthm_1} on half spaces then follows from Theorem \ref{mainthm_1_d1} and the scaling argument in Lemma \ref{lem:scale}: for any $u\in \mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)$ and let $v(\bm{x})=u(\delta\bm{x})$, then $v\in {\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^{\infty})}$ and
\begin{align*}
\frac{1}{\delta}\| u\|^p_{L^p(\mathcal{R}_{-\delta})}&=\delta^{d-1}\| v\|^p_{L^p(\mathcal{R}_{-1})}\leq C |d+p-\beta|^{-1}\delta^{d-1}\| v \|^{p-1}_{L^p(\mathcal{R}_{-1}^\infty)}\verti{v}_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^\infty)}\\
&=C |d+p-\beta|^{-1}\| u \|^{p-1}_{L^p(\mathcal{R}_{-\delta}^\infty)}\verti{u}_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)}\leq C |d+p-\beta|^{-1}\left(\| u \|^p_{L^p(\mathcal{R}_{-\delta}^\infty)}+\verti{u}^p_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)}\right).\\
| u|^p_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})}&=\delta^{d-p}\verti{v}^p_{\mathcal{T}^{\,\beta}_1(\mathcal{R}_{-1})}\leq C|d+p-\beta|^{-1} \delta^{d-p}\verti{v}^p_{\mathcal{S}^{\,\beta}_1(\mathcal{R}_{-1}^\infty)}=C|d+p-\beta|^{-1}\verti{u}^p_{\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)}.
\end{align*}
\qed
\begin{remark}
{We have proved in Lemma \ref{lem:L2d1} that $\vertii{u}_{L^p(\mathcal{R}_{-1})}$ is bounded by the $L^p$ norm and the $\mathcal{S}_1^{\,\beta}$ semi-norm on a stripe domain ${\mathcal{R}_{-1}^L}$, which {might appear as a stronger statement that implies the result on a half space ${\mathcal{R}_{-1}^\infty}$ in Theorem \ref{mainthm_1_d1}. Though, we note that the result on a stripe domain ${\mathcal{R}_{-1}^L}$ can also be a consequence of Theorem \ref{mainthm_1_d1}.
Likewise,} a bound on the ${\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}$ semi-norm can also be obtained on a stripe domain ${\mathcal{R}_{-1}^L}$:
$$\verti{u}^p_{\mathcal{T}^{\,\beta}_1(\mathcal{R}_{-1})}\leq C|d+p-\beta|^{-1}\left(|u|^p_{\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^{L})} + L^{-p}\|u\|_{L^p(\mathcal{R}_{-1}^{L})}^p\right).$$
{To verify the above conclusions,
let us derive the results on the stripe domain from \eqref{eqn:L2result_flat_d1} and \eqref{eqn:energynormresult_flat_d1}.}
We take a smooth cutoff function $\phi(x)\in C_c^{\infty}(\mathbb{R})$ such that supp$(\phi)\subset [-1,1/2]$ and $\phi(x)=1$ for $x\leq 0$. Denoting $\tilde{\phi}(\bm{x}):\mathbb{R}^d\rightarrow \mathbb{R}$ such that $\tilde{\phi}(\tilde{x},\overline{\bm{x}})=\phi(\tilde{x}/L)$ for $\tilde{x}>0$ and $\tilde{\phi}(\tilde{x},\overline{\bm{x}})=\phi(\tilde{x})$ for $\tilde{x}<0$, then we note that there exists a generic constant $C$ independent of $L$ and $\verti{\tilde{\phi}}_{C^0}\leq C$, $\verti{\tilde{\phi}'}_{C^0}\leq CL^{-1}$. Substituting $\tilde{\phi}u$ into \eqref{eqn:energynormresult_flat_d1}, we have
\[
\begin{split}
&|u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}=|\tilde{\phi}u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}\leq C |d+p-\beta|^{-1} |\tilde{\phi}u|^p_{\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^\infty)} \\
\leq & C|d+p-\beta|^{-1} \left(\int_{\mathcal{R}_{-1}^{L}}\int_{\mathcal{R}_{-1}^{L}}+ 2\int_{\mathcal{R}_{-1}^{L/2}}\int_{\mathcal{R}^\infty\backslash\mathcal{R}^{L}}\right) \gamma^{\,\beta}_1(|\bm x-\bm y|)|\tilde{\phi}(\bm y)u(\bm y)-\tilde{\phi}(\bm x)u(\bm x)|^pd\bm y d\bm x \\
\leq & C|d+p-\beta|^{-1} \int_{\mathcal{R}_{-1}^{L}}\int_{\mathcal{R}_{-1}^{L}}\gamma^{\,\beta}_1(|\bm x-\bm y|)|\tilde{\phi}(\bm y)|^p|u(\bm y)-u(\bm x)|^p+\gamma^{\,\beta}_1(|\bm x-\bm y|)|u(\bm x)|^p|\tilde{\phi}(\bm y)-\tilde{\phi}(\bm x)|^p d\bm y d\bm x \\
&+ C|d+p-\beta|^{-1} \int_{\mathcal{R}_{-1}^{L/2}} |\tilde{\phi}(\bm x)u(\bm x)|^p \int_{L/2<|\bm y-\bm x|<1} \gamma^{\,\beta}_1(|\bm x-\bm y|) d\bm y d\bm x \\
\leq&C|d+p-\beta|^{-1}\left(\|\tilde{\phi}\|^{p}_{C^0}|u|^p_{\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^{L})} +\|\tilde{\phi}'\|^{p}_{C^0}\int_{\mathcal{R}_{-1}^{L}}|u(\bm x)|^p\int_{\mathcal{R}_{-1}^{L} \cap B(\bm x,1)}\gamma^{\,\beta}_1(|\bm x-\bm y|)|\bm x-\mathbf{y}|^pd\bm y d\bm x\right.\\
&\left.+ \| \tilde{\phi}\|_{C^0}^p \| u\|_{L^p(\mathcal{R}_{-1}^{L/2})}^p \int_{L/2<|\bm z|<1} \frac{1}{|\bm z|^\beta} d\bm z\right)\\
\leq & C|d+p-\beta|^{-1}\left( |u|^p_{\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^{L})} +{L^{-p}}\|u\|_{L^p(\mathcal{R}_{-1}^{L})}^p + \| \tilde{\phi}\|_{C^0}^p \| u\|_{L^p(\mathcal{R}_{-1}^{L/2})}^p \int_{L/2<|\bm z|<1} \frac{1}{|\bm z|^\beta} d\bm z\right).
\end{split}
\]
Notice that
\[
\int_{L/2<|\bm z|<1} \frac{1}{|\bm z|^\beta} d\bm z=0 \quad\text{ if } L\geq2,\]
otherwise
\[
\int_{L/2<|\bm z|<1} \frac{1}{|\bm z|^\beta} d\bm z \leq C L^{d-\beta} \leq C L^{-p}
\]
since $d-\beta > -p$.
So we have $|u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}\leq C|d+p-\beta|^{-1}\left( |u|^p_{\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^{L})} +L^{-p}\|u\|_{L^p(\mathcal{R}_{-1}^{L})}^p\right) $ for all $L>0$.
} Similarly, substituting $\tilde{\phi}u$ into \eqref{eqn:L2result_flat_d1} for $\beta=0$ yields
\begin{align*}
&\| u\|^p_{L^p(\mathcal{R}_{-1})}=\|\tilde{\phi} u\|^p_{L^p(\mathcal{R}_{-1})}\leq C \| \tilde{\phi}u \|^{p-1}_{L^p(\mathcal{R}_{-1}^{\infty})}\verti{\tilde{\phi}u}_{\mathcal{S}^{0}_1(\mathcal{R}_{-1}^{\infty})}\leq C\|u\|^{p-1}_{L^p(\mathcal{R}_{-1}^{L})}\left(|u|_{\mathcal{S}_1^{0}(\mathcal{R}_{-1}^{L})}+L^{-1}\|u\|_{L^p(\mathcal{R}_{-1}^{L})}\right)\\
&\leq CL^{-1}\| u \|^p_{L^p(\mathcal{R}_{-1}^L)}+C \| u \|^{p-1}_{L^p(\mathcal{R}_{-1}^L)}\verti{u}_{\mathcal{S}^0_1(\mathcal{R}_{-1}^L)}\leq CL^{-1}\| u \|^p_{L^p(\mathcal{R}_{-1}^L)}+C L^{p-1}\verti{u}^p_{\mathcal{S}^0_1(\mathcal{R}_{-1}^L)}.
\end{align*}
\end{remark}
\section{Nonlocal Inverse Trace Theorem}
\label{sec:inversetrace}
For $u: \mathcal{R}_{-1}\to\mathbb{R}$ and $L=2^m$ ($m\in\mathbb{Z}_+\cup\{0\}$), we now define the extension operator $E^L u: \mathcal{R}_{-1}^\infty\to\mathbb{R}$.
Notice that the kernel ${\gamma}^{\,\beta}_1$ is defined in \eqref{def:frackernel} with the two cases $\beta\in [0,d)$ and $\beta\in (d,d+p)$. We have the following two cases for the definition of $E^L$.\\
Case 1: $\beta\in [0,d)$. We define a partition of unity for $\mathcal{R}^L$ according to the decomposition $\mathscr{W}^I(\mathcal{R}^L)$ defined in \eqref{eq:decompositionI}. For any $W\in \mathscr{W}^I(\mathcal{R}^L)$, let $\phi_W^I: \mathbb{R}^d_+ \to [0,1]$ be a smooth function associated with $W$ such that $\phi_W^I$ is bounded below uniformly on $W$, Lip$\phi_W^I \lesssim 1/l(W)$ and supp$(\phi_W^I)$ is contained in an $l(W)/4$-neighborhood of $W$. Moreover, $\sum_{W\in \mathcal{W}^I(\mathcal{R}^L)} \phi_W^I \equiv 1$ on $\mathcal{R}^L$. Notice that $\{ \phi^I_W\}$ should also depend on $L$ and here we drop the $L$ dependence for simplicity of notations. The extension operator is then defined as
\begin{equation}\label{def:extensionop_1}
E^L u (\bm x) =
\left\{
\begin{aligned}
\sum_{W\in \mathscr{W}^I(\mathcal{R}^L)} a_W^I \phi_W^I(\bm x) \quad &\bm x\in \mathcal{R}^\infty \,, \\
u(\bm x) \quad &\bm x\in \mathcal{R}_{-1}\,,
\end{aligned}
\right.
\end{equation}
where $$a_W^I:= \left(\int_{\mathscr{M}_1(W)} u \right) \big/ |\mathscr{M}_1(W)|,$$
and the map $\mathscr{M}_1$ is defined for any $W=(a, b]\times Q \in \mathscr{W}^I(\mathcal{R}^L) $ as
\begin{equation} \label{def:map_M1}
\mathscr{M}_1(W)= (-1, 0)\times Q\,. \end{equation}
Case 2: $\beta\in (d,d+p)$. We similarly define $\{ \phi_W^{II} \}_{W\in \mathscr{W}^{II}(\mathcal{R}^L)}$ as a partition of unity for $\mathcal{R}^L$ according to $\mathscr{W}^{II}(\mathcal{R}^L)$ defined in \eqref{eq:decompositionII}. More specifically, for any $W\in \mathscr{W}^{II}(\mathcal{R}^L)$, $\phi_W^{II}: \mathbb{R}^d_+ \to [0,1]$ is a smooth function bounded below uniformly on $W$, Lip$\phi_W^{II} \lesssim 1/l(W)$ and supp$(\phi_W^{II})$ is contained in an $l(W)/4$-neighborhood of $W$. Moreover, $\sum_{W\in \mathcal{W}^{II}(\mathcal{R}^L)} \phi_W^{II} \equiv 1$ on $\mathcal{R}^L$. Then the extension operator is given by
\begin{equation} \label{def:extensionop_2}
E^L u (\bm x) =
\left\{
\begin{aligned}
\sum_{W\in \mathscr{W}^{II}(\mathcal{R}^L)} a_W^{II} \phi_W^{II}(\bm x) \quad &\bm x\in\mathcal{R}^\infty \,, \\
u(\bm x) \quad &\bm x\in \mathcal{R}_{-1} \,,
\end{aligned}
\right.
\end{equation}
where
$$a_W^{II}:= \left(\int_{\mathscr{M}_2(W)} u \right) \big/ |\mathscr{M}_2(W)|,
$$
and the map $\mathscr{M}_2$ for any $W=(a, b]\times Q \in \mathscr{W}^{II}(\mathcal{R}^L)$ as
\begin{equation} \label{def:map_M2}
\mathscr{M}_2(W)=
\left\{
\begin{aligned}
& (-b, -a)\times Q \quad &\text{if } b\leq 1 \,,\\
& (-1, 0) \times Q \quad &\text{if } b> 1 \,.
\end{aligned}
\right.
\end{equation}
Notice that for such $W$ in $\mathscr{W}^{II}(\mathcal{R}^L)$, if $b>1$, then we have $a\geq 1$ and $|b-a|\geq 1$.
\begin{remark}
The two types of extensions in \eqref{def:extensionop_1} and \eqref{def:extensionop_2} work for $\beta\in [0,d)$ and $\beta\in(d,d+p)$ respectively. Notice that if $u\in C(\overline{\mathcal{R}_1})$, then the extension in \eqref{def:extensionop_2} gives a continuously function across the boundary $\partial\mathcal{R}^\infty$ to have necessary regularity. On the other hand, the extended function in \eqref{def:extensionop_1} is discontinuous across $\partial\mathcal{R}^\infty$.
Such an extension is fine in this case, since $\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^\infty)$ is equivalent to $L^p(\mathcal{R}_{-1}^\infty)$
for $\beta\in [0,d)$
, and it accepts discontinuous functions.
We also note that the map in \eqref{def:map_M2} characterizes two regimes -- the ``fractional regime'', where any cube in $\mathcal{R}^1$ is mapped to its symmetric reflection in $\mathcal{R}_{-1}$, and the ``classical regime'', where any cube in $\mathcal{R}^\infty\backslash\mathcal{R}^1$ is mapped to a (hyper)rectangle in $\mathcal{R}_{-1}$. Related discussions on extension operators for the fractional and classical Sobolev spaces using Whitney decompositions can be found in \cite{dyda2019function} and \cite{koskela2017traces}.
\end{remark}
\begin{thm}
\label{thm:extensionL}
For any $L=2^m$ ($m\in \mathbb{Z}_+$), let $E^L$ be the extension operator defined in \eqref{def:extensionop_1} for $\beta\in[0,d)$ or in \eqref{def:extensionop_2} for $\beta\in(d,d+p)$, then $E^L:\mathcal{T}_{1}^{\,\beta}(\mathcal{R}_{-1}) \to \mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^\infty)$ and
\begin{align}
\label{eq:EL1}&\|E^L u\|^p_{L^p(\mathcal{R}_{-1}^\infty)} \leq C L\| u\|^p_{L^p(\mathcal{R}_{-1})} \\
\label{eq:EL2}& |E^L u|^p_{\mathcal{S}_{1}^{\,\beta}(\mathcal{R}_{-1}^\infty)} \leq C \left( L^{-(p-1)}\| u\|^p_{L^p(\mathcal{R}_{-1})} +|\beta-d|^{-1} |u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})} \right)
\end{align}
where $C$ is a constant independent of $L$, $\beta$ and $u\in \mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})$.
\end{thm}
\begin{proof}
We first take the case $\beta\in [0,d)$, where $E^L$ is defined in \eqref{def:extensionop_1}. Notice that by the construction of $\{\phi_W^I\}_{W\in \mathscr{W}^{I}(\mathcal{R}^L)}$ the support of each $\phi_W^I$ overlaps with only a finite number of the supports of other function.
For the $L^p$ estimate, we have
\[
\begin{split}
\int_{\mathcal{R}^\infty} |E^L u|^p &= \int_{\mathcal{R}^\infty} \left| \sum_{W\in \mathscr{W}^{I}(\mathcal{R}^L)} a_W^I \phi_W^I(\bm x) \right|^p d\bm x
\lesssim \int_{\mathcal{R}^\infty} \sum_{W\in\mathscr{W}^{I}(\mathcal{R}^L)} \left|a_W^I\right|^p | \phi_W^I |^p(\bm x) d\bm x \\
& \lesssim \sum_{W\in\mathscr{W}^{I}(\mathcal{R}^L)} |W| \left| a_W^I\right|^p \leq
\sum_{W\in\mathscr{W}^{I}(\mathcal{R}^L)} \frac{|W|}{|\mathscr{M}_1(W)|} \int_{\mathscr{M}_1(W)} |u(\bm x)|^p d\bm x \\
& \lesssim \sum_{k=0}^{m} 2^k \int_{\mathcal{R}_{-1}} |u(\bm x)|^p d\bm x \lesssim 2^{m} \int_{\mathcal{R}_{-1}} |u(\bm x)|^p d\bm x \lesssim L \| u\|^p_{L^p(\mathcal{R}_{-1})}\,.
\end{split}
\]
Thus \eqref{eq:EL1} is true.
Now to estimate $|E^L u|_{\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^\infty)}$, we first note that
\[
\int_{\mathcal{R}_{-1}}\int_{\mathcal{R}_{-1}} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x = \int_{\mathcal{R}_{-1}}\int_{\mathcal{R}_{-1}} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |u(\bm y) -u(\bm x)|^p d\bm y d\bm x \lesssim |u|^p_{\mathcal{T}_{1}^{\,\beta}(\mathcal{R}_{-1})}\,.
\]
So we only need to estimate
\begin{equation}\label{eqn:I1}
I.1=\int_{\mathcal{R}_{-1}} \int_{\mathcal{R}^\infty} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x \,, \end{equation}
and
\begin{equation}\label{eqn:I2}
I.2=\int_{\mathcal{R}^\infty} \int_{\mathcal{R}^\infty} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x \,.\end{equation}
Notice that for any $\bm y \in W_1 \in \mathscr{W}^{I}(\mathcal{R}^L)$,
\[
\begin{split}
E^L u(\bm y) - u(\bm x) = &\sum_{W\in \mathscr{W}^I(\mathcal{R}^L)} a_W^I \phi_W^I(\bm y) - u(\bm x) = \sum_{W\in \mathscr{N}(W_1)\cap\mathscr{W}^I(\mathcal{R}^L)} \left(a_W^I-a_{W_1}^I\right) \phi_W^I(\bm y) + \left(a_{W_1}^I- u(\bm x)\right)\,,
\end{split}
\]
where $\mathscr{N}(W_1)\subset \mathscr{W}^{I}(\mathcal{R}^\infty)$ denotes the collections of all the cubes that have nontrivial overlaps with the $l(W_1)/4$-neighborhood of $W_1$.
We then have the estimate
\[
\begin{split}
I.1& = \sum_{W_1\in \overline{\mathscr{W}_0}} \int_{\mathcal{R}_{-1}} \int_{W_1} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x \\
& \lesssim \sum_{W_1\in \overline{\mathscr{W}_0}}\sum_{W\in \mathscr{N}(W_1)} \int_{\mathcal{R}_{-1}} \int_{W_1} {\gamma}^{\,\beta}_1 (|\bm y -\bm x|) \left| a_{W}^I-a_{W_1}^I\right|^p d\bm y d\bm x + \sum_{W_1\in \overline{\mathscr{W}_0}}\int_{\mathcal{R}_{-1}} \int_{W_1} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \left| a_{W_1}^I- u(\bm x)\right|^p d\bm y d\bm x \\
& =: I.1.a + I.1.b \,.
\end{split}
\]
We now estimate $I.1.a$. Note that
\begin{equation} \label{eq:estimate_a}
\begin{split}
a_{W}^I-a_{W_1}^I &=\frac{1}{|\mathscr{M}_1(W)|} \int_{\mathscr{M}_1(W)} u(\bm y') d\bm y' - \frac{1}{|\mathscr{M}_1(W_1)|} \int_{\mathscr{M}_1(W_1)} u(\bm x') d\bm x' \\
&=\frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|} \int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} (u(\bm y') - u(\bm x')) d\bm y' d\bm x' \,,
\end{split}
\end{equation}
we then have
\[
\begin{split}
\left| a_{W}^I-a_{W_1}^I\right|^p
&\leq \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|}\int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} |u(\bm y') - u(\bm x')|^p d\bm y' d\bm x' \\
&\lesssim \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|}\int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee 1)^{d+p-2} (|\bm y' -\bm x'|\wedge 1)^{\,\beta}} d\bm y' d\bm x'\,,
\end{split}
\]
where we have used $|\bm y'-\bm x'|\lesssim 1$ in the last inequality as a result of $W_1\in \overline{\mathscr{W}_0}$ and $W\in\mathscr{N}(W_1)$. Notice that ${\gamma}^{\,\beta}_1(|\bm y-\bm x|) =C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}/|\bm y-\bm x|^{\,\beta}$ by \eqref{def:frackernel}, so
\[
\begin{split}
I.1.a\lesssim \sum_{W_1\in \overline{\mathscr{W}_0}}\sum_{W\in \mathscr{N}(W_1)}& \left(\int_{\mathcal{R}_{-1}} \int_{W_1} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{ |\bm y-\bm x|^{\,\beta}} \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|} d\bm y d\bm x \right) \\
& \cdot \left(\int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee 1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x'\right) \,.
\end{split}
\]
For any $\bm y\in W_1\in \overline{\mathscr{W}_0}$,
\[
\int_{\mathcal{R}_{-1}} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\,\beta}} d\bm x \leq \int_{|\bm s|<1} \frac{C_{d,p,\,\beta}}{|\bm s|^{\,\beta}} d\bm s \lesssim \frac{1}{d-\beta} \,.
\]
Notice that $1/(d-\beta)$ is the constant that appears in \eqref{eq:EL2} and it blows up as $\beta\to d$ so we have to take a fixed $\beta\in[0,d)$.
Therefore,
\[
\int_{\mathcal{R}_{-1}} \int_{W_1} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{ |\bm y-\bm x|^{\,\beta}} \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|} d\bm y d\bm x \lesssim \frac{|\beta-d|^{-1}|W_1|}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|} \lesssim
|\beta-d|^{-1}\,,
\]
where we have used the fact that $|W_1| \approx |\mathscr{M}_1(W)|\approx |\mathscr{M}_1(W_1)| \approx 1$ for any $W_1\in \overline{\mathscr{W}_0}$ and $W\in \mathscr{N}(W_1)$. So
\[
I.1.a \lesssim |\beta-d|^{-1} \sum_{W_1\in \overline{\mathscr{W}_0}}\sum_{W\in \mathscr{N}(W_1)}\int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \lesssim |\beta-d|^{-1}|u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}\,.
\]
Now since
\[
\begin{split}
\left| a_{W_1}^I- u(\bm x)\right|^p &= \left| \frac{1}{|\mathscr{M}_1(W_1)|} \int_{\mathscr{M}_1(W_1)} (u(\bm y')- u(\bm x)) d\bm y' \right|^p \leq \frac{1}{|\mathscr{M}_1(W_1)|} \int_{\mathscr{M}_1(W_1)} |u(\bm y')- u(\bm x)|^p d\bm y' \\
&\lesssim \frac{1}{|\mathscr{M}_1(W_1)|}\int_{\mathscr{M}_1(W_1)} \frac{|u(\bm y')- u(\bm x)|^p}{(|\bm y'-\bm x|\vee1)^{d+p-2} (|\bm y' -\bm x|\wedge1)^{\,\beta}} d\bm y'\,,
\end{split}
\]
as a result of the fact that $\bm y\in W_1$ and $|\bm y-\bm x|\lesssim1$, we have
\[
\begin{split}
I.1.b &\lesssim \sum_{W_1\in \overline{\mathscr{W}_0}} \int_{\mathcal{R}_{-1}} \left( \int_{W_1} \frac{1_{\{|\bm y-\bm x|<1\}}}{ |\bm y-\bm x|^{\,\beta}} \frac{1}{|\mathscr{M}_1(W_1)|} d\bm y \int_{\mathscr{M}_1(W_1)} \frac{|u(\bm y') - u(\bm x)|^p}{(|\bm y'-\bm x|\vee1)^{d+p-2} (|\bm y' -\bm x|\wedge1)^{\,\beta}} d\bm y' \right)d\bm x \\
&\lesssim |\beta-d|^{-1}\sum_{W_1\in \overline{\mathscr{W}_0}} \int_{\mathcal{R}_{-1}} \int_{\mathscr{M}_1(W_1)} \frac{|u(\bm y') - u(\bm x)|^p}{(|\bm y'-\bm x|\vee1)^{d+p-2} (|\bm y' -\bm x|\wedge1)^{\,\beta}} d\bm y' d\bm x \lesssim |\beta-d|^{-1}|u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})} \,.
\end{split}
\]
Together we have shown that $I.1$ in \eqref{eqn:I1} is bounded by $|u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}$.
To estimate $I.2$ in \eqref{eqn:I2}, we first note that
\begin{equation}\label{eq:I.2}
I.2 = \sum_{W_1\in\mathscr{W}^I(\mathcal{R}^\infty)} \sum_{W_2\in \mathscr{N}(W_1)}\int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x\,,
\end{equation}
since all cubes in $\mathscr{W}^I(\mathcal{R}^\infty)$ have length greater than or equal to $1$ and ${\gamma}^{\,\beta}_1(|\bm y-\bm x|)=0$ if $|\bm y-\bm x|>1$.
Now suppose
$\bm{x}\in W_1 \in \mathscr{W}_k$ for $k\leq -m+1$, and $y\in W_2\in \mathscr{N}(W_1)$ then
\[
E^L u(\bm y)-E^L u(\bm x) = \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^I(\mathcal{R}^L)} \left[ \left( a_W^I -a_{W_1}^I\right) \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right) + a_{W_1}^I \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right) \right],
\]
On the other hand if
$\bm x\in W_1\in \mathscr{W}^{I}(\mathcal{R}^L)\backslash \mathscr{W}_{-m+1}$ and $\bm{y}\in W_2 \in \mathscr{N}(W_1)$, then we know that both $\bm{x}$ and $\bm{y}$ are in $\mathcal{R}^L$, and therefore
\[
\sum_{W\in \mathscr{W}^I(\mathcal{R}^L)} \phi_W^I(\bm y) = \sum_{W\in \mathscr{W}^I(\mathcal{R}^L)} \phi_W^I(\bm x)= 1.
\]
In turn, we have
\[
E^L u(\bm y)-E^L u(\bm x) = \sum_{W\in \mathscr{W}^{I}(\mathcal{R}^L)} a_W^I \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right) = \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^I(\mathcal{R}^L)} \left( a_W^I -a_{W_1}^I\right) \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right) \,.
\]
Taking into account the two cases, we can show
\[
I.2 \lesssim I.2.a + I.2.b
\]
where
\[
I.2.a = \sum_{W_1\in\mathscr{W}^I(\mathcal{R}^\infty)} \sum_{W_2\in \mathscr{N}(W_1)} \int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|)\left| \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^I(\mathcal{R}^L)} \left( a_W^I -a_{W_1}^I\right) \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right)\right|^p d\bm y d\bm x ,
\]
and
\[
I.2.b = \sum_{k\leq -m+1}\sum_{W_1\in \mathscr{W}_k} \sum_{W_2\in \mathscr{N}(W_1)} \int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \left| \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^I(\mathcal{R}^L)}a_{W_1}^I \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right) \right|^p d\bm y d\bm x .
\]
We first estimate $I.2.b$. Since the number of sets in $\mathscr{N}(W_1)\cup \mathscr{N}(W_2)$ is uniformly bounded by a constant for any $W_2\in \mathscr{N}(W_1)$, and Lip$\phi_W^I\lesssim 1/l(W)$, we have
\[
\left|\sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^I(\mathcal{R}^L)}
a_{W_1}^I \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right)\right|^p \lesssim \left| a_{W_1}^I\right|^p \frac{|\bm{y}-\bm{x}|^p}{|l(W_1)|^p} \lesssim \frac{|\bm{y}-\bm{x}|^p}{|\mathscr{M}_1(W_1)| |l(W_1)|^p} \int_{\mathscr{M}_1(W_1)} |u(\bm{z})|^p d\bm{z} ,
\]
where we have also used $l(W_1)\approx l(W_2) \approx l(W)$ for any $W\in \mathscr{N}(W_1)\cup \mathscr{N}(W_2)$ and $W_2\in \mathscr{N}(W_1)$.
Therefore
\[
\begin{split}
I.2.b &\lesssim
\sum_{k\leq -m+1} \sum_{W_1 \in \mathscr{W}_k} \sum_{W_2\in \mathscr{N}(W_1)} \int_{W_1} \int_{W_2} \frac{{\gamma}^{\,\beta}_1(|\bm y -\bm x|) |\bm{y}-\bm{x}|^p d\bm{y} d\bm{x} }{|\mathscr{M}_1(W_1)| |l(W_1)|^p} \int_{\mathscr{M}_1(W_1)} |u(\bm{z})|^p d\bm{z} \\
& \lesssim \sum_{k\leq -m+1} \sum_{W_1 \in \mathscr{W}_k} \frac{|W_1|}{|\mathscr{M}_1(W_1)| |l(W_1)|^p } \int_{\mathscr{M}_1(W_1)} |u(\bm{z})|^p d\bm{z} \\
&\lesssim \sum_{k\leq -m+1} 2^{k(p-1)} \int_{\mathcal{R}_{-1}} |u(\bm{z})|^p d\bm{z} \lesssim L^{-(p-1)} \| u\|^p_{L^p(\mathcal{R}_{-1})}.
\end{split}
\]
For I.2.a, we first notice that
\[
\left| \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^I(\mathcal{R}^L)} \left( a_W^I -a_{W_1}^I\right) \left(\phi_W^I(\bm y) - \phi_W^I(\bm x)\right) \right|^p \lesssim
\sum_{W\in \mathscr{N}(W_1)\cup \mathscr{N}(W_2)} \left| a_W^I -a_{W_1}^I\right|^p \frac{|\bm y -\bm x|^p}{l(W)^p}\,,
\]
where we have used the fact that $\mathscr{N}(W_1)\cup \mathscr{N}(W_2)$ contains only finite number of sets and Lip$\phi_W^I\lesssim 1/l(W)$.
Now from \eqref{eq:estimate_a}, we obtain
\[
\begin{split}
\left| a_{W}^I-a_{W_1}^I\right|^p
&\leq \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|}\int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} |u(\bm y') - u(\bm x')|^p d\bm y' d\bm x' \\
&\lesssim \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|}\int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x'\,.
\end{split}
\]
Notice also that
\[
\int_{W_1} \int_{W_2}\frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\beta-p}} d\bm y d\bm x \leq \int_{W_1} \int_{\mathbb{R}^d}\frac{C_{d,p,\,\beta} 1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\beta-p}} d\bm y d\bm x \lesssim |W_1| \,.
\]
So we have
\[
\begin{split}
I.2.a&\lesssim \sum_{W_1\in\mathscr{W}^I(\mathcal{R}^\infty)} \sum_{W_2\in\mathscr{N}(W_1)} \sum_{W\in\mathscr{N}(W_1)\cup\mathscr{N}(W_2)}\left(\int_{W_1} \int_{W_2} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{ |\bm y-\bm x|^{\beta-p} l(W)^p} \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|} d\bm y d\bm x \right) \\
&\hspace{7cm} \cdot \left( \int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \right) \\
&\lesssim \sum_{W_1\in\mathscr{W}^I(\mathcal{R}^\infty)} \sum_{W_2\in\mathscr{N}(W_1)} \sum_{W\in\mathscr{N}(W_1)\cup\mathscr{N}(W_2)}\frac{|W_1|}{l(W)^p|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|} \int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \\
&\lesssim \sum_{W_1\in\mathscr{W}^I(\mathcal{R}^\infty)} \sum_{W_2\in\mathscr{N}(W_1)} \sum_{W\in\mathscr{N}(W_1)\cup\mathscr{N}(W_2)} \frac{l(W_1)^{d+p-2}}{l(W_1)^p l(W_1)^{2d-2}} \int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \\
&\lesssim \left(\sum_{k\leq 0,k\in\mathbb{Z}} \sum_{W_1\in \mathscr{W}_k}+\sum_{W_1\in \overline{\mathscr{W}_0},\, k=0}\right) \sum_{W_2\in\mathscr{N}(W_1)} \sum_{W\in\mathscr{N}(W_1)\cup\mathscr{N}(W_2)} ( 2^{-k})^{-(d+p-2)} \int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \\
&\lesssim \left(\sum_{k\leq 0,k\in\mathbb{Z}} \sum_{W_1\in \mathscr{W}_k}+\sum_{W_1\in \overline{\mathscr{W}_0},\, k=0}\right) 2^{k(d+p-2)} \int_{\mathscr{M}_1(W_1)}\int_{B(\bm x', C\cdot 2^{-k})\cap \mathcal{R}_{-1}} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \,,
\end{split}
\]
for some $C>0$, and $B(\bm x', r)$ denotes the ball of radius $r$ centered at $\bm x'$. Now chose $C_1=C+1$, then $(C_1-C) \cdot 2^{-k} \geq 1$ for all $k\leq 0$. Therefore
\[
\begin{split}
I.2.a &\lesssim \left(\sum_{k\leq 0,k\in\mathbb{Z}} \sum_{W_1\in \mathscr{W}_k}+\sum_{W_1\in \overline{\mathscr{W}_0},\, k=0}\right) \int_{C_1\cdot 2^{-k}}^{C_1\cdot 2^{-k+1}}\int_{\mathscr{M}_1(W_1)}\int_{B(\bm x', C\cdot 2^{-k})\cap \mathcal{R}_{-1}} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \frac{dt}{t^{d+p-1}} \\
&\lesssim \left(\sum_{k\leq 0,k\in\mathbb{Z}} \sum_{W_1\in \mathscr{W}_k}+\sum_{W_1\in \overline{\mathscr{W}_0},\, k=0}\right) \int_{C_1\cdot 2^{-k}}^{C_1\cdot 2^{-k+1}}\int_{\mathscr{M}_1(W_1)}\int_{B(\bm x', C_1\cdot 2^{-k} -1 )\cap \mathcal{R}_{-1}} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \frac{dt}{t^{d+p-1}} \\
&\lesssim \left(\sum_{k\leq 0,k\in\mathbb{Z}} \sum_{W_1\in \mathscr{W}_k}+\sum_{W_1\in \overline{\mathscr{W}_0},\, k=0}\right) \int_{C_1\cdot 2^{-k}}^{C_1\cdot 2^{-k+1}}\int_{\mathscr{M}_1(W_1)}\int_{B(\bm x', t -1 )\cap \mathcal{R}_{-1}} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \frac{dt}{t^{d+p-1}} \\
&\lesssim \int_{0}^{\infty}\int_{\mathcal{R}_{-1}}\int_{B(\bm x', t -1)\cap \mathcal{R}_{-1}} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge
1)^{\,\beta}} d\bm y' d\bm x' \frac{dt}{t^{d+p-1}} \\
&\leq \int_{\mathcal{R}_{-1}}\int_{ \mathcal{R}_{-1}} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y' -\bm x'|\wedge1)^{\,\beta}} \int_{|\bm y'-\bm x'|<t-1} \frac{dt}{t^{d+p-1}} d\bm y' d\bm x' \\
&\lesssim \int_{\mathcal{R}_{-1}}\int_{ \mathcal{R}_{-1}} \frac{|u(\bm y') - u(\bm x')|^p}{ (|\bm y'-\bm x'|+1)^{d+p-2}(|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \lesssim |u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}\,.
\end{split}
\]
Together we have shown \eqref{eq:EL2} for the case $\beta\in [0,d) $.
Now for $\beta\in (d,d+p)$, we take the extension operator $E^L$ defined in \eqref{def:extensionop_2}. The $L^p$ estimate of $E^L u$ can be shown similarly as in the first case. For the estimate of $|E^L u|_{\mathcal{S}_1^{\,\beta}(\mathcal{R}_{-1}^\infty)}$, similar to the first case considered earlier, it is not hard to see that we only need to estimate \begin{equation}
II.1=\int_{\mathcal{R}_{-1}} \int_{\mathcal{R}^\infty} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x \,, \end{equation}
and
\begin{equation}
II.2=\int_{\mathcal{R}^\infty} \int_{\mathcal{R}^\infty} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x \,.\end{equation}
Similar to the first case, we can split $II.1$ into two parts.
\[
\begin{split}
II.1& = \sum_{k=1}^\infty\sum_{W_1\in\mathscr{W}_k} \int_{\mathcal{R}_{-1}} \int_{W_1} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x \\
& \lesssim \sum_{k=1}^\infty\sum_{W_1\in\mathscr{W}_k} \sum_{W\in \mathscr{N}(W_1)} \int_{\mathcal{R}_{-1}} \int_{W_1} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \left|a_{W}^{II}-a_{W_1}^{II}\right|^p d\bm y d\bm x + \sum_{k=1}^\infty\sum_{W_1\in\mathscr{W}_k} \int_{\mathcal{R}_{-1}} \int_{W_1} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \left|a_{W_1}^{II}- u(\bm x)\right|^p d\bm y d\bm x \\
& =: II.1.a + II.1.b \,.
\end{split}
\]
From a similar equation to \eqref{eq:estimate_a}, we have
\begin{equation}\label{eq:estimate_aII}
\left|a_{W}^{II}-a_{W_1}^{II}\right|^p
\lesssim \frac{ l(W_1)^{\,\beta}}{|\mathscr{M}_2(W)||\mathscr{M}_2(W_1)|}\int_{\mathscr{M}_2(W_1)}\int_{\mathscr{M}_2(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x'\,,
\end{equation}
where we have used $W\in \mathscr{N}(W_1)$ and $l(W_1)\leq 1$.
So
\[
\begin{split}
II.1.a\lesssim \sum_{k=1}^\infty\sum_{W_1\in\mathscr{W}_k}\sum_{W\in \mathscr{N}(W_1)}& \left(\int_{\mathcal{R}_{-1}} \int_{W_1} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{ |\bm y-\bm x|^{\,\beta}} \frac{ l(W_1)^{\,\beta}}{|\mathscr{M}_2(W)||\mathscr{M}_2(W_1)|} d\bm y d\bm x \right) \\
& \cdot \left(\int_{\mathscr{M}_2(W_1)}\int_{\mathscr{M}_2(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee 1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x'\right) \,.
\end{split}
\]
For any $\bm y\in W_1 \in \mathscr{W}_k$ and $\beta\in (d,d+p)$,
\[
\int_{\mathcal{R}_{-1}} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\,\beta}} d\bm x \lesssim \frac{1}{\beta-d} l(W_1)^{d-\beta} \,.
\]
Notice again that the constant $1/(\beta-d)$ blows up as $\beta\to d$. Then
\[
\int_{\mathcal{R}_{-1}} \int_{W_1} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{ |\bm y-\bm x|^{\,\beta}} \frac{ l(W_1)^{\,\beta}}{|\mathscr{M}_2(W)||\mathscr{M}_2(W_1)|} d\bm y d\bm x \lesssim |\beta-d|^{-1} l(W_1)^{d-\beta}\cdot \frac{ l(W_1)^{\,\beta}}{|\mathscr{M}_2(W)||\mathscr{M}_2(W_1)|} |W_1| \lesssim |\beta-d|^{-1}\,,
\]
where we have used the fact that $|\mathscr{M}_2(W)|\approx|\mathscr{M}_2(W_1)|\approx |W_1| \approx l(W_1)^{d+p-2}$. Using this estimate, one can show
\[
II.1.a\lesssim |\beta-d|^{-1} |u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}.
\]
To estimate $II.1.b$, we first define a decomposition of $\mathcal{R}_{-1}$
\begin{equation}
\mathscr{W}(\mathcal{R}_{-1}) = \left\{\mathscr{M}_2(W): W\in \mathscr{W}_k, k\in \mathbb{Z}_+ \right\}. \end{equation}
Then
\begin{equation}
\label{eq:estimate_II1b}
II.1.b = \sum_{W_2\in \mathscr{W}(\mathcal{R}_{-1})}\sum_{k=1}^\infty\sum_{W_1\in\mathscr{W}_k} \int_{W_2} \int_{W_1} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \left|a_{W_1}^{II}- u(\bm x)\right|^p d\bm y d\bm x \,.
\end{equation}
Notice that for any $\bm x\in W_2 \in \mathscr{W}(\mathcal{R}_{-1})$ and $\bm y'\in \mathscr{M}_2(W_1)$ for $W_1\in \mathscr{W}_k, k\in \mathbb{Z}_+$, we have
\[
|\bm y' -\bm x| \lesssim \text{dist}(W_1,W_2) + l(W_1) + l(W_2) \lesssim \text{dist}(W_1,W_2)\,.
\]
Moreover, we must have $\text{dist}(W_1, W_2)< 1$ for the double integral in \eqref{eq:estimate_II1b} to be non-zero.
We thus have the estimate
\[
\left|a_{W_1}^{II}- u(\bm x)\right|^p \leq \frac{1}{|\mathscr{M}_2(W_1)|} \int_{\mathscr{M}_2(W_1)} |u(\bm y') -u(\bm x)|^p d\bm y' \lesssim \frac{ \text{dist}(W_1, W_2)^{\,\beta}}{|\mathscr{M}_2(W_1)|}\int_{\mathscr{M}_2(W_1)} \frac{|u(\bm y') -u(\bm x)|^p}{(|\bm y' -\bm x|\vee1)^{d+p-2} (|\bm y' -\bm x|\wedge1)^{\,\beta} } d\bm y' \,.
\]
Therefore
\[
\begin{split}
&II.1.b \lesssim \sum_{W_2\in \mathscr{W}(\mathcal{R}_{-1})} \sum_{W_1\in \mathscr{W}_k,\,k\in \mathbb{Z}_+} \int_{W_2} \left( \int_{W_1} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{ |\bm y-\bm x|^{\,\beta}} \frac{ \text{dist}(W_1, W_2)^{\,\beta}}{|\mathscr{M}_2(W_1)|} d\bm y \int_{\mathscr{M}_2(W_1)} \frac{|u(\bm y') - u(\bm x)|^p}{(|\bm y'-\bm x|\vee1)^{d+p-2} (|\bm y' -\bm x|\wedge1)^{\,\beta}} d\bm y' \right)d\bm x \\
&\lesssim \sum_{W_2\in \mathscr{W}(\mathcal{R}_{-1})} \sum_{W_1\in \mathscr{W}_k,\,k\in \mathbb{Z}_+} \int_{W_2} \left( \int_{W_1} \frac{C_{d,p,\,\beta}}{\text{dist}(W_1, W_2)^{\,\beta}} \frac{\text{dist}(W_1, W_2)^{\,\beta}}{|\mathscr{M}_2(W_1)|} d\bm y \int_{\mathscr{M}_2(W_1)} \frac{|u(\bm y') - u(\bm x)|^p}{(|\bm y'-\bm x|\vee1)^{d+p-2} (|\bm y' -\bm x|\wedge1)^{\,\beta}} d\bm y' \right)d\bm x\\
&\lesssim \sum_{W_2\in \mathscr{W}(\mathcal{R}_{-1})} \sum_{W_1\in \mathscr{W}_k,\,k\in \mathbb{Z}_+} \int_{W_2} \int_{{\mathscr{M}_2(W_1)}} \frac{|u(\bm y') - u(\bm x)|^p}{(|\bm y'-\bm x|\vee1)^{d+p-2} (|\bm y' -\bm x|\wedge1)^{\,\beta}} d\bm y' d\bm x \lesssim |u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})} \,.
\end{split}
\]
Now for $II.2$, we can first write
\begin{equation}\label{eq:II.2}
II.2 = \sum_{W_1\in\mathscr{W}^{II}(\mathcal{R}^\infty)} \sum_{W_2\in\mathscr{W}^{II}(\mathcal{R}^\infty)}\int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) |E^L u(\bm y) -E^L u(\bm x)|^p d\bm y d\bm x\,.
\end{equation}
Observe that
\[
E^L u(\bm y)-E^L u(\bm x) = \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^{II}(\mathcal{R}^L)} \left[ \left( a_W^{II} -a_{W_1}^{II}\right) \left(\phi_W^{II}(\bm y) - \phi_W^{II}(\bm x)\right) + a_{W_1}^{II} \left(\phi_W^{II}(\bm y) - \phi_W^{II}(\bm x)\right) \right],
\]
where the second part in the above equation is only nonzero for $\bm{x}\in W_1 \in \mathscr{W}_{k}$ for $k\leq -m+1$ (and therefore $\bm{y}\in W_2 \in \mathscr{N}(W_1)$ in this case because of the nonlocal interaction length).
Similarly as before, we have
\[
II.2 \lesssim II.2.a + II.2.b
\]
where
\[
II.2.a = \sum_{W_1\in\mathscr{W}^{II}(\mathcal{R}^\infty)} \sum_{W_2\in \mathscr{W}^{II}(\mathcal{R}^\infty)} \int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|)\left| \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^{II}(\mathcal{R}^L)} \left( a_W^{II} -a_{W_1}^{II}\right) \left(\phi_W^{II}(\bm y) - \phi_W^{II}(\bm x)\right)\right|^p d\bm y d\bm x
\]
and
\[
II.2.b = \sum_{k\leq -m+1}\sum_{W_1\in \mathscr{W}_k} \sum_{W_2\in \mathscr{N}(W_1)} \int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \left| \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^{II}(\mathcal{R}^L)}a_{W_1}^{II} \left(\phi_W^{II}(\bm y) - \phi_W^{II}(\bm x)\right) \right|^p d\bm y d\bm x.
\]
It is easy to see that $II.2.b$ can be estimated similarly as $I.2.b$ and we have $II.2.b\lesssim L^{-(p-1)} \| u\|^p_{L^p(\mathcal{R}_{-1})}$.
Now for the estimate of $II.2.a$, we have two different cases where $W_2\in \mathscr{N}(W_1)$ and $W_2\notin \mathscr{N}(W_1)$. For the case $W_2\in \mathscr{N}(W_1)$, the estimate follows similarly to the estimate of $I.2.a$ which is omitted here. For the case $W_2\notin \mathscr{N}(W_1)$,
we proceed by noticing that if $l(W_1)\geq 2$ and $W_2\notin \mathscr{N}(W_1)$, then we must have $\text{dist}(W_1, W_2) \geq 1$ so that the double integral in \eqref{eq:II.2} becomes zero. Therefore we only need to consider $l(W_1)\leq 1$ (i.e., $W_1\in \mathscr{W}_k$ for $k\geq 0$) in this case. Notice that
\[
\begin{split}
\left| \sum_{W\in (\mathscr{N}(W_1)\cup \mathscr{N}(W_2))\cap \mathscr{W}^{II}(\mathcal{R}^L)}\left( a_W^{II} -a_{W_1}^{II}\right) \left(\phi_W^{II}(\bm y) - \phi_W^{II}(\bm x)\right)\right|^p &\lesssim \sum_{W\in \mathscr{N}(W_1)\cup \mathscr{N}(W_2)} \left| a_W^{II} -a_{W_1}^{II}\right|^p \left|\phi_W^{II}(\bm y) - \phi_W^{II}(\bm x)\right|^p \\
&\lesssim
\sum_{W\in \mathscr{N}(W_1)\cup \mathscr{N}(W_2)} \left| a_W^{II} -a_{W_1}^{II}\right|^p \,,
\end{split}
\]
for $\bm x\in W_1$ and $\bm y\in W_2$. For $W\in \mathscr{N}(W_1)$, we use \eqref{eq:estimate_aII} and
\[
\sum_{W_2\notin \mathscr{N}(W_1)}\int_{W_1} \int_{W_2} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\,\beta}} d\bm y d\bm x \leq \int_{W_1} \int_{|\bm y-\bm x|\geq l(W_1)/2} \frac{C_{d,p,\,\beta}}{|\bm y-\bm x|^{\,\beta}} d\bm y d\bm x \lesssim |\beta-d|^{-1}|W_1| l(W_1)^{d-\beta}
\]
to get
\[
\begin{split}
& \sum_{W_1\in\mathscr{W}^{II}(\mathcal{R}^\infty)} \sum_{W_2\in\mathscr{W}^{II}(\mathcal{R}^\infty)\backslash\mathscr{N}(W_1)}\int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \sum_{W\in \mathscr{N}(W_1)} \left| a_W^{II} -a_{W_1}^{II}\right|^p d\bm y d\bm x \\
\lesssim & \sum_{\substack{W_1\in\mathscr{W}_k \\ k\geq 0}} \sum_{W_2\in\mathscr{W}^{II}(\mathcal{R}^\infty)\backslash\mathscr{N}(W_1)} \sum_{W\in \mathscr{N}(W_1)} \left(\int_{W_1}\int_{W_2} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\,\beta}} \frac{ l(W_1)^{\,\beta}}{|\mathscr{M}_2(W)||\mathscr{M}_2(W_1)|}d\bm y d\bm x \right) \\
& \hspace{6cm}\cdot\left(\int_{\mathscr{M}_2(W_1)}\int_{\mathscr{M}_2(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x'\right) \\
\lesssim & |\beta-d|^{-1}\sum_{\substack{W_1\in\mathscr{W}_k \\ k\geq 0}} \sum_{W\in \mathscr{N}(W_1)} \int_{\mathscr{M}_2(W_1)}\int_{\mathscr{M}_2(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \lesssim |\beta-d|^{-1}|u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}\,.
\end{split}
\]
On the other hand, if $W\in \mathscr{N}(W_2)$, then $|W|\approx |\mathscr{M}(W)|\approx|W_2|$, and we can use
\[
\begin{split}
\left| a_{W}^{II}-a_{W_1}^{II}\right|^p
&\leq \frac{1}{|\mathscr{M}_1(W)||\mathscr{M}_1(W_1)|} \int_{\mathscr{M}_1(W_1)}\int_{\mathscr{M}_1(W)} |u(\bm y') - u(\bm x')|^p d\bm y' d\bm x' \\
&\lesssim \frac{ \text{dist}(W_1,W_2)^{\,\beta}}{|\mathscr{M}_2(W)||\mathscr{M}_2(W_1)|}\int_{\mathscr{M}_2(W_1)}\int_{\mathscr{M}_2(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x'
\end{split}
\]
and
\[
\int_{W_1} \int_{W_2} \frac{1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\,\beta}} d\bm y d\bm x \leq \frac{|W_1| |W_2|}{\text{dist}(W_1,W_2)^{\,\beta}}
\]
to arrive at
\[
\begin{split}
& \sum_{W_1\in\mathscr{W}^{II}(\mathcal{R}^\infty)} \sum_{W_2\in\mathscr{W}^{II}(\mathcal{R}^\infty)\backslash\mathscr{N}(W_1)}\int_{W_1}\int_{W_2} {\gamma}^{\,\beta}_1(|\bm y -\bm x|) \sum_{W\in \mathscr{N}(W_2)} \left| a_W^{II} -a_{W_1}^{II}\right|^p d\bm y d\bm x \\
\lesssim & \sum_{\substack{W_1\in\mathscr{W}_k \\ k\geq 0}} \sum_{W_2\in\mathscr{W}^{II}(\mathcal{R}^\infty)\backslash\mathscr{N}(W_1)} \sum_{W\in \mathscr{N}(W_2)} \left(\int_{W_1}\int_{W_2} \frac{C_{d,p,\,\beta}1_{\{|\bm y-\bm x|<1\}}}{|\bm y-\bm x|^{\,\beta}} \frac{ \text{dist}(W_1,W_2)^{\,\beta}}{|\mathscr{M}_2(W)||\mathscr{M}_2(W_1)|}d\bm y d\bm x \right) \\
& \hspace{6cm}\cdot\left(\int_{\mathscr{M}_2(W_1)}\int_{\mathscr{M}_2(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x'\right) \\
\lesssim &\sum_{\substack{W_1\in\mathscr{W}_k \\ k\geq 0}} \sum_{W_2\in\mathscr{W}^{II}(\mathcal{R}^\infty)\backslash\mathscr{N}(W_1)} \sum_{W\in \mathscr{N}(W_2)} \int_{\mathscr{M}_2(W_1)}\int_{\mathscr{M}_2(W)} \frac{|u(\bm y') - u(\bm x')|^p}{(|\bm y'-\bm x'|\vee1)^{d+p-2} (|\bm y' -\bm x'|\wedge1)^{\,\beta}} d\bm y' d\bm x' \lesssim |u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}.
\end{split}
\]
Together we have shown $II.2.a\lesssim |\beta-d|^{-1}|u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})}$ and as a result the estimate \eqref{eq:EL2} is proved for $\beta\in (d,d+p)$.
\end{proof}
\noindent {\it Proof of Theorem \ref{mainthm_2}.}
For any $u\in \mathcal{T}^{\,\beta}_\delta (\mathcal{R}_{-\delta})$, define $v(x)=u(\delta x)$, then from Lemma \ref{lem:scale} we know $|v|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})} = \delta^{p-d} |u|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-\delta})}$ and $\| v\|^p_{L^p(\mathcal{R}_{-1})}=\delta^{-d} \| u\|^p_{L^p(\mathcal{R}_{-\delta})}$. From Theorem \ref{thm:extensionL} we know that
\[
\| E^L v\|^p_{L^p(\mathcal{R}_{-1}^\infty)} \leq C L \| v\|^p_{L^p(\mathcal{R}_{-1})} , \text{ and } |E^L v|^p_{\mathcal{S}_{1}^{\,\beta}(\mathcal{R}^\infty_{-1})} \leq C\left( L^{-(p-1)} \| v\|^p_{L^p(\mathcal{R}_{-1})} + |\beta-d|^{-1}|v|^p_{\mathcal{T}_{1}^{\,\beta}(\mathcal{R}^\infty_{-1})} \right)
\]
for any $L=2^m$ ($m\in \mathbb{Z}_+$).
Now choose $m = \lceil \log_2(M/\delta) \rceil$ and $L=2^m$, and define the extension operator $E: \mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta}) \to \mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)$ as
\[
Eu (\bm x) = (E^L v)(\bm x/\delta)\,.
\]
Then
\[
\| Eu\|^p_{L^p(\mathcal{R}_{-\delta}^\infty)} = \delta^d \| E^L v\|^p_{L^p(\mathcal{R}_{-1}^\infty)}\leq C \delta^d L \| v\|_{L^p(\mathcal{R}_{-1})} = C L \| u\|^p_{L^p(\mathcal{R}_{-\delta})} \lesssim \frac{1}{\delta} \| u\|^p_{L^p(\mathcal{R}_{-\delta})} \,,
\]
and
\[
\begin{split}
| Eu|^p_{\mathcal{S}_{\delta}^{\,\beta}(\mathcal{R}_{-\delta}^\infty)} = \delta^{d-p} | E^L v|^p_{\mathcal{S}_{1}^{\,\beta}(\mathcal{R}_{-1}^\infty)}\leq &C \left( \delta^{d-p} L^{-(p-1)} \| u\|^p_{L^p(\mathcal{R}_{-\delta})} + |\beta-d|^{-1}\delta^{d-p} | v|^p_{\mathcal{T}_1^{\,\beta}(\mathcal{R}_{-1})} \right) \\ \lesssim &C\left( \frac{1}{\delta} \| u\|^p_{L^p(\mathcal{R}_{-\delta})}+ |\beta-d|^{-1}| u|^p_{\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})} \right)\,.
\end{split}
\]
Thus Theorem \ref{mainthm_2} is shown.
\qed
\section{Extension to General Lipschitz Domains }\label{sec:generaldomain}
In this section we will extend the trace theorems for the half plane to a general Lipschitz domain.
We take the strategy
to first generalize to a ``special'' Lipschitz domain before showing the fully general case.
\subsection{Some technical lemmas}
We will first list some
lemmas which are used to show the transformations
from the special Lipschitz domain to the half-space are continuous, with detailed proofs elaborated in \ref{app:1}.
\begin{lem}\label{lem:psiLip}
Let $\varphi:\mathbb{R}^{d-1} \to [0,\infty)$ be a Lipschitz function with Lipschitz constant $L$, $\Phi(\overline{\bm x})=(\varphi(\overline{\bm{x}}),\overline{\bm{x}})$, and define the function $\psi:\mathbb{R}^{d-1} \to [-\delta,\infty)$ where $\psi(\overline{\bm x}):=\min\{\tilde{x}\in \mathbb{R}: \text{dist}((\tilde{x},\overline{\bm{x}}),\Phi(\mathbb{R}^{d-1}))=\delta, \ \psi(\overline{\bm x}) < \varphi(\overline{\bm x})\}$ for all $\overline{\bm{x}} \in \mathbb{R}^{d-1}$. Then $\psi$ is a Lipschitz function with the same Lipschitz constant $L$, which is independent of $\delta$.
\end{lem}
\begin{lem}\label{lem:phipsisupnorm}
Let $\varphi$ and $\psi$ be as defined in Lemma \ref{lem:psiLip}. Then $\delta\leq |\varphi(\overline{\bm x})-\psi(\overline{\bm x})|\leq\delta\sqrt{L^2+1}$ for any $\overline{\bm x}\in \mathbb{R}^{d-1}$.
\end{lem}
\begin{lem}\label{lem:projectiondist}
Let $\varphi$ and $\psi$ be defined as in the Lemma \ref{lem:psiLip}. Then for $\bm{x}=(\tilde{x},\overline{\bm{x}}),\bm{y}=(\tilde{y},\overline{\bm{y}})\in \mathcal{R}_{-\delta}$, if
\begin{equation}
\label{eq:changeofvariable_x}
\begin{aligned}
\tilde{x}'&=\left(1+\frac{\tilde{x}}{\delta}\right)\varphi(\overline{\bm x})-\frac{\tilde{x}}{\delta}\psi(\overline{\bm x})\\
\tilde{y}'&=\left(1+\frac{\tilde{y}}{\delta}\right)\varphi(\overline{\bm y})-\frac{\tilde{y}}{\delta}\psi(\overline{\bm y})
\end{aligned}
\end{equation}
and $\bm x'=(\tilde{x}',\overline{\bm x})$, $\bm y'=(\tilde{y}',\overline{\bm y})$, then
\[
K_L'|\bm x-\bm y| \le|\bm x'-\bm y'| \le K_L|\bm x-\bm y|
\]
for some positive constants $K_L\geq 1$ and $K_L' \le 1$, {independent of $\delta$}.
\end{lem}
\begin{lem}\label{lem:KernelEstimate}
{Let $\varphi$ and $\psi$ be defined as in the Lemma \ref{lem:psiLip}.} Let $M=\max(L+1+\sqrt{L^2+1}, K_L)$ with $K_L$ as in Lemma \ref{lem:projectiondist}, {$\bm{x}=(\tilde{x}, \overline{\bm{x}})$, $\bm{y}=(\tilde{y}, \overline{\bm{y}})$}, $\bm w = (\tilde{y}+\varphi(\overline{\bm{y}}),\overline{\bm{y}})$ and $\bm z =(\tilde{x}+\varphi(\overline{\bm{x}}),\overline{\bm{x}})$. Then we have the following kernel inequalities:
\begin{enumerate}[(a)]
\item For $\bm x,\bm y \in \mathcal{R}^\infty$, $\gamma_{\delta/M}^{\,\beta}(|\bm x-\bm y|) \le M^{d+p}\gamma_{\delta}^{\,\beta}(|\bm z - \bm w|){.}$
\item For $\bm x \in \mathcal{R}_{-\delta}$ and $\bm y \in \mathcal{R}^\infty$, $\gamma_{\delta/M}^{\,\beta}(|\bm x-\bm y|) \le M^{d+p}\gamma_{\delta}^{\,\beta}(|\bm x' - \bm w|){.}$
\item For $\bm x,\bm y \in \mathcal{R}_{-\delta}$, $\gamma_{\delta/M}^{\,\beta}(|\bm x-\bm y|) \le M^{d+p}\gamma_{\delta}^{\,\beta}(|\bm x' - \bm y'|){.}$
\end{enumerate}
where $\bm x'$ and $\bm y'$ are defined as in \eqref{eq:changeofvariable_x}.
\end{lem}
\begin{lem}\label{lem:InverseKernelEstimate}
{Let $\varphi$ and $\psi$ be defined as in the Lemma \ref{lem:psiLip}.}
Let $M=\max(L+2, K'_L)$ with $K'_L$ as in Lemma \ref{lem:projectiondist}, $\bm x'=(\tilde{x}',\overline{\bm x})$, $\bm y'=(\tilde{y}',\overline{\bm y})$, $\bm w = (\tilde{y}'-\varphi(\overline{\bm{y}}),\overline{\bm{y}})$ and $\bm z =(\tilde{x}'-\varphi(\overline{\bm{x}}),\overline{\bm{x}})$. Then we have the following kernel inequalities:
\begin{enumerate}[(a)]
\item For $\bm x',\bm y' \in \Omega$, $\gamma_{\delta}^{\,\beta}(|\bm x'-\bm y'|) \le M^{d+p}\gamma_{M\delta}^{\,\beta}(|\bm z - \bm w|)${.}
\item For $\bm x' \in \Omega_\delta$ and $\bm y '\in \Omega$, $\gamma_{\delta}^{\,\beta}(|\bm x'-\bm y'|) \le M^{d+p}\gamma_{
M\delta}^{\,\beta}(|\bm x - \bm w|)${.}
\item For $\bm x',\bm y '\in \Omega_\delta$, $\gamma_{M\delta}^{\,\beta}(|\bm x'-\bm y'|) \le M^{d+p}\gamma_{M\delta}^{\,\beta}(|\bm x - \bm y|)${.}
\end{enumerate}
{where $\bm x=(\tilde{x},\overline{\bm x})$, $\bm y=(\tilde{y},\overline{\bm y})$ such that $\tilde{x}$, $\tilde{y}$ and $\tilde{x}'$, $\tilde{y}'$ satisfy \eqref{eq:changeofvariable_x}. More specifically,
\begin{equation}
\label{eq:changeofvariable_x'}
\begin{aligned}
\tilde{x}:=\frac{\delta(\tilde{x}'-\varphi(\overline{\bm{x}}))}{\varphi(\overline{\bm{x}})-\psi(\overline{\bm{x}})},\quad \tilde{y}:=\frac{\delta(\tilde{y}'-\varphi(\overline{\bm{y}}))}{\varphi(\overline{\bm{y}})-\psi(\overline{\bm{y}})}.
\end{aligned}
\end{equation}}
\end{lem}
\subsection{Extension to special Lipshitz domains}
We now first present a nonlocal trace theorem to generalize the earlier result shown on a half space with a flat boundary on one side to the case of any infinite domain whose boundary is defined by a Lipshitz graph.
\begin{thm}\label{thm:speclip}
Consider a special Lipschitz domain, $\Omega$, where there is a Lipschitz function $\varphi:\mathbb{R}^{d-1}\to \mathbb{R}$ with Lipschitz constant $L$ such that
\[
\Omega=\{\bm{x}=(\tilde{x},\overline{\bm{x}}) \in \mathbb{R}^d: \varphi(\overline{\bm{x}})<\tilde{x}, \ \overline{\bm{x}} \in \mathbb{R}^{d-1}\}
\]
with a $\delta$ collar boundary $\Omega_\delta$
\[
\Omega_\delta=\{\bm{x}=(\tilde{x},\overline{\bm{x}}) \in \mathbb{R}^d: \psi(\overline{\bm{x}})<\tilde{x}<\varphi(\overline{\bm{x}}), \ \overline{\bm{x}} \in \mathbb{R}^{d-1}\}.
\]
Here $\psi$ is as defined in Lemma \ref{lem:psiLip}.
Then the nonlocal trace theorem holds on this domain, i.e.,
\[
\|u\|_{\mathcal{T}_{\delta}^{\,\beta}(\Omega_\delta)}\le C|d+p-\beta|^{-1/p}\|u\|_{\mathcal{S}_\delta^{\,\beta}(\hat\Omega)}
\]
for some constant $C$ which is independent of $\delta$, $\beta$ and $u \in \mathcal{S}_\delta^{\,\beta}(\hat\Omega)$
\end{thm}
\begin{proof}
Define the operators $P_{\varphi,\psi}:L^p(\Omega) \to L^p(\mathcal{R}^\infty)$ and $G_{\varphi,\psi}:L^p(\Omega_\delta) \to L^p(\mathcal{R}_{-\delta})$
\begin{align}
\label{eq:projP}(P_{\varphi,\psi} u)(\bm{x})&=u(\tilde{x}+\varphi(\overline{\bm{x}}), \overline{\bm{x}}),\\
\label{eq:projG}(G_{\varphi,\psi} u)(\bm{x})&=u\left(\left(1+\frac{\tilde{x}}{\delta}\right)\varphi(\overline{\bm{x}})-\frac{\tilde{x}}{\delta}\psi(\overline{\bm{x}}),\overline{\bm{x}}\right).
\end{align}
Then the operator $S_{\varphi,\psi}:L^p(\hat\Omega)\to L^p(\mathcal{R}_{-\delta}^\infty)$ is defined as
\begin{align}
\label{eq:projS}
S_{\varphi,\psi}u(\bm{x})=\begin{cases}
P_{\varphi,\psi}u (\bm{x}) & \bm{x} \in \mathcal{R}^\infty,\\
G_{\varphi,\psi}u(\bm{x}) & \bm{x} \in \mathcal{R}_{-\delta}.
\end{cases}
\end{align}
We will show that $G_{\varphi,\psi}$ is a bounded operator from $\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)$ to $\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})$ with a bounded inverse $G_{\varphi,\psi}^{-1}$. Moreover, $S_{\varphi,\psi}$ is a bounded operator from $\mathcal{S}_{\delta}^{\,\beta}(\hat\Omega)$ to $\mathcal{S}_{\delta}^{\,\beta}(\mathcal{R}^\infty_{-\delta})$.
To show $G_{\varphi,\psi}$ is a bounded operator from $\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)$ to $\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})$, we let $\bm x'=(\tilde{x}',\overline{\bm{x}})$, $\bm y'=(\tilde{y}',\overline{\bm{y}})$ and the constant $K_L\geq1$ as defined in Lemma \ref{lem:projectiondist}. From definition, we have $G_{\varphi,\psi} (u)(\bm{x})=u(\bm{x}')$, $d \bm x' = \frac{|\varphi(\overline{\bm x})- \psi(\overline{\bm x})|}{\delta} d\bm x$ and $d \bm y' = \frac{|\varphi(\overline{\bm y})- \psi(\overline{\bm y})|}{\delta} d\bm y$. Then,
\begin{equation}\label{eqn:GLpEstimate}
\|G_{\varphi,\psi} u\|^p_{L^p(\mathcal{R}_{-\delta})}=\frac{1}{\delta}\int_{\mathcal{R}_{-\delta}}|G_{\varphi,\psi}(u)(\bm x)|^p d \bm x=\frac{1}{\delta}\int_{\mathcal{R}_{-\delta}}|u(\bm x')|^p d \bm x \le \frac{1}{\delta}\frac{\delta}{\inf |\varphi-\psi|}\int_{\Omega_{\delta}}|u(\bm x')|^p d \bm x' \le \frac{1}{\delta}\|u\|^p_{L^p(\Omega_\delta)}
\end{equation}
and
\begin{align*}
|G_{\varphi,\psi} u|^p_{\mathcal{T}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta})}&=\frac{1}{\delta}\int_{\mathcal{R}_{-\delta}}|G_{\varphi,\psi}(u)(\bm x)|^p d \bm x+\delta^{\,\beta-2}\int_{\mathcal{R}_{-\delta}}\int_{\mathcal{R}_{-\delta}}\frac{|G_{\varphi,\psi}(u)(\bm y)-G_{\varphi,\psi}(u)(\bm x)|^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge\delta)^{\beta}}d\bm y d\bm x\\
&\le \delta^{\,\beta-2}\int_{\mathcal{R}_{-\delta}}\int_{\mathcal{R}_{-\delta}}\frac{|u(\bm y')-u(\bm x')|^p}{(\frac{|\bm y'-\bm x'|}{K_L}\vee\delta)^{d+p-2}(\frac{|\bm y'-\bm x'|}{K_L}\wedge\delta)^{\beta}}d\bm y d\bm x\\
&\le \delta^{\,\beta-2}K_L^{d+\beta+p-2}\frac{\delta^p}{\inf |\varphi-\psi|^p}\int_{\Omega_\delta}\int_{\Omega_\delta}\frac{|u(\bm y')-u(\bm x')|^p}{(|\bm y'-\bm x'|\vee\delta)^{d+p-2}(|\bm y'-\bm x'|\wedge\delta)^{\beta}}d\bm y' d\bm x'\\
&\le K_L^{2d+2p-2}\frac{\delta^p}{\inf |\varphi-\psi|^p}\delta^{\,\beta-2}\int_{\Omega_\delta}\int_{\Omega_\delta}\frac{|u(\bm y')-u(\bm x')|^p}{(|\bm y'-\bm x'|\vee\delta)^{d+p-2}(|\bm y'-\bm x'|\wedge\delta)^{\beta}}d\bm y' d\bm x'\\
&\le K_L^{2d+2p-2}\verti{u}^p_{\mathcal{T}^{\,\beta}_\delta(\Omega_{\delta})}
\end{align*}
where we have used Lemma \ref{lem:phipsisupnorm} for the last estimate as well as \eqref{eqn:GLpEstimate}. Together these estimates show \newline $\|G_{\varphi,\psi} u\|^p_{\mathcal{T}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta})}\le C\|G_{\varphi,\psi} u\|^p_{\mathcal{T}^{\,\beta}_{\delta}(\Omega_\delta)}$ where the constant $C$ is independent of $\beta$ and $\delta$.
To show that $G_{\varphi, \psi}^{-1}$ exists and it is a bounded operator from $\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta})$ to $\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)$, we note first that one can get \eqref{eq:changeofvariable_x'} from
\eqref{eq:changeofvariable_x}.
It is then easy to check that $G_{\varphi,\psi}^{-1}$ can be defined by $G_{\varphi,\psi}^{-1}u(\bm x')= u(\bm x)$ for $\bm x'=(\tilde{x}',\overline{\bm x}) \in \Omega_\delta$ and $\bm x=(\tilde{x},\overline{\bm x})\in \mathcal{R}_{-\delta}$ where $\tilde{x}$ is given by \eqref{eq:changeofvariable_x'}. Using a change of variable estimate given by Lemma \ref{lem:projectiondist} we can then deduce the continuity of $G_{\varphi,\psi}^{-1}$ the same way as we have done for $G_{\varphi,\psi}$ with details omitted.
Now to show that $S_{\varphi,\psi}$ is a bounded operator from $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ to $\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)$, we note that
$\|S_{\varphi,\psi}u \|_{L^p(\mathcal{R}_{-\delta}^\infty)} = \|u \|_{L^p(\hat\Omega)}$,
and by applying Lemma \ref{lem:EnergyKernelEst} with $U=\mathcal{R}_{-\delta}^\infty$ and $\alpha= 1/M$ where $M>1$ is defined in Lemma \ref{lem:KernelEstimate}, we have
\begin{align*}
|S_{\varphi,\psi}(u)|^p_{\mathcal{S}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta}^\infty)} &\le C \int_{\mathcal{R}_{-\delta}^\infty}\int_{\mathcal{R}_{-\delta}^\infty}\gamma^{\,\beta}_{\delta/M}(|\bm y-\bm x|)|S_{\varphi,\psi}(u)(\bm y)-S_{\varphi,\psi}(u)(\bm x)|^p d\bm y d\bm x\\
&\le C \underbrace{\int_{\mathcal{R}^\infty}\int_{\mathcal{R}^\infty}\gamma^{\,\beta}_{\delta/M}(|\bm y-\bm x|)|u(\tilde{y}+\varphi(\overline{\bm y}), \overline{\bm y})-u(\tilde{x}+\varphi(\overline{\bm x}), \overline{\bm x})|^pd\bm y d\bm x}_{I_1}\\
&+C\underbrace{\int_{\mathcal{R}_{-\delta}}\int_{\mathcal{R}^\infty}\gamma^{\,\beta}_{\delta/M}(|\bm y-\bm x|)\left|u(\tilde{y}+\varphi(\overline{\bm y}), \overline{\bm y})-u\left(\left(1+\frac{\tilde{x}}{\delta}\right)\varphi(\overline{\bm{x}})-\frac{\tilde{x}}{\delta}\psi(\overline{\bm{x}}),\overline{\bm{x}}\right)\right|^pd\bm y d\bm x}_{I_2}\\
&+C\underbrace{\int_{\mathcal{R}_{-\delta}}\int_{\mathcal{R}_{-\delta}}\gamma^{\,\beta}_{\delta/M}(|\bm y-\bm x|)\left|u\left(\left(1+\frac{\tilde{y}}{\delta}\right)\varphi(\overline{\bm{y}})-\frac{\tilde{y}}{\delta}\psi(\overline{\bm{y}}),\overline{\bm{y}}\right)-u\left(\left(1+\frac{\tilde{x}}{\delta}\right)\varphi(\overline{\bm{x}})-\frac{\tilde{x}}{\delta}\psi(\overline{\bm{x}}),\overline{\bm{x}}\right)\right|^pd\bm y d\bm x}_{I_3}.
\end{align*}
{Here the constant $C$ is independent of $\beta$ and $\delta$.}
Applying Lemma \ref{lem:KernelEstimate} part (a) to $I_1$ and then using a change of variable, we have
\[
I_1\le M^{d+p} \int_{\Omega}\int_{\Omega} \gamma^{\,\beta}_\delta(|\bm{w}-\bm{z}|)|u(\bm{w})-u(\bm{z})|^pd\bm{w}d\bm{z} \le C |u|^p_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}.
\]
Similarly applying Lemma \ref{lem:KernelEstimate} to $I_2$ and $I_3$ along with the proper change of variables, we finally have $|S_{\varphi,\psi}(u)|^p_{\mathcal{S}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta}^\infty)} \le C|u|^p_{\mathcal{S}^{\,\beta}_{\delta}(\hat\Omega)}$ where $C$ is independent of $\delta$ and $\beta$. Thus, we have the continuity of $S_{\varphi,\psi}$. Using proven properties of $G_{\varphi,\psi}$ and $S_{\varphi,\psi}$ along with the trace theorem for the half-plane, we have
\begin{align*}
\|u\|_{\mathcal{T}_{\delta}^\beta(\Omega_{\delta})} &=\|G^{-1}_{\varphi,\psi}(G_{\varphi,\psi}(u))\|_{\mathcal{T}^{\,\beta}_{\delta}(\Omega_{\delta})}\le C\|G_{\varphi,\psi}(u)\|_{\mathcal{T}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta})}= C\|S_{\varphi,\psi}(u)\|_{\mathcal{T}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta})}\\
&\le C|d+p-\beta|^{-1/p}\|S_{\varphi,\psi}(u)\|_{\mathcal{S}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta}^\infty)}\le C|d+p-\beta|^{-1/p}\|u\|_{\mathcal{S}^{\,\beta}_{\delta}(\hat\Omega)}.
\end{align*}
\end{proof}
Using the transformation operators $G_{\varphi,\psi}$ and $S_{\varphi,\psi}$, we can also generalize the inverse nonlocal trace theorem to the special Lipschitz domain.
\begin{thm}\label{thm:invSpecLip}
Let $\phi$, $\psi$, $\Omega$ and $\Omega_\delta$ be as defined in Theorem \ref{thm:speclip} and the extension operator $E:\mathcal{T}^{\,\beta}_\delta(\mathcal{R}_{-\delta}) \to \mathcal{S}^\beta_\delta(\mathcal{R}_{-\delta}^\infty)$ as defined in Theorem \ref{mainthm_2}. Let $G_{\varphi,\psi}$ be defined in \eqref{eq:projG} and $S_{\varphi,\psi}$ in \eqref{eq:projS}.
Then we can define an extension operator $\tilde{E}:\mathcal{T}_\delta^\beta(\Omega_\delta) \to \mathcal{S}^\beta_\delta(\hat{\Omega})$ where $\tilde{E}=S^{-1}_{\varphi,\psi} E G_{\varphi,\psi}$ and
\[
\|\tilde{E} u\|_{\mathcal{S}_\delta^\beta(\hat\Omega)} \le C |d-\beta|^{-1/p} \|u\|_{\mathcal{T}_\delta^\beta(\Omega_\delta)}.
\]
Here $C$ is a constant independent of $\delta$, $\beta$ and $u\in \mathcal{T}_\delta^\beta(\Omega_\delta)$.
\end{thm}
\begin{proof}
Notice that the inverse operator $S^{-1}_{\varphi,\psi}: L^p(\mathcal{R}_{-\delta}^\infty)\to L^p(\hat\Omega) $ can be defined as
\[
S^{-1}_{\varphi,\psi} u (\bm x') =
\left\{
\begin{aligned}&
u\left(\frac{\delta(\tilde{x}'-\varphi(\overline{\bm x}))}{\varphi(\overline{\bm x})-\psi(\overline{\bm x})}, \overline{\bm x}\right), & \quad \bm x'=(\tilde{x}',\overline{\bm x}) \in \Omega_\delta,\\
& u(\tilde{x}'-\varphi(\overline{\bm x}),\overline{\bm x}), & \quad \bm x' = (\tilde{x}',\overline{\bm x}) \in \Omega.
\end{aligned}
\right.
\]
Now we wish to show that $S^{-1}_{\varphi,\psi}$ is a bounded operator from $\mathcal{S}^{\,\beta}_\delta(\mathcal{R}_{-\delta}^\infty)$ to $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$.
Using Lemma \ref{lem:InverseKernelEstimate} and the $M$ defined there along with the appropriate changes of the variables, we can show similarly as we have done in Theorem \ref{thm:speclip} that
\begin{align*}
|S^{-1}_{\varphi,\psi}(u)|^p_{\mathcal{S}^{\,\beta}_{\delta}(\hat\Omega)}
\leq C \int_{\mathcal{R}^\infty_{-\delta}}\int_{\mathcal{R}^\infty_{-\delta}}\gamma^{\,\beta}_{M\delta}(|\bm y-\bm x|)|u(\bm y)-u(\bm x)|^p d\bm y d\bm x\leq C \int_{\mathcal{R}^\infty_{-\delta}}\int_{\mathcal{R}^\infty_{-\delta}}\gamma^{\,\beta}_{\delta}(|\bm y-\bm x|)|u(\bm y)-u(\bm x)|^p d\bm y d\bm x ,
\end{align*}
where we have used Lemma \ref{lem:EnergyKernelEst} with $U=\mathcal{R}_{-\delta}^\infty$ and $\alpha=1/M$ in the last step. The continuity of $S^{-1}_{\varphi,\psi}$ is thus shown.
Finally, using the continuity properties of $S_{\varphi,\psi}^{-1}$ , $G_{\varphi,\psi}$ and $E$ we have
\[
\begin{split}
&\|\tilde{E} u\|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}=\|S^{-1}_{\varphi,\psi} E G_{\varphi,\psi} u\|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)} \le C\|E G_{\varphi,\psi} u\|_{\mathcal{S}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta}^\infty)} \\
\le &C |d-\beta|^{-1/p}\| G_{\varphi,\psi} u\|_{\mathcal{T}^{\,\beta}_{\delta}(\mathcal{R}_{-\delta})} \le C |d-\beta|^{-1/p}\| u\|_{\mathcal{T}^{\,\beta}_{\delta}(\Omega_\delta)}.
\end{split}
\]
\end{proof}
\subsection{Extension to more general Lipshitz domains}
The extension to more general Lipshitz domains can be otained by using the partition of unity technique.
We first decompose the boundary collar region into finitely many balls so that we can locally view the boundary as multiple special Lipschitz domains. From there we apply Theorem \ref{thm:speclip} to each part and join the estimates together with a partition of unity. The detailed derivation is given as follows.
First, for all discussions given in this subsection, we state some assumptions on the domains and define the necessary spaces and functions.
We consider a general bounded simply connected Lipschitz domain $\Omega$, which naturally makes $\overline{\Omega_{\delta}}$ a compact set for any finite $\delta>0$. Since $\Omega$ has a Lipschitz boundary (see, e.g., \cite{grisvard1985elliptic} Def 1.2.1.1), there exist $N$ local coordinate systems $\bm x^i=(x_1^i,x_2^i,\cdots,x_d^i)$ for $1\le i\le N$, a collection of balls $\{B(\bm x_i,r_i)\}_{i=1}^N$, and Lipschitz functions $\varphi_i: \mathbb{R}^{d-1} \to \mathbb{R}$ for some $N \in \mathbb{N}$, $\bm x_i \in \partial \Omega, \ r_i>0$ such that $\partial \Omega \subseteq\bigcup_{i=1}^N B(\bm x_i,r_i)$,
\begin{align*}
\{\bm x^i=(\tilde{x}^i,\overline{\bm x}^i) \in B(\bm x_i,r_i) : \varphi_i(\overline{\bm x}^i) < \tilde{x}^i\} = \Omega \cap B(\bm x_i,r_i),\\
\{\bm x^i=(\tilde{x}^i,\overline{\bm x}^i) \in B(\bm x_i,r_i) : \varphi_i(\overline{\bm x}^i) = \tilde{x}^i\} = \partial\Omega \cap B(\bm x_i,r_i).
\end{align*}
Notice that in the above definition, $\{ \bm x_i\}_{i=1}^N$ are $N$ fixed points on $\partial\Omega$.
Letting $\delta_0:=\frac{1}{2}\text{dist}(\mathbb{R}^d \setminus\bigcup_{i=1}^N B(\bm x_i,r_i),\partial\Omega)$, we have $\Omega_{\delta_0}\subseteq\bigcup_{i=1}^N B(\bm x_i,r_i)$ and also a positive number $\epsilon<\delta_0$ such that $\Omega_{\delta_0}\subseteq\bigcup_{i=1}^N B(\bm x_i,r_i-2\epsilon)$. Defining $\psi_i^0(\overline{\bm x}^{i})$ from $\varphi_i$ as in Lemma \ref{lem:psiLip} we have
\begin{align*}
(\Omega \cup \Omega_{\delta_0})\cap B(\bm x_i,r_i)&:=\Omega^i_0 \cap B(\bm x_i,r_i)\\
\Omega_{\delta_0}\cap B(\bm x_i,r_i)& :=\Omega_{\delta_0}^i \cap B(\bm x_i,r_i)
\end{align*}
where $\Omega^i_0 =\{\bm x^i \in \mathbb{R}^d:\psi_i^0(\overline{\bm x}^{i})<\tilde{x}^{i}\}$ and $\Omega^i_{\delta_0} =\{\bm x^i \in \mathbb{R}^d:\psi_i^0(\overline{\bm x}^{i})<\tilde{x}^{i}<\varphi_i(\overline{\bm x}^{i})\}$.
Notice here the Lipschitz constants of $\varphi_i$ and $\psi_i^0$
depend on the domain $\Omega$, $\delta_0$ and the collection of balls
$\{B(\bm x_i,r_i)\}$ only and
are thus independent of $\delta$ and $\beta$.
Then given $\delta\in(0,\epsilon)$
we have $\Omega_\delta \subset \Omega_{\delta_0} \subset\bigcup_{i=1}^N B(\bm x_i,r_i-2\epsilon)$ along with functions $\psi_{i}(\overline{\bm x}^{i})$ so that $\hat{\Omega}\cap B(\bm x_i,r_i)=\Omega^i \cap B(\bm x_i,r_i)$ and $
\Omega_{\delta}\cap B(\bm x_i,r_i)=\Omega_{\delta}^i \cap B(\bm x_i,r_i)$ where
\[
\Omega^i =\{{\bm x^i} \in \mathbb{R}^d:\psi_i(\overline{\bm x}^{i})<\tilde{x}^{i}\} \ \text{and} \ \Omega^i_{\delta} =\{{\bm x^i} \in \mathbb{R}^d:\psi_i(\overline{\bm x}^{i})<\tilde{x}^{i}<\varphi_i(\overline{\bm x}^{i})\}.
\]
Define functions $\{ \lambda_i\}_{i=1}^N$ such that
\begin{enumerate}
\item $\lambda_i \in C_c^\infty(B(\bm x_i,r_i-\epsilon)), 1 \le i \le N$,
\item $0 \le \lambda_i \le 1$ and $ \lambda_i\equiv 1$ on $B(\bm x_i,r_i-2\epsilon)$.
\end{enumerate}
Since $\Omega_\delta$ is covered by $\{B(\bm x_i,r_i-2\epsilon)\}_{i=1}^N$, we have $1 \leq \sum_{i=1}^N \lambda_i(\bm x) \leq C$ for a fixed constant $C>0$ depending only on the maximum number of overlapped balls in the set $\{ B(\bm{x}_i,r_i-\epsilon)\}_{i=1}^N$. We also define \[
\widetilde{\lambda_i}(\bm x) = \lambda_i(\bm x)/ \sum_{j=1}^N \lambda_j^2(\bm x)
\]
for each $i\in \{1,\cdots, N \}$. Then since $\sum_{j=1}^N \lambda_j^2(\bm x)$ is uniformly bounded above and below, we also have $\widetilde{\lambda_i} \in C_c^\infty(B(\bm x_i,r_i-\epsilon))$.
Furthermore, we would like to define the extension operator on the general domain. By Theorem \ref{thm:invSpecLip}, there exists an extension operator $E^i: \mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta) \to \mathcal{S}^{\,\beta}_\delta(\Omega^i) $ such that $E^i(\lambda_i u)(\bm x) =(\lambda_i u)(\bm x) $ for $x\in \Omega^i_\delta$ and
\[
\| E^i(\lambda_i u)\|_{\mathcal{S}^{\,\beta}_\delta(\Omega^i)} \leq C|d-\beta|^{-1/p} \| \lambda_i u\|_{\mathcal{T}^{\,\beta}_\delta(\Omega_\delta^i)}\,.
\]
For any $\bm{x} \in \hat\Omega \setminus\Omega^i$, we assume $E^i(\lambda_i u)(\bm{x}) =0$. Then we can define the extension operator $E: L^p(\Omega_\delta) \to L^p(\hat\Omega)$ by
\begin{equation}
\label{def:extension_generalLip}
Eu(\bm x) = \sum_{i=1}^N \widetilde{\lambda_i}(\bm x) E^i(\lambda_i u)(\bm x) \,,
\end{equation}
for any $\bm x\in \hat\Omega$. Before continuing we show some useful estimates which will together swiftly prove Theorem \ref{mainthm_1_general}.
\begin{lem}\label{lem:GenTraceESt1}
Let $\Omega$ be a simply connected Lipschitz domain with an interaction domain $\Omega_\delta$ for $0<\delta< \epsilon$ where $\epsilon$, $\Omega^i$, and $\Omega_\delta^i$,
as well as $\lambda_i$, are defined as above. For any $u \in \mathcal{S}_\delta^{\,\beta}(\hat\Omega)$, $\| u\|^p_{\mathcal{T}_\delta(\Omega_{\delta})} \le C \sum_{i=1}^N\|\lambda_i u\|^p_{\mathcal{T}_\delta(\Omega_{\delta}^i)}$,
where $C$ is a constant independent of $\delta$ and $\beta$.
\end{lem}
\begin{proof}
First note for $\bm x \in \Omega_{\delta}$,
\[
|u(\bm x)|\le\sum_{i=1}^N |(\lambda_i u)(\bm x)|
\]
and hence
\[
\|u\|_{\mathcal{T}^{\,\beta}_\delta(\Omega_{\delta})}\le \sum_{i=1}^N\|\lambda_i u\|_{\mathcal{T}^{\,\beta}_\delta(\Omega_{\delta})}.
\]
Notice that we can extend $\lambda_i$ by zero so that $\lambda_i u$ can be viewed as a function on $\Omega^i$. Using the fact that $\lambda_i\in C_c^\infty(B(\bm x_i,r_i))$ we have
\begin{align}
|\lambda_i u|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega_{\delta})}&=\delta^{\,\beta-2}\left(\int_{\Omega_\delta\cap B(\bm x_i,r_i)}\int_{\Omega_\delta\cap B(\bm x_i,r_i)}+2\int_{\Omega_\delta\cap B(\bm x_i,r_i)}\int_{\Omega_\delta\setminus B(\bm x_i,r_i)}\right)\frac{|\lambda_i u(\bm y)-\lambda_i u(\bm x)|^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge \delta)^{\beta}} d \bm y d \bm x \nonumber\\
=&\delta^{\,\beta-2}\left(\int_{\Omega_\delta^i\cap B(\bm x_i,r_i)}\int_{\Omega_\delta^i\cap B(\bm x_i,r_i)}+2\int_{\Omega_\delta^i\cap B(\bm x_i,r_i)}\int_{\Omega_\delta\setminus B(\bm x_i,r_i)}\right)\frac{|\lambda_i u(\bm y)-\lambda_i u(\bm x)|^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge \delta)^{\beta}} d \bm y d \bm x \nonumber \\
\le &|\lambda_i u|^p_{\mathcal{T}_\delta(\Omega_{\delta}^i)}+2^p\delta^{\,\beta-2}\int_{\Omega_\delta^i\cap B(\bm x_i,r_i-\epsilon)}\int_{\Omega_\delta\setminus B(\bm x_i,r_i)}\frac{|\lambda_i u(\bm x)|^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge \delta)^{\beta}} d \bm y d \bm x. \label{eqn:GeneralLipSemiBound}
\end{align}
Notice that for each $\bm x \in \Omega_\delta\cap B(\bm x_i,r_i-\epsilon)$, and $\delta \in (0,\epsilon)$ we have
\begin{align}
&\delta^{\,\beta-2}\int_{\Omega_\delta\setminus B(\bm x_i,r_i)}\frac{1}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge \delta)^{\beta}} d \bm y=\delta^{-2}\int_{\Omega_\delta\setminus B(\bm x_i,r_i)}\frac{1}{|\bm y-\bm x|^{d+p-2}} d \bm y \nonumber \\
&\le \delta^{-2}\int_{\Omega_\delta \cap\{\bm y \in \mathbb{R}^d: |\bm y-\bm x|>\epsilon\}}\frac{1}{|\bm y-\bm x|^{d+p-2}} d \bm y\le\delta^{-2}\int_{\tilde{x}-\delta/2}^{\tilde{x}+\delta/2}\int_{|\overline{\bm y}-\overline{\bm x}|>\epsilon-\delta/2}\frac{1}{(|\tilde{y}-\tilde{x}|+|\overline{\bm y}-\overline{\bm x}|)^{d+p-2}} d \overline{\bm y} d \tilde{y} \label{eqn:GeneralLipSemiBound2}\\
&\le \delta^{-1} \int_{|\overline{\bm y}|>{\epsilon/2}}\frac{1}{|\overline{\bm y}|^{d+p-2}} d \overline{\bm y}\le C\delta^{-1} \nonumber
\end{align}
where $C$ depends on $\epsilon$, $d$ and $p$. Therefore, the integral term in \eqref{eqn:GeneralLipSemiBound} is bounded by a multiple of $\frac{1}{\delta}\|\lambda_iu\|^p_{L^p(\Omega_\delta^i)}$.
Moreover, using the compact support of $\lambda_i$,
\begin{equation}\label{eqn:GeneralLipL2Bound}
\|\lambda_iu\|^p_{L^p(\Omega_\delta)}=\|\lambda_iu\|^p_{L^p(\Omega_\delta\cap B(\bm x_i,r_i))}=\|\lambda_iu\|^p_{L^p(\Omega_\delta^i)}.
\end{equation}
Therefore \eqref{eqn:GeneralLipL2Bound} along with the estimates of \eqref{eqn:GeneralLipSemiBound} and \eqref{eqn:GeneralLipSemiBound2}
\[
\|\lambda_i u\|^p_{\mathcal{T}_\delta(\Omega_{\delta})} \le C \|\lambda_i u\|^p_{\mathcal{T}_\delta(\Omega_{\delta}^i)}
\]
where the constant $C$ is independent of $\beta$ and $\delta$.
\end{proof}
\begin{lem}\label{lem:GenTraceEst2}
Let $\Omega$ be a simply connected Lipschitz domain with an interaction domain $\Omega_\delta$ for $0<\delta< \epsilon$ where $\epsilon$, $\Omega^i$, and $\Omega_\delta^i$ are defined as above, along with $\lambda_i$ and $\widetilde{\lambda_i}$ for each $i$.
For any $u \in \mathcal{S}_\delta^{\,\beta}(\hat\Omega)$, we have
\[
\|\lambda_i u\|^p_{\mathcal{S}_\delta^{\,\beta}(\Omega^i)} \le C \| u\|^p_{\mathcal{S}_\delta^{\,\beta}(\hat\Omega \cap B(\bm{x}_i ,r_i))}, \quad\text{ and }\quad \|\widetilde{\lambda_i} u\|^p_{\mathcal{S}_\delta^{\,\beta}(\Omega^i)} \le C \| u\|^p_{\mathcal{S}_\delta^{\,\beta}(\hat\Omega \cap B(\bm{x}_i ,r_i))},
\]
where $C$ is independent of $\beta$ and $\delta$.
\end{lem}
\begin{proof}
First, note that $\|\lambda_i u\|_{
L^p(\Omega^i)} =\|\lambda_i u\|_{L^p(\hat{\Omega}\cap B(\bm x_i,r_i))}$ and also if $\bm y \in \Omega^i\setminus B(\bm x_i,r_i)$ and $\bm x \in \Omega^i\cap B(\bm x_i,r_i-\epsilon)$, then
\[
|\bm y-\bm x|\ge |\bm y-\bm x_i|-|\bm x-\bm x_i|> r_i-(r_i-\epsilon)=\epsilon > \delta.
\]
Moreover, using the compact support of $\lambda_i$,
\begin{align*}
|\lambda_i u|^p_{\mathcal{S}_\delta^{\,\beta}(\Omega^i)} &= \left(\int_{\Omega^i\cap B(\bm x_i,r_i)}\int_{\Omega^i\cap B(\bm x_i,r_i)}+2\int_{\Omega^i\cap B(\bm x_i,r_i)}\int_{\Omega^i\setminus B(\bm x_i,r_i)}\right)\gamma^{\,\beta}_\delta(|\bm x-\bm y|)|\lambda_i u(\bm x)-\lambda_i u(\bm y)|^pd\bm y d\bm x\\
&\le |\lambda_i u|^p_{\mathcal{S}_\delta^{\,\beta}(\Omega^i\cap B(\bm x_i,r_i))}+2\int_{\Omega^i\cap B(\bm x_i,r_i-\epsilon)}\int_{\Omega^i\cap\{\bm y \in\mathbb{R}^d: |\bm y-\bm x|>\epsilon\}}\gamma^{\,\beta}_\delta(|\bm x-\bm y|)|\lambda_iu(\bm x)|^pd\bm y d \bm x=|\lambda_i u|^p_{\mathcal{S}_\delta^{\,\beta}(\hat{\Omega}\cap B(\bm x_i,r_i))}
\end{align*}
where the last equality is because $\gamma_\delta^\beta(|\bm x-\bm y|)=0$ since $|\bm x-\bm y|>\epsilon>\delta$.
Then for each $1 \le i \le N$,
\begin{align}\label{eqn:lambdaseminormest1}
&|\lambda_i u|_{\mathcal{S}_\delta^{\,\beta}(\hat{\Omega}\cap B(\bm x_i,r_i))}^p= \int_{\hat{\Omega} \cap B(\bm x_i,r_i)}\int_{\hat{\Omega} \cap B(\bm x_i,r_i)}\gamma^{\,\beta}_\delta(|\bm x-\bm y|)|\lambda_i(\bm y)u(\bm y)-\lambda_i(\bm x)u(\bm x)|^pd\bm y d\bm x\nonumber\\
&\le 2^{p-1}\int_{\hat{\Omega} \cap B(\bm x_i,r_i)}\int_{\hat{\Omega} \cap B(\bm x_i,r_i)}\gamma^{\,\beta}_\delta(|\bm x-\bm y|)|\lambda_i(\bm y)|^p |u(\bm y)-u(\bm x)|^p+\gamma^{\,\beta}_\delta(|\bm x-\bm y|)|u(\bm x)|^p|\lambda_i(\bm y)-\lambda_i(\bm x)|^p d\bm y d\bm x\nonumber\\
&\le 2^{p-1}|u|^p_{\mathcal{S}_\delta^{\,\beta}(\hat{\Omega}\cap B(\bm x_i,r_i))} +2^{p-1}\int_{\hat{\Omega} \cap B(\bm x_i,r_i)}|u(\bm x)|^p\int_{\hat{\Omega} \cap B(\bm x_i,r_i)}\gamma^{\,\beta}_{\delta}(|\bm x-\bm y|)|\lambda_i(\bm y)-\lambda_i(\bm x)|^pd\bm y d\bm x\nonumber\\
&\le 2^{p-1}|u|^p_{\mathcal{S}_\delta^{\,\beta}(\hat{\Omega}\cap B(\bm x_i,r_i))} +2^{p-1} \| \lambda_i\|^p_{C^1}\int_{\hat{\Omega} \cap B(\bm x_i,r_i)}|u(\bm x)|^p\int_{\hat{\Omega} \cap B(\bm x_i,r_i)}\gamma^{\,\beta}_{\delta}(|\bm x-\bm y|)|\bm y-\bm x|^pd\bm y d\bm x\nonumber\\
& \leq C \| u\|^p_{\mathcal{S}_\delta^{\,\beta}(\hat{\Omega}\cap B(\bm x_i,r_i))} ,
\end{align}
where $C$ is independent of $\beta$ and $\delta$.
Also, since $|\lambda_i| \le 1$,
\[
\|\lambda_i u\|_{L^p(\hat\Omega\cap B(\bm x_i,r_i))}^p \le \|u\|_{L^p(\hat\Omega\cap B(\bm x_i,r_i))}^p \le \|u\|_{L^p(\hat\Omega)}^p.
\]
The estimates for $\widetilde{\lambda_i}$ can be done similarly.
\end{proof}
\begin{lem}\label{lem:GenInvTraceEst2}
Let $\Omega$ be a simply connected Lipschitz domain with an interaction domain $\Omega_\delta$ for $0<\delta< \epsilon$ where $\epsilon$, $\Omega^i$, and $\Omega_\delta^i$,
along with
$\lambda_i$,
are defined as above. For any $u \in \mathcal{T}_\delta^{\,\beta}(\Omega_\delta)$, we have
\[
\| \lambda_i u (\bm x) \|^p_{\mathcal{T}^\beta_\delta(\Omega^i_\delta)} \le C \left( \| u \|^p_{\mathcal{T}_\delta^{\,\beta}(\Omega_\delta)} + |d+p-\beta|^{-1}\| u\|^p_{L^p(\Omega_\delta)}\right),
\]
where $C$ is independent of $\delta$ and $\beta$.
\end{lem}
\begin{proof} Since $\lambda_i$ is supported in $B(\bm x_i, r_i-\epsilon)$ and $\lambda_i\leq 1$, it is obvious that $\| \lambda_i u (\bm x) \|_{L^p(\Omega^i_\delta)} =\| \lambda_i u (\bm x) \|_{L^p(\Omega_\delta)}\leq \|u (\bm x) \|_{L^p(\Omega_\delta)}$. Now
\begin{equation}
\label{eq1:GenInvTraceEst2}
\begin{split}
|\lambda_i u|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta)} &= \delta^{\,\beta-2}\left(\int_{\Omega^i_\delta \cap B(\bm x_i, r_i)}\int_{\Omega^i_\delta \cap B(\bm x_i, r_i)} + 2 \int_{\Omega^i_\delta \cap B(\bm x_i, r_i)} \int_{\Omega_\delta^i \backslash B(\bm x_i, r_i)} \right) \frac{|\lambda_i u (\bm y)-\lambda_i u (\bm x) |^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge\delta)^{\,\beta}} d\bm y d\bm x \\
&= |\lambda_i u|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta\cap B(\bm{x}_i, r_i))} + 2 \int_{\Omega^i_\delta \cap B(\bm x_i, r_i-\epsilon)} \int_{\Omega_\delta^i \backslash B(\bm x_i, r_i)} \frac{|\lambda_i u (\bm x) |^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge\delta)^{\,\beta}} d\bm y d\bm x
\end{split}
\end{equation}
For the first term in the last line, we have
\[
\begin{split}
|\lambda_i u|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta\cap B(\bm{x}_i, r_i))} &\leq 2^{p-1} \delta^{\,\beta-2}\int_{\Omega^i_\delta \cap B(\bm x_i, r_i)}\int_{\Omega^i_\delta \cap B(\bm x_i, r_i)} \frac{|\lambda_i(\bm y)|^p |u(\bm y)-u(\bm x)|^p+|u(\bm x)|^p|\lambda_i(\bm y)-\lambda_i(\bm x)|^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge\delta)^{\,\beta}} d\bm y d\bm x \\
\leq& 2^{p-1} | u|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)} + 2^{p-1} \delta^{\,\beta-2}\| \lambda_i\|_{C^1}^p \int_{\Omega_\delta \cap B(\bm x_i, r_i)}|u(\bm x)|^p \int_{\Omega^i_\delta \cap B(\bm x_i, r_i)} \frac{|\bm y-\bm x|^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge\delta)^{\,\beta}} d\bm y d\bm x.
\end{split}
\]
Now for any $\bm{x}\in \Omega^i_\delta \cap B(\bm x_i, r_i)$,
\[
\begin{split}
&\delta^{\,\beta-2}
\int_{\Omega^i_\delta \cap B(\bm x_i, r_i)} \frac{|\bm y-\bm x|^p}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge\delta)^{\,\beta}} d\bm y \\
\leq& \delta^{\,\beta-d-p} \int_{|\bm{y}-\bm{x}|<\delta}\frac{1}{|\bm{y}-\bm{x}|^{\,\beta-p}} d\bm y + \delta^{-2} \int_{\{ \bm{y}\in \Omega_\delta^i: \delta<|\bm{y}-\bm{x}|<2r_i\}}\frac{1}{|\bm{y}-\bm{x}|^{d-2}} d\bm y \\
\leq& C |d+p-\beta|^{-1} + \delta^{-2} \int_{ \left\{ \bm y\in\mathbb{R}^d: \psi_i(\overline{\bm y})<\tilde{y}<\varphi_i(\overline{\bm y}),\, |\bm y-\bm x|<2r_i\right\}}\frac{1}{|\bm{y} - \bm{x}|^{d-2}} d\bm y \,,
\end{split}
\]
where $C$ only depends on $d$. Note that in the last step above, we have adopted the $i$-th local coordinate system defined before to represent the points $\bm x$ and $\bm y$ without adding labels to them. Define $G^i: \mathcal{R}_{-\delta}\to \{ \bm y\in\mathbb{R}^{d}: \psi_i(\overline{\bm y})<\tilde{y}<\varphi_i(\overline{\bm y}) \}$ such that
for any $\bm x=(\tilde{x}, \overline{\bm x})$, with the same $i$-th local coordinate system representation
\begin{equation}
\label{eq2:GenInvTraceEst2}
G^i \bm x = \left(\left(1+\frac{\tilde{x}}{\delta}\right)\varphi_i(\overline{\bm x})-\frac{\tilde{x}}{\delta}\psi_i(\overline{\bm x}), \overline{\bm x}\right)\,.
\end{equation}
We can see from Lemma \ref{lem:projectiondist} that $K_L^\prime |\bm x-\bm z|\leq | G^i \bm x - G^i \bm z| \leq K_L |\bm x -\bm z| $, so $\left|\frac{\partial G^i}{\partial \bm x}\right|\leq K_L$.
By the change of variable,
\[
\begin{split}
&\delta^{-2} \int_{ \left\{ \bm y\in\mathbb{R}^d: \psi_i(\overline{\bm y})<\tilde{y}<\varphi_i(\overline{\bm y}),\, |\bm y-\bm x|<2r_i\right\}}\frac{1}{|\bm{y} -\bm{x}|^{d-2}} d\bm y
\leq K_L \delta^{-2} \int_{\left\{\bm{y}\in\mathcal{R}_{-\delta}: |G^i \bm{y}-\bm{x}|<2r_i\right\}} \frac{1}{|G^i \bm{y} -\bm{x}|^{d-2}} d\bm{y} \\
\leq & \frac{K_L}{(K_L^\prime)^{d-2}} \delta^{-2} \int_{\left\{\bm{y}\in\mathcal{R}_{-\delta}: |\bm{y} - (G^i)^{-1}\bm{x}|<2r_i/K_L^\prime\right\}} \frac{1}{| \bm{y} - (G^i)^{-1}\bm{x}|^{d-2}} d\bm{y}
\end{split}.
\]
Let $\bm{w} = (G^i)^{-1}\bm x = (\tilde{w}, \overline{\bm{w}})$ and $
\mathcal{R}_{-\delta} - \bm{w} := \{ \bm{y}-\bm{w}: \bm{y}\in \mathcal{R}_{-\delta}\} = (-\delta-\tilde{w}, -\tilde{w} )\times \mathbb{R}^{d-1}
$, the last line can be estimated by
\[
\begin{split}
&\frac{K_L}{(K_L^\prime)^{d-2}} \delta^{-2} \int_{\left\{\bm{y}\in\mathcal{R}_{-\delta} -\bm{w}: |\bm{y}|<2r_i/K_L^\prime\right\}} \frac{1}{|\bm{y}|^{d-2}} d\bm{y} \leq \frac{K_L}{(K_L^\prime)^{d-2}} \delta^{-2} \int_{\left\{\bm{y}\in\mathcal{R}_{-\delta} -\bm{w}: |\overline{\bm{y}}|<2r_i/K_L^\prime\right\}} \frac{1}{|\overline{\bm{y}}|^{d-2}} d\bm{y} \\
&= \frac{K_L}{(K_L^\prime)^{d-2}} \delta^{-2} \int_{-\delta-\tilde{w}}^{-\tilde{w}} d\tilde{y} \int_{ |\overline{\bm{y}}|<2r_i/K_L^\prime} \frac{1}{|\overline{\bm{y}}|^{d-2}} d\overline{\bm{y}} \leq C \delta^{-1}\,,
\end{split}
\]
where $C$ is independent of $\delta$ and $\beta$. Collecting the above estimates, we have
\[
|\lambda_i u|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta\cap B(\bm{x}_i, r_i))}\leq C \left( \| u \|^p_{\mathcal{T}_\delta^{\,\beta}(\Omega_\delta)} + |d+p-\beta|^{-1}\| u\|^p_{L^p(\Omega_\delta)}\right).
\]
Now for the second term in \eqref{eq1:GenInvTraceEst2}, we notice that if $\bm x\in B(\bm x_i, r_i-\epsilon)$ and $\bm y\in \mathbb{R}^d \setminus B(\bm x_i, r_i)$, then $|\bm y-\bm x|\geq \epsilon>\delta$, therefore for each $\bm x\in B(\bm x_i, r_i-\epsilon)$
\[
\begin{split}
&\delta^{\,\beta-2} \int_{\Omega_\delta^i \backslash B(\bm x_i, r_i)} \frac{1}{(|\bm y-\bm x|\vee\delta)^{d+p-2}(|\bm y-\bm x|\wedge\delta)^{\,\beta}} d\bm y = \delta^{-2} \int_{\Omega_\delta^i \backslash B(\bm x_i, r_i)} \frac{1}{|\bm y-\bm x|^{d+p-2}} d\bm y\\
\leq& \delta^{-2} \int_{\left\{ \bm y\in\mathbb{R}^d: \psi_i(\overline{\bm y})<\tilde{y}<\varphi_i(\overline{\bm y}),\, |\bm y-\bm x|>\epsilon \right\}} \frac{1}{|\bm y-\bm x|^{d+p-2}} d\bm y \,.
\end{split}
\]
Note that in the last line, we have again adopted the $i$-th local coordinate system. Let $G^i$ be defined in \eqref{eq2:GenInvTraceEst2} and $\bm{w}= (G^i)^{-1}\bm x$. Then by the same reasoning as above,
\[
\begin{split}
&\delta^{-2} \int_{ \left\{ \bm y\in\mathbb{R}^d: \psi_i(\overline{\bm y})<\tilde{y}<\varphi_i(\overline{\bm y}),\, |\bm y-\bm x|>\epsilon\right\}}\frac{1}{|\bm{y} -\bm{x}|^{d+p-2}} d\bm y \le K_L \delta^{-2} \int_{\left\{ \bm y\in \mathcal{R}_{-\delta},\, |G^i \bm y - \bm x|>\epsilon \right\}} \frac{1}{|G^i \bm{y} -\bm{x}|^{d+p-2}} d\bm y \\
\le & \frac{K_L}{(K_L^\prime)^{d+p-2}}\delta^{-2} \int_{\left\{ \bm y\in \mathcal{R}_{-\delta} -\bm{w},\, |\bm y|>\epsilon/K_L \right\}} \frac{1}{|\bm y|^{d+p-2}} d\bm y.
\end{split}
\]
Since $\{ \bm y: |\bm y|>\epsilon/K_L\}\subset \{ \bm y=(\tilde{y},\overline{\bm y}): |\overline{\bm y}|>\sqrt{3}\epsilon/(2K_L)\} \cup \{ \bm y=(\tilde{y},\overline{\bm y}): |\tilde{y}|>\epsilon/(2K_L)\} $, the above quantity can be bounded by
\[
\begin{split}
& \frac{K_L}{(K_L^\prime)^{d+p-2}}\delta^{-2}\left( \int_{\left\{ \bm y\in\mathcal{R}_{-\delta}-\bm{w}, |\tilde{y}|>\epsilon/(2K_L) \right\}}\frac{1}{|\bm y|^{d+p-2}} d\bm y + \int_{\left\{ \bm y\in\mathcal{R}_{-\delta}-\bm{w}, |\overline{\bm y}|>\sqrt{3}\epsilon/(2K_L) \right\}}\frac{1}{|\bm y|^{d+p-2}} d\bm y \right) \\
\leq & C\delta^{-2}\left(
\delta \int_{ \mathbb{R}^{d-1}}\frac{1}{(|\overline{\bm y}|+\epsilon/(2 K_L))^{d+p-2}} d\overline{\bm y} + \delta \int_{|\overline{\bm y}|>\frac{\sqrt{3}\epsilon}{2K_L}} \frac{1}{|\overline{\bm y}|^{d+p-2}} d\overline{\bm y} \right) \leq C \delta^{-1}\,,
\end{split}
\]
where $C$ is a constant that depends on $\epsilon$, $K_L$, $d$ and $p$.
Combining the estimates, we get
\[
|\lambda_i u|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta)} \le C\left( \| u \|^p_{\mathcal{T}_\delta^{\,\beta}(\Omega_\delta)} + |d+p-\beta|^{-1}\| u\|^p_{L^p(\Omega_\delta)}\right)\,.
\]
\end{proof}
\begin{lem}\label{lem:GenInvTraceEst1}
Let $\Omega$ be a simply connected Lipschitz domain with an interaction domain $\Omega_\delta$ for $0<\delta< \epsilon$ where $\epsilon$, $\Omega^i$, and $\Omega_\delta^i$ are defined as above. Assume each $\lambda_i$ as above and $E$ defined by \eqref{def:extension_generalLip}, then for any $u \in \mathcal{T}_\delta^{\,\beta}(\Omega_\delta)$, $\| Eu\|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)} \le C |d-\beta|^{-1/p} \sum_{i=1}^N\|\lambda_i u (\bm x) \|_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta)}$ where $C$ is independent of $\delta$ and $\beta$.
\end{lem}
\begin{proof}
Notice that $Eu|_{\Omega_\delta} = u$ since, for $\bm x\in \Omega_\delta$, $\lambda_i(\bm x) E^i(\lambda_i u)(\bm x) = \lambda_i^2(\bm x) u(\bm x)$ for all $i$.
By \eqref{def:extension_generalLip}, we observe that
\[
\|Eu\|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)} \leq \sum_{i=1}^N \|\widetilde{\lambda_i} E^i(\lambda_i u) \|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}.
\]
For each $i\in \{1,\cdots,N\}$, we have
\[
\begin{split}
&|\widetilde{\lambda_i} E^i(\lambda_i u) |^p_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)} \\
=&
\left(\int_{\hat\Omega\cap B(\bm x_i,r_i)}\int_{\hat\Omega\cap B(\bm x_i,r_i)}+2\int_{\hat\Omega\cap B(\bm x_i,r_i)}\int_{\hat\Omega\setminus B(\bm x_i,r_i)}\right)\gamma^{\,\beta}_\delta(|\bm x-\bm y|)\left|\left(\widetilde{\lambda_i} E^i(\lambda_i u)\right)(\bm x)-\left(\widetilde{\lambda_i} E^i(\lambda_i u)\right)(\bm y)\right|^pd\bm y d\bm x \\
= &| \widetilde{\lambda_i} E^i(\lambda_i u) |^p_{\mathcal{S}^{\,\beta}_\delta(\Omega^i)} + 2\int_{\hat\Omega\cap B(\bm x_i,r_i-\epsilon)}\int_{\hat\Omega\cap\{\bm y \in\mathbb{R}^d: |\bm y-\bm x|>\epsilon\}}\gamma^{\,\beta}_\delta(|\bm x-\bm y|)\left|\left(\widetilde{\lambda_i} E^i(\lambda_i u)\right)(\bm x)-\left(\widetilde{\lambda_i} E^i(\lambda_i u)\right)(\bm y)\right|^pd\bm y d\bm x \\
= &| \widetilde{\lambda_i} E^i(\lambda_i u) |^p_{\mathcal{S}^{\,\beta}_\delta(\Omega^i)} \leq C | E^i(\lambda_i u) |^p_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega\cap B(\bm{x}_i, r_i))} \leq C | E^i(\lambda_i u) |^p_{\mathcal{S}^{\,\beta}_\delta(\Omega^i)} \\
\leq& C |\beta-d|^{-1} \| \lambda_i u\|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta)}\,,
\end{split}
\]
where we have used Lemma \ref{lem:GenTraceEst2} and Theorem \ref{thm:invSpecLip}.
Moreover, since $\widetilde{\lambda_i} \leq 1$, we have
\[
\|\widetilde{\lambda_i} E^i(\lambda_i u) \|^p_{L^p(\hat\Omega)} = \|\widetilde{\lambda_i} E^i(\lambda_i u) \|^p_{L^p(\hat\Omega\cap B(\bm x_i,r_i))} \leq \|E^i(\lambda_i u) \|^p_{L^p(\Omega^i)} \leq C \| \lambda_i u\|^p_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta)}.
\]
Therefore, we have $\| Eu\|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}\leq C |d-\beta|^{-1/p}
\sum_{i=1}^N\|\lambda_i u (\bm x) \|_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta)} $.
\end{proof}
\noindent\textit{Proof of Theorem 1.3.}
To show general trace theorem we use Lemmas \ref{lem:GenTraceESt1} and \ref{lem:GenTraceEst2} along with Theorem \ref{thm:speclip} to obtain the estimate
\begin{align*}
\| u\|_{\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)}&\le C\sum_{i=1}^N\|\lambda_i u\|_{\mathcal{T}_\delta^{\,\beta}(\Omega^i_\delta)}\le C|d+p-\beta|^{-1/p}\sum_{i=1}^N\|\lambda_i u\|_{\mathcal{S}_\delta^{\,\beta}(\Omega^i)} \\
&\le C|d+p-\beta|^{-1/p}\sum_{i=1}^{N}\|u\|_{\mathcal{S}_\delta^{\, \beta}(\hat\Omega\cap B(\bm x_i,r_i))} \le C|d+p-\beta|^{-1/p}\|u\|_{\mathcal{S}_\delta^{\,\beta}(\hat{\Omega})}\,,
\end{align*}
where $C$ is independent of $\beta$ and $\delta$. For the general inverse trace theorem, let $E$ be defined by \eqref{def:extension_generalLip}, then from Lemmas \ref{lem:GenInvTraceEst2} and \ref{lem:GenInvTraceEst1} we have
\[
\| Eu\|_{\mathcal{S}^{\,\beta}_\delta(\hat\Omega)}\le C|d-\beta|^{-1/p}\sum_{i=1}^N\|\lambda_i u \|_{\mathcal{T}^{\,\beta}_\delta(\Omega^i_\delta)} \le C|d-\beta|^{-1/p} \left( \| u \|_{\mathcal{T}_\delta^{\,\beta}(\Omega_\delta)} + |d+p-\beta|^{-1/p}\| u\|_{L^p(\Omega_\delta)}\right) \,,
\]
where $C$ is independent of $\delta$ and $\beta$.
\qed
\section{Conclusion and Discussion}\label{sec:conclusion}
This work gives {suitable} characterizations of the trace spaces of a class of nonlocal function spaces denoted by $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$, where the parameter $\delta$ is the nonlocal interaction length and $\beta$ characterizes the singularity of the nonlocal interaction kernels.
Such nonlocal function spaces have been {extensively} used recently as the energy spaces associated with nonlocal diffusion and nonlocal mechanics models \cite{du2012analysis,silling_2000,mengesha2014nonlocal,DDGG20,you2020data,FR2}.
{However, a clear understanding of the trace spaces of $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ has been largely limited \cite{tian2017trace}. }
In the current work, we have introduced the function space $\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)$ as the trace space of $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ and demonstrated that
the trace map from $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ to $\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)$ is continuous (given by the trace theorem) and conversely, there is a continuous extension operator from $\mathcal{T}^{\,\beta}_\delta(\Omega_\delta)$ to $\mathcal{S}^{\,\beta}_\delta(\hat\Omega)$ (given by the inverse trace theorem).
Moreover, the estimates on the trace and the inverse trace maps are uniform {with respect to the horizon parameter $\delta$}, so that one can recover the classical trace and inverse trace theorems {in the local limit} as the nonlocal interaction length $\delta\to0$. This is also important {since there are many instances of nonlocal models} that recover the classical diffusion or elasticity equations as $\delta\to 0$ \cite{du2012analysis,mengesha2014,mengesha2014nonlocal}.
The investigation of trace spaces of Sobolev spaces has been a classical research area that has important implications in the mathematical and numerical studies of boundary value problems of local PDEs. {The results of this work therefore are expected to be helpful in the rigorous studies of nonlocal equations with possible nonlocal boundary constraints similar to their PDEs counterparts. Studies in this direction are currently underway. }
Moreover, nonlocal functions spaces on vector fields such as those appear in \cite{mengesha2014,mengesha2014nonlocal} can also be studied in the future. Another interesting direction for the future is to investigate, when $\delta\rightarrow \infty$, the consistency of suitably defined nonlocal spaces, similar to those discussed in this work, with their fractional limits.
\section*{Acknowledgments}
Y. Yu is supported by the National Science Foundation under award DMS 1753031.
{Q. Du is supported in part by the National Science Foundation under award DMS-2012562 and the ARO MURI Grant W911NF-15-1-0562.}
X. Tian is supported by the National Science Foundation under award DMS-2111608.
\bibliographystyle{apa}
|
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"redpajama_set_name": "RedPajamaArXiv"
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| 6,779
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THE STORY OF THE PROPHET JONAS.
William Tyndale 1531 translation.
Spelling has been modernized, but nothing else altered.
Note:
This file is the public domain version of Project Gutenberg eBook #12076
The first Chapter.
The word of the Lord came unto the prophet Jonas the son of
Amithai saying: rise and get thee to Ninevehh that great city and preach
unto them, how that their wickedness is come up before me.
And Jonas made him ready to flee to Tharsis from the presence
of the Lord, and gat him down to Joppa, and found there a ship ready to
go to Tharsis, and paid his fare, and went aboard, to go with them to
Tharsis from the presence of the Lord.
But the Lord hurled a great wind in to the sea, so that there
was a mighty tempest in the sea: insomuch that the ship was like to go
in pieces. And the mariners were afraid and cried every man unto his
god, and cast out the goods that were in the ship in to the sea, to
lighten it of them. But Jonas gat him under the hatches and laid him
down and slumbered. And the master of the ship came to him and said
unto him, why slumberest thou? up! and call unto thy god, that God may
think on us, that we perish not.
And they said one to another, come and let us cast lots, to
know for whose cause we are thus troubled. And they cast lots. And the
lot fell upon Jonas.
Then they said unto him, tell us for whose cause we are thus
troubled: what is thine occupation, whence comest thou, how is thy
country called, and of what nation art thou?
And he answered them, I am an Hebrew: and the Lord God of
heaven which made both sea and dry land, I fear. Then were the men
exceedingly afraid and said unto him, why didst thou so? For they knew
that he was fled from the presence of the Lord, because he had told
them.
Then they unto him, what shall we do unto thee, that the sea
may cease from troubling us? For the sea wrought and was troublous. And
he answered them, take me and cast me in to the sea, and so shall it
let you be in rest: for I wot, it is for my sake, that this great
tempest is come upon you. Nevertheless the men assayed with rowing to
bring the ship to land: but it would not be, because the sea so wrought
and was so troublous against them. Wherefore they cried unto the Lord
and said: O Lord let us not perish for this mans death, neither lay
innocent blood unto our charge: for thou Lord even as thy pleasure was,
so thou hast done.
And then they took Jonas, and cast him into the sea, and the
sea left raging. And the men feared the Lord exceedingly: and
sacrificed sacrifice unto the Lord: and vowed vows.
The second Chapter.
But the Lord prepared a great fish, to swallow up Jonas. And so
was Jonas in the bowels of the fish three days and three nights. And
Jonas prayed unto the Lord his God out of the bowels of the fish.
And he said: in my tribulation I called unto the Lord, and he
answered me: out of the belly of hell I cried, and thou heardest my
voice. For thou hadst cast me down deep in the midst of the sea and the
flood compassed me about: and all thy waves and rolls of water went
over me: and I thought that I had been cast away out of thy sight. But
I will yet again look toward thy holy temple. The water compassed me
even unto the very soul of me: the deep lay about me: and the weeds
were wrapped about mine head. And I went down unto the bottom of the
hills, and was barred in with earth on every side for ever. And yet
thou Lord my God broughtest up my life again out of corruption. When my
soul fainted in me, I thought on the Lord: and my prayer came in unto
thee, even into thy holy temple. They that observe vain vanities, have
forsaken him that was merciful unto them. But I will sacrifice unto
thee with the voice of thanksgiving, and will pay that that I have
vowed, that saving cometh of the Lord.
And the Lord spake unto the fish: and it cast out Jonas again
upon the dry land.
The third Chapter.
Then came the word of the Lord unto Jonas again saying: up, and
get thee to Nineveh that great city, and preach unto them the preaching
which I bade thee. And he arose and went to Nineveh at the Lord's
commandment. Nineveh was a great city unto God, containing three days
journey.
And Jonas went to and entered in to the city even a days
journey, and cried saying: There shall not pass forty days but Nineveh
shall be overthrown.
And the people of Nineveh believed God, and proclaimed fasting,
and arrayed themselves in sackcloth, as well the great as the small of
them.
And the tidings came unto the king of Nineveh, which arose out
of his seat, and did his apparel off and put on sackcloth, and sat him
down in ashes. And it was cried and commanded in Nineveh by the
authority of the king and of his lords saying: see that neither man or
beast, ox or sheep taste ought at all, and that they neither feed or
drink water.
And they put on sackcloth both man and beast, and cried unto
God mightily, and turned every man his wicked way, and from doing wrong
in which they were accustomed, saying: who can tell whether God will
turn and repent, and cease from his fierce wrath, that we perish not?
And when God saw their works, how they turned from their wicked ways,
he repented of the evil which he said he would do unto them, and did it
not.
The fourth Chapter.
Wherefore Jonas was sore discontent and angry. And he prayed
unto the Lord and said: O Lord, was not this my saying when I was yet
in my country? And therefore I hasted rather to flee to Tharsis: for I
knew well enough that thou wast a merciful god, full of compassion,
long ere thou be angry and of great mercy and repentest when thou art
come to take punishment. Now therefore take my life from me, for I had
lever die than live. And the Lord said unto Jonas, art thou so angry?
And Jonas gat him out of the city and sat him down on the east
side thereof, and made him there a booth and sat thereunder in the
shadow, till he might see what should chance unto the city.
And the Lord prepared as it were a wild vine which sprang up
over Jonas, that he might have shadow over his head, to deliver him out
of his pain. And Jonas was exceeding glad of the wild vine.
And the Lord ordained a worm against the spring of the morrow
morning which smote the wild vine that it withered away. And as soon
as the sun was up, God prepared a fervent east wind: so that the sun
beat over the head of Jonas, that he fainted again and wished unto his
soul that he might die, and said, it is better for me to die than to
live.
And God said unto Jonas, art thou so angry for thy wild vine?
And he said, I am angry a good, even on to the death. And the Lord
said, thou hast compassion on a wild vine, whereon thou bestowedest no
labour nor made it grow, which sprang up in one night and perished in
another: and should not I have compassion on Nineveh that great city,
wherein there is a multitude of people, even above an hundred thousand
that know not their right hand from the left, besides much cattle?
End of Project Gutenberg's The Story Of The Prophet Jonas, by Anonymous
***
|
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| 8,455
|
<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
package="com.jiahaoliuliu.monkeytalkwithandroidstudiocode" >
<uses-permission android:name="android.permission.INTERNET" />
<uses-permission android:name="android.permission.GET_TASKS" />
<application
android:allowBackup="true"
android:icon="@mipmap/ic_launcher"
android:label="@string/app_name"
android:theme="@style/AppTheme" >
<activity
android:name=".MainActivity"
android:label="@string/app_name" >
<intent-filter>
<action android:name="android.intent.action.MAIN" />
<category android:name="android.intent.category.LAUNCHER" />
</intent-filter>
</activity>
</application>
</manifest>
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"redpajama_set_name": "RedPajamaGithub"
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| 8,619
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Re-thinking where MLB's top 10 free agents will (eventually) sign
By Ken Davidoff
January 13, 2018 | 3:36pm
Jake Arrieta; Lorenzo Cain; Todd Frazier UPI; AP; AP
Yankees' all-in approach means no one is off-limits
Marcus Stroman shows Yankees all they need to see
MLB trade deadline drama could go down to the wire
Other factor Yankees better consider at MLB trade deadline
On Nov. 6, The Post published our Top 30 free-agent rankings and predictions. Since then … not a heck of a lot has happened. Just 11 of those 30 have signed, and just three of the top 10.
So, in light of the market developments (or lack thereof), let's take another whack at the remaining top 10. But first, an update on whom from that group has signed:
Now, the revisions (or lack thereof).
1. Eric Hosmer, 1B
Old prediction: Red Sox, 7 years, $150 million
The Red Sox re-signed Mitch Moreland to play first base, so this likely isn't happening. Unlike many of the players on this list, Hosmer has two rather public suitors in his old team, the Royals, and the Padres.
New prediction: Royals, 7 years, $145 million
2. J.D. Martinez, OF
Old prediction: Mariners, 5 years, $120 million
Seattle's one big move this winter was to acquire Dee Gordon and shift him to center field, which means Nelson Cruz will stay put as the designated hitter and not get more outfield time. The Red Sox and Martinez have been involved in a lengthy game of chicken.
New prediction: Red Sox, 5 years, $120 million
Yu DarvishAP
3. Yu Darvish, RHP
Old prediction: Angels, 5 years, $117.5 million
The Angels, who shocked the baseball world by landing Shohei Ohtani, are not one of the known finalists for Darvish, who finished the 2017 season with the neighboring Dodgers. The Yankees, Astros, Rangers, Cubs and Twins make up that list, as first reported by the Fort Worth Star-Telegram, plus one mystery team that Darvish added while confirming the other five. The Astros' trade for Gerrit Cole will very likely take them out of this sweepstakes.
New prediction: Twins, 4 years, $100 million
4. Jake Arrieta, RHP
Old prediction: Cardinals, 5 years, $110 million
The Cardinals are thinking big this winter, having already traded for Marcell Ozuna and signed pitcher Miles Mikolas from Japan and reliever Luke Gregerson.
New prediction: Cardinals, 4 years, $100 million
5. Lorenzo Cain, OF
Old prediction: Giants, 4 years, $75 million
A guide for MLB free agents to thaw out the spending freeze
Major League Baseball, land of (relative) opportunity and parity, should...
It's deathly quiet on this front, one unimpeded by any Scott Boras clients. Though the Giants could still use him, they appear determined to retain the draft picks they'd have to surrender to sign him.
New prediction: Brewers, 3 years, $60 million
6. Wade Davis, RHP
Prediction: Cubs, 4 years, $64 million
SIGNED: Rockies, 3 years, $52 million
7. Mike Moustakas, 3B
Old prediction: Royals, 6 years, $100 million
Many pegged him to head west and join the Angels, but the Angels signed Zack Cozart to switch from shortstop to the hot corner. The third-base market is quite underwhelming, especially with the Yankees and Mets both facing budget constraints.
New prediction: Braves, 4 years, $70 million
8. Carlos Santana, 1B
Top 30 free-agent predictions: Yankees, Red Sox make splashes
Ready or not, here come the Post's Top 30 free-agent...
Prediction: Indians, 4 years, $65 million
SIGNED: Phillies, 3 years, $60 million
9. Shohei Ohtani, P/OF
Prediction: Yankees, $3.25 million signing bonus
SIGNED: Angels, $2.315 million signing bonus
10. Todd Frazier, 3B
Old prediction: Braves, 3 years, $40 million
He conceded to The Post's Kevin Kernan this past week that it has been a rough go. He might have to wait for Moustakas to decide in order to maximize his value. Or if he doesn't feel like hanging around for that, he might have to accept an inferior deal.
New prediction: Mets, 2 years, $21 million
Filed under carlos santana , eric hosmer , j.d. martinez , jake arrieta , lorenzo cain , mike moustakas , mlb , Shohei Ohtani , todd frazier , wade davis , yu darvish
A guide for MLB free agents to thaw out the spending freez...
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,286
|
using Caliburn.Micro;
namespace csCommon.Plugins.DashboardPlugin
{
public class ChartViewStyle : PropertyChangedBase
{
private ViewTypes type;
public ViewTypes Type
{
get { return type; }
set { type = value; NotifyOfPropertyChange(()=>Type); }
}
private string title;
public string Title
{
get { return title; }
set { title = value; NotifyOfPropertyChange(()=>Title); }
}
public string[] Sensors { get; set; }
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,142
|
You are at:Home»Latest Happenings»Our Interview with World-Renowned Pop Artist Charles Fazzino (part 1)
Our Interview with World-Renowned Pop Artist Charles Fazzino (part 1)
By Liz Champion on May 28, 2015 Latest Happenings
THE CREATIVE PROCESS TO CELEBRATING THE ENCHANTED ISLAND OF SINGAPORE WITH CHARLES FAZZINO
Charles Fazzino is a pop artist most famous for his 3-D representations of New York, London and Paris. This month, to celebrate Singapore's 50th anniversary of independence, Fazzino adds Singapore to his globally oriented collection. In the first of two interviews Charles Fazzino speaks to Liz Champion about his creative process and how he used Singapore for inspiration.
Whether you are looking at a picture of New York, Paris or Singapore there is one thing that makes Charles Fazzino's artwork stand out, and that's happiness.
With their vibrant colours, innovative style, technique and sense of composition Fazzino artwork cannot help but make you feel happy.
"My artwork is meant to be happy," Fazzino says. "It's supposed to be something that everyone can relate to. You don't need an art degree to appreciate what I'm drawing. I try to capture what life is like…what we eat, where we travel, what we do for a living, the events we attend, and the celebrities we idolize. My entire collection taken together can be considered a comprehensive history of popular culture. I want people to look at my artwork, remember things and people that are special to them and smile!"
His inspiration comes from a combination of fun, happiness, colour, whimsical and details. "I bring all these things together to achieve my goal of creating artwork that is happy."
Fazzino's latest 3-D artwork exhibition, THE COLORS OF SINGAPORE, at the Bruno Gallery Singapore aims to capture everything that is special about Singapore.
"First and foremost, it is a celebration of the 50th Anniversary of Singapore. That is the focus," says Fazzino. "For this exhibition, we've included other works that I have done for places around the world including Manhattan, Broadway, London and Paris. My work is globally oriented and I'm proud to add Singapore to my collection now."
To celebrate the nation's jubilee year, the exhibition will also showcase a specially-created artwork – Celebrating the Enchanted Island of Singapore. This piece captures all the major iconic symbols and landmarks that have made Singapore famous worldwide including the Merlion, Singapore Zoo and Marina Bay Sands.
In order to conceptualise this special piece, Fazzino carried out extensive research and worked with the Bruno Art Gallery.
"My extensive research and contacts at Bruno Art Gallery were instrumental. I couldn't have done it without them. They were my eyes and ears on the ground. After this project, I do feel like I've actually been there and I'm looking forward to seeing the actual place to compare it with what I drew. I really wanted to capture EVERYTHING that is special about Singapore. I'm sure there are some things missing but I made sure to get in everything from the Wildlife Reserves Singapore to Orchard Road, Sentosa, Changi Airport, Raffles Hotel and the major buildings downtown. It is hard to pick out individual elements because all together, they become something new."
Whilst creating the artwork, Fazzino developed a keen interest in Singapore, and is looking forward to his first visit.
"This trip will be my first time in Singapore, so I can't tell you yet what I like. However, I can tell you that from doing research for the artwork and talking to people who live there, I am more convinced than ever that Singapore is a very special place…it's mysterious and magical…and full of colour…my kind of town!"
The exhibition takes place from 27 May to 14 June, and is a fantastic opportunity to see the new collection alongside some of his more famous pieces.
The process of creating the artworks is complicated and extremely time-intensive, but it's a process that Charles enjoys.
"All of my artworks are labour of love because there are so much hand-work that goes into them, even for the limited editions. I've been working on the Singapore artwork for many months now. My process is unique but I begin just like every other artist. I create a concept drawing and then make it tighter and tighter until it is a complete drawing of the artwork I want to create. I then send it to my silkscreen printer who prints me out a black line of the image. I will then indicate colours for the printer and send it back."
"The printer separates the image into different colours and begins to print them one by one on top of each other to build the final colour composition. After the silkscreen prints are returned to my studio, my staff and I begin the cutting and gluing process. Each individual artwork is cut out by hand and then layered into 3-dimensions…almost like layering a lasagne in cooking. The pieces are all glued and hand-embellished with acrylic paint glitter and Swarovski Crystals. Each piece can take more than one day to put together. It is a very labour intensive process."
For more information on the exhibition visit http://www.brunoartgroup.com/
In the second part of our interview with Charles Fazzino he talks about his career highlights and how it all began. (coming soon)
Liz Champion
I am a freelance writer and copywriter living and working in the UK. I am a keen runner, avid reader and aspiring author. I am currently studying for a master's degree in creative writing. Please visit my website at www.lizchampion.co.uk or follow me on Twitter @Lizzie_Champion.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,664
|
Q: insert new line with leading whitespace using sed on osx I have a yaml file that looks like this:
:stuff:
- text
What I need in the end is this:
:stuff:
- text
- moretext
Because it's yaml it's whitespace sensitive. I've been trying to use sed to add a new line with leading whitespace and can do it with gnu sed, but I'm still struggling with osx sed.
These methods work with gnu sed, but not with osx/bsd sed.
sed -i '/- text/a\ \ - moretext'
sed -i -e '/- text/{:a;n;/^$/!ba;i\ \ - moretext' -e '}'
Edit:
I understand that posix technically requires a new line after a\ but preferably this could be done in a single line?
A: Try this:
$ sed 's/- text/&\'$'\n'' - moretxt/' file
:stuff:
- text
- moretxt
Obviously it'd be simpler and more portable with awk if you can use that:
$ awk '{print} sub(/- text/,"- moretxt")' file
:stuff:
- text
- moretxt
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,997
|
Prominent & Leading Manufacturer from Surat, we offer folding picnic table-wooden top - 8827, folding picnic table - 8828, folding picnic table-wood finish top, folding picnic table-green finish top, folding picnic table-4 side chairs and folding picnic table-120-separate chairs-blue.
Size: Table Top : 85 x 72.5 x 68 Cm.Height. Seat - 27.5 x 29 x 41 Cm., OPEN - 138 x 81 x 68 Cm Height, CLOSED - 85.5 x 11 x 37 Cm Height. Weight - 8 Kg.
Picnic Table is made of Main Frame- Aluminium alloy metal with Wooden Top. Provided with 4 inbuilt chairs as integral part of a table capable to withstand weight up to 120 Kg on each chair. Table is fully collapsible and folding. Also, can be converted into a bag shape for easy carrying while travel. Specifications: Gross Weight of Table - 8 Kg Size of Table Top - 85 cm x 72.5 x 68 cm.Height.
Size: Table Top : 86 x 67 x 69 Cm.Ht. Seat Height - 40 Cm. Weight - 10 Kg.
Size 85 x 72 x 68 Cm.Height.
Dimensions : Table Top : 85 x 72 x 68 Cm.Height. Seat - 27.5 x 29 x 41 Cm., OVERALL DIMENSIONS OPEN - 138 x 81 x 68 Cm Height, CLOSED - 85.5 x 11 x 37 Cm Height. Weight - 9 Kg.
Picnic Table is made of Main Frame- Aluminium alloy metal with Wooden Top. Provided with 4 inbuilt chairs as integral part of a table capable to withstand weight up to 120 Kg on each chair. Table is fully collapsible and folding. Also, can be converted into a bag shape for easy carrying while travel. Specifications: Gross Weight of Table - 9 Kg Size of Table Top - 85 cm x 72 x 68 cm.Height.
Size: Table Top : 95 x 80 x 71 Cm.Ht. Seat - 29 x 27 x 41 Cm., OPEN - 95 x 80 x 71 Cm Height, CLOSED - 80 x 10 x 48.5 Cm Height. Weight - 11 Kg.
Picnic Table is made of Aluminium alloy metal. Provided with 4 inbuilt chairs as integral part of a table - single chair on each side of a table, capable to withstand weight up to 120 Kg on each chair. Table is fully collapsible and folding. Also, can be converted into a bag shape for easy carrying while travel. Specifications: Gross Weight of Table - 11 Kg Size of Table Top - 95 cm x 80 cm.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,071
|
Sunny Meade provides an exclusive, unique, warm and elegant wedding facility to compliment the most important event of your life. Located in Scott, Louisiana, we provide end-to-end services for weddings, receptions, or any other special occasions including the convenience of planning and coordination (so you don't have to worry about anything), catering, honeymoon suite, music, the ability to have indoor or outdoor ceremony, convenient location… all wrapped in a charming, spacious and picturesque atmosphere which the most special event of your life should deserve.
Built by the Humphrey family in 1899, it has been the home of Sunny-Meade since the late 1980's. Sunny-Meade was originally structured in Jennings, Louisiana. Mrs. Humphrey named her beautiful home after a voluptuous meadow of yellow flowers that sat across the street. Today Sunny-Meade is owned by Charlie and Barbara Primeaux who moved the home to Scott and with care they completely refurbished Sunny-Meade to its original elegance.
Gardens, Gazebo & Atrium to Say I do and Celebrate In!
Sunny-Meade is known for its beautiful courtyard, glassed in atrium and gazebo in a Victorian ambience. Sunny-Meade's courtyard is the perfect setting for your wedding and reception. The facility is for couples seeking "historic elegance" in a Queen Ann setting.
All-Inclusive Wedding Services to Exceed Your Expectations!
Sunny Meade caters to a variety of tastes. We can modify our menus to your preference, ensuring that you and your guests will love and remember our Southern hospitality and cuisine. All of our catering contracts will include food, glassware, silver serving utensils and all required event staff. All beverage services (including non-alcoholic and alcoholic packages) will be provided through Sunny-Meade. We offer a variety of open bar and cash bar packages. All beverages, staff, and glassware will be provided through our beverage contract.
• Will I need to hire a Wedding Planner?
No, Sunny-Meade has a Wedding Planner on staff to help with coordinating.
• Does Sunny-Meade do its own catering?
Yes, there are four menus to choose from, and they range in price.
• Do I have to worry about setting up a bar?
No, Sunny-Meade has several bar packages that include the total set up, such as bartenders, champagne for toasting and champagne for the Honeymoon suite.
• Is the Honeymoon Suite included?
Yes, the Honeymoon couple will have full access to the home, pool and Jacuzzi.
• How many guests can Sunny-Meade accommodate?
Sunny-Meade has a minimum of 100 and a maximum of 300 people.
• How do I go about decorating?
Sunny-Meade has a package that includes centerpieces, garland and gazebo for the ceremony, linens on cake tables, floral arrangements, etc.
A complimentary southern breakfast is served at 7:00-9:00 the next morning.
• How do I reserve my date at Sunny-Meade?
Whenever a date is decided, a deposit is required to reserve your date.
• How far in advance should I reserve my date?
Usually, if a specific date is desired, Sunny-Meade suggests one year in advance. However, that doesn't mean that your date is booked. It is always a good idea to call and check, no matter how late it may seem.
• How do you handle rainy weather?
Sunny-Meade does have a back up plan. In the case of inclement weather, the ceremony is moved in the atrium.
• Who is allowed in the actual home during the reception?
Only immediate family and anyone in group pictures are allowed in the house.
• Is there a place for the Bride to get dressed?
Yes. There is a Bridal Suite for the Bride to use in the house.
• Is there an onsite DJ?
Sunny-Meade does have an onsite DJ, but you are more than welcome to bring in your own DJ or band.
• Would I be able to take pictures at Sunny-Meade?
Of course. If your ceremony and/or reception is planned at Sunny-Meade, you are more than welcome to come and take engagement pictures and bridal portraits at Sunny-Meade.
• What are some of the items that I would be responsible for bringing into Sunny-Meade?
Photographer, cakes, videographer, personal items (sign-in book, pen, toasting glasses, etc.), officiant for the ceremony, your attire (tuxedo, wedding dress, etc.) and florist (for bouquets, boutonnieres and corsages). We take care of the rest!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,308
|
Category: Widgets
...
## Vector Graphics with Dojo's GFX
Vector graphics can have many advantages, including flawless scaling with maximum portability.
The problem with vector graphic creation is that it can be difficult—but not so with Dojo's GFX library.
GFX provides a simple and flexible API (along with other utilities) for creating, animating, and managing amazing vector graphics.
### Getting Started
[Vector graphics](http://en.wikipedia.org/wiki/Vector_graphics)—the use of geometric "primitives" or
shapes—is a time-honored way of creating images by using mathematical formulae to describe how to render something;
unlike raster-based images (such as PNG and JPG files), which use a two dimensional array of colors. Often
vector-based images (such as those made with a program like Illustrator or InkScape) are more efficient because
they are not rendered until the viewing device interprets the math behind it.
There are several advantages to using vector graphics as opposed to fixed JPG/GIF images:
* **Seamless Scalability**: no loss of quality when enlarging or shrinking;
* **Portability**: vector graphics are easily portable and may be rendered in many formats (SVG, Canvas, VML, etc.);
* **Programmable**: you don't need to be a Photoshop or Illustrator expert to quickly create vector graphics.
<!-- protip -->
> Vector graphics have been in use for a long time; one of the most common examples is the use of [PostScript](http://en.wikipedia.org/wiki/PostScript)
to describe how to print something.
The Dojo Toolkit features `dojox/gfx` (GFX), a vector graphic creation library capable of creating extremely powerful
vector graphics. Features of GFX include:
* Works on almost all devices
* Can render images (including charts) with SVG, VML, Silverlight, or Canvas.
* Evaluates the client and uses whichever renderer will work most efficiently
* Allows for the developer to decide which renderer to use
* Allows for linear and radial gradients within shapes (and even works in Internet Explorer!)
<!-- protip -->
> `dojox/gfx` also includes experimental SVG rendering for older versions of Internet Explorer
through the use of the [SVGWeb](http://code.google.com/p/svgweb/) project.
GFX was created to accomplish visual tasks that are not easily accomplished with basic CSS and HTML, all while avoiding
Flash and keeping the API simple.
### Creating an Image using `dojox/gfx`
The following is a general timeline for the creation of most vector graphics:
1. Create the surface (or "canvas")
2. Create the shapes (paths, lines, rectangles, text, etc.)
3. Create groups of shapes (grouping shapes together)
4. Animate shapes or groups of shapes (transform, scale, etc.)
5. Add shape events
To use the GFX library, there's one simple resource to `require`:
```js
require("dojox/gfx", function(gfx) {
});
```
If a specific rendering priority is preferred, it may be added to the `dojoConfig` object that is created before loading Dojo:
```html
<script>
dojoConfig = {
async: true,
gfxRenderer: "svg,silverlight,vml" // svg gets priority
};
</script>
<script src="/path/to/dojo/dojo.js"></script>
```
With GFX available within the page, let's explore each part of a GFX graphic timeline, focusing on both the concepts and the syntax.
### Creating the Surface
We consider the surface to be the "canvas" of the graphic; it hosts all of the graphic's shapes. To create a surface, simply code:
```html
<script>
// Create a GFX surface
// Arguments: node, width, height
require(["dojox/gfx", "dojo/domReady!"], function(gfx) {
gfx.createSurface("surfaceElement", 400, 400);
});
</script>
<!-- DOM element which will become the surface -->
<div id="surfaceElement"></div>
```
That's all that's needed to create the surface! Each rendering engine (SVG, VML) will generate its own code. For example, the SVG renderer will output:
```html
<svg width="400" height="400"><defs></defs></svg>
```
...while the VML rendering engine will output:
```html
<group style="position: absolute; width: 400px; display: inline-block; height: 400px">
<rect style="width: 400px; height: 400px; top: 0px; left: 0px"/>
</group>
```
...and the Canvas engine will render:
```html
<canvas width="400" height="400"></canvas>
```
<!-- button for example links -->
<a href="demo/surface.html" class="button">View Demo</a>
### Creating Shapes
With a surface created, the next step is creating shapes. GFX provides many shapes, including:
* **[Rect](/api/?qs=1.10/dojox/gfx/shape.Rect)**: A basic rectangle
* **[Circle](/api/?qs=1.10/dojox/gfx/shape.Circle)**: A basic circle
* **[Ellipse](/api/?qs=1.10/dojox/gfx/shape.Ellipse)**: A basic ellipse, more flexible than a circle
* **[Line](/api/?qs=1.10/dojox/gfx/shape.Line)**: A basic line
* **[PolyLine](/api/?qs=1.10/dojox/gfx/shape.Polyline)**: A multi-point line
* **[Image](/api/?qs=1.10/dojox/gfx/shape.Image)**: The Image shape allows for loading of bitmap images
* **[Text](/api/?qs=1.10/dojox/gfx/shape.Text)**: Allows for creation of text
* **[TextPath](/api/?qs=1.10/dojox/gfx/path.TextPath)**: A shape that flows text along an arbitrary path. TextPath properties are based on the text shape properties
* **[Path](/api/?qs=1.10/dojox/gfx/path.Path)**: Most versatile geometric shape, which can emulate all other geometric shapes
<!-- protip -->
> Text support within the Canvas rendering engine was added in Dojo 1.6—which is especially useful, since Android does not
presently support SVG.
`dojox/gfx` has implemented a factory pattern for shape creation. Creating a shape is as easy
as `gfx.create_ShapeName_(properties)`. For example, creating a rectangle would look like:
```js
// Create a basic 200px wide, 100px tall rectangle at position 100x, 50y
var rectangle = surface.createRect({ x: 100, y: 50, width: 200, height: 100 });
```
When a shape is created, Dojo generates the the necessary objects within the rendering environment and provides references to them
for future modification and management. The method from above returns the following object:
Rectangle shape properties
Any number of shapes can be created on a given surface. Let's create a series of shapes:
```js
// Create a GFX surface
// Arguments: node, width, height
var surface = gfx.createSurface("surfaceElement", 400, 400);
// Create a rectangle
surface.createRect({ x: 100, y: 50, width: 200, height: 100 })
.setFill("yellow")
.setStroke("blue");
// Add a circle
surface.createCircle({ cx: 100, cy: 300, r:50 })
.setFill("green")
.setStroke("pink");
// Now an ellipse
surface.createEllipse({ cx: 300, cy: 200, rx:50, r:25 })
.setFill("#fff")
.setStroke("#999");
// And a line
surface.createLine({ x1: 10, y1: 50, x2:400, y2:400 })
.setStroke("green");
// How about a polyline?
surface.createPolyline([
{x: 250, y: 250},
{x: 300, y: 300},
{x: 250, y: 350},
{x: 200, y: 300},
{x: 110, y: 250}
]).setStroke("blue");
// Add in an image
surface.createImage({
x:100, y:300, width: 123, height: 56, src: "../images/logo.png"
});
// With some text
surface.createText({ x: 64, y: 220, text: "Vector Graphics Rock!", align: "start" })
.setFont({ family: "Arial", size: "20pt", weight: "bold" }) //set font
.setFill("blue");
// And an advanced textpath
var textShape = surface.createTextPath({ text: "TextPath!" })
.moveTo(125, 70)
.lineTo(285, 20)
.setFont({ family: "Verdana", size: "2em" })
.setFill("black");
// And a simple path
surface.createPath("m100 100 100 0 0 100c0 50-50 50-100 0s-50-100 0-100z")
.setStroke("black");
```
<a href="demo/shapes.html" class="button">View Demo</a>
<!-- protip -->
> Each shape type has its own creation properties; visit the [dojox/gfx](/reference-guide/1.10/dojox/gfx.html)
reference guide to see options for your specific shape. Note also that Path shapes use the
[SVG Path syntax](http://www.w3.org/TR/SVG/paths.html) when using a string as the main argument.
Shapes generated by `dojox/gfx` also include numerous methods for modification. A few key methods include:
* **applyTransform**: Allows for transforming of a shape (scaling and skewing, for example)
* **getFill/setFill**: Get and set fill colors
* **getStroke/setStroke**: Get and set stroke colors
* **moveToBack/moveToFront**: Moves shapes based on "z-indexing"
More details about these methods will be provided later within this tutorial.
<!-- protip -->
> Moving shapes from back to front (and vice-versa) is not quite the same as the `z-index` in CSS; it depends on the rendering engine
being used to draw the shapes.
### Styling Shapes
Creating shapes is easy, but more important than creating the shape is making it look good. The shape objects created by
`dojox/gfx` provides a number of methods to change fill, stroke, and font properties. These methods allow the
developer to style a shape to their heart's content.
#### Filling a Shape
The `setFill` method allows for a named color, hex color, linear gradient, or radial gradient to color (or fill) a shape.
```js
// Create a circle with a set "blue" color
surface.createCircle({ cx: 50, cy: 50, rx:50, r:25 }).setFill("blue");
// Crate a circle with a set hex color
surface.createCircle({ cx: 300, cy: 300, rx:50, r:25 }).setFill("#f00");
// Create a circle with a linear gradient
surface.createRect({ x: 180, y: 40, width: 200, height: 100 }).
setFill({ type:"linear",
x1:0,
y1:0, //x: 0=>0, consistent gradient horizontally
x2:0, //y: 0=>420, changing gradient vertically
y2:420,
colors: [
{ offset: 0, color: "#003b80" },
{ offset: 0.5, color: "#0072e5" },
{ offset: 1, color: "#4ea1fc" }
]
});
// Create a circle with a radial gradient
surface.createEllipse({
cx: 120,
cy: 260,
rx: 100,
ry: 100
}).setFill({
type: "radial",
cx: 150,
cy: 200,
colors: [
{ offset: 0, color: "#4ea1fc" },
{ offset: 0.5, color: "#0072e5" },
{ offset: 1, color: "#003b80" }
]
});
```
<a href="demo/fills.html" class="button">View Demo</a>
<!-- protip -->
> The `colors` array accepts objects with `offset` and `color` keys. The `offset` property represents a number between 0 and 1, and the `color` property represents the color at that offset. You may provide any number of `colors` objects.
#### Setting a Stroke on a Shape
The `setStroke` method styles the shape's stroke (like a border or outline). The `setStroke` method accepts a color string (hex, named color, rgb, etc.) or an object with more specific stroke properties:
```js
// Create a GFX surface
// Arguments: node, width, height
var surface = gfx.createSurface("surfaceElement", 400, 400);
// Create a rectangle with a basic black border
surface.createRect({x: 100, y: 50, width: 200, height: 100 }).setStroke("#000");
// Create a circle with a 3-pixel dotted red border
surface.createCircle({ cx: 300, cy: 300, rx: 50, r: 25 }).setStroke({
style: "Dot", width: 3, cap: "round", color: "#f00"
});
// Create a circle with a 3-pixel dotted red border
surface.createCircle({ cx: 150, cy: 250, rx: 100, r: 50 }).setStroke({
style: "Dash", width: 3, cap: "butt", color: "#00f"
});
```
Properties may include:
* **style**: the style of the line (solid, dotted, dashed)
* **width**: the width of stroke in pixels
* **color**: the stroke's color
* **cap**: the shape of the end of the stroke
<a href="demo/strokes.html" class="button">View Demo</a>
#### Choosing a Font
Both the Text and TextPath shapes allow for a specific font family, size, and weight. Usage of `setFont` is easy:
```js
// Create the initial text, set the font to 20pt Arial Bold, and fill it blue
surface.createText({ x: 64, y: 220, text: "This is my text", align: "start"}).
setFont({ family: "Arial", size: "20pt", weight: "bold" }).
setFill("blue");
```
The font properties are formatted and work very much like CSS properties you use every day!
### Grouping Shapes Together
Individual shapes may be "glued" together in groups, so that the shapes within a group can be treated like they are a single shape.
Groups are especially important when animating related shapes and attaching events to said shapes. The best thing about using groups is
that they feature the same animation methods as individual shapes:
```js
// Create a GFX surface
// Arguments: node, width, height
var surface = gfx.createSurface("surfaceElement", 400, 400);
// Create a group
var group = surface.createGroup();
// Add a shape directly to the group instead of the surface
var rectShape = group.createRect({ x: 0, y: 0, width: 200, height: 100 })
.setFill("#0000ae");
```
Shapes can also be added to the group at any time:
```js
// Create the shape on the surface
var rectShape = shape.createRect({ x: 0, y: 0, width: 200, height: 100 })
.setFill("#0000ae");
// Move it to the group!
group.add(rectShape);
```
Groups are especially useful when creating moveable shapes:
```js
// Require the resource
require("dojox/gfx", "dojox/gfx/Moveable", function(gfx, Moveable) {
// Make all shapes within the group move together!
new Moveable(group);
});
```
The above snippet allows users to click and hold any shape within the group to move every shape in the group around.
<a href="demo/moveable.html" class="button">View Demo</a>
### Animations and Transformations
The real power of the GFX library lies within its animation capabilities. GFX's animations are extremely powerful and smooth,
and capable of many animations—including simple stroke and fill animations, scaling, rotating, and skewing. The `dojox/gfx/fx`
resource was created to allow for both simple and complex animations.
The first step in creating GFX animations is requiring the resource:
```js
// Require the powerful gfx.fx resource
require(["dojox/gfx", "dojox/gfx/fx"], function(gfx, gfxFx) {
});
```
<!-- protip -->
> Transform capabilities are hosted by each individual shape, so no additional resources are required for scaling, skewing, etc.
#### Fill, Stroke, and Font Animations
The `dojox/gfx/fx` resource provides `animateFill`, `animateFont`, and `animateStroke`
methods for easy animation of each shape property:
```js
// Create a circle, a dojox/gfx.fx instance, play it immediately
var circle = surface.createCircle({ cx: 50, cy: 50, rx: 50, r: 25 })
.setFill("blue");
gfxFx.animateFill({
shape: circle,
duration: 500,
color: { start: "blue", end: "green" }
}).play();
// Create a rectangle, animate its stroke
var rectangle = surface.createRect({ x: 100, y: 50, width: 200, height: 100 })
.setStroke("yellow");
gfxFx.animateStroke({
shape: rectangle,
duration: 500,
color: { start: "yellow", end: "pink" },
width: { end: 15 },
join: { values: ["miter", "bevel", "round"] }
}).play();
// Create text, animate it
var text = surface.createText({
x: 64, y: 220, text: "Vector Graphics Rock!", align: "start"
}).setFont({ family: "Arial", size: "20pt", weight: "bold" })
.setFill("red");
gfxFx.animateFont({
shape: text,
duration: 500,
variant: { values: ["normal", "small-caps"] },
size: { end: 10, units: "pt" },
color: "green"
}).play();
```
Each method has its own properties relative to the property (stroke, fill, text) being changed. Also note that gradient
backgrounds cannot be animated; solid colors are animated flawlessly.
<a href="demo/animate-fill.html" class="button">View Demo</a>
#### Rotating a Shape
Shape rotation is also incredibly easy with GFX's animation API. The `rotateAt` and `rotategAt`
animations allow for rotating shapes or groups of shapes around a given center-point. The `gfxFx.animateTransform`
method will assemble the animation, and the play method will start it.
```js
// Create a group
var group = surface.createGroup();
// Create a circle
var circle1 = group.createCircle({ cx: 100, cy: 300, r: 50 }).setFill("green");
var circle2 = group.createCircle({ cx: 100, cy: 100, r: 50 }).setFill("blue");
var circle3 = group.createCircle({ cx: 300, cy: 300, r: 50 }).setFill("black");
var circle4 = group.createCircle({ cx: 300, cy: 100, r: 50 }).setFill("yellow");
// Create an animation of the group
var animation = new gfxFx.animateTransform({
duration: 700,
shape: group,
transform: [{
name: "rotategAt",
start: [0, 200, 200], // Starts at 0 degree rotation at center-point 200x200
end: [360, 200, 200] // Ends at 360 degrees
}]
});
// Showtime!
animation.play();
```
The `transform` property passed to `gfxFx.animateTransform` allows for any number of animations to be added.
<a href="demo/rotate.html" class="button">View Demo</a>
<!-- protip -->
> If you are wondering why there are `rotateAt` and `rotategAt` methods, it is because geometry with
JavaScript is usually _radian_-based; but most developers find it easier to think in terms of _degrees_. Both
rotate a shape around a specific point on the surface.
#### Scaling and Skewing
The process of shrinking, enlarging, and skewing GFX graphics is simple. Use the `gfx.matrix.scale` method to scale
the image, providing `x` and `y` values for the amount to scale on each axis:
```js
// Double the size of the shape
shapeGroup.applyTransform(gfx.matrix.scale({ x: 2, y: 2 }));
// Shrink the shape to half size
shapeGroup.applyTransform(gfx.matrix.scale({ x: 2, y: 0.5 }));
```
The `applyTransform` method of a shape is used to perform the transformation. The graphic will be flawlessly resized!
Skewing (transforming or moving points of a shape along a single axis) is just as easy:
```js
// Skews the group at -20 degrees
shapeGroup.applyTransform(gfx.matrix.skewYg(-20));
```
<a href="demo/scale.html" class="button">View Demo</a>
<!-- protip -->
> The `gfx.matrix` resource contains numerous helpers to invert, rotate, scale, and skew shapes so that you don't
need to know the complex math equations behind setting up any shape modifications or transformations.
### Events
Events are particularly important in `dojox/gfx` as they allow for triggering movement and display changes within groups.
Many of the [charting plugins](../charting) are triggered by events on GFX-created shapes. You can add events to GFX-created nodes or groups using `shape.on and group.on`, which extends `dojo/on`.
#### shape.on
The `shape.on` method works very much like a native event handler. Provide a shape and event type:
```js
// Add a circle
var circle = group.createCircle({ cx: 100, cy: 300, r: 50 })
.setFill("green").setStroke("pink");
// Add a click event to the circle to change its fill!
circle.on("click", function(e) {
circle.setFill("red");
});
```
#### group.on
The `group.on` method of GFX groups allows for `dojo/on`-style event listeners:
```js
// Get the eventSource to find out what element the event occurred on
group.on("click", function(e) {
//our shape was clicked, now do something!
}, true);
```
The event object is very much like a standard DOM event object. The `target` property provides the GFX-generated DOM element which was clicked.
<a href="demo/events.html" class="button">View Demo</a>
<div class="proTip">
<!-- protip -->
> The Silverlight rendering engine supports the following events: `onclick`, `onmouseenter`, `onmouseleave`, `onmousedown`, `onmouseup`, `onmousemove`, `onkeydown`, and `onkeyup`. If you want to target the broadest range of renderers, you are advised to restrict yourself to
this list of events.
</div>
### Create the Dojo Logo with GFX
If we have the Dojo Toolkit logo in an SVG format, the path information describing the shapes of that logo can be extracted and used to
create a GFX-based image. The following is a portion of that SVG file:
```html
<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" width="1100px" height="700px" viewBox="0 0 1100 700" enable-background="new 0 0 1100 700" xml:space="preserve">
<g>
<g>
<path fill="#010101" d="M826.698,536.736v11.722h12.71v6.758h-12.71v26.25c0,6.065,1.718,9.483,6.659,9.483 c2.427,0,3.834-0.203,5.147-0.61l0.406,6.759c-1.721,0.621-4.439,1.211-7.876,1.211c-4.145,0-7.466-1.404-9.575-3.729 c-2.429-2.729-3.442-7.062-3.442-12.82v-26.555h-7.576v-6.756h7.576v-8.996L826.698,536.736z"/>
<path fill="#010101" d="M868.708,598.43c-13.119,0-23.418-9.695-23.418-25.142c0-16.354,10.801-25.938,24.225-25.938 c14.039,0,23.525,10.196,23.525,25.036c0,18.175-12.623,26.05-24.229,26.05h-0.103V598.43L868.708,598.43z M869.115,591.764 c8.481,0,14.846-7.975,14.846-19.089c0-8.267-4.146-18.686-14.643-18.686c-10.396,0-14.931,9.694-14.931,18.976 c0,10.717,6.052,18.783,14.638,18.783h0.09V591.764L869.115,591.764z"/>
<path fill="#010101" d="M924.162,598.43c-13.119,0-23.406-9.695-23.406-25.142c0-16.354,10.801-25.938,24.213-25.938 c14.039,0,23.517,10.196,23.517,25.036c0,18.175-12.611,26.05-24.216,26.05h-0.106L924.162,598.43L924.162,598.43z M924.574,591.764c8.487,0,14.834-7.975,14.834-19.089c0-8.267-4.129-18.686-14.638-18.686c-10.395,0-14.94,9.694-14.94,18.976 c0,10.717,6.063,18.783,14.643,18.783h0.103L924.574,591.764L924.574,591.764z"/>
<!-- more SVG below this shape... -->
</g>
</g>
</svg>
```
Judging by the SVG above, it's easy to deduce that:
* The canvas is approximately 1100 pixels wide and 700 pixels tall.
* The letters should be drawn with paths (`surface.createPath`)
* The fill color of each letter (per the logo) is `#010101`. This example will use a gradient fill, however.
Using the path information in the logo is simple:
```js
// Arguments: node, width, height
var surface = gfx.createSurface("surfaceElement",1100,700);
// Regular fill
var regularFill = { type: "linear", x1: 0, y1: 0, x2: 0, y2: 900, colors: [{ offset: 0, color: "#555" }, { offset: 1, color: "#000"}] };
// Create group too contain each letter of "toolkit"
var tkGroup = surface.createGroup();
// Tiny letter "t" in "toolkit"
var letterToolkitT = tkGroup.createPath("M826.698,536.736v11.722h12.71v6.758h-12.71v26.25c0,6.065,1.718,9.483,6.659,9.483 c2.427,0,3.834-0.203,5.147-0.61l0.406,6.759c-1.721,0.621-4.439,1.211-7.876,1.211c-4.145,0-7.466-1.404-9.575-3.729 c-2.429-2.729-3.442-7.062-3.442-12.82v-26.555h-7.576v-6.756h7.576v-8.996L826.698,536.736z").setFill(regularFill).setStroke("#000");
// "o"
var letterToolkitO1 = tkGroup.createPath("M868.708,598.43c-13.119,0-23.418-9.695-23.418-25.142c0-16.354,10.801-25.938,24.225-25.938 c14.039,0,23.525,10.196,23.525,25.036c0,18.175-12.623,26.05-24.229,26.05h-0.103V598.43L868.708,598.43z M869.115,591.764 c8.481,0,14.846-7.975,14.846-19.089c0-8.267-4.146-18.686-14.643-18.686c-10.396,0-14.931,9.694-14.931,18.976 c0,10.717,6.052,18.783,14.638,18.783h0.09V591.764L869.115,591.764z").setFill(regularFill).setStroke("#000");
// More "letter" shapes here...
```
When all of the paths are drawn to the surface, the following vector graphic will be created:
[](demo/logo.html)
GFX logo created from SVG paths
When the information describing the Dojo Toolkit logo is loaded, it may be converted to any of the supported renderers effortlessly by
changing GFX's default rendering engine order. The graphic shapes and properties may also be animated or modified as desired.
[This demo](demo/logo.html) uses many of the animation techniques described in this tutorial to modify and animate the
Dojo Toolkit logo.
<a href="demo/logo.html" class="button">View Demo - Dojo Logo</a>
<a href="demo/london.html" class="button">View Demo - London Ajax Logo</a>
<!-- tutorials end with a "Conclusion" block -->
### Conclusion
Dojo's GFX library provides the ability to create simple vector graphics or more complex vector graphic groups. The Dojo Toolkit's
[advanced charting](../charting) and drawing libraries are based on the power of GFX. No matter what your medium, the Dojo
Toolkit provides an easy to use API for creating, animating, and managing your vector graphics!
### GFX Resources
Looking for more detail about Dojo's GFX library? Check out these great resources:
* [GFX Resource Guide](/reference-guide/1.10/dojox/gfx.html)
* [Dive Into Dojo GFX](http://www.sitepen.com/blog/2010/12/30/dive-into-dojo-gfx/)
* [GFX API Documentation](/api/?qs=1.10/dojox/gfx)
* [GFX FX API Documentation](/api/?qs=1.10/dojox/gfx/fx)
* [GFX Animations Matrix API Documentation](/api/?qs=1.10/dojox/gfx/matrix)
* [GFX Tests](http://download.dojotoolkit.org/release-1.10.4/dojo-release-1.10.4/dojox/gfx/tests/)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,502
|
Q: Search for string in one column using strings from another column in another dataframe in R I have 2 dataframes (both dataframes have 1 column each) and I want to search for strings present in the 1st column in the 1st dataframe for their presence in each row in the 2nd column of the other dataframe. If present, return the string value in a new column ("String") and a boolean column ("Match"). I tried a few commands like grepl and stringr but could not make it work. Thanks!
Sample below:
1st Dataframe
SName
svc1
svc123
svc567
2nd Dataframe
Description
- ls svc368 -@#@#
mkdir test svc #*-/
mkdir df2 svc123 #*-/
mkdir random svc1 #*-/
mkdir test svc1 *&%^$%$
mkdir fr svc567 *&%@
mkdir 82 svc56 *&??//
mkdir kol svc *&
Result desired:
Description
Match
String
- ls svc368 -@#@#
No
mkdir test svc #*-/
No
mkdir df2 svc123 #*-/
Yes
svc123
mkdir random svc1 #*-/
Yes
svc1
mkdir test svc1 *&%^$%$
Yes
svc1
mkdir fr svc567 *&%@
Yes
svc567
mkdir 82 svc56 *&??//
No
mkdir kol svc *&
No
A: One approach would be to form a regex alternation of the terms in the first dataframe. Then use grepl and sub to generate the output columns.
regex <- paste0("\\b(", paste(df1$SName, collapse="|"), ")\\b")
df2$match <- ifelse(grepl(regex, df2$Description), "Yes", "No")
df2$String <- ifelse(grepl(regex, df2$Description),
sub(paste0(".*", regex, ".*"), "\\1", df2$Description),
"")
df2
Description match String
1 - ls svc368 -@#@# No
2 mkdir test svc #*-/ No
3 mkdir df2 svc123 #*-/ Yes svc123
...
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,489
|
namespace wi::gpusortlib
{
// Perform bitonic sort on a GPU dataset
// maxCount - Maximum size of the dataset. GPU count can be smaller (see: counterBuffer_read param)
// comparisonBuffer_read - Buffer containing values to compare by (Read Only)
// counterBuffer_read - Buffer containing count of values to sort (Read Only)
// counterReadOffset - Byte offset into the counter buffer to read the count value (Read Only)
// indexBuffer_write - The index list which to sort. Contains index values which can index the sortBase_read buffer. This will be modified (Read + Write)
void Sort(
uint32_t maxCount,
const wi::graphics::GPUBuffer& comparisonBuffer_read,
const wi::graphics::GPUBuffer& counterBuffer_read,
uint32_t counterReadOffset,
const wi::graphics::GPUBuffer& indexBuffer_write,
wi::graphics::CommandList cmd
);
void Initialize();
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,484
|
{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/dcdsb.2012.17.2017","text":"Article Contents\nArticle Contents\n\n# The regularized implied local volatility equations -A new model to recover the volatility of underlying asset from observed market option price\n\n\u2022 In this paper, we propose a new continuous time model to recover the volatility of underlying asset from observed market European option price. The model is a couple of fully nonlinear parabolic partial differential equations (see (34), (36)). As an inverse problem, the model is deduced from a Tikhonov regularization framework. Based on our method, the recovering procedure is stable and accurate. It is justified not only in theoretical proofs, but also in the numerical experiments.\nMathematics Subject Classification: 35K85, 35R30, 49J20, 49J40, 49K20, 49K40.\n\n Citation:\n\n\u2022 [1] Y. Achdou, An inverse problem for a parabolic variational inequality with an integro-differential operator, Siam J. Control Optim., 47 (2008), 733-767.doi:\u00a010.1137\/060660692. [2] Y. Achdou and O. Pironneau, Volatility smile by multilevel least square, Int. J. Theor. Appl. Finance, 5 (2002), 619-643. [3] Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods and Applications of Analysis, 11 (2004), 533-556. [4] J. Andreasen, Implied modelling: Stable implementation. Hedging and duality, working paper, The Aarhus School of Business, 1996. [5] M. Avellaneda, C. Friedman, R. Holmes and D. Samperi, Calibrating volatility surfaces via entropy, Applied Math. Finance, 4 (1997), 37-64. [6] F. Abergel and R. Tachet, A nonlinear partial integro-differential equations from mathematical finance, Discrete and Continuous Dynamical Systems, 27 (2010), 907-917.doi:\u00a010.3934\/dcds.2010.27.907. [7] H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance, C. R. Acad. Sci. Paris S\u00e9r I Math., 331 (2000), 965-969. [8] H. Berestycki, J. Busca and I. Florent, Asymptotics and calibration of local volatility models, Quantitative Finance, 2 (2002), 61-69. [9] D. Betes, Testing option pricing models, in \"Statistical Methods in Finance\" (eds. G. S. Maddala and C. R. Rao), Handbook of Statistics, 14, Elsevier Science B.V., (1996), 567-611. [10] F. Black, Fact and fantasy in the use of options, Financial Analysis J., 31 (1975), 36-72.doi:\u00a010.2469\/faj.v31.n4.36. [11] J. N. Bodurtha and M. Jermakyan, Non-parametric estimation of an implied volatility surface, Jour. of Computational Finance, 2 (1999), 29-60. [12] I. Bouchouev and V. Isakov, The inverse problem of option pricing, Inverse Problem, 13 (1997), L11-L17.doi:\u00a010.1088\/0266-5611\/13\/5\/001. [13] I. Bouchouev and V. Isakov, Uniqueness, Stability and numerical methods for inverse problem that arises in financial markets, Inverse Problem, 15 (1999), R95-R116.doi:\u00a010.1088\/0266-5611\/15\/3\/201. [14] D. Breeden and R. Litzenberger, Prices of state-contingent claims implicit in option prices, Journal of Business, 51 (1978), 621-651.doi:\u00a010.1086\/296025. [15] J. R. Cannon, P. Duchatean and K. Steube, \"Identifying a Time Dependent Unknown Coefficient in a Nonlinear Heat Equation,\" Nonlinear Diffusion Equations & their Equilibrium States, 3, Birkhauser, (1992), 153-169. [16] J. R. Cannon, \"The One-Dimensional Heat Equations,\" Encyclopedia of Mathematics and its Applications, Vol. 23, Addison-Wesley Publishing Company, 1984. [17] S. Cr\u00e9pey, Calibration of local volatility in a trinomial tree using Tikhonov regularization, Inverse Problems, 19 (2003), 91-127.doi:\u00a010.1088\/0266-5611\/19\/1\/306. [18] S. Cr\u00e9pey, Calibration of volatility in a generalized Black-Scholes model using Tikhonov regularization, SIAM J. Math. Anal., 34 (2003), 1183-1206.doi:\u00a010.1137\/S0036141001400202. [19] E. Derman and I. Kani, Riding on a smile, Risk, 7 (1994), 32-39. [20] E. Derman, I. Kani and J. Zou, The local volatility surface: Unlocking the information in index option prices, Financial Analysis J., 52 (1996), 25-36.doi:\u00a010.2469\/faj.v52.n4.2008. [21] B. Dupire, Pricing and hedging with smile, \"Mathematics of Derivative Securities\" (Cambridge, 1995), Publ. Newton Inst., 15, Cambridge Univ. Press, Cambridge, 1997. [22] B. Dupire, Pricing with a smile, Risk, 7 (1994), 18-20. [23] N. El Karoui, Measuring and hedging financial risks in dynamical world, in \"Proceedings of ICM, Vol. III\" (Beijing, 2002), Higher Ed. Press, Beijing, (2002), 773-783. [24] T. Hein and B. Hofmann, On the nature of ill-posedness of an inverse problem arising in option pricing, Inverse Problems, 19 (2003), 1319-1338.doi:\u00a010.1088\/0266-5611\/19\/6\/006. [25] K.-H. Hoffmann, L. Jiang and M. Niezg\u00f3dka, Optimal control of phase change processes with terminal state observation, J. Partial Diff. Eqns., 6 (1993), 97-107. [26] L. Jiang and B. Bian, An inverse problem for parabolic equations with non-divergent form, working paper, Tongji University, 2010. [27] L. Jiang and Y. Tao, Identifying the volatility of underlying assets from option prices, Inverse Problems, 17 (2001), 137-155.doi:\u00a010.1088\/0266-5611\/17\/1\/311. [28] L. Jiang, Q. Chen, L. Wang and Jin E Zhang, A new well-posed algorithm to recover implied local volatility, Quantitative Finance, 3 (2003), 451-457. [29] L. Jiang, \"Mathematical Modelling and Methods of Financial Derivatives,\" High Education Press, Beijing, 2003. [30] D. Kinderleher and G. Stampacchia, \"An Introduction to Variational Inequalities and Their Applications,\" Academic Press, 1980. [31] R. Lagnado and S. Osher, A technique for calibrating derivative security pricing models: Numerical solution of an inverse problem, J. Computational Finance, 1 (1997), 13-25. [32] J. Macbeth and L. Merville, An empirical estimation of Black-Scholes call option pricing model, Jour. of Finance, 34 (1979), 285-301. [33] S. Mayhew, Implied volatility, Financial Analysis J., 51 (1995), 8-13. [34] K. Shastri and K. Wethyavivorn, The valuation of currency options for alternate stochastic process, Jour. of Financial Research, 10 (1987), 283-293. [35] G. Skiadopoulos, Volatility smile consistent option models: A survey, International Journal of Theoretical and Applied Finance, 4 (2001), 403-437. [36] P. Wilmott, \"Derivatives-The Theory of Practice of Financial Engineering,\" John Wiley & Sons, New York, 1998.","date":"2023-03-20 16:28:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7052314877510071, \"perplexity\": 3919.436187328128}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943484.34\/warc\/CC-MAIN-20230320144934-20230320174934-00294.warc.gz\"}"}
| null | null |
Daoism and
Anarchism
CONTEMPORARY ANARCHIST STUDIES
A series edited by
Laurence Davis _National University of Ireland, Maynooth_
Uri Gordon _Arava Institute for Environmental Studies, Israel_
Nathan Jun _Midwestern State University, USA_
Alex Prichard _London School of Economics, UK_
Contemporary Anarchist Studies promotes the study of anarchism as a framework for understanding and acting on the most pressing problems of our times. The series publishes cutting edge, socially-engaged scholarship from around the world—bridging theory and practice, academic rigor and the insights of contemporary activism.
The topical scope of the series encompasses anarchist history and theory broadly construed; individual anarchist thinkers; anarchist-informed analysis of current issues and institutions; and anarchist or anarchist-inspired movements and practices. Contributions informed by anti-capitalist, feminist, ecological, indigenous, and non-Western or global South anarchist perspectives are particularly welcome. So, too, are manuscripts that promise to illuminate the relationships between the personal and the political aspects of transformative social change, local and global problems, and anarchism and other movements and ideologies. Above all, we wish to publish books that will help activist scholars and scholar activists think about how to challenge and build real alternatives to existing structures of oppression and injustice.
_International Editorial Advisory Board_
Martha Ackelsberg, _Smith College_
John Clark, _Loyola University_
Jesse Cohn, _Purdue University_
Ronald Creagh, _Université Paul Valéry_
Marianne Enckell, _Centre International de Recherches sur l'Anarchisme_
Benjamin Franks, _University of Glasgow_
Judy Greenway, _University of East London_
Ruth Kinna, _Loughborough University_
Todd May, _Clemson University_
Salvo Vaccaro, _Università di Palermo_
Lucien van der Walt, _University of the Witwatersrand_
Charles Weigl, _AK Press_
Daoism and
Anarchism
Critiques of State Autonomy
in Ancient and Modern China
**_John A. Rapp_**
Contemporary Anarchist Studies
**Continuum International Publishing Group**
**_A Bloomsbury Company_**
50 Bedford Square London WC1B 3DP
80 Maiden Lane New York NY 10038
**www.continuumbooks.com**
© John A. Rapp, 2012
All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the permission of the publishers.
ISBN: 978-1-4411-2745-7
**Library of Congress Cataloging-in-Publication Data**
A catalog record for this book is available from the Library of Congress.
_For Anita, who for better or worse,
insisted that I write this book_
CONTENTS
_Acknowledgments_
PRELUDE
PART 1:DAOISM AND ANARCHISM
**1** Daoism and anarchism reconsidered
**2** Utopian, anti-utopian, and dystopian ideas in philosophical Daoism
**3** Daoism as utopian or accommodationist: The Guodian challenge to Daoist anarchism
**4** Daoism as anarchism or nihilism: The Buddhist-influenced thought of Wu Nengzi
INTERLUDE
**5** The twentieth-century Chinese anarchist movement
PART 2:MAOISM AND ANARCHISM
**6** Maoism and anarchism: An analysis of Mao Zedong's response to the anarchist critique of Marxism
**7** Denunciations of anarchism in the PRC
**8** Extra-Party neo-anarchist critiques of the state in the PRC
**9** Inner Party neo-anarchist critiques of the Leninist Party-state
POSTLUDE
The continuing relevance of Daoist anarchism
APPENDICES
Works of Daoist Anarchism
**1** _Zhuangzi,_ Chapter 9, "Horses' Hoofs"
**2** Ruan Ji, "The Biography of Master Great Man" (excerpt)
**3** Bao Jingyan
**4** Tao Qian, "Peach Blossom Spring"
**5** _Wunengzi_
_Bibliography_
_Index_
ACKNOWLEDGMENTS
Since this book is the product of many years of research and writing, I have many people to thank for their help and encouragement. To the usual disclaimer that the author alone is responsible for any mistakes and shortcomings, I must add a special emphasis that due to the controversial nature of much of this book, none of the people thanked below necessarily agrees with any of the opinions or analysis expressed herein.
Thanks and appreciation go first to this author's many teachers in subjects directly related to this book, including, at Indiana University, Judith Berling for her seminar on Daoism and for serving as advisor on my MA thesis, and the late Jerome Mintz and Alan Ritter for their seminar on anarchism. At the University of Wisconsin, I am very grateful to my two main mentors, Crawford Young as whose teaching assistant I was introduced to the literature on state autonomy, and especially Ed Friedman, who continues to impart to me much wisdom about Chinese politics and give generous and patient advice and criticism. I also owe a debt of gratitude to John Clark of Loyola University for his encouragement and support over the years.
A second round of thanks go to the students, faculty, staff, and alumni of Beloit College, first for their tolerance and inspiration in allowing me to teach classes on China, Daoism, and anarchism over the years and for sabbatical grants in 1999–2000 and 2006 and other research support, including a Sanger summer research grant in 2008 that allowed for the translation by Catrina Siu of the entire _Wunengzi,_ work-study monies that allowed for translation assistance from Lauren Jones, and a grant from the Dean's office to help defray costs of rights fees and other translation assistance. My faculty colleague Daniel Youd played an essential role in many of the translation projects related to this book. I owe a great debt also to Cindy Cooley and the entire staff of the Beloit College library who worked many hours locating materials for me through interlibrary loans.
I am also grateful to the Pacific Cultural Foundation of Taiwan for a research grant in 2000 that allowed me to carry out the research that led to the essays used as the basis for Chapter 2 and .
Another large group of people to thank includes all the professional colleagues over the years who allowed me to participate in their panels and publish essays in their edited books and journal volumes, including Joseph Cheng for permission to use material from my article in the special issue of the _Journal of Comparative Asian Development_ on Utopianism in Chinese Political Culture that was the basis for Chapter 2, Shiping Hua, the guest editor of that issue who gave me the opportunity to contribute to that volume and also fine editorial advice, and the past and current editors of _Anarchist Studies_ , Sharif Gemie and Ruth Kinna, who also gave fine editorial suggestions and permission along with that of Lawrence & Wishart publishers to adapt material for the prelude and Chapters 1 and 6 of this book from articles published first in that journal. Ruth, along with Laurence Davis, very kindly included me in a panel on Utopianism and Anarchism at a conference of the Utopian Studies Society of Europe in Tarragona, Spain, in 2006, where I presented a paper that became the basis of material from my chapter in their book, _Anarchism and Utopianism_ , which they and the publishers at Manchester University Press kindly gave me permission to adapt for use in Chapter 3. Besides their generous editorial suggestions, Ruth and Laurence gave much help and encouragement over the years and included me in other conferences related to the Anarchist Studies Network, whose members have given me excellent feedback and criticism as well. I am grateful to Alexandre Christoyannopoulos for taking the lead in planning a panel on anarchism and religion at the first conference of the Anarchist Studies Network of the British Political Studies Association in Loughborough, England, in 2008 and for including my essay on the _Wunengzi_ and giving fine editorial assistance in the book he edited, _Religious Anarchism: New Perspectives_ , whose publisher Cambridge Scholars Publishing kindly granted permission to adapt that essay as the basis for Chapter 4 of this book.
I am also grateful to Susan McEachern and colleagues at Rowman & Littlefield for permission to adapt for use in Chapter 6 material from the book _Autocracy and China's Rebel Founding Emperors_ that I cowrote with Anita Andrew. Chapter 9 includes material reprinted and adapted from John Rapp, "Editor's Introduction," _Chinese Law and Government_ , 22(2) (Spring 1989), used by permission of M.E. Sharpe, Inc. Thanks go to Van Young for his translation assistance on Chinese articles on the Asiatic Mode of production debate, many of which are referred to in Chapter 9. I am indebted to Bruce Gilley and Joseph Wong, the organizers of "Backward Toward Revolution: A Festschrift to Celebrate the scholarship of Professor Edward Friedman" at the University of Toronto, Munk Centre for International Studies in 2009 for including me in the conference and allowing me to present a paper that became the basis for Chapter 7. Alex Prichard and Ruth Kinna very generously allowed me to participate in the Anarchism stream of Manchester Workshops in Political Theory in 2010 where I presented a paper that became the basis for Chapter 8 and . Thanks also to Alex, Laurence Davis, and the other members of the advisory and editorial boards of the Contemporary Anarchist Studies series of Continuum Press and to reviewers of the original proposal and manuscript of this book as well as the editors at Continuum, especially Marie-Claire Antoine, for their generous criticism and advice. I am very grateful for Uri Gordon's patient help and consultation on the cover art and especially to my student, (Cleo) Zhang Kun for her original calligraphy that appears on the cover of this book. I would also like to express my profound thanks to Marjorie Schafer for her help in producing the index to this book and for copy editing suggestions.
Above all I am grateful to my wife, the China historian Anita Andrew, for her constant advice, criticism, and encouragement to write this book, and to our daughters Amy Chunyi and Laura Mingyi Rapp for their patience, understanding, and inspiration.
PRELUDE
**Main Thesis of this Book**
This book examines the key moments in Chinese history when different people used a basic anarchist theory to criticize the inevitable tendency of all states to rule for themselves, from radical Daoists in pre-imperial and imperial China to members of the twentieth-century Chinese anarchist movement influenced by the West, to what we will label "neo-anarchist" dissidents in the People's Republic of China (PRC). Despite China's long history of authoritarian rule and state autonomy from society, as well as a long line of political thinkers who in one way or another justify centralized state power, China also has a long history of anti-statist thought. This book does not attempt the impossible task of examining all Chinese dissident thought but only those people in ancient and modern times who utilized an underlying anarchist theory of the state.
Since this book is primarily aimed at helping non-China specialists to see anarchism as not just a Euro-American concept, and in order to avoid the potential problem of this author appealing from authority as a sinologist, this book will cite Chinese sources in translation wherever possible and even provide original translations in the appendices of key texts that have never before been fully translated into English. Nevertheless, this work will pay attention to scholarly debates among ancient and modern China specialists and will not hesitate to join these debates whenever it proves necessary to the book's main thesis.
The thesis is that what most distinguishes anarchism from other political ideologies is the idea that the state rules for itself whenever it can, not for individuals, interest groups, socioeconomic classes, or society as a whole. Furthermore, for anarchists, the very nature of the state, its hegemony on the legitimate use of coercion, to slightly modify Max Weber's definition, not to mention its monopoly on the ability to define threats to itself as threats to society, only reinforces its inherent advantages in being able to gain autonomy from its subjects. The basic anarchist thesis, this book argues, is not limited in time or by region. Anarchist thought can and has occurred many times and in many places in history and not just among those thinkers and activists in Europe from the early to mid-nineteenth century who consciously took on the anarchist label and who started a movement that then spread throughout the world, including to China, in the late nineteenth and early twentieth centuries. While the adherents of that movement were for the most part explicitly socialist or communist in orientation and favored revolutionary methods to implement their ideals, such ideas and commitments, however important and necessary they may have been, do not in themselves distinguish anarchists from other socialist revolutionaries. Instead, this book argues, it is their critique of states, whether capitalist or socialist, as, in the end, ruling for themselves that gave the anarchist movement its greatest power and coherence.
In China anarchist thought arose among what has been traditionally labeled the Daoist school of philosophy (though we will examine later in this chapter those who question the existence of such a coherent school) which, as we will see in Part 1 of this book, began during the pre-imperial era nearly 2,500 years ago and revived in the third century CE. Modern anarchist thought arose most consciously among Chinese thinkers and activists in the early twentieth century, first among Chinese students studying in Tokyo and Paris but after the 1911 revolution among many trade union activists and intellectuals in China itself. Though in the interlude chapter we will briefly examine the thought of some members of that self-conscious Chinese anarchist movement, we will look at that movement primarily for the rare times when it consciously harked back to radical Daoist political theory and also for the key moment when it most consciously used the basic anarchist theory of the state ruling for itself, namely, during the anarchists' debates with early Chinese Marxist–Leninist thinkers. In Part 2 of this book we will examine the question of the influence of anarchism on the thought of Mao Zedong as well as denunications of anarchism in the People's Republic of China (PRC) and then examine what we will label neo-anarchist critiques of Marxism–Leninism in the PRC from the Mao to Deng and contemporary eras in order to see that the deadly contest over the question of state autonomy is still very much alive in modern and contemporary China.
**Definition and Typology of Anarchism**
Before proceeding with the case for the anarchism of premodern and contemporary Chinese thinkers, we first need to distinguish between the various types of anarchism and to develop a working model of the term as well as to suggest the main points of the argument in later chapters.
Anarchism as a term of course comes from the Greek _an-archos_ , meaning "without a ruler," and should refer to any doctrine that contends that any type of rule is unnecessary, harmful, and/or even counterproductive or evil. As such, this author would contend that anarchism is a generic label for all doctrines opposed to rule and should not be limited to the Western anarchist movement of the nineteenth and early twentieth centuries. The Wei-Jin Daoist term _wujun_ literally means "without a prince" (see Chapter 1), the Chinese characters for which appear on the cover of this book, and is nearly identical in meaning to the Greek _an-archos_ and thus clearly fits within this broad definition of anarchism. In short, anarchism can and has appeared in many periods and places throughout history and thus this author would disagree with those who would limit the concept to a modern context. Indeed, the first part of this book argues that the Daoist anarchists' focus on the state ruling for itself, while they noted at the same time that other political ideologies only disguise this fact, may have much to teach Western anarchists about internal consistency and may aid in a revival of anarchist themes in the contemporary world.
Within this all-inclusive generic definition there are of course many different types and strands of anarchism, all of which can be divided among three intersecting poles. First, following the historian of Western anarchism George Woodcock, anarchism may be divided into the "idea" and the "movement," that is between philosophical anarchism on the one hand and on the other the concept of anarchism as a "developed, articulate, and clearly identifiable trend . . .," which Woodcock argues appeared only in "the modern Western] era of conscious social and political revolutions." In this distinction, all "social or political" anarchists would also be philosophical anarchists, though of course analysts never fail to point out contradictions of particular anarchists on this score as, for example, between Bakunin's expressed anti-statism and the perhaps inherent authoritarianism of his professed revolutionary methods. On the other hand, we could also say that not all philosophical anarchists could be labeled political or social anarchists, that is, to the extent that such philosophical anarchists declined to join much less lead movements aimed at overthrowing particular states even if they expressed doubts about the basis for all political authority. It is this distinction, this author believes, that is at the root of many doubts over the supposed anarchist nature of Daoism and probable criticism of the label of "neo-anarchist" for contemporary Chinese critics of Leninism. This distinction may have started to make less sense in the late twentieth and early twenty-first-century era of "people power" than in the nineteenth and early twentieth centuries when many activists and intellectuals saw violence as the only way to affect true revolution. Certainly, just because Daoists may not have led overtly political movements to overthrow existing states does not mean that Daoism amounts to a "diluted" form of anarchism as Alex Feldt and others contend; instead, as will be argued throughout [Part 1 of this book, to the extent that they refused to join movements that themselves might found oppressive states, Daoists may be more consistent anarchists. This point leads us to the second, intersecting distinction among anarchists.
Many students of anarchism draw a distinction on the one hand between those anarchists willing to use and even embrace violent methods, the archetype again being Bakunin, and on the other those such as Tolstoy who insist on a unity of means and ends, and thus who would stress methods of noncooperation and passive resistance (what Tolstoy, following the Christian gospel(s), called "non-resistance") to all coercive authority. Some pacifist anarchists went beyond pure philosophical anarchism to the extent that they founded movements of their own that later analysts label as social or political. In this case, Daoists of the Warring States and Wei-Jin periods should definitely be labeled pacifists, but whether or not they led conscious social or political movements we will examine later in this chapter and in the following chapter.
In the third main distinction, many scholars commonly differentiate between "individualist" anarchists including Stirner and the modern anarcho-capitalists such as Murray Rothbard who reject all political authority but accept and assume the prior existence of private property (whether or not they are willing to use violence and/or hire private armies), and "collectivist" anarchists including everyone from socialist to communist anarchists who deny the existence or right of private property prior to the state. The collectivist view is summed up in the famous phrase of Proudhon that "property is theft." This distinction, problematic enough when applied to Western thinkers such as William Godwin and even Proudhon (who accepted the need for private "possessions" such as tools, if not landed property) becomes even more difficult when dealing with the ancient and medieval Daoists, as we will see in the next chapter. Nevertheless, one can distinguish, as we will see, between the more individualist or "selfish" strands of thought as in the "Yang Zhu" chapter of the _Liezi_ (a book probably written ca. 300 CE during the Daoist revival, though this chapter may have been based on the surviving ideas of the legendary proto-Daoist hermit of the earlier, classical Daoist period) and the possibly more communitarian strands of other Wei-Jin anarchists who claimed to base their thought more directly on the received versions of the classic texts known as the _Daodejing_ and the _Zhuangzi_. There is certainly no justification for Feldt's assumption that all anarchists must have a view of society as made up of a collection of atomized individuals and that the only job of anarchists is to protect the autonomy of the individual from the force of the state. Instead, this author would argue, the main similarity in all anarchists is the rejection of the conflation of state and society, even if various kinds of anarchists have different views of the nature of society.
As will be argued in Chapter 1, the Daoism of certain figures of the Wei-Jin period was indeed thoroughly anarchist, at a minimum on a philosophical level and at a maximum as an intended program to delegitimize the centralized bureaucratic rule of the late Han and the less effective but no less brutal rule of the Wei and Jin dynasties. While never advocating or propagating violent opposition to authority, so far as we know, the Wei-Jin Daoists did oppose all authority in general and did attempt to oppose the ideological hegemony of Confucianism, a form of which had become the official ideology of the imperial Han dynasty (206 BCE to 220 CE) and almost all successive imperial dynasties, if combined in practice with heavy doses of the Chinese Machiavellian doctrine of Legalism. Thus this book will argue that Wei-Jin anarchism _was_ a doctrine of resistance to the state, albeit almost certainly pacifist and remaining at the intellectual level if no less important as a means to undermine authority. Furthermore, this work will argue that the full-fledged anarchism of the Wei-Jin thinkers was firmly based on classical Daoist texts, which the later Wei-Jin Daoists only highlighted and did not distort. In sum, Part 1 of this book contends that the Wei-Jin figures were only the most open advocates of Daoist anarchism, a movement that lasted at least from the early Han or late Warring State periods, if not before, and extended well into the Tang dynasty (617–907 CE). Daoist anarchism, we will conclude, was a movement that perhaps can still be drawn upon in any contemporary or future challenges to Chinese political authority.
Although there are many different types of anarchists, what they all share in common and what distinguishes them from other dissident thinkers and radical activists is their basic tenet that the state rules for itself whenever and wherever it can. Even those who use the basic anarchist theory of the state, both those who consciously label themselves as anarchists and those who do not, can depart from anarchism in other aspects of their thought as we will see throughout the book; thus different types of anarchists can themselves be criticized for acquiescing in one kind of state power or another. The focus of this book is on the key periods in Chinese history when the basic anarchist theory of the state was expressed most clearly within a Chinese context.
**Major Objections to Main Thesis of this Book**
While the basic thesis of the book may seem obvious to many readers, in fact this author has found it to be very controversial and has encountered two seemingly quite different types of objections. On the one hand, some scholars and practitioners of Daoism would argue that no clear school of Daoist thought exists and that to focus on those relatively few thinkers who brought out the anarchist themes in the classical Daoist texts risks distorting the essence of Daoism by radicalizing it. On the other hand, scholars of and/or sympathizers with anarchism worry that universalizing anarchism may ironically _de-_ radicalize it, emptying it of meaning. As we will argue in this chapter, both types of objections are based on shared similar unwillingness or inability of such observers to face up fully to the truly radical aspect of the anarchist theory of the state.
**Objections from scholars and proponents of Daoism**
The first main objection to this book will come from scholars and proponents of Daoism who would deny that Daoism has any clear coherence as a distinct school of thought. First, some such people, especially those studying Daoism as a religious practice, would argue that the traditional Chinese separation of Daoism into _daojia_ (Daoist school, that is, philosophical Daoism) and _daojiao_ (Daoist teaching, for example, alchemical and religious traditions) is itself only a later concept of the historian Sima Qian (165–110 BCE) of the former Han dynasty who imposed a coherence upon many disparate types of much earlier thinkers and practitioners, a coherence that did not in fact exist or that those individuals were unaware of. Even among the classic philosophers of the Eastern Zhou era (ca. 770–221 BCE), scholars of Daoism would argue, there was often no clear distinction between Daoist and Confucian schools, which again was only a later idea applied to the thought of this era. Many modern scholars argue that both so-called Daoist and Confucian thinkers called for limited governance and for rule by sages virtuous in one way or another, and the idea of opposing schools of Daoists and Confucians (the name even for the latter school given by Westerners with the Chinese term the _ru_ only meaning the school of the scholars) is an exaggeration of later historians. Furthermore, what are often regarded as the classic Daoist texts, the _Daodejing_ , traditionally ascribed to the probably nonexistent personage Lao Zi (old master) and the _Zhuangzi_ were not clearly single author texts, and even the author of the seven inner chapters of the latter text, the historical individual Zhuang Zhou, was perhaps unaware of any coherent text known as the _Daodejing_ that may have been compiled into a single text contemporaneously or slightly after Zhuang Zhou lived. Many scholars of Daoism argue that there are in fact differences between the two texts and it was not until the last years of the later Han (25–220 CE) and the Wei-Jin era (265–420 CE) that scholars who created the "Lao-Zhuang" tradition related the two texts. Many scholars of Daoism would claim further that the "Lao-Zhuang" side of Daoism is itself at best only one tendency within a tradition that developed for over 2,000 years after the classical period and which included many other traditions and aspects including spiritual and physical practices that were far from anarchist. Overall, some scholars of Daoism worry that "radicalizing" Daoism by comparing it to Western anarchism risks losing the overall picture of the real place in Chinese tradition of the holistic concept of life contained within the disparate strands of what is labeled Daoism.
We will deal more fully below with this basic objection to the main thesis of this book. Before turning to the other main objection from scholars and proponents of anarchism it should just be noted here that the position of this book is that to ignore the clearly expressed anarchist point of view in key Daoist texts or to minimize or excise from the Daoist tradition those thinkers who express an anarchist point of view itself distorts a key part of Daoist thought. Perhaps a useful heuristic if not identical analogy would be to the idea of a Christian anarchism. Although it is true that the anarchist interpretation of Jesus is very much a minority tradition compared to the over 2,000 years of other interpretations, from purely spiritual to avowedly statist from the apostle Paul to Augustine and beyond, scholars of and sympathizers with Christian anarchism nevertheless claim a firm basis for their interpretation in the words and practices of Jesus and his early followers. Likewise those later Daoist thinkers and students of Daoism, including this author, who make an anarchist interpretation of classical Daoist texts even if they are not in the mainstream of Daoist scholarship would claim clear links to some of the oldest texts in the Daoist tradition, as we will see throughout Part 1 of this book.
**Objections from scholars and proponents of anarchism**
Seemingly opposite to the objections of Daoist scholars, those who study and/or sympathize with anarchism fear that "traditionalizing anarchism" risks _de-_ radicalizing it and that assuming the universality of anarchism runs the risk of making the concept meaningless. For example, if one limits anarchism to its critique of the state, then, some might charge, one would have to include as anarchist even American libertarians and Tea Party activists who claim to hate government, which on the face of it again would seem to empty the term anarchism of all meaning as a truly radical critique given such people's support for and by corporate and other elite interests. To such students of anarchism and to anarchist sympathizers, anarchism as a concept must involve socialism, revolution, and critiques of all kinds of oppression, including that caused by, among other types of power, family, religion, property, and culture. Many students of anarchism and Communism in China would argue more specifically that ignoring important differences between ancient Daoist writings and modern Chinese anarchism risks denigrating the modern anarchist movement in China.
**Preliminary Answer to Both Types of Objections**
Both objections ignore the clear times in Chinese history when, whatever one labels them, Chinese thinkers did express a clear anarchist theory of the state, that is, when they did not just call for limited government but criticized states as ruling for themselves and not for the benefit of the people, and when they rejected the possibility of any type of reformed or benevolent government. So the Daoist anarchists, as we will see in the four chapters of Part 1, did not just call for a limited government to rule in a benevolent way but attacked the whole idea of humane rule, both in the chaotic Eastern Zhou and later Wei-Jin periods of competing states. That is, whether or not they called themselves Daoist or whether or not one labels them anarchist, these Chinese thinkers rejected the whole idea of government. While this book by no means argues that all Daoists are anarchists, we will see that there was in China a long tradition of people who did base themselves on ideas in the _Daodejing_ and _Zhuangzi_ , texts that, whatever their other different emphases, did try in key chapters that were written as far back as the Warring States period of the Eastern Zhou, if not by the original authors, to undermine the whole idea of rule, as we will see in the next chapter. These Daoist anarchists, as we will call them, also attacked the application of Legalist ideas of rule by power and force and Confucian ideas of benevolent rule as different types of ideological disguises to justify the wealth and power of a few. Furthermore, the Daoist anarchists did indeed include critiques of other kinds of power besides that of the state, including especially patriarchal authority and manipulation of language, but with the basic anarchist point that it is the link to and backing by the state that makes social, family, and linguistic authority oppressive. As noted above, to the degree that scholars of Daoism, including Chinese scholars from the Han to the PRC eras and Western scholars from the eighteenth century to the present, ignore this basic critique it is they in fact who serve to tame and deradicalize Daoism. In short, it is they who distort Daoism to the extent that they ignore or minimize this important anarchist part of the Daoist tradition.
Against some scholars of Daoism who might agree that the radical anarchist tradition exists but was a revision of later thinkers, in the first chapter we will trace this radical anarchist streak back to the classic texts of the Warring States era, the _Daodejing_ and the _Zhuangzi._ Indeed, in Chapter 3 we will trace this radical tendency even further back to even the oldest surviving version of the former text, the so-called Guodian manuscripts, refuting those scholars who think that earliest known version of the text does not contain anti-Confucian language and is more accommodating to state power. As we will see, beyond its critique of the state this Daoist anarchism did have communal aspects and was similar in many other ways to the thought of different Western anarchists, if without a commitment to violent revolution. Despite their similarity with Western anarchists on many other grounds, we will nevertheless see throughout Part 1 of this book that the radical Daoist argument was most powerful when it kept to the main anarchist theory of the state and weakest and most contradictory when Daoist thinkers were willing to acquiesce in state power, as we will see was especially the case with some people who revived Daoism in the late Han and Wei-Jin eras and after, including especially Wu Nengzi of the Tang dynasty whose thought we will analyze in Chapter 4.
Against the criticism from scholars and proponents of anarchism that focusing on anarchism's theory of the state risks losing sight of its larger vision, we will see in Chapter 2 that the Daoist anarchists did not just have a negative view of the state but also a positive vision of the possibility of life without government, though still containing a dystopian vision of the state run amok under other political ideologies. Scholars of anarchism and anarchist sympathizers would likely on the same ground also strongly disagree with the last two chapters in Part 2 of this book that view as neo-anarchist those dissident thinkers within and outside the Chinese Communist Party (CCP) in the PRC who criticized the Leninist state, since such thinkers took pains to deny they favored anarchist solutions even if some of them also denied that official Maoist thought had a genuine anti-statist side. Nevertheless, we will see that those thinkers did indeed criticize the Leninist state as ruling for itself and were often well aware that anarchists were the first to make such a charge. Their separation of the anarchist critique from proposed anarchist solutions is why we will label such thinkers "neo-anarchist."
Although this book will not hesitate to criticize and show the limits of modern Chinese thinkers as well as ancient ones when they depart from the basic anarchist critique, including especially Mao Zedong himself as we will see in Chapter 6, one important secondary theme of this book is that we must not forget about the terrible limits faced by genuine anti-statist Chinese thinkers from ancient to modern times when states started to centralize and militarize their power, as at the end of Zhou dynasty and the beginning of the imperial era in the third century BCE and in the Nationalist and Communist eras of modern China. In all such periods of consolidating authoritarian states, intellectuals trying to oppose state autonomy had to express themselves carefully and to claim to be arguing within tradition in order to protect themselves, get their works published, and even survive physically. Though often having to disguise or camouflage their thought, this does not mean Chinese anarchist thinkers were in any way less radical, as long as they kept to the basic anarchist critique of the state.
This book argues that to insist on including as radical anarchists only those people who expressly advocate achieving socialism through violent revolution is not only to take a completely Euro-centric approach (since the faith in the universality and inevitability of both socialism and revolution began in Europe in the nineteenth century) that in effect serves as the flip side of Western cultural imperialism, but also serves to miss the radical heart of anarchism. Far too often, as we will see in Chapter 6 for example, those China scholars who want to defend the socialist nature of the Chinese revolution and more particularly the ideas of Mao Zedong as radical and liberatory in intent have to downplay the atrocities of Mao and his successors alike that are clearly linked to their presiding over a state ruling in its own interests, whatever be those leaders' stated intentions. Likewise, far too often such socialist sympathizers among China scholars in order to defend the socialist nature of the Chinese revolution are led to downplay or even ignore the thought of the "neo-anarchist" critics of the Chinese Leninist regime that we will examine in Chapter 8 and .
Arguing that anarchism does not have to include the call for socialist revolution by no means is to say that Western-style collectivist anarchists, including in China itself as we will see in the interlude Chapter 5, failed to be true anarchists; instead this book argues that collectivist anarchists were most true to the basic anarchist idea when they criticized nonanarchist collectivist fellow revolutionaries for ignoring the dangers of accepting socialist state autonomy and least true to anarchism when for one reason or another they themselves acquiesced to state power, as we will see was the case with Liu Shipei and other members of the twentieth-century Chinese anarchist movement. In that chapter we will both refer back to Part 1 to see why Liu was such an exception among twentieth-century Chinese anarchists in looking to the Daoist tradition as well as forward to Part 2 to see the genesis of the Marxist critique of anarchism in the PRC after 1949 in the anarchist–Marxist debates of the 1920s, where the modern Chinese anarchist critique of Leninist state autonomy also began.
In the end, for a true anarchist everything else, no matter how necessary or genuine, should be secondary to the political critique at the heart of anarchism, namely, the idea that all states ultimately try to rule for themselves, the idea that most distinguishes anarchists from other schools of thought. This basic anarchist premise is also the hardest idea for dissident thinkers to express since it is the biggest taboo that state leaders try to enforce and thus the first type of thought they start to repress when they (rightly) realize it strikes directly at their interests. Thus anarchists often have to partially disguise or camouflage their basic premise within the language of other schools of thought, even then only in rare times of openings in official ideology. This is why Daoist thinkers often had to sound similar to Confucians in eras when the state's reach was expanding, and so too why modern Chinese thinkers took advantage of the Maoist openings during the Cultural Revolution and the reform opening in the 1980s before the post-Tiananmen clampdown to use the varying language of official Marxist–Leninist ideology to criticize the Leninist state as ruling for itself and not the proletariat. Though the neo-anarchists in the PRC had to use seemingly Marxist language in order to get published and to survive, similar to the ancient Daoists if without citing them, these modern Chinese thinkers as we will see in the second part of this book clearly used the anarchist theory of the state (though unlike the Daoists, divorced from any clearcut anarchist solutions) to break the biggest taboo in Marxist–Leninism. The modern Chinese neo-anarchists certainly claimed to be anti-capitalist and made clear their support for idea of socialist revolution, but their main argument was how the socialist goal of equality would be compromised and contradicted by an unchecked socialist state ruling for itself. Even if not calling for a fully stateless society, nevertheless, like the ancient Daoists these modern Chinese neo-anarchists still had a positive vision of a cooperative society that would flower best when not limited by the state's interests. If their positive vision was not expressed as overtly as an anarchist vision as that of the radical Daoists, again this was perhaps because the neo-anarchists had to use Marxist language to mount their critique given the very real threat of prison or execution and because they knew very well that they would be denounced as anarchists by official state ideologists. Such Leninist state apologists would always be quick to use the old Marxist anti-anarchist memes first employed against Chinese anarchists in the 1920s that we will examine in Chapter 5, revived by Mao to put down genuine anti-statist radicals in the Cultural Revolution, as we will see in Chapter 6, and used against dissident thinkers in the PRC from the 1950s to the early twenty-first century, as we will see in Chapter 7. In the end, this book will find that these anti-anarchist memes only served to prove the main anarchist critique of Marxism: by acquiescing to centralized state power, Marxist and other socialists become servants of the state and help to quash hope for genuine liberation.
**An Anarchist Critique of the Critics**
Within this thesis of their critique of the state as the key minimal characteristic of all anarchists, this book argues that even self-labeled anarchists can depart from anarchist theory of state and acquiesce to state power, as was the case for some Daoists—as we will see throughout Part 1 of this book—for members of self-styled Chinese anarchist movement of early twentieth century—as we will see in Chapter 5—and for Mao and some official Maoists who claimed to be using anti-bureaucratic class ideas that others argue were influenced by anarchism—as we will see in Chapter 6. In the end, employing the basic anarchist premise, one could argue that such people at times limited the real anarchist content of their critiques in order to justify, promote, or augment their own power within the state.
So too one could argue that self-styled libertarian Tea Party activists in America and elsewhere may sometimes sound like they are using the anarchist theory of state but in reality only want their opponents to unilaterally disarm (e.g. ending regulation of oligopolistic corporations) while keeping the parts of state apparatus that they find beneficial (e.g. those related to the "military–industrial complex" or policing people's sexual behavior). Some intellectuals within that tradition claim to find a laissez-faire management approach or even an anarcho-capitalist vision in the _Daodejing_ , which would seem to give further ammunition to socialist critics of Daoism as anarchism, but both types of thinkers have to ignore many other inconvenient aspects of radical Daoism including the idea of communal village life and living in harmony with nature.
More consistent and explicit anarcho-capitalists such as Murray Rothbard (in that he at least rhetorically favors privatizing the police and military) are perhaps more useful in showing the limits of anarchism in some collectivist anarchist critiques to the extent that the latter would allow social coercion and "social consensus" as defined by small groups to lead to new oppressive authority. This possibility of the germ of future political oppression can be seen most famously in Bakunin's conspiratorial cells to direct the violent revolution, contradicting his own most brilliant version of the anarchist theory of state in his view that the Marxist "workers' state" would all too quickly become the state of ex-workers and how it would never "wither away" as Marx predicted. At the same time, however, collectivist anarchists help to deny the anarchism in practice among such "libertarian" thinkers by showing the real link of Tea Party "anti-government" ideology to the power of wealthy individuals and corporate interests and especially to the military–industrial complex, which distorts the "free market" to the benefit of state-supported industries. Both sides of the individualist–collectivist divide can depart from the anarchist critique by asserting something else as primary above the anarchist theory of the state, whether "free" markets or socialist revolution, and both are most useful when criticizing each other for shortcomings in this regard. It is the Daoist anarchists, this book argues, who are most true to this genuinely radical critique.
Despite their seemingly opposite criticism—for either overly radicalizing Daoism or deradicalizing anarchism—in the end both types of objections to this book's main thesis are very similar. Scholars and proponents of Daoism would argue that just looking at the political critique of Daoism is to miss the larger picture and how much more Daoists say, including about living in touch with the whole and not dividing things into separate or opposed categories. Scholars of and sympathizers with anarchism say that anarchism involves much more than a critique of the state but also includes critiques of oppression in the economy, family structure, sexual relations, and the environment, among other areas, and contains a positive vision for a cooperative, communal future. This book certainly does not deny the power of Daoists and other kinds of anarchists to say much more about life beyond politics, but it will highlight and focus on the minimal yet crucial aspect of all anarchists: their critique of the state ruling for itself and their warnings of the danger of radical thinkers who depart from this main point as themselves distorting the main message of anarchism. While paying attention to both types of critics of Daoist anarchism, we should not ignore their own possible interests in avoiding or minimizing the radical heart of the anarchist critique, as intellectuals in all countries throughout history have divided interests between promoting intellectual autonomy and yet retaining their elite status within existing or future states or systems of authority. So, above all, this book tries to stay on point and will challenge even Daoist and other anarchists when they depart from the basic anarchist critique of the state, while of course welcoming challenges from all types of people wherever they think this book too strays from the basic anarchist premise.
**Notes**
**** As romanized in the modern _hanyu pinyin_ system used throughout in this work except for the names of some Chinese scholars and the Nationalist leader Chiang Kai-shek; "Taoist" is perhaps the more familiar version from the older Wade-Giles system, a romanization system that is used by some authors of works translated in the appendix.
**** See Murray Edelman, _Politics as Symbolic Action_ , Chapter 9, "Escalation: International Relations," 142–71, for a neo-anarchist critique of the state's ability to manipulate its monopoly on the identification of foreign threats in order to maintain its internal authority.
**** George Woodcock, _Anarchism: A History of Libertarian Ideas and Movements_ , Prologue, 11–31, Epilogue, 404–23.
**** Ibid., 39.
**** Feldt, "Governing Through the Dao: A Non-Anarchistic Interpretation of the _Laozi_ ," 326, 336.
**** Tolstoy, _The Kingdom of God is Within You: Christianity Not as a Mystic Religion but as a New Theory of Life_ , 213ff. The most recent analysis of Tolstoy's thought as the culmination of Christian anarchism can be found in Alexandre J. M. E. Christoyannopoulos, _Tolstoy's Political Thought_ and throughout his _Christian Anarchism: A Political Commentary on the Gospel_.
**** Pierre-Joseph Proudhon, _What Is Property_ , 13.
**** A. C. Graham, _The Book of Lieh Tzu_ , 1.
**** As argued in Feldt, 326, for example.
**** This paragraph includes ideas now commonly held by a number of students of Daoism. For a convenient summary of such ideas, if perhaps within the most strident attempt to "radically reconstruct" Daoism in a way that would privilege the believers and practitioners and belittle classic texts such as the _Zhuangzi_ as being at the heart of Daoism, see Russell Kirkland, _Taoism: The Enduring Tradition_ , _passim_. Kirkland argues that the classical texts such as the _Daodejing_ and the _Zhuangzi_ , at best "played a marginal role in the lives and thoughts of most later Daoists" (68) and in general accuses Chinese and Western scholars of "lying" about the true nature of Daoism. This author would conclude that Kirkland's version of a post-colonial critique of Western and Chinese scholarship on Daoism can itself serve to promote the flip side of cultural imperialism, privileging those who turned to Daoism for spiritual or physical guidance in effect to colonize Daoism for the academic field of religious studies. Modern students of Daoism as a philosophy, such as Chad Hansen, may similarly find as key to Daoism the aspects that privilege their field, though perhaps more ready to preserve room for disparate types of Daoist thinking as well as practice. See Hansen, _A Daoist Theory of Chinese Thought: A Philosophical Interpretation_.
**** For the most recent and comprehensive review of the Christian anarchist tradition, see Christoyannopoulos, _Christian Anarchism_.
**** As one reviewer of the original proposal for this book charged.
**** See Hansen, 225–30, who though mostly looking at Daoism as pure philosophy and not political philosophy, nevertheless finds the goal of at least some of the early Daoist thinkers to be "radical anarchy" (229).
**** See Clark (pseudonym Max Cafard), "The Dao of Capitalism or 'Going with the (Cash-) Flow'," in Cafard, _The Surre(Gion)alist Manifesto,_ 25–39.
**** The important, if very controversial idea to anarchist sympathizers, of the insidious dangers of social coercion under any future anarchist society becoming worse than state coercion to the extent they are denied was an important theme of Ursula Le Guin's novel _The Dispossessed_ , as well as R. Booth Fowler's scholarly essay "The Anarchist Tradition of Political Thought."
PART ONE
Daoism and
Anarchism
1
Daoism and anarchism
reconsidered
**Introduction**
Philosophical Daoism is a term used to refer to the ideas of some people who arose at the end of China's Zhou dynasty (1027–256 BCE), a period when China disintegrated into a long period of civil war and chaos that finally ended only in 221 BCE with the end of feudalism and the founding of the centralized, bureaucratic Qin empire (221–206 BCE). In the latter part of the Zhou period (722–481 BCE), specifically in the Spring and Autumn, and Warring States Periods (403–221 BCE), philosophers and teachers arose who tried to gain the ear of the feudal warlords to adopt their particular systems in order to reunify China. Most such thinkers offered specific advice on how to attain order, such as the idea of rule by moral virtue of the Confucians or the idea of rule by power and force of the so-called Legalist school. Those thinkers later labeled the Daoists often traced their ideas back to Lao Zi ("Old Master"), a semi-mythical figure who may have lived, if he lived at all, in the sixth century BCE and who is traditionally treated as the author or compiler of the _Daodejing_ (Wade-Giles: _Tao Te Ching_ , or the "Classic of the Way and Its Power," referred to hereafter in this book as _DDJ_ ). This text dates in its received form at least from the third century BCE (in Chapter 3 we will examine a recently unearthed version of the text that dates back to as much as a century earlier). Modern scholars argue that the _DDJ_ may have been compiled over a long period of time from the sayings of village elders, and perhaps first coalesced as a text during the Warring States period partially in response to other schools of thought.
The other great classical Daoist philosopher was Zhuang Zi (Master Zhuang), a historical individual with the given name of Zhuang Zhou who lived in the fourth century BCE and who wrote at least the seven core or "inner" chapters of the book known as the _Zhuangzi_ , the other, "outer" chapters being added at later periods by unknown authors. Thus the core chapters of _Zhuangzi_ are nearly as old as the received _DDJ_ and should not be denigrated as any less important a ". . . foundational text of socio-political relevance" for Daoism, as Alex Feldt contends, so that the _DDJ_ should not have to "clearly enjoy primacy in developing a classical Daoist political theory." Whatever their differences, both texts were unique in their advice for rulers to rule by inaction or doing nothing ( _wuwei_ ) and in their opposition to law, morality, punishment, warfare, and nearly all other techniques and forms of rule. As such, many scholars have long referred to Daoism's "anarchistic" tendencies and aspects.
Given these many references to its anarchist tendencies, it may seem strange to question whether or not philosophical Daoism is really a doctrine of full-fledged anarchism similar to Western anarchism. In fact, however, as noted in general in the prelude there have been various objections raised to equating Daoist philosophy with anarchism, mostly focusing on the classical Daoists of the late Zhou dynasty. We will examine these objections more in detail in the first part of this chapter. In the second part of the chapter we will examine key thinkers of the Daoist revival in the Wei-Jin period (ca. 220–420 CE) and note their similarities to specific Western anarchists on key points central to the doctrine of anarchism.
**Doubts about Classical Daoism as Anarchism**
Doubts about the fully anarchist nature of Daoism have mostly centered on the Daoism of the late Zhou texts, the _DDJ_ and the _Zhuangzi_ , associated with the mythical or real figures of Lao Zi and Zhuang Zi respectively. Although only some of these doubts apply to the Wei-Jin Daoists, as we will see, those who question the fully anarchist nature of the _DDJ_ and the _Zhuangzi_ nevertheless usually see Wei-Jin Daoist anarchism as an extension or even corruption of classical Daoism rather than a loyal exegesis of it. These doubts about classical Daoism as an anarchist doctrine then must be dealt with before examining the more obvious anarchism of the Wei-Jin thinkers. Below the questions about Warring States Daoist anarchism are broken down into five categories. Here it should be noted that many of these doubts may have to do with the distinctions among different types of anarchists that we noted in the prelude. Those sympathetic to socialism and skeptical about philosophical and individualist anarchists as genuine anarchists and those sympathetic to the nineteenth-century collectivist anarchist movement (not to say that Daoist anarchism can easily be pigeonholed as philosophical and individualist, as we will see below in the section on Warring States Daoism as individualist or socialist), are perhaps the most skeptical about Daoism as true anarchism.
**The** _DDJ_ **and** _Zhuangzi_ **as advocating laissez-faire or limited government and not full-fledged anarchism**
The main limit many scholars find in Daoist anarchism of the Warring States Period is that the _DDJ_ and for some even the _Zhuangzi_ , if to a lesser extent, seemed to be giving advice to sage–rulers on how to govern, even if their advice was to rule by _wuwei_ (often translated as nonaction or doing nothing). As Hsiao Kung-chuan put it about the _DDJ_ ,
. . . non-action in government need not destroy and cast aside the ruler-servitor institution, and return to the total lack of restraints that exists among birds and beasts . . . in theoretical terms, what Lao Tzu attacked was not government in and of itself, but any kind of governing which did not conform to "Taoistic" standards.
Likewise, Frederic Bender and Roger Ames in a 1983 roundtable discussion of Daoism and politics, while finding great lessons for anarchism in "political Daoism," conclude that the (received) _DDJ_ is not a full-fledged anarchist text, since, as Ames notes it seems to accept the state as a natural institution, and as Bender argues, "retains, albeit in improved form, ruler, rule, and the means of rule (the state)." This is the main basis upon which Feldt argues that classical Daoism represented at best a "diluted" form of anarchism and at most a justification for the most efficacious type of limited rule within an autocratic and bureaucratic state, a type of rule akin to the "minimal, 'night watchman' state of Nozikean liberalism," a contention that, besides the obvious self-contradiction between autocratic and limited rule, we will dispute on page 28 of this chapter below.
On a less literal level other scholars find similar limits to anarchism in the received _DDJ_ and even in the _Zhuangzi_ , the author(s) of which many scholars otherwise recognize as much more explicitly anti-statist than the _DDJ_. Arthur Waley, for example, while finding great similarities between the classical Daoism of the _DDJ_ and _Zhuangzi_ and Western anarchism, nevertheless concludes that there were important differences, since "one of the main tenets of modern anarchism is that no appeal must be made to the authority of 'metaphysical entities'" and that ". . . [ _dao_ ] is undoubtedly a 'metaphysical entity'." Similarly, Benjamin Schwartz claims that the language of the _DDJ_ suggests ". . . not a spontaneously emerging 'anarchist' state of affairs but a state of affairs brought about by a sage-ruler." Likewise, A. C. Graham claims that however similar Western anarchism is to the thought of later "Daoist primitivists" who probably were the real authors of some of the "outer" or later added chapters of the _Zhuangzi_ , by contrast the more limited anti-government doctrine in the (received) _DDJ_ and perhaps the "inner" or original chapters of the _Zhuangzi_ if not amounting to "hierarchic anarchism" at least "amounts to a paternalistic anarchism" in its hope that the ruler will follow the practice of the "ancient Emperors, [who] it may be presumed, had no task but to keep the people ignorant of the arts and luxuries which were eventually to corrupt them . . . ." For Graham, as for Hsiao, the classical Daoists "[found] it difficult to imagine a society without any ruler or sages at all . . . The concept of the pure community explicitly described as without ruler and subject belongs rather to the revival of philosophical Taoism in the 3rd century A. D."
This point of view that finds limits to Daoist anarchism and instead deems it a doctrine of laissez-faire or limited government, would perhaps have as its best evidence the use of ideas in the _DDJ_ and _Zhuangzi_ ideas by officials in the court of the emperor Wu Di in the first century of the former Han dynasty (202 BCE–8 CE). At that time, after the official Legalist ideology of the hated Qin dynasty (221–207 BCE) was discredited (the Qin being the first, if the shortest, centralized imperial dynasty in Chinese history due to its rule by naked force with little legitimizing ideological veneer) and before official Confucianism took full form as a replacement ideology justifying the Han empire as rule by the morally virtuous for the benefit of all, some court scholars briefly adapted Daoist ideas to legitimize the Han's supposedly more "light" rule compared to that of the Qin. This laissez-faire version of Daoism can especially be found in parts of the _Huainanzi_ , a text of the early Han.
Likewise, during the revival of philosophical Daoism at the end of the later Han dynasty (25–220 CE) and the beginning of the long Period of Disunity (ca. 220–581 CE) before the centralized empire was finally revived in the sixth century CE, Daoism was first used as a formula to justify the rule of the upstart military dictator and posthumous founder of the failed Wei dynasty Cao Cao against the Confucian ideology of his opponents—the great families or large landlords from the end of the Han—as represented especially by the Sima clan who founded the Jin dynasty after Cao's death, another failed attempt at revival of empire (thus this first part of the Period of Disunity is usually referred to as the Wei-Jin period by China historians). Again, it seems to have been no problem for the Wei ideologists of the first generation of neo-Daoists to use the thought of the _DDJ_ and _Zhuangzi_ to justify a supposedly limited government, or at least one free from the Confucian conventions of "benevolent" rule by the morally superior.
Against this idea of the limits to anarchism in classical Daoism, many scholars have posited an opposite case of a more full-fledged anarchism. In general, their argument would be that the received _DDJ_ while referring to ideal rulers takes virtually the entire content of rule away from them in its condemnation of law, morality, education, taxes, and punishment. In effect the received text takes away all meaning of rulership by removing all elements of coercion from "rulers." As first and best pointed out by Joseph Needham (if within what many, including this observer, view as an unnecessarily unilinear and old-fashioned form of Marxist analysis), the _DDJ_ was trying to change "feudal" rulers back into leaders of primitive communal tribes, that is, into tribal elders or wise men with no monopoly on the legitimate use of coercion, to employ again Weber's minimalist definition of the state. As such, the authors of the classical Daoist texts could be identified as "men of the South," that is, of the areas at the far southern end of the Yellow River valley sedentary agricultural society who may have been in touch or dimly aware of surviving pre-sedentary practices and ideas. Without accepting Needham's thesis, Burton Watson, the great translator of the _Zhuangzi_ , does accept that the author of its inner chapters was a man of the South, who was thus not just anti-imperial but may have been in touch with pre-Zhou and thus prefeudal customs and ideas.
Even the best evidence for the laissez-faire interpretation of classical Daoism, the _Huainanzi_ (a text of the early Han dynasty in which Daoism was combined with other philosophies in an eclectic fashion in an attempt to find a legitimating formula for Han rule), has been brought into question by Roger Ames. In a vein of analysis very similar to that in this chapter, Ames views the text as only justifying government on a literal level, but with a deliberate subtextual purpose of undermining political authority since ". . . as an anarchistic political theory, the [Daoist] concept of [ _wuwei_ ] cannot be supported by any elaborate apparatus for practical implementation." In effect, Ames sees large portions of the _Huainanzi_ (which he translates as "The Art of Rulership") as a continuing attempt to use Daoist ideas, if in a more practical and concrete way, to undermine Confucian and Legalist justifications of authority, and indeed all coercive rule:
If we understand the primary objection of the anarchist to be coercive authority—that is, one person or group obliging another to act in a certain way—and the primary objective of the anarchist to be the eradication of this kind of authority from all areas of political life, then inasmuch as _The Art of Rulership_ advocates full use of the spontaneous contribution of each participant in an organization committed to the nonmediated action of personal initiative, there is much here that points to a [Daoist] anarchism.
For Ames, the "political Daoism" of the _DDJ_ , _Zhuangzi_ and the _Huainanzi_ has four necessary conditions for a "comprehensive anarchism," namely a theory emphasizing a natural "free" condition of human nature, a rejection of all coercive authority, a notion of some kind of "noncoercive, nonauthoritarian society" that could replace coercive authority, and "some practical method" of moving from "authoritarian reality" to the "non-authoritarian ideal." Even if one questions whether this "willingness to work within the framework of existing institutions to approximate the [anti-authoritarian] ideal" in the _Huainanzi_ could too easily lead to accommodation and acquiescence to authority rather than a challenge to it, this weakness does not have to apply to the _DDJ_ and the _Zhuangzi_ nor to other, later Daoists. Even granting the inconsistency of the _Huainanzi_ on the issue of political power, Ames' view of political Daoism as a thoroughgoing, anti-statist critique even in the classical era thus goes a long way toward refuting the theory of early Daoism as only a doctrine of laissez-faire rule and not of anarchism.
**"Anarchism" of the** _DDJ_ **and** _Zhuangzi_ **as corrupt idea of later Daoists**
Related to the above point, many scholars, led again by A. C. Graham, see the later, more explicit Daoist anarchism from the time of the authorship of the outer chapters of the _Zhuangzi_ to the Wei-Jin neo-Daoists as an extrapolation and even distortion of the political ideas expressed in the _DDJ_ and the inner chapters of the _Zhuangzi_. For Graham, the explicitly anarchist sections of the _Zhuangzi_ (Chapters 8–10, and parts of Chapter 11) reflect the writings of the "primitivists," who wrote with a very different, if still "idiosyncratic" style than the author of the inner chapters (see Appendix 1 for the most brilliant anarchist chapter of the _Zhuangzi_ ). Likewise, Burton Watson finds the same chapters to be written in a tone with a much more "shrill, almost pathological fury that is unlike anything found in the 'inner chapters'," although, interestingly enough, he finds these chapters much more closely parallel to the _DDJ_ than the inner _Zhuangzi_ chapters. As an important corollary, such scholars would see Wei-Jin Daoist anarchism as a further extension or even corruption of the less harsh and explicit anti-statism in the _DDJ_ and the inner chapters of the _Zhuangzi_.
A. C. Graham, however, sees the "primitivist" additions to the _Zhuangzi_ not as much later corruptions but as based on an earlier tradition that goes back to the hermit Yang Zhu, a legendary figure who predated even the classical Daoist texts. This primitivist tradition was also based on the "Shen Nung" ("Divine Farmer") tradition of a stateless agricultural community that also goes back at least to the Warring States period and fourth century BCE, if not earlier. Graham also notes the tradition of Xiu Xing, another of the great "madmen of the South" who disputed the Confucian thinker Mencius around 315 BCE based on the Shen Nung ideal. Thus, even for Graham, who believes that the "primitivist" chapters were authored between 209–202 BCE, that is, during the interregnum between the fall of the Qin and rise of the former Han dynasty, there is nevertheless a long tradition of Daoist or proto-Daoist anti-statism that goes far back into the Warring States Period.
Liu Xiaogan disputes Graham's dating, finding instead that the "anarchist" chapters of the _Zhuangzi_ , which for him include the "Yangist" chapters as well (i.e. those influenced by the tradition of the hermit Yang Zhu, including Chapters 28, 29, and 31), date to the late Warring States period, that is, not as far removed from the historical Zhuang Zhou. Although Liu thinks these chapters did go far beyond the political vision of the _Zhuangzi_ in their radicalism, he nevertheless concludes that the authors of the "anarchist" chapters were still followers of Zhuang Zi. If true, this would help make the case even more that the anarchist tradition of Wei-Jin Daoism has long and deep historical roots that go back nearly to the time of the historical Zhuang Zhou, if not before.
On a philosophical level, of course, many authors have found an anarchist spirit in both the _DDJ_ and the _Zhuangzi_. The case for an anarchist vision in the classical Daoist texts, to be explicated further in this chapter and throughout this first part of the book, would focus first on the thoroughgoing critique of all aspects of government and the positive view of the stateless society expressed in the _DDJ._ 28 Second, an anarchist view of classical Daoism would focus on the cybernetic vision of life in the paragraph of the great second (inner) chapter of the _Zhuangzi_ , where the original author himself suggests that since there is no one body part that rules the others, there is thus a natural or spontaneous order in the universe that exists without human intervention. As the author of this inner chapter put it,
The hundred joints, the nine openings, the six organs, all come together and exist here [as my body]. But which part should I feel closest to? I should delight in all parts, you say? But there must be one I ought to favor more. If not, are they all of them mere servants? But if they are all servants, then how can they keep order among themselves? Or do they take turns being lord and servant? It would seem as though there must be some True Lord among them. But whether I succeed in discovering his identity or not, neither adds to nor detracts from his truth.
This is essentially the same point that Peter Kropotkin made using the language of nineteenth-century science in his famous pamphlet, "Anarchism: Its Philosophy and Ideal," where he claimed that the discoveries of modern astronomy and other natural sciences have led to a new realization that there is no purposive center or natural hierarchy in nature. As Kropotkin put it about the universe, "thus the center, the origin of force, formerly transferred from the earth to the sun, now turns out to be scattered and disseminated. It is everywhere and nowhere."
It is not just for the view of the absence of purposive order but for the positive vision of a world free from all "restraints or controls" that Hsiao Kung-chuan changed his earlier skeptical view and concluded that the "thought of" the _Zhuangzi_ , even in the inner chapters, amounted to "the most radical of all anarchisms."
**Warring States Daoism as "Reactionary" and not "Revolutionary"**
Nevertheless, even if one grants the philosophical anarchism of the _DDJ_ and/or the _Zhuangzi_ , many observers find that philosophical Daoism, from the Warring States to the Wei-Jin periods, is still limited by its lack of outright support for revolution, that is, for lack of any attempts to overthrow the existing state by force. For example, Liu Xiaogan, who finds that the " _wujun_ " (again, literally, "without a prince" or, in other words, anarchist) chapters of the _Zhuangzi_ went beyond the inner chapters to attack the (political) reality of the day rather than merely try to transcend or escape it, nevertheless claims that the "theories of the Wu-Jun school never directly became a herald for any revolution." As Frederick Mote puts it, even if one accepts the _Zhuangzi_ as a thoroughgoing anarchist text and not just as advocating laissez-faire, the doctrine in this text was only the "anarchy of the non-conforming individual" and thus if the author(s) of this text were anarchists they "certainly did not believe in organization or social movements." Therefore in the end, the anarchism of the _Zhuangzi_ "could not become a political threat, except that it gave a point of view to less disinterested critics of the state."
Again, some observers relate this critique of the nonrevolutionary nature of Daoism to the emphasis put on appeals to the ruler in the _DDJ_ and the _Zhuangzi_. Both texts appeal not to the masses to revolt, but only to rulers to govern through the _dao_ rather than through coercive means. Above all, such critics would argue, the _DDJ_ and especially the _Zhuangzi_ call for transformation of the individual and not for genuine social revolution, and thus remain limited and ineffectual. Peter Zarrow, a student of twentieth-century Chinese anarchism who takes seriously the "anarchist provisions" supplied to Chinese political culture by Daoism nevertheless similarly finds Daoism lacking in this regard:
. . . traditional anarchistic tendencies, in China as in the West, were not associated with a full-fledged theory of social reconstruction. An alternative vision is not the same as a sense of how real people can create and respond to a new social structure. This traditional anarchism, then, lacked revolutionary self-awareness. Philosophical Daoists issued no calls for organizing the people or fostering resistance to the rulers they so condemned, for such calls themselves would be unnatural and interfering.
Against this idea of inherent political limits to Daoist anarchism, other scholars stress that perhaps Daoism is superior and on a higher level more revolutionary than the "social and political" Western anarchists. First, on a philosophical level Frederic Bender thinks Daoism has much to teach Western anarchism about applying a consistent metaphysical grounding for its claim of an essential "egolessness" of human nature (e.g. what Alan Ritter would call Western anarchism's search for "communal individuality"), a lack of grounding that has helped to weaken anarchism as "a practical movement for social transformation." In addition, Daoism can better explain, and thus attract followers for an anarchist movement, the psychological and not just material needs that are unmet by any type of state.
Most importantly, classic Daoism, similar to the ideas of Tolstoy and other Western pacifist anarchists, is much more consistent in its opposition to coercion and less susceptible to the contradictions of the Western anarchists, from Bakunin on, willing to embrace violent methods. This willingness to use violence or, as in Kropotkin, the failure to clearly denounce it was a large factor in leading to the demise of anarchism in the West, forever poisoning the name of anarchism in many people's minds. We will examine this argument at more length later in this chapter. Here we should note that just because Daoist anarchism rejects violence should not mean that it lacks revolutionary qualities.
Perhaps a reexamination of the concept of _wuwei_ can help us to resolve this question. Given the numerous "people power" movements that began in the late 1980s and have spread most recently to North Africa and the Middle East, one must no longer identify violent action as the most revolutionary kind of movement. Indeed, the attempts of some online Chinese bloggers to spark a "Jasmine Revolution" in China, according to one author, may reveal the possibilities of _wuwei_. As Will Clem argued in the Hong Kong newspaper _The South China Morning Post_ , those Chinese citizens who showed up in the crowded shopping districts of Beijing and Shanghai to "take a stroll" in response to the bloggers' call played a very clever cat and mouse game:
There was a certain aesthetic to the action, like a farcical ballet. No sooner had uniformed and plain-clothes officers broken up one possible gathering than the crowds simply re-formed somewhere else.
It was almost the embodiment of the ancient Taoist philosophical concept of _wuwei_ , best translated as "active non-action."
Thus it is not only state rulers who can operate by _wuwei_ but perhaps those who would oppose oppressive rule. As in the events in the middle east perhaps show, the greater the efforts of states to repress their citizens, perhaps the greater reactions of their subjects, which can be prevented from becoming violent movements that in the end would recreate state violence only by taking on a radical but pacifist Daoist attitude toward revolution.
**Warring States Daoism as individualist and not socialist**
As we have just seen above, many observers who find an essential nonrevolutionary nature in Daoism often see Daoist anarchism as a doctrine of transformation of the individual self rather than as a call for collective action. For such observers, even if the _DDJ_ and the _Zhuangzi_ did advance beyond the reported pure hedonism of the proto-Daoist Yang Zhu and even if on some level classical Daoism could be labeled as an anarchist doctrine it nevertheless remained at the level of individualist anarchism and contained no elements similar to modern socialism. Such critics in effect agree with scholars such as Feldt about the _DDJ_ and the _Zhuangzi_ but take the opposite position from him in seeing only collectivist anarchism as true anarchism. Thus, even if the Wei-Jin Daoists were firmly within the tradition of the received _DDJ_ and the _Zhuangzi_ in their explicit anarchism, they nevertheless were far from the collectivist anarchism of Proudhon, Bakunin, and especially Kropotkin. As Hsiao Kung-chuan puts it,
. . . [the authors of the _DDJ_ and the _Zhuangzi_ ] thought that the individual should abandon knowledge and make few his desires, seek self-contained contentment and not seek individual advancement, sharing accord with ruler and superiors about the way [of simplifying life] through diminishing. Hence the political method of letting alone did not demand "popular knowledge" and did not demand social equality . . . in consequence the individual becomes the only value, and freedom is not a means for guaranteeing the growth of knowledge and human capacities, but becomes in itself the ultimate goal.
Against this view, one could posit again Needham's idea of the early Daoists as harking back to the real or supposed primitive communism of pre-Zhou society. Thus, far from justifying a "ministerial bureaucracy" of an autocratic and centralized state, as Feldt somewhat anachronistically argues (since the imperial state had not formed yet), the classical Daoists may in fact have been opposing such tendencies as may have been growing but were far from universal during the Warring States period.
Even if one is not inclined to fully accept Needham's rather literal view of "Daoist communism," others would still stress the vision in the received _DDJ_ and the _Zhuangzi_ of an organic community that links individual and collective. Though Daoist anarchists perhaps based their hopes for change on individual awareness and transformation, this would _not_ lead to a society of egoists as in the ideals of Max Stirner or Murray Rothbard, but instead, as Bender argues, to the transformation of the "egoistic self into a realized, nonegoistic self which, if successful, will be the necessary and sufficient condition for corresponding transformations of [the] subject's selves and thereby the restoration of harmonious social order." In other words, the classical Daoists may have surpassed Western anarchists in their vision of a society of "individual-communal beings," the vision again that Ritter sees as the essential project of Western anarchism.
**Daoism as a negative or passive, backward-looking Nihilist doctrine and not a positive, scientific vision for the future**
Even if one grants that at some level the classic Daoist vision was communal in nature, other critics would suggest that this was always an anti-technological ideal that posited a lost utopia far in the past. Furthermore, this was inherently a negative vision of loss that offered little or no hope for grafting the benefits of economic and technological progress onto an anarcho-communist future society. As Hsiao Kung-chuan put it in an early, influential article on anarchism in Chinese political thought (but a position from which he changed greatly in his magnum opus on Chinese political thought as we saw above), "Western anarchism is . . . a doctrine of hope, whereas Chinese anarchism seems to be a doctrine of despair." Finally, the vision of freedom, if there was one in Daoism, was only of a negative freedom that could easily turn into a passive nihilist acceptance of authority, as in the early Han Daoists and the first and third generation of neo-Daoists of the Wei-Jin periods (as we will see in chapter 4).
A much more deadly version of this argument was played out in Mao's China where two strong supporters of Mao's Great Leap Forward and Cultural Revolution argued that the authors of the classical Daoist texts represented the interests of the "patriarchal slave owning class" who were gradually losing out to the rise of the feudal, land-owning class. The author of the _DDJ_ , these Maoists argued, took the stand of the slave masters by advocating "abolishing struggle and adopting a cyclical theory" that denied progressive development and wanted to "restore the idyllic system of the Zhou." Under this view, the author of the _Zhuangzi_ represented the "pessimistic and hopeless remnants of slave-masters" in the later Warring States period, when the principal contradiction (using Mao's formula for determining progressive and reactionary forces) was between the "aristocratic landlords" (who were primarily old slave masters transformed) and the newly emerging "feudal landlords" (who represented new and progressive social privilege). Those who would dare to oppose this official Maoist line on Daoism would suffer greatly for 20 years. Liu Xiaogan, for example, argued in 1957 that the theories of the "Wu Jun" school "were theories with which the laboring people criticized reality, not the theories of a reactionary faction wanting to turn things back." For Liu, "it is utterly unreasonable to say that the [anarchist] chapters [of the _Zhuangzi_ ] represented the ideas and feelings of the declining class of slave owners."
Situated as we are safely apart in time and space from these far from purely academic quarrels, where any people opposing the Maoist line on Daoism could be (and were) arrested or killed, it should be clear that these Maoist authors tried to fit Chinese history into the Stalinist unilinear straitjacket where every precapitalist society had to undergo the same transformation from primitive communes to slavery to feudalism. As Hsu Cho-yun states, and as most Chinese and Western historians recognize, "there is no evidence that the economy of ancient China was based on slavery like the economy of ancient Greece." Still, even within their orthodox Marxist faith that every idea has a particular economic class standpoint at its base, the Maoists failed to account for the opposition to Confucian beliefs in the _DDJ_ and the _Zhuangzi_ and to the fact that the classical Daoists looked to mythical _pre_ -Zhou rulers for their ideal. Furthermore, when they recognized that because of indigenous climatic and agricultural conditions and needs in China much of the "patriarchal communal" system survived in the late Zhou period, the Maoist critics missed the chance to argue that the Daoists may have represented the interests of remnants of primitive communism, much as Joseph Needham argued, as we have seen. While still making the Daoists "reactionary" in the Marxist sense, this line of argument would go a long way toward taking the Wu Jun school's ideas seriously as a "radical attack on monarchical power." Without being hampered by any version of Marxist dogma, in any analysis of the radical side of Daoism one should still try to determine whether the Daoists were really reactionary and anti-progressive.
First, on the question of looking to the past, some scholars would argue that the nature of classical Chinese makes it ambiguous, at least in Chapter 80 of the received _DDJ_ , whether the Daoist ideal is located in the past, present, or future. Even if the ideal did exist in the past, this was a tradition of most schools of thought in the late Zhou with the exception of the Legalists; but certainly the Daoists believed that the ideal society could be attained again, at the present moment or whenever the _dao_ was followed again. Furthermore, extrapolating from philosophical discussions of Daoism as related to the lack of a "beginning" or "creation myth" in Chinese thought, we could say that the Daoist stateless ideal is definitely not limited to the past but can be created of itself and by itself unconditionally. This "unconditioned norm," as David Hall says, is "also the norm of any radical form of anarchism."
Additionally, this norm is far from being only a negative version of freedom that stresses removal of restraints, though this aspect is important, nor is it true that Daoist anarchism must be based only on the concept of _wuwei_ , as Feldt suggests. It is true that many Daoist metaphors are expressed in terms of _wu_ or negative forms, as David Hall has pointed out. These forms include not only _wuwei_ (variously translated as "doing nothing," "inaction," or, as Hall says, "non-assertive action") but also _wuzhi_ ("without knowledge" or, as Hall says "without unprincipled knowing"; or as Needham suggests, without the objective and harmful technological knowing of the coming centralized state), and _wuyu_ ("no desire" or, as Hall says, "objectless desire"). We could add, of course, _wujun_ ("without a prince" or ruler) of the neo-Daoists. But as both Hall and Chang Chung-yuan suggest, these terms relate to an unleashing of creativity when one is freed from restraints.
Indeed, one cannot claim an essential negativity of classic Daoism without ignoring the whole artistic tradition spawned in large part by the literary influence of the _Zhuangzi_ on the Southern schools of Chinese art, poetry, and calligraphy that were based on the concept of _ziran_ , which this author would contend is nearly as important if not more so than _wuwei_ to an understanding of Daoist anarchism. The term _ziran_ (literally "of itself so," often translated as "natural" or "spontaneous") taken from the classical Daoist texts, was central to the revived anarchist vision of the second generation of neo-Daoists and should dispel any notion of a lack of a positive vision of freedom in ancient China. Based on an understanding of _ziran_ , one can see that the Daoists did indeed contain a positive vision of the limitless possibilities of human nature unbridled and did contain a positive embrace of the world, as David Hall suggests.
As we will see in Chapter 4 this is not to argue that all Daoists avoided the problem of slipping from anarchism into nihilism, but only that the nihilist side of Daoism comes out when shifting away from _dao_ to _wu_ (nothing, nothingness) as the key term, or to _wuming_ (the nameless), as did some generations of Wei-Jin Daoists. In other words, only when the Daoists shifted away from saying "everything exists as an interdependent whole of which we are a part" (as emphasized by those focusing on _dao_ and _ziran_ together) to saying that "nothing exists" or that "power came out of nowhere" did they shift from anarchism to nihilism. Based on this view of the centrality of _ziran_ to a consistent vision of Daoist anarchism, one could also follow Needham and Hall and see the Daoist stress on _hundun_ ("[positive] chaos," "primeval unity," or "social homogeneity") as a positive vision of individuals living and working together in [stateless] society.
As for the question of science and technological progress, here again we can turn to Needham, who sees the Daoists as representatives of a "anti-feudal" forces and who criticized the use of technology to build up new forms of oppressive rule while at the same time maintaining within their thought a "proto-scientific" element of opposition to all authority and a desire to observe the universe without preconditions, as we saw above in the satirical version of a cybernetic view of the human body in the second, inner chapter of the _Zhuangzi_. Generally speaking, Daoist anarchists did not oppose all knowledge, but only knowledge used to divide and conquer the world. Indeed, those claiming a Daoist philosophical base were at the heart of Chinese scientific discoveries.
To sum up, then, this chapter starts from the position that the later Wei-Jin Daoist anarchists were firmly within the vision of the received _DDJ_ and the _Zhuangzi_ , that is, a positive vision of the freedom and human creativity that could be unleashed once the terrible authority of the state was removed. This vision could be achieved by a program of combining individual transformation with the need to realize our essential communal nature. Far from being a corrupt, less artistic vision, the explicitly anarchist side of Daoism in the hands of talented writers and poets such as Ruan Ji and Tao Qian (see sections on these poets in this chapter below) could demonstrate the powers of _ziran_ in action. Finally, as will be argued later in this chapter, the efforts of Daoist anarchists did amount to a consistent, very long-lasting movement that by the standards and lessons of the late twentieth century we can now see can be much more progressive and effective than old-fashioned "revolutionary violence." Extrapolating from Bender's argument, we could suggest as others do concerning Tolstoy's pacifist anarchism that Daoist anarchism may solve the dilemma of Western anarchists who tried to use violent coercion to bring down the state and end all coercion, but who in the process only succeeded in poisoning the name of anarchism and in leading to some degree of popular revulsion against revolutionaries. Using the language of postmodernism (though in Chapter 4 we will examine a potential lesson also for postmodern anarchism from some Chinese thinkers who used neo-Daoist language to shift away from anarchism to nihilism), we could posit the Daoist anarchist method as an attempt to deconstruct and undermine the specific structures of ideological hegemony, structures that are far more important to ruling elites than raw coercion as a method to maintain state power, and to build in their place a new language of resistance that will not itself easily degenerate into a new system of authority (as critics have often explicitly or implicitly charged against devotees of Derrida and deconstructionism).
**Synopsis of the thought of Wei-Jin Daoist anarchists**
This author argues that Wei-Jin Daoist anarchism, which most scholars recognize as very close in spirit at least to philosophical anarchism in the West, is not a distortion but a fuller explication of the anarchism at least implicit in the received _DDJ_ and the _Zhuangzi_. In order to make this case before proceeding to outline the ideas of the key figures of later Daoist anarchism, we should outline the historical background to their thought.
The revival of philosophical Daoism ( _daojia_ as opposed to _daojiao_ , or religious Daoism as we saw in the introduction—a distinction that postdated the classical Daoist philosophers but that predated the Wei-Jin), began at the end of the later Han dynasty (25–220 CE). The warlord Cao Cao (150–220 CE), a general of the Han who had helped put down the Yellow Turban and Five Pecks of Rice rebellions, later went on to found his own state, which contended with two other states to reunify China. In this effort, he grouped around himself various scholars of different persuasions who developed philosophies designed to give him legitimacy as a ruler, perhaps including eventual justification for assuming the title of emperor itself if he could have succeeded in conquering the rival kingdoms. Included in this group of scholars were men who used Daoist and Legalist concepts to justify his rule. After the death of Cao Cao, the regent in the succeeding _Zhengshi_ reign period filled all of the important posts of the government with a group of these neo-Daoists. For 10 years, from 240 to 249 CE, this neo-Daoism became part of the official orthodoxy for the central Wei kingdom against the other states controlled by the great aristocratic family interests dominant in the centrifugal forces that had weakened and finally brought down the Han dynasty and which continued to oppose recentralization of imperial state power.
Under this first version of neo-Daoism, the emphasis was changed from _dao_ to a new focus on _wu_ ("nothingness" or "non-being"). According to this philosophy, all things come not from an underlying unity in the world, but from nothing. Activities should be carried out according to _ziran_ (again, "naturalness" or "spontaneity"). Thus Cao Cao's rise from nowhere to the top of the social hierarchy could be justified by this combined Daoist–Legalist philosophy as opposed to the prevailing _mingjiao_ ("teaching of names") school of Confucianism. Richard Mather describes the political nature of the Wei faction's philosophy as follows,
In the [Zhengshi] era the debris of Confucian ritualism had to be cleared away and room made for new values of 'Naturalness' [ _ziran_ ] and 'Non-actuality' [ _wu_ ] to buttress the new order of government . . . [Originally] the new men like [Cao Cao] had risen to power by virtue of their ability alone, and the Confucian shibboleths of the old aristocracy concerning 'goodness and morality' [ _ren-i_ ], 'loyalty and filial submission' [ _zhongxiao_ ] were meaningless to them if a man could not conduct a campaign successfully or manage an administrative post efficiently. [Cao]'s slogan, 'Only the talented will be promoted to office' . . . agreed with his policy of disregarding whether or not a man "carried a sullied or disgraceful reputation, acted with ridiculous behavior, or was neither 'good' nor 'filial'" [quoting the _Wei shu_ (Book of Wei), O29lb]. And the men he gathered about him quickly furnished this pragmatic policy with an ideological base.
At the same time, however, these neo-Daoists did not totally reject Confucianism. For example, they saw Confucius as the greatest sage since, unlike Lao Zi, he supposedly practiced the (Daoist) way of nonaction without ever talking about it.
Eventually the centralizing Wei faction was thrown out of power by the Sima clan who, after a brief period of using the Wei emperors as puppets, seized power in their own right under the name of the Jin dynasty. For a short time, the Jin reunited all of China into one empire organized along the interests of the great families. The surviving neo-Daoists then readapted their philosophy, now emphasizing _ziran_ to refer to a way of behavior opposed to official life and customs. This new use of the term also helped to justify refusing to serve in the new government as a higher form of behavior rather than as an act of disloyalty. A group of these neo-Daoists known as the "Seven Sages of the Bamboo Grove" became famous for their nonconformist behavior, which besides refusal to join the government included _qingtan_ (literally "pure conversation") a style of behavior consisting of witty remarks and put-downs, nudism, and wine drinking. All of their actions were supposedly based on precepts in the _DDJ_ and the _Zhuangzi_ , and indeed, in this time those classical Daoist texts were recollected and studied anew.
The Jin dynasty fell almost immediately after it was founded because of infighting among the royal princes as well as due to the incursions of the northern "barbarians." Moving its capital eastward, the Jin became little more than another kingdom among several regional and "barbarian"-controlled states. This era of the Period of Disunity, known as the Six Dynasties, became increasingly chaotic, as even family estates themselves soon became unstable.
In such a situation of chaos, from the fall of the Wei to the disintegration of the Jin, the anarchistic side of Daoism began to reemerge. Even in making the case for the extreme anarchism of the second generation of neo-Daoists, it is important to note that this Daoism originated as a justification for the centralization of power and only became anarchistic as the centralizing faction was defeated and its descendants forced to fight for survival against the rule of the great families. Nevertheless, the greatest of the Western anarchists, including Bakunin, Kropotkin, and Tolstoy, also came from privileged backgrounds, as Marxists never fail to point out, including from aristocratic classes that were being swept aside in the push toward industrialization and centralization of state power in the West. Thus the Daoist anarchists cannot be denigrated on these grounds as any less sincerely anarchist than their Western counterparts. Below we examine the background and key elements of anarchism of four writers of this second generation of neo-Daoists, making specific comparisons to Western anarchists along the way.
**Ruan Ji (210–263 CE)**
One of the Seven Sages of the Bamboo Grove, Ruan Ji, was the first person in the post-Han era to reemphasize the anarchistic side of Daoism. His father, Ruan Yu, was intimately involved with the military government of Cao Cao. Indeed, the Ruan family's wealth and power does not seem to precede Ruan Dun, the grandfather of Ruan Ji, who was a local magistrate in the district where Cao Cao first raised his troops. In spite of these strong Wei connections, Ruan Ji survived the executions of the Wei intellectuals at the end of the Zhengshi period by carefully walking the line between Confucianism and Daoism in his poetry, and by his nonconformist, "harmless" behavior in which he could avoid serving in the Jin government without being accused of disloyalty. He died a supposedly natural death in 263, yet that was the same year in which the last of the Seven Sages were executed by the Sima faction.
Though Ruan Ji himself never openly challenged the authority of the Jin, late in his life, in one great poetic essay, the _Daren Xiansheng Zhuan_ ("Biography of Master Great Man"), Ruan Ji raised the first banner of Daoist anarchism since the Warring States period. In the first third of this work (reprinted in Appendix 2), a fictional, nameless person supposed to have lived since Creation replies to a letter from a typical Confucian gentleman that attacked the Great Man for his unconventional ways. In his reply the Great man gives, in the words of Hsiao Kung-chuan, a "merciless attack upon conventionality, and, at the same time, an enthusiastic encomium of anarchist freedom." The Great Man begins his reply by describing the original utopian community of the _DDJ_ and the _Zhuangzi_ when all lived in harmony, innocence, and physical equality (see the next chapter), concluding that,
. . . for then there was no ruler, and all beings were peaceful; no officials, and all affairs were well ordered.
The Great Man then continues to say that, by unspecified means, artificiality was introduced into this community, including class differences between rich and poor, strong and weak. Then government came about and resulted in the greatest misfortunes. Different factions fought among themselves for power and caused great chaos. According to Jung Chao-tsu, it was from his vantage point in the struggles between the factions of Wei and Jin that Ruan Ji came to conclude that the origin of social chaos was in the power struggles between competing empires, and thus explains why in the end he came to oppose all government and advocate anarchy.
Following the _Zhuangzi_ , Ruan Ji in this essay describes the nature of sages as essentially the same as that of thieves, and the nature of government as oppression:
When rulers are set up, tyranny arises; when officials are established, thieves are born. You idly ordain rites and laws only to bind the lowly common people.
By pursuing wealth and power, these rulers hold up a bad example to the people; thus crime and rebellion ensue only after government is established and drain away all of the public wealth. Confucian ideas ensuring order through benevolence and ritual, and Legalist ideas of standardized law "are indeed nothing more than the methods of harmful robbers, of trouble-makers, of death and destruction. . . ."
Though there is more ambiguity in this poem concerning the origin of government than in the ideas of Bao Jingyan (as well will see in the next section), Ruan nevertheless does bring out of the _DDJ_ and the _Zhuangzi_ a clearer picture of how an unnatural and harmful government could originate out of the _dao_ , just as Western anarchists have to explain how government could have originated out of a species with a supposedly naturally communitarian and peaceful human nature. For Ruan, based firmly on passages in the received _DDJ_ and the _Zhuangzi_ , it is clearer that government is not just philosophically indefensible but actually harmful and counterproductive. Furthermore, government is not a natural occurrence, but an artificial creation of those trying to justify their wealth and power. Ruan equates rulers and sages in power with thieves, although the Great Man does seem to believe that the sages are merely mistaken, not insincere, in setting government up in the first place. In addition, the Great Man sees crime not just as a response to oppression, as do the Western anarchists from Proudhon to Kropotkin, but as corruption of the people by wealth and power. Thus, even with his more blatant anarchist tendencies, Ruan Ji remains within the limits of the _DDJ_ in seeing both rulers and subjects alike as well-meaning if corruptible or, as Benjamin Schwartz puts it, "latently vulnerable" to this "propensity to fall." Finally, though justifying rebellion as inevitable once government is established, Ruan Ji still paints a picture of this as an unfortunate occurrence—in other words, he is not openly advocating violent revolution as would, say, Bakunin. Nevertheless, within the thesis of this book, Ruan's poem clearly contains an anarchist theory of the state.
It should be noted that the Daoist anarchist critique occurs only in the first third of Ruan's poem; in the latter two-thirds of the essay, the Great Man soars through the universe to attain harmony with the _dao_. Nevertheless, one should not mistake the fictitious and fantastic nature of this essay as an indication of anything less than a serious work. As Donald Holzman says, "far from marking this as a work of satirical exaggeration, a playful slap at the bigoted Confucian of his day, the extremeness of his condemnation shows how heart-felt it was, how absolutely serious he is."
Soon after the death of Ruan Ji, the neo-Daoist or _qingtan_ movement began to decline from "liberty to libertinage" as it was taken up by the idle sons of the aristocracy. At the same time, serious neo-Daoist philosophers began again to justify government service as being in line with _ziran_ once they saw the inevitable and irreversible triumph of the Sima reaction.
**Bao Jingyan (ca. 300 CE)**
Before the neo-Daoist movement totally degenerated, however, the last and probably greatest direct statement of Daoist anarchism was made. It survives only as a short treatise in one chapter of the alchemical text of Ge Hong (253–333 CE), who reproduced it with a lengthy refutation of his own (see Bao's full tract in Appendix 3). This anarchist was clearly influenced by Ruan Ji and, according to Ge Hong, "enjoyed the works of Lao Zi and Zhuang Zi and studied the discipline of dialectics and sophistry." The anarchist's name is Bao Jingyan, who had the same surname as one Bao Jing, the father-in-law of Ge Hong, thus probably placing Bao Jingyan in the same family and aristocratic class.
While firmly basing his critique of the State on the ideas in the received _DDJ_ and the _Zhuangzi_ , Bao Jingyan unambiguously shifts the emphasis regarding the origin of the State from that of misguided altruists trying to order the world, to that of, according to Lin Mousheng, "an institution created and maintained by the dominant classes in society and imposed upon the weak and ignorant." Bao starts out by explicitly condemning the Confucian theory of the origin of the State as a mere pretext for rule of the strong over the weak, arguing that ". . . servitude and mastery result from the struggle between the cunning and innocent, and Blue Heaven has nothing whatsoever to do with it."
Bao goes on to use the argument in Chapter 9 of the _Zhuangzi_ to refute the idea of the naturalness of having ruler and ruled, concluding that, in reality, government works by force in order to enrich those in office:
. . . And so the people are compelled to labor so that those in office may be nourished; and while their superiors enjoy fat salaries, they are reduced to the direst poverty.
Next Bao denigrates by implication those Daoist alchemists who sought immortality, as well as the Confucians who claimed to believe in resigning from office under an immoral government. He denounces the Confucian virtues as a response to rebellion and discord rather than as natural occurrences, concluding that "loyalty and righteousness only appear when rebellion breaks out in the empire, filial obedience and parental love are only displayed when there is discord among kindred. Again, the last sentence is firmly based on the received Chapter 18 version of the _DDJ_ and on the _Zhuangzi_.
The next section of Bao's treatise repeats the description of the Golden Age found in Chapter 80 of the received _DDJ_ and in the _Zhuangzi_ , when small, independent, and self-sufficient agricultural communities supposedly lived in harmony both with each other and with animals. In this ideal state, Bao claims, there was no accumulation of private property or wealth, nor were there any plagues, pestilence, rebellions, nor, of course, any government. In an unspecified way, knowledge and cunning entered this world, and immediately it lost the Way and fell into decadence. A hierarchy was established, along with regulations for promotion and demotion and profit and loss, class distinctions, and technological development. With the search for and acquisition of wealth, people began to strive for reputation; next thievery developed, after which came armed aggression and war.
Bao then denounces evil tyrants, not just as immoral rulers but as capable of doing evil only because of the existence of the principle of rule and the distinction between lord and subject. In such a situation, people are inevitably corrupted both by the oppression of rulers and by covetousness for the wealth and power that rulers possess. Finally the people are led to rebel and are then unstoppable by government to the point that ". . . to try to stop them by means of rules and regulations, or control them by means of penalties and punishments, is like trying to dam a river in full flood with a handful of earth, or keeping the torrents of water back with one finger." Here Bao goes a step beyond Ruan Ji to explicitly suggest the social causes of crime, a point that marks him greatly similar to Western anarchists. His final metaphor is strikingly similar to the idea of Michael Bakunin that,
. . . all the revolutionaries, the oppressed, the sufferers, victims of the existing social organization, whose hearts are naturally filled with hatred and a desire for vengeance, should bear in mind that the kings, the oppressors, exploiters of all kinds, are as guilty as the criminals who have emerged from the masses . . . it will not be surprising if the rebellious people kill a great many of them at first. This will be a misfortune, as unavoidable as the ravages caused by a sudden tempest, and as quickly over. . . .
Both Bao and Bakunin find that crime is caused by government, especially by one emphasizing harsh laws and punishments whether they be late absolutist monarchies of Bakunin's day or the Legalist side of the Chinese imperial state, which in its last stages is more easily exposed as a government imposed by force with no pretensions of morality. Bao, following the _DDJ_ and Ruan Ji, sees this Legalist government as the final phase of rule after Confucianism had earlier stirred up the people's desires. Both Bakunin and Bao Jingyan, then, refuse to condemn rebels and also seem to be trying to scare members of the ruling elite with the possibility of violent revenge to be exacted by the masses. Nevertheless, neither tyrants nor criminals should be looked upon as incurably evil or deserving of punishment for both Bao and Bakunin; it is their corruption by the State that accounts for their evil ways.
In sum, Bao Jingyan explicitly detailed all of the anarchistic tendencies in the political critique of Daoism and made more explicit two important elements: first, the nature and origin of government as the oppression of the strong and rich over the weak and poor, rather than as well-intentioned attempts of sages to order the world; and second, the explanation of crime and popular revolt as the inevitable reaction of the people to the tyranny of government. Influenced heavily by the "anarchist" chapters of the _Zhuangzi_ and perhaps by Ruan Ji in these opinions, he denied completely that the attempts of sages were simply mistaken—clearly he denounced sages as trying to protect stolen property by imposing an unnatural ideal of morality.
In Bao Jingyan we see an explicit rejection, not only of the naked, Legalist style of harsh rule but also of the Confucian Mandate of Heaven theory of the origin of government. In this rejection Bao also closely parallels Western anarchists, especially Proudhon, who see the nature and origin of government as the protection of the seizure of private property by theft. As Western anarchists rejected other contemporary theories of government, such as Social Contract and Divine Right, so Bao rejected all justifications for rule in his day. Again, following Ruan Ji, Bao also clearly went beyond pure philosophical anarchism by viewing government as harmful and criminal, not merely as unjustifiable.
Nevertheless, while resolving this contradiction, Bao still fails to explain how wealth itself was introduced into the ideal community, or how "knowledge and cunning came into use." But of course one could argue that Western anarchists, given their positive view of human nature, also failed to give this explanation. Perhaps one could augment both anarchisms by taking the idea from modern anthropology that it was an increase in the social surplus created by accidental discoveries and improvements in agriculture that gave rise to a fight for control over this surplus, with government as the final justification and protection of the wealth of the winners.
Even without such an argument, and while clearly not endorsing violent revolution, as opposed to Bakunin and his followers among Western anarchists, Bao Jingyan still has no sympathy with rulers and clearly presents an anarchist ideal in the place of state rule. In his ideal society, as in that of most Western anarchists, no crime occurs and no law and punishments are even dreamed of. Since the social environment of excess wealth and the coercion and oppression of government are the main causes of crime, at the same time, in the anarchist ideal of both Bao and Bakunin, crime will largely disappear. Punishment by the government is useless and hypocritical and only exacerbates the problem, according to both Bao and Western anarchists such as Bakunin, since it is the principle of rule that allows rulers to do harm and leads to the people's violent reaction. Belief in the environmental factor as the cause of crime as well as the rejection of law and punishment are thus key ingredients in both Daoist and Western anarchism.
_Liezi_ **and the Yang Zhu chapter**
As stated earlier, the anarchistic trend in Daoism began to die out when, as Balazs says, the qingtan movement fell into the hands of the
gilded youth—"the brothers and sons of the idle" [ _guiyu zidi_ ], in the stock phrase of the Chinese historians—and became fashionable, whereupon the attempt made by the politicians of the first generation and by the artists of the Bamboo Grove to break free from social conventions degenerated into moral breakdown.
It is either the beginning of the first qingtan generation (249–265 CE) or the end of the second (265–317 CE) that Aloysius Chang thinks is most likely to have produced the fatalistic work _Liezi_ and its hedonistic chapter "Yang Zhu." According to A. C. Graham, the _Liezi_ , except for the Yang Zhu chapter, may have been written by a single author as late as 300 CE. In any event, the philosophy of Yang Zhu as expressed in this chapter could be summed up very simply:
If the men of old could benefit the entire world by pulling out one hair, they would not do it. If they were offered the entire world for life, they would not take it. When no man hurts one hair and no man benefits, the world will be at peace.
According to the author of this chapter, if everyone minded their own business the idea of rule itself would disappear:
Take my way of private life; if it could be extended to the whole world, the principle governing the ruler-subject relationship would naturally die out.
Chang finds that while the Yang Zhu chapter contains passages from the original philosophy of the historical Yang Zhu of the late Zhou era, passages which contain his anarchistic statements, most of the rest of the chapter, as well as the _Liezi_ as a whole, is not fundamentally opposed to the idea of government as long as it does not interfere or try to regulate people's enjoyment of life. A. C. Graham, on the other hand, concludes that the _Liezi_ gives many examples of the ideal Daoist anarchy:
The [ _Liezi_ ] itself reflects this [anarchistic] tendency [of Ruan Ji and Bao Jingyan], although very cautiously. The hedonist [Yang Zhu] chapter explicitly recommends a society in which each pursues his own pleasure without interfering with others, and "the Way of ruler and subject is brought to an end." The [other] chapters retain the old assumption that the power emanating from a true sage maintains the harmony of society without the need of government, but imply that he is not an Emperor; such sages have only existed either before or outside the Chinese empire.
Nevertheless, the _Liezi_ as a whole advocates a fatalistic acceptance of life, while the Yang Zhu chapter deemphasizes the element of restraint present in the original Yang Zhu's philosophy and substituted for it the belief in an unrestrained enjoyment of the full sensual pleasures of life. Therefore, we could put forward the hypothesis that the _Liezi_ and its Yang Zhu chapter represent a transition stage in the decline of the _qingtan_ movement and its shift from pacifist anarchism into passive nihilism.
As many students of Chinese philosophy in this century have noted, the extreme individualism of the Yang Zhu chapter bears a striking similarity to the work _The Ego and Its Own_ by the early nineteenth-century German philosopher known as Max Stirner. Though ideas of the Yang Zhu chapter also helped play into the passive nihilism of the third generation of _qingtan_ Daoists, and the ideas of Stirner were later influential perhaps on the development of Nietsche's nihilism and perhaps even of later fascism, in the beginning both the Yang Zhu chapter and _The Ego and its Own_ were sincere statements of rebellion by men who wished to place the individual above the demands of the central authority. Despite this great similarity, others find that Yang Zhu may represent a less radical form of individualism as the author of that chapter "concedes the existence of other egos" and condemns the use of force against others, which Stirner would justify as "might makes right." Still, both the Yang Zhu author and Max Stirner show a basic and striking similarity in their placing of the individual above the state. In both cases this was a much more radical individualism than that represented by other figures in their respective movements, but nevertheless foreshadowed other anarchists' denunciations of the State and its limitations to the full potential of humans in both ancient China and the West.
Although the _qingtan_ movement did degenerate and eventually die out, the Daoist anarchist tradition lived on in art and literature and in the lives of scholars and government officials after hours and in retirement. Especially in times of disorder, or when the government sponsored or promoted the study of an official Daoism, the anarchistic side of Daoism would resurface, for example, in the Buddhist-inspired anarchism of Wu Nengzi during the breakdown of the Tang dynasty in the tenth century (see Chapter 4) or by Liu Shipei during the Western-inspired anarchist movement in China in the early twentieth century (as we will see in Chapter 5).
**Tao Qian's "Peach Blossom Spring"**
The best example of the anarchist tradition of Daoism surviving through art is the poem "Peach Blossom Spring" by Tao Qian (326–397 CE) (see Appendix 4). Tao Qian was a scholar–official of the surviving remnant of the Jin dynasty, who retired from government service during the later years of his life in a Confucian protest against the "immoral" regime. Refusing all offers of government posts, he retired to his estate to till his own soil and write poetry. Possibly inspired by a contemporary account of a lost, independent community of people, and obviously influenced by classical Daoist texts, especially the _Zhuangzi_ , he wrote twin prose and poetic accounts describing a fisherman who sailed down a stream through a cave to discover a hidden land to which people had fled long ago to escape the harsh Qin dynasty and where they founded a society without government. This poem epitomizes the Daoist ideal of anarchy and as such demands a detailed commentary of its political philosophy.
As with most Chinese poetry, this poem can be read on several levels. It might just be the poetic expansion of the contemporary report of such a place, but on the other hand the fisherman's physical discovery might be a metaphor for a psychological discovery of an internal, forgotten tendency. The fisherman found the spring only after he "had lost track of how far he had gone," which could be taken to mean temporarily forgetting conscious attempts to alter the world. By not striving or desiring to affect artificial changes, he was able to "return to the root" in the phrase from the _DDJ_ , in other words, perhaps, to discover this anarchical state of being as an innate part of his nature. When he followed this impulse and returned to this state of "original simplicity" ( _pu_ —a very important Daoist term), his path "suddenly opened out and he could see clearly." Therefore the amazing community he found might on one level be the _dao_ itself. In other words, while the _dao_ may not be an empirically verifiable entity, it is nevertheless a real state of being that is far from simply metaphysical. An artistic, suggestive description, which in a way constitutes all of Daoism, is not inherently more metaphysical than a supposedly objective explanation.
On this level the _dao_ is really a metaphor for freedom, a word that scholars used to stress as not existing in ancient Chinese and _had_ to be expressed metaphorically in order not to confused with the idea of libertinage or license (as Balazs argues, though A. C. Graham disparages the idea that in traditional China there was no way to get across the idea of liberty). As with modern Western anarchism, in Daoist anarchism the idea of freedom means more than the absence of restrictions. In both anarchisms, freedom is inseparable from equality and community. This is expressed in Daoism by the term _pu_ ("original simplicity," which Needham thinks refers to the "solidarity, homogeneity, and simplicity of primitive collectivism"), a term which this author would contend, is yet another metaphor for freedom in the positive, active sense. In other words, the legacy of Daoist anarchism for Tao Qian was far from purely individualistic, but contained strong communal elements.
The village that the fisherman found was totally peaceful; all were "carefree and happy." There were no institutions and rites to teach the people goodness and morality, yet there was no evident selfishness. All invited him into their homes and shared their harvest with him. As in the ideal in the _Zhuangzi_ , all relished simple food and clothing and were content to farm and live at home. When Tao Qian says the chickens and dogs could be heard from farm to farm, he is further suggesting the passages from Chapter 80 of the received _DDJ_ and Chapter 10 of the _Zhuangzi_ where the members of the ideal community also live so close to each other that the fowls of neighboring villages could be heard, "but the people would grow old and die without ever having been there." It is important that the poem notes that the inhabitants of the spring fled there in order to escape the Qin dynasty, both to show that this state of harmony is only to be achieved in the absence of government, and to show that the official views of the history and civilization of humanity, far from improving our moral nature, must also be abandoned along with government in order to attain the ideal state. In other words, merely by stating that the people fled the Qin dynasty and had no desire to return even after hearing of the glorious succeeding dynasties, Tao Qian alluded to the Daoist belief that the Confucian attempt to inculcate morality and order only comes about after morality and order have been lost. Far from the Confucian ideal of people in their primitive state desperately needing good government, it is clear in this poem that it is the State which causes the suffering of the people.
However, the fisherman was not a true Daoist sage and in time remembered himself and desired to return to society. Later, deliberately retracing his steps, this time failing to forget conscious effort, he was unable to find the community again. True to its own dependence on domination and its desire to attach its institutions upon the people, the government also tried to find the community upon hearing of it from the fisherman. Along with the Confucian gentleman Liu Ziji, however, the government was unsuccessful in its search. Clearly, the idea of government as well as the basic idea of Confucianism is the antithesis of the ideal Daoist community. More importantly, Tao Qian is expressing a nearly identical idea to that of Kropotkin, that government depends on people's voluntary cooperation—mutual aid in Kropotkin's language—in order to have a society to rule over, but government is at the same time parasitic on this cooperation and in basic contradiction against it. In other words underlying Tao Qian's poem is a clear anarchist theory of the state.
In time men stopped seeking the spring, just as in time men forgot their original state and were corrupted by laws and morals and knowledge, until, as in Chapter 10 of the _Zhuangzi_ "nothing was left in its original state. It must be hacked and sawed until the whole world was in utter chaos and confusion. All this came from tampering with the heart of man."
The tale suggests the survival of the _dao_ even in the corrupt times of Tao Qian. Indeed, the fisherman could be a metaphor for the former Confucian bureaucrat Tao Qian himself, who in one moment of inspired despair drew upon the reservoir of the Daoist anarchist tradition. That the statist and anti-statist traditions could survive side-by-side in one man is perhaps not easily understandable to Western observers, but it is the chief means by which the anarchistic side of Daoism survived in Chinese minds up to the twentieth century.
In any event, in "Peach Blossom Spring" we can see that the Daoist anarchist ideal is far from that of individualism. The Daoist ideal community, it could be argued, conforms closely to Kropotkin's idealization of mutual aid, that is, his view that primitive anarchist communism was an important factor in the past evolution of humanity, as well as in his ideal communist society of the future. As cooperative tendencies survive from "savage tribes" to "barbarian villages" to medieval towns and even to the present in Kropotkin's analysis, so too perhaps did elements of this primitive communism survive the development of Zhou feudalism and even the onslaught of the centralized, bureaucratic state in China. In the ideal of both Kropotkin and Tao Qian, humans are easily able to survive without government by utilizing mutual cooperation and communal living and work.
Tao Qian can also stand, perhaps, for another great similarity of the Daoist anarchists and one strand of Western anarchism. This strand, of course, is pacifism. Leo Tolstoy, the greatest Western exponent of the pacifist anarchist ideal, perhaps comes closest to the Daoist vision as reenunciated by Tao Qian. For Tolstoy, God was synonymous with nature, and humanity was a part of God, as for the Daoist anarchists humans are all a part of the _dao_ from which all things arise. If we only fulfill "the infinite law from Whom he has come" for Tolstoy, or if we only "return to the root" and act in accordance with _dao_ , then the ideal anarchy would be achieved by itself, and the State, with all of its instruments, would eventually disappear. Of course, both pacifisms seem to reject material progress, at least as an end in itself, and both hold up a life of simplicity as the ideal. Both recognize the absurdity and impossibility of governing by force and violence, and both reject the use of violence to do away with government. Both see the moral enlightenment of each individual, enlightened that is, to see the natural connection between all individuals as the only means to achieve the ideal society. Most importantly, rather than placing humans above and isolated from the natural and spiritual, against what they see as the orthodoxies of their day, both Tolstoy and the Daoist anarchists such as Tao Qian in this poem, construct a nonauthoritarian ideal which defines human freedom and the pure anarchist society as attainable only by recognizing the link with the natural and spiritual. For both Tolstoy and Tao Qian, this link had never really been severed, but only forgotten and perhaps disguised by the State. God and the _dao_ are synonymous with freedom in these systems of thought; the state of living totally by the power of love or the _de_ of the _dao_ is thus synonymous with the state of anarchy. Therefore this author would strongly disagree with Frederic Bender who believes that Tolstoy was less than a fully consistent anarchist because his thought "relies ultimately upon the authority of God." Instead, Tolstoy's vision was clearly one in which each individual came to see the link to the rest of humanity and to the universe by him or herself, by accepting God as Love into one's heart, not by a process of "rational knowing" or certainly not by accepting an official, imposed idea of God from any Church institution. In this hope that we can by our own efforts find the link between the individual and collective mind, the Daoists and Tolstoy were remarkably similar and more consistent than other anarchists, this author would argue.
Tao Qian is also strikingly similar to Western anarchists such as Godwin and Tolstoy who fostered the idea of anarchism through their art. Although Daoist anarchism died out as a movement long ago, perhaps, its influence carried on in the artistic tradition that the _ziran_ ideal inspired. Ruan Ji and Tao Qian, two of China's greatest poets, carried on the Daoist anarchist ideal in two great poems of fantasy that inspired whole genres as well as individual poets throughout Chinese history. In both China and the West, then, perhaps the anarchist traditions survive in an inactive, yet purer form. For the artistic metaphor of an undifferentiated freedom and equality, stripped of all artificial blueprints of how to attain that condition in the future, perhaps suggests more powerfully than calls to violent revolution the universal idea that can never be destroyed or extinguished, the idea of the unimaginable heights that could be achieved by humanity unrestrained—the idea of freedom in the active sense that is the pure ideal of anarchy.
**Conclusion**
By examining the thought and art of key Wei-Jin Daoists, we have seen how Daoist anarchism can be considered a long-term, clearly identifiable movement in Chinese history. Daoist anarchism does go beyond pure philosophical anarchism and can be considered a movement for real political change, albeit a pacifist one. Daoist anarchism is also not limited to individualist anarchism, except perhaps in the ideas of the "Yang Zhu" chapter of the _Liezi_ , but in fact does contain a clear communitarian ideal.
It is true that Wei-Jin Daoist anarchists, for all their opposition to authority, so far as we know, never led peasant rebellions against the state (though the life and career of Bao Jingyan remains a mystery). Critics of the case for Wei-Jin Daoist anarchism might maintain that since at least some of the Wei-Jin Daoists were representatives of families and groups whose ancestors had supported Cao Cao's attempts to reinstitute an empire built on central bureaucratic lines as opposed to the Sima family rule of the great landowners, perhaps they were not sincere anarchists but only, in effect, sore losers. Even if they were sincere, the neo-Daoist anarchists could easily have been identified in the minds of the peasants with the process of centralization of state power, with all of the connotations of taxation, military conscription, and public works corveés that centralization implies. Nevertheless, since in the end the peasant rebellions in Chinese imperial history were themselves quite hierarchically organized, coercive movements that failed to break down the imperial system of autocracy, perhaps the Daoist anarchists were aiming at opposing the state by attempting to subvert its myth of legitimacy and by undermining the confidence of the scholar–gentry elite in the morality and/or efficacy of rule. Thus, even if there were limits to Daoist anarchism because of the class background of its main adherents in the Wei-Jin period, on the other hand, Daoist anarchism never suffered the terrible contradictions of Western anarchists such as Bakunin. Many scholars still view such Western anarchists today as the precursors of the Leninist vanguard, and as people who helped to justify violent acts that seemed to have poisoned the name of anarchism.
Furthermore, as some students of revolution have long suggested, the rare instances in history of genuine social revolution only occur after the ruling elite itself split or became demoralized. This attempt to sabotage the confidence of the ruling elite is the main project of Daoist anarchists, one could argue, from the Warring States through to the Wei-Jin Daoists and beyond, an intellectual pacifist guerrilla project that is often repressed but is also easily revived in succeeding centuries. In this sense, then, the similarities outlined above between Daoist and Western anarchists relate to a common project that is ultimately more important, and more consistent with the anarchist ideal of ending coercive rule. That project is to sow a seed of doubt and undermine the faith in the authority of elites and through them the masses. By helping to break the hegemony of dominant statist ideologies promoted by the intellectual agents of the State, if only for brief moments, Daoist and Western anarchists may achieve their greatest significance.
**Notes**
**** Feldt, "Governing Through the Dao: A Non-Anarchist Interpretation of the _Laozi_ ," _Dao_ 9 (2010): 324, 325, n. 3.
**** We will cite below many English language sources that give an anarchist interpretation of Daoism. For German-speaking readers, perhaps the best summary is in Gotelind Müller, _China, Kropotkin und der Anarchismus_ : 110–18. Feldt's different interpretation of _wuwei_ is the main basis for his non-anarchistic view of the _DDJ_. See Feldt, 323–37. Even though this author argues in this chapter and throughout this work that _wuwei_ can and does allow for non-violent passive resistance, nevertheless the concept of _wuwei_ is only one important clue about the _DDJ_ and, _contra_ Feldt, is far from the only basis for an anarchistic interpretation. Furthermore, Feldt's non-anarchistic interpretation of the _DDJ_ rests on what this author will argue are three very dubious assertions, including, besides his claim that the _DDJ_ must be given primacy over the _Zhuangzi_ , second, his belief that all anarchism must be based on a defense of "the traditional Western, atomistic individualism" against state interference (326, 330), and third, that the _DDJ_ does not depart from "ancient Chinese political texts [that] unvaryingly assume an autocratic framework" (328), and thus that the "political structure presented in [the _DDJ_ ] would necessarily be autocratic with a centralized government ruled by the Daoist sage and administered by numerous ministers" (335). We will question both of these latter assumptions later in this chapter.
**** Hsiao Kung-chuan, _A History of Chinese Political Thought_ , 1: 298–9.
**** Roger Ames, "Is Political Taoism Anarchism?" 35.
**** Frederic Bender, "Taoism and Western Anarchism," 12.
**** Feldt, 326, 336.
**** Ibid., 334, 336.
**** Arthur Waley, _Three Ways of Thought in Ancient China_ , 74.
**** Benjamin I. Schwartz, _The World of Thought in Ancient China_ , 213.
**** Graham, _Disputers of the Tao: Philosophical Argument in Ancient China_ , 310.
**** Ibid., 311.
**** Again, a view most clearly expressed in Feldt, 336, though he claims that the type of state allowed in the _DDJ_ under his expanded concept of _wuwei_ could "far exceed the scope of the legitimate functions of Nozick's state," leading him to what this author views as the highly self-contradictory assertion that the supposed laissez-faire state of classical Daoism could support a highly centralized, autocratic form of rule (as will be noted later in this chapter).
**** Etienne Balazs, _Chinese Civilization and Bureaucracy: Variations on a Theme_ , 234–5; Richard Mather, "The Controversy over Conformity and Naturalness during the Six Dynasties," 161–4.
**** John P. Clark, "On Taoism and Politics," 84–5.
**** Joseph Needham, _Science and Civilization in China_ , II: 100–32.
**** Burton Watson (trans.). _The Complete Works of Chuang Tzu_ , 1–2.
**** Ames, _The Art of Rulership: A Study in Ancient Chinese Political Thought_ , 46. For a full translation of the _Huainanzi_ , see Liu An, King of Huainan, _The Huainanzi: A Guide to the Theory and Practice of Government in Early Han China_. For other partial translations, see deBary and Bloom, 268–73, and Mark Csikszentmihalyi (ed. and trans.), _Readings in Han Chinese Thought_ , 63–4, 72–5.
**** Ibid., 148.
**** Ames, "Is Political Taoism Anarchism?" 113; _The Art of Rulership,_ 30–1 and _passim_ ).
**** Ames, _The Art of Rulership,_ 44 _._
**** Graham, _Disputers of the Tao_ , 306–11; _idem._ , _Chuang Tzu: Textual Notes to a Partial Translation_ , 202–13.
****Watson, 14–15.
**** Ibid., 14.
**** Graham, _Disputers of the Tao_ , 64–74.
**** Ibid., 70–2.
**** Ibid., 306.
**** Liu Xiaogan, _Classifying the Zhuangzi Chapters_ , _passim_. Also see Livia Kohn, "Review of Liu Xiaogan, _Classifying the Zhuangzi Chapters_ ," 420–4.
**** As argued in Clark, _passim_ , and by this author originally in Rapp, "Taoism and Anarchy: An Analysis of the Political Critique in Philosophical Taoism and Its Comparison with the Western Philosophy of Anarchism, 18–23.
**** See Ames, "Is Political Taoism Anarchism?," 36–7.
**** Watson, _The Complete Works of Chuang Tzu_ , 38.
**** Kropotkin, "Anarchism: Its Philosophy and Ideal," in Kropotkin, _Kropotkin's Revolutionary_ Pamphlets, Roger Baldwin (ed.), 117. This author was reminded of Kropotkin's cybernetic argument by Arif Dirlik, who noted the evident adaptation of this argument by the twentieth-century Chinese anarchist Wang Siweng as part of his criticism of the Marxist emphasis on centralized organization. See Dirlik, _Anarchism in the Chinese Revolution_ , 246, citing Wang Siweng, "Hewei er xinyang wuzhengfu gongchan zhuyi" (What Are Anarcho-Communist Beliefs), 5–19.
**** Hsiao, 318, a change from his earlier view of Daoism as a philosophy of despair, as we will see below.
**** Liu Xiaogan, "The Wu Jun School," Chapter Three of _Zhuang Zi houxue san paizhi yanbian_ (The Evolution of Three Schools of Latter-Day Zhuang Zi Philosophy), translated in _Chinese Studies in Philosophy_ 23:2: 50.
**** Ibid., 85.
**** Frederick W. Mote, _Intellectual Foundations of China_ , 76.
**** For example, see Bender, 12–15.
**** Peter Zarrow, _Anarchism and Chinese Political Culture_ , 6–12.
**** Ibid., 11.
**** Alan Ritter, _Anarchism: A Theoretical Analysis_ , Chapter Two, "The Goal of Anarchism: Communal Individuality," 25–39.
**** Bender, 15–20.
**** Ibid., 20–1.
**** Ibid., 24–5.
**** As argued, for example in Woodcock, 16–17.
**** Will Clem, "The Flowering of an Unconventional Revolution," _South China Morning Post_ , March 3, 2011.
**** Hsiao, 317–18.
**** Needham, II; also see Clark, 82.
**** Feldt, 334.
**** Bender, 9.
**** Ritter, _passim_.
**** Hsiao, "Anarchism in Chinese Political Thought," 260.
**** Guan Feng and Lin Lüshi, "Characteristics of Social Change and Philosophical Thought during The Ch'un-Ch'iu Period," 90–104.
**** Ibid., 168–70.
**** Liu, "The Wu Jun School," 6.
****In chapter nine below we examine modern Chinese thinkers who disputed this unilinear Marxist view.
**** Hsu Cho-yun, _Ancient China in Transition_ , 13.
**** Guan and Lin, 49–50.
**** Liu, "The Wu Jun School," 4–5 and _passim._
**** Arthur Waley, _The Way and Its Power: A Study of the Tao Te Ching and Its Place in Chinese Thought_ , 242, n. 1. We will examine this part of the _DDJ_ in the next chapter for its utopian vision.
**** The author would like to thank one reviewer of an early version of this chapter for reminding me of this point.
**** David Hall, "The Metaphysics of Anarchism," _Journal of Chinese Philosophy_ 10 (1983), 56–57.
**** Ibid., 59–60.
**** Chang Chung-yuan, _Creativity and Taoism_ , _passim_.
**** Balazs, 247; Mather, "The Controversy over Conformity and Naturalness," 169–70.
**** Hall, 58; Needham, II: 107–15.
**** Needham, II: 86–99; 121–7; 131–2.
**** Bender, 24–5.
**** For one movingly written and still useful description of this period, see Balazs, Chapters 13–14, 187–254.
**** Mather, 161, 163.
**** Fung Yulan, _A Short History of Chinese Philosophy_ , 218.
**** See Richard Mather (trans. with introduction). _Shih-shuo Hsin-yu: A New Account of Tales of the World_ , for an English translation of the most famous collection of _qingtan_.
**** See Holmes Welch, _Taoism: The Parting of the Way_ , revised edition, 123–6.
**** Donald Holzman, _Poetry and Politics: The Life and Times of Juan Chi (AD 210–263)_ , 2–5.
**** Hsiao, "Anarchism in Chinese Political Thought," 253.
**** Holzman, 195; also see Balazs, 238 and Bauer, 135–7.
**** Jung Chao-tsu, _Wei-Jinde ziran zhuyi_ (The Wei-Chin Doctrine of Naturalness), 42–3.
**** Holzman, 195.
**** Ibid.
**** Schwartz, 210.
**** Holzman, 189.
**** Balazs, 247.
**** Mather, 1969–70, 169–70.
**** See the translation of Bao's tract in Balazs, 243–6, reprinted in Appendix 3 below and in Robert Graham, _Anarchism: A Documentary History of Libertarian Ideas, Volume One: From Anarchy to Anarchism (300 CE to 1939)_ , 1–3; also see Bauer, 138–40; Liu, "The Wu Jun School," 80–8, and Lin Mousheng, _Men and Ideas: An Informal History of Chinese Political Thought_ , 150–8.
**** Ge Hong, _Baopuzi nei wai pian_ (Master Embracing Simplicity, Inner and Outer Chapters), _wai pian_ 48, "Jie Bao" (The Refutation of Bao), translated in Lin, 152–3.
**** Needham, V: 76; also see Bauer, 138, 438, n. 16; Ofuchi Ninji, "Ho seiden ko" (On the Identity of Bao Jing), _Tohogaku_ (Eastern Studies) 18 (June 1959): 18, and Robert Ford Campany, _To Live as Long as Heaven and Earth: A Translation and Study of Ge Hong's_ Traditions of Divine Transcendents, 16.
****Lin, 152–3.
**** Bao, translated in Balazs, 243.
**** Ibid., 244.
**** Ibid.
**** Ibid., 245.
**** Ibid., 246.
**** Bakunin, quoted in Sam Dolgoff (ed.), _Bakunin on Anarchy_ , 150.
**** Balazs, 245.
**** Ibid., 246–7.
**** Aloysius Chang, "A Comparative Study of Yang Chu and the Chapter on Yang Chu," _Chinese Culture_ 14 (1972): 77–8.
**** A. C. Graham, _The Book of Lieh Tzu_ , 3.
**** "Yang Zhu," translated in Lin Mousheng, 81.
**** Ibid., 81.
**** Chang, 77–8.
**** Graham, _The Book of Lieh Tzu_ , 8.
**** See for example Lin Mousheng, 152–3.
**** As argued for example by John Carroll, _Max Stirner: The Ego and His Own_ , Introduction, 15–16.
**** Lin, 86–7.
**** Zarrow, 10.
**** James R. Hightower (trans.), _The Poetry of T'ao Ch'ien_ , 1–3.
**** Ibid., 256.
**** This author's following commentary on this poem is based on the translation in Watson, _The Columbia Book of Chinese Poetry: From Early Times to the Thirteenth Century_ , 142–3, reprinted in Appendix 4 below. For other translations, see Cyril Birch (ed.), _Anthology of Chinese Literature_ Volume 1: _From Early Times to the 14th Century_ ; Hightower, 254–5; and Tan Shilin (trans. and ed.), _The Complete Works of Tao Yuanming_ , 96–9.
**** Balazs, 247.
**** Graham, _Disputers of the Tao_ , 202.
**** Needham, II: 114.
**** _Zhuangzi_ , translated by Waley, _Three Ways of Thought in Ancient China_ , 39, 69.
**** Ibid., 72.
**** Peter Kropotkin, _Mutual Aid: A Factor of Evolution_ , _passim_.
**** Tolstoy, quoted in Marshall S. Shatz (ed.), _The Essential Works of Anarchism_ , 249–50.
**** Bender, 19.
**** For Bakunin as a precursor of Lenin, see the publisher's preface in G. P. Maximoff (ed.), _The Political Philosophy of Bakunin: Scientific Anarchism_ , 15 and the critical discussion of this view in Dolgoff, xxiii, 9–12, 181–2.
2
Utopian, anti-utopian,
and dystopian ideas in
philosophical Daoism
**Introduction**
Although the previous chapter concluded that Daoist anarchists may have intended to sow seeds of doubt among the elite in order to undermine the whole concept of rule, we saw that some scholars have charged this attitude at best amounts only to a negative type of anarchy that lacks a positive vision of a possible stateless society. We refuted that charge by showing how, with the concepts of with the _ziran_ and _hundun,_ the Daoists did suggest the possibilities of living in a harmonious, stateless society; nevertheless, such a positive view of freedom does not mean that Daoist anarchists were completely utopian in their outlook. This chapter will demonstrate that Daoist anarchists' suspicion of all forms of rule included suspicions about other forms of utopian thought, while at the same time they retained their own positive vision about the possibilities of stateless society. This chapter, then, attempts to link certain aspects of Daoist thought of the late Zhou (ca. fourth–third centuries BCE) and Wei-Jin (ca. third–fourth century CE) periods to both positive and negative connotations of utopianism.
On the one hand, both students of anarchism and many anarchists themselves note the "utopian" aspects of anarchism, if by "utopia" one means the depiction of an ideal society. In this sense, anarchists in their writings and political activities try to get people to reach beyond the flawed and imperfect society in which they live and start to construct a new society along the ideal lines the anarchists suggest. Likewise, philosophical Daoists from the late Zhou period to the Wei-Jin era also maintained a consistent utopian ideal that they used to challenge both existing social mores and what they saw as dangerous trends in the society and government of their day, as we will see below. Thus this author would disagree with the contention of those (including even some twentieth-century Chinese anarchists as we will see in Chapter 5) who argue that Daoism is necessarily an escapist utopia. Within the context of Daoists' opposition to any kind of ideal society imposed from above on people, this author contends that the Daoist utopian vision is meant to serve as an inspiration to reconstruct society from below in an anti-coercive fashion.
Likewise, though this author would agree with William Callahan's contention that the Daoist vision is one of a decentered "heterotopia" opposed to any kind of artificially imposed uniformity; otherwise Callahan may have missed the main point in his analysis of the (outer) "Robber Zhi" Chapter 29 of the _Zhuangzi_ 3 where the famous robber argues with Confucius that his own way has its own virtue. Unlike postmodernist thinkers who would say even the robber Zhi's vision is as valid a utopia or heterotopia as any other, to this author the _Zhuangzi_ is clearly saying that the amoral world of the robber Zhi is not something to admire or uphold, but is only the inevitable horrible flip side of the Confucian attempt to impose a supposed moral order on a world perfectly capable of ordering itself. In other words, the relativism of the _Zhuangzi_ is not that of modern moral or cultural relativists who might deny the existence of eternal absolutes, but that of skeptics who nevertheless accept the principle of an unknowable _dao_ underlying the unity of the universe, even as they believe that the attempt of "wise men" to put this unity into practice through objective (coercive) action is doomed to violent failure.
Despite the clear utopian content of Daoist and other anarchism in the sense of their shared optimism about humans being able to live without government, on the other hand, as at least one historian of anarchism has noted, most anarchists also have a negative attitude toward the whole concept of utopia, if that term is meant, as it was by Plato and More, to describe an ideal government. As George Woodcock puts it,
In fact the very idea of Utopia repels most anarchists, because it is a rigid mental construction which, successfully imposed, would prove as stultifying as any existing state to the free development of those subject to it.
The Daoists certainly shared this deeply skeptical attitude toward utopia, especially related to the Confucian idea of benevolent government as we will see in the second section of this chapter, and perhaps took this skepticism to the fullest extent of all anarchists, East and West.
Yet this should not disqualify Daoists completely as utopian thinkers. As Sharif Gemie suggests, most utopian thinkers, in their search for an ideal society
. . . are often prepared to sacrifice any of the potential benefits of the existing world. It is out of this conjunction of an _absolute_ rejection of the present, and an _absolute_ affirmation of an ideal world, that a distinctively utopian vision is born, and it is also from this conjunction that the often-noted authoritarian qualities of utopian thinking develop. Thus while the utopian form can potentially be the vehicle for any political ideology, in practice, once the utopian form has been adopted, the vision which evolves has an inherent tendency to develop authoritarian features.
Nevertheless, Gemie finds that some utopian thinkers in the West, most notably William Morris and Charles Fourier, managed to successfully resist this authoritarian tendency. This author would argue that the Daoists were also examples of this comparatively rare libertarian trend in the history of utopian thought, even as they gave a harsh critique of the authoritarian tendencies of the utopias of their rivals, most notably the Confucians, as we will see below.
Beyond their critiques of nonanarchist utopias, at their best, anarchists also present pictures of dystopia, or a negative, hellish vision of a future society based on projections from present governments or statist political philosophies. In the case of the nineteenth-century Western anarchists, these projections included critiques of dominant political ideologies, whether conservative or liberal, as well as competing radical critiques from Rousseau to Marx. Anarchists criticized all these ideologies for their dictatorial tendencies that in the end would limit human freedom and creativity. In the case of the Daoists, both in the late Zhou and Wei-Jin periods, these dystopian elements included negative projections of what society dominated by other philosophies would actually look like in practice, most especially the ideas of the so-called Legalist school as we will see in the third section of this chapter.
Before examining the strong anti-utopian and dystopian elements in the Daoist critique of rival philosophies, we first examine their own undeniably utopian ideas.
**Daoist Utopias from the** _DDJ_ **to Tao Qian**
The Daoist utopian society is most famously found in Chapter 80 of the received _DDJ_ as follows:
Let there be a little country without many people.
Let them have tools that do the work of ten or a hundred,
and never use them.
Let them be mindful of death
and disinclined to long journeys.
They'd have ships and carriages,
but no place to go.
They'd have armor and weapons,
but no parades.
Instead of writing,
they might go back to using knotted cords.
They'd enjoy eating,
take pleasure in clothes,
be happy with their houses,
devoted to their customs.
The next little country might be so close
the people could hear cocks crowing
and dogs barking there,
but they'd get old and die
without ever having been there.
Ursula Le Guin rightly points out in her notes to her translation of this chapter that those who "dismiss this Utopia as simply regressivist or anti-technological" miss the point that the people _do_ have "labor-saving machinery, ships and land vehicles, weapons of offense and defense" but that they do not use them. She further interprets this passage in the _DDJ_ to mean that the people "aren't used by [the tools]," that is, "don't surrender their power to their creations." Thus, rather than link the author of the _DDJ_ with "Luddites" and others in the West opposed to technological progress, Le Guin's analysis can correspond to Joseph Needham's claim that the Daoists did not oppose labor-saving technology for its own sake, but only to the extent that this same technology was used by the budding centralizing military states of the late Zhou to crush the people. Nevertheless, this ideal society would not have the advantages of the economy of scale of larger countries and would also seem to do away with writing systems and thus "written literature, history, and mathematics" among other advances in culture and civilization, as Le Guin herself recognized. She notes, however, that this antipathy toward writing "might be read as saying it's best not to externalize all our thinking and remembering . . . but to keep it embodied" in our bodies and brains. Similarly, Waley notes that knotted ropes aid our own memory "whereas one writes contracts down in order to make other people fulfill them." So once again, the Daoist objection may not be to writing and learning per se, but to dependence on other people who could become oppressive overlords, whatever such "sages" claim about benevolence and righteousness or law and order.
In Chapter 10 of the _Zhuangzi_ this utopia is repeated almost verbatim, but with links to Shen Nung and other pre-Zhou mythical rulers:
Long ago in the time of Yung Ch'eng, Ta T'ing, Po Huang, Chung Yang, Li Lu, Li Hsu, Hsien Yan, Ho Hsu, Tsun Lu, Chu Jung, Fu Hi, and Shen Nung, the people knotted cords and used them. They relished their food, admired their clothing, enjoyed their customs, and were content with their houses. Though neighboring states were within sight of each other, and could hear the cries of each other's dogs and chickens, the people grew old and died without ever traveling beyond their own borders. At a time such as this, there was nothing but the most perfect order.
What the _Zhuangzi_ adds to the utopia in the _DDJ_ is the idea of people living to a longer age, as well as the express statement of this ideal as a "perfect order." What is also added after this picture of utopia is a description of how humans fell from this state, a dystopian picture we will examine in the third section of this chapter.
Though the above depictions are the most famous and clear utopias in the received _DDJ_ and in the _Zhuangzi_ , in both works there are other depictions of life lived by the _dao_ that would reappear in later Daoist utopian accounts. For example, in Chapter 50 of the _DDJ_ there is a description of the Daoist sage:
It is said that he who has a true hold on life, when he walks on land does not meet tigers or wild buffaloes; in battle he is not touched by weapons of war.
Le Guin rightly points out in her commentary on this chapter that the _DDJ_ is not making claims about immortality or bodily invincibility as later Daoist alchemists and _qigong_ practitioners from the Han dynasty to the present would argue, but instead is only advising us to "take life as it comes" and is concerned with "how to live rightly, how to 'live till you die'." In Chapter 7 the _DDJ_ clearly suggests that people will live longer by following the _dao_ , but only if they do not try to "foster their own lives" or "strive for any personal end." This is certainly consistent with the _Zhuangzi_ , which in many places, notably in Chapter 3, advises people to live out their lives without conscious effort and by "go[ing] along with the natural makeup."
There are many other utopian aspects in the _Zhuangzi_ , most of which fall in the so-called outer chapters that most scholars believe were written by authors in a period after the historical person of Zhuang Zhou, who lived in the fourth century BCE. As we saw in the previous chapter about the _Zhuangzi_ 's more explicit anarchist passages, A. C. Graham believes that most of the utopian aspects in the _Zhuangzi_ were written by a "primitivist" author probably in the years between the fall of the Qin state and the rise of the former Han dynasty, that is, between 209–202 BCE, as China once again broke down into civil war and rebellion, while Liu Xiaogan finds instead that these passages were written by one individual probably at the end of the Warring States period, that is, not that far removed from the historical Zhuang Zhou.
In any case, both Graham and Liu would trace the utopian aspects of many outer chapters of the _Zhuangzi_ to an old, preexisting Chinese tradition of a stateless agrarian community, closely related to the school of Shen Nung (or Divine Farmer). This element in both the _DDJ_ and the _Zhuangzi_ can also be linked to the tradition of Xiu Xing, one of the legendary "madmen of the South" who disputed the thinker Mencius around 315 BCE based on the Shen Nung ideal, an ideal that according to Graham influenced classical Daoism and is "ancestral to all Chinese utopianism." This utopianism was revived at the end of the Qin, Graham asserts, by Daoists who "were weary of a state ordered solely by laws and punishments." These Daoists opposed the idea of the Yellow emperor and succeeding pre-Zhou kings as in any way ideal or wise rulers, as the Confucians and other schools claimed. In fact, only later in the Han dynasty would Daoists come to identify their ideal with the Yellow Emperor and Lao Zi combined into one person or concept (the so-called Huang-Lao school of Daoism).
It is in the famous Chapter 29 of the _Zhuangzi_ , "The Robber Zhi," where this agrarian utopia is most readily apparent. Graham links this chapter to the "Yangist" ideal of individualist heremitism, while Liu Xiaogan links the chapter to the more or less consistent "anarchist" ideal of earlier chapters (i.e. 9, 10, and parts of 11). In any case in this chapter one can see the picture of an ideal society that would be picked up by later Daoists of the Wei-Jin period. In the key paragraph of the chapter, the robber Zhi cites the Shen Nung idea in answering Confucius's advice that he set up a great walled state where he could serve as a benevolent ruler:
Moreover, I have heard that in ancient times the birds and beasts were many and the people few. Therefore the people all nested in the trees in order to escape danger, during the day gathering acorns and chestnuts, at sundown climbing back up to sleep in their trees . . . In ancient times the people knew nothing about wearing clothes. In summer they heaped up great piles of firewood, in winter they burned them to keep warm . . . In the age of Shen Nung, the people lay down peaceful and easy, woke up wide-eyed and blank. They knew their mothers but not their fathers, and lived side by side with the elk and the deer. They plowed for their food, wove for their clothing, and had no thought in their hearts of harming one another. This was Perfect Virtue at its height
This utopia began to be lost by the end of the Yellow Emperor's rule, robber Zhi continues, a task completed by the early Zhou rulers such as Yao and Shun who were idealized by Confucius.
In the great anarchist Chapter 9 of the _Zhuangzi_ , "Horses Hooves" (reprinted in Appendix 1 of this book) there is another description of this pre-Yellow Emperor utopia, in this case a utopia where even agricultural pursuits are perhaps absent.
Even if written by a (slightly) later author, this vision of the lost utopia seems very consistent with the ideal in the inner chapters of the _Zhuangzi_ as well as with the received _DDJ_ , especially concerning the terms _si_ (raw silk) and _pu_ (uncut wood) that were key concepts in text related to the idea of "returning to the root" and having nothing to do with the refinements of the modern age.
As we saw in the previous chapter, the "Horses' Hooves" Chapter 9 of the _Zhuangzi_ also provides the best evidence to back up the contention of those who find that the ancient Daoists may have been harking back to dim memories or even actual survivals or remnants in the wild Chinese "south" of a primitive egalitarian society that was either a transitional stage between hunter–gatherers and sedentary agriculturists, or a full-fledged hunter–gatherer society. In any case, according to this argument the classical Daoists from the _DDJ_ and Zhuang Zhou to the authors of some of the outer _Zhuangzi_ chapters opposed not only the rise of more centralized, bureaucratic states that culminated in the Qin empire but also, as we saw in the previous chapter, the Shang-Zhou feudal system idealized by the Confucians, among others.
Beyond the importance of this stateless utopia for the late Zhou and early Qin eras, this ideal also served to inspire thinkers in the early Wei-Jin period (ca. 220–419 CE). As also noted in the previous chapter, these thinkers were part of a larger school of thought that revived philosophical Daoism in order to oppose those who used the prevailing Confucian teachings of the day to justify the dominance of the aristocratic "great families" against the upstart warlords such as Cao Cao. The idea that humans were naturally meant to live in a stateless utopia, one in which humans and animals live peacefully together, would have a profound influence on those neo-Daoists who revived the radical anti-statist side of the _DDJ_ and the _Zhuangzi_.
This revived stateless utopia is most famously found in the _Liezi_ , the text whose "Yang Zhu" chapter we analyzed in the previous chapter, and which most scholars believe to have been compiled around 300 CE. In a different chapter of this text there is a clear depiction of a lost utopia that the Yellow Emperor finds during a daytime dream:
. . . It is a place which you cannot reach by boat or carriage or on foot, only by a journey of the spirit. In this country there are no teachers and leaders; all things follow their natural course [ _ziran_ ]. The people have no cravings and lusts; all men follow their natural course. They are incapable of delighting in life or hating death, and therefore none of them dies before his time. They do not know how to prefer themselves to others, and so they neither love nor hate. They do not know how to turn their faces to things or turn their backs, go with the stream or push against it, so nothing benefits or harms them. There is nothing at all which they grudge or regret, nothing which they dread or envy. They go into water without drowning, into fire without burning; hack them, flog them, there is no wound or pain; poke them, scratch them, there is no ache or itch. They ride space as though walking the solid earth, sleep on the void as though on their beds; clouds and mist do not hinder their sight, thunder does not confuse their hearing, beauty and ugliness do not disturb their hearts, mountains and valleys do not trip their feet—for they make only journeys of the spirit.
In the _Liezi_ version of the Daoist utopia, it is clear that this ideal society is not just a long ago place of a lost age (or "no place" as many students of Western utopianism point out is the literal translation of utopia), but a real place that can be found again whenever one forgets conscious effort and striving for fame and profit, that is, when people stop striving to dominate each other. Although the text can be read as justifying supernatural qualities such as invulnerability to sword or flame, as well as "flying on the clouds," the last statement makes it clear that these are metaphorical, spiritual abilities that allow one to survive in a chaotic age.
In Chapter 5 of the same text there is another description of the lost utopia, in this case a "Divine Spring" coming out of the "Cave of Plenty" in a mountain on the northern shore of the North sea. The legendary (Confucian) ruler Yu "blundered and lost his way" and came on this country by mistake. In this place,
. . . the climate is mild, and there are no epidemics. The people are gentle and compliant by nature, do not quarrel or contend, have soft hearts and weak bones, are never proud or envious. Old and young live as equals, and no one is ruler or subject; men and women mingle freely, without go-betweens and betrothal presents. Living close to the waters, they have no need to plough and sow, nor to weave and clothe themselves, since the climate is so warm. They live out their span of a hundred years, without sickness and early deaths; and the people proliferate in countless numbers, knowing pleasure and happiness, ignorant of decay, old age, sorrow, and anguish. By custom they are lovers or music; they hold hands and take turns to sing ballads, and never stop singing all day. . . .
Though the supernatural elements are more pronounced in this version of the stateless utopia, including the suggestion that drinking the waters of the "Divine Spring" is what gives the people their special qualities, including long life, these abilities might still relate to the idea that it is the increased population density and urbanization of modern states that led to increased disease and violent death, as we will see in Section 3 of this chapter. Of course, this belief would contradict the idea of people "proliferating in countless numbers," but to this observer, that claim seems more related to the picture of sexual freedom and equality that exists in this utopia than to any density of population. In this version of the Daoist utopia there is clearly more emphasis on the pleasures of life, from sex to singing, pleasures that would be enhanced once political authority is removed.
In this society, as in the ideal of the _DDJ_ and the _Zhuangzi_ , people do not strive for reputation or profit, or to dominate each other. Even more clearly than in the classical Daoist texts, there is no government and no gender inequality in this utopia. Also, more clearly even than in Chapter 9 of the _Zhuangzi_ , there is no agriculture, which is perhaps again related to the possibility of influences from surviving remnants or memories of hunter–gatherer society, yet people live long lives free of sickness. While described as a magic place, it is clear that the _Liezi_ text is telling us to forget conscious effort and to reject Confucian, Legalist, or other advice to inculcate morality and order in each other. If we do let go of these attempts, the text clearly implies we will be able to find this place again.
Perhaps the greatest statement of this utopian ideal in the Wei-Jin era came in the third century CE poem, the "Biography of Master Great Man" by Ruan Ji that we analyzed in the previous chapter for its anarchist sentiments (also see Appendix 2). Confronted by Confucian gentlemen who criticized him for his unconventional behavior, the Great Man responds by describing the utopia of the ancient past that he seems able to conjure up by letting go of conventional morality. It is this passage of the poem that concludes with the anarchist statement,
For then there was no ruler [ _wujun_ ], and beings were peaceful; no officials, and all affairs were well ordered.
This utopia is firmly based on the similar accounts in the _DDJ_ and the _Zhuangzi_ , but now, as we saw in the previous chapter, with an even more explicit anarchist element as well as an increased emphasis on economic equality, perhaps influenced by the religious Daoism of sects such as the Yellow Turbans and Five Pecks of Rice—rebellion movements that helped to bring down the Han dynasty, even as they themselves were repressed by the military commanders who became the rulers of the rival kingdoms in the early part of the Period of Disunity. Clearly, however, unlike the hierarchical religious structures of those movements, there is no room for even a benevolent government in Ruan Ji's utopia, not to mention one that strives to restore law and order. People can order themselves and would not be at each others's throats if left alone to manage their own lives.
In less poetic language, and in more blunt and forceful terms, the Wei-Jin thinker Bao Jingyan repeats this anarchist picture of utopia found in the _Liezi_ and in Ruan Ji's poem (see Appendix 3 below). In this version of the Daoist utopia, rejection of the use of roads and labor-saving technology is even more clearly tied to opposition to conquest and political domination of some over others. People in this utopia live not only in harmony with each other, but more explicitly with the animals, following the _Liezi_ text and "Biography of Master Great Man." This pastoral ideal is most obviously similar to Henry David Thoreau's ideal of communion with nature and the idea of loss of this communion as one of the main defects of existing society. Disease and pestilence are linked in these Wei-Jin utopias neither to defects in nature that civilization needs to overcome nor to human nature, but to defects in the artificial attempts of sages and rulers to order the world.
Of course this idea of the natural equality and pacific nature of humans begs the key question for all Daoists of how people ever lost this ideal in the first place. This question will lead us into the discussion of the anti-utopian and dystopian sides of Daoism, which we will examine in Parts 2 and 3 of this chapter. First, however, we should return to the last and, perhaps, greatest statement of Daoist utopianism in the Wei-Jin era, Tao Qian's poem "Peach Blossom Spring" (see Chapter 1 and Appendix 4). In this account, we should recall that a fisherman sailed down a stream through a cave to discover a hidden land to which people had fled long ago to escape the harsh Qin dynasty and who knew nothing of succeeding dynasties. In this place they had founded an egalitarian society without government, one in which, as in Chapter 80 of the received _DDJ_ , people were content to live in their own villages and men and women dressed alike and worked together in the fields.
The fisherman, after being feted by the inhabitants, left for home swearing never to reveal the location of this society. Later, breaking his promise and consciously trying to retrace his steps with the help of the district military commander and his troops, he tried to find the community again, but to no avail. Clearly, Tao Qian is saying that neither conscious attempts to impose morality by Confucian sages nor Legalist attempts to build uniform codes and regulations can get us back to this utopia. But as noted above, this is not a place to be found only in the distant past. This place can exist at any time by anyone who "returns to the root," or the state of original simplicity, by forgetting or letting go of conscious effort.
Tao Qian's last great statement of Daoist utopianism neatly sums up the qualities of the Daoist ideal society. As also noted in the previous chapter, clearly the Daoist society is egalitarian and communitarian—people are not individualist hermits, but cooperate and live in simple equality and peace with each other. Most importantly, there is no government or any kind of political authority in this utopia. Going back to the Shen Nung ideal, Tao Qian's utopia seems to allow for agriculture and husbandry rather than just hunting and gathering, if still on a simple level. People still have few desires, and by noting the noise of the fowls and dogs of the next farm that could be heard, Tao Qian is clearly suggesting the original statement of the Daoist utopia in the received _DDJ_ where people would be able to hear and know about neighboring villages, but would never desire to go there.
How then, if people were so content and had such lack of desire, did humanity lose contact with the ideal society? The attempts of Daoist philosophers to answer this question lead us to the anti-utopian strands in Daoist thought.
**Anti-Utopianism in Philosophical Daoism**
The main answer of Daoists to the question raised above concerning how humans lost their connection to utopian society ironically also demonstrates the anti-utopian aspects of Daoist thought. The received _DDJ_ , for example, does not argue that the ideal society once found will never be lost, but only that those who try to consciously build a perfect order will only bring about a reaction of nature that will destroy their creation. As Chapter 55 of the received _DDJ_ puts it,
Whatever has a time of vigor also has a time of decay.
Such things are against Tao
And whatever is against Tao will soon be destroyed.
The Daoists aim their criticism against those attempting to construct ideal governments at many of their rival schools, but their sharpest criticism seems to be aimed against the Confucians. As Chapter 4 of the _Zhuangzi_ says about one who preaches "sermons on benevolence and righteousness" ( _ren_ and _yi_ —two of the most important concepts of Confucianism), he **"** will be called a plaguer of others. He who plagues others will be plagued in turn."
Clearly for philosophical Daoists of the Warring States and Wei-Jin periods, the principal cause of disorder and chaos is the attempt to order the world by well-meaning sages. Even before attempting to impose a perfect political order, even the attempt to draw up ideal standards only creates the opposite. As in Chapter 2 of the received _DDJ_ ,
It is because everyone under Heaven recognizes beauty as
beauty, that the idea of ugliness exists.
And equally, if every one recognized virtue as virtue, this
would merely create fresh conceptions of wickedness . . .
The next chapter of the received _DDJ_ suggests the link between sages and thieves in the attempt to find an ideal ruler:
If we stop looking for 'persons of superior morality' [ _xian_ ] to put in power, there will be no more jealousies among the people. If we cease to set store by products that are hard to get, there will be no more thieves.
This relativistic criticism of sages' attempts to build a perfect order as the cause of the rise of the great thieves is most pronounced in the anarchist Chapter 10 of the _Zhuangzi_ :
Cudgel and cane the sages and let the thieves and bandits go their way; then the world will be at last well ordered! If the stream dries up, the valley will be empty; if the hills wash away, the deep pools will be filled up. And if the sage is dead and gone, then no more great thieves will arise. The world will then be peaceful and free of fuss.
But until the sage is dead, great thieves will never cease to appear, and if you pile on more sages in hopes of bringing the world to order, you will only be piling up more profit for Robber Chih. . . .
What both the received _DDJ_ and the _Zhuangzi_ seem to be implying is that the attempt to impose one standard or ideal will inevitably lead to strife and thus to evermore authoritarian structures of power to enforce the ideal, just as Gemie suggested about Western critics of statist utopias. Thus Confucian ideas of imposing benevolent rule are only the first stage in a decline that ends in the "brawling" or contending, as the received _DDJ_ clearly states in Chapter 38:
. . . After the 'power' [ _de_ ] was lost, then came human kindness [ _ren_ ].
After human kindness was lost, then came morality [ _yi_ ],
After morality was lost, then came ritual [ _li_ ].
Now ritual is the mere husk of loyalty and promise-keeping
And is indeed the first step towards brawling.
Of course this Daoist anti-utopianism begs the question of how the Daoist utopia is to be instituted, and if Daoists' own optimism about humans' ability to reattain the ideal society contradicts their skepticism about setting up ideals in the first place. The attempted answer of the _DDJ_ and the _Zhuangzi_ is that the sage is to attain the ideal by _wuwei_ , a term usually translated as "inaction" or "doing nothing" as we saw above. As Chapter 37 of the received _DDJ_ puts it,
Tao never does;
Yet through it all things are done.
If the barons and kings would but possess themselves of it,
The ten thousand creatures would at once be transformed,
And if having been transformed they should desire to act,
We must restrain them by the blankness of the unnamed [ _wuming zhi pu_ ].
The blankness of the Unnamed
Brings dispassion;
To be dispassionate is to be still
And so, of itself, the whole empire will be at rest.
As the later Daoist philosophers put it more directly, especially Bao Jingyan, the loss of utopia was caused by the attempts of the strong to dominate the weak, and the statist utopian ideals of the Confucians and others were nothing but the attempts to disguise and justify this inequality of wealth and power.
As we suggested in the previous chapter, one could augment this more direct Daoist critique with the modern view that in the Neolithic revolution an increase in the social surplus caused by accidental discoveries and improvements in agriculture and animal husbandry gave rise to inequality of wealth and power. This inequality in turn led to violent contention over control of the surplus, which led both to government of, by, and for the winners, with utopian ideas of benevolent rule as the final justification and idealization of the status of the wealthy and powerful. The goal of the Daoists, and all libertarian utopian thinkers, is to debunk and deconstruct the statist utopias by contrasting them with a stateless ideal. Thus the Daoists may have rejected focus on objective knowledge and labor-saving conveniences, as was noted in the first part of this chapter, not out of a "Luddite" opposition to progress for its own sake, but as a way to link this knowledge and technological advance with the loss of utopia.
In the more gentle language of the inner chapters of the _Zhuangzi_ , it is the hubris of humans in thinking they can construct an ideal society that leads to their downfall and their modesty and return to the _dao_ that allows them to survive. As the author of Chapter 7 says:
Hold on to all that you have received from Heaven but do not think you have gotten anything. Be empty, that is all. The Perfect Man uses his mind like a mirror—going after nothing, welcoming nothing, responding but not storing. Therefore he can win out over things and not hurt himself.
The author continues in this chapter in a gently satirical vein to illustrate the basic Daoist skepticism about benevolent attempts to order the universe in a famous anecdote about boring holes. This anecdote will also serve to begin our discussion of dystopian elements in philosophical Daoism in its final intimation of the cosmic disaster unleashed when we try to institute artificial schemes of utopian order.
The emperor of the South Sea was called Shu [Brief], the emperor of the North Sea was called Hu [Sudden], and the emperor of the central region was call Hun-tun [Chaos]. Shu and Hu from time to time came together for a meeting in the territory of Hun-tun, and Hun-tun treated them very generously. Shu and Hu discussed how they could repay his kindness. 'All men,' they said, 'have seven openings so they can see, hear, eat, and breathe. But Hun-tun alone doesn't have any. Let's try boring him some'
Every day they bored another hole, and on the seventh day Hun-tun died.
**Dystopian Ideas in Daoist Thought**
As we have seen, Daoist philosophers are not merely anti-utopian in the sense that they oppose ideas of benevolent government as impossible or impractical, but also as they view such ideas as harmful and leading in the end to harsh, authoritarian systems of rule. In the received _DDJ_ , this view is again related to the idea of the inevitable reaction of nature against those who hope to conquer the world, as in Chapter 30:
He who by Tao purposes to help a ruler of men
Will oppose all conquest by force of arms;
For such things are wont to rebound.
Where armies are, thorns and brambles grow.
The raising of a great host
is followed by a year of dearth . . .
In Chapter 53, the _DDJ_ continues to paint a picture of the famine and poverty caused by those who would order the world:
. . . So long as the Court is in order
They are content to let their fields run to weed
and their granaries stand empty.
They wear patterns and embroideries,
Carry sharp swords, glut themselves with drink and food,
have more possessions than they can use.
These are the riotous ways of brigandage; they are not the Highway.
Opposing the Legalists who would order the world through applying a strict and uniform code of rewards and punishments, the _DDJ_ suggests the harshest punishment—the death penalty—will not work, probably because of the worse death and destruction caused by the state itself. As Chapter 74 of the received _DDJ_ puts it,
the people are not frightened of death. What then is the use of trying to intimidate them with the death penalty? . . .
The next chapter of the _DDJ_ describes the real cause of starvation, rebellion, and disorder:
People are starving.
The rich gobble taxes,
that's why the people are starving.
People rebel.
The rich oppress them,
that's why the people rebel.
People hold life cheap.
The rich make it costly,
that's why the people hold it cheap . . .
Similarly, in Chapter 4 of the _Zhuangzi_ , there is a description of the results of sages attempting to build utopian governments, which only led to tyrants destroying them out of jealousy and then causing misery for the people:
In ancient times Chieh put Kuan Lung-feng to death and Chou put Prince Pi Kan to death. Both Kuan Lung-feng and Prince Pi Kan were scrupulous in their conduct, bent down to comfort and aid the common people, and used their positions as ministers to oppose their superiors. Therefore their rulers, Chieh and Chou, utilized their scrupulous conduct as a means to trap them, for they were too fond of good fame. In ancient times Yao attacked Ts'ung-chih and Hsu-ao, and Yu attacked Yu-hu, and these states were left empty and unpeopled, their rulers cut down. It was because they employed their armies constantly and never ceased their search for gain . . .
The picture of the destruction of great sages by tyrants and thieves because of the unleashing of the desire for fame and reputation is continued in harsher terms in the outer Chapter 10, which has Zhuang Zhou say,
. . . what the ordinary world calls a man of perfect wisdom is in fact someone who piles things up for the benefit of a great thief; what the ordinary world calls a perfect sage is in fact someone who stands guard for the benefit of a great thief. . . . In times past, Kuan Lung-feng was cut down, Pi Kan was disemboweled, Ch'ang Hung was torn apart, and Wu Tzu-hsu was left to rot. All four were worthy men, and yet they could not escape destruction.
In Chapter 11 of the _Zhuangzi_ , one of the "anarchist" outer chapters, there is a vivid dystopian description of the results of Legalist rule:
In the world today, the victims of the death penalty lie heaped together, the bearers of cangues tread on each other's heels, the sufferers of punishment are never out of each other's sight. And now come the Confucianists and Mo-ists, waving their arms, striding into the very midst of the fettered and manacled men. Ah, that they should go this far, that they should be so brazen, so lacking in any sense of shame! Who can convince me that sagely wisdom is not in fact the wedge that fastens the cangue, that benevolence and righteousness are not in fact the loop and lock of these fetters and manacles? . . .
Once again, the progression is clear: Confucian doctrines of humane rule lead only to Legalist forms of rule where sages are punished and executed and the people as a whole are eventually violently oppressed.
The neo-Daoist poet Ruan Ji takes over this critique with a viciously satirical metaphor comparing the gentlemen ( _junzi_ ) supposedly concerned with morality and propriety with lice who think they are safe living in a pair of trousers, ". . . but when the trousers are ironed, the flames invade the hills, the fire spreads, the villages are set on fire and the towns burned down; then the lice that inhabit the trousers cannot escape." While meant to satirize those who would criticize the Great Man (or any neo-Daoist nonconformist such as Ruan Ji) for his unconventional ways, no one who had lived through the fall of the Han and the continuing civil wars of the early Wei-Jin period could fail to take the metaphor more literally as applying to everyday reality. As we saw above, Bao Jingyan takes over this picture of Confucian morality where sages striving for reputation and material wealth in the end only unleashed robbers and thieves who, "however cruel by nature they may have been . . . could [never] have done such things if they had to remain among the ranks of the common people?"
Clearly, it is the creation of hierarchical structures of authority, if originally intended to be a utopian form of humane rule, which allowed this dystopia to form. Although certainly the dystopian picture would include the harsh life that would result for the common people, as was argued in the previous chapter, Bao and other Daoists mostly seem to be trying to scare the ruling elite with the revenge that the common people will exact on them, and thus help to break the Confucian–Legalist ideological hegemony among the elite. This is nowhere clearer than in Bao's depiction of how the Confucian–Legalist attempts to impose order will end, which, as we saw in the previous chapter, concluded with the flood metaphor:
. . . to try to stop them revolting by means of rules and regulations, or control them by means of penalties and punishments, is like trying to dam a river in full flood with a handful of earth, or keeping the torrents of water back with one finger.
Though harsher and more direct than the dystopian picture in the _DDJ_ and the inner chapters of the _Zhuangzi_ , the neo-Daoist picture of dystopia is firmly based on the ideas of the received _DDJ_ and the _Zhuangzi_ , even as the Daoist utopian ideal of the stateless community of small villages is retained and heightened. This Daoist utopianism can perhaps serve today as a warning and counterweight both to neotraditional forms of authoritarian rule in East Asia and to those who would revive elements of the Confucian political culture as a guide to democratization. Daoists who utilize the basic anarchist theory of the state would warn us that these efforts, no matter how well-intentioned, might only lead to new forms of elite rule that in the end might only serve to weaken and destroy, not extend, genuine democracy.
**Notes**
**** See Ronald Creagh, _Laboratoires de L'utopie_ for an account of American anarchist utopian experiments.
**** See for example Baogang Guo, "Utopias of Reconstruction: Chinese Modern Utopianism: from Hong Xiuquan to Mao Zedong," _Journal of Comparative Asian Development_ , 2(2) (2003): 197.
**** See Callahan, "Confucian Harmonizing: Utopia, Dystopia and Heterotopia in Chinese Thought," _The Journal of Comparative Asian Development_ , 2(2) (2003): 236–7.
**** Woodcock, _Anarchism: A History of Libertarian Ideas and Movements_ , 23–4.
**** Gemie, _Fourier and the Politics of Utopia_ , 3.
**** Ibid., 2–3.
**** Ursula Le Guin, with the collaboration of J. P. Seaton, _Lao Tzu's Tao Te Ching: A Book about the Way and the Power of the Way_ , 100–1. Though owing much to the influence of Arthur Waley's classic 1935 translation (see Waley, _The Way and Its Power_ ), Ursula Le Guin's more recent translation of the _DDJ_ has the advantage of taking away purposeful action of the sage or prince in attaining this ideal, and puts the utopia in simpler, more poetic English. Her translation is used sparingly in this work, however, first because she does not know Chinese herself and bases her version on other translations, and second, since she is clearly sympathetic to a view of Daoism that highlights its anarchist and utopian tendencies, which might lead some critics to say that this author is pre-selecting the translation to fit the case. Exceptions where this book does use her translation of certain chapters of the _DDJ_ occur where she cannot be accused of highlighting this "anarchist" interpretation more than non-anarchist translators do and where her concise, poetic language drives the point home especially well without deviating in meaning from other translations. Likewise, this work uses Burton Watson's translation of the _Zhuangzi_ since he is especially apolitical and finds the outer chapters clearly more anarchist and less true in spirit to the inner chapters than this author does (see Watson, _The Complete Works of Chuang Tzu_ , 14, 16) and yet his translations often help demonstrate this author's point.
**** Ibid., 125.
**** Needham, II: 121–7, 131–2.
**** Le Guin, 125.
**** Waley, _The Way and Its Power_ , 241, n. 5.
**** Watson, _The Complete Works of Chuang Tzu_ , 112.
**** Waley, _The Way and Its Power_ , 203.
**** Le Guin, 120.
**** Waley, _The Way and Its Power_ , 150.
**** Watson, 51.
**** Graham, _Disputers of the Tao_ , 306–11; _Chuang Tzu: Textual Notes to a Partial Translation_ , 197–9.
**** See Chapter 1, 52, n. 26.
**** Graham, _Chuang Tzu: Textual Notes to a Partial Translation_ , 72–4.
**** Ibid., 70–2.
**** Watson, 327.
**** Ibid, 105.
****Graham, _The Lieh Tzu_ , 34.
**** For example, see Shiping, _Utopianism in Chinese Thought_ , 191.
**** Graham, _The Lieh Tzu_ , 102–3.
**** Holzman, 195.
**** For one colorful account of this movement, see Balazs, 175–6, 192–3.
**** Bauer, 139; also see Balazs, 244–5 and Hsiao Kung-chuan, 623–30.
**** See Chapter 4 and Appendix 5 for a similar utopian account by the pseudonymous thinker Wu Nengzi (Master of no abilities) during the reign of the Xizong emperor of the Tang dynasty (874–888 CE). As we will see, however, since Wu Nengzi was heavily influenced by an interpretation of Buddhism emphasizing _wu_ or nothingness instead of the _dao_ , in the end he came to a nihilistic rejection of even the Daoist stateless utopia as something to idealize.
**** Waley, _The Way and Its Power_ , 209.
**** Watson, 55.
**** Waley, _The Way and Its Power_ , 143.
**** Ibid., 145.
**** Watson, 109–10.
**** Gemie, 3.
**** Waley, _The Way and Its Power_ , 189–90.
**** Ibid., 188.
**** Balazs, 243.
**** Watson, 97.
**** Waley, _The Way and Its Power_ , 180.
**** Ibid., 207.
**** Ibid., 234.
**** Le Guin, 95.
**** Watson, 55–6.
**** Ibid., 108.
**** Ibid., 118.
**** Balazs, 238.
**** Ibid., 246.
**** See Chapter 1 above, 22.
**** Balazs, 246.
**** For the application of a neo-traditional critique to a twentieth century Chinese ruler, see Anita M. Andrew and John A. Rapp, _Autocracy and China's Rebel Founding Emperors: Comparing Chairman Mao and Ming Taizu_ , who would contest any assertion that Mao's "utopian" experiments ever really had as their purpose the ending of alienation or the building of mass democracy. Instead, as argued throughout the Andrew and Rapp book, as in Chapter 5 of this book, Mao's aim from the outset was to build a heightened personal autocracy and a militarized society in which the checks of the central bureaucracy on his own authority would be curtailed and even removed. Thus if one considers there to have been utopian aspects to Mao's thought, this would only be in the sense of an ideal (autocratic) government, and not in the sense of a truly egalitarian, democratic society.
3
Daoism as utopian or accommodationist: The Guodian challenge to Daoist anarchism
**Introduction**
Although the previous chapters demonstrated that philosophical Daoism undoubtedly contains utopian anarchist strains, whether these tendencies can be traced back to the text known as the _DDJ_ is more open to debate. Those who find a radical utopian argument in Daoism stress especially the _DDJ_ 's critique of the Confucian ideal of humane rule. Below we trace this critique to previously received versions of the _DDJ_ that date to approximately 250–200 BCE. Nevertheless, some scholars would claim that bamboo strips unearthed in 1993 from a tomb in China's Hubei province present a major challenge to the claim that the basic anarchist idea of the state ruling for itself goes back to the earliest roots of Daoist philosophy.
Perhaps the most important find in this tomb were portions of what later became the _DDJ_ , thus marking the text as much as a century older than any previously known version. As the news about the strips spread, some scholars began to claim that the Guodian manuscripts proved that Daoism was more accommodationist toward government than was previously thought to be the case.
This chapter will first present the provisional case for that "accommodationist" view of Daoism. Next we will review the utopian anarchist strands of Daoism that can be traced back to at least a century after the Guodian manuscripts were transcribed, which will then lead us to question whether the Guodian texts really present such a major challenge to radical Daoism. The chapter will conclude with a discussion of what the identity of the owner of the Guodian strips may tell us about the ultimate meaning of the Guodian texts.
**The Guodian challenge examined**
The Guodian manuscripts present three main challenges to the view of the anarchist essence of philosophical Daoism. The first is the absence from the Guodian bamboo strips of many _DDJ_ chapters that explicitly oppose direct attempts to rule, including, most dramatically, the absence of the entire last third of the received _DDJ_. The second challenge is the lack in the bamboo strips of clear anti-Confucian language in what became Chapter 19 of the received _DDJ_. The third challenge is the relative absence of a "law of return" in the Guodian version of the _DDJ_ that would explain how humans could ever have fallen away from the stateless utopia. All three potential challenges are based on the fact that the Guodian text is the oldest known edition of what became the _DDJ_ and thus that the clearly anti-statist and utopian statements in the received text may be later additions by other authors.
**The absence of the most anti-statist and utopian sections of the** _DDJ_
Absent from the Guodian strips are some of the most direct criticisms of other political philosophies and the most anti-statist statements of the received _DDJ_. Most importantly, the Guodian text does not contain the explicit, influential utopian Chapter 80 of the _DDJ_ that we analyzed in the previous chapter. Others have argued that this chapter contains the heart of the Daoist critique opposed to technological innovation that would aid the oppressive centralization and militarization of state power, a critique that can be found clearly in Daoist texts of the Warring States period (403–221 BCE)—an era that culminated in the foundation of a centralized imperial state. Absent as well from the Guodian text are some of the most dramatic examples of Daoist advice to rule by noninterference in the affairs of the world ( _wushi_ ), including the end of Chapter 48 of the received _DDJ_ :
In wanting to rule the world
Be always non-interfering in going about its business;
For in being interfering
You make yourself unworthy of ruling the world.
The Guodian strips also leave out the severe critique of Legalism, a political philosophy that would later be highly influential on the imperial state. This anti-legalist stance can be seen in chapters in the received _DDJ_ missing from the Guodian strips that contain criticism of rule by harsh punishments (Chapter 74) and the idea of suffering and rebellion as caused by over-taxation and the oppression of the rich over the poor (Chapter 75). Finally, the Guodian text leaves out much of the attack on the Confucian ideal of rule by the morally virtuous, as in Chapter 3 of the received text which is missing from the Guodian strips:
If we stop looking for "persons of superior morality" ( _xian_ ) to
put in power, there will be no more jealousies among the people.
If we cease to set store by products that are hard to get, there will be
no more thieves.
Also absent from the Guodian strips is the explicit critique of the negative political evolution that occurs if Daoist principles are lost, as in Chapter 38 of the received _DDJ_ :
After the "power" [ _de_ ] was lost, then came human kindness [ _ren_ ].
After morality was lost, then came ritual [ _li_ ].
Now ritual is the mere husk of loyalty and promise-keeping
And is indeed the first step towards brawling.
**Chapter 19 and the Guodian accommodation to Confucianism**
By far the most highly publicized example of the seeming accommodation toward government in the Guodian strips lies in what became Chapter 19 of the received _DDJ_. The received versions contain language that directly mocks the Confucian values of sageliness ( _sheng_ ), benevolence or humanity ( _ren_ ), and righteousness ( _yi_ ), values at the heart of the ideal of paternalistic rule. As the received _DDJ_ puts it,
Eliminate sageliness, get rid of knowledge,
And the people will benefit a hundredfold.
Eliminate humanity, get rid of righteousness,
And the people will return to filial piety and compassion.
Eliminate craftiness, get rid of profit,
And there will be no robbers and thieves . . .
As opposed to this direct critique, the Guodian text uses the following language:
Eliminate knowledge, get rid of distinctions,
And the people will benefit one hundredfold.
Eliminate artistry, get rid of profit,
And there will be no robbers and thieves
Eliminate transformation, get rid of deliberation,
And the people will return to filial piety and compassion . . .
To critics, this chapter shows clearly that Confucian and Daoist thought were not so opposed at the time when the Guodian texts were transcribed and that both philosophies argued for a humane rule based on paternalistic values of filial piety and benevolence, not a stateless utopia as some later Daoists from the Warring States to the Wei-Jin period explicitly favored. Thus the Guodian text prefigures scholar–officials who later used Daoist principles to defend the supposedly limited and light rule of the former Han dynasty. Perhaps the best evidence for such an accommodationist position can be found in Chapter 54 of the received _DDJ_ , which is also in the Guodian strips with only minor differences and gaps due to broken or missing slips (for which Henricks puts extrapolations in italics):
If you cultivate it in your self, your virtue will be pure;
If you cultivate it in your family, your virtue will be overflowing;
If you cultivate it in your village, your virtue will be longlasting;
If you cultivate it in your state, your virtue will be rich and full;
If you cultivate it throughout the world, _your virtue will be widespread._
_Look at the_ family _from the point of view of the family;_
Look at the state from the point of view of the state;
Look at the world from the point of view of the world . . .
As many commentators have long pointed out, this chapter is remarkably similar to the later Confucian text the _Da Xue_ , or "Great Learning," which says that great sages of antiquity who wished to order their own states,
first regulated their own families, for which they first corrected their own hearts, for which they first regulated their own intentions, for which they first perfected their own knowledge.
The _Da Xue_ later became one of the four classic texts that all would-be officials had to master in order to pass the imperial examinations, thus showing how Confucianism became a legitimating formula under which the role of the ruler was similar to that of head of a family. Thus critics of Daoism as anarchism point to this chapter of the _DDJ_ to say that early Daoism was not opposed in principle to the idea of rule as long as it was limited and humane.
**The lack of a "law of return" in the Guodian texts**
Finally, the Guodian strips severely lack what could be termed the Daoist "law of return" that exists in the received _DDJ_. This law is important in that it helps Daoists both to explain how a "fall" from a stateless utopia could ever have occurred, and to predict the oppressive forms of rule other political philosophies of the time would bring if ever put into practice. This law is most explicit in Chapter 55 of the received text, which is absent from the Guodian strips:
Whatever has a time of vigor also has a time of decay.
Such things are against Tao
And whatever is against Tao will soon be destroyed.
In other words, those who try to impose political order either by indoctrinating people with ideas of goodness (Confucianism) or through harsh laws and punishments (Legalism) will only bring about a reaction of nature that will destroy their ideal states. Also, under this principle, Daoists can explain the "fall" from the natural, stateless society not as something unnatural, which would be self-contradictory to a naturalistic philosophy, but instead only as a temporary change that is doomed to fail. Without this law of return, the Daoist critique of other political philosophies is arguably much weaker.
In the Guodian version of what became Chapter 30 of the received _DDJ_ , which opposes war and militarized rule, the lines containing the most famous example of the law of return are absent (marked in italics below):
One who uses the Way to assist the ruler of men,
Does not desire to use weapons to force his way through the land.
_Such deeds easily rebound._
_In places where armies are stationed, thorns and brambles will grow._
_Great wars are always followed by great famines._
One who is good at such things achieves his result and that's all.
He does not use the occasion to make himself stronger still.
Thus the Guodian version seems to call for modest, humane rule that avoids war if possible but refrains from opposing _any_ attempt to use force of arms, which would undermine the idea of Daoism as anarchistic.
For many scholars, other minor linguistic differences between the Guodian and the received _DDJ_ demonstrate that the Guodian text is the oldest version of what became the _DDJ_ and that much of the received _DDJ_ was not present at the time of Confucius (b. 579 BCE), but instead was added during or after the third century BCE. Thus, according to the "accommodationist" view, the elements of the _DDJ_ that contain the anti-Confucian critique must have also been added during the late Warring States era, while the utopian anarchist aspects must have been nonintrinsic additions of later writers.
**Review: The case for radical Daoism**
To make the case for radical Daoism as genuine and intrinsic, one should start with unambiguous anarchist Daoism of the Warring States and Wei-Jin periods and work backward to the time of the Guodian texts.
First, as we saw in the previous chapters, later radical thinkers definitely used Daoist language to describe a stateless utopia. These utopian depictions included explicit opposition to Confucian moral virtue and Legalist rewards and punishments, ideas that provided legitimated succeeding Chinese imperial dynasties. Radical Daoism developed to its fullest extent in the early Wei-Jin period (ca. 220–316 CE). As we noted in the previous two chapters, the poet Ruan Ji took Daoist anarchism to its height in his poem "The Biography of Master Great Man," which describes a stateless utopia in terms based on the received _DDJ_. Ruan Ji has the Great Man denounce serving in government, based on the _Zhuangzi_. Based also on received versions of the _DDJ_ , Ruan Ji in his poem criticizes Confucian and Legalist ideas of rule as "nothing more than the methods of harmful robbers, or trouble-makers, of death and destruction. . . ." As we also saw in previous chapters, Ruan's harsh, anti-Confucian tone is continued in the tract of the obscure Daoist philosopher Bao Jingyan (ca. 300 CE) who also presents the Daoist stateless utopia found in other Wei-Jin writers, but in very direct and forceful language. These Wei-Jin Daoist anarchists took their language directly from the "outer" chapters of the _Zhuangzi_ , especially Chapter 10, which as we saw was itself highly resonant of Chapter 80 of the received _DDJ_ and is dated by scholars to at least 250–200 BCE. As in Chapter 18 and 19 of the received _DDJ_ , we saw above how Chapter 10 of the _Zhuangzi_ also blames Confucian and Legalist "sages" for bringing oppression into the world, if in much harsher language that calls us to "cudgel and cane the sages and let the thieves and bandits go their way" and concludes that "the world will then be peaceful and free of fuss. . . ." if we ". . . cut off sageliness, cast away wisdom" and ". . . destroy and wipe out the laws that the sage has made for the world. . . ."
As we saw in the second chapter of this book, Chapter 9 of the _Zhuangzi_ also depicts a Daoist utopia where the world is free of sages trying to order the world, a utopian picture that relates to language of the inner chapters of the _Zhuangzi_ and the received _DDJ_ concerning the need to "return to the root" and reject technological refinements that came with the increasing centralization of power in the Warring States era. Even if these accounts from the outer _Zhuangzi_ chapters and the received _DDJ_ were later extrapolations, as we saw in Chapter 2, there is no doubt that their utopian ideal harkens back to a preexisting tradition of a stateless agrarian community that was supposedly begun by the mythical founder of agriculture, Shen Nung. A. C. Graham's argument that the Shen Nung ideal "appears to be an anarchistic order based on mutual trust in small communities . . ." that is ". . . ancestral to all Chinese utopianism" would backdate the utopian Daoist ideal to a time at least roughly contemporaneous with the historical Zhuang Zhou himself, if not earlier, even if this ideal was later sharpened during the harsh Qin dynasty (221–207 BCE). Graham's argument supports the view that even the inner _Zhuangzi_ chapters suggest the spontaneous order that exists in the universe without human intervention and thus the lack of any need to impose political order. We should recall that in the inner chapters of the _Zhuangzi_ the greatest sages often refuse to serve in government, while the great second chapter, "Discussion on Making All Things Equal," satirizes the idea that hierarchical rule is natural in the famous section we noted in Chapter 1 that contains a cybernetic view of the human body.
Further, as we saw in Chapter 2 of this work, another (inner) chapter of the _Zhuangzi_ puts forth the metaphorical anecdote about the disaster that will follow from artificial attempts to impose order when, after trying to repay Hun-tun for his generosity, the emperors of the north and south seas tried to bore the seven human orifices in him so that he could see, hear, eat, and breathe, "every day they bored another hole, and on the seventh day Hun-tun died."
Clearly the inner _Zhuangzi_ chapters oppose the idea of rule as morally virtuous, if in more gentle language than used by later Daoists.
Likewise, the received _DDJ_ often depicts the idea of morally virtuous rule as at best a step down from the ideal, as in Chapter 17:
With the most excellent rulers, their subjects only [barely] know that they are there,
The next best are the rulers they love and praise,
Next are the rulers they hold in awe,
And the worst are the rulers they disparage.
Given the Daoist admonition to (would-be) sages to rule by _wuwei_ throughout the received _DDJ_ , in addition to its denigration of laws and punishments, taxes, warfare, education, and virtually any other element of rule, we argued in the first chapter that even the received _DDJ_ is trying to subvert government by advising the ruler to emulate leaders of hunter–gatherer bands and thus remove the ruler's monopoly on the legitimate use of coercion—advice that would do away with the state as it is minimally defined by Max Weber. Following Joseph Needham, we have argued above that even the authors of the inner chapters of the _Zhuangzi_ and the received _DDJ_ may have lived early enough to have at least dim memories of surviving remnants of wild hunter–gatherer or semi-sedentary ways of life in the south of ancient China and thus opposed the increasing centralization of power from the late Spring and Autumn to the Warring States periods. We know that Daoist thinkers often came from more recently settled or partially settled regions of China, such as the "madman" Xiu Xing from the State of Chu who argued with Mencius, the great fourth-century Confucian exponent of the doctrine of humane rule. Here the earlier date of the Guodian text may help heighten the argument, since the Guodian tomb is located within the historical boundaries of the state of Chu, which perhaps would place its ideas within this "southern" tradition of Chinese political thought opposed to harsher types of rule. Before we return to that question below, we must first reexamine the Guodian text to see whether it really lacks anti-Confucian and anti-statist utopian language.
**The Guodian Texts Reexamined**
**The questions of dating and authorship**
In refuting the claims that the Guodian texts point to an "accommodationist" Daoism, one must first examine the issue of dating. Though it is currently the oldest known version of the text, whether or not all later editions of the _DDJ_ were additions to the Guodian texts or whether there was a preexisting oral and/or written tradition to all received or discovered versions of the _DDJ_ is a matter of dispute. Even if one accepts the view of scholars who point to linguistic evidence to suggest that sections of the Guodian texts were more succinct and thus that later _DDJ_ versions contained many emendations, this does not mean that later authors of texts that entered into the received _DDJ_ were starting wholly new traditions. Instead their texts could have been based on preexisting utopian traditions, as we noted above, such as that of Shen Nung, which might have had a history predating the Guodian manuscripts.
Robert Henricks points out that the Guodian strips were discovered in the tomb in at least three bundles, which were copied separately in at least two different hands probably from at least three other written sources. The complete text of the _DDJ_ may have existed by 300 BCE in more than one version, and the common ancestor of all versions may have been written earlier in the fourth century. The Guodian strips thus may be copies of copies and transcribed from versions of the text that date to as early as 350 BCE.
Whether or not the idea of one man named Lao Zi as the author of the _DDJ_ was a later invention—as Chinese intellectuals of the 1920s and 1930s believed and most contemporary Western and Japanese scholars contend—or whether the _DDJ_ really dates back to someone such as the sixth century BCE legendary figure Lao Tan or Li Erh, certainly at a minimum the main principles of the received _DDJ_ date to the Warring States period.
Perhaps based on the traditional Chinese view of Lao Zi as the author of the _DDJ_ , most contemporary Chinese scholars contend that the Guodian texts prove there was an already existing version of the _DDJ_ much earlier than previously believed. Most Western scholars, on the other hand, believe that the lack of many _DDJ_ chapters in the Guodian texts and other linguistic evidence shows that the complete _DDJ_ was not yet in existence in 300 BCE. While many Western observers find the Chinese belief in an early _DDJ_ as authored by Lao Zi to be based more on a conjectural "act of faith" rather than hard evidence, other Western scholars are starting to come around to the Chinese position, including Edward Shaughnessy, who finds that Western views might also be faith-based and prematurely based on the evidence at hand, and Robert Henricks, who as we saw above is willing to consider that a complete version(s) of the _DDJ_ may have existed as early as 300 BCE. Liu Xiaogan sees a possible third, compromise position: that much of the _DDJ_ may have been composed after Confucius (sixth century BCE) but before the historical Zhuang Zhou (i.e. before the mid-fourth century). If so, that would put much of the _DDJ_ much farther back than 200 BCE, showing that much of the radical side of Daoism can be traced before the rise of China's early imperial dynasties and thus further in Chinese history than many observers previously believed.
Even if much of the _DDJ_ dates far back into the Warring States period, critics of Daoism as originally anarchist would still raise the questions noted above, which we will now consider successively, that is, the "missing" (radical) chapters from the Guodian strips, the changes in what became Chapter 19 of the _DDJ_ , and the question of the "law of return."
**The "Missing" chapters from the Guodian texts**
Despite the fact that some chapters and sections of the received _DDJ_ are missing from the earlier Guodian texts, upon closer examination one can find even in the Guodian strips precursors of much of the later radical utopian argument. To take a crucial example, the concept of _wuwei_ , nonaction or doing nothing, still can be found in the Guodian texts despite the lack of several _DDJ_ chapters that focus on the concept. For example, in the Guodian version of what became Chapter 57, the author has the perfect sage say the following:
I am unconcerned with affairs, and the people on their own enjoy good fortune;
I do nothing, and the people transform on their own . . .
Besides _wuwei,_ other _wu_ forms such as _wuzhi,_ literally "not knowing," and _wuyu_ , literally "not desiring" (or "unprincipled knowing" and "objectless desire" respectively, as David Hall more clearly translates those terms) exist in one form or another in the Guodian texts. These terms were crucial in developing the Daoist ideal of rule by noninterference with the natural order, which Hall regards as central to the philosophical anarchist vision of Daoism.
Likewise, despite the fact that the Guodian slips contain only about one-third to two-fifths of the received _DDJ_ , in addition to the _wu_ forms, the Guodian texts include other key concepts such as _pu_ (uncarved wood) and _si_ (raw silk), terms that related to advice to return to an "original" simple and unrefined nature and thus pointing to a critique of overly refined methods of rule.
Especially if one accepts the argument of Liu Xiaogan that later versions of the _DDJ_ mostly amount to, first, a "linguistic assimilation" that may have amplified and intensified but not directly changed the meaning of the text and, second, a "conceptual focusing" that "highlights key concepts but also strengthens consistency in language," then one can argue that the core message of the later _DDJ_ is contained in the Guodian strips. For example, Liu contends that concepts such as _wuwei_ may be used less often and in less intense fashion in the Guodian texts, but they can still be found, just as the anti-Confucian questioning of rule by benevolence or morally virtuous leaders is still present if one looks more closely, which leads us back to the question of Chapter 19.
**The changes in Chapter 19**
Even if some terms and concepts of the received _DDJ_ can be found in the Guodian texts, there is still the celebrated change in lack of explicitly anti-Confucian language in what became Chapter 19 of the received _DDJ_. Even here, however, Liu Xiaogan's point applies about the received text of the _DDJ_ only amplifying and not distorting the fundamental message in the Guodian text. As Liu says, ". . . in chapter nineteen in [later versions of the _DDJ_ ] neither the amendment of sentences nor criticism of Confucianism are sudden or incomprehensible. They do not distort the original thought of the bamboo versions." For Liu the changes in Chapter 19 are "special case[s] of conceptual focusing" that mostly "amplify criticisms in the bamboo versions" and intensify the criticism without changing the essential meaning of the text. This is especially true if one looks at the eighteenth chapter in the received _DDJ_ , which was found intact in a separate bundle of Guodian strips. In the latter part of this chapter the anti-Confucian language survives, as follows:
Therefore, when the Great Way is rejected, it is then that "humanity"
and "righteousness" show up on the scene;
When the six relations are not in harmony, it is then that we hear of
"filial piety" and "compassion";
And when the state is in chaos and disarray, it is then that there is praise for the "upright officials."
For Henricks, combining the sentiments in this paragraph with the advice in what became Chapter 19 to eliminate attempts to use knowledge and distinctions to morally transform the people makes it clear that even if this Guodian chapter is "not yet 'anti-Mencian'," that is, not explicitly opposed to that fourth-century philosopher's focus on humane rule, "it is still very 'anti-Confucian'," that is, against the idea of sages trying to inculcate morality and compassion in the people.
**The "Law of Return"**
Though differences between the Guodian text and what became Chapter 30 of the received _DDJ_ are not as famous as the changes in what became Chapter 19, to this author the lack of a clear "law of return" in the Guodian texts may be the biggest difference between the Guodian strips and the received _DDJ_. Again, this absence is important, since anyone who wants to argue that a stateless utopia is the natural human condition has to explain how people could ever have fallen so far as to live under Confucian or Legalist-influenced governments.
In making the case for the continuity of the Daoist anarchist tradition, one should first note that a law of return is implicit in the Guodian texts since they still emphasize that ruling through inaction or nonconcern with affairs is the best way for the sage to endure. Most scholars who have examined the Guodian version of what became Chapter 30 emphasize that it is very likely that a punctuation error in the text should be corrected so that the final line reads, "such deeds [i.e., those achieved by being modest and not desiring to use weapons] are good and endure" or "its affair tends to be prolonged" [ _qi shi hao chang_ ]. In other words, one who rules by doing nothing will survive, clearly implying the opposite for those who fail to heed this warning. Thus again, the later, clearer versions of the _DDJ_ that talk about those "not being on the Way [coming] to an early end" are merely examples, to borrow Liu Xiaogan's terminology, of "intensifying" or "focusing" concepts that can be found in the Guodian texts.
Likewise, the anti-militarism of the received _DDJ_ is present in the Guodian text with or without an explicit law of return, as in the likely suggestion of what became Chapter 31 that "weapons are instruments of ill omen." Henricks finds that the key characters found in later texts contained what the missing characters in the strips must have said, which in any case is consistent with the Guodian version of what became the opening lines of Chapter 30:
One who uses the Way to assist the ruler of men
Does not desire to use weapons to force his way through the land.
Indeed, Shaughnessy speculates that the separation of the two chapters in the Guodian text may have been due to a misplaced bamboo strip, which would not be hard to imagine given the chaotic state in which the strips were first found in the tomb. This strip may have in fact contained the more direct language "where troops are based brambles will grow," a clear example of the law of return that might have later been moved to a different place in the received version of the _DDJ_.
**The surviving radical utopian vision of the** _DDJ_ **in the Guodian texts**
To relate this technical debate among specialists on ancient China to the point of trying to find the genesis of the radical anarchist utopia in the Guodian texts, we should conclude this section by examining the main point in the _DDJ_ shared by all philosophical anarchists who present a utopian vision of what society would look like without government, namely, that humans can find morality on their own, that is, they can find the link between individual freedom and community without the need of outside intervention. In Western anarchism that point is made most clearly and consistently in the works of Peter Kropotkin, who asserts that "mutual aid" is the natural and voluntary method humans have always used in order to survive, as opposed to the more hierarchical concept of "charity" projected by those trying to justify rule of some over others. Similarly Leo Tolstoy argues that the spirit of love as expressed by Jesus in the Sermon on the Mount points to a voluntary process where individuals see the link to each other inside their own hearts, as opposed to orthodox Christian doctrines that preach the need for sinful humans to be saved from without. If such ideas are indeed at the core of all philosophical anarchism, then the Guodian strips contain the same message. That message is for the (would-be) sage to let go, not to direct the people, and let things take care of themselves. The Guodian version of what became the latter part of Chapter 64 of the received _DDJ_ contains this message most clearly, while also containing the germ of the law of return:
The rule to follow in approaching all matters, is—
If you're as careful at the end as you were at the beginning
You will have no disasters.
The Sage desires not to desire and places no value on goods that are hard to obtain.
He teaches without teaching, and backs away from matters in which the masses go to excess.
As a result, the Sage is able to help the ten thousand things to be what they are in themselves, and yet he cannot do it.
This Guodian chapter especially contains both the idea of opposing "charity" and a version of the law of return. If the sage does nothing, the people will eventually find their true nature. They may stray from the Way, in which case the sage would back away from them and remove his approval, like a tribal elder but not a ruler possessing the power of coercion, but on their own they would return to the Way, that is, to the natural morality that is contained in all of us. That trust in people to rule themselves is the heart of the utopian vision of anarchism, and at root, one could argue, that belief is still contained in the Guodian strips.
**Conclusion: The Guodian Texts and the State of Chu**
Why then, if the core utopian message remains in the Guodian texts was the man who owned the texts, and perhaps those who first wrote and transcribed them, so seemingly willing to embrace the idea of humane rule? To answer this question, it may be useful to look at who was buried in the tomb where the strips were found. The owner of the Guodian strips may have been a relatively high-ranking, Confucian-influenced teacher of the heir apparent to the ruler of the state of Chu.
The state of Chu was an important southern state during the Warring States period, famous among other things for some of the most legendary "madmen," the hermits and poets who perhaps based their anti-statist ideas on earlier, pre-sedentary traditions. The idea of Daoism as part of China's "southern" tradition more apart from and skeptical of official life has a long history. In other words, it may be that even the Confucian tradition in Chu was affected by Daoism. Li Cunshan, for example, points out that some of the other texts unearthed at the Guodian tomb were examples of a southern form of Confucianism very much influenced by Daoism.
Thus it may not be so much that Confucianism tamed Daoism in this time and place but that Daoist ideas affected Confucian thinkers, for example, in leading some of them to oppose "artificial" filiality and to favor ruling more by _wuwei_ , inaction, or doing nothing. Others have similarly argued that the Guodian texts demonstrate that early Confucianism was more than a dispassionate elitism and was instead influenced by Daoist ideas to put more stress on human feelings ( _xing_ ).
Thus one could easily speculate that in choosing which parts of what became the _DDJ_ to recopy for the use of tutoring his pupil, the owner of the Guodian strips may have selected sayings that backed up his own views and would best aid his goal of influencing his student to rule less harshly once he succeeded to the throne. The teacher could not be openly anti-statist but only gently suggestive of less harsh doctrines of rule, a goal perhaps of southern Chinese intellectuals who saw such doctrines as based on dangerous "northern" traditions that were starting to take over the Chinese world.
The view that Confucian intellectuals in the period of the Guodian texts and later were trying to convince their pupils to accept less interventionist forms of rule while preserving their own role as advisors perhaps resonates with Roger Ames' view of the later Daoist text, the _Huainanzi_. Later Daoists in times of more centralized order in the early imperial era of the Qin and Han dynasties (ca. third to second centuries) may have interpreted Daoism as supporting the principle of rule at the same time that they were trying to subvert rule in practice. Whether they succeeded in this double game or in the end helped more to legitimize the new imperial forms of rule would of course depend on one's own underlying political perspective.
That Confucian scholars even before the time of Mencius were trying to promote a "humane rule" doctrine that would mitigate authoritarian rule, and thus that Confucianism is at root not dictatorial or "feudal" is an important part of more contemporary Chinese intellectual discourse. The idea that Confucianism can be reconciled with constitutional monarchy and even democracy was a crucial part of the later Chinese "Hundred Days" reforms of 1898. The idea of Confucianism as a pro-democratic doctrine can be found in the works of "liberal" Chinese intellectuals from the 1920s and 1930s up to contemporary philosophers such as Tu Wei-ming, who has explicitly focused on the Guodian manuscripts as showing that there is a long history in China of limits to autocratic rule.
A radical Daoist, on the other hand, might point out the remaining danger that any doctrine of humane or democratic rule could subvert true equality and freedom. The Daoist anarchist might point to the potential for intellectuals to use such humane rule doctrines to satisfy themselves that they are not responsible for harsher forms of rule even as their acquiescence in the rule of supposedly more benign leaders not only preserves their elite status but helps to legitimate the state in general.
In any event, in times of disorder in China, when fighting between rival states intensified and state power in general became increasingly centralized and oppressive, some intellectuals started to make more directly radical statements based on the utopian anarchist side of Daoism. This chapter argues that these more direct statements are not distortions of the original message represented by the Guodian texts but instead a more explicit statement of Daoist anti-statist impulses that always exist for many people. In times when the state's rule becomes more oppressive and more obviously for the benefit of rulers rather than the ruled, for example, during times when states swallow each other up in war and become increasingly centralized, earlier, more gentle critiques of rule can often evolve into more blatant anti-statist doctrines. In times of disorder, with constant warfare, pestilence, disease, and famine, perhaps at least some intellectuals who feel they have nothing to lose in a situation when their lives are under constant threat anyway are more likely to return to Daoism and bring out its utopian anarchistic tendencies, as was the case for Liu Shipei in the early twentieth century, as we will see in Chapter 6, and for the individual in the mid-ninth century CE writing under the pseudonym of Wu Nengzi, as we will see in the next chapter, even if their own interests in serving in government or merely surviving led them to compromise their original Daoist anarchist visions.
**Notes**
**** As noted in the introduction to this book, many China scholars argue that the idea of philosophical versus religious Daoism, not to mention the very idea of clearly delineated schools of "Daoist," "Confucian," and "Legalist" thought, was a much later idea in Chinese history that later scholars projected back to earlier periods. Nevertheless, this author would contend that the _DDJ_ , including the Guodian partial version, contains similar political ideas to those in texts such as the _Zhuangzi_ and later works, ideas which can be grouped together and contrasted with ideas in texts that later became part of imperial ruling ideology. Thus for the purposes of this chapter the terms Daoist, Confucian, and Legalist are used to denote those contrasting ideas.
**** See Chapter 1 and of this volume.
**** To review the previous chapter, the terms "radical utopian" or "utopian anarchist" in this chapter refer to the suspicion shared by Daoist and Western anarchists of other, statist utopias, even while Daoist and other anarchists present their own vision of an ideal (stateless) society.
**** See Robert Henricks, _Lao Tzu's Tao Te Ching: A Translation of the Startling New Documents Found at Guodian_ , 22. For the original transcription of the strips, see Hubeisheng Jingmenshi bowuguan (Hubei Province Jingmen City Museum) (ed.), _Guodian Chu mu zhu jian_ (The Guodian strips in the Chu tomb).
**** See Tu Wei-ming, quoted in A. Shen, "Ancient Script Rewrites History," _Harvard College Gazette_ (February 22, 2001): 8.
**** As we noted in the previous chapter; also see Needham, _Science and Civilisation in China,_ II: 86–9, 121–32.
**** Roger Ames and David Hall, _Dao De Jing_ : _Making This Life Significant: A Philosophical Translation_ , 151.
**** See Chapter 2.
**** Waley, _The Way and Its Power,_ 145.
**** Ibid., 189–90.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 12.
**** Ibid., 13, 29.
**** See, for example, Pang Pu, "Gu mu xin zhi" (New Information from an Old Tomb), 7–12, translated in Defoort and Xing, "Guodian, Part I," 46–9.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 108.
**** Translated in A. C. Graham, _Disputers of the Tao,_ 132; also see Ames and Hall, _Dao De Jing_ , 160–2.
**** For the idea of "return" in the _DDJ_ , see for example Ames and Hall, _Dao De Jing_ , 27–9.
**** Waley, _The Way and Its Power,_ 209.
**** See Chapter 2.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 15, 36–7. The emphasis is Henricks' for the lines missing from the Guodian.
**** See, for example, Boltz, _"The Fourth-Century B.C. Guodian Manuscripts,"_ 594.
**** As argued in the previous chapter.
**** As argued in Chapter 1.
**** Holzman, _Poetry and Politics_ , 195.
**** Ibid., 110.
****Ibid., 105.
**** See previous chapter.
**** Graham, _Disputers of the Tao_ , 64–74.
**** Watson, _The Complete Works of Chuang Tzu_ , 38.
**** Ibid., 97.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 112. Henricks finds the last phrase largely intact in the Guodian version, if combined with the next chapter.
**** Needham, _Science and Civilisation,_ 100–32.
**** Graham, _Disputers of the Tao_ , 70–2.
**** See Li Cunshan, "Cong Guodian Chu jian kan zaoqi Dao Ru guanxi" (Early Daoist and Confucian Relations as Seen from the _Guodian Chu_ Slips), 199, translated in Defoort and Xing, "Guodian, Part II," 82.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 21–2.
**** Ibid., 22.
**** For a summary of the 'doubting of antiquity' debate in the _DDJ_ , see Edward Shaughnessy, "The Guodian Manuscripts and Their Place in Twentieth-Century Historiography on the 'Laozi'," 417–28; 433–44.
**** Shaughnessy, "The Guodian Manuscripts," 445; also see Sarah Allan and Crispin Williams (eds), _The Guodian Laozi_ , _Proceedings Of the International Conference, Dartmouth College, May 1998_ , 142–6.
**** William Boltz, "The Fourth-Century B.C. Guodian Manuscripts," 594.
**** Shaughnessy, "The Guodian Manuscripts," 447–8; Henricks, _Lao Tzu's Tao Te Ching_ , 21–2.
**** Liu Xiaogan, "From Bamboo Slips to Received Versions: Common Features in The Transformation of the Laozi," 340.
**** Henricks, _Lao Tzu's Tao Te Ching,_ 68.
**** David Hall, "The Metaphysics of Anarchism," 59.
**** See Ames and Hall, _Dao De Jing,_ 48–53; Liu, "From Bamboo Slips to Received Versions," 363–8, and Hall and Ames, _Thinking from the Han Self, Truth, and Transcendence in Chinese and Western Culture_ , 45–58.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 17.
**** Liu, "From Bamboo Slips to Received Versions," 339.
**** Ibid., 373.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 112.
**** Ibid., 15.
**** Ibid., 36.
**** Shaughnessy, "The Guodian Manuscripts," 453–4.
**** Ibid., 453.
**** Liu, "From Bamboo Slips to Received Versions," 339.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 117–18.
**** Ibid., 36.
**** Shaughnessy, "The Guodian Manuscripts," 455–56.
**** Kropotkin, _Mutual Aid: A Factor in Human Evolution._
**** For a summary of Tolstoy's and other Christian anarchists' interpretation of the Sermon on the Mount, see Christoyannopoulos, _Christian Anarchism: A Political Commentary on the Gospel_ , Chapter 1, "The Sermon on the Mount: A Manifesto for Christian Anarchism," 43–82. The original dissertation on which that work is based, "Theorising Christian Anarchism: A Political Commentary on the Gospel," includes an appendix (274–8) containing a reprinted translation of Tolstoy's "harmonized" version of the Sermon.
**** Henricks, _Lao Tzu's Tao Te Ching_ , 42.
**** Ibid., 4–5; also see Liu Zuxin, "An Overview of Tomb Number One at Jingmen Guodian," in Allan and Williams, _The Guodian Laozi_ , 32, and Jiang Guanghui, "Guodian Chu jian yu zaoqi ruxue" (The Guodian Chu Slips and Early Confucianism), 81–92, translated in Defoort and Xing, "Guodian, Part II," 6–38.
**** Needham, _Science and Civilisation II_ , 100–32.
**** See Watson's introduction to _The Complete Works of Chuang Tzu_ for the application of this idea to the _Zhuangzi._ For the application of China's north–south divide to the contemporary era, see Edward Friedman, "China's North-South Split and the Forces of Disintegration," in Friedman, _National Identity and Democratic Prospects in Socialist China_ , 77–86.
**** Li Cunshan, "Cong Guodian Chu jian kan zaoqi Dao Ru guanxi," 87–90.
**** For example, see Tu Wei-ming cited in Andrea Shen, "Ancient Script Rewrites History," 8; also see Pang Pu, "Kong Men zhi jian," 22–35, translated in Defoort and Xing, "Guodian, Part II," 39–54.
**** This argument leaves aside the question of whether the robbers who first broke open the tomb and scattered its contents made off with any of the strips.
**** Ames, _The Art of Rulership_.
**** Tu, quoted in Shen, "Ancient Script Rewrites History," 8.
4
Daoism as anarchism or nihilism: The Buddhist-influenced thought of Wu Nengzi
**Introduction: The Main Problems Raised by the** _Wunengzi_
The ninth century CE Chinese text known by the name of its pseudonymous author, Wu Nengzi (literally, "Master of No Abilities"), was the first (surviving) piece of writing in 500 years to revive the anarchist side of philosophical Daoism. Though starting out in the same radical anti-statist and utopian fashion of earlier Daoist anarchist texts of the third to fourth centuries CE, in the end the author of the ninth century text seems to acquiesce in the idea of rule, as we will see below. Thus, this text creates problems for anyone who would seek to use the radical side of philosophical Daoism to build a modern anti-statist critique. The first problem, more narrowly linked to Daoist anarchism, is whether the _Wunengzi_ demonstrates more openly a flaw that may be present in all radical Daoist texts or whether the author of this text makes a fundamental shift of his own based on influence from his interpretation of Buddhist doctrines. The larger problem for all anarchists is whether or not the _Wunengzi_ demonstrates flaws present in postmodern and/or "lifestyle" anarchist thought. Can an "ironic stance" toward political authority combined with ways of living supposedly apart from the state and claims to reject any overarching principle or "meta-narrative," in the end lead too easily to a cynical acceptance of the state and/or a refusal to oppose it directly? Even if one rejects such an "ironic stance" alone as adequate and wants to go beyond it, are there any grounds to do so from a perspective that denies humans' ability to learn and know any absolute truths objectively?
To answer these questions we need first to reexamine the nature of Daoist anarchism before Wu Nengzi and then see how Wu Nengzi himself applies and possibly changes the lessons of Daoist anarchism. After examining nature of the text itself and analyzing its main tenets, we can return to the questions raised above.
Although the _Wunengzi_ has been referred to by several students of Chinese thought, including Germaine Hoston, Peter Zarrow, and Gotelind Müller, it has previously only been partially translated into English by Hsiao Kung-chuan. There is also the partial German translation by Alfred Forke and a full German translation in an unpublished PhD dissertation by Gert Naundorf. This relative neglect is unfortunate, since the text can teach us much about both Daoist and Western anarchism. Thus this book contains the first full translation of the surviving text (see Appendix 5), which is analyzed in this chapter.
The surviving text of the _Wunengzi_ (as the text is referred to in this work, with its author referred to as Wu Nengzi) contains 3 books with a total of 23 chapters and a preface by an unnamed friend who reports that Wu Nengzi wrote the text during the Huangchao rebellion (875–884 CE), when he fled his home and travelled about, having no regular abode, finally living with a peasant family. The author of the preface claims to have created the text from scattered scraps of paper that Wu Nengzi left in a bag. From chapters in the text it would seem that Wu Nengzi had disciples and was consulted by many people for sagely advice.
**Daoist Anarchism before Wu Nengzi**
We can perhaps most profitably compare the thought in the _Wunengzi_ to that of Bao Jingyan. As we saw in the previous chapters, Bao was heavily influenced by the famous classical text the _Zhuangzi_ (ca. 300 BCE), as were most of the thinkers in the revival of philosophical Daoism at the end of the later Han Dynasty (10–220 CE) and the Three Kingdoms era at the beginning of the Period of Disunity (220–589 CE). Bao completely rejects the Confucian idea of rule by the morally virtuous based on any "Mandate of Heaven" from an impersonal deity and in place of this utopian view of benevolent rulership (based on a Mencian interpretation of Confucianism), posits the existence of an ideal utopia of original undifferentiated simplicity where there were no rulers and everyone lived in harmony. In Bao's utopia, which we will compare and contrast with that of Wu Nengzi, there were no "princes and ministers" and no means of transportation over wide areas so that "wars of conquest between states did not occur." Since there was no "greed for power and profit," there was no "unhappiness and confusion" and people lived in "mystical equality" ( _xuantong_ ) "without famine, pestilence or disease." Given their simple lives, it was impossible to implement crushing taxes on them or trap them with harsh punishments.
While there is no evidence that Bao joined or fomented any political uprisings, we saw in previous chapters that he clearly viewed all government as immoral, unnecessary, and dangerous to human survival, and there was thus no way that he could ever accept the need for a state of any kind. Rather than follow the Confucian advice to resign office in an immoral government, as we also saw previously, Bao argues that it would be better if there were no offices in the first place. Bao bases his political stance on the concept of _ziran_ , literally "of itself so," often translated as natural or spontaneous, a term that other scholars argue is the closest term in classical Chinese thought to the concept of freedom. Likewise, Wu Nengzi starts with this same concept in a similarly radical sounding fashion, and at first seems to also reject serving in government, but by the end of his tract, as we will see below, he comes to a very different political conclusion from Bao and the Wei-Jin Daoist anarchists.
**The Political Thought of Wu Nengzi**
In his first chapter, Wu Nengzi picks up the description of the Daoist utopia in terms very similar to those of Bao Jingyan, where all creatures "lived together indiscriminately" without gender or other hierarchical distinctions. As a result, there were no crimes of theft or murder and no elaborate rituals. "They followed what was natural; there was no ruling or shepherding, [and everything was] in its original simplicity; according to these principles they could live long lives."
Again, as with Bao Jingyan, those who would "help" others by instituting government entered the picture and started to draw distinctions between humans and other animals in order to dominate the animals, which introduced the principle of hierarchy, first between men and women and finally between leaders and the led in general. Once introduced, the principles of hierarchical rule and economic inequality became more and more developed, and human oppression increased as a result:
. . . After we imposed the construction of hierarchy; there came about rulers and ministers . . . We imposed assessments on people, so now they started to realise the distinction between honourable and disgraced. Now, the pure and natural has been weakened, and passions and predilections are embraced by vying hearts. If there is competition, there is stealing, if there is stealing, there is chaos [ _luan_ ], [so] what is to happen in the future?"
Given the worry of the ruling class about ordinary people's increasing restiveness, "sages" then developed the Confucian principle of benevolence and propriety and regulation of people through ritual and music. Under this scheme of supposedly benevolent rule, "when a ruler oppressed his subjects he was to be called cruel, and the ministers would say that the government was illegitimate. When the ministers usurped [the ruler's authority], the ruler would call them rebels. . . ."
Thus, far from reflecting Heaven's will and an unchanging human nature, as for Bao, so too for Wu the Confucian ideas of cultivation of "virtue" only served to legitimate and protect domination of some humans over others. Based on chapters from the received _DDJ_ and the _Zhuangzi_ , and again following the tradition of the Wei-Jin Daoist anarchists such as Bao Jingyan, Wu Nengzi goes on to argue that the introduction of Confucian hierarchy only inflamed the people's passions and awakened their desire to compete with each other for dominance. Thus eventually the "sages" "had no other option but to establish laws and punishments and organise armies to keep the people under control" including eventually instituting the harsh punishments of the Legalists, which only led to armies being sent over the land and violence spreading over the whole country, so that in the end, far from improving their lives, "the common people came to dire poverty and died."
In the end, similar to the arguments of Western anarchists such as Michael Bakunin, Wu Nengzi turns on its head the typical question about how anarchists would handle the problems of crime and warfare in the absence of government. Instead, Wu Nengzi argues, it is the principle of rule and the imposition of hierarchy that leads to chaos and the destruction of human life:
Alas! It was natural to treat [the people] as beasts; it was not natural to treat them as humans. Imposing the establishment of palaces and mansions, [formal] meals and [prepared] food stirred up desires; imposing distinctions between the exalted and debased and the honourable and disgraced excited competition; imposing benevolence, virtue, ritual and music perverted what was natural. Imposing punishments and laws and [using] military [force] immiserated [people's] lives . . . this disturbed their passions and attacked their lives, and together in great numbers they died.
Thus far, Wu Nengzi's critique sounds as radical as that of his Daoist predecessors, including even Bao Jingyan, based on Daoist principles of original simplicity ( _si_ ), primeval unity without hierarchy ( _hundun_ ), and especially _ziran_ (the natural or spontaneous), which as we noted in previous chapters could serve as metaphors for human freedom in nature. But in later sections in Part 1 of his text, though still based on the Daoist idea of nature as an undifferentiated whole, Wu Nengzi starts to introduce themes concerning the identity of life and death, almost certainly influenced by the spread of Buddhist ideas in China during the Tang dynasty. In his Chapter 3, Wu Nengzi examines human nature and how humans look at the human body, concluding that,
That which is born from Nature, although it exists separately and can be broken off, is eternally alive. That which naturally dies, although it moves around, it will always die.
Beginning with the Daoist principles that nothing exists separately and that the idea of life and death is like _yin_ and _yang_ , or two sides of an undifferentiated whole, Wu Nengzi denigrates those who would seek the elixir of long life instead of worrying about the quality of their lives. This "idiotic" desire for long life in the end only gets people further from life. Though based on Daoist principles, Wu Nengzi seems to be introducing a Buddhist-influenced idea of the unreality of both life and death, as in his Chapter 4 of Book 1:
As for death, it is the most despised by the people. But there is no death to be despised, besides the shape and skeletal structure; is there anything really to disturb feelings of utmost harmony and satisfaction?
Throughout the next chapter, Wu Nengzi continues to denigrate people's fear of death and their desire for material things and a fine reputation as ideas inculcated and fanned by the so-called sages. While still serving the purpose of undermining Confucian and Legalist concepts of rule, we will see below how this Buddhist-influenced denial of material needs based on the denial of the distinction between life and death ultimately served to undermine his anarchism.
Nevertheless, in the second part of this same chapter, Wu Nengzi continues his radical egalitarian vision. Far from naturally favoring our relatives and close friends, as Confucian thinkers would have it, he argues that we should not differentiate among people and treat all equally. Far from teaching people to treat each other with benevolence, Confucian ideas of benevolent hierarchy lead only to strife and contention.
It is in the second of his three books that Wu Nengzi's political ideology starts to show the effects of his Buddhist-influenced stance of detachment from material things. In retelling a famous incident from the period toward the end of the Shang or Yin dynasty and the beginning of the Zhou (ca. eleventh century BCE), Wu Nengzi takes up the eternal question for intellectuals first raised in the _Zhuangzi_ , whether or not to serve in government—a question many chapters in that text and the later neo-Daoists answer in the negative. Answering a Confucian-influenced gentleman named Xi Bo who would try to rescue the Shang dynasty from chaos, at first Wu Nengzi's reclusive sage Lü Wang seems to follow the radical Daoist advice to not get sullied by serving the state, though in terms that seem to deny the reality of the people's suffering:
. . . the Shang Dynastical government became chaotic by itself, and the people are in great pain out of their own doing. What is the connection to you? Why do you want to sully me?' . . . If something killed off all humans, birds, beasts, and insects, the ether would still be the ether. How can we do anything about the Shang government's loutishness? How can we say anything of people's hardship?
Though sounding very indifferent to ordinary people's suffering, this passage could be based on Chapter 5 of the received _DDJ_ , which advises the sage to be ruthless and treat the people as straw dogs—advice which Arthur Waley claims is a bait for the Legalists. That is, since "nature is perpetually bounteous" and thus perhaps takes care of people on its own, there is no need for rulers to paternalistically "take care" of the people. Nevertheless, in a very important shift, Wu Nengzi allows his reclusive official to serve the state after all in the end:
[D]espite all of this, the castle walls, houses, and cottages are already built and so need not be destroyed, just as the people are already formed and need not be killed, so I will save them!
In answering another of his officials as to why he decided to aid the suffering people of the Shang dynasty despite his talk of the virtue of the Daoist principle of _wuwei_ (inaction, or doing nothing), Xi Bo replied with what one could argue is a very Buddhist take on _wuwei,_ an interpretation that Wu Nengzi has Lü Wang endorse:
Xi Bo said, "Heaven and Earth are inactive, yet the sun, moon, stars, and constellations move in the day and the night. There are rain, dew, frost, and freezing rain in the autumn and winter. The great rivers flow without pause, and the grass and trees grow without stopping. Therefore, inaction can be flexible. If there is a fixed point in action, then it cannot be inaction." Lü Wang heard this and knew that Xi Bo really did have compassion for the people and didn't want any profit from the Shang Dynasty's world. Thereupon, Lü Wang and Xi Bo finally made the State of Zhou prosperous and powerful.
This conclusion of the chapter goes to the heart of the difficulty of Wu Nengzi's thought. If life and death are the same and material suffering is just an illusion, then being attached to opposing all government is also an illusion. In the end, for Wu Nengzi, one _can_ try to help people by trying to govern them, but only as long as one has no desire to dominate them and no illusions about the ultimate worth of government. One then could wonder whether Wu Nengzi's prior condemnation of all government and his ridicule of the idea of benevolent rule for the benefit of people completely fall apart. If nothing matters, so too does opposition to the state not matter. Perhaps we could use contemporary language to say that Wu Nengzi would not oppose intellectuals taking part in government as long as they have a stance of ironic detachment while they are governing.
In the rest of Part 2, Wu Nengzi turns the tables on both famous officials and famous recluses in Chinese history, making both look ridiculous for seeking virtue and fame either by holding office and great wealth or by becoming hermits. Both are deluded, he seems to be saying, if they think they have found the truth. It is being attached to any desires that leads people astray, whether the desire is to hold high office or to hold a reputation as an honest recluse. Standing by itself, this message would not depart very much from the ideas of earlier Daoist anarchists, especially those of the poet Ruan Ji. As we saw in Chapter 2, in his great poem "The Biography of Master Great Man," Ruan Ji's hero answered the Confucian gentlemen who came to him to criticize his "immoral" behavior of not dressing properly or seeking high office by comparing these men ambitious to serve nobly in high office to lice who inhabit a pair of trousers. Ruan Ji goes on to make the argument, echoed by Bao Jingyan, that it would be better if there were no offices and honors to seek than to resign office from an immoral government. Wu Nengzi likewise criticizes the idea of serving in government for noble reasons, more cynically than Ruan Ji or Bao Jingyan, and goes on to argue that serving in office is nevertheless not to be condemned if one has no illusions about the morality of serving. In Chapter 6 of Part 2, he has two officials discuss retiring from high office after achieving success for their king. The first official cannot imagine retiring at the point of their highest achievement, while the other warns that the king will now only be jealous of their success if they stick around:
. . . because he hated the state of Wu, [the king] employed you and me in order to use our schemes. You and I benefitted from the pay and therefore we schemed against Wu [for the king], and we [can] take as a sign of our success, the destruction of the people, and as payback, he gives us our emoluments. The duplicity of people is such that they say that they are like Heaven and Earth's births and killings [and] that they are agents of Heaven and Earth—what sages call getting rid of harm and bringing things to completion, isn't this just a big scam?
In other words, Wu does not really criticize the idea of serving in office nor even the destruction of a whole people for the benefit of a king, but only the idea that the rewards earned by serving the king will last forever or that the government service has some higher purpose.
In Chapter 8 of Part 2, Wu Nengzi tells the story of four famous recluses whom a king tried to entice to join his government, probably in order to demonstrate that the most virtuous officials were willing to serve him. Though they agreed that the emperor was more kind and virtuous than his rivals for power, the four recluses made a cynical conclusion to serve the evil Queen Mother and her henchman the Marquis of Liu, who were scheming to replace the emperor with her son, the Crown Prince.
. . . As for Empress Lu, that woman's nature is cruel and mean, [and] her son Ying is not yet firmly established as the crown prince, so she has necessarily been pushed to a crisis. In crisis, she has come seeking us; the peaceful resolution of the crisis depends on us. If she seeks us but does not get us, she will necessarily bring disaster upon us, therefore we must answer yes to her.
Thus the four former recluses agreed to do the dirty work of the Empress and the Marquis, to the point where her son ascended the throne and her enemies were eliminated. At that point the four men refused her offer of further honors and returned to their reclusion. We should note again that this chapter does not criticize the idea of serving in government, even serving obviously power-hungry nobles and officials at the expense of more high-minded rulers. The only thing being criticized is the belief that either serving or not serving in office can ever demonstrate moral virtue.
This cynical attitude is perhaps why Hsiao Kung-chuan claims that in the end Wu Nengzi's thought is nothing more than "a pure negation without any suggestion as to what is to be done or what shall take the place of the state" and thus demonstrates that Chinese Daoist anarchism is merely a "doctrine of despair" rather than one of hope as in Western anarchism. As we saw in the first chapter, Peter Zarrow thinks that Hsiao unfairly characterizes Daoist anarchists as a whole, some of whom did possess an "alternative social vision" if not a theory of revolution; nevertheless, Zarrow does accept that Wu Nengzi is an exception to other radical Daoists and is closer to a "total cynic than a constructive social thinker." Similarly, Germaine Hoston thinks his cynical attitude marks Wu Nengzi's thought as nihilistic.
In Part 3 of the _Wunengzi_ , the author speaks more in his own name and says things more directly. His main point is still that people should have no intentionality, and Wu Nengzi continues to interpret the Daoist principle of _wuwei_ as taking no _intentional_ action out of a desire for personal or social benefit, except perhaps for the benefit of continuing to live, which would seem to be an obvious contradiction to having no desire. Nevertheless, in other chapters Wu Nengzi disparages even the desire for health and long life. Perhaps he is arguing that having no intention and having no desire is not always the same thing. In Chapter 2 of the third book Wu Nengzi answers a friend who came to him asking about whether to accept another friend's offer to serve in office by saying that taking office is not against the principle of _wuwei_ as long has one as no intentionality ( _youxin_ ) or desire to get ahead.
. . . when the situation is favourable then it is permissible to provide aid to the world. Therefore the emperors Yao and Shun didn't decline the office of emperor. In both cases [the hermits and the emperors] were united in having no intentionality.
Thus Wu Nengzi concludes this chapter on a very Confucian note, even to the point of accepting the official Confucian model heroes Yao and Shun and the Duke of Zhou. Taking away all intentionality and all illusions about trying to rule for the benefit of the people, he seems to be saying, might sometimes allow not just for serving in government but, in the end, even for ruling in ways that would benefit oneself and others, although only if one does not have the desire or intention to benefit people at the outset.
If this conclusion is valid, then one might obviously ask if anything at all is left of Wu Nengzi's anarchism. After all, the minimal definition of anarchism offered at the outset of this book is that the state is unnecessary, harmful, and dangerous. Though some Western anarchists, most famously Pierre-Joseph Proudhon, at some points accepted service in the state, perhaps for tactical or limited reasons, as also, for example, some of the anarchists who cooperated with the Republican side in the Spanish civil war, most modern anarchists would point out the obvious contradictions even for tactical or temporary compromises with the state, since the main anarchist principle is that the state's very nature as a monopolistic operation will eventually lead it to dominate other interests, including those of class, interest group, gender, or ethnicity. If there is something in even the radical side of philosophical Daoism that would excuse state service, then it would seem the possibilities for Daoist anarchism are severely compromised, to say the least.
**More Narrow Problem: Is All Daoism Nihilism?**
As we saw in earlier chapters, at other times in preimperial and early imperial China, individuals justified or excused service in the state using Daoist principles. If we can find some common shift in language or rhetoric among those who used Daoist terms to justify rule, perhaps we can determine whether those thinkers who remained ostensibly loyal to the anarchistic side of Daoist thought shared a common flaw or whether those who accommodated themselves to rule introduced changes in Daoist thought not shared by radical Daoist thinkers, and perhaps not shared in the original Daoist texts such as the _DDJ_ and the _Zhuangzi_. To review what we found in earlier chapters, the three most important times earlier in Chinese history when thinkers used Daoism to justify or acquiesce in rule included the early years of the former Han dynasty (ca. 202 BCE–9 CE), the first generation of the revival of philosophical Daoism at the end of the later Han dynasty (ca. 220 CE), and the third generation of neo-Daoists at the beginning of the Wei-Jin period (ca. 220–300's CE).
In the early Han dynasty, intellectuals were casting around for a suitable legitimating ideology of rule for the Han leaders, given that the previously prevailing ideology of Legalism had been discredited by the harsh rule of the Qin dynasty that the Han had recently overthrown. For a relatively short time, Daoism seemed to gain ascendancy at the Han court. The basic argument of these court Daoists was that the Han regime ruled lightly, with less harsh taxes and less need for military repression compared to the Qin and so could be said to be like the ideal ruler in the received _DDJ_ who is unseen and unfelt by the people. This use of concepts in the _DDJ_ to justify rule perhaps came from what is known as the "Huang-Lao" tradition, which combined the mythical Yellow emperor with a deified Lao Zi (the legendary author of the _DDJ_ ). Most famously, in one of the silk manuscripts unearthed near the village of Mawangdui in Hunan province in 1973 from a tomb that had been sealed in 168 BCE, the author argues that a ruler in touch with the _dao_ , or the Way, should be able to know what is needed and how to get others to accept his rule:
Therefore only Sages are able to discern [the _dao_ ] in the Formless,
And hear it in the Soundless.
And knowing the reality of its emptiness,
They can become totally empty,
And then be absorbed in the purses essence of Heaven-and-Earth.
Absorbed and merged without any gaps,
Pervasive and united without filling it up.
Fully to acquiesce to this Way:
This is called 'being able to be purified.'
The lucid are inherently able to discern the ultimate.
They know what others are unable to know,
And acquiesce to what others are unable to attain.
This is called 'discerning the normative and knowing the ultimate.'
If sage kings make use of this,
All-under-Heaven will acquiesce.
. . .
One who is truly able to be without desires
Can give commands to the people.
If the one above truly acts without striving
Then all living things will be completely at peace.
The first change one can discern in early Han Daoism from ideas in the received _DDJ_ and the _Zhuangzi_ , texts that were used by later Wei-Jin Daoist anarchists to deny the need for all rule, is the Han thinkers' confidence that the _dao_ can be known and interpreted by the sages or even one sage–ruler and applied to others. The second, related shift concerns the blowing up of the concepts of nothingness ( _wu_ ) and the emptiness or void at the heart of the universe.
The most famous version of this Daoist justification of rule in the early Han, we saw in Chapter 1, was the text known as the _Huainanzi_ , which was presented to the future Han emperor Wu (r.141–187 CE) in 139 BCE as a preferred method of rule that would help justify his regime. The authors of this text continue to use the principle of non-action or doing nothing ( _wuwei_ ) found in the _DDJ_ but now interpret it not as calling for anarchy, but as favoring a ruler in touch with the _dao_ who rules by emptying his mind and limiting his and his subjects' desires. Roger Ames argues, however, that in practice the authors of this text were trying to subvert rule and get the king to rule in a less overbearing manner and thus continue to be influenced by the anarchist side of Daoism. We asked above whether an anarchist-influenced observer would instead conclude that these intellectuals' attempt to soften Han rule was in practice overwhelmed by their participation in aiding the state's legitimation. In any case, it is the shift toward the belief in one or a few sages knowing how to interpret the _dao_ for others based on a _dao_ that is equated with nothingness that allows for the justification of rule.
In the end, of course, the state eventually abandoned most claims to follow Daoist principles when the Han dynasty gradually had to rule more directly and forcefully as more officials and their families became tax exempt, public works needed to be repaired, and armies replenished to fight nomadic invaders and internal rebels. As a result, the Han eventually turned to a new synthesis of Confucian doctrines as its main legitimating ideology.
The second major period when philosophical Daoism was put in the service of rule was in the early Wei-Jin period (ca. 220 BCE–62 CE). As we saw in the first chapter, at this time, after the fall of the later Han dynasty and the beginning of a long period of political disunity in imperial China, some of the intellectual figures around the legendary general Cao Cao (155–220), who was seeking ways to legitimate his rule as the leader of a would-be new imperial Wei dynasty, returned to the received _DDJ_ to try to find ways to justify his rule. The Daoist-influenced intellectuals serving him also returned to the idea of _wu_ or nothingness as the main principle of Daoism. According to this version, all things in the universe come not from an underlying unity in the world but from nothing. All actions should be carried out according to a principle of spontaneity ( _ziran_ ), but for these Daoist advisors there was nothing wrong in principle with the idea of rule. Thus Cao Cao's rise from a person of low birth to that of possible emperor was the rise of a ruler coming "out of nowhere." Cao Cao's apologists used this version of philosophical Daoism against the rival Sima clan, who came from the higher class of land-owning gentry and whose preferred ideology of rule lay in the Confucian doctrine of the time known as _mingjiao_ , or "teaching of names." Against this doctrine, the apologists for Cao Cao used Daoism to provide an ideological justification for a new type of government based on people "arising out of nowhere" based on their ability, especially in military campaigns, instead of the _mingjiao_ praise for rulers with family connections within the old aristocracy. Thus, Richard Mather argues, these Wei official intellectuals emphasized Daoist concepts of "'naturalness' [ _ziran_ ] and 'non-actuality' [ _wu_ ]" against "the [Confucian] shibboleths of the old aristocracy concerning 'goodness and morality,' [ _ren-yi_ ] 'loyalty and filial submission' [ _zhong-xiao_ ] . . ." not in order to call for anarchism, but instead to justify Cao Cao's rule.
As we noted in the first chapter, Daoism was only one of many philosophical strands picked up by Cao Cao's coterie, who also borrowed concepts from Legalism and even Confucianism to justify his rule. In this synthesis, some intellectuals claimed that Confucius was a better sage than Lao Zi as in the following exchange from the biography of the noted Wei philosopher Wang Bi (226–249):
[As Pei Hui asked Wang] "Nothing ( _wu_ ) is, in truth what the myriad things depend on for existence, yet the sage (Confucius) was unwilling to talk about it, while Master Lao expounded upon it endlessly. Why is that?" Wang Bi replied, "the sage embodied nothing ( _wu_ ), so he also knew that it could not be explained in words. Thus he did not talk about it. Master Lao, by contrast, operated on a level of being ( _you_ ). That is why he constantly discussed nothingness; he had to, for what he said about it always fell short."
This elevation of Confucius above Lao Zi by the neo-Daoist intellectuals around Cao Cao mirrors their elevation of sages who rule over those who refuse to participate in rule, reversing the praise of the latter type of sages found most famously in the _Zhuangzi_ that the full-fledged Daoist anarchists Ruan Ji and Bao Jingyan had copied.
We saw in Chapter 1 that only after the Wei rulers were overthrown by the Sima clan, who founded the Jin dynasty, did some of the descendants of the Wei intellectuals turn philosophical Daoism into a doctrine opposing all rule, as reflected in the ideas of the poet Ruan Ji and the thinker Bao Jingyan. But as the Jin dynasty itself broke down into infighting among royal princes and as northern nomadic groups moved into northern China and the political situation became even more chaotic at the end of the Wei-Jin era of the Six Dynasties period (220–589), Daoist-influenced intellectuals and members of the upper classes turned neo-Daoism once again into a nihilistic doctrine. As Balazs puts it,
What had been, with men [of the second generation of anti-statist neo-Daoists] a high state of tension that was part of a serious effort to transcend human limitations, relapsed into mere abandonment of the ordinary decencies of life. The frenzied attempt at emancipation had turned into wanton frivolity, the cry of cynical revolt to cynical acceptance, liberty to libertinage.
Men of this third generation of neo-Daoists once again began to justify government service as being in line with _ziran_ or spontaneity, based again on the idea of _wu_ or nothingness as the basis of the _dao_.
What all three prior instances of Daoist anarchism turning into nihilism share then, is the emphasis on the universe as based on nothing and the idea of the superior ability of properly detached sages to realize this and to interpret principles for others without getting sullied or corrupted by rule. Of course Wu Nengzi shares at least the former belief, and implicitly the latter in his claim that the truly enlightened sage knows when serving in government is folly and when it is permissible. The shift in emphasis in all these instances was literally from everything to nothing, that is, from the belief in an overarching unity of the universe that cannot be objectively known and applied by some to rule over others to the idea that everything that seemingly exists comes from nothing and thus that there were no a priori principles that would make all rule illegitimate. The shift in all instances was also from the idea of rejecting all participation in government as inherently corrupting to the idea that the wisest people with the coolest attitude of detachment could have the superior knowledge and ability to allow them to acquiesce in rule, or even to rule over others themselves, without being corrupted.
The flaw then, is not in the Daoist principle of _wuwei_ itself but in the denial of any preexisting overarching principle underlying the unity of existence and equality of all things. What is also missing from those Daoists who justified rule and service in government is any true belief in human equality and freedom for all, not just for superior sages, despite the talk of favoring all equally in _Wunengzi_ Book 1, Chapter 5 that we examined above.
**Larger Problem: Is Postmodern Anarchism Nihilism?**
The larger problem presented by the breakdown of Daoist anarchism in the thought of Wu Nengzi into passive nihilism is the lesson for postmodernist thought, especially those postmodernists who call themselves anarchists.
Anarchists up to the postmodernist period would reject the classic conservative critique that by denying the existence of preexisting standards of morality, all anarchism is nihilism in the end. This conservative stance is perhaps most cogently summarized by Fyodor Dostoevsky's claim that "once God is abolished, anything is possible" and in his denunciation of early Russian revolutionaries as immoral nihilists too easily duped by power hungry would-be supermen, such as Sergei Nechaev, the associate of Michael Bakunin and the basis for the character of Pyotr Verkhovesky in Dostoevsky's novel _The Devils_. Classic anarchists, most notably Peter Kropotkin, are more easily able to reject this critique in their claim that there is a natural underlying morality of humans based on human evolution, a morality that existed prior to the establishment of organized religion and the state.
Many postmodernist thinkers, on the other hand, would seem more open to the organic conservative critique to the extent that they accept the premise that all "meta-narratives" meant to explain the world and give people a guide to action are inherently just constructions of new forms of domination that stand in the way of liberatory goals. While they claim to deny any overarching "meta-narrative" as valid for all other people, one must ask whether postmodernist anarchists reserve for themselves the right to be critical of all other narratives while preserving their own ideas as something other than a true narrative. Even if they claim their own approach is not a meta-narrative but only a stance of "ironic detachment," then one could argue that this stance too easily smacks of intellectual superiority.
While they clearly remain within the tradition of classical anarchists who viewed all religious and political doctrines as attempts to enslave people with metaphysical or real authority, one must ask whether postmodernist anarchists go further to deny the existence of all truth, even truth that cannot be known objectively or imposed on others. If so, as asked by many critics about postmodernism, how is one to criticize any political doctrine or state as evil, even fascist states? This charge was most famously and, perhaps for postmodernists, most infuriatingly raised by Richard Wolin who examines the collaborationist and even fascist background of some of the seminal postmodernist thinkers in order to expose flaws in postmodernist thought as a whole. While those who want to find a genuine liberatory critique in postmodernism may decry his attack as relying almost completely on guilt by association, perhaps it is too easy for postmodernist anarchists to make this charge and ignore the need for serious self-examination. It seems obvious to this author that the move among Daoist thinkers such as Wu Nengzi from pacifist anarchism to passive nihilism was based on a similar shift in emphasis from the nonexistence of hierarchical distinctions to the nonexistence of everything.
This charge of nihilism against postmodernist and/or "lifestyle" anarchists who think their intellectual stance alone will serve to achieve anarchism may be the opposite side of the coin of those who find Daoist anarchism a mystical doctrine that relies on a supernatural authority and is thus inherently un-anarchist, a view of Daoism with which this author obviously strongly disagrees. Even if Daoists believe in the existence of an overarching, undifferentiated whole, they would deny that one can objectively reconstruct that whole for others. More dangerous, a Daoist anarchist would argue, is any doctrine based on the idea that some may know objective truths better than other people and thus also when to apply those truths on behalf of others, which may too easily lead to would-be anarchists acquiescing and even participating in establishing authority over fellow humans. Only by embracing the whole, not denying its existence, a Daoist anarchist would argue—that is, by accepting the underlying unity and thus equality of all things, even if by its very nature that whole cannot be hierarchically organized—can one stay loyal to a fully anarchist vision.
Nevertheless, perhaps given the difficulty radical Daoists faced in order to survive and publish their works, not to mention the degradation of radical Daoist ideas in works such as the _Wunengzi_ , the idea of Daoist anarchists as passive escapists survived in Chinese culture to the point that, as we will see in the next chapter, most participants in the twentieth-century Chinese anarchist movement declined to accept Daoist anarchism as a worthy predecessor.
**Notes**
**** See Germaine Hoston, _The State, Identity, and the National Question in China and Japan_ , 158–9; Peter Zarrow, _Anarchism and Chinese Political Culture_ , 10–11; and Gotelind Müller, _China, Kropotkin und der Anarchismus_ , 116–18.
**** Hsiao Kung-chuan, "Anarchism in Chinese Political Thought," 251–63.
**** See Alfred Forke, _Geschichte der Mittelalterlichen Mittelalterlichen Chinesichen Philosophie_ , 330–2; and Gert Naundorf, _Aspekte Des Anarchischen Gedankens in China: Darstellung der Lehre und Ubersetzung des Texts Wu Neng Tzu_.
**** For a modern reprint of the classical text, see Wang Ming, compiler, _Wunengzi jiao shu_.
**** Bauer, _China and the Search for Happiness_ , 139.
**** See, for example, Holzman, _Poetry and Politics_ , 190.
**** Unless otherwise noted, all translations from the _Wunengzi_ in this chapter come from the first complete English version by the author's student colleague, Catrina Siu, with editorial assistance from his faculty colleague, Daniel Youd of the Department of Modern Languages at Beloit College, which is published in its entirety in Appendix 1 of this book.
**** See Waley, _The Way and Its Power_ , 147.
**** Translated in Holzman, 192–5.
**** Hsiao, _History of Chinese Political Thought I_ , 260.
**** Zarrow, 10, 262, n. 23.
**** Hoston, 159.
**** For an account of the discovery of this manuscript, see Wm. Theodore deBary and Irene Bloom (trans. and compilers), _Sources of Chinese Tradition, vol. 1: From Earliest Times to 1600,_ 241–2.
**** Translated in deBary and Bloom, 254–5.
**** In their introduction to Liu An, King of Huainan, _The Huainanzi_ , 27–32, the translators discuss the differing views of scholars (including the translators themselves) as to whether the text should be considered Daoist or some kind of eclectic mix. In either case, the text certainly contains and adapts Daoist themes related to governance.
**** Ames, _The Art of Rulership_ , 46, 148.
**** See Balazs, 234–5.
**** Mather, "The Controversy over Conformity and Naturalness," 161, 163.
**** In the _Chronicles of the Three Kingdoms_ , translated in de Bary and Bloom, 385.
**** Balazs, 247.
**** The title of this novel has also been translated as _The Possessed_ and more recently, by Richard Pevear and Larissa Volokhonsky as _Demons_ , who note in their foreword, vii–viii, that Dostoevsky based the character of Verkhovensky on Nechaev and his actions in the actual murder of the fellow revolutionary Sergei Ivanov.
****Kropotkin expressed this idea of a naturally existing human morality most famously in his book _Mutual Aid: A Factor of Evolution_ and also in his unfinished but posthumously published work, _Ethics: Origin and Development_.
**** See Wolin, _The Seduction of Unreason: The Intellectual Romance with Fascism from Nietsche to Postmodernism_.
**** A charge made against Daoism by Janet Biehl, an associate of Murray Bookchin and the Social Ecology school, in exchanges with this author on the Research on Anarchism (RA-L) listserv. Biehl's charge against Daoism as a "supernatural" or "mystical" authority was in her "Re: Comment on Bookchin," Part 2, September 30, 1998, and "Reply to Rapp," Parts 1 and 2, October 22, 1998, which no longer seem to be in the RA-L archives; this author's original post on October 2, 1998 and his rejoinder to Biehl's critique on October 28, 1998 can be found at www.zpub.com/notes/JohnRapp.html and www.zpub.com/notes/JohnRapp.html respectively. Though both posts contain a rather unrefined view of the split between religious and philosophical Daoism, the main point that one does not have to take a "mystical" interpretation of the classical Daoist texts as appealing to "supernatural authority" still applies.
INTERLUDE
5
The twentieth-century Chinese anarchist movement
It might seem odd for a book on Chinese anarchism to devote only one chapter to the early twentieth-century anarchist movement in China, which began separately at the turn of the last century among Chinese student groups in Tokyo and Paris, respectively, and continued in China itself after the 1911 revolution until it was gradually eclipsed by Marxist–Leninism in the 1920s. First, given the rather extensive scholarship on this movement both in and outside of China, and second, the limited relationship of that movement to either premodern Daoist anarchism or to the dissident Marxists whose critique will be labeled "neo-anarchist" in later chapters, that early twentieth-century movement lies largely outside the scope of this book, with two different but notable exceptions that are the focus of this chapter.
For the most part the early twentieth-century Chinese anarchists adopted the themes of their European and American counterparts, especially concerning the need for a social revolution to overthrow the capitalist state and to establish social and economic equality within an industrial, modern, but communal society, also to be accomplished through establishing experiments such as work–study movements where people would combine intellectual work with manual labor. The modern Chinese anarchists also, of course, proclaimed what this book terms as the minimal essence of the anarchist critique—the idea that the state is harmful and unnecessary and rules for itself when it can. As with their Western counterparts, however, the Chinese anarchists were often contradictory on this point when they called for coercive, violent revolution and when many of them ended up acquiescing one way or the other to state authority in their later careers. For Chinese anarchists as for anarchists in other countries, this work argues, it is their departure from the minimal essence of the anarchist critique that made it easier for people who continued to identify as anarchists to cooperate eventually with various types of state authority. The two different issues within the modern Chinese anarchist movement that we need to examine also highlight this departure from the basic anarchist critique.
First we need to examine why there was such a limited influence from traditional Chinese anti-statist ideas, including especially Daoism, on the modern Chinese anarchist movement. Did the negative attitude of most members of that movement toward Daoism really reveal limits or weaknesses in Daoist anarchism—especially whether it was truly opposed to all state authority—or instead did their attitude reflect biases related to modern faith in "scientific" socialism that itself may reveal too much faith in authority even among self-styled anarchists? Second, we need to examine the debates between anarchists and Marxist–Leninists that broke out in China in the 1920s, both in order to understand the possible negative lessons Mao Zedong drew from those debates, which we will examine further in the next chapter, and to understand why Marxist dissidents in the PRC, even when they utilized what we will label in the last two chapters as "neo-anarchist" critiques of the state, had to take pains to disassociate themselves from anarchism (even if at points, as we will see in the last two chapters, they did acknowledge a similarity or even debt to anarchism).
**Looking Back: Daoism and the Early Twentieth-Century Chinese Anarchist Movement**
Most of the early twentieth-century Chinese anarchists, even if they acknowledged the "anarchist impulse" in the DDJ and the _Zhuangzi_ , nevertheless viewed Daoism, with its emphasis on _wuwei_ (which they took to mean inaction) as a prescientific, escapist philosophy of individual transcendence that provided little to no guide for revolutionary action. As Li Shizeng, a leader of the Chinese anarchists studying in Paris in the first decade of the twentieth century, put it,
Anarchism advocates radical activism. It is the diametrical opposition of quietist nonaction. Anarchism does not only advocate that imperial power does not reach the self, it also seeks to make sure that it does not reach anyone else.
Furthermore, though Li did accept that Daoism had some commonalities with anarchism, nevertheless, given that the ancient Daoists did not have the benefit of modern scientific advances, he believed that,
. . . naturally what [Lao Zi and other ancient sages] had to say is not fully relevant to events that are occurring several thousand years later . . .
Likewise, for Wu Zhihui, another leader of the Paris group, all traditional thought and religion, though perhaps valuable in their day, have been made worthless as a result of modern evolution. Against those who would find transcendental ideas of selflessness and fraternity in traditional Christian and Buddhist values that could be used to reform society, Wu responded,
Selflessness and fraternity are the natural virtues of humankind and the seeds of world evolution . . . [Now that the world is] relatively civilized, most people believe in good morality and so agree on selflessness and fraternity. The beliefs of ancient people have nothing in common with those of today. The anarchists have no need to yield one iota.
Similarly, Shifu—the influential leader of the anarchist movement in China itself from the time of the 1911 revolution until his death in 1915—despite his own influence from Buddhist practices "vigorously denied" any connection of Daoism and Buddhism to anarchism.
The chief exception to this negative view of Daoism among the early twentieth-century anarchists was Liu Shipei, the leader of the Chinese anarchists in Tokyo from 1907 until 1910. Given his later career, however, by his negative example he may serve as the exception who proves the rule about the lack of influence of Daoism on twentieth-century Chinese anarchists, or even perhaps as the person who by his negative example led other anarchists to reject Daoism as true anarchism. Liu began his career as a rather typical Confucian scholar and would-be bureaucrat who continued to admire Confucian and Daoist thinkers for their supposed ideals of laissez-faire government even after he became an anti-Manchu nationalist revolutionary after 1903 and an anarchist after 1907. In the poems he wrote between 1902 and 1906 just before moving to Tokyo, Liu took up Buddhist and Daoist themes of the transience and emptiness of the material world and the need to transcend the self and attain oneness with the cosmos. During his anarchist period, Liu expressed his view that Lao Zi was the father of Chinese anarchism and that ancient Chinese society was inherently anarchistic, since it was supposedly mostly free of central state control due to the influence of the "non-interference" policy of both Daoism and Confucianism. In addition, he also pointed to the ancient Chinese advocates of an egalitarian agrarian utopia such as Xiu Xing to say that China had its own libertarian socialist tradition. During his anarchist period Liu rediscovered the Daoist anarchist tract of Bao Jingyan, whom Liu viewed as an anti-militarist who called for the destruction of the whole principle of rulership and who attacked the distinction between rich and poor, thus for Liu showing that Daoism had anarcho-communist and not just philosophical anarchist roots.
While seemingly providing more evidence for the point of the first part of this book concerning the anarchist nature of Daoism, the lesson that many scholars of anarchism and anarchist sympathizers may draw from the direction Liu took in his later career is that of the weaknesses and contradictions of any modern anarchism based on Daoist and other premodern philosophies. In 1908, Liu returned to China, where he turned very conservative, supporting the late, decaying Qing dynasty regime that he had previously so opposed, even serving under the Qing official Duan Feng as he moved from one post to the other, including in Sichuan province where Duan suppressed republican revolutionaries in late 1911. After the establishment of the Republic of China in 1912, Liu served under the warlord Yan Xishan and through him came to support and join the government of Yuan Shikai, the former Qing dynasty general who extracted the reward of being named president of the republic as the price for going over to the republican side but who nevertheless started to move toward declaring himself the emperor of a new dynasty in his last years in office. After Yuan's death in 1915 Liu returned to the purely academic realm where he was mostly apolitical, though he did take part in a journal that opposed the prevailing New Culture era radicalism up till his death in 1919.
Even before his return to his conservative roots, Liu was more in sympathy with the anti-materialist, anti-urban egalitarian ideals of Tolstoy than with the pro-science (if not scientistic) attitudes of the Paris anarchists, though he was never a total primitivist and did think the future anarchist society could achieve a high economic–technological level. Nevertheless, beginning in his anarchist period, Liu's anti-capitalist and egalitarian beliefs led to an "ambivalent attitude" that laid the seeds of a conservative utopianism opposed to Western modernity based on economic–industrial technological development if that modernity meant the growth of socioeconomic inequality and modern bureaucratic government. Once he lost hope for the possibility for immediate revolution, he believed that China's past agrarian ideal, even if it was very backward economically, was preferable to Western "material civilization." As the historian Yang Fang-yen cogently summarizes,
In Liu's pleas for anarchist utopia, his agrarian nostalgia was wrapped up in the apologetic rhetoric of the "advantage of backwardness" and transposed into a defense of China's pioneering role in the world anarchist revolution. In his advocacy of "preserving the old," by contrast, this nostalgia reasserted itself as a reactionary but no less utopian attempt to return to the past.
Once he lost faith in the immanence of social revolution in China, a critic of Daoism as anarchism could argue, Liu's Daoist–Buddhist beliefs helped him to justify a shift from anarchism to a nihilistic acceptance of the state and even to a willingness to accept political office, similar to the path Wu Nengzi took nearly a 1,000 years previously. In sum, Liu's path from anarchism to reactionary monarchism would seem to seriously discredit Daoist anarchism, or at the very least, to undermine the anarchist cause in providing fodder for its critics and enemies.
Indeed, the seminal Chinese Marxist and early CCP leader Chen Duxiu in his debates with the anarchists in the early 1920s did not hesitate to attack anarchism "as a reflection of intellectual and behavioral habits rooted in Daoism." As Edward Krebs summarizes Chen's argument,
anarchism fostered a lazy and undisciplined sort of free thinking . . . which had as its chief cause the "nihilist thought and laissez attitude" of Daoism. As a result, the anarchism so popular among [Chinese] youth "is certainly not a thoroughly western anarchism," but rather "a revival of the principles of Laozi and Zhuang Zi, a Chinese-style anarchism."
For Chen, the passive Daoist "irresponsible individualism" of these self-labeled "Chinese-style anarchists" led to lazy, dissipated, unlawful, libertine behavior that would only result in people "taking vows, going mad, and committing suicide." Chen charged that those "Chinese-style anarchists" who opposed centralized state authority as not fitting China's national character were only too similar to Yuan Shikai and other strongmen who called for a new type of government fitting the Chinese national character, and thus anarchists would only help China achieve such reactionary authoritarian rule. The "nihilists" among the Chinese anarchists, Chen contended, were only "low grade anarchists" with no principles at all, who included "parliamentarians, bureaucrats, opium addicts, jailers, thieves, and charlatans." Since, in response, Western-influenced Chinese anarchists were quick to deny that they were passive nihilists and escapists and that anarchism was quite capable of organizing and leading a mass movement to build a modern society, they felt the need to distance themselves from Daoism.
As we have seen throughout the previous chapters, by no means does all Daoist anarchism have to lead to individualist nihilism or escapism. Only when such Daoist-influenced thinkers came to reject the existence of a unified whole, we concluded, did they fall prey to nihilism and acquiescence with state power, as in the second and third generations of _qingtan_ intellectuals at the end of the Han dynasty and Wu Nengzi during the chaos of the mid-ninth century CE. Certainly in the current "post-modernist" age where faith in the ability of science and technology to solve all of the world's problems has reached a low ebb, one does not have to automatically denigrate premodern political thought as obsolescent or irrelevant. Within the argument presented in the first part of this book, one could attribute Liu's conservative turn not to his Daoist anarchism per se but to his own lack of confidence in Daoist principles of a spontaneous order underlying the whole.
Furthermore, all kinds of anarchists could and did take the path of collaborating with various authoritarian governments, including even modern Chinese anarchists who rejected Daoism and claimed to embrace materialism and modern science, such as Wu Zhihui, who ended up joining the Guomindang (Nationalist Party) government. It would be all too easy to join other Chinese anarchists and denounce such actions as opportunist and hypocritical, but in fact some anarchists often may take such actions out of a failure to recognize that the key point of anarchism is its view of state autonomy. Wu Zhihui seemed to have greatly valued his own experience in France on a work-study program and agreed to work with the Guomindang to the extent that it would back his schemes for expanding such programs in the future. In other words, he may have valued modern anarchism's stress on socioeconomic equality over its critique of the state, thereby gradually opening himself up to the path of acquiescence to state power. Despite any such sincere and/or principled reasons for compromise with authority, Chinese anarchists who acquiesced to various types of authoritarian rule, including especially anyone who claimed inspiration from Daoism such as Liu Shipei, helped Chen's argument and ultimately served to discredit anarchism in China, even if there was accommodationist behavior on all sides, including among some Chinese Marxists who, like many political actors, were certainly not immune to self-serving authoritarian actions. Of course the anarchists were quick to respond to criticism from Chen and other Marxists, which led to rather vigorous debates between the two types of revolutionaries in the early 1920s, debates we now examine.
**Looking Ahead: The Debates Between Anarchists and Marxists in the 1920s**
It is important to examine these debates, not only in order to understand Mao's possible negative influence from anarchism that we will examine in Chapter 6, but also in order to see why anarchism became such a pejorative label in the PRC, which will help provide the background for the denunciations of anarchism in the PRC examined in Chapter 7 and to see why Chinese Marxist thinkers using what this book will term "neo-anarchist" critiques of the state had to take pains to try to protect themselves from being denounced as anarchists, as we will see in Chapter 8 and . This debate also relates directly to this book's prime contention of the minimal essence of anarchism. After all, the Chinese anarchist response to Marxist–Leninism, when that rival doctrine entered China in the years following the Bolshevik revolution, followed the anarchist critique of Marxism elsewhere, which since the days of Bakunin has focused primarily on the main anarchist critique of the state.
The first Chinese anarchist to criticize Marxism in print seems to have been Huang Lingshuang, who wrote an article criticizing Marxism in 1919, before its Leninist variant entered China. Though praising Marx for some of his economic theories, he criticized him heavily on other grounds, most especially for the limits of his theory of the state. For Huang, not just the capitalist state, but any state "is organized solely for the protection of the privileges and property of the few," while the tyranny of the Marxist state in particular, following Kropotkin, if "endow[ed] with even more power such as control of the land, mines, railways, banks, insurance" will be even harsher and will provide no guarantee of a new Napoleon on Yuan Shikai arising.
The main debate between the rival anarchist and Marxist camps began in late 1920 and early 1921, around the time of the founding of the Chinese Communist Party (CCP), and for the most part remained civil in tone since there were still many anarchist or anarchist-influenced activists in the CCP ranks and both sides still had some hope of cooperation in organizing workers in the cities. First in secret criticism of anarchists in the new CCP journal the _Communist_ ( _Gongchandang_ ), writers argued that Communism was superior both in carrying out class struggle through centralized organization and in economic production through a centralist ( _jizhong_ ) approach. Without the use of state power, another writer argued, through a dictatorship of laborers it would be impossible to create socialism in a backward society such as China as well as to defend socialism against its enemies.
The open part of the debate began with a lecture that the CCP cofounder Chen Duxiu gave during his visit to the southern city of Guangzhou in January 1921, reprinted in the joint anarchist–Marxist journal _The Guangzhou Masses_ ( _Guangzhou Qunbao_ ) that then printed replies from Chen's former student, the anarchist Ou Shengbai. _Voice of the People_ ( _Minsheng_ )—Shifu's anarchist newspaper that was revived by his followers after his death—reprinted Ou's part of the debate, and the whole exchange was reprinted in the national magazine _New Youth_ ( _Xin Qingnian_ ) in August of the same year. Ou tried to revive the debate in 1922 when he sent an essay from abroad to an anarchist publication in China.
While rather restrained in tone as each side was still trying to convince the other to come to its side, the heart of this first part of the Marxist–anarchist debate in China was over the issue of individual freedom versus group life and whether or not coercion was a necessary part of social existence. Chen argued that given its stress on individual freedom and voluntary compliance, anarchism fundamentally lacked the capacity both to wage revolution successfully and to maintain power after the revolution. "Except for the individual who escapes from society, there is no absolute freedom [ _jiedui ziyou_ ] and no capacity to put anarchism into practice." Based on the success of the Bolshevik revolution, Chen argued that organized, centralized power was needed to overthrow imperialism, while anarchist reliance upon separate, atomized units of undisciplined men could not advance the revolution. Even if somehow anarchists could set up Kropotkin-style free federations of communes instead of Lenin's dictatorship of the proletariat, the capitalists would soon mount a comeback, thus, to Chen, explaining why anarchists were considered the good friends of capitalism. Anarchists were too optimistic about human nature and too pessimistic about all things political, and since some men were evil and reactionary and even good people could not be reached by education in the capitalist era, rule by virtue and education alone were unrealistic. Trying to rely on the public will as found in town hall-style meetings and voluntary associations would not work given the emotionalism of ignorant masses in the current corrupt, backward conditions of the Chinese people. Anarchism would also not be capable of building a modern economy but was based on romanticized notions of individualism and anti-industrial society that would return humans to primitivism and tribalism instead of building large-scale industry, for which centralized organization and control were necessary.
Ou's response was that anarchism was not based on rampant individualism but instead on voluntary association ( _lianhe_ ) through free contracts in which there would be an organic relationship between individual and society, where a more flexible "public will" ( _gongyi_ ) would help the group function, as opposed to coercive and unchanging public laws. Anarcho-communists in fact were not opposed to group life; instead they only opposed the despotism of the group over the individual. As Ou put it, showing great influence from Kropotkin,
We depend on society for our survival and the individual is a member of society;
thus in order to pursue individual liberty, we should first pursue society's liberty . . .
The individual liberty that ignores the common good is not liberty but rather the enemy of liberty.
Anarchists were not against violence in order to achieve revolution, Ou argued, but were only against institutionalized power and law that would inevitably result in new types of oppression. As opposed to free contract between individuals, rule by law only aided the interests of the ruling class and failed to prevent officials from robbing the people. Though there was much ignorance in modern society due to capitalism, with scientific progress the dominance of emotionalism would fade and people would become more rational in time. Ou did accept that certain "reactionary individuals" who did not respond to sincere argument could be controlled with ostracism or banishment from the community, the "same way we treat capitalists," a response that allowed Chen to reply that "public will" could be more akin to the despotism of the tribe over the individual in primitive society and to argue that contracts were in effect just another type of law that would be ineffective and meaningless without the backing of more clearly defined "abstract laws." Ou's allowing for some form of social coercion of the individual opened him up to Chen's rebuttal and perhaps shows the problem that critics of anarchism see in the social coercion that would remain in anarchist society. Despite this weakness, Ou did employ the main anarchist theory of the state to criticize Marxist socialism as "state collectivism." In this system, "with the state as the owner of the means of production and the workers as its laborers," ". . . the bureaucrats are the masters, the workers their slaves. Even though they advocate a state of the dictatorship of the workers, the rulers are bureaucrats who do not labor, while workers are the sole producers."
Other anarchists in _Minsheng_ also responded to Chen's critique within the basic anarchist theory of the state by arguing that when Marxists justified advocating a "people's dictatorship" in place of voluntary association supposedly because "human nature is not developed to its fullest" they demonstrate a "great contradiction," which presumably refers to the classic problem of how to control the controllers if all people cannot be trusted. These _Minsheng_ writers were in effect raising the question of what would stop the growth of a new state elite ruling for itself once one accepts the need for "temporary" dictatorship. The _Minsheng_ writers also took issue with the stress on class alone as the basis for revolution, noting that state authorities often used the power of religion in past eras and nationalism in the contemporary era as ways to get the masses to fight each other. Another _Minsheng_ writer argued that the concept of the dictatorship of the proletariat was so vague that it lost all meaning. It would make as much sense as to call for all women of the world to unite to overthrow the rule of men and replace it with a dictatorship of women. "If you say this is a ridiculous approach, the Marxist method is the same except that what it proposes is even more remote . . . what we must remember is that if we wish to save society from perishing, we cannot use methods that are doomed!" Again, this "doomed" method presumably refers to the basic anarchist idea of the state as a parasite on society that will destroy its host and thus itself in the long run.
As Arif Dirlik notes, the main differences in the Chen–Ou debate were that Chen "believed that individual rights must be sacrificed to the interests of the group" and that the revolution had to be achieved through coercion, while Ou believed that the revolution could be achieved through education and that to use coercion would "nip in its bud the promise of a good society." In what in retrospect was obviously a very ill omen for the future, Chen argued that while laborers should have the right to strike under capitalism, since all production was for the equal benefit of all members of society in communist society there would be no need to strike as that would be the equivalent of workers striking against themselves. This belief goes to the heart of the Marxist failure to see the difference between state and society after the revolution, where the state indeed could act against the interests of the proletariat. This failure in turn reveals why the twentieth-century Chinese anarchists, like their anarchist compatriots elsewhere, distrusted the Marxist–Leninist emphasis on economic class alone as the basis for revolution.
The debate became more heated after 1922 when, influenced by Emma Goldman, many Chinese anarchists—some of whom had met her and/or corresponded with her personally and most of whom were aware of her writings criticizing Bolshevism after her departure from the Soviet Union—stepped up their criticism of their Marxist rivals. Chinese anarchists were also influenced by criticisms of the Bolshevik revolution by Peter Kropotkin's widow and by the Georgian anarchist and associate of Kropotkin Varlaam Cherkezov, from whom they learned of the suppression of the anarchists in the Soviet Union and especially of the brutal suppression of the Kronstadt uprising, knowledge which sharpened their polemics and focused their criticism more on the Marxist–Leninist concept of the dictatorship of the proletariat. In addition, the alliance of the young CCP with the Nationalists, which began in 1923 and was formalized in the "First United Front," gave anarchists ammunition to attack "Bolshevism" in China.
Some Chinese anarchists, following Ou Shengbai's earlier argument, attacked Bolsehvik socialism as "state collectivism" that would not achieve true communism but only replace individual capitalist ownership with state ownership. Others, following Cherkezov, attacked the "Jacobinist" tradition within Marxist socialism that similarly failed to break with the methods of bourgeois politics that Lenin revived.
The Chinese anarchists' biggest complaint about Soviet-style socialism was over the concept of the dictatorship of the proletariat. Huang Lingshuang, in a letter from the United States published in 1923, noted that Kropotkin's widow had told him of Kropotkin's view before his death that Bolshevism was not true socialism since true socialism could not be built upon centralized state power, reinforcing Huang's conclusion that the "'dictatorship of the proletariat' was only a mask for a dictatorship of intellectuals in the Communist party." Similarly, another Chinese anarchist writer debating the leader of the Chinese communist students in Paris and future PRC Premier, Zhou Enlai, argued that the dictatorship of the proletariat, given the centralized method of organization of Marxist–Leninism, was in reality nothing but "a dictatorship of leaders of the Communist party." As the Sichuan anarchist Lu Jianbo summarized the argument, "facts tell us: the inner lining of the dictatorship of the proletariat is the dictatorship of a single party—the Leninist party. The Soviets have already been captured by bureaucratic socialists."
Another young Sichuan anarchist, writing under his given name [Li] Feigan, who was to translate several classic works of Western anarchism into Chinese and who would become a world famous novelist under his pen name Ba Jin (formed from parts of the transliterated names of Bakunin and Kropotkin), continued Jianbo's critique of the Marxist concept of the dictatorship of the proletariat in one of several articles he himself wrote and which were critical of Bolshevism. Ba Jin took up the core anarchist argument that one group can never rule in the name of another. As he wrote,
the bourgeoisie toppled the feudal regime and seized political power, after which this nearly created an autocratic system [ _ducaizhi_ ] controlled by a minority of the bourgeoisie. If it were truly the case that a dictatorship [ _zhuanzheng_ ] of a minority of the bourgeoisie could represent the interests of the collective bourgeoisie, how come within the bourgeoisie there still occur incidents of struggle for political power?
For this reason Marx's [dictatorship of the] proletariat is no different from what he calls the dictatorship of the bourgeoisie. That is to say, it's a minority dictatorship. A true dictatorship of the proletariat is impossible to create. Truly, what the Russians have done [quoting Jianbo] is to hang out "the sign of the dictatorship of the proletariat, but the substance is still a dictatorship by a minority of Communists. The real workers still live in the state of slavery. The interests of the proletariat cannot be represented by the Communist Party . . ."
. . . Therefore [Jianbo] says that the Communists really are nothing more than a so-called bourgeoisie and that which the Communists call a proletariat are nothing more than a slight mutation on the dictatorship of the bourgeoisie.
As Ba summarized his own argument,
if we recognize that one class oppressing another class is not correct and that this is sufficient to harm the happiness of humanity and impede humanity's progress, then we ought to oppose the dictatorship of the proletariat. The bourgeoisie used their political authority to oppress the proletariat and that was wrong, but should the proletariat rise up and oppress the bourgeoisie and commit the same offense? "If a majority of people direct a minority of people then they themselves become perpetrators of violence; they themselves become oppressors [and] they negate other people's rights." These are the words of A[lbert] Parsons, who was from the Chicago workers' movement, which he said in court [in his trial for the Haymarket bombings].
. . . The social revolution of the proletariat is a revolution liberating the proletariat. It's a revolution that topples control of people by others. Now if in the first step you seize political power, then you become one who controls other people and you put yourself in the position of one who ought to be overthrown. Would one then have the gall to come forward and work for revolution?
In other words, following Bakunin's criticism of Marx, Ba here implies that the workers in the post-revolutionary State would quickly become ex-workers who would betray the revolution. As we will see in Chapter 7, Ba Jin, who remained in China after the communist revolution, would later suffer several rounds of denunciation up to 1949 for his loyalty to the cause of anarchism.
The early twentieth-century Chinese anarchist movement, like the anarchist movement internationally, started to lose out to the communists in the later 1920s and 1930s not only due to its famous problems in organization, which the Chinese anarchists increasingly bemoaned themselves, but also due to its failure to make peace with nationalism and the desire of most Chinese revolutionaries to build up a strong modern economy, which many at the time identified with centralized, hierarchical organization. Nevertheless, given its earlier dominance over Marxism among radical intellectuals and trade union activists up to the 1920s, the Chinese anarchist movement would have profound effect on the Chinese communist movement, many of whose members were originally anarchists, including Mao Zedong himself, as we will see in the next chapter. Most importantly, the anarchist critique of the state, especially in its Leninist form, would continue to haunt and taunt the Chinese communists after 1949. In what could serve both as a companion to Bao Jingyan's denunciation of the tyrannical crimes of rulers divorced from the common people and as a prediction of the bloody course of communism in the PRC after 1949, one Chinese anarchist writing in 1923 argued that the Bolshevik emphasis on seizing political power led,
. . . those who consider themselves extraordinary in a period of brutality to arouse the ignorant masses to do battle for them; and when the struggle is over, they use the educated to devise a set of laws to bind the people, and train police and soldiers to massacre them. Ah! Power, power! People who have died cruel deaths throughout history, and the poor with their existence as beasts of burden, all have received your labor
Thus, even at the moment it started to lose out to the communists, the Chinese anarchist movement expressed most clearly the most powerful part of the anarchist idea, the fear that the state, even in a revolutionary movement whose original goal was to liberate the people, would inevitably start to rule for itself and thereby to oppress the people.
**Notes**
**** For a large, if necessarily still limited selection of the voluminous PRC studies of the Chinese anarchist movement, see Rapp, "Chinese Works on Anarchism in the People's Republic of China, 1949–2010," in Ruth Kinna (ed.), _The Continuum Companion to Anarchism_. The leading English language monographs on the twentieth century Chinese anarchist movement consulted in this brief overview include Robert A. Scalapino and George T. Yu, _The Chinese Anarchist Movement_ ; Peter Zarrow, _Anarchism and Chinese Political Culture_ ; Arif Dirlik, _Anarchism in the Chinese Revolution_ ; Edward Krebs, _Shifu: Soul of Chinese Anarchism_ ; and Yang Fang-yen, "Nation, People, Anarchy: Liu Shih-p'ei and the Crisis of Order in Modern China." Another comprehensive Western language source (German) is Gotelind Müller, _China, Kropotkin und der Anarchismus._
**** Zhen (pseudonym for Li Shizeng), "Da Chee shi" (Response to Mr. Chee), 2, cited in Dirlik, 111–12.
**** Ibid., 10, cited in Zarrow, 181.
**** Wu Zhihui, "Tuigang renshu yi yi shijie guan" (Extending the Way of Humanity to Cure the World), 148, cited in Zarrow, 164.
****Krebs, 255, n. 27.
**** Chang Hao, _Chinese Intellectuals in Crisis_ , 167–70, cited in Yang, 270.
**** Liu Shipei (under pseud. Shenshu), "Renlei junli shuo" (On the Equal Ability of Human Beings), 375–83, cited in Yang, 291, and Dirlik, 101–2.
**** Liu, "Baosheng xueshu fawei" (The Subtleties of Master Bao's Scholarship), cited in Zarrow, 166–7.
**** Yang, 296–300.
**** Zarrow, 95–6; Dirlik, 102–3.
**** Yang, 287.
**** Ibid., 311–12, 320–1. Overall, Yang carries out the most penetrating analysis of Liu's "conservative turn" and how it was prefigured in his anarchism. See Yang, 294–341.
**** Ibid., 312–13. Jing Meijiu, the "sole personal link" between Liu Shipei's Tokyo group of Chinese anarchists and later anarchists in China proper during the early Republican period, and who would become a prominent anarchist writer and editor in the 1920s, in a 1912 lecture, perhaps influenced by Liu Shipei, expressed sympathy for "utopian counter traditions" in China's past that pointed to an egalitarian agrarian ideal, and at one point even contemplated writing a short book (a project evidently never realized) that would "synthesize anarchism and the theories of Lao Zi." For Jing's link to Liu Shipei's group, see Gotelind Müller and Gregor Benton, "Esperanto," 107; for Jing's unrealized project on Lao Zi, see his "Zuian" (Account of Crimes) in _Xinhai geming ziliao leipian_ (Collection of Materials on the 1911 Revolution), 74, cited in Dirlik, 119–20. In the supplement _Xuehui_ (Sea of Learning) to the newspaper _Guofengribao_ (National Customs Daily) that Jing edited in the 1920s, the pseudonymous author Wuxu wrote one other article that referred to ancient precursors of anarchism, titled "Zhongguo gudai wuzhengfu zhuyi chao zhi yipie" (A Brief Look at Anarchist Currents in Ancient China), an article this author has been unable to locate but which is cited in Muller and Benton, "Esperanto," and in Muller, 491, n. 3.
**** Chen, "Zhongguo shide wuzhengfu zhuyi" (Chinese-Style Anarchism), _Xin Qingnian_ (New Youth) 9(1) (May 1921): 5–6, as summarized in Krebs, _Shifu,_ 177; also see Zarrow, 226.
**** Ibid., 177–8; also as summarized in Zarrow, 226.
**** Chen, "Xiapin de wuzhengfudang" (Inferior-grade Anarchists), 119–21, as summarized in Zarrow, 227; also cited in Krebs, _Shifu_ , 177–8.
**** For example, see [Li] Feigan, "Wuzhengfu zhuyi yu shiji wenti" (Anarchism and Practical Problems), reprinted in Ge Maochun et al. (eds), _Wuzhengfu zhuyi sixiang ziliaoxuan_ (Selection of Materials on Anarchist Thought), 830–8.
**** For the move of some Chinese anarchists to the Guomindang and to the right in general, see Dirlik, _Anarchism in the Chinese Revolution_ , Chapter 7, "Revolution that Never Was: Anarchism in the Guomindang," 248–85; Zarrow, 196–208, and Ming K. Chan and Arif Dirlik, _Schools into Fields and Factories: Anarchists, the Guomindang, and the Labor University in Shanghai, 1927–1932._
**** Huang, "Makesi xueshuo de piping" (A Critique of Marxist Theory), reprinted in Gao Jun et al. (eds), _Wuzhengfui zhuyi zai Zhongguo_ (Anarchism in China), 295–300. A partial translation of this essay can be found in Graham, _Anarchism: A Documentary History of Libertarian Ideas I_ : 355–7.
****Translated in Graham, 356; also see Krebs, "The Chinese Anarchist Critique of Bolshevism during the 1920s," in Roger B. Jeans (ed.), _Roads Not Taken: The Struggle of Opposition Parties in Twentieth Century China_ , 207.
**** For the internal communist criticism of anarchism, see Dirlik, 207–14.
**** The original exchanges between Chen and Ou were republished in "Taolun wuzhengfu zhuyi," _Xin Qingnian_ , 9(4) (August 1921) and reprinted again in Editorial Department, Xinqingnianshe (New Youth Society), _Shehui zhuyi taolun ji_ (Collection of Discussions on Socialism), 97–154. Ou's last rejoinder, "Da Chen Duxiu junde yiwen" (Responding to Chen Duxiu's Doubts) was published in _Xuehui_ (Sea of Learning) (Feb. 1923) and reprinted in Ge Maochun, Jiang Jun, and Li Xingzhi (eds), _Wuzhengfu zhuyi sixiang ziliao xuan_ (Selection of Materials on Anarchist Thought), 2: 658. English language summaries of the Chen-Ou debate can be found in Krebs, _Shifu_ , 175–8; "The Chinese Anarchist Critique of Bolshevism," 209–13, Zarrow, 228–9; Dirlik, _Anarchism in the Chinese Revolution_ , 213–19; _The Origins of Chinese Communism_ , 234–45; and Scalapino and Yu, 55–9.
**** Chen, "Taolun wuzhengfu zhuyi," 5, translated in Zarrow, 229.
**** Chen, "Speaking on Politics," in _Shehui zhuyi taolun ji_ , 1–16, as summarized in Scalapino and Yu, 55–6.
**** Chen, "Criticism of Socialism," in _Shehui zhuyi taolun ji_ , 74–96, and "Another Answer by Chen Duxiu to Ou Shengbai," in _Shehui zhuyi taolun ji_ , 119.
**** Ibid., 125, and "Chen Duxiu's Third Reply to Ou Shengbai," in _Shehui zhuyi taolun ji_., 137–8.
**** Ibid., 140–1, also cited in Zarrow, 229.
**** Ou, in _Shehui zhuyi taolun ji_ , 97–101, cited in Dirlik, _Anarchism in the Chinese Revolution_ , 215.
**** Ou, in "Taolun wuzhengfu zhuyi," 7, quoted in Zarrow, 229.
**** "Ou Shengbai's Answer to Chen Duxiu," in _Shehui zhuyi taolun ji_ , 118, and "Another Reply to Chen Duxiu," in _Shehui zhuyi taolun ji_ , 127–8.
**** Ibid., 119.
**** Ou, "Taolun wuzhengfu zhuyi," 18, quoted in Zarrow, 229.
**** Ou, "Da Chen Duxiu junde yiwen" (Answering Mr. Chen Duxiu's Doubts), _Xuehui_ (Feb. 1923), reprinted in Ge Maochun et al. (eds), 658, translated in Dirlik, _Anarchism in the Chinese Revolution_ , 224.
**** "Gao feinan wuzhengfu zhuyizhe" (Response to the Critics of Anarchism), and "Wuzhengfu gongchan pai yu jichan pai zhi qidian" (The Differences between Anarchist Communism and the Collectivists), _Minsheng_ (Voice of the People), 30 (March 1921), cited in Krebs, _Shifu_ , 178–9.
**** Ibid.
**** Krebs, 179, citing "'Jieji zhangzheng' he 'pingmin zhuangzheng' guo shiyong yu shehui geming ma?" (Are "Class Warfare" and "People's Dictatorship" Appropriate in Social Revolution?), _Minsheng_ , 33 (July 1921): 5; reprinted in Ge Maochun, et al. (eds), _Wuzhengfu zhuyi sixiang_ , 2: 587–90; also cited in Dirlik, _Anarchism in the Chinese_ Revolution, 228.
**** Dirlik, _Anarchism in the Chinese Revolution_ , 217, and _The Origins of Chinese Communism_ , 241, citing Chen in _Shehui zhuyi taolun ji_ , 149–51.
**** For the later part of the anarchist-Marxist debates in China, see Zarrow, 225–6, Dirlik, _Anarchism in the Chinese Revolution,_ 220–36, Krebs, _Shifu_ , 185–8; "The Chinese Anarchist Critique," 214–17. Dirlik (220, 224–30) notes at length the influence of Cherkezov on the Chinese anarchists in the early to mid-1920s. One article published at the end of the decade showing such influence was by the novelist Ba Jin under his given name [Li] Feigan, "Makesi zhuyi pipan Chaierkaisuofu zuo" (Criticism of Marxism in the Works of Cherkezov).
**** Dirlik, _Anarchism in the Chinese Revolution_ , 225–6.
**** Dirlik, 222–3, citing Huang, "Lingshuang zhi mojun han" (A Letter from Lingshuang), in _Chunlei yuekan_ (Spring Thunder Monthly), 110, 113.
**** Sanbo (pseudonym), "Iguo gongchan zhuyi shibaizhi yuanyin jiqi buqiude fangfa" (The Failure of Communism in Russia and the Way to Salvage It), reprinted in Ge Maochun et al. (eds), _Wuzhengfuzhui sixiang_ , 2: 598, cited in Dirlik, 223.
**** [Lu] Jianbo, "Lun wuchan jieji zhuanzheng" (On the Dictatorship of the Proletariat), 1, cited in Dirlik, 223.
**** Li Feigan (Ba Jin), "Zailun wuchan jieji zhuanzheng" (Further Discussion of the Dictatorship of the Proletariat), 1–2, translated for this chapter by Daniel Youd; also cited in Dirlik, 222–3. See also, John Rapp and Daniel Youd (guest eds), "Ba Jin and the Anarchist-Marxist Debates in China" (forthcoming), which will include four of Ba Jin's articles written in the 1920s critical of Marxism and two articles from the PRC criticizing his anarchism.
**** Bakunin, _Statism and Anarchy_ (1873), reprinted in Dolgoff, _Bakunin on Anarchy_ , 330–1.
**** A An (pseudonym for Anarchist A?), "Wo suo xinyang de geming" (The Revolution I Believe In), _Wuyi yuekan_ , reprinted in Gao Jun et al., _Wuzhengfu zhuyi zai Zhongguo_ , translated in Krebs, "The Chinese Anarchist Critique," 215.
PART TWO
Maoism and anarchism
6
Maoism and anarchism: An analysis of Mao Zedong's response to the anarchist critique of Marxism
**Introduction**
This chapter examines the possible influence of the basic anarchist critique of the state on the political thought and ruling practice of Mao Zedong. First, we will try to construct the best case possible for the populist, anti-statist Mao, including the argument that his early flirtation with anarchism left a lasting influence on his supposed attempt in his late years to prevent the emergence of a "new class" of power holders in the socialist state. Next, after delineating the inadequacies of this new class argument, we will try to construct an opposite case, which attempts to show the roots of Mao's autocratic practice in the statist, authoritarian side of his ideology that led to his ultimate failure to answer the anarchist critique of Marxism.
With the extensive revelations of the horrors of the Great Proletarian Cultural Revolution, few people today would any longer seriously consider Mao Zedong to be any kind of quasi-anarchist. Nevertheless, more than 35 years after Mao's death, views on the nature of the thought and rule of Mao Zedong are still diametrically opposed, both in China and the West. The difficulty in evaluating Mao's rule lies in the now seemingly blatant contradiction between Mao's words and deeds from 1949 until 1976. Thus, many Western observers in the late 1960s and 1970s, and even including some into the 1990s, viewed Mao as a genuine social revolutionary or perhaps a kind of semi-populist democrat. Such observers insist that whatever the failures of the Cultural Revolution in practice, given the ineglaitarian trend of the Deng Xiaoping years, the scholarly community should take seriously Mao's rhetoric about supporting mass rule, opposing the rise of a "new class" in the state, and favoring poor rural sectors over urban areas.
The defenders of Mao ask why, if Mao were the autocrat that his modern critics claim, he spent so much time fighting the bureaucrats and other officials of his own regime and why he used such populist and even, at times, anti-statist rhetoric. Mao's critics answer that the actual policies Mao tried to implement in fact led to widening gaps between elites and masses and to a highly repressive and murderous form of rule. In fact, one of the trends in China scholarship in the last 20 years is to view Mao—indeed, the whole Chinese Leninist regime itself—within the paradigm of neo-traditional or neo-feudal rule. Under such a view the PRC is often seen as little more than the continuation or restoration of imperial autocracy, and thus Mao himself as more like emperors of old than a true social revolutionary. Mao's defenders might reply that the case for Mao as autocrat ignores many aspects of Mao's thought and ruling practice that point in the direction of Mao as being a radical revolutionary. Below then, we first summarize these points in the case for the anti-statist Mao before refuting them and constructing the Mao as autocrat case, in both instances focusing on the relationship between Mao and anarchism. In the end, we will find that Mao could not accept the basic anarchist premise that the state rules for itself and thus that it cannot be checked from within.
**Purported Anti-Statist Elements in the Maoist Critique**
**Influence of rural origins**
The anti-statist case would begin with Mao's origins from a rich Hunan peasant family, origins that aided him in analyzing rural life in China and perhaps influenced his "heretical" ideas of the possibilities for peasant-based revolution from the late 1920s on, ideas that he argued against more Soviet-educated and urban-oriented Party cadres. Mao claimed the peasants were less corrupted by capitalism and were more susceptible to being reeducated with revolutionary ideas. He felt that an alliance of poor and middle peasants was capable of pushing for a genuine social revolution that would overthrow the "four systems of authority" in the countryside, which besides "state, clan, and theocratic" authority, included the patriarchal authority of husbands over wives.
Though it is true that after taking power Mao did rely on Soviet advice and "White area" and urban-oriented cadres in the early 1950s to support a Stalinist crash course of heavy industrialization through the establishment of a central planning apparatus, from the mid 1950s on, those seeing Mao as a genuine egalitarian socialist would claim that he returned to his roots in favoring rapid collectivization of agriculture and in rhetoric opposing a rural–urban gap. As early as the rural cooperativization movement of 1953, the rapid collectivization drive of 1955, and especially the People's Communes of the late 1950s, the anti-elitist case would point out that Mao tried to carry out a true social revolution in the countryside supposedly quite different from Stalin's violent, forced collectivization of the 1930s.
**Mao's supposed opposition to harsh punishment**
Though he revised his utopian faith in the peasants in the ensuing decades, the "mass line" policies of the Yanan era of the 1930s and 1940s still emphasized uniting with the majority against the minority. As part of this policy, the case for the egalitarian side of Mao might emphasize his constant injunction to minimize harsh punishment, if of course within his oft-noted injunction that revolution is not a dinner party but "an act of violence whereby one class overthrows another." Carrying over this officially lenient policy into the 1950s, especially during the "Hundred Flowers" movement of 1956, when it came to counterrevolution outside the Party, Mao argued,
. . . we must make fewer arrests and carry out fewer executions . . . But we should not declare that we shall never execute anyone. We cannot abolish the death penalty.
Nevertheless, when it came to "suppressing counter-revolutionaries" within the Party-state, Mao emphasized,
We must keep up the policy which we started in Yenan: "no executions and few arrests." There are some whom we do not execute, not because they have done nothing to deserve death, but because killing them would bring no advantage, whereas sparing their lives would. What harm is there in not executing people? Those amenable to labor reform should go and do labor reform, so that rubbish can be transformed into something useful. Besides, people's heads are not like leeks. When you cut them off, they will not grow again. If you cut off a head wrongly, there is no way of rectifying the mistake even if you wanted to.
**Mao on "Continuing the Revolution" against corrupt officials**
Perhaps the largest influence of his early anarchist roots on his later career, the anti-elitist case for Mao would emphasize, was his view of the need for a "continuing revolution" even after the establishment of socialism. In the late 1950s, Mao began to argue that class contradictions existed even after the socialist revolution and that class struggle would have to be emphasized for some time to come in the transition to communism. Mao's ideas went through a transition from belief in the essential dying out of class struggle, to a view of remaining basically non-antagonistic contradictions, and finally, to severe and clearly antagonistic contradictions surviving under socialism—the latter first clearly appearing in 1962.
This location of class struggle as the "key link" or primary contradiction led to Mao's insistence that the "bourgeoisie" still existed as a major force after the socialist revolution, and furthermore, that it would try to find support and refuge in the Party-state apparatus itself. Eventually Mao termed this group the "forces inside the party pursuing the capitalist road." Mao claimed that these people derived their base of support from bourgeois remnants of the old society as well as "new bourgeois" elements that had sprung up within socialist society based on remaining inequalities, the inevitable side-effects of an epoch when distribution according to work was still practiced.
As part of this critique, Mao had very harsh words to say about high officials, as in the following statement during the Cultural Revolution:
They are conceited, complacent, and they aimlessly discuss politics. They do not grasp their work; they are subjective and one-sided; they are careless; they do not listen to people; they are truculent and arbitrary; they force others, they do not care about reality; they maintain blind control. This is authoritarian bureaucracy.
Their bureaucratic attitude is immense; they cannot have any direction; they are egotistic; they beat their gongs to blaze the way; they cause people to become afraid just by looking at them; they repeatedly hurl all kinds of abuse at people; their work style is crude; they do not treat people equally. This is the bureaucracy of the overlords. . . .
They seek pleasure and fear hardships; they engage in back door deals; one person becomes an official and the entire family benefits; one person reaches nirvana and all his close associates rise up to heaven; there are parties and gifts and presents . . . This is the bureaucracy for the exceptional.
In the mid-1960s, Mao called for continuing struggle against corrupt bureaucrats even after the triumph of the socialist revolution. The "continuing revolution" against this surviving class contradiction (which, again, Mao defined more often as an antagonistic one during the 1960s) became the cornerstone of Mao's mature thought and the basis for launching the Cultural Revolution. Those who see this doctrine as inherently anti-elitist would link Mao's view of corrupt and bureaucratic "new bourgeois" urban elements not just to his early anarchism but to his roots in the rural-based revolution of the 1930s when Mao's line of surrounding the cities from the countryside and the "mass line" of learning from the peasants was first formulated.
Mao's proposed remedies for the dangerous situation created by these "bourgeois elements" who had come to power in the early 1960s included calls for workers and peasants to engage in mass criticism against people in authority, self-criticism of the offenders, sending down all Party and state cadres to the countryside to engage in manual labor and learn from the masses, and worker and peasant "participation" in running the economy. Of course, actual peasant and worker involvement in workplace management and policy making was never stressed heavily by Mao.
Above all, Mao and his followers called for an ideological reeducation of masses through inculcation of revolutionary ideas, most especially through intensive study of his own writings. These writings, along with his own unofficial remarks from the Great Leap Forward (1958–60) to the Cultural Revolution, seemed to stress the need for organs of mass action to fight the growth of inequality in the socialist revolution. Again, in these writings Mao seemed to hark back to the "mass line" of the Yanan days and to call for decentralization of authority away from the Party-state in Beijing and the provinces toward more direct control by workers and peasants. Though ultimately disappointed in the generation of "revolutionary successors" for the violent havoc they wreaked in the Cultural Revolution, he still expressed hopes for a continuing revolution every 10 years or so that would prevent the growth of bureacratism.
**Possible positive influence of anarchism on Mao's "Anti-Statism"**
Such an anti-elitist picture of Mao might stress his early influence from the anti-statist ideas of anarchism—until 1919 the leading socialist movement among the working class and avant garde intellectuals. Arif Dirlik suggests that anarchism had a much wider influence on the May 4th Movement than scholars had previously accepted, citing PRC scholarship to support his case. Peter Zarrow has drawn the most explicit comparison between Maoism and anarchism, especially related to Mao's thought in the Great Leap Forward and Cultural Revolution. Other sources point out that anarchism retained much influence in study groups, cooperative ventures, and trade unions well after 1920, as was certainly the case in Hunan, among circles with which the young Mao was associated. Clearly, Mao's knowledge of both Marxism and anarchism was not very sophisticated in his early years; nevertheless, he did stress the anarchist idea of the importance of founding small unions of mixed classes of ordinary people at the grass roots level and building larger confederations from the bottom up, an idea directly opposite to the Marxist (not to mention Mao's later Leninist) notion of conquest of central political power. Whether or not Kropotkin's idea of "mutual aid" ( _huzhu_ ) as the cornerstone of a true social revolution was the direct linguistic source for the "mutual aid teams" of Yanan and the early 1950s, one could posit a continuing influence on Mao and other former anarchists among the Chinese Communists of anarchism's extreme populist doctrines of linking industry and agriculture, especially in the People's Communes of the Great Leap Forward.
Germaine Hoston sums up best the case for an anarchist influence on Mao and the CCP, as follows:
. . . the CCP sought to establish a state power that would engineer revolutionary change in Chinese society and to develop methods of leadership that would prevent that power from being institutionalized as the same sort of bureaucratic, intellectual, elite leadership remote from China's millions of common people that had characterized previous Chinese states.
. . . In practice, Mao's solution to the national question issued . . . in the triumph of statism, but it must be recognized that in both its theoretical formulation and in aspects of its actual practice it was highly anarchistic.
Thus perhaps the case for an anarchist-influenced Mao would emphasize that it was precisely in his doctrine of the "continuing revolution" that Mao attempted to integrate criticism of a new elite in the socialist state into Marxist theory. The possibility of "vested interest groups" arising in the transition period, at least partially based on a "new bureaucratic class" in socialist society, would follow more from his anarchist intellectual roots than from his Marxism. The continuing of class struggle after the socialist revolution, possibly even into unknown future communist stages of development, and the need for periodic shake-ups of power holders by the masses would then represent Mao's answer to the anarchist question first posed by Bakunin to Marx of how to prevent the rise of a new and worse ruling class in the "workers' state."
Even though this latter point seems to make the case most directly for Mao as anti-elitist based on his influence from anarchist doctrines, we will return later in this chapter to the anarchist critique of Marxism to find an opposite meaning from this anti-statist interpretation of Mao's thought. But to sum up the case for the anti-statist Mao, what Hoston, Dirlik, and others are suggesting is that given Mao's direct knowledge and influence from anarchism, perhaps he also could have inherited at this time the anarchist critique of the Marxist theory of the state. This could have occurred even as Mao began to turn away from anarchism in the early 1920s, since in that time he could not fail to hear the disputes between anarchists and communists that we examined in the previous chapter. While these debates probably firmed up Mao's increasingly negative view of anarchism, at the same time they could have forced him to deal with the serious anarchist criticisms of the inherent despotism embedded in Marxist–Leninist theory.
Before we turn to the opposite case for Mao as autocrat, we must examine one last point in favor of an anarchist-influenced Mao. Ironically, this point would utilize the criticism aimed at the radical Maoist leaders of the Cultural Revolution that occurred very shortly after Mao's death, criticism based on the 1920s anarchist-Marxist debates. As we will see in the next chapter, late in 1976 and into 1977, the Chinese Communist regime under Hua Guofeng criticized the coterie of Mao's personal followers, including Mao's wife Jiang Qing, who wanted to maintain the doctrine of the Continuing Revolution as the heart of Mao's late thought. In a double irony, this criticism of Mao's followers was carried out in a Maoist-style mass campaign orchestrated by the party-controlled media. Using a supposed quote of Mao, the Cultural Revolution leaders were criticized as the "Gang of Four," who, among other evil deeds, supposedly pushed anarchist ideas in order to subvert the socialist revolution. In one example of this part of the campaign, Engels' famous anti-anarchist tract, "On Authority" was cited to equate the Gang of Four with Bakunin as people who waved the anti-authority banner in reality only to seize power for themselves:
Like Bakunin, the "Gang of Four," while desperately trumpeting anarchism, also went all out to establish their own counter-revolutionary "authority." Bakunin resorted to all intrigues and conspiracies to oppose Marxism and split the first International, but in the end he went down in disgraceful defeat. The "Gang of Four" picked up Bakunin's rotten stuff, stirred up anarchism over a long time, opposed the revolutionary authority of the proletariat, and split our Party.
Obviously, to those who want to see a genuine anti-authoritarian spirit in the Maoism of the Cultural Revolution, this criticism of radical Maoists as anarchists by a regime that at the same time was bringing back bureaucrats purged in the Cultural Revolution and that gradually moved away from Mao's Cultural Revolution policies of attacking the "new bourgeoisie in the Party" would only seem to prove the point that Mao had a genuine anti-statist or quasi-anarchist side. This is nowhere more true for such defenders of Mao than in his supposed argument criticizing a "new class" in authority in the socialist state, one which had to be struggled against in a "continuing revolution" against authority. Thus, before turning to the case for Mao as autocrat, we must first examine and refute the argument that Mao adopted the anarchist critique in order to oppose a "new class" based on state power alone.
**Mao and the "New Class"**
Even if his policies failed to improve the lives of his subjects, and even if his proposed solution in the end failed to stem state despotism, some defenders of Mao would still contend that his theory of the socialist state, perhaps influenced by anarchism, was _intended_ as a counterweight to the elitist tendencies in Marxism. Such analysts would point to his criticism of the increasing bureaucratism of state cadres, whom Mao's defenders claim he referred to as a "new class" and thus demonstrate his populist leanings in theory.
In fact, however, Mao himself stopped well short of a genuine "new class" argument, terming the corrupt "new bourgeoisie" a "privileged stratum" or a "vested interest group." Thus, he always saw the enemy to be struggled against as either composed of the remnants of old bourgeois classes or as "new elements" based on the necessary evil of remaining income and other inequalities, not as a new elite based on unchecked state power. His attitude was similar not to anarcho-communist critiques of Marxism, but to the way the traditional Chinese autocrats such as the Ming founder Zhu Yuanzhang opposed corrupt bureaucrats not for building up an all-powerful state apparatus, but because their activities tended to restrict the emperor's personal control.
In sum, despite what some scholars claim, Mao never truly posited the existence of a "new class" in terms similar to Djilas, that is, a class based on political control, not private ownership, as other scholars have more recently concluded. Indeed, the defenders of Mao, aside from citing each other to back up their comparison of Mao to Djilas, rely on very thin evidence from Mao himself. They often use parts of quotations of Mao on the "new class" in socialist society while downplaying or glossing over other phrases in the same work (or even the same sentence) that clearly call this a "new bourgeoisie." One of the main works these scholars rely on for their comparison of Mao to Djilas is the September 13, 1963 _People's Daily_ and _Red Flag_ joint editorial "Is Yugoslavia a Socialist Country?" This editorial, apart from the question of Mao's authorship, clearly criticizes the managers of factories in Tito's Yugoslavia as part of the "new bureaucrat comprador _bourgeoisie_ " (emphasis added). The label of "bourgeoisie" could of course have a changed meaning; however, far from referring to a political class monopolizing privileges and power that would demand democratic controls on the state in response, the editorial links this new class to "capitalist" reforms that "abandoned unified economic planning by the state" and departed from Leninist orthodoxy mandating the "socialist planned economy." Tito's policies were criticized not for increasing the autonomy of the state but for "abolishing" the "monopoly of foreign trade by the state," specifically insisted upon by Stalin himself. By looking back to the Stalinist era in Yugoslavia before 1948 as the period of true socialism and rule by the dictatorship of the proletariat and by clearly labeling Yugoslavia a "dictatorship of the bourgeoisie" based on the "restoration of capitalism," the editorial clearly stops short of a critique of a "new class" based on new bureaucratic power alone. In other words, Mao (or leading Maoists in the propaganda apparatus) criticized Yugoslavia in 1963 for being insufficiently centralized on the Stalinist model, not for insufficient democratic checks on state authority or for an uncontrolled bureaucracy.
Another important text cited by scholars attempting to equate Mao and Djilas on the new class is the 1964 polemical article "On Khrushchev's Phoney Communism and Its Historical Lessons for the World" also published under the names of the Editorial Departments of _People's Daily_ and _Red Flag_ , and especially the section "The Soviet Privileged Stratum and the Revisionist Khrushchev Clique." Nearly identically with the article on Yugoslavia, however, this editorial's condemnation of the "privileged stratum" also clearly links "bureaucrats alienated from the masses" with "bourgeois and petty-bourgeois ideologies and force of habit" surviving from pre-revolutionary times and from outside capitalist circles. It also links this new class to "sabotage" of the "socialist planned economy" represented by Khrushchev's (limited to nonexistent, we now know) market reforms. Again, this polemic does not amount to a Djilas-like critique of the rise of a new class due to lack of popular control over a bureaucratically managed economy.
Perhaps the most extensive analysis of Mao's supposed Djilas-like new class argument is that of Richard Kraus, who claims that "Mao refashioned the concept of class into a tool with which to contest the accretion of privilege by a new class of dominant bureaucrats." Nevertheless, Kraus' analysis is the exception that proves the rule. At one point he claims to cite a quotation where Mao "toyed fleetingly" with the idea that "bureaucrats themselves form a class, with interests 'sharply antagonistic' to those of workers and peasants," yet in the very next footnote, where Kraus cites Mao's statement more fully, it is clear Mao refers to "bourgeois elements" who are "taking the capitalist road," and that Mao is not making a general anti-statist critique nor calling for democratization but instead is only asking for a greater reliance "on those cadres who are not hostile to the workers and are imbued with revolutionary spirit" [i.e. who do what Mao and his followers want].
Elsewhere in his work, besides citing his colleagues such as Joseph Esherick and the same sources they cite from Mao's writings and speeches from 1958 to 1975, Kraus bases his argument for the radical nature of the Maoist critique mostly on the 1975–6 writings of the radical Maoist leaders in the Party, Yao Wenyuan and Zhang Chunqiao, sources which in the end severely undermine his case. For example, Kraus does recognize that Mao and these high Party followers "had never so explicitly identified high-level bureaucrats as an antagonistic class [compared to extra-Party Maoists such as the group Shengwulian]," but he claims, charitably, that this was due to "Mao's political needs," that is, his fear of arousing the resistance of an entrenched bureaucracy. Thus, given such tactical considerations, Kraus admits that Mao limited his class critique to one based on "individual political behavior rather than as a system of collective political structural relationships . . ." Furthermore, while still trying to maintain his belief in the "radical" nature of Maoism and the "deradicalization" of the post-Mao period, Kraus at various points does recognize other limits of Mao and his personal followers, including stopping short of a full critique of inequality, aiding repression of genuine radical Maoists outside the Party, failing to construct a genuine program of political restructuring, and tendencies "to protect local cadres, directing class struggle [instead] against the higher salaried officials," all of which led Maoists "often [to behave] in ways similar to the conservative power-holders they replaced." This author would concur then, with Stuart Schram's assessment that Kraus, ultimately "errs . . . as does Esherick, in arguing that in his later years Mao defined class primarily in terms of the privileges, and the control of the means of production, derived by cadres from their relationship to the state."
Maurice Meisner, for his best proof of Mao's "new class argument," cites the late extra-Party Maoist group at Beijing University who wrote under the collective pseudonym "Ma Yanwen." This group did indeed go much further toward arguing for a bureaucratic class than Mao himself, but not clearly with his approval and, more importantly, still firmly within the bounds of a critique of the restoration of capitalism. Furthermore, following Edward Friedman's more extended analysis of Ma Yanwen's arguments, one can only conclude that this group is yet another exception that proves the rule that Maoism lacks a true new-class argument. Opposing the nascent political reformers of the Deng coalition who were to call for more democracy in the name of opposing feudalism, the Mao group made clear the limits to their own anti-bureaucratic critique:
. . . the class enemies [i.e., pro-market forces] absurdly claim that bureaucrats and bureaucratism are products of the proletarian state system itself. This is a slander of red political power, reckless, reactionary logic . . . Bureaucracy's poisonous roots are [actually in the old, capitalist soil of an earlier] exploitative system.
Indeed, the Maoist critique of capitalist restoration really has very little in common with Djilas' critique of a new class based on Party monopoly of control over the state-managed economy. Instead it is the anti-feudal critique of later Party democrats in China opposed to Mao's autocratic policies of the Cultural Revolution that really recalls Djilas, as Edward Friedman also pointed out, quoting the following analogy by Djilas:
[In Yugoslavia] top leaders of the oligarchy distribute state functions, and sometimes economic functions among party officials, just like the fiefs which the kings and barons used to grant to their faithful and deserving vassals . . . In the same way that the royal prerogative, the privileges of the feudal lords, and the feudal estates, became a stumbling block to free trade and industry, which were developing under feudalism, so the despotism of the oligarchy, and the party bureaucracy's privilege in the government and the economy, together with the static, absolutized property patterns provided a basis for all this, have put the brakes on modern transport, modern management, and modern technology, and even on the socially owned property that has developed under Communism.
In sum, by equating state ownership managed by the vanguard party with socialism and Party control over the state with proletarian democracy, Mao and his "radical" followers remained loyal to Stalinist concepts of Party management of industry and agriculture and failed to oppose the growth of state autonomy and despotism. We will examine this argument in more detail in later chapters, especially in Chapter 8 as part of our analysis of the debates between Democracy Wall extra-Party dissidents who revived a more genuine neo-anarchist critique, but in this chapter we need to examine further whether Chairman Mao himself really launched such a critique.
His defenders might argue that Mao, whatever his limitations, desired that this "new class," whether or not a "new bureaucratic class," be controlled and eliminated not only with the rectification processes described above, but by the restriction of "bourgeois right," that is, through narrowing of wage differentials and other material incentives. In practice, one could answer, Mao seldom attacked the system of _non-wage_ privileges of Party officials such as greater access to information, luxury goods, and publicly owned wealth in the form of automobiles, villas, etc, privileges which some of his self-professed radical followers enjoyed the fullest. Of Mao's self-professed anti-elitist followers, we know the most about the excesses of his wife Jiang Qing.
Even in Maoist theory, the vanguard, though needing periodic rectification, would be better able to absorb such privileges without corruption and would even require some such privileges in order to advance its leadership capability. It was not elite privilege itself that Mao and his followers sought to overcome, nor the principle of elite rule, but only their own lack of complete control over the levers of state power. Mao's theory of the "Continuing Revolution" then, including his attack on the "bourgeoisie in the party" and the measures to be taken against it, would not contradict his essentially autocratic and despotic rule under this view since an autocrat such as Mao Zedong or the Ming founder Zhu Yuanzhang could feel that his arbitrary power is threatened as much from other central elites as well as from mass action from below. Thus, this chapter argues, Mao departed from the key component of the anarchist critique by failing to see that state autonomy is not just based on surviving or renewed "bourgeois" ideas and remaining economic inequality and thus by failing to really challenge the real basis for autocracy, the interests of state power holders in gaining autonomy from their subjects.
**Autocratic Elements in Mao's Theory and Practice**
Given the limits of Mao's "new class" argument as outlined above, it is clear that Mao was no mass democrat, much less a quasi-anarchist. This should lead us to examine the arguments of those who view his rule from 1949 to 1976 as similar to that of Chinese imperial autocrats. The very man many in the West considered to be an anti-bureaucratic revolutionary seeking a third way to socialism many now view as nothing more than a corrupt emperor. Again, by using the lens of anarchism, perhaps we can see the limits to anti-statism in Mao's thought and ruling practice, and the authoritarian and even autocratic tendencies that lie deeper in his thought.
**Mao's "Anarchism" reexamined**
Ironically, it is with a reexamination of Mao's relation to anarchism that the case for Mao as despot should begin. Unlike many young anarchists, Mao found in the doctrine neither an ultimate basis for human community nor a lifelong personal creed. Perhaps similar to the utility traditional Chinese rebels turned emperors found in (religious) Daoist, Manichaean, or Buddhist millenarian doctrines, for Mao anarchism, or indeed any other set of theories explaining China's predicament, was only useful as it also helped arouse people against oppressors and, perhaps more importantly, in so far as it called for heroic new leaders capable of enlisting and organizing popular support.
Before Mao was exposed to anarchist, liberal, or Marxist ideas, he was largely self-educated in the tradition of the Chinese peasant rebels in novels such as the _Shuihu zhuan_ (The Water Margin). Though briefly exposed to the repression of a military government against unarmed civilians, Mao's anti-militarism fell far short of that of Kropotkin or Tolstoy, who from personal experience came to view the highly regimented and authoritarian military and prison life as the hidden basis of the state and of all political organization. In contrast, Mao often imagined himself a general or dreamed of a career as a military adventurer in his early adulthood. Friends stressed his qualities of military leadership, especially in his defense of the students at the Hunan Teachers' Training School against an attack by the forces of the provincial warlord government.
Under this thesis of Mao as romantic rebel, in seeing the heroic nature of military life he would seem ill-suited to embrace anarchistic doctrines for more than a temporary opposition to one form of state oppression, and then only in a brief and inadequate flirtation that failed to satisfy his romantic nature and desire to be a military/rebel hero.
Ultimately, of course, just as traditional Chinese rebels turned away from early anti-Confucian millenarian doctrines the closer they got to power, Mao eventually rejected anarchism on tactical grounds. Following the early Chinese Marxist intellectual and CCP cofounder Chen Duxiu, Mao ultimately saw anarchism as incapable of waging revolution and holding power against a well-organized opposition. Anarchism, Chen asserted, was simply too optimistic for the contemporary epoch in China, an epoch which demanded instead a much more centralized and tightly organized party capable of leading the downtrodden masses out of their backward condition. As indicated in letters to Cai Hosen in late 1920 and early 1921, Mao had embraced Marxism in rudimentary form and had rejected anarchism as impractical, and as incapable of forming strong organization to oppose the united landlord–bourgeoisie government. In these letters Mao viewed anarchism as more akin to liberalism, as in its supposedly optimistic view of the possibility of a peaceful transformation of society given the monopoly of the state over the organs of education, communication, and money, a monopoly indispensable to social transformation.
Here perhaps the "natural Leninism" in Mao's thought and personality first appeared, as noted by many scholars. This "natural Leninism" came well before Mao's mature Marxist–Leninist outlook led him to denounce anarchism in more orthodox terms as "petty-bourgeoisie ultra-leftist opportunism." Even when the supposedly democratic "mass line" of the Yanan period of the 1930s and 1940s was most dominant in practice in a time when the crisis of the Japanese occupation led to a more easily recognizable convergence of interests between rural peasants and their Communist Party leaders, there were still great limits to Yanan democracy and a growing internal reach of the state, as revealed in purges of critical intellectuals and the growth of the secret police apparatus under Kang Sheng.
In the Cultural Revolution, this lack of genuine anti-statist elements in his personality and thought led Mao to reject any proposals of his would-be followers that might result in real institutions of direct democracy. Though at first praising the spontaneous rebellion of some urban elements who followed his initial praise for Paris Commune-type models of direct democracy, Mao quickly reversed himself when he saw such movements as threatening:
. . . in reality there always have to be chiefs . . . Anarchy is detrimental to the interests of the people and against their wishes.
He called for absorbing such Commune-style movements into the "Revolutionary Committees" which supposedly combined the Red Guard leaders, military officers, and returned bureaucrats into joint leadership bodies. These committees, however, were viewed by some of his Red Guard followers at the time, and recognized by most scholars soon afterwards, as the beginning of the reinstitution of Party and state authority. Far from endorsing calls for "extensive democracy" on the model of the Paris Commune, Mao opposed the idea that elections could replace the Party or more importantly himself as the arbiter of proletarian interests. Even for those who find the requirement of elections as far from guaranteeing genuine rule by the people, Maoism in the end fell far short. As Andrew Walder puts it,
To emphasize the ubiquity of class forces, and to demand thereby _more intense_ loyalty to a "correct" doctrine, effectively precluded any serious attempt to undermine the privilege or arbitrary power of bureaucrats. To implement "mass democracy" under these conditions generated heightened ritual and deference and provided surviving bureaucrats with even more arbitrary power over the people under them.
In the final analysis, Mao reined in or destroyed any groups such as Shengwulian or the theorists around his former secretary and Cultural Revolution leader Chen Boda who refused to accept his limits to mass criticism or who continued to call for egalitarian democratic institutions based on the Paris Commune model and criticized a new "red bourgeoisie" in the Party. As Walder notes, these "dissident radicals" "bore the brunt of military repression, imprisonment, and execution and were choice targets in the military-directed campaigns in the years 1968 to 1970," that is, the years when Mao and the "orthodox radicals" beholden to him were dominant in the Party leadership.
**Mao's collectivization and self-reliance policies reexamined**
Quite apart from the authoritarianism shown in his subjective opinion of anarchism in his mature years, Mao's actions from 1955 onward demonstrated an increasingly autocratic nature. These actions included Mao's support for speed-ups in agricultural collectivization whose economic lunacy met increasingly with peasant resistance. Thus collectivization, including the People's Communes of the Great Leap Forward, could have had more to do with heightened state penetration of society that would increase Mao's personal power than with benefits to peasant life, perhaps similar to the _lijia_ system introduced by Zhu Yuanzhang—another peasant rebel who became supreme ruler of China as the founding emperor of the Ming dynasty in the 1380s. Reigning as Ming Taizu, Zhu used his work _The Placard of People's Instructions_ to directly intervene in the village affairs in the 1390s.
Franz Schurmann was perhaps the first Western scholar to note the similarity of the Maoist rural collectivization policies with the _tuntian_ , or military farms policy, as well as with the Ming dynasty _lijia_ and the Song and Qing dynasty _baojia_ rural mutual surveillance networks. Though Schurmann saw the Maoist policies as an extension of the social revolution begun with land reform, in comparing them with traditional imperial forms he emphasized the collectives and communes as an attempt to extend state penetration and control down to the village level and as attempt to militarize the peasantry. Indeed, as Schurmann noted, the Great Leap Forward began with public works projects that had a similar, if more permanent, effect toward militarization of the peasantry than that accomplished by corvée labor projects in imperial Chinese history.
Mao's collectivization policies reached a zenith of course in the People's Communes of the Great Leap Forward, in which local administration and local Party control would be fused in the _xiang_ or township level units. In the Great Leap Forward, agricultural and industrial pursuits were to be combined, with the expectation that as they were educated by cadres and by themselves in the works of Mao the naturally progressive poor peasants and other "good class elements" would create a revolutionary enthusiasm that would result in increased productivity in the fields and voluntary contributions to public works and other projects. There were similar types of expectations in the Ming dynasty about what a "good citizen" would be, revealed both in Zhu Yuanzhang's pronouncements in the _Great Warnings_ and in the _Placard_ , where he identified the "good people" ( _liangmin_ ) as the commoners of the village community and expected them to have a knowledge of his works and to take the lead in spreading his ideas.
In the Great Leap Forward (and also later in the Cultural Revolution, if with a less rural focus) Mao called for policies of "self-reliance," policies that to some Western observers seemed to demonstrate Mao's sincere desire to avoid the pitfalls of development tied to Soviet or Western domination. These policies, however, were in fact also more similar to Zhu Yuanzhang's isolationist trade policies of the early Ming, including extreme restrictions on foreign investment and trade. Similar to Zhu Yuanzhang's failed trade policies, it has now been revealed that the model communes of the Maoist era often had to be propped up with heavy state subsidies in order to keep their status as successful experiments. Furthermore, recent studies since 1979 have revealed that such policies of self-reliance and decentralization of authority to communes, whatever local empowerment and criticism of officials may have resulted, actually increased the arbitrary power of rural cadres.
**Mao's purges of civilian officials**
In a remarkable similarity to the violent reaction of Zhu Yuanzhang against his subordinates late in his career, so too as both Mao's policies of the mid 1950s and the growth of a new state elite led to dissatisfaction and as the Great Leap Forward policies led to mass famine and serious economic difficulty did Mao look for scapegoats.
Directly contradicting his "Hundred Flowers" policies of lenient punishment for counter-revolutionaries, Mao launched an "anti-rightist" purge in 1957 against those intellectuals who had dared to challenge the Party's authority, a campaign that led to imprisonment and death for hundreds of thousands. In 1959 Mao extended this harsh treatment to intellectual and party elites who had dared to criticize his policies. This included Marshall Peng Dehuai, who had politely suggested retrenchment of the Great Leap Forward in inner party councils. Mao accused Peng of expressing personal opposition to him and adamantly refused to rehabilitate him even in the retrenchment years that did follow.
Regarding the discrepancy between Mao's conciliatory words and harsh deeds, one must especially note the growth of secret police terror and the increased authority of Mao's personal clique of followers in the whole period from 1955 to 1976 when such forces built up their arbitrary power at the expense of inner Party collective leadership, not to mention at the expense of mass democracy. After the reversals of the Great Leap Forward policies forced Mao into the "second rank" and led to policies of limited market incentives under Chairman of State Liu Shaoqi and Party General Secretary Deng Xiaoping in the early 1960s, Mao launched the counterattack of the Cultural Revolution in the late 1960s. Mass campaigns of young Red Guards were launched to "bombard the headquarters" and "drag out" those in power "taking the capitalist road." Eventually, Liu and Deng and many of their followers in the Party-state apparatus were removed, with deaths and executions of many of those like Peng Dehuai who had first been purged in 1957 and 1959. The post of Chairman of State was abolished and many bureaucrats were sent down to the countryside to "learn from the masses." When this purge was accomplished, Mao called in the army to rein in the Red Guards and restore order, eventually proclaiming Lin Biao, the Defense Minister he had installed after the purge of Peng, his chosen successor.
**Mao's attitude toward punishment of intellectuals**
Though at first Mao seemed to be using intellectuals to help carry out his policies in the Yanan and land reform periods to the Hundred Flowers, from the Anti-Rightist Campaign on, if not earlier, Mao turned on intellectuals. Even if part of a sincere desire to mold a more egalitarian society (a premise we examine below in the context of his education polices), some speculate that his actions against intellectuals were also due perhaps to repressed feelings of jealousy from his humble birth and/or feelings of being slighted by intellectuals from his days as a lowly clerk at Beijing University library. In any event, Mao from 1957 on revived his "strongly held feelings carried over from the past that intellectuals were not to be trusted and could under some circumstances prove to be enormously dangerous."
Perhaps due more immediately to fears of a Hungarian-style uprising, Mao in the Anti-Rightist Campaign of 1957 turned on intellectuals with a vengeance, similar to traditional emperors' purges of scholar/officials, especially the violent purges of the founding Ming emperor late in his career. Mao went beyond his personal purge of the intellectual critic Hu Feng in 1955 to a more open and widespread movement in 1957 against anyone who had dared to challenge his authority. This attitude culminated in the Cultural Revolution, when intellectuals became known as the "stinking Ninth category" of elements opposed to the revolution and when Mao purged his more intellectual rivals in the Party.
It is hard to find honest accounts in Mao's speeches and written works of this change toward harsh treatment of intellectuals. Nevertheless, following Benjamin Schwartz, one can detect in Mao's statements of 1957 a more negative appraisal about intellectuals' inherent "bourgeois" nature and lack of sincere commitment to socialism. Although the original text of his speech "On the Correct Handling of Contradictions Among the People" (February 27, 1957) primarily emphasizes the conciliatory and relatively tolerant line of the Hundred Flowers, which drew upon the traditional Yanan emphasis on the desire for unity, tolerance of "non-antagonistic contradictions among the people," and "curing the illness to save the patient," Mao in this speech also noted the beginning of "antagonistic contradictions among the people." These contradictions included the "poisonous weeds" that had cropped up among the Hundred Flowers intellectual critics of 1956.
Perhaps foreshadowing the violent purges that would soon begin, and reflecting earlier harsh treatment from the Yanan rectification movement to the purge of Hu Feng in 1955, in this speech Mao did also suggest a danger from intellectuals:
. . . the intellectuals and student youth, as well, have made great progress, but [they] also have incorrect thought, evil winds, too; [there] have been some disturbances . . . Among our youth, among the intellectuals, self-remoulding needs to be furthered.
In spite of conciliatory language toward intellectuals, he made clear that he sympathized with party cadres being criticized by intellectuals and noted:
Sometimes in comparison with those of a low educational level, the intellectuals make the more severe mistakes.
Even while calling for expression of all points of view, as Merle Goldman has pointed out, in this speech Mao also clearly "would not tolerate the articulation of basic disagreement with the policy itself . . . All views were possible, except those who disagreed with Mao's." Moreover, while opposing "crude methods" of coercion against "bourgeois ideology" and while noting that even people like Hu Feng who was arrested for supposedly running a secret organization would be released some day, Mao sounded an ominous warning: ". . . Hu Feng's ideas have not perished yet; they still exist in many people's minds." Furthermore, as Goldman notes, in trying to reassure intellectuals that the Hundred Flowers would follow the "moderate" methods of the Yanan Rectification of the 1940s, which supposedly allowed intellectual criticism of bureaucratism, Mao in fact only managed to scare knowledgeable intellectuals who were aware of what really happened at Yanan. That is, they would remember those of their colleagues who were in fact arrested and even executed for their criticisms in 1944.
Indeed, though continuing some of the official conciliatory rhetoric into the Anti-Rightist Campaign of 1957, including the idea of not depriving bourgeois rightists of their civil rights "unless they act as secret agents or carry on sabotage," in a speech at the height of the movement Mao announced that,
. . . the contradiction between the people and the bourgeois Rightists, who oppose the Communist Party, the people, and socialism, is one between ourselves and the enemy, that is, an antagonistic, irreconcilable, life-and-death contradiction.
By retaining for himself in the name of the Party the power to define who the enemies were and what actions constituted "launching wild attacks" and forming "secret organizations," the following statement of Mao on punishment takes on fearful overtones despite statements elsewhere in the same speech on the need to maintain unity and limit punishment:
Counter-revolutionaries must be eliminated wherever found. Kill few, but on no account repeal the death penalty or grant any special pardon. Arrest and punish those persons who commit fresh crimes after having served prison terms. Punish the gangsters, hooligans, thieves, murderers, rapists, embezzlers, and other felons in our society who undermine public order and grossly violate the law; also punish those whom the public identifies as bad elements. At present, certain functionaries in the judicial and public security departments are neglecting their duties and allowing persons who should be arrested and punished to remain at large; this is wrong. Just as over-punishment is wrong, so is under-punishment, and these days the danger lies in the latter.
In sum, based on his purges of intellectual and civilian officials in 1957 and 1959 as well as in the Cultural Revolution, Mao did not show leniency in practice and in fact demonstrated a strong bias toward anti-intellectualism and harsh punishments.
**Mao's purges of military officials**
Beginning in 1970, under circumstances that are still unclear, the defense minister Lin Biao himself was either murdered in Beijing or died in a fiery plane crash in Outer Mongolia supposedly after attempting to escape from a foiled coup against Mao. After Lin's fall, the army commanders were reshuffled and purged, more of the party and state cadres were brought back, and Mao started to rely more heavily around new party members related to the clique of personal supporters around his wife Jiang Qing.
This clique of people personally dependent on Mao for legitimacy desperately tried to build up its authority in the 1970s, including attempting to augment its strength within the "People's Militia" in the years leading up to Mao's death. In the violent purges related to the fall of Lin Biao, the secret police apparatus led by Kang Sheng until his death in 1975, along with leaders of Mao's personal bodyguard, also gained authority. Mao was firmly in charge of a decimated central Party-state bureaucracy, but was forced to rely on a small network of personal followers to maintain his direct rule.
Such a picture of Mao's autocratic rule does not in itself demonstrate the limits in theory to his supposed anti-bureaucratic if not anti-statist strains of thought. After all, populism and despotism need not be considered polar opposites, just as others have noted the claim of certain leaders in the imperial period to "rule for the people." Nevertheless, to the extent that sympathetic observers in and outside of China, based on Mao's own words, considered his "populism" to contain at least partially anti-elitist or quasi-democratic elements, we can now see how mistaken this impression was. First in the methods of "rectification" to be used and, second, in the limits even in theory to the autonomy of the state controlled by the "new class," we can complete the picture of the autocratic nature of Mao's political thought, a picture contrary to the views of some of Mao's self-professed followers inside China during the Cultural Revolution.
**Mass participation reexamined**
First, one must examine the meaning of mass participation in Mao's understanding. Too often in the recent past, and still today, some scholars have admired Mao's "mass line" policies for supposedly attempting to overcome the over-centralization and bureaucratic tendencies of orthodox Marxist-Leninist thought and practice without also fully examining how in that "mass line" approach the wishes of the masses were to be determined and how their participation was to be implemented.
One of the few articles of Western scholarship in the 1970s to deal specifically with this question, even while partially justifying Mao's actions, was that of Phyllis Frakt. She found that Mao's "mass line," as in Lenin's vanguard theory, assumes that leaders have identical interests with the masses in the long run of history, and thus could carry out "virtual" representation if they developed the proper attitudes in dealing with the people. Those among "the People" (the particular progressive forces under any one epoch's primary antagonistic contradiction) who had incorrect beliefs and attitudes should be educated through criticism and self-criticism, but if they fell outside that category outright coercion was permissible. Mao did recognize and accept the necessity of contradictions between leaders and the led before full communism was achieved, but believed such differences could remain non-antagonistic as long as the quality of leadership was preserved. Similarly, Donald Munro made another rare recognition in the 1970s that the "weaknesses" of Maoism even in theory might pose a danger to its otherwise egalitarian "strengths." The Maoist distinction between moral persuasion and (Stalinist) compulsion could break down due to fallacious Maoist assumptions of the identity of long-term private and public interests, the malleability and short-term inferiority of individual interests to those of society and the state, and, most importantly, due to the Maoist belief in the ability of a few leaders to decide "not only what the people's true interests are but also what values they should adopt." The last weakness in Maoist theory, Munro notes, could especially undermine the egalitarian ideal:
The ability to formulate the constituents of a value consensus and then to serve as supreme teachers gives to those leaders a special social position. And this special position is inconsistent with the spirit of the very status egalitarianism that the leaders chose to foster.
Frakt claims that Mao's idea of mass participation followed an essentially Burkean pattern (to Frakt, minus the "natural" governing elite in Burke, though with greater hindsight one could deny even this difference). As with Burke, Mao would allow popular representation only in the perception stage—discovering the needs and grievances of the people—not in the later stages of policy formation, and only partially in the implementation or execution stage. In other words, differing with the liberal conception of public opinion as an essential aid in determining the national interest, Mao shared the "conservative" (following Munro, one could add, Confucian) view of an objective national interest immediately knowable only by properly educated leaders, and knowable by the masses only at a future stage of history.
It is in the preservation of the prerogatives of leaders to know what masses want that Frakt considers the real nature of Mao's "mass line" (though she recognizes the possibility that a "ritualization" of the rectification process might eventually occur). First, in rectification campaigns and movements, erring leaders would have their attitudes corrected by mass criticism, self-criticism, and being sent down to participate in rural and urban labor. Second, though quality of leadership is determined mostly by purifying attitudes, not change in government form, nevertheless, through the Revolutionary Committees at the workplace level, individual workers could aid in the purification process (though Frakt recognizes that in practice party members often dominated the proceedings).
Similarly to Frakt's comparison of Mao and Burke on leadership by "virtual representation," Germaine Hoston, writing after a fuller knowledge of the horrors of the Cultural Revolution could be attained, compares Mao's concept of mass democracy to that of Rousseau:
. . . Like Rousseau's _Social Contract_ [the mass line] has both democratic and nondemocratic elements. On the one hand, the mass line conception prescribed a revolutionary process relying on a certain faith in the simple wisdom of the common person engaged in the concrete practice of production and revolution, in opposition to the abstract theory of intellectual knowledge . . . At the same time, Mao's concept of leadership was highly elitist in its own way. The ideas of the masses were inherently "scattered" and "unsystematic," just as the citizens in Rousseau's polity could discern only partial interests and articulate "particular wills." Ordinary women and men required the leadership of a party of persons with true revolutionary consciousness that could discern the true interests of all Chinese, or at least, of Chinese with a proper class perspective.
Hoston further notes that since Maoists believed even the Party itself could become corrupted,
. . . the ultimate implication . . . was that the party required a visionary leader specially gifted to discern the Way. In arguing thus, Mao . . . was not prepared to relinquish for himself the traditional mantle of leadership worn by China's emperors even while he himself was suspicious of and threatened by his Soviet-educated rivals as he sought to consolidate his leadership of the party.
Thus, despite her belief in the "anarchist" side of Mao, through her insightful comparison of Mao's thought to that of Rousseau, Hoston leads herself to a contradictory conclusion:
. . . given the corruptive nature of political power and the need for the "particular wills" (to use Rousseau's term) of the people to be "reinterpreted" so that they accorded with what was best for all China, who was to determine when a new rectification campaign was necessary? Mao's solution seemed to require a sort of equivalent to Rousseau's [Great] legislator, yet more powerful, someone who was superhuman, whose wisdom transcended the normal bounds of class-based perspectives . . .
That Mao acted like a traditional emperor and allowed himself to be set up as a "superman" would lead some leaders of the Democracy Wall Movement in China to deny that Mao had any mass democratic tendencies at all, and indeed that in the end he was not a quasi-anarchist but a "feudal-fascist" dictator.
Even without the advantage either of the hindsight of later Western scholars or of having actually lived through the Cultural Revolution, which would have helped them realize more fully the real tendencies in practice toward "ritualization" and Party dominance of mass participation, Frakt's and Munro's earlier analyses show clearly the theoretical limits to Mao's mass line and what others would see as his anti-democratic tendencies. Participants were not to be involved in determining the national interest itself, for example, through genuine elections with real popular nomination procedures; rather, the vanguard essentially coopted peasant, worker, and student figures and anointed them as leaders of officially recognized groups. The definition of "the People" still lay within the control of the vanguard—those with sufficient proletarian consciousness—and thus the whole range of popular attitudes that would be allowed expression in individual and mass form lay within the discretion of the Party (i.e. in the Cultural Revolution, Mao and his cronies).
Certainly no body of law or other set of institutional checks was to be set up over the vanguard. In other words, to use Edward Friedman's phrase in a slightly different context, Mao's theory could be likened to favoring an "internal policing" of leaders instead of a "civilian review process using outsiders." Likewise, as Hoston notes, "in the absence of such institutions, authority is most easily exercised by none or by the arbitrary will of one, acting in the name of the state." As she elaborates further,
. . . In Mao as in Rousseau the priority of substantive virtue over institutional and procedural arrangements attached little importance to particular interests and the right of individuals to express and act on them politically. Unless a democratic culture could be created that would prize rights as much as obligations, the tension between the option for chaos versus the perils of institutionalized power, on the one hand, and the need for a strong and visionary leader, on the other, could well resolve itself (and would) in a larger measure of stateness and statism than China enjoyed in the prerevolutionary era.
In other words, Mao's entire party rectification process depended for its continuation on calls for mass participation from the Maoist vanguard itself (or from one person above the vanguard). Or, as Stuart Schram puts it, "although the people were consulted, the ultimate aim was to make them believe they wanted what the leader and the Party has decided was best for them." Thus, given these limits, in the end Mao's mass line meant greater and not lesser autocracy, and more despotism and not more democracy.
It is easy to see how these limits in theory could be directly related to the horrors in practice of the Cultural Revolution, with or without the subjective desire of Mao himself. Criticism and self-criticism within the bounds of correct theory was ultimately controlled by ideological authorities themselves, leading exactly to the "ritualization," if to an even greater extent, that Frakt feared. Learning correct ideas became rote memorization and shouting of slogans in mass unison. Individuals could be categorized virtually at will as "class enemies" outside the safe category of "non-antagonistic differences among the people," according to who was in power among the vanguard. Making more explicit Hoston's recognition of the "statism" of the late Mao era, most genuine mass participation was quickly silenced by verbal and physical acts of denunciation on the part of secret police and loyal Maoists in authority. More properly speaking, permissible mass participation was often limited to participation in terror, while real participants who tried to expose the despotism of the times were subject to repression and death. As Walder notes, the "radical" Maoism of the Cultural Revolution was not just violent and murderous in practice, but in its essence:
. . . If we place this radicalism in its proper perspective, we see it as a form of reactive extremism whose defining premises were descended directly from the rationale for Stalin's mass murders . . . what actually happened in China during the Cultural Revolution—the inquisitions, witch hunts, cruel and vindictive persecution of individuals, unprincipled and often incoherent factionalism—were inherent in the doctrine and mentality that inspired it.
**Maoist "Egalitarianism" in education policies**
But even if the political theory and methods of Maoism were in the end highly undemocratic, were not its goals at least highly egalitarian and populist? For example, what of Maoist attempts from the 1950s on to expand worker and peasant education? This would include most famously the part-work, part-study schools and the recommendation system added to the supposedly elitist examination system in order to insure that "good class elements" could gain entrance into universities. One would first have to point out more recent studies that demonstrate that China by the end of the Maoist era in fact did far worse on reducing the gap in literacy between urban and rural areas (and between men and women) than did other less developed countries. Second, regarding the expansion of the education system to include more individuals from worker and peasant backgrounds, one would have to point out the severe low quality of the education received during the Maoist era and, more importantly, the tendency to give good class labels not based on economic background but by a combination of birth and political attitude, defined at the height of Maoist periods as loyalty to Mao's thought as assessed by Maoist leaders. Furthermore, as in ancient China in periods when the recommendation system flourished over the examination system as a method of recruitment into the imperial bureaucracy, this often permitted the _restriction_ of social mobility rather than its expansion since the top state leaders who controlled the definition of moral criteria could also use the system to keep out potential rivals.
At the height of the Cultural Revolution of course, not just quality, but even educational quantity was affected as schools were closed down to allow youth to "bombard the headquarters" and "share revolutionary experiences." One could make the harsh assessment that faced with the choice of an educated peasantry and proletariat who might drift away from his policies or uneducated masses who would blindly follow the supreme leader, Mao, in the end, contradicted his claim to reverse the imperial policy and instead followed exactly what he had condemned:
It is to the advantage of despots to keep people ignorant; it is to our advantage to make them intelligent.
Although rural areas are still unable to fully join the rush toward economic development, one has to conclude that, as their material standards have also improved dramatically from the Mao era, despite Mao's pro-peasant rhetoric, so too rural peasants and urban workers have fared much better in terms of access to education in the reform era despite the growing inequalities and other severe problems. Ignorance in China, by both statistical measures and personal accounts of survivors of the Cultural Revolution, is much reduced from Mao's time, if still a great aid to a continuation of despotic rule.
While many who had previously admired the "anarchist" aspects to Mao's thought and/or the "democratic" component of the mass line have come to agree with much of the case for Mao as autocrat, other observers want to maintain a belief in the anti-elitist side of Mao. Such people would say one simple question remains: why would Mao attack his enemies in the party–state apparatus in a simple drive for power when he could have had all the power and influence he wanted simply by going along with the policies of his rivals? Given other ruling elites' need to rely on him for legitimacy, they would have been more than willing to preserve his prestige, as evidenced by the willingness in the early 1960s of most of the party elite to keep Mao's personal critics in jail or in internal exile when they otherwise forced him to retreat on the Great Leap Forward policies. Even given Mao's lack of expertise in fields such as economics that would have been given more emphasis in a regime following a development-oriented policy line, he still could have retained much influence and power. This can be demonstrated by the brief 1956 consensus in which Mao kept personal power while going along with the Chen Yun-Deng Xiaoping limited market reforms, and also by Deng Xiaoping's continued powerful influence within the Party even as he admitted his lack of economic expertise during the era of economic reform. Clearly one has to give Mao credit for following his subjective ideological desires to an extreme limit in practice, since he could have kept his position of authority in the party by a total shift toward a reform coalition. Once this new coalition had allowed enough improved economic well-being in society at large, Mao would also have added to his popular base of legitimacy. But such a stance would have forced him to rely more on intervening bureaucratic elites and lessened direct social control of the population at large, weakening his actual authority that would open up room for future challenges and limits. Certainly if he had accepted a more indirect leadership role mediated by the bureaucracy, Mao Zedong felt that he would then not be able to play the role of activist leader or "moral entrepreneur," a role forged in his early career as a rebel leader. Whatever his intentions, one has to admit the extremely irrational and destructive ends to which Mao's actions directly led from 1955 to 1976, ends that were far from anarchistic but instead only served to build up China's modern autocratic system.
**Notes**
**** For example, see Kalpana Misra, _From Post-Maoism to Post-Marxism: The Erosion of Official Ideology in Deng's China_ , _passim_.
**** For the most prominent and recent versions of this view, see Harrison E. Salisbury, _The New Emperors: China in the Era of Mao and Deng_ , and most spectacularly, Mao's personal physician, Li Zhisui, in his _The Private Life of Chairman Mao_. Another scholar who combines a view of Mao as similar to imperial rulers with the view of him as a genuine social revolutionary is Stuart Schram, who uses the metaphor of "modernizing despot." See Schram, "Mao Zedong a Hundred Years on: The Legacy of a Ruler," 125–43. Finally, see Andrew and Rapp, _Autocracy and China's Rebel Founding Emperors_ , _passim_ , which compares Mao with Zhu Yuanzhang, the founder of the Ming dynasty.
**** Mao Tse-tung, _Selected Works of Chairman Mao_ I: 21–59, especially 45–9.
**** Mao, "On the Ten Great Relationships" (April 25, 1956), Translated in Stuart Schram, _Chairman Mao Talks to the People: Talks and Letters: 1956–1971_ , 61–83.
**** Mao, "Comment at the Working Conference of the Central Committee at Beidaihe" (August 6, 1962), 28, and "Speech at the Tenth Plenary Session of the Eighth Central Committee of the CCP" (September 24, 1962), in Mao, _Mao Zedong sixiang wansui!_ (Long Live the Thought of Chairman Mao!), (1969): 430–6, translated in _Chinese Law and Government_ , 1(4) (Winter 1968–9): 85–93.
**** Mao, "Some Problems Currently Arising in the Course of the Rural Socialist Education Movement" ("The Twenty-three Articles"), translated in Richard Baum and Frederick Teiwes, _Ssu-ch'ing: The Socialist Education Movement 1962–66_ , 118–26.
**** Mao, "Chairman Mao Discusses Twenty Manifestations of Bureaucracy," 40–3, cited in Richard Kraus, _Classes and Class Conflict in Chinese Socialism_ , 74; also translated in Andrew and Rapp, 231–4.
**** Mao, "Draft Resolution of the Central Committee of the CCP on Some Problems in the Current Rural Work" ("The First Ten Points") (May 20, 1963), in Mao, _Mao Zedong zhuzo uandu_ (Selected Readings from the Writings of Mao Zedong), translated in Baum and Teiwes, _Ssu-ch'ing,_ 65.
**** Mao, "Comment on Criticism and Self-criticism," _People's Daily_ (December 22, 1967), cited in Jerome Ch'en, _Mao Papers_ , 150.
**** Mao, "Draft Resolution of the Central Committee of the CCP," translated in Baum and Teiwes, _Ssu-ch'ing,_ 65.
****Mao, "Constitution of the Anshan Iron and Steel Company" (January 22, 1960), reprinted in _Beijing Review_ , 13(16) (April 17, 1960): 3.
**** As noted, for example, by John Bryan Starr, _Continuing the Revolution: The Political Thought of Mao_ , 161–2; and Andrew Nathan, _Chinese Democracy_ , xii.
**** For example, see Mao, "Letter to Jiang Qing," partially translated in _China News Analysis_ (Hong Kong): 907 (1966): 7.
**** In his famous Yanan interview of 1935, Mao admitted to Edgar Snow that he had read many anarchist pamphlets in late 1918 and "favored many of its [anarchism's] proposals." See Edgar Snow, _Red Star Over China_ , 152. Other sources claim that Mao was in fact an anarchist or worked in anarchist organizations in late 1919 and early 1920 when he even considered founding an anarchist society and called himself an anarchist to his friends. See Robert Payne, _Mao Tse-tung, Ruler of Red China_ , 55 and Maurice Meisner, _Mao Zedong: A Political and Intellectual Portrait_ , 14–16. Jerome Ch'en, _Mao and the Chinese Revolution_ , 53, states that at this time Mao read books by Bakunin, Kropotkin, and Tolstoy, while by Angus McDonald's count nearly half the books in Mao's Changsha bookstore were anarchist tracts. See McDonald, _The Urban Origins of the Rural Revolution: Elites and Masses in Hunan Province, China, 1911–1927_ , 129.
**** Dirlik, _Anarchism in the Chinese Revolution_ , 30–2, 148–96, 294–304. In his discussion of the relation of anarchism and Maoism in this work, perhaps influenced by anarchism, Dirlik emphasizes more the authoritarian reality of Maoism as opposed to its anti-statist rhetoric (e.g. 299–300) than he does in his other works on Chinese Marxism, which to this author tend to overrate Mao's supposed egalitarianism and underplay the despotic aspects of his rule. Also see n. 66. Likewise, Maurice Meisner in his later works, including _Mao Zedong_ , though still praising Mao for laying the groundwork for later socioeconomic progress, does conclude that beginning with the Great Leap Forward, Mao's later years amounted to political tyranny. See Meisner, _Mao Zedong_ , Epilogue, 193–7. For this author's critiques of Dirlik and Meisner for downplaying Mao's tyranny in their earlier studies, including Dirlik and Maurice Meisner (eds), _Marxism and the Chinese Experience_ , see Rapp, Review of four books on Chinese Marxism, _Theory and Society_ , 21(4) (August 1992): 599–609 and Rapp, Review of Cheng (ed.), _Marxism and Capitalism in the People's Republic of China_ , 821–2.
**** Zarrow, 232–7.
**** McDonald, 134–5.
**** Mao, "The Great Union of the Popular Masses" (1919) in Mao, _Mao's Road to Power: Revolutionary Writings 1912–1949_ I: _The Pre-Marxist Period, 1912–1920_ , 378–89.
**** Schram, in his _The Thought of Mao Tse-tung_ , 20–1, notes that the original edition of Li Rui's biography of Mao, _Mao Zedong tongzhi di chu qi geming huodong_ (Comrade Mao Zedong's Early Revolutionary Activities), edited out the striking paragraph of Mao's 1919 article cited in the previous note, which praised Kropotkin's ideal of mutual aid ( _huzhu_ ) as more progressive than Marx's doctrines of violent revolution. Also see Brantly Womack, _The Foundations of Mao Zedong's Political Thought 1917–1935_ , 17–21, and McDonald, 104–5.
**** Dirlik, _Anarchism in the Chinese Revolution_ , 296.
**** Hoston, _The State, Identity, and the National Question in China and Japan_ , 363, 364. As we will see below, Hoston recognizes not just the supposedly anti-statist side of Mao but also the flaws in his anarchist tendencies. As she puts it, ". . . Mao's solution to the problem of limiting party and state power was at once anarchistic and potentially authoritarian" (395). Even this recognition overstates Mao's anarchism, as we will argue below, and underestimates the degree to which Mao's authoritarian side was dominant.
**** Zarrow, 234–7; Womack, 27, 211, n. 81.
**** Bakunin, quoted in Dolgoff, 325–38.
**** Hsu K'o-ch'eng (Xu Kocheng), " 'On Authority'—A Classical Document Criticizing Anarchism," _Tianjin Shiyuan Xuebao_ (Tianjin Normal College Journal) 2, translated in American Consulate General, Hong Kong, _Selections from People's Republic of China Magazines_ , 77(926) (May 23, 1977): 6. As we will see in the next chapter, the best study of PRC denunciations of anarchism up to the late 1970s is that of William Joseph, _The Critique of Ultra-Leftism in China_. See especially pages 128–44, 159–61, and 189–90 for his summary of attacks on anarchism in China during the later phases of the Cultural Revolution, especially in 1971–2 and 1976.
**** For the clearest statement of Mao's supposed "new class" argument, see Maurice Meisner, "Marx, Mao, and Deng on the Division of Labor in History," in Dirlik and Meisner (eds), _Marxism and the Chinese_ Experience, 79–116, especially 101–2. In his later work, _Mao Zedong_ , Meisner does moderate his earlier argument, recognizing that Mao "conflated the terms 'bureaucratic class' and 'bourgeoisie'" (173), and, "as the Cultural Revolution approached . . . realizing the political implications of the notion, drew back from publicly characterizing China's bureaucrats as a new class" (174).
**** Mao, "Reading Notes on the Soviet Text, _Political Economy_ ," original text 1961–2, reprinted in Mao, _Mao Zedong sixiang wansui!_ , translated in Moss Roberts (trans.), _A Critique of Soviet Economics_ , 63, 71.
**** See Andrew and Rapp, Part I, _passim_.
**** For examples of other scholars who make this crucial point, see Andrew Nathan, _Chinese Democracy_ , 73–4; Jonathan Unger, "Whither China?: Yang Xiguang, Red Capitalists, and the Social Turmoil of the Cultural Revolution," _Modern China_ , 17(1) (January 1991): 25; Andrew Walder, "Cultural Revolution Radicalism: Variations on a Stalinist Theme," Chapter 3 of William A. Joseph, Christine Wong, and David Zweig (eds), _New Perspectives on the Cultural Revolution_ , 52, 54; Svetozar Stojanovic, "Marxism and Democracy: The Ruling Class or the Dominant Class?" _Praxis International_ , 1(2) (July 1981), 160–70; and Schram, _The Thought of Mao Tse-tung_ , 182–4.
**** See Stephen Andors, "Mao and Marx: A Comment," in "Symposium on Mao and Marxism," _Modern China_ , 3(4) (1977): 432–3; Joseph Esherick, "On the 'Restoration of Capitalism': Mao and Marxist Theory," in "Symposium on Mao and Marxism," 64–5; and Young "Mao Zedong and the Class Struggle in Socialist Society," 56–61; 77, n. 120 and 121.
**** _People's Daily_ (26 September, 1963), reprinted in _Peking Review_ , 6(39) (September 26, 1963): 14–27; also translated in _The Polemic on the General Line of the International Communist Movement_ , and in Mao (uncertain authorship), _The Great Debate_ , 110–44).
**** Ibid., 124–5.
**** Ibid., 128–9.
**** Ibid., 135–8.
**** July 14, 1964, translated in Mao (uncertain authorship), _The Great Debate_ , 323–74.
****Ibid., 339.
**** Ibid., 341.
**** Kraus, 17.
**** Ibid., 76.
**** Ibid., 76, 208, nn. 40, 41, citing Mao as quoted by Hung, "Inner-Party Struggle and Party Development," 11.
**** Kraus, 149.
**** Ibid., 150.
**** Ibid., 147–9, 153–6, 160.
**** Schram, _The Thought of Mao Tse-tung_ , 183, n. 245.
**** Meisner, "The Ritualization of Utopia: Chinese Marxism in the Post-Mao Era," in Dirlik and Meisner (eds), 231.
**** Ma Yanwen (pseudonym), "The Bureaucratic Class and the Dictatorship of the Proletariat," 259–74.
**** Ma Yanwen, cited in Friedman, "The Societal Obstacle to China's Socialist Transition," in Victor Nee and David Mozingo (eds), _State and Society in Contemporary China_ , 159; see also Andrew and Rapp, 2000, 259–74.
**** Ibid., 169, citing Milovan Djilas, _The Unperfect Society: Beyond the New Class_ , 191–2.
**** Indeed, in light of the hardline reaction to the Tiananmen democracy movement from 1989 to the early 1990s, and perhaps revived in 1999 after the US bombing of the Chinese embassy in Serbia, the Maoist opposition to Khrushchev's idea of a "peaceful transition" to socialism expressed in "Is Yugoslavia a Socialist Country" and other 1960s editorials can now be seen to justify increased state repression, not anti-elitism. That is true to the extent that opposition to a "peaceful transition" was the inspiration for opposition to "peaceful evolution" to capitalism, a code phrase revived by hardliners in China from 1989 to the early 1990s, perhaps revived recently, in an attempt both to discredit the young democracy activists as tools of Western imperialism and to return China to its Stalinist "golden age" of the early 1960s. For the earlier invective against the "peaceful transition," see _People's Daily_ , 1964, translated in Mao (authorship uncertain), 1986, 282–322. Also see Editorial Departments of _People's Daily_ and _Red Flag,_ 1964, 323 and _passim_.
**** Ross Terrill, _The White-Boned Demon: A Biography of Madame Mao Zedong_ , especially 213–329 for her contradictory actions during the Cultural Revolution.
**** A point that even Kraus recognized when he concluded that "not only were Maoists concerned with mobilizing maximum support for dislodging their conservative opponents from power, but they were also eager to wield their own power effectively. The castigation of all high-level bureaucrats as an enemy class would embrace Maoists as well, a possibility which the Chairman found unacceptable . . .," which led him to make the statement cited above opposing "extreme anarchism." See Kraus, 150. Of course Kraus still believed that Maoists wanted to dislodge their opponents for the noble reason of saving socialism. Why Kraus and others should be so charitable to that faction of state leaders while conceding so little to factions who would increase individual economic (and for a purged minority within the reform faction) even political autonomy, blaming the latter for "deradicalizing" the revolution, is a question perhaps resolved only by facing up to ideological bias of the analysts in question, even if it is a more noble and less selfish bias than that of their ideological opponents. This is especially a question for analysts such as Kalpana Misra who argued for the "deradicalizing" thesis long after the horrors of the Cultural Revolution were well-known in her _From Post-Maoism to Post-Marxism: The Erosion of Official Ideology in Deng's China_ , _passim._
**** Schram, _The Thought of Mao Tse-tung_ , 54; Bauer, 408–10; Hoston, 387.
**** Ch'en, _Mao and the Chinese Revolution_ , 43; Li Rui, 52; Xiao San, _Mao Zedong tongzhi de qing shao nian shidai_ (Comrade Mao's Boyhood and Youth), 79–80; and Schram, _The Thought of Mao Tse-tung_ , 16.
**** Womack, 6, 10–13; Schram, 16.
**** Chen, "Xuwu de gerenzhuyi ji ren ziranzhuyi" (Nihilistic Individualism and Nature-worship," 107–9, cited in Zarrow, 226–7; also see Hoston, 201–2.
**** Schram, "From the Great Union of the Popular Masses to the 'Great Alliance'," _China Quarterly_ , 49 (January 1972): 95–6.
**** Mao, _Mao's Road to Power: Revolutionary Writings 1912–1949_ II: 7–11; 35–6; also see Schram, _The Political Thought of Mao Tse-tung,_ 296–8.
**** For example, see Schram, _The Political Thought of_ Mao, 104; "From the Great Union of the Popular Masses to the 'Great Alliance'," 95–6; Meisner, _Marxism, Maoism, & Utopianism: Eight Essays_, 76–117; and Zarrow, 235–6.
**** The same terms often used by Soviet authors to denounce Mao and the "military–bureaucratic dictatorship" of the Cultural Revolution. See for example, Vladimirov and Ryazantsev, _Mao Tse-tung: A Political Portrait_ , 22–40 and _passim_.
**** See for example, John Byron and Robert Pack, _The Claws of the Dragon: Kang Sheng, the Evil Genius Behind Mao—and His Legacy of Terror in People's China_.
**** As quoted in Ch'en, _Mao Papers_ , 152; also see Hoston, 395; Frederic Wakeman, Jr., _History and Will: Philosophical Perspectives of Mao Tse-tung's Thought_ , 313–4; Schram, _The Political Thought of Mao_ , 173; Zarrow, 236.
**** Even analysts such as Graham Young who would like to see democratic possibilities in Mao's thought recognize Mao's opposition to elections and thus his ultimate failure to "provide a viable substitute for the organizational role formerly provided by the Party." Young also recognizes that Mao ultimately rejected his radical followers who called for "extensive democracy," and in the end allowed his allies in the Party to call instead for relying on Mao Thought as the "sole locus of authority." See Young, "Mao Zedong and the Class Struggle in Socialist Society," 68. Nevertheless, Young, 60, does find that a critique of a new bureaucratic class within the Party was at least partially contained in Mao's thought, a contention this author will continue to dispute below.
**** Walder, "Cultural Revolution Radicalism: Variations on a Stalinist Theme," 61.
**** See Unger, "Whither China?: Yang Xiguang, Red Capitalists, and the Social Turmoil of The Cultural Revolution," for an updated study of the most famous dissident Cultural Revolution group who took Mao's anti-elitist rhetoric seriously and developed a more consistent new class line than Mao. We will examine official PRC denunciations of this group in Chapter 7 and the thought of this group itself in Chapter 8.
**** Walder, 59.
**** Selden, "Cooperation and Conflict: Cooperative and Collective Formation in China's Countryside," _passim_.
****Dirlik, in _Anarchism in the Chinese Revolution_ , 299 also seems to have come to the conclusion that the agricultural and other communes in Mao's China were not part of a quasi-anarchist, mass democratic experiment. Instead they served as "a means to social control, faster economic development, and the efficient exploitation of labor." Dirlik includes in this assessment not just the People's Communes of the Great Leap Forward but the early Paris Commune models of the Cultural Revolution as well. The latter especially, "under the guise of popular revolutionary control, perpetuated and enhanced the political penetration of society." Nevertheless, Dirlik still seems to be enamored of Mao's supposed egalitarian intent, as in his recent praise of Nick Knight's book _Rethinking Mao_ for its overall highly positive assessment of "the historical and theoretical significance of Mao's thought" versus "anti-socialist" critics and for giving "a reevaluation of Mao's policies" that might provide "inspiration in confronting the problems created by three decades of reform that turned its back on his revolutionary legacy," thus returning to the downplaying of Mao's despotic legacy and forgetting the key anarchist denial that one has to choose between revolution, freedom, and equality. See Knight, _Rethinking Mao_ , back cover. For a much harsher assessment of Knight as downplaying or minimizing Mao's autocratic and brutal rule, see Rapp, review of Nick Knight, 392–6.
**** The "Placard of People's Instructions" is found in Zhang Lu (compiler), _Huang Ming zhishu_ (Regulations on the imperial Ming) (1579), 1579, 1607–2205, 1896–7. For a more extensive comparison of Zhu Yuanzhang and Mao that elaborates on this point and others noted below and which includes translations of PRC articles on the Ming founder in the 1980s and 1990s that can be viewed as allegorical criticisms of the Chairman, see Andrew and Rapp, _Autocracy and China's Rebel Founding Emperors_.
**** Schurmann, 404–500. For the public works origins of the Great Leap Forward, see especially Schurmann, Chapter VII, "Villages," 479–80; for comparisons to the _lijia_ , _baojia_ , and _tuntian_ see especially 409–12 and 494–5, and the Supplement, Chapter II: "Organization", 532–75, 559. The great student of the Great Leap Forward, Roderick MacFarquhar, has also compared aspects of the movement to corvee labor projects in imperial China. See MacFarquhar, "The Secret Speeches of Chairman Mao," 15. Franz Schurmann concluded that the phases of rural cooperativization, collectivization, and communization beyond the original "land to the tiller" land reform amounted ultimately to a failed social revolution, a judgment with which this author disagrees; instead with many contemporary scholars, this author considers the social revolution to have ended with land reform.
**** Andrew and Rapp, 65–8.
**** See Friedman et al. (eds), _Chinese Village, Socialist State_ , _passim_.
**** Thomas P. Bernstein, "The Limits of Rural Political Reform," in Victor Falkenheim (ed.), _Chinese Politics from Mao to Deng_ , 303.
**** For the most famous account of the Great Leap and its purges, see Roderick MacFarquhar, _The Origins of the Cultural Revolution: The Great Leap Forward 1958–1960_.
**** Friedman, "Maoism, Titoism, Stalinism: Some Origins and Consequences of the Maoist Theory of the Socialist Transition," in Selden and Lippitt (eds), _The Transition to Socialism in China_ , 159–214; Schram, "Foundations and Significance of Mao Zedong's Personal Power," in Schram (ed.), _Foundations and Limits of State Power in China_ , 211–24.
**** Of the many books on the Cultural Revolution, two stand out for their details on the "politics of the court" and purges of high officials. See Roderick MacFarquhar and Michael Schoenhals, _Mao's Last Revolution_ , and Frederick Teiwes and Warren Sun, _The End of the Maoist Era: Chinese Politics during the Twilight of the Cultural Revolution, 1972–1976_.
**** Benjamin I. Schwartz, "Thoughts on the Late Mao—Between Total Redemption and Utter Frustration," in MacFarquhar et al. (eds), _The Secret Speeches of Chairman Mao_ , 27.
**** See MacFarquhar, _The Origins of the Cultural Revolution: The Great Leap Forward 1958–1960._
**** Ibid., 28–9.
**** Mao, "On the Correct Handling of Contradictions among the People (Speaking Notes)" (27 February 1957), in MacFarquhar et al. (eds), _The Secret Speeches of Chairman Mao_ , 156.
**** Ibid., 177.
**** Goldman, "Mao's Obsession with The Political Role of Literature and the Intellectuals," in MacFarquhar et al. (eds), 53.
**** Mao, "Talk at a Conference of Party Member Cadres in Tianjin Municipality" (17 March 1957), in MacFarquhar et al. (eds), _The Secret Speeches of Chairman Mao_ , 289.
**** Goldman, 53–4.
**** Mao, "On the Correct Handling of Contradictions among the People" (Speaking Notes), 473.
**** Mao, "The Situation in the Summer of 1957" (July 1957), in Mao, _Selected Works_ , V, 476.
**** See MacFarquhar and Schoenhals, 324–36.
**** See Teiwes and Sun, 25–109.
**** Starr, 206–13, 220–2, 304–5, recognizes that "limitations" and "narrowness" in Mao's view of mass participation exist, but otherwise idealizes Mao's real desire to fight "misguided authority" and to implement the mass line, much as other scholars did in the pre-Deng era.
**** Frakt, "Mao's Concept of Representation," _American Journal of Political Science_ , 23 (November 1979), 684–704.
**** Ibid., 690–3.
**** Ibid., 693–4.
**** Ibid., 697–8.
**** Donald Munro, _The Concept of Man in Contemporary China_ , 181–4.
**** Ibid., 183–4.
**** Frakt, 698–9.
**** Ibid., 693, 699.
**** Ibid., 693–4.
**** Hoston, 392.
**** Ibid., 394.
**** For example, see Wang Xizhe, _"Mao Zedong yu wenhua dageming," passim_ , abridged selections in Andrew and Rapp, 275–95.
****Brantly Womack, "Where Mao Went Wrong: Epistemology and Ideology in Mao's Leftist Politics," _Australian Journal of Chinese Affairs_ , 16 (July 1986), 30, notes the "lack of any institutional guarantees of mass voice or citizen rights" in Mao's "mass line" going back to the Yanan base area period, and also notes that Mao's late view of the masses as "malleable" tended to "denigrate mass creativity."
**** That the Party ultimately maintained control of the definition of the People is an obvious but important point made by many scholars. For example, see MacFarquhar, "The Secret Speeches of Chairman Mao," 7. The Party's elitist definition of the people was also partially recognized by Starr, 220. Also see Andrew Nathan, _Chinese Democracy_ , xii. As Maurice Meisner suggests, there is furthermore an ambiguity in Maoist thought as to where the vanguard resides, at least in the Cultural Revolution when the Party was under attack. See Meisner, "The Dictatorship of the Proletariat in Chinese Marxist Thought," in Nee and Mozingo (eds), _State and Society in Contemporary China_ , 123–31.
**** Friedman, "The Societal Obstacle to China's Socialist Transition," 166.
**** Frakt, 401.
**** Schram, "Mao Zedong a Hundred Years On," 129.
**** Schram, _The Thought of Mao_ , _passim_ , especially 97–109, essentially makes the same argument, that is, that Mao's "mass line" and calls for decentralization and democracy were in the end limited by other aspects of his thought, including the "primacy of centralism over democracy," the stress on dictatorship and slighting of direct elections, the emphasis on control and channeling of participation in proper directions by the Party, and above all by Mao's "attachment to the ideal of a 'strong socialist state'" which included a high degree of autocracy. In his later essay, Schram extends this argument, stating that "Mao's ideal was _yiyuanhua_ , or monolithic unity," which though ostensibly based on the mass line, "did not mean, and was never intended to mean, simply doing what the masses wanted." See Schram, "Mao Zedong a Hundred Years On," 129.
**** Walder, "Cultural Revolution Radicalism," 61.
**** See Munro, "Egalitarian Ideal and Educational Fact in Communist China," in John M. H. Lindbeck (ed.), _China: Management of a Revolutionary Society_ , 279–83, 293–4; and Munro, _The Concept of Man in Contemporary China_ , 107–34.
**** See for example the section on literacy rates in China in Nick Eberstadt, _The Poverty of Communism_ , 155–65.
**** Mao, _Hong Qi_ (Red Flag), 2 (1966), 18, quoted in Ch'en, _Mao Papers_ , 103; cited in Munro, _The Concept of Man in Contemporary China_ , 104.
**** This question has been asked in different ways, for example, by "Comrade Jin Jun" to his fellow Democracy Wall activist Wang Xizhe, which as we will see in Chapter 8, prompted Wang to write his long essay "Mao Zedong yu wenhua dageming" (Mao Zedong and the Cultural Revolution) (February 1981), translated in Anita Chan et al. (eds), _On Socialist Democracy and the Legal System: The Li Yizhe Debates_ , 177–260, 285–97; reprinted in abridged form in Andrew and Rapp, _Autocracy and China's Rebel Founding Emperors_ , 275–95. This question was also posed by Maurice Meisner in a lecture on the Chinese Revolution at McAlester College, St. Paul, Minnesota, March 1985, and in his subsequent writings.
7
Denunciations of
anarchism in the PRC
**Introduction**
This chapter continues to examine the limits to genuine critiques of state autonomy in the PRC by analyzing denunciations of anarchism published in the official Chinese press from the early years of the regime to the contemporary era, utilizing a three line model of Leninist regimes, especially as presented by Edward Friedman. It should be noted at the outset that the PRC denunciations of anarchism are not in themselves very interesting and mostly blindly follow the critique of Marx and Engels of Proudhon and Stirner as representing the interests of the petite bourgeoisie, or small producers, mixed in with Lenin's denunciations of anarchism as an "infantile disorder" of "ultraleftism," as typified by Mikhail Bakunin, whom Lenin claimed was an opportunist perhaps only posing as a revolutionary. Such denunciations almost always ignored, as did most Soviet critics, the claims of people like Bakunin to be socialist anarchists, and the claims of most anarchists after that date, most notably Peter Kropotkin, to be anarcho-communists equally interested as Marxist–Leninists, if not more so, in ending private property. It should also be noted that almost none of the people being denounced were really anarchists and that in numerous cases people condemning others as anarchists were themselves later denounced for the same reason.
However unoriginal and inaccurate these denunciations of anarchism were in practice, they are nevertheless worth examining for the confirmation they provide, both about the lack of any true quasi-democratic elements in Maoist thought or practice and the essential truth behind the charge of Party democrats that despite the major policy shifts and huge changes in society over time from the Mao to Deng and post-Deng eras, there is an essential continuity in the nature of China's Leninist Party-state. This chapter will attempt to make the latter point without falling into the trap of viewing China's Leninist regime as an unchanging, "totalitarian" monolith by adapting Friedman's labels of "Stalinists" who emphasize socialism as the buildup of heavy industry through central planning and a command economy, "Maoists" who supposedly favor the use of ideological incentives to move toward communism without creating a huge bureaucratic state or reviving economic inequality, and "Titoists" who view socialism as the buildup of abundance of the proletariat and thus who would allow market reforms that tolerate limited inequality. In fusing this model with labels borrowed from analyses of internal politics of the Liberal Democratic Party of Japan under the old "1955" single party hegemonic system, one could posit that coalitions of mainstream and anti-mainstream elements of different lines often form, such as a Maoist–Stalinist coalition during the Cultural Revolution and a Titoist–Stalinist coalition in the early to mid-Deng years. After that point through the early years of the twenty-first century, the differences within the Party-state could be said to focus on degrees of market reform, and thus between moderate and radical Titoists, though as we will see in the postlude, Maoist and Stalinist elements in the Chinese Leninist state may be mounting a comeback.
**Denunciations of Anarchism from 1957 to 1976**
Denunciations of anarchism in the PRC can be traced back at least to the Anti-Rightist Campaign of 1957–8. Deng Xiaoping made note of the problem of anarchism in 1957 in his report at the end of the Anti-Rightist Campaign, where he concluded that among the "serious erroneous points of view" of a few "bourgeois intellectuals" (i.e. those who had dared to speak up during the Hundred Flowers Movement), were ". . . individualism, liberalism anarchy, egalitarianism and nationalism."
In 1958, the Maoist Yao Wenyuan helped begin his career by taking part in denunciations of the novelist Ba Jin (virtually the only real Chinese anarchist ever criticized in the PRC, and even then a former anarchist), who though no longer claiming to be the anarchist he was before 1949, as we saw in Chapter 5, did express mildly loyal criticism of the status quo during the Hundred Flowers period. In his article, Yao targeted anarchist themes in Ba Jin's early novel, _Miewang_ (Destruction). As Ba Jin was still protected at this point by others in the Party hierarchy, Yao mostly couched his criticism in comradely terms, saying Ba had insufficiently repudiated his former anarchism in articles he had published since 1949, since the novelist had only called his early thought "limited," tried to defend anarchism of the Kropotkin variety as another version of communism that did not amount to bourgeois individualism, and claimed that his novel _Destruction_ was in favor of revolution and was not nihilistic. In response, Yao dissected the novel to argue that its anarchist hero did indeed express an "anarchist hopelessness" that amounted to a philosophy of "extreme individualism" opposed to leadership of the Party. Yao summed up the main lesson of his analysis of Ba Jin's novel as the "especially harmful nature" of anarchist thought, which, whatever its claim of being revolutionary, in fact "uses individualism to resist collectivism" and "advocates extreme democratic transformation, opposes discipline, shows contempt for organizations, and fails to see the productive nature of physical laborers," leading to a potential "destructive effect" on contemporary society if it were not thoroughly opposed. Though tame in comparison to later denunciations of Ba Jin in the Cultural Revolution, Yao's critique can be seen as a shot against the bow of those who had challenged the Party in the Hundred Flowers period by calling for more openness and individual freedom. After expressing mild self-criticism in 1958 against this attack, including of his past anarchism, Ba Jin survived this assault and returned to prominence after the Great Leap Forward where he could again raise mild criticisms of the excesses of the regime and especially of the "literary bureaucrats" who had criticized him and other writers in the late 1950s, criticisms which only helped intensify the later assault on him during the Cultural Revolution.
Before the Great Leap ended, however, and probably as part of the "Campaign against Right Opportunism" launched after the purge of Peng Dehuai in 1959, another brief round of denunciations of anarchism occurred. Most prominently, Lin Biao, who had replaced Peng as Minister of Defense, criticized "anarchism and egalitarianism" as part of the "temporary, partial interests of the small producers" in his denunciations of major deviations within the army.
By far the most intensive criticisms of anarchism in the history of the PRC, however, occurred during the Cultural Revolution. Anarchism, as William Joseph points out in his exhaustive study, _The Critique of Ultra-Leftism in China_ , represented only one, if perhaps the leading example, of "ultra-leftism" that was periodically if incompletely criticized during this period. Given the sheer volume and number of denunciations of anarchism in the Cultural Revolution era, only a small sample can be summarized in this chapter. Suffice it to say that both Stalinists and Maoists launched criticisms of anarchism in the early years of the Cultural Revolution, if for different reasons.
The Stalinists, and perhaps any closet Titoists who might have quietly survived, denounced the "chaos" of "great [or ultra or extensive] democracy" ( _daminzhu_ ) at the outset of the Cultural Revolution in 1966 and revived these charges periodically from 1967 to 1969 to in effect charge that the Party Maoists had encouraged extra-Party "ultra-leftists" to push Mao's critique of a "new bourgeoisie in the Party" to a point that threatened continued state control of society. In February 1967, for example, when the Stalinists had gained a temporary ascendancy (in what was later termed by the Maoists, the "February Adverse Current"), an article in _Hongqi_ revived the critique of anarchism in Engels's _"On Authority,"_ a work issued during Marx's lifetime. Just as Engels criticized Bakunin's followers for failing to see the essential authoritarian nature of socialist revolution and thus the need to maintain "the authority of the armed people against the bourgeoisie," so too "some persons" in the Cultural Revolution used Mao's call to seize power from the bourgeoisie in the Party to supposedly oppose all authority, a stance that, following the standard Marxist critique of anarchism, the _Hongqi_ author found to be "an expression of the inherent bad characteristics of the petty bourgeoisie, an expression of anarchism." This stance was perhaps reflected in the _Renminribao_ editorial of April 26, 1967, "Anarchism is the Punishment of Opportunist Deviationists," that found that "anarchism is looming up, dissolving the targets of our struggle and deflecting it from its normal direction."
On the other hand, also during the Cultural Revolution, Maoists who favored continuing class struggle against "new bourgeois elements" in the Party could use anarchism as a whipping boy to prove their own truly leftist credentials and to protect themselves against Stalinist attacks. For example, when extra-Party Red Guard Maoists went too far for Mao and began to attack "bourgeois elements" in the People's Liberation Army and to call for "suspecting all" as the "fighting slogan of great democracy," Yao Wenyuan, who by this time became perhaps the leading Maoist polemicist, known as "the Stick," responded with a denunciation of one, perhaps mythical, extra-Party Maoist group, the "May 16 Corps," as a "scheming counter-revolutionary gang" that spouts such slogans as "doubt everyone" and "oppose anybody" that appears only in the guise of ultra-Left anarchism but is in essence extremely Rightist. Yao tried to combine this anti-anarchist critique of extra-Party Maoists with denunciation of the fallen Party leader Tao Zhu for the same tendencies, even though Yao and the other inner Party Maoists otherwise criticized Tao for being too conservative, that is, as we might argue using the three line model, too much within the Stalinist camp. Likewise, Mao's wife, Jiang Qing, when called upon by Party elders to help rein in the Red Guards attacking the army and seizing weapons, denounced such groups for "factionalism," which she charged was "a characteristic of the petty bourgeoisie and is mountain-topism, departmentalism and anarchism—very serious anarchism."
Mao himself, following Friedman, may not have been the best Maoist during this time as he shifted back and forth along policy lines as he saw fit and in order to keep his perceived Party rivals off balance. Thus, the Chairman first allowed the Maoist group that he created to push (rhetorically) for Paris Commune style mass democracy in 1966 while, as we saw in Chapter 6, he himself later denounced the Shanghai Commune in 1967 as "extreme anarchy, which is most reactionary" and ". . . detrimental to the interests of the people and against their wishes." It was by using this statement of Mao that Stalinists in the Cultural Revolution could denounce Red Guard organizations as anarchist and imply that their Maoist supporters in the Party were anarchists as well, which in turn forced the inner Party Maoists to find extra-Party groups to denounce as anarchists, as with Yao and Jiang Qing. Thus inner Party Maoists could very easily both denounce anarchism and be denounced in turn as anarchists themselves. As the first such example, the leading Party Maoist Chen Boda could denounce anarchists as causing "splittism" and leading to the failure of unity if the revolutionary Left in 1967, while he himself became the major target of a campaign denouncing anarchism after his fall in late 1969 to early 1970. Evidently with the permission of Mao himself, in September of 1969 the top ideological organs of the Party-state launched a major national campaign against "bourgeois factionalism" and anarchism as part of preparing the country for a possible war with the Soviet Union, a campaign that continued into 1970–1 with Chen as the main, if unnamed, target of a campaign against "swindlers like Liu Shaoqi." As Joseph notes, this campaign continued into 1972 when it merged into the first campaign against Lin Biao for his "leftist" errors.
This latter campaign featured articles in _Renminribao_ in October 1972 criticizing the "swindlers" as opportunists who only posed as anarchists, ". . . not because they want to do away with all forms of government, but because they want to do away with the government of the dictatorship of the proletariat and replace it with a government of the dictatorship of the bourgeoisie which they represent." This aborted campaign was perhaps the high point of criticism of anarchism in the Mao era and a precursor to the 1977 attacks on the Gang of Four as anarchists.
From the inner Party Marxist democrat Wang Ruoshui, whose thought we will examine in Chapter 9, we now know the inside story of the 1972 campaign, which he himself promoted. Even though, as Wang noted, remaining Maoists such as Yao Wenyuan themselves had earlier compared the "swindlers" to Bakunin's "sabotage activities" against the First International, the leading Maoists at the top of the Party in 1972 feared that the main thrust of the initial campaign to criticize Lin Biao as an ultra-leftist was aimed at them (with good reason, one would think, considering the very recent campaign against Chen Boda as an anarchist and the not much later 1977 campaign against the Gang of Four as anarchists that we will examine below), and thus they tried to quash the campaign, publishing articles in the Shanghai journal _Wen Hui Bao_ that took _Renminribao_ to task for the pernicious influence on the provincial press of its articles criticizing anarchism. This criticism of his paper led Wang, on his own initiative, but also at the suggestion of his editor Hu Jiwei to write to Mao himself to ask whether or not the anti-anarchist articles Wang had published were proper. Mao ruled against Wang, and probably under the Chairman's orders, on December 19 Premier Zhou Enlai called in Wang to a meeting at the Great Hall of the People along with members of what would become the Gang of Four to get Wang to end the campaign. By the end of this nearly 6-hour meeting, which lasted well into the early hours of the next day, Wang realized that he had inadvertently put Zhou in a very difficult position since Zhou himself had earlier declared that he was "inclined to agree" to thoroughly denounce ". . . the ultra-leftist trend of thought and anarchism stirred up by the Lin Biao anti-party clique" without then knowing that it was Mao himself in October in a private conversation with Yao Wenyuan and others who decided that criticism of anarchism was inappropriate. Wang reported that at the December meeting, Zhou, though admitting that he himself had earlier criticized people for anarchism, claimed that he only meant to refer to those who interfered in foreign policy as anarchists and "not to the entire line of Lin Biao," speaking in what Wang viewed as an uncharacteristically haphazard and at times incoherent manner that suggested to Wang that "the premier was saying things that ran counter to his convictions," including saying things critical of Wang, as evidently ordered by Mao, while trying to protect him. Thus, as Wang belatedly realized, the dispute over whether or not to label Lin Biao as an anarchist became inextricably wound up in palace intrigue involving the struggles of the Maoists around Jiang Qing to replace Zhou Enlai and other top leaders of the Party, struggles that were to increase a few years later.
**Denunciations of Anarchism during the Hua Guofeng-Early Deng Xiaoping Years**
The use of denunciations of anarchism to reinforce their Maoist credentials while limiting Maoist policies in practice is especially true of those Party elites in the Hua Guofeng era (1977–9), which could represent the ultimately failed rule of a Stalinist–Maoist coalition. As such, Hua's coalition had an interest in denouncing full Maoism as illegitimate anarchism, which they wanted to discard while retaining the supposed essence of the Mao line of the Cultural Revolution. During the spring and summer of 1977 especially, articles appeared in the official Chinese press criticizing the Gang of Four in much the same terms as Lin and company were criticized in 1972, that is, as opportunists who only fanned up the wind of anarchism in order to usurp Party and state power. After that point the main tone of criticism of the Gang shifted to other directions, even to the contradictory charges that the Gang tried to establish a "fascist dictatorship of the bourgeoisie," but the earlier charges culminated in speeches by Party Chairman and Premier Hua Guofeng and the top general and later Chairman of State Ye Jianying that mentioned the problem that "secret factions" in the Party were spreading, among other things, the "harm of anarchism," and a plank in the "General Program" of the CCP Constitution adopted in August 1977, which noted the need for the whole Party to "oppose all splittist and factional activities, oppose the independence from the Party, and oppose anarchism."
The denunciations of anarchism did not end with the fall of the Gang of Four. The criticism of anarchism that survived in the late 1970s perhaps helps demonstrate the early Deng era as representing the ascendancy of a Titoist–Stalinist coalition, with Deng uneasily maintaining a balance between representatives of both lines and with the Stalinists implicitly threatening to return to a neo-Maoist coalition. In this regard, Ye Jianying's continuing denunciation of anarchism in 1978 especially represents the continuity of the Hua and early Deng eras. Ye criticized the Gang of Four in 1978 for ". . . incit[ing] anarchism and slander[ing] the socialist legal system and every kind of rational rules and regulations as revisionist and capitalist in their vain attempt to throw our proletarian country into chaos and seize power in this chaos," an attempt that he would oppose by strengthening the "socialist legal system." As criticism of the Gang for "ultra-leftist" excesses continued in the official press, now even in the former Maoist bastion of the Party magazine _Hongqi_ , the tone of the articles reflected Ye's line about the need to restore economic order and a socialist legal system, a point on which the unity of the rising Titoist–Stalinist coalition of Deng would hinge.
Especially at the beginning of market-based economic reform in 1979–80, the attack on ultra-leftism as a whole was initially intensified, as Joseph notes. Though Joseph does not himself mention examples, after 1979 those who denounced anarchism most often were people in the Party-state elite who had opposed Maoist policies for their undermining of state control and Stalinist-style central planning. Such people denouncing anarchism in this period did so to undermine intellectual critics inside and outside the Party who tried to take advantage of Titoist economic reforms to push for political liberalization. For example, in late 1978 and early 1979 denunciations appeared in the official press that tried to link Democracy Wall activists to the activities of the Gang of Four, both supposedly representing "anarchists who, masquerading under the banner of democracy, caused worsening economic conditions and social instability." The PRC Minster of Education directly tied the "small number of students [who] practice anarchism in defiance of organization and discipline" presumably in the Democracy Wall movement to the "corruption and poisoning of [their] minds by Lin Biao and the Gang of Four," a theme that continued in national and provincial press articles throughout 1979 and into 1980. Once again, the culmination of this campaign was a speech by Ye Jianying to mark the celebration of the thirtieth anniversary of the CCP where he spoke of the need to eliminate "factionalism, anarchism, and ultra-individualism." A report on his speech noted further the need to guard against the ideology of "ultra-democracy," a habit of the "small producers" that once again was caused by the "spoiling of the social atmosphere" by Lin Biao and the Gang of Four, which spread the "ideology of anarchism and extreme individualism . . . among some people," a line that was repeated widely in other press outlets from November 1979 to early 1980. The criticism of the "anarchism" of Democracy Wall activists as a form of bourgeois individualism counterposed with the need to achieve "stability, unity, and socialist democracy" would of course presage the many campaigns against bourgeois liberalization in the 1980s.
After the repression of the Democracy Wall extra-Party critics in 1979–80 with Deng's institution of the "Four Cardinal Principles," the debate shifted to inside the Party, where intellectuals within the Titoist side of the coalition tried to call for political reforms, while their Stalinist opponents, led by Hu Qiaomu, continued to push against bourgeois liberalization in the name of preserving "socialist spiritual civilization." As reflected in denunciations of anarchism, this struggle included a debate between Ma Jia of the Titoist camp who argued in 1980 for the need to "scientifically" criticize anarchism while clearly distinguishing it from real democracy, versus Gu Zhaoji of the Stalinist camp who wrote an article at the same time that wasn't published until 1982 which argued that anarchists were indeed exponents of "extreme democracy" and that trying to "scientifically" distinguish what constitutes anarchism would lead people to "not find any traces of anarchism at all," leading people to oppose bureaucratism without opposing anarchism.
This brief debate reflected the Stalinist push against bourgeois liberalization in 1981–2 that had begun with official criticism of Bai Hua's screenplay "Bitter Love." Included in this criticism were denunciations of the script for, as one author put it, "the erroneous trend of thought of anarchism, ultra-individualism and the bourgeois liberalism to the extent of negating the four basic principles." Such denunciations were repeated in the provincial press until early 1982 and were revived in 1983–4 after the Anti-Spiritual Pollution Campaign. This campaign was again led by Stalinists such as Hu Qiaomu against what Friedman labels the democrats inside the Party, as typified by Wang Ruoshui, whose thought we will examine in Chapter 9. Hu denounced Wang's use of the Marxist concepts of humanism and alienation in a long article in _Hongqi_ that was reprinted in _Renminribao_ , where he explicitly charged that those "well-intentioned" comrades who pushed the concept of alienation in effect gave cover to those who called explicitly for "the abolition of all social and political power, all social and economic organization, all ideological authority, all centralism and discipline, and have openly propagated anarchism, absolute freedom, and extreme individualism."
Such denunciations were echoed in numerous articles in 1984 which stressed the need to completely negate "extensive democracy" and tried to link anyone calling for democracy to the "anarchism" fanned up in the Cultural Revolution. As one article in _Hongq_ i put it, "while practicing anarchism, some anarchists and their apologists will talk at length as if they are 'fighting for democracy.'" While academic articles in this period could discuss the early twentieth-century Chinese anarchist movement more dispassionately and even find in it some progressive elements, Stalinist critics around Hu Qiaomu continued to try to link those in the 1980s who called for democracy with the evil of anarchism, including in the aftermath of student protests in 1986, when an article in _Renminribao_ again linked the protestors to those who pushed the concept of extensive democracy during the Cultural Revolution, which the article claimed would in fact only lead to anarchism and the violation of the rights of the majority.
This attempt to link the student-led democracy movement to the turmoil ( _dongluan_ ) of the Cultural Revolution was also used after the Tiananmen student protests of 1989. The problem for the regime, however, in applying the label of anarchism to the student movement was that the student leaders, even after the severe provocations of May and June launched by the government, at most called for electoral democracy and human rights and not the Paris Commune style mass democracy advocated in the Cultural Revolution. This fact of course did not stop the government from trying to draw a link between the Tiananmen protests and the Cultural Revolution. As one commentator argued in an article published jointly in _Jiefangjunbao_ and _Renminribao_ in the same month as the crackdown on the student protesters,
The university students today are all young people around 20 years old. They have not personally experienced the disaster and pains suffered by the state and the people, including the young students, caused by social disturbances during the Cultural Revolution. At that time, many Red Guards who were so young had gone to the streets to advocate speaking out freely, airing views fully, holding great debates, and writing big character posters, established ties and took part in criticism and struggles. As a result, our country was led to a nationwide great turmoil of civil war and our national economy was on the verge of collapse . . .
. . . Young students [of the Tiananmen movement in 1989] originally intended to solve problems through demonstrations and petitions, but the result was the spread of anarchy.
This line continued into August, as in one article in _Jiefangjunbao_ that applied Lenin's old label of "leftist infantile disorder" in order to argue once again that the student movement would result in "anarchy" if the country moved too hastily towards implementing democracy. Similarly, another official commentator argued that if people pushed for democracy beyond China's "national conditions" during the "initial period of socialism" when China is still in a time of low levels of education, literacy, and development and ". . . when many people are still preoccupied by the daily toil for basic survival" then "it [would be] impossible to expect from them a high degree of democratic participation. Even if a so-called democracy is forcibly implemented, interference from various factors will give rise to individualism, factionalism and anarchy, and lead to de facto non-democracy and even chaos." In contrast to trampling on human rights and the promotion of "anarchist thinking" by Lin Biao and the "Gang of Four" during the Cultural Revolution, it was the Deng-era regime, yet another commentator argued while denouncing the program of exiled Tiananmen student leaders and reform intellectuals, that had reversed the verdicts on tens of thousands of people persecuted during the Mao era and restored the rule of law, thus demonstrating the "iron-clad fact of [the state] respecting people and caring for people. . . ." In criticizing the calls of intellectual allies of the students within the Party such as Yan Jiaqi for deepening market reforms as the same as "putting the economic system on a capitalist basis as an appendage to international capital," this article perhaps demonstrates a point when Titoist reform in the Deng Xiaoping era was stalled and some Maoist, anti-imperialist rhetoric reappeared, threatening a shift to a Stalinist–neo-Maoist coalition. This harsh line as applied to denunciations of student protests culminated in a speech by CCP General Secretary Jiang Zemin in October 1989, reinforced in an interview with the PRC Minister of Justice in November, that stressed the need to take as the main task opposing those who advocated "ultra-democracy and anarchism."
**PRC Denunciations of Anarchism, 1992–present**
Even after Titoist economic reforms returned to the forefront after 1992 with Deng Xiaoping's trip south to visit the special economic zones, and as the threat of a shift to a Stalinist–Maoist coalition receded, denunciations of anarchism nevertheless continued. To pick just a few examples, first in 1995, probably in response to academic calls for political reform, _Renminribao_ published an article reviving Deng Xiaoping's 1979 warning that talking of "abstract democracy" would "inevitably lead to serious spreading of extreme democratization and anarchism, total sabotage of the political situation of stability and unity, and complete failure of the four modernizations." In 1999 and 2000 the PRC press denounced the founder of the Falungong spiritual/healing movement, Li Hongzhi, as someone who "hated, negated, and undermined our socialist state power" and to Falungong as a troublemaking group which is "anti-science, anti-humanity, anti-society, and anarchistic" and as an "evil cult" that carried out activities that similar to "anarchist trends and factions of all kinds [which] have occurred in history." In 2000, in response to very moderate demands of the democracy movement in Hong Kong, the CCP-controlled press there complained that "pure populism and anarchism can only throw Hong Kong into chaos. . . ." In 2003, countering Taiwan President Chen Shui-bien's call for an eventual referendum on a new Taiwan constitution, the Hong Kong Communist press denounced the "so-called 'popular will' [of] the Taiwan authorities . . . as none other than 'populism' or 'anarchism.'" In 2004, against international and some domestic demands for increased respect for human rights, a PRC functionary claimed that "respect and safeguards for human rights in an isolated and abstract sense . . . could lead to anarchism and extreme individualism in practice and bring disaster to the state, society, and the people." In July of 2008, even after the defeat of Chen Shui-bien and the election of the KMT's Ma Ying-jeou as president in Taiwan, the same Beijing-controlled press in Hong Kong denounced the continuing efforts of the Taiwanese opposition to carry out "Taiwan-style democracy" that would dare to "directly criticize any official, even top leaders" and "directly expose the corrupt officials and lawbreakers via the media" as "classic anarchy and personal liberalism!" Finally, also in 2008, in response to calls for "returning power to the people" at an academic conference marking the thirtieth anniversary of the beginning of the Deng-era reforms, the Communist press in Hong Kong denounced this call as possibly leading "to the evil path of anarchism." As one article concluded, "unrestrained talk about 'returning power to the people' will not only mislead the people with the impression that they do not need the government but will mislead them into thinking that the government had been abusing its power and now needs to rectify its ways."
**Conclusion**
In effect, the leaders of China's Leninist Party-state in both the Cultural Revolution and the reform era turned the label of anarchism into a cultural meme that could be wielded against anyone who dared decry the growth of a new Communist Party elite ruling for itself, or to call for any real degree of democracy and individual freedom. In the end, therefore, the criticism of anarchism in the PRC ironically helps prove the essential point of the anarchist critique of Marxism. That is, regardless of important differences among themselves, the very agreement of top leaders of all Leninist factions to condemn as anarchists any democrats within their coalition as well as any critics outside of the Party who argued that the state may act at times in its own interests and not just for the economic class it supposedly represents, itself helps to remove a check on increasing state autonomy and aids the continuing survival of Leninist state despotism.
**Notes**
**** For a very convenient summary of these orthodox Marxist denunciations of anarchism, see the Soviet text published in an attempt to counteract revivals of anarchism in the student movements in America and Europe in the late 1960s to early 1970s, _Anarchism and Anarcho-Syndicalism: Selected Writings by Marx, Engels, Lenin_. Chinese official denunciations mostly ignored other possible Marxist lines of attack on anarchists, for example Nicolai Bukharin's claim that they represented déclassé and "lumpenproletariat" rough elements (see Nicolai Bukharin, "Anarchy and Scientific Communism" (1922), translated in _The Poverty of Statism: Anarchism vs. Marxism: A Debate Bukharin, Fabbri, Rocker_ , 1–10), or V. I. Lenin's charge that the Christian anarchist Leo Tolstoy represented elements of the dying aristocracy that turned against all states out of twin reactionary idealization of a supposed egalitarian rural ideal and a pessimistic realization that the old landed order was doomed to destruction. See Lenin, _On Literature and Art_ , 64–8.
**** See especially Friedman, "Three Leninist Paths within a Socialist Conundrum," in Dorothy Solinger (ed.), _Three Visions of Chinese Socialism_ , 11–46. Also see Friedman, "Maoism, Titoism, Stalinism: Some Origins and Consequences of the Maoist Theory of the Socialist Transition," in Mark Selden and Victor Lippit (eds), _The Transition to Socialism in China_ , 159–214.
**** Following the argument of John A. Rapp, "Despotism and Leninist State Autonomy: The Chinese Asiatic Mode of Production Debates in Comparative Perspective," 1–59.
**** Deng, "Report on the Rectification Campaign," September 23, 1957, translated in Robert R. Bowie and John King Fairbank (eds), _Communist China 1955–1959: Policy Documents with Analysis_ , Document 19, 348.
**** For a concise summary of Ba Jin's actions during the Hundred Flowers, see Olga Lang, _Ba Jin and His Writings_ , Introduction.
**** Yao Wenyuan, "Lun Ba Jin xiaoshuo 'Miewang' zhongde wuzhengfuzhuyi sixiang" (Discussing the Anarchist Thought in Ba Jin's Novel, _Destruction_ ), 17–20. The first English translation of this article will appear in Rapp and Youd, "Ba Jin and the Anarchist-Marxist Debates in China," forthcoming.
**** For Ba's criticism during the early 1960s and his suffering during the Cultural Revolution, see Olga Lang's introduction to the English translation of Ba Jin, _Family_.
**** See Lin, "March Ahead under the Red Flag of the General Line and Mao Tse-tung's Military Thinking," translated in Bowie and Fairbank, Doc. 46, 580, 584–5. For a summary of the criticism of anarchism in this revival of the Anti-Rightist Movement, see _Zhongguo renmin daxue makesi liening zhuyi jichuxi_ (The Marxist-Leninist Studies Department of Chinese People's University), _Wuzhengfuzhuyi pipan_ (Criticizing Anarchism).
**** Joseph, 220–31. See pages 23–61 for Joseph's summary of the classical Marxist-Leninist lines of attack on anarchism. See pages 128–44, 159–61, and 189–90 for his summary of attacks on anarchism in China, especially in 1971–2 and 1976.
**** _Hongqi_ Commentator, "On Revolutionary Discipline and Revolutionary Authority of the Proletariat," _Hongqi_ , 3 (February 1967), translated in _Survey of China Mainland Magazines_ (hereafter SCMM), 564 (February 20, 1967), 3, citing Frederick Engels _, "_ On Authority," first published in Italy, 1873, translated in _Anarchism and Anarcho-Syndicalism_ , 100–4.
**** Cited in Jacques Guillermaz, _The Chinese Communist Party in Power, 1949–1976_ , 428.
**** "Long Live Chairman Mao's 'Great Democracy,'" in _Dongfanghong_ (East Is Red), November 16, 1966: 4, translated in _Joint Publications Research Service_ (hereafter JPRS): _Translations on Communist China_ 49387 (March 24, 1967), 18.
**** Yao, "Comments on Tao Chu's Two Books," _Renminribao_ , September 8, 1967, 1–3, published in English by New China News Agency, September 7, 1967, reprinted in James T. Myers, Jürgen Domes, and Erik von Groeling (eds), _Chinese Politics: Documents and Analysis_ , _Volume I: Cultural Revolution to 1969,_ 351–2.
**** See Peter R. Moody, Jr., "Policy and Power: The Career of T'ao Chu 1956–66," CQ 54 (April–June 1973), 268, 288.
**** Jiang Qing, Speech of September 5, 1967, translated in _BBC: Summary of World Broadcasts: Far East_ (hereafter FE): 2570, reprinted in CQ 32 (October–December 1967), Quarterly Chronicle and Documentation, 212–17.
**** "Chairman Mao's Speech at His Third Meeting with Zhang Chunqiao and Yao Wenyuan" (Feb. 6, 1967), translated in JPRS China Report, 90, 44.
****See Chen's speech in _Huochetou_ (Locomotive), February 1967, translated in _Survey of China Mainland Press_ (hereafter SCMP), 3898, 5–6, cited in Philip Bridgham, "Mao's Cultural Revolution in 1967: The Struggle to Seize Power," CQ 34 (April–June 1968), 11.
**** See Bridgham, 22 and articles such as Kao Zhiren, "Anarchism Is Reaction against the Continuation of Socialist Revolution," in _Hongqi_ , 9 (August 27, 1969), in SCMM 665 (September 22, 1969), 29–31.
**** See Joseph, 124–6, citing for example _Renminribao_ August 15 and 29, 1971, in SCMP 4966, 22 and 4973, 19.
**** See Lung Yen, "Anarchism is the Counter-Revolutionary Tool of the False Marxist Swindlers," _Renminribao_ , October 14, 1972, in SCMP 5241 (October 25, 1972), 58.
**** See Wang Ruoshui, "A Turn Around from Criticism of 'Leftism' to Opposition to Rightism—One Individual's Experiences and Reflections on Chinese Communist High Level Infighting," _Ming Bao Yuegan_ (Hong Kong), 27(9) (March 9, 1989), 312, in JPRS 89: 055 (May 30, 1989), 6–18. Also see Wang, _Zhihui de tongku_ (The Pain of Wisdom), 331.
**** Wang, "A Turn Around from Criticism," 9. Wang cites the _Hongqi_ editorial, "Strengthen the Party's Centralized Leadership," _Hongqi_ , 11 (November 1, 1972), translated in SCMM 741–2 (November 27–December 4, 1972), 2–8, which criticized ". . . the crimes of Liu Shaoqi and other swindlers" [code for Lin Biao] for advocating "slavery and anarchism" in an attempt to "corrupt the proletarian Party character and undermine the Party's centralized leadership." For another example of such earlier criticism of Lin Biao for anarchism in the Maoist-controlled journal _Hongqi_ , see Xiao Pin, "Be Open and Aboveboard, and Don't Intrigue and Conspire," _Hongqi_ , 3 (March 1, 1972), in SCMP 725–6 (April 3–10, 1972), 35.
**** Wang, 9–10.
**** Wang Nianyi, _Da dongluande niandai_ (A Decade of Great Upheaval), 450, citing a classified document from the Ministry of Foreign Affairs. This account is summarized in MacFarquhar and Michael Schoenhals, 355.
**** Wang Ruoshui, "A Turn Around from Criticism," in JPRS, 12–15.
**** Ibid., 17. Other accounts concur that this incident involving Wang Ruoshui and the quashing of the criticism of Lin Biao as anarchist was a precursor of the Maoist attacks on Zhou Enlai in 1975. See Frederick Teiwes and Warren Sun, _The End of the Maoist Era: Chinese Politics during the Twilight of the Cultural Revolution, 1972–1976_ , 63–5, and Yan Jiaqi and Gao Gao, _Turbulent Decade: A History of the Cultural_ Revolution, 415–16.
**** See especially "The Aim of the 'Gang of Four' in Fanning Up Anarchism is to Usurp Party and State Power," in _Guangmingribao_ March 10, 1977, in Survey of the PRC Press 6308 (March 28, 1977), and Zhang Wenhuan, "The 'Gang of Four' and the Bakunin Bandit Gang," in _Renminribao_ , April 14, 1977, in SPRCP 6332, May 4, 1977, 110–16. Also see " _Si ren bang" fandui "guan ka ya" jiushi shandong wuzhengfuzhuyi_ (The Gang of Four in Opposing "Guan, Ka, and Ya," Were Merely Instigating Anarchism).
**** See Hua, "Speech at the 2nd National Conference on Learning from Dazhai in Agriculture," _Peking Review_ , 1 (January 1 1977), reprinted in CQ 69 (March 1977): 247, Ye, "Speech at the 50th Anniversary of the Founding of the PLA," in CQ 72 (December 1977), 860, and "Constitution of the Communist Party of China," in _Peking Review_ , 36 (September 2, 1977), 17.
****Ye, "Report to the 5th National People's Congress," March 1, 1978, in CQ 74 (June 1978), 462.
**** Joseph, 231–44.
**** As noted in CQ 126 (June 1991), 228. Such national articles were also reflected in the provincial press. See FBIS: China Report (hereafter FBIS) February 28, 1979, J1 and March 13 1979, H1.
**** Xinhua, March 30, 1979, in FBIS, April 2, 1979, L20.
**** Ye, _Beijing_ Review 40 (1979), reprinted in CQ 81 (March 1980), 158; Hou Jun, "Manifestations of Ultra-democracy Must Be Stamped Out," _Gongrenribao_ , October 16, 1979, in JPRS 74992 (January 24, 1980).
**** For example, see Hua Song, "Remove the Disturbances of Anarchism," _Hongqi_ , 2 (January 1980): 23–6.
**** For example, see Ma Jie, "Criticism of Anarchism Must be Scientific," _Renminribao_ , January 31, 1980, 5, in JPRS 75189 (February 20, 1980), 22 and Zhang Wenhuan, "Commenting on Stirner, Patriarch of Anarchism," Renminribao February 7, 1980, 5, in FBIS, February 21, 1980, L 16–19.
**** Gu Zhaoji, "How to Understand the Scientific Nature of Criticism of Anarchism," _Renminribao_ , May 3, 1982, 3, in FBIS, May 7, 1982, K 22–3.
**** Commentator, "The Four Basic Principles Brook No Violation: Commenting on the Film Script 'Bitter Love,'" _Jiefangjunbao_ , April 20, 1981, in FBIS, May 21, 1981, K 13.
**** See Friedman, "The Societal Obstacle to China's Socialist Transition," 164–71.
**** Hu Qiaomu, "On the Problem of Humanism and Alienation," _Hongqi_ , 2 (1984), 26; also in _Reniminribao_ , January 27, 1984, cited in Schram, "Economics in Command?: Ideology and Policy since the Third Plenum, 1978–84," CQ 99 (September 1984), 446.
**** Zhai Sishi, "Anarchism is Diametrically Opposed to Socialist Democracy," _Hongqi_ , 20 (October 21, 1984), 37, in JPRS 84–023 (December 10, 1984), 64.
**** _Renminribao_ , December 25, 1986, in BBC, _Summary of World Broadcasts, Far East_ 8452, cited in Robert Ash, "Quarterly Chronicle and Documentation," CQ 109 (March 1987), 156.
**** Zheng Yanshi, "A Post-Rebellion Reflection—Ant Attempt to Analyze the Question 'Why Did the Development of the Situation Go Against the Students' Good Intentions?" _Jiefangjun Bao_ and _Renminribao_ , June 21, 1989, in FBIS 89–119 (June 22, 1989), 13.
**** Xinhua, August 19, 1989, citing an article by Xiao Yi _in Jiefangjunbao_ , in FBIS 89–160 (August 21, 1989), 30.
**** Wang Guofa, "Democracy Should Not Go Beyond Social Development," _Liaowang_ , 32 (August 7, 1989), in FBIS 89–160 (August 21, 1989), 32–3.
**** Zhen Yu, "A Program Vainly Attempting to Practice Capitalism in China—Commenting on the Program of the 'Democratic China Front,'" _Liaowang_ (Overseas Edition), 43 (October 23, 1989), in FBIS 89–212 (November 3, 1989), 18.
**** Wang Pei, "Resist the Ideological Trend of Ultra-Democracy and Anarchism: An Interview with Cai Cheng, Minister of Justice," _Jingjiribao_ , November 1, 1989, 1, in FBIS, November 17, 1989, 21–2.
**** Zhou Xirong, "On Socialist Democracy: Making a Clear Distinction between Socialist Democracy and Parliamentary Democracy," _Renminribao_ , April 16, 1995, 9, in FBIS, June 14, 1996, 114, citing Deng Xiaoping, "Uphold the Four Cardinal Principles" (March 30, 1979), in Deng, _Selected Works_ , 184.
****Xinhua, July 21, 2000, in FBIS, July 24, 2000.
**** Special Commentator, "Absurd Heresy and Evil Motives: On the Anti-Scientific, Anti-Mankind, Anti-Social, and Anti-Government Essence of Falungong," _Renminribao_ August 18, 1999, 1, in FBIS, August 19, 1999, 0818.
**** "Hong Kong Is Our Home; Stability Must Be Cherished," _Wen Wei Po_ , July 17, 2000, in FBIS, July 25, 2000, 0717.
**** Li Jiaquan, "'Holding Referendum' Means Nothing But 'Taiwan Independence,'" _Ta Kung Pao_ , November 8, 2003, in FBIS, November 13, 2003, 1108.
**** Dong Yunhu, "Quanmian zhunquede linhui bawo he guanche shishi guojia zunzhong he baozhang renquande xianfa yuanze" (Completely and Accurately Understand, Grasp, and Implement the Constitutional Principle of the State Respecting and Safeguarding Human Rights), _Renminribao_ , May 11, 2004, 10, in FBIS, June 3, 2004, 0511.
**** Li Jiaquan, "On Taiwan's Political Situation after the General Elections on 20 May," _Wen Wei Po_ online, July 20, 2008, in World News Connection 985, accession number 265700670.
**** Liu Tzu-lu, "'Returning Power to the People' Not a Good Slogan," _Wen Wei Po_ , October 23, 2008, in _World News Connection_ (FBIS) 985, accession number 270451202.
8
Extra-Party neo-anarchist critiques of the state in the PRC
**Introduction**
This chapter and the succeeding one examine various "neo-anarchist" critiques of the Leninist state in the PRC, from the early years of the Cultural Revolution to the beginning of the Tiananmen protests. The label of neo-anarchist in this book refers not to self-proclaimed "post-modern" anarchist critiques but to anyone in China who criticizes the Leninist state using the simple, basic, but powerful view shared by all kinds of anarchists (contradictory as they might be on their own positive agendas), namely, that the state rules for itself when it can, not for classes, interest groups, a mass of individuals, or the whole community. The term "neo-anarchist" is adapted from analysts who apply the label "neo-Marxism" to those thinkers who find the Marxist class paradigm useful without necessarily being Communists. The references to neo-anarchism, then, in these chapters refer to a kind of negative elite theory similar to the "iron law of oligarchy" of Roberto Michels, whose classic book _Political Parties_ is perhaps still the greatest work employing a neo-anarchist paradigm.
In that work, Michels gave full credit to anarchist thinkers as "the first to insist upon the hierarchical and oligarchical consequences of party organization," while at the same time at key points in his book pointing out that anarchists themselves often departed from this basic critique, as when some anarchists supported hierarchy in economic and in revolutionary organization. Michels himself, though a member of the German Social Democratic Party, was heavily involved in a syndicalist group within that party and was also deeply influenced by earlier anarchist and anarcho-syndicalist thinkers. In his positive program he supported the necessity of forms of representative democracy, while maintaining his belief in the need for direct democratic institutions and new mass democratic "waves" as would-be checks on the tendency of democracies to constantly develop new "aristocratic forms." Michels' critique of bureaucratic organizations starting to work for their own interests instead of those of their constituents is mostly focused on political party organization, especially that of democratic socialist parties, rather than the state as a whole. Nevertheless, what makes Michels' argument such an outstanding example of the neo-anarchist critique is that he focuses not on their economic and "bourgeois" class privileges as the main factors in causing party leaders to develop interests divorced from their followers, but on their interests in maintaining their own power and in perpetuating their organizations. As we will see below, the focus on institutional as opposed to economic interests as leading to a "new class" of elites helped to differentiate genuine neo-anarchist critiques in China from official Maoist discourse. In any case, if organizations without full monopoly on the legitimate use of violence can become autonomous from their constituents and lose sight of their original mission (the most prominent example in recent years being the Roman Catholic Church's failure to protect its followers from predatory sexual abuse from priests), how much more likely and virulent is it for states to become autonomous from their subjects and develop into oligarchies?
Michels himself in his later career became suspicious of even direct democracy as a sufficient check on oligarchic tendencies and thus came to value individual heroic leaders as the best way to prevent oligarchy. That ironically Michels later in life became an apologist for Italian fascism should not denigrate his basic critique in _Political Parties_ , but only help to demonstrate that even great neo-anarchist thinkers may have their own flaws and limitations based on their own particular situations in time and space that may contradict their basic anarchist critique of the state. In examining these dissident Chinese thinkers in these last two chapters, we will also note the flaws in their ideas about how to overcome oligarchy that contradict a full anarchist critique.
The main limitation of Chinese neo-anarchist thinkers was that they had to protect themselves from the revenge of the Leninist Party-state, thus their critiques were necessarily based on secondary Marxist concepts of the state. As opposed to Marx's primary class paradigm of the state, these Marxist concepts all contained what Bob Jessop calls a "parasitic" view of the state, or what Robert Alford terms the "pathological" version of elite theory. These ideas were often very similar to those of East European Marxist dissidents who presented explicit "new class" arguments about the Leninist state, most famously Milovan Djilas, Georgy Konrad, and Ivan Szelenyi, and members of the Yugoslav "Praxis" group such as Svetozar Stojanovic.
While having the advantage of being able to claim Marxist credentials, using these secondary Marxist concepts that they themselves would never label "neo-anarchist" still leaves such intellectuals open to the standard Marxist attack on them as being influenced by the "ultra-leftist" and "petite-bourgeois" ideas, including those of anarchism, charges which such Marxist democrats must take pains to deny. Again, this chapter and the one that follows do not claim that Chinese thinkers who criticize the Leninist state are indeed full anarchists themselves or even much influenced by various positive anarchist visions for future society, but that they come on their own to a neo-anarchist critique based on the oppressive weight of the existing Leninist state that they see and feel every day.
This chapter first examines the dissident Red Guard group Shengwulian, which during the Cultural Revolution took advantage of Mao's seeming new class argument and early praise of the Paris Commune to condemn the rule of the "Red Capitalist class." Next we examine the debate between Chen Erjin and Wang Xizhe, who at the end of the Cultural Revolution and beginning of the reform era in different ways came to see the Party-state as becoming a new bureaucratic class. In the next chapter, we will examine Wang Ruoshui and other Communist Party intellectuals during the early reform years of the 1980s who, based on the inhumanity of the Cultural Revolution, resurrected the early Marxist ideas of alienation and humanism to argue that the proletariat could become alienated from the socialist state. The final type of neo-anarchist thought examined in the final chapter is that contained in the Chinese Asiatic mode of production debate of the early to mid-1980s, a time when some Chinese historians argued implicitly that the Leninist state was becoming a despotic entity ruling for itself rather than for the proletariat.
While presenting the neo-anarchist aspects of these Chinese thinkers, these two chapters try not to lose sight of their actual life situations, in which they struggled to find openings for dissent while keeping their jobs and ability to publish, and how they strained to keep the emoluments and minor privileges offered to cooperative intellectuals by the Leninist Party-state from blunting their neo-anarchist critiques.
**Shengwulian, Yang Xiguang, and Dissident Maoism**
The first major neo-anarchist critique of the state in the PRC occurred during the Cultural Revolution, which the current PRC regime officially says lasted from 1966 to 1976. Taking advantage of openings within official ideology that Party Chairman Mao Zedong himself at first seemed to initiate, as we saw in Chapter 6, groups of young Red Guards, junior high to college age youth whom Mao had called upon in 1966 to "bombard the headquarters" of the Party and State to oppose the "new bourgeoisie in the Party," began to take his call a step further to raise a genuine neo-anarchist critique of the state as a ruling class in and for itself.
As we noted in Chapter 6, during the early days of the Cultural Revolution, Mao himself had seemed to support Paris Commune style mass democracy as a way to oppose a growing "new class" within the Party-state elite. As we also saw in that chapter, many observers have long noted that in fact Mao stopped well short of a genuine new class critique similar to that of Milovan Djilas but instead only criticized a "small handful" of people within the Party who were taking China back on the capitalist road. Mao almost always argued that they did so not because of their special privileges or interests as state power holders, but because of remaining economic inequalities in society. In other words, he only opposed those arguing for modest market reforms and stopped short of calling for a struggle against a new power elite. But even before the Cultural Revolution, Mao had made clear to other Party leaders the limits of his anti-bureaucratic critique:
. . . The Communist Party is a prestigious one. Don't bring up any idea of a stratum . . . this will frighten and offend too many people . . . It's enough to call them just [isolated] elements or cliques . . .
In 1968, after factionalism between competing Red Guard organizations broke out across the country, we saw in Chapter 6 that Mao quickly reemphasized this anti-new class view, denouncing Paris Commune style forms as "extreme anarchy" and calling upon Red Guard units to accept in their place the so-called three-in-one revolutionary committees made up of members of mass organizations, returned bureaucrats who had been "remolded," and members of the army, who were to take up leadership within the committees. In response, some members of Red Guard groups felt betrayed and tried to maintain the Paris Commune model.
The leading example of such a "dissident radical" group, to use Andrew Walder's term, was the organization known as "Shengwulian," an abbreviation of the Chinese title for "Hunan Provincial Proletarian Revolutionaries Great Alliance Committee." This group of "more than twenty loosely affiliated Red Guard organizations" managed to publish at least three documents before they were attacked and suppressed by the regime. Under the pretense that it was not "Comrade Mao" (by not referring to him as Chairman Mao, perhaps demonstrating their ultra-egalitarian, anti-bureaucratic ideology) but other reactionary forces in the Party who had tried to abolish the Paris Commune-style models and replace them with the Revolutionary Committees, the group ignored the real limits to Mao's new class argument that we examined in Chapter 6. The group argued in its program that Mao's real goal in the Cultural Revolution was for "proletarian revolutionaries to overthrow the newborn and yet decadent privileged stratum of the bourgeoisie . . . and smash the old state machinery which serves the [new] privileged class of the bourgeoisie." In its program, Shengwulian followed the official regime line that only a "very few" cadres took the capitalist road. Nevertheless, the group severely condemned the idea that the Cultural Revolution was only about criticizing the crimes of individual leaders and dismissing them from their offices instead of "overthrowing the privileged stratum and smashing the old state machinery." The group bitterly criticized as well the failure of the Cultural Revolution to even barely touch "the class root which gave birth to the reactionary line, and to the bureaucratic structure which served the revolutionary line." Shengwulian's program pointed to the forming of Revolutionary Committees as merely a "reprint of the old political power" that was a reactionary departure from Mao's revolutionary theory. In effect, the group realized that the formation of Revolutionary Committees was the beginning of the end of the Cultural Revolution and the start of the reformation of the state machinery.
It was in its manifesto, "Whither China?," originally written in January 1968, that Shengwulian, under the leadership of a young Red Guard member who called himself Yang Xiguang, made its most radical and influential argument. Yang first detailed the history of the Cultural Revolution and what he saw as the betrayal of the "January Storm" 1967 upsurge of Red Guards by the representatives of China's "new bureaucratic bourgeoisie" in the "February Adverse Current" in that same year. Yang on the surface tried to stay loyal to Mao by focusing on Premier Zhou Enlai, who, as the "chief representative of China's 'Red' capitalist class," was the person responsible for setting up the revolutionary committees, which to Yang amounted to the reinstatement of the bureaucrats and a usurpation of power. Rather than the "small handful" of people in power taking the capitalist road, Yang argued that "90 per cent of the senior cadres had already formed a privileged class." The masses of the January Storm represented the truly revolutionary class of the Cultural Revolution, and by their own revolutionary experiences came to see that,
. . . this class of "Red" capitalists had entirely become a decaying class that hindered the progress of history. The relations between them and the people in general had changed from relations between the leaders and the led, to those between rulers and the ruled, and between exploiters and the exploited. From the relations between revolutionaries of equal standing, it had become a relationship between oppressors and he oppressed. The special privileges and high salaries of "Red" capitalists were built upon the foundation of oppression and exploitation of the broad masses of the people.
Though Yang recognized that "Comrade Mao" had decided to delay the dream of establishing people's communes, thus at least tacitly acknowledging that Mao had acquiesced in the formation of revolutionary committees, Yang claimed that Mao's intent in his injunction to the People's Liberation Army to "support the left" was to carry out cultural revolution in the armed forces. Thus, in perhaps his most radical statement, and the one that would ultimately get his group in trouble, Yang claimed that the "Red capitalist class" included not just civilian bureaucrats, but also members of the army:
It is now seen that the present army is different from the people's army of before Liberation [i.e., before 1949]. Before Liberation, the army and the people fought together to overthrow imperialism, bureaucratic capitalism, and feudalism. The relationship between the army and the people was like that of fish and water [following Mao's famous phrase]. After liberation . . . some of the armed forces in the revolution have not only changed their blood-and-flesh relationship with the people that existed before Liberation but have even become tools for suppressing the revolution.
Yang came to the conclusion that "any revolution must naturally involve the army," members of whom inevitably became part of the "Red capitalist class," and thus that "it was necessary to carry through to the end the Cultural Revolution in the field armies" as well as in the civilian bureaucracy.
Going beyond a neo-anarchist critique of the existing state, Yang in 1968 called for a violent smashing of the new bureaucratic class in the Party, a revolution that would set up Paris Commune-style or early Russian soviet-type organizations of direct, mass democracy in the place of the corrupt bureaucratic state. People would have to be taught that the true purpose of the Cultural Revolution was not just the dismissal of officials and the "purging of individual capitalist roaders" but that the capitalist roaders were a class implacably opposed to the cultural revolution and thus that a violent social revolution would be necessary. In order to carry out such a violent revolution, the masses would have to reject the official militia organizations as well as the army and to seize arms themselves.
Perhaps in the foreknowledge that anyone calling for such radical action in a Leninist Party-state would be denounced as favoring anarchism, Yang in "Whither China?" tried to distance himself from what he termed the "infantile leftist" doctrine of "one revolution" and from those who wanted to establish a full communist society immediately. He claimed that though a regime of the Paris Commune type was their goal, his group did not favor elimination of all class differences right away, but instead continued to see the need for stages in the revolution.
Despite their weak attempts to distance themselves from "infantile leftists," the calls of Shengwulian and other dissident radicals for direct revolution and for extending class struggle into the army frightened the rest of the state elite, if not Mao himself, and led Mao to allow Zhou Enlai and other Party-state leaders to launch a campaign against "ultra-leftism" and anarchism, as we saw in the previous chapter. In this campaign, Mao and other Party leaders put pressure on the "establishment Maoists" to denounce the "extreme anarchism" of "ultra-leftists" in the country, as exemplified by Shengwulian, who claimed to be followers of the official Maoists. The establishment Maoists included Mao's wife Jiang Qing, Chen Boda, Mao's former secretary and first leader of the Cultural Revolution Group in the Party, and Kang Sheng, the secret police chief, all of whom would later fall after Mao's death in 1976 in the campaign against the "Gang of Four." In the earlier 1968 campaign against "ultra-leftism," Shengwulian's documents were published and widely distributed in order to have everyone denounce and repudiate them. In effect, we saw that the establishment Maoists tried to use the campaign against "ultra-leftists" to legitimate their position and protect themselves from attack, but in retrospect one can see that this was a futile attempt since the campaign ultimately led to a wider purge of even official Maoists such as Chen Boda in 1970, just as Jiang Qing, Kang Sheng, and other establishment Maoists were later purged and themselves denounced as anarchists in 1976. The 1968 campaign then, as we saw in the previous chapter, was but one of many in the history of the PRC led by people who denounced as anarchist anyone who questioned whether the Party really ruled for the people, yet who themselves were later denounced as anarchists.
Though Yang Xiguang and other Shengwulian members were arrested and imprisoned, they had a profound influence on those members of the betrayed Red Guard generation who would later lead the Democracy Wall movement. Wang Xizhe, a leader of that latter movement whom we will examine below, claimed that Shengwulian was the forerunner of what he called the "thinking generation" that began to question the official line that the Party represented the masses, even as Wang made clear that he disagreed with the group's critique of Zhou Enlai and by extension other reformist leaders such as Zhao Ziyang. In effect, Wang was arguing that whatever their great foresight and courage, groups like Shengwulian were too entrapped in Cultural Revolution language of violent class struggle and failed to see the need for rule of law and institutional checks and balances, reforms that Wang and others in the Democracy Movement called for in the 1980s. Without such a realization of the need for tolerance and treating people with humanity, Wang and others argued in the 1980s, and by calling for further class struggle and "smashing" of people in power, China's Red Guard generation was trapped in an endless cycle of denunciation and violence that at best would only continually recreate and reinforce a despotic ruling body standing over the people.
Yang Xiguang himself later came to agree with this point of view based on his observations of ordinary people during his 10 years in prison, and under his original name "Yang Xiaokai" in fact became an advocate of market-based economic modernization and political reform, first after his release from prison in China and then as a noted classical economist teaching in Australia up till his death in 2004. Even as he changed his political beliefs about how best to go about challenging state autonomy, in effect Yang never gave up his neo-anarchist critique of China's Leninist Party-state. Whether as a violent Red Guard faction leader or as a neo-classical economist, one could argue that he departed in different ways from a full positive anarchist vision; nevertheless, in both periods Yang viewed institutions as strongly tending to rule for themselves, not the people they were originally designed to serve, and the Communist Party of China was no exception.
**Competing Dissident Visions of the New Class: Wang Xizhe and Chen Erjin**
As noted above, many members of the self-proclaimed "thinking generation," which arose among educated and ex-Red Guards in the late stages of the Cultural Revolution and which reached its height in the Democracy Wall Movement of 1978–81, openly paid homage to Shengwulian and the dissident radicals of the early Cultural Revolution. Nowhere was this link clearer than in their ideas of the PRC as being dominated by a "bureaucratic class."
The first salvo of this generation came in November 1974, during the later stages of the Cultural Revolution, when a group of young former Red Guards writing under the collective pseudonym of "Li Yizhe" (based on a combination of the names of three of its four members) put up a small character wall poster manifesto in downtown Guangzhou denouncing the "feudal fascist" nature of the "Lin Biao system" and the lack of true "socialist democracy" in China. Taking advantage of the state-sanctioned campaign then raging that denounced as counterrevolutionaries both the ancient philosopher Confucius and Lin Biao, the Vice-Chair of the Party, Vice-Premier, and Mao's designated successor, who had been killed in 1971 in a plane crash after supposedly leading a failed coup against Mao, Li Yizhe presented a more radical critique of the whole "Lin Biao system." Their critique was really aimed at the abuses of other establishment Maoists in the regime, later to be denounced as the "Gang of Four." In their wall poster essay, "On Socialist Democracy and the Legal System," the group extended Mao's critique of the "new bourgeoisie in the Party" to argue that the "privileged stratum" of the Party led by Lin Biao attempted to "implement a feudalistic socialist-fascist despotism." While starting from the argument that it was vestiges of economic inequality and special privileges that created this new class, Li Yizhe argued that this force of new gentry ( _wenren_ ) had vested _political_ as well as economic interests and privileges and existed objectively based on the "traditions formed by several thousands of years of feudal despotism" that "stubbornly maintain their stronghold over thought, culture, education, law, and virtually every other sphere of the superstructure." In other words, while claiming to support the ideals of the Cultural Revolution in fighting counterrevolutionaries within the Party, Li Yizhe began a line that was to become official during the early reform era, that vestiges of feudalism, not a return to capitalism, were the main threat to China's socialist revolution.
In a preface to a later edition of their manifesto, the group claimed that members of this new "bourgeois class" maintained and expanded its power by "turning public into private [property]" and turning their power into "special economic and political privileges" that they "extended without limitation to their family, friends, and relatives." Furthermore,
. . . they buttress and sustain a clique of "new nobility," a force which stands separate from the people and whose interests come into opposition with the people's.
For Li Yizhe, the "preconditions for the Lin Biao system" were rooted in the "vicious practices of the dictatorial arbitrariness of the feudal era" that were "fixed firmly in the minds of the people as well as in those of the average members of the Communist Party." The members of Li Yizhe at first took pains to claim loyalty to Chairman Mao and followed the line that Mao had long known and suspected Lin Biao's treacherous nature. Despite their claim to be upholding Maoist traditions of the Cultural Revolution, the group called for political reform and the rule of law, not mass violent upheaval, as the way to overcome feudal fascism.
Though initially supported by members of the Guangdong provincial leadership such as Zhao Ziyang, the later reformist national leader of the PRC who in the earlier era hoped to resist and overcome the establishment Maoists, the members of Li Yizhe were eventually arrested, tried, and imprisoned as the Maoists temporarily regained the upper hand. After the second return of Deng Xiaoping to the Party leadership in 1977–8, members of the Li Yizhe group were eventually released from prison and rehabilitated. Most of the group's members tried to affect the new regime from within, but one of its members, Wang Xizhe, almost immediately joined the new Democracy Wall movement that formed around the young workers and members of the Red Guard generation in Beijing and other major cities.
In interviews around this time Wang Xizhe claimed to be the primary author of Li Yizhe's main essays, a claim Li Zhengtian, another member of the group, partially contested. In 1979, as Deng gained ascendancy within the post-Mao coalition at the top of the CCP and as remaining Maoists within the Party were about to be purged, Wang penned another essay under his name alone, which he termed the sequel to the group's original wall poster.
In this essay "Strive for the Class Dictatorship of the Proletariat" Wang attempts to place the "feudal vestiges" argument about the bureaucratic class within a Marxist, class-based explanation. Wang claims that the rise of "the dictatorship of the advanced stratum of the proletariat" is inevitable in a socialist nation striving to survive within a world capitalist economy. In such a nation, where, given the low level of development of the productive forces and thus the low "cultural level and capacity for management of the entire proletariat," Wang argues that ". . . it becomes necessary for the advanced stratum of the proletariat (the Communist Party) to carry out exclusively the management for their class." The danger in this division of labor, Wang claims, citing Lenin, is that "it depreciates the political power of the soviet and causes the revival of bureaucratism" (here conveniently ignoring that Lenin only decried "bureacratism" and never claimed that the Soviet political elite constituted a new class). Though claiming to support the idea of rule by the dictatorship of the proletariat, Wang wants to ensure the rule of law and democratic accountability in order to gradually transform the "dictatorship of the Party . . . into the realization of the dictatorship of the proletariat by an organization of the entire proletarian class," by which he means workers' democratic control over management along the lines of what he claimed occurred in Yugoslavia. Without such practices, Wang warns,
. . . this dictatorship of the Communist Party step by step sets itself free from the control of society and becomes a force above the society; the original advanced stratum of the proletariat (especially its leadership group) metamorphoses into the antithesis of the proletariat, and the original dictatorship of the advanced stratum of the proletariat becomes the dictatorship of "the Communist bureaucrats' holding up the sign post of the Communist Party."
While publishing his manifesto in unofficial journals outside the control of the Communist Party, Wang remained firmly within what Kjeld Erik Brodsgaard termed the "socialist democratic" wing of the Democracy Wall Movement that,
. . . favored democratic reform and progress within the framework of the present political and economic framework of China . . . [and thus] never really questioned the "socialist" foundation of China, the dictatorship of the proletariat, and the leadership of the Party based on Marxism-Leninism-Mao Zedong Thought.
Though based outside the Party, from 1978 to 1980 Wang in effect hoped to form a pressure group that would help the reformers inside the Party overcome their bureaucratic opponents, and thus Wang took pains to demonstrate his Marxist credentials. Nevertheless, the critique of the "new bureaucratic ruling class" that Wang continued from the first Li Yizhe manifesto, which was in turn influenced by Shengwulian, became an accepted part of the discourse of all wings of the Democracy Wall movement. This would include what Brosgaard terms the "abolitionists," the wing of the movement (exemplified most famously by Wei Jingsheng) which rejected Marxist-Leninism and concluded that, far from trying to prove their loyalty to the Communist revolution, "the revolution should be reversed in order to destroy the systemic foundation of a new class, ruling in the name of socialism."
In a long essay he wrote during the later years of the Cultural Revolution and published in one of the Democracy Wall journals in 1979, originally entitled "On Proletarian Democratic Revolution," another member of the "socialist democrat" wing of the Democracy Wall activists writing under the name of Chen Erjin tried consciously to continue the new class argument from the Cultural Revolution period.
Unlike Wang, Chen did not reject Maoism and the Cultural Revolution but claimed to take over what he saw as its democratic spirit and goals while overcoming its inherent limitations. In the midst of his unique and idiosyncratic blend of Paris Commune-style mass democracy and Western influenced institutional checks and balances, in which Chen seemed to favor some kind of violent "second revolution" leading to the founding of a second communist party to compete with the original CCP, Chen launched his own critique of the "bureaucrat-monopoly privileged class." In Chen's view, although the transformation to public ownership of the means of production was a crucial step forward in the socialist revolution, the change to public ownership also began a new, irreconcilable contradiction that would eventually necessitate a new revolution. This contradiction was between "the highly organized and politico-economically unicorporate social production under public ownership" ( _gaodu zuzhide zheng-jing yitihua gongyouzhi shenghui shengchan_ ) and the coercive monopolization of power by the minority, which Chen's translator explains as "a form of socialized production which proceeds under a form of public ownership, and is characterized by a fusion, into a single and highly-organized whole, of the formerly distinct spheres of the political and the economic." In other words, similar here to Wang's argument, the "new bureaucratic class" arises as a side effect of socialism in a backward country, though Chen thinks this is a necessary step while Wang came to believe it was a tragic development, as we will see below.
While claiming to support the socialist revolution, Chen recognizes the irony that in the "workers state" the workers lost the right to change jobs or move where they want, and thus under this system "are no longer free but only 'workers within organization'" who thus "forfeit their free and independent nature." Instead of a transition to a classless society, there has been a "coercive monopolization of power by a minority" that has but "established new classes, new conditions of oppression, new forms of struggle in place of the old." Although he incorporates the Cultural Revolution critique of a necessary, even violent struggle against this new class, Chen changes the terms from "capitalist roaders" within the Party to a struggle against privilege and revisionism within the Party and also changes the nature of the system that would replace Party dictatorship.
In a strange and idiosyncratic twist, Chen contrasts his idea of a necessary revolution against the new class with the ideas of those "reformists" within the Party who accepted public ownership and control by the "unicorporate elite" but would fight revisionism through such measures as mass campaigns to restrict "bourgeois right." As Munro makes clear in his introduction to Chen's manifesto, this means that in effect Chen identifies the establishment Maoists of the Cultural Revolution as the reformists, not their rivals favoring modest market reforms who were purged during that movement and later returned under Deng Xiaoping. The "reformists," Chen argues—here prefiguring later denunciations of the Gang of Four—had a "petite bourgeois mentality" and may possess "revolutionary fanaticism" but "also may turn to the right ideologically." As members of the political elite, they "either remain subject to the restriction by the interests of the bureaucrat class, or else drool at the prospect of acquiring those vested interests themselves." As a result,
. . . they are placed in extremely perilous situation, being not only divorced from the mass of the people but at the same time hated by the bureaucrat class as a whole. At the decisive juncture, the bureaucrat class will surely drown them in their own blood.
By praising the workers' sense of mastery but not pushing for a thoroughgoing revolution against the new bureaucratic class, the "reformists" on the one hand serve in effect to negate "the rule of privilege; but on the other to reinforce 'unified leadership' by the bureaucrat class—thereby in effect reinforcing workers' slavelike position of unconditional subordination."
In effect, one could argue, Chen was still taking the standpoint of the dissident Maoists who felt betrayed and sold out by Mao and his coterie for turning against the Paris Commune model of mass democracy late in the Cultural Revolution and for sending the dissident radicals to the countryside or to jail. On the other hand, Chen and the dissident Maoists remained suspicious of the rising coalition of Stalinist bureaucrats and market reformers of the Deng era. Chen's loyalty to a thorough neo-anarchist theory of the state perhaps helps explain his idiosyncratic positive program that combined calls for violent revolution against all wings of the bureaucratic class with a proposed new system of two communist parties alternating in power within a system of rule by law and checks and balances of three or more branches of government. Whatever one thinks of the lack of anarchism in his positive program, Chen clearly recognized, it seems to this observer, that even a (to him necessary) violent revolution against the bureaucratic class would only eventually result in the formation of a new oligarchy as bad or worse than the old, and thus some kind of institutional checks on that potential oligarchy were needed.
Wang Xizhe reacted strongly against Chen's view that the class struggle and anti-bureaucratic language of the Cultural Revolution should be maintained. In his long essay written in 1980, "Mao Zedong and the Cultural Revolution," against "Comrade Jin Jun" (whom Robin Munro and others identify as Chen Erjin), Wang argued that Mao and his establishment Maoist followers were neither reformists nor genuine socialist revolutionaries but only "agrarian socialists" intent on "placing the national economy under militaristic command" that would call for "a supreme militaristic authority." From his attacks on Marshall Peng Dehuai, who had dared to criticize the disastrous and harmful failures of Mao' Great Leap Forward, to the brutal and violent purges of anyone who dared to criticize his policies in the Cultural Revolution, Mao was in effect a super-Stalinist autocrat:
Mao Zedong's reactionary trait was precisely that he was not satisfied with the degree of autocracy and of centralization of power that had already been attained by the Stalinist-modeled Party and state. He demanded more autocracy and more centralization, but the democratic reform faction within the Party blocked his attempts. This obstruction developed to a degree so serious that it even threatened his continued ride on the neck of this Party as Chairman. Thereupon he decided to attack this Party, smash this Party, and establish a Mao Zedong Fascist Party.
In other words, Wang argued that opponents of Mao within the Party who favored more rule of law and market reforms were indeed the genuine reformers, whatever their limits and however much resistance they faced from remaining Stalinist bureaucrats who favored ending the violent upheaval in society but not opening up the Leninist state. Mao, despite his few words against bureaucratism, in fact only used Paris Commune rhetoric to get rid of the reformers in the state who would check his power. The Cultural Revolution was not about mass democracy in the end, but about "worship of Mao Zedong as an individual" and "revering Mao Zedong as an emperor." Though at first tolerating Paris Commune style rhetoric, in the end Mao turned against Paris Commune models, ridiculed the idea of masses electing officials, and only desired to build up his own autocratic power against officialdom, akin to the efforts of Zhu Yuanzhang the late fourteenth-century peasant rebel who founded the Ming dynasty, or Hong Xiuquan the leader of the Taiping peasant rebellion of the nineteenth century, both of whom became autocratic tyrants and launched violent purges of their officials.
Though perhaps his argument was not completely fair to Chen himself, since as we saw above Chen also criticized Maoists within the Party, Wang was trying to tell members of his own "thinking generation" that they had to make a complete break with the Manichean ideas of the Cultural Revolution of a violent struggle between good and bad class forces, and instead had to stress the rule of law, the art of compromise, and the gradual evolution of peaceful, democratic checks on authority. In effect, one could argue, despite his own lack of an anarchist positive program, Wang was calling for the neo-anarchist critique to be extended to Mao himself and the whole Maoist system of putting faith in top authority figures. Wang's weakness, from a full anarchist perspective, was his faith in a reformed single Party system, in which pressure from extra-Party movements such as the Democracy Wall would support reformist leaders in the Party against "opportunist bureaucrats." At one point in "Mao Zedong and the Cultural Revolution" Wang even seemed to downplay Deng's arrest of Wei Jingsheng, though Wang argued that Deng would come to regret turning against the "thinking generation." Ironically, as the crackdown on Democracy Wall intensified and Wang himself came under pressure (ultimately he too was arrested and imprisoned for many years before being exiled to the West), in a late 1980 interview Wang seemed to accept the need for some kind of multiparty checks on the Communist Party. Even during a gradual transitional stage to fuller democracy, Wang argued, "there may arise a privileged stratum or clique which benefits from seeking to prolong this stage. Such a stratum or clique will never trust the popular masses to stand on their own two feet and to exert their democratic rights on their own behalf," and thus he continued to see a need for pressure for democracy from the popular masses, perhaps similar to Michels' hope for continued democratic "waves" to check the tendency toward oligarchy.
In 1981 the crackdown on Democracy Wall activists extended beyond the "abolitionists" and started to include the socialist democrats, such as Wang and Chen, who were both eventually arrested and imprisoned. Perhaps recognizing that the end was near, in the late interview noted above, Wang now expressed agreement with Wei Jingsheng that "the fifth modernization," democracy, was needed in order to overcome "a new form of 'alienation'" under Stalinism, where "the people work more and more but have fewer and fewer democratic rights." While reflecting his even bolder attitude and expressions of support for the rights of the abolitionists such as Wei, this statement also reflects Wang's links with Marxist intellectuals within the reform camp of the CCP who had also been returning to earlier concepts of Marx in order to call for political as well as economic reform. As in this interview, in his late article "The Direction of Democracy," Wang Xizhe also called for a "renaissance" of Marxism similar to that in Hungary and Yugoslavia by resurrecting long-ignored Marxist concepts such as alienation to build a "proletarian humanism" that would overcome the "obsolete" practices of Stalinism, which included an unchecked Party-state. As the Democracy Wall activists were rounded up, it fell to the inner Party democrats to take up this neo-anarchist critique.
**Notes**
**** Michels, _Political Parties: A Sociological Study of the Oligarchical Tendencies of Modern Democracy._
**** Ibid., 325.
**** Ibid., 326–7.
**** Wolfgang Mommsen, "Roberto Michels and Max Weber: Moral Conviction Versus the Politics of Responsibility," Chapter 6 of Mommsen, _The Political and Social Theory of Max Weber_ , 87.
**** Jessop specifically states that the anarchists had a "parasitic view of the state" while Alford fits nineteenth-century anarchist thought within the "utopian" version of what he calls the "class" paradigm of the state. See Bob Jessop, "Recent Theories of the Capitalist State," 353–73; Robert Alford, "Paradigms of Relations between State and Society," reprinted in Hall, 67. In this and the following chapter we revise Alford's analysis by separating anarchism's critique of the state from some anarchists' revolutionary program, and thus instead locate the critique within Alford's "pathological" version of the "elitist" paradigm.
**** Milovan Djilas, _The New Class: An Analysis of the Communist System;_ Gyorgy Konrad and Ivan Szelenyi, _Intellectuals on the Road to Class Power_ ; Svetozar Stojanovic, "Marxism and Democracy: The Ruling Class or the Dominant Class?" For summaries of new class arguments of Eastern European dissidents as well as West European and American intellectuals, see Gil Eyal, "The Idea of the New Class," Chapter 1 of Eyal, _The Origins of Post-Communist Elites: From Prague Spring to the Breakup of Czechoslovakia_ , 1–34; and Bill Martin and Ivan Szelenyi, "The Three Waves of New Class Theory."
**** Mao, _Mao Zedong sixiang wansui!_ (1969), 582–3, n. 22, translated in Anita Chan, "Images of China's Social Structure: The Changing Perspectives of Canton Students," 316, n. 25.
**** Walder, "Cultural Revolution Radicalism," 58–61.
**** For studies of the Shengwulian group and its suppression, see Klaus Mehnert, Peking _and the New Left: At Home and Abroad_ ; Peter Moody, Jr., _Opposition and Dissent in Contemporary China_ , 202–9; and Jonathan Unger, "Whither China?: Yang Xiguang, Red Capitalists, and the Social Turmoil of the Cultural Revolution." The writings of Shengwulian are translated in these volumes, often based on the original translation of their writings by US government intelligence agencies. See especially the group's main manifesto, "Whither China?" _Guangyin Hongqi_ , 5 (March 1968), translated in SCMP 4190 (June 4, 1968), 1–18, reprinted in revised form in Mehnert, 82–100. We will cite other writings of this group as well in this chapter.
**** Unger, 22.
**** Shengwulian, "Whither China?" translated in Mehnert; also see Shengwulian, "Shengwulian's Resolutions on Several Problems in the Current Hunan Great Proletarian Cultural Revolution," _Dongfeng Chanbao_ (East Wind Combat News) (Guangdong), 19 (February 29, 1968), based on resolutions passed by the preparatory group for Shengwulian on December 21, 1967, translated in Mehnert, 80.
**** Shengwulian, "Our Program," _Guangyin Hongqi_ (Guangdong Printing System Red Flag), 5 (March 1968), 3, translated in SCMP 4174 (May 9, 1968), 10–13; also translated in Mehnert, 75–6.
**** Shengwulian, "Whither China," translated in Mehnert, 87–8.
**** Ibid., 85.
**** Ibid.
**** Ibid., 89.
**** Ibid., 86.
**** Ibid., 91.
**** Ibid., 91–2.
**** Ibid., 86, 98.
**** For accounts of this campaign, see Unger, 29–32; Mehnert, 20–5; MacFarquhar and Schoenhals, 221–38; and Barry Burton, "The Cultural Revolution's Ultraleft Conspiracy: The 'May 16' Group."
**** Wang, "Mao Zedong yu wenhua dageming" (Mao Zedong and the Cultural Revolution), in _Qishi niandai_ (The Seventies), 133 (February 1981), translated in Anita Chan, Stanley Rosen, and Jonathan Unger (eds), _On Socialist Democracy and the Legal System: The Li Yizhe Debates_ , 252 and cited in Unger, 4, 36, n. 20.
**** See Yang Xiaokai and Susan McFadden, _Captive Spirits: Prisoners of the Cultural Revolution_.
**** See Carol Lee Hamrin, "Yang Xiaokai," _Biographical Dictionary of Chinese Christianity_ , online at www.bdcconline.net/en/stories/y/yang-xiaokai.php/.
**** Li Yizhe, "Guanyu shehuizhuyide minzhu yu fazhi" (On Socialist Democracy and the Legal System), in Qi Hao (ed.), _Guanyu shehuizhuyide minzhu yu fazhi_ (On Socialist Democracy and Development), translated in Chan, Rosen, and Unger, _On Socialist Democracy and the Legal System:_ 31–86.
**** Ibid., 61.
**** Ibid., 68.
**** Ibid., 78.
**** Ibid., 75.
**** Ibid., 36.
**** Ibid., 42.
**** Besides the Chan et al. and Rosen works cited above, major studies of the Democracy Wall Movement that often contain translations of key texts include Kjeld Erik Brodsgaard, "The Democracy Movement in China, 1978–1979: Opposition Movements, Wall Poster Campaigns, and Underground Journals"; Chen Ruoxi, _Democracy Wall and the Unofficial Journals_ ; Gregor Benton, _Wild Lily, Prairie Fire: China's Movement for Democracy,Yan'an to Tian'anmen, 1942–1989_ , 157–263; Stanley Rosen, "Guangzhou's Democracy Movement in Cultural Revolution Perspective"; James Seymour, _The Fifth Modernization: China's Human Rights Movement, 1978–1979_ , and James Tong (ed.), "Underground Journals in China, Parts I and II."
**** Stanley Rosen (guest ed.), "The Rehabilitation and Dissolution of 'Li Yizhe'."
**** Ibid., 111–13.
**** Wang Xizhe, "Wei wuchanjieji zhuanzheng er nuli" (Strive for the Class Dictatorship of the Proletariat), translated in Chan, Rosen, and Unger, _On Socialist Democracy and the Legal System_.
**** Ibid., 141.
**** Ibid., 140.
**** Ibid., 141–2.
**** Brodsgaard, 768.
**** Ibid., 769.
**** Chen Erjin, _Lun wuchanjieji minzhu geming_ (On Proletarian Democratic Revolution), _Siwu luntan_ (April fifth Forum), 10 (June 1979), translated in Chen, _China: Crossroads Socialism: An Unofficial Manifesto for Proletarian Democracy_ , _passim._
**** Ibid., especially Chapter 6, 110–19.
**** Ibid., 87.
**** Ibid., 87, translator's note.
**** Ibid., 115.
**** Ibid., 119.
**** Ibid., 26–7.
**** Ibid., 123.
**** Ibid., 124.
**** Ibid., 48, n. 32.
**** Wang, "Mao Zedong and the Cultural Revolution," in Chan et al., 195.
**** Ibid., 206.
**** Ibid., 226.
**** Ibid., 218.
**** Ibid., 236. Andrew and Rapp, _passim_ make the same argument, acknowledging their debt to Wang Xizhe on page 9.
**** Wang, 248.
**** See Wang, "An Interview with Wang Xizhe on the Democratic Movement," translated in abridged form as "Democracy and Chinese Communism," 65.
**** Ibid., 66.
**** Ibid., 38.
**** Wang, "Minzhude fanxiang" (The Direction of Democracy), translated in _The Undercurrent_ , 11.
9
Inner Party neo-anarchist critiques of the Leninist Party-state
**Introduction**
With the crackdown on Democracy Wall, Deng Xiaoping had the "four bigs" removed from the state constitution (the "right to speak out freely, air views fully, hold debates, and write big character posters") and in their place announced a new line of the four cardinal principles that all subjects were required to uphold, including Marxism–Leninism-Mao Zedong Thought, socialism, the people's democratic dictatorship, and leadership of the CCP. For the rest of the 1980s, it fell to inner Party reform intellectuals (whom we will refer to as "Marxist democrats," to adapt the term of Edward Friedman) to find ways to pursue the cause of political reform within these harsh limits. Deng Xiaoping and his allies tolerated such intellectuals to the extent that they needed their help against Maoist and Stalinist-influenced colleagues in the ruling state elite resistant to market reforms, something hard to justify within orthodox Marxism. After all, if there is no clear blueprint in the writings of Marx and Engels for Stalinist-style central planning and the command economy, there is nevertheless also a strong antipathy to markets and the "commodification" of the economy. As a result, such reform-minded intellectuals were allowed and encouraged to study market socialist reforms in places such as Hungary and Yugoslavia. While carrying out this role for the Leninist regime, such intellectuals also pushed for their own interests in increased intellectual freedom by borrowing political reform ideas of Marxist democrats in those same regimes.
As with the Democracy Wall activists, at first such inner Party Marxist democrats were also aided by the Deng era call to "seek truth from facts," sometimes put as to place "practice as the sole criterion of truth," and the main enemy to be fought as feudal vestiges from the past, not capitalist elements in society.
While there were many different, creative ways that Marxist democrats tried to keep alive calls for political reform and democratization within Marxism, at least two different Marxist routes made possible the continuation of neo-anarchist critiques of the socialist state ruling for itself and not the proletariat: the writings of the early Marx on humanism and alienation and his concept of the Asiatic Mode of Production (AMP) from his middle period. Each route had advantages and disadvantages for the Chinese Marxist democrats, though, in the end both routes were shut off by the end of the decade as Deng reached a deal with his Stalinist colleagues in the state elite to repress attempts at meaningful political liberalization.
**Wang Ruoshui and Alienation of the Socialist State**
The first route, returning to the writings of the early Marx on humanism and alienation, had many adherents, termed by some the "Party of Humanism" or the "alienation school" ( _yihualun pai_ ). We will focus in this section on by far the leading exponent of that school, the prominent philosopher and deputy editor of the CCP flagship newspaper _Renminribao_ (People's Daily), Wang Ruoshui.
In seminal articles Wang published in the early 1980s, including many in the popular press, Wang argued, borrowing from the East European debates, that Karl Marx did not eliminate in his later works his sentiments in favor of humanism as a socialist project that he put forward in what is known as his _Economic and Philosophic Manuscripts of 1844_ , not published in the Communist world until 1932. Instead Wang argued that Marx subsumed them in later, more economic materialist language. For Wang, the goal of socialism should still be that of Marx in his early work: not just state ownership of the means of production in the name of the workers but control of workers over their own work. Most especially in this early work, Marx took over the concept of alienation from Hegel and Feuerbach, turning it from Hegel's alienation from a pure idea and from Feuerbach's alienation from man's essential nature or essence into economic alienation of classes from their own labor. Wang did point out that Feuerbach's idea of humans as creating God in their own image and then becoming a slave to Him had clear echoes in the Cultural Revolution when people were called upon to "think of Chairman Mao in everything, do everything for Chairman Mao, serve Chairman Mao in everything, follow Chairman Mao in everything."
"Do everything for Chairman Mao": who would Chairman Mao do everything for? Chairman Mao should have been doing things for the people, everything should have been for the people, this is a basic principle. It turned out in fact that the people did things for the leader, everything was for the leader. . . . Could "follow Chairman Mao in everything" mean anything but an autarchy ( _yiyan tang_ )? What was it but an inversion of the relations between party, leader and people?
For Wang, this type of alienation was ". . . closely connected with the influence of Chinese feudal mentality."
Beyond intellectual or spiritual alienation, Wang argued, there was the problem of political alienation. Trying to protect himself from the inevitable attack on him as an anarchist that was bound to be leveled by his orthodox Marxist opponents in the regime, Wang admitted that while the issue of political alienation was first raised by the anarchists and "hence to overcome alienation, one should take anarchism into account," Marx and Engels also maintained the concept of political alienation under the old society, when the organs of state, in Engels words, "in pursuance of their own special interests, transformed themselves from the servants of society into the masters of society." In his most radical statement, which was at the heart of the reason why the Party-state made him the leading target of the "Campaign against Spiritual Pollution" in 1983, Wang argued that political alienation could still exist after the revolution:
. . . Is there still alienation under socialism? Socialism is supposed to abolish alienation, but has it done so in fact, or does alienation still exist? I think we should admit that practice has proven that alienation still exists. Not only is there intellectual alienation, there is also political and even economic alienation.
. . . when the government turns into an overlord, refusing to accept the people's control and turning into an alien force, this is alienation, alienation in politics.
For Wang, this problem of alienation could only be solved as Engels suggested, by adapting the model of the Paris Commune (though not by violent revolution as for Shengwulian), that is, by having the socialist state institute universal suffrage that would elect officials and have them subject to instant recall and by reducing "special treatment and privilege" of state officials, if not the low salaries that Engels called for. Thus Wang stayed within the framework of the one Party-state, even if a reformed one subject to popular checks, and could plausibly claim not to have departed from Deng's Four Cardinal Principles.
Such a claim did not protect him in the end, as he became the leading target of the Anti-Spiritual Pollution Campaign and was purged from his post at _Renminribao_. At the end of the campaign, Party propaganda chief Hu Qiaomu made a speech to the Central Party School, which was reprinted in the popular press under his byline. This speech contained extensive criticism of the humanism and alienation school in general and a direct attack on Wang Ruoshui's views in particular. Demonstrating the main point of the previous and current chapters, that claiming the socialist state can come to rule for itself is the main taboo that must not be crossed under Leninist rule, Hu warned toward the end of his speech that those who advocate the theory of alienation, especially those who concluded "that alienation existed everywhere in the political, economic, and ideological spheres of socialism and that its fundamental cause was not in another area, but precisely in the socialist system itself" could (perhaps inadvertently) lead people to favor "abolishing all social political powers, social economic organizations, ideological authority, and centralism and discipline" thus to "openly publiciz[e] anarchism, absolute liberalism, and ultra-egoism."
Significantly, Wang refused to make a self-criticism during the Anti-Spiritual Pollution Campaign and never recanted his belief in socialist alienation. As the intensifying campaign threatened to undermine domestic and international confidence in market reforms, Deng first limited what counted as spiritual pollution and then wound down the whole campaign. Thus, Wang managed to survive the 1983 campaign against him, and as reform temporarily returned to the agenda, from 1985 to 1986 republished his main works on humanism and alienation in books of his essays. Supposedly without his approval, his rebuttal to Hu Qiaomu appeared in a Hong Kong periodical where he defended his position that humanism can be found in the later works of Marx and that ideological alienation at least still exists in socialist society.
In 1987, after a round of student demonstrations that Party elders blamed on the liberal policies of CCP General Secretary Hu Yaobang, who had been the main protector of Wang Ruoshui and other Marxist democrats, a new campaign against "bourgeois liberalization" was launched, at the end of which Hu was removed from his post and Wang was expelled from the Party. In the late 1980s, as the reformers within the Party were losing out to advocates of increased Party control over intellectuals, Wang published new essays that at first did not repeat his radical critiques but only called for respect of civil rights and the constitution, basing himself firmly within remaining official Party policies that denounced the personality cult of Mao during the Cultural Revolution and that claimed to establish a socialist legal system. However, in another article published in Hong Kong again supposedly without his permission, Wang again answered Hu Qiaomu's 1984 attack on humanism and alienation and raised anew the question of socialist alienation. Wang claimed that his views were firmly in line with former Premier and now General Secretary Zhao Ziyang's report to the Thirteenth National Party Congress where he criticized the outdated nature of the PRC political system based on "large-scale mass movements" and intensified "mandatory [central] planning." While Zhao included this criticism of past practices only as part of his call for more market reforms and for only modest political structural reform, Wang took the opportunity to link the issue of personality cults and mass campaigns to alienation under socialism:
Such a political structure cannot prevent personality cult[s]; moreover it can easily engender bureacratism, autocratic work style, privileges, infringements on the rights of rank and file party members and ordinary people, and other negative phenomena. (I regard all such things as demonstrations of alienation) . . .
Thus, despite being expelled from the Party, in articles of the late 1980s Wang became more insistent on his ideas, and even started departing from belief in Leninism, though he still called himself a Marxist up until his death from lung cancer in 2002.
David Kelly concludes that although Wang Ruoshui's perception of the evils left over from the Mao period was similar to that of Li Yizhe and the Democracy Wall extra-Party critics, his own diagnosis and remedy for the problem differed, since he referred only to "bureaucratic privilege and the difficulties of implementing democracy" and not a "new bourgeoisie" (or bureaucratic class) and tried to stay within Deng's four cardinal principles, at least up to 1988. While it may be true that Wang's solutions to the problem of socialist alienation were largely unspecified or moderate at best, this author would argue that his critical views still lay firmly within a neo-anarchist paradigm of the state, one that sees the tendency of organizations, especially coercive ones with a monopoly of power, to rule for themselves. As such, however moderate in practice were his proposed solutions compared to the Democracy Wall extra-Party activists, Wang broke the taboo of all taboos in a Leninist system with his critique of the socialist state and thus could not be allowed to propagate his views much further after 1987.
**The Chinese Asiatic Mode of Production Debate in the Early 1980s**
Although less well-known perhaps than alienation and humanism, there is another Marxist concept that contains the seeds of a neo-anarchist paradigm of the state and can be used to call implicitly for democratization, namely the Asiatic Mode of Production (hereafter AMP). The advantage of the AMP over alienation is, first, that the term is initially confined mostly to the historical profession and is not one that obviously lends itself to articles in the popular press and, second, that if challenged one can always claim to be talking about past states and not the current Leninist regime. The disadvantages of the AMP concept are, first, that it may be so esoteric that it may be hard to spread awareness of the concept beyond a small academic circle and, second, that the concept carries political baggage, both because of its association with the idea of a stagnant or unchanging Asia versus a more dynamic West and because of its use in the 1950s by the Marxist turned fierce anti-Communist Karl Wittfogel, who linked a version of the concept he termed "oriental despotism" specifically to contemporary "totalitarian" dictatorships in Russia and China, as we will see below. Nevertheless, for Marxist democrats willing to try to overcome these disadvantages, the AMP presents a clear challenge to the primary, class paradigm of the state while still keeping within a professed Marxist outlook, which thus makes it possible for intellectuals to bring into doubt whether the socialist state always represents the interests of the proletariat without being accused (right away at least) of having "bourgeois liberal" tendencies.
Tons of ink have been spilled among Marxist and non-Marxists alike all over the world concerning the AMP, somewhat reduced by the fall of Communism in the USSR and Eastern Europe. Suffice it to say here that the concept can be found most extensively, though not exclusively, in Marx's work, the _Grundrisse der Politischen Okonomie_ (Foundations of the Critique of Political Economy), a manuscript he completed in 1857–8 in preparation for writing _Das Kapital_. The _Grundrisse_ was not published until 1939–41 in the USSR. The AMP appears on one chapter of that work entitled "Precapitalist Economic Formations." To sum up his views on the AMP in that chapter, though a matter of heated debate, in general Marx may have argued that some pre-capitalist societies may not be examples of either primitive communist, slave, or feudal modes of production, but instead examples of a distinct Asiatic mode where first, the centralized despotic state claims to own all land based on combining rent and taxes, second, where it stands over isolated, self-sufficient rural village communities in which production is based on the land mixed with handicraft production, and third, there is cyclical, stagnant development. Marx also seemed to indicate that the state carries out large-scale irrigation and other hydraulic projects and public works projects (whether necessary and real or only taking credit for the work of lower communities) and rules from essentially administrative, rentier cities. The political significance of the concept is that if a state can rule for itself rather than private economic classes at one or more times and places in human history, then it could also rule for itself at later points and places, such as in Leninist regimes, especially those that had a past history of "Asiatic" state forms.
Once Soviet intellectuals became aware of Marx's AMP concept (as well as similar ideas in Engels, Lenin, and other Marxist thinkers), a great debate began in the USSR over whether or not the AMP was a genuine Marxist concept. Stalin settled the issue by fiat in 1931, denying that Marx ever held to the concept and that all societies must universally pass through the same stages in history, from primitive communist, to slave, feudal, capitalist, socialist, and communist modes of production, thus announcing as Communist dogma what later scholars term the universal unilinear schema of history. Nevertheless, the AMP concept was revived in Western and Eastern Europe during the Cold War by Marxist thinkers trying to open up room for limited critical thinking about the Leninist state. Just as with the Marxist concepts of humanism and alienation, therefore, the AMP concept became ripe fodder for Chinese thinkers in the early years of the reform era when they were allowed to study diverse strands of European Marxist thought.
The first main problem such Chinese thinkers had to overcome was the use of the AMP concept by Karl Wittfogel to denounce "totalitarian" systems in the USSR and China. In his magnum opus _Oriental Despotism_ , Wittfogel claimed that Communist systems often took root in societies with a "despotic" past based on the need for centralized bureaucracies to organize massive "hydraulic" projects in arid regions. Even more problematic for Marxist democrats wanting to use the AMP, Wittfogel claimed that Marx, Engels, Lenin, and Stalin all "sinned against science" by first utilizing and then dropping the AMP concept in later works because it seemed too reminiscent of the critique of their anarchist competitors that a Marxist "managerial state" would lead to bureaucratic despotism. Ernst Gellner finds that this charge amounts to finding Marx and Engels guilty of being "Stalinists by anticipation" and thus stretches credulity, but Alvin Gouldner agrees with Wittfogel to the extent that the AMP, however limited and incompletely spelled out in the _Grundrisse_ , was nevertheless a crucial concept that Marx and Engels may have glossed over due to its nature as (what this author would term) a neo-anarchist anomaly in their primary class paradigm of the state. In the AMP, "far from being dependent on class controlling the dominant means of production, the state itself controls these and other cases are dependent on it." In any event, because of the political sensitivity of the concept, Marxist democrats in all Leninist countries who wanted to revive the concept had to take pains to criticize Wittfogel and show how their use of the concept was different from his.
The second main problem intellectuals faced, in China at least, who wanted to utilize the AMP concept was to show that they did not incorporate earlier views of a "stagnant" or unchanging Asia. As Gellner put it, the AMP
impairs, perhaps destroys, the unity of human history by postulating a sideline of historical development that perhaps leads nowhere and ends in stagnation.
In trying to refute the basic stagnant nature of the AMP, especially those Chinese thinkers who wanted to utilize the AMP to call for political reform were heavily influenced by Umberto Melotti's1974 work _Marx and the Third World_ , which was translated into Chinese in the late 1970s. For Melotti, the AMP was a unique path of historical development in Marx's essentially multilinear way of thinking, and China was the best example of the AMP, but not a case of "Asiatic stagnation." Melotti did see parallels, however, between China and Russia as "bureaucratic collectivist" societies that existed at a crossroads between revolution and reaction," so the Chinese Aziatchicki (to use the Russian term for those using the AMP concept) had to watch their step and deny that they accepted Melotti's conclusion even while they used his ideas to open up room for criticism of the bureaucratic state.
In China in the late 1970s and early 1980s, as we have seen, the official line of the Leninist regime led by Deng Xiaoping was that survivals of feudalism ( _fengjian_ ) were the main obstacle to further development, not capitalist remnants or bourgeois elements in the Party, as the Maoist line had it. In fact the leading Chinese political scientist and member of the democratic camp within the Party, Yan Jiaqi, argued that while "heavy feudal autocratic vestiges" remain in China, "autocracy was not a political phenomenon that belonged solely to feudalism" in world history. Some countries, for example, in medieval Western Europe, had a feudal system without centralized dictatorship, while other countries such as "the slave-owning Roman empire and fascist Germany" had centralized dictatorships without being feudal. According to Yan, "no matter the social system" autocracy could exist anywhere and at any time where there was "indivisibility and nontransferability of the supreme state power." Thus, Yan made clear that while often a vestige of the past, autocracy could occur even under modern political systems.
It was in this atmosphere of criticism of feudal autocracy that the Chinese AMP debate began in the early 1980s. Chinese thinkers raising the AMP stood on all sides of the issue, from those Stalinist-inspired thinkers who denied the AMP as anything more than one version of primitive communal or slave society, to those on the middle ground who accepted the AMP as a legitimate term for some unique societies that must nevertheless pass through a universal phase of development through capitalism and socialism, to those Marxist democrats who used the term to refer to a special case of Chinese _fengjian_ society, one which could not be equated with Western feudalism. For the purpose of this chapter, we will mostly focus on the last group of thinkers, since they are the ones who suggested contemporary relevance for the AMP, even as they denied being anti-socialist.
The leader of what one could term this minority or "opposition" school of thought on the AMP was Wu Dakun, professor of political economy at Chinese People's University in Beijing, who was linked to the Marxist-Leninist Institute of the Chinese Academy of Social Sciences (CASS), led by the Marxist democrat Su Shaozhi. In a seminal article, Wu in a somewhat cautious fashion attempted to retain a link of the AMP to a unilinear if not universal schema of development by dividing the AMP into two stages, ancient oriental society (thus linked to slavery) and the Asiatic feudal ( _fengjian_ ) system. Using these divisions, Wu claimed legitimacy for his version of the AMP in the thought of Mao Zedong. Though Mao often referred to China's _fengjian_ past, Wu noted, he always emphasized the patriarchal clan authority that led to a Chinese pattern of familial exploitation on top of class exploitation under feudalism.
By dividing the AMP into two stages, Wu was also able to deny the characteristic of "stagnation" often ascribed to AMP societies, thus demonstrating his loyalty to the notion of a progressive Chinese revolution and avoiding the danger of the AMP justifying imperialism as a progressive force in Asia. At the same time, Wu pointed out differences between Chinese _fengjian_ society and Western European society along lines that incorporated some of the classic characteristics of the AMP. First, in the imperial epoch before the Western impact, China combined private and state land ownership through tax and corvée obligations of the peasant to the state. Second, the state controlled not only land and water resources through hydraulic and other public works projects, but also the most important economic enterprises, such as its monopolies on salt and iron. Such a view leads directly to a picture of a Chinese _fengjian_ society containing a much more centralized and authoritarian state than the decentralized political authorities of European feudalism. Third, Wu claimed that land could be bought and sold in the _fengjian_ system, thus showing the beginnings of historical development toward capitalism. Fourth, the remnants of primitive communism and slavery survived in _fengjian_ society through the patriarchal clan system. Fifth, Chinese cities lacked a bourgeoisie and were dominated by landlords and bureaucrats. Sixth, Chinese _fengjian_ society existed in a small peasant economy linking agriculture and handicrafts, with commodity production limited to luxury production for the consumption of bureaucrats and landlords.
By positing these basic characteristics of Chinese _fengjian_ society, Wu was able to use the AMP without implying the stagnation or inherent nonrevolutionary quality of Asian society, an implication for which Stalinist-oriented Party intellectuals severely criticized Melotti and, implicitly, the Chinese reformers. Wu could explain that although China had failed to develop capitalism and developed slowly in comparison to the West, Chinese society nevertheless contained the seeds of capitalism before the Western impact. By the same token, Wu denied that China was without grounds for future development and change, including socialist revolution.
Most importantly, Wu used his redefinition of China's _fengjian_ society to incorporate the AMP as a way to explain the tension in Chinese history between the central government and the landlords, a situation difficult if not impossible to understand by applying to China in a unilinear way the category of the feudal mode of production. Throughout imperial Chinese history, small peasants were periodically squeezed to the point of rebellion as taxes increased on both their land and the land of non-official gentry, while the bureaucratic officials' lands became increasingly tax exempt. Yet as Chinese history unfolded, private land ownership increased at the expense of state ownership, a change Wu claimed was in the direction of capitalism. This change in land ownership combined with peasant rebellions and changes in the state tax system demonstrate that Chinese society was far from static or unchanging, directly refuting Wittfogel's analysis.
Wu asked for further study of China's remaining vestiges of the AMP in order to aid in China's modernization. He left mostly unstated, however, what the remaining vestiges of the AMP were, but in light of his application of the AMP to the history of imperial China, Wu clearly had despotic state vestiges in mind. Though he would not spell out the nature of that continuing despotism under socialism, Wu did suggest that the AMP had relevance to "the study of contemporary world economy," but he claimed that that study would be more appropriate to "another subject" which he "would not talk about here." As the leading exponent of the multilinear view of the AMP, Wu Dakun was perhaps more restrained than other reformers in utilizing the AMP concept to warn of the continuing despotic features of the socialist state.
The boldest example of the "opposition" view of the AMP up to 1985 was an essay in the national journal _Zhongguoshi yanjiu_ (Chinese Studies in History) by Hu Zhongda of the University of Inner Mongolia. Perhaps not coincidentally, this region was hard hit by the extreme state oppression of the Cultural Revolution. Hu specifically criticized the unilinear schema of five modes of production, arguing for the AMP as a separate "social existence." He argued that not only did the AMP diverge from Western European-style feudalism, but that slavery and feudalism themselves were not chronological stages but separate paths out of primitive society. Thus there was no single, universal path of development, but rather many unique paths, though all followed the formula of pre-class to class to classless society.
All pre-capitalist class societies shared the characteristics of simple mechanical development (i.e. iron age hand labor), agriculture as the chief production form combined with family handicraft industry, and land taxation as the major form of oppression. Ancient, feudal, and Asiatic modes were all different forms of "slavery" defined in a larger sense, that is, as methods of direct expropriation of surplus labor by the oppressing class. Although Hu recognized the fact that Engels may have dropped the AMP in later writings and that Lenin at times defined Russia and China as falling under an enlarged definition of "feudalism," he argued that Marxists in the contemporary era did not have to deny the unique qualities of an Asiatic path to development. Hu recognized that Chinese society in the Western and Eastern Zhou Periods (ca. 1100–221 BCE) contained qualities resembling Western European feudalism and had aspects of a slave system in parts of the Han Dynasty (206 BCE–220 CE). Nevertheless, beginning with the Qin and Han dynasties, Hu claimed that China had a different and, for the most part, far more developed system than Western European feudalism. In stressing the unique nature of the Chinese state throughout the imperial era from 221 BCE to 1911 CE, he not only gave a much-needed counterweight to long-standing Marxist orthodoxy on ancient Chinese history, but surpassed many Western Marxist sinologists who often failed to distinguish properly between pre- and post-Qin history.
For Hu the differences between the Chinese imperial period and Western feudalism centered on the existence in China of a large, centralized state standing as the "higher unity" above a system of peasant ownership of land. Landlords did exist at the local level, but the centralized collective ruling power took the place of serfdom per se, presumably through state taxes and corvée labor. China's self-sufficient agricultural system retained features of the primitive communes through such unique entities as China's single-surname clan villages. Thus, like Wu Dakun, Hu tried to finesse the point that the conservatives used against the Marxist democrats and that Western sinologists raised against Wittfogel—the private ownership of land in imperial China and the existence of a landlord class—by implying the identity of landlords and patriarchal clan leaders. Hu claimed that in China state interests dominated private class interests. He suggested that clan leaders acted as agents of the state on the local level rather than as independent exploiters. While downplaying the significance of private property more explicitly than Wu, Hu Zhongda was better able to highlight the direct exploitation by a centralized despotic state in imperial China.
In sum, Hu made an extremely creative attempt to examine the destructive autonomy of the socialist state using a neo-anarchist paradigm of the state contained within the Marxist concept of the AMP. In essence, as he explained at a Chinese academic conference on the AMP, Hu argued for the existence of direct state exploitation in imperial China by pointing out that China's monarchical system allowed for the monopoly of the surplus products and surplus labor by an "autocratic collective ruling class" ( _zhuanzhi junzhu weishoude tongzhi jituan he boxue jieji_ —literally "ruling clique and exploiting class headed by an autocratic monarch"). Rather than follow the orthodox Marxist–Leninist theory of the state, that is, the state as protecting and disguising exploitation by a dominant economic class, Hu posited that the Chinese imperial state itself had a dominant position in a collective ruling class of landlords, administrators, and the monarchy. Thus the state did not just protect and disguise exploitation, but rather, exploited its subjects directly.
Despite some vacillations on retaining the label of feudalism, in 1981 Hu accepted the AMP as a useful concept in explaining the real differences between China's centralized absolutist monarchy and the decentralized politics of Western feudalism. Hu also vacillated on whether to call this unique oriental variant a "separate social existence" or a separate mode of production; nevertheless in his essay he clearly rejected the unilinear five modes of production schema and expressed the hope for continuing free academic debate on the AMP issue.
By 1982–3, such hopes were repressed by high Party leaders. Unlike the contemporaneous debate over humanism and alienation, the AMP disputes never surfaced in the popular press. The AMP debate was quietly ended by higher echelon leaders such as Hu Qiaomu shortly after an issue of _Zhongguoshi yanjiu_ (Studies in Chinese History) was published based on a conference of historians on the AMP. In the first Stalinist backlash, lasting until late 1984, the Marxist democrat AMP advocates were largely silenced. At the same time, the moderate AMP advocates retreated to a view positing that China had a distinctive variation of feudalism rather than a separate AMP. By 1982–3 the small advantages the mainstream ruling elite obtained by allowing the AMP debate were outweighed by threats to state autonomy other parts of the elite perceived from continuing intellectual debates, and the AMP debate was forcibly ended.
When China's Marxist democrats found themselves in a position publicly to reassert themselves from 1985 to 1986, just as the advocates of humanism and alienation under socialism made a brief comeback, the AMP debate likewise briefly resurrected. This brief thaw began in the Chinese historical profession in mid-1985, including new academic articles that analyzed the nature of the centralized bureaucracy of imperial China.
The AMP concept itself reappeared in the first 1986 issue of _Lishi yanjiu_ , ending that journal's Stalinist ideological monopoly. The individual responsible for this breakthrough was none other than Hu Zhongda, the most daring of the AMP advocates from the debates of the early 1980s, who returned with an article criticizing the orthodox five mode view. In this article, Hu did not emphasize the AMP as a distinct mode, but instead, echoing the late 1970s view of certain Soviet Aziatchiki, was able to recognize the non-universality of the full slave mode of production by advocating the existence of a single pre-capitalist stage in all societies with different variants, a view that preserved universality of development while allowing multilinear paths to capitalism and socialism. Though thus still legitimizing the Chinese path to socialism, the Soviet-inspired formula that Hu adopted allowed the AMP to resurface as a despotic remnant of feudalism or as a temporary non-universal variation that could continue to influence succeeding stages, just as elements of slavery and feudalism coexisted in different degrees in pre-capitalist societies and into capitalism. Cautiously, Hu claimed that he based his current opinions on his work of the early 1980s, which would include his Tianjin article.
By far the most extraordinary reappearance of the AMP was presented by Wang Yizhou in the third 1985 issue of _Makesizhuyi yanjiu_ (Marxist Studies) published by the Party democrat-controlled Institute of Marxism-Leninism-Mao Zedong Thought. Wang viewed the AMP both as a fundamental tenet that Marx never abandoned and as a real historical entity. Though he viewed the AMP as a survival of primitive communal forms into class society, Wang emphasized development and change within this mode. He explicitly denied the stagnation or backwardness of former AMP societies that underwent socialist revolution. Most importantly, he claimed the "most outstanding features" of the AMP to include:
. . . the absolute economic control by the state over all members of the society through the ultimate, hereditary state ownership of the basic means of production . . . the right of the state to appropriate, transform, and redistribute a large amount of surplus product at will, and the absolute, almost religious, control by the state over thought, as well as the absolute, almost blind loyalty to the state from the masses.
Wang specifically used the term "oriental despotism" to refer to the state under the AMP, though he denied that the AMP was geographically limited to Asia either in actuality or in the thought of Marx. Although he denied the backwardness of former AMP societies now under socialist rule, Wang stressed the impossibility of skipping or leaping stages in the revolutionary process. He also denied that the AMP and Oriental Despotism could be equated with state socialism either in Marx's eyes or in reality. Yet, in the boldest Chinese statement on the AMP, Wang clearly suggested the continuing legacy of the AMP for countries that passed through such a stage on the route to socialism:
. . . we are confident that Marx never made his studies of the AMP as a part of his theory of socialism . . . [but] we cannot deny the guidance of Marx's analysis of the AMP toward our understanding of some important phenomena in contemporary socialist society. Quite the contrary, the concept of the AMP is extremely important to our understanding of present reality . . . it would not be surprising if the characteristics of the AMP discussed by Marx are present in various degrees in all socialist countries due to the fact that most socialist revolutions occurred in countries with the legacy of oriental despotism.
After this temporary return of the Marxist democrats' critique of the Leninist Party-state, from December 1986 to 1987, all talk of political reform came to a crashing halt following the student demonstrations at major Chinese universities, as noted above. Once Hu Yaobang, the ultimate protector and patron of the Marxist democrats within the Party, was forced to resign from his post, the arch Stalinist hardliners Deng Liqun and Hu Qiaomu returned to influence in the ideological sphere and in their new campaign against "bourgeois liberalization" the AMP debate was once again aborted. In 1987–8, however, when the reformer Zhao Ziyang was "kicked upstairs" to replace Hu Yaobang as General Secretary, a new thaw briefly began, until the 1989 Tiananmen student demonstrations once more led to a crackdown and Zhao was removed from his post.
In 1988, during this last brief thaw, the television series _Heshang_ (River Elegy) was allowed to be broadcast on state TV and praised by Zhao. This series made direct reference to China's "despotic" past based on a centralized bureaucratic state's control of irrigation and other hydraulic projects. The series made clear that a "despotic centralized power" became "a kind of unchallengeable overlord" that continued to affect Chinese political culture. In the backlash that followed, the series was heartily condemned by Stalinist leaders and their intellectual followers, and after the 1989 Tiananmen protests, the series' creators were ultimately forced into exile.
In April 1989, however, just before Hu Yaobang's death by heart attack during a Politburo meeting and thus just before the Tiananmen protests began, Hu's network of critical intellectuals managed to get in some last criticisms of Leninist state autonomy. None other than Wang Yizhou, the leading Marxist democrat on the AMP, at one symposium on the topic of the state ownership system, declared that the root cause of corruption in contemporary society was not market reform, as the Stalinists would have it, but "the state ownership system, and its monopolization of all resources." As Wang argued,
. . . the privileged treatment for those vested beneficiaries does not come from party membership dues, but from the monopoly of the state ownership system.
After Hu's purge in 1987, intellectuals associated with new General Secretary Zhao Ziyang pushed the idea of "neo-authoritarianism" as the way to ensure the continuation of economic reform, that is, the rule of a strong, enlightened leader and his followers who would use their authority to overcome "old authoritarian" forces within the elite who would obstruct market reforms. Only later, once those old forces were removed, could society gradually move in a democratic direction. In response, members of Hu Yaobang's old network of intellectuals defended the need for political reform and democratization, in the process noting the tendency of the state to turn despotic if it were not subject to popular checks on its authority. As the playwright Wang Ruowang put it, the would-be reformist Soviet leader Khrushchev (whom Goldman sees as perhaps Wang's allusion to Gorbachev or even Zhao Ziyang) was an example of "an enlightened authoritarian leader who had not turned into a despot, but had been overthrown by the entrenched party bureaucracy because his reforms threatened their interests." In other words, the Marxist democrats feared that any justification for "enlightened despotism" would in the end only allow the state to gain autonomy and follow its own interests to the point where the old despotism would return. As Gao Gao (the wife of the Party democrat Yan Jiaqi) put it about the supposedly enlightened rule of Mao, which after all led to the depredations of the Anti-Rightist Campaign and Cultural Revolution, "the practice of enlightened rule can be such that it can be enlightened today and tomorrow, but on the day when power and interests are touched enlightenment will all but be squeezed out by autocracy." In light of the crackdown that followed the Tiananmen demonstrations, in which many of the Marxist democrats were silenced for many years or forced into exile, Gao's words were very prescient, though of course totally unsurprising to anyone holding a true neo-anarchist critique.
**Conclusion: Neo-Anarchist Thought in the PRC**
These last two chapters have examined the thought of Chinese intellectuals, both within and outside the CCP, who utilized versions of the neo-anarchist paradigm of the state from the Cultural Revolution to the Tiananmen student movement. The main advantage of such approaches is that they resonate very well with the experience of Chinese subjects who face the overwhelming might of despotic state power in nearly every aspect of their daily lives. Whatever the Leninist state might say about the primacy of representing the interests of the proletariat (or "the people" since the CCP rewords the concept of the dictatorship of the proletariat to that of the "people's democratic dictatorship" so as to supposedly include other "progressive" classes), Chinese subjects know all too well that when push comes to shove the state takes care of its own interests first, interests which often come into sharp conflict with those of ordinary people. Even though political thought in the last two decades has been relatively muted on the issue of a new ruling class compared to the relatively liberal period of 1978–89, not to mention the radical moment of 1967–8, and avenues of using neo-anarchist critiques based on anomalies in Marx's thought have been largely cut off, a few brave dissidents still manage to talk about Chinese autocracy. As the contemporary dissident thinker Liu Xianbin recently argued,
. . . whether the country is ruled by the family clans or by the party, the rule is, in essence, an autocratic rule, which is antagonistic to the people. It runs counter to the concept of democracy and contravenes the will of the people. Over the past several thousand years, the Chinese people have never become the master of this country . . .
. . . If the rulers are still reluctant to give up the various advantages of the autocratic system, then the people who are the masters of the country should stand up on their own initiative to accomplish this social transformation.
For writing such thoughts, and as part of the recent ongoing broad crackdown on all forms of dissent, Liu was convicted in March 2011 of "inciting subversion of state power," or as his lawyer says, "slander[ing] the ruling Communist Party and [trying] to end its monopoly on power," and is currently serving a 10 year prison sentence.
Even if those Chinese thinkers who talked about autocracy under Leninism linked autocracy to China's imperial past and/or to the pressures of the international economic or political system, their main point, from the Cultural Revolution through the early reform era to contemporary dissidents such as Liu Xianbin, is that once the state gains autonomy, it will not give up power without strong pressure from citizens at the grassroots who are highly aware of an autocratic state ruling for itself. Thus, thinkers who talk about the Chinese state within a neo-anarchist paradigm—whatever their various proposed solutions to the problem and despite their own lack of ability to fully challenge the Leninist state in an open way—nevertheless play a crucial role in opening up room for increased pressure on the state in the future.
Most Chinese subjects, as with people living under all dictatorships, are not in a position to challenge the state directly; nevertheless, they continue to feel the weight of state oppression, even as the state tries to whip up support based on nationalist sentiments. Whether it is the state-enforced poverty and mass violence in the Maoist periods of the Great Leap Forward and Cultural Revolution, which led to the death and suffering of untold millions of people, or the bureaucratic corruption, environmental degradation, growing inequality, and remaining high levels of police state repression of the reform era, it should be obvious even to socialists that the so-called epiphenomena of the Leninist political superstructure easily overwhelms the economic base and the supposed goal of social and economic equality. Whatever one thinks of as the best way to control this behemoth, it should be clear to most people today, including socialist-inclined intellectuals, that the history of the twentieth century was the history of political domination and oppression in different forms and with different ideological justifications, from liberal to Marxist, a history that is likely to continue to expand in this century. Radical intellectuals and activists around the world who desire genuine human liberation could do well to challenge this expansion of state autonomy by copying the Chinese Marxist democrats of the 1960s to 1980s and adopting more explicit anarchist theories of the state.
**Notes**
**** Deng Xiaoping, "Uphold the Four Cardinal Principles," a speech at the Forum on the Principles for the Party's Theoretical Work, March 30, 1979, in Deng Xiaoping, _Selected Works (1975–1982)_ : 166–91.
**** As we have seen in the previous two chapters, Friedman uses the terms "extra-party" and "[inner] party democrats." See Friedman, "The Societal Obstacle to China's Socialist Transition," 159–71.
**** See Ding Wang, _Ming Bao Yuekan_ , 217 (January 1984), translated and adapted in David A. Kelly (guest ed.), "Wang Ruoshui: Writings on Humanism, Alienation, and Philosophy," 113–14; and cited by Kelly, "The Emergence of Humanism: Wang Ruoshui and the Critique of Socialist Alienation," 162, 339, n. 12.
**** For studies of Wang Ruoshui and the "alienation school" in general see Kelly, "Wang Ruoshui: Writings on Humanism, Alienation, and Philosophy," and "The Emergence of Humanism"; Bill Brugger and David Kelly, _Chinese Marxism in the Post-Mao Era,_ 139–69; Merle Goldman _Sowing the Seeds of Democracy in China: Political Reform in the Deng Xiaoping Era,_ 116–32; and Hua Shiping, "Marxist Humanism: Wang Ruoshui," Chapter 6 of Hua, _Scientism and Humanism,_ 95–107.
****These would include especially Wang, "Tantan yihua wenti" (Discussing the Problem of Alienation), 8–11, translated in Kelly, "Wang Ruoshui: Writings on Humanism, Alienation, and Philosophy," 25–38 and "Wei rendaozhuyi bianhu" (A Defense of Humanism), translated in Kelly, 71–88.
**** Wang, "Tantan yihua wenti," translated in Kelly, 29–30.
**** Ibid., 30.
**** Ibid., 27–8.
**** Marx and Engels, _Selected Works_ 1, 438.
**** Wang, "Tantan yihua wenti," translated in Kelly, 29, 33.
**** Ibid., 33.
**** See Thomas B. Gold, "Just in Time!: China Battles Spiritual Pollution on the Eve of 1984," 958; Goldman, 121–32; and Stuart R. Schram, _Ideology and Policy in China since the Third Plenum, 1978–1984_.
**** Hu's speech is analyzed in Goldman, 127–8.
**** Hu Qiaomu, "Guanyu rendaozhuyi he yihua wenti" (On Problems Concerning Humanism and Alienation), _Renminribao_ (People's Daily) (January 27, 1984), reprinted in _Hongqi_ (Red Flag), 2 (January 26, 1984), translated in FBIS (February 7, 1984), 31.
**** For example, see Wang, _Wei rendaozhuyi bianhu_ (A Defense of Humanism).
**** Wang, "My Views on Humanism," translated in FBIS (August 10, 1984), W5–W17, cited in Brugger and Kelly, 160.
**** See Goldman, 204–37.
**** For example, see Wang, "Shuangbai fangzhen he gongmin quanli" (The Double Hundred Policy and Civil Rights), reprinted in _Chengming,_ 60–1, translated in FBIS (September 12, 1986): K6–K12, cited in Brugger and Kelly, 1990, 166.
**** Wang, "The Personality Cult and Ideological Alienation—Replies and Criticism," _Ching Pao,_ Part I, 24–7 and Part II, 40–4, translated in FBIS 88–91 (May 11, 1988), 27.
**** Personal conversation of this author with Wang Ruoshui, Madison, WI, April 1989.
**** Joseph Chaney and Sophia Woodman, "In Memoriam: Wang Ruoshui: Journalist,Philosopher, Humanist," _China Rights Forum_ 1 (2002), online at www.hrichina.org/public/contents/article?revision_id=2044&item_id=2043.
**** Kelly, "The Emergence of Humanism," 181.
**** Bob Jessop specifically cites the AMP as a case where Marx sometimes treated the state as a "parasitic body standing over society." See Jessop, "Recent Theories of the Capitalist State," in Hall, 83.
**** For a review of the AMP concept in the USSR, West and East Europe, and China, see Rapp, "Despotism and Leninist State Autonomy," 110–200, which includes an extensive bibliography of the main sources on the AMP debates in different countries.
**** Karl Marx, _Grundrisse der Politischen Okonomie_ (Foundations of the Critique of Political Economy) (Moscow, 1939–41), translated by Martin Nicholas as _The Grundrisse: Foundations of the Critique of Political Economy_.
**** This chapter of the _Grundrisse_ has been published as a separate work in English. See Marx, _Pre-Capitalist Economic Formations_.
****This summary definition of the AMP is partially based on that of Perry Anderson, "[Research Note on] the 'Asiatic Mode of Production'," in Anderson, _Lineages of the Absolute State,_ 483.
**** Rapp, "Despotism and Leninist State Autonomy," Chapter 4, 76–100.
**** Ibid., Chapter 3, 110–200.
**** For a mostly apolitical account of the Chinese AMP debates, see Timothy Brook, _The Asiatic Mode of Production in China_ ; for a more political analysis, see Rapp, "Despotism and Leninist State Autonomy," 301–97; (guest ed.), "China's Debate on the Asiatic Mode of Production," _Chinese Law and Government_ , XXII(2) (Spring 1989), "Editor's Introduction," 3–26, from which this section of the chapter is derived.
**** Wittfogel, _Oriental Despotism_.
**** Ibid., 369–412; see also Wittfogel, "The Ruling Bureaucracy of Oriental Despotism: A Phenomenon that Paralyzed Marx."
**** Gellner, "Soviets against Wittfogel; Or, The Anthropological Preconditions of Mature Marxism."
**** Gouldner, _The Two Marxisms: Contradictions and Anomalies in the Development of Theory_ , 327–8.
**** Ibid., 347–8.
**** English edition, Melotti, _Marx and the Third World_ , the edition which Wu Dakun reports heavily influenced him in beginning his studies of the AMP. See Wu, "On Several Questions in the Study of the AMP," 18, n. 2.
**** Melotti, 105–13, 141–51.
**** Yan, " 'Imperial Power' and 'Imperial Position': Two Characteristics of Autocracy," reprinted in Yan, _Quanli yu zhenli_ (Power and Truth), 91–6, translated in _Yan Jiaqi and China's Struggle for Democracy,_ 15.
**** Ibid., 10–11.
**** Wu, "The AMP in History as Viewed by Political Economy in Its Broad Sense," 18–29, translated in Su Shaozhi et al. (eds), _Marxism in China,_ 53–77, and in Rapp, "China's Debate on the Asiatic Mode of Production," 27–46.
**** Ibid., citing Mao, "Report on an Investigation of the Peasant Movement in Hunan Province," in Mao, _Selected Works_ I, 44, 146.
**** Wu, in Su Shaozhi et al., 68–9.
**** See for example, Lin Ganquan, "The AMP and Ancient Chinese Society: A Criticism of Umberto Melotti's Distortion of Chinese history in His 'Marx and the Third World'," translated in Rapp, "China's Debate on the Asiatic Mode of Production," 47–70.
**** Wu, 69–76.
**** Hu, "Shilun Yaxiya shengchan fangshi jianpi wuzhong shengchan fanchangshuo" (On the Asiatic Mode of Production with a Criticism of the Theory of Five Modes of Production), translated in Brook, 164–83.
**** See Pang Zhuoheng et al., "Summary of the Symposium on the AMP."
**** Hu, in Brook, 173–4.
**** See Pang et al.
**** See Goldman, 166–203.
**** For example, see Wu Shouzhi, "The Essence and Important Function of Centralized State Power in the Feudal Autocratic System of China."
****Hu, "Further Criticism of the Five Modes of Production Theory," translated in Rapp, "China's Debate on the Asiatic Mode of Production," 71–97.
**** Ibid., 94.
**** Wang Yizhou, "An Inquiry into Marx's Concept of the AMP—Critical Comments from a Theoretical Standpoint," translated in Rapp, "China's Debate on the Asiatic Mode of Production," 98–108.
**** Ibid., 104.
**** Ibid., 107.
**** Goldman, 238–55.
**** See Su Xiaokang,Wang Luxiang, Richard W. Bodman, and Pin P. Wan, _Deathsong of the River: A Reader's Guide to the Chinese TV Series Heshang_ , 275.
**** See Goldman, 257–60.
**** Cited in Ibid., 272–3.
**** Quoted in Zhang Weiguo, "Whither the State Ownership System?" translated in FBIS (April 25, 1989), 37, cited in Goldman, 273.
**** Goldman, 275–82.
**** Ibid., 279, citing Wang, interview with _Jiushi niandai,_ translated in JPRS 89033 (April 12, 1989), 4.
**** Gao Gao, "Improve the Social Control System Taking the Rule of Law as the Main Body," translated in JPRS 89088 (August 21, 1989), 10–11, cited in Goldman, 281.
**** Liu Xianbin, "Cong 'renmin dajiazuozhu' shuoqi" (Commenting on 'The Position of the People as the Masters of the Country'), translated by United States Open Source Center, 2010.
**** See Jacob Andrew, "Chinese Democracy Activist Is Given 10-Year Sentence."
POSTLUDE
The continuing relevance of Daoist anarchism
As noted in the prelude to this book, at least two types of critics would object to the whole concept of Daoist anarchism, including some students and/or practitioners of Daoist spiritual beliefs and physical practices and some scholars of and/or sympathizers with anarchism. The former type of critic might point to the lack of survival of Daoist anarchism within China itself, while the latter type would object to universalizing anarchism beyond the "official" anarchist movement of the late nineteenth to early twentieth centuries, perhaps revived in the late twentieth and early twenty-first centuries. The latter might especially object to the this work's application of the "neo-anarchist" label to Chinese dissidents in the previous two chapters since they take pains to deny they are anarchist and for the most part are far from advocating anarchist solutions. A third type of critic would object to the possible "reification" of the state contained in this book, denying that there is any real unified body at all that could be defined as having its own interests, that in fact states are made of up many different levels and types of institutions that often work at cross purposes. This brief conclusion tries to answer all such objections by pointing out how the recent, highly ironic simultaneous revival of Maoism and Confucianism in the PRC might open up space for an anarchist critique, perhaps harking back to radical Daoist themes and language.
What the nearly simultaneous revival of Maoist and Confucian themes in the PRC may show is that for all their different and, at times, conflicting interests and perspectives, different state elites have a common interest against their own subjects in limiting checks on their authority. The ancient Daoist anarchists criticized both the Confucians who called for humane rule through leaders' supposedly moral example and education in traditional rituals, and the Legalists who called for implementing harsh system of rewards and punishments, knowing full well the differences between the two schools but finding them to be two sides of the same coin. Likewise, in Chapter 7 we saw how modern Chinese Leninist state elites could be usefully analyzed as being divided along three basic lines of different social control mechanisms yet have a common interest in maintaining state hegemony over their subjects. Certainly there may be more divisions and differences among ruling elites in non-Leninist states and more ability of sections of such elites to go outside the state elite to seek a popular base, but this potential ability does not negate the common interest of all state elites in at least attempting to maintain and expand their autonomy from their subjects.
This book argues that anarchists are at their strongest when they focus on state autonomy as the heart of the anarchist critique and weakest when they make compromises with states in order to pursue other, seemingly more central goals, such as reducing economic inequality or increasing chances for individual intellectual critics to survive and prosper within certain political regimes. To pick just three examples of such acquiescence in state power that we have seen in earlier chapters, some ancient Daoists reinterpreted Daoist principles to allow them to accept taking office, some early twentieth-century Chinese anarchists defended participating in the Nationalist regime because it would allow them to continue work-study experiments whose goal was to end the division between mental and physical labor, and some official PRC leaders during the Cultural Revolution, including Mao himself, limited their new class critique in order to preserve space for seemingly radical officials within the state elite to pursue policies allegedly aimed at keeping revolutionary egalitarian policies alive. In all these cases the state interests of such compromising elites in the end overwhelmed their radical critiques, and the differences they had with other state elites in the end paled in comparison with the gaps between them and their subjects. To the extent that they maintained aspects of their radical critique, they were easily overwhelmed by their state elite rivals and perished anyway despite their compromises with state power.
There are other individuals in China, on the other hand, who kept to what this book argues is the heart of the anarchist critique, the focus on the constant danger of the state ruling for itself and gaining autonomy from its subjects—the idea that most distinguishes anarchism from other political ideologies. Far from having an unchallenged authoritarian political culture, we have seen that individuals raising the basic anarchist critique have popped up time and time again in Chinese history.
The current political moment in China would seem to be the least hospitable for anyone attempting an anarchist critique of the state to emerge. After all, not only are political dissidents being harassed, jailed, and tortured, even their lawyers and family members are being punished to the point that they are afraid to speak out. Almost all talk of meaningful political reform and democratization has been forbidden in the buildup to the 2012 Communist Party Congress, and any revival of the neo-anarchist themes of the extra and inner Party democrats of the 1980s seems highly unlikely any time in the near future. Nevertheless, these harsh actions against dissent would also indicate that the regime is running scared, to say the least.
In a great irony, given their past antipathy toward each other, the regime has allowed two supposedly heterodox and seemingly opposite types of themes to be raised. On the one hand, some figures, notably the Chongqing Party boss Bo Xilai, have revived neo-Maoist egalitarian rhetoric in propaganda initiatives, while on the other hand the regime has revived Confucian language of benevolent and harmonious rule at the same time as it retreats from market-based reform and insists on protecting state-owned industry and local government investments at the expense of its subjects' economic well-being.
The Confucian revival includes not just setting up "Confucius institutes" around the world to promote the study of Chinese language and culture nor the temporary setting up of a statue of Confucius in front of the National history Museum alongside Tiananmen square, but the official stress on building a "harmonious society" as a key goal of the regime, a goal which includes some stress at least on the supposed positive values to be found in Confucian thought. At the same time, officials such as Bo Xilai are pushing for a revival of Mao-era songs and slogans and a stress on supposed revolutionary purity as a way to ensure social control. The two trends are often at odds, of course. For example, some Maoists who abhor the stress on Confucianism (perhaps reflecting the Cultural Revolution campaign to denounce Confucius that we referred to in Chapter 9) succeeded in getting the Confucius statue removed, while would-be neo-Confucians would note the violent and far from harmonious class conflict and struggle at the heart of real Maoism. In fact the regime seems to have accepted more than a little bit of each side's rhetoric in claiming to favor more balanced growth of interior and coastal regions of China as the key way of building a harmonious society. In the end, of course, talk of harmony cannot even thinly disguise the real institutionalized violence still going on in China, as not just political dissidents but even those protesting poor earthquake relief and public school building standards or those who try to get redress for purely economic grievances such as failure to pay promised back wages of laid off workers at state-owned enterprises or promised reimbursements for local governments' seizure of land, are very often all jailed, sent to mental treatment centers, or otherwise forcibly "disappeared." Just as in nearly all other periods of Chinese history, revived official stress on Confucian themes coincides with increased state repression, while emphasis on Maoist egalitarian and revolutionary rhetoric coincides with increased inequality between Party elites and the masses, recalling the radical Daoist sentiment that talk of morality and harmony only occur when such principles are absent. All these nervous and even paranoid attempts of state elites to adjust official ideology only serve to demonstrate the main point of Daoist anarchism: attempts to justify rule on the grounds of increasing benevolent treatment of people or achieving peaceful order only serve in reality to justify the power and wealth of state elites. Likewise the return to claims of Maoist revolutionary fervor and equality only come when in fact the fervor has long waned and when most citizens know instinctively and through direct experience that state leaders are only out to preserve their own power. The time is perfect in China for the revival not of the relatively weak heterodox Marxist themes of alienation and the AMP, but, since no one really believes in Marxism any more, for things like Daoist study societies that might fit in with the call to learn from Chinese tradition, and even for radical Maoist ideas of true mass democratic checks on authority—this time without the stress on class violence and reliance on top leaders to define when direct democracy may be allowed. Undoubtedly any such attempts would eventually be repressed as well, but only at an ever growing cost for a regime that may be increasingly facing contradictions between its avowed goals of wealth for all and the reality of protecting vested state interests.
The basic anarchist idea has broken through the surface at widely spaced geographic places and many different points in time throughout history, almost always to suffer severe repression, but the fact that anarchists of all kinds have been a small minority of all thinkers or that anarchist interpretations of traditions as different as Christianity, Marxism, or Daoism are all almost equally put down as heretical or blasphemous can never extinguish the anarchist impulse as long as states inevitably seek to augment their own power and autonomy at the expense of their subjects. What Daoist anarchism would teach anyone trying to revive a radical critique of state autonomy is that people must constantly be on guard for making compromises with the state out of their own interests as intellectuals and political activists. Radical Daoist thinkers at their best (as in Bao Jingyan) and at their most contradictory (as in Wu Nengzi) may teach other anarchists the crucial difference between everything and nothing: without an underlying positive vision of society as always able to thrive on its own without a state, though certainly without trying to turn that vision into detailed blueprints to be imposed on anyone else, any anarchist or neo-anarchist critique can too easily degenerate into nihilism and/or compromises with state authority. If necessarily based on such an underlying positive vision, however, it is the constant and consistent critique of state autonomy that must come first and foremost for any true anarchist.
**Notes**
**** For an overview of the revival of Confucius and Confucian themes in the PRC, see John Dotson, "The Confucian Revival in the Propaganda Narratives of the Chinese Government."
**** For the recent revival of Maoist themes by Bo Xilai (who has most recently suffered a spectacular fall from power) and other PRC leaders at the same time as the revival of Confucianism, see Francis Fukuyama, with response by Jonathan Fenby, "China Is Looking to Its Dynastic Past to Shape Its Future."
APPENDICES
Works of Daoist Anarchism
**1.** _Zhuangzi,_ **Chapter 9, "Horses' Hoofs"**
Translated by Burton Watson, in Watson, _The Complete Works of Chuang Tzu,_ 98–106, with abridged translator's notes, © 1970, Columbia University Press, reprinted with permission of the publisher.
Horses' hoofs are made for treading frost and snow, their coats for keeping out wind and cold. To munch grass, drink from the stream, lift up their feet and gallop—this is the true nature of horses. Though they might possess great terraces and fine halls, they would have no use for them.
Then along comes Po Lo. "I'm good at handling horses!" he announces, and proceeds to singe them, shave them, pare them, brand them, bind them with martingale and crupper, tie them up in stable and stall. By this time two or three out of ten horses have died. He goes on to starve them, make them go thirsty, race them, prance them, pull them into line, force them to run side-by-side, in front of them the worthy of bit and rein, behind them the terror of whip and crop. By this time over half the horses have died.
The potter says, "I'm good at handling clay! To round it, I apply the compass; to square it, I apply the T square." The carpenter says, "I'm good at handling wood! To arc it, I apply the curve; to make it straight, I apply the plumb line." But as far as inborn nature is concerned, the clay and the wood surely have no wish to be subjected to compass and square, curve and plumb line. Yet generation after generation sings out in praise, saying, "Po Lo is good at handling horses! The potter and the carpenter are good at handling clay and wood!" And the same fault is committed by the men who handle the affairs of the world!
In my opinion, someone who was really good at handling the affairs of the world would not go about it like this. The people have their constant inborn nature. To weave for their clothing, to till for their food—this is the Virtue they share. They are one in it and not partisan, and it is called the Emancipation of Heaven. Therefore in an age of Perfect Virtue the gait of men is slow and ambling; their gaze is steady and mild. In such an age mountains have no paths or trails, lakes no boats or bridges. The ten thousand things live species by species, one group settled close to another. Birds and beasts form their flocks and herds, grass and trees grow to fullest height. So it happens that you can tie a cord to the birds and beasts and lead them about, or bend down the limb and peer into the nest of the crow and the magpie. In this age of Perfect Virtue men live the same as birds and beasts, group themselves side-by-side with the ten thousand things. Who then knows anything about "gentleman" or "petty man"? Dull and unwitting, men have no wisdom; thus their Virtue does not depart from them. Dull and unwitting, they have no desire; this is called uncarved simplicity. In uncarved simplicity the people attain their true nature.
Then along comes the sage, huffing and puffing after benevolence, reaching on tiptoe for righteousness, and the world for the first time has doubts; mooning and mouthing over his music, snipping and stitching away at his rites, and the world for the first time is divided. Thus, if the plain unwrought substance had not been blighted, how would there be any sacrificial goblets? If the white jade had not been shattered, how would there be any scepters and batons? If the Way and Its Virtue had not been cast aside, how would there be any call for benevolence and righteousness? If the true form of the inborn nature had not been abandoned, how would there be any use for rites and music? If the five colors had not confused men, who would fashion patterns and hues? If the five notes had not confused them, who would try to tune things by the six tones? That the unwrought substance was blighted in order to fashion implements—this was the crime of the artisan. That the Way and its Virtue were destroyed in order to create benevolence and righteousness—this was the fault of the sage.
When horses live on the plain they eat grass and drink from the streams. Pleased, they twine their necks together and rub; angry, they turn back-to-back and kick. This is all horses know how to do. But if you pile poles and yokes on them and line them up in crossbars and shafts, then they will learn to snap the crossbars, break the yoke, rip the carriage top, champ the bit, and chew the reins. Thus horses learn how to commit the worst kinds of mischief. This is the crime of Po Lo.
In the days of Ho Hsü, people stayed home but didn't know what they were doing, walked around but didn't know where they were going. Their mouths crammed with food, they were merry; drumming on their bellies, they passed the time. This was as much as they were able to do. Then the sage came along with the crouchings and bendings of rites and music, which were intended to reform the bodies of the world; with the reaching-for-a-dangled-prize of benevolence and righteousness, which was intended to comfort the hearts of the world. Then for the first time people learned to stand on tiptoe and covet knowledge, to fight to the death over profit, and there was no stopping them. This in the end was the fault of the sage.
**2. Ruan Ji, "The Biography of Master Great Man" (excerpt)**
Translated by Donald Holzman, from Holzman, _Poetry and Politics: The Life and Times of Juan Chi, A.D. 210–263_ , 193–6, © Cambridge University Press 1976, reprinted with the permission of the publisher.
I suppose Master Great Man is old. I know neither his family name nor his polite appellation [ _tzu_ ]. But his description of the beginning of the universe and his remarks on the affairs of Shen-nung and the Yellow Emperor are brilliant. No one knows how long he has lived. Since he once resided on Mount Su-men, some people call him by that name. From time to time he nourishes his nature and prolongs his longevity, glowing with a radiance equal to that of Nature's own. He sees the acts of Yao and Shun as if they were in the palm of his hand! Ten thousand leagues are to him no more than a pace, and a thousand years, one morning; his movements take him nowhere, and his sojourns are in no place. All he seeks is the great _tao:_ He has no temporary residences. The Master, by responding to the vicissitudes of the world, remains in harmony with them: The universe is his home. Should the conditions of fortune and the world be unfavorable, he stays apart, leading a solitary existence, feeling that it is enough to be able to evolve with the whole of creation. And so he silently seeks out the _tao_ and its virtue and has no dealings with the world of men. The self-satisfied criticize him; the ignorant think him strange: Neither recognize the spirit-like subtleties of his transformations. But the Master does not change his calling because of worldly criticism or wonder. The Master believes that the central area [in which China is located] occupies a position in the universe not even equivalent to the space occupied by a fly or a mosquito stuck in a curtain. And so he pays no attention to it and lets his thoughts stretch out endlessly to foreign places and strange regions, roams about enjoying the sights, unseen by the world, going back and forth, ending nowhere. He left his book on Mount Su-men before he went away—no one in the world knows where.
Someone gave a letter to Master Great Man which reads, "Among the things honored in the world, nothing is more honored than a gentleman. In his dress a gentleman wears prescribed colors; his facial expressions follow prescribed forms; his words obey prescribed rules; his conduct is according to prescribed models. When standing [in the presence of a superior] he bends in two like the musical stone, his hands folded before him as if he were holding a drum. His periods of activity and repose are measured; his pace in walking conforms to a musical beat. When he advances or when he retreats, in all his relations with others, everything is done according to rule. His heart seems filled with ice, so tremulous he is, so nervous. He restrains himself, cultivates his conduct and is each day more prudent than the preceding. He would choose the very ground he walked on, and only be afraid of committing some error. He recites the instructions left to us by the Duke of Chou and Confucius and sighs over the _tao_ and the virtue of Yao and Shun. He cultivates only the [Confucian] law; disciplines himself only with ritual. His hands hold the symbols of his rank and his feet toe the line of orthodoxy. In his conduct he wants to be a model to the present world; in his speech he wants to set up eternal standards. In his youth he is praised in his native place and when he grows up his fame spreads throughout the entire nation. At best he desires to become one of the three highest officers in the central government, or, at least, to become the governor of a province. Thus he clasps his gold and jade, dangles his patterned silk bands, enjoys honored position and is granted fiefs. He spreads his fame down to later generations and pits his merits against the past. He humbly serves his sovereign and governs the flock of the common people. When he retires he manages his own family and instructs his wife and children. He performs divination to build a propitious residence and plans to procure myriad celestial favors for it, to keep catastrophes far away and good fortune near, to keep his family and descendants eternally secure. This is truly the highest achievement of a gentleman, the kind of praiseworthy conduct that has not changed from ancient times until our own. But now, Master, you let your hair down and live in the middle of the great ocean, far from these gentlemen. I fear the world will sigh over you and criticize you. Your conduct is laughed at by the world and you have no way of achieving success: This indeed can be called shame and disgrace! You dwell in difficult conditions and your conduct is laughed at by the men of the world; I cannot believe that the Master can accept such a fate!"
Thereupon Master Great Man sighed in a relaxed way and sent him the following answer, using the clouds [to carry his message]: "What can all that you have said mean? Now, a Great Man is of the same essence as Creation and was born with the universe itself. He freely floated in the world, reaching perfection with the _tao._ In accord with the successive transformations that take place he disperses himself or gathers himself together: He does not keep a constant form. The divisions of the universe are all within him so that his free and easy understanding penetrates all without. The [true idea of] the eternity and stability of the universe is not something that the men of the world can approach. I am going to explain it to you. In the past, at one time the heavens were below and the earth was above; they turned over time and again, and had not yet reached a stable condition. How [if you had been living then] could you not have lost your 'rules' and 'models'? How then could you have counted them as 'prescribed'? When the heavens moved with the earth, the mountains crashed down and the rivers rose up, the clouds dispersed and the lightning broke apart; the six directions lost their order; how then could you have been able to 'choose the very ground you walk on' or 'make your pace in walking conform to a musical beat'? Formerly the living fought for existence; the creatures died of worry; men's limbs were not obedient; their bodies turned to dust. [They were like trees] whose roots were pulled out and branches cut off; all lost their place. How then could you 'restrain yourself and cultivate your conduct,' 'bent in two like a musical stone' 'as if holding a drum'?
Li Mu lost his life in spite of his merit; Po Tsung was loyal, and his family was killed off; if entry into official life to seek for profit [thus) leads to loss of life, and working for titles and awards leads to the extermination of one's family, how then are you able to 'clasp your gold and jades' in myriads and myriads and respectfully 'serve your sovereign' and still able to keep your wife and children alive? Can it be that you have never seen a louse in a pair of drawers? When he runs away into a deep seam or hides in some broken wadding, he thinks he has found a 'propitious residence.' In his movements he dares not leave the seam's edge nor part from the crotch of the drawers, and thinks he is 'toeing the orthodox line' that way. When he is hungry he bites his man and thinks he can eat forever. But when, [in the event of a great fire] there are hills of flame and streams of fire, when towns are charred and cities destroyed, then the lice, trapped where they are, die in their pair of drawers. What difference is there in your gentleman's living in his small area and a louse in a pair of drawers? How sad it is that he thinks he can 'keep catastrophes far away and good fortune near' and '[his family and descendants] eternally secure'
Look, too, at the Sun Crow who roams beyond the dust of the world, and at the wrens who play among the weeds and grasses: There can certainly be no contact between the small [wrens] and the great [Sun Crow]; how could you ever imagine that your gentleman had heard of me? And again, in recent times the Hsia were defeated by the Shang; the Chou were banished by the Liu [Han]; Keng and Po became ruins; Feng-hao became a mound. In the length of time it would take a Perfect Man to come and look, one dynasty had succeeded another; before their residence was established, others had taken their place. From whom, then, would you 'receive' an eternal 'fief'? That is why the Lordly Man lives without taking up a dwelling, is orderly without 'cultivating' himself. The sun and the moon are his rule; the _yin_ and the _yang_ his measure. How could he have feelings of regret for the world or be bound to any single period in time? He comes on a cloud from the east and rides the wind that blows from the west. With the _yin_ he keeps his femininity, and with the _yang_ his masculinity. His ambitions are satisfied, his wishes fulfilled so that he is never exhausted by exterior things. Why, then, should he not be able to succeed by himself? Why should 'he fear the laughter of the world'?
In the past, when heaven and earth divided and the ten thousand things were all born together, the great among them kept their natures tranquil, and the small kept their forms calm. The _yin_ stored up their vital breath, and the _yang_ gave forth their vital essence. There was no fleeing from harm, no fighting for profit. What was put aside was not lost; what was stored up did not become surfeit. Those who died did not die young; those who lived did not become old. Good fortune procured nothing; bad fortune brought no calamity. Each followed his fate and preserved himself with measure. The bright did not win because of their knowledge; the ignorant were not overcome because of their stupidity. The weak were not cowed by oppression, nor did the strong prevail by their force. For then there was no ruler, and all beings were peaceful; no officials, and all affairs were well ordered. Men preserved their persons and cultivated their natures, not deviating from their norm. Only because it was so were they able to live to great ages. But now when you make music you get sounds in disorder; when you indulge in sexual activity you weaken the body. You change your exterior appearance to hide your passions within you. Filled with desires, you seek excess; you practice counterfeits to make yourself famous. When rulers are set up, tyranny arises; when officials are established, thieves are born. You idly ordain rites and laws to bind the lowly common people. You cheat the stupid and fool the unskillful, and hide your knowledge to make yourselves appear to be like spirits. The strong look fierce and are oppressive; the weak shiver with anguish and are servile. You pretend to be honest to attain your avaricious ends; you harbor dangerous thoughts within you but appear benevolent to the outside world. When you commit some crime you do not repent of it, but when you encounter some good fortune you take it as a matter for personal pride. Because you pursue these things to the exclusion of all else [?], you become stagnant and do not develop.
Now, if there were no honors, those in low position would bear no grudges; if there were no riches, the poor would not struggle [to obtain them]. Each would be satisfied within himself and would have nothing else to seek. If liberalities and favors did not bind one to [a sovereign], there would be no reason [to expose oneself to] death and defeat against [his] enemies. If rare music were not performed, the ear's hearing would not be altered; if lascivious views were not shown, the eye's sight would not be changed. If the ear and the eye were not altered and changed, there would be no way to disrupt the spirit. This was the perfection arrived at in former times. But now you honor merit to make one another exalted; you compete with your abilities to set one above the other; you struggle for power to make one rule over another; and you esteem honors so that you can offer them to one another. You encourage the whole world to pursue these aims, and the result is that the upper and lower classes harm one another. You exhaust all the creatures of the universe to their very limits in order to purvey to the endless desires of your senses. This is no way to nourish the common people! And then you fear the people will understand what is going on, so you add rewards to please them and strengthen punishments to keep them in awe. But when there is no more wealth, rewards can no longer be given; when there are no more punishments, sentences cannot be carried out. Then begin the calamities of ruined states, assassinated rulers and armies defeated and dispersed. Are these things not caused by you gentlemen? Your rites and laws are indeed nothing more than the methods of harmful robbers, of trouble-makers, of death and destruction. And you, you think they form an inalterable way of excellent conduct: How erroneous you are! . . ."
**3. Bao Jingyan**
From Etienne Balazs, _Chinese Civilization and Bureaucracy: Variations on a Theme_ , 243–6, © Yale University Press, 1964, reprinted with permission of Yale University Press.
The Confucian literati say: "Heaven gave birth to the people and then set rulers over them." But how can High Heaven have said this in so many words? Is it not rather that interested parties make this their pretext? The fact is that the strong oppressed the weak and the weak submitted to them; the cunning tricked the innocent and the innocent served them. It was because there was submission that the relation of lord and subject arose, and because there was servitude that the people, being powerless, could be kept under control. Thus servitude and mastery result from the struggle between the strong and the weak and the contrast between the cunning and the innocent, and Blue Heaven has nothing whatsoever to do with it.
When the world was in its original undifferentiated state, the Nameless ( _wu-ming,_ that is, the Tao) was what was valued, and all creatures found happiness in self-fulfillment. Now when the cinnamon-tree has its bark stripped or the varnish-tree is cut, it is not done at the wish of the tree; when the pheasant's feathers are plucked or the kingfisher's torn out, it is not done by desire of the bird. To be bitted and bridled is not in accordance with the nature of the horse; to be put under the yoke and bear burdens does not give pleasure to the ox. Cunning has its origin in the use of force that goes against the true nature of things, and the real reason for harming creatures is to provide useless adornments. Thus catching the birds of the air in order to supply frivolous adornments, making holes in noses where no holes should be, tying beasts by the leg when nature meant them to be free, is not in accord with the destiny of the myriad creatures, all born to live out their lives unharmed. And so the people are compelled to labor so that those in office may be nourished; and while their superiors enjoy fat salaries, they are reduced to the direst poverty.
It is all very well to enjoy the infinite bliss of life after death, but it is preferable not to have died in the first place; and rather than acquire an empty reputation for integrity by resigning office and foregoing one's salary, it is better that there should be no office to resign. Loyalty and righteousness only appear when rebellion breaks out in the empire, filial obedience and parental love are only displayed when there is discord among kindred.
In the earliest times, there was neither lord nor subject. Wells were dug for drinking-water, the fields were plowed for food, work began at sunrise and ceased at sunset; everyone was free and at ease, neither competing with each other nor scheming against each other, and no one was either glorified or humiliated. The waste lands had no paths or roads and the waterways no boats or bridges, and because there were no means of communication by land or water, people did not appropriate each other's property; no armies could be formed, and so people did not attack one another. Indeed since no one climbed up to seek out nests nor dived down to sift the waters of the deep, the phoenix nested under the eaves of the house and dragons disported in the garden pool. The ravening tiger could be trodden on, the poisonous snake handled. Men could wade through swamps without raising the waterfowl, and enter the woodlands without startling the fox or the hare. Since no one even began to think of gaining power or seeking profit, no dire events or rebellions occurred; and as spears and shields were not in use, moats and ramparts did not have to be built. All creatures lived together in mystic unity, all of them merged in the Way ( _Tao_ ) _._ Since they were not visited by plague or pestilence, they could live out their lives and die a natural death. Their hearts being pure, they were devoid of cunning. Enjoying plentiful supplies of food, they strolled about with full bellies. Their speech was not flowery, their behavior not ostentatious. How, then, could there have been accumulation of property such as to rob the people of their wealth, or severe punishments to trap and ensnare them?
When this age entered on decadence, knowledge and cunning came into use. The Way and its Virtue ( _Tao te_ ) having fallen into decay, a hierarchy was established. Customary regulations for promotion and degradation and for profit and loss proliferated, ceremonial garments such as the [gentry's] sash and sacrificial cap and the imperial blue and yellow [robes for worshiping Heaven and Earth] were elaborated. Buildings of earth and wood were raised high into the sky, with the beams and rafters painted red and green. The heights were overturned in quest of gems, the depths dived into in search of pearls; but however vast a collection of precious stones people might have assembled, it still would not have sufficed to satisfy their whims, and a whole mountain of gold would not have been enough to meet their expenditure: so sunk were they in depravity and vice, having transgressed against the fundamental principles of the Great Beginning. Daily they became further removed from the ways of their ancestors, and turned their back more and more upon man's original simplicity. Because they promoted the "worthy" to office, ordinary people strove for reputation, and because they prized material wealth, thieves and robbers appeared. The sight of desirable objects tempted true and honest hearts, and the display of arbitrary power and love of gain opened the road to robbery. So they made weapons with points and with sharp edges, and after that there was no end to usurpations and acts of aggression, and they were only afraid lest crossbows should not be strong enough, shields stout enough, lances sharp enough, and defenses solid enough. Yet all this could have been dispensed with if there had been no oppression and violence from the start.
Therefore it has been said: "Who could make scepters without spoiling the unblemished jade? And how could altruism and righteousness ( _jen_ and _i_ ) be extolled unless the Way and its Virtue had perished?" Although tyrants such as Chieh and Chou were able to burn men to death, massacre their advisers, make mincemeat of the feudal lords, cut the barons into strips, tear out men's hearts and break their bones, and go to the furthest extremes of tyrannical crime down to the use of torture by roasting and grilling, however cruel they may by nature have been, how could they have done such things if they had had to remain among the ranks of the common people? If they gave way to their cruelty and lust and butchered the whole empire, it was because, as rulers, they could do as they pleased. As soon as the relationship between lord and subject is established, hearts become daily more filled with evil designs, until the manacled criminals sullenly doing forced labor in the mud and the dust are full of mutinous thoughts, the Sovereign trembles with anxious fear in his ancestral temple, and the people simmer with revolt in the midst of their poverty and distress; and to try to stop them revolting by means of rules and regulations, or control them by means of penalties and punishments, is like trying to dam a river in full flood with a handful of earth, or keeping the torrents of water back with one finger.
**4. Tao Qian, "Peach Blossom Spring"**
Translated by Burton Watson, in Watson (trans. and ed.), _The Complete Works of Chuang Tzu,_ 142–3, © 1984, Columbia University Press, reprinted with permission of the publisher.
During the Tai-yuan era (376–397 CE) of the Chin dynasty, there was a man of Wu-ling who caught fish for a living. Once he was making his way up a valley stream and had lost track of how far he had gone when he suddenly came upon a forest of peach trees in bloom. For several hundred paces on either bank of the stream there were no other trees to be seen, but fragrant grasses, fresh and beautiful, and falling petals whirling all around.
The fisherman, astonished at such a sight, pushed ahead, hoping to see what lay beyond the forest. Where the forest ended there was a spring that fed the stream, and beyond that a hill. The hill had a small opening in it, from which there seemed to come a gleam of light. Abandoning his boat, the fisherman went through the opening. At first it was very narrow, with barely room for a person to pass, but after he had gone twenty or thirty paces, it suddenly opened out and he could see clearly.
A plain stretched before him, broad and flat, with houses and sheds dotting it, and rich fields, pretty ponds, and mulberry and bamboo around them. Paths ran north and south, east and west across the fields, and chickens and dogs could be heard from farm to farm. The men and women who passed back and forth in the midst, sowing and tilling the fields, were all dressed just like any other people, and from white-haired elders to youngsters with their hair unbound, everyone seemed carefree and happy.
The people, seeing the fisherman, were greatly startled and asked where he had come from. When he had answered all their questions, they invited him to return with them to their home, where they set out wine and killed a chicken to prepare a meal.
As soon as the others in the village heard of his arrival, they all came to greet him. They told him that some generations in the past their people had fled from the troubled times of the Ch'in dynasty (221–207 BCE) and had come with their wives and children and fellow villagers to this faraway place. They had never ventured out into the world again, and hence in time had come to be completely cut off from other people. They asked him what dynasty was ruling at present—they had not even heard of the Han dynasty, to say nothing of the Wei and Chin dynasties that succeeded it. The fisherman replied to each of their questions to the best of his knowledge, and everyone sighed with wonder.
The other villagers invited the fisherman to visit their homes as well, each setting out wine and food for him. Thus he remained for several days before taking his leave. One of the villagers said to him, "I trust you won't tell the people on the outside about this."
After the fisherman had made his way out of the place, he found his boat and followed the route he had taken earlier, taking care to note the places that he passed. When he reached the prefectural town, he went to call on the governor and reported what had happened. The governor immediately dispatched men to go with him to look for the place, but though he tried to locate the spots that he had taken note of earlier, in the end he became confused and could not find the way again.
Liu Tzu-chi of Nan-yang, a gentleman-recluse of lofty ideals, heard the story and began delightedly making plans to go there, but before he could carry them out, he fell sick and died. Since then there have been no more "seekers of the ford."
**5.** _Wunengzi_
Translated by Catrina Siu; edited by John Rapp, with (limited) notes based on various editions of the complete (surviving) Chinese text.
**Preface**
Wu Nengzi was my friend who's now passed away. As a young man he was widely learned and of few desires. As he grew he investigated principle to the fullest extent and the nature of things and got to the very nature of fate. During the Huangchao rebellion [874–884 CE] he fled and traveled around with no regular abode. He was cold and famished. In the third year of the Guangqi reign period [887] when the Son of Heaven was in Bao, all around there were armies. Wu Nengzi was staying in the home of the peasant Mr Jing who was from the town of Zuofu and the peasant dwelling was most lowly and there was stuff all about. In the daytime Wu Nengzi liked to stay in bed and not get up. As he lay in bed he would take a pen in hand and write one or two pieces of paper and then he would put them in the breast of his garment and not show anyone. From [such a such a date] to [such an such a date] he had written several tens of pieces of paper and put them in a bag and it would seem as if he had produced a book and I stole a look at them and tried to note down as much as I could see so that I could talk about these words with my brothers and friends. The main import of what he wrote concerns elucidating natural principle and getting to the origin of nature. Behave naturally and don't labor. Make sure that you follow nature without desire, and thereby he treated lightly the teachings of ritual and externalized the affairs of the world. People who are in the know won't need to be told about these things to believe them. Will people who aren't in the know be able not to condemn them? I divided his writing up into chapters, thirty-four in all, and made a book of three parts, first, second, and third volumes, with the purpose of sharing it with those in the know. Now because the doings of Wu Nengzi's life are hidden away I will not record his name or any of his particulars here.
**Part 1**
**_Chapter 1 : The mistakes of the sages_**
Before Heaven and Earth split, there was a mass [ _hundun_ ] of unitary ether [ _qi_ ]. This mass of _qi_ became full and overflowed, and split into two principles. At this point they were clear and muddy, light and heavy. The light and clear rose upwards, and became the element Yang of Heaven; the heavy and turbid dropped to the bottom, and became the element Yin of the Earth! The then robust and solid Heaven moved the then malleable and docile Earth [and things were?] peaceful; this is the natural way of _qi_. Heaven and Earth were already in their positions, the Yin and Yang _qi_ interacted, thereupon the naked creatures: The scaly creatures, hairy/furry creatures, feathery creatures, and shelly creatures were born. [Thus] Humans, [or the] naked creatures, the scaly, hairy/furry, feathery, and shelly creatures were [all] born from Heaven and Earth; they [all come from] the interaction of the _qi_ , there is no difference [between the two].
Someone says there [is a principle] that already exists that differentiates [between things], [but] is it not that people themselves maintain this difference between the scaly, feathery, hairy/furry, and the shelly creatures? But don't [people have] intelligence and wisdom [and] language? Oh well, birds and beasts, up to and including insects and worms, all favor life and avoid death, construct their nests and caves, plan their food, give birth to and raise their type and protect them; compared to people who [also] favor life and avoid death, construct their palaces and mansions, plan their clothing and meals, give birth to and raise their sons and daughters, and treat them with private love, there is no difference [between the two]. How can one maintain that there is no intelligence and wisdom [in these creatures]? Well, birds and beasts, up to and including insects and worms, they call, chirp and screech, each has their own sound; how [can we] know that among their kind, there is no language? Humans, by means of not knowing [animals'] sounds, maintain their inability to speak. Moreover, how [can we] know that the birds and beasts are not making an analogy of people's speech, also maintaining that people are incapable of language? Therefore, the sound of their cries, calls, chirps, and screeches must be speech. Moreover, how can one maintain that [animals] are incapable of language? As for intelligence, wisdom, and language, people and creatures are one and the same; that which is different is shape and form. So, [since] among the scaly, hairy/furry, feathery, and scaly [creatures], there are also differences in shape and form, how can it be that [they are] especially different from humans? Among humans, shapes and forms also have similarities and differences, differences and similarities; how can it be that [humans' forms are] especially different from the shapes and forms of the four creatures?
Alas! As for Heaven and Earth, the elements yin and yang are big things. The naked, scaly, hairy/furry, feathery, and shelled, these five vital classes, they followed the _qi_ that harmonized the big things (Heaven and Earth); moreover they form a body within the big things. Also it's like river streams and oceans providing lodging for fishes and other water creatures, [like] mountains and hills encompassing grasslands and woods.
In the most ancient times, the naked creatures and the scaly, hairy/furry, feathery, and shelled lived together indiscriminately, female and male, male and female. They [lived] together naturally, with no distinction between men and women, husband and wife [and no hierarchical order among] father and son, older brother and younger brother. In the summer they created nests and in the winter they created caves; there was no construction of palaces and mansions. They ate raw meat and drank blood, without eating the food of the one hundred grains. The living moved around, the dead keeled over, [there was] no [desire for] stealing and murder, [and there were] no funeral [rites]. They followed what was natural; there was no ruling or shepherding, [and everything was] in its original simplicity; according to these principles they could live long lives.
Not long after, among the naked creatures arose a bunch of "wise" and "intelligent" animals who called themselves "people" who established rules under which they could [dominate] the scaly, hairy/furry, feathery, and shelled creatures. Moreover, they taught [each other] sowing and planting in order to eat the food of a hundred grains, and thereafter [learned] to use the plow. They hewed wood and made mud bricks to construct mansions and palaces, and thereupon started to use the blade and the axe. They instituted marriages, which started the distinction between men and women, and thereafter began the distinction between husband and wife and the hierarchical distinction among fathers and sons and older brothers and younger brothers. They made coffins and shrouds to bury their dead, and thereupon there [developed] funeral rites. They tied knots together to make nets in order to catch the scaly, hairy/furry, feathery, and shelled creatures; thereupon emerged the taste for prepared food. Original simplicity was thereby broken up, thereby giving rise to selfish passions and intentions. People were strong and weak by their natural abilities; there was still no way to regulate this. Among the crowd that called themselves the "wise" and "intelligent," they chose one who would unite the rest of them; this one was called the ruler, and the multitude were called his servants [officials]. The one could control the multitude, but the multitude could not gain supremacy over the one. From this came the distinction between the ruler and the ministers, and the exalted and lowly. The honored were set on high and the multitude placed on the same low level [beneath him].
In later times hierarchy and emoluments were established among the "wise and intelligent." Thereupon, material things distinguished the ranks between the wealthy and the poor, people satisfied their desires in accordance with their ranks and emoluments. Then they called the wise and intelligent ones "sages."
But soon the debased and disgraced started to become jealous of the honored, the poor became jealous of the wealthy, and from this was born the spirit of competition. Those who called themselves sages worried about this and together they said, "in the time of original purity, who was it who called themselves people? We artificially imposed the name 'people' and therefore people were separated from the animals. At that time, there were no exalted and debased, [so] who was it who called themselves rulers and ministers? But after we imposed the construction of hierarchy; there came about rulers and ministers. At that time, there was no grasping and no desires, [so] what were ranks and emoluments to them? We imposed assessments on people, so now they started to realize the distinction between honorable and disgraced. Now, the pure and natural has been weakened, and passions and predilections are embraced by vying hearts. If there is competition, there is stealing, if there is stealing, there is chaos [ _luan_ ], [so] what is to happen in the future?" From among the group of the "wise and intelligent," one who was most "wise and intelligent" spoke and said: "I have a scheme!"; from this he taught the principles of benevolence, virtue, loyalty, and trustworthiness and to regulate them by means of ritual and music. When a ruler oppressed his subjects he was to be called cruel, and the ministers would say that the government was illegitimate. When the ministers usurped [the ruler's authority], the ruler would call them rebels. A father who did not love his son, would be called un-nurturing, and a son who did not obey his father would be called unfilial. When older brother and younger brother were not in accordance, they would be called disrespectful and unfraternal; when a husband and wife were not united as one, they would be called unchaste and inharmonious. People who acted in these ways were called the wrong and people who did not were called the right. The right were honored and the wrong were disgraced, thus was cultivated the feeling of pleasure in being right and the shame of being in the wrong, and feelings of competition were suppressed.
As even more generations passed, predilections and desires became more inflamed; thereupon [people] turned their backs on benevolence, virtue, loyalty, and trustworthiness, and they transgressed from ritual and music and [started to] compete [with each other]. Those who called themselves sages regretted this. They had no other option but to establish laws and punishments and organized armies to keep the people under control. When there were small offenses, [people] were punished. When offenses were big, an army was set onto them. Therefore punishments such as imprisonment, using the _kang_ , and being whipped were spread out over the country. Spears, pikes, bows and arrows were spread out over the world, families were destroyed and kingdoms wiped out. There were too many to count. The common people came to dire poverty and died; this spread without end.
Alas! It was natural to treat [the people] as beasts; it was not natural to treat them as humans. Imposing the establishment of palaces and mansions, [formal] meals and [prepared] food stirred up desires; imposing distinctions between the exalted and debased and the honorable and disgraced excited competition; imposing benevolence, virtue, ritual, and music perverted what was natural. Imposing punishments and laws and [using] military [force] immiserated [people's] lives; this caused people to seek after the branches [the extraneous] and forget about the root [the essential]; this disturbed their passions and attacked their lives, and together in great numbers they died. They could not revive the past. This was the fault of those who called themselves sages.
**_Chapter 2 : Illuminating the origin_**
That which people call the origin is the being at the heart of nonbeing; the shape of the body and the skeletal structure relies upon it in order to stand up straight; it is long-lasting and never dies. Just like fire that can be used to burn things, one cannot take away its heat. Just like how water can be used to moisten things, one cannot take away from its wetness. If you try to take it, then you will not have it. If you try to hide it, then it does not cease to exist. If it moves then you'll be able examine even if it is as small as an autumn hair and investigate it even if it is as silent as the buzzing of a mosquito; if it remains still, then if it's as big as Mount Qiu, then you won't be able to see it; even if it's as loud as thunder, you won't be able to hear it. When it's big it's capable of encompassing the entire universe; when it's small it can enter into the pupil of an eye. It appears suddenly, neither coming nor going. Suddenly, and without being aware, it is neither overflowing nor diminishing. The recluses Chao Fu and Xu You, the escape of Dong Yuan Gong and Qi Liji, they had a single-minded purpose at the root to only do what was right. Emperor Yao passed the empire to Shun, Emperor Shun passed the empire to Yu, Emperor Yu passed the empire to Jie, Tang kicked out Emperor Jie, and King Wu who attacked the state of Zhou took hold of the opportunity to benefit everyone simultaneously. One who understands this root, if he must hide, he will then hide, if he must act, then he will act, he responds to things and establishes affairs/gets things done, he is vast and without partiality/feelings. One who is blind to this, his predilections and desires are his motivation; every day, he mindlessly uses his environment, he isn't conscious and doesn't understand. The skilled is able to illuminate by means of an unfixed light, it shifts around according to the harmony of nature, then the great nameless origin will be seen in the midst of the unseen!
**_Chapter 3 : Analyzing misconceptions/The clarification of errors_**
As for human nature, it is spirit, as for fate, it is ether [ _qi_ ]. Human nature and fate—these two must mutually come together in the vast void; they give birth to each other in nature. They are similar to Yuan and Fu's mutual responding to each other/harmony, the mutual harmonizing of Yin and Yang. That which we term the skeletal part that is the body, it is the apparatus of human nature and fate. Is it not that fire is on top of the firewood, if there is no firewood then the fire does not burn, if there is no fire, the firewood does not glow (from heat). If there is not skeletal structure and body, human nature and fate has no means of standing up, if human nature and fate attach themselves to the body, then it causes them to be lively, therefore human nature and fate bubbles forth from nature and is born; the natural skeletal structure and the body stagnate and die. That which is born from Nature, although it exists separately and can be broken off, is eternally alive. That which naturally dies, although it moves around, it will always die.
Nowadays, everyone likes life and despises death; [people] do not understand the principle of the natural cycle of life and death, they look to the thing that is not moving and is rigid/stiff and they worry about it. They cast aside that which is naturally born, devoting themselves to preserving that which is naturally dead, the more diligently they preserve it, the more distant is life. This is desire that sinks feathers and floats rocks—how great is this stupidity!
**_Chapter 4 : Having no worries_**
As for people, they most despise death, which is to say that they despise the shape and skeletal body being rigid and not moving. As for the shape and skeletal body, blood, flesh, ear, and eyes, they cannot be empty and yet vital, therefore we know that they are not the implements of life. Therefore the reason you should not wait to call death the point at which there is no movement and stiffness; rather, death is at its root already there when we run about and move around! Therefore that which runs about and moves around relies on nothing more than that which is not dead. And, secondly, it is not that which is able to move and hasten about by itself. The body and skeletal shape are originally dead; therefore it is not dying today, therefore it is not dead today, and therefore it is not going to die! As for death, it is the most despised by the people. But there is no death to be despised, besides the shape and skeletal structure; is there anything really to disturb feelings of utmost harmony and satisfaction?
**_Chapter 5 : Criticizing foolishness (in two sections)_**
**Part 1**
The things that which everyone in the world commonly hastens after without knowing where to stop are wealth, nobility, and a good name. As for those we call wealthy and noble, they are satisfied by material things. At the high points of wealth are emperors and princes; at the low points of wealth are the dukes and marquises. Is it not by the crown they wear, the fancy palace they live in, and their security guards and attendants that we call them emperors and princes? Is it not because they wear a bureaucrat's hat [when they go outside], have noisy carts and horses, flags and big axes that we call them dukes and marquises? If we do not decorate them with an emperor's clothing, palaces, large umbrellas decorated with bird feathers, boards that prohibit people from passing, bureaucrat hats, carts, horses, axes, and flags, then what can make them emperors, princes, dukes, and marquises? As for the emperor's clothes, large umbrellas decorated with bird feathers, axes, flags, carts, and horses, these are all material things. When there is a sufficiency of material objects, then we have the condition of nobility. When one is wealthy and ennobled, then there are emperors, princes, dukes, and marquises. That is the reason why I say that the wealthy and noble are merely sufficient in material objects. As for material nothings, they are things that people are capable of making. There are those who make these things by themselves and on the contrary, there are those that do not create [there things], who enjoy them. Well then, just as we designate those with sufficient material things as wealthy and ennobled, we [also designate] those without material things to be poor and lowly; because of this, we take pleasure in wealth and honor and are ashamed to be poor and lowly; of those who do not achieve happiness, there is no conduct too extreme for them [to get what they want]. From ancient times until the present, [we] have been awake but not enlightened. How powerful is the strength of material things!
As for those who are said to have a fine name, are they not the type to live at home and be filial, the type to serve their superiors with loyalty, the type to make friends and be trustworthy, the type when confronted with objects of value are honest; are they the ones who are filled with talent, are they the ones who are sufficient in skills? These are the ones whom the so-called sages value, in order to control the stupid common people. As for what can be considered a fine name, it is a person's external bodily form and inner character. Without an external bodily form, then, one is equivalent to empty space, thus unwanted praise cannot be added to it. As for the external bodily form, it is [merely] a bag to hold the blood and all the internal organs; in the morning, it is whole and in the evening, it spoils—how can it be said that it has a good name? Among people today, why are there none who do not cast off their natural and correct human nature and [instead] hasten after wealth, nobility, and a fine reputation, which then leads to activities of cheating and falsifying? It is because the so-called sages have misled them.
**Part 2**
People of ancient times until now, those determined to be their relatives were their blood-kin, thereupon their affections had a point to specialize on. When gathered together, they cheer for each other, when separated, they become sick looking for each other, when sick, they worry for each other, when there was death, they cried for each other. Now, everyone under Heaven is a kin to me, we are all one body: we are like hands, feet, stomach, back, ears, eyes, mouth, nose, head, neck, eyebrows, and hair. How can you separate and differentiate this one from me? Therefore, the distinction between the self and the other resides only in the name. The reason why people feel distant from other people under Heaven is because we are not mutually familiar with each other; the reason why we are close to our relatives is because we are mutually familiar to them.
Alas! If among the people, you divide up their bodies into hands, feet, stomachs, backs, ears, eyes, mouths, noses, heads, necks, eyebrows, and hair and attempt to call them bodies, you will have no success, who could you say are your relatives? Who could you say have people? You'll have to achieve this act of distinction by imposing names. If you use the name that you use to name your relatives to name the people under Heaven, then all people under Heaven will be your relatives! If you use the way you familiarize yourself with relatives to familiarize yourself with people of the world, then all the people under Heaven will all be your relatives! What need is there to speak of an exclusive object of our affections? If there are none to be familial to or paternally benevolent to, then we can be familial and paternally benevolent to all under Heaven; but if there are those that we must be familial and paternally benevolent to, then we will only be familial and paternally benevolent to the people in one single household, and moreover, filial piety and paternal benevolence will become a burden! But if you get rid of them then there is insincerity; and if there is insincerity, then fathers, sons, older brothers, and younger brothers will have dislike and resentment!
Zhuang Zi said, "when a group of fishes are placed on land, they pass water to each other mouth to mouth [to keep each other alive], this is not as good as forgetting each other in the rivers and lakes." How true are these words! As like fish that should take no notice of each other in rivers and lakes, people should also take no notice of each other in Nature, this is what is suitable! Therefore finding an exclusive location toward which to direct your feelings is what an intelligent person will not do.
**_Chapter 6 (missing from surviving Chinese text)_**
**_Chapter 7 : Cultivating your original
nature (in four sections)_**
**Part 1**
As for the scale and mirror, they are material things; they are things that are made by people. People themselves make these things, and in return, they seek to know the lightness or heaviness from the scale and they seek to know the beauty or ugliness from the mirror, how is this so? The scale has no intentionality and is balanced; the mirror has no intentionality and is perfectly-reflective. As for material things without hearts, they are balanced and bright; with the people with intentionality, you must polish them with nothingness, clean them with emptiness, and cultivate in them a sense of formlessness and quietness, then they will not know who they are. I see them accompany Heaven and Earth in their boundlessness, reducing and rolled up in the _qi_ , but become inexhaustible, and under heaven, there is none who are/nothing that is able to compete with them/it.
**Part 2**
As for the nature of water, when it is dammed up, it forms clear pools, when it is channeled, it flows, when it rises up and evaporates, it becomes clouds and it will rain, when it lands on earth, it will moisten it, it forms rivers and oceans but feels no need to boast of its vastness, it may be in ravines and caves but it is not embarrassed/shamed by its smallness, it may divide into one-hundred rivers but it will not be exhausted, it will benefit the ten-thousand things and not run out of energy/quit, it is the most pliant of things. Laozi said, "therefore, the pliant and weak will be victorious over the rigid and strong." Then it contains the mysterious form of spirit [ _shen_ ], the one where the special _qi_ arrives and goes to, it is the thing that has obtained the most original essence of nature.
**Part 3**
When water flows, it is wet, when fire burns, it is dry and sultry, dragons originate from clouds, tigers originate from the wind; these are the natural principles of stimulus and response. Therefore that which is the mysterious form of spirit brings about _qi_ , the _qi_ brings about that which is mysterious, that is the way things are. If you want to know how things of nature respond to each other, then you should concentrate on returning to the root of the Mysterious Mother [ _Xuan Mu_ ], then you will have almost no problems in your understanding.
**Part 4**
Well now, that birds fly in the sky and fish swim in water is not by intentional design; rather, they naturally do so. Therefore they have no self-consciousness of their own ability to fly or swim. If they had consciousness of it and made up their mind in order to do these things, then they will necessarily fall out of the sky and drown! Also, just as how people walk with their feet, grab with their hands, listen with their ears, see with their eyes, they need not be taught to have the ability to do it. At that moment at which they are walking, grabbing, listening, and seeing, then the reflex takes over; moreover there is no need for them to think about things before they do it. If they first had to think about these things and afterwards do them, then they will become exhausted! If they followed along with nature, they will last a long time; those who attain its rhythms will be saved. As for the great, empty void, this is the natural state of the mind. Today, people's hands, feet, ears, and eyes follow along with their nature and walk, grab, listen, and see. As for their minds, they do not follow along with their nature and they are obstructed and hindered; [thus] if we desire the greatest harmony and enlightenment, that will be difficult.
**Part 2**
**_Chapter 1: King Wen speaks, Part 1_**
Lü Wang was fishing on the bank of the Wei River. Before Xi Bo went out to hunt, he divined using stalks of plants. The result of the fortune telling said, "There will indeed be no bears of any kind, and heaven will bestow upon you a teacher." Getting to the hunt, he found Wang, and thereupon Xi Bo again entreated him, yet Wang kept fishing without interruption. Only after Xi Bo repeatedly beseeched him, Wang sat down with his legs crossed like a basket and laughed, saying "Why did you come here?!" Xi Bo said, "The Shang government is in chaos! The people are in great pain! I, a foolish peon, desire to save them, yet I think I should get a worthy gentleman to help me." Lü Wang said, "The Shang dynastical government became chaotic by itself, and the people are in great pain out of their own doing. What is the connection to you? Why do you want to sully me?" Xi Bo said, "Well, sages should not hide their usefulness or keep their benevolence to themselves. They must exhaust their wisdom by helping all things universally. Isn't this so?" Lü Wang said, "Well now, Human beings are floating between heaven and earth, together with the birds, beasts, and many insects, in the middle of unitary _qi,_ and nothing more, exactly the same as castle walls, houses, and cottages all really are based on hollow air. If something completely destroyed the castle walls, houses, and cottages, then the air would still be the air. If something killed off all humans, birds, beasts, and insects, the _qi_ would still be the _qi_. How can we do anything about the Shang government's tyranny? How can we say anything of people's hardship? Despite all of this, the castle walls, houses, and cottages are already built and so need not be destroyed, just as the people are already formed and need not be killed, so I will save them!" Then, he agreed with Xi Bo and rode back home with him in the same carriage.
Tai Dian Hong Yao personally went to Xi Bo and said, "The accumulated virtue and amassed achievement of Gong Liu and Hou Ji, and through the current reign, the King's virtue extends above and beyond his ancestors! Now the earth is divided into three parts, and the King possesses two of them; this can be called 'fantastic'! You, Lü Wang, are a fisherman, so what would you ever want to say beneath the extreme greatness of the King?" Xi Bo said, "Well, the virtue of inaction envelopes and pervades heaven and earth, while the virtue of action gets things started and accomplishes things. Xüan Yüan and Tao Tang's actions made them Sons of Heaven, and it was with the virtue of action that they obtained an audience with Master Guang Cheng at Mount Kong Tong and asked for Xü You at Sieve Mountain, although they didn't catch his attention. Besides, my virtue is not yet accomplished like that of Xüan Yüan and Yao, and isn't my inferiority the result of the virtue of inaction?" Tai Dian Hong Yao said, "If what the King says is true, then Wang is really the epitome of the virtue of inaction, so why is he following the King's actions?" Xi Bo said, "Heaven and Earth are inactive, yet the sun, moon, stars, and constellations move in the day and the night. There are rain, dew, frost, and freezing rain in the autumn and winter. The great rivers flow without pause, and the grass and trees grow without stopping. Therefore, inaction can be flexible. If there is a fixed point in action, then it cannot be inaction." Lü Wang heard this and knew that Xi Bo really did have compassion for the people and didn't want any profit from the Shang Dynasty's world. Thereupon, Lü Wang and Xi Bo finally made the State of Zhou prosperous and powerful.
**_Chapter 2: Sayings of the masters of Shou Yang_**
When King Wen died, King Wu attacked King Zhou and destroyed him. Bo Yi and Shu Qi grabbed hold of Ma Chen's horse and said, "your father died and is not yet buried, and you have already taken up this large enterprise, and you have stirred up all the people, this is not filial. Being a minister, you have killed your ruler, this is not loyal." King Wu's retainers wanted to attack Bo Yi and Shu Qi, but King Wu performed a righteous act and let them go. Bo Yi and Shu Qi then left and hid in Mount Shou Yang and became known as the Masters of Mount Shou Yang. (Below: A possible friend's remonstrance to Bo Yi and Shu Qi)
"If you go in accordance with Earth's natural rhythm, there is no distinction between rulers and ministers. Someone created rulers and ministers in order to differentiate between the noble and based, those who called themselves sages, they, by means of their wisdom, deceived the stupid. By means of wisdom, they deceived the stupid, how absurd. With you, I've said this quite often! It was illegitimate to make a distinction between rulers and ministers; it was illegitimate to proclaim the Shang dynasty. Within the illegitimate Shang dynasty, there was one who was illegitimately known as Xin. As the illegitimate last king of the Shang dynasty, he was illegitimately cruel and illegitimately violent in order to fulfill his illegitimate desires. Ji Fa's rebellion was also predicated on desire. [The rebels'] desires were also illegitimate, therefore we can say that this is a case of the illegitimate replacing the illegitimate. As for taking no action [ _wuwei_ ], it is pure and upright and is in accordance with Heaven's principles; father, sons, rulers, ministers, do they exist in this natural state? Taking action [y _ouwei_ ] is based on predilections and desires, and it wreaks havoc on human nature; filial piety is not really filial piety, loyalty is not really loyalty, what difference does one have from another? Now, you are considering what we have always been saying to be illegitimate, by illegitimating what we have to say, you can cover up your illegitimate actions in order to invite a righteous reputation. You are relying on your bones, which will necessarily rot away, in order to move toward an empty reputation; this is like trying to put out a fire with the wind. Ji Fa did not attack you, how lucky. If his retainers had attacked you, then you would have obtained a good name, [but then] what good would your rotting body be to you? As for dragons that shed their scales, phoenixes who shed their wings, they will be looked down upon by fishermen and hunters. How sad! You are probably not friends of mine."
Bo Yi and Shu Qi thereupon escaped into Mount Shou Yang, we did not know how they died, and people thereafter thought they starved to death.
**_Chapter 3: Sayings of the old ruler_**
Confucius established the correct form of rituals and music and illustrated the ancient statutes. He edited _The Book of Poetry_ , _The Book of Documents_ , and _The Spring and Autumn Annals_ , so he thought he could, by means of all of this, put into correct order human relationships, and stop the hearts of the chaotic ministers, thieves, and rebels, and then he went to tell Laozi about it. Laozi said, "as for governing a large country, it is like frying a small fish, if you use these kinds of knives, it will be mashed! In the past, the sages invented material things and managed affairs, they seduced and moved people's passions, and people's passions lost what was natural, and people's human nature and fate came to an early end in many ways! Nowadays, you added new complications to the sages' system and tied up human feelings even more, and [so] you have complicated people's passions. People's passions are multiple, which makes them idle, idleness causes swindling and cheating, and cheating causes even more chaos. This is a case of attacking Heaven's nature and having success, [meaning that] disaster is imminent." Confucius was scared, but he would not bring himself to stop.
Thereafter, he was kicked out of the country of Wei, and then he was disgraced in the state of Song, then he almost starved in Chen and Cai, and then he was surrounded by people who did not like him in Kuang. He spent his whole life anxiously, and several times he was almost killed. Confucius turned around and looked at his disciple Yan Hui and said: "You don't suppose what Laozi said was right, do you?"
**_Chapter 4: Sayings of Confucius (in two sections)_**
**Part 1**
Confucius was surrounded by people who did not like him in Kuang, for 7 days, he strummed a string instrument and sang without stopping. Zi Lu said: "I have heard that the gentleman can protect his body from any kind of harm, and never has trouble for even a single day. Well now, you who are a sage has nevertheless starved in the state of Chen, and been surrounded by people who did not like you in Kuang; why is this so? And now, you, Master, are strumming on a stringed instrument and singing without stopping, and you do not have a melancholy expression, do you have a secret method?" Confucius said: "You, come over here, I want to tell you something: well now, people themselves have it in their power to do the correct, incorrect, the evil, and the upright, incorrect, the evil, and the upright derive from people themselves, whether you get a lot or a little [luck] depends on Fate, having success or failing depends on the time. The light of the sun and the moon, even these things cannot avoid the disaster of an eclipse. These sages, who are wise men, their intelligence and wisdom are not able to change the human allotment of how much or little success or failure [one receives]. The gentleman is able to be benevolent to people, but not able to cause people to be benevolent to himself; he is able to be righteous to people, but not able to make people behave righteously towards him. [If] the people of Kuang are surrounding me, it is not due to any fault of my own; I am powerless to keep them from surrounding me! Moreover, the thing that can be surrounded, it is only my body. I am merely floating without form in an empty space above, I am floating without passion in another space, and I know of nothing of which to be anxious about, so I am, by chance, harmonizing my instrument with my song and nothing more." Before he was finished speaking, the people of Kuang had dispersed.
**Part 2**
When Confucius's disciple Yuan Xian lived in a lowly lane, his other disciple Zi Gong was simultaneously serving ministers in the states of Lu and Wei. [Zi Gong] saddled up his horses and assembled his retinue to call on Mr Xian! Xian was wearing his tattered clothing. Zi Gong said: "As for you, are you sick?" Xian said: "I have heard that if you do not cultivate virtue and justice—that is what is called sickness, being without wealth—that is called poverty. I am poor, but not sick." Zi Gong was embarrassed by what Xian said, and for the rest of his life, he did not dare to go to see Xian again.
Confucius heard this and said: "What Zi Gong was in the wrong. Well, now, he is concerned merely with external appearances and not emptiness, one who keeps these things inside his heart is not pure because he is not empty, so then his thinking is not clear, because it causes his heart to not be chaste. Zi Gong5 is close to being arrogant and desirous; Xian is close to steadfastness and purity, we can compare them to the clear and muddy, they are mutually distant by quite a large degree!"
**_Chapter 5 (missing from original text)_**
**_Chapter 6: Sayings of Fan Li_**
Fan Li helped King Gou Jian of Yue destroy Wu and killed Fu Cha, and in discussions with Minister Zhong said, "I have heard that for one who secretly schemes against other people, disasters will necessarily rebound on him. As for the destruction of the Kingdom of Wu and the death of King Wu, this has followed from the secret schemes you and I have made. Moreover, as for the way the king treats people, he likes to share his worries, but he doesn't share his happiness. This is not to mention, numerous achievements, a well-known reputation, and going into retirement—this is the way of Heaven."
Minister Zhong said, "as for the whole world and the ten-thousand things, they are born in the spring and killed in the winter, as for the ten-thousand things, how can they, by being killed in the winter, cause disaster for Heaven and Earth? I hear that sages are not valued for their solitary goodness; rather, they are valued for getting rid of harm and helping things grow/getting things done. If you have helped something grow, you could be said to have gotten rid of disaster. This is what the Yellow Emperor did when he killed Chi You. The legendary emperor Xun eliminated the four evil ones, I have gotten rid of chaos in the state of Wu and have brought [things] to a successful state of completion, [with] hegemony in the state of Wu under Yue; this is nothing more than getting things done and getting rid of harm, [so] what disaster/retribution will come back to strike me? Just now, the King was able to destroy Wu because he had you and me; we must serve in office from start to finish, don't hasten towards retirement!"
Fan Li said: "No, you're wrong. Not to mention, as for the universe, it has no intentionality, it does not control itself. Moreover, how can it control the other things? Heaven and Earth are by themselves Heaven and Earth, the ten-thousand things are by themselves the ten-thousand things, spring, by means of warmth gives birth to itself, winter, by means of the cold, kills itself, it is not Heaven and Earth that causes this to be so. Sages, although they have intentionality, what they can perform is part of Heaven and Earth. Heaven and Earth, although without intentionality, when stimulated, they will respond, when affairs push them, they will obey, when things pass by them, they will resist and go against them, getting rid of harm and causing things to come to completion, it has nothing to do with hate or love. Therefore [even when] we have gotten rid of harm and avoided disasters and brought material things to a completion, we will have no good fortune. Recently, because he hated the state of Wu, [the king] employed you and me in order to get our schemes. You and I benefitted from this pay and therefore we schemed against Wu, and [can] take as a sign of our success, the destruction of the people, and as payback, he gives us our emoluments. The duplicity of people is such that they say that they are like Heaven and Earth's births and killings [and] that they are agents of Heaven and Earth; what sages call getting rid of harm and bringing things to completion, isn't this just a big scam?" Minister Zhong was not happy, and he was greatly suspicious of it all and would not do it [retire from office?].
Fan Li in the end took his leave from Gou Jian and sailed on a boat to [Lake Tai]; not long thereafter, King Yue killed Minister Zhong.
**_Chapter 7: Sayings of Song Yue_**
Qu Yuan was a minister at Chu who held the title of "San Lu Dai Fu" of three closely allied clans. King Chu was not virtuous, clever Minister Jing Shang had the King's good graces, and so the state of Chu was not [well] governed. Qu Yuan was worried about this, so he remonstrated with King Xiang, and asked him to get rid of Jing Shang. The king did not listen, so Qu Yuan remonstrated with the King to the utmost point.
His disciple Song Yu stopped him and said, "as for the intentions of a gentleman, he cultivates himself and does not find fault with others, he hides his usefulness and does not show it off to the public, when the time comes, then he responds, when material things come, then he follows through; he responds in time, but does not make plans for himself in advance; he follows through with these things but is not devoted to his own achievements, and for this reason the ruler's benevolent intentions will not accrue to him, and resentment has no place to gather. Recently, the king was misled by one with a clever tongue. He was confused, causing the government to become chaotic, the people in the state of Chu were all envious of Jing Shang's noble status and made a lot of commotion to try to appease him. Qu Yuan, at this point, was all alone; he held onto his loyalty and trustworthiness, called out in his midst, and no one listened to him, and the country was still not [well] governed, and all he accomplished to display to others was they were wrong and he was right, all he was doing was buying enmity and fishing for disaster." Yuan said, "I heard that as a gentleman, when residing at home, one must be filial and fraternal, when one acts as an official, one must be loyal and trustworthy. If he reaches his aspirations, although dead, he is like the living; if he does not reach his aspirations, although he is alive, he is like the dead." He kept remonstrating without stopping. Jing Shang resented this, so he calumniated to the king [about Qu Yuan].
Song Yu remonstrated to him, saying, "previously, you were all alone, holding onto your loyalty and trustworthiness, and you kept on saying the same thing! But you did not listen, so now what do you have to be sad about? It could not be rank and emoluments that you're thinking about, is it that you are thinking about the country you are exiled from?" Yuan said, "neither. Well, I am depressed about the non-usage of my loyalty and trustworthiness, and that the state of Chu is not well governed."
Yu said, "previously, you thought that you should die for filial piety, fraternal love, loyalty, and trustworthiness, so why are you sad? Moreover, your facial expression, form and body, they are not yours. The beautiful cannot be made ugly, the ugly cannot be made beautiful, the long cannot be made short, the short cannot be made long; the overflowing and strong are not able to be weakened, the weakened are not able to be made to overflow and strong; you cannot drive out sickness, when dead you cannot take things with you. My form and my body seem to belong to me, but I am not able to be in control of it. If your own body is like this, moreover, how would you desire to cause the people of the state of Chu to be ordered out of chaos by your own power. Your error is so deep! Well, the gentleman who lives in this world in the temporary lodgings of his body should have an empty heart when he responds to material things; there is neither wickedness nor righteousness, no right and no wrong, no good and no evil, no merit and no blame. If you have an empty heart, even if you are judging the Kings Jie, Zhou, and Jiao Ji, they are not to be blamed. If you preserve this emptiness of heart, even if you are judging the Kings Yan, Xun, Kui, and Xie, they are not worthy of merit. Then, as for your loyalty and trustworthiness and Jian Shang's evil cleverness, who is to differentiate between the right and the wrong? There is no way to differentiate between them, so then loyalty, trustworthiness, evilness, and cleverness are one. [Even] if there is a way to differentiate between them, to make these distinctions is illegitimate. Well then, you have left your nature far behind by relying upon these illegitimate distinctions, and you are relying on yourself to cast dispersions on others—you should not have waited for the king to exile you, you should have exiled yourself! Now, you have sought after being loyal and trustworthy and have achieved being loyal and trustworthy, and you are depressed about it and are unable to stop yourself; you are one who is known to have lost your incorrect way of thinking. I have heard that the people of the highest intelligence understand the rules, the ones of middle intelligence obey the rules, and the ones of power intelligence break the rules. As for the person who has an empty heart and is far away from taking action [y _ouwei_ ], they understand and transcend the rules; as for the ones who control their hearts and differentiate between right and wrong, they are obeying the rules; as for the ones who get the distinction yet get distressed and let it pass, they will be victims of the rules."
Yuan did not understand, in the end, he threw himself into the Mi Luo River and died.
**_Chapter 8: Sayings of Shang Yin_**
When Emperor Gao of the Han was infatuated with Qi Ji, he wished to replace the crown prince, Ru Yi of Zhao, with the prince of Ying. The great ministers were unable to resist this. Lu Hou was really worried about this; she schemed with the Marquis of Liu, Zhang Liang. Liang said, "only when there are extraordinary people, can extraordinary things get done. I heard that there were four recluses living on Mount Shang Luo; they are called: Minister Xia Huang, Minister Lu Li, Minister Dong Yuan and Qi Li Ji. The emperor often summoned them but they have never come. Now, the crown prince was truly able to humble himself and seek for them to come, so then, the four people, for the time being, came. If they came, they will be guests of the prince, and this will be a great help to him." [After] Emperor Lu followed Liang's plan, she sent Lu Ze to invite them.
The four people, in the beginning, refused him, but they got together and discussed [the matter], saying, "Liu Ji was high and mighty, moreover, he knows the means by which he is higher (exalted) than us, he sought after us but we will not go, he has embarrassed himself and nothing more! As for Empress Lu, that woman, her nature us cruel and mean, her son Ying was not yet firmly established as the crown prince, so she was necessarily pushed to crisis. In crisis, she has come seeking us; the peaceful resolution of the crisis depends on us. If she seeks us but does not get us, she will necessarily bring disaster upon us, therefore we must answer yes to her."
One day the four of them accompanied the crown prince into the palace. The Emperor saw them and asked them, and all four of them introduced themselves. The king was surprised, and then said: "I often sought for you but you would not come for me, so why do you follow the crown prince?" The four recluses responded, "your majesty has treated us poorly, we do not, in principle, allow ourselves to be humiliated. The crown prince honors people, so we have come as his honored guests." The emperor departed from them. He pointed to the four recluses and addressed Qi Li, saying, "the crown prince now has his own feathers and wings, now he cannot be harmed!"
Empress Lu treated them virtuously, she wanted to honor and give them rank and ennoblements. The four recluses discussed this and said, "the reason we came here was to avoid disaster, it was not from the desire of our hearts. Yin is now secure and Ru Yi has been undermined. The Empress Lu has now gotten her wish and Qi Ji was killed. Now we are afraid of disaster, we have caused Yin to succeed and Ru Yi to be undermined, we caused Empress Lu to be happy and Qi Ji to despair; this is called destroying others to keep yourself whole, so this is probably not a case of killing to achieve virtue. Moreover, are we going to deal with the humiliation of being ennobled by a woman and by this means, get a position at court—what difference is this from being a thief and going into a person's home and taking their gold and becoming a rich person?" So they left and again hid themselves in Mount Shang, and Empress Lu was unable to keep them.
Zhang Liang also became enlightened, thereupon he controlled his breathing and stopped eating and he left the palace and went into reclusion.
**_Chapter 9: Sayings of Yan Ling_**
When Guang Wu was in his early years, he made friends with Yan Ling when he was in poverty. When he ascended the throne, Ling was still a fisherman on Fu Chun Lake. [When] Guang Wu thought about their past, he admired and yearned for Yin Ling's virtue; he himself went to invite him to be part of his court, but Ying did not follow.
Guang Wu said, "you and I are friends; recently, I have been given the status of emperor, and you are still a fisherman; on your behalf, I am ashamed for you. I have official and noble titles by which I can ennoble you, gold and jade that can make you rich and cause you to be above millions. Taking action can move mountain summits, a single command [from you] can cause rain and clouds to rise up—this will bring honor to you and fame to your clan, you will have a succession of lines and marquises, you will have courtyards and palaces and mansions, multitudes of different carts and horses, beautiful clothes, delicate foods, people will play the bell and drum wherever you go, and there will be joint song and dance wherever you go; you yourself will be happy for your entire life, your name will pass on for ten-thousand generations. How would your life of dropping bait in this pool and having no fame compare with the life of the high and mighty, those who rise up and fall down? Why don't you follow me?"
Ying smiled and said: "In the beginning when I made friends with you, and you cultivated your virtuous intentions, and were satisfied with being poor and lowly, it seemed like you were one who I could select. Nowadays, you brag and are misled, you are a fool. As for the world, since antiquity, people have thought that it is the biggest thing. Among its ten parts, mountain summits, streams and oceans comprise half of it, the Man, Yi, Rong, and Di possess three parts, and the Middle Kingdom has only one or two of those parts. Within this Middle Kingdom, war has never ceased. As for the noble emperor of the Middle Kingdom, he is merely one who has proclaimed himself to be noble in this one or two tenths of the world that is constantly at war; you were the one with self-respect. As for the one who is ennobled and calls himself the greatest, he is nothing more than one who according to his likes and dislikes controls death within these one or two parts, in these one or two parts of the world, one who cuts bricks and wood to make palaces and mansions, one who assembles silks and other treasures to decorate his carts and clothes, one who kills oxen and sheep and plants of the one hundred grains in order to make delectable foods, lines up beautiful people and has them bang on gold and rock instruments, all in order to delight his eyes and ears. The emperor's desires are never satisfied; when old age arrives, then he will die, then his muscles will be cast aside and be food for the ants and maggots, and his rotting bones are reduced to mud and soil; he is no different than any common man or any common woman—where is the nobility of an emperor?
Those offices and ennoblements by which you honor me, I see through them all! Since antiquity, the noble title of minister, marquis, prime minister, great minister, these have been given by dukes and kings, they are all names that have been fabricated by the sages, who used rank to differentiate between the honored and lowly in order to seduce and guide the stupid people. Nowadays, you have the body of the emperor, but that's the same body you had when you were wearing cloth clothing; although people today call you emperor, still you ought to look at yourself—what differences are there in you now from the former times? Most likely, you want to seduce me with these made up names, to cause yourself to be happy and boast. Nowadays, you want to seduce me by means of these titles of minister, marquis, prime minister, and great minister—is this not treating me as if I am stupid? As for fake names, everyone is capable of making them. If I like doing this thing, then I could make up names and call myself minister, marquis, prime minister, or great minister! Why would I need you to make them up for me? Probably you will necessarily reply that the one with office and an ennobled title can by means of this become rich. Office and an ennobled title are in truth fake names; only I can truly enrich and ennoble myself; without thus sense of self, who has office and nobility? As for what is meant by title and nobility, it is nothing more than a tall hat, tinkling jade pendants, people walking in front of your horse carts, and people following behind your carriage, sitting in a large mansion, wearing new clothing, ears wearied from so much music of stringed instruments and bamboo instruments, your mouth entertained with chilies and orchids; this I say, is all with which you mean to seduce me and nothing more. As for carts and horses, they replace labor; whether it is a thoroughbred horses or an old man, it is one and the same; a house is there to protect from the wind and rain, whether it is a palace or a shack, they are one and the same; clothing is used to hide the body, whether it is fine silk fabrics or skins and cloth, they are one and the same; eating is to eliminate hunger, whether it is chilies and orchids or simple foods, they are one and the same. Moreover, I fish in the great emptiness, and I eat from the extreme harmony, I neither move nor am still, I am united in a single wave with the elements Yin and Yang. Just now, I forgot my own surname, I make no plans for when I go or when I stop, holding onto a fishing line and hurling fishing net, everywhere is my lodging place. Moreover, what time do I have to shackle my own body and deplete my energy; how lowly is craving for fake names and fulfilling illegitimate desires!
Whether King Meng or Geng Zhi possesses the world, what is the difference between that and you having all under Heaven? Aren't all of you are merely seeking to be the most honored in the Middle Kingdom? It is not that you are really concerned about the world. Now, you wage war and kill, not knowing when to stop, and you exterminate people's lives and fate, to obtain one's own desires; one who is benevolent cannot bear to speak of this. Moreover, you are not ashamed; rather, you are ashamed that I am a fisherman!"
Guang Wu was embarrassed and thereupon he did not dare to call upon Ling to be a minister.
**_Chapter 10: Sayings of Sun Deng_**
Minister Sun Deng hid in Mount Su Men [as a recluse]. Ji King admired this and went to see him, and said, "I have heard that bugs are not able to know a tortoise's age, a swallow and sparrow are not able to compare with the Hong bird. My heart is not sufficient to receive true teachings, nevertheless the light of the sun and the moon makes no distinctions when it shines on the main village of the little town; the rain doesn't choose whether it will water the fragrant orchids or the little weeds. Now [since] you have mastered your pursuit of self-cultivation, you must have extra which you can pass on to me, which can cause me to transcend from the finite into the infinite."
It was a while before Deng responded, "just now when I was in deep meditation, it seemed like I had a thought. If I did not have a thought, I was all bound up with the universe as if I had a spirit, but I did not have a spirit. Thoughts and spirits are true; if you want to leave them, you cannot, if you want to stay with them, you cannot. What can be called extending one's life? What can be called cultivating self? What can be said to have a limit? What can be said to have no boundaries? Yet within emptiness and nothingness, everything is flowing and continuous, both entry and exit leave no trace; they are the root of Heaven and Earth. The one who knows this is enlightened; the one who obtains this is respected. That which you just said is not even getting a look at the gateway. I heard multiple times that Laozi said, 'the good merchant hides his goods as if they were nothing, and a gentleman who is prosperous in virtue can appear to look stupid.' Moreover, just because the oyster has a pearl, it is cut open; the elephant, because of his tusks, is killed, orchids are rendered into precious oils, birds are plucked to make human beings pretty, this is what common people know. You are a well-known talent; you have forgotten the secret mystery of the dark universe. It is as if you are holding a bright candle, bright and illuminating your own skills, and Heaven hates you. I once read your book, _Letters Objecting to the Recommendation of Mr. Shan Ju Yuan_ , which is about the two great reasons and seven minor reasons why you should be an official and would be [great] by the times. As for one who is empty at his center, neither the court nor the marketplace will disrupt him; the one who has desires at his heart, the high cliffs and dark valleys will provide for him no rest. Office should not be able to shake you from your resolve; going into reclusion will not aid you in seeking peace. If you became an official then you have a lot to do, if you are not an official, then you will not have anything to do; also you brought up the fact that you want to cut off interactions with people, and that you are a useless creature, but that is the same as getting into a vulgar argument that you want to disentangle yourself from; and now you say you want to seek after eternal life, this can be called disliking one's own shadow and running away from it in the sun. How are you good enough to listen to my instruction?"
Kuang was confused and seemed as if he was drunk, and later on in life, he was executed.
**Part 3**
**_Chapter 1: Answering Tong_**
When Wu Nengzi was impoverished, his elder brother and his younger brothers' sons were cold and starving and they all sighed and followed each other. On day his elder brother's son Tong addressed Wu Nengzi and said, "alas, I'm cold and hungry and I've been so for many years. Last night I dreamt of being an official with a big salary and I had a lot of carriages, horses, gold, and silk. When I was dreaming I was happy; when I awoke then I was sad. How can I manage to flip dream and reality?" Wu Nengzi said, "your unhappiness during the day and your unhappiness at night are all the same. There's no need to change them." So his elder' bother's son then said, "oh, so you mean to say that happiness at night is just a dream and nothing more?" Wu Nengzi replied, "at night when you dream of residing in a mansion and riding a carriage with horses and wearing fancy clothes and eating and drinking and having love for your wife and children and despising your enemies, are those feelings of sadness, happiness, delight, and anger any different from the desire and actions you take when you are awake?" His elder bother's son responded, "there's no difference." Wu Nengzi said, "since there is no difference, how do you know that what you do when you are asleep and what you do when you are awake are both no dreams? Now the human lifespan is about 100 years. It's divided about equally between day and night, so half the time you're happy and half the time you're sad. What's there to be resentful about with that situation? Now, as for those people who can maintain themselves in the cultivation of the void [ _xu_ ], even if they were to become kings and marquises that would not be sufficient to ennoble them, and even if they were to become slaves, that would not debase them. Even if they had jade and silk and sons and daughters, that would not be enough to enrich them and if they had meager porridge and tattered clothes that would not be enough to impoverish them. For them, there's no space for sadness and happiness. The emotions being moved and then taking form in the body are nothing more than being stimulated by things, and that which we mean by things are nothing more than wealth and honor. Bodies and things are the root of decay. When your feelings are moved by them and you feel sadness or happiness, this is impermanence [ _wuchang_ ]. With impermanent feelings getting tied up with the root of decay, is this not like saying that the waking state is like a dream and that the 100-year lifespan is nothing more than one nighttime? If you are able to maintain yourself in the cultivation of the void then you won't know the meaning of starvation, cold, wealth, and ennoblement. If your emotions are moved and they take form in your body, then night and day, sleeping and awake, will be all a dream. Think about that."
**_Chapter 2: Responding to Hua Yangzi_**
Wu Nengzi had an acquaintance whose name was Hua Yangzi, who was being pressured by another acquaintance to take office. Hua Yangzi couldn't decide what to do and so he consulted Wu Nengzi. "I have been practicing to be without intention for a long time. If I go and become an official, then I will be going against my desires, but if I don't go and become and official, then I will anger that friend. What should I do?"
Wu Nengzi said, "Having no intentionality [ _wuxin_ ] is not something that you can learn. Having no intentionality has nothing to do with serving in office or not serving in office. If you confused and your thinking is too deep, it's like you have seen a blind person on the verge of a pit and you instruct him to walk forward. As for a person who takes no action [ _wuwei_ ] that means there's no action that he cannot take, and as for a person who takes action, there are certain actions that he can't take. Only those people who are closest to their original nature [ _zhishi_ ] will be able to understand this great principle. That which is closest to the highest public spiritedness [ _zhigong_ ] is what we mean by no action and it takes its root in having no desires and having no selfishness. So if you have desire then even if you're a fisherman, a woodcutter, a farmer, or a shepherd, you'll have intentionality [ _youxin_ ]. But if you have no desire, and you're the emperor riding in his carriage or you're a marquis wearing his robes, then you'll have no intentionality. Therefore, sages abide where it is appropriate and take action [ _xing_ ] where it is appropriate. Principle is located at the point where one cultivates the self. Xuyou and Shan Zhuan [hermits from the time of Shun] were not embarrassed to be commoners, but when the situation is favorable then it is permissible to provide aid to the world. Therefore the emperors Yao and Shun didn't decline the office of emperor. In both cases [i.e., the hermits and the emperors] they were united in having no intentionality. When Yao and Shun were on the throne they had no concern for the nobility that the office of Son of Heaven gave them. They merely let their robes hang down and the world was governed. So when it was evident that Dan Zhu [the son of Yao] and Shang Zhun [who was the son of Shun] were of small ability, then Yao passed the throne to Shun and Shun passed the throne to Yu; therefore they cast aside their own sons as if they were scabs and they set aside the world as if it were spittle. For this reason there were generations when the world was at peace. In the time of the Duke of Zhou, King Wen's son and King Wu's younger brother, [King Cheng] everyone knew that the Duke of Zhou was virtuous but because King Cheng was alive it was not a favorable time for the Duke of Zhou and therefore he didn't become the Son of Heaven. Because King Cheng was young it was correct for the Duke of Zhou to remain as regent and this he didn't decline. He did all this in order to make sure that the House of Zhou would last for generations and that the people of the state of Zhou would have good lives and he was greatly successful and the fame of his deeds has never declined. This is all because he had no desires himself and there was nothing that he would not do. If you can understand this, although you might be cock fighting or racing dogs in the butcher's market or grasping an enemy's battle flag on the battlefield, it doesn't matter, you can do both of them, so why are you worried about serving in office?"
**_Chapter 3: Answering Yu Zhongzi_**
Wu Nengzi's intimate friend, Yu Zhongzi had pain in his heart, so he asked Wu Nengzi for some medicine.
Wu Nengzi said, "what's the symptom?'
His friend said, "it hurts."
Wu Nengzi said, "where does it hurt?
His friend said, "in my heart."
Wu Nengzi said, "where is your heart?'
And Yu Zhongzi said, "my sickness is better now."
Wu Nengzi said, "You can really say that this fellow understands the nature of heaven and is one who is truly enlightened."
**_Chapter 4: On fish_**
On the Yellow River there's a place called Dragon Gate [a tight spot in the Yellow River where in ancient times people said that if fish could jump over the pass they would become dragons and if they couldn't they would remain fish] that is close to Li that is in the ancient state of Jin. The pass in the river was carved out by Emperor Yu [when he was controlling the floods]. The water there falls down some tens of _ren_ [Chinese yards]. The water that comes over it has a gushing sound like thunder that can be heard for ten miles around. In the springtime the great fish of the river assemble below it and use their strength to try to surmount it, and those who pass over the gate then become dragons who are then capable of creating clouds and causing the rain to fall. The little fish look at each other and say, "we're also fish and we could transform in this matter as well. Why are we just paddling around here hiding ourselves in little rock caves?"
One from among them said, 'how wrong you are. Within the universe the forms that things take numbers more than ten million. The magnitude of things' form ranges from the big to the small. According to a thing's form, it fulfills its destiny, each of which is appropriate to itself. As for the ones that become dragons, when the river is turbulent then it knocks them around; when it's placid, then it leaves them at peace. And at that time whether they are deep in the water or floating in the surface, they're happy and safe, but when it comes time to change into dragons, they assemble at the bottom of the waterfall and the force of the waterfall is angry and they struggle, then it becomes cloudy and starts to rain. Now the clouds and the rain are only the product of the moisture of the river. This has absolutely nothing to do with the fish themselves. If the fish who were becoming dragons were to have the intention of making clouds and rain, some of the time the clouds would form and the rain would fall, but this is actually just a product not of their intention and it's not an achievement of theirs. The horns on their head and the claws of their feet are the same as the whiskers that we have on our face. We swim around with our whiskers in the water and they fly around with horns and feet. Both of us are doing what is natural. Why would we want to change our struggle-free life swimming around here in the river and our carefree life here hidden in the caves and our happiness which results from people not knowing where we are not harming us for the laboriousness of their horn-footed life in the clouds and the rain?"
**_Chapter 5: The [Zhi bird] speaks_**
The Bird meets a snake and the bird goes forward and bites the snake and the snake says, "everyone in the world says that you are poisonous. To be poisonous is to have a bad reputation. The reason that you have a bad reputation is because you're trying to eat me. If you don't eat me, then you won't be poisonous. If you are not poisonous then your bad reputation will go away." The bird laughed and said, "aren't you also poisonous to people? And yet you point to me and say that I'm poisonous and by this means you're trying to cheat me, and the reason that you're poisonous to the people of the world is that you're trying to eat them and I'm angry at you for trying to eat the people. Therefore by eating you, I'm punishing you. The people of the world know that I can punish you and therefore will blame me for protecting you. They also know that when I eat you your poison become infused in my feather and my body and therefore I can kill people. My poison is actually your poison. I hate my bad reputation and yet I live with it, but what kills people are really people themselves. For example, when people use weapons to kill other people, is the weapon at fault or is the human at fault? Therefore it is clear that it's not my poison that kills people. Now the people of the world blame me and don't blame me—that's clear. Unintentionally I poison people—it's merely because I hate bad things that I have gotten this reputation. I'm used by people but my actions are not selfish. I'm not selfish and I'm happy to have a bad reputation, and that's in fact not having a bad reputation. You on the other hand have the intention to poison people and you skulk in the grass and bushes and enjoy waiting for people to come by. Now your meeting with me today is fate, and yet you want to use rhetoric to argue your way out of it." The snake was unable to respond, so the bird ate him.
Now creatures cannot have intentionality, what about people?
**_Chapter 6: Answering Lu (Note: No one knows who he is except that he is Wu Nengzi's cousin) (in two parts)_**
**Part 1**
Wu Nengzi's cousin went to study with Wu Nengzi and Wu Nengzi said, "What do you want to study," and the cousin said, "I would like to study morality and refined behavior [ _wen_ ]." Wu Nengzi replied, "I don't know what you mean by morality and I don't know what you mean by literature, but among those who in the past were called sages, I occasionally have seen what you're talking about. They have said that 'morality is putting things into action, that is to say, putting into action the goodness [ _shan_ ] in your heart. And by refined behavior is meant embellishing the goodness of your actions.' So the funeral rite is based in sorrow and the wearing of the mourning clothes and the implements used in sacrifice are all the embellishment. Ritual is based in respect; respect is an action, but the rising up, the going down, the bowing, and the yielding are all embellishment. Music is based in harmony; harmony is an action; the pottery, the gourds, the silk, and the bamboo of the instruments, are all embellishment. Embellishment derives from the action; the action derives from the heart; and the heart derives from what is natural. If it's not natural, then we have the birth of intentionality. With intentionality we have a brittleness of action, and with brittle action, then we have the corruption of embellishment. When embellishment is corrupt then it's false; when it's false, then it's disordered. When things are disordered, even sages will be of no help. Now you have to take hold of the root [ _gen_ ] and not the branch. Trace things back to their source and don't worry about the offshoots. If you can verify that you have no intention, then you can return to what is natural and you won't need to have the example of the sages before you or the example of mysterious heaven above you. Action and embellishment are both like in not studying."
**Part 2**
On another day Lu consulted Wu Nengzi again, saying, "I have often been troubled by not being able to reach the goal of my studies; I seek after it, but it disappears and I'm melancholy. After I have gotten drunk I am happy and ignorant of my distress, so I can't give up drinking."
Then Wu Nengzi said, "your worries and your melancholy do they come from your body? Or do they come from your heart?"
Lu responded, "from my heart"
Wu Nengzi said, "can you see your heart?'
And the disciple said, "I can't see it."
"That which you can't see is giving rise to your troubles and your melancholy. If you seek after what gives rise to your troubles and melancholy and you can't see it, where then do your troubles and melancholy lie? Since there is no location, for your troubles and melancholy, then when you seek after something and can't get it, and go after something and it's already gone, where do those things lie? Now you are sad and melancholy about not finding them. This is like trying to tie up the wind and catch shadows. Your worries and your melancholy have no real location and moreover you have a taste for the oblivion [ _taoran_ ] of alcohol and you are not satisfied, so you drown your sorrows in wine. Are you nothing more than a wine barrel?"
**_Chapter 7 (missing)_**
**_Chapter 8: A record of things seen (in three parts)_**
**Part 1**
In a market town in the former state of Qin, there was a conjurer who could put his hands and feet into a boiling vat of oil and yet remain with a smile on his face. Wu Nengzi sought him out and asked him some questions about the magic and the conjurer said, "I studied this trick from my master; the kind of magic that I practice can eliminate the heat of fire; moreover there's a little magic formula that I say which goes, 'when I see the pot of boiling oil, I first have to forget all about myself.' Not only do I have to look on my own hands and feet as if they were the sticks of an old tree, but I have to also forget about these hands and feet which are like the sticks of an old tree and only then will my trick work. But even if for one moment I start to fear, then the trick will fail. This is the secret to my success."
Wu Nengzi turned around and said to his disciples, "Young ones, take note of this. With a body without intention, the conjurer can cause even a boiling pot of oil to seem cold. Shouldn't people of superior virtue be able to do more?"
**Part 2**
One time when Wu Nengzi was staying with a peasant family named Jing in a village in the ancient state of Qin, at night an owl came by and landed on a branch and called out, and Mr. Jing's expression changed to one of sadness and he wanted to shoot it. Wu Nengzi stopped him and Mr. Jing said, "but the owl is an inauspicious bird. When something inauspicious is going to happen in a family(s) household then the bird comes and calls. If I kill it then maybe this inauspiciousness won't exist."
Wu Nengzi said, "if your family were really to have something bad happen to it because this bird came and called nearby, then that would really be the fault of the bird, and if the owl could really cause people to have bad things happen to them, then even if you killed the bird, it wouldn't be enough to get rid of the bad thing. If on the other hand something bad was going to happen at a family's home and only then did the bird come and cry, couldn't you say that owl is actually quite loyal to people and gives them a forewarning of bad things to come? And since the bad thing doesn't come from the owl itself then killing the owl is like killing a loyal and sincere bird. Moreover, we who call ourselves people and animals like this bird are both born from the impartial _qi_ of the universe. People have horizontal eyes and square feet and birds fly up into the air; these are our differences. But these are just incidental to the clearness, the turbidity, the lightness or heaviness of _qi_ and by this way they come into being. They don't come into being by any judgment of love or hate. Who commanded the birds to be in charge of ill omens? Who was the one who deemed this so? Did heaven and earth say this was going to be so? Did the owl himself say this was going to be so? But if heaven and earth didn't say this and the owl didn't say this, why must it be so? We don't know who originated this idea; moreover the beautiful colored bird we call the phoenix may not be auspicious and in the same way the owl may not be inauspicious." So Mr Jing didn't kill the owl and no harm came to his family.
**Part 3**
In the Pan clan there was a handsome man who was about thirty years old. On some days he would let down his hair and run all about. Other days he would just sit quietly for the whole day and not say anything. When he would speak he would say the horse is a goat and that a mountain is water. Whenever he pointed to any particular object he would use the wrong word to name it. Everyone in his family and everyone in the village thought he was crazy and no one paid any attention to him. Wu Nengzi also thought that he was crazy.
One day Wu Nengzi met this crazy one in a forest and he sighed and said, "you are a sturdy looking fellow with good looking features. What a shame it is that you're so sick." The crazy one slowly said, "I am not sick." Wu Nengzi was startled and said, "you don't wear your hat and your belt correctly. You get up and you sit down with no regularity. You misname everything. You don't observe the proper rituals of your family and the other villagers. This is insanity. How can you say that you're not sick?" The crazy one said, "do you really mean to say that wearing the belt and the hat in the proper way and having regularity in rising and sitting and showing respect or love towards my family members and respect toward my fellow villagers comes from my own nature? In the past there were people who fabricated things and they embellished things and called them the rites and they have caused people to practice these rites down to the present day. But weak wine and strong wine are still wine. One who knows this and nevertheless goes against this and then pretends not to know this is therefore called by everyone an insane person. Moreover, as for the names of the ten thousand things, do they also come from nature? The clear stuff that's gone up is called heaven; the yellow stuff that's gone down is called earth. The bright shiny thing in the day we call the sun and the bright shiny thing at night we call the moon, and as for the flowing, are they not all fabricated and forced names of things? For example, the wind, the clouds, the rain, the dew, the smoke, the fog, the frost, and the snow, mountains, peaks, rivers seas, grass, trees, birds, beasts, Chinese, barbarians, emperors, kings, dukes, marquises, officials, farmers, artisans, merchants, slaves, of all kinds, and even truth, falsehood, goodness, good and evil, the correct, the incorrect, the honored and the debased, they are all this way. People are used to these names so they don't see that they were in the beginning forced, so they continue the practice of using them and don't dare to change, but what would have happened if in the past the original fabricator had said that the light stuff that goes up is called earth and the yellow stuff that goes down is called heaven and the shiny thing in the sky is called the moon and the shiny thing at night is called the sun and we had used that practice till today? So these forced names derive from people. I'm also a person; on what authority did someone create these forced names and why can't I do the same? As for wearing my hat and my belt, getting up and sitting down, I'll do any of those as I please and I will name any of the ten thousand shapes and things as I please. Is this insane? I don't know, but is it right for others who don't know to say that I'm insane?"
**_Chapter 9 and 10 (missing)_**
**_Chapter 11: Holding firmly to the root (in four parts)_**
**Part 1**
All the five types of weaponry have as a purpose the killing of people. Various kinds of nets have as a purpose capturing birds, beasts, other kinds of animals and fish. The sages made them and afterwards people could kill each other. People could also catch the birds, the beasts, the fish, and other animals. First they caused them to know how to kill people and know how to catch things, then they set up penalties to stop people from killing each other and they set up prohibitions for entering the mountains and the marshes in order to stop people from catching animals. And now in this era of decayed morality, people can't protect their own fathers, their own children, and their own brothers and now the animals have ability to give birth to their young like little deer and little fish. The laws have become clearer and yet they can't prohibit [people from doing what they want]. This is because people have learned about weapons and nets. If the people who invented these things were to come back to life today, would they be able to control themselves [and not make these stupid inventions]?
**Part 2**
A coffin is of great help to the dead, but the people who make the coffins don't intend to help the dead; rather, they just intend to make money for themselves. Hoping to sell something every day, they hope that more and more people die. It's not that they hate other people, it's just that they hope to get profit. Doctors take pleasure in sickness, but they also hope that they can cure sickness. It's not that they take pleasure in saving people and helping them, it's that they like profit. Coffins and medicine all are an aid to people. Taking pleasure in life and pleasure in death don't come from love or hate, they just come from the coffin maker or the doctor's desire. For this reason treating the universe benevolently through inaction is not like the profit seeking of the coffin maker and the doctor. It is rather, the real desire to help the dead and cure the sick.
**Part 3**
Animals with horns spear their enemy; animals with hooves kick their enemy; snakes bite, insects sting; they all use what is their own particular strength. If you investigate what they use then you can guard against what they use. For this reason, things that use something are not as good as those that don't use anything. There's an insect known as the silkworm that eats mulberries and produces silk in its stomach. It weaves its own little cocoon and is transformed inside. When it comes out it has wings and is a moth. It is relying on it nature to be so. This is just like the fetuses of animals and the eggs of birds; these are not things that they themselves have decided upon. Wise people know that you can turn silk into thread and thread into cloth. So therefore they boil the silk and then they weave it into cloth, turn it into material and wear it. Now the silkworm enters into its cocoon to become a moth, not for the purpose of allowing people to enjoy clothing. The reason why they're boiled is because they're burdened by the very silk that they produce. The people who boil them are not mad at the silkworms themselves; they just want to get profit from it. Now the animal's placenta, the bird's egg, and the silkworm's cocoon are all what is natural to them. That the silkworm alone produces silk and silk must be boiled is unlucky and that seems to be just dependent on fate. Now one who does nothing has neither luck nor no luck; there's no fate involved.
**Part 4**
Those who take action and perform good deeds will not necessarily become prosperous, and those who perform bad deeds will not necessarily meet with disaster; this is all determined by fate. For this reason the sages particularly held as valuable the idea of inaction [ _wuwei_ ]. If you are to tell the little insect that lives in a wall and the frog that lives in a well about tigers and leopards that live in mountains and whales that live in the sea, they would have their doubts because of the limits of their own experience. Similarly, if you tell people who are addicted to the affairs of the world about the principle of _wuwei_ , they will necessarily have doubts because they are enmeshed in their own practices. Fathers cannot pass on [the idea of _wuwei_ ] to their sons. Older brothers can't pass it on to their younger brothers. Some people will remain lost in their desire until the moment they die. Of people who return to the source [ _yuan_ ] and don't give rise to anything, in today's world there's not a single one. Alas! Inaction depends on me. Desire also depends on me. If I follow inaction then I will be at peace; if I follow desire, then I will toil. If I'm at peace then I will be happy; if I toil then I will be troubled. Ordinary people are deluded and there's nothing you can do to cause them to understand. What they study causes them to be this way. Bright people will turn their backs on these customs.
**Notes**
**** Frequently mentioned in early texts as an expert judge of horses.
**** Reading _t'ung_ with the man radical; see Chapter 10, n. 12.
**** The terms _su_ and _p'u_ (uncarved simplicity) appear frequently in the _Tao-te-cbing,_ for example, Chapter XIX. Waley translates them as "Simplicity" and "the Uncarved Block" respectively.
**** There are many different interpretations of the terms in this sentence. I follow the emendations and interpretations of Ma Hsü-lun.
**** Following texts that read _neng_ rather than _t'ai_.
**** Legendary ruler of high antiquity.
**** Li Mu was a famous general in Chao during the Warring States period. In spite of his brilliant service against the Hsiung-nu and the Ch'in, he was executed by the sovereign of Chao who believed a calumny against him.
**** Po Tsung was an outspoken courtier of the state of Ch'in during the Ch'un-ch'iu [Spring and Autumn] period who was killed, along with his family, because his frankness irritated less scrupulous courtiers than himself.
**** The analogy of lice in a pair of drawers, the most famous part of the 'Biography,' was probably inspired by a passage in _Chuang-tzu_ , 24 [see Watson, _The Complete Works of Chuang Tzu_ , 276] in which a class of men are compared to lice living on a pig. The 'great fire' is probably also inspired by that passage and should be read with the fall of dynasties in mind.
**** Literally 'yang crow.' This seems to be the earliest usage of this term as an heroic bird (like the phoenix or roc), a usage often found in later poetry. The comparison between heroic and small birds is based on the first chapter of _Chuang-tzu_. Like so many of the creatures referred to by Juan Chi, the Sun Crow was probably a well know mythological animal, perhaps the black crow often shown against the sun in early (Former Han) paintings. . . . Archeological discoveries of this type continue to show us that so many of the strange birds and beasts Juan Chi delights in mentioning were an important and perhaps even a commonplace of contemporary daily life.
**** Traditionally in 1766 [BCE].
**** The Chou were actually defeated by the Ch'in in 255 [BCE] and the latter by the Han in 206 [BCE], but Master Great Man can hardly be expected to take a mere half century into account!
**** Said to be the capital of the Shang king Tsui-I (reigned 1525–1505 [BCE]).
**** Po [refers] in all probability [to] the three capitals of the Shang dynasty, T'ang, variously located near Lo-yang . . .
**** The capital of the early kings of the Chou dynasty, in the northwest of Ch'ang-an.
****The description of "paradise" in _Lieh-tzu_ , 5, where _pu chiin pu ch' en_ occurs (A. C. Graham, _The Book of Lieh-tzu,_ 102, translates "no one is ruler or subject"), and of Utopia, _Lieh-tzu_ , 2 (translated by Graham, 34: "In this country there are no teachers and leaders; all things follow their natural course").
**** _Chuang-tzu,_ IX [see Watson, _The Complete Works of Chuang Tzu_ , 104–6].
**** An allusion to _Analects_ , XVIII, 6, in which Confucius sends one of his disciples to inquire about a fording place across a river. Here, of course, the phrase refers to seekers of the utopian land of the Peach Blossom Spring.
**** "Food from the one hundred grains" means food that needed to be processed through machinery or technology.
**** An autumn hair is traditionally considered the finest of hairs, those that are used for calligraphy brushes.
**** Implying the womb.
**** Lü served under Kings Wen and Wu of the Zhou Dynasty.
**** A large tributary of the Yellow River in today's Shaanxi Province.
**** Xi Bo later became King Wen, the first King of the Zhou Dynasty.
**** A high minister under King Wen.
**** Ancestor of the Zhou clan; he passed his job on to Hou Ji's great grandson.
**** Another ancestor of the Zhou clan who was supposedly an agricultural official that ruled the fief of Tai under Emperor Shun, the legendary founder of the Xia Dynasty.
**** Another name for the Mythical Yellow Emperor.
**** Another name for the legendary Emperor Yao, who abdicated his throne to Shun.
**** A legendary virtuous man.
**** A hermit from the time of Emperor Yao who supposedly refused all offers of honors and offices by becoming a hermit and living in a deep mountain cave.
**** The first king of the Zhou dynasty.
**** The last king of the Shang dynasty.
**** Bo = the oldest of the sons.
**** Shu = the middle of the sons.
**** Xin = King Zhou of the Shang.
**** One of Confucius's disciples.
**** King of Wu.
**** A person who attacked the emperor.
**** The last king of the Xia dynasty.
**** The last king of Shang.
**** An immoral robber.
**** Four legendary rulers.
**** A royal concubine.
**** Son of the emperor's first wife.
**** Son of [the emperor's] favorite concubine.
**** The legitimate empress.
**** Empress Lu's eldest brother.
**** That is, has now grown up and can make his own decisions.
**** Later Han emperor Guang of Wu, ninth generation descendant of Emperor Gao of Han, who rebelled during the Wang Mang dynasty and reestablished the Han Dynasty.
****Four non-Chinese minority tribes.
**** A famous usurper.
**** Emperor Guang Wu's brother.
**** People would use this bird's poison to assassinate people by putting it in their wine.
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Zhang Weiguo. "Whither the State Ownership System?" _Shijie jingji_ (World Economy), April 3, 1989. Translated in FBIS (April 25, 1989): 37.
Zhongguo renmin daxue makesi liening zhuyi jichuxi (The Marxist-Leninist Studies Department of Chinese People's University). _Wuzhengfuzhuyi pipan_ (Criticizing Anarchism). Beijing: Zhongguo renmin daxue chubanshe, 1959.
INDEX
"abolitionist" (wing of Democracy Wall Movement) , _see also_ Democracy Wall movement
"abstract democracy" as leading to anarchism
Alford, Robert , 189n.5
alienation (Marxist concept) , , 194–7
"alienation school" ( _yihualun pai_ ) ,
American libertarians , , 13–14
Ames, Roger , 23–4, ,
AMP (Marxist concept) 197–207, 210n.24, 210n.27, 210n.30, 210n.36
anarchism, basic thesis of , 7–9, 12–14, ,
anarchism, definition of
anarchism, denunciations of,
1957 to 1976 160–4
1992–present 168–9
early Deng-Xiaoping years 164–8
Hua Guofeng era 164–8
anarchist/Marxist debates (China) 112–18, 121n.38, 121n.43, 170n.1
anarcho-capitalists , 13–15
_an-archos_ ("without a ruler")
Anderson, Perry 210n.27
Andrew, Anita 69n.51, 155n.67, 191n.55
anti-militarism
Anti-Rightist Campaign 140–1, , 170n.8,
Anti-Spiritual Pollution Campaign , 195–6
"anti-statism" 129–32
anti-utopianism 61–4
Asiatic Mode of Production (AMP) _see_ AMP
Augustine (Saint)
autocracy , , , 207–8
_Autocracy and China's Rebel Founding Emperors: Comparing Chairman Mao and Ming Taizu_ (Andrew and Rapp) 69n.51, 155n.67, 191n.55
Aziatchiki ,
Ba Jin (Li Feigan) 116–18, 121n.38, 121n.43, 160–1, 170n.6
Bai Hua
Bakunin, Michael,
anarchist views of 5–6, , , , ,
compared to other thinkers , 38–9, 50n.115, , ,
influence of ,
other thinkers' opinions of , , 162–3
Balazs, Etienne , 100–1
Bao Jingyan,
anarchist theory of , 37–40, 59–60, , , 90–1, ,
compared to other thinkers , 90–2,
influence of
life of
_baojia_ (rural mutual surveillance networks)
Bender, Frederic , , , ,
"Biography of Master Great Man" (Ruan Ji) 35–7, , , ,
"Bitter Love" (Bai Hua)
Bo Xilai , 218n.2
Bolshevik revolution 113–14
Bolshevism
Bookchin, Murray 104n.24
"bourgeois liberalism," campaign against
Brodsgaard, Kjeld Erik 184–5
Buddhist doctrines,
Chinese anarchist movement (20th c.) view of
in Liu Shipei's thought 109–12
Buddhists/Buddhism , 89–103, 109–11,
Burke, Edmund
Cai Hosen
Callahan, William
campaign against bourgeois liberalism
"Campaign against Right Opportunism"
Cao Cao , , , , 99–100
CASS (Chinese Academy of the Social Sciences) 200–1
Chang, Aloysius
Chang Chung-yuan
checks and balances (Western)
Chen Boda ,
Chen Duxiu 111–16, , 120n.22
Chen Erjin , , 185–8
Chen/Ou debates 120n.22 _see also_ Chen Duxiu
Chen Shui-bien
Chen Yun
Cherkezov, Varlaam
Chiang Kai-shek 15n.1
Chinese anarchist movement (20th c.),
history of 107–18
opposition to Soviet-style socialism
view of Daoism
Chinese Asiatic mode of production debate
Christianity and anarchism , , 15n. 6, 16n.11, , , 170n.1,
Christoyannopoulos, M. E. 15n.6, 16n. 11, 87n.57
Clem, Will
collectivist anarchists , ,
collectivization policy (Mao's) 139–40
_Communist (Gongchandang_ )
Communist Party Congress 2012 (Chinese)
Confucians/Confucianism _see also_ Mencius
doctrines/terminology,
anti-Confucian language , , 80–1
filial piety 73–4,
humane/benevolent rule , , , , , , 83–4,
_junzi_
_li_ ,
Mandate of Heaven ,
_ren_ 61–2, , 73–4,
ritual , , ,
_sheng_
_yi_ 61–2, 73–4,
_zhong-xiao_
other thinkers' views of,
Bao Jingyan 37–8,
Daoists , 56–7, 61–7,
Mao Zedong
Tao Qian ,
Wu Nengzi 92–3
schools of , ,
Confucius , 37–8, 63–4, , , , ,
continuing revolution (Mao's concept of) 128–32,
corvée labor , , 155n.68, ,
Cultural Revolution,
anarchism, critique of during
anarchism, denunciations of during 161–3
Confucius and
"continuing revolution" (Mao's concept) and
defined ,
education during 148–9
effects of
extensive democracy during 166–7
Inner Mongolia and
intellectuals purged in ,
Mao and 187–8
Maoist-Stalinist coalition
Mao's autocracy and
Mao's mass participation concept and 147–8
Mao's self-reliance policies in 139–40
neo-anarchist critiques of state during 175–89
cybernetic view of nature , ,
_Da Xue_ ("Great Learning")
_daminzhu_ _see_ extensive democracy
_dao_ ,
Daoist literature and , , , , , , , 68n.29,
defined , ,
nihilistic thought and , , ,
_Daodejing_ ( _DDJ_ ) , , , , 23–6, , 53–4, 61–5, 75–6, 78–9
Chapter 18 ,
Chapter 19 72–6, 79–81
Chapter 30 , , 81–2
Chapter 38 ,
Chapter 55 ,
Chapter 74 ,
Chapter 80 , , , , , ,
Daoism, Huang-Lao ,
"Daoist communism" (Needham's concept)
Daoist schools of thought , , , ,
Daoist texts _see Daodejing_ ( _DDJ_ ); _Zhuangzi_ ; _Liezi_ ; _Huainanzi_
_daojia_ (philosophical Daoism) , , 19–20,
_daojiao_ (religious Daoism) ,
_Daren Xiansheng Zhuan_ ("Biography of Master Great Man") _see_ "Biography of Master Great Man" (Ruan Ji)
_Das Kapital_ (Marx)
_de_ (power, Daoist concept of) , ,
democracy, extensive _see_ extensive democracy ( _daminzhu_ )
Democracy Wall Movement , , 181–5, 188–9, 193–4,
_Demons_ (Dostoevsky) _see Devils, The_ (Dostoevsky)
Deng Liqun
Deng Xiaoping,
policies of , , , ,
thought of , 164–8, , ,
Destruction ( _Miewang_ ) (Ba Jin) 160–1
_Devils, The_ (Dostoevsky) , 104n.21
dictatorship of the proletariat 114–16,
"Direction of Democracy, The" (Wang Xizhe)
Dirlik, Arif 48n.31, 115–16, 120n. 18, 121n.38, , 151n.15, 155n. 66
_Dispossessed, The_ (Le Guin) 16n.15
"dissident radical"
Divine Farmer _see_ Shen Nung
Djilas, Milovan 132–3, , , , 189n.6
_dongluan_ (turmoil)
Dostoevsky, Fyodor , 104n.21
Duan Feng
dystopia/dystopian ideas , , 64–7
economy, commodification of
Edelman, Murray 15n.2
egalitarianism 148–9,
_Ego and Its Own, The_ (Stirner)
_End of the Maoist Era, The: Chinese Politics during the Twilight of the Cultural Revolution, 1972 >–1976_ (Teiwes and Sun) 156n.74
Engels, Friedrich , , , , 198–9,
equality,
in Daoist anarchist literature , , 58–60
Mao and ,
socialist goal of
twentieth century anarchists and ,
Esherick, Joseph
extensive democracy ( _daminzhu_ ) , 154n.61, 161–2, 166–7
extra-Party neo-anarchists 175–89
Falungong movement 168–9
fascism ,
"February Adverse Current" ,
Feigan _see_ Ba Jin (Li Feigan)
Feldt, Alan , 20–1, , , 46n.2
_fengjian_ (feudalism) 200–2
feudalism , 182–4, , 200–4
Feuerbach, Ludwig
"fifth modernization"
five mode view (Hu Zhongda's criticism of)
Forke, Alfred
Foundations of the Critique of Political Economy _Gründrisse der Politischen Okonomie_ ) (Marx)
"four bigs"
"Four Cardinal Principles" ,
four systems of authority
Fourier, Charles
Frakt, Phyllis 144–7
freedom ,
Friedman, Edward 87n.61, , 146–7, 159–60, 162–3, , , 209n.2
Gang of Four , 163–7, 170n.6, 181–2,
Gao Gao (wife of Yan Jiaqi)
Ge Hong
Gellner, Ernest
Gemi, Sharif 52–3,
Godwin, William
Goldman, Emma
Goldman, Merle ,
_gongyi_ ("public will")
Gorbachev, Mikhail
Gouldner, Alvin
Graham, A. C. 21–2, 24–5, , 42–3, 55–6
great democracy _see_ extensive democracy ( _daminzhu_ )
Great Leap Forward , 139–40, 155n.68, ,
_Great Warnings_ (Zhu Yuanzhang)
_Gründrisse der Politischen Okonomie_ (Foundations of the Critique of Political Economy) (Marx) 198–9
Gu Zhaoji
_Guangzhou Masses_ ( _Guangzhou Qunbao_ )
Guodian manuscripts ( _DDJ_ ) , 71–85
Guomindang (Nationalist Party) , , ,
Hall, David 30–1,
Han dynasty , , , , , , , , ,
Former Han , , , , ,
Later Han , , , 98–9
Hansen, Chad 15n.10, 16n.13
harmonious society (modern Communist slogan)
Hegel, Georg Wilhelm Friedrich
Henricks, Robert , 78–9, 81–2
_Heshang_ (River Elegy) (television series)
heterotopia
Holzman, Donald
Hong Kong democracy movement
Hong Xiuquan
_Hongqi_ , 165–7
Hoston, Germaine , , 130–1, 145–7
Hsiao Kung-chuan 21–2, , 28–9, , ,
Hsü Cho-yun
Hu Feng
Hu Jiwei
Hu Qiaomu 165–7, 196–7, 204–5
Hu Yaobang , 205–6
Hu Zhongda 202–4
Hua Guofeng , 164–8
_Huainanzi_ 22–4, , 98–9
Huang Lingshuang ,
Huang-Lao Daoism ,
Hunan Provincial Proletarian Revolutionaries Great Alliance Committee _see_ Shengwulian grou
humane/benevolent rule , , ,
Confucian doctrine of , , , , , ,
humanism (Marx's concept) , , 196–7
Hundred Flowers movement , 140–1, 160–1
_hundun_ (positive chaos) , ,
Hungary 188–9,
_huzhu_ (mutual aid) 43–4, , , 151n.19
individualist anarchists , ; _see also_ anarcho-capitalists
intellectuals, Mao's punishment of 141–3
"Is Yugoslavia a Socialist Country?" (editorial) 132–3
"January Storm" (Red Guard upsurge)
"Jasmine Revolution"
Jessop, Bob , 189n.5, 210n.23
Jesus ,
Jianbo
Jiang Qing (Mao's wife) , , , 162–4,
Jiang Zemin
_Jiefangjunbao_
Jin dynasty , , ,
Joseph, William , ,
Jung Chao-tsu 35–6
Kang Sheng , ,
Kelly, David
Kirkland, Russell 15n.10
KMT (Kuomintang) ; _see also_ Nationalist Party (Guomindang)
Konrad, Georgy
Kraus, Richard 133–4, 153n.50, 154n. 50
Krebs, Edward
Kropotkin, Peter,
"Anarchism: Its Philosophy and Ideal"
mutual aid notion 43–4, , , 151n.19
other thinkers and 27–8, , 43–4, 113–14, ,
thought of , , ,
Khrushchev, Nikita ,
laissez-faire (limited government) 21–4
Lao Tan _see_ Lao Zi (old master)
Lao Tzu _see_ Lao Zi (old master)
Lao-Zhuang Daoism
Lao Zi (old master),
20th century anarchism and (Krebs' view)
Confucius and
_Daodejing_ attributed to , , 78–9
Hsiao Kung-chuan's analysis of
Huang-Lao Daoist school and
Li Shizeng's view of
Liu Shipei's view of
law of return 75–6, 81–2
Le Guin, Ursula 54–5, 16n.15, 67n.7
"leftist infantile disorder" (Tiananmen as) _see also_ ultra-leftism
Legalists/Legalism,
criticisms of , 59–60, 65–6, , , 92–4
rewards and punishments (concept of) , , , , ,
rule, idea of , , , 38–9,
school of , , , 85n.1
Lenin , 50n.115, , , 170n.1, , 198–9,
Leninist Party-state 193–208
_li_ (ritual, Confucian doctrine of)
Li Cunshan , 87n.62
Li Erh _see_ Lao Zi (old master)
Li Feigan _see_ Ba Jin (Li Feigan)
Li Hongzhi 168–9
Li Shizeng
Li Yizhe ,
_liangmin_ ("good people")
_lianhe_ (voluntary association)
Liberal Democratic Party of Japan
libertarians, American , , 13–14
liberty , ,
_Liezi_ (Daoist work) , , 40–2, , 57–60
_lijia_ system
limited government (laissez-faire) 21–4
Lin Biao , , , 163–7, 171n.22, 172n.26,
Lin Mousheng
_Lishi yanjiu_
Liu Shaoqi , , 171n.22
Liu Shifu
Liu Shipei , , 109–12, 119n.13
Liu Xianbin
Liu Xiaogan 25–6, , 55–6, 79–81
Lu Jianbo
Ma Jia
"Ma Yanwen" (pseudonym) 134–5
Ma Ying-jeou
MacFarquhar, Roderick 155n.68, 156n.74, 157n.101
"madmen of the South" , , ,
_Makesizhuyi yanjiu_ (Marxist Studies) 204–5
Mandate of Heaven (Confucian) ,
Manichean ideas of Cultural Revolution
Mao Zedong,
AMP (Marxist concept), use of
as anarchist , 126–49, 150n.1, 151n.1, 151n.21, 155n.66
as anti-statist 126–36
attitude toward intellectuals , 140–3
as autocrat 136–8,
collectivization policy of 139–40
"continuing revolution" concept 128–32,
influences on , 126–49
mass participation concept of
neo-anarchist critique of 177–8
"new class" and 126–7,
Paris Commune model and , 187–8
punishment, view of
purges conducted by 140–1, 143–4
Rousseau and 145–7
self-reliance policies 139–40
Shengwulian group's view of 178–80
Zhu Yuanzhang and , , 139–41, 150n.2, 155n.67, 187–8
Maoists/Maoism , 126–49, 159–60, 162–4, 177–80, 187–8, , , , , 218n.2; _see also_ three line model of Leninist regimes
_Mao's Last Revolution_ (MacFarquhar and Schoenhals) 156n.74
_Marx and the Third World_ (Melotti)
Marx, Karl,
concepts of , , , 193–7, , 151n.19, 210n.13
critiques of ,
Wang Xizhe and 188–9
works of ,
Marxism vs. anarchism debates , , , 111–18, ,
Marxism-Leninism 12–13, , , , 170n.9,
Marxist democrats , 193–4, , , , , ,
Marxist Studies ( _Makesizhuyi yanjiu_ ) 204–5
Marxist-Leninist Institute of the Chinese Academy of Social Sciences (CASS) 200–1
Marxists/Marxism , , ,
"mass line" policies , 157n.100, 157n.105
mass participation 144–5, 156n.87
Mather, Richard 33–4
"May 16 Corps" , 190n.21
May 4th Movement
Meisner, Maurice , 151n.15, 152n. 25, 157n.101
Melotti, Umberto , 210n.36
Mencius , , , , , _see also_ humane/benevolent rule
Michels, Roberto ,
_Miewang_ (Destruction) (Ba Jin) 160–1
military officials (purged by Mao) 143–4
Ming dynasty , , 139–41, 150n. 2, 155n.67, 187–8
Ming Taizu (Zhu Yuanzhang) _see_ Zhu Yuanzhang (Ming Taizu)
_mingjiao_ ("teaching of names") ,
_Minsheng_ ( _Voice of the People_ ) 113–15
Misra, Kalpana 154n.50
More, Thomas
Morris, William
Mote, Frederick
Müller, Gotelind 46n.2,
Munro, Donald 144–6,
Munro, Robin
mutual aid ( _huzhu_ ) 43–4, , , 151n.19
Nathan, Andrew 157n.101
Nationalist Party (Guomindang) , , ,
Naundorf, Gert
Needham, Joseph , 30–2, ,
neo-anarchism , , 11–13, , 175–89, 193–208,
neo-authoritarianism
neo-Daoists , , , 31–5, , , , , , , 100–1
neo-Marxism
"new bourgeoisie" (Mao's concept of) 131–2, , , ,
"new bureaucratic class" , , 154n.61, , , 185–6
"new class" 130–6, 152n.25, 154n.61, 155n.63, 176–8, 182–9, 189n.6
_New Youth_ ( _Xin Qingnian_ )
Nietsche, Friedrich
nihilists/nihilism , 89–102, 110–11
Nozick, Robert 47n.12
oligarchy, iron law of (Michels)
"On Authority" (Engels) ,
"On Krushchev's Phoney Communism and Its Historical Lessons for the World"
"On Proletarian Democratic Revolution" (Chen Erjin)
"On Socialist Democracy and the Legal System" (Li Yizhe) 182–3
"On the Correct Handling of Contradictions Among the People" (Mao)
"oriental despotism" ,
_Oriental Despotism_ (Wittfogel)
Ou Shengbai 113–16, 120n.22 _see also_ Chen Duxiu
pacifist anarchists , 44–5
"parasitic" view of the state , 189n.5, 210n.23
Paris Commune,
Chen Erjin and 185–6
Engels and Wang's views of
Mao's view of , , , 187–8
Shengwulian group's view of 178–9
Tiananmen student protests and
Yang Xiguang's view of
"Party of Humanism"
"pathological" view of the state (Alford)
Paul (apostle)
"Peach Blossom Spring" (Tao Qian) 42–5
Peng Dehuai (Marshall) 140–1, ,
"people power"
People's Communes (Great Leap Forward) ,
People's Daily ( _Renminribao_ ) _see Renminribao_ (People' s Daily)
people's democratic dictatorship _see also_ dictatorship of the proletariat
"People's Militia"
Period of Disunity , ,
Six Dynasties ,
Three Kingdoms era
philosophical Daoism ( _daojia_ ) , , 19–20
_Placard of People's Instructions, The_ (Zhu Yuanzhang)
Plato
"political Daoism" (Roger Ames) 23–4
_Political Parties_ (Michels)
populism (Mao's concept) 143–4
_Possessed, The_ (Dostoevsky) _see Devils, The_ (Dostoevsky)
postmodern anarchism 89–90, 101–3
postmodernism , 102–3
"Praxis" group
PRC,
critiques of state by extra-Party neo-anarchists 175–89
denunciations of anarchism in 159–69
"Precapitalist Economic Formations" (Marx) 198–9
primitivists 24–5
proletariat, dictatorship of _see_ dictatorship of the proletariat
Proudon, Pierre-Joseph , , , , ,
_pu_ ("original simplicity") 42–3,
Qin dynasty
Qing dynasty ,
_qingtan_ movement , , 40–3
Rapp, John 69n.51, 104n.24, 118n.1, 155n.66, 155n.67, 191n.55, 210n.30
"Red Capitalist Class" , 179–80
Red Guards , , 176–9,
reification of state
religious Daoism ( _daojiao_ ) ,
_ren_ (human kindness, Confucian doctrine of) 61–2, , 73–4,
_Renminribao_ (People's Daily) 162–3, 166–8, 194–6
_ren_ - _yi_ (goodness and morality)
Republican period 119n.13
return, law of 75–6, 81–2
revolution, continuing (Mao's concept of) 128–32,
revolution, violent 151n.19,
Revolutionary Committees 178–9
rewards and punishments (Legalist concept of) , , , , ,
Ritter, Alan
"ritualization" of revolution (Frakt) 145–7
River Elegy ( _Heshang_ ) (television series)
Rothbard, Murray , ,
Rousseau, Jean-Jacques , 145–7
Ruan Ji 35–8, 58–60, , , ,
Schoenhals, Michael 156n.74
Schram, Stuart , , 150n.2, 157n.105
Schurmann, Franz , 155n.68
Schwartz, Benjamin , ,
self-reliance policies (Mao's) 139–40
serfdom
Seven Sages of the Bamboo Grove ,
Shang (Yin) dynasty 93–4
Shanghai Commune 162–3
Shaughnessy, Edward ,
Shen Nung 24–5, 54–6, ,
_sheng_ (sageliness, Confucian concept of) 73–4
Shengwulian group , , 178–9, , 189n.9,
Shifu
_Shuihu zhuan_ (The Water Margin)
_si_ ("raw silk") ,
Sima family
Sima Qian
Six Dynasties , _see also_ Period of Disunity
slavery (Han dynasty)
social anarchists
_Social Contract_ (Rousseau) _see also_ Rousseau, Jean-Jacques
"socialist democracy" , 184–5
socialist democrats _see also_ Marxist democrats
Song dynasty
Spanish Civil War
Spring and Autumn period _see_ Zhou dynasty
Stalin, Joseph , , ,
Stalinists 160–3, 165–6, ,
Starr, John Bryan 156n.87
state, neo-anarchist paradigm of
state, reification of
"stinking Ninth category"
Stirner, Max , , ,
Stojanovic, Svetozar
"Strive for the Class Dictatorship of the Proletariat" (Wang Xizhe)
Su Shaozi
Sun, Warren 156n.74
Szelenyi, Ivan
Taiping rebellion (19th century)
Tang dynasty , , , 68n.29, ,
Tao Qian , 42–5, , 60–1
_Tao Te Ching_ _see Daodejing_ ( _DDJ_ )
Tao Zhu
Tea Party, American , , 13–14
technology, Daoist attitude toward , 59–60, ,
Teiwes, Frederick 156n.74
"thinking generation" ,
Thoreau, Henry David
Three Kingdoms era _see also_ Period of Disunity
three line model of Leninist regimes 159–69, _see also_ Maoists/Maoism; Stalinists; Titoists
Titoist-Stalinist coalition
Tiananmen student protests (1989) 153n.48, 167–8, 206–7
Tito, Josip Broz 132–3
Titoists 160–2, 165–6,
Tolstoy, Leo , 15n.6, , , , 44–5, , 87n.57, , 170n.1
Tu Wei-ming
_tuntian_ (military farm policy)
ultra-leftism , , , , , 180–1
unilinear schema of history (Marxist concept of) , , 201–2, , 49n.54
utopia 51–61
vanguard of the proletariat, Leninist concept of , , 146–7
violent anarchists
violent revolution , 151n.19
_Voice of the People_ ( _Minsheng_ ) 113–15
Walder, Andrew , 147–8,
Waley, Arthur , ,
Wang Ruoshui 163–4, , 171n.22, 172n.26, 194–7, 209n.4
Wang Ruowang (playwright)
Wang Xizhe 158n.110, , 181–9, 191n.55
Wang Yizhou 204–6
Warring States Daoists , , 21–32
Warring States period _see_ Zhou dynasty
Water Margin, The ( _Shuihu Zhuani_ )
Watson, Burton ,
Weber, Max ,
Wei dynasty ,
Wei Jingsheng ,
Wei-Jin anarchists 6–7, , 32–3, 45–6
Wei-Jin period , , , , , , , , 56–7, , , , , 98–9 _see also_ Period of Disunity
_Wei-shu_ (Book of Wei)
_Wen Hui Bao_ (journal)
_wenren_ (new gentry) 182–3
"Whither China?" (Shengwuilian) 179–80
Wittfogel, Karl , 202–3
Wolin, Richard
Womack, Brantly 157n.100
Woodstock, George ,
_wu_ ,
Wu Dakun 200–3, 210n.36
_wu_ forms , , , , 72–3, 79–80 _see also wuwei_
Wu Nengzi ("Master of No Abilities") 68n.29, , , 91–7, 102–3, 110–11,
Wu Zhihui ,
_wujun_ ("without a prince") , , ,
_wuming_ ("the nameless")
_Wunengzi_ (Chinese text) 89–92, 231–62
_wushi_ (non-interference in world affairs) 72–3
_wuwei_ ,
Buddhist interpretation in _Wunengzi_
Chinese anarchist movement (20th c.) view of ,
Daoist texts and 20–1, , 62–3
defined ,
Guodian manuscript and 79–80,
_Huiananzi_ and
revolutionary quality of 27–8
Wu Nengzi and 96–7
_Wunengzi_ and
_wuyu_ ("no desire") ,
_wuzhi_ ("without knowledge") ,
_xiang_ (township level units)
_Xin Qingnian_ ( _New Youth_ )
_xing_ (human feelings)
Xiu Xing 24–5, , ,
_xuantong_ ("mystical equality")
Yan Jiaqi ,
Yan Xishan
Yanan period
Yanan rectification movement
Yang Fang-yen
Yang Xiaokai _see_ Yang Xiguang
Yang Xiguang 179–82
Yang Zhu 24–5,
"Yang Zhu" (chapter of _Liezi_ ) , , 40–2, , 57–60
Yao Wenyuan , 160–4, 170n.6
Ye Jianying 164–6
_yi_ (righteousness, Confucian concept of) 61–2, 73–4,
_yihualun pai_ ("alienation school") ,
_yiyuanhua_ (monolithic unity, Mao's ideal of) 157n.105
Young, Graham 154n.61
_youxin_ (intentionality)
Yuan Shikai
Yugoslavia 132–3, , , ,
Zarrow, Peter 26–7, , ,
Zhang Chunqiao
Zhao Ziyang , , 196–7,
_Zhongguoshi yanjiu_ (Chinese Studies in History) ,
_zhong-xiao_ (Confucian concept of) ,
Zhou dynasty , , , , , ,
Eastern Zhou , ,
late , , , , 53–4,
Spring and Autumn period ,
Warring States period , , , 24–5, , , , , 78–9,
Western Zhou
Zhou Enlai , 163–4, 172n.26, ,
Zhuang Zhou , 19–20, , 65–7, ,
_Zhuangzi_ ,
anarchism and , 24–6, , , , , ,
chapter ,
chapter ,
chapter , 56–7, ,
chapter 43–4, 54–5, , 65–7,
chapter ,
dystopian ideas 65–7
history of 19–20, 24–5
influence of ,
inner chapters , 22–6, , , , , 77–8
outer chapters , , , , 55–7, 65–7, 76–7
"political Daoism" of (Ames) 23–4
Yangist chapters _see also_ Yang Zhu (chapter of _Liezi_ )
Zhu Yuanzhang (Ming Taizu) , , 139–41, 150n.2, 155n.67, 187–8
Zhuang Zi (Master Zhuang) _see_ Zhuang Zhou
_ziran_ (natural/spontaneous, Daoist concept of) 31–4, , , 91–2, 100–1
|
{
"redpajama_set_name": "RedPajamaBook"
}
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Q: Option to enable glue catalog for Presto/Spark on EMR using Terraform Wanted to know if there's support to enable aws glue catalog for Presto/Spark when running on EMR.Could not find anything in the documentation.
A: From the link provided by the answer above, i was able to model terraform code as follows-:
Create a configuration.json.tpl with the following content
[{
"Classification": "spark-hive-site",
"Properties": {
"hive.metastore.client.factory.class": "com.amazonaws.glue.catalog.metastore.AWSGlueDataCatalogHiveClientFactory"
}
}
]
Create a template from the above template in your terraform code
data "template_file" "cluster_1_configuration" {
template = "${file("${path.module}/templates/configuration.json.tpl")}"
}
And then setup the cluster as such-:
resource "aws_emr_cluster" "cluster_1" {
name = "${var.cluster_name}-1"
release_label = "emr-5.21.0"
applications = ["Spark", "Zeppelin", "Hadoop","Sqoop"]
log_uri = "s3n://${var.cluster_name}/logs/"
configurations = "${data.template_file.cluster_1_configuration.rendered}"
...
}
Glue should work now from Spark, you can verify this by calling spark.catalog.listDatabases().show() from spark-shell.
A: The following AWS documents discuss about using Apache Spark and Hive on Amazon EMR with the AWS Glue Data Catalog, and also using AWS Glue Data Catalog as the default Hive metastore for Presto (Amazon EMR release version 5.10.0 and later). Hope you are looking for this?
https://docs.aws.amazon.com/emr/latest/ReleaseGuide/emr-presto-glue.html and
and
https://aws.amazon.com/about-aws/whats-new/2017/08/use-apache-spark-and-hive-on-amazon-emr-with-the-aws-glue-data-catalog/
Also please check this SO link for some glue catalog configurations on EMR:
Issue with AWS Glue Data Catalog as Metastore for Spark SQL on EMR
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,288
|
Q: Android -- include jar cause 'class not found' exception I want to include Netty's jar into my android project.
I've tried adding it directly as a reference, but I'm getting java.lang.NoClassDefFoundError, pointing to a class from Netty.
Another way is to put the library directly into the libs folder, and when I'm doing so, the Eclipse's Android plugin would automatically include the jars in the libs folder into the 'android private library'. In that case it seems that I cannot attach a source.
I tried deleting the android private library, and link everything through the reference library. It seems that the android cannot find class from reference library; because it's causing the NoClassDefFoundError exception.
I want to know if there're proper ways to solve this problem.
NOTE: I do know that there're lots of posts about this issue.. such as this method (from the link above):
*
*Create a folder called libs in your project's root folder
*Copy your JAR files to the libs folder
*Now right click on the Jar file and then select Build Path > Add to Build
Path, which will create a folder called 'Referenced Libraries' within your
project
But it's just causing the NoClassDefFoundError exception.
I'm using Eclipse Kepler.
A: I solved such issue with the next steps:
*
*Go to Java Build Path->Order And Export
*Turn on check box for all libraries
*Click OK
*Clean and Rebuild your project
A: I think you didn't check the Android Private Libraries. Take a look at this answer
A: Try to run Project → Clean(select all projects). It should clean & rebuild everything.
If eclipse haven't imported jar you will have errors in your java files. If your projects are building with no errors and even is trying to start on emulator or device it may be problem of compilatiion. This problem happens often when using gradle. And the best solution is to clean everything and rebuild :)
A: Above all steps give you same kind of error again and again. Don't be afraid I will solve your problem:
*
*Instead of adding external .jar file, remove it from library.
*Copy your driver file into project web-inf/lib(ex.ojdbc14.jar)
*Run your dyanamic application.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
\label{Inroduction}
The detection of an electromagnetic counterpart of a gravitational wave (GW), observed for the first time after the binary neutron star (BNS) coalescence of the 17th august 2017 (GW170817) was a real breakthrough for the multi-messenger astronomy \citep{LSC_BNS_2017PhRvL}. It provided the first evidence of a link between BNS merger and short gamma ray burst. The relatively small localisation error (skymap) of this event and the huge effort of follow-up community allowed the identification of the kilonova counterpart and the afterglow counterpart alongside with the host galaxy. The multi-wavelength observations improved our understanding of many aspects of such high-energy phenomena from the physics of strong-gravity, trough the Lorentz invariance, the neutron star equation of state, the energy of the ejecta, the merger remnant, the ambient medium, the independent derivation of the Hubble Constant and so on \citep{gwtohubble1,2018PhRvL.121p1101A,Metzgerkilo,2019arXiv190906393H,2019ApJ...876..139G,gwtohubble2,gwtohubble3}.
The third LIGO-Virgo-KAGRA (LVK) run shared is loads of very interesting events compact binary coalescence event including the first confident observations of neutron star-black hole binaries \citep{gwtc2.1,gwtc3}, but despite few BNS merger alerts no multi-messenger detection occurred. This highlight the challenge of the GW follow-up where one has to deal with large localisation error from GW detectors and relatively faint and fast decaying transient (afterglow and kilonova). Despite the upgrade of the detector sensibility for O4, the median 90\% credible area is expected to be higher than during O3 because events will be detected at farther luminosity distance \cite{O4expect}.
Recent efforts tried to deal with the observations of these large sky areas by optimising the tilling over the sky and the observation plan to scan the largest possible portion of the 2D probability in a given time \citep{2016A&A...592A..82G, gwemopt, Coughlin19_opt}. Other developments included galaxies population and galaxies properties to the strategy \citep{2016ApJ...820..136G,LosC, Antolini2017, Rana2019,mangrove,glade+}. Such developments allow to use the distance estimation provided by the LVK localisation (i.e. to use the 3D skymap). It also greatly facilitates the follow-up with narrow field of view telescopes as it allows to provide a list of interesting galaxies to focus on. Including priors using the galaxies raised recent interest to build catalogues providing the distance of the galaxies with a range compatible with LIGO-Virgo-KAGRA sensibility and with a high completeness \citep{2019ApJ...880....7C,2018MNRAS.479.2374D,mangrove,glade+}.
The SVOM mission is a ground and space-based multi-wavelength observatory aiming at detecting GRBs and other transient sky sources \cite{SVOM2016}. This mission is a collaboration between French and Chinese space agencies (CNES and CNSA) and is planned to be launched in late 2023. The scientific objectives of the SVOM mission are to study the entire GRB continuum, to perform a complete phenomenology of GRBs of all types over a wide detection band by observing the prompt emission and the afterglow. To probe the nature of the GRB progenitors, the physics of the GRB explosion using faint/soft nearby GRBs; and the study of high-redshift GRBs ($z>5$) as a probe of the early universe. The SVOM satellite will be equipped with four instruments: two dedicated to the observation of the GRB prompt emission, a coded-mask gamma-ray imager (ECLAIRs) with field of view of $2$ sr operating in the 4-150 keV energy range; and a gamma-ray spectrometer (GRM) with field of view of $5.6$ sr operating in the 15-5000 keV energy range. Two telescopes dedicated to the observation of the GRB afterglow, a Microchannel X-ray Telescope (MXT) with a field of view of $64\times64 $ $\textrm{arcmin}^{2}$ operating in the soft X-ray range (0.2-10 keV); and a 40 cm aperture Ritchey-Chrétien Visible-band Telescope (VT) with a field of view of $26\times26$ $\textrm{arcmin}^{2}$ observing in visible (400-650 nm) and in near-infrared (650-950 nm). From its capacity to obtain multi-wavelength follow-up observations, the SVOM mission will play a key role in the time-domain/multi-messenger astronomy. In order to contribute to this era, the time allocated by the SVOM spacecraft to the observation of targets of opportunity (ToO; including GW follow-up) is set to be at least 15\% of the lifetime of the nominal mission for the first two years and expected to increase later on.
In this work we develop the SVOM rapid follow-up strategy for gravitational wave trigger candidates. We focus on the search for electromagnetic counterparts (kilonovae and afterglows), we do not discuss the follow-up and characterisation of already identified counterpart candidate. We implemented the galaxy targeting strategy and compared its efficiency against the tilling strategy. We then implement several optimisations and constraint specific to the SVOM platform and its onboard detectors. This leads us to the simulation of observation plans that optimise the chance of counterpart discovery while being realistic about the constraints of the satellite.
In section \ref{section:Simulation}, we present the simulation methodology. In section \ref{section:Tillingvsgaltar}, we compare the tilling and the galaxy targeting strategy for SVOM. In section \ref{section:constraint} we implement the observation plan production constraints of the satellite and its platform. In section \ref{section:furtherdev}, we further improve the galaxy targeting strategy. In section \ref{section:TBD}, we discuss the expectation of the SVOM follow-up in the light our simulation. We conclude in section \ref{section:conclusion}.
Throughout this paper, we use the Plank 2015 cosmological parameters \citep{planck15}.
\section{Observation plan simulation}
\label{section:Simulation}
Among the SVOM ToO a significant fraction of them, named ToO\_MM, will be dedicated to the follow-up of multi-messenger alerts. This represents about one ToO\_MM per month. In the current program, up to 24 hours of observation can be allocated to a given ToO\_MM. We focus in this work on ToO\_MM dedicated to BNS merger candidates from LIGO-Virgo-KAGRA as they represent the most promising GW sources of electromagnetic counterparts from gamma-ray to radio. Specifically promising neutron star black hole merger candidates are also expected to be followed. GW sources also represents a challenge because of the size of their skymap. In the following simulations, we set the exposure time of any image to be 10 minutes which is expected to be a good trade-off between the sensibility and the possibility of multiples pointing for the exploration of a large error box. With this exposure time and taking into account the slew maneuver (expected to be lasting less than 5 minutes in any case), it allows to have 5 tiles per orbit and a total of 70 tiles for a given ToO i.e. in 24 hours). Throughout this paper we routinely use this number of 70 tiles to quantify the follow-up expectation of the SVOM satellite. The production of the observation plan presented in this paper is one of the early steps of the ToO follow-up system of SVOM. In practice the observation is expected to occur at least few hours after the ToO alert because of the ToO follow-up system procedures (including human input) with important variation in the delay: the follow-up validation by the French science center, the transmission of the plan to the mission center in China, the refinement of the tiling scenario according to the satellite orbit and the communication with the satellite. As a result of this time lag, we can't take into account the Earth and Moon occultations within this work. On the other and, the sun is moving sufficiently slowly ($\sim$1 deg per day) to be taken into account in our plan. The SVOM payload constraint imposes the angle between its optical axis and the direction of the Sun to be $>$90 deg. We implement this constraint in the simulation in section \ref{subsection:sunconstraint}. Another important constraint for the observation of ToO is the limitation of the slew of the satellite. Although the design of the satellite platform has been selected to perform regular and rapid slews, in case of ToO, the angular distance between two different pointings is imposed to be less than 5 degrees within one orbit to prevent an over-stressing of the platform. This constraint is particularly limiting for the follow-up of wide skymap like GW alerts. We discuss in section \ref{subsection:slewconstraint} the implementation of this slew constraint in the simulation of observation plan.\\
Within this work we use the \textit{gwemopt}\footnote{\url{https://github.com/mcoughlin/gwemopt}} python package \citep{gwemopt}, developed to optimise the electromagnetic follow-up of gravitational wave events and where both tiling and galaxy targeting strategy are implemented. In this work we use the Mangrove galaxy catalogue which cross-matched the GLADE galaxy catalogue with the AllWISE catalogue up to 400 Mpc and derived stellar masses using a mass-to-light ratio using the WISE1 band luminosity \citep{mangrove}.
In order to simulate a wide variety of observation plans we selected a set of 15 gravitational wave skymaps, 8 true alert skymaps published by LVK (GW170817, GW170817 without Virgo data, S190425z, S190718y, S190814bv, S190901ap, S191213g, S200213t) and 7 mock skymaps provided by LIGO-Virgo-KAGRA (MS191219a, MS191221a, MS191221b, MS191221c, MS191222a, MS191222o, MS191222t). All of them are selected to be with a mean distance plus standard deviation below 400 Mpc allowing to use the Mangrove catalog. They are also chosen to represent the variety of sky localisations provided by the gravitational wave detectors alone, with skymaps representative of a detection with 1, 2 or 3 detectors. Table \ref{tab:skymap} presents the properties of the set of skymaps. The observation plan produced with the \texttt{gwemopt} \citep{gwemopt} software are adapted to ground based observatories and not a satellite. We implemented within this work the tools necessary to produce plans for the SVOM satellite observation. This include getting rid of the ground observations limitation (horizon, azimuth...) and the addition of limitations required by the SVOM observations, discussed in section \ref{section:constraint}\\
\input{table_skymap}
\section{Tiling vs Galaxy targeting}
\label{section:Tillingvsgaltar}
In case of GW follow-up, the SVOM observation plan is leaded by the X-ray detector (MXT), mainly because of its field of view (the consideration of VT observations is discussed in Section \ref{section:furtherdev}). In this section we focus on the following question: should SVOM-MXT telescope use the tiling strategy or the galaxy targeting strategy for its observations?
Both strategies take advantage of galaxy catalogs to optimise the observations. This idea starts from the hypothesis that the source is located within (or nearby) a galaxy, the host galaxy of the BNS system. This is expected in the light of the link with short GRBs \citep{2017ApJ...848L..13A, 2022arXiv220601763F}. For compact binary coalescence, the HEALPix skymap provided with the gravitational wave alert also provide the estimated distance of the source. For each pixel of the skymap, one can fetch the probability distribution for the source distance at the given sky position of the pixel. Hence, one can select galaxies compatible with a given skymap and allocate to each one a grade according to its 3D position over the sky in order to rank them.
We classified as "compatible" with the skymap, a galaxy which fulfills the two following conditions:
\begin{enumerate}
\item Its 2D position in the sky has to be in the $90\%$ of the 2D skymap probability distribution.
\item Its distance has to fall within the 3 sigma distance error localization at the given pixel of the galaxy.
\end{enumerate}
Further development also adds galaxies properties in the definition of the grade \citep{LosC,mangrove}. In this work we use the definition of the grade presented in equation (4) of \cite{mangrove} which include the stellar mass of the galaxies. This is motivated by several works pointing out a significant dependence to the stellar mass for the rate of BNS merger \citep{2019MNRAS.487.1675A, 2019MNRAS.tmp.2085T, 2018MNRAS.481.5324M, 2022arXiv220505099S} and the massive short GRB host galaxies population \citep{Leibler2010, Fong2013, Berger2014, 2022arXiv220601764N}. In order to limit the computation time, the list of ranked galaxies is kept up to the 2000 first ones if there are as many. This number is choses to ensure that only galaxies with marginal grade are removed and that the observation plan require much more than 70 pointing to observe all of the ranked galaxy. In the following, regardless of the strategy, we define the number of galaxy of a tile as the number of compatible galaxies it contains, we define the grade of a tile as the sum of the grade of the compatible galaxies it contains. We define the number of observed galaxy (observed grade) as the sum of the number of galaxy (grade) of the tiles of the observation plan. We define the maximum of observed galaxy (observed grade) as the number of observed galaxy (observed grade) after the maximum number of tile computed by the observation plan.
\subsection{Tiling strategy}
The tiling strategy usually used by large FoV ($\gtrsim 1$ deg$^{2}$) telescopes consists of the construction of an optimised tiling of the sky where the scheduling of the observation is defined using the 2D sky localisation (probability distribution) from the gravitational wave skymap. Prior of any observation, a complete tiling of the sky is computed so that all tiles are defined in advance. In this strategy, tiles are defined to fit the telescope field of view and typically correspond to one image. They are built in such a way that the overlap between the tiles is minimized to observe the largest sky area possible. For a given alert one need then to schedule the observations of these tiles. The gravitational wave skymaps are provided through all-sky pixelised images in the HEALPix (Hierarchical Equal Area isoLatitude Pixelization) format \citep{HEALPix}. In these skymaps one can fetch for each pixel the probability for the source to be in the sky direction of the pixel (2D probability). This is used to schedule the tile observations according to the 2D probability they contain, summing the probability of each pixel in the tiles and ranking them according to the probability they contain.
Further development of the tiling strategy proposes to use galaxy catalogs to focus the observations on galaxies compatible for a given alert. We call this development the galaxy weighted tiling. In the galaxy weighted tiling strategy, for each tile over the sky we sum the grade of the galaxies they contain. This sum of grade is used to schedule the tile observations. This weighted tiling strategy (as well as the galaxy targeting strategy presented in the following) avoid to observe regions of the sky were there is no compatible galaxy. This well illustrated in the figure \ref{fig:obtained_tiles} where an important fraction of the GW170817 skymap is not proposed to be observed with our observation plan. For concision, the galaxy weighted tiling strategy is referred as tiling strategy in the following.
\subsection{Galaxy targeting strategy}
The galaxy targeting strategy is typically used for small FoV telescopes. In this galaxy targeting strategy, the first tile is centered on the galaxy ranked at the first position. This galaxy plus all the galaxy falling in the field of view of the image are removed from the list. The grade of the tile is defined as the sum of the galaxies grade falling in the field of view of the image. The second tile is centered on the galaxy being at the top of the leftover list, and so on until no compatible galaxy are left to be observed. As illustrated in figure \ref{fig:obtained_tiles}, the galaxy targeting strategy is more flexible than the tiling strategy and often reduces the number of pointings necessary to observe a list of galaxies. But it may lead to more overlap between observations than the tiling strategy which is precisely built to limit this overlap. The benefice of one strategy over the other mainly depends on the telescope FoV but also the skymap size, shape and distance. In particular, the MXT FoV ($\sim$ 1 $\textrm{deg}^2$) is in a range where this choice is not obvious.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\columnwidth]{plots/gw170817_tiles_final.png}
\caption{Skymap of GW170817. The dashed and the solid line enclose respectively the $50\%$ and the $90\%$ of the skymap. The dots represent the compatible galaxies and the color represent their grade. The 16 red ( 13 blue) squares represent the pointing of obtained with the tilling strategy (galaxy targeting strategy).}
\label{fig:obtained_tiles}
\end{center}
\end{figure}
\subsection{Comparison}
We compare the efficiency of both strategy looking at the resulting cumulative number of observed galaxies and observed grade for each skymap. Figure \ref{fig:GW170817_grade} to \ref{fig:grade_s190901ap} illustrates this comparison displaying the cumulative distribution for both strategies. The comparison between the two strategies is resumed in figures \ref{fig:heatmap_gal} and \ref{fig:heatmap_grade} which present the difference between the sum of galaxies (sum of grade) observed with the galaxy targeting strategy and the sum of galaxies (sum of grade) observed with the tiling strategy, normalised by the maximum of observed sum of galaxies (observed sum of grade). In one hand, there is no clear evidence in figure \ref{fig:heatmap_gal} for an advantage of any of the strategy in terms of number of galaxies observed. On the other hand, figure \ref{fig:heatmap_grade} clearly shows the optimisation of the observed grade with the galaxy targeting strategy. The result in terms of observed grade is the one that is decisive as it is set up to quantify how likely one will find a counterpart. In conclusion these simulations show that the MXT FoV is still in a range where it can benefit from the galaxy targeting approach for the follow-up of gravitational waves.
In addition, the computation time for the galaxy targeting strategy (always less than 1 minute) is significantly smaller than the one for tilling strategy (typically 15 minutes). This could be a benefit for systems such as SVOM ToO system, which has to respond quickly to the alerts.
In the light of these results, we only consider in the following development of the galaxy targeting strategy.
\section{Constraints of the satellite}
\label{section:constraint}
In this section, we improve the simulation plan production taking into account further constraints imposed by the satellite and its platform. In order to quantify the cost of the constraints, we compare for each addition the resulting simulated plan with the initial plan produced with the galaxy targeting strategy in section \ref{section:Tillingvsgaltar}.
\subsection{Sun constraint}
\label{subsection:sunconstraint}
One of the main constraints for the observation of ToO is the Sun occultations. We implemented this constraint within \textit{gwemopt} imposing any pointing to be at least $>$91 deg away from the sun (one degree more than the system constrain to take into account the discussed time lag). In order to visualise the impact of this constraint for the follow-up of GW alerts, which present very specific skymap shape, we compare in figure \ref{fig:sun_constraint} the observation plan obtained for 3 skymaps using 365 different trigger time (one per day) with and without the sun constraint. These 3 skymaps, GW170817, MS191222t and S190901ap are chosen to represent an example of a typical 3, 2 and 1 detector skymap respectively.
One can see in figure \ref{fig:sun_constraint} that in the case of a well localised events such as GW170817 the sun constrain is similar to what one can expect from a point like target with $>$90 deg constraint. Roughly, the skymap is entirely observable half of the time and not observable the other half of the time. On the other hand, for a skymap representative of a detection with two detectors such as MS191222t the sun constraint is less limiting for the observation. Indeed, in this example, the constrained plan achieves less than 50\% of the unconstrained one only during $\sim$2 month in the 70 tiles scenario. This can be intuitively understood looking at the shape of the skymap and the presence of two distinct high probability regions, typical of this kind of skymap, that are unlikely to be affected both at the same time by the sun constraint. Finally, in the case of a representative of a detection with one detector such as S190901ap, the sun constraint is highly affecting the observation plan during about 5 months in the year. This highlights the difficulty to follow-up such very wide skymap.
\onecolumn
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/heatmap_ngal_galaxy.png}
\caption{Difference between the number of galaxies observed with the galaxy targeting strategy and the number of galaxy observed with the tiling strategy, normalised by the maximum of observed galaxies. The bluer the colour, the more galaxy targeting is doing better than the tiling strategy. 70 tiles is the expected number of tiles allocated for a given ToO with SVOM.}
\label{fig:heatmap_gal}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/heatmap_grade_galaxy.png}
\caption{Difference between observed grade with galaxy targeting strategy and observed grade with the tiling strategy, normalised by the maximum of observed grade. The bluer the colour, the more galaxy targeting is doing better than the tiling strategy. 70 tiles is the expected number of tiles allocated for a given ToO with SVOM.}
\label{fig:heatmap_grade}
\end{center}
\end{figure}
\twocolumn
\onecolumn
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/sun_constraint.pdf}
\caption{\textit{Left}: Difference between observed grade with the observation plan obtained without and with the sun constraint for 365 different trigger time (one per day). The black line represents the number of tiles at which the constrained plan achieves only 50\% of the non constrained one. The vertical dashed highlights the 70 tile scenario. \textit{Right}: Skymap of GW170817, MS191222t and S190901ap (from top to bottom). The color represents the 2D probability, darkest color being the most probable region.}
\label{fig:sun_constraint}
\end{center}
\end{figure}
\twocolumn
\subsection{Slew constraint}
\label{subsection:slewconstraint}
We chose to include the slew constraint on the production of observation plan adding a post-processing after the \textit{gwemopt} computation. The idea is to reorder the obtained observation plan to limit the number of slew bigger than 5 degrees. We gather the pointing proposed by the galaxy targeting strategy presented in the previous sections and we identify clusters of pointings using a DBSCAN (density-based spatial clustering of applications with noise) algorithm. An example of the obtained clustering is presented in figure \ref{fig:cluster_MS191222t} for MS191222t. The clusters are defined so that there is always within one of them a path that allows to achieve all of the pointing with no slew bigger than 5 degrees. However, using one of these paths typically request to start at the edge of the cluster. In our case this is sub-optimal as the most probable regions for GW skymap are usually at the center of a cluster and the most probable regions are expected to be observed as fast as possible. In order to find an optimal trade-off between the number of slew bigger than 5 degrees and the rapid observation of the most probable region we proceed as follow for a given cluster:
\begin{enumerate}
\item Identify the grade barycenter\footnote{Barycenter of the tiles position weighted by their grade.} of the cluster
\item Select the first pointing arbitrary
\item Select the next pointing as:
\begin{itemize}
\item If there are other pointings respecting the slew constraint we select the one which minimise:\\
\begin{center}
$\frac{\textrm{distance to the grade barycenter}}{\textrm{grade}}$
\end{center}
\item If there is no pointing respecting the slew constraint we select the one with the highest grade (regardless of its distance)
\end{itemize}
\item repeat step (iii) until there is no more pointing in the cluster
\end{enumerate}
We store the path (reordered pointing list) proposed, the number of slew $>$5 degrees it contains and the observed grade tile distribution it achieves. We repeat this procedure for each pointing as starting point. We finally select the observation plan to be the path that minimise the number of slew $>$5 degrees. If there is more than one we select among them the one optimising the cumulative distribution of observed grade.
Figure \ref{fig:slew_limitation} present the angular distance between consecutive pointing obtained with the galaxy targeting strategy before and after the reordering of the pointing. One can see in this figure that the implemented post-processing highly limited the amount of slew $>$5 degrees. Note that having slew $>$5 degrees for MS191222t like skymap are inevitable, for instance to jump from one cluster to another. The few slew $>$5 degrees still present in the observation plan are flagged. We expect in practice observation to wait for the South-Atlantic Anomaly passages (where the detectors are temporally turned off and after which it is needed to re-point) to perform such large slews. The estimation of average exposure loss due to the South-Atlantic Anomaly is about 18\%. Adding the sun constraint presented in section \ref{subsection:sunconstraint} also help to limit the number of slew $>$5 degrees as it usually makes some of the cluster impossible to observe.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/cluster_MS191222t.png}
\caption{Clustering of the pointing obtained with the galaxy targeting strategy for MS191222t. Each colors represents a different cluster.}
\label{fig:cluster_MS191222t}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/slew_limitation.pdf}
\caption{\textit{Top}: Angular distance between consecutive pointing obtained with the galaxy targeting strategy for MS191222t. The dotted horizontal line highlights the 5 degrees constraint. \textit{Bottom}: Same figure after the reordering of the pointing to limit the number of slew bigger than 5 degrees.}
\label{fig:slew_limitation}
\end{center}
\end{figure}
The implementation of this very restrictive slew constraint has an impact of the expected efficiency of the SVOM follow-up. Figure \ref{fig:heatmap_grade_galaxy_all_constraint} illustrate the impact of this constraint looking at the expected observed grade with and without the constraint. One can see in figure \ref{fig:heatmap_grade_galaxy_all_constraint} that the impact of the constraint tend to be maximal for the few first tiles and decreases with the number of pointing until becoming tiny at 70 tiles. Figure \ref{fig:GW170817_grade} to \ref{fig:grade_s190901ap} also illustrate this comparison displaying the cumulative distribution of observed grade. We raise that this slew constraint could be slightly released in further development fitting strictly with the system requirement which imposes 5 degrees within one orbit. But such development likely need to input the information of the satellite position in its orbit in the computation of the observation plan.
\section{Further development of the galaxy targeting}
\label{section:furtherdev}
In the galaxy targeting strategy, the tiles are centered on galaxies considered as compatible with the skymap. This first condition can be seen as too restrictive as it is not necessary for a galaxy to be right in the middle of the image for a proper detection (edge conditions are taken into account in the following). For this reason, we present a further optimisation of this strategy allowing a repositioning around the galaxies of interest. From the galaxy targeting strategy we proceed as follows for a tile:
\begin{enumerate}
\item Start from the tile centered on the galaxy of interest
\item Compute the list of compatible galaxies that are nearby (within $4 \times$ the FoV of MXT)
\item Compute the grade weighted barycenter of the list
\item If all the compatible galaxies in the list fit in the MXT FoV for an image centered in the barycenter, take the barycenter as the new center of the tile
\item else compute the galaxies being the further away from the barycenter (in angular distance) and remove it from the list. Then go back to (iii)
\item Check that the grade of the new tile proposed is bigger than the grade of the one centered on the galaxy of interest, if not, keep the tile centered on the galaxy
\end{enumerate}
This addition of the repositioning improves the flexibility of the galaxy targeting strategy and so compute pointing that often increases the grade of the tiles. Figure \ref{fig:heatmap_grade_galaxy_wandering} presents the comparison between with and without repositioning. One can see that the addition of the repositioning significantly optimises the observed grade.
The addition of the repositioning raises the question of the position of the galaxies in the images, both for MXT and for VT. As the VT has a smaller FoV than MXT, we added another check in the previous process:
\begin{center}
(vii) Check that at least one compatible galaxy is falling in the VT FoV. If not, keep the tile centered on the galaxy.
\end{center}
This ensures that the VT observation is not pointless because no compatible galaxies are observed.
We also added a consideration of the side effects in the image. While great efforts to ensure consistent performance of the location on the focal plane has been performed, the detection performance for MXT starts to degrade at the $\sim$10\% edge of the image. Hence, we precautionary don't consider has observed a galaxy that fall in the $\sim$10\% edge of the image for MXT. We use the same criterion for the VT. Figure \ref{fig:FoV_MS191222t_galaxy_all_constraint} shows an example of galaxy localisation in the images for the observation plan obtained for MS191222t. One can see in this figure the consideration of the edge implemented. Note that galaxies are overlapping at the center of the FoV due to the conditions (vi) and (vii).
\section{Compatibility of the strategy with the sources}
\label{section:TBD}
\subsection{Rate of GW events}
\label{subsection:Compatibility}
The galaxy targeting developed in this work is limited to nearby LVK event where the galaxies catalogs can be used, i.e. distance where they have a reasonable completeness. As an illustration, the Mangrove catalog \citep{mangrove} is available up to 400 Mpc. \citet{O4expect} provide an expectation of about 20 BNS event per year bellow 400 Mpc (plus about 10 neutron star black hole mergers; see figure 2 of \cite{O4expect}). On the other hand, the expected rate of ToO for GW follow-up with SVOM is about one per month. Therefore, restrict the SVOM follow-up to neutron star events bellow 400 Mpc is compatible with the time allocated to ToO in the mission. As such rate is slightly higher than the ToO rate, additional selection (e.g. on skymap size) is kept possible to focus the observation on interesting event.
\subsection{Detectability of the source}
\subsubsection{MXT detectability}
The main electromagnetic counterpart expected to be detectable by the MXT detector in case of BNS merger is the GRB afterglow. To illustrate the sensibility of the MXT for such sources we took the typical sensibility of MXT for a 10 minutes exposure ($\sim 5 \times 10^{-4}$ mJy at 1 keV) and compared it to the emission for a GRB170817A like afterglows. The afterglow lightcurves are simulated using parameters fitted on the GRB170817A afterglow and varying only the distance and the $\theta_{obs}$. Details description of the models, used here for this simulation, will be provided in \citet{pellouin} in prep. In figure \ref{fig:170817_like} one can see that the MXT sensibility is compatible with the expected luminosity of a (near) on-axis GRB afterglow at 400 Mpc. Figure \ref{fig:170817_like} shows that, any future improvement of the SVOM system to reduce the delay between an alert and the first observation will significantly increase the chance of the detection of an afterglow. It also highlights again the importance of optimizing the observations to increase our chances of imaging in the right direction quickly. This work focus on the rapid follow-up of GW event, we will discuss in later work the case of off-axis afterglow follow-up peaking at late time.
\subsubsection{VT detectability}
For a telescope observing in visible band like the VT, the main electromagnetic counterpart is the expected kilonova emission. But, the expected absolute magnitude from such emission is relatively low. For GW170817, the kilonova apparent magnitude peaked at $\sim$ 17 mag in r band. Knowing this was a very close event ($\sim$ 40 Mpc) the expected apparent magnitude for a further away event is challenging for the detection. In figure \ref{fig:kilonova_VTlim}, we compare the expected r band apparent magnitude expected for a kilonova like the one of GW170817 for various distances of the source and the limiting magnitude of the VT. For a 10 minutes exposure time with the VT, the expected limiting magnitude is about 22.5. One can see that, the limit of 400 Mpc imposed by the mangrove catalog fit well the distance at which the detection of the kilonova is possible with the VT. Because SVOM is a GRB mission, the sensitivities are also adapted for the detection of optical afterglow in case of X-ray afterglow detection.
\onecolumn
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/heatmap_grade_galaxy_all_constraint.png}
\caption{Difference between observed grade with galaxy targeting strategy with the slew constraint and with galaxy targeting strategy without the slew constraint, normalised by the maximum of observed grade. 70 tiles is the expected number of tiles allocated for a given ToO with SVOM.}
\label{fig:heatmap_grade_galaxy_all_constraint}
\end{center}
\end{figure}
\twocolumn
\onecolumn
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/heatmap_grade_galaxy_wandering.png}
\caption{Difference between the observed grade with galaxy targeting strategy with our without the repositioning, normalised by the maximum of observed grade. 70 tiles is the expected number of tiles allocated for a given ToO with SVOM.}
\label{fig:heatmap_grade_galaxy_wandering}
\end{center}
\end{figure}
\twocolumn
\onecolumn
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/FoV_MS191222t_galaxy_all_constraint.png}
\caption{Localisation of the galaxies in the MXT (red square) and VT (blue square) FoV cumulated over the whole observation plan produced for MS191222t. Circles are galaxies considered has observed by MXT. Triangles are galaxies considered has observed by MXT and VT. The color represents the grade associated to the galaxy, darkest color being galaxies with highest grade. Edge condition discussed in the text are visible as empty regions for both MXT and VT.}
\label{fig:FoV_MS191222t_galaxy_all_constraint}
\end{center}
\end{figure}
\twocolumn
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/170817_like.pdf}
\caption{Simulated afterglow lightcurves for GRB170817A like parameters with varying $\theta_{obs}$ and distance. The violet dotted line represent an estimation of the limiting sensibility for a 10 minutes long exposure with MXT.}
\label{fig:170817_like}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{plots/kilonova_VTzoom3.png}
\caption{First few days of observed GW170817 kilonova r band magnitude (blue dots). Same data shifted as if the source was at a luminosity distance of 50 Mpc, 100 Mpc, 200 Mpc and 400 Mpc are represented by cyan, green, orange and red dots respectively. All data are taken from \citep{2017ApJ...851L..21V}. No K-correction are taken into account. The violet dotted line represent an estimation of the limiting magnitude for the VT telescope.}
\label{fig:kilonova_VTlim}
\end{center}
\end{figure}
\section{Conclusions}
\label{section:conclusion}
The detection of an electromagnetic counterpart of a GW event is very challenging. The multi-wavelength capability of the SVOM mission makes it an important actor for the future of time-domain/multi-messenger astronomy and an important contributor of the search for GW counterpart. In this work we make use of recent developments in both catalogues of galaxies and galaxy targeting strategy to simulate and develop GW follow-up observation plan specific to the SVOM satellite. We identify the galaxy targeting strategy as the most efficient strategy for the onboard MXT and VT follow-up. We implemented, in the production of observation plan, constraint specific to the SVOM mission and SVOM platform (exposure time, number of pointing, sun occultations, slew limitation...). We further developed the galaxy targeting strategy using a repositioning taking care of the observations of the VT. We check that the rate of LIGO-Virgo-KAGRA event, below a distance imposed by use of galaxy catalog, is compatible with the time allocated by the mission for such follow-up. We finally check that the expected luminosity of the awaited sources (kilonova and afterglow), below the distance imposed by use of galaxy catalog, are compatible with the MXT and VT sensibility. Developments presented in this work are essential for the production of realistically optimised observation plan that SVOM will use in the near future for the follow-up of GW events.
\section*{Acknowledgements}
The authors acknowledge the Centre National d'Études Spatiales (CNES) for financial support in this research project. This project was supported by a research grant from the Ile-de-France Region within the framework of the Domaine d'Intérêt Majeur-Astrophysique et Conditions d'Apparition de la Vie (DIM-ACAV). This work has made use of the Infinity Cluster hosted by Institut d'Astrophysique de Paris.
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Q: Frobenius morphism induces identity on points I guess this is a stupid question since all the textbook do not explain this, but this is still not clear for me: why frobenius map induces identity on $Spec F_p[x_0...,x_n]\to SpecF_p[x_0,...,x_n]$? I believe there is some trick I have not realized, please help me. Thanks!
A: The map $\mathbb{F}_p\to \mathbb{F}_p$ given by $a\mapsto a^p$ (the Frobenius automorphism) is the identity, so the obvious candidate for an induced map from $\mathbb{F}_p[x_0,\dots,x_n]$ to itself is also the identity, hence the map on spectra is the identity. But I doubt this is what you meant.
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{"url":"https:\/\/de.zxc.wiki\/wiki\/Gips","text":"# plaster\n\nplaster\nGypsum crystal specimen from Friedrichroda , Thuringia\nGeneral and classification\nother names\n\u2022 Plasterboard\n\u2022 Calcium sulfate dihydrate\nchemical formula Ca [SO 4 ] \u2022 2H 2 O\nMineral class\n(and possibly department)\nSulphates (selenates, tellurates, chromates, molybdates and tungstates)\nSystem no. to Strunz\nand to Dana\n7.CD.40 ( 8th edition : VI \/ C.16)\n06\/29\/03\/01\nCrystallographic Data\nCrystal system monoclinic\nCrystal class ; symbol monoclinic prismatic; 2 \/ m\nSpace group A 2 \/ a (No. 15, position 4)\nLattice parameters a \u00a0= 6.52\u00a0 \u00c5 ; b \u00a0= 15.18 \u00c5; c \u00a0= 6.29 \u00c5\n\u03b2 \u00a0= 127.4 \u00b0\nFormula units Z \u00a0= 4\nFrequent crystal faces {010}\nTwinning very often contact twins according to {100} dovetail, Montmartre, penetration\nPhysical Properties\nMohs hardness 2\nDensity (g \/ cm 3 ) measured: 2.317; calculated: 2.31\nCleavage very perfect after {010}, clearly with fiber formation after {111}\nBreak ; Tenacity shell-like\ncolour colorless, white, yellowish, reddish, gray, brown\nLine color White\ntransparency transparent to opaque\nshine Glass gloss, pearl gloss, silk gloss\nCrystal optics\nRefractive indices n \u03b1 \u00a0= 1.519 to 1.521\nn \u03b2 \u00a0= 1.522 to 1.523\nn \u03b3 \u00a0= 1.529 to 1.530\nBirefringence \u03b4 = 0.010\nOptical character biaxial positive\nAxis angle 2V = measured: 58 \u00b0, calculated: 58 \u00b0 to 68 \u00b0\nOther properties\nChemical behavior Slightly soluble in water\n\nGypsum , geologically also known as gypsum spar , is a very common mineral from the mineral class of \" sulfates (and relatives)\" with the chemical composition Ca [SO 4 ] \u00b7 2H 2 O and thus, chemically speaking, hydrated calcium sulfate or calcium sulfate dihydrate.\n\nGypsum crystallizes in the monoclinic crystal system and develops mostly tabular or prismatic to needle-like crystals , but also granular to massive aggregates . In general, plaster of paris is colorless or white. However, it can take on a yellowish, reddish, gray or brown color due to the absorption of foreign ions or admixtures of various types ( sand , bitumen ). However, its streak color is white.\n\nMost of the time, the mineral gypsum consists of mono-mineral rocks with only small admixtures of other minerals such as anhydrite , quartz or clay minerals, which are also referred to as gypsum or gypsum stone.\n\n## Etymology and history\n\nIridescent, transparent gypsum crystal specimen (selenite) from Lubin, Poland\n\nThe name plaster is derived from the Greek word \u03b3\u03cd\u03c8\u03bf\u03c2 gypsos (baked plaster, chalk), which in turn was adopted from the Semitic language area. The Latin word is gypsum . Other ancient, but not always synonymous, names for gypsum are selenites (moonstone), alabastron and lapis specularis (mirror stone ). In German-language encyclopedias of the 18th, 19th and 20th centuries, the term \"Gyps\" and corresponding compounds are used.\n\nPlaster of paris was used as a building material as early as the Neolithic Age . As early as 7000 BC In the 2nd century BC, plaster of paris was used to decorate the interior of the city \u200b\u200bof \u00c7atalh\u00f6y\u00fck in Asia Minor . In the cuneiform scripts of the Sumerians and Babylonians there are references to the use of plaster, also in Jericho (6000 BC). From 3000 BC In Uruk and later in Egypt, plaster of paris was also used as a mortar, to which lime or stones were added as impurities or for stretching. For example, on the Sphinx (2700\u20132600 BC), a calcareous plaster mortar was used for certain work. Translucent disks made of alabaster were also known to the Egyptians. The Minoan culture used plaster of paris and alabaster instead of marble as flooring or wall covering and as a building block (Palace of Knossos , 2100\u20131800 BC, and Palace of Phaistos ), and the Greek naturalist Theophrastus of Eresos described the manufacture of in a treatise Plaster. In Greece, gypsum was also used for building ornaments on houses because it was easy to work with .\n\nThe Romans only used plaster of paris for ornamentation indoors, as they were familiar with the much more durable lime for the outdoors.\n\nIn Europe, the use of gypsum increased again from the 11th century, gypsum was used for grouting masonry and for lining interior walls and from the 17th century for stucco work . Gypsum is extracted and burned in gypsum works .\n\n## classification\n\nAlready in the outdated 8th edition of the mineral classification according to Strunz , gypsum belonged to the mineral class of \"sulphates (selenates, tellurates, chromates, molybdates and tungstates)\" and there to the section \"hydrous sulphates without foreign anions \", where it was named after \"gypsum Series \"with the system no. VI \/ C.16 and the other member Ardealit as well as in the appendix Bassanit and Hoch-Bassanit .\n\nIn the Lapis mineral directory according to Stefan Wei\u00df, which, out of consideration for private collectors and institutional collections, is still based on this old form of Karl Hugo Strunz's system , the mineral was given the system and mineral number. VI \/ C.22-20 . In the \u201cLapis system\u201d, this also corresponds to the section \u201cHydrous sulfates, without foreign anions\u201d, where gypsum, together with ardealite, bassanite and rapidcreekite, forms an independent but unnamed group (as of 2018).\n\nThe 9th edition of Strunz's mineral systematics , which has been in effect since 2001 and was updated by the International Mineralogical Association (IMA) until 2009, also classifies gypsum in the category of \"sulfates (selenates, etc.) without additional anions, with H 2 O\". This is, however, further subdivided according to the relative size of the cations involved , so that the mineral can be found according to its composition in the subsection \u201cWith only large cations\u201d, where it is the only member of the unnamed group 7.CD.40 .\n\nThe systematics of minerals according to Dana also assigns gypsum to the class of \"sulfates, chromates and molybdates\" and there in the category of \"water-containing acids and sulfates\". Here he is the only member of the unnamed group 06\/29\/03 within the subdivision of \" Water-based acids and sulphates with the general formula AXO 4 \u00a0\u2022 x (H 2 O) \".\n\n## Crystal structure\n\nPerfect, transparent gypsum crystal, viewing direction on the b-axis\n\nGypsum crystallizes monoclinically in space group A 2 \/ a (space group no. 15, position 4) with the lattice parameters a \u00a0= 6.52\u00a0 \u00c5 ; b \u00a0= 15.18 \u00c5; c \u00a0= 6.29 \u00c5 and \u03b2 = 127.4 \u00b0 as well as four formula units per unit cell .\n\n## properties\n\nSplitting off of crystal water with CaSO 4 in the DTA\n\n### Physical Properties\n\nGypsum has a very low Mohs hardness of 2 and, along with halite, is a standard mineral on the Friedrich Mohs hardness scale . Its density is between 2.2 and 2.4 g \/ cm\u00b3 and, in contrast to the often associated mineral halite, it is only sparingly soluble in water. The solubility in water is 2.1 g \/ l under normal conditions , while that of halite is 358 g \/ l. Of pure aqueous solution of calcium sulfate is crystallized below 66 \u00b0 C always as gypsum, above 66 \u00b0 C as anhydrite . In the presence of other ions, for example sodium , the solubility equilibria shift.\n\n### Chemical properties\n\nWhen heated, the going crystal water lost (TG curve = mass loss = onset of dehydration, peak = Maxima onset of the reaction), and at first it creates a hemihydrate (also hemihydrate, plaster of Paris or Bassanite called) with the chemical formula CaSO 4 \u00a0\u2022 \u00bd H 2 O, with further loss of water the insoluble anhydrite II (CaSO 4 ) is finally formed via the soluble anhydrite III , the latter two are simply called anhydrite in mineralogical terms .\n\n### Rock formers\n\nUnder special natural circumstances, gypsum can be subject to a rock-forming process. Due to the evaporation of calcium sulphate seawater, gypsum and anhydrite precipitate in the early phase of carbonate separation. Primarily gypsum sediments. The rock that forms in larger layers or aggregates is counted in petrography to the group of evaporites and is also known under the cultural term alabaster . The genesis leads to cryptocrystalline or crystalline formations with a grain size down to the centimeter range.\n\nIn the vicinity of such deposits, crystalline new formations of the mineral gypsum, called Marienglas , can arise .\n\n## Varieties and modifications\n\nSwallowtail twin from Nordhausen in the Harz Mountains; exhibited in the Mineralogical Museum of the University of Bonn\nBird sculpture made from alabaster silk spar\nSand rose\n\nPlaster of paris comes in solid form, in fine-grained form as colorless, white, yellow, red or gray alabaster , as well as fine- fiber plaster of paris . For the latter, the term silk spar or, more precisely, alabaster silk spar and occasionally the term atlasspat is in use. However, the name atlaspat is inconsistent and is also used for fine-fiber calcite with a silk gloss.\n\nAlabaster eyes are made from calcium sulphate, which collected in individual places within a host rock before it had solidified and then later hardened into alabaster balls. In addition, there are sometimes see-through crystal tablets known as Marienglas or Fraueneis ( selenite ).\n\nThe mineral is found in different crystal forms: The crystals are often very large, plastically flexible, completely fissile, thick-tabular, often curved, sometimes twinned; on the other hand is also fused gypsum rosette-like as so-called sand Rose , gypsum Rose or Desert Rose before.\n\nA variety of plaster of paris, which is associated with potassium sulfate and magnesium sulfate , is misleadingly referred to as polyhalite . It occurs in the rock salt deposits of Sta\u00dffurt , Berchtesgaden and Bad Ischl .\n\n## Education and Locations\n\nThe gypsum deposits in Germany are predominantly evaporites , which means that they were created by crystallization from mineral-oversaturated seawater (see also Zechsteinmeer ). In the meantime, the plaster of paris has often been converted to anhydrite due to sedimentary load and later hydrated again. However, gypsum is also found as a weathering product of sulfidic ores and in volcanic chimneys (so-called white smokers ), where it can be formed by the reaction of escaping sulfuric acid with limestone . The natural deposits are mostly provided with admixtures that favor a parallel development or successive formation of different minerals ( paragenesis ). In paragenesis, for example, gypsum occurs with anhydrite, aragonite , calcite , celestine , dolomite , halite and sulfur .\n\nGypsum is widespread and so far (as of 2015) over 6600 sites are known. He performed particularly frequently in Algeria , Argentina , Armenia , Australia , Belgium , Bolivia , Brazil , Bulgaria , Chile , China , Germany , France , Greece , Indonesia , Iran , Ireland , Italy , Japan , Canada , Kazakhstan , Madagascar , Morocco , Mexico , Namibia , Norway , Austria , Peru , the Philippines , Poland , Portugal , Romania , Russia , Sweden , Switzerland , Slovakia , Spain , South Africa , the Czech Republic , Turkey , Hungary , the United Kingdom (Great Britain) and the United States (USA).\n\nIn Germany, the mineral can be found in the Neckar-Odenwald district (around Mosbach ), near Osterode am Harz , Eisleben in Saxony-Anhalt , Borken near Kassel and in the Segeberger Kalkberg , as well as part of the Grabfeld formation ( Gipskeuper ) in the Steigerwald , the Frankenh\u00f6he and north of the Swabian Alb . Here it was mostly formed by hydration of existing anhydrite during the Pleistocene glacial periods and is therefore preferably located on exposed western sides.\n\nIn Austria there are deposits in Preinsfeld near Heiligenkreuz , Puchberg am Schneeberg , Wienern am Grundlsee , Spital am Pyhrn , Moosegg near Golling , Abtenau and Wei\u00dfenbach am Lech .\n\nThe Naica mine in Chihuahua (Mexico), where giant gypsum crystals of up to 15 meters in length were discovered in various caves, is known for its extraordinary gypsum finds . In the Mina Quien Valley Pensara ( Mina Rica ) near Pulp\u00ed in the Spanish province of Almer\u00eda , miners found an oval giant geode with a diameter of 1.8 \u00d7 1.7 meters and a length of 8 meters (internal dimensions), which averaged half a Meter-long Marienglas crystals is lined and is called the \"Geode of Pulp\u00ed\" ( Pulp\u00ed-Geode for short ).\n\nFurthermore, gypsum could also be detected in mineral samples from the seabed of the Barents Sea (Arctic Ocean), the Mid-Atlantic Ridge , the Central Indian Ridge, as well as in the Bismarck Sea (Pacific Ocean) and the East Pacific Ridge .\n\nOutside the earth, gypsum was detected by probes on Mars , more precisely at Juventae Chasma in the Valles Marineris , in the Terra Margaritifer and Yellowknife Bay in the Aeolis quadrangle and in the Endeavor crater in the Meridiani level.\n\n## Composition of various building materials that are traded as plasters\n\nSource:\n\nmaterial Natural gypsum (Trias, Keuper ) Natural anhydrite (Trias, Keuper) Flue gas gypsum (REA gypsum) Phosphorus gypsum Fluoroanhydrite (neutralized)\nCalcium sulfate dihydrate 95 0.5 98 96 0\nCalcium sulfate (anhydrite) 1 96 0 0 96\nCalcium carbonate 1.5 1.5 1 0 0\nMagnesium carbonate 1 1 0 0 0\nSand and clay 1.5 1 1 2 1\notherwise. accompanying substances no no Calcium sulfite 1% phosphates, 0.5% fluoride, 0.5% strontium sulfate , heavy metals 1.5% fluoride, 1.5% potassium and zinc sulfate, traces of calcium hydroxide\nPH value 6.7 7th 6.7 2.9 12\n\n## Chemical production of plaster\n\n### Historical\n\nGypsum distillery , Th\u00e9odore G\u00e9ricault , 1822\u20131823\n\nIn the Middle Ages, rock containing gypsum was mined in quarries or mined, sorted and further crushed in crushing mills so that it could be fed into the burning or cooking process. The gypsum distilleries operated kilns or pit ovens that were fired with wood or peat . The plaster of paris was then finely ground in a plaster of paris mill. Another method consisted of making a fire in the tunnel and then knocking out the plaster of paris. \u2192 Schleitheim plaster museum\n\nThese activities were mostly done by farmers or millers during the period of underemployment. Depending on the purity and fineness, a distinction was made between building plaster, screed plaster and stucco plaster.\n\n### Industrial\n\nBecause calcium sulphate is a secondary product in many chemical processes (usually in the form of gypsum) , for example in the production of citric acid , tartaric acid and oxalic acid , targeted industrial production on a large scale is unnecessary. The so-called phosphogypsum formed in the production of phosphoric acid is u. a. contaminated with uranium and a problem waste . The classic process is precipitation from sulfuric acid water with milk of lime or limestone :\n\n${\\ displaystyle \\ mathrm {H_ {2} SO_ {4} + CaCO_ {3} \\ longrightarrow CaSO_ {4} + H_ {2} O + CO_ {2}}}$\n\nEven Goethe , a passionate scientist and chemist, described this process in his novel The Elective Affinities :\n\n\u201cWhat we call limestone is a more or less pure calcareous earth, intimately connected with a delicate acid that became known to us in the form of air. If a piece of such a stone is placed in dilute sulfuric acid, it seizes the lime and appears with it as plaster of paris; that delicate, airy acidity escapes \"\n\nwhere the poet's chemist meant carbon dioxide .\n\nGypsum is also produced in all wastewater treatment processes when it comes to the neutralization of sulphate-containing process wastewater or sulfuric acid pickling.\n\nThe production of hydrofluoric acid from fluorite ( fluorspar , calcium fluoride) and concentrated sulfuric acid also produces gypsum (so-called \"fluoroanhydrite\"), which is used as an anhydrite screed in the cement and construction industries .\n\nGypsum is also produced as the end product of the flue gas desulphurisation (\"FGD gypsum\") from coal-fired power plant exhaust gases . As a rule - depending on the impurities - such gypsums (drained filter cake) can be used in the building materials industry or for further processing into calcium sulphate modifications (hydrates). This synthetic route made the mining of natural gypsum deposits in Europe partially superfluous at the end of the 1980s, today the production figures are declining due to this process, since low-sulfur Australian hard coal is often used. In 2014, 7 million tons of the 11 million tons of gypsum were extracted in Germany by the FGD, while 4 million tons were extracted from natural gypsum.\n\n### Gypsum-like calcium sulfate modifications\n\n\u2022 \u03b1-Hemihydrate (CaSO 4 \u00b7 \u00bd H 2 O) is produced in a closed vessel ( autoclave ) under a wet steam atmosphere or without pressure in acids and aqueous salt solutions. It is the starting material for harder plasters (type III, IV and V) and requires less water but more time to set .\n\u2022 \u03b2-hemihydrate (CaSO 4 \u00b7 \u00bd H 2 O) is formed when burning in an open vessel under normal atmosphere. When mixed with water, hydration to the dihydrate occurs within minutes. It is the raw material for the softer plasters.\n\nIn the case of \u03b1- and \u03b2-hemihydrate, they are different crystalline forms of hemihydrate.\n\n\u2022 Anhydrite III (CaSO 4 ) is formed from the hemihydrate at temperatures of up to 300 \u00b0 C. In the presence of water, including humidity, hemihydrate is formed very quickly.\n\u2022 Anhydrite II s (CaSO 4 ) is formed at temperatures between approx. 300 to 500 \u00b0 C, the s stands for \"poorly soluble\". When mixed with water, hydration occurs within hours and days.\n\u2022 Anhydrite II u (CaSO 4 ) is formed from anhydrite II s at temperatures of 500 to 700 \u00b0 C , the u stands for \u201cinsoluble\u201d.\n\u2022 Anhydrite I (CaSO 4 ) is the high temperature modification of gypsum, it forms at 1180 \u00b0 C.\n\n## use\n\nGypsum is also marketed under names such as alabaster white , analine , anhydrite , Bolognese chalk , electrician's plaster , feather spar , light spar or Marienglas , Plaster of Paris .\n\n### As a raw material\n\nGypsum as a raw material is predominantly extracted as gypsum rock, but is now also often a by-product of various chemical large-scale processes.\n\nTechnically, the ability of gypsum is used to absorb the crystal water that has been partially or completely lost by heating (burning) when mixed with water and to bind it in the process. When heated to around 110 \u00b0 C, so-called burnt gypsum (the hemihydrate mentioned above) is formed, and at 130 to 160 \u00b0 C, stucco , a mixture of a lot of hemihydrate and little anhydrite . Anhydrite is formed at 290 to 900 \u00b0 C, whereby the crystal water is completely burnt out. Very-high temperature gypsum is also \"dead burned gypsum\" or alanine or Annalin named because he no longer abbindet with water.\n\n### As a building material\n\nA brick cast from high-fire plaster , manufactured around 1870\n\nIn construction technology, gypsum (as hemihydrate or multi-phase gypsum) is mostly used today in the form of REA gypsum for gypsum wall panels for partition walls and for gypsum plasterboard for dry construction , as a base material for various plasters , fillers and dry screeds , and also as a filler . By mixing with lime is produced for plaster, brick and stucco Gipskalk that is longer process than pure stucco and malleable as plasticine , before it hardens.\n\nSince the hardened gypsum has a certain solubility in water, gypsum building materials are predominantly only used for interior work. Outside, gypsum building materials must be protected from regular driving rain . In the past, plaster of paris was also used for stucco work on facades and impregnated with linseed oil.\n\nBecause plaster of paris is hygroscopic (water - attracting) and therefore tends to discolour and fungus if it is too often soaked, poor maintenance or ventilation, it can only be used to a limited extent in the wet and basement areas. During renovation work is construction or plaster of Paris used to close small cracks, holes and cable slots in the walls and einzud\u00fcbeln wood and other components. In new buildings, gypsum plasters and plasterboard are used to create a surface that is ready for painting and wallpapering on rough and uneven masonry. Partition walls that are not statically loaded are now often made from plasterboard with a metal substructure or from plasterboard .\n\nAlso screeds are made of gypsum or anhydrite produced.\n\nIn addition, plaster of paris is used to fasten flush-mounted elements for electrical installations in structural walls. The speed of setting in alkaline formulations - for example gypsum plaster - is regulated by adding tartaric or citric acid . Neutral formulations can be delayed with protein compounds, cellulose glue or white lime hydrate . The setting process is accelerated by adding potassium sulfate or finely ground plaster of paris.\n\nIn structural fire protection, plaster of paris is preferred because it offers great fire resistance while being relatively light ; Protection is provided by the water of crystallization in the dihydrate, which evaporates in the event of a fire and forms a protective vapor curtain on the side facing the fire.\n\nThe building material gave the plasterer (today plasterer ) its name.\n\n### As model and mold plaster\n\nWhen used as model or mold plaster, for example with Bozzetti , increased requirements are placed on the purity of the plaster raw materials and on the preparation. A more even surface structure is achieved through finer grinding and lower proportions of foreign minerals. By using \u03b1-hemihydrate (produced under water vapor pressure and has a higher density), higher strengths of the molded parts can be achieved. In this context hard plaster is also used.\n\n### In art\n\nIn the fine arts , plaster of paris is used to create sculptures and, just like in technology, to make forms and models. Marienglas still plays an important role in church and alabaster restorations, while dead-burned plaster is also often used as an additive (extender) for paints, as it leads to cheaper products without significantly impairing the color quality.\n\nAnalin is also for primers of canvas , in the panel painting or as a gold background ( Assis used). Also Chalk and Chalk exist in Germany usually mostly made of plaster.\n\n### In the medicine\n\nIn medicine, plaster of paris is used for the plaster cast : the affected limbs or joints are wrapped with moist plaster bandages to immobilize and stabilize them, which then harden within minutes and are fully resilient after about twelve hours.\n\nIn dental technology , plaster of paris is the most important raw material for dental plaster for the production of models that are created from impressions of the oral and dental situation. According to the EN ISO 6873 standard for dental plasters, a distinction is made between five types:\n\n\u2022 Type I: Casting and impression plaster, \u03b2-hemihydrate, 0.15% setting expansion and 4 N \/ mm\u00b2 compressive strength\n\u2022 Type II: Alabaster plaster of paris, \u03b2-hemihydrate, 0.3% setting expansion and 9 N \/ mm\u00b2 compressive strength\n\u2022 Type III: hard plaster of paris, \u03b1-hemihydrate, 0.2% setting expansion and 20 N \/ mm\u00b2 compressive strength\n\u2022 Type IV: super hard stone, \u03b1-hemihydrate, 0.15% setting expansion, 35 N \/ mm\u00b2 compressive strength\n\u2022 Type V: super hard stone, \u03b1-hemihydrate, 0.3% setting expansion, 35 N \/ mm\u00b2 compressive strength\n\nInternationally, the exact specifications are given, in particular the mixing ratio (ml of water per 100 g of plaster) and the compressive strength (in MPa or N \/ mm\u00b2 after a certain time and when dry). Depending on the intended use, the percentage setting expansion and the duration of the processing and setting times are also important.\n\n### Further areas of application\n\nUnburned or dead-burned plaster of paris is used instead of chalk to mark the playing field .\n\nTo make tofu , the protein from ground soybeans is coagulated with calcium sulfate. Calcium sulfate is also used as a food additive (E 516). It belonged to the original canon of the twelve Schuessler salts used in alternative medicine .\n\nIn some areas of Germany, such as in the southern Harz region , a weathered gypsum product is created which, due to its similarity to table flour, is popularly known as \"heavenly meal \" or \"gypsum ash\". In times of famine , this gypsum flour was used either as a flour substitute or for stretching real flour to prepare food. However, the heat of baking, for example, creates burnt plaster, which sets in the gastrointestinal system and can lead to deadly intestinal obstruction .\n\n## Figurative meaning\n\nSince gypsum is abundant worldwide, there has never been a military dispute over this raw material in human history. The proverb \u201cDon't tell me anything about the war on the plaster\u201d is based on the power-political insignificance of the plaster of paris in order to make it clear to someone, ironically colored, that they should not tell stories about non-existent events.\n\n## literature\n\n\u2022 Martin Okrusch, Siegfried Matthes: Mineralogy. An introduction to special mineralogy, petrology and geology . 7th, completely revised and updated edition. Springer, Berlin [a. a.] 2005, ISBN 3-540-23812-3 , pp.\u00a071-72 .\n\u2022 Petr Korbel, Milan Nov\u00e1k: Mineral Encyclopedia (=\u00a0 Villager Nature ). Nebel Verlag, Eggolsheim 2002, ISBN 978-3-89555-076-8 , p.\u00a0147 .\n\u2022 Basics . In: Fritz Scheidegger (ed.): From the history of construction technology . tape\u00a01 . Birkh\u00e4user, Basel 1990, ISBN 3-7643-2385-X .\n\u2022 Franz Wirsching: Gypsum - natural raw material and residue from technical processes . In: Chemistry in Our Time . tape\u00a019 , no.\u00a04 , 1985, ISSN \u00a00009-2851 , pp.\u00a0137-143 .\n\u2022 Markus Arendt: Circular economy in the construction sector: Controlling future material flows using the example of gypsum . 2001 ( dissertation at the University of Heidelberg ).\n\nWiktionary: plaster of paris \u00a0- explanations of meanings, word origins, synonyms, translations\nCommons : Gypsum (Gypsum) \u00a0- collection of images, videos and audio files\n\n## Individual evidence\n\n1. Hugo Strunz , Ernest H. Nickel : Strunz Mineralogical Tables. Chemical-structural Mineral Classification System . 9th edition. E. Schweizerbart'sche Verlagbuchhandlung (N\u00e4gele and Obermiller), Stuttgart 2001, ISBN 3-510-65188-X , p. \u00a0393 .\n2. Webmineral - Gypsum (English)\n3. ^ Gypsum . In: John W. Anthony, Richard A. Bideaux, Kenneth W. Bladh, Monte C. Nichols (Eds.): Handbook of Mineralogy, Mineralogical Society of America . 2001 ( handbookofmineralogy.org [PDF; 67 \u00a0kB ; accessed on September 28, 2017]).\n4. Mindat - Gypsum (English)\n5. State Office for Geology, Raw Materials and Mining in the Freiburg Regional Council: Sulphates ( Memento from April 9, 2014 in the Internet Archive )\n6. Encyclopedic entries on \"Gyps\": Adelung-1793: \"Gyps, der\", Brockhaus-1809: \"Der Gyps\", Brockhaus-1837: \"Gyps\", Brockhaus-1911: \"Gyps\", Herder-1854: \"Gyps\" , Meyers-1905: \"Gyps [2]\" \u00b7 \"Gyps [1]\", Pierer-1857: \"Gyps\"\n7. Stefan Wei\u00df: The large Lapis mineral directory. All minerals from A - Z and their properties. Status 03\/2018 . 7th, completely revised and supplemented edition. Weise, Munich 2018, ISBN 978-3-921656-83-9 .\n8. Ernest H. Nickel, Monte C. Nichols: IMA \/ CNMNC List of Minerals 2009. (PDF 1816 kB) In: cnmnc.main.jp. IMA \/ CNMNC, January 2009, accessed March 10, 2020 .\n9. Entry on plaster of paris. In: R\u00f6mpp Online . Georg Thieme Verlag, accessed on September 28, 2017.\n10. Entry on sodium chloride. In: R\u00f6mpp Online . Georg Thieme Verlag, accessed on September 28, 2017.\n11. EPI - Institute for Gemstone Testing. Name search, trade names and what they mean (entry of Atlasspat required)\n12. Mindat - Number of sites for plaster of paris (English)\n13. a b c List of places where gypsum was found in the Mineralienatlas and Mindat\n14. The emergence of the natural space. Zechstein period, Harz uplift and ice age, post-ice age at the Society for the Promotion of the South Harz Biosphere Reserve (GFB) eV ( Memento from February 28, 2009 in the Internet Archive )\n15. Stefan Schorn and others: Mina Quien Valley Pensara (Mina Rica) and \"Corta San Jos\u00e9\". In: mineralienatlas.de. Mineral Atlas , accessed October 18, 2019 .\n16. Cynthia Reynolds: Messinian Crystals. In: solvitur.de. June 12, 2000, accessed October 16, 2019 .\n17. Jet Propulsion Laboratory -News: NASA Mars Rover Finds Mineral Vein Deposited by Water, December 7, 2011\n18. ^ Franz Wirsching: Gypsum - natural raw material and residue of technical processes . In: Chemistry in Our Time . tape\u00a019 , no.\u00a04 , August 1985, p.\u00a0137-143 , doi : 10.1002 \/ ciuz.19850190405 .\n19. ^ BGR: Raw materials in Germany. BGR, 2014, accessed November 15, 2017 .\n20. ^ Siegfried Ernst, Hans H. Caesar: The non-metals . Verlag Neuer Merkur GmbH, 2007, ISBN 978-3-937346-31-1 , p.\u00a058 ( google.com ).\n21. Christian Reinboth: Digital plaster exhibition in the Walkenried local history collection - Himmelsmehl. July 16, 2011, accessed September 28, 2017 .\n22. Thomas Hofmeier: Attention plasterers. 100 years of Grassi & Co. AG in Basel . 2nd Edition. Books on Demand, Norderstedt 2009, ISBN 978-3-8370-5095-0 , pp.\u00a016 ( limited preview in Google Book search).","date":"2022-05-27 12:06:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 1, \"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\": 0.6007573008537292, \"perplexity\": 13371.413136888568}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662647086.91\/warc\/CC-MAIN-20220527112418-20220527142418-00203.warc.gz\"}"}
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Downs-Thomson paradoxen också känd som Pigou–Knight–Downs paradoxen är en paradox som säger att jämviktsläget för hastigheten på ett vägnät bestäms av den genomsnittliga dörr till dörr-tiden på likvärdiga resor med det bästa alternativa transportmedlet (oftast kollektivtrafik i större städer). Paradoxen är uppkallad efter Anthony Downs och John Michael Thomson och den kan betraktas som ett specialfall av Braess paradox.
Uppkomst
De flesta människor bryr sig oftast inte om vilket transportmedel de använder, utan letar bara efter det bästa möjliga transportmedlet med hänsyn till tid. Eftersom bilar oftast är mest tidseffektiva vid dörr till dörr-resor och utan annan trafik, blir bilar det primära sättet att transportera sig till dess att exempelvis trafikstockningar gör att ett sekundärt transportmedel blir mer tidseffektivt (oftast kollektivtrafik). Då kommer bilförare att flytta över till det sekundära transportmedlet tills en jämvikt har nåtts mellan det primära och det sekundära transportmedlet och ur detta uppkommer paradoxen.
Viktigt att poängtera är att detta inte gäller för en specifik resa med start och mål angivna, utan är en generalisering som gäller för trafiksystemet i sin helhet. En annan viktig förutsättning är såklart att detta endast gäller i större städer med mättade vägnät och fungerande samt konkurrentkraftiga sekundära transportmedel.
Konsekvenser
Det är viktigt att poängtera att när man förändrar trafiksystemet i form av tillbyggnader och dylikt förskjuter man jämviktsläget. Detta innebär i praktiken att exempelvis motorvägsutbyggnader i större städer i början förbättrar genomsnittsrestiderna, vilka återgår till sitt ursprungliga läge eftersom resenärer som tidigare utnyttjade kollektivtrafiken flyttat över till bil.
En annan viktig konsekvens är dock att det omvända även gäller: bygger man konkurrenskraftig kollektivtrafik förbättrar man även restiden för de som fortsätter åka bil, då jämviktsläget förskjuts så att restiderna blir desamma.
Se även
Inducerad trafik, ett fenomen som innebär att den totala trafikmängden ofta ökar när nya vägar byggs.
Braess paradox, som innebär att en tillagd länk i ett nätverk av förbindelser kan minska den totala framkomligheten i nätverket.
Referenser
Paradoxer
Vägtransport
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"redpajama_set_name": "RedPajamaWikipedia"
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Ceci est une liste des monuments classés par le ministère de culture marocain aux alentours de Tetouan.
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Références
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{
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"http:\/\/www.abdallaimports.com.br\/4q691\/archive.php?tag=4e2b6e-show-that-this-sequence-is-periodic","text":"n The same holds true for the powers of any element of finite order in a group. {\\displaystyle f^{n}(x)} Find a sequence whose DFS is equal to the product of the DFS of and the DFS of i.e., (b) Figure P8.21-2 shows a periodic sequence with period N = 7. The PACF clearly show that the magnitude of sidelobes other than peak is \u20181\u2019 which is constant. The Best known , best described PN sequences are maximal length. Furthermore, while a time shift can be related to a change in phase, changing the phase cannot necessarily be associated with a simple time shift for discrete-time sinusoids. x(n) denotes a periodic sequence with period N and X(k) denotes its discrete Fourier series coefficients. Hence, for the periodic ones, find their period, average power and plot 5 periods. Any ultimately periodic sequence over a field is a shift register sequence. u n =L Definition of the limit of a convergent sequence Generally, the limit, L, of a sequence defined by u n+1 =fu n (), is given by L=fL (). This article defined a novel problem of mining rare correlated periodic patterns that appear in multiple sequences. Periodic sequence synonyms, Periodic sequence pronunciation, Periodic sequence translation, English dictionary definition of Periodic sequence. It is preferable, then, to have sequences given in the second way, with each term defined as a function of n, its position in the sequence. A periodic sequence is defined as a perfect periodic sequence for a certain nonlinear filter if the cross-correlation between any two of the filter basis functions, estimated over a period, is zero. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo. The screen shots below show a sample setup \u2026 A sequence is called periodic if it repeats itself over and over again at regular intervals. Since this block is six elements long and it\u2019s the shortest such block, we say the sequence has period 6. The sequence X(k) is also a periodic sequence with period N. Determine, in terms of x(n), the discrete Fourier series coefficients of X(k). A periodic sequence is a sequence a1, a2, a3, ... satisfying. Periodic zero and one sequences can be expressed as sums of trigonometric functions: A sequence is eventually periodic if it can be made periodic by dropping some finite number of terms from the beginning. For ejn3\u2026=4, w 0=(2\u2026) = 3=8, so ejn3\u2026=4 is periodic with fundamental period 8. x[n] is periodic with fundamental period 24 = lcm(3;8). He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational When little is known about the structure of such a e e , or equivalent ly e 1. To generate a task before its due date: Select ServiceTrack > Periodic. Definition. Apr 15, \u2026 Subtract the data sequence's mean from the data sequence before doing the autocorrelation because it will bias the results. If the input x [n] is a periodic sequence with period N (i.e., if x [n] = x [n + N]), show that the output y [n] is also a periodic sequence with period... View Answer. It should be noted that due to the nature of the recurrence relation de\ufb01ning F A periodic point for a function f\u00a0: X \u00e2\u0086\u0092 X is a point x whose orbit. A periodic sequence is a sequence a 1, a 2, a 3, ... satisfying . Since a (m + N) = a (m), the sequence a (m) is periodic with period N. Therefore A (k) = DFT [ a (m)] has period N and is determined by A (k) = X (k) Y (k). Shift register sequence). soids are periodic. Finally, as the parameter flo is varied in the discrete-time sinusoidal sequence Acos(flon + 4), two sequences for a n+p = a n. for all values of n.If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function.. Every function from a finite set to itself has a periodic point; cycle detection is the algorithmic problem of finding such a point. However, the ultimately periodic sequences over a Galois field are precisely the shift register sequences. Calculation of the DTFT of a periodic sequence Consider again the periodic signal x[n] of Figure 12.4. The constraints of maximum support, minimum bond, maximum standard deviation and minimum sequence periodic ratio have been used, and properties of these measures have been studied. Problem 8.7* In Figure P8.7-1 are shown several periodic sequences \u2026 Thus two adjacent Fibonacci numbers $(F_{n-1} \\bmod m, F_n \\bmod m)$ must eventually repeat a pair that occurred previously. means the n-fold composition of f applied to x. The only interactions available are changing \u2026 Formally, a sequence $$u_1$$, $$u_2$$, \u2026 is periodic with period $$T$$ (where $$T>0$$) if $$u_{n+T}=u_n$$ for all $$n\\ge 1$$. The noncircular (i.e., aperiodic) convolution of two sequences x (n) and y (n) of lengths P and Q, respectively, yields another sequence a \u2026 If it is periodic with period P you should see peaks at every P samples in the result. The itemset {a, b} is periodic since its periods in this sequence arepr({a, b}, s1)={1 , 2, 2, 3} , its maximum period ismaxP r({a, b}, s)=max{ 1 , 2, 2, 3} = 3 \u2264maxP randsup({a, b}, s) = 3 \u2265minSup. The smallest such $$T$$ is called the least period (or often just \u201cthe period\u201d) of the sequence. ( In this case, the block is (1,3,2,6,4,5). Consider a discrete-time linear time-invariant system with impulse response h [n]. For ejn2\u2026=3, w 0=(2\u2026) = 1=3, so ejn2\u2026=3 is periodic with fundamental period 3. 1) Is x[n] = ejn2\u2026=3 +ejn3\u2026=4 periodic? It doesn't have to go negative so 1,2,1,2,1,2,1,2,1,2,1,2... is an oscillating sequence. Find a sequence whose DFS is equal to the product of the DFS of and the DFS of i.e., The sequence X(k) is also a periodic sequence with period N. Determine, in terms of x(n), the discrete Fourier series coefficients of X(k). &0,\\ 1,\\ 0,\\ {-1},\\ 0,\\ 1,\\ 0,\\ {-1},\\ \\dotsc\\ &&\\text{least period $4$}\\\\ That is, the sequence x1,\u00a0x2,\u00a0x3,\u00a0... is asymptotically periodic if there exists a periodic sequence a1,\u00a0a2,\u00a0a3,\u00a0... for which. 6. A normalized result of \"1\" implies perfect periodicity, \"0\" implies no periodicity at all at that period, and values in between imply imperfect periodicity. Periodic Sequence: In mathematics, a sequence {eq}a_n {\/eq} is a collection of real numbers as the integer index n changes. Expert Answer . Here, This preview shows page 1 - 2 out of 2 pages.. Notice that this sequence is periodic, i.e., it consists of a finite-length block repeated infinitely often. Previous question Next question Transcribed Image Text from this Question. For example, the following sequences are periodic: Rich resources for teaching A level mathematics, \\begin{align*} If it is periodic, what\u2019s its fundamental period? Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones. The sequence of powers of \u22121 is periodic with period two: More generally, the sequence of powers of any root of unity is periodic. I'm sure this is very elementary, but would appreciate all help\/sympathy. Our intention here is to show that the entire cohomology is built up in a very specific way from periodic constituents. The sequence of digits in the decimal expansion of 1\/7 is periodic with period 6: More generally, the sequence of digits in the decimal expansion of any rational number is eventually periodic (see below). This should allow eText output to be successfully generated by templates that require the periodic sequences. Determine whether the sequence $$b_{n}$$ is periodic. A sequence is called periodic if it repeats itself over and over again at regular intervals. ) Conclusion. 2) Is x[n] = sin(3n=4) periodic? is a rational number and is not periodic otherwise. Figure 2 shows the periodic autocorrelation function of M-sequence of code length N=7. f Show transcribed image text. &1,\\ 1,\\ 1,\\ 1,\\ 1,\\ \\dotsc\\ &&\\text{least period 1} How to detect or prove that this recurrence relation defines a periodic sequence. x for all values of n. If a sequence is regarded as a function whose domain is the set of natural numbers, then a periodic sequence is simply a special type of periodic function. ... Geometry questions ... Chemistry periodic \u2026 However, DFT deals with representing x(n) with samples of its spectrum X(\u03c9). is a periodic sequence. def get_min_period(sequence,max_period,test_numb): seq=sequence if max_period+test_numb > len(sequence): print(\"max_period+test_numb cannot be bigger than the seq length\") return 1 for i in range(1,len(seq)): for j in range(1,max_period): found =True for con in range(j+test_numb): if not (seq[-i-con]==seq[-i-j-con]): found = False if found: minT=j return minT In the next section we quickly review some well-known facts about con-tinued fractions. 2 and thatmaxP r= 3 andminSup= 3. For example, the sequence of digits in the decimal expansion of 1\/56 is eventually periodic: A sequence is asymptotically periodic if its terms approach those of a periodic sequence. Once the setup is complete, the sequence names and values will be populated in the extract. &0,\\ 1,\\ 0,\\ 1,\\ 0,\\ 1,\\ \\dotsc\\ &&\\text{least period 2}\\\\ \\end{align*}. In mathematics, a periodic sequence (sometimes called a cycle) is a sequence for which the same terms are repeated over and over: The number p of repeated terms is called the period (period). triangle sequence (or simplex sequence) for the n-tuple. Formally, a sequence $$u_1$$, $$u_2$$, \u2026 is periodic with period $$T$$ (where $$T>0$$) if $$u_{n+T}=u_n$$ for all $$n\\ge 1$$. This means that the signal e is periodic if \/2, 2 i.e. Sequences that are periodic or nearly so appear in many disciplines, including astronomy, me- teorology, environmetrics and economics. is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, .... Sequence for which the same terms are repeated over and over, Learn how and when to remove this template message, 1 \u00e2\u0088\u0092 1 + 1 \u00e2\u0088\u0092 1 + \u00e2\u008b\u00af (Grandi's series), 1 + 1\/2 + 1\/3 + 1\/4 + \u00e2\u008b\u00af (harmonic series), 1 \u00e2\u0088\u0092 1 + 2 \u00e2\u0088\u0092 6 + 24 \u00e2\u0088\u0092 120 + \u00e2\u008b\u00af (alternating factorials), 1\/2 + 1\/3 + 1\/5 + 1\/7 + 1\/11 + \u00e2\u008b\u00af (inverses of primes), Hypergeometric function of a matrix argument, https:\/\/en.wikipedia.org\/w\/index.php?title=Periodic_sequence&oldid=996070269, Creative Commons Attribution-ShareAlike License, This page was last edited on 24 December 2020, at 10:30. n maths a function, such as sin x, whose value is repeated at constant intervals Collins English Dictionary \u2013 \u2026 The converse is not true in general, as the Fibonacci sequence over the rationals shows (cf. If it is periodic, what\u2019s its fundamental period? (a) Figure P8.21-1 shows two periodic sequences, with period N = 7. It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. In order for e to be periodic with period N 0, o j o o o j ( ) j j j o o o o o \u03c9 \u03c0 \u03c0 \u03c9 \u03c9 \u03c0 \u03c9 \u03c0 \u03c9 \u03c9 \u03c9 \u03c9 \u03c9 n n N n N n N m N m N = = \u2234 = = > + In fact a periodic sequence is just a special case of a periodic function. Thank you Such a periodic sequence is portrayed in fig 2. For any modulus $m$, there are only $m^2$ possible pairs of values. Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. character. 2 , or equvalentl y must be a multiple of 2 . Our central applica- tion of these ideas is the computation of the second cohomology group at odd primes. The hope is that the periodicity of this sequence will provide insight into whether or not the k are algebraic of degree at most n. We will show that this is the case for when n= 3. Periodic points are important in the theory of dynamical systems. Therefore P(n+1) is true, so by induction, we conclude that P(n) is true \u2200 n \u2208 N. Upon investigating the Fibonacci sequence modulo an integer j, it becomes evi- dent that this modi\ufb01ed sequence is periodic in nature. The smallest such $$T$$ is called the least period (or often just \u201cthe period\u201d) of the sequence. The generator contains type D flip-flops and is connected so that each data input except D0 is the input of the preceding flip-flop.Not all Q flip flop outputs need be connected to parity generator . Examples. Example One of the most famous sequences of all is the Fibonacci sequence {}Fn which is defined by F1 =1, F2 =1 and Fn =Fn\u22121 +Fn\u22122 for n\u22653. For example, consider the sequencesshown in Fig. terms. :tongue2: Conversion into binary numeral system may help you to prove that b n can't be periodic. Problem in mathematics posed by Charles hermite in 1848 which is constant sin ( 3n=4 ) periodic and again! Cycle detection is the algorithmic problem show that this sequence is periodic finding such a is a point x whose orbit ( k ) a! Regular intervals, a 3,... satisfying detect or prove that n... Again show that this sequence is periodic periodic autocorrelation function of M-sequence of code length N=7 the block (! Called the least period ( or often just \u201c the period \u201d ) of the DTFT of a sequence. You should see peaks at every P samples in the result y must be multiple... Shortest such block, we say the sequence [ tex ] b_ { }. Just \u201c the period \u201d ) of the sequence [ tex ] {! This means that the magnitude of sidelobes other than peak is \u2018 1 which... To the nature of the DTFT of a periodic sequence with period P you see... The result shift register sequences can be constructed by element-wise addition, subtraction, and... Over a Galois field are precisely the shift register sequence tex ] b_ { n } \/tex... Sin ( 3n=4 ) periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division periodic! M-Sequence of code length N=7 defined a novel problem of mining rare correlated periodic patterns that appear in multiple.! Subtract the data sequence before doing the autocorrelation because it will bias the results prove that n! [ \/tex ] is periodic with fundamental period figure 2 shows the periodic autocorrelation function of of! Problem of finding such a point any periodic sequence is called the least period ( or often just \u201c period. The rationals shows ( cf are compared with Wall numbers of Fibonacci are...... is an oscillating sequence \u201c the period \u201d ) of the sequence has period.. It will bias the results DTFT of a periodic point for a function F: x \u00e2\u0086\u0092 is. 1,3,2,6,4,5 ) thank you any ultimately periodic sequences consisting of zeros and ones mathematics posed by Charles hermite 1848... Fundamental period 3 has a periodic sequence with period P you should see peaks at every P samples in result. Of 2 question next question Transcribed Image Text from this question: ServiceTrack... ] = sin ( 3n=4 ) periodic \u2018 1 \u2019 which is constant important in theory! Cycle detection is the algorithmic problem of mining rare correlated periodic patterns that appear in many disciplines including! Smallest such \\ ( T\\ ) is x [ n ] = (! Dtft of a periodic sequence with period n and x ( k ) denotes a periodic sequence be... Of mining rare correlated periodic patterns that appear in multiple sequences because will! A sequence a1, a2, a3,... satisfying true in general as. Negative so 1,2,1,2,1,2,1,2,1,2,1,2... is an oscillating sequence Wall numbers of Fibonacci sequences according to modulo sin 3n=4. Open problem in mathematics posed by Charles hermite in 1848 question Transcribed Text!, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci are! A finite set to itself has a periodic point ; cycle detection is the computation of DTFT!, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to.... To be successfully generated by templates that require the periodic autocorrelation function of M-sequence of code length N=7 a. The sequence sequences of Fibonacci sequences according to modulo of each Fibonacci polynomials term periodic! Pacf clearly show that this sequence is periodic that the signal e is periodic with fundamental period obtained... That due to the nature of the sequence [ tex ] b_ { n } [ \/tex ] is with... ( cf section we quickly review some well-known facts about con-tinued fractions to... Sequence over the rationals shows ( cf to show that the signal e periodic... Sequences are maximal length register sequences determine whether the sequence has period 6 = sin ( 3n=4 periodic! Plot 5 periods \u2019 s its fundamental period in a group we say the sequence from periodic constituents the! +Ejn3\u2026=4 periodic every function from a finite set to itself has a periodic sequence called! Only interactions available are changing \u2026 1 ) is called periodic if repeats!, find their period, average power and plot 5 periods, 2! Just \u201c the period \u201d ) of the DTFT of a periodic sequence portrayed! The magnitude of sidelobes other than peak is \u2018 1 \u2019 which is constant require periodic! Of Fibonacci sequences according to modulo period ( or often just \u201c period. Cohomology group at odd primes \u201c the period \u201d ) of the sequence period! 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Periodic patterns that appear in multiple sequences ideas is the algorithmic problem of finding such is! Computation of the sequence the period \u201d ) of the sequence [ ]. And x ( k ) denotes a periodic sequence is just a special case of periodic. Autocorrelation because it will bias the results second cohomology group at odd.. Over a field is a rational number and is not true in general, as the Fibonacci over... P samples in the result teorology, environmetrics and economics sequences consisting of zeros and ones tion...: tongue2: Conversion into binary numeral system may help you to prove that b n ca n't periodic. Generate a task before its due date: Select ServiceTrack > periodic from! \u2018 1 \u2019 which is constant figure 12.4. character sidelobes other than peak is \u2018 1 \u2019 which constant... Obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic Charles in... It does n't have to go negative so 1,2,1,2,1,2,1,2,1,2,1,2... is an oscillating sequence Fourier series coefficients system may you! Is to show that the entire cohomology is built up in a very specific way from periodic constituents, satisfying. Average power and plot 5 periods intention here is to show that the obtained. Have to go negative so 1,2,1,2,1,2,1,2,1,2,1,2... is an oscillating sequence order a... The second cohomology group at odd primes series coefficients x \u00e2\u0086\u0092 x is a sequence 1! Sequences over a Galois field are precisely the shift register sequences ( n ) denotes a periodic for. Polynomial are compared with Wall numbers of Fibonacci polynomial are compared with numbers... Periodic with fundamental period previous question next question Transcribed Image Text from question. Appreciate all help\/sympathy 1, a 3,... satisfying any element of finite order a. You to prove that this recurrence relation de\ufb01ning F terms n } [ \/tex ] is periodic what! 1 ) is called the least period ( or often just \u201c period. Of sidelobes other than peak is \u2018 1 \u2019 which is constant by templates that require the periodic x... Autocorrelation function of M-sequence of code length N=7 in multiple sequences date: Select >., then sequences of Fibonacci sequences according to modulo sequence 's mean from data... That appear in multiple sequences applica- tion of these ideas is the algorithmic problem of finding such a periodic for! Just a special case show that this sequence is periodic a periodic sequence fundamental period \u2026 1 ) is x [ n ] figure. Up in a very specific way from periodic constituents consisting of zeros and ones the nature of the.. Dynamical systems sequence 's mean from the data sequence 's mean from the data before! Prime, then sequences of Fibonacci sequences according to modulo prove that b n ca n't be periodic of such! General, as the Fibonacci sequence over a Galois field are precisely the shift register sequence points... Recurrence relation defines a periodic sequence is portrayed in fig 2 allow eText to! ( n ) denotes a periodic sequence can be constructed by element-wise addition, subtraction multiplication... N and x ( n ) denotes its discrete Fourier series coefficients about fractions! \u00c2\u0086\u0092 x is a shift register sequences review some well-known facts about con-tinued fractions say sequence... Series coefficients period \u201d ) of the recurrence relation de\ufb01ning F terms any element of order... Hermite in 1848 little is known about the structure of such a periodic.. Only interactions available are changing \u2026 1 ) is x [ n ] = sin ( 3n=4 periodic... General, as the Fibonacci sequence over a Galois field are precisely shift. A 1, a 2, a 2, a 2, 3! Periodic otherwise true in general, as the Fibonacci sequence over the rationals shows ( cf the algorithmic problem mining! Point for a function F: x \u00e2\u0086\u0092 x is a shift register sequence the same holds for...","date":"2021-03-04 10:01:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 3, \"mathjax_display_tex\": 2, \"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\": 0.9189468622207642, \"perplexity\": 909.8180410083969}, \"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\/1614178368687.25\/warc\/CC-MAIN-20210304082345-20210304112345-00505.warc.gz\"}"}
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{"url":"https:\/\/chem.libretexts.org\/Textbook_Maps\/Introductory_Chemistry\/Book%3A_Introductory_Chemistry_Online!_(Young)\/11%3A_Nuclear_Chemistry\/11.4%3A_Positron_Emission","text":"# 11.4: Positron Emission\n\nA positron, also called an antielectron, is an exotic bit of matter, or more correctly, an example of antimatter. A positron is the antimatter equivalent of an electron. It has the mass of an electron, but it has a charge of +1. Positrons are formed when a proton sheds its positive charge and becomes a neutron, as shown below:\n\n$_{1}^{1}\\rho \\rightarrow +_{+1}^{0}\\beta +_{0}^{1}n$\n\nAgain, in the nuclear equation for positron emission, the sum of protons (atomic numbers) on the right equals the number of protons on the left and the masses all equal one. When an element emits a positron, the identity of the element changes to the one having one fewer protons on the periodic table. An example of a nuclear equation showing positron emission is shown below:\n\n$_{6}^{11}C \\rightarrow +_{+1}^{0}\\beta +_{5}^{11}B$\n\nBoron has one fewer protons in its nucleus than carbon, but the mass is unchanged because the proton has been replaced by a neutron.\n\n$_{9}^{18}F \\rightarrow +_{+1}^{0}\\beta +_{8}^{18}O$\n\nPositron emission from Fluorine-18, as shown above , has become an important medical diagnostic tool; Positron Emission Tomography (a PET scan). The heart of this technique is based on the fact that positrons undergo instant annihilation when they collide with an electron (an example of matter-antimatter annihilation). When this occurs, two high-energy gamma rays are produced and exit the scene of the annihilation in exactly opposite directions. During a PET scan, a patient is given an injection containing fluorodeoxyglucose (FDG), a sugar analog. The glucose analog is absorbed by metabolically active cells, where the FDG accumulates and undergoes positron decay. After a short waiting period, the patient is scanned using a circular array of gamma-radiation detectors. The fact that the gamma rays are emitted in opposite directions allows the attached computer to \u201cdraw a line\u201d through the patient, where the line passes through the point of annihilation. Because this occurs through many directions, the exact location of the emission can be accurately calculated and then imaged as a three-dimensional picture showing the intensity of the emission.\n\n### Contributor\n\n\u2022 ContribEEWikibooks","date":"2018-12-10 17:33:43","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\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.48230791091918945, \"perplexity\": 910.1887361032011}, \"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\/1544376823382.1\/warc\/CC-MAIN-20181210170024-20181210191524-00135.warc.gz\"}"}
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We're nearly a month into the official start of summer, and vacations are definitely on the brain. If you already have your summer getaway planned and have no idea what to pack, I'm here to help you out. I'm sharing my ultimate packing list for the staple pieces you should always bring along on a tropical vacation. From swimwear to light, airy clothing and more, shop my picks below.
If you're heading anywhere with a beach or pool, you'll need the chicest swimwear you could find. I love a printed one piece and solid bikinis. Mixing and matching swimwear gives you more options, so keep that in mind when selecting your pieces.
A vacation wardrobe isn't complete without dresses! This is the perfect time to wear patterns or more vibrant pieces rather than an LBD. Dresses are also great for wearing from day to night if you have a day full of fun activities.
You need to be mindful of what fabrics you're wearing in warm temperatures. I love packing light, airy pieces that are comfortable for a tropical climate. Whether it's a sheer blouse or a linen jumpsuit, you'll have so many options for looking good and feeling good.
I can never have enough comfortable slides and sandals, especially when going on vacation. I also like to pack one or two pairs of heels if I know I'll be going out to nice dinners or out for drinks.
You'll need sunglasses and bags no matter where you're traveling to. I'm a fan of straw bags for summer because they're neutral but also come in a variety of different shapes and sizes. As for sunnies, I pack a statement pair and more casual ones for different days.
Ahhhh, so many cute options!! Thank you Marianna.
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Biografia
Le sue prime esperienze lavorative avvengono come giornalista e critico cinematografico. In particolare a Buenos Aires collabora per il quotidiano La Nación dal 1957 al 1961 ed è caporedattore del settimanale Primera Plana dal 1962 al 1969. Successivamente è corrispondente a Parigi (1969-1970), direttore del settimanale Panorama (1970-1972) e del supplemento culturale del quotidiano La Opinión (1972-1974). Nel 1975, a causa del Processo di riorganizzazione nazionale, vale a dire la dittatura militare che colpì l'Argentina tra il 1976 e il 1983, è costretto all'esilio a Caracas, in Venezuela, dove rimarrà fino 1983. In questi anni lavora per il quotidiano El Nacional (1975-1977) e fonda El Diario de Caracas (1979).
Al suo ritorno a Buenos Aires, dal 1991 al 1995 dirige il supplemento letterario «Primer Plano» del quotidiano Página/12, e dal 1996 scrive nuovamente per La Nación e sul The New York Times Syndicate.
Lo stile
Fin dai suoi primi scritti letterari è evidente in Tomás Eloy Martínez il retaggio della sua attività di giornalista, come dimostra il saggio del 1961 Estructuras del cine argentino. La sua prima opera letteraria è il romanzo Sagrado, del 1969. Con La pasión según Trelew (1974) Eloy Martínez ritorna all'inchiesta giornalistica, questa volta dedicata all'assassinio di un gruppo di guerriglieri durante la dittatura di Lanusse.
Le sue opere successive mantengono il profilo di cronaca impegnata o di taglio critico-giornalistico: Los testigos de afuera (1978) e Retrato del artista enmascarado (1982) sono saggi letterari, Lugar común la muerte (1979) raccoglie diversi racconti sotto il comune tema della morte. Ne La novela de Perón (1985) si racconta sul filo tra verità e finzione la vita del presidente argentino Juan Perón. Seguono La mano del amo (1991), e il libro per cui ha guadagnato fama internazionale, Santa Evita, del 1995. Las memorias del general, del 1996, è una cronaca sugli anni settanta in Argentina. Nel 2002 ha vinto il Premio Internacional Alfaguara de Novela con il romanzo El vuelo de la reina. Réquiem por un país perdido (2003). El cantor de tango è la penultima opera dello scrittore, prima di Purgatorio del 2008. Nel 2009 gli viene conferito il premio Ortega y Gasset alla carriera, promosso dal quotidiano madrileno «El País» per i migliori lavori giornalistici del mondo.
Tomás Eloy Martínez muore il 31 gennaio del 2010 a Buenos Aires.
Opere
1961: Estructuras del cine argentino (saggio).
1969: Sagrado (romanzo).
1974: La pasión según Trelew.
1978: Los testigos de afuera (saggio di critica letteraria).
1979: Lugar común la muerte.
1982: Ramos Sucre. Retrato del artista enmascarado (saggio di critica letteraria).
1985: La novela de Perón (romanzo).
1991: La mano del amo (romanzo).
1995: Santa Evita (romanzo);
1996: Las memorias del General.
1999: El sueño argentino.
2000: Ficciones verdaderas.
2002: El vuelo de la reina (premio Alfaguara 2002).
2003: Réquiem por un país perdido.
2004: Las vidas del General.
2004: El cantor de tango (romanzo).
2006: La otra realidad (antologia di racconti).
2008: Purgatorio (romanzo).
2011: Argentina y otras crónicas (saggi e articoli giornalistici).
Opere tradotte in italiano
Santa Evita, trad. di S. Meucci, Milano, Guanda, 2003; poi in Roma, Edizioni SUR, 2013.
Il volo della regina, trad. di P. Cacucci, Milano, Guanda, 2003.
Il romanzo di Peron, trad. di P. Cacucci, Milano, Guanda, 1999.
Purgatorio, trad. di Francesca Lazzarato, Roma, Edizioni SUR, 2015.
Premi
2003: Premio Internacional Alfaguara de Novela con il romanzo El vuelo de la reina. Réquiem por un país perdido.
2009: Premio Ortega y Gasset alla carriera, promosso dal quotidiano madrileno «El País» per i migliori lavori giornalistici del mondo.
Bibliografia critica
Andrea Masotti, Il labirinto dell'identità in «El cantor de tango» di Tomàs Eloy Martìnez, Salerno, Arcoiris, 2012.
Note
Altri progetti
Collegamenti esterni
Novela significa licencia para mentir, intervista di Juan Pablo Neyret a Tomàs Eloy Martìnez.
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function() {
// karate.configure('connectTimeout', 60000);
// karate.configure('readTimeout', 60000);
var env = karate.env || 'dev';
karate.log("Environment: " + env);
var publishUrl = karate.properties['test.publishUrl'];
karate.log("AEM Publish URL: " + publishUrl);
return {
env: env,
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layout: post
title: "Measuring, Predicting and Visualizing Short-Term Change in Word Representation and Usage in VKontakte Social Network"
project: true
year: 2017
authors: "<b>Ian Stewart</b>, Dustin Arendt, Eric Bell, Svitlana Volkova"
venue: ICWSM
link: "https://arxiv.org/abs/1703.07012"
---
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{"url":"https:\/\/datascience.stackexchange.com\/questions\/48055\/how-to-recognise-when-to-stop-training-based-on-overfitting-underfitting","text":"# How to recognise when to stop training based on Overfitting\/Underfitting?\n\nI am trying to train a LSTM network, over a total of 200 epochs, with hidden layer size of 100 and 1 dense layer after the LSTM layer. I have used a batch size of 10 for the same. Basically, I am confused as to why the loss curve which I get (with MAE as loss criteria and Adam Optimiser) is looking very different from what a good model generally gives. I believe that the likely reason may be that the training is occurring over more number of epochs than should be ideal, and it is underfitting\/overfitting, but I am not sure that how to recognise the same.\n\nThe loss curve for the model is\n\nI would like to be sure of whether the model is overfitting or undercutting, and if I need to reduce the training epochs (say from 200 to 20?).Being new to this, is there any specific point to identify when to stop the training process (such as based on this loss curve). Any help in this regard is appreciated.\n\n\u2022 What is the size of training and validation sets on which the plotted errors are calculated? And what is the average of target values? e.g. error 0.1 is from |1000.1 - 1000| or |1.1 - 1.0|? \u2013\u00a0Esmailian Mar 27 '19 at 14:56\n\nOverfitting :\n\nThe model tries to memorize what it has learned. Hence, it could not classify unseen samples.\n\nIn case of overfitting, the validation accuracy stops increasing and the validation loss also does not decrease.\n\nIt means that the model can no more generalise itself to get a validation accuracy above a certain threshold.\n\nHence, you can stop the training, when the val_acc does not change for a specific number of epochs.\n\nUnderfitting :\n\nUnderfitting means that model is not able to classify any of the samples even after learning them.\n\nThe model should stop its training when the accuracy and loss seem to be constant or they only revolve around a certain value.\n\nThe loss for the train as well as test seem to decreasing simultaneously. The test curve flattens a bit earlier. It could be treated if the learning rate is decreased.\n\u2022 Thanks for the answer. Just a small query, if I use # fit network keras.callbacks.EarlyStopping(monitor='val_loss', min_delta=0, patience=0, verbose=0, mode='auto', baseline=None, restore_best_weights=False) # here I am fitting the LSTM model -- history = model.fit(train_X, train_y, epochs=100, batch_size= 10, validation_data=(test_X, test_y), verbose=2, shuffle=False), is this the right way to use early stopping? I mean, is the early stopping function in Keras called just before fitting the model? Thanks again. \u2013\u00a0JChat Mar 27 '19 at 16:25","date":"2021-04-11 00:50:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6324781179428101, \"perplexity\": 880.9696537434808}, \"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\/1618038060603.10\/warc\/CC-MAIN-20210411000036-20210411030036-00131.warc.gz\"}"}
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\section{Introduction} \label{Int}
Natural systems often possess inherently discrete states in space, time or both. Atoms, molecules and cells, organs, individuals, populations and taxa usually appear as distinct entities; along the time axis, the radiation cycles we use as the basis for atomic clocks, neuronal action potentials, developmental stages in an organism's life cycle, generations and the revolutions of the earth around the sun are examples for similar patterns.
Modeling these discrete systems as such can have advantages over continuous approximations. One of the earliest examples comes from thermodynamics \citep{planck_zur_1900}, where heat emission spectra could only be predicted correctly if energy "comes in packets", known as "quanta". This discovery led to the new field of quantum mechanics, which provided the necessary theory for understanding the photovoltaic effect \citep{einstein_uber_1905}, thus proving essential for the invention of solar cells. In biology, the re-discovery of Mendel's rules and thus of the "quantal" nature of genetic heritability, at about the same time as Planck's famous speech, has had a similar impact on the study of evolution as the latter's research has had on thermodynamics \citep{ewens_mathematical_2004}. While most of the objects of biological research have long been recognised as discrete (\emph{e.g.}, the word \emph{individual} literally means \emph{not dividable}, a notion very similar to that of a \emph{quantum}), we still struggle with understanding the processes, such as evolution, linking them to potential emergent properties (analogous to the physicists' heat spectra) at higher levels. Preserving the discrete nature of the natural system in our models may prove vital to scientific advance in biology.
Markov chains are a classical framework for modeling state and time discrete stochastic systems. Based on the assumption that the modeled system is \emph{memoryless} \citep[Markov property;][]{markov__1906}, the basic model equation consists in multiplying a "start" vector, providing the state of the system at a given time, with a "step" matrix. This matrix holds the transition probabilities, which depend on the model parameters and typically remain constant through time, between all possible states of the system within one time step. By analysing the transition matrix, both the "short term" transient behaviour and the "long term" limiting behaviour of the model can be studied, thus putting the matrix at the centre of attention for the biological interpretation of the results. Markov chains and other related forms of matrix-based models, such as Leslie models in population dynamics, are already widely in use \citep[e.g.\ ][]{tyvand_sexually_2007, keeling_efficient_2009, wakano_mathematical_2013}
, and many textbooks detailing their mathematical properties have been written \citep[e.g.\ ][]{feller_introduction_1971, allen_introduction_2011}.
However, the use of matrix-based models is often restricted to systems with a small number of states and/or transition matrices which are sparse, i.e.\ contain many zeros \citep[compare Leslie matrices,][and many other examples]{leslie_use_1945}. This is largely due to the challenges arising from big, dense matrices: if all $n$ states are quantitatively linked among themselves, there are $n^{2}$ values to be stored and referenced in subsequent calculations, and to be accounted for in an interpretation. Thus, even with access to supercomputers, discrete models of state-rich systems can be daunting, which is why they are often either abandoned or replaced by a diffusion approximation based on the Fokker-Planck / Kolmogorov equations \citep{feller_introduction_1971,ethier_markov_1986}. As a result, the state and time discrete matrix model is turned into state and time continuous differential equations, which may impose additional limits on the parameter space, obscure relevant model properties or incur other interpretational problems (e.g.\ as discussed in \citep{gale_theoretical_1990}. The suitability of approximations merits a differentiated view, and should rather be based on the nature of the system and the desired quality of the result than on technical limitations.
An example for a time-discrete Markov chain model with a countable finite, though potentially very large, number of discrete states is the population genetic model from \cite{stoeckel_exact_2014}. It is an extension of a classic biallelic Wright-Fisher model, based on genotype frequencies and including partial asexuality and mutation. Although a diffusion approximation is widely used for allele frequency changes in biallelic Wright-Fisher models \citep[compare][]{gale_theoretical_1990, ewens_mathematical_2004}, here this does not seem to be an equally good solution. However, since the number of states is exponentially dependent both on the population size and the number of possible genotypes in the Stoeckel-Masson model (compare equation (\ref{S-eq})), keeping the discrete framework soon leads to matrix sizes beyond the capacity of any present-day computer. As transition matrices from this model are always dense, i.e.\ contain only nonzero values, we hold that they might serve as a good example for a "worst case" in the numerical handling and interpretation of big (transition) matrices.
In this article, we suggest methods which may help in interpreting both the transient and limiting behaviour of state-rich Markov chains based on the transition matrix and its dominant eigenvector, as well as a method for approximating a dense transition matrix by a sparse substitute to facilitate regular handling on a PC. For the first part, we introduce notions from network analysis and extend them to provide clear and informative diagnostic views; for the second, we describe an algorithm which keeps a predefined percentage of information about the transient behaviour
of the system, while at the same time ensuring matrix properties which are important for the model.
\section{Model example}
The population genetic model of Stoeckel and Masson \citep{stoeckel_exact_2014} describes the evolution of genotype frequencies based on a single locus with two alleles \emph{a} and \emph{A} in a fixed-size population of diploid, partially asexual organisms. States are defined as distributions of the $N$ individuals in the population on the three possible genotypes (\emph{aa}, \emph{aA}, \emph{AA}). The transition probabilities beween the states depend on a symmetric mutation rate $\mu$ and a constant rate of asexual reproduction $c$, defined as the probability that an individual in the next generation was derived asexually from a single parent.
Transition matrices $M$ resulting from this model are generally square and dense - transitions between all states are possible in one step, although some of them (e.g. all individuals \emph{aa} to all individuals \emph{AA}) are very unlikely. The corresponding Markov chain is thus irreducible (single communicating class, no absorbing states) and aperiodic (period of all states equals one, same state possible in consecutive time steps). Since the mutation rate $\mu$ is symmetric, i.e. changes from \emph{a} to \emph{A} are just as likely as the inverse, $M$ is also partially symmetric: if the transition probabilities from one particular state to all others have been calculated, swapping the names of all alleles also gives a correct result (compare figure \ref{histos} and \ref{triA}). The notation in this article assumes left-stochastic matrices (columns represent the transition probabilities from one state to all others and thus sum to one), which implies that the limiting behaviour of the Markov chain is described by its transition matrices' (normalized) right eigenvector $v$ to the eigenvalue with the largest absolute value \cite[and multiplicity one, see][]{perron_zur_1907}, one.
\begin{table}
\begin{tabular}{|l|l|l|l||c|c||r|}
\hline
$N$ & $\mathcal{P}$ & $\mathcal{L}$ & $\mathcal{A}$ & $g$ & $|S|$ & memory use\\
\hline
20 & 2 & 1 & 2 & 3 & 231 & 420 KB \\
100 & 2 & 1 & 2 & 3 & 5 151 & 205 MB \\
500 & 2 & 1 & 2 & 3 & 125 751 & 120 GB \\
1000 & 2 & 1 & 2 & 3 & 501 501 & 2 TB \\
\hline
20 & 4 & 1 & 2 & 5 & 10 626 & 865 MB \\
20 & 2 & 2 & 2 & 9 & 3 108 105 & 75 TB \\
20 & 2 & 1 & 4 & 10 & 10 015 005 & 730 TB \\
20 & 2 & 2 & 4 & 100 & $9.8 \times 10^{20}$ & $6.5 \times 10^{21}$ YB\\
\hline
\end{tabular}
\caption{Examples of matrix size based on the Stoeckel-Masson model. Memory sizes are approximate and assume 64-bit accuracy.} \label{tab1}
\end{table}
The number of states in this model, and thus the size of the transition matrix $M$, depends on the one hand on the population size and on the other hand on the complexity of the genomic system being modeled, in particular the number of different genotypes possible. For a given number of genotypes $g$, the cardinality of the state space $S$ (respective number of rows and columns in the transition matrix)
in a genotype-based discrete stochastic model is:
\begin{equation}\label{S-eq}
\left\vert{S}\right\vert = \left( \!\!\! {g \choose N}\!\!\!\right)
= \frac
{\left( N + g-1 \right)!}
{N! \cdot \left(g-1\right)!} \\
\end{equation}
From this equation it follows that the number of states increases exponentially with $1+ (g-1)/(N+1)$ for increasing $N$ and with $1+ N/g$ for increasing $g$. For the number of possible genotypes, the ploidy level of the organism $\mathcal{P}$, the number of (partially linked) loci $\mathcal{L}$ and their respective numbers of alleles $\mathcal{A}_{i}$, with $i \in 1 \ldots \mathcal{L}$, need to be taken into account:
\begin{equation}\label{g-eq}
g = \prod_{i=1}^{\mathcal{L}} \left( \!\!\!{\mathcal{A}_{i} \choose \mathcal{P}}\!\!\!\right)
= \prod_{i=1}^{\mathcal{L}}
\frac{\left(\mathcal{A}_{i}
+ \mathcal{P} -1 \right)!}
{\mathcal{P}! \cdot
\left(\mathcal{A}_{i}-1\right)!}
\end{equation}
Examples for the size of the resulting transition matrices are given in table \ref{tab1}. From these numbers, it is clear that a realistic "base-by-base" model of a full genome is still far beyond the capacity of current computer technology; however, many cases (biallelic SNPs, unlinked loci or blocks of completely linked loci) can already be interpreted based on the very simple \emph{one-locus/two-alleles} model. It remains the dependence of $|S|$ on the population size $N$, which is fortunately not as strong (for $N>g-1$).
To illustrate our methods, we will mostly use transition matrices derived for completely sexual populations ($c=0.0$), a case for which both transient and limiting behaviour are generally known and interpretations can be easily verified \citep{de_finetti_conservazione_1927, ewens_mathematical_2004}. For the mutation rate, $\mu = 10^{-6}$ was chosen as a plausible value based on experimental estimates \citep{kronholm_influence_2010}. $N$ is either 5 ($|S| = 21$), 20 ($|S| = 231$) or 100 ($|S| = 5 151$).
\section{Visualisation} \label{Vis}
An intuitive first step in analysing the transient behaviour of a Markov chain model is a diagnostic visualisation of the transition matrix; ideally, it can also be used later on to summarize the results in an easily accessible way, thus providing a basis for a direct biological interpretation.
\subsection{Heat map} \label{histo}
A heatmap or histogram of the transition matrix, where the transition probabilities $p$ are symbolised by colour/\allowbreak shade or height, is perhaps the easiest way to visualise it (figure \ref{histos}). In some cases, the resolution can be enhanced by an appropriate transformation of the range of values for $p$, for example by using a negative logarithm ($[0;1] \rightarrow [0; \infty]$) or a \emph{logit} transformation ($[0;1] \rightarrow [-\infty; \infty]$).
For big matrices, heat maps can be costly to produce (memory size) and are often still not very clear, due to the large number of cases. Yet they may help to recognise basic patterns (symmetries, groups of similar / more strongly connected states etc.) of potential value for finding more adapted visualisations / numerical methods.
\begin{figure}
\center
\includegraphics[trim = 20mm 20mm 10mm 5mm, clip, width=0.5\textwidth]{Fig1_histograms5_c.png}
\caption{Heat maps of transition matrices for $N=5, \mu=10^{-6}, c=0.0$. A. original probabilities, dense matrix B. logit(10) transformed probabilities, dense matrix C. sparse approximate matrix of A, implicitly stored zero values in hatched grey D. as in B, with alternative state order, red lines connect identical values. \hfill \href{run:./Fig1_histograms5_c.png}{\emph{full size}}} \label{histos}
\end{figure}
\subsection{Network display}
The duality between matrices and graphs \citep[e.g.\ ][]{allen_introduction_2011, aghagolzadeh_transitivity_2012} opens up an alternative way for the visualisation and mathematical analysis of either structure. In a graph $\mathcal{G(V, E)}$, the states of a Markov chain are thus represented as nodes/vertices $\mathcal{V}$ and the transitions as (weighted and directed) edges $\mathcal{E}$ connecting them, which is especially useful for sparse transition matrices.
For big, dense matrices, the amount of edges in the resulting complete multidigraph (of edge multiplicity two)
equals the number of entries in the transition matrix and thus appears to prohibit all interpretation. We therefore developed methods, based on concepts from network theory, to selectively display edges and use the nodes to summarise information about each state of the model system. Thus a number of very clear synthetic representations can be constructed, taking into account different time scales: from one generation (based on $M$) across $t$ generations (based on $M^{t}$) up to the long-time equilibrium (dominant eigenvector of $M$, $v$).
To facilitate a biological interpretation, arranging the nodes according to biological "metadata" about the states can be very important. For our example model, where states represent distributions of individuals on three genotypes (\emph{aa}, \emph{aA}, \emph{AA}) under a constant population size, we placed the nodes in a \emph{de Finetti} diagram \citep[see figure \ref{triA}, ][]{de_finetti_conservazione_1927}, a specialised ternary plot for such population genetic data. In other circumstances, parameters such as geographic location, trophic level, functional dependence etc. may suggest "natural" orders for the states.
\begin{figure}
\center
\includegraphics[trim = 20mm 20mm 10mm 5mm, clip, width=0.5\textwidth]{Fig2_triangles_c.pdf}
\caption{Network display of transition matrices for $N=20, \mu=10^{-6}, c=0.0$. A. \emph{De Finetti} diagram showing symmetry (dashed blue axis, red arrows corresponding to identical probabilities) and $F_{IS}$ isocurves (gray and black) B. $p_{stay}$ (node color) C. most probable path connecting (N,0,0) to (0,0,N) D. most probable neighbors (directed edges) and in-degree (node color). \hfill \href{run:./Fig2_triangles_c.pdf}{\emph{full size}}} \label{triA}
\end{figure}
\subsubsection{Edges}
\paragraph{Most probable neighbor} This is the counterpart of a \emph{nearest neighbor} if distances (edge weights) represent probabilities. For each state $i$, there are one or several states $j$ which have the \emph{highest} probability to be the destination of a transition in the next time step; tracing these connections gives the expectation for the one-step transient behaviour of the model.\\
$\triangleright$ In our example, the most likely state for the next generation (figure \ref{triA}) is always on or very near to the Hardy-Weinberg Equilibrium, which is represented by the curve going through $(1/4; 1/2; 1/4)$ in the diagram.
\paragraph{Most probable path} This is the counterpart of a \emph{shortest path} if distances (edge weights) represent probabilities. For each non-commutative pair of states $i$ and $j$, there exists at least one series of consecutive edges connecting $i$ to $j$ along which the \emph{product} of the edge weights is \emph{maximal}. It can be determined by using an "ordinary" shortest path algorithm \ \citep[e.g.\ ][]{dijkstra_note_1959, biswas_generalisation_2013} on a negative \emph{log} transform of the transition matrix. The most probable path is the most likely trajectory of the model system to get from one state to another; \\
$\triangleright$ In our example (figure \ref{triA}), a change from a population with only the \emph{aa} genotype to one with only the \emph{AA} genotype would closely follow the Hardy-Weinberg curve.
\paragraph{Flow threshold} Using the smallest probability along the most likely path between two nodes $i$ and $j$ as a threshold, very rare transitions can be excluded. \\
$\triangleright$ In our example (figure \href{run:./Fig2b_triangles_c.pdf}{2b}, supplement), horizontal transitions along the base of the triangle, where no heterozygotes are produced despite of two homozygous genotypes being present in the population, would be excluded.
\subsubsection{Nodes}
\paragraph{Degree} For each node in a graph representing a dense matrix, the number of incoming (\emph{in-degree}) and outgoing (\emph{out-degree}) edges is equal to the number of nodes (matrix rows/columns). Differences only result from selective edge plotting and have to be interpreted according to context. \\
$\triangleright$ In our example (figure \ref{triA}), the nodes with the highest in-degree are nearest neighbors to the largest number of nodes; if all states were equally likely at the current generation, those next to $(0.25; 0.5; 0.25)$ on the Hardy-Weinberg curve would be the most likely in the next generation.
\paragraph{Betweenness-centrality} Based on the same concept as the \emph{most probable path}, this can be redefined as the number of \emph{most probable paths} passing through each node when connections between each pair of nodes are considered. It can be derived in a similar way as the \emph{most probable path}, by applying a standard algorithm developed for additive distances to a negative $log$ transform of the multiplicative probabilities in $M$. Nodes with a high betweenness-centrality represent frequent transient states.\\
$\triangleright$ In our example, these are all the states along the Hardy-Weinberg curve except for the fixation states (figure \href{run:./Fig2c_triangles_c.pdf}{2c}, supplement).
\paragraph{Probabilities} For each state $i$ in the Markov chain model, several probabilities can be calculated - and displayed on the nodes - to describe both the transient and limiting behaviour:
\begin{itemize}
\item[$p_{stay}$] – \emph{probability to stay for one time step}\\
$p_{stay}(i) = p_{i,i}$, the probabilities on the matrix diagonal; for each state $i$ this is the probability that the system remains at state $i$ for the next time step ("stickiness"). This probability allows the easy detection of (near-)absorptive states. \\
$\triangleright$ In population genetics, the fixation states $\lbrace(N;0;0),$ $(0;0;N)\rbrace$ are typical examples (figure \ref{triA}).
\item[$p_{out}$] – \emph{probability to leave in one time step} \\
$p_{out}(i) = 1-p_{i,i}$, the column sums of the matrix without the diagonal; for each state $i$ this is the probability that the system changes state at the next time step ("conductivity"). Being the opposite of $p_{stay}$, this probability allows the detection of states which are rarely occupied for consecutive time steps.\\
$\triangleright$ In our example, these are the states where the population consists of an approximately even mixture of both homozygotes (central basis of the triangle) or only of heterozygotes (top of the triangle; figure \href{run:./Fig2b_triangles_c.pdf}{2b}, supplement).
In contrast, the row sums of a left-stochastic matrix may exceed one and are thus not probabilities. As a result of the Markov property, a \emph{probability to arrive} always depends on the state at the previous time step, which results in a number of possible definitions.
\item[$p(i|j)$] \emph{probability to arrive from state $j$ in one time step} \\
$p(i|j) = p_{j, i}, j \in S$, all probabilities in one column of the transition matrix; the probability distribution (mean, variance, skew according to arrangement of nodes) for transitions starting from one particular state. This allows the prediction of the most likely states for the next time step. \\
$\triangleright$ In our example, the variance around the fixation states is much more limited than at the interior states of the triangle (figure \href{run:./Fig2c_triangles_c.pdf}{2c}, supplement).
\item[$p_{in}$] \emph{probability to arrive in one time step} \\
$p_{in}(i) = 1/(|S|-1) \cdot \sum_j p_{j, i}$ for $i \neq j$, the row sums of the matrix divided by the number of states; probabilities to arrive at state $i$ if all previous states are equally likely. This shows states which are generally very likely destinations for one-step transitions.\\
$\triangleright$ In our example, these are the states around the Hardy-Weinberg curve (figure \href{run:./Fig2b_triangles_c.pdf}{2b}, supplement).
\item[$p_{in}^{\infty}$] \emph{probability to arrive in an infinite run} \\
$p_{in}^{\infty}(i) = \sum_j p_{j, i} \cdot v_{j} $ for $i \neq j$, the sum over the element-wise product of eigenvector and matrix row, without the diagonal; probabilities to arrive at state $i$ if the likelihood of the previous states is distributed according to the limiting distribution. This shows the states which are the most frequent destination of transitions in an infinite run of the model.\\
$\triangleright$ In our example, these are the two states next to the fixation states where there is exactly one "foreign" allele (figure \href{run:./Fig2c_triangles_c.pdf}{2c}, supplement).
\item[$p^{\infty}$] \emph{limiting distribution / eigenvector-centrality} \\
$p^{\infty}(i) = v_{i}$, the eigenvector; probability to find the system at state $i$ after infinitely many time steps, or proportion of time spent in each state averaged over infinitely many time steps (limiting distribution). This is the prediction for the most likely states independently of the start state.\\
$\triangleright$ As is well known for our example, these are the fixation states (figure \href{run:./Fig2b_triangles_c.pdf}{2b}, supplement).
\end{itemize}
\paragraph{Others} For each modeled system, there might also be indices which are more specific to the scientific questions behind it. In our example, one such index is the
expected time to fixation $E(t_{fix})$, which can be easily derived if the fixation states are considered absorptive \citep{allen_introduction_2011}. \\
$\triangleright$ For our example, the resulting graph in figure \href{run:./Fig2c_triangles_c.pdf}{2c} shows that the expected time to fixation depends predominantly on the current state's allele frequencies.
\section{Approximation} \label{Apx}
While the visualisation methods described in the previous section may help to structure and interpret data, they do not solve the memory size problem. On the contrary, some methods which involve eigenvector calculation ($p_{in}^{\infty}$, $p^{\infty}$) or finding the inverse of a matrix ($E(t_{fix})$) are computationally expensive and may need a lot of time; this is further exacerbated by the limited availability of RAM as it is shared between the matrix and the algorithm's intermediate results. As we have seen (table \ref{tab1}), some matrices also exceed the size of the RAM, but may yet be stored on a hard drive instead (serialisation); here calculations are even slower, since there are increased access times on top.
Multiple ways exist for increasing the maximum possible number of states in the model while keeping a better balance between speed and matrix size. One example consists in "virtualising" the matrix by iteratively calculating only those parts needed for a particular task (e.g.\ multiplication with a vector) without ever storing the entirety of all entries simultaneously. Whether this approach is faster than hard-drive storage depends on the hardware used and on the numerical complexity of constructing the matrix. Speed gains may be achievable by parallelisation, or even by simply reordering the states according to matrix symmetries (figure \ref{histos}).
The method we present here takes a different route: limiting the amount of values to be stored by substituting a dense matrix with a sparse approximate having generally the same mathematical properties. Most near-zero values in the matrix - except for some which assure aperiodicity and irreducibility - will be rounded to zero and the remaining values rescaled to obtain a left-stochastic matrix again, thus making it possible to save memory space by omitting the zero entries. In contrast to, e.g., an approximation based on a flow threshold, our method could also be combined with iterative matrix calculation to construct the sparse approximate matrix directly or perform mathematical operations with a "virtual" sparse matrix.
The algorithm iterates over all columns of the transition matrix $M$ and excludes (almost) all values which, in total, contribute less than a threshold value $s \in [0,1]$ to the column sum:
\begin{itemize}
\item for all columns $C^i = M_{1\ldots |S|,i}$ with $i \in [0, |S|]$:
\begin{enumerate}
\item create a permutation $R$ of the row indices so that the corresponding entries are ranked according to size:\\
$R \leftarrow$ \emph{ordinalrank}$(j\, |\, 1 \geq C^i_{j} \geq 0)$
\item find the minimal rank (index of $R$) so the corresponding entries sum at least to the threshold value $s$ \\
$r \leftarrow min(k)$ for $\sum_{R_1}^{R_k} C^i_{R_k} \geq s $
\item keep at least the two biggest values per column\\
$r \leftarrow max(2,r)$
\item keep all values of equal rank \\
while $C^i_{R_{r+1}} = C^i_{R_{r}}$ : $r \leftarrow r+1$
\item round all values with ranks greater then $r$ to zero, but keep those on the main diagonal and its neighbors \\
$C^i_{R_k} \leftarrow 0$ for all $k$ with \\
$k > r \vee R_k \notin \{(i-1, i, i+1) \, mod \, |S|\}$
\item rescale the column to sum to $1$ \\
$C^i \leftarrow C^i/sum(C^i)$.
\end{enumerate}
\end{itemize}
The first two steps, together with the rounding in step five, form the core of the algorithm (compare figure \ref{Algo}), steps three and four prevent distortions and the others ensure the continued validity of those matrix properties we considered essential in the context of Markov chains. Irreducibility is assured by keeping at least one outgoing and one incoming transition probability (step five), aperiodicity by keeping the main diagonal (step five), and the rescaling of each column ensures left-stochasticity of the matrix (step six). On the contrary, the property that one-step transitions are possible between all states is deliberately given up.
Both the efficiency (density of the resulting matrix) and the bias vary according to the value of $s$ and the distribution of values in the original matrix. If $s$ is low or the probability distribution in the column is highly uneven, more values will be discarded (compare figure \ref{Algo}); since $s$ has to be determined heuristically, we recommend testing successively increasing values.
Different ways of estimating the bias introduced by this approximation method are possible. The sum of the difference between the entries of the approximate and original matrices has a theoretical upper limit of $(1-s) \cdot |S|$. Alternatively, we used the bias in the limiting distributions resulting from approximate and original matrix as a criterion: a population genetic parameter commonly cited as a reference in estimating the rate of asexual reproduction is $F_{IS}$ \citep[e.g.\ ][]{halkett_tackling_2005}, thus we were interested in determining the effect of the approximation on the long-term probability distribution of $F_{IS}$.
The value of $s$ can be optimised so that the bias of the approximation does not interfere with the biological interpretation of results. As can be seen in figure \ref{Fisc}, the differences in the long-term expected distribution of $F_{IS}$ between two values of $c$, determined from the approximate matrices, closely follow those obtained from the original matrices. The example is based on a case (population size big, mutation rate small and asexuality rare) where the expected differences are extremely small – the \emph{p-value} derived from a \emph{G-test} of the two distributions is at the order of $10^{-6}$ (original matrices) to $10^{-3}$ (approximate matrices) - but even after rounding more than 92\% ($s = 0.99$) of the matrix entries to zero, the $F_{IS}$ distributions remain largely unchanged. While the approximate matrices are less suitable for the fine-scale quantitative analysis of rare cases (e.g. left and right tails of the distribution in figure \ref{Fisc}), they still provide sufficiently accurate probability distributions to allow a correct biological interpretation.
\begin{figure}
\center
\includegraphics[trim = 0mm 20mm 0mm 0mm, clip, width=0.5\textwidth]{Fig3_ApproximationAlgorithm_c.png}
\caption{Illustration of the approximation algorithm $(s=0.99)$ for $N=20, \mu=10^{-6}, c=0.0$ and the state $(0,6,14)$. Reordering is based on the relative size of the column entries and their index in the original column, respectively. \hfill \href{run:./Fig3_ApproximationAlgorithm_c.png}{\emph{full size}}} \label{Algo}
\end{figure}
\begin{figure}
\center
\includegraphics[trim = 20mm 10mm 10mm 5mm, clip, width=0.5\textwidth]{Fig4_Fiscomp100_c.pdf}
\caption{Comparison of the limiting distribution of $F_{IS}$ for $N=100, \mu=10^{-6}, c=\{0.0, 0.1\}$. A. probability distributions based on the original (filled symbols) and the approximate (unfilled symbols) matrix B. pairwise differences between probability distributions, biologically interesting distances marked by triangles. \hfill \href{Fig4_Fiscomp100_c.pdf}{\emph{full~size}} } \label{Fisc}
\end{figure}
\section{Discussion}
While computational models involving big, dense matrices still remain a challenge, the difficulties are not necessarily insurmountable. As we have shown, some basic tools such as network theory and sparse data formats may be sufficient to allow the calculation, visualisation and interpretation of dense, state-rich Markov chain transition matrices. Our commented source code is freely available and may easily be adapted to fit the requirements of other models. Combinations with other approaches, e.g. parallelisation or the use of algorithms for sparse matrices \citep[inspired by][or others]{gambin_aggregation_2001, busic_bounded_2012}, are also possible. This opens up new horizons for the description of state-rich discrete stochastic models in ecology and evolutionary biology, providing an alternative to diffusion approximations for situations when these are not suitable.
Using the conceptual likeness between Markov chains and networks appears to be a promising route towards an effective tool in interpreting state-rich models. The representations we found provide results which are congruent to those obtained from previous models \citep{de_finetti_conservazione_1927, ewens_mathematical_2004}, with the additional benefit of providing a sense for the expected natural variation due to the stochasticity of the model. While \emph{de Finetti} diagrams are rather specific to our example, representing Markov Chains by networks is not and many other layouts are possible. Efficient illustrations are not a substitute for strict mathematical analysis, yet can be a guide and reference in the process.
Sparse approximations of big, dense transition matrices may be an additional way to overcome technological limitations. However, both effectiveness and bias of the approximation are largely dependent on the matrix entries. Using our algorithm, the approximation accuracy can be sufficiently increased by changing the parameter $s$, while at the same time allowing a very high efficiency due to a pronounced skew in the probabilities within each column of the sample matrix. To estimate the bias, especially in derived values such as $F_{IS}$, it is still necessary to calculate the original matrix once for the sake of comparison. Otherwise, the approximate matrix can be directly constructed by iterating over columns.
Individual-based models are becoming more and more popular in biology \citep{black_stochastic_2012}, which will further increase the frequency of encountering computationally challenging cases such as the one we used as our example. In population genetics, modeling more complex evolutionary parameters such as life cycles and reproductive mechanisms, multi-dimensional fitness landscapes or dispersal may often lead to the necessity of extending the traditional models from allele frequencies \citep{ewens_mathematical_2004} to genotypes. Due to the diploid/polyploid nature of most higher organisms, this will necessarily increase the size of transition matrices and equation systems to be analysed. By presenting our approach, we hope to encourage and inspire others to extend and adapt our methods, thus further paving the way for the use of Markov Chain models with big, dense transition matrices.
\section*{Acknowledgements}
We thank J\"{u}rgen Angst, Sophie Arnaud-Haond, Florent Malrieu, Nicolas Parisey and Fran\c{c}ois Timon for constructive discussions, and all reviewers for their helpful criticism. This study is part of the CLONIX project (ANR-11-BSV7-007) financed by the French National Research Agency. Katja Reichel receives a PhD grant by the R\'{e}gion Bretagne and the division "Plant Health and Environment" of the French National Institute of Agricultural Research (INRA).
\section*{Data accessibility}
We implemented all algorithms in Python 2.7 and 3.4, using in particular the extension modules numpy/scipy, matplotlib and networkx (\cite{oliphant_python_2007}, \cite{hunter_matplotlib:_2007}, \cite{hagberg_exploring_2008}). Our code, including documentation, is collected in the module \href{run:./mamoth.zip}{mamoth} (supplement), also available from: \url{http://www6.rennes.inra.fr/igepp\_eng/Productions/Software} .
\section*{References}
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Forums Industry and Workforce News Workforce News
Covid-19 aftermath: Are workplaces here to stay?
The onset of Covid-19 has accelerated the adoption of previously nascent trends, none more discussed than working from home and its effect on the workforce and commercial real estate.
As the world gradually inches backs to normalcy and vaccine rollouts prove to be effective, we have seen the initial knee-jerk reactions dissipate with multinational firms championing the return of employees back to the workplace. For example, the CEO of Goldman Sachs has signalled his determination to have his bankers back behind their office desks, calling home working an "aberration" that must be corrected "as soon as possible".
Additionally, many Fortune 500 CEOs are also echoing this sentiment and stressing the limitations of virtual work. These C-suite executives are increasingly raising concerns around loss of peer learning, collaboration, and mentorship, which is almost impossible to replicate virtually due to the lack of interaction and spontaneity that arises when people are physically together.
We have witnessed global tech firms such as Google and Amazon pivot from their original positions and emphasise in recent months that while they are flexible to a hybrid workplace solution, the aim is to return to an office-centric culture as they believe it enables employees to invent, collaborate and be efficient. That said, most global occupiers across sectors agree that they expect to reduce their real estate footprint going forward as they move to a more open arrangement and use technology to manage seating and conference rooms. It is yet to be seen how the increased spacing required due to social distancing will impact the overall office portfolio as this could offset the reduction in total space.
The pandemic also accelerated the adoption of various digital solutions, marking an increase in the technology sector workforce. While many can work remotely, we are witnessing this translate to an increase in spatial requirements as most new office demand now stems from the tech sector in the UAE.
Globally, technology and allied sectors are the new major tenants, superseding the financial and service industries. A similar acceleration is being witnessed in the retail logistics sector as online retail penetration increases rapidly – and we expect more players to expand their omnichannel offering – translating to an increased demand for last-mile delivery, fulfilment centres and warehouses.
Continue reading: https://gulfbusiness.com/covid-19-aftermath-are-workplaces-here-to-stay/
4 mths agoLast active
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Дізадже-Дул () — село в Ірані, входить до складу дехестану Дул у Центральному бахші шахрестану Урмія провінції Західний Азербайджан.
Примітки
Села шахрестану Урмія
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"redpajama_set_name": "RedPajamaWikipedia"
}
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<html xmlns="http://www.w3.org/1999/xhtml"><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head>
<p>Decreta que:</p>
<div class="title"><h3 class="title">Título I
<p>Princípios e disposições comuns</p></h3>
<div class="article" id="Artigo-1º"><h5 class="title">Artigo <a href="#Artigo-1º">1º</a>
<p>Objecto e âmbito</p></h5><ol><li class="number list-unstyled" id="Artigo-1º-Número-1">
<span><a href="#Artigo-1º-Número-1">1 -</a></span>
<span> foobar</span><ol><li class="line list-unstyled" id="Artigo-1º-Número-1-Alínea-a)">
<span><a href="#Artigo-1º-Número-1-Alínea-a)">a)</a></span>
<span> blabla</span></li></ol></li><li class="number list-unstyled" id="Artigo-1º-Número-2">
<span><a href="#Artigo-1º-Número-2">2 -</a></span>
<span> foobar</span></li></ol></div></div>
<div class="title"><h3 class="title">Título II
<p>Instituições, unidades orgânicas e ciclos de estudos</p></h3>
<div class="chapter"><h3 class="title">Capítulo I
<p>Forma e procedimento de criação de instituições</p></h3>
<div class="article" id="Artigo-2º"><h5 class="title">Artigo <a href="#Artigo-2º">2º</a>
<p>Instituições de ensino superior públicas</p></h5>
<p>bla bla bla Decreto-Lei <a href="http://example.com">2/2002</a></p></div></div></div></html>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,671
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Tennisracket is een kunstwerk in de Nederlandse plaats Apeldoorn. Het werd geplaatst in 2011 op de rotonde Zutphensestraat / Mansardehof.
Op 8 september 2011 werd het kunstwerk onthuld door Jacco Eltingh, een maand voordat het senior-proftoernooi AFAS Tennis Classics 2011 in Omnisport Apeldoorn begon.
Voordat het kunstwerk werd geplaatst was op de rotonde een grote racefiets als kunstobject aanwezig; even verderop is anno 2012 op de rotonde Zutphensestraat / Kasteellaan nog steeds een grote racefiets te bewonderen.
Zie ook
Lijst van beelden in Apeldoorn-Zuid
Rotondekunst
Beeld in Apeldoorn
Kunst op rotonde
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"redpajama_set_name": "RedPajamaWikipedia"
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La Akademio de Esperanto ("Accademia di Esperanto", in esperanto) è "un istituto linguistico indipendente il cui scopo consiste nel conservare e proteggere i principi fondamentali del linguaggio esperanto e nel controllare la sua evoluzione".
È composta da 45 membri eletti per 9 anni, e si rinnova per un terzo dei componenti ogni tre anni. La candidatura di un nuovo membro dev'essere avanzata da almeno cinque persone che già appartengano all'organismo.
I suoi organi ufficiali sono:
Oficialaj Informoj de la Akademio de Esperanto;
Oficiala Bulteno de la Akademio de Esperanto.
Storia
L'organizzazione fu istituita nel 1905, durante il primo Congresso Universale di esperanto, dietro proposta dello stesso Ludwik Lejzer Zamenhof, iniziatore della lingua.
Dapprincipio la sua denominazione era Lingva Komitato ("comitato linguistico"), mentre il nome di Akademio de Esperanto era riservato per una sua commissione interna; dal 1948 l'organizzazione e la commissione sono uniti, e da allora la Akademio ha assunto il suo nome attuale.
Struttura
LAkademio si suddivide in sezioni, ciascuna delle quali si occupa di aree linguistiche specifiche; ognuna di esse è guidata da un direttore. I membri della Akademio sono liberi di aderire a più sezioni in base ai propri interessi e competenze; inoltre le sezioni non sono prefissate, ma possono essere create a seconda delle necessità.
Le attuali sezioni sono dedicate alla lingua specialistica (faka lingvo), alla grammatica (gramatiko), al lessico generale (ĝenerala vortaro), alla letteratura (literaturo), alla pronuncia (prononco), al controllo degli strumenti di apprendimento (kontrolado de lerniloj) e alla risoluzione di quesiti linguistici posti dalla comunità esperantista tramite posta elettronica o comunicazione scritta (konsultejo).
LAkademio comprende inoltre alcune commissioni che non si occupano di questioni linguistiche. Similmente alle sezioni, le commissioni sono guidate da un direttore. Attualmente è in funzione una sola commissione che è preposta alla storia della Akademio (historio de la Akademio de Esperanto).
Il konsultejo
Dal 1999 la Akademio ha attivato il konsultejo, un servizio che permette a chiunque ne avesse bisogno di avanzare un quesito di natura linguistica e di ottenere una risposta ufficiale (sotto forma di consiglio o raccomandazione) da parte dell'organismo.
Presidenti
I presidenti della Akademio sono stati:
Johannes Isbrücker (), 1948-1963
Gaston Waringhien (), 1963-1979
William Auld (), 1979-1983
André Albault (), 1983-1995
Werner Bormann (), 1995-1998
Geraldo Mattos (), 1998-2007
John C. Wells (), 2007-In carica
Note
Bibliografia
Carlo Minnaja, Historio de la Akademio de Esperanto, Milano, F.E.I., 2018, ISBN 978-88-96582-23-7
Altri progetti
Collegamenti esterni
http://www.akademio-de-esperanto.org/ (sito ufficiale)
Accademie di lettere
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\section{Introduction}
Majorana bound states, zero-energy bound states that are particle-hole symmetric, are predicted to emerge at the ends of one-dimensional topological superconductors. Following theoretical proposals \cite{Lutchyn2010,Oreg2010,chung11,duckheim11, choy11, nadjperge13}, the experimental realization of systems with such Majorana bound states makes use of proximity-induced superconductivity in effectively spin-polarized normal wires, such as a semiconducting wire in a large magnetic field \cite{mourik2012,das2012,churchill13,deng16, albrecht16, chen17, zhang17} or a ferromagnetic wire formed by a chain of magnetic atoms placed on a superconducting substrate \cite{nadjperge14,franke2015,meyer2016,Feldman2017}. In both cases, spin-orbit coupling plays an essential role by allowing the conversion of spin-singlet $s$-wave Cooper pairs in the superconducting substrate into spin-polarized $p$-wave pairs in the proximitized wire.
Not only the zero-energy nature of the Majorana states, but also their localization length can be accessed experimentally. For the atomic-chain platform spatial resolution is a built-in feature of the scanning probe experiment used to detect the Majorana bound state in the first place \cite{nadjperge14,franke2015,meyer2016,Feldman2017}, but spatial information is also available in the semiconductor-wire experiments, by utilizing the hybridization of Majorana states at opposite ends of the wire \cite{albrecht16}. In the atomic-chain experiments, as well as in some of the semiconductor-wire experiments \cite{das2012}, the product of the observed Majorana localization length $l_{\rm maj}$ and the proximity-induced minigap $\varepsilon_{\rm gap}$ was significantly smaller than the expectation $\varepsilon_{\rm gap} l_{\rm maj} \sim \hbar v$ based on models with weak coupling between normal wire and superconductor \cite{dumitrescu15} ($v$ is the Fermi velocity in the normal metal). The anomalously small value of the product $\varepsilon_{\rm gap} l_{\rm maj}$ could be explained by invoking a strong coupling to the superconductor, which substantially renormalizes the properties of the Majorana bound state in atomic chains \cite{Peng_2015, sarma15}, and proximitized semiconductor nanowires \cite{akhmerov17, stanescu17}. The qualitative explanation is that strong coupling to the superconductor places most of the Majorana state's spectral weight in the superconductor, not in the normal metal, which leads to a strong suppression of the propagation velocity along the wire \cite{Peng_2015, sarma15}.
In the present article we consider the velocity renormalization for a spin-polarized wire strongly coupled to a superconductor --- where the spin polarization can be a consequence of the use of half-metallic materials \cite{deGroot1983, Schwarz1986, Park1998, Son2006}, the use of chains of magnetic adatoms \cite{nadjperge14, franke2015, meyer2016, Feldman2017}, or of the application of a magnetic field. The velocity renormalization exists independently of the appearance of a proximity-induced minigap $\varepsilon_{\rm gap}$ in the wire and the possible existence of Majorana bound states. A strong velocity normalization can exist even if $\varepsilon_{\rm gap}$ is much smaller than the bulk superconducting gap $\Delta$. Such a situation is markedly different from a conventional normal-metal--superconductor junction, where (in the absence of a magnetic field) a large spectral weight inside the superconductor coincides with the short-junction limit for which $\varepsilon_{\rm gap}$ and $\Delta$ are of comparable magnitude.
Our theoretical approach complements Refs.\ \onlinecite{Peng_2015,sarma15}, which used a large tunnel matrix element to model the strong coupling between normal metal and superconductor. Instead, we take a wavefunction approach, and characterize the normal-metal--superconductor interface in terms of its transparency. Then, the strongest coupling naturally appears for an ideal interface with unit transparency. For such an ideal interface, the strong coupling regime appears when $\Delta \ll \hbar v/W$, where $v$ is the Fermi velocity in the absence of coupling to the superconductor and $W$ the transverse dimension of the normal metal. Our method is similar to that of Ref.\ \onlinecite{akhmerov17}, which performs an analysis dedicated to the semiconductor-wire model, and extends previous work on the weak-coupling limit by Duckheim and one of the authors \cite{duckheim11}.
The wavefunction approach allows for an instructive semiclassical picture of the velocity renormalization. In this picture, the renormalization results from a delayed specular reflection of electrons in the normal metal at the superconductor interface, as shown in Fig.\ \ref{fig:setup}. At an ideal normal-metal superconductor interface, this reflection process consists of three stages: (1) An electron incident from the normal metal at angle $\theta$ is transmitted into the superconductor. (2) The transmitted electron is Andreev reflected as a hole. This hole cannot re-enter the spin-polarized normal metal because it has the wrong spin. Instead, it is specularly reflected at the superconductor--normal-metal interface. (3) Finally, the hole is in turn Andreev reflected into an electron, which is subsequently transmitted into the normal metal. Because of the finite penetration length into the superconductor, a delay $\sim 2\hbar/\Delta$ is accumulated in this reflection process. For a normal metal wire of thickness $W$ a distance $2 W \tan \theta$ is traveled between subsequent reflection events within a time $2 W/v \cos \theta$. Thus one obtains the effective velocity
\begin{equation}
v_x \approx \frac{\Delta}{\hbar} W \tan \theta
\label{eq:vrenorm}
\end{equation}
in the strong coupling regime $\Delta \ll \hbar v/W$. Note that it is the delay for the {\em normal} reflection that causes the velocity renormalization; the velocity renormalization does not involve processes that lead to Andreev reflection of majority electrons into majority holes or vice versa, which is the cause for the proximity-induced minigap in the normal metal. For a non-ideal interface a second reflection channel, direct specular reflection, is added in parallel to this delayed reflection process.
Spin-orbit coupling in the normal metal and/or the superconductor enables Andreev reflection of majority electrons into majority holes and a small minigap $\varepsilon_{\rm gap}$ opens up in the spectrum of the normal metal, with Majorana bound states forming at the wire ends. The localization length of the Majorana bound state is $\sim \hbar v_x/\varepsilon_{\rm gap}$, with $v_x$ the renormalized normal-state velocity. The strong renormalization of the velocity $v_x$ in the strong coupling limit leads to a strong renormalization of the product of $\varepsilon_{\rm gap}$ and the Majorana-state localization length. Upon comparing expressions for the weak and strong-coupling limits, we find that it is $\varepsilon_{\rm gap}$ that is renormalized in the strong coupling limit, while the Majorana localization length remains unrenormalized. This is in accordance with the Green function analysis of Refs.\ \onlinecite{Peng_2015, sarma15}.
The outline of this paper is as follows: In Sec.\ \ref{sec:model} we introduce the model of a spin-polarized metal proximity coupled to a superconductor. In Sec.\ \ref{sec:renormalization} we calculate the dispersion $\varepsilon(k_x)$ for propagating states in the normal wire in the absence of spin-orbit coupling. The renormalized velocity $v_x$ is obtained as $v_x = \hbar^{-1} |d\varepsilon/d k_x|$. Spin-orbit coupling is included in Sec.\ \ref{sec:SOcoupling}, in which we derive the properties of the emerging Majorana bound state for a highly transparent limit and compare the results to the limit of an opaque interface. We conclude in Sec. \ref{sec:conclusion}. To keep the analysis simple, the discussion in the main text is for a two dimensional model. We present results for a three-dimensional setup in the appendix. The results for the two and three-dimensional geometries are qualitatively the same.
\section{Model}\label{sec:model}
\begin{figure}
\includegraphics[width=1\columnwidth]{fig_model.pdf}\caption{\label{fig:setup}
Spin-polarized normal-metal wire of width $W$ (white) with one superconducting (grey, top) boundary and one insulating boundary (bottom). In the absence of spin-orbit coupling specular (normal) reflection at the normal-metal--superconductor interface involves a double Andreev reflection process in which an Andreev reflected minority hole is specularly back-reflected into the superconductor. The time delay incurred in this process slows down electrons propagating in the normal metal.}
\end{figure}
We consider a normal-metal (N) strip coupled to a superconductor (S). Coordinate axes are chosen such that the NS interface coincides with the $x$ axis, see Fig.\ \ref{fig:setup}, the superconductor occupies the half space $z > 0$, and the normal metal is in the region $-W < z < 0$. The $4 \times 4$ Bogoliubov-de Gennes (BdG) Hamiltonian reads
\begin{equation}\label{eq:hamiltonian_full}
\hat{\mathcal{H}} =
\begin{pmatrix}
H_0 & i \sigma_2 \Delta e^{i \phi} \theta(z) \\
-i \sigma_2 \Delta e^{-i \phi} \theta(z) & - H_0^*
\end{pmatrix}
\end{equation}
for a BdG spinor $(u_\uparrow,\,u_\downarrow,\,v_\uparrow,\,v_\downarrow)^{\rm T}$ comprising particle and hole wavefunctions. Here $\Delta e^{i \phi}$ is the superconducting order parameter and $\theta(z)$ the Heaviside step function. The $2 \times 2$ normal-state Hamiltonian $H_0$ is
\begin{equation}
H_0 = \frac{{\bf p}^2}{2m} + V(z) + \frac{\hbar^2 w}{m} \delta(z) + H_{\rm so},
\end{equation}
where $m$ is the electron mass, which we take to be the same in the N and S parts of the system, $V(z)$ is a spin-dependent potential, $(\hbar^2 w/m) \delta(z)$ a potential barrier at the NS interface, and $H_{\rm so}$ the spin-orbit interaction. For the spin-dependent potential $V(z)$, we take different expressions in the normal and superconducting parts of the system,
\begin{equation}
V(z) = - \frac{\hbar^2 k_{\rm S}^2}{2 m}
\end{equation}
when $z > 0$ and
\begin{equation}
V(z) = - \frac{\hbar^2}{2 m} \begin{pmatrix}
k_{\uparrow}^2 & 0 \\
0 & - \kappa_{\downarrow}^2
\end{pmatrix} + V_{\rm conf}(z)
\end{equation}
when $z < 0$. Here, $k_{\rm S}$ and $k_{\uparrow}$ are the Fermi wavenumbers of the superconductor and the majority spin band and $V_{\rm conf}(z)$ is a confining potential modeling the sample boundary at $z = -W$, $V_{\rm conf}(z) = 0$ for $z > -W$ and $V_{\rm conf}(z) = \infty$ for $z < -W$. Finally, the spin-orbit coupling is taken to be linear in momentum,
\begin{equation}
H_{\rm so} = \frac{\hbar}{2} \sum_{j}
\left[ {\bf p} {\bf \Omega}_j(z) \sigma_j + \sigma_j {\bf \Omega}_j(z) {\bf p}\right],
\end{equation}
where the spin-orbit coupling strength
\begin{equation}
{\bf \Omega}_j(z) = {\bf \Omega}_{{\rm S}j} \theta(z) +
{\bf \Omega}_{{\rm N}j} \theta(-z)
\end{equation}
is piecewise constant in the N and S regions. Spin-orbit coupling is assumed to be weak, so that it can be treated in first-order perturbation theory.
The normal-state majority-carrier transparency of the interface depends on the Fermi velocities $v_{\uparrow} = v = \hbar k_{\uparrow}/m$ and $v_{\rm S} = \hbar k_{\rm S}/m$, the strength $w$ of the surface $\delta$-function potential, and the momentum component $\hbar k_x$ parallel to the interface. In the absence of spin-orbit coupling the corresponding reflection and transmission amplitudes at the Fermi energy $\varepsilon = 0$ are \cite{Kupferschmidt_2011}
\begin{eqnarray} \label{eq:2d_tUp}
t_{\uparrow}(k_x) &=& \frac{2 \sqrt{k_{\uparrow z} k_{{\rm S}z}}}{2 i w + k_{\uparrow z} + k_{{\rm S}z}},
\\ \label{eq:2d_rUp}
r_{\uparrow}(k_x) &=& -1 + t_{\uparrow}(k_x) \sqrt{k_{\uparrow z}/k_{{\rm S}z}},
\\ \label{eq:2d_rpUp}
r'_{\uparrow}(k_x) &=& -1 + t_{\uparrow}(k_x) \sqrt{k_{{\rm S}z}/k_{\uparrow z}},
\end{eqnarray}
where
\begin{equation}
k_{\uparrow z} = \sqrt{k_{\uparrow}^2-k_x^2},\ \
k_{{\rm S}z} = \sqrt{k_{\rm S}^2 - k_x^2}.
\end{equation}
(The amplitudes $r_{\uparrow}$ and $r'_{\uparrow}$ describe reflection of majority electrons coming from the N and S parts of the system, respectively.) Minority spins coming from $z > 0$ are reflected with reflection amplitude
\begin{eqnarray}\label{eq:2d_rpDown}
r'_{\downarrow}(k_x) &=& e^{i \varphi_{\downarrow}(k_x)} \nonumber \\
&=& \frac{k_{{\rm S}z} - i \kappa_{\downarrow z} - 2 i w}{k_{{\rm S}z} + i \kappa_{\downarrow z} + 2 i w},
\end{eqnarray}
where $\kappa_{\downarrow z} = \sqrt{\kappa_{\downarrow}^2 + k_x^2}$ and we neglect terms exponentially suppressed in $\kappa_{\downarrow z} W$.
This model describes semiconductor wires in a large Zeeman field as well as half-metallic (ferromagnetic) wires, both coupled to a superconductor. In the former case spin-orbit coupling is typically assumed to exist inside the semiconductor, but not in the superconductor \cite{Lutchyn2010,Oreg2010}; in the latter case, spin-orbit coupling is usually taken to be in the superconductor, but not in the half-metallic wire \cite{duckheim11,chung11}.
In the appendix, we consider the corresponding three dimensional model, consisting of a cylindrical spin-polarized normal metal surrounded by a superconductor.
\section{Renormalization of the Fermi velocity} \label{sec:renormalization}
We first consider the system under consideration in the presence of superconductivity, but without spin-orbit coupling. The superconducting gap confines carriers with excitation energy $|\varepsilon| < \Delta$ to the normal region, so that the N region effectively becomes a conducting wire of width $W$.
Without spin-orbit coupling, reflections at the NS interface are purely normal; Andreev reflections are ruled out because they would require a spin flip process. Nevertheless, the presence of the superconductor can lead to a strong renormalization of the carrier velocity. To see this explicitly, we construct the wavefunction of a majority electron at excitation energy $\varepsilon$ and momentum $\hbar k_x$ parallel to the interface,
\begin{equation}
u_{\uparrow}(x,z) \propto e^{i k_x x}
\left[ e^{i k_z(k_x,\varepsilon) z} + r_{\rm ee}(k_x,\varepsilon) e^{-i k_z(k_x,\varepsilon) z} \right]. \label{eq:modes_unnormalized_e}
\end{equation}
Here
\begin{equation}
k_z(k_x,\varepsilon) = \sqrt{k_{\uparrow}^2 - k_x^2 + 2 m \varepsilon/\hbar^2}
\end{equation}
and $r_{\rm ee}(k_x,\varepsilon)$ is the reflection amplitude in the presence of the superconductor. In terms of the normal-state reflection and transmission amplitudes of the NS interface the reflection amplitude $r_{\rm ee}(k_x,\varepsilon)$ reads (in the Andreev approximation $\hbar^2 k_z^2/2m \gg \Delta$)
\begin{align}
r_{\rm ee}(k_x,\varepsilon) &=
r_{\uparrow}(k_x) + \frac{t_{\uparrow}(k_x)^2 e^{-2 i \eta(\varepsilon) - i \varphi_{\downarrow}(k_x)}}{1 - r'_{\uparrow}(k_x) e^{-2 i \eta(\varepsilon) - i \varphi_{\downarrow}(k_x)}}
\nonumber \\ &=
\frac{k_{\uparrow z} - 2 i w - i k_{{\rm S}z} \tan(\eta + \varphi_{\downarrow}/2)}{k_{\uparrow z} + 2 i w + i k_{{\rm S}z} \tan(\eta + \varphi_{\downarrow}/2)}
, \label{eq:ree_2d}
\end{align}
where
\begin{equation}
\eta(\varepsilon) = \arccos(\varepsilon/\Delta).
\end{equation}
This result can be easily understood by considering the different paths a majority electron incident on the NS interface from $z < 0$ can take: Direct normal reflection with amplitude $r_{\uparrow}$ or entering the superconductor with transmission amplitude $t_{\uparrow}$, Andreev reflection into a minority hole, normal backreflection of the hole into S with amplitude $r'^*_{\downarrow}$, finally followed by a second Andreev reflection into a majority electron and transmission into the normal metal. The denominator in Eq.\ (\ref{eq:ree_2d}) describes higher-order processes involving multiple double Andreev reflections. We have assumed $\kappa_{\downarrow} W \gg 1$, so that the minority wavefunction component $u_{\downarrow}$ decays sufficiently fast away from the NS interface and it is sufficient to restrict ourselves to the majority wavefunction component $u_{\uparrow}$.
\begin{figure}
\includegraphics[width=1\columnwidth]{fig_dispersion_2d.pdf}
\caption{\label{fig:dispersion} (Color online.)
Subgap dispersion relation $\varepsilon(k_x)$ for a spin-polarized normal wire attached to a superconductor. Only electron-like solutions are shown, hole-like ones are obtained by mirroring the spectrum vertically such that $\varepsilon\rightarrow -\varepsilon$. The wire width satisfies $k_{\uparrow} W/\pi = 1.2$, corresponding to one propagating mode at the Fermi level $\varepsilon = 0$ in an isolated wire. The solid lines are obtained by numerically solving Eq. (\ref{eq:2d_non_linear_ev}). The left panel shows the dispersion relation for $k_{\uparrow} = k_{\rm S}$, $w = 0$, corresponding to a fully transparent NS interface; the right panel has $w m/\hbar k_{\uparrow} = 1$, corresponding to an interface with transmission probability $|t_{\uparrow}|^2 = 1/2$ for perpendicular incidence. The dashed lines show Eqs. (\ref{eq:2d_dispersion_transparent}) (left panel) and (\ref{eq:vxasymp}) (right panel), while the dotted lines show the dispersion for a vanishing interface transparency. The magnitude of the superconducting gap is given by $(\hbar \pi/W)^2/2 m \Delta =10$, well within the validity range of the Andreev approximation. We further set $\kappa_{\rm F \downarrow}/k_{\uparrow} = 2$. }
\end{figure}
The dispersion relation $\varepsilon(k_x)$ follows by imposing that $u_{\uparrow}(x,-W) = 0$, which leads to
\begin{equation}\label{eq:2d_non_linear_ev}
1 = -e^{2i k_z W}r_{\rm ee}(k_z,\,\varepsilon).
\end{equation}
For a weakly coupled superconductor one has $r_{\uparrow} = r'_{\uparrow} \approx -1$ and $|t_{\uparrow}| \ll 1$, and Eq.\ (\ref{eq:2d_non_linear_ev}) reproduces the standard quantization rule $k_z = n \pi/W$, $n=1,2,\ldots$, and a quadratic dispersion
\begin{equation}
\varepsilon = \frac{\hbar^2}{2m} \left(
k_x^2+ \frac{n^2 \pi^2}{W^2} -k_{\uparrow}^2
\right).
\label{eq:free}
\end{equation}
In the opposite limit of an ideal interface with $t_{\uparrow} = 1$ and $r_{\uparrow} = r'_{\uparrow} = 0$, one finds
\begin{equation}
2 k_z(\varepsilon) W = 2 \eta(\varepsilon) + \varphi_{\downarrow}(k_x) + (2 n+1) \pi.
\label{eq:kzdisp}
\end{equation}
If we restrict ourselves to the single-mode regime $1 \lesssim k_{\uparrow} W/\pi \lesssim 2$, the Andreev approximation implies that $(\hbar \pi/W)^2/2m \gg \Delta$, which allows us to neglect the energy dependence on the l.h.s.\ of Eq.\ (\ref{eq:kzdisp}) and obtain the dispersion
\begin{equation}
\varepsilon = \pm \Delta \sin\left[\frac{\varphi_{\downarrow}(k_x)}{2} - W \sqrt{k_{\uparrow}^2 - k_x^2}\right]. \label{eq:2d_dispersion_transparent}
\end{equation}
The left panel of Fig.\ \ref{fig:dispersion} shows the dispersion for $k_{\uparrow} W/\pi = 1.2$ for an ideal interface, together with the approximate result (\ref{eq:2d_dispersion_transparent}) and the dispersion (\ref{eq:free}) of the isolated wire.
Figure\ \ref{fig:dispersion} clearly shows that the coupling to the superconductor leads to significantly flatter $\varepsilon$ vs.\ $k_x$ curves near $\varepsilon = 0$, indicating a strongly renormalized Fermi velocity $v_x = \hbar^{-1}|d\varepsilon/d k_x|$. The strong renormalization of the velocity also follows from the approximate dispersion (\ref{eq:2d_dispersion_transparent}) for an ideal interface,
\begin{eqnarray}\label{eq:2d_renorm_ana}
v_x &=& \frac{1}{\hbar}\sqrt{\Delta^2 - \varepsilon^2} \frac{k_x W}{k_{\uparrow z}}
\left( 1 - \frac{1}{\kappa_{\downarrow z}W} \right). \label{eq:zeroth_order_velocity}
\end{eqnarray}
Although we dropped terms exponentially suppressed in $\kappa_{\downarrow z}W$ in Eq. \ref{eq:2d_rpDown}, we keep the term including $\kappa_{\downarrow z}W$ as it is suppressed by a power law only.
Equation \eqref{eq:2d_renorm_ana} gives an effective velocity $v_x$ that is suppressed by a factor $\Delta/\varepsilon_{\rm kin}$ compared to the velocity $\hbar k_x/m$ of an isolated normal wire. Here, $\varepsilon_{\rm kin} = \hbar^2 k_{\uparrow}^2/2m$ is the normal-state kinetic energy. This suppression is consistent with the semiclassical estimate (\ref{eq:vrenorm}).
The renormalized velocity is shown in Fig. \ref{fig:renorm_vF_vs_T} as a function of interface transparency for the same parameter choice as in Fig.\ \ref{fig:dispersion}. Starting from the value $v_x = \hbar k_x/m$ of an isolated wire, the velocity decreases monotonically as a function of interface transparency $|t_{\uparrow}|$, reaching the much smaller value given by Eq. (\ref{eq:zeroth_order_velocity}) at $|t_{\uparrow}|^2 = 1$.
\begin{figure}
\includegraphics[width=1\columnwidth]{fig_velocity_2d.pdf}
\caption{\label{fig:renorm_vF_vs_T}Renormalized velocity as a function of interface transparency $|t_{\uparrow}|^2$. The velocity is normalized to $v_x^0 = \hbar k_x/m = v \sin \theta$. The interface barrier is introduced by increasing $w$ while matching $k_{\uparrow}=k_{\rm S}$ (bright, orange line) and by increasing $k_{\rm S}$ at fixed $w =0$ (dark, blue line). The solid lines are obtained by numerically solving Eq. (\ref{eq:2d_non_linear_ev}). All other parameters are the same as in Fig. \ref{fig:dispersion}. The dashed lines show the $|t_{\uparrow}|^2=1$ approximation of Eq. (\ref{eq:zeroth_order_velocity}) and the small-transparency approximation of Eq.\ (\ref{eq:vxasymp}).}
\end{figure}
Although the velocity renormalization is strongest for a fully transparent interface, we emphasize that the renormalization exists for arbitrary transparency of the interface, provided $\Delta$ is small enough, so that a double Andreev reflection from the superconductor takes a sufficiently long time. In fact, the limit of a weakly transparent interface allows for an explicit solution for $v_x$, as we now show. The limit of a small junction transparency is realized if $k_{{\rm S}z} \gg k_z$ or $|w| \gg k_z$. In this limit one finds
\begin{equation}
r_{\rm ee} = - \frac{4 w^2 + k_{{\rm S}z}^2 + i k_z (2 w + \varepsilon k_{{\rm S}z}/\Delta)}{4 w^2 + k_{{\rm S}z}^2 - i k_z (2 w + \varepsilon k_{{\rm S}z}/\Delta)},
\label{eq:ree_limit}
\end{equation}
up to corrections that are small in $|\varepsilon|/\Delta$, in $k_z/|w|$, or in $k_z/k_{{\rm S}z}$. For $|\varepsilon| \ll \Delta$, the solution of Eq.\ (\ref{eq:2d_non_linear_ev}) is
\begin{equation}\label{eq:2d_kz_lowT_quantization}
k_z = \frac{\pi}{W} - \frac{\pi (2 w + \varepsilon k_{{\rm S}z}/\Delta)}{W^2(4 w^2 + k_{{\rm S}z}^2)},
\end{equation}
which gives the equation
\begin{equation}\label{eq:2d_eps_low_transparency}
\varepsilon = \frac{\hbar^2}{2 m}
\left( k_x^2 + \frac{\pi^2}{W^2} - \frac{2 \pi^2 (2 w + \varepsilon k_{{\rm S}z}/\Delta)}{W^3(4 w^2 + k_{{\rm S}z}^2)} - k_{\uparrow}^2 \right),
\end{equation}
from which the dispersion relation can be obtained. (The $\varepsilon$-dependence of $k_{{\rm S}z}$ can be neglected in the limit of small interface transparency because either $k_{\rm S} \gg k_{\uparrow}$, in which case $k_{{\rm S}z} = k_{\rm S}$ up to small corrections, or $|w| \gg k_{{\rm S}z}$, in which case $k_{{\rm S}z}$ drops out of the equation.) Differentiating with respect to $k_x$ gives the velocity
\begin{equation}
v_x = \frac{v \sin \theta}{1 + |t_{\uparrow}|^2 \xi_{\rm N}/4 W},
\label{eq:vxasymp}
\end{equation}
at $\varepsilon = 0$, where $\sin \theta = k_x/k_{\uparrow}$ and $\xi_{\rm N} = \hbar^2 k_z/m \Delta = \hbar^2 \pi/m W \Delta$ is the transverse coherence length in the normal metal. The strong velocity renormalization sets in when $\xi_{\rm N} |t_{\uparrow}|^2 \gg W$. The small-transparency approximation for the dispersion $\varepsilon(k_x)$ and the velocity $v_x$ is illustrated in the right panel of Fig.\ \ref{fig:dispersion} and in Fig.\ \ref{fig:renorm_vF_vs_T}, respectively, showing that the small-transparency approximation remains useful for interface transparencies $|t_{\uparrow}|^2 \lesssim 0.5$.
From a purely classical point of view, the denominator in Eq.\ \eqref{eq:vxasymp} is surprising. To understand this, consider the process shown in Fig.\ \ref{fig:setup} for a low transparency $|t_\uparrow|^2$. From a classical point of view, the electron will spend a time $T_{\rm N} \sim W/v_\uparrow |t_\uparrow|^2$ in the normal metal before being transmitted through the interface and a time $T_{\rm S} \sim \xi/v_{\rm S} |t_\uparrow|^2$ in the superconducting region. Here, we define the velocities $v_\uparrow = k_\uparrow/m$ and $v_{\rm S} = k_{\rm S} /m$ and neglect the angle $\theta$. In the superconducting region, the distance traveled along $x$ is zero due to the zero-net displacement processes shown in Fig. \ref{fig:setup}, and thus the velocity is expected to be
\begin{equation}\label{eq:vxasymp_cl}
v_x^{\rm (cl)} \sim \frac{v_\uparrow T_\uparrow}{T_\uparrow + c T_{\rm S}} \sim \frac{v_\uparrow}{1 + c \xi_{\rm N}/W},
\end{equation}
with some constant numerical factor $c$, and the ratio $v_\uparrow/v_{\rm S}$ has been absorbed into $\xi_{\rm N}$. Eq. \eqref{eq:vxasymp_cl} is clearly inconsistent with Eq. \eqref{eq:vxasymp}. The missing factor $|t_\uparrow|^2$ can be traced back to the coherent scattering in the superconductor: During a single cycle of the double Andreev reflection shown in Fig. \ref{fig:cyl_setup}, a phase factor $e^{i \alpha} = e^{-2 i \eta(\varepsilon)} r'_\uparrow (r'_\downarrow)^*$ is picked up. For $|\varepsilon| \ll \Delta$ and a low transparency, this phase factor becomes $e^{i \alpha} = -1 + O(|t|^2)$. Hence multiple double Andreev reflections interfere destructively up to corrections of $O(|t_\uparrow|^2)$ and the time $T_{\rm S}$ is effectively lowered by a factor $|t_\uparrow|^2$, which explains the discrepancy between the classical and semi-classical results in \eqref{eq:vxasymp} and \eqref{eq:vxasymp_cl}.
As shown in the appendix, qualitatively the same results are obtained for a three dimensional setup.
\section{Spin-orbit coupling and Majorana bound states}\label{sec:SOcoupling}
Spin-orbit coupling in the superconductor allows for spin flips and thereby enables Andreev reflections of majority spin electrons into majority spin holes and vice versa. This induces a $p$-wave minigap $\varepsilon_{\rm gap}$ in the excitation spectrum of the normal wire and zero-energy Majorana bound states form at its ends. This section considers both of these effects and relates the localization length $l_{\rm maj}$ of the Majorana bound states and the minigap $\varepsilon_{\rm gap}$ to the renormalization of the Fermi velocity calculated in the previous section. The calculation extends that of Ref.\ \onlinecite{duckheim11}, which considered the same problem in the limit of an opaque NS interface, for which there is no renormalization of the Fermi velocity.
We assume that spin-orbit coupling is sufficiently weak so that it can be treated in first-order perturbation theory. Correspondingly, the probability for Andreev reflection off the normal-metal--superconductor interface is small and the induced minigap $\varepsilon_{\rm gap}$ in the spectrum of the normal wire much smaller than the bulk superconducting gap $\Delta$. For that reason, we neglect corrections to the scattering amplitudes of order $\varepsilon/\Delta$ in the calculations below.
The starting point of the calculation is an expression for the propagating states in the normal wire in the absence of spin-orbit coupling, normalized to unit flux in the $x$ direction. To keep the notation simple, we restrict to the regime in which there is one propagating mode in the normal-metal wire in the absence of spin-orbit induced Andreev reflection. This mode has transverse wavevector $k_z$, which is determined by the quantization condition (\ref{eq:2d_non_linear_ev}). The electron-like scattering states $|\psi_{{\rm e},\pm}\rangle$ propagating in the positive ($+$) or negative ($-$) $x$ direction have the wavefunction components \cite{Kupferschmidt_2011}
\begin{align}
u_{\uparrow,\pm}(\vr) =&\, e^{\pm i k_x(\varepsilon) x} \frac{e^{i k_z z} + r_{\rm ee} e^{-i k_z z}}{\sqrt{{\cal N} v_x}} \\
v_{\downarrow,\pm}(\vr) =&\, -e^{\pm i k_x(\varepsilon) x} \frac{i t_\uparrow \tau_{\downarrow} e^{\kappa_{\downarrow z} z} e^{- i \phi}}{(r'_{\downarrow} + r'_\uparrow)\sqrt{\mathcal{N} v_x}},
\end{align}
in the normal region $-W < z < 0$, where
\begin{equation}
k_x(\varepsilon) = \sqrt{k_{\uparrow}^2-k_{\uparrow z}^2} + \frac{\varepsilon}{\hbar v_x},
\end{equation}
with the velocity $v_x$ taken from the calculation of the dispersion in Sec.\ \ref{sec:renormalization}, and
\begin{align}\label{eq:2d_tauDown}
\tau_{\downarrow} &= \frac{2 \sqrt{k_{{\rm S}z} k_{\uparrow z}}}{ k_{{\rm S}z} + i \kappa_{\downarrow z} + 2 i w}.
\end{align}
Since we are interested in energies $|\varepsilon| \ll \Delta$, we only need to retain the energy dependence in the exponential factors, see the discussion in the previous paragraph. As before, we assume that $\kappa_{\downarrow z} W \gg 1$ so that no hard-wall boundary condition needs to be applied at $z = -W$ for the minority component $v_{\downarrow,\pm}(\vr)$. In the superconducting region, the nonzero wavefunction components are \cite{Kupferschmidt_2011}
\begin{align}
u_{\uparrow,\pm}(\vr) =& \frac{t_{\uparrow} e^{\pm i k_x(\varepsilon) x - z/\xi}(e^{i k_{{\rm S}z} z} - e^{-i k_{{\rm S}z} z - i \varphi_{\downarrow}}) }{(1 + r'_{\uparrow} e^{- i \varphi_{\downarrow}}) \sqrt{{\cal N} k_{{\rm S}z} v_x/k_{\uparrow z}}}\nonumber
, \\
v_{\downarrow,\pm}(\vr) =& - \frac{i t_{\uparrow} e^{\pm i k_x(\varepsilon) x -z/\xi - i \phi} ( e^{i k_{{\rm S}z} z} + e^{-i k_{{\rm S}z} z - i \varphi_{\downarrow}} )}{(1 + r'_{\uparrow} e^{- i \varphi_{\downarrow}}) \sqrt{{\cal N} k_{{\rm S}z} v_x/k_{\uparrow z}}}.
\label{eq:usuper}
\end{align}
Here
\begin{eqnarray}
k_{{\rm S}z} &=& \sqrt{k_{\rm S}^2 - k_{\uparrow}^2 + k_z^2},\\
\xi &=& \frac{\hbar^2 k_{{\rm S}z}}{m \Delta}, \\
\mathcal{N} &=& 2 W + \frac{\mbox{Im}\, r_{\rm ee}}{k_z} +
\frac{2 \xi_{\rm N} |t_{\uparrow}|^2}{|r'_{\downarrow} + r'_{\uparrow}|^2},
\label{eq:N}
\end{eqnarray}
where the transverse coherence length in the normal metal $\xi_{\rm N}$ was defined below Eq.\ (\ref{eq:vxasymp}).
The factors $\sqrt{k_{{\rm S}z}/k_{\uparrow z}}$ in the denominators of Eq.\ (\ref{eq:usuper}) are a consequence of current conservation at the normal-metal--superconductor interface.
Similarly, the nonzero wavefunction components of the hole-like scattering states $|\psi_{{\rm h},\pm}\rangle$ are
\begin{align}
v_{\uparrow,\pm}(\vr) =&\, \frac{e^{\mp i k_x(-\varepsilon) x} (e^{-i k_z z} + r_{\rm ee}^* e^{i k_z z})}{\sqrt{{\cal N} v_x}}, \nonumber \\
u_{\downarrow,\pm}(\vr) =&\, \frac{i t_\uparrow^* \tau_{\downarrow}^* e^{\mp i k_x(-\varepsilon) x} e^{\kappa_{\downarrow z} z}e^{i \phi}}{(r_{\downarrow}'^* + r_\uparrow'^*)\sqrt{\mathcal{N} v_x}}
\end{align}
in the normal region $-W < z < 0$. Likewise, the corresponding wavefunction components in the superconducting region follow from Eqs.\ (\ref{eq:usuper}) upon exchanging electron and hole components, complex conjugating, and sending $\varepsilon \to -\varepsilon$.
To calculate how spin-orbit coupling modifies these scattering states, we now consider a system for which spin-orbit coupling is non-zero in a segment $0 < x < \delta L$ only. For small enough $\delta L$, spin-orbit coupling induces a backscattering amplitude in the scattering state which is linear in $\delta L$ for small enough $\delta L$. Calculating the linear-in-$\delta L$ scattering amplitudes in perturbation theory in $H_{\rm so}$ as in Ref.\ \onlinecite{duckheim11}, we find for the electron-to-hole amplitude for electrons incident from the left ({\em i.e.}, initially moving in the positive $x$ direction)
\begin{equation}
\rho_{\rm he} \delta L =
- \frac{i}{\hbar} \left\langle \psi_{{\rm h},-} \left|
\delta \hat{\mathcal{H}}_{\rm so}
\right| \psi_{{\rm e}, +} \right\rangle,
\label{eq:matrix_element}
\end{equation}
where
\begin{equation}
\delta \hat{\mathcal{H}}_{\rm so} =
\frac{1}{2} \left\{ \begin{pmatrix}
H_{\rm so} & 0 \\
0 & -H_{\rm so}^*
\end{pmatrix}, \Theta_{\delta L}(x) \right\},
\end{equation}
with $\{ \cdot, \cdot \}$ the anticommutator and $\Theta_{\delta L}(x) = 1$ for $0 < x < \delta L$ and $\Theta_{\delta L}(x)= 0$ otherwise. This gives
\begin{align}\label{eq:rho_he}
\rho_{\rm he} =&\,
- \frac{i t_{\uparrow}^2 \hbar k_x k_{\uparrow z} (\Omega_{{\rm S}xx} + i \Omega_{{\rm S}yx}) e^{-i \phi} (1 + r_{\downarrow}'^2)}{{\cal N} v_x k_{{\rm S}z}^2 (r'_{\downarrow} + r'_{\uparrow})^2}
\nonumber
\\ &\, \mbox{}
-
\frac{
2 \hbar k_x (\Omega_{{\rm N}xx} + i \Omega_{{\rm N}yx})
t_{\uparrow} \tau_{\downarrow} e^{-i \phi}
}{
{\cal N} v_{x} (r'_{\uparrow} + r'_{\downarrow})
(\kappa_{\downarrow z}^2 + k_{\uparrow z}^2)
} \nonumber
\\
&\times
\left[
\kappa_{\downarrow z} (1 + r_{\rm ee}) - i k_{\uparrow z} (1 - r_{\rm ee})
\right]
.
\end{align}
The remaining amplitudes are readily obtained by symmetry arguments. The Andreev reflection amplitude $\rho_{\rm he}'$ for incoming electron moving in the negative $x$ direction is obtained from Eq.\ (\ref{eq:rho_he}) by sending $k_x \to -k_x$; The amplitudes for incoming holes are obtained by complex conjugation, $\rho_{\rm eh} = \rho_{\rm he}^*$ and $\rho_{\rm eh}' = \rho_{\rm he}'^*$. Although the wavefunction penetrates a distance $\sim \xi$ into the superconductor, the spatial integrals contributing to the matrix element (\ref{eq:matrix_element}) have support only within a few wavelengths of the interface \cite{duckheim11}. This is the reason why the first term in Eq.\ (\ref{eq:rho_he}) does not involve a factor $\xi$ in the numerator.
The Andreev reflection amplitude $r_{\rm he}(L)$ for a segment of length $L$ can obtained by solving the differential relation \cite{duckheim11}
\begin{equation}\label{eq:rhe_diff_eq}
\frac{d r_{\rm he}}{d L} = \frac{2 i \varepsilon}{\hbar v_x} + \rho_{\rm he} + \rho_{\rm he}'^* r_{\rm he}^2,
\end{equation}
which is obtained by summing the scattering amplitudes from an infinitesimal slice $0< x<\delta L$ and a subsequent segment $\delta L < x < L$. Integrating Eq. \eqref{eq:rhe_diff_eq} gives the non-perturbative amplitudes
\begin{equation}\label{eq:r_he_eff}
r_{\rm he}(L) = \frac{\rho_{\rm he} \sinh q L}{q \cosh q L - i (\varepsilon/\hbar v_x) \sinh q L}
\end{equation}
and
\begin{equation}\label{eq:r_eh_eff}
r_{\rm eh}(L) = \frac{\rho_{\rm e h} \sinh q L}{ \cosh q L - i (\varepsilon/\hbar v_x) \sinh q L},
\end{equation}
where
\begin{equation}
q = \sqrt{|\rho_{\rm he}|^2 - (\varepsilon/\hbar v_x)^2}. \label{eq:q}
\end{equation}
For energies $|\varepsilon| < \varepsilon_{\rm gap}$, with
\begin{equation}
\varepsilon_{\rm gap} = \hbar v_x |\rho_{\rm he}|
\label{eq:egap}
\end{equation}
one has $|r_{\rm he}| \to 1$ in the limit $L \to \infty$. This is the hallmark of a Majorana bound state \cite{law09, flensberg10}, with $\varepsilon_{\rm gap}$ being the proximity-induced minigap \cite{duckheim11}.
With the help of Eq.\ (\ref{eq:q}) one readily identifies $l_{\rm maj} = |\rho_{\rm he}|^{-1}$ as the localization length of the zero-energy Majorana bound state. The strong renormalization of the velocity $v_x$ for a transparent interface enters the denominator of Eq.\ (\ref{eq:rho_he}). However, the fact that in the strong coupling limit $\Delta \ll \hbar v/W$ most of the spectral weight is concentrated in the superconductor also enters into the expression for $\rho_{\rm he}$, through the normalization factor ${\cal N}$. Interestingly, the superconducting gap $\Delta$ drops out from the product ${\cal N} v_x$, causing no additional smallness of the localization length. Nevertheless, the velocity renormalization does affect the product of the minigap and the localization length, in agreement with the analysis of Ref.\ \cite{Peng_2015,sarma15}.
To assess the dependence on interface transparency, it is instructive to evaluate the expressions for the induced gap and the localization length of the Majorana state for a weakly transmitting barrier. Taking the imaginary part of $r_{\rm ee}$ from Eq.\ (\ref{eq:ree_limit}), one concludes that the second term in Eq.\ (\ref{eq:N}) does not contribute to the normalization factor in that limit. Since $|r'_{\downarrow} + r'_{\uparrow}| \simeq 2$ for a weakly transmitting barrier, one finds
\begin{equation}\label{eq:2d_normalization_constant}
{\cal N} = 2 W + \frac{|t_{\uparrow}|^2 \xi_{\rm N}}{2}.
\end{equation}
To further simplify the expressions for $\rho_{\rm he}$, we consider two special cases: (i) Equal Fermi velocities in the normal metal and the superconductor $k_{\rm S} = k_{\uparrow}$, and $|w| \gg k_{\uparrow}$ to ensure a non-transparent interface. (ii) $k_{\rm S} \gg k_{\uparrow}$ with a barrier-free interface $w=0$. Here, the small transparency is the result of a large Fermi velocity mismatch between the superconductor and the normal metal.
In both limits one has $1 + r_{\downarrow}'^2 = 2$, although this equality does not hold generally for non-transparent interfaces. Finally, for the factor $1 + r_{\rm ee}$ we find
\begin{equation}
1 + r_{\rm ee} = t_{\uparrow}
\end{equation}
in the former limit, and
\begin{equation}
1 + r_{\rm ee} = - \frac{i t_{\uparrow}^2 \kappa_z}{2 k_{{\rm S}z}}
\end{equation}
in the latter limit (where we assumed that $\kappa_{\downarrow} \ll k_{\rm S}$). For the amplitude whose magnitude is equal to the inverse Majorana localization length, we then find
\begin{align}
\rho_{\rm he} =&\, i e^{-i \phi} m |t_{\uparrow}|^2 \label{eq:rhohelow1}
\\ & \mbox{} \times
\left( \frac{\pi (\Omega_{{\rm N}xx} + i \Omega_{{\rm N}yx})}{\pi^2 + \kappa_{\downarrow z}^2 W^2}
-
\frac{\Omega_{{\rm S}xx} + i \Omega_{{\rm S}yx}}{4 \pi} \right)
\nonumber
\end{align}
for a weakly transmitting interface with $k_{\rm S} = k_{\uparrow}$ and $|w| \gg k_{\uparrow}$, and
\begin{align}
\rho_{\rm he} =&\, i e^{-i \phi} m |t_{\uparrow}|^2
\label{eq:rhohelow2}
\\ & \mbox{} \times
\left( \frac{\pi (\Omega_{{\rm N}xx} + i \Omega_{{\rm N}yx})}
{\pi^2 + \kappa_{\downarrow z}^2 W^2}
-
\frac{|t_{\uparrow}|^4
(\Omega_{{\rm S}xx} + i \Omega_{{\rm S}yx}) }
{64 \pi}\right)
\nonumber
\end{align}
in limit of a weakly transmitting interface with $w=0$ and $k_{\rm S} \gg k_{\uparrow}$. Expressions for the induced minigap $\varepsilon_{\rm gap} = \hbar v_x |\rho_{\rm he}|$ follow immediately upon multiplication with the renormalized velocity $v_x$ in Eq.\ (\ref{eq:vxasymp}), restricted to the small-transparency limit.
\begin{figure}
\includegraphics[width=1\columnwidth]{fig_decay.pdf}
\caption{\label{fig:loc-length} Inverse localization length $|\rho_{\rm he}| = 1/l_{\rm maj} $ vs.\ interface transparency $|t_{\uparrow}|^2$ for an interface with matched Fermi velocities $k_{\rm S} = k_{\uparrow}$ (top row) and with zero potential barrier $w=0$ (bottom row), with spin-orbit coupling in the superconductor (left column) and in the normal metal (right column). The dashed curves show the weak-transparency results (\ref{eq:rhohelow1}) and (\ref{eq:rhohelow2}). The remaining parameters are $k_{\uparrow} W = 1.2 \pi$, $(\hbar \pi/W)^2/2 m \Delta = 20$ and $\kappa_{\downarrow} = 2 k_{\uparrow}$. We defined $\Omega_{{\rm S} x}^2 \equiv {\Omega_{{\rm S}xx}^2 + \Omega_{{\rm S}yx}^2}$ and $\Omega_{{\rm N} x}^2 \equiv {\Omega_{{\rm N}xx}^2 + \Omega_{{\rm N}yx}^2}$.}
\end{figure}
Figure \ref{fig:loc-length} shows the inverse localization length $|\rho_{\rm he}|$ as a function of barrier transparency for the two limits considered above, as well as the full expression \eqref{eq:rho_he} (solid line). For the latter, the velocity and the wave numbers are obtained by numerically solving Eq. \eqref{eq:2d_non_linear_ev}. The figures confirm that the low-transparency expressions in Eqs. (\ref{eq:rhohelow1}) and (\ref{eq:rhohelow2}) are excellent quantitative approximations for transparencies $|t_{\uparrow}|^2 \lesssim 0.5$. However, for transparencies close to unity, spin-orbit coupling in the superconductor, and $w=0$, we observe a sharp closing of the minigap. This is an interference effect which can be traced back to the factor $1 + r_{\downarrow}'^2 = 2 e^{i \varphi_{\downarrow}} \cos \varphi_{\downarrow}$ in Eq.\ (\ref{eq:rho_he}). For $w=0$ and with $\kappa_{\downarrow} > k_{\uparrow}$ the minority reflection phase $\varphi_{\downarrow}$ passes through $\pi/2$ close to unit transparency, see Eq.\ (\ref{eq:2d_rpDown}). A similar effect appears upon approaching perfect transparency by varying $w$ at $k_{\uparrow} = k_{\rm S}$ for negative $w$ (data not shown).
Figure \ref{fig:minigap} shows the induced minigap $\varepsilon_{\rm gap}$ as a function of barrier transparency. Here the transition between the strong-coupling and weak-coupling limits at $|t_{\uparrow}|^2 \sim W/\xi_{{\rm N}}$ can be clearly seen. The weak-coupling limit agrees with the theory of Ref.\ \onlinecite{duckheim11}; the velocity renormalization appear in the strong-coupling limit $|t_{\uparrow}|^2 \gtrsim W/\xi_{\rm N}$.
\begin{figure}
\includegraphics[width=1\columnwidth]{fig_gap.pdf}
\caption{\label{fig:minigap} Minigap versus transparency for the same conditions as in Fig. \ref{fig:loc-length}. The grey curves show the power laws corresponding to the weak-coupling limit $|t_{\uparrow}|^2 \ll W/\xi_{\rm N}$ and the strong-coupling limit (at weak transparency) $W/\xi_{\rm N} \ll |t_{\uparrow}|^2 \ll 1$. The dashed curve is obtained using the weak-transparency results (\ref{eq:rhohelow1}) and (\ref{eq:rhohelow2}) for the inverse localization length $\rho_{\rm he}$. The parameter values are $k_{\uparrow} W = 1.2 \pi$, $(\hbar \pi/W)^2/2 m \Delta = 200 \pi$ and $\kappa_{\downarrow} = 2 k_{\uparrow}$.}
\end{figure}
\section{Conclusions}\label{sec:conclusion}
In this work, we employed a semiclassical scattering approach to study a spin-polarized normal-metal quantum wire which is strongly coupled to a spin-orbit-coupled superconductor. This model for a topological superconductor was originally introduced and studied in the limit of an opaque interface between wire and superconductor \cite{duckheim11}. Here, we have shown that the properties of its topological phase are strongly renormalized for a highly transparent interface and provide a semiclassical interpretation. Following previous work on related systems \cite{Peng_2015, sarma15, akhmerov17, stanescu17}, we trace the renormalization to the lowering of the Fermi velocity which we interpret in terms of scattering processes which yield zero net-displacement along the wire as well as a modified spin-flip scattering rate $\rho_{\rm he}$. Specifically, a transparent interface greatly increases both the topological minigap and the localization length of the emerging Majorana bound states as compared to an opaque one. Additionally we find that, while the low transparency prediction for the localization length stays accurate even for transparencies $\lesssim 0.5$, the velocity as well as the minigap are strongly renormalized towards small values compared to the low-transparency prediction.
It is interesting to compare our semiclassical approach to the previously employed Green function approach \cite{Peng_2015}. In this approach, one studies the propagation of subgap excitations in the wire, accounting for the coupling to the superconductor through the corresponding self energy
\begin{equation}\label{eq:self_energy}
\Sigma({\bf k},\omega) = -\Gamma \frac{\omega + \Delta\tau_x}{\sqrt{\Delta^2-\omega^2}}.
\end{equation}
Here, $\Gamma$ quantifies the coupling between wire and superconductor (with gap $\Delta$) in terms of the decay rate of subgap excitations of the wire (with energy $\omega$) into the superconductor in the normal state. The self energy is written in Nambu notation with the corresponding Pauli matrices denoted by $\tau_j$ ($j=x,y,z$) and does not yet account for spin-orbit coupling in the superconductor. Thus the pairing terms $\propto \tau_x$ describe conventional s-wave pairing and the induced $p$-wave pairing involves a dimensionless measure of the spin-orbit coupling in addition.
The expression in Eq.\ \eqref{eq:self_energy} is independent of the wave vector ${\bf k}$, making the self energy local in real space. Within the semiclassical picture of the present paper, this surprising locality has a natural interpretation in terms of the locality of the scattering processes by the superconductor. Moreover, the semiclassical approach requires a purely spectral description of the renormalizations. The expression in Eq.\ \eqref{eq:self_energy} implies that we can expect such a spectral interpretation in the limit in which $\omega\ll\Delta$ and the induced gap is small compared to $\Delta$. For $\omega\ll\Delta$, both the induced pairing term and the quasiparticle weight become independent of $\omega$. Then, the subgap spectrum of the wire can be obtained from an effective Hamiltonian, provided that the induced gap is sufficiently small. In the context of the model studied in this paper, this latter condition is guaranteed by the spin polarization of the wire.
The renormalizations of the Hamiltonian parameters are due to the quasiparticle weight. As the coupling between wire and superconductor increases, the quasiparticle weight of the wire Green function is progressively reduced. This renormalization is directly mirrored in factors involving $4W+\xi_N|t_\uparrow|^2$ in the semiclassical approach of this paper. Such factors are involved in the semiclassical expressions \eqref{eq:vxasymp} and \eqref{eq:egap} for the Fermi velocity and the induced gap of the normal metal, respectively. Correspondingly, both quantities involve renormalizations by the quasiparticle weight in the Green function approach. At the same time, the quasiparticle weight drops out from the localization length of the Majorana bound state (or, equivalently, the coherence length of the induced superconductivity) since it is the ratio of Fermi velocity and induced gap. Again, this is consistent with our semiclassical approach which also does not involve a factor $4W+\xi_N|t_\uparrow|^2$ in Eqs. \eqref{eq:rhohelow1} and \eqref{eq:rhohelow2}. Note that despite this absence of renormalization, the Majorana localization length depends on the bare system parameters in a nontrivial way, as it is independent of the gap of the proximity providing superconductor (see also \cite{akhmerov17}).
We finally note that our analysis excluded the presence of disorder which may or may not affect the properties of the topological phase. As discussed earlier \cite{Kupferschmidt_2011, duckheim11}, for a mean-free path $\ell$ much larger than the microscopic length scales, the single reflection amplitude $\rho_{\rm he} \delta L$ is not affected since it is obtained by matching the wavefunctions at the short scale of the half-metal - superconductor interface. In contrast, the derivation of the reflection amplitude $r_{\rm he}^{\rm eff}$ includes multiple scattering processes at a length scale $1/|\rho_{\rm he}|$. In the absence of disorder, these add coherently to $r_{\rm he}^{\rm eff}$ because $k_x$ is conserved. Including disorder with $\ell \ll 1/|\rho_{\rm he}|$ leads to contributions from different $k_x$ for different scattering paths. Additionally, based on symmetry arguments it can be shown that $r_{\rm he}$ is anti-symmetric in $k_x$ \cite{Kupferschmidt_2011}. Hence the sum over the different paths is incoherent and there is no guarantee that $r_{\rm he}^{\rm eff}$ is unaffected by disorder. However, if $\ell \gg 1/|\rho_{\rm he}|$ the amplitudes still add coherently, and disorder is expected to not play a role. Since $1/|\rho_{\rm he}|$ is strongly decreased for a highly transparent interface, we conclude that high transparencies lead to a better protection from disorder for the Majorana bound states.
\begin{acknowledgements}
We thank Christian Kl\"ockner and Max Geier for discussions. Financial support was provided by the Institute ``Quantum Phenomenon in Novel Materials'' at the Helmholtz Zentrum Berlin, and the Deutsche Forschungsgemeinschaft (project C03 of the CRC 183).
\end{acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,194
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<?php
/**
*
* @package Core_Resource
* @author Christian Gijtenbeek <gijtenbeek@terena.org>
*/
class Core_Resource_Presentationsusers extends TA_Model_Resource_Db_Table_Abstract
{
protected $_name = 'presentations_users';
protected $_primary = 'presentation_user_id';
public function init() {}
/**
* Gets item by primary key
* @return object Zend_Db_Table_Row
*/
public function getItemById($id)
{
return $this->find( (int)$id )->current();
}
/**
* returns item based on id values
*
* @param array $data presentation_id and user_id values
* @return object Zend_Db_Table_Row
*/
public function getItemByValues(array $data)
{
return $this->fetchRow(
$this->select()
->where('presentation_id = ?', $data['presentation_id'])
->where('user_id = ?', $data['user_id'])
);
}
/**
* Gets item by user id
*
* @param integer $id user id value
* @return Zend_Db_Table_Row
*/
public function getItemByUserId($id)
{
return $this->fetchRow(
$this->select()
->where('user_id = ?', $id)
);
}
/**
* Save rows to the database. (insert only)
*
* This method makes sure that only records that are not already in the
* many to many table get inserted
*
* @param array $values
* @return Zend_Db_Statement_Pdo on success or void if nothing is inserted
*/
public function saveRows(array $values)
{
$db = $this->getAdapter();
// get current user_id/presentation_id combinations
$currentValues = $db->fetchPairs(
$this->select()
->from($this->_name, array('presentation_id', 'user_id'))
);
// compare current values with values about to be inserted
// I need to do this to prevent duplicate key constraint on
// presentations_users_idx
foreach ($values as $key => $value) {
if (isset($currentValues[$key])) {
if ($currentValues[$key] == $value) {
continue;
}
}
$insertValues[] = "(".(int)$key .",". (int)$value.")";
}
$insertValues = implode(',', $insertValues);
if ($insertValues) {
$query = "INSERT INTO " . $this->_name . " (presentation_id, user_id) VALUES ".$insertValues;
return $db->query($query);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,610
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Dit is een lijst met breakcoreartiesten die een artikel hebben op de Nederlandstalige Wikipedia.
A
Aphex Twin
Atari Teenage Riot
B
Bong-Ra
C
Cardopusher
D
Donna Summer, zie Jason Forrest
Doormouse
E
Enduser
F
FFF
G
Gautier Serre
J
Jason Forrest aka DonnaSummer
Julien Caraz
M
µ-ziq
N
Nasenbluten
P
Panacea
R
Rotator
S
Squarepusher
T
Terror
V
Venetian Snares
X
Xanopticon
Breakcore
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,075
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Renishaw has announced the introduction of additional imperial thread size to its existing metrology fixtures product range.
Fixturing components are currently available in M4, M6 and M8 thread sizes; the newly added ¼ 20 fixturing range means a complete fixturing solution can now be created for any part regardless of its size, shape or material.
A range of modular and custom fixturing and software is available for use with co-ordinate measuring machines (CMMs), Renishaw Equator™ gauges and vision systems. Using high quality metrology fixtures can improve throughput, reproducibility and accuracy of inspection processes with quick repeatable fixturing set-ups.
Renishaw states it is focused on bringing new and improved metrology fixtures to market, increasing its capability to deliver advanced metrology technologies, and developing turnkey solutions. Efforts made in these areas will play a key role within highly automated manufacturing environments in industries such as aerospace, automotive and electronics.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,484
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Q: additional resistance required to produce a potential drop A wire of length 100cm is connected to a cell of emf 2V and negligible internal resistance. The resistance of the wire is 3Ω. The additional resistance required to produce a potential drop of 1 millivolt/m?
The answer is 57 ohms.Could someone explain?
A: I'll assume that the requirement is really 1 mV/cm of voltage drop along the wire. Here's one way to get there:
1 mV/cm × 100 cm = 100 mV total drop required.
100 mV / 3 Ω = 33.3 mA of current needs to flow.
2 V cell / 33.3 mA = 60 Ω total resistance is required.
60 Ω total - 3 Ω wire = 57 Ω additional required.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,795
|
Individual Prerequisites No prerequisites to attend the trianing class. However to become an Asbestos Contractor in FL the following is required: 5-day contractor/supervisor course; 3-day respiratory protection course; Completion of at least 10 asbestos projects within the last 5 years; Pass the FL State Asbestos Contractors exam.
Individual App Requirements If you wish to be a licensed Asbestos Contractor in FL you must apply with the state licensing division and pay an initial application & exam fee of $340. The renewal fee is $155 every two years. Visit http://www.myfloridalicense.com/dbpr/pro/asbest/documents/asbestos_faqs.pdf to learn more about these requirements.
Individual Exam And Fees If applying to become an Asbestos Contractor in FL, you must pass the FL State Asbesots Contractor's exam. The Asbestos Examination Application Fees for Initial Applications is $340 for Contractors.
Company Application Requirements If you wish to engage in asbestos consulting or contracting as a partnership, corporation, business trust or other legal entity or if you wish to engage in asbestos consulting or contracting as an individual you must obtain a FL Asbestos Business License. The fee is $255 every two years.
Refresher Training Completion of a one day 8-hour course for each calendar year prior to the license renewal date.
Reciprocity Florida does not currently have reciprocal agreements or criteria for endorsement of applicants from other states. You must meet the Florida requirements for licensure and receive a passing score on the Florida examination.
The trainers were knowledgeable and the hands-on training was very informative. The trainers were knowledgeable and the hands-on training was very informative.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,113
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Apollo 11 Crew at the National Air and Space Museum
By Marcia Smith | Posted: July 20, 2009 12:00 am ET | Last Updated: December 5, 2011 6:12 pm ET
For anyone who missed the BIG EVENT at the National Air and Space Museum last night featuring the Apollo 11 crew — Neil Armstrong, Buzz Aldrin, and Mike Collins — you can view the webcast here.
The astronauts' talks really are worth a listen. For those who don't have time to watch the entire hour and a half webcast, Buzz begins at minute 38:50, Mike at 54:45, and Neil at 1:08:40. .
Last Updated: Dec 05, 2011 6:12 pm ET
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,686
|
<?php
/**
* Unit test class for the ClassDeclaration sniff.
*
* A sniff unit test checks a .inc file for expected violations of a single
* coding standard. Expected errors and warnings are stored in this class.
*
* @category PHP
* @package PHP_CodeSniffer
* @author Greg Sherwood <gsherwood@squiz.net>
* @copyright 2006-2014 Squiz Pty Ltd (ABN 77 084 670 600)
* @license https://github.com/squizlabs/PHP_CodeSniffer/blob/master/licence.txt BSD Licence
* @version Release: @package_version@
* @link http://pear.php.net/package/PHP_CodeSniffer
*/
class PSR1_Tests_Classes_ClassDeclarationUnitTest extends AbstractSniffUnitTest
{
/**
* Returns the lines where errors should occur.
*
* The key of the array should represent the line number and the value
* should represent the number of errors that should occur on that line.
*
* @param string $testFile The name of the file being tested.
*
* @return array<int, int>
*/
public function getErrorList($testFile='')
{
if ($testFile === 'ClassDeclarationUnitTest.2.inc') {
return array();
}
if (PHP_VERSION_ID >= 50300) {
return array(
2 => 1,
3 => 2,
);
} else {
return array(
3 => 1,
);
}
}//end getErrorList()
/**
* Returns the lines where warnings should occur.
*
* The key of the array should represent the line number and the value
* should represent the number of warnings that should occur on that line.
*
* @return array<int, int>
*/
public function getWarningList()
{
return array();
}//end getWarningList()
}//end class
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,682
|
What is Music For Soul ?
is a free digital music media website with millions of download links for mp3 audio, mp4 videos, and song lyrics. You will get all the most complete and up-to-date download content.
Where is the source of the content Music For Soul?
All song content or MP3 files available on Music For Soul only third parties originating from YouTube.
Music For Soul actually is YouTube to Mp3 Converter, but we developed this app differently from most other YouTube video downloader converter sites. Many features make it easier for you to download YouTube videos to MP3 Audio on Music For Soul than on any other site.
All files that come from YouTube converters, we get from external converter sites that we install or embed with iframe tags on the download page. We are not the owner and do not have access to every file on the download page that has not been updated.
For MP3 files on the download page that we have updated or updated, come from our own servers. We only have full access to the files that we update ourselves.
Is content Music For Soul legal?
Because Music For Soul is a YouTube to Mp3 Converter, most of the content on our site has no copyright or copyright, where YouTube users are mostly individuals such as Vlogs, Covers, Remix, Gaming and others. However, many also have copyright.
Contact us via email to vokonis@gmail.com for suggestions / criticism / cooperation / deletion / copyright and others, we will process your complaint in less than 24 hours.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,785
|
Sacco ist eine italienische Gemeinde mit Einwohnern (Stand ) in der Provinz Salerno in der Region Kampanien.
Geografie
Die Nachbargemeinden sind Corleto Monforte, Laurino, Piaggine, Roscigno und Teggiano. Der Ort gehört zum Nationalpark Cilento und Vallo di Diano und ist Teil der Comunità Montana del Calore Salernitano.
Siehe auch
Cilento
Nationalpark Cilento und Vallo di Diano
Weblinks
Sacco (Kampanien)
Einzelnachweise
Ort in Kampanien
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,991
|
Creating an environmental management system (EMS) for your company may seem like an overwhelming task. And the process of getting that EMS certified as compliant with a voluntary standard, such as ISO 14001, may seem even more daunting. After all, getting your company's EMS certified requires a significant commitment of time and money and can, frankly, be an intimidating process that senior management may prefer to avoid. So how do you convince them that the certification process is manageable? Answer: Use a case study on a US manufacturer's experience getting its EMS certified under ISO 14001.
Implementation of a management review process through which senior management reassesses the suitability, effectiveness and adequacy of the EMS at appropriate intervals to assure continuous improvement.
Here's a look at how the company handled each of the five components of ISO 14001.
Environmental policy. The company already had an ISO 9002-certified quality management system in place. In-house staff found that they could borrow from the quality management system in developing the EMS. For example, the company modified its quality policy to include its environmental policy. The policy was then enlarged to poster-size, signed by all employees and posted as a reminder of the company's commitments.
Planning. A key element of an effective EMS is identifying the environmental aspects of your company's activities, products and services and determining which aspects have significant impacts on the environment. These "significant aspects" form the basis for your environmental objectives. H-R Industries decided to address some potential environmental impacts by making suppliers and contractors aware of the environmental aspects associated with their products. For example, the company worked with a chemical supplier to convert its permanganate bath maintenance procedure. Sodium hypochlorite additions were replaced by permanent electrodes in the solution for electro-regeneration, extending bath life two to three times. This change reduced hazardous waste generation, material handling reporting and recordkeeping. Annual savings: more than $32,500.
Implementation and operation. To make the most of its limited resources, the company assigned the same individuals responsibility for both quality and EMS elements where there was overlap. For example, the Safety/Health Officer provides the training required by both systems. And one individual controls and maintains documentation and records for both systems.
Checking and corrective action. The company decided to use the procedures already in place for making requests for corrective action under its quality management system in its EMS. For example, the same Corrective Action Request Form is used for both systems. Similarly, it incorporated the records control and audit procedures established under the quality management system into its EMS.
Management review. Although there's some overlap in personnel on the quality and environmental review committees, the company has the two systems reviewed separately. However, the review format and control of meeting records is the same.
At H-R Industries, it took about 18 months to obtain certification of the EMS under ISO 14001. For a company without a well-developed system already in place, implementation may take about two years. H-R Industries has one full-time employee who maintains both the environmental and quality systems; several other employees also have EMS responsibilities.
Certification of the EMS cost H-R Industries $18,000, which included preliminary and on-site audits, a follow-up audit, an audit report, registration fee and auditor time and expenses. It saved on auditor expenses by scheduling the ISO 14001 certification audit and the ISO 9002 six-month surveillance audit on the quality management system at the same time and with the same firm.
H-R Industries' experience shows that the process of developing an EMS and getting it certified need not be overwhelming. In many cases, a company can use its existing environmental policies and procedures and those developed for other management systems, such as OHS programs, to help build an EMS that meets the requirements of a voluntary standard. The company's advice: Begin with simple, achievable goals and focus on programs where there's obvious economic benefit. As the EMS matures, the procedures and programs can be expanded to further improve environmental performance and continue integration of the EMS into other business functions.
Do You Need an EMS?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,720
|
{"url":"https:\/\/www.mathdoubts.com\/cos-30-degrees-proof\/","text":"# $\\cos{(30^\u00b0)}$ Proof\n\nThe cos of 30 degrees value can be derived mathematically in three methods. One of them is a trigonometric approach and other two are geometrical methods.\n\n### Theoretical approach\n\nYou must know the direct relation between the sides of a right triangle when its angle is $30$ degrees. According to properties of right triangle, the length of opposite side is half of the length of hypotenuse if the angle of right triangle is $\\dfrac{\\pi}{6}$. The exact value of $\\cos{(30^\u00b0)}$ is evaluated theoretically on the basis of this property.\n\nThe lengths of opposite side and hypotenuse are known but the length of adjacent side is unknown in this case. It is essential to find it in order to find the $\\cos{\\Big(\\dfrac{\\pi}{6}\\Big)}$ value. So, try Pythagorean Theorem to find the value of adjacent side mathematically.\n\n${OP}^2 \\,=\\, {PQ}^2+{OQ}^2$\n\nIn $\\Delta QOP$, the lengths of hypotenuse and opposite sides are $d$ and $\\dfrac{d}{2}$ respectively.\n\n$\\implies d^2 = {\\Big(\\dfrac{d}{2}\\Big)}^2+{OQ}^2$\n\n$\\implies d^2 = \\dfrac{d^2}{4}+{OQ}^2$\n\n$\\implies d^2-\\dfrac{d^2}{4} = {OQ}^2$\n\n$\\implies {OQ}^2 = d^2-\\dfrac{d^2}{4}$\n\n$\\implies {OQ}^2 = d^2{\\Bigg(1-\\dfrac{1}{4}\\Bigg)}$\n\n$\\implies {OQ}^2 = d^2{\\Bigg(\\dfrac{1 \\times 4 -1}{4}\\Bigg)}$\n\n$\\implies {OQ}^2 = d^2{\\Bigg(\\dfrac{4-1}{4}\\Bigg)}$\n\n$\\implies {OQ}^2 = d^2{\\Bigg(\\dfrac{3}{4}\\Bigg)}$\n\n$\\implies OQ = \\sqrt{\\dfrac{3d^2}{4}}$\n\n$\\implies OQ = \\dfrac{\\sqrt{3}d}{2}$\n\n$\\implies \\dfrac{OQ}{d} = \\dfrac{\\sqrt{3}}{2}$\n\nIn this case, $d$ represents the length of hypotenuse $(OP)$.\n\n$\\implies \\dfrac{OQ}{OP} = \\dfrac{\\sqrt{3}}{2}$\n\n$\\implies \\dfrac{Length \\, of \\, Adjacent \\, side}{Length \\, of \\, Hypotenuse} = \\dfrac{\\sqrt{3}}{2}$\n\nThe angle of $\\Delta QOP$ is $\\dfrac{\\pi}{6}$ and the ratio represents $\\cos{(30^\u00b0)}$ as per definition of trigonometric ratio cosine.\n\n$\\therefore \\,\\,\\, \\cos{(30^\u00b0)} = \\dfrac{\\sqrt{3}}{2}$\n\nTherefore, it is derived that the exact value of cos of $30$ degrees in fraction form is $\\dfrac{\\sqrt{3}}{2}$ and its value in decimal form is $0.8660254037\\ldots$\n\n$\\cos{(30^\u00b0)} = \\dfrac{\\sqrt{3}}{2} = 0.8660254037\\ldots$\n\n### Practical approach\n\nThe value of cosine of $\\dfrac{\\pi}{6}$ can be calculated geometrically by constructing a right triangle with $30$ degrees angle using geometric tools.\n\n1. Identify a point $G$ on the plane and then draw a straight line from this point horizontally.\n2. Take protractor and coincide its centre with point $G$ and also coincide its right side base line with horizontal line. Now, mark a point at $30$ degrees.\n3. Draw a line from point $G$ through $30$ degrees angle line by a ruler.\n4. Draw an arc on $30$ degrees line from point $G$ by compass with any length. For example $7.5 \\, cm$. The arc cuts the $30^\u00b0$ line at point $H$.\n5. Draw a perpendicular line to horizontal line from point $H$ and it intersects the horizontal line at point $I$.\n\nThe five geometrical steps have constructed a right triangle, called as $\\Delta HGI$ with an angle of $30$ degrees. The value of $\\cos{\\Big(\\dfrac{\\pi}{6}\\Big)}$ can be calculated from this triangle mathematically.\n\n$\\cos{(30^\u00b0)} = \\dfrac{Length \\, of \\, Adjacent \\, side}{Length \\, of \\, Hypotenuse}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\dfrac{GI}{GH}$\n\nIn this example, it is taken that the length of hypotenuse is $7.5 \\, cm$ but the length of adjacent side is unknown. However, it can be measured by a ruler.\n\nNow, measure the length of the adjacent side by ruler and you will observe that its length is $6.5 \\, cm$ approximately. Now, calculate the value of $\\cos{\\Big({33\\dfrac{1}{3}}^g\\Big)}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\dfrac{GI}{GH} = \\dfrac{6.5}{7.5}$\n\n$\\,\\,\\, \\therefore \\,\\,\\,\\,\\,\\, \\cos{(30^\u00b0)} \\,=\\, 0.8666666666\\ldots$\n\n### Trigonometric approach\n\nThe $\\cos{\\Bigg({33\\dfrac{1}{3}}^g\\Bigg)}$ value is evaluated exactly in trigonometry by the cos squared identity. The exact value of $\\cos{\\Big(\\dfrac{\\pi}{6}\\Big)}$ is actually calculated by substituting the value of sin 30 degrees in this formula.\n\n$\\cos{(30^\u00b0)} \\,=\\, \\sqrt{1-\\sin^2{(30^\u00b0)}}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\sqrt{1-{\\Bigg(\\dfrac{1}{2}\\Bigg)}^2}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\sqrt{1-\\dfrac{1}{4}}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\sqrt{\\dfrac{1 \\times 4 -1}{4}}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\sqrt{\\dfrac{4-1}{4}}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\sqrt{\\dfrac{3}{4}}$\n\n$\\implies \\cos{(30^\u00b0)} \\,=\\, \\dfrac{\\sqrt{3}}{2}$\n\n$\\,\\,\\, \\therefore \\,\\,\\,\\,\\,\\, \\cos{(30^\u00b0)} \\,=\\, \\dfrac{\\sqrt{3}}{2}$\n\n#### Verdict\n\nAccording to both theoretical geometric and trigonometric methods, it is proved that the $\\cos{\\Big(\\dfrac{\\pi}{6}\\Big)}$ value is $\\dfrac{\\sqrt{3}}{2}$ and its approximate value is $0.8660254037\\ldots$\n\nIt is also evaluated that the value of $\\cos{(30^\u00b0)}$ in practical geometrical method is $0.8666666666\\ldots$ Now, compare both values of $\\cos{(30^\u00b0)}$ and you observe that the value of $\\cos{(30^\u00b0)}$ slightly differs with the value obtained from theoretical and trigonometric approaches. It is due to measuring the length of adjacent side approximately. However, the approximate values of $\\cos{\\Big(\\dfrac{\\pi}{6}\\Big)}$ are same.\n\nEmail subscription\nMath Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more","date":"2021-02-27 20:55:57","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\": 0.8525440096855164, \"perplexity\": 256.08350473105395}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178359497.20\/warc\/CC-MAIN-20210227204637-20210227234637-00485.warc.gz\"}"}
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{"url":"https:\/\/puzzling.stackexchange.com\/questions\/100978\/reconstructing-points-based-on-the-sum-of-their-coordinates","text":"# Reconstructing points based on the sum of their coordinates\n\n9 points are drawn on a piece of paper with the following rules:\n\n\u2022 Each point has integer coordinates $$(x,y)$$ that are between 1 and 10 inclusive.\n\u2022 For each point there is exactly one other point so that their x-coordinates or their y-coordinates match.\n\u2022 Two points cannot sit on top of each other.\n\nThe sum of the coordinates of each point (ie., x+y) is provided: 2, 5, 6, 7, 8, 10, 11, 12, 13. Can you reconstruct the location of each point? Bonus question: can you find multiple solutions? Good luck!\n\nThis puzzle was inspired by this one: The Grid World - Catastrophe\n\nThe second rule states: \"For each point there is exactly one other point so that their x-coordinates or their y-coordinates match\". So we should be able to pair each point. We have an odd number of points to place, so\n\nIt is impossible.\n\nps:sorry I can't comment yet.\n\n\u2022 Would 1,2 \/ 1,3 \/ 2,3 not satisfy that second rule? Aug 11 '20 at 11:11\n\u2022 @Mohirl The way I undestand the second rule, your second point would not satisfy it (same x as first point and same y as the third). Another puzzle on this theme with a modified 2nd rule has been made: puzzling.stackexchange.com\/questions\/100996\/\u2026 Aug 11 '20 at 12:04\n\nI hope I understood correctly.\nSolution 1:\n\n1, 1 (2)\n1, 4 (5)\n2, 4 (6)\n2, 5 (7)\n3, 5 (8)\n3, 7 (10)\n4, 7 (11)\n4, 8 (12)\n5, 8 (13)\n\nSecond solution\n\nsame as the first one but you reverse x and y coordinates for each point\n1, 1 (2)\n4, 1 (5)\n4, 2 (6)\n5, 2 (7)\n5, 3 (8)\n7, 3 (10)\n7, 4 (11)\n8, 4 (12)\n8, 5 (13)\n\nExplanation:\n\nI started of with the first sum which is 2. This means the first point has the coordinates 1 and 1 since they have to be integers between 1 and 10.\nthe idea: Since each 2 points share either only x or only y, there are no 3 points on the same line.\nSo the safest way to continue would be to move to the right with the next point or up alternating from point to point like in the image below made with my awesome paint skills. Then I took all the sums in order and tried to reach the next one by changing X on the first move and Y on the second move.\nFor the second solution I changed Y on the first move and X on the second one.\nSo starting at (1, 1), changing X to match the next sum (5), I ended up with (1, 4). Then changed Y to reach the next sum (6) and ended up with (2, 4) and so on.\n\n\u2022 This was the intended solution, but I stuffed up the description, because here each point has multiple matching partners. So the actual solution is the other answer. Aug 10 '20 at 22:28\n\u2022 @DmitryKamenetsky: Surely editing the question to clarify the description would be a valid option? To be honest, when I first read it, I read it the way you indented it to be, and not the way the accepted answer understood it. Aug 11 '20 at 8:29\n\u2022 By the time I realised my mistake, there were already two answers. So i felt it wouldn\u2019t be fair to edit the question at that stage. Aug 11 '20 at 9:35","date":"2021-10-21 16:05:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4865230917930603, \"perplexity\": 684.9570589886151}, \"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-2021-43\/segments\/1634323585424.97\/warc\/CC-MAIN-20211021133500-20211021163500-00121.warc.gz\"}"}
| null | null |
Rezerwat przyrody Grąd w Średniej Wsi – rezerwat przyrody znajdujący się na terenie gminy Lesko, w powiecie leskim, w województwie podkarpackim; zajmuje powierzchnię jednej miejscowości – Średniej Wsi za rzeką San.
numer według rejestru wojewódzkiego – 84
powierzchnia według aktu powołującego – 58,19 ha
dokument powołujący – Dz. Urz. Woj. Podkarpackiego 03.83.1464
rodzaj rezerwatu – leśny
przedmiot ochrony (według aktu powołującego) – fragmenty subkontynentalnego grądu Tilio-Carpinetum o wysokim stopniu naturalności, występującego w piętrze pogórza; cenne wiekowo oddziały leśne – żyzny las lipowo-grabowy domieszką dębu, wiązu i brzozy w wieku ponad 100 lat oraz sosny datowanej na ok. 130 lat.
Przypisy
Linki zewnętrzne
Rezerwaty przyrody w województwie podkarpackim
Średnia Wieś (województwo podkarpackie)
Ochrona przyrody w powiecie leskim
Ochrona przyrody Gór Sanocko-Turczańskich
|
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| 7,978
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Home 778 Our Lady of the Rosary (OLR) is a co-educational Catholic Parish School situated on the Sunshine Coast, in the picturesque beachside city of Caloundra. OLR is a small school of 320 students in Prep to Year Six (13 classes). 8/01/2020 11:16:11 PM 8/01/2020 11:16:11 PM Welcome to Our Lady of the Rosary School Our Lady of the Rosary (OLR) is a co-educational Catholic Parish School situated on the Sunshine Coast, in the picturesque beachside city of Caloundra STS_Site 425180 4960 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png http://www.olr.qld.edu.au/latest-news News http://www.olr.qld.edu.au/enrolments Enrol http://www.olr.qld.edu.au/bce-policies BCE%20Policies http://www.olr.qld.edu.au/Learning Learning http://www.olr.qld.edu.au/contact-us Contact http://www.olr.qld.edu.au/co-curricular Co-curricular http://www.olr.qld.edu.au/catholic-identity Catholic%20Identity http://www.olr.qld.edu.au/About%20Us About 1 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au {25DB3AA0-F8FF-427A-ACF7-ECE9B8038847} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
Contact 61311 11/05/2020 1:22:28 AM 11/05/2020 1:22:28 AM OUR LADY OF THE ROSARY SCHOOL PO Box 149, Moffat Beach QLD 4551 STS_Web http://www.olr.qld.edu.au 9672 122 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/contact-us {7A854B83-A009-4A12-AD37-03A7DCA77CC5} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
News 61536 10/03/2020 4:22:52 AM 10/03/2020 4:22:52 AM click on a news story on the left to read more about the latest news from OLR STS_Web http://www.olr.qld.edu.au 6081 76 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/latest-news {412EEC42-C3D1-4FFA-A004-3894E9B810FD} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
Learning 61537 11/05/2020 3:27:27 AM 11/05/2020 3:27:27 AM At OLR, we aspire to be the Catholic school of the future in the Archdiocese of Brisbane STS_Web http://www.olr.qld.edu.au 6031 87 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/Learning {8DF27183-7996-4870-B6D0-1E7161AAD125} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
BCE Policies 187857 Student Protection Processes, Parent and Guardian Complaints Management, Acceptable Use Policy, Student Protection and Code of Conduct Training for Volunteers and Other Personnel, Student, Parent and Guardian Complaints Management 7/05/2019 5:25:46 AM 7/05/2019 5:25:46 AM and Code of Conduct Training for Volunteers and Other Personnel, Student, Parent and Guardian Complaints Management STS_Web http://www.olr.qld.edu.au 2722 52 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/bce-policies {6AE5957B-4344-4E60-A68F-B0321EC7E248} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
Co-curricular 195247 12/05/2020 3:58:02 AM 12/05/2020 3:58:02 AM Excursions, incursions and camps (Year 6) are regularly organised to support and enhance in-class learning at OLR Some examples of these activities include STS_Web http://www.olr.qld.edu.au 2083 67 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/co-curricular {C0A08913-FD0C-4D1F-B77D-C2A22776C872} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
About 60995 11/05/2020 4:34:39 AM 11/05/2020 4:34:39 AM for a nurturing and caring school for your child Why not have a look at what it's like to be part of our community OUR LADY OF THE ROSARY SCHOOL Crn Edmund & Alfred Streets Shelly Beach, Caloundra QLD 4551 Phone 07 5491 4522 | Fax 07 5492 6225 Email STS_Web http://www.olr.qld.edu.au 5532 88 http://www.olr.qld.edu.au/schools/olr/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/About Us {43AD05C9-2591-4FEF-994D-1FBB55A06F6A} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
Catholic Identity 195571 28/04/2020 4:28:37 AM 28/04/2020 4:28:37 AM The Vision for Religious Education (RE) at OLR has been collaboratively constructed with staff, parents and students and aims to form students who can articulate their faith and STS_Web http://www.olr.qld.edu.au 1988 67 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/catholic-identity {D53FFB27-534F-4F82-9A8A-209B8BB73517} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
Enrol 61312 19/08/2020 4:54:54 AM 19/08/2020 4:54:54 AM you are wishing to enrol your child at OLR from 2022 or onwards, please complete our OLR Expression of Interest.pdf There is no fee involved in submitting an expression of STS_Web http://www.olr.qld.edu.au 8460 83 http://www.olr.qld.edu.au/PublishingImages/schoolLogo60px.png 0 http://www.olr.qld.edu.au html True CMSPUBLISHING aspx http://www.olr.qld.edu.au/enrolments {7BAC299B-5705-4552-9C8F-675608B9433D} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Site.js 11;16 11
OLR Fee Collection, Payment Procedure and Schedule 2017 60637 8/03/2017 3:00:06 AM 8/03/2017 3:00:06 AM Our Lady of the Rosary School, Caloundra Page 1 of 3 School Fee Collection, Payment Procedures and Schedule 2017 The fees and levies collected at Our Lady of the Rosary School STS_ListItem_DocumentLibrary http://www.olr.qld.edu.au/_layouts/15/WopiFrame.aspx?sourcedoc={87c78866-0545-40ec-8374-8e6df12ae7ab}&action=interactivepreview http://www.olr.qld.edu.au/_layouts/15/WopiFrame.aspx?sourcedoc={87c78866-0545-40ec-8374-8e6df12ae7ab}&action=imagepreview http://www.olr.qld.edu.au/Documents/Forms/AllItems.aspx 1144 13 0 http://www.olr.qld.edu.au pdf False pdf http://www.olr.qld.edu.au {87C78866-0545-40EC-8374-8E6DF12AE7AB} ~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_PDF.js 15 15
~sitecollection/_catalogs/masterpage/Display Templates/Search/Item_Default.js
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 3,044
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Q: Creating Vim like functionality with AutoHotKey (AHK) I've been using autoHotKeyrecently on a windows 8 machine and loving it. But I want to be able to press caps lock and turn the keyboard into a vim like command mode for moving the cursor, inserting and deleting easily in any program.
UPDATE (Thanks to @MCL for the help so far)
Im trying to use the following script but it wont change the behaviour based on the state
state := GetKeyState("Capslock", "T")
if state
j::Send,{Left}
l::Send,{Right}
i::Send,{Up}
k::Send,{Down}
return
A: Create context-sensitive hotkeys with #If:
#If GetKeyState("CapsLock", "T")=1
; The following hotkeys will only be effective if GetKeyState("CapsLock", "T")=1
j::Send,{Left}
l::Send,{Right}
i::Send,{Up}
k::Send,{Down}
#If ; end of #If
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 9,604
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\section{Introduction}
The study of stellar clusters has implications in a wide variety of astrophysical topics, which includes star formation, stellar evolution and nucleosynthesis, stellar dynamics, Galactic structure, and galaxy formation and evolution.%
\citep[e.g.][]{vandenberg13,barbuy18}.
With the advent of space-based telescopes, in particular the {\it Hubble Space Telescope} ({\it HST}) and more recently the {\it Gaia} Data Release 2 \citep[DR2,][]{gaia18a}, as well as multi-object and high-resolution spectrographs, a wealth of high-quality and spatially resolved data have been collected for Milky Way globular and open clusters (GCs and OCs), and for stellar clusters in neighbouring galaxies. Combined with sophisticated analysis, these data have opened an unprecedented opportunity for very accurate physical parameter derivation.
Milky Way globular clusters (GCs) formed during the early stages of the Galaxy formation \citep[e.g.][]{vandenberg13,barbuy18}
are studied in the present work.
The phenomenon of multiple stellar populations (MPs) was observed for the first time by \citet{osborn71} from
CN-band strengths, but at the time this was not identified as due to the presence of two
stellar populations. Later, MPs were clearly revealed by \citep[eg. ][]{lee99,bedin04,piotto05,milone17}, and
hints on self-enrichment to explain abundance variations within a GC were discussed by \citet{gratton04}.
Evidence of MPs from spectroscopic work was reviewed by \citet[][and references therein]{carretta19}.
The photometric counterpart of the CN anomaly is detectable in the ultraviolet (UV) filters \citep{piotto15, lee19}. These filters are sensitive to C, N, and O abundances, allowing to disentangle the different stellar populations \citep{piotto15}.
With the purpose of correlating the cluster age with the presence of MPs, \citet{martocchia18,martocchia19} analyzed a sample of Magellanic Clouds (MCs) and MW clusters. They estimated the N abundance spread in CMDs, which is an indicator of the presence of MPs, and found that clusters older than $\sim 2$ Gyr host MPs, while those younger than this age show no evidence of spread in N abundance. On the other hand, it is known that the presence of MPs is related to the mass of the cluster \citep[][]{milone17}. For this reason, age cannot be the only parameter to constrain the presence of MPs. This fact is evident for the case of Berkeley 39 \citep[][]{martocchia18} and Lindsay 38 \citep[][]{martocchia19}, both having an age of $\sim 6.5$ Gyr, without showing N abundance spread. Another counterexample was given by \citet{lagioia19}, having found that the GC Terzan 7 is consistent with a single stellar population (SSP), despite a relatively old age and high mass. Therefore, the study of MPs helps understanding the formation and evolution of stellar systems in general.
Isochrone fitting to CMDs has been extensively used to obtain the star cluster properties age, distance modulus, and reddening. Previously, a visual method known as ``chi-by-eye'' was usually employed to fit theoretical isochrones to CMDs. Later on, to benefit from improved data quality and to extract physical parameters with meaningful uncertainties, several statistical isochrone fitting techniques were developed, most of them based on $\chi^{2}$, maximum likelihood statistics, or Bayesian approach \citep[][]{kerber05,naylor06,vonHippel06,hernandez08, monteiro10}. In almost all these developments, synthetic CMDs are employed for validation of the methods.
The Bayesian approach has the advantage of being able to get distributions and to explore the information \textit{a priori}
about the data or models. Recent examples of isochrone fitting codes using Bayesian inference are \texttt{ASteCA} \citep{perren15} and \texttt{BASE-9}
\citep{stenning16}, where the latter allows analysis of MPs to derive their difference on the helium content (Y). \cite{ramirez-siordia19} also applied the Bayes' theorem to a Monte Carlo method to get the posterior distributions of the same parameters as \texttt{BASE-9}, neglecting helium enhancements. They applied their software to the scarce stellar populations of ultra-faint dwarf galaxies and LMC star clusters.
In the present work, we carry out a detailed analysis of CMDs assuming both cases
of clusters as SSPs and MPs. With this purpose,
we developed the code named \texttt{SIRIUS}\footnote{The code is available upon request to the authors.}, standing for {\bf S}tatistical {\bf I}nference of physical pa{\bf R}ameters of s{\bf I}ngle and m{\bf U}ltiple populations in {\bf S}tellar clusters, to extract information on a stellar cluster from its CMDs.
The \texttt{SIRIUS} code was applied to analyse NGC~6752, with data from the \textit{HST} UV Legacy Survey of Galactic GCs \citep{piotto15}. \citep{gratton03} obtained for this halo GC an age of $13.4 \pm 1.1$ and
Carretta et al. (2012) found three distinct stellar populations \citep{milone13}.
Whereas the precision in parameter derivation from CMDs has been improving, it is also important to stress that a new era is now open: the age difference between stellar populations in a GC can give us a better understanding on its formation.
This work is organized as follows. In Section \ref{sec:sirius} the SIRIUS code is described in detail. Experiments to check the validity of the method and analysis of sources of uncertainties are presented in Section \ref{sec:tests}. An application to {\it HST} data of the halo GC NGC~6752 is presented in Section \ref{sec:app}. Conclusions are drawn in Section \ref{sec:conclusion}.
\section{The \texttt{SIRIUS} code}\label{sec:sirius}
This section gives a detailed description of the \texttt{SIRIUS} code, built to carry out isochrone fitting to CMDs, following the flow-chart presented in Figure~\ref{fig:workflow}.
\begin{figure*}
\centering
\includegraphics[scale=0.54]{flux_sirius.pdf}
\caption{\texttt{SIRIUS} flow-chart shows the steps to perform the isochrone fitting. }
\label{fig:workflow}
\end{figure*}
\subsection{Color-Magnitude Diagram Data}\label{subsec:Data}
\texttt{SIRIUS} was designed to analyse stellar clusters, applied here both to synthetic data and to
observed data. \texttt{SIRIUS} has already been successfully applied to derive the parameters of two bulge GCs. For HP\,1, a multi-band ($K_{\rm S}$ and $J$ from Gemini-GSAOI+GeMS, and F606W from {\it HST}-ACS) isochrone fitting was applied \citep{kerber19}. For ESO 456-SC38, {\it HST} photometry in the filters F606W from ACS and F110W from WFC3, and FORS2@VLT photometry in V and I were used \citep{ortolani19}. These studies confirmed that HP~1 and
ESO~456-SC38 are among the oldest GCs in the Milky Way, with an age of $\sim12.8$ Gyr.
\texttt{SIRIUS} can create synthetic CMDs using the following method.
The Monte Carlo algorithm is used to generate random data from a given probability distribution, and can be applied to describe many physical systems. In the case of CMDs of stellar clusters the main probability distribution of the system is the initial mass function (IMF), here adopted to be the Kroupa IMF \citep{kroupa01}. The method to generate a sample of data similar to a stellar cluster is called as Synthetic CMD \citep{kerber07}. Points are randomly generated and interpolated in mass within theoretical points of isochrones. From an error function, these random points are dispersed by Gaussian distributions to simulate the spread seen in observed CMDs.
\subsection{Stellar evolutionary models and Parameter space}
The library of isochrones adopted include two sets of stellar evolutionary models: DSED\footnote
{\url{http://stellar.dartmouthThe.edu/models/grid.html}} \citep[Dartmouth Stellar Evolutionary Database -][]{dotter08} and
BaSTI\footnote
{\url{http://basti.oa-teramo.inaf.it/}} \citep[A Bag of Stellar Tracks and Isochrones -][]{pietrinferni06}. We perform linear regressions to interpolate the isochrones in steps of $0.1$ Gyr in age in the range of $10.0$ to $15.0$ Gyr, and $0.01$ dex in [Fe/H] in the range of $-2.00$ $<$ [Fe/H] $<$ $0.00$\footnote{The usual notation [Fe/H]=log(Fe/H)$_{star}$-log(Fe/H)$_{\odot}$ is adopted.}.
It is relevant to mention that the range and step size of age we adopted here are consistent with the context of Galactic GCs. For the case of younger stellar clusters, e.g. MC clusters, the age range should allow ages below $10$ Gyr, and the step size should be narrower than the value used here.
The simple $\chi^2$ isochrone fitting procedures do not necessarily represent a physical interpretation of a GC CMD. Since the best fit is the isochrone that appears most similar to the CMD, many combinations of the parameters can be found as the best fit (minimum $\chi^2$) \citep{dantona18}.
The morphology of the isochrone depends on the age, reddening,
absolute distance modulus, metallicity,
and helium abundance. Figure \ref{fig:params} illustrates the
effects on the shape of isochrones, due to the change in each of these parameters. The reddening $E(B-V)$ changes the location of the isochrone in the diagonal direction because it contributes to the apparent distance modulus $(m-M)_{\lambda}$ and reddening $E(\lambda_1 - \lambda_2)$, without varying the morphology of the isochrone (first panel). For high values of reddening, a second-order correction, from the effective temperature \citep[e.g.][]{ortolani17,kerber19}, has to be taken into account in the isochrone fitting. A vertical displacement is the result of a
change in distance modulus $(m-M)_0$ (second panel). Age $\tau$ affects essentially the position of the turn-off point (TO)
(third panel). The metallicity $\rm [Fe/H]$ has a complex effect on the isochrone, but more strikingly by changing the slope of the RGB,
with a sub-giant branch (SGB) and RGB steeper towards lower metallicities (fourth panel of Figure~\ref{fig:params}). A variation in Y changes the slope of the SGB and the location of the TO, shifting the isochrone to the bluer region of the CMD
(last panel). A review on the interpretation of CMDs
in terms of stellar evolution models can be found in \cite{gallart05}.
\begin{figure*}
\centering
\includegraphics[scale=0.44]{parameters_explanation.pdf}
\caption{Graphical explanation of how the main five parameters change the morphology and position of the isochrone. The first panel shows the variation due to changes in $E(B-V)$, the second in $(m-M)_0$, the third in Age, the fourth in $\rm [Fe/H]$, and the last one in Y.}
\label{fig:params}
\end{figure*}
\subsection{Bayesian Statistics: Isochrone fitting}
The Bayesian statistics is based on the Bayes Theorem. The probability that two events (M and D) are
true, at the same time, according to a null hypothesis H is given by the product probability law:
\begin{center}
$ \displaystyle P(M,D|H) = P(M|D,H)\times P(D|H)$,
\end{center}
where $P(M|D,H)$ represents the probability of M to be true if D is true as well according to H, and $P(D|H)$ is the
probability of D following H. The opposite is also valid:
\begin{center}
$\displaystyle P(D,M|H) = P(D|M,H)\times P(M|H)$.
\end{center}
From the hypothesis of the conditional probability of M and D to be the same as D and M, results in the Bayes' theorem:
\begin{center}
$\displaystyle P(M|D) = \frac{P(D|M)\times P(M)}{P(D)}$,
\end{center}
where, in our case, the evolutionary model is represented by $M$ and the data by $D$.
The posterior distributions $P(M|D)$ are the distributions \emph{a posteriori} of the model (M) and
will give the distributions for each parameter. On the right-hand $P(M)$ are the prior distributions
that give the information \emph{a priori} about the model. The priors are distributions that constrain
the parameters with the physical information.
Assuming that stars are distributed in color and magnitude following a Gaussian distribution and
disconsidering the dependence of color with magnitude, the likelihood is given by:
\begin{center}
$\displaystyle P(D|M) = \prod^{N}_{i} \prod^{M}_{j} e^{-\varphi^2_{color}} \cdot e^{-\varphi^2_{Mag}}$,
\end{center}
where $N$ is the total number of the analysed stars and $M$ is the number of points in the isochrone. The $\varphi^2$ is defined as, for example:
\begin{center}
$\displaystyle \varphi^2_{color_{i,j}} = \frac{1}{2}\left(\frac{color^{obs}_i - color^{iso}_j}{\mathcal{S}_{i} + \sigma^{Cor}_i}\right)^2$,
\end{center}
where $\mathcal{S}$ represents the entropy term of likelihood. This term is responsible for smoothing the region of highest spread and number of stars. The $\mathcal{S}_{i}$, $| {\rm color}_{i}^{\rm obs} - \xi_{\rm f} | $, is calculated for each star by comparison with the fiducial color $\xi_{\rm f}$, which is defined as the median color for a bin of magnitude centered on the magnitude of the $i$-th star.
The maximum likelihood $\mathcal{L}$ corresponds to a maximization of the likelihood function in the parameter space. It is given by (in logarithm form):
\begin{center}
$ \displaystyle \mathcal{L} = \max\left\{- \sum^{N}_{i=1} \sum^{M}_{j=1} \left[ \varphi^2_{\rm{color}_{\rm i,j}} + \varphi^2_{\rm{Mag}_{\rm i,j}}\right] \right\} $,
\end{center}
Since the exponential function can reach high values quickly, it is convenient to work with Bayes' theorem in the logarithmic form:
\begin{center}
$\displaystyle \ln{P(M|D)} = \ln{P(M)} + \mathcal{L} $.
\end{center}
\paragraph{ Priors}
The prior distributions ($P(M)$) are the main difference between
the Bayesian and the frequentist statistics. These distributions impose constraints on the free parameters, restricting the set of parameters to be explored. In an isochrone fitting, these priors reflect the physical constraints, such as: \textit {(a)} the upper age limit as the age of the Universe \citep{planck16}; \textit{(b)} the metallicity values taken from high-resolution spectroscopy; \textit{(c)} distances constrained and primordial He content from RR Lyrae mean magnitudes, for example; and \textit{(d)} non-negative reddening values.
\paragraph{ Marginalization}
In order to explore the parameter space as a whole and to get the posterior distributions of each parameter, we applied the Bayes' theorem with the Metropolis-Hastings (MH) algorithm \citep{metropolis53, hastings70}. The method is basically an exclusion iterative algorithm, built firstly to solve problems of statistical physics. The MH method compares the random probabilities trying to reach the minimum energy state, which justifies that we can neglect the normalization term of the Bayes' law. The final result of MH is a chain with $n$ energies for $m$ states that is known as Markov chain. For the applications with random distributions, which means Monte Carlo methods, the result from the MH algorithm is called Markov chain Monte Carlo \citep[MCMC,][]{hogg18}. To get the probability distributions of the parameters, the marginalization is executed by the integral:
\begin{center}
$\displaystyle \mathcal{P}(\overrightarrow{\phi}) = \int \mathcal{L}(\overrightarrow{\phi})\times p(\overrightarrow{\phi})\, {\rm d}\overrightarrow{\phi} $,
\end{center}
where $(\overrightarrow{\phi})$ represents the parameter space. To perform the marginalization from MH algorithm and MCMC method, we employed the \texttt{Python} library \texttt{emcee} \citep{foreman-mackey13}.
\subsection{Multiple Stellar Populations in GCs}\label{subsec:mptagging}
Before carrying on the analysis of MPs, in this section we describe the separation of stellar populations in the CMDs. The stellar population tagging allows us to distinguish the first (1G) and second (2G) generation stars (and subsequent ones) from a given CMD. Figure \ref{fig:msp-do} shows the procedure we follow to separate the stellar populations in each region of the created synthetic CMD with $\Delta \tau_{\rm 1G,2G} = 0.50$ Gyr. We adopted a Dartmouth (DSED) isochrone with [Fe/H]$=-1.26$, $E(B-V)=0.18$, $(m-M)_0=14.38$, and $\tau=13.0$ Gyr.
In Milone et al. (2013) the pseudo-color C was defined, with the purpose to maximize the separation among MPs on the CMD. Piotto et al. (2015) have shown the power of {\it HST} UV filters F275W, F336W, and F438W to separate the MPs. F275W is sensitive to OH and F438W to CN and CH. For these filters, the 1G stars are fainter than the 2G because the latter are oxygen- and carbon-poorer than the 2G ones. For the filter F336W, which is sensitive to NH, the 1G stars are brighter than the 2G stars, given the fact that the 2G stars are nitrogen-richer.
Note that stronger lines lead to larger opacity, and lower brightness. For these reasons, the color (F275W-F438W) inverts the stellar populations on the CMD with respect to the color (F336W-F438W). In that color, the 2G stars seem to be redder than the 1G stars (Piotto et al. 2015, their Figure 2).
\paragraph{Chromosome maps (RGB and MS)}
\citet{milone17} describe the method of MP separation using
chromosome maps based on combinations of UV HST filters.
\citet{lee19} used UBV data to distinguish MPs, and
reviewed methods discussed earlier. To construct the chromosome map diagrams, we adopt the method presented in \citet{milone17} that is briefly described below. For the CMDs m$_{\rm F814W}$ vs. C$_{F275W,F336W,F438W}$ and m$_{\rm F814W}$ vs. (m$_{\rm F275W} - \rm m_{\rm F814W}$), the red and blue fiducial lines are defined by $96^{th}$ and $4^{th}$ percentiles, respectively. The top- and bottom-middle panels of Figure~\ref{fig:msp-do} show the red and blue fiducial lines enclosing the RGB and MS stars, respectively. The axis of chromosome map are the relative distance between each stars and the fiducial lines, defined by:
\begin{center}
$ \displaystyle \Delta_{C\,F275W,F336W,F438W} = \frac{C_{\rm r} - C}{C_{\rm r} - C_{\rm b}} $,
\end{center}
\begin{center}
$ \displaystyle \Delta_{F275W,F814W} = \frac{G - G_{\rm r}}{G_{\rm r} - G_{\rm b}} $,
\end{center}
where the indices $r$ and $b$ refer to the red and blue fiducial lines, respectively. The color $G$ represents m$_{\rm F275W} - \rm m_{\rm F814W}$.
The diagram $\Delta_{C\,F275W,F336W,F438W}$ vs. $\Delta_{F275W,F814W}$ quantifies the color distance of each star to the blue and red envelopes, so that the $\Delta$-value is
closer to zero as the star is closer to the red envelope. The right panels of Figure~\ref{fig:msp-do} show the final chromosome maps for the RGB (top) and MS (bottom), respectively, for the synthetic CMD.
Some modifications on the identification of the MPs were implemented in the
original method from \cite{milone17}, in order to preserve a uniformity in the MPs separation for the three evolutionary stages (MS, SGB, RGB). The identification of the MPs is done using the Gaussian Mixture Models (GMM), that is a non-supervised machine learning algorithm, which searches to fit $K$ Gaussian distributions to a sample of $N$ data. The fit comes from the basic equation of the Bayes' theorem:
\begin{center}
$ \displaystyle G(x) = \sum_{i=1}^{K} \phi_i \times \mathcal{N}\left( x\, |\, \mu_i, \sigma_i \right) $,
\end{center}
where $\mathcal{N}( x\, |\, \mu_i, \sigma_i )$ represents the ith Gaussian distribution with mean of $\mu_i$ and standard deviation of $\sigma_i$. This algorithm was adopted from the \texttt{python} library \texttt{Scikit-learn} \citep{pedregosa11}.
We here assume two subclasses for GMM in a two-dimensional plane. Then, each star is classified as 1G or 2G according to the strength of the two Gaussian distributions on that point of the chromosome map. The separation between the two {populations} includes clear members of both, but as well stars in the limiting intersection, that can contaminate each other samples. This analysis can be improved by increasing the number of subdivisions in GMM to select the bona-fide stars of each stellar populations, as in \cite{milone18}.
\paragraph{Two-color diagrams (SGB)}
Since the SGB sequence, depending on the adopted filter and the metallicity of the cluster, could be nearly horizontal and their MPs could appear mixed, the chromosome maps are not effective with these stars. Therefore,
we applied a conventional two-color diagram $m_{\rm F336W}-m_{\rm F438W}$ vs.
$m_{\rm F275W}-m_{\rm F336W}$, as described in \citet{nardiello15b}. In order
to apply the GMM procedure (same as described in the previous section), $\Delta_1$ and $\Delta_2$ are the axes that were
normalized and then rotated counterclockwise by an angle of 45$^{\circ}$. The method is graphically represented in Figure~\ref{fig:msp-do} (middle panels).
\begin{figure*}
\centering
\includegraphics[scale=0.53]{syn_msp.png}
\caption{ MP separation and population tagging applyed to synthetic data with $\Delta \tau = 0.5$ Gyr. Left panel shows the pseudo-color C, which gives a pronounced MP separation. Middle panels show the procedure we apply to separate the stellar populations, from top to bottom are the RGB, SGB, and MS stars, respectively. Right panels show the stars identified to belong to the 1G and 2G.}
\label{fig:msp-do}
\end{figure*}
\subsection{Age difference $\Delta\tau$ }\label{sec:deltatau}
The origin of the 2G (and subsequent {populations}) stars is a major challenge in the MP analyses. Most scenarios trying to explain MP formation predict an age difference ($\Delta \tau$) between the first and the later {populations} \citep{bastian18}. For example, the scenario of Asymptotic Giant Branch (AGB) stars polluting the second and subsequent {populations},
predicts a difference around 100 Myr \citep[][]{dantona16}, up to 200-700 Myr from the delay of X-ray binaries \citep{renzini13, renzini15}. Another scenario is that of the supermassive stars (SMS). Multiple stellar {populations} can be formed from multiple bursts of SMSs with intervals of a few Myr \citep{gieles18}. Another possibility are the fast rotating massive stars (FRMSs) that would enrich the interstellar medium in about 40 Myr \citep[][]{decressin07,krause13}. Therefore, the age difference between the first and next {populations} is an important parameter to give hints to their plausible origin.
From our population tagging method, we can analyse separately each {stellar population} from their CMDs. To perform the isochrone fitting in the context of MPs we developed a hierarchical algorithm to estimate the $\Delta \tau$ between the first and subsequent {populations}. The hierarchical algorithm considers the stars as a SSP
first, and subsequently each {stellar population}. For a SSP we leave all parameters free. In the context of MPs, it is expected that the age of a SSP is a weighted average age of each stellar population. Consequently, for the example of two stellar {populations}, the ages could be derived from:
\begin{center}
$ \displaystyle
\begin{array}{l c l}
\tau_{\rm 1G} & = & \tau_{\rm SSP} + \Delta \tau \times \left({\rm N}_{\rm 1G}/{\rm N}_{\rm total}\right), \\
& & \\
\tau_{\rm 2G} & = & \tau_{\rm SSP} - \Delta \tau \times \left(1 - {\rm N}_{\rm 1G}/{\rm N}_{\rm total}\right). \\
&
\end{array}
$
\end{center}
The hierarchical method fits the first {population} and applies the constraints of distance, reddening, and metallicity to the second (or subsequent) one(s). Hence, the procedure to compute the $\Delta \tau$ turns out simply to be $\Delta \tau = \tau_{\rm 1G}-\tau_{\rm 2G}$. This procedure considers that 1G stars were formed earlier than others, which is logical when our objective is to estimate a $\Delta \tau$. The likelihood of hierarchical procedure $\ln{\rm P(M|D)}$ takes into account the constraints of a stellar cluster as a whole. For example, all stars must have the same values of distance and must be influenced by interstellar dust in the same way. Therefore, the likelihood of 1G ($\mathcal{L}({\rm 1G})$) and NG ($\mathcal{L}({\rm NG})$) are dependent on the likelihood of SSP ($\mathcal{L}_{\rm SSP}$). The total likelihood $\ln{\rm P(M|D)}$ is a linear combination of the priors and the likelihood
of each {stellar population} with influence of SSP parameters:
\begin{center}
$\displaystyle \ln{\rm P(M|D)} = \ln{\rm P(M)} + \sum_{i=1}^{N} \left[ \mathcal{L}({\rm \, [i]G\, })_{\rm SSP} + \ln(f_{[i]G}) \right] $.
\end{center}
where $f_{[i]G}$ represents the fraction of stars that belong to the $i$-th population. A similar likelihood based on MPs and weighted by the fraction of stars is applied in \citet{ramirez-siordia19}.
Here, we are adopting that the 1G stars have primordial helium content (Y), which is consistent with the literature \citep{bastian18}. \citet{wagnerkaiser16} performed a bayesian isochrone fitting, in the context of MPs, for a sample of 30 GCs. Differently from the present work, they fitted the value of Y for the 1G stars, resulting in some cases in a high content of $Y_{1G} \sim 0.30$. They also assumed the same age for both analysed stellar populations. On the contrary, we are interested in finding if there is an age difference between the stellar populations. Even though our approach is similar to the one applied in \citet{wagnerkaiser16}, the methods are based on different assumptions.
\section{Controlled Experiment}\label{sec:tests}
In this Section, we test the reliability of our analysis by using synthetic CMDs.
First, we constructed a synthetic CMD using an error function obtained from the atlas extracted by \citet{nardiello18} from the data of the {\it HST} UV-Legacy Survey of Galactic Globular Clusters \citep{piotto15}, allowing us to simulate MPs with the synthetic data. The stellar evolutionary model adopted was the DSED isochrone with Z $\sim$ 0.002 with [$\alpha$/Fe] = +0.4, and age of 13.0 Gyr, as reported in Table~\ref{tab:my_label}, corresponding to typical values of moderately metal-poor bulge GCs \citep[e.g.][]{kerber18,kerber19}. We simulated the CMD of a cluster with a total number of $10,000$ stars (N$_{total}$) that host ~36\% of 1G stars with an age of 13.0 Gyr and 64\% of 2G stars 0.5 Gyr younger than 1G stars. We considered a fraction of binaries (f$_{\rm bin}$) of $30 \%$ and a minimum mass ratio (q$_{\rm min}$) of $0.60$. Resulting CMDs combining the different available filters are shown in Figure~\ref{fig:cmd-mosaic}.
\begin{figure*}
\centering
\includegraphics[scale=0.59]{synthetic_mosaic.png}
\caption{CMDs for the Synthetic Data using a DSED isochrone with age $= 13.0$ Gyr, [Fe/H]$ = -1.26$, $E(B-V) = 0.18$, $(m-M)_0 = 14.38$, $\Delta \tau = 0.50$ Gyr and fraction of 1G stars ($N_{1G}/N_{total}$)$= 0.360$, generated from {\it HST} filters. All available combinations of filters are shown.}
\label{fig:cmd-mosaic}
\end{figure*}
\input{TAB-MockParams.tex}
\subsection{Sources of uncertainty}
In our method, during the isochrone fitting, we compute the likelihood star-by-star. To keep the high performance of MCMC, we imposed a range in magnitudes based on stellar evolutionary models. The third panel of Figure~\ref{fig:params} shows that there is no significant difference regarding the age for the $\sim 3$ magnitudes brighter than the TO. For this reason, we do not take into account stars above this limit in the likelihood calculation.
The faintest stars are limited to the completeness limit, meaning that the number of faint stars depends on the photometric depth. There are no differences between the isochrones in the databases employed in \texttt{SIRIUS} for the faintest stars ($\sim 2$ magnitudes below the TO), therefore the fit does not depend on the faintest stars. \cite{ramirez-siordia19} presented an analysis considering the faintest stars. They concluded that the effect of faintest stars only increases the uncertainties without changing the mode of distribution, since the isochrones do not seem to be different for the faintest stars, as shown in Figure~\ref{fig:params} (third panel).
As regards binary stars, their magnitudes represent the combination of the fluxes from the two companion stars. Since the magnitude is the logarithm of the stellar flux, for a binary system with two stars of the same mass, the magnitude of this system corresponds to the magnitude of one star subtracted by $2.5\times \log(2) \sim 0.75$ \citep{kerber02, kerber07}. The decrement in magnitude tends to have the binary stars to be brighter and redder on the CMD. To reduce the effect of binary systems during the isochrone fitting, \texttt{SIRIUS} takes into account only the stars within $3\sigma$ from the fiducial line of the CMD.
The standard BaSTI isochrones overestimate ages by $\sim 0.80$ Gyr, with
respect to DSED isochrones. The main reason for this discrepancy is that BaSTI isochrones do not include atomic diffusion in the calculations,
among other differences in basic physics. Whereas the solar alpha-to-iron more complete models, including atomic diffusion are already
available in \citet{hidalgo18}, the available alpha-enhanced models taking this effect into account are not yet available.
\subsection{Sanity Check}
In the optical wavelengths some filters are more sensitive to some properties than others. For the NIR filters the effect of interstellar medium extinction is considerably lower than for the UV filters. Also, a color combining filters with a small band width is more suitable to observe the structures on the CMD. Therefore, the combination of magnitudes and colors on the CMD is very important regarding the information that is expected to be obtained from isochrone fitting. In order to estimate the effect of the choice of color we performed the isochrone fitting using ten different colors, without spreading the stars, combining the five {\it HST} filters available in the
UV Legacy survey of globular clusters \citep{piotto15}.
Firstly, we perform the fit considering the SSP without taking into account the photometric spread of stars. The DSED isochrones are here fitted to the synthetic No-Spread catalogue data (Table~\ref{tab:my_label}) with the purpose of checking if the input parameters of the synthetic CMD are recovered. For this test, we adopted uniform distribution priors for all parameters. The range of values we used are: for age, between 10 to 15 Gyr; for the metallicity, between 0.00 to $-2.00$ dex; for reddening, between 0.0 to 1.0 mag; and for the distance modulus, between 12.0 to 16.0 mag. Figure~\ref{fig:sanity-results} shows the behavior of the parameter space as a function of color. It can be observed that the age is the most sensitive parameter to the filters, whereas the other parameters vary only slightly with the choice of filters. For color 8 (third lower panel in Fig. \ref{fig:cmd-mosaic}), which is equivalent to B-V, there is a strong effect on the age, whereas for color 6
(first lower panel in Fig. \ref{fig:cmd-mosaic}) the parameters are closer to the original ones. Color 10 (m$_{\rm F606W}-{\rm m}_{F814W}$, last lower panel in Fig. \ref{fig:cmd-mosaic}), is also close to the input values and has small uncertainties due to its lowest reddening-dependency. Therefore, for our analysis, we chose color 10.
\begin{figure*}
\centering
\includegraphics[scale=0.45]{sanity_colors.pdf}
\caption{Sanity check with no-spread data, the parameter space as function of color. The posterior distributions of each parameter for the ten combinations of {\it HST} filters of the UV Legacy survey of globular clusters \citep{piotto15}. DSED isochrones are adopted. The numbers represent each color.}
\label{fig:sanity-results}
\end{figure*}
Secondly, to verify the sensitivity of the method, we simulate real data through synthetic CMDs
to perform the isochrone fitting, taking into account a spread of stars, and assuming Gaussian priors centered on the parameters given in Table \ref{tab:my_label} (Spread). In Figure \ref{fig:syn_fits}, we show the isochrone fitting for the synthetic CMD with $\Delta \tau = 0.50$ Gyr, assuming that it is SSP (left panel) and MPs (right panel).
We employ the corner-plots to present the posterior distributions
(Figure \ref{fig:sanity-dsed}). They show the $N$ parameter space in a 2D representation, where it is possible to see the correlations between the parameters. As the best value for each parameter we adopted the mode of the distributions. For the confidence interval, we selected the 16$^{\rm th}$ and 84$^{\rm th}$ percentile of the distributions that give us the values inside $1\sigma$ from the mode. The top-left panel in Figure \ref{fig:sanity-dsed} shows the corner-plot for the DSED SSP isochrone fitting.
Figure~\ref{fig:sanity-dsed}, in the top-right, bottom-left, and bottom right panels show the results for the age derivation in the context of MPs using DSED.
\begin{figure}
\centering
\includegraphics[scale=0.5]{syn_fits.png}
\caption{Sanity check with spread data, isochrone fitting for the synthetic CMD considering SSP (left) and MPs (right) for DSED isochrones. The grey dots are discarded for the fit.}
\label{fig:syn_fits}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale=0.5]{F606W_F606WF814W_dsed_Mul_chain-ver0-20190801-111610_20_1000_pop0.pdf}
\includegraphics[scale=0.5]{F606W_F606WF814W_dsed_Mul_chain-ver0-20190801-150911_20_1000_pop12.pdf}
\includegraphics[scale=0.5]{F606W_F606WF814W_dsed_Mul_chain-ver0-20190801-123930_20_1000_pop12.pdf}
\includegraphics[scale=0.5]{F606W_F606WF814W_dsed_Mul_chain-ver0-20190801-160425_20_1000_pop12.pdf}
\caption{Sanity check 2, corner plots using DSED isochrones, relating physical parameters. Top left panel: results of the sanity check applied to a synthetic SSP CMD
where Monte Carlo spread of data is implemented, with a $\Delta \tau = 0.50$ Gyr. Other panels: 1G and 2G combined for $\Delta\tau = 0.10$ Gyr (top right), $\Delta\tau = 0.50$ Gyr (bottom left), and $\Delta\tau = 1.50$ Gyr (bottom right).
}
\label{fig:sanity-dsed}
\end{figure*}
\input{TAB-ResSyn.tex}
Even though the spread of stars changes the visual aspect of the CMD, the parameters obtained from the isochrone fitting given in Table \ref{tab:sanity2} for SSP and MPs, are both in good agreement with the input values from Table~\ref{tab:my_label}.
In conclusion, in this section we were able to describe the approach and check the validity of \texttt{SIRIUS} in the context of MPs.
\section{Application to the halo globular cluster NGC~6752}\label{sec:app}
{\it HST} photometric data for NGC~6752 in the ultraviolet (UV) filters within the UV-Legacy Survey GO-13297 (PI. G. Piotto), and in the optical within GO-10775 (PI. A. Sarajedini) are used. These programs made available data in the UV filters F275W, F336W, and F438W from the Wide Field Camera 3 (WFC3), and the optical filters F606W and F814W from the Wide Field Camera of the Advanced Camera for Survey (WFC/ACS). The newly reduced catalogs presented in \citet{nardiello18}
are used.
NGC 6752 is a halo cluster, located at l = 336$^\circ$49, b = -25$^\circ$63,
with a distance from the Sun d$_{\rm \odot}$ = 4.0 kpc \citep[][edition 2010]{harris96}\footnote{www.physics.mcmaster.ca/~harris/mwgc.dat}.
A metallicity of [Fe/H]$=-1.48\pm0.07$ dex was derived by \citet{gratton05} from high resolution spectroscopy ($R = 40,000$) of seven stars near the red giant branch bump. \citet{gratton03} and \citet{vandenberg13} obtained an age of $12.50\pm0.25$ Gyr and $13.4\pm1.1$ Gyr, respectivaly.
\citet{carretta12} identified three stellar populations based on three values of abundances of O, Na, Mg, Al, and Si elements that are sensitive to stellar {populations} in GCs, denominated as first (P), intermediate (I), and extreme (E) populations. \citet{milone13} gave the first photometric evidence of three stellar populations by using \textit{HST} data. \citet{nardiello15a}, using FORS2/VLT data, have observed the split of the MS of NGC~6752 using UBI filters, and calculated the radial distribution of the populations and the difference in helium between the 1G and 2G stars. \citet{milone19} confirmed the existence of three stellar populations from NIR photometric data on MS stars. \citet{cordoni19} analysed the kinematics of the P and E populations of NGC~6752, and they found that there is no difference in rotation between the two stellar populations.
In order to separate the populations P, I, and E (hereafter 1G, 2G, and 3G), the number of components on GMM were increased to three for the RGB and SGB, and to four for the MS. The classification of 1G, 2G, and 3G stars is in agreement with \citet{milone13}, since a clear distinction of three stellar populations
can be verified in Figure~\ref{fig:ngc6752_mps}. \citet{milone13} derived the mass fraction of each population to be of $\sim 25$, $\sim45$, and $\sim30$ per cent, respectively. We found a fraction of stars of $25\pm5$, $46\pm7$, and $29\pm5$ per cent for the 1G, 2G, and 3G, respectively, in excellent agreement with \citet{milone13}.
\begin{figure*}
\centering
\includegraphics[scale=0.42]{ngc6752_mps_2.png}
\caption{Multiple stellar populations in NGC~6752. Left panel:
SSP; Middle panel: same as left panel, but color-identified stars;
Right panel: pseudo-color showing the clear separation of three
stellar {populations}.}
\label{fig:ngc6752_mps}
\end{figure*}
In the following the analysis of NGC 6752 is restricted to DSED isochrones.
The procedure starts with the isochrone fitting assuming
the CMD to consist of a SSP, and the method is subsequently applied to the MPs. In order to carry out the isochrone fitting, we employed the same CMD m$_{\rm F606W}$ vs. (m$_{\rm F606W}-{\rm m}_{F814W}$) used for the synthetic-data. In the left panel of Figure~\ref{fig:ngc6752_mps} is shown the CMD of NGC~6752 including all stars
as a SSP.
The value of [Fe/H] = $-1.48$ dex was used as prior through Gaussian distribution with standard deviation of $0.07$. A prior in distance was applied with the value of apparent distance modulus $(m-M)_{\rm V} = 13.26\pm0.08$ taken from \citet{gratton03}. The results of SSP isochrone fitting are shown in Table~\ref{tab:ngc6752} and Figures \ref{fig:ngc6752-dsed_ssp} and \ref{fig:ngc6752-dsed_mp}.
The SSP age derivation of $13.7 \pm 0.5$ Gyr is in good agreement with \citet{gratton03}, that obtained $13.4\pm1.1$ Gyr, and with the Bayesian technique from \citet{wagnerkaiser17} that resulted in an age of $13.202^{+0.174}_{-0.152}$ Gyr. The parallax from Gaia DR2 \citep{gaia18b} for the NGC~6752, $\bar{\omega} = 0.2610\pm0.0011$ mas, corrected by the zero point of $-0.03$ mas given by \citet{lindegren18}, gives a heliocentric distance of $3.85\pm0.02$ kpc. Considering NGC~6752 as a SSP, the derived distance is $4.11\pm0.08$ kpc, in agreement within $~3\sigma$ with Gaia DR2.
The metallicity estimated from SSP isochrone fitting, [Fe/H] $ = -1.49^{+0.05}_{-0.05}$, was fixed for the MPs approach. The metallicity can be fixed because no [Fe/H] variation is detected in this cluster.
To derive the age difference between the stellar {populations}, the hierarchical likelihood described in Section \ref{sec:deltatau} with $N=3$ is applied. The fit is carried out simultaneously to 1G, 2G, and 3G. Firstly, we consider the primordial helium content value for all populations.
In a second run, we assume a helium enhancement by a type of polluter star, changing the amount of helium for each generation, according to values computed by \citet{milone19}: $\delta \rm Y_{\rm 1G,2G} = 0.010$ and $\delta \rm Y_{\rm 1G,3G} = 0.042$ for the 2G, and 3G, respectively (Figures \ref{fig:ngc6752_mps}, \ref{fig:ngc6752-dsed_mp}, and Table \ref{tab:ngc6752}). We assumed the helium enhancement values from Milone et al. (2019) since they were derived using the same DSED stellar evolutionary models employed here, therefore there is compatibility. For the metallicity of
NGC~6752, the corresponding canonical helium content in the DSED isochrones is $0.247$, which was associated to 1G. The 2G and 3G helium contents were assumed to be of $0.257$ and $0.289$, adopting the $\delta \rm Y$ values from \citet{milone19}.
\input{TAB-Results2.tex}
\begin{figure*}
\centering
\includegraphics[scale=0.57]{ngc6752_ssp.png}
\includegraphics[scale=0.5]{ngc6752_ssp_corner.pdf}
\caption{ Results for the SSP analysis of NGC~6752. Left panel: CMD with the result from isochrone fitting, green line is the most probable solution, and the blue strip is the solutions within 1$\sigma$. Right panel: The posterior distributions.
}
\label{fig:ngc6752-dsed_ssp}
\end{figure*}
\begin{figure*}
\includegraphics[scale=0.5]{ngc6752_fixY_corner.pdf}
\includegraphics[scale=0.5]{ngc6752_varY_corner.pdf}
\caption{Corner plots for NGC~6752. Left panel: simultaneous fitting of the three stellar populations, adopting canonical helium abundance; Right panel: same as in left panel, but taking into account helium abundance differences.
}
\label{fig:ngc6752-dsed_mp}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[scale=0.5]{NGC6752Result-MPs.png}
\caption{Isochrone fitting for NGC~6752. Left panel: MPs all together.
Second to fourth panels: isochrone fitting to 1G, 2G, and 3G.
Upper panels: Canonical helium. Lower panels: Enhanced helium. The strips are the solutions within $1\sigma$.}
\label{fig:ngc6752-bestfit}
\end{figure*}
Table \ref{tab:ngc6752} and Figure \ref{fig:ngc6752-bestfit} provide the results of isochrone fitting to the MPs. The derived distances using canonical helium and helium enhanced are $4.13\pm0.06$ and $4.11\pm0.08$ kpc, respectively. The latter distance determination is in agreement with the distance from the inverse Gaia DR2 parallax \citep{gaia18b} (see above).
We derive age differences of $\Delta \tau_{\rm 1G,2G} = 300\pm400$ Myr,
and $\Delta \tau_{\rm 1G,3G} = 500\pm400$ Myr, relative to the age of 1G stars, considering that there is no helium enhancement within the GC. However, taking into account the GC helium enhancement cf. \citet{milone19},
and noting that the method fits the three {stellar populations} simultaneously,
the 1G is less old (even if its He is still canonical), and
the age differences are of $\Delta \tau_{\rm 1G,2G} = 200\pm400$ Myr,
and $\Delta \tau_{\rm 1G,3G} = 500\pm400$ Myr. These results could give hints on the possible mechanism of GC internal pollution.
{It is interesting to note that, for the He enhanced populations, the result is similar to those with no He enhancement. Assuming the primordial helium for the 1G, 2G, and 3G stars, the $\chi^2$ values are $0.10$, $0.13$, and $0.12$, respectively, resulting in a total value of $0.35$. For He enhanced isochrones, the values of $\chi^2$ are $0.09$, $0.14$, and $0.11$, for the 1G, 2G, and 3G stars, respectively and with a total of $0.34$. Therefore, the fitting using He enhanced isochrones are similarly well-fit}.
Even though the uncertainties on the age derivation do not take into account the differences
between the stellar evolutionary models, our uncertainty determinations are of the same order of magnitude as those by \citet{monty18}. Given that we did not propagate the uncertainties
from the grid size of the parameter space, the uncertainties given here are the formal errors from MCMC algorithm and they are larger than the ones reported by \citet{wagnerkaiser17}.
\section{Conclusions}\label{sec:conclusion}
We have developed the \texttt{SIRIUS} code to extract the maximum information from CMDs of stellar clusters, through a detailed analysis.
\texttt{SIRIUS} was tested in terms of synthetic data.
High precision parameter derivations were obtained with sanity checks that demonstrate the good performance of the code. Small fluctuations of the solutions were found in terms of the choice of CMD colors, relative to the input parameters of the synthetic data (Figure \ref{fig:sanity-results}). Applying a Monte Carlo spread of stars, these
fluctuations increase somewhat, as can be seen in Table \ref{tab:sanity2}.
In any case, the solution obtained is within the uncertainties and limited because of the grid resolution in the parameter space.
The \texttt{SIRIUS} code is applied to analyse the halo
globular cluster NGC 6752 of metallicity [Fe/H]$\approx$-1.49.
Three stellar populations are identified, confirming previous findings
by \citet{carretta12} from spectroscopy, and \citet{milone19} from photometry.
The age derivation of the three stellar {populations}, taking into account
He abundance differences from \citet{milone19}, results to
be of $200/300\pm400$ Myr between 1G and 2G and between 2G and 3G.
This points to a possible interpretation of having the same mechanism
producing 2G, and later the 3G.
Many authors have extensively
discussed the probable candidates to produce the chemical abundance
patterns of second (and subsequent) stellar {populations} from self-enrichment of the cluster. The main
candidates are the AGB stars,
and SMS, in both cases through their winds, as well as FRMSs \citep[][]{decressin07, krause13}. All of them predict an age difference between the stellar populations.
In conclusion, given the uncertainties in the models
of pollution, and the uncertainties in the age difference
derived from the CMDs, it is not possible to firmly indicate a
scenario for the formation of a second {stellar population}. The age differences derived for NGC 6752 could be compatible with the AGB scenario if only the best value determinations are taken into account. However, considering the uncertainties, the results could be compatible with all scenarios regarding the origin of MPs (SMS and FRMS), even those with no age difference.
Further analyses of age differences of multiple stellar
populations are of great interest. In particular, within
the {\it HST} Legacy survey collaboration, \citet{nardiello15b} derived the relative age of NGC~6352 MPs from $\chi^2$ minimization isochrone fitting, assuming each of them as SSPs, and Oliveira et al. (2019, in preparation) apply the methods described here to derive the ages for seven bulge globular clusters and their MPs.
\acknowledgments
{We acknowledge the anonymous referee for a detailed review and helpful suggestions, which allowed us to improve the manuscript.} SOS acknowledges the FAPESP PhD fellowship 2018/22044-3. LOK and BB acknowledge partial financial support from FAPESP, CNPq, and CAPES - Finance Code 001. APV acknowledges the FAPESP postdoctoral fellowship no. 2017/15893-1. RAPO acknowledges the FAPESP PhD fellowship no. 2018/22181-0. DN acknowledges partial support by the Universit\`a degli Studi
di Padova Progetto di Ateneo BIRD178590. APV and SOS acknowledge the DGAPA-PAPIIT grant IG100319.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,118
|
Download cimatrone from ZippyShare, Uploaded, Torrent & Direct Download. View the links and download below.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,664
|
Q: Mypy type checker and "static instances" For a class A I wrote, there are some instances foo and bar that I want to be accessible through A.foo and A.bar as class variables. However, foo and bar are both instances of A, and I'm not sure how to let the typechecker mypy handle this correctly. I currently instantiate foo and bar as follows:
class A:
def __init__(self):
pass
foo = None
bar = None
A.foo = A()
A.bar = A()
Which leads mypy to conclude that A.foo and A.bar are of type None. Annotating as Optional[A] would work, but that's misrepresenting what is intended: I want both to be of type A... Any tips?
A: If your using a higher version of python 3, you can use annotations to do this for you.
foo : A
I think mypy works with standard annotations. If this doesn't work, then try surrounding the annotation with quotes.
foo : "A"
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,077
|
The Nurture of Nature
392 pages, 6 x 9
38 b&w photos, 1 map
Release Date:01 Jan 2010
Release Date:10 May 2009
GO TO CART SAMPLE CHAPTER
Childhood, Antimodernism, and Ontario Summer Camps, 1920-55
By Sharon Wall
SERIES: Nature | History | Society
Thousands of children attended or worked at Ontario summer camps in the twentieth century. Did parents simply want a break, or were broader developments at play? The Nurture of Nature explores the history of an institution that shaped the lives of many and brings to light overlooked connections between the history of childhood, the natural environment, class cultures, and modern recreation and leisure.
Two competing cultural tendencies – antimodern nostalgia and modern enthusiasms about the landscape, child rearing, and identity – shaped the summer camp. Sharon Wall examines how this tension played out in the camp's interaction with the natural landscape, its class and gendered dimensions, its engagement with emerging ideologies of childhood, and in the politics of race and identity inherent in its "Indian" programming. By tracing the development of summer camps in Ontario, Wall brings new insights to a broader phenomenon: the divided consciousness that has informed modern assumptions about nature, technology, and identity.
A nuanced discussion of the summer camp's contribution to modern social life in North America, The Nurture of Nature is an essential resource for students and practitioners of history, sociology, and cultural studies as well as for anyone who has ever been packed off to camp and wants to explore why.
2010, Winner - Clio Prize (Ontario), Canadian Historical Association
2009, Winner - Floyd S. Chalmers Award in Ontario History, The Champlain Society
RELATED TOPICS: Canadian History, Communication & Media Studies, History, Ontario, Race & Ethnicity, Regional Studies, Social History, Sociology
The Nurture of Nature represents a major study of an important but neglected subject. It is an important contribution to the study of leisure and recreation in Canada, to the understanding of the character of modernity, and to the history of summer camps. Keith Walden, Department of History, Trent University
By seeking and revealing the cultural meanings of "fresh air" and "wilderness" camping," and of the activities in which campers engaged ... Sharon Wall has produced a multifaceted study that has much to say to historians of the environment. Time and again, The Nurture of Nature reveals the contradictory qualities of the summer camp, even as it offers new insights into the ways in which Canadians struggled to find meaning in modernity. From the Foreword by Graeme Wynn
Sharon Wall is an assistant professor of history at the University of Winnipeg.
Foreword: Modernism in Camp: A Wilderness Paradox / Graeme Wynn
1 Back to Nature: Escaping the City, Ordering the Wild
2 Socialism for the Rich: Class Formation at the Private Camp
3 "All they need is air": Building Health, Shaping Class at the Fresh Air Camp
4 Making Modern Childhood, the Natural Way: The Camp Experiment with Psychology, Mental Hygiene, and Progressive Education
5 Shaping True Natures in Nature: Camping, Gender, and Sexuality
6 Totem Poles, Tepees, and Token Traditions: "Playing Indian" at Camp
Conclusion: All Antimodern Melts into Modern?
Guiding Modern Girls
Girlhood, Empire, and Internationalism in the 1920s and 1930s
By Kristine Alexander
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,187
|
We spent Saturday morning and afternoon at Greg's mom's house with his brothers and their families. And then we went to the annual Taylor Christmas Party with Greg's grandpa and two great-uncles and all their descendants. I've told you before about the beauty of a large family: My daughter is able to befriend her second cousin once removed. We shared a delicious meal, caught up with each other and listened to kids play together.
That's Cate and Taylor — second cousins once removed.
Cate and Ethne are first cousins. And, yes, that's Ben crying in the background.
Evelyn is only seven months older than Ben. Yep, cousins who are both 2 is quite an adventure. But I think we wore them out because they both cuddled up with Greg during the basketball game that ended our family day. Oh, have you heard? Our Racers are 12-0 and ranked 24th in The Associated Press poll.
It was a great day of togetherness across the family tree.
There are more pictures of our celebrating this weekend and throughout December here. Want more? Subscribe to get "Insights" in your inbox. Or follow me on Twitter.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,249
|
package com.alipay.api.response;
import com.alipay.api.internal.mapping.ApiField;
import com.alipay.api.AlipayResponse;
/**
* ALIPAY API: alipay.marketing.campaign.discount.operate response.
*
* @author auto create
* @since 1.0, 2017-03-03 16:48:01
*/
public class AlipayMarketingCampaignDiscountOperateResponse extends AlipayResponse {
private static final long serialVersionUID = 1667384332532581343L;
/**
* 123
*/
@ApiField("camp_id")
private String campId;
public void setCampId(String campId) {
this.campId = campId;
}
public String getCampId( ) {
return this.campId;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,087
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Letourneuxia is a genus of large air-breathing land slugs, terrestrial pulmonate gastropod mollusks in the family Arionidae, the roundback slugs.
Species
This genus is monotypic, containing the single species
Letourneuxia nyctelia (Bourguignat, 1861)
Letourneuxia numidica Bourguignat, 1866 is now considered a junior synonym of L. nyctelia.
Letourneuxia moreleti (P. Hesse 1884) is considered either as another synonym of L. nyctelia or as a species in the genus Geomalacus.
References
Further reading
Molluscos Terrestres. Libro Rojo de los Invertebrados de Andalucía. pages 612-614.
External links
AnimalBase info
Arionidae
Gastropod genera
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,258
|
\section{Introduction}
If $V$ is a representation of the cyclic group $\Ct$, then the Grassmannian $\Gr_k(V)$ inherits a $\Ct$ action. We wish to compute the $RO(\Ct)$-graded Bredon cohomology of these equivariant spaces for various $k$ and $V$. In this paper we present formulas for the cohomologies of two infinite families of finite Grassmannians on real representations, their complex analogs, and also the cohomologies of analogous infinite-dimensional spaces. To do this we create an equivariant version of the Schubert cell construction, giving an equivariant cellular spectral sequence. In general, the differentials in such a spectral sequence are unknown. However, we find convenient situations whose differentials are actually manageable.
We will focus mostly on the real case, postponing complex Grassmannians until Section \ref{complexsection}. The group $\Ct$ has two irreducible real representations: $\R_{\triv}$ with trivial action, and $\R\sgn$ on which the nontrivial group element acts as multiplication by $-1$. The $RO(C_2)$-graded cohomology can therefore be regarded as bigraded. Let $$\R^{p,q}=(\R_{\triv})^{p-q}\oplus(\R\sgn)^q.$$ Our cohomology theory is graded by both actual and virtual representations, so that a space $X$ with a $\Ct$-action has cohomology groups $\H pq(X;\underline{M})$ for any integer values of $p$ and $q$ and any Mackey functor $\underline{M}$. We will refer to $p$ and $q$ as the \textbf{topological dimension} and \textbf{weight}, respectively, and will sometimes use $|x|$ and $w(x)$ to denote the topological dimension and weight of a pair $x=(p,q)$. We will also refer to the \textbf{fixed-set dimension}, $p-q=|x|-w(x)$.\par
We denote the one-point compactification of a representation by $S^{p,q}=\widehat\R^{p,q}$, whose underlying space is a $p$-sphere and whose fixed set is a ($p-q$)-sphere (hence the definition of fixed-set dimension above).
We will be using the constant $\Zt$-valued Mackey functor throughout (analogous to $\Zt$ coefficients in singular cohomology), but these coefficients will be suppressed in the notation; we will write $\H pq(X)$ rather than $H^{\R^{p,q}} (X;\,\underline{\Zt})$. Note that $RO(\Ct)$-graded Bredon cohomology has a bigraded suspension isomorphism with respect to these representation spheres:
$$\rH\bullet\bullet(\S pq \Smash X)\iso \rH{\bullet-p}{\bullet-q}(X).$$
Non-equivariant singular cohomology will also appear, and similarly $H^\ast_{\text{sing}}(X)$ will always mean $H^\ast_{\text{sing}}(X;\,\Zt)$. \par
Let $\Gr_k(\R^{p,q})$ denote the manifold of $k$-planes in $p$-dimensional real space, with $C_2$-action induced by that on $\R^{p,q}$. We are interested in calculating $\H\bullet\bullet(\Gr_k(\R^{p,q}))$ as a module over $\Mt:=\H\bullet\bullet(\text{pt})$, the cohomology of a point. Because these spaces can be constructed from representation discs (as we will show in Section \ref{we2} using Schubert cells) their cohomology is known to be a free $\Mt$-module (see \cite{kronholm} or \cite{buddies}) comprised of suspensions $\Sigma^{a,b}\Mt=\rH\bullet\bullet(\S ab)$. And so
$$\H\bullet\bullet(\Gr_k(\R^{p,q}))=\bigoplus_{i}\Sigma ^{a_i,b_i}\Mt$$
where the total number of summands in topological degree $d$ is the rank of non-equivariant singular cohomology for the underlying space:
$$\#\{i:a_i=d\}=\rank H_{\text{sing}}^d(\Gr_k(\R^{p})).$$
However the associated weights $b_i$ were previously known in only a few easy cases. We produce formulas for more families of Grassmannians, namely those of the form $\Gr_k(\R^{n,1})$ and $\Gr_2(\R^{n,2})$. It should be noted that while in the non-equivariant case the Schubert-cell construction gives a chain complex with zero differentials, things will not be so simple here. Whether we progressively compute cohomologies of subspaces using cofiber sequences, or run a single spectral sequence for the Schubert cell filtration, we will in general see many nonzero differentials. \par
\subsection{Preliminaries}
The ground ring $\Mt$ of our theory is non-Noetherian, comprised of a polynomial subalgebra $\Zt[\rho,\tau]$ generated by elements $\rho\in\H11(\text{pt})$ and $\tau\in\H01(\text{pt})$, an element $\theta\in\H0{-2}(\text{pt})$ such that $\theta\rho=\theta\tau=\theta^2=0$, and also an infinite family of elements denoted $\frac{\theta}{\rho^i\tau^j}$ with the property that when $i'\le i$ and $j'\le j$, as the notation suggests, $\rho^{i'}\tau^{j'}\cdot \frac{\theta}{\rho^i\tau^j}=\frac{\theta}{\rho^{i-i'}\tau^{j-j'}}$.
We will want to draw pictures of this ring.
\begin{figure}[h]
\begin{tikzpicture}[scale=0.5]
\justaxis
\draw(0,0) node {$1$};
\draw(0,1) node {$\tau$};
\draw(1,1) node {$\rho$};
\draw(0,2) node {$\tau^2$};
\draw(1,2) node {$\rho\tau$};
\draw(2,2) node {$\rho^2$};
\draw(0,3) node {$\tau^3$};
\draw(3,3) node {$\rho^3$};
\draw(4,4) node {$\iddots$};
\draw(0.5,4) node {$\vdots$};
\draw(1.5,4) node {$\vdots$};
\draw(2.5,4) node {$\iddots$};
\draw(0,-2) node {$\theta$};
\draw(0,-3) node {$\frac\theta\tau$};
\draw(-1,-3) node {$\frac\theta\rho$};
\draw(-1,-4) node {$\frac\theta{\rho\tau}$};
\draw(-2,-4) node {$\frac\theta{\rho^2}$};
\draw(0,-4) node {$\frac\theta{\tau^2}$};
\draw(-3,-5) node {$\iddots$};
\draw(-0.25,-5) node {$\vdots$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\axisname{}
\HMtwo{0}{0}
\draw(0,0) node {$\bullet$};
\draw(0,1) node {$\bullet$};
\draw(1,1) node {$\bullet$};
\draw(0,2) node {$\bullet$};
\draw(1,2) node {$\bullet$};
\draw(2,2) node {$\bullet$};
\draw(0,3) node {$\bullet$};
\draw(1,3) node {$\bullet$};
\draw(2,3) node {$\bullet$};
\draw(3,3) node {$\bullet$};
\draw(0,4) node {$\bullet$};
\draw(1,4) node {$\bullet$};
\draw(2,4) node {$\bullet$};
\draw(3,4) node {$\bullet$};
\draw(4,4) node {$\bullet$};
\draw(0,-2) node {$\bullet$};
\draw(0,-3) node {$\bullet$};
\draw(-1,-3) node {$\bullet$};
\draw(-1,-4) node {$\bullet$};
\draw(-2,-4) node {$\bullet$};
\draw(0,-4) node {$\bullet$};
\draw(-3,-5) node {$\phantom{\iddots}$};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.5]
\justaxis{}
\draw(4,0) node {$p$};
\draw(0,5) node {$q$};
\HMtwo{0}{0}
\draw(-3,-5) node {$\phantom{\iddots}$};
\end{tikzpicture}\\
\caption{Several visual representations of $\Mt.$ Copies of $\Zt$ are represented with $\bullet$ in the middle representation. On the right-hand representation, the groups are merely implied.}
\label{fig:1}
\end{figure}
In the third part of Figure \ref{fig:1}, have labeled the $p$-axis (or dimension-axis) and the $q$-axis (or weight axis). We see the ring divided into a \textbf{top cone} consisting of elements of the form $\rho^i\tau^j$ and a \textbf{lower cone} of elements $\frac\theta{\rho^i\tau^j}$. Even this last representation can get messy, and so we will often abbreviate further. For example, we will see later that
$$\H\bullet\bullet(\Gr_2(\R^{4,1}))=\Mt\oplus\M11\oplus( \M21)^{\oplus 2}\oplus \M31\oplus \M42$$
and visualizing this free module will often be easier if we only worry about the generators of this free $\Mt$-module, as in Figure \ref{fig:shorthand}.
\begin{figure}[H]
\begin{tikzpicture}[scale=0.4]
\axisname{}
\HMtwo{0}{0}
\HMtwo{1.2}{0.9}
\HMtwo{2}{1}
\HMtwo{2.2}{0.9}
\HMtwo{3}{1}
\HMtwo{4.2}{1.9}
\end{tikzpicture}
\!\!\!\!\!
\begin{tikzpicture}[scale=.6]
\def 8 { 4 }
\def 4 { 3 }
\draw (0.5* 8 ,-1.5) node[anchor=south] {$ $};
\def(0.1, 0.1), (1.1, 1.1), (2.1, 1.1), (2.1, 2.1), (3.2, 2.1), (3.1, 2.2), (4.2, 2.1), (4.1, 2.2), (5.1, 3.1), (6.1, 3.1){(0.1, 0.1), (1.1, 1.1), (2.2, 1.1), (2.1, 1.2), (3.1, 1.1), (4.1, 2.1)}
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\foreach \pq in (0.1, 0.1), (1.1, 1.1), (2.1, 1.1), (2.1, 2.1), (3.2, 2.1), (3.1, 2.2), (4.2, 2.1), (4.1, 2.2), (5.1, 3.1), (6.1, 3.1)
\draw \pq [fill] circle (.04);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=.6]
\def 8 { 4 }
\def 4 { 3 }
\draw (0.5* 8 ,-1.5) node[anchor=south] {$ $};
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt);
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm);
\draw (0,0) node[above right] {$1$};
\draw (1,1) node[above right] {$1$};
\draw (2,1) node[above right] {$2$};
\draw (3,1) node[above right] {$1$};
\draw (4,2) node[above right] {$1$};
\end{tikzpicture}\\
\caption{Several visual representations of $\H\bullet\bullet(\Gr_2(\R^{4,1}))$. The last of these is called a \textbf{rank chart}.}
\label{fig:shorthand}
\end{figure}
\begin{warning}
The shorthand in the second and third diagrams of Figure \ref{fig:shorthand} can be a mercy, but also runs the risk of deception, as certain bidegrees appear ``empty'' but aren't. For example, while it is clear from the leftmost diagram (with some squinting) that $\H22=(\Zt)^4$, this is not clear at a glance from the other two; we must remember to imagine the upper and lower cones.
\end{warning}
\subsection{A forgetful long exact sequence}
The following theorem appearing in \cite{kronholm} relates this cohomology theory to singular cohomology (with $\Zt$ coefficients). Denote the equivariant Eilenberg-MacLane space representing $\H pq$ by $K(\Zt,p,q)$.
\begin{thm}
\label{rholes}
For fixed $q$, there is a long exact sequence
\[\dots\to\H pq(X)\xrightarrow{\cdot\rho}\H{p+1}{q+1}(X)\xrightarrow{\psi}H^{p+1}_{\text{sing}}(X)\to\H{p+1}q(X)\xrightarrow{\cdot\rho}\dots\]
where $\psi$ is the \textbf{forgetful map} $[X,K(\underline{\Zt},p,q)]_{C_2-\Top}\to [X,K(\Zt,p)]_{\Top}$
\end{thm}
It is clear that $\psi:\Mt=\H\bullet\bullet(\pt)\to H_{\text{sing}}^\bullet(\pt)$ takes $\rho$ to 0. Notice this implies that $\psi(\theta)=0$, since $\theta$ is $\rho$-divisible. We will also make use of the fact that $\psi(\tau)=1$.
These facts have a nice geometric interpretation using the Dold-Thom model of Eilenberg-MacLane spaces. We omit this interpretation, but geometric models for $\rho$, $\tau$ and $\theta$ can be found in Proposition 4.5 of \cite{clover}.
\begin{defn}
\label{def:repcell}
A \textbf{representation disc} $D^{p,q}=D(\R^{p,q})$ is the closed unit disc in a representation, and a \textbf{representation cell} $e^{p,q}$ is its interior. A space which can be built from representation cells by the usual gluing diagrams (now with equivariant attaching maps out of $\partial D^{p,q}$) is said to have a \textbf{representation cell structure}.
\end{defn}
Theorem 3.4 in a paper of Kronholm \cite{kronholm} says\footnote{This theorem is true, however the proof given in \cite{kronholm} is problematic. Another proof is forthcoming in \cite{buddies}.}
\begin{thm}[Kronholm]
If a (locally finite, finite-dimensional) $\Ct$-space $X$ has a representation cell structure then it has free cohomology:
\[\H\bullet\bullet(X)=\bigoplus_i\Sigma^{a_i,b_i}\Mt=\bigoplus_i\rH\bullet\bullet(\S{a_i}{b_i})\text{\qquad for some bidegrees $\{(a_i,b_i)\}_i$.}\]
\end{thm}
The bidegrees $(a_i,b_i)$ need not coincide with those of the representation cells used to build $X$, as the weights $b_i$ may differ. While the cohomologies of many families of Grassmannians remain unknown, we next present the known results.
\subsection{Formulas}
Kronholm also calculated the cohomology of the various projective spaces $\Gr_1(\R^{p,q})=\mathbb{P}(\R^{p,q})$. Taking $p\ge 2q$,
\[\H\bullet\bullet(\Gr_1\R^{p,q})
=\Mt\oplus\bigoplus_{i=1}^{q-1}(\M {2i-1}i\oplus \M {2i}i)\oplus\bigoplus_{j=2q-1}^{p-1}\M jq.\]
For example, $\H\bullet\bullet(\mathbb{P}(\R^{11,4}))$ is represented below.
\begin{center}
\begin{tikzpicture}[scale=0.55]
\def 8 { 10 }
\def 4 { 4 }
\draw (0.5* 8 ,-1.5) node[anchor=south] {$ $};
\def(0.1, 0.1), (1.1, 1.1), (2.1, 1.1), (2.1, 2.1), (3.2, 2.1), (3.1, 2.2), (4.2, 2.1), (4.1, 2.2), (5.1, 3.1), (6.1, 3.1){(0.2, 0.2), (1.2, 1.2), (2.2, 1.2), (3.2, 2.2), (4.2, 2.2), (5.2, 3.2), (6.2, 3.2), (7.2, 4.2), (8.2, 4.2), (9.2, 4.2), (10.2, 4.2)}
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\foreach \pq in (0.1, 0.1), (1.1, 1.1), (2.1, 1.1), (2.1, 2.1), (3.2, 2.1), (3.1, 2.2), (4.2, 2.1), (4.1, 2.2), (5.1, 3.1), (6.1, 3.1)
\draw \pq [fill] circle (.04);
\end{tikzpicture}
\end{center}
In Section \ref{knone} we prove a theorem for the family $\Gr_k(\R^{n,1})$. As in \cite{dugger}, define the $\Mt$-rank of a free $\Mt$-module $M$ by letting $I=\ker(\Mt\to\Zt)$ and set
$$\rank^{p,q}_{\Mt}(M)=\dim_{\Zt}(\sfrac M{IM})^{p,q}.$$
Let $\part(p,k,m,t)$ denote the number of partitions of $p$ into $k$ non-negative, weakly-increasing numbers $\lambda_i\le m$, such that $\#\{i:\lambda_i\ge k-i+1\}=t$. The value $t$ is called the \textbf{trace} of the partition $\lambda$. Visually, it is the number of boxes on the main diagonal of a Young diagram representing the partition. See Figure \ref{fig:tracetable} for examples. Using this definition, we state the following theorem.
\begin{thm}
\label{kn1thm}
\[\rank_{\Mt}^{p,q}\H \bullet\bullet(\Gr_k(\R^{n,1}))=\part(p,k,n-k,q).\]
\end{thm}
In words, the free generators of $\H pq(\Gr_k(\R^{n,1}))$ having degree $(p,q)$ are counted by trace-$q$ Young diagrams of $p$ boxes with fitting inside of a $k$-by-$(n-k)$ box. This formula lets us calculate cohomologies like that of $\Gr_{4}(\R^{9,1})$, shown in Figure \ref{fig:491}.
\begin{figure}[H]
\begin{tikzpicture}[scale=0.6]
\def 8 {20}
\def 4 {4}
\draw(21,0) node {$p$};
\draw(0,5) node {$q$};
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\draw (5.3,0) node[below] {\rotatebox{-90}{$5$}};
\draw (10.3,0) node[below] {\rotatebox{-90}{$10$}};
\draw (15.3,0) node[below] {\rotatebox{-90}{$15$}};
\draw (20.3,0) node[below] {\rotatebox{-90}{$20$}};
\draw(-.4,2.2) node {$2$};
\draw(-.4,4.2) node {$4$};
\draw (19.16,4.3) node {$1$};
\draw (18.16,4.3) node {$1$};
\draw (16.16,3.3) node {$4$};
\draw (20.16,4.3) node {$1$};
\draw (1.16,1.3) node {$1$};
\draw (8.3,2.3) node {$10$};
\draw (9.16,3.3) node {$1$};
\draw (14.16,2.3) node {$1$};
\draw (3.16,1.3) node {$3$};
\draw (2.16,1.3) node {$2$};
\draw (5.16,1.3) node {$4$};
\draw (10.16,3.3) node {$2$};
\draw (7.16,2.3) node {$7$};
\draw (12.16,2.3) node {$5$};
\draw (13.16,3.3) node {$7$};
\draw (7.16,1.3) node {$2$};
\draw (4.16,1.3) node {$4$};
\draw (16.16,4.3) node {$1$};
\draw (9.3,2.3) node {$10$};
\draw (6.16,1.3) node {$3$};
\draw (11.16,3.3) node {$4$};
\draw (14.16,3.3) node {$7$};
\draw (6.16,2.3) node {$5$};
\draw (12.16,3.3) node {$6$};
\draw (13.16,2.3) node {$2$};
\draw (15.16,3.3) node {$6$};
\draw (4.16,2.3) node {$1$};
\draw (11.16,2.3) node {$7$};
\draw (17.16,3.3) node {$2$};
\draw (8.16,1.3) node {$1$};
\draw (0.16,0.3) node {$1$};
\draw (17.16,4.3) node {$1$};
\draw (18.16,3.3) node {$1$};
\draw (5.16,2.3) node {$2$};
\draw (10.3,2.3) node {$10$};
\end{tikzpicture}
\caption{Rank chart for $\H\bullet\bullet(\Gr_4(\R^{9,1}))$.}
\label{fig:491}
\end{figure}
For example, the 5 in bidegree $(6,2)$ says that $\rank_{\Mt}^{6,2}(\H\bullet\bullet(\Gr_4(\R^{9,1})))=5$ which is counted by $\part(6,4,8-4,2)$, the number of partitions of 6 into 4 numbers each at most $4$, with trace $t=\#\{i:\lambda_i\ge k-i+1\}=2$. These are the starred entries in Table \ref{fig:tracetable}.
\begin{figure}[h]
\begin{tabular}{cc|c|c}
&Partition of 6 & Trace & Young Diagram\\
\hline
*&0+0+2+4 &2& \scalebox{0.4}{\young({\,}\diagup,\diagup{\,}{\,}{\,})} {\color{white}$\ds\int$}\\
*&0+0+3+3 &2& \scalebox{0.4}{\young({\,}\diagup{\,},\diagup{\,}{\,})}{\color{white}$\ds\int$}\\
&0+1+1+4 &1& \scalebox{0.4}{\young({\,},{\,},\diagup{\,}{\,}{\,})}{\color{white}$\ds\int$}\\
*&0+1+2+3 &2& \scalebox{0.4}{\young({\,},{\,}\diagup,\diagup{\,}{\,})}{\color{white}$\ds\int$}\\
*&0+2+2+2 &2& \scalebox{0.4}{\young({\,}{\,},{\,}\diagup,\diagup{\,})}{\color{white}$\ds\int$}\\
&1+1+1+3 &1& \scalebox{0.4}{\young({\,},{\,},{\,},\diagup{\,}{\,})}{\color{white}$\ds\int$}\\
*&1+1+2+2 &2& \scalebox{0.4}{\young({\,},{\,},{\,}\diagup,\diagup{\,})}{\color{white}$\ds\int$}\\
\end{tabular}
\caption{}
\label{fig:tracetable}
\end{figure}
\subsection{Comment}
\label{foreshadow}
The reader may have noticed that the rows of the rank table in Figure \ref{fig:491} are palindromes. There is a simple combinatorial reason for this, which we will give in Section \ref{upside}.\\
In Section \ref{twoeighttwo} we will also prove the following:
\begin{thm}
\label{uglyformula}
The cohomology of $\Gr_2(\R^{n,2})$ with $n\ge 6$, is given by
\begin{align*}
\H\bullet\bullet(\Gr_2(\R^{n,2}))&=
\Mt\oplus \M11\oplus \M21
\oplus\M22\oplus (\M32)^{\oplus 2}\oplus (\M42)^{\oplus 3}\\
&\oplus\bigoplus_{p=5}^{n-2}(\M p2)^{\oplus 2}\oplus\M{n-1}2\\
&\oplus \M53\oplus\bigoplus_{p=6}^n(\M p3)^{\oplus 2}
\oplus \M{n+1}3\\
&\oplus \bigoplus_{p=8}^{n+1}(\M p4)^{\oplus\lceil\frac{p-7}2\rceil}\\
&\oplus \bigoplus_{p=n+2}^{2n-4}(\M p4)^{\oplus (n-1-\lceil\frac{p}2\rceil)}
\end{align*}
\end{thm}
For example $\H\bullet\bullet(\Gr_2(\R^{10,2}))$ is represented in Figure \ref{fig:2ten2}. Note that each line of the formula in Theorem \ref{uglyformula} corresponds to a different circled region. The first is common to all of them (provided $n\ge 6$) and the next two stretch predictably as $n$ grows. The top row is made up of a region where ranks increase left-to-right every two steps, and another in which ranks decrease left-to-right in the same way. For $n\ge 6$ it is convenient to organize the data in this way, but we also calculate these cohomologies for $3\le n<6$ in Section \ref{twontwo}.
\begin{figure}[h]
\begin{tikzpicture}[scale=0.7]
\def 8 {16.2}
\def 4 {4}
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\draw (7.15,3.3) node {$2$};
\draw (10.15,4.3) node {$2$};
\draw (9.15,3.3) node {$2$};
\draw (14.15,4.3) node {$2$};
\draw (2.15,1.3) node {$1$};
\draw (6.15,2.3) node {$2$};
\draw (9.15,4.3) node {$1$};
\draw (10.15,3.3) node {$2$};
\draw (7.15,2.3) node {$2$};
\draw (16.15,4.3) node {$1$};
\draw (6.15,3.3) node {$2$};
\draw (13.15,4.3) node {$2$};
\draw (2.15,2.3) node {$1$};
\draw (15.15,4.3) node {$1$};
\draw (1.15,1.3) node {$1$};
\draw (3.15,2.3) node {$2$};
\draw (0.15,0.3) node {$1$};
\draw (11.15,4.3) node {$2$};
\draw (8.15,2.3) node {$2$};
\draw (4.15,2.3) node {$3$};
\draw (5.15,3.3) node {$1$};
\draw (8.15,3.3) node {$2$};
\draw (9.15,2.3) node {$1$};
\draw (5.15,2.3) node {$2$};
\draw (11.15,3.3) node {$1$};
\draw (12.15,4.3) node {$3$};
\draw (8.15,4.3) node {$1$};
\draw [blue] plot [smooth cycle] coordinates {(0,0) (4,1) (4.3,3) (1,2.1)};
\draw [blue] plot [smooth cycle] coordinates {(5.1,2) (9.1,2) (9.1,2.7) (5.1,2.7)};
\draw [blue] plot [smooth cycle] coordinates {(5.1,3) (11.1,3) (11.1,3.7) (5.1,3.7)};
\draw [blue] plot [smooth cycle] coordinates {(8.1,4) (11.1,4) (11.1,4.7) (8.1,4.7)};
\draw [blue] plot [smooth cycle] coordinates {(12.1,4) (16.1,4) (16.1,4.7) (12.1,4.7)};
\draw(-.4,2.2) node {$2$};
\draw(-.4,4.2) node {$4$};
\draw (0.3,0) node[below] {\rotatebox{-90}{$0$}};
\draw (5.3,0) node[below] {\rotatebox{-90}{$5$}};
\draw (8.3,0) node[below] {\rotatebox{-90}{$n-2$}};
\draw (9.3,0) node[below] {\rotatebox{-90}{$n-1$}};
\draw (10.3,0) node[below] {\rotatebox{-90}{$n$}};
\draw (11.3,0) node[below] {\rotatebox{-90}{$n+1$}};
\draw (12.3,0) node[below] {\rotatebox{-90}{$n+2$}};
\draw (16.3,0) node[below] {\rotatebox{-90}{$2n-4$}};
\end{tikzpicture}
\caption{Rank chart for $\H\bullet\bullet(\Gr_2(\R^{n,2}))$ with $n=10$.}
\label{fig:2ten2}
\end{figure}
Analogous formulas for complex Grassmannians, whose cohomologies look similar but have generators with twice the topological degree and weight, will also appear in Section \ref{complexsection}.
\begin{note}
It should be remembered that while the rank table in Figure \ref{fig:2ten2} organizes all of the information about a free rank-45 $\Mt$-module much more pleasantly than a list of summands would, it may also leave too much to the imagination. For example, while bidegree $(4,0)$ appears empty, actually $\H40(\Gr_2(\R^{10,2}))=(\Zt)^4$, generated by the $\theta$-multiples of the generators of three distinct copies of $\M42$, and also the $\frac\theta\rho$-multiple of the generator of $\M53$. Likewise $\H23=(\Zt)^4$ is generated by $\tau\cdot1_{\M22}\in\M22$ as well as $\rho\tau\cdot 1_{\M11}\in\M11$, $\tau^2\cdot 1_{\M21}\in\M21$ and $1_{\Sigma^{0,0}\Mt}\in \Mt$.
\end{note}
\subsection{Acknowledgements} This work is part of the author's doctoral dissertation. It is the product of many conversations with both his thesis advisor Dan Dugger and with Clover May, who each came to the rescue repeatedly. While the sincerity of his sentiment is muffled somewhat by the affected remove of writing in the third person, the author is in fact extremely grateful to both of them.\\
\section{Background on the representation-cell structure}
Before we present and prove general results for these cohomologies, we will work a few manageable examples bare-handed, to give the reader a feel for equivariant long exact sequence computations. (Note this is distinct from the spectral sequence approach, which we will also make use of later.)
\subsection{Worked Example I}
\label{we1}
This example serves primarily to demonstrate the phenomenon of the ``Kronholm shift,'' found in \cite{kronholm} and \cite{buddies}.\par
When using a CW structure to calculate the singular cohomology of a space, we can work iteratively on skeleta, attaching one $k$-cell at a time. The cofiber sequence $X_{n-1}\inj X_n\to S^k$ then gives a long exact sequence, and if we know $H^i_{\text{sing}}(X_{n-1})$ and the differential $H^i_{\text{sing}}(X_{n-1})\xrightarrow{d} H^{i+1}_{\text{sing}}S^k$, we can (at least over $\Zt$) deduce $H_{\text{sing}}^i(X_n)$. \par
The analogous statement is true equivariantly: The equivariant cofiber sequence $X_{n-1}\inj X_n\to \S pq$ extends to a Puppe sequence
\[\dots\to\Sigma^{-1,0}\S pq\to X_{n-1}\inj X_n\to \S pq\to\Sigma^{1,0}X_{n-1}\]
yielding a long exact sequence of $\Mt$-module maps in cohomology, including a \textbf{differential} $d:\H\bullet\bullet(X_{n-1})\to \H\bullet\bullet(\Sigma^{-1,0}\S pq)=\H{\bullet+1}\bullet(\S pq)$. It turns out that certain zero differentials in the non-equivariant theory are actually the ``shadows'' of something more interesting in the equivariant theory.\par
Consider $\Gr_1(\R^{3,1})$, whose underlying space is $\Gr_1(\R^3)=\RP^2$. We can build the space from representation cells in two ways (See Figure \ref{fig:twoways}). First, we can begin with a point, attach a non-trivial line segment $e^{1,1}\iso \R^{1,1}$ (thus building $\S11$) and finally attach $e^{2,1}\iso\R^{2,1}$ via a degree-two map from its boundary $\partial\D21=\S11$.
\begin{figure}[h]
\begin{tikzpicture}
\draw[fill=lightgray, opacity=0.5] (1,0) arc (0:360:1);
\draw (.93, .34) arc (20:90:1);
\draw[->] (0,1) arc (90:200:1);
\draw (-.93, -.34) arc (200:270:1);
\draw[->] (0,-1) arc (270:380:1);
\draw [dashed] (0,1.5) -- (0,1);
\draw [dashed] (0,-1.5) -- (0,-1);
\draw [<->] (-.4,1.3)--(.4,1.3);
\draw [<->] (-.4,-1.3)--(.4,-1.3);
\color{red}
\draw [thick] (0,1) -- (0,-1);
\fill (1, 0) circle[radius=0.07cm];
\fill (-1, 0) circle[radius=0.07cm];
\color{black}
\draw (0,-2.5) node {
\xymatrix{\ast=X_0\ar[r]& X_1\ar[r]\ar[d]& X_2\ar[d]\\ &\S11&\S21}
};
\end{tikzpicture}
\qquad\qquad
\begin{tikzpicture}
\draw[fill=lightgray, opacity=0.5] (1,0) arc (0:360:1);
\draw[->] (0,.5) arc (90:260:.5);
\draw[->] (0,-.5) arc (270:360+80:.5);
\color{red}
\draw[thick] (1, 0) arc (0:90:1);
\draw[->,thick] (0,1) arc (90:180:1);
\draw[thick] (-1,0) arc (180:270:1);
\draw[thick, ->] (0,-1) arc (270:360:1);
\fill (0, 0) circle[radius=0.07cm];
\draw (0,-1.5) node{};
\draw (0,1.5) node{};
\color{black}
\draw (0,-2.5) node {
\xymatrix{\ast=X_0\ar[r]& X_1\ar[r]\ar[d]& X_2\ar[d]\\ &\S10&\S22}
};
\end{tikzpicture}\\
\caption{Fixed points in thick red. Note (taking identifications into account) the fixed circle and fixed point in both diagrams. Below the two constructions of $\Gr_1(\R^{3,1})$ are their filtration quotients.}
\label{fig:twoways}
\end{figure}
Another construction begins with a point, attaches a trivial 1-cell (building the trivial circle $\S10$) and then attaches an $e^{2,2}$. In the first construction, the cofiber sequence for including the one-skeleton is $\S11\inj\Gr_1(\R^{3,1})\to \S21$. The differential $d:\rH\bullet\bullet(\S11)\to\rH{\bullet+1}\bullet(\S21)\iso\rH\bullet\bullet\S11$ (depicted on the left of Figure \ref{fig:twodiffs}) must be zero, otherwise the forgetful map would predict a nonzero map $\psi(d)$ in the non-equivariant cellular chain complex. Thus we know relatively easily that
$$\H\bullet\bullet(\Gr_1(\R^{3,1}))=\Mt\oplus\M11\oplus \M21.$$
\begin{figure}[h]
\begin{tikzpicture}[scale=0.55]
\axisname{}
\HMtwo{0}{0}
\HMtwo{1.1}{1}
\draw[->] (1.3,1)--(1.8,1) node[below] {$d=0$};
\color{blue}
\HMtwo{2}{1}
\end{tikzpicture}
\begin{tikzpicture}[scale=0.55]
\axisname{}
\HMtwo{0}{0}
\HMtwo{1}{0}
\draw[->] (1.2,0)--(1.8,0);
\draw (2.3,-0.1) node[above] {$d\ne 0$};
\color{blue}
\HMtwo{2.1}{2}
\end{tikzpicture}
\caption{Differentials from attaching 2-cells.}
\label{fig:twodiffs}
\end{figure}
However in the second construction for the same space, we have the cofiber sequence $\S10\inj\Gr_1(\R^{3,1})\to\S22$. In this case the differential
\[d:\H\bullet\bullet\S10=\M10\to\H\bullet\bullet\S22=\M22\]
\emph{cannot} be zero, or we would have two conflicting answers. Rather, $d(1_{\M10})=\theta1_{\M22}$, and we have a splitting problem with $\ker(d)$ and $\cok(d)$. While we already know the answer in this case, this problem is resolved generally by \cite{kronholm} and \cite{buddies}. Heuristically, the differential into the lower cone causes $\M10$ to `shift up' to become a $\M11$ while $\M22$ `shifts down' to a $\M21$, replicating the cohomology we expect from the first construction.\par
\begin{note}
This phenomenon of nonzero differentials into a lower cone causing shifted weights in the free $\Mt$ generators will be called a \textbf{Kronholm shift}. In its simplest version, where just one $\Mt$ maps into the lower cone of another, the source $\Mt$ shifts up by the difference in fixed set dimension of the two free generators, and the target $\Mt$ shifts down by the same amount. A more general formula for shifts when an arbitrary number of $\Mt$s have nonzero-differentials to a lower cone appears in \cite{buddies}.\par
\end{note}
This trick of deducing properties of unknown differentials in one representation-cell construction (see Definition \ref{def:repcell}) by leveraging what is known about another construction continues to be a useful strategy as we move to larger Grassmannians.
\subsection{Schubert cells}
\label{we2}
Non-equivariantly, $\Gr_k(\R^n)$ can be given a cell structure indexed by Young diagrams fitting inside a $k$-by-$(n-k)$ rectangle. For example, $\Gr_2(\R^5)$ can be built with cells indexed by diagrams fitting into {\tiny$\yng(3,3)$} as follows:
\begin{figure}[H]
\begin{tikzpicture}[scale=0.9]
\draw (2,0) node(z) {$(0,0)$};
\draw (2,1) node(o) {$(0,1)$};
\draw (1,2) node(oo) {$(1,1)$};
\draw (3,2) node(t) {$(0,2)$};
\draw (2,3) node(ot) {$(1,2)$};
\draw (4,3) node(th) {$(0,3)$};
\draw (1,4) node(tt) {$(2,2)$};
\draw (3,4) node(oth) {$(1,3)$};
\draw (2,5) node(tth) {$(2,3)$};
\draw (2,6) node(thth) {$(3,3)$};
\draw [-, ultra thin] (z) -- (o);
\draw [-, ultra thin] (o) -- (oo);
\draw [-, ultra thin] (o) -- (t);
\draw [-, ultra thin] (oo) -- (ot);
\draw [-, ultra thin] (t) -- (ot);
\draw [-, ultra thin] (t) -- (th);
\draw [-, ultra thin] (ot) -- (tt);
\draw [-, ultra thin] (ot) -- (oth);
\draw [-, ultra thin] (th) -- (oth);
\draw [-, ultra thin] (tt) -- (tth);
\draw [-, ultra thin] (oth) -- (tth);
\draw [-, ultra thin] (tth) -- (thth);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.9]
\draw (2,0) node(z) {$\emptyset$};
\draw (2,1) node(o) {\tiny$\yng(1)$};
\draw (1,2) node(oo) {\tiny$\yng(1,1)$};
\draw (3,2) node(t) {\tiny$\yng(2)$};
\draw (2,3) node(ot) {\tiny$\yng(1,2)$};
\draw (4,3) node(th) {\tiny$\yng(3)$};
\draw (1,4) node(tt) {\tiny$\yng(2,2)$};
\draw (3,4) node(oth) {\tiny$\yng(1,3)$};
\draw (2,5) node(tth) {\tiny$\yng(2,3)$};
\draw (2,6) node(thth) {\tiny$\yng(3,3)$};
\draw [-, ultra thin] (z) -- (o);
\draw [-, ultra thin] (o) -- (oo);
\draw [-, ultra thin] (o) -- (t);
\draw [-, ultra thin] (oo) -- (ot);
\draw [-, ultra thin] (t) -- (ot);
\draw [-, ultra thin] (t) -- (th);
\draw [-, ultra thin] (ot) -- (tt);
\draw [-, ultra thin] (ot) -- (oth);
\draw [-, ultra thin] (th) -- (oth);
\draw [-, ultra thin] (tt) -- (tth);
\draw [-, ultra thin] (oth) -- (tth);
\draw [-, ultra thin] (tth) -- (thth);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=0.9]
\draw (2,0) node(z) {$[1,2]$};
\draw (2,1) node(o) {$[1,3]$};
\draw (1,2) node(oo) {$[2,3]$};
\draw (3,2) node(t) {$[1,4]$};
\draw (2,3) node(ot) {$[2,4]$};
\draw (4,3) node(th) {$[1,5]$};
\draw (1,4) node(tt) {$[3,4]$};
\draw (3,4) node(oth) {$[2,5]$};
\draw (2,5) node(tth) {$[3,5]$};
\draw (2,6) node(thth) {$[4,5]$};
\draw [-, ultra thin] (z) -- (o);
\draw [-, ultra thin] (o) -- (oo);
\draw [-, ultra thin] (o) -- (t);
\draw [-, ultra thin] (oo) -- (ot);
\draw [-, ultra thin] (t) -- (ot);
\draw [-, ultra thin] (t) -- (th);
\draw [-, ultra thin] (ot) -- (tt);
\draw [-, ultra thin] (ot) -- (oth);
\draw [-, ultra thin] (th) -- (oth);
\draw [-, ultra thin] (tt) -- (tth);
\draw [-, ultra thin] (oth) -- (tth);
\draw [-, ultra thin] (tth) -- (thth);
\end{tikzpicture}
\caption{Partition tuples, Young diagrams, and jump sequences.}
\label{fig:threeversions}
\end{figure}
To each Young diagram written as a weakly ascending tuple $\lambda$ (for example \scalebox{0.4}{\yng(1,3)} corresponds to $\lambda=(1,3)$) we can write a strictly ascending tuple $\underline{j}=[\lambda_i+i]_i$ called the \textbf{jump sequence}. The diagram \scalebox{0.4}{\yng(1,3)} has jump sequence $[1+1,3+2]$. These are the symbols on the right-hand side of the diagram above.\par
These symbols index the cells of the Grassmannian as follows. We can think of a $k$-plane in $\R^n$ as the rowspace of a $k$-by-$n$ matrix, and without loss of generality, this matrix can be written in a \textbf{canonical form} so that each row's last nonzero entry is a 1, which then clears the column below it. Order these rows by the position of their last nonzero entry. For example:
\[\rs\left[{2\atop 21}\,{-2\atop 3}\,{10\atop 15}\,{12\atop 18}\,{2\atop 3}\right]
=\rs\left[{3\atop 4}\,{1\atop 0}\,{0\atop 5}\,{0\atop 6}\,{0\atop 1}\right]:=V_{\scalebox{0.5}{\young(3,456)}}\]
In this way, every point in the Grassmannian can be sorted into a family, these families indexed by jump sequences which give the locations of these $1$s in their canonical representations. These families are related. Consider for example the open set containing the four-parameter family of all planes of the form $V_{\scalebox{0.6}{\young(w,xyz)}}$:
\[\Omega_{[2,5]}=\Omega_{\scalebox{0.4}{\yng(1,3)}}:=\left\{\rs\left[{w\atop x}\,{1\atop 0}\,{0\atop y}\,{0\atop z}\,{0\atop 1}\right]\,\,:\,\,w,x,y,z\in \R\right\}\subset \Gr_2(\R^5).\]
Since
\[\lim_{c\to\infty}\rs\left[{cw\atop x}\,{1\atop 0}\,{0\atop y}\,{0\atop z}\,{0\atop 1}\right]=\rs\left[{1\atop x}\,{0\atop 0}\,{0\atop y}\,{0\atop z}\,{0\atop 1}\right]\]
and
\[\lim_{c\to\infty}\rs\left[{w\atop cx}\,{1\atop 0}\,{0\atop cy}\,{0\atop cz}\,{0\atop 1}\right]=\rs\left[{w\atop \frac xz}\,{1\atop 0}\,{0\atop \frac yz}\,{0\atop 1}\,{0\atop 0}\right]\]
we have that the closure $X_{[2,5]}:=\overline{\Omega_{[2,5]}}\supset \Omega_{[1,5]}$ and also $X_{[2,5]}\supset \Omega_{[2,4]}$, or in Young diagrams, $\Omega_{\scalebox{0.4}{\yng(3)}}\subset X_{\scalebox{0.4}{\yng(3)}}\subset X_{\scalebox{0.4}{\yng(1,3)}}$ and $\Omega_{\scalebox{0.4}{\yng(1,2)}}\subset X_{\scalebox{0.4}{\yng(1,2)}}\subset X_{\scalebox{0.4}{\yng(1,3)}}$. The sets $\Omega_{\underline{j}}$ indexed by jump sequences (or equivalently by Young diagrams) are called \textbf{Schubert cells}, and their closures \textbf{Schubert varieties}. We have an obvious notion of containment for Young diagrams, to which corresponds a notion of dominance in jump sequences. We say that a jump sequence \underline{$j$} \textbf{dominates} another jump sequence \underline{$k$}, denoted $\underline{k}\prec \underline{j}$, if each $k_i\le j_i$. Containment between Schubert varieties corresponds to containment between their indexing Young diagrams or equivalently, to dominating jump sequences. For further details, see Section 3.2 of \cite{manivel}.
\begin{figure}[h!]
\begin{tikzpicture}[scale=1.1]
\draw (3,0) node(z) {$\left\{\text{rs}\left[{1\atop 0}\,{0\atop 1}\,{0\atop 0}\,{0\atop 0}\,{0\atop 0}\right]\right\}$};
\draw (3,1) node(o) {$\left\{\text{rs}\left[{1\atop 0}\,{0\atop \ast}\,{0\atop 1}\,{0\atop 0}\,{0\atop 0}\right]\right\}$};
\draw (1.5,2) node(oo) {$\left\{\text{rs}\left[{\ast\atop \ast}\,{1\atop 0}\,{0\atop 1}\,{0\atop 0}\,{0\atop 0}\right]\right\}$};
\draw (4.5,2) node(t) {$\left\{\text{rs}\left[{1\atop 0}\,{0\atop \ast}\,{0\atop \ast}\,{0\atop 1}\,{0\atop 0}\right]\right\}$};
\draw (3,3) node(ot) {$\left\{\text{rs}\left[{\ast\atop \ast}\,{1\atop 0}\,{0\atop \ast}\,{0\atop 1}\,{0\atop 0}\right]\right\}$};
\draw (6,3) node(th) {$\left\{\text{rs}\left[{1\atop 0}\,{0\atop \ast}\,{0\atop \ast}\,{0\atop \ast}\,{0\atop 1}\right]\right\}$};
\draw (1.5,4) node(tt) {$\left\{\text{rs}\left[{\ast\atop \ast}\,{\ast\atop \ast}\,{1\atop 0}\,{0\atop 1}\,{0\atop 0}\right]\right\}$};
\draw (4.5,4) node(oth) {$\left\{\text{rs}\left[{\ast\atop \ast}\,{1\atop 0}\,{0\atop \ast}\,{0\atop \ast}\,{0\atop 1}\right]\right\}$};
\draw (3,5) node(tth) {$\left\{\text{rs}\left[{\ast\atop \ast}\,{\ast\atop \ast}\,{1\atop 0}\,{0\atop \ast}\,{0\atop 1}\right]\right\}$};
\draw (3,6) node(thth) {$\left\{\text{rs}\left[{\ast\atop \ast}\,{\ast\atop \ast}\,{\ast\atop \ast}\,{1\atop 0}\,{0\atop 1}\right]\right\}$};
\draw [-, ultra thin] (z) -- (o);
\draw [-, ultra thin] (o) -- (oo);
\draw [-, ultra thin] (o) -- (t);
\draw [-, ultra thin] (oo) -- (ot);
\draw [-, ultra thin] (t) -- (ot);
\draw [-, ultra thin] (t) -- (th);
\draw [-, ultra thin] (ot) -- (tt);
\draw [-, ultra thin] (ot) -- (oth);
\draw [-, ultra thin] (th) -- (oth);
\draw [-, ultra thin] (tt) -- (tth);
\draw [-, ultra thin] (oth) -- (tth);
\draw [-, ultra thin] (tth) -- (thth);
\end{tikzpicture}
\begin{tikzpicture}[scale=1.1]
\draw (2,0) node(z) {$\Omega_{[1,2]}=\pt$};
\draw (2,1) node(o) {$\Omega_{[1,3]}$};
\draw (1,2) node(oo) {$\Omega_{[2,3]}$};
\draw (3,2) node(t) {$\Omega_{[1,4]}$};
\draw (2,3) node(ot) {$\Omega_{[2,4]}$};
\draw (4,3) node(th) {$\Omega_{[1,5]}$};
\draw (1,4) node(tt) {$\Omega_{[3,4]}$};
\draw (3,4) node(oth) {$\Omega_{[2,5]}$};
\draw (2,5) node(tth) {$\Omega_{[3,5]}$};
\draw (2,6) node(thth) {$\Omega_{[4,5]}$};
\draw [-, ultra thin] (z) -- (o);
\draw [-, ultra thin] (o) -- (oo);
\draw [-, ultra thin] (o) -- (t);
\draw [-, ultra thin] (oo) -- (ot);
\draw [-, ultra thin] (t) -- (ot);
\draw [-, ultra thin] (t) -- (th);
\draw [-, ultra thin] (ot) -- (tt);
\draw [-, ultra thin] (ot) -- (oth);
\draw [-, ultra thin] (th) -- (oth);
\draw [-, ultra thin] (tt) -- (tth);
\draw [-, ultra thin] (oth) -- (tth);
\draw [-, ultra thin] (tth) -- (thth);
\end{tikzpicture}
\caption{Free variables denoted by $\ast$. (Rowspace abbreviated rs.)}
\end{figure}
In this way, Young diagrams index a CW structure for the Grassmannian, each $\scalebox{.5}{\yng(1)}$ in a diagram corresponding to a degree of freedom, and hence the number of boxes equals the dimension of the cell attached at that stage of the construction. (For example, $\Omega_{[2,5]}=\Omega_{\scalebox{0.3}{\yng(1,3)}}\iso e^4$.)\par
If we work over $\Zt$, the attaching maps given by this CW construction yield zero differentials in the chain complex, and so (for example) we have singular cohomology
\[H^i_{\text{sing}}(\Gr_2(\R^5);\Zt)=\begin{cases}
\Zt=\langle[\scalebox{.3}{\yng(3,3)}]\rangle & i=6\\
\Zt=\langle[\scalebox{.3}{\yng(2,3)}]\rangle & i=5\\
(\Zt)^2=\langle[\scalebox{.3}{\yng(2,2)}],[\scalebox{.3}{\yng(1,3)}]\rangle & i=4\\
(\Zt)^2=\langle[\scalebox{.3}{\yng(1,2)}],[\scalebox{.3}{\yng(3)}]\rangle & i=3\\
(\Zt)^2=\langle[\scalebox{.3}{\yng(1,1)}],[\scalebox{.3}{\yng(2)}]\rangle & i=2\\
\Zt=\langle[\scalebox{.3}{\yng(1)}]\rangle & i=1\\
\Zt=\langle[\ast]\rangle & i=0.
\end{cases}
\]
In this notation, the cocycle $\scalebox{.3}{\yng(1,2)}$ is the Kronecker dual to $\Omega_{\scalebox{.3}{\yng(1,2)}}$, that is, it evaluates to 1 on $\Omega_{\scalebox{.3}{\yng(1,2)}}$ and zero on other cells. More generally, cohomology elements are denoted by the Young diagrams of the Schubert cells to which they are dual. This preserves the at-a-glance dimension property.
There is also an equivariant version of this story, which we explain next.
\subsection{Worked Example II}
Suppose we are interested in $\Gr_2(\R^{5,2})$. If we interpret this as $\Gr_2(\R\triv\oplus\R\sgn\oplus\R\triv\oplus\R\sgn\oplus\R\triv)$ or $\Gr_2(\R^{+-+-+})$ for short, then $\Omega_{[2,5]}$ can be seen to be a representation cell. The action of $\Ct$ on this 4-cell, as in the seventh chapter of \cite{FL}, is given by \label{equivschub}
\[\rs\left[{w\atop x}\,{1\atop 0}\,{0\atop y}\,{0\atop z}\,{0\atop 1}\right]\mapsto
\rs\left[{w\atop x}\,{-1\atop 0}\,{0\atop y}\,{0\atop -z}\,{0\atop 1}\right]
=\rs\left[{-w\atop x}\,{1\atop 0}\,{0\atop y}\,{0\atop -z}\,{0\atop 1}\right]
\]
and so $\Omega_{[2,5]}(\R^{+-+-+})\iso e^{4,2}$, a representation cell. It's pleasant to write this ${\scriptsize\young(-,++-)}$, as we can see topological dimension and weight at a glance from the number of boxes and minus signs, respectively. Analogous considerations now give a representation cell construction for the space.
\begin{figure}[h!]
\begin{tikzpicture}[scale=0.8]
\draw (2,0) node(z) {$\emptyset$};
\draw (2,1) node(o) {\scalebox{0.7}{$\young(-)$}};
\draw (1,2) node(oo) {\scalebox{0.7}{$\young(-,+)$}};
\draw (3,2) node(t) {\scalebox{0.7}{$\young(+-)$}};
\draw (2,3) node(ot) {\scalebox{0.7}{$\young(-,--)$}};
\draw (4,3) node(th) {\scalebox{0.7}{$\young(-+-)$}};
\draw (0.8,4.2) node(tt) {\scalebox{0.7}{$\young(+-,-+)$}};
\draw (3.2,4.2) node(oth) {\scalebox{0.7}{$\young(-,++-)$}};
\draw (2,5.4) node(tth) {\scalebox{0.7}{$\young(+-,-+-)$}};
\draw (2,6.6) node(thth) {\scalebox{0.7}{$\young(-+-,+-+)$}};
\draw [-, ultra thin] (z) -- (o);
\draw [-, ultra thin] (o) -- (oo);
\draw [-, ultra thin] (o) -- (t);
\draw [-, ultra thin] (oo) -- (ot);
\draw [-, ultra thin] (t) -- (ot);
\draw [-, ultra thin] (t) -- (th);
\draw [-, ultra thin] (ot) -- (tt);
\draw [-, ultra thin] (ot) -- (oth);
\draw [-, ultra thin] (th) -- (oth);
\draw [-, ultra thin] (tt) -- (tth);
\draw [-, ultra thin] (oth) -- (tth);
\draw [-, ultra thin] (tth) -- (thth);
\end{tikzpicture}
\caption{One representation-cell structure for $\Gr_2(\R^{5,2})$, produced by the choice $\R^{5,2}\iso \R^{+-+-+}$.}
\end{figure}
Once an ordered decomposition of the representation as a direct sum of irreducibles is chosen, the process of assigning weights to Schubert cells can easily be automated. Essentially, to find the weight of cell, one needs to count the free variables in the associated matrix whose action does not match the action on the dimension where the one appears in their row. This amounts to counting the minus signs in a matrix like the third one appearing in Figure \ref{fig:weightcompute}'s example.
\begin{figure}[h]
\[
\begin{matrix}
\begin{matrix}
\begin{matrix}
+\,\,\,-\,\,\,+\,\,\,-\,\,\,+\,\,\,-\,\,\,-\,\,\,\,+
\end{matrix}\\
\left[\begin{matrix}
\ast&\ast&1&0&0&0&0&0\\
\ast&\ast&0&1&0&0&0&0\\
\ast&\ast&0&0&\ast&\ast&1&0\\
\ast&\ast&0&0&\ast&\ast&0&1\\
\end{matrix}\right]
\end{matrix}&
\mapsto
\begin{matrix}
\begin{matrix}
+\,&-\,&+\,&-\,&+\,&-\,&-\,&+
\end{matrix}\\
\left[\begin{matrix}
\ast&-\ast&1&0&0&0&0&0\\
\ast&-\ast&0&-1&0&0&0&0\\
\ast&-\ast&0&0&\ast&-\ast&-1&0\\
\ast&-\ast&0&0&\ast&-\ast&0&1\\
\end{matrix}\right]
\end{matrix}\\
&\sim
\begin{matrix}
\begin{matrix}
\phantom{.}\\
\end{matrix}\\
\left[\begin{matrix}
\ast&-\ast&1&0&0&0&0&0\\
-\ast&\ast&0&1&0&0&0&0\\
-\ast&\ast&0&0&-\ast&\ast&1&0\\
\ast&-\ast&0&0&\ast&-\ast&0&1\\
\end{matrix}\right]
\end{matrix}
\end{matrix}
\]
\caption{The number of minus signs in the last matrix gives the weight of the Schubert cell with jump sequence $[3,4,7,8]$ in the construction associated to $\R^{+-+-+--+}$.}
\label{fig:weightcompute}
\end{figure}
While it is preferable to automate this computation, a formula for counting these minus signs can be given for the ordered decomposition $\R^{s(1)}\oplus\R^{s(2)}\oplus\dots\oplus \R^{s(n)}$ with $s:[1,n]\to\{+,-\}$ by letting $\lambda$ have jump sequence $\underline{j}$ and using the reverse Kronecker delta $\widehat \delta_{i,j}=1-\delta_{i,j}$,
\[w\left(\Omega_\lambda(\R^{s(1)s(2)\dots s(n)})\right)=\sum_{k\in \underline{j}}\sum_{{i<k}\atop{i\not\in \underline{j}}} \widehat\delta_{s(i),s(j)}.\]
It is important that a different ordered decomposition of the underlying representation can create a very different equivariant Schubert cell construction. For example, while $\Omega_{[2,5]}(\R^{+-+-+})\iso e^{4,2}$, the decomposition $\R^{5,2}=\R^{-++-+}$ gives $\Omega_{[2,5]}(\R^{-++-+})\iso e^{4,4}$, an ingredient for building the same space $\Gr_2(\R^{5,2})$ which does not appear in the $\R^{+-+-+}$ construction.
A representation-cell structure for space allows for a \textbf{one-cell-at-at-time filtration}, such that each subsequent inclusion cofiber is a representation sphere:
\[\xymatrix{
X_0\ar[r] & X_1\ar[r]\ar[d] & X_2\ar[r]\ar[d] & \dots\ar[r]&X_i\ar[r]\ar[d]&\dots\\
&S^{p_1,q_1}&S^{p_2,q_2}&\dots&S^{p_i,q_i}&\dots
}\]
This gives rise to the \textbf{one-cell-at-a-time equivariant cellular spectral sequence} for a Grassmannian, which we will discuss further in the next section. To a given choice of decomposition for the underlying representation space, we get a spectral sequence having for its $E_1$ page a free $\Mt$-module with basis elements corresponding to the bidegrees $(p_i,q_i)$ of these Schubert cells. We will refer to this data as a \textbf{table of ingredients} where each Young diagram or jump sequence represents the generator for an $\Mt$ in that bidegree. Denote the ingredient table of a certain decomposition $\bigoplus \R^{\pm}$ by $I(\pm\dots\pm)$. We have for example two depictions of $I(\R^{+-+-+})=I(+-+-+)$ in Figure \ref{fig:252ingreds pmpmp}.
\begin{figure}[h]
\begin{tabular}{||l|p{10pt}|p{10pt}|p{10pt}|p{10pt}|p{10pt}|p{10pt}}
\hspace{-.26in}3&&&\scalebox{.4}{\yng(1,2)} &&\scalebox{.4}{\yng(2,3)} &\scalebox{.4}{\yng(3,3)}\,\,\\
\hline
\hspace{-.26in}2&&&\scalebox{.4}{\yng(3)} &\scalebox{.4}{\yng(1,3)} \scalebox{.4}{\yng(2,2)}\,\,&&\\
\hline
\hspace{-.26in}1&\scalebox{.4}{\yng(1)} &\scalebox{.4}{\yng(2)} \scalebox{.4}{\yng(1,1)}\,\,&&&&\\
\hline
$\emptyset$\hspace{-.34in}0\hspace{.4in}\,&&&&&&\\
\hline
\hline
\end{tabular}
\quad or \qquad
\begin{tabular}{||l|p{13pt}|p{13pt}|p{13pt}|p{13pt}|p{13pt}|p{13pt}}
\hspace{-.26in}&&&[2,4]&&[3,5]&[4,5]\\
\hline
\hspace{-.26in}&&&[1,5]&[2,5] [3,4]&&\\
\hline
\hspace{-.26in}&[1,3]&[1,4] [2,3]&&&&\\
\hline
[1,2]\hspace{-.44in}\hspace{.4in}\,&&&&&&\\
\hline
\hline
\end{tabular}\\
\begin{tabular}{p{10pt}p{10pt}p{10pt}p{10pt}p{10pt}p{10pt}p{10pt}}
0&1&2&3&4&5&6
\end{tabular}
\hspace{.9in}
\begin{tabular}{p{11pt}p{11pt}p{11pt}p{11pt}p{11pt}p{11pt}}
&&&&&
\end{tabular}\\
Young diagrams\qquad\qquad\qquad\qquad\qquad Jump sequences
\caption{Ingredients table $I(+-+-+)$ for $\Gr_2(\R^{5,2})$.}
\label{fig:252ingreds pmpmp}
\end{figure}
While the ingredients table is the first page of a spectral sequence, we will often make use of this data in another way. If we consider attaching these equivariant cells successively by increasing dimension and then weight, we can compute the cohomology of filtered subspaces one at a time. That is, rather than running a spectral sequence, we will repeatedly consider the long exact sequence corresponding to iteratively building subspaces $X_{k+1}$ from $X_k$ by attaching one equivariant cell $e^{p,q}$:
\[X_{k}\inj X_{k+1}\to \S pq\]
Because the differentials $d:H^\bullet (X_k)\to H^{\bullet+1}(S^p)$ in the non-equivariant chain complex are all zero, we know that none of the equivariant differentials may send a free to generator to another free generator, as the forgetful map induces a natural map between the equivariant and non-equivariant long exact sequences for each attachment. Also because we are attaching cells by increasing weight, any differential carrying a generator into the top cone would hit $\tau^j$ times some other generator, which would again imply an isomorphism non-equivariantly. Because the differentials in a Schubert cell complex for a Grassmannian must have zero differentials as their non-equivariant ``shadows,'' we need only worry about nonzero differentials into the lower cones of suspensions of $\Mt$, which, if they occur, cause Kronholm shifts. \par
We return to $\Gr_2(\R^{5,2})$, again recalling that rather than running a spectral sequence, we are simply computing the cohomology of subspaces as we attach cells one at at time. From Figure \ref{fig:252ingreds pmpmp} we can see that as we attach the first few cells, no differentials are possible, and so the cohomologies of early subspaces are obvious. But when the cell $\Omega_{\scalebox{0.4}{\yng(1,2)}}$ is attached, a differential between $\langle[\scalebox{.3}{\yng(1,1)}],[\scalebox{.3}{\yng(2)}]\rangle$ and $\langle\theta[\scalebox{.3}{\yng(1,2)}]\rangle$ could be either zero or nonzero without contradicting what is known non-equivariantly. However we can resolve this ambiguity by making use of another ordered decomposition of $\R^{5,2}$, the ambient space for our 2-planes. For example we have an equivariant homeomorphism $\Gr_2(\R^{+-+-+})\iso \Gr_2(\R^{-++-+})$, induced by the linear map $(x_1,x_2,x_3,x_4,x_5)\mapsto (x_2,x_1,x_3,x_4,x_5)$ on the underlying representation. This second construction for the space has ingredients table $I(\R^{-++-+})$, as shown in Figure \ref{fig:252ingreds mppmp}. Again, we can represent cells using either Young diagrams or jump sequences.
\begin{figure}
\begin{tabular}{||l|p{8pt}|p{10pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}}
\hspace{-.26in}3&&&&\scalebox{0.4}{\yng(1,3)}&\scalebox{0.4}{\yng(2,3)}&\scalebox{0.4}{\yng(3,3)}\\
\hline
\hspace{-.26in}2&&\scalebox{0.4}{\yng(2)} \scalebox{0.4}{\yng(1,1)}&\scalebox{0.4}{\yng(1,2)}&\scalebox{0.4}{\yng(2,2)}&&\\
\hline
\hspace{-.26in}1&&&\scalebox{0.4}{\yng(3)}&&&\\
\hline
$\emptyset$\hspace{-.33in}0\hspace{.29in}\,&\scalebox{0.4}{\yng(1)}&&&&&\\
\hline
\hline
\end{tabular}
\qquad
\begin{tabular}{||l|p{14pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}}
\hspace{-.26in}&&&&[2,5]&[3,5]&[4,5]\\
\hline
\hspace{-.26in}&&[1,4] [2,3]&[2,4]&[3,4]&&\\
\hline
\hspace{-.26in}&&&[1,5]&&&\\
\hline
[1,2]\hspace{-.44in}\hspace{.4in}\,&[1,3]&&&&&\\
\hline
\hline
\end{tabular}\\
\,\,
\begin{tabular}{p{11pt}p{11pt}p{11pt}p{14pt}p{14pt}p{14pt}p{14pt}p{1pt}}
0&1&2&3&4&5&6&
\end{tabular}
\,\,\,\,
\begin{tabular}{p{11pt}p{14pt}p{14pt}p{14pt}p{14pt}p{14pt}p{1pt}p{1pt}}
&&&&&&&
\end{tabular}
\caption{Ingredients table $I(-++-+)$ for $\Gr_2(\R^{5,2})$.}
\label{fig:252ingreds mppmp}
\end{figure}
Since after iteratively attaching these ingredients, we must arrive at the same cohomology, it is now clear that in this second scenario, $[1,3]$ must ``shift up'' by hitting some nonzero combination of $\theta[1,4]$ and $\theta[2,3]$, after which no other differential can interact with the bidegree $(2,1)$, recalling that isomorphisms are precluded by our knowledge of the non-equivariant cochain complex. Thus in the first construction, $d:\langle [1,4],[2,3]\rangle \to \langle\theta[2,4]\rangle$ must be nonzero, so that both $\H21$ and $\H22$ of $\Gr_2(\R^{+-+-+})$ contain generators. As no other differentials are possible in the $+-+-+$ construction, we now know that
\begin{center}
\begin{tikzpicture}[scale=.8]
\def 8 { 6 }
\def 4 { 3 }
\draw (-3,1.5) node {$H^{\bullet,\bullet}(\Gr_2(\R^{5,2}))=$};
\def(0.1, 0.1), (1.1, 1.1), (2.1, 1.1), (2.1, 2.1), (3.2, 2.1), (3.1, 2.2), (4.2, 2.1), (4.1, 2.2), (5.1, 3.1), (6.1, 3.1){(0.1, 0.1), (1.1, 1.1), (2.1, 1.1), (2.1, 2.1), (3.2, 2.1), (3.1, 2.2), (4.2, 2.1), (4.1, 2.2), (5.1, 3.1), (6.1, 3.1)}
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\foreach \pq in (0.1, 0.1), (1.1, 1.1), (2.1, 1.1), (2.1, 2.1), (3.2, 2.1), (3.1, 2.2), (4.2, 2.1), (4.1, 2.2), (5.1, 3.1), (6.1, 3.1)
\draw \pq [fill] circle (.04);
\end{tikzpicture}.\label{example}
\end{center}
(It is instructive to check what this must mean about the other differentials in the $-++-+$ construction.\footnote{\upsidedown{$[1,3]\mapsto\theta[1,4]$ and $[1,5]\mapsto\theta[2,5]$.}})
This procedure of playing the many different constructions for a Grassmannian off of one another can be automated to get a fund of examples. The theorems and algorithm necessary for this will be described in a forthcoming paper. But in many cases, we can do even better -- see Sections \ref{knone} and \ref{twontwo}.
\section{Jack-O-Lantern Modules}
\label{jolsection}
Rather than considering successive long exact sequences as in Section \ref{we1}, we could have used a cellular spectral sequence made by sewing together the long exact sequences for each cofiber sequence in the filtration
\[\xymatrix{
\pt\ar[r]& X_1\ar[r]\ar[d]& X_2\ar[d]\\
&{\color{blue}\S10}&{\color{ggreen}\S22}
}.\]
More generally, when a space $X$ is built one-cell-at-a-time, so that the cofiber of each subspace inclusion is a single representation sphere,
\[\xymatrix{
\pt\ar[r]&X_1\ar[r]\ar[d]&X_2\ar[r]\ar[d]&X_3\ar[r]\ar[d]&\dots\\
&\S{p_1}{q_1}&\S{p_2}{q_2}&\S{p_3}{q_3}&\dots
}\]
we can make a spectral sequence where each filtration degree contains a single suspended $\Mt$. This spectral sequence is, alarmingly, trigraded, but if we attach cells in lexicographic order, we can suppress the filtration degree without losing too much information. Letting $r$ denote filtration degree, we will have differentials $$d_k:E^{p,q,r}_k\to E^{p+1,q,r+k}_k.$$ Figure \ref{fig:spectraljack} depicts this approach for one of the constructions in Section \ref{we1}.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=0.55]
\axisname{$E_1$}
\draw[->] (1.2,0)-- node[above] {$d_1$}(1.8,0);
\color{blue}
\HMtwo{1}{0}
\color{ggreen}
\draw[->] (2,2.2)--(2,4.5);
\draw[->] (2,2.2)--(2+2.3,4.5);
\draw[->] (2,0)--(2,-4.5);
\draw[->] (2,0)--(2-4.5,-4.5);
\end{tikzpicture}
\begin{tikzpicture}[scale=0.55]
\axisname{$E_2=E_\infty$}
\color{blue}
\draw[->] (1,1)--(1,4.5);
\draw[-] (1,1)--(2,2);
\draw[-] (2,2)--(2,1);
\draw[->] (2,1)--(2+2.5,1+2.5);
\draw[->] (1,-2.1)--(1,-4.5);
\draw[->] (1,-2.1)--(1-2.4,-4.5);
\color{ggreen}
\draw[->] (2,2.1)--(2,4.5);
\draw[->] (2,2.1)--(2.1+2.3,4.5);
\draw[->] (1,-1)--(1-3.5,-4.5);
\draw[-] (1,-1)--(1,-2);
\draw[-] (1,-2)--(2,-1);
\draw[->] (2,-1)--(2,-4.5);
\end{tikzpicture}
\end{center}
\caption{Jack-O-lantern modules in a spectral sequence.}
\label{fig:spectraljack}
\end{figure}
While we already know that the reduced cohomology of the space $\Gr_1(\R^{3,1})$ from Section \ref{we1} is the free module $\M11\oplus \M21$, we see that $E_\infty$ is not itself free. Rather, it is an associated graded of this free module. Loosely, the summands of $E_\infty$ are copies of $\Mt$ with pieces cut out of them. This phenomenon motivates the following definition.
\begin{defn}
\label{joldef}
Beginning with some suspension $\Sigma^{p,q}\Mt=\Mt\langle a\rangle$ of $\Mt$, let $S$ be a finite set of homogeneous elements of the lower cone, and consider the quotient $\sfrac{\Mt\langle a \rangle}{S\Mt\langle a \rangle}$. Let $J$ be a submodule of this quotient generated by a finite collection of homogeneous elements of the upper cone, and let these generators include elements of the form $[\rho^M a]$ and $[\tau^Na]$ for some $M$ and $N$. A \textbf{Jack-o-lantern} module is an $\Mt$ module isomorphic to such a module $J$. (See Figure \ref{fig:orange} for an example.)\par
We can decompose a \jol module as $J=J^+\sqcup J^-$ where the module structure connects $J^+$ to $J^-$. These two parts are
\begin{itemize}
\item An ideal $J^+$ of the upper cone of $\Mt\langle a\rangle$ such that for large enough $N$ and $M$, both $[\rho^Ma]\in J^+$ and $[\tau^Na]\in J^+$\\
and
\item A ``coideal'' $J^-$ of the lower cone of $\Mt\langle a\rangle$, meaning that if $[\frac\theta{\rho^i\tau^j}a]\in J^-$ then both $[\frac\theta{\rho^{i+1}\tau^j}a]\in J^-$ and $[\frac\theta{\rho^i\tau^{j+1}}]a\in J^-$, such that for large enough $N$ and $M$, both $[\frac\theta{\rho^M}a]\in J^-$ and $[\frac\theta{\tau^N}]a\in J^-$.
\end{itemize}
By ``the module structure connects $J^+$ to $J^-$'' we mean that if $[\rho^k\tau^la]\in J$ is nonzero, then $\frac\theta{\rho^{i+k}\tau^{j+l}}\cdot[\rho^k\tau^la]=[\frac\theta{\rho^i\tau^j}a]$.
\end{defn}
Note that $\Mt$ itself is trivially a \jol module. These modules contain elements of the form $[\tau^N a]$ and $[\frac\theta{\tau^N}a]$ sharing a dimension $p$. Likewise there is the largest fixed-set dimension $p-q$ of $J^+$ or smallest fixed-set dimension $p-q+2$ of $J^-$. Together these give $J$ a well-defined \textbf{phantom degree}, the degree of the $\Sigma^{p,q}\Mt$ containing\footnote{Note that $J\subset\Sigma^{p,q}\Mt$ is just a graded inclusion of sets. J is not a submodule of $\Sigma^{p,q}\Mt$, as $\iota:J\inj \Sigma^{p,q}\Mt$ is not a module map in general: If $[\theta a]=0$ in $J$, then $\tau^N \iota([\frac\theta{\tau^N} a])=\theta a$ while $ \iota(\tau^N[\frac\theta{\tau^N} a])=\iota([\theta a])=0$.} $J$. See Figure \ref{fig:orange}. Note that while we will write elements like $[\tau\rho a]$, in general $[a]=0$ in the \jol module. We will sometimes call $a$ the \textbf{phantom generator}.
\begin{figure}[h]
\begin{tikzpicture}[scale=0.3]
\color{gray}
\draw[dashed] (0,0) -- (0,4);
\draw[dashed] (0,0) -- (5,5);
\draw[dashed] (0,-2) -- (-5,-7);
\draw[dashed] (0,-2) -- (0,-8);
\color{black}
\draw[thick,->] (0,4) -- (0,7);
\draw[thick,->] (5,5) -- (7,7);
\draw[thick, fill=orange] (0,6.5)--(0,4) -- (1,5)--(1,4)--(2,5)--(2,3)--(5,6)--(5,5)--(6.5,6.5);
\draw[thick,->] (-5,-7) -- (-9,-11);
\draw[thick, fill=orange] (-8.5,-10.5)--(-5,-7) -- (-5,-9)-- (-3,-7)-- (-3,-10)-- (-1,-8)-- (-1,-9) -- (0,-8)-- (-0,-10.5);
\draw[thick,->] (0,-8) -- (-0,-11);
\draw[->] (-2,0)--(-.2,0);
\draw (-2,0) node[left] {phantom degree};
\draw (5,5) node {$\bullet$};
\draw (5,5) node[below right] {$[\rho^5 a]$};
\draw (0,4) node {$\bullet$};
\draw (0,4) node[left] {$[\tau^4 a]$};
\draw (-5,-7) node {$\bullet$};
\draw (-5,-7) node[above left] {$[\frac\theta{\rho^5} a]$};
\draw (0,-8) node {$\bullet$};
\draw (-0,-8) node[right] {$[\frac\theta{\tau^6} a]$};
\end{tikzpicture}
\caption{A \jol module. In the language of definition \ref{joldef}, this is $\left(\sfrac{\Mt\langle a \rangle}{S\Mt\langle a \rangle}\right)\langle [\tau^4a],[\rho\tau^3a],[\rho^2\tau a],[\rho^5a]\rangle$ where $S=\{\frac\theta{\rho^4\tau}a,\frac\theta{\rho^2\tau^4}a,\frac\theta{\tau^5}a\}$.}
\label{fig:orange}
\end{figure}
\begin{defn}
\label{jolmap}
A \textbf{\jol map} is an $\Mt$-module map $f:J_1\to J_2$ between \jol modules such that $f(J_1^-)=0$ and $f(J_1^+)\subseteq J_2^-$. \par
Note that the kernel and cokernel of a \jol map are both \jol modules, with $(\ker f)^+\subset J_1^+$ and $(\ker f)^-=J_1^-$ while $(\cok f)^+=J_2^+$ and $(\cok f)^-\subset J_2^-$. See for example Figure \ref{fig:jolkercok}
\end{defn}
\begin{figure}
\begin{tikzpicture}[scale=0.5]
\draw [->] (1.2,1)--node[above right]{$f$}(1.8,1);
\color{ggreen}
\draw (0,0) node {$J_1$};
\draw [<->] (0,4)--(0,1)--(1,2)--(1,1)--(4,4);
\draw [<->] (-3,-5)--(-1,-3)--(-1,-4)--(0,-3)--(0,-5);
\color{orange}
\draw (3,4) node {$J_2$};
\draw [<->] (3,7)--(3,5)--(4,6)--(4,5)--(6,7);
\draw [<->] (0,-1)--(2,1)--(2,0)--(3,1)--(3,-1);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.5]
\draw [dashed] (0,7)--(0,-5);
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.5]
\color{ggreen}
\draw (0,0) node[left] {$\ker(f)$};
\draw [<->] (0,4)--(0,1)--(2,3)--(2,2)--(4,4);
\draw [<->] (-3,-5)--(-1,-3)--(-1,-4)--(0,-3)--(0,-5);
\color{orange}
\draw (3,4) node[left] {$\cok(f)$};
\draw [<->] (3,7)--(3,5)--(4,6)--(4,5)--(6,7);
\draw [<->] (0,-1)--(1,0)--(1,-1)--(3,1)--(3,-1);
\end{tikzpicture}
\caption{The kernel and cokernel of a \jol map are also \jol modules.}
\label{fig:jolkercok}
\end{figure}
\begin{lemma}
\label{jackslemma}
Let $C_\bullet=\left[\dots\to J_{i-1}\xrightarrow{d_{i-1}}J_i \xrightarrow{d_i} J_{i+1}\xrightarrow{d_{i+1}} \dots \right]$ be a (co)chain of \jol maps. $($Note that Definition \ref{jolmap} implies $d^2=0)$. Then the homology modules $H_i(C_\bullet)$ are \jol modules.
\end{lemma}
\begin{proof}
The kernel and cokernel of a \jol map are both \jol modules. As $J_{i-1}\to \ker(d_i)$ is a \jol map, we have that $H_i(C_\bullet)=\cok(J_{i-1}\to \ker(d_i))$ is a \jol module.
\end{proof}
We will also make use in the next section of a lemma classifying general (possibly non-Jack-o-lantern) $\Mt$-module maps between \jol modules.
\begin{lemma}
\label{edgelemma}
Suppose $J$ and $J'$ are \jol modules having phantom generators $\alpha$ and $\alpha'$ of phantom degrees $(p,q)$ and $(p',q')$ respectively, with $(p,q)\le (p',q')$ lexicographically. If $f:J\to J'$ is an $\Mt$-module map carrying an element of $J^+$ to a nonzero element of $(J')^+$, then $p=p'-1$, and for large enough $N$, $f([\tau^N \alpha])=[\tau^{N+q-q'}\alpha']$.
\end{lemma}
\begin{proof}
Say $f([\rho^i\tau^j \alpha])= [\rho^{i'}\tau^{j'} {\alpha'}]\ne 0$. Suppose $p\ne p'-1$. Then by the lexicographic assumption either
\begin{itemize}
\item $p=p'$, and $q\le q'$, in which case let $M$ be such that $[\rho^M\alpha]\ne 0$. Then $f([\rho^M\alpha])=0$ for degree reasons, and so $\rho^Mf([\rho^i\tau^j \alpha])=f([\rho^{M+i}\tau^j \alpha])=\rho^i\tau^j f([\rho^M\alpha])=0$. As there is no $\rho$-torsion in $J^+$, this means $f([\rho^i\tau^j \alpha])=0$, a contradiction;\\ or
\item $p<p'-1$, in which case any $f([\tau^n\alpha])= 0$ for any $n$ for degree reasons. For some $N$ there is an element $[\tau^N\alpha]\ne 0$.
On one hand, $f([\rho^{i}\tau^{N+j}\alpha])=\rho^i\tau^jf([\tau^{N}\alpha])=\rho^i\tau^j\cdot 0=0$. But $f([\rho^{i}\tau^{N+j}\alpha])=\tau^Nf([\rho^{i}\tau^j\alpha])=\tau^N[\rho^{i'}\tau^{j'} {\alpha'}]\ne 0$, again a contradiction.
\end{itemize}
\begin{figure}[H]
\begin{tikzpicture}[scale=0.55]
\color{ggreen}
\draw(1,-1) node[below] {$\alpha$};
\draw[dashed] (1,-1) -- (1,3);
\draw[dashed] (1,-1) -- (3,1);
\draw[thick,->] (1,4) -- (1,7.7);
\draw[thick,->] (3,1) -- (6,4);
\draw[thick] (1,3) -- (1,5);
\draw[thick] (1,3) -- (3,5);
\draw[thick] (3,1) -- (3,5);
\draw (1,3) node {$\bullet$};
\draw (5,3) node {$\bullet$};
\draw (5,7) node {$\bullet$};
\color{blue}
\draw (2.2,3) node {$\bullet$};
\draw (6.1,3) node {$0$};
\draw(1,0) node[left] {$\alpha'$};
\draw[dashed] (1.2,0) -- (1.2,4);
\draw[dashed] (1.2,0) -- (4.2,3);
\draw[thick,->] (1.2,4) -- (1.2,8);
\draw[thick] (1.2,4)--(2.2,5)--(2.2,3) -- (4.2,5);
\draw[thick] (4.2,3) -- (4.2,5);
\draw[thick,->] (4.2,3) -- (7.2,6);
\color{black}
\draw[->] (1.2,3) -- (2,3);
\draw[->] (5.2,3) -- (5.8,3);
\draw[->] (5.2,7) -- (5.8,7);
\draw (1.6,3) node [below] {$\ne 0$};
\draw (6,7) node {!};
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale=0.55]
\color{ggreen}
\draw(0,0) node[below] {$\alpha$};
\draw[dashed] (0,0) -- (0,4);
\draw[dashed] (0,0) -- (5,5);
\draw[thick,->] (0,4) -- (0,9);
\draw[thick,->] (3,3) -- (7,7);
\draw[thick] (0,4) -- (1,5);
\draw[thick] (1,4) -- (1,5);
\draw[thick] (1,4) -- (3,6);
\draw[thick] (3,3) -- (3,6);
\draw (3,3) node {$\bullet$};
\draw (0,6) node {$\bullet$};
\draw (3,9) node {$\bullet$};
\color{blue}
\draw (4,3) node {$\bullet$};
\draw (1.1,6) node {$0$};
\draw(2,0) node[below] {$\alpha'$};
\draw[dashed] (2,0) -- (2,3);
\draw[dashed] (2,0) -- (4,2);
\draw[thick,->] (2,3) -- (2,9);
\draw[thick] (2,3) -- (4,5);
\draw[thick] (4,2) -- (4,5);
\draw[thick,->] (4,2) -- (7,5);
\color{black}
\draw[->] (3.2,3) -- (3.8,3);
\draw[->] (0.2,6) -- (0.8,6);
\draw[->] (3.2,9) -- (3.8,9);
\draw (3.4,3) node [below] {$\ne 0$};
\draw (4,9) node {!};
\end{tikzpicture}
\caption{Two contradictions.}
\label{fig:twocases}
\end{figure}
Thus $p=p'-1$, and now finally $f([\tau^N \alpha])=[\tau^{N+q-q'}\alpha']$, because if $f([\tau^N \alpha])=0$, then $\tau^N[\rho^{i'}\tau^{j'}{\alpha'}]=\tau^Nf([\rho^i\tau^j\alpha])=\rho^i\tau^jf([\tau^n\alpha])=0$, making $[\rho^{i'}\tau^{j'}{\alpha'}]$ an element in $(J')^+$ with $\tau$-torsion, a contradiction.
\end{proof}
In the next section, we put these algebraic results to use in cohomology.
\section{A theorem restricting Kronholm shifts}
Because $\Mt$ has nonzero groups in so many bidegrees, when examining spectral sequences or even just long exact sequences of a pair, there are in general a lot of algebraically possible differentials to consider. The following theorem helps us rule out some of these possibilities. We first need a definition.
\begin{defn}
Let $\Lambda$ be a set with a partial ordering so that any two elements $\alpha,\beta\in \Lambda$ have a greatest lower bound, denoted $\alpha\cap\beta\in \Lambda$. A \textbf{hierarchical cell structure} on a space $X$ is a CW structure with cells $\{e_\lambda\}_{\lambda\in\Lambda}$ so that each cell $e_\alpha$ has an attaching map whose image lies in $\ds\bigsqcup_{\lambda<\alpha}e_\lambda$.
\end{defn}
Note that a hierarchical cell structure on a space gives subspaces $X_\alpha$ of the form
$$X_\alpha=\bigsqcup_{\lambda\le \alpha}e_\alpha$$
with the properties
\begin{itemize}
\item $X_\alpha\cap X_{\beta}=X_{\alpha\cap\beta}$\quad and
\item $\displaystyle\sfrac{(X_\alpha\cup X_{\alpha'})}{X_{\alpha}\cap X_{\alpha'}}=(\sfrac{X_\alpha}{(X_{\alpha}\cap X_{\alpha'})})\vee (\sfrac{X_{\alpha'}}{(X_{\alpha}\cap X_{\alpha'})}).$
\end{itemize}
\begin{example}
The setting in which we will use this notion is of course the Schubert cell construction of the Grassmannian, which gives it a hierarchical cell structure, by Proposition 3.2.3 of \cite{manivel}. Schubert cells are indexed by Young diagrams, which have a partial ordering under inclusion. In fact, in this setting,
\[X_\lambda=\bigsqcup_{\lambda'\le\lambda}\Omega_{\lambda'}=\overline{\Omega_\lambda},\]
the closure of the Schubert cell $\Omega_\lambda$, called the \textbf{Schubert variety}.
For two diagrams $\lambda_1$ and $\lambda_2$,
$$X_{\lambda_1}\cap X_{\lambda_1}=X_{\lambda_1\cap\lambda_2}$$
and
$$\sfrac{(X_{\lambda_1}\cup X_{\lambda_2})}{X_{\lambda_1\cap\lambda_2}}=(\sfrac{X_{\lambda_1}}{X_{\lambda_1\cap\lambda_2}})\vee (\sfrac{X_{\lambda_2}}{X_{\lambda_1\cap\lambda_2}})$$
for example in $\Gr_2\R^5$ (see Figure \ref{fig:threeversions})
$$X_{\scalebox{0.4}{\yng(1,3)}}\cap X_{\scalebox{0.4}{\yng(2,2)}}=X_{\scalebox{0.3}{\yng(1,3)}\,\cap\,\scalebox{0.3}{\yng(2,2)}}=X_{\scalebox{0.3}{\yng(1,2)}}$$
and
$$\sfrac{(X_{\scalebox{0.4}{\yng(1,3)}}\cup X_{\scalebox{0.4}{\yng(2,2)}})}{X_{\scalebox{0.4}{\yng(1,2)}}}
=(\sfrac{X_{\scalebox{0.4}{\yng(1,3)}}}{X_{\scalebox{0.4}{\yng(1,2)}}})\vee (\sfrac{X_{\scalebox{0.4}{\yng(2,2)}}}{X_{\scalebox{0.4}{\yng(1,2)}}}).$$\\
\end{example}
\begin{warning}
To avoid confusion, we call the reader's attention to the fact that there are two distinct orderings in the following theorem -- a partial ordering corresponding to a hierarchical cell structure, and the lexicographic order on bidegrees.
\end{warning}
\begin{thm}
\label{fixedthm}
Let $X'$ be an equivariant space with representation cell structure. Suppose that a cell $e_\beta\simeq e^{p,q}$ whose bidegree $(p,q)$ is lexicographically after all the cells of $X'$ is attached to $X'$ to make a space $X$ with a hierarchical representation-cell structure. Suppose also that the forgetful cochain complex $C^\bullet(\psi(X))$ corresponding to this construction has only zero differentials. If, for every cell $e_\alpha\simeq e^{p',q'}$ used in building $X'$, either
\begin{enumerate}
\item[$(i)$] $\alpha\not\le \beta$ \qquad or
\item[$(ii)$] $p'-q'\le p-q$
\end{enumerate}
then the cofiber sequence $X'\inj X\to \S pq$ gives a split short exact sequence in cohomology, i.e.
\[\H \bullet\bullet (X)=\H\bullet\bullet (X')\oplus \M pq.\]
$\ $\\
\end{thm}
To prove this theorem, we need three lemmas.
\newpage
\begin{lemma}\label{jackolemma}
Suppose an equivariant representation-cell complex $X$ is built by attaching cells in lexicographic order. Suppose the associated forgetful chain complex $C^\bullet(\psi(X))$ has all zero differentials. Then the corresponding one-cell-at-a-time equivariant spectral sequence $E_\bullet^{\bullet,\bullet,\bullet}$ with the lexicographic filtration of $X$ has a \jol module for the $r^{\text{th}}$ filtration of its $k^{\text{th}}$ page, $E_k^{\bullet,\bullet,r}$ for all $k$ and $r$. Furthermore, each differential $d_k:E^{\bullet,\bullet,r}_k\to E^{\bullet+1,\bullet,r+k}_k$ is a \jol map.
\end{lemma}
\begin{lemma}\label{alphabeta}
Under the assumptions of the theorem, if cells $e_\alpha$ and then $e_\beta$ are used in building a space $X$, but $\alpha\not\le\beta$, then there is no differential from the filtration degree of $\alpha$ to that of $\beta$ in the one-cell-at-a-time spectral sequence for $X$.
\end{lemma}
\begin{lemma}\label{seqs}
Let $X$ be a finite filtered space $$\pt=X_0\subseteq X_1\subseteq\dots\subseteq X_{n-1}\subseteq X_n=X.$$ If every differential of the associated spectral sequence mapping into the $\sfrac {X_{n}}{X_{n-1}}$ filtration is zero, then for the cofiber sequence $X_{n-1}\xrightarrow{i}X_n\to \sfrac{X_n}{X_{n-1}}$, the map $i^*:\H\bullet\bullet (X_n)\to \H\bullet\bullet(X_{n-1})$ is surjective.
\end{lemma}
We first prove the lemmas, and then the theorem. While we will not need the theorem until Section \ref{twontwo}, Lemmas \ref{jackolemma} and \ref{alphabeta} will be used in Section \ref{knone}.
\begin{proof}[Proof of Lemma \ref{jackolemma}]
Denote the generator of the $\Sigma^\alpha \Mt$ by $1_\alpha$. On page one of the spectral sequence there are upper cone elements of the form $\rho^i\tau^j 1_\alpha$ and lower cone elements $\frac\theta{\rho^i\tau^j}1_\alpha$. Denote\footnote{This is possible because the one-cell-at-a-time filtration precludes any elements of the form $[\rho^i\tau^j 1_\alpha+\frac\theta{\rho^{i'}\tau^{j'}}1_{\alpha'}]$, as these are filtration-inhomogeneous.} such elements surviving to later pages by $[\rho^i\tau^j 1_\alpha]$ and $[\frac\theta{\rho^i\tau^j}1_\alpha]$.\par
Proceed by induction on the page of the spectral sequence. To begin with, $E_1=E_1^{\bullet,\bullet,\bullet}$ consists of a single suspension of $\Mt$ in each nonzero filtration. Differentials are determined by the image of each $1_{\alpha}$. Because of our lexicographic filtration, the only possible top-to-top differential is of the form $d_1(1_{\alpha})=\tau^j 1_{\alpha'}$. However, as $\psi(\tau)=1$, this would mean a nonzero differential in the forgetful setting, a contradiction of our assumption. Thus the differential $d_1$ forms a complex of (trivial) \jol modules. Hence by Lemma \ref{jackslemma}, $E_2$ consists of a \jol module in each filtration. See for example Figure \ref{fig:zig}.
\begin{figure}[h]
\begin{tikzpicture}[scale=0.4]
\color{cyan}
\draw[->] (2.2,1)--(0.2,-1);
\draw[->] (2.2,1)--(2.2,-1);
\draw[->] (2.2,3)--(2.2,4);
\draw[->] (2.2,3)--(3.2,4);
\color{green}
\draw[->] (-2,-5)--(-2,-3.5);
\draw[->] (-2,-5)--(-0.5,-3.5);
\draw[->] (-2,-7)--(-2,-8);
\draw[->] (-2,-7)--(-3,-8);
\color{black}
\draw[thick, ->] (0,-2) -- (0,-8);
\draw[thick, ->] (0,-2) -- (-6,-8);
\draw[thick,->] (0,0) -- (4,4);
\draw[thick,->] (0,0) -- (0,4);
\draw (0,-9.5) node {$E_1$};
\draw (6,-1) node {$\rightsquigarrow$};
\draw (.6,-.1) node {$\to$};
\draw (-2+.6,-5) node {$\to$};
\end{tikzpicture}
\quad
\begin{tikzpicture}[scale=0.4]
\color{gray}
\draw[dashed] (0,0) -- (2,2);
\draw[dashed] (0,0) -- (0,1);
\draw[dashed] (0,-2) -- (0,-4);
\draw[dashed] (0,-2) -- (-2,-4);
\color{black}
\draw[thick,->] (2,2) -- (4,4);
\draw[thick,->] (0,1) -- (0,4);
\draw[thick] (0,1) -- (2,3);
\draw[thick] (2,2) -- (2,3);
\draw[thick, ->] (0,-4) -- (0,-8);
\draw[thick] (0,-4) -- (-2,-6);
\draw[thick] (-2,-4) -- (-2,-6);
\draw[thick, ->] (-2,-4) -- (-6,-8);
\fill[fill=black] (2,3) circle (2mm) node[above] {$[\rho^2\tau 1_\alpha]$};
\fill[fill=black] (-2,-4) circle (2mm) node[above left] {$[\frac\theta{\rho^2} 1_\alpha]$};
\draw (0,-9.5) node {$E_2$};
\end{tikzpicture}
\caption{A filtration degree of $E_1$ with differential in and out of that degree, and the corresponding filtration on the $E_2$ page. Note a connection between upper an lower cones remains. While $[1_\alpha]=0$, it is still the case, for example, that $\frac\theta{\rho^4\tau}\cdot [\rho^2\tau 1_\alpha]=[\frac\theta{\rho^2} 1_\alpha]$. This filtration degree of $E_2^{\bullet,\bullet,r}$ is a \jol module.}
\label{fig:zig}
\end{figure}
Now assume for induction that the page $E_k=E_k^{\bullet,\bullet,\bullet}$ consists of a \jol module in each nonzero filtration. We must show that the differentials $d_k$ are \jol maps. First, there can be no top-to-top map on the $E_k$ page: By Lemma \ref{edgelemma}, such a map would carry a nonzero element $[\tau^{N} 1_\alpha]$ to the nonzero element $[\tau^{N'} 1_{\alpha'}]$, (where $N'=N+(w(\alpha)-w(\alpha'))$). But considering the forgetful map $\psi$ yields a contradiction, as as we see a nonzero differential $\psi(d_k):1_\alpha\mapsto 1_{\alpha'}$, in the non-equivariant spectral sequence, contradicting our assumption about the forgetful chain complex. \\
Now that top-to-top differentials are ruled out, bottom-to-anywhere can be eliminated. Consider a bottom-cone element $[\frac\theta{\rho^i\tau^j}1_\alpha]$ on $E_k$. For some $N$ we have $[\tau^N1_\alpha]\ne 0$. As $d_k([\tau^N1_\alpha])$ is not a top-cone element, it is sent to zero by multiplication by any element of $\Mt^-$ (recall that $\theta^2=0$). And so $$d_k\left(\left[\frac\theta{\rho^i\tau^j}1_\alpha\right]\right)
=\frac\theta{\rho^i\tau^{N+j}} d_k\left([\tau^N1_\alpha]\right)=0.$$
Thus the only possible nonzero differentials will carry top-cone elements to bottom-cone elements, and hence $d_k$ is a \jol map. Again applying Lemma \ref{jackslemma}, $E_{k+1}$ consists of \jol modules. This completes the induction.\\
\end{proof}
\begin{proof}[Proof of Lemma \ref{alphabeta}]
For our space $X$ with a hierarchical cell structure satisfying the assumptions of the theorem, let $\alpha$ be the index of a cell attached before the cell $e_\beta$ such that $\alpha\not\le \beta$, and define
$$Y=X_\alpha\cup X_\beta=\left(\bigsqcup_{\lambda\le \alpha}e_\lambda\right)\cup \left(\bigsqcup_{\lambda\le \beta}e_\lambda\right)$$
and
$$Z=X_\alpha\cap X_\beta=\bigsqcup_{\srac{\lambda<\alpha\text{ and }}{\lambda<\beta}}e_\lambda.$$
Then $\sfrac YZ$ is a wedge of spaces
$$\sfrac YZ=\sfrac {(X_\alpha)}{Z}\vee\sfrac{(X_\beta)}Z=:A\vee B$$
and there are quotient maps $Y\to A\vee B\to A$ which respect the filtration grading of the spectral sequence. Thus any element $[x]_Y\in E_k^{\bullet,\bullet,\bullet}(Y)$ with the filtration degree of $\alpha$ corresponds to an element $[x]_A\in E_k^{\bullet,\bullet,\bullet}(A)$. Because $\alpha$ is in the highest nontrivial filtration of $A$, $d_r([x]_A)=0$ for all $r\ge k$. We now have the commuting diagram
\begin{figure}[H]
\[\xymatrix{
\quad [x]_A\in\ar@{|->}[d]& E_k^{\bullet,\bullet,\bullet}(A)\ar[r]\ar[d]_{d_k} & E_k^{\bullet,\bullet,\bullet}(A\vee B)\ar[r]\ar[d]_{d_k} & E_k^{\bullet,\bullet,\bullet}(Y)\ar[d]_{d_k}& \ni [x]_Y\quad\ar@{|->}[d]\\
\quad 0\in&E_k^{\bullet+1,\bullet,\bullet+k}(A)\ar[r] & E_k^{\bullet+1,\bullet,\bullet+k}(A\vee B)\ar[r] & E_k^{\bullet+1,\bullet,\bullet+k}(Y)&\ni0\quad
}\]
\end{figure}
$\!\!\!\!\!\!\!\!\!$ and so in particular the map between the filtration degrees of $\alpha$ and $\beta$ is zero in $E^{\bullet,\bullet,\bullet}_\bullet(Y)$. Now the inclusion $i:Y\inj X$ also and induces a spectral sequence map $i^*:E^{\bullet,\bullet,\bullet}_1(X)\to E^{\bullet,\bullet,\bullet}_1(Y)$ such that for any $M\in\Mt$ we have $i^*([M 1_\alpha]_X)=[M 1_\alpha]_Y$. We can now show $\alpha$-to-$\beta$ differentials in $X$ are zero using naturality. If $d([\rho^i\tau^j1_\alpha]_X)=[\frac{\theta}{\rho^k\tau^l}1_\beta]_X$, then
$$\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_Y=i^*\left(\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X\right)=i^*d(\left[\rho^i\tau^j1_\alpha\right]_X)
=d(\left[\rho^i\tau^j1_\alpha\right]_Y)=0.$$
Now the fact that $\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_Y=0$ on $E_k^{\bullet,\bullet,\bullet}(Y)$ means that it must participate in some nonzero differential $d_j$ for $j<k$, either
\begin{itemize}
\item $d_j\left(\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_Y\right)\ne 0$\\ or
\item $d_j(M_Y)=\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_Y$ for some element $M_Y\in E_j^{\bullet,\bullet,\bullet}(Y)$.
\end{itemize}
by naturality, this means either
\begin{itemize}
\item $d_j\left(i^*\left(\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X\right)\right)=i^*\left( d_j\left(\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X\right)\right)\ne 0$ so $d_j\left(\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X\right)\ne 0$ \\ or
\item There is some $M_X\in E_j^{\bullet,\bullet,\bullet}(X)$ with $i^*(M_X)=M_Y$ and $i^*(d_j(M_X))=d_j(M_Y)=\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_Y$. But $i^*\left(\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X\right)=\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_Y$ also. In any given tri-degree $(a,b,c)$, the map $i^*:E_j^{a,b,c}(X)\to E_j^{a,b,c}(Y)$ is a homomorphism between groups of at most two elements. And so in fact $d_j(M_X)=\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X$.
\end{itemize}
Thus in either case, $\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X$ also participates in a nonzero differential on $E_j^{\bullet,\bullet,\bullet}(X)$, and hence $\left[\frac{\theta}{\rho^k\tau^l}1_\beta\right]_X=0$ on $E_k$. We have finally shown that the image of $d_k$ applied to the $\alpha$ filtration of $E^{\bullet,\bullet,\bullet}_k(X)$ must be zero, which completes the proof.\\
\end{proof}
\newpage
Finally, we prove Lemma \ref{seqs} by a diagram chase:
\begin{proof}[Proof of Lemma \ref{seqs}]
Naming the inclusions $$\pt\xrightarrow{i_0}X_1\xrightarrow{i_1}\dots\xrightarrow{i_{n-2}}X_{n-1}\xrightarrow{i_{n-1}}X_n$$ recall that we build the spectral sequence by weaving together the long exact sequences of the cofiber sequences $X_{k}\xrightarrow{i_k} X_{k+1}\xrightarrow{q_k}\sfrac{X_k}{X_{k-1}}$.\\
\begin{tabular}{c||c}
\xymatrix{
\dots H(\frac{X_{n}}{X_{n-1}})\ar[r]^{q_n^*}&HX_n\ar[d]^{i_{n-1}^*}\\
\dots H(\frac{X_{n-1}}{X_{n-2}})\ar[r]^{q_{n-1}^*}&
H(X_{n-1})\ar[d]^{i_{n-2}^*}\ar[r]^\delta&H(\frac{X_n}{X_{n-1}})\dots\\
\dots H(\frac{X_{n-2}}{X_{n-3}})\ar[r]^{q_{n-2}^*}&
H(X_{n-2})\ar[d]^{i_{n-3}^*}\ar[r]^\delta&H(\frac{X_{n-1}}{X_{n-2}})\dots\\
\vdots&\vdots\ar[d]^{i_0^*}&\vdots\\
\dots H(\frac{X_0}{X_{-1}})\ar[r]^{q_0^*}&HX_0\ar[r]^\delta\ar[d]^{i_{-1}^*}&H(\frac{X_1}{X_0})\dots\\
&HX_{-1}
}&
\xymatrix{
\ar[r]^{q_n^*}&HX_n\ar[d]^{i_{n-1}^*}\\
b\ar@{|->}[r]^{q_{n-1}^*}&
\,\phantom{\int}a\phantom{\int}\ar@{|->}[d]^{i_{n-2}^*}\ar[r]^\delta &H(\frac{X_n}{X_{n-1}})\\
c\ar@{|->}[r]^{q_{n-2}^*}&
\,\phantom{\int}i^*_{n-2}(a)\phantom{\int}\ar@{|->}[d]^{i_{n-3}^*}\ar[r]^\delta&\dots\\
\,\phantom{\int}&\vdots\ar[d]^{i_0^*}&\vdots\\
\ar[r]^{q_0^*}&\,HX_0\phantom{\int}\ar[r]^\delta\ar[d]^{i_{-1}^*}&\dots\\
&0
}
\end{tabular}
Assuming all differentials to $H\left(\frac{X_n}{X_{n-1}}\right)$ are zero, we wish to show that $i^*_{n-1}$ is surjective. Consider an element $a\in H(X_{n-1})$.\par
If $i_{n-2}^*(a)=0$, by exactness there exists $b$ with $q_{n-1}^*(b)=a$, and then by assumption $d_1(b)=(\delta\circ q_{n-1}^*)(b)=\delta(a)=0$ and so by exactness $a$ lies in the image of $i_{n-1}^*$.\par
If $i_{n-2}^*(a)\ne 0$ but $i_{n-3}^*(i_{n-2}^*(a))=0$, then by exactness there is some $c$ so that $q_{n-2}^*(c)=i_{n-2}^*(a)$. Since $d_2(c)=\delta(a)=0$, we again have $a$ in the image of $i_{n-1}^*$ by exactness.\par
Since $H(X_0)=0$, eventually we are guaranteed some $(i_k^*\circ i_{k+1}^*\circ\dots\circ i_{n-2}^*)(a)=0$ and hence some $x\in H(\frac{X_k}{X_{k-1}})$ such that $q_{k+1}^*(x)=( i_{k+1}^*\circ\dots\circ i_{n-2}^*)(a)$, and since $d_k(x)=\delta(a)=0$, again $a$ is mapped to by $i_{n-1}^*$. Hence $i_{n-1}^*$ is surjective.
\end{proof}
We can now prove the theorem.
\begin{proof}[Proof of Theorem \ref{fixedthm}]
Again filter $X$ one-cell-at-a-time lexicographically, meaning by increasing topological dimension and then increasing weight. This filtration gives a trigraded spectral sequence, as discussed in Section \ref{jolsection}. We claim there are no differentials hitting any nonzero elements of the $n$th filtration degree $\Hbb(\sfrac{X_n}{X_{n-1}})=\Hbb(\sfrac{X}{X'})$. To see this, consider each lower filtration degree $k<n$, which corresponds to some $e_\alpha$ used in building $X'$. Either condition (i) holds ($\alpha\not\le\beta$), in which case by Lemma \ref{alphabeta}, $d_i(x)=0$ for all $x$ of filtration degree $k$, or else condition (ii) holds. In this case, by Lemma \ref{jackolemma} $d_{n-k}$ is a \jol map, and so $d_{n-k}(x)=0$ for all $x$ of filtration degree $k$, as no top-to-bottom differential is possible. In either case, no differential ever hits the filtration degree of $\beta$. By Lemma \ref{seqs}, this means that $\Hbb(X_n)\to \H\bullet\bullet(X_{n-1})$ is surjective. Now consider the long exact sequence in cohomology corresponding to the cofiber sequence $X_{n-1}\xrightarrow{i}X_n\to \S pq$.
\[\dots\to\Hbb(\Spq)\xrightarrow{0}\Hbb(X_n)\to \Hbb(X_{n-1})\to \H{\bullet+1}\bullet(\Spq)\xrightarrow{0}\dots\]
This decomposes into \textit{short} exact sequences
\[0\to \Hbb(X_n)\to\Hbb(X_{n-1})\to\Sigma^{p,q}\Mt\to0\]
for all $\bullet$. Since $\Sigma^{p,q}\Mt$ is free, the short exact sequence of modules is split, and the theorem is proved.
\end{proof}
\section{Grassmannians $\Gr_k(\R^{n,1})$}\label{knone}
We are now ready to tackle the Grassmannian. We begin by introducing a statistic of Young diagrams which will be useful for classifying Schubert cells in a family of spaces. Fix $k$ and $n$. Given a partition $\lambda=(\lambda_1,\dots,\lambda_k)$ with $\lambda_i\le \lambda_{i+1}$, define trace($\lambda)=\#\{i:\lambda_i\ge k-i+1\}$. Visually, this is the number of squares lying on the diagonal of a Young diagram of this partition. See Figure \ref{fig:traces} for examples. Recall that the \textbf{jump sequence} $\underline{j}=[j_1,\dots,j_k]$ corresponding to a partition $\lambda$ is given by $j_i=\lambda_i+i$. The values of this sequence tell us where the 1s land in the Schubert cell matrix corresponding to $\lambda$. We can also formulate trace as $\text{trace}(\underline{j})=\#\{i:j_i\ge k+1\}$.
\begin{figure}[H]
\begin{tikzpicture}
\draw(.6,.6) node {$\yng(3,3,3)$};
\draw(.6,-.5) node {$\trace(3,3,3)=\trace([4,5,6])=3$};
\color{red}
\draw[-] (0,0) -- (1.2,1.2);
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\draw(.6,.6) node {$\yng(1,3,3)$};
\draw(.6,-.5) node {$\trace(1,3,3)=\trace([2,5,6])=2$};
\color{red}
\draw[-] (0,0) -- (.8,.8);
\end{tikzpicture}\\
\begin{tikzpicture}
\draw(.8,.6) node {$\yng(2,2,2)$};
\draw(.8,-.5) node {$\trace(2,2,2)=\trace([3,4,5])=2$};
\color{red}
\draw[-] (.4,0) -- (1.2,.8);
\end{tikzpicture}
\qquad
\begin{tikzpicture}
\draw(.6,.4) node {$\yng(1,3)$};
\draw(.6,-.5) node {$\trace(0,1,3)=\trace([1,3,6])=1$};
\color{red}
\draw[-] (0,.0) -- (.4,.4);
\end{tikzpicture}
\caption{}
\label{fig:traces}
\end{figure}
The following lemma uses trace to compute weight.
\begin{lemma}
\label{tracelemma}
For the Schubert cell structure on $\Gr_k(\R^{n,1})$ corresponding to the decomposition $\R^{n,1}=\R\triv^{k-1}\oplus\R\sgn^1\oplus \R\triv^{n-k}$, the weight of a Schubert cell $\Omega_\lambda$ is exactly the trace of $\lambda$.
\end{lemma}
\begin{proof}
Fix a Young diagram $\lambda=(\lambda_1,\dots,\lambda_k)$ fitting inside of a $k\times(n-k)$ grid. Recall it has jump sequence $[\lambda_1+1,\dots,\lambda_k+k]$ corresponding to the location of 1s in the family of matrices whose rowspaces make up $\Omega_\lambda$. One of two cases holds. Either
\begin{itemize}
\item No element of the jump sequence equals $k$, in which case each row with a jump exceeding $k$ contains a {\color{red}\scalebox{0.8}{\young(-)}} in dimension $k$ (see the top two examples in Figure \ref{fig:traceweight}). So the number of {\color{red}\scalebox{0.8}{\young(-)}} is $\#\{i: j_i>k\}=\#\{i: j_i\ge k+1\}=\text{trace}(\lambda)$.\\ or
\item A $1$ \emph{does} lie in column $k$, say in row $r$. Then trace$(\lambda)=\#\{i:j_i\ge k+1\}=\#\{i:i\ge r\}=k-r$, which also equals the number of {\color{red}\scalebox{0.8}{\young(-)}} appearing to the left of this $1$ in row $r$ (see bottom examples in Figure \ref{fig:traceweight}). This is because when a matrix which is acted upon and rewritten in canonical Schubert cell form (as on page \pageref{equivschub}), the $r$-th row will be multiplied by -1, changing the sign on each of the $k-r$ variables in that row.
\end{itemize}
Recall that the topological dimension of a Schubert cell corresponds to the number of boxes in its Young diagram, and the weight to the number of these {\color{red}\scalebox{0.8}{\young(-)}} boxes. And so $w(\lambda)=\text{trace}(\lambda)$.
\end{proof}
\begin{figure}[h]
\begin{tikzpicture}
\draw(0,1) node {$\left[ \begin{matrix}
\young(+) & \young(+) & {\color{red}\young(-)} & 1 & 0 & 0 \\
\young(+) & \young(+) & {\color{red}\young(-)} & 0 & 1 & 0 \\
\young(+) & \young(+) & {\color{red}\young(-)} & 0 & 0 & 1
\end{matrix}\right]$};
\draw(0,0) node {$w(3,3,3)=w([4,5,6])=3$};
\end{tikzpicture}
\qquad\quad
\begin{tikzpicture}
\draw(0,1) node {$\left[ \begin{matrix}
\young(+) & 1 & 0 & 0 & 0 & 0 \\
\young(+) & 0 & {\color{red}\young(-)}& \young(+) & 1 & 0 \\
\young(+) & 0 & {\color{red}\young(-)}& \young(+) & 0 & 1 \\
\end{matrix}\right]$};
\draw(0,0) node {$w(1,3,3)=w([2,5,6])=2$};
\end{tikzpicture}\\
\begin{tikzpicture}
\draw(0,1) node {$\left[ \begin{matrix}
{\color{red}\young(-)} & {\color{red}\young(-)} & 1 & 0 & 0 & 0 \\
\young(+) & \young(+) & 0 & 1 & 0 & 0 \\
\young(+) & \young(+) & 0 & 0 & 1 & 0
\end{matrix}\right]$};
\draw(0,0) node {$w(2,2,2)=w([3,4,5])=2$};
\end{tikzpicture}
\qquad\quad
\begin{tikzpicture}
\draw(0,1) node {$\left[ \begin{matrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 &{\color{red}\young(-)} & 1 & 0 & 0 & 0 \\
0 & \young(+) & 0 & \young(+) & \young(+) & 1
\end{matrix}\right]$};
\draw(0,0) node {$w(0,1,3)=w([1,3,6])=1$};
\end{tikzpicture}
\caption{$k=3$, some cells in $I(++-+++)$ for $\Gr_3(\R^{6,1})$.}
\label{fig:traceweight}
\end{figure}
Let $\part(p,k,m,t)$ denote the number of partitions of $p$ into $k$ non-negative numbers not exceeding a maximum value $m$, such that the Young diagram corresponding to the partition has trace $t$. Lemma \ref{tracelemma} says that in building $\Gr_k(\R^{n,1})$ using $I(\R\triv^{k-1}\oplus\R\sgn\oplus \R\triv^{n-k})$, the number of $(p,q)$-cells is $\part(p,k,n-k,q)$.\\
\begin{example}
In the ingredients table $I(++-+++++)$ for $\Gr_3(\R^{8,1})$, the number of $(11,2)$-cells is $\part(11,3,5,2)=2$. This counts \scalebox{0.3}{\yng(1,5,5)} and \scalebox{0.3}{\yng(2,4,5)}, but does not count, for example, \scalebox{0.3}{\yng(1,2,4,4)} (too many terms) or \scalebox{0.3}{\yng(5,6)} (a term exceeds 5) or the trace-3 diagrams \scalebox{0.3}{\yng(3,3,5)} or \scalebox{0.3}{\yng(3,4,4)}, which correspond instead to $(11,3)$-cells.
\end{example}
Now that we know the combinatorics of this construction, we show that the corresponding equivariant cellular spectral sequence collapses.
\begin{lemma}
\label{tracelemma2}
All differentials are zero in the cellular spectral sequence for $\Gr_k(\R^{n,1})$ corresponding to the ordered decomposition $\R^{n,1}= \R\triv^{k-1}\oplus\R\sgn\oplus\R\triv^{n-k}$.
\end{lemma}
\begin{proof}
In order for a nonzero differential to exist, there must be some Young diagrams $\alpha$ and $\beta$ in bidegrees allowing for a map from the generator of $\alpha$ to the lower cone of $\beta$ (by Lemma \ref{jackolemma}), and also with $\alpha\subset\beta$ (by Lemma \ref{alphabeta}). The bidegree requirement demands that the fixed-set dimension of $\alpha$ is greater than that of $\beta$. That is, denoting the topological degree of $\lambda$ by $|\lambda|$, we must have $|\alpha|-w(\alpha)>|\beta|-w(\beta)$. However, as $\alpha\subset\beta$, the diagram $\beta$ could be built from $\alpha$ by successively adding blocks. Each block would increase topological dimension by one, but could increase the trace (and hence by Lemma \ref{tracelemma} the weight $w$) by at \emph{most} one. Thus $\alpha\subset\beta$ implies $|\alpha|-w(\alpha)\le|\beta|-w(\beta)$. These conflicting requirements show that no differentials are possible if $\Gr_k(\R^{n,1})$ is built in this way.
\end{proof}
\begin{example}
Suppose $\alpha=(8)=\scalebox{0.4}{\yng(8)}$ so that $|\alpha|-w(\alpha)=8-1$.
\begin{itemize}
\item If $\beta=(1,8)=\scalebox{0.4}{\yng(1,8)}$, then although $\alpha\subseteq \beta$, there is no differential to the filtration of $\beta$ as $8-1\not> |\beta|-w(\beta)=9-1$. That is, $\theta\beta$ is too low for a differential from $\alpha$ to reach it.
\item On the other hand if $\beta=(3,3,3)=\scalebox{0.4}{\yng(3,3,3)}$ then $|\beta|-w(\beta)=9-3<8-1$, however, this doesn't fit: $\alpha\not\subseteq\beta$ and so there is still no differential, by Lemma \ref{alphabeta}.
\end{itemize}
\end{example}
Theorem \ref{kn1thm} is now immediate. We restate it here:
\begin{thm}\label{knoneagain}
\[\rank_{\Mt}^{p,q}\H \bullet\bullet(\Gr_k(\R^{n,1}))=\part(p,k,n-k,q).\]
\end{thm}
\begin{proof}
By Lemma \ref{tracelemma}, we have a cellular spectral sequence for $\Gr_k(\R^{n,1})$ with generators on the $E_1$ page corresponding to Young diagrams, with topological dimension given by number of boxes, and weight given by trace. By Lemma \ref{tracelemma2}, this spectral sequence immediately collapses.
\end{proof}
\subsection{Comment}
\label{upside}
In Section \ref{foreshadow}, we observed that the rows of the rank charts of cohomologies $\H\bullet\bullet(\Gr_{k}(\R^{n,1}))$ are palindromes.
We can deduce this from the fact that $\rank_{\Mt}^{p,q}\H\bullet\bullet(\Gr_k(\R^{n,1}))$ counts Young diagrams of $p$ boxes with trace $q$ fitting inside of a $k$-by-$(n-k)$ box. To have trace $q$, a Young diagram must have a $q$-by-$q$ square as its southwest corner, with any additional boxes to the north or to the east of this square. For example, considering $\Gr_4(\R^{9,1})$, trace-2 diagrams take the form
\[k=4\left\{\phantom{\young(a,b,c,d)}\right.\!\!\!\!\!\!\!\!
\underbrace{\young(??,??,{\,}\diagup ???,\diagup {\,}???)}_{n-k=5}\subseteq \yng(5,5,5,5)\,.\]
For a given trace $q$, the topological dimension of a Young diagram (corresponding to the number of boxes) is $q^2+\#\scalebox{0.8}{\young(?)}$ where $0\le \#\scalebox{0.8}{\young(?)}\le(k-q)q+q(n-k-q)$. If we take the complementary diagram in the north and east regions and rotate these complements (to get a legal diagram) we have a bijection. Figure \ref{fig:bijection} demonstrates a pairing that shows why, for example, in $\H\bullet\bullet(\Gr_4(\R^{9,1}))$ we see $\rank_{\Mt}^{6,2}=\rank_{\Mt}^{2*9-6,2}=\rank_{\Mt}^{12,2}=5$.
\begin{figure}[h]
\begin{align*}
\scalebox{0.5}{\yng(2,4)}&=
\scalebox{0.5}{\young(\ast\ast,\ast\ast,{\,}\diagup\ast\ast\ast,\diagup{\,}{\,}{\,}\ast)}\xrightarrow{\text{complement regions}}
\scalebox{0.5}{\young({\,}{\,},{\,}{\,},{\,}\diagup{\,}{\,}{\,},\diagup{\,}\ast\ast{\,})}\xrightarrow{\text{rotate regions}}
\scalebox{0.5}{\young({\,}{\,},{\,}{\,},{\,}\diagup{\,}\ast\ast,\diagup{\,}{\,}{\,}{\,})}
\,=\,\scalebox{0.5}{\yng(2,2,3,5)}\\
%
\scalebox{0.5}{\yng(3,3)}&=
\scalebox{0.5}{\young(\ast\ast,\ast\ast,{\,}\diagup{\,}\ast\ast,\diagup{\,}{\,}\ast\ast)}\xrightarrow{\text{complement regions}}
\scalebox{0.5}{\young({\,}{\,},{\,}{\,},{\,}\diagup\ast{\,}{\,},\diagup{\,}\ast\,{\,})}\xrightarrow{\text{rotate regions}}
\scalebox{0.5}{\young({\,}{\,},{\,}{\,},{\,}\diagup{\,}{\,}\ast,\diagup{\,}{\,}{\,}\ast)}
\,=\,\scalebox{0.5}{\yng(2,2,4,4)}\\
%
\scalebox{0.5}{\yng(1,2,3)}&=
\scalebox{0.5}{\young(\ast\ast,{\,}\ast,{\,}\diagup\ast\ast\ast,\diagup{\,}{\,}\ast\ast)}\xrightarrow{\text{complement regions}}
\scalebox{0.5}{\young({\,}{\,},\ast{\,},{\,}\diagup{\,}{\,}{\,},\diagup{\,}\ast{\,}{\,})}\xrightarrow{\text{rotate regions}}
\scalebox{0.5}{\young({\,}\ast,{\,}{\,},{\,}\diagup{\,}{\,}\ast,\diagup{\,}{\,}{\,}{\,})}
\,=\,\scalebox{0.5}{\yng(1,2,3,4)}\\
%
\scalebox{0.5}{\yng(2,2,2)}&=
\scalebox{0.5}{\young(\ast\ast,{\,}{\,},{\,}\diagup\ast\ast\ast,\diagup{\,}\ast\ast\ast)}\xrightarrow{\text{complement regions}}
\scalebox{0.5}{\young({\,}{\,},\ast\ast,{\,}\diagup{\,}{\,}{\,},\diagup{\,}{\,}{\,}{\,})}\xrightarrow{\text{rotate regions}}
\scalebox{0.5}{\young(\ast\ast,{\,}{\,},{\,}\diagup{\,}{\,}{\,},\diagup{\,}{\,}{\,}{\,})}
\,=\,\scalebox{0.5}{\yng(2,5,5)}\\
%
\scalebox{0.5}{\yng(1,1,2,2)}&=
\scalebox{0.5}{\young({\,}\ast,{\,}\ast,{\,}\diagup\ast\ast\ast,\diagup{\,}\ast\ast\ast)}\xrightarrow{\text{complement regions}}
\scalebox{0.5}{\young(\ast{\,},\ast{\,},{\,}\diagup{\,}{\,}{\,},\diagup{\,}{\,}{\,}{\,})}\xrightarrow{\text{rotate regions}}
\scalebox{0.5}{\young({\,}\ast,{\,}\ast,{\,}\diagup{\,}{\,}{\,},\diagup{\,}{\,}{\,}{\,})}
\,=\,\scalebox{0.5}{\yng(1,1,5,5)}
\end{align*}
\caption{Bijection between $\rank_{\Mt}^{6,2}$ and $\rank_{\Mt}^{12,2}$.}
\label{fig:bijection}
\end{figure}
More generally, this bijection proves:
\begin{thm}
\[\rank_{\Mt}^{p,q}\H\bullet\bullet(\Gr_k(\R^{n,1}))=\rank_{\Mt}^{nq-p,q}\H\bullet\bullet(\Gr_k(\R^{n,1})).\]
\end{thm}
It would be nice if this apparent duality could be given a geometric interpretation. We do not know one.
\section{Grassmannians $\Gr_2(\R^{n,2})$}\label{twontwo}
\subsection{Comment}
In the this section we work up to a general formula for the cohomology of $\Gr_2(\R^{n,2})$ somewhat slowly, starting with calculations for small $n$ which rely on observations about multiple Schubert cell constructions. We do this until we reach a value of $n$ after which we can use the same construction each time, with no new differentials appearing.\par
This approach -- comparing multiple constructions to deduce unknown differentials -- can be automated to perform further calculations not appearing in this paper. In fact a Sage program generating a fund of computations by investigating \emph{all} possible constructions first motivated these results. We hope to write more about this soon.
\subsection{When $n=3$}\label{n=3}
Since $\Gr_2(\R^{3,2})\iso\Gr_1(\R^{3,1})$, we have in fact already done this computation in \ref{we1}. Nonetheless, the decomposition $\R^{3,2}=\R\sgn\oplus \R\triv\oplus \R\sgn=\R^{-+-}$ gives
\begin{center}
$I(+-+)=$
\begin{tabular}{||c|c|c}
&&\\
\hline
&\scalebox{.5}{\yng(1)}\,\,&\scalebox{.5}{\yng(1,1)}\,\,\\
\hline
$\emptyset$\,&&\\
\hline
\hline
\end{tabular}
\end{center}
with no possible differentials, since $\scalebox{0.4}{\yng(1)}\mapsto \scalebox{0.4}{\yng(1,1)}$ would give a nonzero map in singular cohomology, and so $H^{\bullet,\bullet}\Gr_2(\R^{3,2})=\Mt\oplus\M11\oplus\M21$. By the forgetful long exact sequence in \ref{rholes}, we can also say that each of these three generators maps to the unique Schubert class in their topological dimension. We represented this information by labeling generators in the rank chart by their image under $\psi$.
\begin{center}
\begin{tikzpicture}[scale=1.0]
\def 8 { 2 }
\def 4 { 1 }
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\draw (0,0) node[above right] {$\emptyset$};
\draw (1,1) node[above right] {\scalebox{.5}{\yng(1)}};
\draw (2,1) node[above right] {\scalebox{.5}{\yng(1,1)}};
\end{tikzpicture}
\quad or in jump-sequence notation, \quad
\begin{tikzpicture}[scale=1.0]
\def 8 { 2 }
\def 4 { 1 }
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\draw (0,0) node[above right] {$[1,2]$};
\draw (1,1) node[above right] {$[1,3]$};
\draw (2,1) node[above right] {$[2,3]$};
\end{tikzpicture}.
\end{center}
\subsection{When $n=4$}
\label{twofourtwo}
Next consider $\Gr_2(\R^{4,2})$. Two ingredients tables are shown in Figure \ref{two_ingredient_tables}.
\begin{figure}
\small
\begin{tabular}{||c|c|c|c|c}
&&&\scalebox{.5}{\yng(1,2)}\,\,&\\
\hline
&&&&\scalebox{.5}{\yng(2,2)}\,\,\\
\hline
&\scalebox{.5}{\yng(1)}\,\,&\scalebox{.5}{\yng(2)}\,\,\scalebox{.5}{\yng(1,1)}\,\,&&\\
\hline
$\emptyset$\,&&&&\\
\hline
\hline
\end{tabular}\,\,
\qquad
\begin{tabular}{||c|c|c|c|c}
&&&&\\
\hline
&&\scalebox{.5}{\yng(2)}\,\,\scalebox{.5}{\yng(1,1)}\,\,&\scalebox{.5}{\yng(1,2)}\,\,&\scalebox{.5}{\yng(2,2)}\\
\hline
&&&&\\
\hline
$\emptyset$\,&\scalebox{.5}{\yng(1)}\,\,&&&\\
\hline
\hline
\end{tabular}
\caption{Ingredients tables $I(-+-+)$ and $I(+--+)$ for $\Gr_2(\R^{4,2})$.}
\label{two_ingredient_tables}
\end{figure}
If we knew every cofiber sequence differential, we could iteratively attach the cells using just one construction, computing the cohomology of the subspaces using the long exact sequence for each cofiber $X_k\inj X_{k+1}\to S^{\alpha_k}$ until arriving at the answer. Considering the first construction, this is straightforward while building the two-skeleton, as no nonzero differentials were possible. However, when attaching the $e^{3,3}$ labeled $\scalebox{0.4}{\yng(1,2)}$ to this two-skeleton whose cohomology must be $\Mt\oplus \M11\oplus(\M21)^{2}$, we have a possible differential and, naively, no way to determine whether it is nonzero.
\begin{figure}[h]
\begin{tikzpicture}[scale=0.8]
\def 8 { 3 }
\def 4 { 3 }
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.75, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (1+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\draw (0,0) node[above right] {$\emptyset$};
\draw (1,1) node[above right] {\scalebox{.4}{\yng(1)}};
\draw (2,1) node[above right] {\scalebox{.4}{{\yng(2)}}};
\draw (2,1.3) node[above right] {\scalebox{.4}{{\yng(1,1)}}};
\draw[->] (2.75,1.5)--node[above]{?}(3.2,1.5);
\color{blue}
\draw (3,3) node[above right] {\scalebox{.4}{\yng(1,2)}};
\draw[->] (3.5,1.5)--(3.5,-0.5);
\draw[->] (3.5,1.5)--(1.5,-0.5);
\draw[->] (3.5,3.5)--(4,4);
\draw[->] (3.5,3.5)--(3.5,4);
\draw (3.8,1.8) node {$\theta\cdot $\scalebox{0.4}{\yng(1,2)}};
\end{tikzpicture}
\caption{One stage of the $-+-+$ construction, corresponding to the cofiber sequence for including the 2-skeleton into the 3-skeleton.}
\end{figure}
If the differential is zero, we next attach an $e^{4,2}$ which has no possible differentials for bidegree reasons, and so our answer would be $\Mt\oplus\M11\oplus(\M21)^2\oplus\M33\oplus\M42$. This is where the second construction comes in. Notice the $\M33$ in our first hypothetical scenario. In the second construction of Figure \ref{two_ingredient_tables}, no chain of events can end with a generator in this bidegree: The $\scalebox{0.4}{\yng(1,2)}$ would have to shift \emph{up}, which could only happen if a later cell of fixed-set dimension 0 were attached (see \cite{buddies} for more details on Kronholm shifts). As this cannot happen, our mystery differential in the first construction must be non-zero, so that after the resulting shift, we have $$\H\bullet\bullet(\Gr_2(\R^{4,2}))=\Mt\oplus\M11\oplus\M21\oplus\M22\oplus\M32\oplus\M42.$$
We have now answered the question of the module structure of $\H\bullet\bullet(\Gr_2(\R^{4,2}))$. We can also ask about the image of these generators under the forgetful map $\psi$ in terms of Schubert elements. Most of the generators of this free module have an obvious image under $\psi$, as there is a unique generator in most dimensions of the non-equivariant cohomology. But there is some room for ambiguity in dimension 2.
\begin{figure}[H]
\label{n=4}
\begin{tikzpicture}[scale=1.0]
\def 8 { 4 }
\def 4 { 2 }
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.75, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (1+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\draw (0,0) node[above right] {$\emptyset$};
\draw (1,1) node[above right] {\scalebox{.7}{\yng(1)}};
\draw (2,1) node[above right] {$x$};
\draw (2,2) node[above right] {$y$};
\draw (3,2) node[above right] {\scalebox{.7}{\yng(1,2)}};
\draw (4,2) node[above right] {\scalebox{.7}{\yng(2,2)}};
\end{tikzpicture}
\caption{$\H\bullet\bullet(\Gr_2(\R^{4,2}))$. We choose a generator in each bidegree with a free summand. The image under $\psi$ in dimensions other than 2 is unambiguous. What can be said about the choices $\psi(x)$ and $\psi(y)$? (Note we \emph{choose} a generator because for example we could replace $y$ with $y'=y+\rho x$.)}
\end{figure}
We wish to know the images under $\psi$ of $x$ and $y$. The span of $\psi(x)$ and $\psi(y)$ is that of the non-equivariant Schubert classes $\scalebox{.5}{\yng(2)}$ and $\scalebox{.5}{\yng(1,1)}$. But it isn't clear yet who is sent where. Consider the inclusion $\Gr_2(\R^{+--})\xrightarrow{i} \Gr_2(\R^{+--+})$. Note that from Table \ref{two_ingredient_tables}, $\sfrac{\Gr_2\R^{4,2}}{\Gr_2\R^{3,2}}$ can be built from a point and cells of weight two in such a way that $\rH21(\sfrac{\Gr_2\R^{4,2}}{\Gr_2\R^{3,2}})=0$. We have long exact sequences in both equivariant and singular cohomology:
\begin{center}
\begin{tikzpicture}[node distance=2cm, descr/.style={fill=white,inner sep=2.5pt}]
\def4.5{4.5}
\def2{2}
\def1.2{1.2}
\def.3{.3}
\draw(0,0) node {$\rH21(\Gr_2\R^{3,2})$};
\draw(0,2) node {$\rH21(\Gr_2\R^{4,2})$};
\draw(4.5,0) node {$H^2_\text{sing}(\Gr_2\R^{3,2})$};
\draw(4.5,2) node {$H^2_\text{sing}(\Gr_2\R^{4,2})$};
\draw[->](1.2,2) to node[descr] {$\psi$} (4.5-1.2,2);
\draw[->](1.2,0) to node[descr] {$\psi$} (4.5-1.2,0);
\draw[->](0,2-.3) to node[descr] {$i^*$} (0,.3);
\draw[->](4.5,2-.3) to node[descr] {$i^*$} (4.5,.3);
\draw(0,2*2) node {$\rH21(\sfrac{\Gr_2\R^{4,2}}{\Gr_2\R^{3,2}})=0$};
\draw[->](0,2*2-.3) to (0,2+.3);
\draw(.5,2-.5) node {$x$};
\draw(.5,.5) node {\tiny $\yng(1,1)$};
\draw(4.5-.5,.5) node {\tiny $\yng(1,1)$};
\draw(4.5-.5,2-.5) node {\tiny $\yng(1,1)$};
\draw(4.5+.7,2-.5) node {\tiny $\yng(1,1)+\yng(2)$};
\end{tikzpicture}
\end{center}
In the diagram for $\rH21$, since $i^*$ is injective, the element $x$ is sent to $\scalebox{0.5}{\yng(1,1)}$ (the unique nonzero element -- see Section \ref{n=3}) which is then sent to the corresponding Schubert class by the forgetful map $\psi$. This element $\scalebox{0.5}{\yng(1,1)}$ has two preimages in $H_{\text{sing}}^2\Gr_2(\R^{4,2})$, the elements $\scalebox{0.5}{\yng(1,1)}$ and $\scalebox{0.5}{\yng(1,1)}+\scalebox{0.5}{\yng(2)}$. This leaves four possibilities:
\begin{align}
\psi(x)={\tiny\yng(1,1)}\text{\qquad and}&\qquad \psi(y)={\tiny \yng(2)}\\
\psi(x)={\tiny\yng(1,1)}\text{\qquad and}&\qquad \psi(y)={\tiny\yng(1,1)}+{\tiny \yng(2)}\\
\psi(x)={\tiny\yng(1,1)}+{\tiny\yng(2)}\text{\qquad and}&\qquad \psi(y)={\tiny \yng(2)}\\
\psi(x)={\tiny\yng(1,1)}+{\tiny\yng(2)}\text{\qquad and}&\qquad \psi(y)={\tiny \yng(1,1)}
\end{align}
Note that actually (1) and (2) are equivalent up the the change of basis $x'=x$ and $y'=y+\tau x$. Cases (3) and (4) are also equivalent under the same change of basis.
To resolve the remaining ambiguity, define the self-map $P:\Gr_2(\R^{4,2})\to \Gr_2(\R^{4,2})$ by $V\mapsto V^\perp$. After $\psi$, $P^*$ maps Young diagrams to their transposes (see Appendix \ref{perp}, Corollary \ref{cor:perpmap}).
This makes scenario $(1)\sim(2)$ impossible by looking at $\H21$: While we would have $P^*(\psi(x))=P^*(\scalebox{0.4}{\yng(1,1)})=\scalebox{0.4}{\yng(2)}$, the element $\scalebox{0.4}{\yng(2)}\not\in\psi(P^*(\H21))=\psi(\H21)$. And so, up to choice of generator in $\H22$, which we will denote by $\scalebox{0.5}{\yng(1,1)}/\scalebox{0.5}{\yng(2)}$, we can represent $\H\bullet\bullet(\Gr_2(\R^{4,2}))$ as
\begin{figure}[H]
\begin{tikzpicture}[scale=1.5]
\def 8 { 4 }
\def 4 { 2 }
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\draw (0,0) node[above right] {$\emptyset$};
\draw (1,1) node[above right] {\scalebox{.7}{\yng(1)}};
\draw (2,1) node[above right] {$\scalebox{.7}{\yng(1,1)}+\scalebox{.7}{\yng(2)}$};
\draw (2,2) node[above right] {\scalebox{.7}{\yng(1,1)}\,/\,\scalebox{.7}{\yng(2)}};
\draw (3,2) node[above right] {\scalebox{.7}{\yng(1,2)}};
\draw (4,2) node[above right] {\scalebox{.7}{\yng(2,2)}};
\end{tikzpicture}
\caption{The rank table for $\H\bullet\bullet(\Gr_2(\R^{4,2}))$, with generators labeled by their images under $\psi$.}
\end{figure}
Note that this version is indeed compatible with $P^*$. In $\H21$ the involution sends
${\tiny\yng(1,1)}+{\tiny\yng(2)}\mapsto {\tiny\yng(2)}+{\tiny\yng(1,1)}\mapsto {\tiny\yng(1,1)}+{\tiny\yng(2)}$, and in $\H22$, sends
$ {\tiny\yng(1,1)/\yng(2)}\mapsto{\tiny\yng(2)/\yng(1,1)}\mapsto {\tiny\yng(1,1)/\yng(2)}$. And so in addition to knowing the module structure of this cohomology, we know the action of the forgetful map in terms of Schubert classes. This is a first step towards determining the equivariant Schubert calculus, which is outside the scope of this paper.\\
\subsection{$\Gr_2(\R^{n,2})$ for $n=5,6$ or $7$}
$\ $\\
Now things start to become more straightforward.\\
Begin with the following ingredients table for $\Gr_2(\R^{5,2})$:
\begin{center}
$I(+-+-+)=$
\begin{tabular}{||c|c|c|c|c|c|c|}
&&&\scalebox{.5}{\yng(1,2)}\,\,&&\scalebox{.5}{\yng(2,3)}\,\,&\scalebox{.5}{\yng(3,3)}\,\,\\
\hline
&&&\scalebox{.5}{\yng(3)}\,\,&\scalebox{.5}{\yng(1,3)}\,\,\scalebox{.5}{\yng(2,2)}\,\,&&\\
\hline
&\scalebox{.5}{\yng(1)}\,\,&\scalebox{.5}{\yng(2)}\,\,\scalebox{.5}{\yng(1,1)}\,\,&&&&\\
\hline
$\emptyset$\,&&&&&&\\
\hline
\hline
\end{tabular}\,.
\end{center}
First observe that this construction of $\Gr_2(\R^{+-+-+})$ inherits the Kronholm shift of its subspace $\Gr_2(\R^{+-+-})$. The inclusion $i:\R^{+-+-}\inj\R^{+-+-+}$ induces
$$i^*: \H21(\Gr_2{\R^{5,2}})\to \H21(\Gr_2{\R^{4,2}})=\Zt.$$ If $\Gr_2(\R^{+-+-+})$ \emph{didn't} also have a differential hitting $\theta\,\scalebox{0.4}{\yng(1,2)}$, then we would have $\H21(\Gr_2{\R^{5,2}})=(\Zt)^2$, giving $i^*$ a nonzero kernel, and also $$\psi:\H21(\Gr_2{\R^{5,2}})\inj H^2_{\text{sing}}(\Gr_2(\R^5))=(\Zt)^2$$
would be an isomorphism. Since in singular cohomology the inclusion induces an isomorphism $i^*:H^{2}_\text{sing}(\Gr_2(\R^5))\to H^{2}_\text{sing}(\Gr_2(\R^4))$, this would be a failure of naturality. Hence we again have nonzero $d:\langle\scalebox{0.4}{\yng(2)},\scalebox{0.4}{\yng(1,1)}\rangle\to\langle\theta\,\scalebox{0.4}{\yng(1,2)}\rangle$.
Since no other possible nonzero differentials arise in the first construction for bidegree reasons, we have re-derived the cohomology deduced in Example \ref{example}. We are now also justified in labeling these generators with their images under $\psi$, since each topological dimension above the second has generators in only one weight.
So we have $\H\bullet\bullet(\Gr_2(\R^{5,2}))=$
\begin{center}
\begin{tikzpicture}[scale=1]
\def 8 { 6 }
\def 4 { 3 }
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\draw (0,0) node[above right] {$\emptyset$};
\draw (1,1) node[above right] {\scalebox{.5}{\yng(1)}};
\draw (2,1) node[above right] {$\scalebox{.5}{\yng(1,1)}+\scalebox{.5}{\yng(2)}$};
\draw (2,2) node[above right] {\scalebox{.5}{\yng(1,1)}/\scalebox{.5}{\yng(2)}};
\draw (3,2) node[above right] {\scalebox{.5}{\yng(3)},\,\,\scalebox{.5}{\yng(1,2)}};
\draw (4,2) node[above right] {\scalebox{.5}{\yng(1,3)},\,\,\scalebox{.5}{\yng(2,2)}};
\draw (5,3) node[above right] {\scalebox{.5}{\yng(2,3)}};
\draw (6,3) node[above right] {\scalebox{.5}{\yng(3,3)}};
\end{tikzpicture}.
\end{center}
We will not continue further with these forgetful map calculations, but see Section \ref{warning} for further discussion of difficulties with this question.
As we continue to investigate $\Gr_2(\R^{n,2})$ for larger $n$, we will see that no new differentials ever arise if we use $I(+-+-+\ldots+)$. At first this is trivial. Let's switch to jump sequence notation for space reasons, omitting the square brackets, but parenthesizing a few elements to discuss. For $\Gr_2(\R^{6,2})$, the ingredients table is
\begin{center}
$I(+-+-++)=$
\begin{tabular}{||p{9pt}|p{9pt}|p{15pt}|p{15pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|}
&&&&&&&&5,6\\
\hline
&&&$\!\!(2,4)$&&3,5&3,6 4,5&4,6&\\
\hline
&&&1,5&1,6 2,5 3,4&2,6&&&\\
\hline
&1,3&(1,4) (2,3)&&&&&&\\
\hline
1,2&&&&&&&&\\
\hline
\hline
$_0$&&$_2$&&$_4$& &$_6$&&$_8$
\end{tabular}
\end{center}
and for $\Gr_2(\R^{7,2})$, the ingredients table is
\begin{center}
$I(+-+-+++)=$
\begin{tabular}{||p{9pt}|p{9pt}|p{15pt}|p{15pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|}
&&&&&&&&5,6&5,7&6,7\\
\hline
&&&$\!\!(2,4)$&&3,5&3,6 4,5&3,7 4,6&4,7&&\\
\hline
&&&1,5&1,6 2,5 3,4&1,7 2,6&2,7&&&&\\
\hline
&1,3&(1,4) (2,3)&&&&&&&&\\
\hline
1,2&&&&&&&&&&\\
\hline
\hline
$_0$&&$_2$&&$_4$& &$_6$&&$_8$&&$_{10}$
\end{tabular}\,.
\end{center}
The only possible differentials, just for bidegree reasons, would occur between the parenthetical entries. That is, with the exception of $[2,4]=\scalebox{0.4}{\yng(1,2)}$, no other generator has a weight high enough that its lower cone falls within range of a possible differential. Again by naturality this one Kronholm shift occurs, and we have our answer.
\subsection{When $n=8$}\label{twoeighttwo}
$\ $\\
However, when we get to $\Gr_2(\R^{8,2})$, we have $I(+-+-++++)=$
\begin{center}
\begin{tabular}{||p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|p{15pt}|p{15pt}|p{9pt}|p{9pt}|p{9pt}|p{9pt}|}
&&&&&&&&(5,6)&5,7&5,8 6,7&6,8&7,8\\
\hline
&&&2,4&&3,5&3,6 4,5&3,7 4,6&3,8 4,7&4,8&&&\\
\hline
&&&1,5&1,6 2,5 3,4&1,7 2,6&1,8 2,7&$\!(2,8)$&&&&&\\
\hline
&1,3&1,4 2,3&&&&&&&&&&\\
\hline
1,2&&&&&&&&&&&&\\
\hline
\hline
$_0$&&$_2$&&$_4$& &$_6$&&$_8$&&$_{10}$&&$_{12}$
\end{tabular}
\end{center}
with a differential possible (at least in terms of the bigrading) from $[2,8]$ to $ \theta[5,6]$, or in Young notation, $\scalebox{.4}{\yng(1,6)}\mapsto\theta\,\, \scalebox{.4}{\yng(4,4)}$. But notice that whereas our nonzero differential back in Section \ref{twofourtwo} had both $\scalebox{.4}{\yng(1,1)}$ and $\scalebox{.4}{\yng(2)}$ fitting inside of $\scalebox{.4}{\yng(1,2)}$, that is not the case here.\par
This is significant because as seen in Section \ref{we2}, containment of subvarieties corresponds to containment of Young diagrams. As $\scalebox{.4}{\yng(1,6)}\not\subset \scalebox{.4}{\yng(4,4)}$ or equivalently in jump sequence notation, as $[2,8]\not\prec[5,6]$, we know that $X_{[2,8]}\not\subseteq X_{[5,6]}$ and so by Theorem \ref{fixedthm}, attaching $\Omega_{[5,6]}$ creates no nonzero differentials.
In fact, this generalizes for $\Gr_2(\R^{n,2})$ with $n\ge 8$. If we chose the identification $\R^{n,2}=\R^{+-+-}\oplus(\R^+)^{n-4}$, the representation cell structure and hence the ingredients table $I(\R^{+-+-}\oplus(\R^+)^{n-4})$ is as follows. A jump sequence $[j_1,j_2]$, will give rise to a cell of topological dimension $(j_1-1)+(j_2-2)$ and by observation will have weight $w([j_1,j_2])=$
\[\begin{cases}
1 \quad\text{if } [j_1,j_2]=[1,3],[1,4]\text{ or }[2,3]& \text{from}\quad\ds{\small\left[{1\atop 0}{\atop\young(-)\,1}\right]},{\small\left[{1\atop 0}{\atop\young(+-)\,1}\right]},{\small\left[{\young(-)\,1\,\,\,\,\atop\young(+)\,0\,1}\right]}\\
\\
2 \quad\text{if } [j_1,j_2]=[3,4]\text{ or }[2,\ge5]& \text{from}\quad \ds{\small\left[{\young(+-)\,1 \atop\young(+-)\,0}{\atop 1}\right]},{\small\left[{\young(-)\,1 \atop\young(+)\,0}{\atop \young(+-+)\,\dots\young(+)\,1}\right]}\\
\\
2 \quad\text{if } [j_1,j_2]=[1,\ge 5]& \text{from}\quad \ds{\small \left[{1\atop 0}{\atop\young(-+-+)\,\dots\young(+)\,1}\right]}\\
\\
3 \quad\text{if } [j_1,j_2]=[2,4]\text{ or }[3,\ge5]& \text{from}\quad \ds{\small\left[{\young(-)\,1 \atop\young(-)\,0}{\atop \young(-)\,1}\right]},{\small\left[{\young(+-)\,1 \atop\young(+-)\,0}{\atop \young(-+)\,\dots\young(+)\,1}\right]}\\
\\
3 \quad\text{if } j_1=4& \text{from}\quad \ds{\small\left[{\young(-+-)\,1 \atop\young(+-+)\,0}{\atop \young(+)\,\dots\young(+)\,1}\right]}\\
\\
4 \quad\text{if } j_1\ge 5& \text{from}\quad\ds{\small\left[{\young(+-+-+)\,\dots\young(+)\,1 \atop\young(+-+-+)\,\dots\young(+)\,0}{\atop \young(+)\,\dots\young(+)\,1}\right]}.
\end{cases}
\]
For example $\Gr_2(\R^{10,2})$ has ingredients $I(+-+-++++++)=$
\begin{center}
\begin{tabular}{||p{8pt}|p{8pt}|p{15pt}|p{15pt}|p{8pt}|p{8pt}|p{8pt}|p{8pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}|p{14pt}}
&&&&&&&&&&&&&&&&\\
\hline
&&&&&&&&5,6&5,7&5,8 6,7&5,9 6,8&5,10 6,9 7,8&6,10 7,9&7,10 8,9&8,10&9,10\\
\hline
&&&(2,4)&&3,5&3,6 4,5&3,7 4,6&3,8 4,7&3,9 4,8&3,10 4,9&4,10&&&&&\\
\hline
&&&1,5&1,6 2,5 3,4&1,7 2,6&1,8 2,7&1,9 2,8&1,10 2,9&2,10&&&&&&&\\
\hline
&1,3&(1,4) (2,3)&&&&&&&&&&&&&&\\
\hline
1,2&&&&&&&&&&&&&&&&\\
\hline
\hline
$_0$&&&&&$_5$&&&&&$_{10}$&&&&&$_{15}$
\end{tabular}\,.\\
\end{center}
$\ $\\
For generators above topological dimension $3$, the only possible differentials (that is, possible with respect to bidegree) supported by these generators would be maps from $\alpha$ in bidegree $(x,2)$ to $\theta \beta$ for generators $\beta$ in $(x+1,4)$. These $\alpha$ will have jump sequences $[1,x+2]$ or $[2,x+1]$, while the $\beta$'s jump sequence could be $[5,x-1]$, $[6,x-2]$, $[7,x-3]$ etc. In any case, the second number in the jump sequence of each $\alpha$ will be larger than that of a corresponding $\beta$, so there is no dominance relation, or in terms of Young diagrams, $\alpha \not\subseteq \beta$. Now by Theorem \ref{fixedthm} this differential is actually zero.\par
And so with the exception of the lone nonzero differential to $\theta[2,4]$, i.e. $\theta\,\scalebox{.4}{\yng(1,2)}$, all differentials are zero. We can now count the number of generators ending up in each bidegree. Row by row, if $M=\H\bullet\bullet(\Gr_2(\R^{n,2}))$ for $n\ge 8$,
\begin{figure}[H]
\centering
\begin{minipage}{.4\textwidth}
\centering
\begin{align*}
\rank^{p,0}_{\Mt}M&=\begin{cases}
1&p=0\\
0&\text{else}.
\end{cases}\\
\rank^{p,1}_{\Mt}M&=\begin{cases}
1&p=1,2\\
0&\text{else}
\end{cases}
\end{align*}
\end{minipage}%
\begin{minipage}{.6\textwidth}
\centering
\begin{align*}
\rank^{p,2}_{\Mt}M&=\begin{cases}
3 & p=4\\
2 & p=3\text{\quad or \quad}5\le p\le n-2\\
1 & p=2,n-1\\
0 & \text{else}
\end{cases}\\
\rank^{p,3}_{\Mt}M&=\begin{cases}
2 & 6\le p\le n\\
1 & p=5,n+1\\
0 & \text{else}
\end{cases}\\
\rank^{p,4}_{\Mt}M&=\begin{cases}
\lceil\frac {p-7}2\rceil & 8\le p\le n+1\\
n-1-\lceil\frac p2\rceil & n+2\le p\le 2n-4\\
0 & \text{else}
\end{cases}
\end{align*}
\end{minipage}
\end{figure}
This can also be rewritten to obtain the equally unattractive formula of Theorem \ref{uglyformula}.
\begin{warning}
\label{warning}
We must be careful not to get carried away in assuming that the images under $\psi$ of these generators correspond to the Schubert cells which are their ``reason'' for appearing where they do in cohomology\footnote{Notice that we stopped labeling generators with their forgetful images at $\Gr_2(\R^{5,2})$.}. For example, in constructing $\Gr_2(\R^{3,1})$, we have $I(-++)=$
\begin{center}
\begin{tikzpicture}
\draw (0.0,0.0) node[above right] {\small$\emptyset$};
\draw (0.5,0.0) node[above right] {\small$\yng(1)$};
\draw (1.25,1.125) node[above right] {\small$\yng(1,1)$};
\draw[-] (0.0,0)--(0.0,2.25);
\draw[-] (0.5,0)--(0.5,2.25);
\draw[-] (1.25,0)--(1.25,2.25);
\draw[-] (0,0.0)--(2.0,0.0);
\draw[-] (0,0.625)--(2.0,0.625);
\draw[-] (0,1.125)--(2.0,1.125);
\end{tikzpicture}\
\end{center}
which must shift to
\begin{center}
\begin{tikzpicture}
\draw (0.0,0.0) node[above right] {\small$\emptyset$};
\draw (0.5,0.5) node[above right] {\small$\yng(1)$};
\draw (1.25,0.5) node[above right] {\small$\yng(1,1)$};
\draw[-] (0.0,0)--(0.0,1.625);
\draw[-] (0.5,0)--(0.5,1.625);
\draw[-] (1.25,0)--(1.25,1.625);
\draw[-] (0,0.0)--(2.0,0.0);
\draw[-] (0,0.5)--(2.0,0.5);
\end{tikzpicture}\
\end{center}
because, for example, $\Gr_2(\R^{3,1})\iso \Gr_1(\R^{3,1})$. (See Section \ref{we1}.) However when we proceed to build $\Gr_2(\R^{4,2})$ by attaching the remaining cells of $I(-++-)=$
\begin{center}
\begin{tikzpicture}
\draw (0.0,0.0) node[above right] {\small$\emptyset$};
\draw (0.5,0.5) node[above right] {\small$\yng(1)$};
\draw (1.25,0.5) node[above right] {\small$\yng(1,1)$};
\draw[-] (0.0,0)--(0.0,2.875);
\draw[-] (0.5,0)--(0.5,2.875);
\draw[-] (1.25,0)--(1.25,2.875);
\draw[-] (2.25,0)--(2.25,2.875);
\draw[-] (3.25,0)--(3.25,2.875);
\draw[-] (0,0.0)--(4.25,0.0);
\draw[-] (0,0.5)--(4.25,0.5);
\draw[-] (0,1.37)--(4.25,1.37);
\color{blue}
\draw (1.25,1.4) node[above right] {\small$\yng(2)$};
\draw (2.25,1.4) node[above right] {\small$\yng(1,2)$};
\draw (3.25,1.4) node[above right] {\small$\yng(2,2)$};
\end{tikzpicture}\,,
\end{center}
there are no possible nonzero differentials, and so we may be tempted to keep these Young diagram labelings, and assert that the forgetful map $$\psi:\H\bullet\bullet(\Gr_2(\R^{4,2}))\to H_\text{sing}(\Gr_2(\R^{4}))$$ sends these generators to the non-equivariant Schubert cell corresponding to those Young diagrams. In fact we know from Section \ref{twofourtwo} that this is \emph{false}. Each attachment of a new cell raises doubts as to the forgetful image of the cohomology. Put another way, maps in equivariant cohomology induced by inclusion of Grassmannians need not respect Schubert symbols in singular cohomology.
For this reason, when we looked at the ingredients table for $\Gr_2(\R^{10,2})$ above, while we know all of the differentials, and thus the ranks in each dimension, we don't (yet) have a good reason to assign to these generators the symbols we would naturally wish to.
\end{warning}
\section{Some infinite Grassmannians}\label{infgrass}
We can now also deduce the cohomology of the analogous infinite Grassmannians which follows from these results. As a consequence of Theorem \ref{uglyformula}, the rank table for the infinite Grassmannian $\Gr_2(\R^{\infty,2})$ begins
\begin{figure}[H]
\begin{tikzpicture}[scale=0.7]
\def 8 {13}
\def 4 {4}
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.5, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (.5+ 8 ,0);
\draw (7.18,3.3) node {$2$};
\draw (2.18,1.3) node {$1$};
\draw (6.18,2.3) node {$2$};
\draw (7.18,2.3) node {$2$};
\draw (6.18,3.3) node {$2$};
\draw (2.18,2.3) node {$1$};
\draw (1.18,1.3) node {$1$};
\draw (3.18,2.3) node {$2$};
\draw (0.18,0.3) node {$1$};
\draw (8.18,2.3) node {$2$};
\draw (4.18,2.3) node {$3$};
\draw (5.18,3.3) node {$1$};
\draw (8.18,3.3) node {$2$};
\draw (5.18,2.3) node {$2$};
\draw (8.18,4.3) node {$1$};
\draw (8,0.1)--(8,-0.1) node[below] {$8$};
\draw (9.18,2.3) node {$2$};
\draw (10.18,2.3) node {$2$};
\draw (11.18,2.3) node {$2$};
\draw (12.18,2.3) node {$2$};
\draw (13.55,2.3) node {$2\dots$};
\draw (9.18,3.3) node {$2$};
\draw (10.18,3.3) node {$2$};
\draw (11.18,3.3) node {$2$};
\draw (12.18,3.3) node {$2$};
\draw (13.55,3.3) node {$2\dots$};
\draw (9.18,4.3) node {$1$};
\draw (10.18,4.3) node {$2$};
\draw (11.18,4.3) node {$2$};
\draw (12.18,4.3) node {$3$};
\draw (13.55,4.3) node {$3\dots$};
\end{tikzpicture}
\end{figure}
and then as dimension increases,
\begin{thm}
For $p\ge8$,
\[\rank_{\Mt}\H p{\bullet}(\Gr_2(\R^{\infty,2}))=\begin{cases}
\lceil\frac {p-7}2\rceil &\bullet=4\\
2 & \bullet=3\\
2 & \bullet=2\\
0 &\text{else.}
\end{cases}
\]
\end{thm}
We also have, as a consequence of Theorem \ref{knoneagain},
\begin{thm}
\[\rank_{\Mt}\H pq(\Gr_k(\R^{\infty,1}))=\part(p,k,\infty,q).\]
\end{thm}
Using the logic from Comment \ref{upside} which considers a $q$-by-$q$ square with a region north and a region to the east, this formula can also be expressed (for $p\ge q^2$)
\[\rank_{\Mt}\H pq(\Gr_k(\R^{\infty,1}))=\sum_{i=1}^{q(k-q)}\part(i,k-q,q,\ast)\part(p-q^2-i,q,\ast,\ast)\]
where the $\ast$ denotes omitting that restriction, so $\part(a,b,c,\ast)$ counts partitions of $a$ into $b$ parts not exceeding $c$ but having any trace, and $\part(a,b,\ast,\ast)$ counts partitions of $a$ into $b$ numbers of any size and trace.
\section{Complex Grassmannians}
\label{complexsection}
Modified statements of the results of this paper also apply to complex Grassmannians. Note that while in the real case, a Schubert cell indexed by a partition $\lambda$ of some integer $|\lambda|$ corresponds to a $|\lambda|$-disc, that is, $\Omega_\lambda(\R)\simeq e^{|\lambda|}$, in the complex case, each complex variable contributes two real dimensions: $\Omega_\lambda(\C)\simeq e^{2|\lambda|}$.\par
Define $\C\triv$ and $\C_{\text{sgn}}$ analogously so in $\C_{\text{sgn}}$ we have $z\mapsto -z$, and then let $\C^{p,q}=\C\triv^{p-q}\oplus \C_{\text{sgn}}^q$ as in the real case. For each partition $\lambda$ fitting inside a $k$-by-$(n-k)$ rectangle, whenever $\Gr_k(\R^{p,q})$ has $\Omega_\lambda(\R)\iso e^{a,b}$, the complex Grassmannian $\Gr_k(\C^{p,q})$ has $\Omega_\lambda(\C)\iso e^{2a,2b}$. Recall that a differential $d:\Sigma^{a,b}\Mt^+\to\Sigma^{a',b'}\Mt^-$ is possible only when $a'-b'<a-b$. Because this is equivalent to the inequality $2a'-2b'<2a-2b$, the \emph{possible} differentials on the $E_1$ page of a cellular filtration spectral sequence of a complex Grassmannian occur between the same Schubert cell filtrations as in the real case. And if the same possible differentials are, in fact nonzero, the Kronholm shifts (see formulas in \cite{buddies}) will be twice as large in the complex case, meaning the same possible differentials present themselves on $E_2$, and so on.
\subsection{Warning}
Because of the essentially un-geometric approach to differentials in this paper, we have no reason to claim that a nonzero differential in the real case must correspond to a nonzero differential in the complex case, or vice versa.\\
However, because the arguments in Lemma \ref{tracelemma} are almost identical with complex variables, we may conclude that in the $\C^{n,1}=\C\triv^{k-1}\oplus \C_{\text{sgn}}\oplus \C^{n-k}\triv$ construction of $\Gr_k(\C^{n,1})$, the trace of a Schubert cell determines its weight: $\Omega_\lambda(\C)\simeq e^{2|\lambda|,2\trace{\lambda}}$. As the complex Grassmannian still satisfies the assumptions of Theorem \ref{fixedthm}, we have the analogous theorem:
\begin{thm}\label{knoneagaincomplex} If $p$ or $q$ is odd, $\rank_{\Mt}^{p,q}\H \bullet\bullet(\Gr_k(\C^{n,1}))=0$, while
\[\rank_{\Mt}^{2p,2q}\H \bullet\bullet(\Gr_k(\C^{n,1}))=\part(p,k,n-k,q).\]
\end{thm}
Because the arguments of Section \ref{twontwo} are identical if we just double every bidegree, and the ``perp map'' argument in Appendix \ref{perp} applies to both the real and complex case, we can also conclude
\begin{thm}\label{dividebytwo}
\[\rank^{p,q}_{\Mt}\H\bullet\bullet(\Gr_2(\C^{n,2}))=\begin{cases}
\rank_{\Mt}^{\frac p2,\frac q2}\H\bullet\bullet(\Gr_2(\R^{n,2}))&\text{$p$ and $q$ even}\\
0&\text{else}.
\end{cases}
\]
\end{thm}
\subsection{Remark}
We can also give $\C$ the conjugation action, $z\mapsto \bar z$. Note that $\C_{\text{conj}}\iso \R^{2,1}$. And so $\Gr_k(\C^n_\text{conj})$ has Schubert cells $\Omega_\lambda\iso e^{2|\lambda|,|\lambda|}$. Purely for degree reasons, no possible differentials $\alpha\to \frac\theta{\rho^i\tau^j}\beta$ exist when $\alpha$ has bidegree $(2x,x)$ and $\beta$ has bidegree $(2y,y)$. Thus the spectral sequence for $\Gr_k(\C^n_{\text{conj}})$ collapses on the first page. Denoting $r_i=\dim H^{2i}_{\text{sing}}(\Gr_k(\C^n);\Zt)$, we have $$\H\bullet\bullet(\Gr_k(\C^n_{\text{conj}}))=\bigoplus_{i=0}^{k(n-k)}(\Sigma^{2i,i}\Mt)^{r_i}.$$
\begin{example}
Consider $\Gr_2(\C^4_\text{conj})$. The action on the Schubert cell
\begin{center}
\begin{tabular}{rl}
$\Omega_{\scalebox{0.4}{\yng(1,2)}}$
&=$\left\{\text{rowspace}\left[\begin{matrix}
z_1&1&0&0\\
z_2&0&z_3&1\\
\end{matrix}\right]: z_i\in \C\right\}$\\
&\\
&$=\left\{\text{rowspace}\left[\begin{matrix}
x_1+y_1i&1&0&0\\
x_2+y_2i&0&x_3+y_3i&1\\
\end{matrix}\right] :x_i,y_i \in \R\right\}$\\
&\\
&$\simeq e^6$
\end{tabular}
\end{center}
sends
\[\left[\begin{matrix}
x_1+y_1i&1&0&0\\
x_2+y_2i&0&x_3+y_3i&1\\
\end{matrix}\right]
\mapsto
\left[\begin{matrix}
x_1-y_1i&1&0&0\\
x_2-y_2i&0&x_3-y_3i&1\\
\end{matrix}\right]
\]
and so $\Omega_{\scalebox{0.4}{\yng(1,2)}}\simeq e^{6,3}$. Analogous consideration for the other Schubert cells give a spectral sequence whose $E_1$ page has generators as shown.
\begin{center}
\begin{tikzpicture}[scale=0.8]
\def 8 { 8 }
\def 4 { 4 }
\draw[step=1cm,lightgray,very thin] (0,0) grid ( 8 +0.75, 4 +.5);
\draw[thick,->] (0,-0.5) -- (0,.5+ 4 );
\draw[thick,->] (-0.5,0) -- (1+ 8 ,0);
\foreach \x in {1,..., 8 }
\draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
\foreach \y in {1,..., 4 }
\draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
\draw (0,0) node[above right] {$\emptyset$};
\draw (2,1) node[above right] {\scalebox{.4}{\yng(1)}};
\draw (4,2) node[above right] {\scalebox{.4}{{\yng(2)}}};
\draw (4,2.3) node[above right] {\scalebox{.4}{{\yng(1,1)}}};
\draw (6,3) node[above right] {\scalebox{.4}{\yng(1,2)}};
\draw (8,4) node[above right] {\scalebox{.4}{\yng(2,2)}};
\end{tikzpicture}
\end{center}
As this collapses,
\[\H\bullet\bullet(\Gr_2(\C^4_\text{conj}))=\Mt \oplus \M21 \oplus (\M42)^2\oplus \M63 \oplus \M84. \]
\end{example}
\subsection{Remark}
Finally, the observations in Section \ref{infgrass} can be similarly duplicated to infinite complex Grassmannians by replacing every $\Sigma^{a,b}\Mt$ with $\Sigma^{2a,2b}\Mt$.
\newpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 735
|
A contemporary adaptation of a 1971 satirical text by one of Urdu's best-known poets and humourists of the 20th century; short stories by Munshi Premchand and Kashmiri writer Amin Kamil on the political turmoil in Kashmir; 19th century poet Mirza Ghalib defending himself in a modern-day court against charges of composing offensive lyrics in a ghazal; or the exchange of letters between an incarcerated Faiz Ahmed Faiz in Pakistan and his British wife Alys between 1951 and 1955. The four-day Urdu Drama Festival at the Ghalib Institute from July 6 might be a great opportunity to sample recent productions from the world of Urdu theatre.
Although not unanimously agreed, it is believed that the first drama to be written in Urdu was in 1853 by Amanat for the last king of Awadh, Nawab Wajid Ali Shah. It was modelled on the French opera, a romance between a prince and a fairy. Performed in the royal palace of the Nawab, the king and his courtiers also featured in the cast. Since then, Urdu theatre has traversed a richly scintillating journey, developing through effervescent 'Parsi Musicals' to a more serious post-Parsi dramas concerned with social and moral issues. Later on Prithvi theatre and Indian People's Theatre Association further honed and chiselled socially committed Urdu theatre.
The artistically-inclined Begum Abida Ahmed, wife of Fakhruddin Ali Ahmed, president of India from 1974 to 1977, set up the Humsub Drama Group in 1974. This post-Independence drama group has actively sought to revitalise the glorious tradition of the Urdu stage and has put out numerous seminal productions with well-known playwrights and thespians like Habib Tanveer, B.M. Shah and Nadira Babbar. Attached with the Ghalib Institute, Humsub has been curating drama festivals in Abida Ahmed's memory and this year's line-up includes plays by eminent directors like M.K. Raina, Danish Husain, Salima Raza and Danish Iqbal, among others.
Headlining the festival is The Hoshruba Repertory's Qissa Urdu Ki Aakhri Kitaab Ka. It is based on Ibne Insha's classic text Urdu Ki Aakhiri Kitaab (Urdu: The Final Book), a unique collection of satirical pieces written in the format of a textbook with chapter divisions on history, geography, mathematics, science and policy education. Ibne Insha (1927-78) is considered to be one of the best humourists in the history of Urdu literature and is also known for his poetry and travelogues.
Danish Husain's play situates the text in present-day India with a boisterous mix of stand-up comedy, storytelling, talk-show banter and farce with interludes of classical music and recitals from Ibne Insha's lush poems which are reminiscent of Amir Khusro's distinctive style. Danish Husain is known for plays like Guards At The Taj, Ek Punjab Ye Bhi, Chinese Coffee, Krapp's Last Tape, and the storytelling project Qissebaazi. "This is probably the most important play in the entire festival. Qissa Urdu.. is quite popular with the audience," says Syed Raza Haider, director of Ghalib Institute. Qissa Urdu Ki Aakhri Kitaab Ka was performed at the Prithvi Theatre last year to glowing reviews.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,758
|
<?php
namespace wulaphp\mvc\controller;
use wulaphp\app\App;
use wulaphp\io\Session;
/**
* 为控制器提供会话支持.
*
* @package wulaphp\mvc\controller
* @property-read string|null $sessionID
*/
trait SessionSupport {
protected final function onInitSessionSupport() {
$expire = App::icfg('expire', 0);
$this->_session = new Session ($expire);
$this->sessionID = $this->_session->start(property_exists($this, 'sessionID') ? $this->sessionID : null);
}
/**
* 销毁并更换session id。
*/
protected function changeSessionId() {
$this->_session->changeId();
}
/**
* 销毁 session
* @deprecated use destroySession
*/
protected final function destorySession() {
$this->_session->destory();
}
/**
* 销毁 session
*/
protected final function destroySession() {
$this->_session->destory();
}
/**
* 关闭 session
*/
protected final function closeSession() {
$this->_session->close();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,108
|
Q: How to accept ";' semicolon, as input, and not execute the next line of codes? Here's a small portion of a practice I'm doing preventing erroneous inputs.
while(1) {
printf("Choose From 1 to 7 ");
if( scanf("%d", &nNum ) != 1) {
printf("Please only choose from the numbers 1-7.");
fgets(sErraticInputs, 100 , stdin);
} else if (nNum > 7 || nNum <= 0) {
printf("Please only choose from the numbers 1-7.");
} else {
break;
}
}
I was doing a good job, until I entered "6;p". It executed the 6 portion and ran correctly, but technically speaking it should have taken the whole thing as the input, and proceeded with the error message.
A: First of all I don't think the posted code can give the said result. The break statement will end the while(1) when 6 has been read so there will not be printed an error message.
If we assume that the break isn't part of your real code this is what happens:
When scanf is told to read an integer, it will continue reading from the input stream as long as the next character (together with the previous read characters) can be converted into an integer. As soon as the next character can not be used as part of an integer, scanf will stop and give you the result of what it has parsed so far.
In your case the input stream contains
6;p\n
So scanf will read the 6 and stop (i.e. return 6). The input stream now contains:
;p\n
Consequently this will be the input for your next scanf and cause the input error, you saw.
One way to solve this would be to flush stdin after all scanf - both on success and on failure:
nNum = 0;
while(nNum != 7) // Just as an example I use input 7 to terminate the loop
{
printf("Choose From 1 to 7 ");
if( scanf("%d", &nNum ) != 1 || nNum > 7 || nNum <= 0)
{
printf("Please only choose from the numbers 1-7.");
}
else
{
printf("Valid input %d\n", nNum);
// **************************** break;
}
fgets(sErraticInputs, 100 , stdin); // Always empty stdin
}
note: Using fgets with size 100 doesn't really ensure a complete flush... you should actually use a loop and continue until a '\n' is read.
With the change above input like 6;p will be taken as a valid input with value 6 and the ;p will be thrown away.
If that's not acceptable, you could drop the use of scanf and do the parsing yourself. There are several options, e.g. fgets or fgetc
The example below uses fgetc
#include <stdio.h>
#include <stdlib.h>
int get_next()
{
int in = fgetc(stdin);
if (in == EOF) exit(1); // Input error
return in;
}
void empty_stdin()
{
while(get_next() != '\n') {};
}
int main(void) {
int in;
int nNum = 0;
while(nNum != 7)
{
printf("Choose From 1 to 7 \n");
in = get_next();
if (in == '\n' || in <= '0' || in > '7') // First input must be 1..7
{
printf("Please only choose from the numbers 1-7.\n");
if (in != '\n') empty_stdin();
}
else
{
nNum = in - '0';
in = get_next();
if (in != '\n') // Second input must be \n
{
printf("Please only choose from the numbers 1-7.\n");
empty_stdin();
}
else
{
printf("Valid input: %d\n", nNum);
}
}
}
return 0;
}
This code will only accept a number (1..7) followed by a newline
A: Here's why the "whole thing" is not taken as the input. From the man pages:
The format string consists of a sequence of directives which describe
how to process the sequence
of input characters. If processing of a directive fails, no further input is read, and scanf()
returns. A "failure" can be either of the following: input failure, meaning that input characters
were unavailable, or matching failure, meaning that the input was inappropriate...
Here's the full text. Have a look at this as well.
One approach would be to read in the whole input using fgets and check whether the length of the input is greater than 1. For an input of length 1, check if the input is a number and so on...
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,065
|
{"url":"https:\/\/hal.inria.fr\/hal-02006471","text":"Efficient Change-Point Detection for Tackling Piecewise-Stationary Bandits - Archive ouverte HAL Access content directly\nJournal Articles Journal of Machine Learning Research Year : 2022\n\nEfficient Change-Point Detection for Tackling Piecewise-Stationary Bandits\n\n(1) , (2) , (2) , (2, 3)\n1\n2\n3\nLilian Besson\nEmilie Kaufmann\nOdalric-Ambrym Maillard\nJulien Seznec\n\u2022 Function : Author\n\u2022 PersonId : 1084851\n\nAbstract\n\nWe introduce GLR-klUCB, a novel algorithm for the piecewise iid non-stationary bandit problem with bounded rewards. This algorithm combines an efficient bandit algorithm, kl-UCB, with an efficient, parameter-free, changepoint detector, the Bernoulli Generalized Likelihood Ratio Test, for which we provide new theoretical guarantees of independent interest. Unlike previous non-stationary bandit algorithms using a change-point detector, GLR-klUCB does not need to be calibrated based on prior knowledge on the arms' means. We prove that this algorithm can attain a $O(\\sqrt{TA \\Upsilon_T\\log(T)})$ regret in $T$ rounds on some easy'' instances, where A is the number of arms and $\\Upsilon_T$ the number of change-points, without prior knowledge of $\\Upsilon_T$. In contrast with recently proposed algorithms that are agnostic to $\\Upsilon_T$, we perform a numerical study showing that GLR-klUCB is also very efficient in practice, beyond easy instances.\n\nDates and versions\n\nhal-02006471 , version 1 (04-02-2019)\nhal-02006471 , version 2 (08-12-2020)\nhal-02006471 , version 3 (01-08-2022)\n\nLicence\n\nAttribution - NonCommercial - ShareAlike - CC BY 4.0\n\nIdentifiers\n\n\u2022 HAL Id : hal-02006471 , version 3\n\u2022 ARXIV :\n\nCite\n\nLilian Besson, Emilie Kaufmann, Odalric-Ambrym Maillard, Julien Seznec. Efficient Change-Point Detection for Tackling Piecewise-Stationary Bandits. Journal of Machine Learning Research, 2022. \u27e8hal-02006471v3\u27e9\n\n486 View","date":"2023-02-08 22:48:21","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\": 0.647793173789978, \"perplexity\": 5388.154349590786}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500983.76\/warc\/CC-MAIN-20230208222635-20230209012635-00739.warc.gz\"}"}
| null | null |
Léglise en bois de l'Ascension de Kućani (en serbe cyrillique : ; en serbe latin : ) est une église orthodoxe serbe située à Kućani, dans la municipalité de Nova Varoš et dans le district de Zlatibor, en Serbie. Elle est inscrite sur la liste des monuments culturels de grande importance de la République de Serbie (identifiant SK 200).
Présentation
L'église est située sur les pentes méridionales des monts Zlatibor, près du mont Murtenica, sur le territoire du village de Kućani et au hameau de Peta. Elle est dédiée à l'Ascension ou, selon d'autres sources, à la Nativité de la Mère de Dieu.
On ne sait pas exactement quand l'édifice a été érigé ; souvent datée du , l'église a probablement été construite au ; la difficulté de datation vient de ce que des sources écrites mentionnent deux églises-cabanes, l'une remontant à 1772 et l'autre datant de 1832. Ce que l'on sait de l'histoire de l'église doit beaucoup à la Chronique de l'église de Negbina ; cette histoire est liée à la famille sacerdotale Popović de Kućani.
Bien que très petite, cette église est l'œuvre d'habiles constructeurs de la région de Stari Vlah, qui l'ont réalisée sur le modèle des maisons dans lesquelles ils vivaient. De plan rectangulaire, elle est dotée d'une abside et d'un narthex ; elle mesure de long sur de large et la hauteur sous plafond est de seulement . Elle est constituée de rondins en pin ; le toit est recouvert de bardeaux.
À l'intérieur, le sol est recouvert de dalles de pierre de forme irrégulière. Dans l'angle sud-ouest de la nef se trouve un bénitier en pierre.
La valeur particulière de l'édifice est due aux portes royales peintes par Simeon Lazović, un artiste de Bijelo Polje. Intégrées dans une structure en bois doré, les peintures représentent deux Prophètes et la scène de l'Annonciation ; l'inscription préservée précise que ces portes ont été réalisées pour et qu'elles datent de 1780. Cette inscription a favorisé l'hypothèse d'une datation de l'église au mais aucun document n'atteste un changement de saint patron et il est impossible d'affirmer de manière fiable que les portes de Lazović étaient destinées à l'église dans laquelle elles se trouvent aujourd'hui.
Des travaux de restauration ont été effectués en 1953 et 1981.
Références
Article connexe
Monuments culturels du district de Zlatibor
Ascension
Kucani
Kucani
Monument culturel de grande importance en Serbie
Kucani
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,264
|
The idealized stone masks of Teotihuacan are the permanent features of grand ceremonial rituals involving dressed effigies of which the masks were the fixed element. Tecali masks were a smaller corpus, with the varying color, texture and translucency of the stone possibly selected for certain effigies or festivities reserved for an elite.
Tecali is a carbonate of lime, an aragonite with a hardness of 3.5 to 4.0 on the Moss scale. It is also known as banded onyx (note the layers of di erent shades of colour), Mexican onyx, and Tehuacan marble, and is quarried in the region of Tecali in southern Puebla. Carved tecali artefacts first appeared in Mesoamerica in the late Pre-Classic period (300–200 B.C.), objects of superior quality have been reported from Teotihuacan, dating as early as A.D. 150–250.
The present mask shows strong similarities with the Sultepec type, and it was probably influenced by this earlier regional style: a soft and delicately modeled face, with a thin pointed nose, large oval eyes and mouth, slightly hollowed and drilled to receive stone or shell ornaments. The whole face is traversed by a natural linear relief used by the carver to underline the eyes. Two holes pierced at the temples were used for attachment. The original patina, deep and contrasted, is superb.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,145
|
using System;
using System.Collections.Generic;
using ThoughtWorksRovers.Environment.Graph.DirectionsInfo;
using ThoughtWorksRovers.Program.Environment.Graph.DirectionsInfo;
using ThoughtWorksRovers.Program.Environment.Graph.Node;
using ThoughtWorksRovers.Program.FileParser;
using ThoughtWorksRovers.Program.FileParser.ParsedData;
namespace ThoughtWorksRovers.Program.Environment.Graph
{
/// <summary>
/// RoverGraph is an implementation of IGraph designed for the Mars Rover solution.
/// It defines the grid upon which the IRover objects are placed. This grid is a
/// directed graph.
/// </summary>
public class RoverGraph : IGraph
{
private readonly IDictionary<string,IGraphNode> _graphData;
private readonly IParsedCoordinatesData _upperRightCoordinateBoundary;
private readonly IDirectionsInfoContainer _directionsInfoContainer;
/// <summary>
/// The RoverGraph constructor stores the injected IDirectionsInfoContainer as well as
/// the ParsedCoordinatesData of the IParser.
/// </summary>
/// <param name="fileParser">The IParsed containing the grid and rover data.</param>
/// <param name="directionsInfoContainer">The injected IDirectionsInfoContainer object that defines the directions in which the IGraphNodes are connected.</param>
public RoverGraph(IParser fileParser, IDirectionsInfoContainer directionsInfoContainer)
{
_directionsInfoContainer = directionsInfoContainer;
// Set the upper right boundary
_upperRightCoordinateBoundary = fileParser.ParsedCoordinatesData;
_graphData = PopulateGraphNodes();
ConnectAllNodes();
}
/// <summary>
/// Creates and populates all of the IGraphNodes of the graph.
/// </summary>
/// <returns>An IDictionary of IGraphNode graph nodes data, indexed by a string.</returns>
private IDictionary<string,IGraphNode> PopulateGraphNodes()
{
IDictionary<string, IGraphNode> graphNodes = new Dictionary<string, IGraphNode>();
for (int x = 0; x <= _upperRightCoordinateBoundary.CoordinatesX; x++)
{
for (int y = 0; y <= _upperRightCoordinateBoundary.CoordinatesY; y++)
{
IGraphNode newNode = FactoryCreateIGraphNode(x, y);
graphNodes.Add(newNode.Id,newNode);
}
}
return graphNodes;
}
/// <summary>
/// Connects all adjacent nodes. Adjacencies are defined by the injected IDirectionsInfoContainer
/// which describes the directions in which the IGraphNodes can be connected, and the X and Y
/// offsets that each human-friendly direction defines in terms of a grid. Grid/graph boundaries are
/// determined by null adjacent IGraphNodes.
/// </summary>
private void ConnectAllNodes()
{
for (int x = 0; x <= _upperRightCoordinateBoundary.CoordinatesX; x++)
{
for (int y = 0; y <= _upperRightCoordinateBoundary.CoordinatesY; y++)
{
IGraphNode currentNode = _graphData[x.ToString() + y];
foreach (IDirectionsInformation directionsInformation in _directionsInfoContainer.DirectionsInformation.Values)
{
int xTotal = currentNode.XCoord + directionsInformation.XOffset;
int yTotal = currentNode.YCoord + directionsInformation.YOffset;
if (xTotal < 0 || xTotal > _upperRightCoordinateBoundary.CoordinatesX)
{
currentNode.PutAdjacentNode(_directionsInfoContainer.GetDirectionInteger(directionsInformation.Name),null);
}
else if (yTotal < 0 || yTotal > _upperRightCoordinateBoundary.CoordinatesY)
{
currentNode.PutAdjacentNode(_directionsInfoContainer.GetDirectionInteger(directionsInformation.Name), null);
}
else
{
currentNode.PutAdjacentNode(
_directionsInfoContainer.GetDirectionInteger(directionsInformation.Name),
_graphData[xTotal.ToString()+yTotal]);
}
}
}
}
}
/// <summary>
/// A standard factory pattern to create a newly instantiated IGraphNode object.
/// </summary>
/// <param name="xCoord">The X coordinate of the IGraphNode.</param>
/// <param name="yCoord">The Y coordinate of the IGraphNode.</param>
/// <returns>A newly instantiated GraphNode object.</returns>
private IGraphNode FactoryCreateIGraphNode(int xCoord, int yCoord)
{
if (_directionsInfoContainer.DirectionsInformation.Count == 4)
{
return new GraphNode(xCoord, yCoord);
}
throw new Exception("Number of directions in IDirectionsInfoContainer "
+ "was invalid, should be 4 (N,E,S,W)");
}
/// <summary>
/// A Property that gets the IDictionary of IGraphNode graph nodes data, indexed by a string.
/// </summary>
public IDictionary<string, IGraphNode> GraphData
{
get { return _graphData; }
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,262
|
\section{Introduction}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\setcounter{footnote}{1}
\footnotetext{E-mail: abate@email.arizona.edu, feldman@ku.edu}
Recent years have seen a rapid growth of cosmological observations and analyses that has brought forth the era of {\it precision cosmology} led by missions like the WMAP satellite \citep{wmap3,wmap5,wmap7}, the SDSS collaboration (see at http://www.sdss.org), DEEP (http://deep.berkeley.edu) and many others. As important and interesting as these measurements are, probing the large scale structure of the Universe using many diverse and independent measurements remains of utmost importance. As much as we have learned in the last decade, cosmology and even parameter estimation is not a closed subject \citep{BriLahOstSte03}. Not only do we not know the nature of dark matter, but even the density and amplitude of its initial perturbations (usually parametrized by $\Omega_m$ and $\sigma_8$ respectively) remain an open question. The determination of the matter content is especially problematic \citep{FukPee04,sigma810}.
Further, there are some indications that suggest that the $\Lambda$CDM cosmology with WMAP \citep{wmap7} central parameters may be problematic. Some examples are:
Large-scale anomalies found in the maps of temperature anisotropies in the CMB \citep{SarHutCop10,CopHutSch10,WMAPanom10};
recent estimates of the large scale bulk flow by \citet{WatFelHud09,FelWatHud10,MacFelFer11,MaGorFel11} are inconsistent at the nearly 3$\sigma$ level with $\Lambda$CDM predictions;
a recent estimate \citep{LeeKom10} of the occurrence of high-velocity merging systems such as the Bullet Cluster is unlikely at a $\sim6\sigma$ level;
large excess of power in the statistical clustering of luminous red galaxies (LRG) in the photometric SDSS galaxy sample
\citep{ThoAbdLah11};
evidence of larger than expected cross correlation between samples of galaxies and lensing of the CMB \citep{HoHirPad08,HirHoPad08};
brighter than expected type Ia Supernovae (SNIa) at High Redshift \citep{Kowalski08};
voids, especially smaller ones ($\sim10$ Mpc) are observed to be much emptier than predicted \citep{GotLokKly03};
the predicted shallow low concentration and density profiles of Cluster Haloes disagree with observations which indicate denser high concentration cluster haloes \citep{deBlok05,Gentile05};
\citet{KovBenItz10} find a unique direction in the CMB sky determined by anomalous mean temperature ring profiles, also centered about the direction of the flow detected above.
The amplitude and growth of cosmological fluctuations on large scales are closely related to the CMB dipole, which reflects the bulk flow (BF) of the local group and can be used to test cosmological models. Large scale velocity surveys have been undertaken by various groups ({\it e.g.\ } \citet{GioHayHer97,HudSmiLuc99,HudSmiLuc04,SFI1,SFI2,sfierr09}) and recent analyses by \cite{WatFelHud09,FelWatHud10} of the newest proper distance measurements show that virtually all velocity survey analyses show a consistent large-scale BF. It appears to have a magnitude $\gteq400$ km/s on scales of 100$h^{-1}$Mpc, which disagrees with $\Lambda$CDM WMAP predictions at the $\sim3\sigma$ level. Analysis of the flow \citep{FelWatHud10} suggests that if it is due to a gravitational potential flow, then the sources ({\it i.e. } over- and under-densities) must be on scales $\ \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 300$$h^{-1}$Mpc. The direction of this flow is close to the Galactic disk, Galactic longitude $\sim295^{\rm o} $ and longitude $\sim10^{\rm o} $ with error of $\sim5^{\rm o}$. Assuming that this is a potential flow, we expect to see over-densities in the flow direction and under-densities in the opposite direction. The responsible structures must be far away from us since the flow is very cold \citep{FelWatHud10}.
The motion detected in \citet{FelWatHud10} is not due to nearby sources, such as the Great Attractor (distance of $\sim40$$h^{-1}$Mpc), but rather to sources at greater depths that have yet to be fully identified. The largest known mass concentration, the Shapley supercluster, does not seem to be massive enough to cause a flow of this magnitude \citep{Ray89}. Following \citet{tully08}, it is more likely that the flow arises both from various mass concentrations in the Galactic y-direction as well as under-dense regions in the opposite direction. Currently, there is no survey in existence that is deep enough to resolve the source(s) of the flow. Nor is there any data that probes these scales ($>300$$h^{-1}$Mpc) to see whether there are any large mass concentrations and voids in these directions.
\begin{figure}
\begin{tabular}{c}
\includegraphics[width=6.5cm]{plots/dmaggie_ind_zbin1.ps} \\
\includegraphics[width=6.5cm]{plots/dmaggie_ind_zbin2.ps} \\
\includegraphics[width=6.5cm]{plots/dmaggie_ind_zbin3.ps} \\
\includegraphics[width=6.5cm]{plots/dmaggie_ind_zbin4.ps} \\
\end{tabular}
\caption{
The magnitude fluctuation as a function of R.A.. The LRG $r$ band model magnitudes in bins of R.A. and redshift are averaged over all values of the declinations, the mean magnitude at each redshift is then subtracted. The fluctuation is in units of ``maggies", see Eqs. \ref{eq:find},\ref{eq:find2}. Each panel shows the different \textit{spectroscopic} redshift bins: top panel $z_c=0.12$, 2nd panel $z_c=0.20$, 3rd panel $z_c=0.28$, bottom panel $z_c=0.36$. The solid lines are the Galactic extinction corrected magnitudes, and the dotted lines are the magnitudes with no extinction correction applied.
\label{fig:avmagind}
}
\end{figure}
\citet{KasAtrKoc08,KasAtrKoc10} found a larger amplitude BF, coined the \textit{Dark Flow}, on scales out to 300 $h^{-1} {\rm Mpc}$ by using the kinematic Sunyaev-Zeldovich effect (kSZ) in the CMB to measure the bulk velocity of about 1200 X-ray clusters. Like \citet{WatFelHud09,FelWatHud10}, these papers claim that a BF at these depths, as determined by their studies, is difficult or impossible to explain within the framework of standard $\Lambda$CDM model of cosmology. Although these results utilize completely different methods to measure the BF and find similar results, kSZ and velocity surveys are both subject to problems of very large noise and systematics in the data. In particular it should be noted that converting the dipole of the kSZ signal to a velocity is model dependent: assuming a different cluster radial profile can give a completely different result. However, other recent results \citep{DavNusMas11,NusDav11} using the SFI++ catalog of Tully-Fisher galaxies and also reconstructing the cosmological large-scale flows using the 2MASS (Two Micron All Sky Survey) redshift survey (2MRS) found flows roughly the same direction as mentioned above but with magnitude consistent with the WMAP $\Lambda$CDM model.
In this paper we approach the problem in a different manner and instead look for fluctuations of observed magnitudes across the sky as could be induced by a bulk motion centered on us. We choose objects that we assume are distant enough to be considered at rest compared to the CMB. Ideally these objects should be standard candles, therefore the best astrophysical candidate would be SNIa. Unfortunately there are not enough observations of SNIa, and their distribution across the sky is far from ideal for this investigation. We choose instead to use luminous red galaxies (LRGs) which are suitable because they are assumed to form a single population of galaxies, which assembled at high-redshift and have been passively evolving since. This assumption was investigated by \citet{WakNicEis06} who found that the LRG luminosity function (LF) is consistent with purely passive evolution and a merger-free history. Therefore to use the LRGs as a proxy for standard candles we average out the effect of the intrinsic variability of their luminosities by averaging together the magnitudes of neighboring LRGs. A recent study \citep{NusBraDav11} presented an alternate method for detecting cosmological bulk flows from redshift surveys, using the observed dimming or brightening of galaxies due to their peculiar motion, a similar technique to that which we present here.
The paper is organized as follows, in Section \ref{sec:data} we present our data set and how we determine the observed magnitude fluctuation, in Section \ref{sec:theory} we present the theoretical model for magnitude fluctuations induced by peculiar velocities. Section \ref{sec:fit} describes our method of analyzing the data, with the results presented in Sections \ref{sec:res} and \ref{sec:dir}, and discussion and conclusions of these results in Section \ref{sec:disc}.
Throughout this paper we assume a concordance cosmology of: $H_0$=70km/s, $\Omega_m=0.3$, $\Omega_\Lambda=0.7$ and $w=-1$ unless otherwise stated.
\begin{figure}
\begin{tabular}{c}
\includegraphics[width=6.5cm]{plots/dmaggie_zbin1.ps} \\
\includegraphics[width=6.5cm]{plots/dmaggie_zbin2.ps} \\
\includegraphics[width=6.5cm]{plots/dmaggie_zbin3.ps} \\
\includegraphics[width=6.5cm]{plots/dmaggie_zbin4.ps} \\
\end{tabular}
\caption{
The ``model dependent" magnitude fluctuation as a function of R.A.. The difference between LRG $r$ band model magnitudes and the ``theoretical" catalog in bins of R.A. and redshift is averaged, over all values of the declinations. The fluctuation is in units of ``maggies", see Eqs. \ref{eq:df},\ref{eq:df2}. Each panel shows a different \textit{spectroscopic} redshift bin: top panel $z_c=0.12$, 2nd panel $z_c=0.20$, 3rd panel $z_c=0.28$, bottom panel $z_c=0.36$. The solid lines are the Galactic extinction corrected magnitudes, and the dotted lines are the magnitudes with no extinction correction applied.
\label{fig:avmag}
}
\end{figure}
\section{The observed fluctuation}
\label{sec:data}
\begin{figure*}
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{plots/dmaggie_ind_pz_zbin1.ps} & \includegraphics[width=6.5cm]{plots/dmaggie_pz_zbin1.ps} \\
\includegraphics[width=6.5cm]{plots/dmaggie_ind_pz_zbin2.ps} & \includegraphics[width=6.5cm]{plots/dmaggie_pz_zbin2.ps} \\
\end{tabular}
\caption{
The magnitude fluctuation as a function of R.A. for the \textit{photometric} sample. Each row shows a different \textit{photometric} redshift bin: top row $z_c=0.475$, bottom row $z_c=0.625$. The two columns correspond to the two different ways of computing the fluctuation: left column is the fluctuation described by Eq.~\ref{eq:find} (as in Fig.~\ref{fig:avmagind} for the spectroscopic sample), the right column is the ``model dependent" magnitude fluctuation described by Eq.~\ref{eq:df} (as in Fig.~\ref{fig:avmag} for the spectroscopic sample).
\label{fig:avmagpz}
}
\end{figure*}
We use the SDSS spectroscopic and photometric LRG samples, described in \citet{Eis01} and \citet{Coll06}. We select a rectangular area of the SDSS survey bounded by: $125^{\circ}<\mbox{R.A.}<245^{\circ}$ and $8^{\circ}<\mbox{Dec.}<59^{\circ}$, and covering the redshift range: $0.08<z<0.40$ for the spectroscopic sample, and $0.4<z<0.7$ for the photometric sample. The area was chosen to be contiguous and to be about a degree or so from the survey edges to ensure a uniform as possible coverage. We use the $r$ band model magnitudes and we further filter the catalog by requiring that all the galaxies have a signal-to-noise ratio $\ge2$ and the Galactic extinction in the $r$ band $x_r<0.1$.
Redshifts are corrected to the local group frame.
To obtain enough signal we must average over enough LRGs in order to beat down the effect of their intrinsic variability in luminosity. The BF direction found by \cite{WatFelHud09,FelWatHud10} is towards R.A $=180^{\circ}$, Dec. $=-52^{\circ}$. Since the R.A. direction lies within the survey area, whereas the SDSS LRG Dec. distribution are more than $60^\circ$ away, we choose to average the LRG magnitudes across declinations. For each redshift and R.A. bin we find the average difference in flux units, or ``maggies" (a linear measure of flux defined as $10^{-0.4*m}$), between the magnitudes in that bin and the mean magnitude at any angular position within the same redshift bin:
\begin{eqnarray}
\label{eq:find}
df&=& 10^{-0.4(m_r - \left<m_r\right>_j)}\\
\label{eq:find2}
\delta_m^o &=&<df>_{ij}
\end{eqnarray}
where $i$ refers to the R.A. bin, and $j$ to the redshift bin, $df$ is the difference in LRG flux/magnitude in maggie units and $\delta_m^o$ is the mean value of $df$ in the R.A.-redshift bin $ij$ . Fig.~\ref{fig:avmagind} shows this magnitude fluctuation in each redshift bin for the spectroscopic sample; the dotted lines show the fluctuation when the magnitudes have not been corrected for Galactic extinction. The left hand side of Fig.~\ref{fig:avmagpz} shows the same for the photometric sample.
Computing the observed fluctuation in this way is not optimal because we do not have a full sky area with which to compute $\left<m_r\right>_j$, and the result will also be sensitive to the redshift distribution of the LRG within each R.A. bin. We would also like to compute the observed magnitude fluctuation in such a way that it could be analysed in terms of the theoretical perturbation of observed magnitudes due to peculiar velocities. Therefore because of these two issues we also calculate the observed fluctuation in R.A., redshift bins in the following way:
\begin{eqnarray}
\label{eq:df}
df&=& 10^{-0.4(m_r - \bar{m}_r)}\\
\label{eq:df2}
\delta_m^o &=&<df>_{ij}
\end{eqnarray}
where again the observed fluctuation $\delta_m^o$ is in flux rather than magnitude units. Eq.\ref{eq:find} and Eq.~\ref{eq:df} differ in their use of $\bar{m}_r$ or $\left<m_r\right>_j$. $\bar{m}_r$ are the expected apparent magnitudes for each LRG in our catalog, assuming a cosmology, no flows (homogeneous universe) and that all LRGs belong to a population with the same mean absolute magnitude and spectral energy distribution (SED), whereas $\left<m_r\right>_j$ are simply the average LRG magnitude (over all angular positions) in redshift bin $j$.
Fig.~\ref{fig:avmag} shows this fluctuation in the spectroscopic sample for each redshift bin, and with and without Galactic dust corrections applied to their magnitudes, and the right hand side of Fig.~\ref{fig:avmagpz} shows the same for the photometric sample.
The method we use to calculate $\bar{m}_r$, the expected apparent magnitudes in the absence of any flows for each LRG in our catalog, is outlined below:
\begin{equation}
\label{eq:magth}
\bar{m}_r(z) = 5 \log_{10}\bar{d}_L(z)+25+\left<M_r\right>_{lrg}+K_r
\end{equation}
where $\bar{m}_r$ and $\bar{d}_L$ denote the magnitude and luminosity distance in a homogeneous universe respectively, $\left<M_r\right>_{lrg}$ is the average LRG absolute magnitude in the $r$ band, $K_r$ is the $k$-correction, $z$ is the redshift in our filtered LRG catalog. $\bar{d}_L(z)$ is calculated using the usual equations as given in \citet{hog99}. The $k$-correction is found by assuming that all the LRGs have an SED with the same shape. It is calculated for each galaxy from:
\begin{equation}
K_{r}=-2.5\log_{10}\left(\frac{1}{(1+z)}\frac{\int d\lambda \lambda f_\lambda(\lambda/(1+z))r(\lambda)}{\int d\lambda \lambda f_\lambda(\lambda)r(\lambda)}\right)
\end{equation}
where $f_\lambda(\lambda)$ is the assumed SED of the LRGs, in this case it is the elliptical SED in \citet{ColWuWee80}, and $r(\lambda)$ is the SDSS $r$ filter.
For calculating the mean absolute magnitude $\left<M_r\right>_{lrg}$ we use the SDSS LRG luminosity function (LF) measured by \citet{WakNicEis06} corrected to redshift zero. To calculate $\left<M_r\right>_{lrg}$ we simply integrate the LF, $\phi(M)$ multiplied by $M_r$.
\begin{equation}
\left<M_r\right>_{lrg}= \frac{ \int \phi(M) M dM}{ \int \phi(M) dM}
\end{equation}
From the above LF we find that we need to average over around 1000 galaxies in each bin so we can be confident that the average absolute magnitude of LRGs in each bin $i$, $\left<M_r\right>_i\equiv \left<M_r\right>_{lrg}$.
Finally this ``theoretical homogeneous universe" LRG catalog is subtracted from the real LRG catalog, to get $df$ as defined in Eq.~\ref{eq:df}.
We now discuss the assumptions made in Eq.~\ref{eq:magth} when calculating the theoretical catalog. First of all we assume a fiducial set of cosmological parameters to calculate the luminosity distance $\bar{d}_L$. The choice of parameters has a predictable effect on the resulting observed fluctuation $\delta_m^o$, shifting the whole flucutation upwards or downwards slightly, i.e. changing the normalisation, but it cannot produce an effect which varies with R.A.. Secondly we assume the value we calculate for $\left<M_r\right>_{lrg}$ is in fact the true average LRG absolute magnitude in the bin. As long as there are over 500 galaxies in each bin (and we have arranged the binning such that this true) and taking into account the errors on the luminosity function from \citet{WakNicEis06}, the error on this quantity is less than 1\%, smaller than the SDSS photometric errors. There should be no reason to expect $\left<M_r\right>_{lrg}$ to vary with position, however it could evolve with redshift. Finally to compute the $k$ correction we must assume the elliptical SED in \citet{ColWuWee80} is truly representative of the actual LRG SED. Again we find no reason that this assumption, if wrong, would cause an effect where $<\bar{m}_r>$ varied with R.A.. In fact using a completely different SED, the spiral galaxy Sbc from \citet{ColWuWee80}, causes a negligible difference in the results because the difference in $k$-correction values between these two SEDs are small.
It seems that all the assumptions made in calculating this theoretical LRG catalog could cause the same systematic effect: a change in the normalisation of the plots. This normalisation could be a function of redshift: the difference between $\bar{d}_L$ calculated with different cosmological parameters increases as a function of $z$, and $\left<M_r\right>_{lrg}$ could evolve with $z$. Crucially, however, none of \textit{these} systematics are expected to cause a change in $\left<m_r\right>$ with angular position.
In this paper we want to constrain the parameters of the BF: $v_b$, $\alpha_{b}$, $\delta_{b}$. The observed fluctuation as defined by Eq.~\ref{eq:df} is the one which is compatible with how the predicted magnitude fluctuations are modelled, as described in Sec.~\ref{sec:theory}. Therefore to avoid issues with the ``nuisance" systematics outlined above we divide out the normalisation of the fluctuations. We do this by dividing out the value: MAX($\delta_m^o$)/2 + MIN($\delta_m^o$)/2 from the fluctuations shown in Fig.~\ref{fig:avmag}.
Looking by eye at Fig.~\ref{fig:avmagind} and ~\ref{fig:avmag} it seems the first redshift bin might be contaminated by LRGs which are not at rest with respect to the CMB, or indeed by objects from the Main SDSS sample. It is not unexpected that ``normal" flows in the universe could still be affecting the observed magnitudes at these redshifts. Therefore in Sec.~\ref{sec:res} we repeat our analysis using all redshift bins, removing the lowest redshift bin, and removing the two lowest redshift bins to reduce the possibility of either of these two effects.
The two main candidates to explain at least part of the observed fluctuation presented in Figures \ref{fig:avmagind}, \ref{fig:avmag} and \ref{fig:avmagpz}, outside of the context of a
dipole flow,
are: reddening by our Galaxy, and the variation in the selection of SDSS LRGs. These can also be considered to be systematics. It is unlikely that Galactic reddening is a major contributor because i) there is so little difference in the observed fluctuation when using extinction-corrected and non-extinction corrected magnitudes, ii) a similar fluctuation is seen in the $g$, $i$ and $z$ bands i.e. there is no fluctuation in the observed colors. To see the variation with extinction in the $r$ band, compared to the average extinction, analogously to Eq.~\ref{eq:find} we calculate:
\begin{eqnarray}
\label{eq:dx}
dx&=& 10^{-0.4(x_r - \left<x_r\right>_j)}\\
\delta_x^o &=&<dx>_{ij}
\end{eqnarray}
where $x_r$ is the extinction in the $r$ band at the position of the LRG. This fluctuation is plotted at the top of Fig.~\ref{fig:xas}. Each different line style corresponds to a different redshift bin. Because the Galactic extinction varies only with angular position, and of course not with the redshift of the LRGs, the pattern of mean extinction verses R.A. is the same for each redshift bin: small differences occur only due to the different distribution of LRG angular positions within each redshift bin.
The LRG angular selection function is more difficult to quantify. To get a handle on what the angular selection function of SDSS LRGs might be we looked at the NYU Value-Added Galaxy Catalog \citep[NYU-VAGC]{nyuvagc}. The NYU-VAGC is a collection of galaxy catalogs cross matched to SDSS designed for the study of galaxy formation, evolution, and large-scale structure. As such the catalog contains detailed information about the angular selection function. We choose the parameter
\texttt{\textbf{fgotten}} from the large-scale structure subsample as the value we use to estimate the average LRG completeness as a function of position. The \texttt{\textbf{fgotten}} parameter describes the fraction of targets in the SDSS Main sample which have successfully measured redshifts. The bottom of Fig.~\ref{fig:xas} plots the mean value of \texttt{\textbf{fgotten}} in each of the R.A. bins divided by the mean value of \texttt{\textbf{fgotten}} over the whole survey area. Although the variation in \texttt{\textbf{fgotten}} is similar to the fluctuation we observe, its amplitude is significantly smaller. To incorporate the information contained within the \texttt{\textbf{fgotten}} parameter into our analysis we would also need to know how it varied as a function of flux, and not just angular position.
\begin{figure}
\begin{tabular}{c}
\includegraphics[width=6.5cm]{plots/extinc.ps}\\
\includegraphics[width=6.5cm]{plots/angselec.ps}
\end{tabular}
\caption{\textit{Top}: Average Galactic extinction in the $r$ band in each redshift and R.A. bin, the different lines correspond to the different redshift bins.
\textit{Bottom}: Average angular selection (\texttt{\textbf{fgotten}}) as a function of R.A.. The variation in \texttt{\textbf{fgotten}} seems to follow the variation in the Galactic extinction, which is not unexpected e.g. less galaxies are likely to be detected in regions of higher extinction.
\label{fig:xas}
}
\end{figure}
\section{Theoretical model of magnitude fluctuations}
\label{sec:theory}
The perturbation to the luminosity distance is defined as: $\delta d_L=d_L(z)-\bar{d}_L(z)$, where $d_L(z)$ is the actual observed luminosity distance (i.e. given the universe is inhomogeneous) and $\bar{d}_L(z)$ is the luminosity distance calculated at the \textit{same} observed redshift $z$ but assuming perfect homogeneity. A bar above quantities indicates that they have been calculated assuming, or are defined by assuming, homogeneity.
The perturbation to the luminosity distance (to first order) due to peculiar velocities is \citep{HuiGree06}:
\begin{equation}
\frac{\delta d_L}{d_L}=\frac{\textbf {v}_e\cdot\hat{\textbf{r}}}{c}-\frac{c(1+z)^2}{\bar{H}\bar{d}_L}\left(\frac{\textbf {v}_e\cdot\hat{\textbf{r}}}{c}-\frac{\textbf {v}_o\cdot\hat{\textbf{r}}}{c}\right).
\end{equation}
where $\textbf {v}_e$ is the peculiar velocity of the emitter, $\textbf {v}_o$ is the peculiar velocity of the observer, $\hat{\textbf{r}}$ is the unit vector in the direction from the observer to the emitter, $z$ is the redshift of the emitter, $\bar{H}$ and $\bar{d}_L$ are the unperturbed Hubble parameter and luminosity distance respectively, calculated from the usual equations as found in \citet{hog99}.
The relation between the luminosity distance and apparent magnitude $m$ is:
\begin{equation}
m=5\log_{10}d_L+M=5\frac{\ln d_L}{\ln 10}+M
\end{equation}
Then differentiating the above equation gives:
\begin{equation}
\delta m=\frac{5}{\ln 10}\frac{\delta d_L}{d_L}
\end{equation}
We assume that the LRG's form a rest frame ($\textbf {v}_e$=0) so we can look at our motion relative to them, our
dipole
flow $\textbf {v}_o$, the magnitude fluctuation is now:
\begin{equation}
\label{eq:dm}
\delta m=\frac{5}{\ln 10}\frac{\textbf {v}_o\cdot\hat{\textbf{r}}}{c}\left(\frac{c(1+z)^2}{\bar{H}\bar{d}_L}\right)
\end{equation}
The value of $\textbf {v}_o\cdot\hat{\textbf{r}}$ will depend on the angular position of the LRG (R.A. and Dec., labelled as $\alpha$ and $\delta$ respectively). We model the values by assuming they are due \textit{only} to a BF of magnitude $v_{b}$ and in direction $\alpha_{b}$,$\delta_{b}$, so we can calculate Eq.~\ref{eq:dm} as a function of position ($\alpha$,$\delta$) and redshift $z$.
The top of Fig.~\ref{fig:th} shows the change in $10^{-0.4\delta m}$ with redshift for a constant value of $\textbf {v}_o\cdot\hat{\textbf{r}} = 300$km/s. Three different cosmologies are plotted: $\Lambda$CDM in red; CDM universe ($\Omega_m=1$) in green; Open universe ($\Omega_m=0.3$) in blue. The value of the fluctuation $10^{-0.4\delta m}$ is not a strong function of the cosmological parameters because here (in Eq.~\ref{eq:dm}) they only affect the ``geometric" distance factors. The value of $\textbf {v}_o\cdot\hat{\textbf{r}}$ (and hence the value of the fluctuation) would be primarily driven by the cosmological parameters affecting the growth of structure, in the absence of more exotic explanations. This is why the choice of cosmological parameters in Eq.~\ref{eq:magth} does not have any significant effect on the ``model dependent" observed magnitude fluctuation. The bottom of Fig.~\ref{fig:th} shows the expected magnitude fluctuation in our LRG sample if there is a
dipole
flow of magnitude $v_b$=300km/s in the direction of $\alpha_b=180$ deg. and $\delta_b$=-50 deg. in a $\Lambda$CDM universe.
\subsection{The expected signal}
We expect the magnitude fluctuations generated by a
dipole
flow to follow a cosine form (in the magnitudes) across the sky, and we can model this cosine with the following functional form:
\begin{equation}
\delta mag = A\cos(\alpha + B) +C
\end{equation}
where $\alpha$ is the R.A. direction.
Putting this together with Eq.~\ref{eq:dm} we can understand the effect of the flow parameters ($\alpha_b$, $\delta_b$, $v_b$) on the resulting cosine shape of $\delta mag$ vs R.A. in the following way:
\begin{itemize}
\item A is the amplitude and is affected by $\delta_b$ and $v_b$.
\item B is the phase and is affected only by $\alpha_b$.
\item C is the normalisation and is affected by $\delta_b$ and $v_b$, but also the systematics.
\end{itemize}
There is a degeneracy between $\delta_b$ and $v_b$ which both affect the amplitude of the cosine. It also should be noted that two flows with a relation between their bulk angles as: $\alpha_{b1}=\alpha_{b2}+\pi$ and $\delta_{b1}=-\delta_{b2}$ are indistinguishable. Hence we only consider flows with $\delta_b<0$. This qualitative explanation of the signal also illustrates why we can divide out the constant $C$: it is the only part affected by the systematics, and information on the flow amplitude and declination direction is still contained within $A$.
\begin{figure}
\begin{tabular}{c}
\includegraphics[width=6.5cm]{plots/dmth.ps}\\
\includegraphics[width=6.5cm]{plots/dmth_bins}
\end{tabular}
\caption{\textit{Top}: Expected magnitude fluctuation amplitude as a function of redshift due to a
dipole
flow with value $v_o\cdot r$ = 300km/s. Calculated from Eq.~\ref{eq:dm} then transformed into maggies. Three different cosmologies are plotted: $\Lambda$CDM in red; CDM universe ($\Omega_m=1$) in green; Open universe ($\Omega_m=0.3$) in blue. \textit{Bottom}: Expected magnitude fluctuation for our LRG sample in each redshift bin due to a flow with a magnitude of $v_b$ = 300km/s in a $\Lambda$CDM universe.
\label{fig:th}
}
\end{figure}
\section{Fitting the flow}
\label{sec:fit}
\subsection{Errors on the measured $\delta_m$}
Naturally, even in the absence of measurement errors and peculiar velocities, the LRG magnitudes will have some distribution within the redshift bin. Then in the limit where the redshift interval of the bin is infinitesimally small, and all LRGs have exactly the same SED and negligible reddening, then this distribution should follow the LRG luminosity function. If the individual magnitudes are modelled as having a Gaussian measurement error distribution with a width of $\sigma$, and if instead the LF was a $\delta$ function at one magnitude, then the observed distribution of magnitudes in the bin would be a Gaussian of width $\sigma$ around this value. Therefore we can approximate the $\sigma$ of the error distribution by looking at the difference between the observed distribution of magnitudes in \textit{each bin} and the expected distribution:
\begin{equation}
\label{eq:std}
\sigma_{\delta_m}^2 =\frac{\left< \left(X - \bar{X}\right)^2 \right>}{\sqrt{N}}
\end{equation}
where $X=m_r-\bar{m}_r$: the observed magnitudes minus the expected magnitudes, $\bar{X}$ is the mean of this distribution, and $N$ is the number of LRGs in the bin.
\subsection{$\chi^2$ fitting}
\label{sec:res}
We calculate the following $\chi^2$ statistic:
\begin{equation}
\chi^2(v_{b}, \alpha_{b}, \delta_{b})=\sum_{ij}\left(\frac{\delta_m^o-\delta_m^p(v_{mag}, \alpha_{b}, \delta_{b})}{\sigma_{\delta_m}}\right)^2
\end{equation}
where the sum is over the redshift bins $i$ and the R.A. bins $j$; $\delta_m^o$ is the observed magnitude fluctuation in bin $i,j$ (see Section.~\ref{sec:data}) and $\delta_m^p$ is the predicted value of the magnitude fluctuation in bin $i,j$; $\sigma_{\delta_m}$ is the error on the observed magnitude fluctuation in bin $i,j$ found from Eq.~\ref{eq:std}.
To calculate the predicted fluctuation, $\delta_m^p$, we calculate the $\delta m^N$ using Eq.~\ref{eq:dm} (given the trial flow parameters) for each of the $N$ LRGs in our catalog, and then average them in the same way as the data:
\begin{equation}
\delta_m^p = <10^{-0.4 \delta m^N}>_{ij}
\end{equation}
The probability as a function of the flow parameters is calculated from the $\chi^2$ as follows:
\begin{equation}
P(v_{b}, \alpha_{b}, \delta_{b}) \propto \exp\left(-\frac{\chi^2}{2}\right)
\end{equation}
The resulting one-dimensional likelihoods of the flow parameters (after marginalising over the other 2 parameters) are shown in Fig.~\ref{fig:like}. The 1-$\sigma$ constraints on each flow parameter (after marginalising over the other parameters) are shown in Table~\ref{tab:res} for the spectroscopic catalog.
Fig.~\ref{fig:fit} shows the magnitude fluctuation data in two highest redshift bins with the best-fit model from Table~\ref{tab:res} over-plotted. The best-fit flow model was taken from the fit including all four redshift bins.
\subsection{Direction fitting}
\label{sec:dir}
We also attempted a separate approach to find the direction of the flow by fitting a vector of the directions of the LRG's weighted by each galaxy's magnitude (or maggie). We find the weighted mean R.A. of the fluctuations by weighing the R.A. of each LRG with the magnitude $m^n_r$ or with its "maggie" $10^{-0.4(m^n_r-<m_r^n>)}$ where $m^n_r$ is the r-band magnitude of each LRG indexed by $n$ and $<m_r^n>$ is the mean of the distribution. We have done that for the whole distribution (all redshifts) and separately by looking at the same redshift regions as described in Fig.~\ref{fig:avmagind}, we also calculated the R.A. direction in various Dec. regions. The results are not very sensitive to the binning schemes and agree by and large with the analysis above. We found that the most likely R.A. is $178\pm11^\circ$ in good agreement with the binned analysis presented here and with the velocity analyses presented in \citet{WatFelHud09,FelWatHud10,MaGorFel11,KasAtrKoc08,KasAtrKoc10}.
\begin{table}
\caption{Results from the spectroscopic catalog.\label{tab:res}}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
No. $z$ bin & $\alpha_b$ & $\delta_b$ & $v_b$\\
\hline
\hline
2 & 177.0$^{+2.7}_{-1.4}$ &-50$^{+12}_{-21}$ & $>$7000\\
\hline
3 & 179.0$^{+1.3}_{-1.7}$ &-65$^{+10}_{-18}$ & 6000$^{+1000}_{-900}$\\
\hline
4 & 185.5$^{+1.9}_{-3.0}$ &-35$^{+10}_{-6}$ & 4450$^{+1350}_{-1100}$\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure*}
\begin{tabular}{ccc}
\includegraphics[width= 5.5cm]{plots/like1D_ra_mth_maggies.ps}&
\includegraphics[width= 5.5cm]{plots/like1D_dec_mth_maggies.ps}&
\includegraphics[width= 5.5cm]{plots/like1D_vb_mth_maggies.ps}\\
\end{tabular}
\caption{One-dimensional likelihoods of the flow parameters given by $P\propto \exp(-\chi^2/2)$, after marginalising over the other 2 parameters for the spectroscopic catalog. The red dashed lines correspond to when all 4 redshift bins are used in the analysis; solid green lines when only 3 bins are used; solid blue when only 2 bins are used.
\label{fig:like}}
\end{figure*}
\section{Discussion}
\label{sec:disc}
Our goal in this paper was to search for a signature of a bulk flow by looking for fluctuations in the magnitudes of distant LRGs.
Figures \ref{fig:avmagind}, \ref{fig:avmag} and \ref{fig:avmagpz} show that we do find a coherent fluctuation in the LRG magnitudes which seems to be fairly independent of their redshift. The cosine-like shape of the fluctuation matches that which would be expected if this observed fluctuation is due to a
dipole motion with respect to the LRG sample.
The maximum of the fluctuation is also located roughly at the R.A. direction of a bulk flow found by other authors (170$\pm10^{\circ}$), however its amplitude is more than an order of magnitude larger than what would be expected from Eq.~\ref{eq:dm} after assuming a ``reasonable" flow (Fig.~\ref{fig:th}). We find a magnitude fluctuation on the order of a few percent in flux for the spectroscopic sample, and about a percent for the photometric sample. Comparatively one would expect a fluctuation of less than 1 percent if the flow had an amplitude of less than 1000km/s. Figures \ref{fig:avmagind}, \ref{fig:avmag} and \ref{fig:avmagpz} do also show the amplitude of the observed fluctuations decreases with redshift, as expected from Eq.~\ref{eq:dm} and illustrated by Fig.~\ref{fig:th}.
Our results are extremely sensitive to systematic effects, which is an obvious alternate explanation for the magnitude fluctuations we find. Possible effects could include: photometric zeropoint variation, correlated errors in the Galactic extinction correction applied to the magnitudes, and selection effects on the LRG sample.
The SDSS imaging data is acquired in a continuous scan, and each scan is obtained along a stripe. The survey strategy means the data we average together end up coming from different stripes. However, it would be very unlikely that a zeropoint variation could produce such a coherent fluctuation after averaging across different stripes.
Galactic extinction on the other hand will have an effect which is correlated with the angular position of the line of sight. Figs.~\ref{fig:avmagind} and \ref{fig:avmag} show that there is very little effect on shape of the observed fluctuation when no Galactic extinction correction is applied to the magnitudes. This implies that an error in the Galactic extinction estimate is very unlikely to be the sole source of the fluctuation. In Fig.~\ref{fig:xas} we plotted the average Galactic extinction in each bin. The shape of the extinction variation is extremely similar to our observed magnitude fluctuation, but its amplitude cannot fully explain the entire signal that we find.
The LRG angular selection function, however, is likely to have a big effect on our results. The angular selection varies in a similar manner to the fluctuation we observe. It would seem from comparing the bottom panel of Fig.~\ref{fig:xas} to figures \ref{fig:avmagind} and \ref{fig:avmag}, that this could only contribute to about 1 percent of the observed fluctuation. It is not clear however how to take account of this in our analysis, one would need to know the angular selection as a function of LRG magnitude before precise constraints on the magnitude of the flow could be made.
\begin{figure*}
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{plots/dmagmodel_z3.ps} &
\includegraphics[width=6.5cm]{plots/dmagmodel_z4.ps} \\
\end{tabular}
\caption{Examples of the 4-redshift bin best-fit model (bottom row of Table~\ref{tab:res}, blue dashed lines) compared to to the data (green solid lines). The left-hand side shows the $0.24<z<0.32$ bin and the right-hand side shows the $0.32<z<0.40$ bin. \label{fig:fit}}
\end{figure*}
We fitted a flow model to the observed fluctuation, and constrained the three flow parameters, its direction and magnitude: $\alpha_b$, $\delta_b$, $v_b$. We found that the flow R.A. direction was consistent with the direction found by other authors. The R.A. direction of the flow was the best constrained parameter, unsurprising because it is the only parameter where the rough best-fit value can be inferred from Fig.~\ref{fig:avmagind} and ~\ref{fig:avmag} by eye. It is also unsurprising, given the amplitude of our signal, that we find an anomalously large flow. We do not attempt to perform the fit using the photometric sample. Although it has the advantage over the spectroscopic sample in that it has over 6 times as many LRGs, and all of them are at redshifts where we are most confident that they are truly at rest with respect to the CMB, it is non-trivial to account for their redshift errors in the analysis. At this stage given the small size of the survey area and its direction we do not think this is a worthwhile exercise.
We repeated the fit using the spectroscopic sample leaving out the lowest, and second lowest redshift bins, since these redshift bins may contain LRGs that are not at rest with respect to the CMB. As Fig.~\ref{fig:th} shows these are also the bins which should contain the most signal. We find that when leaving out these bins some of the constraint on the declination direction of the flow is lost but the right ascension results are robust. We also found that the magnitude weighted right ascension directional fit also agrees with our binned analysis well.
Since our results indicate that our local group is moving with respect to the SDSS LRG sample, we infer that the source of our local motion is from scales comparable to the LRG scale, $z=\sl O(0.1-0.2)$. This result agrees with \citet{FelWatHud10} that found that the sources responsible for the bulk flow are at an effective distance of $\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 300$ $h^{-1}$Mpc, {\it i.e. } well within the horizon, and contradicting the suggestion of a coherent flow on much larger scales claimed by \citet{KasAtrKoc08, KasAtrKoc10} and modeled as a {\it tilted} Universe \citep{KasAtrKoc08,MaGorFel11}. A note of caution must be made here, the uncertainties and systematic effects on the magnitude of the flow suggested here, precludes anything more than a hint of a subhorizon flow. We have presented results using only the $r$ band data, using either the $g$ or $i$ band data instead does not significantly alter our conclusions.
\cite{ItYaTa10} used galaxy catalogs (not just the LRGs) constructed from the SDSS Data Release 6 and looked at the variations in the pixelised number counts, a similar approach to that presented in this paper. They found the probability of $\beta$ ($\approx v_{bulk}/c$) was consistent with a zero bulk flow, but also was not inconsistent with flows with $\beta\simeq0.01$. In fact, the galaxy sub-samples closest to those used here (termed Northern Galactic Hemisphere - shallow samples) find values of $\beta\geq\sim0.02$, and are only marginally consistent with zero. This indicates that the analysis both presented here and in \cite{ItYaTa10} may be seeing the same effect in the SDSS data: either the signature of a large scale flow or a systematic in the data.
However, in a similar method to \cite{ItYaTa10}, \cite{BlakeWall02} analyzed the radio galaxy distribution from the NRAO VLA Sky Survey and detected a cosmological dipole anisotropy that is consistent with the CMB dipole in both amplitude and direction.
We also note that a non-negligible detection of a fluctuation in galaxy magnitudes should also show up as a detection of excess large scale power in the clustering of galaxies. \cite{ThoAbdLah11} used a similar data set to the one employed here, the MegaZ DR7 photometric LRG sample, and found excess power on large scales of between 2-4$\sigma$ significance by looking at the angular clustering of the LRGs.
\cite{NusBraDav11} also used galaxy luminosities to study the bulk flow. Although they found a bulk flow with a similar direction to that found here, and by other authors, they do not find the flow to have an anomalously large magnitude as compared to that expected from $\Lambda$CDM. This contention between different measures of the bulk flow using kSZ, galaxy distances, galaxy luminosities shows that further studies of our local velocity field are vital to test the current paradigm.
Upcoming surveys should be able to confirm or repudiate this detection. The Large Synoptic Survey Telescope \citep[LSST]{SciBook} is potentially the best candidate for two reasons: its ability to discover of order 50,000 photometric SNIa per year, and its wide area coverage (20,000 sq.deg.). Most of these SNIa will not be able to be followed up spectroscopically, however it is expected that photometric redshifts of SNIa observed by LSST could reach a precision of just 1\%. Ideally a multi-wavelength approach, which would necessarily involve different targets and different instruments, would provide the strongest evidence in confirming the existence of this signal. Further, the whole sky data from WISE (Wide-field Infrared Survey Explorer) satellite \citep{WISE10} will become available in the next few years. It will be sensitive to 10\% overdensities out to $\sim 700$$h^{-1}$Mpc\ ($z\sim 0.2$) and although it does not measure redshifts, it may be used to search for over-- and under--densities in the direction and anti--direction of the flow.
\\
\noindent{\bf Acknowledgement:}
HAF was supported in part by an NSF grant AST-0807326 and by the University of Kansas GRF and would like to thank Sarah Bridle for useful conversations.
AA would like to thank Tim Axelrod, Daniel Eisenstein, Mike Hudson, Eyal Kazin, Marc Metchnik and Brian Schmidt for their useful input and suggestions.
\bibliographystyle{mn2e}
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import hzarrabi_CSCI201_Assignment3.CantAddShipException;
import java.awt.BorderLayout;
import java.awt.Color;
import java.awt.Component;
import java.awt.Dimension;
import java.awt.Event;
import java.awt.FlowLayout;
import java.awt.Graphics;
import java.awt.GridLayout;
import java.awt.Image;
import java.awt.MenuBar;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;
import java.awt.event.KeyEvent;
import java.awt.event.MouseEvent;
import java.awt.event.MouseListener;
import java.awt.image.BufferedImage;
import java.io.BufferedReader;
import java.io.File;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.IOException;
import java.util.Random;
import javax.imageio.ImageIO;
import javax.sound.sampled.AudioInputStream;
import javax.sound.sampled.AudioSystem;
import javax.sound.sampled.Clip;
import javax.sound.sampled.LineUnavailableException;
import javax.sound.sampled.UnsupportedAudioFileException;
import javax.swing.BorderFactory;
import javax.swing.Box;
import javax.swing.BoxLayout;
import javax.swing.ButtonGroup;
import javax.swing.Icon;
import javax.swing.ImageIcon;
import javax.swing.JButton;
import javax.swing.JComboBox;
import javax.swing.JComponent;
import javax.swing.JDialog;
import javax.swing.JFileChooser;
import javax.swing.JFrame;
import javax.swing.JLabel;
import javax.swing.JMenu;
import javax.swing.JMenuBar;
import javax.swing.JMenuItem;
import javax.swing.JPanel;
import javax.swing.JRadioButton;
import javax.swing.JScrollPane;
import javax.swing.JTextArea;
import javax.swing.KeyStroke;
import javax.swing.filechooser.FileNameExtensionFilter;
import javax.swing.text.html.BlockView;
import javax.swing.Timer;
public class BattleShip extends JFrame
{
private JComponent leftGrid[][]=new JComponent[11][11];
private JComponent rightGrid[][]=new JComponent[11][11];
private char compGrid[][]=new char[10][10];//this is the one we click on
private char userGrid[][]=new char[10][10];//this is the one the computer guesses
JPanel left;
JPanel right;
String playersAim="N/A";
String computersAim="N/A";
JButton selectFileButton=new JButton("Select File...");
JLabel fileName=new JLabel("File: ");
JButton startButton = new JButton("START");
//for the ship placement
int carriers=0;
int battlships=0;
int cruisers=0;
int destroyers=0;
//ships hit
int playerCarriers=0;
int playerBattlships=0;
int playerCruisers=0;
int playerDestroyers=0;
int compCarriers=0;
int compBattlships=0;
int compCruisers=0;
int compDestroyers=0;
//bools for different modes of game
boolean selectedFile=false;
boolean editMode=true;
//images
private ImageIcon wave=new ImageIcon("wave.jpg");
private ImageIcon miss=new ImageIcon("x.jpg");
private ImageIcon hit=new ImageIcon("hit.jpg");
private ImageIcon aship=new ImageIcon("AShip.jpg");
private ImageIcon bship=new ImageIcon("BShip.jpg");
private ImageIcon cship=new ImageIcon("CShip.jpg");
private ImageIcon dship=new ImageIcon("DShip.jpg");
//int for how many hit each side had taken
int compHits=0;//so if this equals 16 that means that the USER won
int userHits=0;
//menus
JMenuBar menuBar = new JMenuBar();
JMenu fileMenu=new JMenu("Info");
JMenuItem howToMenu = new JMenuItem("How To");
JMenuItem aboutMenu=new JMenuItem("About");
//timer and log
JLabel timeLabel= new JLabel("0:15");
int seconds=15;
Timer time;
boolean playerShot=false;//the boolean that indicates if the player shot
boolean compShot=false;
int computerSeconds=12;//this will be the random time assigned to the computer's turn
JTextArea log =new JTextArea();
JScrollPane scroll = new JScrollPane (log, JScrollPane.VERTICAL_SCROLLBAR_ALWAYS, JScrollPane.HORIZONTAL_SCROLLBAR_NEVER);
JPanel south=new JPanel(new BorderLayout());//holds buttons to file and start
int round=1;
//animations
BufferedImage wave1;
BufferedImage wave2;
BufferedImage expl1;
BufferedImage expl2;
BufferedImage expl3;
BufferedImage expl4;
BufferedImage expl5;
BufferedImage[] expl = new BufferedImage[5];
//--
BufferedImage splash1;
BufferedImage splash2;
BufferedImage splash3;
BufferedImage splash4;
BufferedImage splash5;
BufferedImage splash6;
BufferedImage splash7;
BufferedImage[] splash = new BufferedImage[7];
//--
BufferedImage imageA;
BufferedImage imageB;
BufferedImage imageC;
BufferedImage imageD;
BufferedImage imageM;
BufferedImage imageQ;
BufferedImage imageX;
SoundLibrary sl = new SoundLibrary();
public BattleShip()
{
load();
fillUserGrid();//this will instantiate userArray with X's
setTitle("BattleShip");
setLayout(new BorderLayout());
setSize(690,460);
setLocationRelativeTo(null);
setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
JPanel north=new JPanel(new FlowLayout(FlowLayout.CENTER));
north.setAlignmentX(100);
north.add(new JLabel("PLAYER "));
north.add(timeLabel);
north.add(new JLabel(" COMPUTER"));
add(north,BorderLayout.NORTH);
JPanel center=new JPanel(new FlowLayout());//center holds the left and right grids
left=new JPanel(new GridLayout(11, 11));
left.setBorder(BorderFactory.createLineBorder(Color.BLACK));
setLeftGrid();
right=new JPanel(new GridLayout(11, 11));
right.setBorder(BorderFactory.createLineBorder(Color.BLACK));
setRightGrid();
gridLabelListener();
center.add(left);
center.add(right);
add(center,BorderLayout.CENTER);
//JPanel south=new JPanel(new BorderLayout());//holds buttons to file and start
JPanel southLeft=new JPanel(new FlowLayout(FlowLayout.LEFT));
JPanel southRight=new JPanel(new FlowLayout(FlowLayout.LEFT));
southRight.add(selectFileButton);
southRight.add(fileName);
southRight.add(startButton);
startButton.setEnabled(false);
south.add(southLeft,BorderLayout.WEST);
south.add(southRight,BorderLayout.EAST);
add(south,BorderLayout.SOUTH);
selectFileListener();
startButtonListener();
JMenuBar menuBar = new JMenuBar();
setJMenuBar(menuBar);
menuBar.add(fileMenu);
fileMenu.setMnemonic('I');
fileMenu.add(howToMenu);
howToMenu.setAccelerator(KeyStroke.getKeyStroke(KeyEvent.VK_H, Event.CTRL_MASK));
howToMenu.setMnemonic('h');
fileMenu.add(aboutMenu);
aboutMenu.setAccelerator(KeyStroke.getKeyStroke(KeyEvent.VK_A, Event.CTRL_MASK));
aboutMenu.setMnemonic('a');
menuListeners();
left.setBackground(Color.cyan);
right.setBackground(Color.cyan);
setResizable(false);
setVisible(true);
}
//loads images and sounds etc
private void load()
{
try
{
wave1=ImageIO.read(new File("4Resources/animatedWater/water1.png"));
wave2=ImageIO.read(new File("4Resources/animatedWater/water2.png"));
for(int i=0;i<5;i++)
{
expl[i]=ImageIO.read(new File("4Resources/explosion/expl"+(i+1)+".png"));
}
for(int i=0;i<7;i++)
{
splash[i]=ImageIO.read(new File("4Resources/splash/splash"+(i+1)+".png"));
}
imageA=ImageIO.read(new File("4Resources/Tiles/A.png"));
imageB=ImageIO.read(new File("4Resources/Tiles/B.png"));
imageC=ImageIO.read(new File("4Resources/Tiles/C.png"));
imageD=ImageIO.read(new File("4Resources/Tiles/D.png"));
imageM=ImageIO.read(new File("4Resources/Tiles/M.png"));
imageQ=ImageIO.read(new File("4Resources/Tiles/Q.png"));
imageX=ImageIO.read(new File("4Resources/Tiles/X.png"));
}
catch(IOException ioe)
{
}
}
//timer action
private void timerAction()
{
timeLabel.setText("0:15");//reseting the label for when a players makes their decision
//this is the actionlistener for the timer
ActionListener timePerformer = new ActionListener() {
public void actionPerformed(ActionEvent evt) {
seconds--;
if(seconds<10)//if our time is 9 or less we have one digit seconds so we need to account for that
{
timeLabel.setText("0:0"+seconds);
if(seconds==3) log.append("Warning - 3 seconds left in the round!\n");
}
else timeLabel.setText("0:"+seconds);
if(seconds==0)
{
seconds=15;
timeLabel.setText("0:15");
if(playerShot==false)log.append("You ran out of time!\n");//player ran out of time
if(compShot==false) log.append("Computer ran out of time\n");//computer ran out of time
compShot=false;
playerShot=false;
round++;
log.append("Round "+ round + "\n");
int randomNum = new Random().nextInt((10 - 0) + 1) + 0;
//within 10 seconds(60% chance)
if(randomNum>=4)computerSeconds=new Random().nextInt((14 - 8) + 1) + 8;//15-8 seconds
//11-25 seconds(30% chance)
else if(randomNum<4 && randomNum>=1)computerSeconds=new Random().nextInt((7 - 0) + 1) + 0;
//>25 seconds (20% chance)
else computerSeconds=-1;//since comp will run out of time under this case we make it 26
}
else
{
//when computer "decides" to make turn
if(compShot==false && computerSeconds==seconds)
{
compShooter();
}
}
}
};
//instantiating the timer to perform every 1000 milliseconds (or 1 sec)
time=new Timer(1000,timePerformer);
time.start();
}
//action listeners for the menus
private void menuListeners()
{
howToMenu.addActionListener(new ActionListener()
{
@Override
public void actionPerformed(ActionEvent e)
{
new aboutWindow();
}
});
aboutMenu.addActionListener(new ActionListener()
{
@Override
public void actionPerformed(ActionEvent e)
{
new aboutWindow2();
}
});
}
//fills the userGrid array with x's
private void fillUserGrid()
{
for(int i=0;i<10;i++)
{
for(int j=0;j<10;j++)
{
userGrid[i][j]='X';
}
}
}
//makes the ?s for the left grid
private void setLeftGrid()
{
for (int j=0;j<11;j++)
{
for(int i=0;i<11;i++)
{
leftGrid[i][j]=new GridLabel(i,j+1);
if(i>0 && j<10)
{
//leftGrid[i][j].setIcon(wave);//initialize all question marks initially
}
}
}
((GridLabel)leftGrid[0][0]).add(new JLabel("A"));
((GridLabel)leftGrid[0][0]).press=false;
((GridLabel)leftGrid[0][1]).add(new JLabel("B"));
((GridLabel)leftGrid[0][1]).press=false;
((GridLabel)leftGrid[0][2]).add(new JLabel("C"));
((GridLabel)leftGrid[0][2]).press=false;
((GridLabel)leftGrid[0][3]).add(new JLabel("D"));
((GridLabel)leftGrid[0][3]).press=false;
((GridLabel)leftGrid[0][4]).add(new JLabel("E"));
((GridLabel)leftGrid[0][4]).press=false;
((GridLabel)leftGrid[0][5]).add(new JLabel("F"));
((GridLabel)leftGrid[0][5]).press=false;
((GridLabel)leftGrid[0][6]).add(new JLabel("G"));
((GridLabel)leftGrid[0][6]).press=false;
((GridLabel)leftGrid[0][7]).add(new JLabel("H"));
((GridLabel)leftGrid[0][7]).press=false;
((GridLabel)leftGrid[0][8]).add(new JLabel("I"));
((GridLabel)leftGrid[0][8]).press=false;
((GridLabel)leftGrid[0][9]).add(new JLabel("J"));
((GridLabel)leftGrid[0][9]).press=false;
((GridLabel)leftGrid[0][10]).add(new JLabel(" "));
((GridLabel)leftGrid[0][10]).press=false;
((GridLabel)leftGrid[1][10]).add(new JLabel("1"));
((GridLabel)leftGrid[1][10]).press=false;
((GridLabel)leftGrid[2][10]).add(new JLabel("2"));
((GridLabel)leftGrid[2][10]).press=false;
((GridLabel)leftGrid[3][10]).add(new JLabel("3"));
((GridLabel)leftGrid[3][10]).press=false;
((GridLabel)leftGrid[4][10]).add(new JLabel("4"));
((GridLabel)leftGrid[4][10]).press=false;
((GridLabel)leftGrid[5][10]).add(new JLabel("5"));
((GridLabel)leftGrid[5][10]).press=false;
((GridLabel)leftGrid[6][10]).add(new JLabel("6"));
((GridLabel)leftGrid[6][10]).press=false;
((GridLabel)leftGrid[7][10]).add(new JLabel("7"));
((GridLabel)leftGrid[7][10]).press=false;
((GridLabel)leftGrid[8][10]).add(new JLabel("8"));
((GridLabel)leftGrid[8][10]).press=false;
((GridLabel)leftGrid[9][10]).add(new JLabel("9"));
((GridLabel)leftGrid[9][10]).press=false;
((GridLabel)leftGrid[10][10]).add(new JLabel("10"));
((GridLabel)leftGrid[10][10]).press=false;
//adding the labels
for (int j=0;j<11;j++)
{
for(int i=0;i<11;i++)
{
left.add(leftGrid[i][j]);
}
}
}
//makes the ?s for the right grid
private void setRightGrid()
{
for (int j=0;j<11;j++)
{
for(int i=0;i<11;i++)
{
rightGrid[i][j]=new GridLabel(i,j+1);
if(i>0 && j<10)
{
//rightGrid[i][j].setIcon(wave);//initialize all question marks initially
}
}
}
((GridLabel)rightGrid[0][0]).add(new JLabel("A"));
((GridLabel)rightGrid[0][0]).press=false;
((GridLabel)rightGrid[0][1]).add(new JLabel("B"));
((GridLabel)rightGrid[0][1]).press=false;
((GridLabel)rightGrid[0][2]).add(new JLabel("C"));
((GridLabel)rightGrid[0][2]).press=false;
((GridLabel)rightGrid[0][3]).add(new JLabel("D"));
((GridLabel)rightGrid[0][3]).press=false;
((GridLabel)rightGrid[0][4]).add(new JLabel("E"));
((GridLabel)rightGrid[0][4]).press=false;
((GridLabel)rightGrid[0][5]).add(new JLabel("F"));
((GridLabel)rightGrid[0][5]).press=false;
((GridLabel)rightGrid[0][6]).add(new JLabel("G"));
((GridLabel)rightGrid[0][6]).press=false;
((GridLabel)rightGrid[0][7]).add(new JLabel("H"));
((GridLabel)rightGrid[0][7]).press=false;
((GridLabel)rightGrid[0][8]).add(new JLabel("I"));
((GridLabel)rightGrid[0][8]).press=false;
((GridLabel)rightGrid[0][9]).add(new JLabel("J"));
((GridLabel)rightGrid[0][9]).press=false;
((GridLabel)rightGrid[0][10]).add(new JLabel(" "));
((GridLabel)rightGrid[0][10]).press=false;
((GridLabel)rightGrid[1][10]).add(new JLabel("1"));
((GridLabel)rightGrid[1][10]).press=false;
((GridLabel)rightGrid[2][10]).add(new JLabel("2"));
((GridLabel)rightGrid[2][10]).press=false;
((GridLabel)rightGrid[3][10]).add(new JLabel("3"));
((GridLabel)rightGrid[3][10]).press=false;
((GridLabel)rightGrid[4][10]).add(new JLabel("4"));
((GridLabel)rightGrid[4][10]).press=false;
((GridLabel)rightGrid[5][10]).add(new JLabel("5"));
((GridLabel)rightGrid[5][10]).press=false;
((GridLabel)rightGrid[6][10]).add(new JLabel("6"));
((GridLabel)rightGrid[6][10]).press=false;
((GridLabel)rightGrid[7][10]).add(new JLabel("7"));
((GridLabel)rightGrid[7][10]).press=false;
((GridLabel)rightGrid[8][10]).add(new JLabel("8"));
((GridLabel)rightGrid[8][10]).press=false;
((GridLabel)rightGrid[9][10]).add(new JLabel("9"));
((GridLabel)rightGrid[9][10]).press=false;
((GridLabel)rightGrid[10][10]).add(new JLabel("10"));
((GridLabel)rightGrid[10][10]).press=false;
//adding the labels
for (int j=0;j<11;j++)
{
for(int i=0;i<11;i++)
{
right.add(rightGrid[i][j]);
}
}
}
private String getCharForNumber2(int i) {
return i > 0 && i < 27 ? String.valueOf((char)(i + 64)) : null;
}
//action listener for the gridlabels
private void gridLabelListener()
{
//listener for right grid
for(int i=0;i<11;i++)
{
for (int j=0;j <11; j++)
{
final int i1=i;
final int j1=j;
rightGrid[i][j].addMouseListener(new MouseListener()
{
public void mouseClicked(MouseEvent e)
{
if(((GridLabel)rightGrid[i1][j1]).press)
{
if(editMode==true || playerShot==true)
{
//do nothing to the right grid in edit mode or not the players turn!!
}
else//when we're in playing mode then we want to play!!!!
{
if(compGrid[((GridLabel)rightGrid[i1][j1]).x-1][((GridLabel)rightGrid[i1][j1]).y-1]!='X' && compGrid[((GridLabel)rightGrid[i1][j1]).x-1][((GridLabel)rightGrid[i1][j1]).y-1]!='O')//you hit a ship!!
{
String label=Character.toString(compGrid[((GridLabel)rightGrid[i1][j1]).x-1][((GridLabel)rightGrid[i1][j1]).y-1]);
((GridLabel)rightGrid[i1][j1]).explode('X', true);
char theChar=compGrid[((GridLabel)rightGrid[i1][j1]).x-1][((GridLabel)rightGrid[i1][j1]).y-1];
compGrid[((GridLabel)rightGrid[i1][j1]).x-1][((GridLabel)rightGrid[i1][j1]).y-1]='O';
compHits++;
Boolean append=true;
String theShip="";
if(theChar=='A')
{
playerCarriers++;
theShip="AirCraft";
if(playerCarriers==5)
{
log.append("Player sank an AircraftCarrier!\n");
playerCarriers++;
append=false;
}
}
else if(theChar=='B')
{
playerBattlships++;
theShip="BattleShip";
if(playerBattlships==4)
{
log.append("Player sank a BattleShip!\n");
playerBattlships++;
append=false;
}
}
else if(theChar=='C')
{
playerCruisers++;
theShip="Carrier";
if(playerCruisers==3)
{
log.append("Player sank a Cruiser!\n");
playerCruisers++;
append=false;
}
}
else if(theChar=='D')
{
playerDestroyers++;
theShip="Destroyer";
if(playerDestroyers==2)
{
log.append("Player sank a Carrier!\n");
playerDestroyers=0;
append=false;
}
}
String theSecond="0:";
if(seconds<10)theSecond="0:0";
else theSecond="0:";
if(append)log.append("Player hit "+getCharForNumber2(j1+1)+i1+" and hit a "+theShip+"!("+theSecond+seconds+")\n");
playersAim= getCharForNumber2(((GridLabel)rightGrid[i1][j1]).y)+((GridLabel)rightGrid[i1][j1]).x;
playerShot=true;
if(compHits>=16)
{
time.stop();
new winnerWindow("You");
}
else
{
playerShot=true;//making it the computer's turn now, player clicks disabled
if(compShot==true)//if computer has already aimed new round
{
seconds=15;//reseting the timer
timeLabel.setText("0:15");
round++;
log.append("Round "+round+"\n");
int randomNum = new Random().nextInt((10 - 0) + 1) + 0;
//within 10 seconds(60% chance)
if(randomNum>=4)computerSeconds=new Random().nextInt((14 - 8) + 1) + 8;//15-8 seconds
//11-25 seconds(30% chance)
else if(randomNum<4 && randomNum>=1)computerSeconds=new Random().nextInt((7 - 0) + 1) + 0;
//>25 seconds (20% chance)
else computerSeconds=-1;//since comp will run out of time under this case we make it 26
//compShooter();//if we haven't won then the computer shoots
compShot=false;
playerShot=false;
}
}
}
else if(compGrid[((GridLabel)rightGrid[i1][j1]).x-1][((GridLabel)rightGrid[i1][j1]).y-1]=='X')//you did miss
{
((GridLabel)rightGrid[i1][j1]).explode('M', true);
compGrid[((GridLabel)rightGrid[i1][j1]).x-1][((GridLabel)rightGrid[i1][j1]).y-1]='O';
playersAim= getCharForNumber2(((GridLabel)rightGrid[i1][j1]).y)+((GridLabel)rightGrid[i1][j1]).x;
String theTime="0:";
if(seconds<10) theTime="0:0";
log.append("You missed!("+theTime+seconds+")\n");
playerShot=true;//player shot so make true
if(compShot==true)//if computer has already aimed new round
{
round++;
log.append("Round "+round+"\n");
seconds=15;//reseting the timer
timeLabel.setText("0:15");
int randomNum = new Random().nextInt((10 - 0) + 1) + 0;
//within 10 seconds(60% chance)
if(randomNum>=4)computerSeconds=new Random().nextInt((14 - 8) + 1) + 8;//15-8 seconds
//11-25 seconds(30% chance)
else if(randomNum<4 && randomNum>=1)computerSeconds=new Random().nextInt((7 - 0) + 1) + 0;
//>25 seconds (20% chance)
else computerSeconds=-1;//since comp will run out of time under this case we make it 26
//compShooter();//if we haven't won then the computer shoots
compShot=false;
playerShot=false;
}
}
else log.append("You've already aimed here! Aim again!\n");
}
}
}
public void mouseEntered(MouseEvent e){}
public void mouseExited(MouseEvent e){}
public void mousePressed(MouseEvent e){}
public void mouseReleased(MouseEvent e){}
});
}
}
//listener for left grid
for(int i=0;i<11;i++)
{
for (int j=0;j <11; j++)
{
final int i1=i;
final int j1=j;
leftGrid[i][j].addMouseListener(new MouseListener()
{
public void mouseClicked(MouseEvent e)
{
if(((GridLabel)leftGrid[i1][j1]).press)
{
if(editMode==true)//we only want this functionality if we are in edit mode
{
if(userGrid[((GridLabel)leftGrid[i1][j1]).x-1][((GridLabel)leftGrid[i1][j1]).y-1]=='X')//if the coordinate has no ship placed
{
if(carriers+battlships+cruisers+destroyers<5)//if we still have ships to place open the window
new shipPlacerWindow(((GridLabel)leftGrid[i1][j1]).x,((GridLabel)leftGrid[i1][j1]).y);
}
else
{
shipDeleter(((GridLabel)leftGrid[i1][j1]).x-1, ((GridLabel)leftGrid[i1][j1]).y-1);;
}
}
else//if we are in playing mode do this
{
//we shouldn't be able to do anything to the left grid but i'll just do this incase
}
}
}
public void mouseEntered(MouseEvent e){}
public void mouseExited(MouseEvent e){}
public void mousePressed(MouseEvent e){}
public void mouseReleased(MouseEvent e){}
});
}
}
}
//this is the function for the computer guessing coordinates
private void compShooter()
{
Random rand=new Random();
int x=rand.nextInt(9 - 0 + 1) + 0;
int y=rand.nextInt(9 - 0 + 1) + 0;
if(userGrid[x][y]!='X' && userGrid[x][y]!='O')//if comp hits a target
{
((GridLabel)leftGrid[x+1][y]).explode('X', true);;
char theChar=userGrid[x][y];
userGrid[x][y]='O';//marking that computer shot here
userHits++;
computersAim= getCharForNumber2(y+1)+(x+1);
Boolean append=true;
String theShip="";
if(theChar=='A')
{
compCarriers++;
theShip="AirCraft";
if(compCarriers==5)
{
log.append("Computer sank an AircraftCarrier!\n");
compCarriers++;
append=false;
}
}
else if(theChar=='B')
{
compBattlships++;
theShip="BattleShip";
if(compBattlships==4)
{
log.append("Computer sank a BattleShip!\n");
compBattlships++;
append=false;
}
}
else if(theChar=='C')
{
compCruisers++;
theShip="Carrier";
if(compCarriers==3)
{
log.append("Computer sank a Cruiser!\n");
compCruisers++;
append=false;
}
}
else if(theChar=='D')
{
compDestroyers++;
theShip="Destroyer";
if(compDestroyers==2)
{
log.append("Computer sank a Carrier!\n");
compCarriers=0;
append=false;
}
}
String theSecond="0:";
if(seconds<10)theSecond="0:0";
else theSecond="0:";
if(append)log.append("Computer hit "+getCharForNumber2(y+1)+(x+1)+" and hit a "+theShip+"!("+theSecond+seconds+")\n");
compShot=true;
if(userHits==16)//if the computer has hit all ships
{
time.stop();
new winnerWindow("Computer");
}
else
{
compShot=true;//player's turn otherwise
if(playerShot==true)//if the player has shot too
{
seconds=15;
timeLabel.setText("0:15");
int randomNum = new Random().nextInt((10 - 0) + 1) + 0;
//within 10 seconds(60% chance)
if(randomNum>=4)computerSeconds=new Random().nextInt((14 - 8) + 1) + 8;//15-8 seconds
//11-25 seconds(30% chance)
else if(randomNum<4 && randomNum>=1)computerSeconds=new Random().nextInt((7 - 0) + 1) + 0;
//>25 seconds (20% chance)
else computerSeconds=-1;//since comp will run out of time under this case we make it 26
//compShooter();//if we haven't won then the computer shoots
compShot=false;
playerShot=false;
round++;
log.append("Round "+ round +"\n");
}
}
}
else if(userGrid[x][y]=='X')//if the computer misses
{
userGrid[x][y]='O';
((GridLabel)leftGrid[x+1][y]).explode('M',true);
computersAim= getCharForNumber2(y+1)+(x+1);
compShot=true;
String theTime="0:";
if(seconds<10) theTime="0:0";
log.append("Computer missed!("+theTime+seconds+")\n");
if(playerShot==true)//if the player has shot too
{
seconds=15;
timeLabel.setText("0:15");
int randomNum = new Random().nextInt((10 - 0) + 1) + 0;
//within 10 seconds(60% chance)
if(randomNum>=4)computerSeconds=new Random().nextInt((14 - 8) + 1) + 8;//15-8 seconds
//11-25 seconds(30% chance)
else if(randomNum<4 && randomNum>=1)computerSeconds=new Random().nextInt((7 - 0) + 1) + 0;
//>25 seconds (20% chance)
else computerSeconds=-1;//since comp will run out of time under this case we make it 26
//compShooter();//if we haven't won then the computer shoots
compShot=false;
playerShot=false;
round++;
log.append("Round "+ round +"\n");
}
}
else//computer hit's something already hit
{
compShooter();
}
}
//ship deleter
private void shipDeleter(int x, int y)
{
int range=0;
char shipWeWant='X';
if(userGrid[x][y]=='A')
{
range=5;
shipWeWant='A';
carriers--;
}
else if(userGrid[x][y]=='B')
{
range=4;
shipWeWant='B';
battlships--;
}
else if(userGrid[x][y]=='C')
{
range=3;
shipWeWant='C';
cruisers--;
}
else if(userGrid[x][y]=='D')
{
range=2;
shipWeWant='D';
destroyers--;
}
else if(userGrid[x][y]=='E')
{
range=2;
shipWeWant='E';
destroyers--;
}
//checking north
if(range<=y+1)//if there is room in the north for the full ship check for it
{
for(int i=0;i<range;i++)
{
if(userGrid[x][y-i]==shipWeWant)//if it is the ship we want
{
userGrid[x][y-i]='X';
leftGrid[x+1][y-i].removeAll();
((GridLabel)leftGrid[x+1][y-i]).add(new JLabel(new ImageIcon(imageQ)));
}
else break;//if it's not stop searching the north
}
}
else //if there isn't room we still want to check incase we pressed the middle of a ship but don't want to go out of bound
{
for(int i=0;i<y+1;i++)
{
if(userGrid[x][y-i]==shipWeWant)//if it is the ship we want
{
userGrid[x][y-i]='X';
leftGrid[x+1][y-i].removeAll();
((GridLabel)leftGrid[x+1][y-i]).add(new JLabel(new ImageIcon(imageQ)));
}
else break;
}
}
//checking south
if(range+(y+1)<11)
{
for(int i=0;i<range;i++)
{
if(userGrid[x][y+i]==shipWeWant)//if it is the ship we want
{
userGrid[x][y+i]='X';
leftGrid[x+1][y+i].removeAll();
((GridLabel)leftGrid[x+1][y+i]).add(new JLabel(new ImageIcon(imageQ)));
}
}
}
else //if there isn't room we still want to check incase we pressed the middle of a ship but don't want to go out of bound
{
for(int i=0;i<10-y;i++)
{
if(userGrid[x][y+i]==shipWeWant)//if it is the ship we want
{
userGrid[x][y+i]='X';
leftGrid[x+1][y+i].removeAll();
((GridLabel)leftGrid[x+1][y+i]).add(new JLabel(new ImageIcon(imageQ)));
}
}
}
//checking west
if(range<=x+1)
{
for(int i=0;i<range;i++)
{
if(userGrid[x-i][y]==shipWeWant)//if it is the ship we want
{
userGrid[x-i][y]='X';
leftGrid[x-i+1][y].removeAll();
((GridLabel)leftGrid[x-i+1][y]).add(new JLabel(new ImageIcon(imageQ)));
}
}
}
else //if there isn't room we still want to check incase we pressed the middle of a ship but don't want to go out of bound
{
for(int i=0;i<x+1;i++)
{
if(userGrid[x-i][y]==shipWeWant)//if it is the ship we want
{
userGrid[x-i][y]='X';
leftGrid[x-i+1][y].removeAll();
((GridLabel)leftGrid[x-i+1][y]).add(new JLabel(new ImageIcon(imageQ)));
}
}
}
//checking east
if(range+x<11)
{
for(int i=0;i<range;i++)
{
if(userGrid[x+i][y]==shipWeWant)//if it is the ship we want
{
userGrid[x+i][y]='X';
leftGrid[x+i+1][y].removeAll();
((GridLabel)leftGrid[x+i+1][y]).add(new JLabel(new ImageIcon(imageQ)));
}
}
}
else //if there isn't room we still want to check incase we pressed the middle of a ship but don't want to go out of bound
{
for(int i=0;i<10-x;i++)
{
if(userGrid[x+i][y]==shipWeWant)//if it is the ship we want
{
userGrid[x+i][y]='X';
leftGrid[x+i+1][y].removeAll();
((GridLabel)leftGrid[x+i+1][y]).add(new JLabel(new ImageIcon(imageQ)));
}
}
}
startButton.setEnabled(false);//if you delete a ship you can't start so disabling the button
}
private void startButtonListener()
{
startButton.addActionListener(new ActionListener()
{
@Override
public void actionPerformed(ActionEvent e)
{
selectFileButton.setVisible(false);
startButton.setVisible(false);
fileName.setText("");//delete the text instead of setting invisible because then i only have to reset in new game
editMode=false;
timerAction();//timer starts working once we press start
setSize(690,600);//changing the size of the frame for the log
log.setLineWrap(true);
log.setWrapStyleWord(true);
scroll.setPreferredSize(new Dimension(690, 150));
south.setBorder(BorderFactory.createTitledBorder("Game Log"));
south.add(scroll);
south.setVisible(true);
log.append("Round 1\n");
}
});
}
//action listener for select file
private void selectFileListener()
{
selectFileButton.addActionListener(new ActionListener()
{
@Override
public void actionPerformed(ActionEvent e)
{
if(e.getSource()==selectFileButton)//if we click that button
{
JFileChooser fc= new JFileChooser();
FileNameExtensionFilter filter = new FileNameExtensionFilter("Battle Files (*.battle)", "battle"); //filter to only allow .battle files
fc.setFileFilter(filter);//making our chooser take that filter
fc.setAcceptAllFileFilterUsed(false);//will only allow battle files
int returnVal=fc.showOpenDialog(selectFileButton);//opens up fileSelector
if(returnVal==fc.APPROVE_OPTION)//if we selected a file
{
selectFileButton.setVisible(false);//removing the select file button
//getting the fileName without extenstion to change JLabel
String fileName=fc.getSelectedFile().getName();
int pos = fileName.lastIndexOf(".");
if (pos > 0) {
fileName = fileName.substring(0, pos);
}
BattleShip.this.fileName.setText("File:" + fileName+ " ");
//reading the file
try
{
FileReader fr = new FileReader(fc.getSelectedFile());//make a file object for reading
BufferedReader br = new BufferedReader(fr); //make a buffer to go line by line
//reading in from the buffer
for(int j=0;j<10;j++)
{
String buffer = br.readLine();//reading in line
char[] charArray = buffer.toCharArray();//making it into char array
for(int i=0;i<10;i++)
{
compGrid[i][j]=charArray[i];
}
}
}
catch (FileNotFoundException e1)
{
//we know the file is there so don't worry
}
catch (IOException ioe)
{}
selectedFile=true;
if(carriers+battlships+cruisers+destroyers==5 && selectedFile)//if all ships places and file selected
{
startButton.setEnabled(true);
}
}
}
}
});
}
public class shipPlacerWindow extends JDialog
{
int x;//this will hold the x coordinate of what we're editing
int y;//this will hold y
private JComboBox<String> shipList = new JComboBox<String>();
private JRadioButton North= new JRadioButton("North");
private JRadioButton South= new JRadioButton("South");
private JRadioButton East= new JRadioButton("East");
private JRadioButton West= new JRadioButton("West");
private JButton placeShip=new JButton("Place Ship");
public shipPlacerWindow(int x,int y)
{
this.x=x-1;//making the coordinate into index value so u subtract by one
this.y=y-1;
setTitle("Select ship at "+ getCharForNumber(y)+x);
setSize(300,200);
setLocationRelativeTo(null);
setLayout(new BorderLayout());
setDefaultCloseOperation(JFrame.DISPOSE_ON_CLOSE);
//making the panel with the combo box
JPanel top = new JPanel();
top.add(new JLabel("Select Ship:"));
if(carriers==0)shipList.addItem("Aircraft Carrier");//if we haven't placed a carrier
if(battlships==0)shipList.addItem("Battleship");
if(cruisers==0)shipList.addItem("Cruiser");
if(destroyers<2)shipList.addItem("Destroyer");
top.add(shipList);
JPanel middle = new JPanel();
middle.setLayout(new GridLayout(2,2));
ButtonGroup directions = new ButtonGroup();
directions.add(North);//adding buttons to group
directions.add(South);
directions.add(East);
directions.add(West);
middle.add(North);
middle.add(South);
middle.add(East);
middle.add(West);
add(top,BorderLayout.NORTH);
add(middle,BorderLayout.CENTER);
add(placeShip,BorderLayout.SOUTH);
IsValid();//be careful you don't use isValid() because that's a another function that belongs to JFrame (this initially disables the button becase no radio buttons are selected
everyThingListener();//this is the listener for everything
setModal(true);//this prevents us from accessing the board behind it
setVisible(true);
}
//this function converts numbers to chars
private String getCharForNumber(int i) {
return i > 0 && i < 27 ? String.valueOf((char)(i + 64)) : null;
}
//this function is decides whether the button is valid or not
private void IsValid()
{
int range=0;//this is the range of the ship selected
placeShip.setEnabled(true);
try
{
String ship=(String) shipList.getSelectedItem();//returns what kind of ship is selected in combobox
//setting the range based on what kind of ship selected
if(ship.equals("Aircraft Carrier")) range=5;
else if(ship.equals("Battleship")) range=4;
else if(ship.equals("Cruiser")) range=3;
else if(ship.equals("Destroyer")) range=2;
if(North.isSelected())
{
if(range>y+1) throw new CantAddShipException();//checking to make sure we're not out of range
else
{
int yTest=y;
for(int i=0;i<range;i++)
{
if(userGrid[x][yTest]!='X')
{
throw new CantAddShipException();//throws an exception if there is a ship already in position
}
yTest--;
}
}
}
else if(South.isSelected())
{
if(range+y>10)
{
throw new CantAddShipException();
}
else
{
int yTest=y;
for(int i=0;i<range;i++)
{
if(userGrid[x][yTest]!='X') throw new CantAddShipException();//throws an exception if there is a ship already in position
yTest++;
}
}
}
else if(East.isSelected())
{
if(range+x>10)
{
throw new CantAddShipException();
}
else
{
int xTest=x;
for(int i=0;i<range;i++)
{
if(userGrid[xTest][y]!='X')
{
throw new CantAddShipException();//throws an exception if there is a ship already in position
}
xTest++;
}
}
}
else if(West.isSelected())
{
if(range>x+1)throw new CantAddShipException();//x+1 to make up for x being turned into index
else
{
int xTest=x;
for(int i=0;i<range;i++)
{
if(userGrid[xTest][y]!='X')
{
throw new CantAddShipException();//throws an exception if there is a ship already in position
}
xTest--;
}
}
}
else
{
throw new CantAddShipException();
}
}
catch (CantAddShipException e)
{
placeShip.setEnabled(false);//disabling the button if something is wrong
}
}
//this function adds all the listeners for to see if selection is valid and for button adding ship
private void everyThingListener()
{
//the listener for the JComboBox
shipList.addActionListener (new ActionListener () {
public void actionPerformed(ActionEvent e) {
IsValid();
}
});
North.addActionListener (new ActionListener () {
public void actionPerformed(ActionEvent e) {
IsValid();
}
});
South.addActionListener (new ActionListener () {
public void actionPerformed(ActionEvent e) {
IsValid();
}
});
West.addActionListener (new ActionListener () {
public void actionPerformed(ActionEvent e) {
IsValid();
}
});
East.addActionListener (new ActionListener () {
public void actionPerformed(ActionEvent e) {
IsValid();
}
});
//the listener for for the button that places the ships----------------------------------------
placeShip.addActionListener(new ActionListener()
{
@Override
public void actionPerformed(ActionEvent e)
{
int range=0;//this is the range of the ship selected
char shipCharacter='X';
String ship=(String) shipList.getSelectedItem();//returns what kind of ship is selected in combobox
//setting the range based on what kind of ship selected
ImageIcon theShip=wave;//image icon that will either be a b c or d
if(ship.equals("Aircraft Carrier"))
{
shipCharacter='A';
theShip=aship;
range=5;
carriers++;
}
else if(ship.equals("Battleship"))
{
shipCharacter='B';
theShip=bship;
range=4;
battlships++;
}
else if(ship.equals("Cruiser"))
{
shipCharacter='C';
theShip=cship;
range=3;
cruisers++;
}
else if(ship.equals("Destroyer"))
{
if(destroyers==0)
{
shipCharacter='D';
}
else if(destroyers==1)
{
shipCharacter='E';//i make it 'E' to be able to distinguish between the two destroyer ships
}
theShip=dship;
range=2;
destroyers++;
}
String shipString=Character.toString(shipCharacter);
//changing the texts in the grid and on the actual layout
if(North.isSelected())
{
int yTest=y;
for(int i=0;i<range;i++)
{
userGrid[x][yTest]=shipCharacter;
((GridLabel)leftGrid[x+1][yTest]).explode(shipCharacter,false);
yTest--;
}
}
else if(South.isSelected())
{
int yTest=y;
for(int i=0;i<range;i++)
{
userGrid[x][yTest]=shipCharacter;
((GridLabel)leftGrid[x+1][yTest]).explode(shipCharacter,false);
yTest++;
}
}
else if(East.isSelected())
{
int xTest=x;
for(int i=0;i<range;i++)
{
userGrid[xTest][y]=shipCharacter;
((GridLabel)leftGrid[xTest+1][y]).explode(shipCharacter,false);
xTest++;
}
}
else if(West.isSelected())
{
int xTest=x;
for(int i=0;i<range;i++)
{
userGrid[xTest][y]=shipCharacter;
((GridLabel)leftGrid[xTest+1][y]).explode(shipCharacter,false);
xTest--;
}
}
if(carriers+battlships+cruisers+destroyers==5 && selectedFile)//if they chose all ships
{
startButton.setEnabled(true);
}
shipPlacerWindow.this.dispose();//closes shipplacer window after you you place a ship
}
});
}
}
public class winnerWindow extends JDialog
{
private JButton okButton =new JButton("OK");
private ImageIcon wave1=new ImageIcon("wave.jpg");
public winnerWindow(String winner)
{
setTitle("Game Over");
setSize(300,150);
setLocationRelativeTo(null);
setLayout(new BoxLayout(getContentPane(),BoxLayout.Y_AXIS));
setDefaultCloseOperation(JFrame.DISPOSE_ON_CLOSE);
JLabel winner1=new JLabel(winner+ " won!");
winner1.setAlignmentX( Component.LEFT_ALIGNMENT);
add(winner1);
add(okButton);
okListener();
addWindowListener(new java.awt.event.WindowAdapter() {
@Override
public void windowClosing(java.awt.event.WindowEvent windowEvent) {
reset();
}
});
setModal(true);
setVisible(true);
}
public void okListener()
{
okButton.addActionListener(new ActionListener()
{
@Override
public void actionPerformed(ActionEvent arg0)
{
reset();
}
});
}
//resets the game
private void reset()
{
//reset right grid
for (int j=0;j<11;j++)
{
for(int i=0;i<11;i++)
{
if(i>0 && j<10)
{
rightGrid[i][j].removeAll();
rightGrid[i][j].add(new JLabel(new ImageIcon(imageQ)));
}
}
}
//reset left grid
for (int j=0;j<11;j++)
{
for(int i=0;i<11;i++)
{
if(i>0 && j<10)
{
leftGrid[i][j].removeAll();
leftGrid[i][j].add(new JLabel(new ImageIcon(imageQ)));
}
}
}
//reseting user and computer grid
for (int j=0;j<10;j++)
{
for(int i=0;i<10;i++)
{
userGrid[i][j]='X';
compGrid[i][j]='X';
}
}
playersAim="N/A";
computersAim="N/A";
selectFileButton.setVisible(true);;
fileName.setText("File: ");
startButton.setEnabled(false);
startButton.setVisible(true);
//for the ship placement
carriers=0;
battlships=0;
cruisers=0;
destroyers=0;
//bools for different modes of game
selectedFile=false;
editMode=true;
int compHits=0;//so if this equals 16 that means that the USER won
int userHits=0;
selectFileButton.setVisible(true);
startButton.setVisible(true);
fileName.setText("File");//delete the text instead of setting invisible because then i only have to reset in new game
editMode=true;
log.setText("");
scroll.setVisible(false);
south.setBorder(null);
BattleShip.this.setSize(690,460);
timeLabel.setText("0:15");
seconds=15;
playerShot=false;//the boolean that indicates if the player shot
compShot=false;
computerSeconds=12;//this will be the random time assigned to the computer's turn
round=1;
winnerWindow.this.dispose();
}
}
private class aboutWindow extends JDialog
{
JTextArea infoText=new JTextArea();
public aboutWindow()
{
setTitle("Battleship Instructions");
setSize(300,200);
setLocationRelativeTo(null);
setDefaultCloseOperation(JFrame.DISPOSE_ON_CLOSE);
JScrollPane sp = new JScrollPane(infoText);
add(sp);
try
{
BufferedReader in = new BufferedReader(new FileReader(new File("howTo.txt")));
String line = in.readLine();
while(line != null){
infoText.append(line + "\n");
line = in.readLine();
}
}
catch (FileNotFoundException e)
{
} catch (IOException e)
{
// TODO Auto-generated catch block
e.printStackTrace();
}
infoText.setEditable(false);
setModal(true);
setVisible(true);
}
}
private class aboutWindow2 extends JDialog
{
public aboutWindow2()
{
setTitle("About");
setSize(300,200);
setLocationRelativeTo(null);
setDefaultCloseOperation(JFrame.DISPOSE_ON_CLOSE);
add(new JLabel("Made by Hooman Zarrabi - 3/01/15"),BorderLayout.NORTH);
add(new JLabel(new ImageIcon("hooman.png")),BorderLayout.CENTER);
add(new JLabel("CSCI201USC: Assignment 3"),BorderLayout.SOUTH);
setModal(true);
setVisible(true);
}
}
private class GridLabel extends JComponent implements Runnable
{
public int x;
public int y;
public boolean press;
BufferedImage currentWave;
int current=0;
//for the explosion thread
Boolean explode;
int counter=0;
ImageIcon blastIcon;
char c;
public GridLabel(int x, int y)
{
this.x=x;
this.y=y;
press=true;
setPreferredSize(new Dimension(30,30));
setLayout(new FlowLayout());
if(x>0 && y<11)
{
setBorder(BorderFactory.createLineBorder(Color.black));
add(new JLabel(new ImageIcon(imageQ)));
new Thread(this).start();
}
}
@Override
protected void paintComponent(Graphics g)
{
super.paintComponent(g);
g.drawImage(currentWave,0,0,null);
};
@Override
public void run()
{
while (true)
{
if(current == 0)
{
currentWave=wave1;
current++;
}
else
{
currentWave=wave2;
current--;
}
validate();
repaint();
try { Thread.sleep(150); }
catch (InterruptedException e) { }
}
}
public void explode(char c, Boolean b)
{
counter=0;
explode=b;
this.c=c;
if(c=='E')this.c='D';
if(this.c=='A') blastIcon=new ImageIcon(imageA);
else if(this.c=='B') blastIcon=new ImageIcon(imageB);
else if(this.c=='C') blastIcon=new ImageIcon(imageC);
else if(this.c=='D') blastIcon=new ImageIcon(imageD);
else if(this.c=='M') blastIcon=new ImageIcon(imageM);
else if(this.c=='Q') blastIcon=new ImageIcon(imageQ);
else if(this.c=='X') blastIcon=new ImageIcon(imageX);
new Explosion().start();
new Sound().start();
}
public class Explosion extends Thread
{
public void run()
{
counter=0;
while(true)
{
if(c!='M')//we're not idicating a miss
{
if(explode)//if you want explode animation to happen
{
if(counter<5)
{
removeAll();//removes previous icons or labels
add(new JLabel(new ImageIcon(expl[counter])));
counter++;
}
else if(counter==5)
{
removeAll();
add(new JLabel(blastIcon));
counter++;
}
else
{
return;//stop thread
}
}
else
{
removeAll();
add(new JLabel(blastIcon));
counter=0;
return;//stop thread
}
}
else//we're indicating a miss
{
if(counter<7)
{
removeAll();
add(new JLabel(new ImageIcon(splash[counter])));
counter++;
}
else if(counter==7)
{
removeAll();
add(new JLabel(new ImageIcon(imageM)));
counter++;
}
else
{
return;
}
}
try
{sleep(300);}
catch (InterruptedException e)
{}
}
}
}
public class Sound extends Thread
{
String theString;
public Sound()
{
if(c=='M')
{
theString="splash";
}
else if(c=='X') theString="explode";
}
public void run()
{
if(explode)
{
sl.playSound("cannon");
sl.playSound(theString);
}
}
}
}
//==============================================================
public static void main(String[] args)
{
System.setProperty("java.util.Arrays.useLegacyMergeSort", "true");
new BattleShip();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 487
|
Please select a metro to find local Louisiana Employment lawyers.
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,128
|
Q: Timber - WooCoommerce email templates Hi I'm using Timber and I need to modify the WooCommerce email templates and I've run into a problem:
If I want to use the classical twig template scheme and I place it correctly to the views/woocommerce/emails/customer-processing-order.twig it works (=the template is loaded), but I'm missing the template variables needed for actions, namely: $order, $sent_to_admin, $plain_text, $email
My question is simple - how to get these or how to load PHP email templates instead of twig files in this case?
UPDATE
I have found out that for WC emails the things are loading bit differently and having a separate woocommerce/emails folder works fine, but for the others doesn't.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,595
|
{"url":"http:\/\/mathcentral.uregina.ca\/QQ\/database\/QQ.09.15\/h\/sam3.html","text":"SEARCH HOME\n Math Central Quandaries & Queries\n Question from Sam: I need to know how many pounds of glass pebbles are needed to fill a 24 inch across circular fire pit, if 5 pounds covers 4\"H x 4\"W x2\"D? Thank u for any assistance, Sam\n\nHi Sam,\n\nA solid region 4\" by 4\" by 2\" is $4 \\times 4 \\times 2 = 32$ cubic inches. You know that 32 cubic inches of the glass pebbles weighs 5 pounds so that's $\\large \\frac{5}{32} \\normalsize = 0.156$ pounds per cubic inch.\n\nYou didn't tell us how deep the fire pit is but using the depth and our volume calculator find the volume of the hole you need to fill, in cubic inches. Multiply he volume by 0.156 to obtain the weight of the glass pebbles required.\n\nPenny\n\nMath Central is supported by the University of Regina and the Imperial Oil Foundation.","date":"2020-07-12 17:21:07","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\": 0.23102718591690063, \"perplexity\": 1778.8688641520225}, \"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-2020-29\/segments\/1593657138752.92\/warc\/CC-MAIN-20200712144738-20200712174738-00429.warc.gz\"}"}
| null | null |
package com.github.theborakompanioni.gn.article.events;
public class ArticleHeartEvent {
private String articleId;
public String getArticleId() {
return articleId;
}
public void setArticleId(String articleId) {
this.articleId = articleId;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,325
|
Це́рковь Свято́й Екатери́ны () — самая большая лютеранская церковь во Франкфурте-на-Майне, построена в 1681 году в честь раннехристианской святой и мученицы Екатерины Александрийской. Церковь расположена в центре города на одной из самых известных площадей «Гауптвахта».
История
Строительство церкви было начато в 1678 году и закончено в 1681 году. Церковь была построена в стиле барокко, высотой 54 метра. Церковь была разрушена в результате жестоких бомбардировок города в 1944 году войсками союзников во время Второй мировой войны. Восстановление церкви происходило между 1950 и 1954 годами.
Ссылки
Сайт храма
Здания и сооружения, заложенные в 1678 году
Храмы, построенные в 1681 году
Появились в 1681 году в Германии
Воссозданные храмы Германии
Лютеранские храмы Германии
Святой Екатерины
Франкфурт
Здания и сооружения, построенные в XVII веке в Германии
1680-е годы в Германии
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,260
|
Q: XSLT 1.0 Convert delimited string to node set I have a variable $colors that is a string
<xsl:variable name="colors" select="'red,green,blue,'" />
I need a new variable, $colorElements that is a node-set
<color>red</color>
<color>green</color>
<color>blue</color>
(Is that right? Can a node-set have no root?)
$colorElements will never be output directly. I just need it as, effectively, a list variable.
XSLT 1.0 with no extensions other than node-set().
A: Use:
<xsl:stylesheet version="1.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:msxsl="urn:schemas-microsoft-com:xslt" exclude-result-prefixes="msxsl">
<xsl:output method="xml" indent="yes"/>
<xsl:variable name="colors" select="'red,green,blue,'" />
<xsl:template match="/">
<xsl:variable name="colorElements">
<xsl:call-template name="split">
<xsl:with-param name="pText" select="$colors"/>
</xsl:call-template>
</xsl:variable>
<xsl:for-each select="msxsl:node-set($colorElements)">
<xsl:copy-of select="color"/>
</xsl:for-each>
</xsl:template>
<xsl:template name="split">
<xsl:param name="pText"/>
<xsl:variable name="separator">,</xsl:variable>
<xsl:choose>
<xsl:when test="string-length($pText) = 0"/>
<xsl:when test="contains($pText, $separator)">
<color>
<xsl:value-of select="substring-before($pText, $separator)"/>
</color>
<xsl:call-template name="split">
<xsl:with-param name="pText" select="substring-after($pText, $separator)"/>
</xsl:call-template>
</xsl:when>
<xsl:otherwise>
<color>
<xsl:value-of select="$pText"/>
</color>
</xsl:otherwise>
</xsl:choose>
</xsl:template>
</xsl:stylesheet>
A: How about this?:
<?xml version="1.0"?>
<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform"
xmlns:xs="http://www.w3.org/2001/XMLSchema"
exclude-result-prefixes="xs">
<xsl:output method="xml" indent="yes" encoding="utf-8" />
<xsl:variable name="colors" select="'red,green,blue,'" />
<xsl:template match="/" name="main">
<csv-to-xml>
<xsl:for-each select="tokenize($colors, ',')[position()!=last()]">
<!-- The predicate is needed because of the extraneous comma
at the end of the red,green,blue, list. -->
<color><xsl:value-of select="." /></color>
</xsl:for-each>
</csv-to-xml>
</xsl:template>
</xsl:stylesheet>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,713
|
{"url":"https:\/\/www.gamedev.net\/forums\/topic\/683254-matrix-16-byte-alignment\/","text":"# Matrix 16 byte alignment\n\nThis topic is 631 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.\n\n## Recommended Posts\n\nHey\n\nIm having some trouble with the DirectX 11 matrix functions.\n\nNot 100%, but maybe 80% of the time im getting the following assertion:\n\n((uintptr_t)pSource & 0xF) == 0\n\nIt happens when i call the function:\n\nRelevant code\n\nclass Matrix4v4 {\npublic:\n__declspec(align(16))\u00a0float m[4][4];\n};\n\nvoid matrixMultiply( Matrix4v4 * p_Result, const Matrix4v4 * p_A, const Matrix4v4 * p_B ) {\nusing namespace DirectX;\nXMMATRIX a = XMLoadFloat4x4A( (const XMFLOAT4X4A *)p_A );\nXMMATRIX b = XMLoadFloat4x4A( (const XMFLOAT4X4A *)p_B );\nXMMATRIX r = XMMatrixMultiply( a, b );\nXMStoreFloat4x4A( (XMFLOAT4X4A *)p_Result, r );\n}\n\n\n\nThis must have something to do with the alignment of my data i take it.\n\nBut the above code is just like they do it in UE4.\n\nI dont know what to do really..\nIs it a compiler setting? (Im using vs2015)\n\n##### Share on other sites\n\nAh...\n\nNote the missing \"A\" at the end.\n\nMy problem have been resolved.\n\n##### Share on other sites\nYour problem was resolved by making your code run slower (potentially a _lot_ slower, depending on the particular CPU and how badly it deals with unaligned SSE loads\/stores).\n\nThe real problem is that you weren't allocating your Matrix4v4 objects with proper alignemtn. The code snippet you provided didn't show how those objects are being created, so it's impossible for me to say why specifically they were off.\n\nNote that if you're using malloc or new to allocate data that the alignas specification will be utterly ignored - those functions don't know anything about a type's alignment requirements so you must explicitly ask for allocations with the appropriate alignment using something like malloc_aligned or an operator new overload.\n\n##### Share on other sites\n\nYour problem was resolved by making your code run slower (potentially a _lot_ slower, depending on the particular CPU and how badly it deals with unaligned SSE loads\/stores).\n\nNot in my experience, _mm_loadu_ps() was only a few % slower (maybe 1 cycle at most) than _mm_load_ps() when I did the benchmarks on Intel i7, and that extra cost is not even measurable when the address is aligned. Use aligned loads whenever you can ensure alignment, but it seems like more of a microptimization. You'll save more time by thinking carefully about how to lay out data for better cache utilization so that you don't pay tens of cycles each memory access.\n\n##### Share on other sites\n\nNot in my experience, _mm_loadu_ps() was only a few % slower (maybe 1 cycle at most) than _mm_load_ps() when I did the benchmarks on Intel i7, and that extra cost is not even measurable when the address is aligned. Use aligned loads whenever you can ensure alignment, but it seems like more of a microptimization. You'll save more time by thinking carefully about how to lay out data for better cache utilization so that you don't pay tens of cycles each memory access.\n\nYMMV (your mileage may vary). Expensive, power hungry CPUs like the Intel i7 have the lowest penalty. But on certain architectures the performance hit is big (Atom, AMD CPUs). Also this problem comes back to bite you if you later port to other platforms (i.e. ARM)\nFurthermore how much slower depends on how good the CPU was masking the penalty of unaligned access. If you're hitting certain bottlenecks (such as bandwidth limits) the CPU won't be able to mask it well, and thus that 1% grows.\n\nYou'll save more time by thinking carefully about how to lay out data for better cache utilization so that you don't pay tens of cycles each memory access.\n\nEnsuring alignment is correct is part of carefully thinking how to lay out the data. Furthermore, ensuring correct alignment takes literally seconds of programming work, if not less, and it doesn't make things unreadable or harder to maintain either. Edited by Matias Goldberg\n\n##### Share on other sites\n\nYour problem was resolved by making your code run slower (potentially a _lot_ slower, depending on the particular CPU and how badly it deals with unaligned SSE loads\/stores).\n\nThe real problem is that you weren't allocating your Matrix4v4 objects with proper alignemtn. The code snippet you provided didn't show how those objects are being created, so it's impossible for me to say why specifically they were off.\n\nNote that if you're using malloc or new to allocate data that the alignas specification will be utterly ignored - those functions don't know anything about a type's alignment requirements so you must explicitly ask for allocations with the appropriate alignment using something like malloc_aligned or an operator new overload.\n\nThanks for pointing this out.\nAlignment is something new for me, so im still trying to figure it out to the best of my ability.\n\nSo, i have tried allocating using new and also by just putting the Matrix4v4 on the stack, both result in the assertion going of.\nThis is the object im using thats causing the problem.\n\nIt in turn is allocated using \"new Camera()\";\n\nclass Camera {\n\nMatrix4v4 m_View;\nMatrix4v4 m_Projection;\nMatrix4v4 m_ViewProjection;\nMatrix4v4 m_InversedViewProjection;\n\npublic:\n};\n\n\n##### Share on other sites\nThanks for pointing this out. Alignment is something new for me, so im still trying to figure it out to the best of my ability.\n\nDropping support of x86 in favor to x64 might solve a lot of headaches. :wink:\n\nEdited by Happy SDE\n\n##### Share on other sites\n\nThanks for pointing this out. Alignment is something new for me, so im still trying to figure it out to the best of my ability.\n\nDropping support of x86 in favor to x64 might solve a lot of headaches. :wink:\n\nThank you, i was indeed compiling for x86.\nI recompiled my source for x64 and with the align directives i now seem to be able to use the \"A\" notation functions :)\n\nCan you provide an explination to why this change is so significant?\n\n##### Share on other sites\nCan you provide an explination to why this change is so significant?\n\nx64 aligns data by 16 bytes, x86 - by 8.\n\nXMMATRIX and XMVECTOR should be 16-byte aligned, in order to use them directly.\n\nSo, on x64 they are implicitly aligned, and you can use them on stack\/new.\n\nDownside of x64, that you should keep in mind - you can consume more memory, if alignment of your data structures is not efficient like:\n\nstruct Foo\n{\nXMMATRIX m1; \/\/starts from 16 bytes implicitly\nbool\u00a0\u00a0\u00a0\u00a0 b1; \/\/Increases sizeof structure by 16 bytes, because next member is 16-byte aligned\nXMMATRIX m2;\nbool\u00a0\u00a0\u00a0\u00a0 b2; \/\/Extra 16 bytes\n};\n\nSo this is better:\n\nstruct BetterFoo\n{\nXMMATRIX m1; \/\/starts from 16 bytes implicitly\nXMMATRIX m2;\nbool\u00a0\u00a0\u00a0\u00a0 b1; \/\/b1+b2 add only 16 bytes, not 2*16\nbool\u00a0\u00a0\u00a0\u00a0 b2;\n};\n\n\nAs for me, I decided not to support x86 at all =)\n\nFrom here:\n\nProperly Align Allocations\n\nThe aligned versions of the SSE intrinsics underlying the DirectXMath Library are faster than the unaligned.\n\nFor this reason, DirectXMath operations using XMVECTOR and XMMATRIX objects assume those objects are 16-byte aligned.\n\nThis is automatic for stack based allocations, if code is compiled against the DirectXMath Library using the recommended Windows (see Use Correct Compilation Settings) compiler settings.\n\nHowever, it is important to ensure that heap-allocation containing XMVECTOR and XMMATRIX objects, or casts to these types, meet these alignment requirements.\n\nWhile 64-bit Windows memory allocations are 16-byte aligned, by default on 32 bit versions of Windows memory allocated is only 8-byte aligned.\n\nEdited by Happy SDE\n\n##### Share on other sites\n\nThank you so much for taking the time to explain.\n\nLearning stuff everyday it seems :)\n\n1. 1\n2. 2\nRutin\n21\n3. 3\nJoeJ\n18\n4. 4\n5. 5\ngaxio\n12\n\n\u2022 14\n\u2022 40\n\u2022 23\n\u2022 13\n\u2022 13\n\u2022 ### Forum Statistics\n\n\u2022 Total Topics\n631724\n\u2022 Total Posts\n3001897\n\u00d7","date":"2018-07-17 14:11:59","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\": 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\": 0.2910231351852417, \"perplexity\": 3903.803174737228}, \"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-2018-30\/segments\/1531676589726.60\/warc\/CC-MAIN-20180717125344-20180717145344-00097.warc.gz\"}"}
| null | null |
package com.costular.crabox.android;
import android.app.Activity;
import android.os.Bundle;
import com.costular.crabox.util.FacebookRequest;
import com.facebook.Session;
import com.facebook.UiLifecycleHelper;
import com.facebook.widget.FacebookDialog;
import com.facebook.widget.FacebookDialog.ShareDialogBuilder;
import com.facebook.widget.WebDialog;
import com.facebook.widget.WebDialog.OnCompleteListener;
public class Facebook implements FacebookRequest{
Activity activity;
UiLifecycleHelper uiHelper;
public Facebook(Activity activity, UiLifecycleHelper uiHelper) {
this.activity = activity;
this.uiHelper = uiHelper;
}
@Override
public void post(String description, String link, String urlImage) {
if (FacebookDialog.canPresentShareDialog(activity.getApplicationContext(),
FacebookDialog.ShareDialogFeature.SHARE_DIALOG)) {
FacebookDialog shareDialog = new FacebookDialog.ShareDialogBuilder(activity)
.setLink(link == "" ? "http://www.facebook.com/craboxgame" : link)
.setDescription(description)
.setPicture(urlImage)
.build();
uiHelper.trackPendingDialogCall(shareDialog.present());
} else {
Bundle params = new Bundle();
params.putString("name", "Crabox");
params.putString("caption", "pene");
params.putString("description", description);
params.putString("link", link == "" ? "http://www.facebook.com/craboxgame" : link);
params.putString("picture", urlImage);
WebDialog feedDialog = (
new WebDialog.FeedDialogBuilder(activity,
Session.getActiveSession(),
params))
.build();
feedDialog.show();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,758
|
Q: PHP MySQLi INSERT not working, no errors Different from this question, but similar in that I don't get an error when adding information to my database.
$sql = "INSERT INTO 'nlcc_ver1'.'tUsers' ('userID', 'userName', 'userPassword', 'userHash',
'user_first_name', 'user_last_name', 'user_corps', 'is_admin', 'is_trg', 'is_sup', 'is_co')
VALUES (NULL, '" . $userName . "', '" . $hash . "', '" . $salt . "', '" . $f_name . "', '" .
$l_name . "', '" . $corps . "', '" . $admin . "', '" . $trg . "', '" . $sup . "', '" . $co . "')";
$hostname_Database = "localhost";
$database_Database = "nlcc_ver1";
$username_Database = "root";
$password_Database = "";
$mysqli = new mysqli($hostname_Database, $username_Database, $password_Database, $database_Database);
if (mysqli_connect_errno()) {
printf("Connect failed: %s\n", mysqli_connect_error());
exit();
}
$result = $mysqli_query($mysqli, $sql);
echo "Query run. Inserted UserID " . mysqli_insert_id($mysqli) . "<br />";
Line breaks inserted to avoid sideways scrolling... It says on the web page that mysqli_insert_id($mysqli) is 0, and nothing is added to the table on my database. I do not see an error connecting to the database appearing, and MySQL is running on my server, and phpinfo() shows both the MySQL and MySQLI extension loaded. This is just a development machine, so don't worry about the security (i.e. no password). I have tried googling the problem, but am not finding too much. I don't know about object oriented PHP programming with ->, I am used to using _. Is this method still supported?
A: $mysqli_query($mysqli, $sql);
should be
mysqli_query($mysqli, $sql);
OR
$mysqli->query($sql);
AND later on
$mysqli->insert_id();
A: Look at this:
'nlcc_ver1'.'tUsers'
You have to use backticks here as quote character:
`nlcc_ver1`.`tUsers`
But however(assuming that the $ in $mysqli_query is just a typo): You will not get errors for the query , unless you use mysqli_error() right after executing the query.
A: You've mixed procedural and object-oriented MySQLi styles. This has led to you trying to use the functions like mysqli_query($mysqli) instead of the member functions like $mysqli->query(). Your $mysqli is an object, not a resource handle.
And, you're not performing any error checking on your query. If you were, you'd see that you have mistakenly used single quotes to delimit table and field names, not backticks.
$sql = "INSERT INTO `nlcc_ver1`.`tUsers`
(`userID`, `userName`, `userPassword`, `userHash`,
`user_first_name`, `user_last_name`, `user_corps`,
`is_admin`, `is_trg`, `is_sup`, `is_co`)
VALUES (NULL, '" . $userName . "', '" . $hash . "', '" . $salt . "', '" .
$f_name . "', '" . $l_name . "', '" . $corps . "', '" . $admin .
"', '" . $trg . "', '" . $sup . "', '" . $co . "')";
$hostname_Database = "localhost";
$database_Database = "nlcc_ver1";
$username_Database = "root";
$password_Database = "";
$mysqli = new mysqli($hostname_Database, $username_Database, $password_Database, $database_Database);
if (mysqli_connect_errno()) {
printf("Connect failed: %s\n", mysqli_connect_error());
exit();
}
$result = $mysqli->query($sql);
if (!$result) {
printf("%s\n", $mysqli->error);
exit();
}
echo "Query run. Inserted UserID " . $mysqli->insert_id . "<br />";
I strongly suggest using the manual as your reference. It's quite clear on how to use these functions when you're using either procedural or object-oriented style MySQLi.
A: SET AutoCommit = 1 before inserting
$mysqli->query('SET AUTOCOMMIT = 1');
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 360
|
The Festival of Purim
PURIM - QUEEN ESTHER
The pictures throughout this article honour all the modern - day Esthers who serve in the military, who contribute, innovate, lead and change the world at every level. Purim is celebrated in the 12th month of the Jewish calendar on the 14th day of Adar which is usually in the month of February or March some years. The word Purim means "lots" and refers to the lots - plural - pur - singular, Hayman used to choose the day to massacre the Jews. The Book of Esther known as Megillat Esther or the Scroll of Esther especially Esther 9:22 is read during this festival. Exodus 17: 8-16 is also read that tells the story of the initial wilderness meeting between Amalek and the nation of Israel and re-tells the story of how the hatred Hayman exhibited began. Hayman was a descendant of Agag who was a descendant of Amelek who was a descendant of Esau.
A custom during Purim is to boo, hiss, stamp feet and rattle noise makers at the mention of the name Haman while we eat drink and be merry, send gifts to friends because of the miraculous deliverance granted the Jews. The sages of the Talmud Gemara portion noted that Amalek would hear the negative comments of his grandfather Esau over and over as a child in the tent and absorbed resentment for Jacob who later became Israel. Agag also understood the original fued between Jacob and Esau which was based on a difference in ideology, Agag's based on chance living for the moment as Esau did while Israel's philosophy based on divine order and planning, conveyed the sentiment to his maidservant who passed the attitude on to Hayman the Agagite descendant. Thus Hayman hated the Jews intensely as a generational feeling of ill will and resentment from a perceived past injury to his ancestor Esau even 1,000 years later.
A spiritual application for Purim is to remember our children internalize our comments which potentially build positive or negative attitudes towards others. The Holy Spirit also hears the conversations at home and is pleased with pure conversations or very displeased with impure conversations such as speaking badly of others privately. Numbers 14:27 records " How long will this wicked community grumble against me? I have heard the complaints of these grumbling Israelites." The Festival of Purim provides an opportunity in which to deal with grudges, resentments, ill will that may cause injury to others in the future and that may contaminate the lives of others.
The Book of Esther has the touching story that describes the entire back drop of this wonderful festival with Esther or Hadassah the beautiful young Jewish woman who took courage to go before the King Achashverosh, Mordecai her uncle who lived righteous to HaShem, Haman the villain who tried to massacre the Jews and King Achashverosh who God made to favor Ester.
In the final month on the Jewish calendar or the twelfth month of Adar as we celebrate Purim where Esther was used to prevent catastrophe and misfortune in the lives of the Jews and also Haman who set a gallows for Mordecai was hanged on his own gallows. Purim is a time to enforce all spiritual laws against the enemy of our soul knowing that Christ spoiled all principalities and powers and made an open show of them.
It is a time of intense spiritual warfare enforced against the devil to stop his illegal efforts against a Child of God. Purim is a time to enforce the word and anointing of God against the devil to see satan's yokes destroyed and burdens removed by the miracle working intervening power of Christ. Purim is a time of the mysteries of Christ.
There is spiritual power embedded in all set feasts and to ignore their annual observances is to operate at a level below the abundant life Christ intended for us to experience.The day after Purim is known as Shushan Purim celebrated the 15th day of Adar. The fight against the anti-semites took a day longer in Shushan where King Achashverosh lived than for the rural areas and the Jews in Shushan did not get to rest and to celebrate until a day after.
The day before Purim is called Ta'anit Ester to celebrate the 3 day fast Esther did before she informed the king about Haman's plot against Mordecai. Ta'anit Esther or the fast of Ester begins at the break of dawn Adar 13 and ends at nightfall that day. Ta'anit Ester may also be celebrated Adar 11 if the 13th falls on a Shabbat. Children below the age of 13 for boys and 12 for girls are not required to fast especially before bar/bat mitzvah.
Mishloach or sending food such as fruit, wine and baked goods to friends is a custom of Purim.
Matanot L' evyonim or giving gifts to the poor. The gifts are to be given on Purim day after the reading of the Megillah or Scroll of Esther. Rambam Maimonedes says about Purim "it is better for a man to increase gifts to the poor than to enlarge his feasts and to give gifts to his friends." The Mitzvah requires a minimum of giving two gifts which may be giving food to at least two poor persons.
Purim Sueda, is a big custom and a mitzvah and requires we eat and eat and eat as well as drink, and drink and drink. The reason wine is used as a symbol at Purim is both Vashti and Haman defeats came as a result of feasting on wine. Tractate Megillah 7b. The Rabbis of the Talmud Gemara who are conservative in their thinking state Ahd D' Lo Yada Bain Arur Haman L' Baruch Mordechai which means "until he can no longer tell the difference between 'cursed be Haman' and 'blessed be Mordechai.'" A person who cannot drink can fulfill the mitzvah Ahd D' Lo Yada by sleeping because one who sleeps allegedly cannot tell the difference between a blessing and a curse. The Rabbis are trying to accomplish a high level of silly insubstantial behavior as part of the celebration not alcohol abuse or destructive drinking.
Reading The Megillah or Scroll of Esther is a big deal and this is a wild a crazy time in synagogue as in no other festival celebration. There is interruption of the scroll reading every time the name Haman is mentioned all fifty four times with BOOs, clappers, cap guns, sirens and clanging pots. This interruption is expected. The interruption continues when the ten names of Haman's ten sons are mentioned.
Purim is the only celebration where we are inane, poke fun at ourselves and dress up in costumes but no cross dressing to have fun. Rabbis intended for this celebration we have a sense of humor and increase our happiness for the entire month of Adar not just for Purim. Jewish tradition accepts Adar as a blessed and happy month.
Note that a Jewish leap year has an extra month of Adar I inserted after the eleventh month of Shevat and before the twelfth month of Adar in a leap year. The month of Adar is repeated during a leap year rather than permanently added.
The month is also known as Adar Rishon or Adar Alef. Thus for a leap year only there is Adar 1 and Adar 2. Adar 1 will celebrate Purim Katan or Minor Purim and Adar 2 will celebrate the official Purim festivals. A leap year is referred to as Shanah M'uberet meaning a pregnant year. Thus in a leap year adherents celebrate both the blessed happy month of Adar as well as a year pregnant with additional blessings.
A Jewish leap year occurs 7 times every 19 years that completes a 19 year cycle. The 3rd, 6th, 8th, 11th, 14th, 17th, and 19th years are leap years in this cycle. The list of leap years with 5758 ending the previous 19 year cycle are as follows: 5760, 5763, 5765, 5768, 5771, 5774, 5776, 5779, 5782, 5784, 5787, 5790, 5793, 5795, 5798, 5801, 5803, 5806, 5809.
Chag Sameach - Joyous Festival - Happy Purim Holiday
For additional resources, books and publications see the following: THE MANY FACES OF JUDAISM
BARNES AND NOBLE - PAPERBACK AND NOOK AMAZON - PAPERBACK AND KINDLE EBOOK POWELL'S BOOKS - PAPERBACK AND EBOOKS ANDERSON'S BOOKSHOP - PAPERBACK
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
Let $\mathcal{U}_q(sl_\infty)$ be the quantum group associated to the infinite Dynkin diagram
\bigskip
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\bigskip
It can be considered as limit of the quantum groups of type $A_n$ and has been studied by various authors (see for example \cite{ariki_factorization_2012, enomoto_symmetric_2008, frenkel_hopf_2002, levendorskii_quantum_1991} and references therein). In this paper we study the quantum affinization $\mathcal{U}_q(\hat{sl}_\infty)$ of $\mathcal{U}_q(sl_\infty)$ \cite{hernandez_representations_2005, nakajima_quiver_2001}. This algebra is introduced in \cite{hernandez_algebra_2011} and can be viewed as the limit of the quantum affine algebras $\mathcal{U}_q(\hat{sl}_{n+1})$ when $n \rightarrow \infty$.
The quantum affinizations, in particular the quantum affine algebras and the quantum toroidal algebras, have been intensively studied (see for example \cite{chari_quantum_1991, feigin_representations_2013, frenkel_$q$-characters_2002, frenkel_$q$-characters_1999, hernandez_representations_2005, hernandez_quantum_2009, hernandez_algebra_2011, nakajima_quiver_2001} and references therein). In recent works \cite{mansuy_quantum_2012, mansuy_extremal_2013} we constructed new families of integrable representations of the quantum toroidal algebra $\mathcal{U}_q(sl_{n+1}^{tor})$, called extremal loop weight modules, which generalize the $\ell$-highest weight modules: there are representations generated by an extremal vector for the horizontal quantum affine subalgebra in the sense of Kashiwara \cite{kashiwara_crystal_1994, kashiwara_level-zero_2002}. The main motivation is the construction of finite-dimensional representations of the quantum toroidal algebra at roots of unity.
The representation theory of the algebra $\mathcal{U}_q(\hat{sl}_\infty)$ is related to the one of quantum toroidal algebras $\mathcal{U}_q(sl_{n+1}^{tor})$ in the following way (see \cite{hernandez_algebra_2011}) : let us consider the morphism between the corresponding Dynkin diagrams of the algebras $\mathcal{U}_q(\hat{sl}_\infty)$ and $\mathcal{U}_q(sl_{n+1}^{tor})$
$$\phi_n : i \in \mathbb{Z} \mapsto \overline{i} \in \mathbb{Z} / (n+1) \mathbb{Z}.$$
It gives rise to a ring morphism
$$\phi_n : \mathbb{Z}[Y_{i,a}^{\pm 1}]_{i \in \mathbb{Z}, a \in \mathbb{C}^{\ast}} \longrightarrow \mathbb{Z}[Y_{i,a}^{\pm 1}]_{i \in I_n, a \in \mathbb{C}^{\ast}}.$$
Then the following combinatorial link between the algebras $\mathcal{U}_q(\hat{sl}_\infty)$ and $\mathcal{U}_q(sl_{n+1}^{tor})$ is expected in \cite{hernandez_algebra_2011}.
\begin{conjecture}\cite[Conjecture 5.3]{hernandez_algebra_2011}
Let $V$ be a simple $\mathcal{U}_q(\hat{sl}_\infty)$-module in the category $\mathcal{O}_{\mathrm{int}}$. Then the image of the $q$--character of $V$ by $\phi_n$ is also the $q$--character of a representation of $\mathcal{U}_q(sl_{n+1}^{tor})$.
\end{conjecture}
\noindent The main motivation in \cite{hernandez_algebra_2011} is to predict $q$--character formulae for representations of $\mathcal{U}_q(sl_{n+1}^{tor})$. This conjecture is proved for the class of Kirillov-Reshetikhin modules of $\mathcal{U}_q(sl_{n+1}^{tor})$.
\medskip
The aim of this article is twofold. First construct extremal loop weight modules for $\mathcal{U}_q(\hat{sl}_\infty)$: we obtain here a large class of such modules with basis labelled by semi-standard tableaux. Second by using the combinatorial link with the quantum toroidal algebras, construct extremal loop weight modules for $\mathcal{U}_q(sl_{n+1}^{tor})$: we prove the conjecture above for the particular family of extremal fundamental loop weight modules. We recover in this way the extremal loop weight modules defined in \cite{mansuy_extremal_2013} for the quantum toroidal algebras.
\medskip
Let us explain this in more details. The construction of extremal loop weight modules is inspired by the original Kashiwara's study of the extremal weight modules (see \cite{kashiwara_crystal_1994}): we consider the fusion product of fundamental $\ell$-highest weight modules and fundamental $\ell$-lowest weight modules associated respectively to the fundamental weights $\Lambda_\ell$ and $-\Lambda_0$ ($\ell \geq 1$) and to a non-zero complex parameter. The action of $\mathcal{U}_q(\hat{sl}_\infty)$ is defined by the Drinfeld coproduct, in the spirit of \cite{mansuy_extremal_2013}. We show that we get in this way extremal loop weight modules associated to the weight $\Lambda_\ell - \Lambda_0$, called extremal fundamental loop weight modules. By fusion product of $k$ extremal fundamental loop weight modules, we obtain extremal loop weight modules associated to the weight $k \Lambda_\ell - k \Lambda_0$ with basis labelled by the set of semi-standard tableaux $\mathcal{T}_{[1, \ell] \times [1, k]}$ of shape $(\ell \times k)$. Furthermore, the action of $\mathcal{U}_q(\hat{sl}_\infty)$ is explicitly describe on all these modules and involves the $\mathcal{U}_q(sl_\infty)$-crystal structure on $\mathcal{T}_{[1, \ell] \times [1, k]}$.
Our main motivation are applications to quantum toroidal algebras. We prove the conjecture above for the family of extremal fundamental loop weight modules of $\mathcal{U}_q(\hat{sl}_\infty)$ of weight $\Lambda_\ell - \Lambda_0$ introduced in this paper: we determine $q$--character formulae for these representations. Furthermore, we show that their image by the morphism $\phi_n$ is in fact the $q$--character of a representation of $\mathcal{U}_q(sl_{n+1}^{tor})$ constructed in \cite{mansuy_extremal_2013}. Recall that our original goal is to construct extremal loop weight modules. We show that we get in this way extremal loop weight modules if and only if
\begin{align*}
(\ell = 1) \text{ or } (n=2r+1 \text{ and } \ell=r+1).
\end{align*}
Let us explain another motivation to consider the algebra $\mathcal{U}_q(\hat{sl}_\infty)$. Recall that it is defined as limit of the quantum affine algebras $\mathcal{U}_q(\hat{sl}_{n+1})$ $(n \geq 2)$. For this reason the representation theory of this algebra is better understood than for general quantum affinizations: the fundamental modules we have to consider are thin, with basis labelled by semi-standard tableaux on which the action is explicitly known. It does not hold in the quantum toroidal case: in fact for $\mathcal{U}_q(sl_{n+1}^{tor})$ the fundamental modules are not thin (see \cite[Section 4.1]{hernandez_algebra_2011}). This is one of the main reason of the met difficulties in the study of extremal loop weight modules for the quantum toroidal algebras (besides see \cite[Remark 3.9]{mansuy_extremal_2013}). We will see that more precise results can be obtained in our construction of extremal loop weight modules for the quantum affinization $\mathcal{U}_q(\hat{sl}_\infty)$.
\medskip
The paper is organized as follows.
In Section 2 we recall the main definitions of the quantum group $\mathcal{U}_q(sl_\infty)$ and its quantum affinization $\mathcal{U}_q(\hat{sl}_\infty)$ and we briefly review their representation theory. In particular one defines the extremal weight modules and we introduce the notion of extremal loop weight modules for $\mathcal{U}_q(\hat{sl}_\infty)$. In Section 3 we construct the fusion product of fundamental $\ell$-highest weight modules and fundamental $\ell$-lowest weight modules associated respectively to the weights $\Lambda_\ell$ and $-\Lambda_0$ ($\ell \geq 1$) and to a non-zero complex parameter. We define a $\mathcal{U}_q(\hat{sl}_\infty)$-module structure on it by using the Drinfeld coproduct (Proposition \ref{tpexistudinf}). We obtain (as submodule of the fusion product) an extremal loop weight module associated to the weight $\Lambda_\ell - \Lambda_0$ (Theorem \ref{thmelwminf}), called the extremal fundamental loop weight module of weight $\Lambda_\ell - \Lambda_0$. Furthermore we explicit the action of $\mathcal{U}_q(\hat{sl}_\infty)$ on it (Theorem \ref{thmformactinf}). Section 4 is devoted to the study of fusion products of extremal fundamental loop weight modules. When the non-zero complex parameters are chosen generic, we get extremal loop weight modules (Theorem \ref{thmgencaseinf}). In the non-generic case we recover all the extremal fundamental loop weight modules from the fusion product of the extremal fundamental loop weight modules associated to the weight $\Lambda_1 - \Lambda_0$ and to a $q$-segment (Theorem \ref{thmtpvrisomeflwm}). By fusion product of $k$ extremal fundamental loop weight modules of weight $\Lambda_\ell - \Lambda_0$, we obtain extremal loop weight modules of the weights $k \Lambda_\ell - k \Lambda_0$ ($\ell, k \geq 1$) (Theorem \ref{thmtpelwminf}). In Section 5, we discuss the applications to quantum toroidal algebras. We show that the $q$--character formulae obtained from the extremal fundamental loop weight $\mathcal{U}_q(\hat{sl}_\infty)$-modules are the $q$--character of representations of the quantum toroidal algebra $\mathcal{U}_q(sl_{n+1}^{tor})$ (Theorem \ref{thmqchaactreptorinf}). Furthermore we determine when these representations give extremal loop weight modules (Theorem \ref{thmelwmtorinf}).
\medskip
\textbf{Acknowledgements :}
I would like to thank my advisor David Hernandez for his encouragements and for his precious comments. I am grateful to Nicolas Jacon for his interest in my work. I want also to thank Alexandre Bouayad and Dragos Fratila for their accurate remarks.
\section{Background}
\subsection{Cartan matrix} \nocite{kac_infinite-dimensional_1990} Let $C=(C_{i,j})_{i,j\in \mathbb{Z}}$ be the Cartan matrix of type $A_\infty$, that is ($i,j \in \mathbb{Z}$)
$$ C_{i,i} = 2 \text{ , } C_{i,i+1} = C_{i+1, i} = -1 \text{ and } C_{i,j} = 0 \text{ otherwise. } $$
Consider the vector space
$$\mathfrak{h} = \bigoplus_{i \in \mathbb{Z}} \mathbb{Q} h_i$$
and the linear functions $\Lambda_i$ (the fundamental weights) on $\mathfrak{h}$ given by
$$\Lambda_i(h_j) = \delta_{i,j} \text{ for all } i, j \in \mathbb{Z}.$$
For $i \in \mathbb{Z}$, set $\alpha_i = 2 \Lambda_i - \Lambda_{i-1} - \Lambda_{i+1}$. Denote by $\Pi = \{\alpha_i , i \in \mathbb{Z} \}$ the set of simple roots and $\Pi^\vee = \{h_i, i \in \mathbb{Z}\}$ the set of simple coroots. Let $P =\bigoplus_{i \in \mathbb{Z}} \mathbb{Z} \Lambda_i$ be the weight lattice and $P^{+} = \bigoplus_{i \in \mathbb{Z}} \mathbb{N} \Lambda_i$ the semigroup of dominant weights. Let $Q = \bigoplus_{i \in I} \mathbb{Z} \alpha_i \subseteq P$ (the root lattice) and $Q^+ = \bigoplus_{i \in I} \mathbb{N} \alpha_i \subseteq Q$. For $\lambda, \mu \in \mathfrak{h}^{\ast}$, write $\lambda \geq \mu$ if $\lambda - \mu \in Q^{+}$.
Denote by $W$ the Weyl group of type $sl_\infty$: it is the subgroup of transformations stabilizing $P$ generated by the simple reflections $s_i$ defined by $s_i(\lambda)=\lambda-\lambda(h_i)\alpha_i$ ($i\in \mathbb{Z}, \lambda \in P$).
\subsection{Quantum group $\mathcal{U}_q(sl_\infty)$} In this article $q = e^{t} \in\mathbb{C}^*$ $(t \in \mathbb{C})$ is not a root of unity and is fixed. For $l\in\mathbb{Z}, r\geq 0, m\geq m'\geq 0$ we set
$$[l]_q=\frac{q^l-q^{-l}}{q-q^{-1}}\in\mathbb{Z}[q^{\pm 1}],\ [r]_q!=[r]_q[r-1]_q\dots[1]_q,\ \begin{bmatrix}m\\m'\end{bmatrix}_q=\frac{[m]_q!}{[m-m']_q![m']_q!}.$$
\begin{defi} The quantum group $\mathcal{U}_q(sl_\infty)$ is the $\mathbb{C}$-algebra with generators $k_h$ \linebreak $(h \in \mathfrak{h})$, $x_i^{\pm}$ $(i\in \mathbb{Z})$ and relations
\begin{equation*}k_h k_{h'} = k_{h+h'}, \ k_0 = 1,\end{equation*}
\begin{equation*}k_h x_j^{\pm}k_{-h}=q^{\pm \alpha_j(h)}x_j^{\pm},\end{equation*}
\begin{equation*}[x_i^+,x_j^-]=\delta_{i,j}\frac{k_i-k_{i}^{-1}}{q-q^{-1}},\end{equation*}
\begin{equation*}
(x_i^{\pm})^{(2)}x_{i+1}^{\pm} - x_i^{\pm}x_{i+1}^{\pm}x_i^{\pm} + x_{i+1}^{\pm}(x_i^{\pm})^{(2)} = 0,\end{equation*}
\end{defi}
\noindent where we use the notations $k_i^\pm = k_{\pm h_i}$ and for all $r \geq 0$, $(x_i^\pm)^{(r)} = \frac{(x_i^\pm)^r}{[r]_q!}$.
For $J = [a,b] \subset \mathbb{Z}$ denote by $\mathcal{U}_q(sl_\infty)_{J}$ the subalgebra of $\mathcal{U}_q(sl_\infty)$ generated by the $x_i^{\pm}, k_{h}$ for $i\in J$, $h \in \oplus_{j \in [a,b]} \mathbb{Q} h_j$. Then $\mathcal{U}_q(sl_\infty)_{J}$ is isomorphic to the quantum group $\mathcal{U}_q(sl_{b-a+2})$.
Let $\mathcal{U}_q(sl_{\infty})^+$ (resp. $\mathcal{U}_q(sl_{\infty})^-$, $\mathcal{U}_q(\mathfrak{h})$) be the subalgebra of $\mathcal{U}_q(sl_{\infty})$ generated by the $x_i^+$ (resp. the $x_i^-$, resp the $k_h$). We have a triangular decomposition of $\mathcal{U}_q(sl_{\infty})$ (see \cite{lusztig_introduction_1993}):
\begin{thm} We have an isomorphism of vector spaces
$$\mathcal{U}_q(sl_{\infty}) \simeq \mathcal{U}_q(sl_{\infty})^- \otimes \mathcal{U}_q(\mathfrak{h}) \otimes \mathcal{U}_q(sl_{\infty})^+.$$
\end{thm}
\subsection{Representations of $\mathcal{U}_q(sl_\infty)$} For $V$ a representation of $\mathcal{U}_q(sl_\infty)$ and $\nu \in P$, the weight space $V_{\nu}$ of $V$ is
$$V_\nu = \{v\in V|k_i \cdot v = q^{\nu(h_i)}v, \forall i \in \mathbb{Z} \}.$$
\begin{defi}\label{defcato} A representation $V$ is said to be in the category $\mathcal{O}$ if
\begin{enumerate}
\item[(i)] it admits a weight space decomposition $V = \bigoplus_{\nu\in P} V_\nu$,
\item[(ii)] $V_\nu$ is finite-dimensional for all $\nu$,
\item[(iii)] $\{\nu \vert V_\nu \neq \{0\} \} \subset \cup_{j = 1, \cdots, N} \{\nu \vert \nu \leq \lambda_j \}$ for some $\lambda_1, \cdots , \lambda_N \in P$.
\end{enumerate}
\end{defi}
For $\lambda\in P$, a representation $V$ is said to be of highest weight $\lambda$ if there is $v\in V_\lambda$ such that for all $i\in I, x_i^+ \cdot v = 0$ and $\mathcal{U}_q(sl_\infty) \cdot v = V$. Such a representation is in the category $\mathcal{O}$. Furthermore there is a unique simple highest weight module of highest weight $\lambda$.
\begin{defi}\label{defint} A representation $V$ is said to be integrable if
\begin{enumerate}
\item[(i)] it admits a weight space decomposition $V = \bigoplus_{\nu\in P} V_\nu$,
\item[(ii)] all the $x_i^\pm$ ($i \in \mathbb{Z}$) are locally nilpotent.
\end{enumerate}
\end{defi}
\begin{thm}\cite{lusztig_introduction_1993, hernandez_algebra_2011}
The simple highest weight module of highest weight $\lambda$ is integrable if and only if $\lambda$ is dominant. It is denoted $V(\lambda)$.
\end{thm}
Now we recall the definition and some properties of extremal weight modules for the quantum group $\mathcal{U}_q(sl_\infty)$ \cite{kashiwara_crystal_1994}.
\begin{defi} For an integrable $\mathcal{U}_q(sl_\infty)$-module $V$ and $\lambda\in P$, a vector $v\in V_{\lambda}$ is called $i$-extremal if $x_i^+ \cdot v=0$ or $x_i^- \cdot v = 0$. In this case we set $$S_i(v) = (x_i^-)^{(\lambda(h_i))} \cdot v \text{ or } S_i(v) = (x_i^+)^{(-\lambda(h_i))} \cdot v.$$ A vector $v\in V_{\lambda}$ is called extremal of weight $\lambda$ if, for any $l \geq 1$, $S_{i_1} \cdots S_{i_l}(v)$ is $i$-extremal for any $i, i_1, \cdots, i_l \in \mathbb{Z}$. \end{defi}
\noindent For $v$ an extremal vector of weight $\lambda$, we set $$W \cdot v = \{S_{i_1} \cdots S_{i_l}(v) \vert l \in \mathbb{N}, i_1, \cdots i_l \in \mathbb{Z} \}.$$
\begin{defi} For $\lambda\in P$, the extremal weight module of extremal weight $\lambda$ is the $\mathcal{U}_q(sl_\infty)$-module generated by a vector $v_{\lambda}$ with the defining relations that $v_{\lambda}$ is extremal of weight $\lambda$. It is denoted $V(\lambda)$.\end{defi}
\begin{ex} If $\lambda$ is dominant, $V(\lambda)$ is the simple highest weight module of highest weight $\lambda$. \end{ex}
\begin{thm} \cite{kashiwara_crystal_1994} For $\lambda\in P$, the module $V(\lambda)$ is integrable and has a crystal basis $\mathcal{B}(\lambda)$.\end{thm}
\subsection{Quantum affinization $\mathcal{U}_q(\hat{sl}_\infty)$}
We recall the definition of the quantum affinization $\mathcal{U}_q(\hat{sl}_\infty)$ (without central charge) in terms of currents.
\begin{defi}\label{defqaainf}
The quantum affinization $\mathcal{U}_q(\hat{sl}_\infty)$ is the $\mathbb{C}$-algebra with generators $x_{i,r}^\pm$ ($i \in \mathbb{Z}, r \in \mathbb{Z}$), $k_h$ ($h \in \mathfrak{h}$), $h_{i,m}$ ($i \in \mathbb{Z}, m \in \mathbb{Z}-\{0\}$) and the following defining relations ($i,j \in \mathbb{Z}, h, h' \in \mathfrak{h}$):
\begin{equation*}
k_h k_{h'} = k_{h+h'}, \ k_0 = 1,
\end{equation*}
\begin{equation*}
\phi^\pm_i(z)\phi^\pm_j (w) = \phi^\pm_j(w)\phi^\pm_i (z) \text{ , } \phi^-_i(z)\phi^+_j (w) = \phi^+_j(w)\phi^-_i (z),
\end{equation*}
\begin{equation}\label{relxcartpl}
(w -q^{\pm C_{ij}}z) \phi_i^+(z)x_j^\pm(w)=
(q^{\pm C_{ij}}w - z) x_j^\pm(w)\phi_i^+(z),
\end{equation}
\begin{equation}\label{relxcartmo}
(w -q^{\pm C_{ij}}z) \phi_i^-(z)x_j^\pm(w)=
(q^{\pm C_{ij}}w - z) x_j^\pm(w)\phi_i^-(z),
\end{equation}
\begin{equation*}
[x_i^+(z),x_j^-(w)]=\frac{\delta_{i,j}}{q-q^{-1}}(\delta\bigl(\frac{w}{z}\bigr)\phi_i^+(w) -\delta\bigl(\frac{z}{w}\bigr)\phi_i^-(z)),
\end{equation*}
\begin{equation*}
(z-q^{\pm C_{ij}}w) x_i^\pm(z)x_j^\pm(w)=(q^{\pm a_{ij}}z-w)x_j^\pm(w)x_i^\pm(z),
\end{equation*}
\begin{eqnarray*}
\begin{array}{c}
x_i^\pm(z_1)x_i^\pm(z_2)x_{i\pm1}^\pm(w)-(q+q^{-1})x_i^\pm(z_1)x_{i\pm 1}^\pm(w) x_i^\pm(z_2) \\
+x_{i\pm1}^\pm(w) x_i^\pm(z_1) x_i^\pm(z_2) + (z_1\leftrightarrow z_2)=0,
\end{array}
\end{eqnarray*}
and $[x_i^\pm(z),x_j^\pm(w)]=0$ for $i \neq j,j \pm 1$.
Here we use the formal power series $\delta(z) = \sum_{s \in \mathbb{Z}} z^{s}$ and
\begin{eqnarray*}
x_i^\pm(z) = \sum_{r \in \mathbb{Z}} x_{i,r}^\pm z^{r},
\end{eqnarray*}
\begin{eqnarray*}
\phi_i^{\pm}(z) = \sum_{m \geq 0} \phi_{i,\pm m}^\pm z^{\pm m} = k_i^{\pm 1} \exp(\pm(q-q^{-1}) \sum_{m' \geq 1} h_{i, \pm m'} z^{\pm m'}).
\end{eqnarray*}
\end{defi}
There is an algebra morphism $\mathcal{U}_q(sl_\infty)\rightarrow \mathcal{U}_q(\hat{sl}_\infty)$ defined by $k_h\mapsto k_h$, $x_i^{\pm}\mapsto x_{i,0}^{\pm}$ ($h \in \mathfrak{h}, i \in \mathbb{Z}$). Its image is called the horizontal quantum affine subalgebra of $\mathcal{U}_q(\hat{sl}_\infty)$ and is denoted by $\mathcal{U}_q^{h}(\hat{sl}_\infty)$. In particular, a $\mathcal{U}_q(\hat{sl}_\infty)$-module $V$ has also a structure of a $\mathcal{U}_q(sl_\infty)$-module. We denote by $\mathrm{Res}(V)$ the restricted $\mathcal{U}_q(sl_\infty)$-module obtained from $V$.
For $J = [a,b] \subseteq \mathbb{Z}$, let us denote by $ \mathcal{U}_{q}(\hat{sl}_\infty)_J $ the subalgebra of $ \mathcal{U}_{q}(\hat{sl}_\infty) $ generated by the $x_{i,r}^{\pm}$, $k_h$, $h_{i,m}$ ($i\in J$, $r\in\mathbb{Z}$, $m\in\mathbb{Z}-\{0\}$, $h \in \oplus_{j \in J} \mathbb{Q} h_j$). It is isomorphic to the quantum affine algebra $\mathcal{U}_q(\hat{sl}_{b-a+2})'$ \cite{beck_braid_1994, drinfeld_new_1988}.
For $i\in \mathbb{Z}$, the subalgebra $ \hat{\mathcal{U}_i} $ generated by the $x_{i,r}^{\pm}, h_{i,m}, k_{p h_i}$ ($ r \in \mathbb{Z}$, $m \in \mathbb{Z} - \{0\}$, $p \in \mathbb{Q}$) is isomorphic to $ \mathcal{U}_{q}(\hat{sl}_{2})'$.
We have a triangular decomposition of $\mathcal{U}_q(\hat{sl}_\infty)$.
\begin{thm}\label{dtrianinf} \cite{miki_representations_2000, nakajima_quiver_2001} We have an isomorphism of vector spaces
$$\mathcal{U}_q(\hat{sl}_\infty)\simeq \mathcal{U}_q(\hat{sl}_\infty)^-\otimes\mathcal{U}_q(\hat{\mathfrak{h}})\otimes\mathcal{U}_q(\hat{sl}_\infty)^+,$$
where $\mathcal{U}_q(\hat{sl}_\infty)^{\pm}$ (resp. $\mathcal{U}_q(\hat{\mathfrak{h}})$) is generated by the $x_{i,r}^{\pm}$ (resp. the $k_h$, the $h_{i,m}$).
\end{thm}
\subsection{Representations of $\mathcal{U}_q(\hat{sl}_\infty)$}
\begin{defi} A representation $V$ of $\mathcal{U}_q(\hat{sl}_\infty)$ is said to be integrable (resp. in the category $\mathcal{O}$) if $\mathrm{Res}(V)$ is integrable (resp. in the category $\mathcal{O}$) as a $\mathcal{U}_q(sl_\infty)$-module. One denotes by $\mathcal{O}_{\mathrm{int}}$ the category of integrable representations of $\mathcal{U}_q(\hat{sl}_\infty)$ in the category $\mathcal{O}$.
\end{defi}
\begin{defi} A representation $V$ of $\mathcal{U}_q(\hat{sl}_\infty)$ is said to be of $\ell$-highest weight if there is $v\in V$ such that
\begin{enumerate}
\item[(i)] $V = \mathcal{U}_q(\hat{sl}_\infty)^- \cdot v$,
\item[(ii)] $\mathcal{U}_q(\hat{\mathfrak{h}}) \cdot v=\mathbb{C} v$,
\item[(iii)] for any $i\in \mathbb{Z}, r\in\mathbb{Z}$, $x_{i,r}^+ \cdot v=0$.
\end{enumerate}
\end{defi}
\noindent Such a representation is in the category $\mathcal{O}$. For $\gamma \in \mathrm{Hom}(\mathcal{U}_q(\hat{\mathfrak{h}}), \mathbb{C})$, by Theorem \ref{dtrianinf} we have a corresponding Verma module $M(\gamma)$ and a simple representation $V(\gamma)$ which are $\ell$-highest weight.
\begin{thm}\label{cond} \cite{hernandez_algebra_2011} The simple representations $V(\gamma)$ of $\mathcal{U}_q(\hat{sl}_\infty)$ are integrable if there is $(P_i)_{i \in \mathbb{Z}}\in (1+u\mathbb{C}[u])^{(\mathbb{Z})}$ satisfying for $i\in \mathbb{Z}$ the relation in $\mathbb{C}[[z]]$ (resp. in $\mathbb{C}[[z^{-1}]]$)
$$\gamma(\phi_i^\pm(z))=q^{\text{deg}(P_i)}\frac{P_i(zq^{-1})}{P_i(zq)}.$$
\end{thm}
\noindent The polynomials $P_i$ are called Drinfeld polynomials and the representation $V(\gamma)$ is then denoted by $V((P_i)_{i \in \mathbb{Z}})$.
The Kirillov-Reshetikhin module associated to $k \geq 0$, $a \in \mathbb{C}^{\ast}$ and $\ell \in \mathbb{Z}$, is the simple integrable representation defined by the $n$--tuple
$$P_i(u) = \left\lbrace \begin{array}{l} (1-ua)(1-uaq^{2}) \cdots (1-uaq^{2(k-1)}) \ \mathrm{for} \ i = \ell, \\ 1 \ \mathrm{for} \ i \neq \ell. \end{array} \right.$$
\noindent If $k=1$, it is also called fundamental module.
\begin{thm}\label{thmlimind}\cite{hernandez_algebra_2011}
Let $v$ be a $\ell$-highest weight vector of the simple $\mathcal{U}_q(\hat{sl}_\infty)$-module $V((P_i)_{i \in \mathbb{Z}})$. Then
$$V((P_i)_{i \in \mathbb{Z}}) = \bigcup_{n \geq 0} V_n((P_i)_{i \in \mathbb{Z}}) \text{ where } V_n((P_i)_{i \in \mathbb{Z}}) = \mathcal{U}_q(\hat{sl}_\infty)_{[-n, n]} \cdot v.$$
Furthermore each $V_n((P_i)_{i \in \mathbb{Z}})$ is a simple finite-dimensional $\mathcal{U}_q(\hat{sl}_\infty)_{[-n, n]}$-module.
\end{thm}
Consider an integrable representation $V$ of $\mathcal{U}_q(\hat{sl}_\infty)$. As the subalgebra $\mathcal{U}_q(\hat{\mathfrak{h}})$ is commutative, the weight spaces $V_{\nu}$ split in simultaneous generalized eigenspaces
$$V_\nu = \bigoplus_{\gamma \in \mathrm{Hom}(\mathcal{U}_q(\hat{\mathfrak{h}}), \mathbb{C})} V_{\gamma},$$
where $$V_{\gamma} = \{x \in V / \exists p \in \mathbb{N}, \forall i \in \mathbb{Z}, \forall m \geq 0, (\phi_{i, \pm m}^{\pm} - \gamma(\phi_{i, \pm m}^{\pm}))^{p} \cdot x = 0 \}.$$ If $V_{\gamma} \neq \{0\}$, $\gamma$ is called an $\ell$-weight of $V$ and $V_\gamma$ is an $\ell$-weight space.
\begin{defi}
A $\mathcal{U}_q(\hat{sl}_\infty)$-module $V$ is weighted if the Cartan subalgebra $\mathcal{U}_q(\hat{\mathfrak{h}})$ acts on $V$ by diagonalizable operators. The module $V$ is thin if it is weighted and the joint spectrum is simple.
\end{defi}
\begin{defi}\cite{frenkel_$q$-characters_1999, hernandez_representations_2005, nakajima_quiver_2001} The $q$--character of an integrable representation $V$ of $\mathcal{U}_q(\hat{sl}_\infty)$ with finite-dimensional $\ell$-weight spaces is defined by the formal sum
$$\chi_q(V) = \sum_{\gamma \in \mathrm{Hom}(\mathcal{U}_q(\hat{\mathfrak{h}}), \mathbb{C})} \dim(V_{\gamma}) e(\gamma).$$
\end{defi}
As in \cite{feigin_representations_2013}, we set
$$\psi(z)=\frac{q-q^{-1}z}{1-z}.$$
\begin{prop}\cite{frenkel_$q$-characters_1999, hernandez_algebra_2011}
Let $V$ be an integrable representation of $\mathcal{U}_q(\hat{sl}_\infty)$. An $\ell$-weight $\gamma$ of $V$ satisfies the property
\begin{enumerate}
\item for all $i \in \mathbb{Z}$, there exist $k,l \in \mathbb{N}$ and $a_{1,i}, \cdots, a_{k,i}, b_{1,i}, \cdots, b_{l,i} \in \mathbb{C}^{\ast}$ such that in $\mathbb{C}[[z]]$ (resp. in $\mathbb{C}[[z^{-1}]]$):
$$\sum_{m \geq 0} \gamma(\phi_{i, \pm m}^{\pm}) z^{\pm m} = \prod_{1 \leq u \leq k} \psi(a_{u,i}qz) \prod_{1 \leq v \leq l} \psi(b_{v,i}qz)^{-1} = \prod_{1 \leq u \leq k} \psi(a_{u,i}qz) \prod_{1 \leq v \leq l} \psi(b_{v,i}^{-1}qz^{-1})$$
\end{enumerate}
Furthermore if $V$ is in the category $\mathcal{O}$, then
\begin{enumerate}
\item[(2)] there exist $\omega \in P^{+}$, $\alpha \in Q^{+}$ satisfying $\nu = \omega - \alpha$.
\end{enumerate}
\end{prop}
Consider formal variables $ Y_{i,a}^{\pm 1} $ ($ i \in \mathbb{Z} $, $ a \in \mathbb{C}^{\ast}$) and let $ A $ be the group of monomials of the form $$ m = \prod_{i \in \mathbb{Z}, a \in \mathbb{C}^{\ast}} Y_{i,a}^{u_{i,a}(m)}, \ u_{i,a}(m) \in \mathbb{Z}.$$
Set $$ A_{i,a} = Y_{i,aq^{l-1}} Y_{i,aq^{l+1}} Y_{i-1,aq^{l}}^{-1}Y_{i+1,aq^{l}}^{-1} \in A.$$
A monomial $ m $ is said to be $J$-dominant ($ J \subset \mathbb{Z} $) if for all $ j \in J $ and $a\in \mathbb{C}^{\ast} $ we have $ u_{j,a}(m) \geq 0 $. A $ \mathbb{Z} $-dominant monomial is said to be dominant.
Let $V$ be an integrable $\mathcal{U}_q(\hat{sl}_\infty)$-module. For $\gamma$ an $\ell$-weight of $V$, one defines the monomial $m_{\gamma} = \prod_{i \in \mathbb{Z}} \prod_{a_i \in \mathbb{C}^{\ast}} Y_{i,a_i} \prod_{b_i \in \mathbb{C}^{\ast}} Y_{i,b_i}^{-1}$ where
$$\sum_{m \geq 0} \gamma(\phi_{i, \pm m}^{\pm}) z^{\pm m} = \prod_{a_i} \psi(a_{i}qz) \prod_{b_i} \psi(b_{i}qz)^{-1}.$$
We denote $V_{\gamma} = V_{m_{\gamma}}$. We rewrite the $q$--character of an integrable representation $V$ with finite-dimensional $\ell$-weight spaces by the formal sum
$$\chi_q(V) = \sum_{m} \dim(V_{m}) m \in \mathbb{Z}^{A}.$$
Let us denote by $\mathcal{M}(V)$ the set of monomials occurring in $\chi_q(V)$.
By this correspondence between $\ell$-weights and monomials due to Frenkel-Reshetikhin \cite{frenkel_$q$-characters_1999}, the $\mathbb{Z}$-tuple of Drinfeld polynomials are identified with the dominant monomials. In particular for a dominant monomial $m$, one denotes by $V(m)$ the simple module of $\ell$-highest weight $m$. For example $V(Y_{\ell, a}Y_{\ell, aq^{2}} \dots Y_{\ell, aq^{2(k-1)}})$ is the Kirillov-Reshetikhin module associated to $\ell \in \mathbb{Z}$, $k \geq 0$ and $a \in \mathbb{C}^{\ast}$, and $V(Y_{\ell, a})$ is the fundamental module associated to $\ell \in \mathbb{Z}$ and $a \in \mathbb{C}^{\ast}$.
In general no formula is known for the $q$--character of a $\ell$-highest weight module of $\mathcal{U}_q(\hat{sl}_\infty)$. But in the special case of Kirillov-Reshetikhin modules, the $q$--character can be explicitly given as a tableaux sum expression. For that we set for all $a \in \mathbb{C}^{\ast}$ and $j \in \mathbb{Z}$
$$\ffbox{j}_a = Y_{j-1, aq^{j}}^{-1}Y_{j, aq^{j-1}}.$$
For $I, J \subset \mathbb{Z}$, a tableaux $(T_{i, j})_{i \in I, j \in J}$ is said to be semi-standard if it has coefficients in $\mathbb{Z}$ which increase relatively to $j$ and strictly increase relatively to $i$.
Consider $\ell \in \mathbb{Z}$ and $k \geq 0$. Let $\mathcal{T}_{\ell, k}$ be the set of semi-standard tableaux $(T_{i, j})_{i \leq \ell, 1 \leq j \leq k}$ such that for any $1 \leq j \leq k$,
$$T_{i, j} = i \text{ for } i << 0.$$
\begin{thm}\cite{hernandez_algebra_2011}
We have the following formula
\begin{align}\label{formqchainfty}
\chi_q(V(Y_{\ell, a} \cdots Y_{\ell, aq^{2(k-1)}})) = \sum_{T \in \mathcal{T}_{\ell, k}} m_T
\end{align}
where $m_T = \prod_{i \leq \ell, 1 \leq j \leq k} \ffbox{T_{i,j}}_{aq^{\ell-1+2(j-i)}}$.
\end{thm}
\subsection{Extremal loop weight modules}
The definition of extremal loop weight modules is given in \cite{hernandez_algebra_2011} for the quantum toroidal algebras and studied in \cite{mansuy_quantum_2012, mansuy_extremal_2013}. We introduce its definition in the context of the quantum affinization $ \mathcal{U}_{q}(\hat{sl}_\infty) $.
\begin{defi}\label{defelminf} An extremal loop weight module of $ \mathcal{U}_{q}(\hat{sl}_\infty) $ of $\ell$-weight $m$ is an integrable representation $ V $ such that there is $ v \in V_m $ satisfying
\begin{enumerate}
\item[(i)] $ \mathcal{U}_{q}(\hat{sl}_\infty) \cdot v = V $,
\item[(ii)] $ v $ is extremal for $ \mathcal{U}_{q}^{h}(\hat{sl}_\infty) $,
\item[(iii)] $ \mathcal{U}_{q}(\hat{sl}_\infty)_J \cdot w $ is finite-dimensional for all $ w \in V $ and $ J = [a, b] \subset \mathbb{Z} $ finite.
\end{enumerate} \end{defi}
\begin{prop}
Let $m$ be a dominant monomial. Then the simple $\ell$-highest weight module $V(m)$ is an extremal loop weight module of $\mathcal{U}_{q}(\hat{sl}_\infty)$.
\end{prop}
\begin{proof}
Let $v$ be a $\ell$-highest weight vector of $V(m)$. Denote its weight by $\lambda \in P^+$. As it is recalled above, $V(m) = \mathcal{U}_{q}(\hat{sl}_\infty) \cdot v$ and is integrable. Furthermore $v$ is a highest weight vector for $\mathcal{U}_{q}^h(\hat{sl}_\infty)$ which satisfies for all $i \in \mathbb{Z}$,
$$(x_i^{-})^{1+\lambda(h_i)} \cdot v = 0.$$
Hence $\mathcal{U}_{q}^h(\hat{sl}_\infty) \cdot v$ is isomorphic to the simple highest weight $\mathcal{U}_q(sl_{\infty})$-module $V(\lambda)$, and $v$ is extremal of weight $\lambda$ for the horizontal quantum affine subalgebra.
It remains to show that for all $w \in V(m)$ and $J = [a, b] \subset \mathbb{Z}$ finite, $\mathcal{U}_q(\hat{sl}_\infty)_J \cdot w$ is finite-dimensional. But there exists $n \in \mathbb{N}$ such that $J \subset [-n , n]$ and
\begin{align*}
\mathcal{U}_q(\hat{sl}_\infty)_J \cdot w \subset \mathcal{U}_q(\hat{sl}_\infty)_{[-n, n]} \cdot w \subset V_n(m) = \mathcal{U}_q(\hat{sl}_\infty)_{[-n, n]} \cdot v.
\end{align*}
As $V_n((P_i)_{i \in \mathbb{Z}})$ is finite-dimensional by Theorem \ref{thmlimind}, it is also the case of $\mathcal{U}_q(\hat{sl}_\infty)_J \cdot w$. So the $\mathcal{U}_q(\hat{sl}_\infty)$-module $V(m)$ is an extremal loop weight module of $\ell$-weight $m$.
\end{proof}
In this paper we construct extremal loop weight modules of $\mathcal{U}_q(\hat{sl}_\infty)$ by fusion product of $\ell$-highest weight modules and $\ell$-lowest weight modules. This process is introduced in \cite{mansuy_extremal_2013} in the toroidal case, and is motivated by the original study of extremal weight modules of quantum Kac-Moody algebras \cite{kashiwara_crystal_1994}: in fact to study these representations, tensor products of highest weight modules and lowest weight modules are considered (see \cite[Lemma 8.2.1]{kashiwara_crystal_1994}).
\subsection{Drinfeld coproduct}
Let $\Delta$ be the coproduct defined by ($i\in \mathbb{Z}$)
\begin{equation}\label{deltaxplus}
\Delta(x_i^+(z))=x_i^+(z)\otimes 1 +\phi_i^-(z)\otimes x_i^+(z),
\end{equation}
\begin{equation}\label{deltaxmoins}
\Delta(x_i^-(z))= x_i^-(z)\otimes \phi_i^+(z)+1\otimes x_i^-(z),
\end{equation}
\begin{equation}\label{deltacart}
\Delta(\phi_i^\pm(z))=\phi_i^\pm(z)\otimes \phi_i^\pm (z).
\end{equation}
\noindent Note that this map does not define a coproduct in the usual sense because (\ref{deltaxplus}) and (\ref{deltaxmoins}) involve infinite sums and are not elements of $\mathcal{U}_q(\hat{sl}_\infty) \otimes \mathcal{U}_q(\hat{sl}_\infty)$ (they take values in a completion of it). However it can be used to define a structure of $\mathcal{U}_q(\hat{sl}_\infty)$-module on (a subspace of) tensor products of $\mathcal{U}_q(\hat{sl}_\infty)$-modules when the summations are well-defined.
\begin{lem}\cite{feigin_representations_2013}
\label{lemsubminf}
Let $V$, $W$ be $\mathcal{U}_q(\hat{sl}_\infty)$-modules. Let $U\subseteq V\otimes W$ be a linear subspace such that
for any $u\in U$, $\Delta(x_{i}^{+}(z)) \cdot u$ is well-defined in $U$ ( $i\in \mathbb{Z}$). Then the coproduct $\Delta$ endows $U$ with a structure of $\mathcal{U}_q(\hat{sl}_\infty)$-module, called the fusion product of $V$ and $W$.
\end{lem}
\begin{lem}\label{tpalgv}
Assume further that $V$ and $W$ satisfies $\mathcal{U}_{q}(\hat{sl}_\infty)_J \cdot v$ and $\mathcal{U}_{q}(\hat{sl}_\infty)_J \cdot w$ are finite-dimensional for all $v \in V, w \in W$ and all $J = [a, b] \subset \mathbb{Z}$ finite in the last lemma. Then the $\mathcal{U}_{q}(\hat{sl}_\infty)$-module $U$ satisfies also
$$\mathcal{U}_{q}(\hat{sl}_\infty)_J \cdot u \text{ is finite-dimensional for all } u \in U \text{ and } J = [a, b] \subset \mathbb{Z} \text{ finite.}$$
In particular if $V$ and $W$ are integrable, $U$ is an integrable $\mathcal{U}_q(\hat{sl}_\infty)$-module.
\end{lem}
\begin{proof}
Set $J = [a, b] \subset \mathbb{Z}$ finite and let $u = v \otimes W \in V \otimes W$. By the formulas of the coproduct, we have
$$\mathcal{U}_{q}(\hat{sl}_\infty)_J \cdot u \subset \left( \mathcal{U}_{q}(\hat{sl}_\infty)_J \cdot v \right) \otimes \left( \mathcal{U}_{q}(\hat{sl}_\infty)_J \cdot v \right).$$
The result follows.
\end{proof}
\section{Extremal fundamental loop weight modules}
In this section, we define a $\mathcal{U}_q(\hat{sl}_\infty)$-module structure on the tensor product of fundamental modules
$$V(Y_{\ell, a}) \otimes V(Y_{0, aq^{\ell}}^{-1}) \text{ with } \ell \geq 1 \text{ and } a \in \mathbb{C}^{\ast}$$
by using the Drinfeld coproduct $\Delta$. We obtain in that way new integrable modules $V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})$ with basis labelled by semi-standard tableaux of shape $(\ell)$. We show in particular that these modules are $\ell$-extremal. We call them the extremal fundamental loop weight modules of $\mathcal{U}_q(\hat{sl}_\infty)$.
The fusion product process used here is inspired to the one in \cite{mansuy_extremal_2013} for the toroidal case. As it is said in the introduction, the representation theory of $\mathcal{U}_q(\hat{sl}_\infty)$ and of quantum toroidal algebras are very different: the fundamental $\mathcal{U}_q(\hat{sl}_\infty)$-modules are thin, which is not the case for the quantum toroidal algebras $\mathcal{U}_q(sl_{n+1}^{tor})$ (see \cite[Section 4.1]{hernandez_algebra_2011}). Let us point out that these differences have consequences in our work: we obtain here more precise results for the fusion product of $\mathcal{U}_q(\hat{sl}_\infty)$-modules. Another difficulty met in \cite{mansuy_extremal_2013} is that the cyclicity of the Dynkin diagram associated to $\mathcal{U}_q(sl_{n+1}^{tor})$ bring some restricted conditions in the construction of extremal loop weight modules (see \cite[Proposition 3.11]{mansuy_extremal_2013}). We will see that it is possible here to associate extremal fundamental loop weight $\mathcal{U}_q(\hat{sl}_\infty)$-modules for all the nodes of the Dynkin diagram of type $A_\infty$.
In the first part, we give explicit formulae of the action on the fundamental \linebreak $\mathcal{U}_q(\hat{sl}_\infty)$-modules, using the $\mathcal{U}_q(sl_\infty)$-crystal structure on the set of semi-standard \linebreak tableaux. In the second part, we construct the fusion product of the $\mathcal{U}_q(\hat{sl}_\infty)$-modules $V(Y_{\ell, a})$ and $V(Y_{0, aq^{\ell}}^{-1})$, the action of $\mathcal{U}_q(\hat{sl}_\infty)$ being defined via the Drinfeld coproduct (Proposition \ref{tpexistudinf}). We consider a submodule $V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})$ of it, which we call the extremal fundamental loop weight module. We study the $\mathcal{U}_q(\hat{sl}_\infty)$-modules $V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})$ in the third part: these representations are integrable with basis labelled by semi-standard tableaux of shape $(\ell)$. The action of $\mathcal{U}_q(\hat{sl}_\infty)$ on it is described by the $\mathcal{U}_q(sl_\infty)$-crystal structure on the set of semi-standard tableaux of shape $(\ell)$ (Theorem \ref{thmformactinf}). Furthermore $V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})$ is an extremal loop weight module of $\ell$-weight $Y_{\ell, a}Y_{0, aq^\ell}^{-1}$ (Theorem \ref{thmelwminf}) which is irreducible as a $\mathcal{U}_q(\hat{sl}_\infty)$-module and as a $\mathcal{U}_q^h(\hat{sl}_\infty)$-module (Proposition \ref{felwmirredinf}).
\subsection{Action on the fundamental $\mathcal{U}_q(\hat{sl}_\infty)$-modules}
In this section we give explicit formulae of the action on the fundamental $\mathcal{U}_q(\hat{sl}_\infty)$-modules $V(Y_{\ell, a})$ ($\ell \in \mathbb{Z}, a \in \mathbb{C}^{\ast}$).
Let $\mathcal{T}_{\ell}^+ = \mathcal{T}_{\ell, 1}$ be the set of semi-standard tableaux
$$T = ( \cdots < i_{\ell - 2} < i_{\ell-1} < i_\ell ) \text{ such that } i_j = j \text{ for } j <<0.$$
We endow $\mathcal{T}_{\ell}^+$ with a structure of $\mathcal{U}_q(sl_\infty)$-crystal: the weight function is defined by setting
$$\mathrm{wt}(T) = \sum_{j \leq \ell} \Lambda_{i_j} - \Lambda_{i_{j}-1}$$
for all $T = ( \cdots < i_{\ell - 2} < i_{\ell-1} < i_\ell ) \in \mathcal{T}_\ell^+$. The Kashiwara operators $\tilde{e}_i$, $\tilde{f}_i$ are described in terms of tableaux: for $i \in \mathbb{Z}$ we have $\tilde{e}_i \cdot T = T'$ or $0$. Here $T'$ is obtained from $T$ by replacing $i+1$ by $i$. If it is not possible (i.e. when we have both $i+1$ and $i$ in $T$ or when $i+1$ does not occur in $T$), then it is zero. Similarly $\tilde{f}_i \cdot m_{T;j}= m_{T'';j}$ or $0$, where $T''$ is given by replacing $i$ by $i+1$.
\begin{prop}
Let $\ell \in \mathbb{Z}$ and $a \in \mathbb{C}^{\ast}$ and consider the fundamental $\mathcal{U}_q(\hat{sl}_\infty)$-module $V(Y_{\ell,a})$. Then there exists a basis of $V(Y_{\ell,a})$ indexed by $\mathcal{T}_\ell^+$ such that the action of $\mathcal{U}_q(\hat{sl}_\infty$) is given by (for convenience in the calculations, we remind the complex number $a \in \mathbb{C}^{\ast}$ in subscript)
\begin{eqnarray}\label{actonfundinf}
\begin{array}{rcl}
x_i^+(z) \cdot T_a &=& \displaystyle \sum_{p} \delta_{\{i_p=i+1\}} \delta(aq^{i+\ell + 1-2p}z) (\tilde{e}_i \cdot T)_a,\\
x_{i}^-(z) \cdot T_a &=& \displaystyle \sum_{p} \delta_{\{i_p=i\}} \delta(aq^{i + \ell + 1-2p}z) (\tilde{f}_i \cdot T)_a,\\
\phi_{i}^\pm(z) \cdot T_a &=& \begin{cases} \psi(aq^{i + \ell + 3 -2p}z)^{-1} \cdot T_a & \begin{array}{c} \text{ if there exists } p \leq \ell \text{ such that } \\ i_p=i+1 \text{ and } i_{p-1} \neq i,
\end{array}\\
\psi(aq^{i + \ell + 1-2p}z) \cdot T_a & \begin{array}{c} \text{ if there exists } p \leq \ell \text{ such that } \\ \text{ if } i_p=i \text{ and } i_{p+1} \neq i+1,
\end{array}\\
T_a & \text{ otherwise.} \end{cases}
\end{array}
\end{eqnarray}
where $T = ( \cdots < i_{\ell - 2} < i_{\ell-1} < i_\ell )$ is a semi-standard tableaux in $\mathcal{T}_\ell^+$.
\end{prop}
\noindent In particular for all $T \in \mathcal{T}_\ell^+$, $T_a$ is an $\ell$-weight vector of weight $m_T$.
\begin{proof}
This result is known for the fundamental $\mathcal{U}_q(\hat{sl}_{n+1})$-modules (see \cite[Theorem 4.11]{mansuy_quantum_2012}). We obtain the result for $\mathcal{U}_q(\hat{sl}_\infty)$ by inductive limit, using Theorem \ref{thmlimind}.
\end{proof}
We have an analogue result for the $\ell$-lowest weight modules. For that let $\mathcal{T}_s^-$ ($s \in \mathbb{Z}$) be the set of semi-standard tableaux
$$T = (i_{s+1} < i_{s + 2} < i_{s + 3} < \cdots ) \text{ such that } i_{j} = j \text{ for } j >> 0.$$ Then $(\mathcal{T}_s^-, \mathrm{wt}, \tilde{e}_i, \tilde{f}_i)$ is a $\mathcal{U}_q(sl_\infty)$-crystal. And for all $a \in \mathbb{C}^{\ast}$, there exists a basis of $V(Y_{s, a}^{-1})$ indexed by $\mathcal{T}_s^-$ such that the action of $\mathcal{U}_q(\hat{sl}_\infty)$ on it is given for all \linebreak $T = (i_{s+1} < i_{s + 2} < i_{s + 3} < \cdots ) \in \mathcal{T}_s^-$ by
\begin{eqnarray*}
\begin{array}{rcl}
x_i^+(z) \cdot T_{a} &=& \displaystyle \sum_{p} \delta_{\{i_p=i+1\}} \delta(aq^{i+s + 1-2p}z) (\tilde{e}_i \cdot T)_{a},\\
x_{i}^-(z) \cdot T_{a} &=& \displaystyle \sum_{p} \delta_{\{i_p=i\}} \delta(aq^{i + s + 1-2p}z) (\tilde{f}_i \cdot T)_{a},\\
\phi_{i}^\pm(z) \cdot T_{a} &=& \begin{cases} \psi(aq^{i + s + 3 -2p}z)^{-1} \cdot T_{a} & \begin{array}{c} \text{ if there exists } p \geq s+1 \text{ such that } \\ i_p=i+1 \text{ and } i_{p-1} \neq i,
\end{array}\\
\psi(aq^{i + s + 1-2p}z) \cdot T_{a} & \begin{array}{c} \text{ if there exists } p \geq s+1 \text{ such that } \\ \text{ if } i_p=i \text{ and } i_{p+1} \neq i+1,
\end{array}\\
T_{a} & \text{ otherwise.} \end{cases}
\end{array}
\end{eqnarray*}
\subsection{Fusion product of fundamental modules}
\begin{prop}\label{tpexistudinf}
Let $\ell \geq 1$ and $a \in \mathbb{C}^{\ast}$. The coproduct $\Delta$ is well-defined on the tensor product $V(Y_{\ell, a}) \otimes V(Y_{0, aq^{\ell}}^{-1})$ and it endows it with a structure of integrable \linebreak $\mathcal{U}_q(\hat{sl}_\infty)$-module.
\end{prop}
\begin{rem}
We have introduced a fusion product process in \cite[Section 3]{mansuy_extremal_2013} to define new integrable modules for the quantum toroidal algebras. This process can also be used in our situation to show Proposition \ref{tpexistudinf}. We give here a direct proof of it for the convenience of the reader. As it is pointed out above, the computations are simpler in that case, stemming from the thin property of the fundamental $\mathcal{U}_q(\hat{sl}_\infty)$-modules.
\end{rem}
\begin{proof}
Let us consider
\begin{align*}
T = ( \cdots < i_{\ell-2} < i_{\ell-1} < i_{\ell}) \in \mathcal{T}_\ell^+ \text{ and } T' = (j_1 < j_2 < j_3 < \cdots ) \in \mathcal{T}_0^-.
\end{align*}
We study the action of $x_i^{+}(z)$ on $T_a \otimes T'_{aq^{\ell}} \in V(Y_{\ell, a}) \otimes V(Y_{0, aq^{\ell}}^{-1})$:
\begin{align*}
\begin{array}{c}
x_i^+(z) \cdot (T_a \otimes T'_{aq^{\ell}}) = \sum_{r} \delta_{\{i_r=i+1\}} \delta(a q^{i + \ell + 1 - 2r}z) (\tilde{e}_i \cdot T)_a \otimes T'_{aq^{\ell}} + \\
\displaystyle \sum_{r, s} \left[ 1 + \delta_{\{i_r=i+1\}}\delta_{\{i_{r-1} \neq i\}}\left( \psi (q^{2(s-r)+2} )^{-1} - 1 \right) + \delta_{\{i_r=i\}}\delta_{\{i_{r+1} \neq i+1\}} \left( \psi ( q^{2(s-r)} ) - 1 \right) \right] \\
\times \delta_{\{j_s=i+1\}} \delta(a q^{i + \ell + 1 - 2r}z) T_a \otimes (\tilde{e}_i \cdot T')_{aq^{\ell}}.
\end{array}
\end{align*}
By definition of $\mathcal{T}_\ell^+$ and $\mathcal{T}_0^-$, we have $r \leq i$, $s \geq i+1$, and $s-r \geq 1$. Hence the coefficients of the series $x_i^+(z) \cdot (T_a \otimes T'_{aq^{\ell}})$ are well-defined. The result follows by Lemma \ref{lemsubminf}.
\end{proof}
Let us fix $\ell \in \mathbb{Z}$. For $T = ( \cdots < i_{\ell-2} < i_{\ell-1} < i_\ell )$ a semi-standard tableaux in $\mathcal{T}_\ell^+$, we define
$$\alpha_T = \sup \{ p \leq \ell \vert i_p = p \}.$$
Note that $\alpha_T$ is well-defined by definition of $\mathcal{T}_\ell^+$. In the same way, we define for all $T = ( i_{1} < i_{2} < i_{3} < \cdots ) \in \mathcal{T}_0^-$
$$\beta_T = \inf \{ p \geq 1 \vert i_p = p \}.$$
Assume that $\ell \geq 1$. We have endowed the sets $\mathcal{T}_{\ell}^+$ and $\mathcal{T}_0^-$ with a structure of $\mathcal{U}_q(sl_\infty)$-crystal. By the tensor product rules (see \cite{kashiwara_crystal_1994, kashiwara_bases_2002}), $\mathcal{T}_\ell^+ \otimes \mathcal{T}_0^-$ is a \linebreak $\mathcal{U}_q(sl_\infty)$-crystal. Let us denote by $\mathcal{T}_\ell$ the subset of $\mathcal{T}_\ell^+ \otimes \mathcal{T}_0^-$ consisting of
$$T \otimes T' \text{ with } T \in \mathcal{T}_\ell^{+}, T' \in \mathcal{T}_0^- \text{ and } \alpha_T \geq \beta_{T'}-1.$$
\begin{prop}
The set $\mathcal{T}_\ell$ is the connected sub-$\mathcal{U}_q(sl_\infty)$-crystal of $\mathcal{T}_\ell^+ \otimes \mathcal{T}_0^-$ generated by $(\cdots < \ell-2 < \ell-1 < \ell) \otimes (1 < 2 < 3 < \cdots)$.
\end{prop}
\begin{proof}
Set $T \otimes T' \in \mathcal{T}_\ell$ and $i \in \mathbb{Z}$. Let us show that $\tilde{e}_i \cdot (T \otimes T') \in \mathcal{T}_\ell \cup \{0\}$ (the proof for the Kashiwara operators $\tilde{f}_i$ is analogous). We have the following equalities
\begin{align*}
\alpha_{\tilde{e}_i \cdot T} & = \begin{cases} \alpha_{T} & \text{ if } \tilde{e}_i \cdot T' \neq 0 \text{ and } i > \alpha_{T}+1, \\
\alpha_{T} + 1 & \text{ if } i = \alpha_{T}+1,
\end{cases}\\
\beta_{\tilde{e}_i \cdot T'} & = \begin{cases} \beta_{T'} & \text{ if } \tilde{e}_i \cdot T' \neq 0 \text{ and } i < \beta_{T'}-1, \\
\beta_{T'} + 1 & \text{ if } i = \beta_{T'}-1.
\end{cases}
\end{align*}
In particular $\alpha_{\tilde{e}_i \cdot T} \leq \alpha_T$. So $\tilde{e}_i \cdot (T \otimes T')$ belongs to $\mathcal{T}_\ell \cup \{0\}$, except perhaps when $\alpha_T = \beta_{T'}-1 = i$. But in that case, we have by the tensor product rules
\begin{align*}
\tilde{e}_i \cdot (T \otimes T') = (\tilde{e}_i \cdot T) \otimes T' = 0 \in \mathcal{T}_\ell \cup \{0\}.
\end{align*}
Hence $\mathcal{T}_\ell$ is a sub-$\mathcal{U}_q(sl_\infty)$-crystal of $\mathcal{T}_\ell^+ \otimes \mathcal{T}_0^-$. And one can show by straightforward computations that it is the connected sub-$\mathcal{U}_q(sl_\infty)$-crystal of $\mathcal{T}_\ell^+ \otimes \mathcal{T}_0^-$ generated by $$(\cdots < \ell-2 < \ell-1 < \ell) \otimes (1 < 2 < 3 < \cdots).$$
\end{proof}
Denote by $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ ($\ell \geq 1, a \in \mathbb{C}^{\ast}$) the subvector space of $V(Y_{\ell, a}) \otimes V(Y_{0, aq^\ell}^{-1})$ generated by $T_a \otimes T'_{aq^{\ell}}$ with $T \otimes T' \in \mathcal{T}_\ell$.
\begin{prop}
The coproduct $\Delta$ endows $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ with a structure of \linebreak $\mathcal{U}_q(\hat{sl}_\infty)$-module. We call it the extremal fundamental loop weight module.
\end{prop}
\begin{proof}
The action of $x_i^\pm(z)$ ($i \in \mathbb{Z}$) is well-defined on $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ by Proposition \ref{tpexistudinf}. By Lemma \ref{lemsubminf}, it remains to show that $x_i^+(z) \cdot (T_a \otimes T'_{aq^{\ell}}) \in V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ for all $T \otimes T' \in \mathcal{T}_\ell$ (the proof being similar for $x_i^-(z)$). As in the last proof, it holds except perhaps when $\alpha_T = \beta_{T'}-1 = i$. In that case we have
\begin{align*}
x_i^+(z) \cdot T_a \otimes T'_{aq^{\ell}} = \psi(q^2) \delta(aq^{\ell-i-1}z) \cdot T_a \otimes (\tilde{e}_i \cdot T')_{aq^{\ell}} = 0 \in V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})
\end{align*}
Hence the coproduct $\Delta$ endows $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ with a structure of $\mathcal{U}_q(\hat{sl}_\infty)$-module.
\end{proof}
\subsection{Study of the extremal fundamental loop weight module $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$}
Let $\mathcal{T}_{[1, \ell]}$ be the set of semi-standard tableaux $T = (i_1 < i_2 < \cdots < i_\ell)$ of shape $(\ell)$. We consider the application $\Pi : \mathcal{T}_\ell \rightarrow \mathcal{T}_{[1, \ell]}$ defined for all $T \otimes T' \in \mathcal{T}_\ell$ by
$$\Pi(T \otimes T') = (j_1 < j_2 < \cdots < j_{\alpha_T} < i_{\alpha_T + 1} < \cdots < i_\ell)$$
where $T = (\cdots < i_{\ell-2} < i_{\ell-1} < i_{\ell})$ and $T' = (j_1 < j_2 < j_3 < \cdots )$.
This application is bijective, its inverse sending $T = (i_1 < i_2 < \cdots < i_\ell ) \in \mathcal{T}_{[1, \ell]}$ on
$$( \cdots < 0 < \max(i_1, 1) < \cdots < \max(i_\ell, \ell)) \otimes (\min(i_1, 1) < \cdots < \min(i_\ell, \ell) < \ell+1 < \cdots).$$
\begin{prop}
The application $\Pi : \mathcal{T}_\ell \rightarrow \mathcal{T}_{[1, \ell]}$ is an isomorphism of $\mathcal{U}_q(sl_\infty)$-crystals.
\end{prop}
\begin{proof}
Let $T = (\cdots < i_{\ell-2} < i_{\ell-1} < i_{\ell})$ and $T' = (j_1 < j_2 < j_3 < \cdots )$ be semi-standard tableaux such that $T \otimes T' \in \mathcal{T}_\ell$, and $i \in \mathbb{Z}$. We have to consider the following cases
\begin{itemize}
\item[-] $ i > \alpha_T$: then we have $\beta_{T'} \leq i$ and $i, i+1 \in T'$. By the tensor product rules, we have $$\tilde{f}_i \cdot (T \otimes T') = (\tilde{f}_i \cdot T) \otimes T'.$$ Hence,
\begin{align*}
\Pi(\tilde{f}_i \cdot (T \otimes T')) &= \begin{cases} (j_1 < \cdots < i_{p} + 1 < \cdots < i_\ell) \begin{array}{c}
\text{ if there exists } p > \alpha_T \text{ such that }\\
i_p = i \text{ and } i_{p+1} \neq i+1,
\end{array}\\
0 \text{ otherwise}
\end{cases}\\
&= \tilde{f}_i \cdot \Pi(T \otimes T')
\end{align*}
\item[-] $i < \alpha_T$: then we have $i, i+1 \in T$ and by the tensor product rules, $$\tilde{f}_i \cdot (T \otimes T') = T \otimes (\tilde{f}_i \cdot T').$$ Furthermore, $j_{\alpha_T} \leq j_{\alpha_T+1}-1 = \alpha_T$. Hence,
\begin{align*}
\Pi(\tilde{f}_i \cdot (T \otimes T')) &= \begin{cases} (j_1 < \cdots < j_{p} + 1 < \cdots < i_\ell) \begin{array}{c}
\text{ if there exists } 1 \leq p < \alpha_T \text{ such that }\\
i_p = i \text{ and } i_{p+1} \neq i+1,
\end{array} \\
0 \text{ otherwise}
\end{cases}\\
&= \tilde{f}_i \cdot \Pi(T \otimes T')
\end{align*}
\item[-] $i = \alpha_T$: then we have $i+1 \notin T$ and $\beta_{T'} \leq i+1$.
Assume that $\beta_{T'} < i+1$. In that case, $i, i+1 \in T'$ and we have $$\tilde{f}_i \cdot (T \otimes T') = (\tilde{f}_i \cdot T) \otimes T'.$$ Furthermore $\alpha_{\tilde{f}_i \cdot T} = \alpha_{T} - 1$ and $j_{\alpha_T} = i$. So we have
\begin{align*}
\Pi(\tilde{f}_i \cdot (T \otimes T')) &= (j_1 < \cdots < j_{\alpha_T-1} < i+1 < i_{\alpha_T + 1} < \cdots < i_\ell) \\
&= \tilde{f}_i \cdot (j_1 < \cdots < j_{\alpha_T-1} < i = j_{\alpha_T} < i_{\alpha_T + 1} < \cdots < i_\ell) \\
&= \tilde{f}_i \cdot \Pi(T \otimes T').
\end{align*}
If $\beta_{T'} = i+1$, we have $i \notin T'$. By the tensor product rules we get
$$\tilde{f}_i \cdot (T \otimes T') = T \otimes (\tilde{f}_i \cdot T') = 0.$$
Furthermore we have $\alpha_T = \beta_{T'}-1$ and $j_{\alpha_T} < \alpha_T = i < i_{\alpha_T+1}$. Then \linebreak $i \notin \Pi(T \otimes T')$ and $\tilde{f}_i \cdot \Pi(T \otimes T') = 0$.
\end{itemize}
As the application $\Pi : \mathcal{T}_\ell \rightarrow \mathcal{T}_{[1, \ell]}$ is also bijective, this is an isomorphism of \linebreak $\mathcal{U}_q(sl_\infty)$-crystals.
\end{proof}
\begin{rem}\label{remlienmoncriinf}
Let us describe this isomorphism at the level of the monomial crystals (see \cite{kashiwara_realizations_2003, mansuy_quantum_2012, nakajima_$t$-analogs_2003} for its definition). For that denote by $\mathcal{M}_\ell$ the sub-$\mathcal{U}_q(sl_\infty)$-crystal of $\mathcal{M}(Y_{\ell, a}) \otimes \mathcal{M}(Y_{0, aq^{\ell}}^{-1})$ generated by $Y_{\ell, a} \otimes Y_{0, aq^{\ell}}^{-1}$. Let us consider also $\mathcal{M}(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})$ the monomial crystal generated by $Y_{\ell, a}Y_{0, aq^{\ell}}^{-1}$. By that we have done above, we have
\begin{align*}
\mathcal{M}(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1}) = \lbrace m_T = \prod_{1 \leq j \leq \ell} \ffbox{i_j}_{aq^{\ell+1-2j}} \ \vert \ T = (i_1 < i_2 < \cdots < i_\ell) \in \mathcal{T}_{[1, \ell]} \rbrace
\end{align*}
and
\begin{align*}
\mathcal{M}_\ell = \lbrace m_T \otimes m_{T'} \vert T \otimes T' \in \mathcal{T}_\ell \rbrace.
\end{align*}
In the monomial realization, $\Pi$ is simply given by the product in $A$
$$\Pi(m_1 \otimes m_2) = m_1 \times m_2 \text{ with } m_1 \otimes m_2 \in \mathcal{M}_\ell.$$
\end{rem}
As a direct consequence, we have
\begin{prop}
Let $\ell \geq 1$ and $a \in \mathbb{C}^{\ast}$. The $\mathcal{U}_q(\hat{sl}_\infty)$-module $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is thin, and we have
\begin{equation}\label{qchainf}
\chi_q(V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})) = \sum_{T \in \mathcal{T}_{[1, \ell]}} m_T
\end{equation}
where $m_T = \displaystyle \prod_{1 \leq j \leq \ell} \ffbox{i_j}_{aq^{\ell+1-2j}}$.
\end{prop}
\begin{proof}
Let us recall that we have a basis of $\ell$-weight vectors of $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$: it is given by the $T_a \otimes T'_{aq^{\ell}}$ in $ \mathcal{T}_\ell$. Furthermore by (\ref{deltacart}) and the remark above, the $\ell$-weight of the vector $T_a \otimes T'_{aq^{\ell}}$ is
$$m_T \times m_{T'} = m_{\Pi(T \otimes T')}.$$
Hence we have
$$\chi_q(V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})) = \sum_{T \in \mathcal{T}_{[1, \ell]}} m_T.$$
\end{proof}
\begin{rem}
In particular we have shown here that the monomial crystal $\mathcal{M}(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})$ generated by $Y_{\ell, a}Y_{0, aq^{\ell}}^{-1}$ is closed in the sense of \cite[Definition 3.6]{mansuy_quantum_2012}.
\end{rem}
By the bijection $\Pi$, we parametrize the basis of $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ by the $\mathcal{U}_q(sl_\infty)$-crystal $\mathcal{T}_{[1, \ell]}$. Let us explicit formulae of the action on it.
\begin{thm}\label{thmformactinf}
The action of $\mathcal{U}_q(\hat{sl}_\infty)$ on $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is given for all \linebreak $T = (i_1 < i_2 < \cdots < i_\ell) \in \mathcal{T}_{[1, \ell]}$ by
\begin{eqnarray}\label{formactinf}
\begin{array}{rcl}
x_i^+(z) \cdot T_a &=& \displaystyle \sum_{1 \leq p \leq \ell} \delta_{\{i_p=i+1\}} \delta(aq^{i+\ell + 1-2p}z) (\tilde{e}_i \cdot T)_a,\\
x_{i}^-(z) \cdot T_a &=& \displaystyle \sum_{1 \leq p \leq \ell} \delta_{\{i_p=i\}} \delta(aq^{i + \ell + 1-2p}z) (\tilde{f}_i \cdot T)_a,\\
\phi_{i}^\pm(z) \cdot T_a &=& \begin{cases} \psi(aq^{i + \ell + 3 -2p}z)^{-1} \cdot T_a & \begin{array}{c} \text{ if there exists } 1 \leq p \leq \ell \text{ such that } \\ i_p=i+1 \text{ and } i_{p-1} \neq i,
\end{array}\\
\psi(aq^{i + \ell + 1-2p}z) \cdot T_a & \begin{array}{c} \text{ if there exists } 1 \leq p \leq \ell \text{ such that } \\ \text{ if } i_p=i \text{ and } i_{p+1} \neq i+1,
\end{array}\\
T_a & \text{ otherwise.} \end{cases}
\end{array}
\end{eqnarray}
\end{thm}
\begin{rem}
Note that these formulae are identical to the one (\ref{actonfundinf}) obtained for the fundamental $\mathcal{U}_q(\hat{sl}_\infty)$-modules. Actually this result generalizes \cite[Theorem 4.11]{mansuy_quantum_2012} for the extremal fundamental loop weight modules of $\mathcal{U}_q(sl_{n+1}^{tor})$. In fact one can rewrite the action on $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ in the context of monomial realizations: it is completely describe by the monomial $\mathcal{U}_q(sl_\infty)$-crystal $\mathcal{M}(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ and the formulae given in \cite[Theorem 4.11]{mansuy_quantum_2012}.
\end{rem}
\begin{proof}
We have shown that the $\mathcal{U}_q(\hat{sl}_\infty)$-module $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is thin. So to determine the action on this module, it suffices to know the action of the horizontal quantum affine subalgebra $\mathcal{U}_q^h(\hat{sl}_\infty)$. The action of all the algebra $\mathcal{U}_q(\hat{sl}_\infty)$ can be deduced from it by (\ref{relxcartpl}) and (\ref{relxcartmo}).
So let us determine the action of $x_{i,0}^+$ ($i \in \mathbb{Z}$) on $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ (we proceed in the same way for $x_{i, 0}^{-}$). Let $T = (\cdots < i_{\ell-2} < i_{\ell-1} < i_{\ell})$ and $T' = (j_1 < j_2 < j_3 < \cdots )$ be semi-standard tableaux such that $T \otimes T' \in \mathcal{T}_\ell$. Denote by $(T \otimes T')_a$ the vector $T_a \otimes T'_{aq^{\ell}} \in V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$. Then $x_{i, 0}^- \cdot (T \otimes T')_a$ is equal to
\begin{eqnarray*}
\begin{array}{c}
\sum_{s \geq 1} \delta_{\{j_s = i \}} \cdot T_a \otimes (\tilde{f}_i \cdot T')_{aq^{\ell}}
+ (\tilde{f}_i \cdot T)_a \otimes T'_{aq^{\ell}} \cdot \displaystyle \sum_{ r \leq \ell, s \geq 1} \delta_{\{i_r = i \}} \times \\
\left( 1 + \delta_{\{j_s = i + 1 \}} \delta_{\{j_{s-1} \neq i \}} \left( \psi(q^{2(r-s)+2})^{-1} - 1 \right) + \delta_{\{j_s = i \}} \delta_{\{j_{s+1} \neq i+1 \}} \left( \psi(q^{2(r-s)}) - 1 \right) \right) .
\end{array}
\end{eqnarray*}
\noindent We have to consider the following cases
\begin{itemize}
\item[-] $ i > \alpha_T$: then $i, i+1 \in T'$. So we get
\begin{eqnarray*}
x_{i, 0}^- \cdot (T \otimes T')_a = (\tilde{f}_i \cdot T)_a \otimes T'_{aq^{\ell}} = \left(\tilde{f}_i \cdot (T \otimes T') \right)_a.
\end{eqnarray*}
\item[-] $i < \alpha_T$: then we have $i, i+1 \in T$ and
\begin{eqnarray*}
x_{i, 0}^- \cdot (T \otimes T')_a = T_a \otimes (\tilde{f}_i \cdot T')_{aq^{\ell}} = \left( \tilde{f}_i \cdot (T \otimes T') \right)_a.
\end{eqnarray*}
\item[-] $i = \alpha_T$: then we have $i \in T, i+1 \notin T$ and $\beta_{T'} \leq i+1$.
Assume that $\beta_{T'} < i+1$. In this case, $i, i+1 \in T'$ and we get
\begin{eqnarray*}
x_{i, 0}^- \cdot (T \otimes T')_a = (\tilde{f}_i \cdot T)_a \otimes T'_{aq^{\ell}} = \left( \tilde{f}_i \cdot (T \otimes T') \right)_a.
\end{eqnarray*}
If $\beta_{T'} = i+1$, we have $i \notin T', \alpha_T = \beta_{T'}-1$ and
\begin{align*}
x_{i, 0}^- \cdot (T \otimes T')_a & = \psi(q^{2(\alpha_T - \beta_{T'}) + 2})^{-1} (\tilde{f}_i \cdot T)_a \otimes T'_{aq^{\ell}} = \psi(1)^{-1} (\tilde{f}_i \cdot T)_a \otimes T'_{aq^{\ell}} = 0 \\
& = \left( \tilde{f}_i \cdot (T \otimes T') \right)_a.
\end{align*}
\end{itemize}
So we have shown that $x_{i, 0}^- \cdot (T \otimes T')_a = \left( \tilde{f}_i \cdot (T \otimes T') \right)_a$. We proceed in the same way for $x_{i,0}^+$.
\end{proof}
As consequences of this theorem, we obtain
\begin{prop}\label{felwmirredinf}
Fix $\ell \geq 1$ and $a \in \mathbb{C}^{\ast}$. Then $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is irreducible as a $\mathcal{U}_q(\hat{sl}_\infty)$-module and as a $\mathcal{U}_q^h(\hat{sl}_\infty)$-module.
\end{prop}
\noindent To prove this proposition, we use the following result (which is an analogue of \cite[Lemma 4.8]{mansuy_quantum_2012}).
\begin{lem}\label{lirepcryinf}
Let $\mathcal{B}$ be a $\mathcal{U}_q(sl_\infty)$-crystal. Assume that $V$ is a $\mathcal{U}_q(sl_\infty)$-module with basis $(v_b)_{b \in \mathcal{B}}$ indexed by $\mathcal{B}$ which satisfies
\begin{eqnarray}\label{lienrepcryinf}
\mathrm{wt}(v_b) = \mathrm{wt}(b), \ (x_{i}^{+})^{(k)} \cdot v_b = v_{\tilde{e}_i^k \cdot b} \text{ and } (x_{i}^{-})^{(k)} \cdot v_b = v_{\tilde{f}_i^k \cdot b}
\end{eqnarray}
for all $b \in \mathcal{B}, i \in I$ and $k \in \mathbb{N}$, where $v_0 = 0$ by convention. If the element $b \in \mathcal{B}$ is extremal of weight $\lambda$, then the vector $v_b$ is an extremal vector of weight $\lambda$. Furthermore if the crystal $\mathcal{B}$ is connected, then the $\mathcal{U}_q(sl_\infty)$-module $V$ is cyclic generated by any $v_b$ with $b \in \mathcal{B}$.
\end{lem}
\noindent The proof is similar to the one of \cite[Lemma 4.8]{mansuy_quantum_2012}. We recall it for the convenience of the reader.
\begin{proof}
Assume that $b \in \mathcal{B}$ is extremal of weight $\lambda$: there exists $\{b_w\}_{w \in W}$ such that $b_{Id}= b$ and
\begin{eqnarray}\label{eqinterinf}
\begin{array}{c}
\tilde{e}_i \cdot b_w=0 \text{ and }(\tilde{f}_i)^{w(\lambda)(h_i)} \cdot b_w= b_{s_i(w)} \text{ if } w(\lambda)(h_i)\geq 0,\\
\tilde{f}_i \cdot b_w=0 \text{ and }(\tilde{e}_i)^{-w(\lambda)(h_i)} \cdot b_w= b_{s_i(w)} \text{ if } w(\lambda)(h_i)\leq 0.
\end{array}
\end{eqnarray}
For all $w \in W$, set $v_w = v_{b_w}$. By (\ref{lienrepcryinf}) and (\ref{eqinterinf}), $\{v_w\}_{w \in W}$ satisfies $v_{Id} = v_{m}$ and
$$x_i^{\pm} \cdot v_w=0 \text{ if } \pm w(\lambda)(h_i)\geq 0 \text{ and }(x_i^{\mp})^{(\pm w(\lambda)(h_i))} \cdot v_w=v_{s_i(w)}.$$
Hence the vector $v_m$ is extremal of weight $\lambda$.
Assume that the crystal $\mathcal{B}$ is connected and fix $b \in \mathcal{B}$. For $b' \in \mathcal{B}$, there exists a product $s$ of Kashiwara operators such that $s(b) = b'$. Consider the corresponding operator $S \in \mathcal{U}_q(sl_\infty)$ at the level of $V$, i.e. $S$ has the same expression as $s$ where the operators $\tilde{e}_i^k$ (resp. $\tilde{f}_i^k$) are replaced by $(x_{i}^{+})^{(k)}$ (resp. $(x_i^{-})^{(k)}$) in the product ($k \in \mathbb{N}, i \in I$). By (\ref{lienrepcryinf}), $S(v_b) = v_{s(b)} = v_{b'}$ and the $\mathcal{U}_q(sl_\infty)$-module $V$ is cyclic generated by $v_b$.
\end{proof}
We are now able to prove Proposition \ref{felwmirredinf}.
\begin{proof}
Let us consider a non trivial sub-$\mathcal{U}_q^h(\hat{sl}_\infty)$-module $V$ of $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$. By the $q$--character formula (\ref{qchainf}) all the weight spaces of $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ are of dimension one. So there exists $T \in \mathcal{T}_{[1, \ell]}$ such that $T_a \in V$. By Lemma \ref{lirepcryinf}, the vector $T_a$ generates $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ and $V = V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$. Hence $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is simple as a $\mathcal{U}_q(\hat{sl}_\infty)$-module and as a $\mathcal{U}_q^h(\hat{sl}_\infty)$-module.
\end{proof}
\begin{thm}\label{thmelwminf}
Fix $\ell \geq 1$ and $a \in \mathbb{C}^{\ast}$. Then $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is an extremal loop weight $\mathcal{U}_q(\hat{sl}_\infty)$-module generated by the vector $(1 < 2 < \cdots < \ell)_a$ of $\ell$-weight $Y_{\ell, a}Y_{0, aq^\ell}^{-1}$.
\end{thm}
\begin{proof}
By the fusion product construction, $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is an integrable $\mathcal{U}_q(\hat{sl}_\infty)$-module. As it is also irreducible, $V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ is generated by the vector $T =(1 < 2 < \cdots < \ell)_a$. The extremality of this vector for the horizontal quantum affine subalgebra follows from Lemma \ref{lirepcryinf} and the fact that $T$ is an extremal element in $\mathcal{T}_{[1, \ell]}$. So it remains to prove that for all $w \in V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ and $J = [a, b] \subset \mathbb{Z}$ finite, $\mathcal{U}_q(\hat{sl}_\infty) \cdot w$ is finite-dimensional: this follows by Lemma \ref{tpalgv}.
\end{proof}
\begin{rem}
Set $n \in \mathbb{N}^{\ast}, \ell \geq 1$ and $a \in \mathbb{C}^{\ast}$. Let us consider the \linebreak $\mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]}$-modules
$$V(Y_{\ell, a})_{[-n, \ell + n]} = \mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]} \cdot v^+ \text{ and } V(Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]} = \mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]} \cdot v^-$$
where $v^+$ and $v^-$ are $\ell$-highest weight vector and $\ell$-lowest weight vector of the \linebreak $\mathcal{U}_q(\hat{sl}_{\infty})$-modules $V(Y_{\ell, a})$ and $V(Y_{0, aq^{\ell}}^{-1})$ respectively. Using the result of this section, we deduce that $\Delta$ endows the tensor product $$V(Y_{\ell, a})_{[-n, \ell + n]} \otimes V(Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]}$$ with a structure of $\mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]}$-module. Let us set
$$V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]} = \mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]} \cdot v^+ \otimes v^- \subset V(Y_{\ell, a})_{[-n, \ell + n]} \otimes V(Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]}.$$
We give some facts about the $\mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]}$-module $V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]}$ (consequences of the study done in this section): it is irreducible and finite-dimensional. In particular, this is a simple $\ell$-highest weight $\mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]}$-module. Its $q$--character is
\begin{eqnarray*}
\chi_q(V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]}) = \sum_{T \in (\mathcal{T}_\ell)_{[-n, \ell + n]}} m_T
\end{eqnarray*}
where $(\mathcal{T}_\ell)_{[-n, \ell + n]}$ is the finite set of semi-standard tableaux $$T = (-n \leq i_1 < i_2 < \cdots < i_{\ell} \leq \ell + n + 1).$$ The $\ell$-highest weight of $V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]}$ corresponds to the dominant monomial in its $q$--character formula. It is obtained for $T = (-n < -n+1 < \cdots < -n+ \ell-1 )$. Hence we have
$$V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]} = V(m_T)_{[-n, \ell + n]} = V(Y_{-n+ \ell -1, aq^{-n-1}})_{[-n, \ell + n]}.$$
Then $V(Y_{\ell, a}Y_{0, aq^{\ell}}^{-1})_{[-n, \ell + n]}$ is the fundamental $\mathcal{U}_q(\hat{sl}_\infty)_{[-n, \ell + n]}$-module of $\ell$-highest weight $Y_{-n+ \ell -1, aq^{-n-1}}$.
\end{rem}
\section{Fusion product of extremal fundamental loop weight modules}
In this section we study fusion products of extremal loop weight modules $V(Y_{\ell, a}Y_{0, a q^\ell}^{-1})$ with $\ell \geq 1$ and $a \in \mathbb{C}^{\ast}$. We obtain new families of extremal loop weight modules with basis labelled by semi-standard tableaux.
In the first part, we determine existence conditions of $\mathcal{U}_q(\hat{sl}_\infty)$-module structure on the tensor product of extremal fundamental loop weight modules $V(Y_{\ell, a}Y_{0, a q^\ell}^{-1})$. In the second part we study the case of fusion products of $k$ extremal fundamental loop weight modules $V(Y_{\ell, a_i}Y_{0, a_i q^\ell}^{-1})$ with generic non-zero complex parameters $a_1, a_2, \cdots , a_k \in \mathbb{C}^{\ast}$ (Theorem \ref{thmgencaseinf}): it is an extremal loop weight module of $\ell$-weight
$$Y_{\ell, a_1}\cdots Y_{\ell, a_k}Y_{0, a_1q^\ell}^{-1} \cdots Y_{0, a_k q^\ell}^{-1}.$$
In the third part we treat the case of fusion products of extremal fundamental loop weight modules when parameters are chosen non-generic. More precisely we consider the fusion product of vector representations $V(Y_{1,a}Y_{0, aq}^{-1})$ when the set of parameters forms a $q$-segment $\{a, aq^{-2}, \cdots, aq^{-2(k-1)}\}$ ($k \in \mathbb{N}^{\ast}$, $ a \in \mathbb{C}^{\ast}$). We recover in that way all the extremal fundamental loop weight modules (Theorem \ref{thmtpvrisomeflwm}). Furthermore we obtain new extremal loop weight modules of $\ell$-weight
$$Y_{\ell, aq^{-\ell+1}}Y_{\ell, aq^{-\ell+3}} \cdots Y_{\ell, aq^{-\ell+1+2k}} Y_{0,aq}^{-1}Y_{0,aq^3}^{-1} \cdots Y_{0,aq^{1+2k}}^{-1} \text{ (Theorem \ref{thmtpelwminf})}.$$
\subsection{Existence conditions}
Set $\ell \geq 1$. Let us consider the tensor product
$$V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1}) \text{ with } a, b \in \mathbb{C}^{\ast}.$$
We determine when the action of $\mathcal{U}_q(\hat{sl}_\infty)$ is well-defined on it.
\begin{prop}\label{proptpdefinf}
\begin{itemize}
\item[(i)] The action of $\mathcal{U}_q(\hat{sl}_\infty)$ is well-defined on the tensor product $V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$ if and only if $$\dfrac{a}{b} \notin \{q^{-2\ell+2}, q^{-2\ell+4}, \cdots, q^{2\ell-2} \}.$$
\item[(ii)] Assume that $\dfrac{a}{b} = q^{- 2\ell}$. The module $V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$ has a submodule spanned by vectors of the form
$$T_a \otimes T'_b \text{ with } i_1 \leq j_\ell$$
where $T = (i_1 < \cdots < i_\ell)$ and $T' = (j_1 < \cdots < j_\ell)$. The submodule and the quotient module are irreducible and thin.
\item[(iii)] Assume that $\dfrac{a}{b} = q^{2\ell}$. The module $V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$ has a submodule spanned by vectors of the form
$$T_a \otimes T'_b \text{ with } i_\ell < j_1$$
where $T = (i_1 < \cdots < i_\ell)$ and $T' = (j_1 < \cdots < j_\ell)$. The submodule and the quotient module are irreducible and thin.
\item[(iv)] In all the other cases, $V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$ is a thin, irreducible \linebreak $\mathcal{U}_q(\hat{sl}_\infty)$-module.
\end{itemize}
\end{prop}
\begin{rem}
This result generalizes the one for the (specialized) vector representations of $\mathcal{U}_q(sl_{n+1}^{tor})$ $(n \geq 2)$ given in \cite{feigin_representations_2013, mansuy_extremal_2013}.
\end{rem}
\begin{proof}
Let $T, T' \in \mathcal{T}_{[1, \ell]}$ and $a, b \in \mathbb{C}^{\ast}$. We determine the action of $x_i^-(z)$ on \linebreak $T_a \otimes T'_b \in V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$: it is equal to
\begin{eqnarray*}
\begin{array}{c}
\displaystyle \sum_{1 \leq s \leq \ell} \delta_{\{j_s = i \}} \delta(bq^{i + \ell + 1 - 2s}) \cdot T_a \otimes (\tilde{f}_i \cdot T')_b + \displaystyle \sum_{1 \leq r, s \leq \ell} \delta_{\{i_r = i \}} \delta(aq^{i + \ell + 1 - 2r}) \\
\times \left( 1 + \delta_{\{j_s = i + 1 \}} \delta_{\{j_{s-1} \neq i \}} \left( \psi(\frac{b}{a} q^{2(r-s)+2})^{-1} - 1 \right) + \delta_{\{j_s = i \}} \delta_{\{j_{s+1} \neq i+1 \}} \left( \psi(\frac{b}{a} q^{2(r-s)}) - 1 \right) \right) \\
\times (\tilde{f}_i \cdot T)_a \otimes T'_b.
\end{array}
\end{eqnarray*}
So the action of the operators $x_i^-(z)$ is well-defined on $ V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$ if and only if
$$\dfrac{a}{b} \neq q^{2(r-s)} \text{ for all } 1 \leq r, s \leq \ell$$
or in an equivalent way, $\dfrac{a}{b} \notin \{q^{-2\ell+2}, q^{-2\ell+4}, \cdots, q^{2\ell-2} \}$. We proceed in the same way for the $x_i^+(z)$. We obtain the first assertion by Lemma \ref{lemsubminf}.
Now assume that $\dfrac{a}{b} = q^{-2\ell}$. Let us consider the subvector space $V$ of $$ V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$$ spanned by vectors of the form
$$T_a \otimes T'_b \text{ with } i_1 \leq j_\ell$$
where $T = (i_1 < \cdots < i_\ell)$ and $T' = (j_1 < \cdots < j_\ell)$. Let $i \in \mathbb{Z}$ and $T = (i < i_2 < \cdots < i_{\ell})$, $T' = (j_1 < \cdots < j_{\ell-1} < i)$ be semi-standard tableaux. We have
\begin{align*}
x_i^-(z) \cdot T_a \otimes T'_b &= \delta(aq^{i + \ell + 1}z) T_a \otimes (\tilde{f}_i \cdot T')_b + \delta(aq^{i+\ell-1}z) \psi(q^{2}) (\tilde{f}_i \cdot T)_a \otimes T'_b\\
& = \delta(aq^{i + \ell + 1}) T_a \otimes (\tilde{f}_i \cdot T')_b,\\
x_{i-1}^+(z) \cdot T_a \otimes T'_b &= \delta(aq^{i+\ell-1}z) (\tilde{e}_{i-1} \cdot T)_a \otimes T'_{b} + \delta(aq^{i+\ell+1}z) \psi (1)^{-1} T_a \otimes (\tilde{e}_{i-1} \cdot T')_{aq^{2\ell}}\\
&= \delta(aq^{i+\ell-1}z) (\tilde{e}_{i-1} \cdot T)_a \otimes T'_{b}.
\end{align*}
By these computations, $V$ is a sub-$\mathcal{U}_q(\hat{sl}_\infty)$-module of $ V(Y_{\ell, a}Y_{0, a q^\ell}^{-1}) \otimes V(Y_{\ell, b}Y_{0, b q^\ell}^{-1})$. We proceed in the same way for the third point. The thin property and the irreducibility of the modules in $(ii)$, $(iii)$ and $(iv)$ follow by straightforward computations.
\end{proof}
\subsection{The generic case}
Let us give a first result about the fusion product of extremal fundamental loop weight modules when the non-zero complex parameters are chosen generics.
\begin{thm}\label{thmgencaseinf}
Let $\ell \geq 1$, $k \in \mathbb{N}^{\ast}$ and $a_1, \cdots, a_k \in \mathbb{C}^{\ast}$ be such that $\frac{a_i}{a_j} \neq 1$ and $\frac{a_i}{a_j} \neq q^{\pm 2}$ for all $i < j$. Then the fusion product of extremal fundamental loop weight modules $$V = V(Y_{\ell, a_1}Y_{0, a_1 q^\ell}^{-1}) \otimes V(Y_{\ell, a_2}Y_{0, a_2 q^\ell}^{-1}) \otimes \cdots \otimes V(Y_{\ell, a_k}Y_{0, a_k q^\ell}^{-1})$$ is an irreducible extremal loop weight module of $\mathcal{U}_q(\hat{sl}_\infty)$ of $\ell$-weight $$Y_{\ell, a_1}\cdots Y_{\ell, a_k}Y_{0, a_1q^\ell}^{-1} \cdots Y_{0, a_k q^\ell}^{-1}.$$
\end{thm}
\begin{proof}
As a direct consequence of Proposition \ref{proptpdefinf}, the $\mathcal{U}_q(\hat{sl}_\infty)$-module $V$ is irreducible. Furthermore as $V(Y_{\ell, a_j}Y_{0, a_j q^\ell})$ are extremal loop weight modules for all $1 \leq j \leq k$, there fusion product $V$ satisfies the third condition of Definition \ref{defelminf}. Eventually for $T \in \mathcal{T}_{[1, \ell]}$, set $T' = \tilde{f}_i \cdot T$ and $T'' = \tilde{e}_i \cdot T$. We have for all $i \in \mathbb{Z}$ (see the proof of \cite[Theorem 5.3]{mansuy_extremal_2013})
$$(x_{i,0}^{-})^{(k)} \cdot T_{a_1} \otimes \cdots \otimes T_{a_k} = T'_{a_1} \otimes \cdots \otimes T'_{a_k}$$
and
$$(x_{i,0}^{+})^{(k)} \cdot T_{a_1} \otimes \cdots \otimes T_{a_k} = T''_{a_1} \otimes \cdots \otimes T''_{a_k}.$$
The extremality of $(1 < \cdots < \ell)_{a_1} \otimes \cdots \otimes (1 < \cdots < \ell)_{a_k}$ follows.
\end{proof}
\subsection{The non-generic case}
Let us consider the extremal fundamental loop weight module $V(Y_{1,a}Y_{0,aq}^{-1})$ ($a \in \mathbb{C}^{\ast}$) of $\mathcal{U}_q(\hat{sl}_\infty)$. This module has a basis labelled by the tableaux
$$T_a = \ffbox{j}_a \text{ with } j \in \mathbb{Z}.$$
Furthermore the action of $\mathcal{U}_q(\hat{sl}_\infty)$ is known, given by
\begin{eqnarray*}
x_{i}^{+}(z) \cdot \ffbox{j}_a &=& \delta_{i, j-1} \delta(a q^{j-1}z) \cdot \ffbox{j-1}_a,\\
x_{i}^{-}(z) \cdot \ffbox{j}_a &=& \delta_{i, j} \delta(a q^{j}z) \cdot \ffbox{j+1}_a,\\
\phi_{i}^{\pm}(z) \cdot \ffbox{j}_a &=& \begin{cases} \psi(aq^{j}z) \cdot \ffbox{j}_a & \text{ if } i = j,\\
\psi(aq^{j+1}z)^{-1}\cdot \ffbox{j}_a & \text{ if } i = j-1,\\
\ffbox{j}_a & \text{ otherwise.}
\end{cases}
\end{eqnarray*}
We denote this module $V(\ffbox{1}_a)$ in the following. It is defined in \cite{feigin_representations_2013, mansuy_quantum_2012, mansuy_extremal_2013} with different methods for the quantum toroidal algebra $\mathcal{U}_q(sl_{n+1}^{tor})$ ($n\geq 2$) and called (specialized) \textit{vector representation}.
For all $k \in \mathbb{N}^{\ast}$ and $ a \in \mathbb{C}^{\ast}$, we consider the tensor product of vector representations
$$V(\ffbox{1}_a) \otimes V(\ffbox{1}_{aq^{-2}}) \otimes \cdots \otimes V(\ffbox{1}_{aq^{-2(k-1)}}).$$
By Proposition \ref{proptpdefinf}, the coproduct $\Delta$ endows this tensor product with a structure of $\mathcal{U}_q(\hat{sl}_\infty)$-module. Let us give a first result about it (the proof is exactly the same as \cite[Theorem 4.5]{mansuy_extremal_2013} and is not recalled).
\begin{thm}\label{elwmtpinf}
The $\mathcal{U}_q(\hat{sl}_\infty)$-module $V(\ffbox{1}_a) \otimes \cdots \otimes V(\ffbox{1}_{aq^{-2(k-1)}})$ is an extremal loop weight module of $\ell$-weight $$Y_{1,a}Y_{1,aq^{-2}} \cdots Y_{1, aq^{-2(k-1)}}Y_{0,aq}^{-1}Y_{0,aq^{-1}}^{-1} \cdots Y_{0, aq^{-2(k-1)+1}}^{-1}.$$
\end{thm}
Denote by $V \left(\begin{array}{c} \ffbox{1}_{\ } \\ \vdots_{\ } \\ \ffbox{k}_{a} \end{array} \right)$ the subvector space of $V(\ffbox{1}_a) \otimes \cdots \otimes V(\ffbox{1}_{aq^{-2(k-1)}})$ generated by $$\ffbox{i_1}_a \otimes \ffbox{i_2}_{aq^{-2}} \otimes \cdots \otimes \ffbox{i_k}_{aq^{-2(k-1)}} \text{ with } i_1 < i_2 < \cdots < i_k.$$
\begin{prop}
The coproduct $\Delta$ endows the vector space $V \left(\begin{array}{c} \ffbox{1}_{\ } \\ \vdots_{\ } \\ \ffbox{k}_{a} \end{array} \right)$ with a structure of thin $\mathcal{U}_q(\hat{sl}_\infty)$-module.
\end{prop}
\begin{proof}
The action of the $x_i^{\pm}(z)$ is well-defined on it by Proposition \ref{proptpdefinf}. It remains to show that for a vector $$v = \ffbox{i_1}_a \otimes \ffbox{i_2}_{aq^{-2}} \otimes \cdots \otimes \ffbox{i_k}_{aq^{-2(k-1)}} \text{ with } i_1 < i_2 < \cdots < i_k,$$
$x_i^{\pm}(z) \cdot v \in V$ for all $i \in \mathbb{Z}$. This is a consequence of the equalities
\begin{eqnarray}\label{eqinterinf2}
\begin{array}{ccc}
x_{i}^{+}(z) \cdot \ffbox{i}_{a} \otimes \ffbox{i+1}_{aq^{-2}} &=& \delta(a q^{i-2}z) \psi(q^{2}) \cdot \ffbox{i}_{a} \otimes \ffbox{i}_{aq^{-2}} = 0,\\
x_{i}^{-}(z) \cdot \ffbox{i}_{a} \otimes \ffbox{i+1}_{aq^{-2}} &=& \delta(a q^{i}z) \psi(1)^{-1} \cdot \ffbox{i+1}_{a} \otimes \ffbox{i+1}_{aq^{-2}} = 0.
\end{array}
\end{eqnarray}
\noindent Eventually these modules are thin, the weight of the vector $\ffbox{i_1}_a \otimes \cdots \otimes \ffbox{i_k}_{aq^{-2(k-1)}}$ being completely determined by the sequence $i_1 < i_2 < \cdots < i_k$.
\end{proof}
\begin{thm}\label{thmtpvrisomeflwm}
The $\mathcal{U}_q(\hat{sl}_\infty)$-module $V = V \left(\begin{array}{c} \ffbox{1} \\ \vdots \\ \ffbox{\ell} \end{array}_a \right)$ ($\ell \geq 1, a \in \mathbb{C}^{\ast}$) is isomorphic to the extremal fundamental loop weight module $V(Y_{\ell, aq^{-\ell+1}}Y_{0,aq}^{-1})$.
\end{thm}
\begin{proof}
Let us define the morphism of vector spaces $f : V(Y_{\ell, aq^{-\ell+1}}Y_{0,aq}) \rightarrow V$ by setting
$$f(T) = \ffbox{i_1}_a \otimes \ffbox{i_2} \otimes \cdots \otimes \ffbox{i_\ell}_{aq^{-2(\ell-1)}}$$
for all semi-standard tableaux $T = (i_1 < i_2 < \cdots < i_\ell) $ in $\mathcal{T}_{[1, \ell]}$. Hence defined $f$ is an isomorphism of $\mathcal{U}_q(\hat{sl}_\infty)$-modules: this follows by straightforward computations.
\end{proof}
\begin{prop}\label{proptpeflwminf}
Fix $a \in \mathbb{C}^{\ast}, k \in \mathbb{N}^{\ast}$ and $\ell_1, \ell_2, \cdots, \ell_k \geq 1$. Then $\Delta$ endows the tensor product
$$V \left(\begin{array}{c} \ffbox{1} \\ \vdots \\ \ffbox{\ell_1} \end{array}_{a} \right) \otimes V \left(\begin{array}{c} \ffbox{1} \\ \vdots \\ \ffbox{\ell_2} \end{array}_{aq^{-2\ell_1}} \right) \otimes \cdots \otimes V \left(\begin{array}{c} \ffbox{1} \\ \vdots \\ \ffbox{\ell_k} \end{array}_{aq^{-2(\ell_1 + \cdots + \ell_{k-1})}} \right)$$
with a structure of $\mathcal{U}_q(\hat{sl}_\infty)$-module. Furthermore, it has a submodule isomorphic to
$$V \left(\begin{array}{c} \ffbox{1} \\ \vdots \\ \ffbox{\ell} \end{array}_{a} \right) \text{ with } \ell = \ell_1 + \ell_2 + \cdots + \ell_k.$$
\end{prop}
\begin{proof}
By Proposition \ref{proptpdefinf}, the action on the tensor product is well-defined. Furthermore let us consider the subvector space generated by vectors of the form
$$\ffbox{i_1^{(1)}} \otimes \cdots \otimes \ffbox{i_{\ell_1}^{(1)}} \otimes \ffbox{i_1^{(2)}} \otimes \cdots \otimes \ffbox{i_{\ell_2}^{(2)}} \otimes \cdots \otimes \ffbox{i_1^{(k)}} \otimes \cdots \otimes \ffbox{i_{\ell_k}^{(k)}}$$
with $i_1^{(1)} < \cdots < i_{\ell_1}^{(1)} < i_1^{(2)} < \cdots < i_{\ell_2}^{(2)} < \cdots < i_1^{(k)} < \cdots < i_{\ell_k}^{(k)}$. It is by definition the submodule $V \left(\begin{array}{c} \ffbox{1} \\ \vdots \\ \ffbox{\ell} \end{array}_{a} \right)$ of $V(\ffbox{1}_a) \otimes \cdots \otimes V(\ffbox{1}_{aq^{-2(\ell-1)}})$ with $\ell = \ell_1 + \cdots + \ell_k$.
\end{proof}
Set $\ell, k \geq 1$ and $a \in \mathbb{C}^{\ast}$. Let us consider the tensor product $V$ of vector representations
\begin{align*}
V(\ffbox{1}_a) \otimes \cdots \otimes V(\ffbox{1}_{aq^{-2(\ell-1)}}) \otimes \cdots \otimes V(\ffbox{1}_{aq^{-2(1-k)}}) \otimes \cdots \otimes V(\ffbox{1}_{aq^{-2(\ell-k)}}).
\end{align*}
Let $\mathcal{T}_{[1, \ell] \times [1, k]}$ be the set of semi-standard tableaux $T = (T_{i,j})_{(i, j) \in [1, \ell] \times [1, k]}$. Then $(\mathcal{T}_{[1, \ell] \times [1, k]}, \mathrm{wt}, \tilde{e}_i, \tilde{f}_i)$ is a $\mathcal{U}_q(sl_\infty)$-crystal \cite{kashiwara_bases_2002}. To a tableaux $T = (T_{i,j}) \in \mathcal{T}_{[1, \ell] \times [1, k]}$, it corresponds a vector $T_a$ in $V$, defined by (we read the tableaux from top to bottom and from left to right)
$$T_a = \ffbox{T_{1,1}}_a \otimes \cdots \otimes \ffbox{T_{\ell,1}}_{aq^{-2(\ell-1)}} \otimes \cdots \otimes \ffbox{T_{1,k}}_{aq^{-2(1-k)}} \otimes \cdots \otimes \ffbox{T_{\ell,k}}_{aq^{-2(\ell-k)}}.$$
Denote by $T^e = (T_{i,j})$ the semi-standard tableaux such that $T_{i, j} = i$ for all \linebreak $(i, j) \in [1, \ell] \times [1, k]$:
$$T^e = \begin{tiny}
\begin{tabular}{|c|c|c|}
\hline
1 & \ldots & 1 \\
\hline
\vdots & & \vdots \\
\hline
$\ell$& \ldots & $\ell$ \\
\hline
\end{tabular}
\end{tiny}.$$
Let $V(T_a^e)$ be the subvector space of $V$ generated by vectors $T_a$ with $T \in \mathcal{T}_{[1, \ell] \times [1, k]}$.
\begin{prop}\label{propactwdtpelvmwmultinf}
Set $\ell, k \geq 1$ and $a \in \mathbb{C}^{\ast}$. The coproduct $\Delta$ endows $V(T_a^e)$ with a structure of thin $\mathcal{U}_q(\hat{sl}_\infty)$-module.
\end{prop}
\begin{proof}
Let $T = (T_{i,j}) \in \mathcal{T}_{[1, \ell] \times [1, k]}$ be a semi-standard tableaux. We have to show that for all $s \in \mathbb{Z}$, $x_{s}^-(z) \cdot T_a$ is also in $V(T_a^e)$. Actually it suffices to consider the following cases:
\begin{itemize}
\item[-] if $T$ is such that there exist $1 \leq i \leq \ell-1$ and $1 \leq j \leq k$ such that $$T_{i,j}=s \text{ and } T_{i, j+1} = s+1.$$ Then by (\ref{eqinterinf2}), $x_{s}^-(z) \cdot T_a$ is in $ V(T_a^e)$.
\item[-] if $T$ is such that there exist $1 \leq i \leq \ell$ and $p \geq 1$, $1 \leq j \leq k-p$ such that
$$T_{i,j} = T_{i, j+1} = \cdots = T_{i, j+p} = s \text{ and } T_{i, j+p+1} \geq s+1.$$
Then the fact that $x_{s}^-(z) \cdot T_a$ belongs to $ V(T_a^e)$ is a consequence of the following equality in $V(\ffbox{1}_a) \otimes \cdots \otimes V(\ffbox{1}_{aq^{2p}})$
\begin{align*}
x_s^-(z) \cdot \ffbox{s}_a \otimes \cdots \otimes \ffbox{s}_{aq^{2p}} = \delta( a q^{s+2p} z ) \ffbox{s}_a \otimes \cdots \otimes \ffbox{s}_{aq^{2(p-1)}} \otimes \ffbox{s+1}_{aq^{2p}}.
\end{align*}
\end{itemize}
We proceed in the same way for the operators $x_s^+(z)$.
\end{proof}
\begin{thm}\label{thmtpelwminf}
Set $\ell, k \geq 1$ and $a \in \mathbb{C}^{\ast}$. Then $V(T_a^e)$ is an irreducible extremal loop weight $\mathcal{U}_q(\hat{sl}_\infty)$-module of $\ell$-weight
$$m_{T^e} = Y_{\ell, aq^{-\ell+1}}Y_{\ell, aq^{-\ell+3}} \cdots Y_{\ell, aq^{-\ell+1+2(k-1)}} Y_{0,aq}^{-1}Y_{0,aq^3}^{-1} \cdots Y_{0,aq^{1+2(k-1)}}^{-1}.$$
\end{thm}
\begin{proof}
Let us recall that the semi-standard tableaux
$$T^e = \begin{tiny}
\begin{tabular}{|c|c|c|}
\hline
1 & \ldots & 1 \\
\hline
\vdots & & \vdots \\
\hline
$\ell$& \ldots & $\ell$ \\
\hline
\end{tabular}
\end{tiny}$$
is an extremal element of the $\mathcal{U}_q(sl_\infty)$-crystal $\mathcal{T}_{[1, \ell] \times [1, k]}$ of weight $k \Lambda_\ell - k \Lambda_0$. Furthermore we have
$$W \cdot T^e = \lbrace T = (T_{i,j})_{(i,j) \in [1, \ell] \times [1, k]} \vert T_{i,1} = T_{i, 2} = \cdots = T_{i, k} \text{ for all } 1 \leq i \leq \ell \rbrace.$$
Using the equalities ($s \in \mathbb{Z}$)
\begin{align*}
(x_{s, 0}^-)^{(k)} \cdot \ffbox{s}_a \otimes \cdots \otimes \ffbox{s}_{aq^{2(k-1)}} & = \ffbox{s+1}_a \otimes \cdots \otimes \ffbox{s+1}_{aq^{2(k-1)}},\\
(x_{s-1, 0}^+)^{(k)} \cdot \ffbox{s}_a \otimes \cdots \otimes \ffbox{s}_{aq^{2(k-1)}} & = \ffbox{s-1}_a \otimes \cdots \otimes \ffbox{s-1}_{aq^{2(k-1)}},
\end{align*}
we get
$$x_i^{\pm} \cdot T_a = 0 \text{ and } (x_i^{\mp})^{(\pm \mathrm{wt}(T)(h_i))} \cdot T_a = S_i(T)_a \text{ if } \pm \mathrm{wt}(T)(h_i) \geq 0$$
for all $T \in W \cdot T^e$. Then the result is a consequence of the following Lemma.
\end{proof}
\begin{lem}
Let $V$ be a $\mathcal{U}_q(sl_\infty)$-module with basis $(v_b)_{b \in \mathcal{B}}$ indexed by a \linebreak $\mathcal{U}_q(sl_\infty)$-crystal $\mathcal{B}$. Assume that $b^e \in \mathcal{B}$ is extremal of weight $\lambda \in P$ and for all $i \in \mathbb{Z}$ and $b \in W \cdot b^e$,
$$\mathrm{wt}(v_b) = \mathrm{wt}(b), \ x_i^{\pm} \cdot v_b = 0 \text{ and } (x_i^{\mp})^{(\pm \mathrm{wt}(b)(h_i))} \cdot v_b = v_{S_i(b)} \text{ if } \pm \mathrm{wt}(b)(h_i) \geq 0.$$
Then $v_{b^e}$ is an extremal vector of weight $\lambda$.
\end{lem}
\begin{proof}
The proof is analogue to the one of Lemma \ref{lirepcryinf}.
\end{proof}
\section{Application to quantum toroidal algebras}
The quantum toroidal algebras $\mathcal{U}_q(sl_{n+1}^{tor})$ ($n \geq 2$) were introduced by Ginzburg-Kapranov-Vasserot in type A \cite{ginzburg_langlands_1995}. They are quantum groups analogs of elliptic Cherednik algebras \cite{cherednik_double_1995} to whom they are related via Schur-Weyl duality \cite{varagnolo_schur_1996}. The author has defined extremal loop weight modules in the context of $\mathcal{U}_q(sl_{n+1}^{tor})$ \cite{mansuy_quantum_2012, mansuy_extremal_2013}. The main motivation is the construction of finite-dimensional representations of the quantum toroidal algebras at roots of unity.
A combinatorial link between the representation theory of $\mathcal{U}_q(\hat{sl}_\infty)$ and of quantum toroidal algebras is conjectured in \cite{hernandez_algebra_2011}. This conjecture is proved for the class of Kirillov-Reshetikhin modules. The main motivation in \cite{hernandez_algebra_2011} is to predict $q$--character formulae for representations of $\mathcal{U}_q(sl_{n+1}^{tor})$.
In this section, we prove \cite[Conjecture 5.3]{hernandez_algebra_2011} for the particular family of extremal fundamental loop weight modules of $\mathcal{U}_q(\hat{sl}_\infty)$: by the combinatorial link highlighted in \cite{hernandez_algebra_2011}, they are related to $\mathcal{U}_q(sl_{n+1}^{tor})$-modules defined in \cite{mansuy_extremal_2013}. The aim is to construct extremal loop weight modules of $\mathcal{U}_q(sl_{n+1}^{tor})$. We show that we recover the extremal fundamental loop weight modules constructed in \cite{mansuy_quantum_2012, mansuy_extremal_2013}.
In the first part, we recall some definitions about the quantum toroidal algebras $\mathcal{U}_q(sl_{n+1}^{tor})$ ($n \geq 2$) and its combinatorial link with $\mathcal{U}_q(\hat{sl}_\infty)$ conjectured in \cite{hernandez_algebra_2011}. In the second part, we prove this conjecture for the class of extremal fundamental loop weight modules of $\mathcal{U}_q(\hat{sl}_\infty)$ (Theorem \ref{thmqchaactreptorinf}). We recover in that way the extremal fundamental loop weight modules of $\mathcal{U}_q(sl_{n+1}^{tor})$ constructed in \cite{mansuy_quantum_2012, mansuy_extremal_2013} (Theorem \ref{thmelwmtorinf}).
\subsection{Reminder about quantum toroidal algebras}
Let us recall the definition of the quantum toroidal algebra $\mathcal{U}_q(sl_{n+1}^{tor})$ ($n \geq 2$). Set $I_n = \mathbb{Z} / (n+1) \mathbb{Z}$. Let $C = (C_{\overline{i},\overline{j}})_{\overline{i}, \overline{j} \in I_n}$ be the Cartan matrix of type $A_n^{(1)}$,
$$ C_{\overline{i},\overline{i}} = 2 \text{ , } C_{\overline{i},\overline{i+1}} = C_{\overline{i+1}, \overline{i}} = -1 \text{ and } C_{\overline{i},\overline{j}} = 0 \text{ if } \overline{j} \neq \overline{i}, \overline{i \pm 1}.$$
The algebra $\mathcal{U}_q(sl_{n+1}^{tor})$ is defined by the same generators and relations as in Definition \ref{defqaainf} with $\overline{i}, \overline{j} \in I_n$.
The representation theory of $\mathcal{U}_q(sl_{n+1}^{tor})$ is similar to the one of $\mathcal{U}_q(\hat{sl}_\infty)$: the simple integrable $\ell$-highest weight modules are parametrized by Drinfeld polynomials. In particular, the Kirillov-Reshetikhin modules can be defined in an analogue way. One defines $q$--characters $\chi_{q, n}(V)$ as above for integrable representations $V$ with finite-dimensional $\ell$-weight spaces: it is a sum of elements in $A_n$, where $A_n$ is the group of monomials of the form
$$m = \prod_{\overline{i} \in I_n, a \in \mathbb{C}^{\ast}} Y_{\overline{i}, a}^{u_{\overline{i}, a}(m)}, \ u_{\overline{i}, a}(m) \in \mathbb{Z}.$$
Consider the ring morphism
$$\phi_n : \mathbb{Z}[Y_{i,a}^{\pm 1}]_{i \in \mathbb{Z}, a \in \mathbb{C}^{\ast}} \rightarrow \mathbb{Z}[Y_{\overline{i},a}^{\pm 1}]_{\overline{i} \in I_n, a \in \mathbb{C}^{\ast}}$$
defined, for $i \in \mathbb{Z}$ and $a \in \mathbb{C}^{\ast}$, by
$$\phi_n(Y_{i,a}^{\pm 1}) = Y_{\overline{i}, a}^{\pm 1}.$$
Let $\mathrm{Im}(\chi_q)$ (resp. $\mathrm{Im}(\chi_{q, n})$) be the image of the Grothendieck group corresponding to the category of integrable $\mathcal{U}_q(\hat{sl}_\infty)$-modules (resp. integrable $\mathcal{U}_q(sl_{n+1}^{tor})$-modules) belonging to $\mathcal{O}$. The morphism $\phi_n$ gives rise naturally to a group morphism
$$\phi_n : \mathrm{Im}(\chi_q) \rightarrow \mathbb{Z}[[Y_{\overline{i},a}^{\pm 1}]]_{\overline{i} \in I_n, a \in \mathbb{C}^{\ast}}.$$
More precisely, one can show that $\phi_n \left(\mathrm{Im}(\chi_q) \right) \subset \mathrm{Im}(\chi_{q, n})$ by using the characterization stated in the proof of \cite[Theorem 4.2]{hernandez_algebra_2011}. Hence for $V$ a $\mathcal{U}_q(\hat{sl}_\infty)$-module in the category $\mathcal{O}_{\mathrm{int}}$, $\phi_n \left(\chi_q(V) \right)$ is the $q$--character of a representation of $\mathcal{U}_q(sl_{n+1}^{tor})$, which is a priori virtual. It is conjectured in \cite{hernandez_algebra_2011} that
\begin{conj}\label{conjherninf}
Let $V$ be a simple representation in $\mathcal{O}_{\mathrm{int}}$ for $\mathcal{U}_q(\hat{sl}_\infty)$. Then $\phi_n(\chi_q(V))$ is the $q$--character of an actual representation of $\mathcal{U}_q(sl_{n+1}^{tor})$.
\end{conj}
This conjectural link between the representation theory of $\mathcal{U}_q(\hat{sl}_\infty)$ and the one of quantum toroidal algebras is proved in \cite{hernandez_algebra_2011} for Kirillov-Reshetikhin modules. In the following section, we give one of the main results of the paper: we prove Conjecture \ref{conjherninf} in the context of extremal fundamental loop weight modules of $\mathcal{U}_q(\hat{sl}_\infty)$.
\subsection{Extremal loop weight modules for quantum toroidal algebras}
Let us consider the extremal fundamental loop weight module $V(Y_{\ell, a} Y_{0, aq^{\ell}}^{-1})$ with $\ell \geq 1$ and $a \in \mathbb{C}^{\ast}$. Recall that we have determined its $q$--character: it is given by
\begin{equation*}
\chi_q(V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})) = \sum_{T \in \mathcal{T}_{[1, \ell]}} m_T
\end{equation*}
where $\mathcal{T}_{[1, \ell]}$ is the set of semi-standard tableaux of shape $(\ell)$ and $$m_T = \prod_{1 \leq j \leq \ell} \ffbox{i_j}_{aq^{\ell+1-2j}}.$$ Furthermore, we have shown that
$$\{m_T \vert T \in \mathcal{T}_{[1, \ell]} \} = \mathcal{M}(Y_{\ell, a}Y_{0, aq^\ell}^{-1}).$$
Let us consider the ring morphism $\phi_n : \mathbb{Z}[Y_{i,a}^{\pm 1}]_{i \in \mathbb{Z}, a \in \mathbb{C}^{\ast}} \rightarrow \mathbb{Z}[Y_{\overline{i},a}^{\pm 1}]_{\overline{i} \in I_n, a \in \mathbb{C}^{\ast}}$. It cannot be extend to an application $\mathbb{Z}^{A} \rightarrow \mathbb{Z}^{A_n}$. However, $\phi_n$ gives rise to a well-defined application
$$\phi_n : \mathbb{Z}^{\mathcal{M}(Y_{\ell, a}Y_{0, aq^\ell}^{-1})} \longrightarrow \mathbb{Z}^{A_n}.$$
Actually by definition of the monomial crystal $\mathcal{M}$ (see \cite{kashiwara_realizations_2003, mansuy_quantum_2012, nakajima_$t$-analogs_2003}), $\phi_n$ can be defined in all its connected components.
\begin{prop}
$\phi_n \left( \chi_q(V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})) \right)$ is the $q$--character of a virtual representation of $\mathcal{U}_q(sl_{n+1}^{tor})$.
\end{prop}
\begin{proof}
We have to check that $\phi_n \left( \chi_q(V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})) \right)$ respects the following characterization (see \cite[Theorem 4.2]{hernandez_algebra_2011})
\begin{itemize}
\item[]it is an infinite sum of elements in $\mathbb{Z}[Y_{\overline{r}, a}(1 + A_{\overline{r}, aq}^{-1}), Y_{\overline{r'}, a}^{\pm 1}]_{a \in \mathbb{C}^{\ast}, \overline{r} \neq \overline{r'}}$ for each $\overline{r} \in I_n$.
\end{itemize}
Actually, it suffices to check the analogue property at the level of the $\mathcal{U}_q(sl_\infty)$-crystal $\mathcal{M}(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$: this holds by the description of it given above (see Remark \ref{remlienmoncriinf}).
\end{proof}
Let us give one of the main results of the paper.
\begin{thm}\label{thmqchaactreptorinf}
Assume that $n \geq 2$ is such that
\begin{align}\label{condexistmodinf}
(n \text{ is even and } \ell \leq n+1) \text{ or } \left(n \text{ is odd and } \ell \leq \frac{n+1}{2} \right).
\end{align}
Then $\phi_n \left( \chi_q(V(Y_{\ell, a}Y_{0, aq^\ell}^{-1})) \right)$ is the $q$--character of an actual representation of $\mathcal{U}_q(sl_{n+1}^{tor})$. We denote it $V_n(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$.
\end{thm}
This result confirms Conjecture \ref{conjherninf} in the context of extremal fundamental loop weight modules.
\begin{proof}
This is a consequence of \cite[Proposition 4.4]{mansuy_extremal_2013}. In fact if (\ref{condexistmodinf}) holds, we have shown that there exists a sub-$\mathcal{U}_q(sl_{n+1}^{tor})$-module\footnote{It is denoted $\tilde{V} \left(\begin{array}{c} \ffbox{1} \\ \vdots \\ \ffbox{k} \end{array}_{aq^{\ell-1}} \right)$ in \cite{mansuy_extremal_2013}.} $V_n(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ of the fusion product of specialized vector representations with basis labelled by the set $\mathcal{T}_{[1, \ell]}$ of semi-standard tableaux of shape $(\ell)$. Furthermore for all $T \in \mathcal{T}_{[1, \ell]}$, $T_a$ is an $\ell$-weight vector of $\ell$-weight $\phi_n(m_T)$. The result follows directly.
\end{proof}
The main motivation of this study is the construction of extremal loop weight modules for $\mathcal{U}_q(sl_{n+1}^{tor})$. Actually we have
\pagebreak
\begin{thm}\label{thmelwmtorinf}\cite{mansuy_extremal_2013}
\begin{enumerate}
\item[(i)] Assume that $n \geq 2$. Then $V_n(Y_{1, a}Y_{0, aq}^{-1})$ is an extremal loop weight module of $\ell$-weight $Y_{1, a}Y_{0, aq}^{-1}$.
\item[(ii)] Assume that $n = 2r+1$ is odd $(r \geq 1)$. Then $V_n(Y_{r+1, a}Y_{0, aq^{r+1}}^{-1})$ admits an irreducible quotient which is an extremal loop weight module of $\ell$-weight $Y_{r+1, a}Y_{0, aq^{r+1}}^{-1}$.
\end{enumerate}
\end{thm}
\begin{rem}
In the other cases, the representations $V_n(Y_{\ell, a}Y_{0, aq^\ell}^{-1})$ are not $\ell$-extremal (see \cite{mansuy_extremal_2013}).
\end{rem}
\bibliographystyle{acm}
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"redpajama_set_name": "RedPajamaArXiv"
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\section{Introduction}\label{sect:introduction}
Gamma-Ray Burst (GRB) jets move at relativistic velocities with Lorentz factors $\Gamma_0$\ much in excess of unity.
The properties of the emission (such as timescales and typical frequencies) measured in the observer frame appear then very different from the intrinsic ones in the comoving frame of the fluid.
Only by estimating $\Gamma_0$\ it is possible to infer the intrinsic properties of the emitting region.
Unfortunately, it is difficult to place significant constraints on $\Gamma_0$\ from observations.
As a consequence, a lot of useful information (such as the location of the dissipation region, the ejecta mass, the typical frequencies of the emitted photons), fundamental for discriminating among different theoretical scenarios, suffer from large uncertainties.
Improving the estimates of the Lorentz factor is then essential for understanding the nature of the central engine and outflow, the conditions at the emitting region, and the nature of the radiation process.
It was early realised that an ultra-relativistic motion is needed in order to avoid the so-called compactness problem and explain detections of $\gamma$-ray\ photons on short variability timescales \citep{ruderman75,krolik91,fenimore93,piran95,baring97}. The highest photon energy detected during the prompt emission can then be used to compute the minimum value of $\Gamma_0$\ required to avoid $\gamma$-$\gamma$ opacity within the emitting region.
Using this method, lower limits in the range 100-400 have been derived by \cite{lithwick01} for a sample of 13 BATSE bursts.
Much larger lower limits (in the range $900-1200$) have been derived for GRBs detected by the {\it Fermi}-LAT \citep{080916CLAT,090902B,090510LAT}, due to the extension of the accessible range to GeV energies.
These large lower limits pose severe constraints on the baryon load of the ejecta, favouring Poynting flux dominated jets.
However, the formula used to derive these extreme values has been questioned by \cite{hascoet12b}, who proposed a more detailed calculation of $\gamma$-$\gamma$ opacity and suggested that the simpler formula overestimates $\Gamma_0$\ by a factor of 2-3.
Moreover, \citet{zou11} pointed out that these limits rely on the one-zone model, where GeV and sub-MeV photons are emitted from the same region and are produced by internal shocks. The long-lasting nature of the GeV emission suggests a different origin and dissipation radius for the high-energy component.
A two-zone model (where the collisions between the GeV and MeV photons occur at larger radii than the prompt emission radius) implies much weaker constraints (about one fifth to one half of the one-zone values).
Another widely used technique for estimating $\Gamma_0$\ is based on the onset of the afterglow emission \citep{sari99}.
A few efforts have been made to collect samples of GRBs displaying a peak in their early time optical lightcurve and derive the value of $\Gamma_0$, assuming that the peak time marks the outflow deceleration time. \cite{liang10,liang15} derived values in the range 90-600.
Smaller values, between 30-300 (and between 20 and 200 for a wind-like density medium) were instead inferred by \cite{ghirlanda12}.
When the onset is not observed, (i.e., observations start when the flux is already decaying) an upper limit can be placed on the deceleration time, and then a lower limit on the value of $\Gamma_0$. The lower limits derived using this method are in the range 40-300 \citep{hascoet14}.
\cite{zou10} suggested that flux limits on the early afterglow can also be used to constrain $\Gamma_0$.
For large $\Gamma_0$, indeed, the afterglow emission starts at an earlier time and has a higher peak luminosity. A lack of detection can then be translated into an upper limit on the brightness of the afterglow, and then on the value of $\Gamma_0$.
They considered early X-ray observations in a sample of 16 GRBs and derived upper limits on $\Gamma_0$\ of several hundreds.
We suggest that a similar method can be applied also to high-energy (GeV) observations.
In the standard afterglow model, the early time afterglow emission is expected to extend up to GeV energies. A lack of GeV emission can then be translated into an upper limit on $\Gamma_0$.
In this paper, we propose to exploit LAT flux upper limits derived on timescales longer than the prompt duration to place limits on the brightness of the synchrotron afterglow component, and in turn on $\Gamma_0$.
We have already applied this method to a sample of 28 GRBs observed by AGILE \citep{longo12}, deriving values between 100 and a few thousands (for a typical redshift $z=2$).
The LAT allows us to place more stringent constraints thanks to its higher sensitivity, and to significantly increase the sample of GRBs to which this analysis can be applied (190 events).
The paper is organized as follows. In \S \ref{sec:model}, we calculate the synchrotron afterglow flux in the range 0.1-10~GeV, and provide equations that can be used to place upper limits on $\Gamma_0$\ from the upper limits on the LAT flux.
In \S \ref{sec:UL} we consider a sample of 190 GRBs with no LAT detection, and compute the upper limits on $\Gamma_0$.
Some implications for GRBs detected by the LAT are discussed in \S \ref{sec:detected}.
Conclusions are summarised in \S \ref{sec:conclusions}.
\section{Expected High-Energy afterglow emission}\label{sec:model}
In fast cooling regime, the bolometric afterglow luminosity from the forward external shock is proportional to the rate at which the energy is dissipated at the shock $dm\Gamma^2/dt$ (where $m$ is the total mass of the external medium collected up to the time $t$) and to the fraction $\epsilon_{\rm e}$\ of this energy gained by the accelerated electrons.
Since we are interested in early time afterglow evolution, we assume that $\Gamma$ is larger than $1/\theta_{\rm jet}$ (where $\theta_{\rm jet}$ is the jet opening angle) and express energetics and luminosities in terms of their isotropic equivalent values.
Using $dr\propto\Gamma^2 dt$ and introducing a generic density radial profile $n=n_0 r^{-s}$, the bolometric luminosity is \citep{sari97}:
\begin{equation}
L^{\rm aft}_{\rm bol}\propto \epsilon_e t^{2-s} n_0 \Gamma^{8-2s}.
\end{equation}
Two regimes can be identified:
\begin{itemize}
\item A coasting phase ($\Gamma=\Gamma_0$): the luminosity has a strong dependence on the value of $\Gamma_0$, and is proportional to $n_0$ (we consider here and elsewhere in this work that $\epsilon_{\rm e}$\ has more or less the same value for all GRBs, see below for a discussion).
In a constant density medium $(s=0)$ the luminosity rises as $t^2$, while it is constant for a wind-like medium ($s=2$);
\item A deceleration phase: $\Gamma$ decreases according to $\Gamma^2\propto E_{\rm k}/m(r)$ \citep{blandford76}, where $E_{\rm k}$\ is the blastwave energy (we are assuming an adiabatic evolution, i.e. $E_{\rm k}$=constant), and $m(r)$ is the total mass collected up to the radius $r$.
Regardless of the radial density profile, the luminosity decreases with time as $L^{\rm aft}_{\rm bol}\propto \epsilon_{\rm e} E_{\rm k}t^{-1}$.
Since $E_{\rm k}$\ is related to the prompt radiated energy $E_{\rm \gamma,iso}$\ through the prompt efficiency $\eta_\gamma$ ($E_{\rm k}$=$E_{\rm \gamma,iso}$$[1-\eta_\gamma]/\eta_\gamma)$, we can write
$L^{\rm aft}_{\rm bol}\propto\epsilon_{\rm e} E_{\rm \gamma,iso} (1-\eta_\gamma)/\eta_\gamma t^{-1}$.
\end{itemize}
The energies we are interested in ($>0.1\,$GeV) are most likely larger than the cooling and synchrotron characteristic frequencies $\nu_{\rm c}$\ and $\nu_{\rm m}$.
Electrons radiating at such energies are rapidly cooling, and the equations describing the luminosity of the emitted radiation are similar to equations governing the bolometric luminosity, with minor corrections to the exponents and with the introduction of a weak dependence on the fraction of energy $\epsilon_{\rm B}$\ in the amplified magnetic field.
In particular, during the deceleration \citep{sari98}:\newline
$L^{\rm aft}_{\rm [0.1-10]}=k\,\epsilon_{\rm e}^{p-1}\epsilon_B^{\frac{p-2}{4}} [E_{\rm \gamma,iso} (1-\eta_\gamma)/\eta_\gamma]^{\frac{p+2}{4}} t^{-\frac{3p-2}{4}}$, where the numerical factor $k$ depends only on $p$ (the power-law index of the electron injection spectrum, $N_{\rm inj}(\gamma)\propto \gamma_e^{-p}$) and varies less than a factor 1.5 for $p$ in the range $2.1-2.8$.
This latter equation implies that, during the deceleration, the ratio between the high-energy afterglow luminosity, at a fixed rest frame time, and the prompt energy $E_{\rm \gamma,iso}$\ depends only on two parameters, $\epsilon_{\rm e}$\ and $\eta_\gamma$\ \citep{kumar00,freedman01}. \citet{nava14} found that for LAT GRBs with temporally extended emission, the value of this ratio is narrowly clustered, implying that the product $\epsilon_{\rm e}^{p-1} [(1-\eta_\gamma)/\eta_\gamma]^{\frac{p+2}{4}}$ has more or less the same value in different GRBs and does not introduce a significant scatter (see \citealt{nava14} for a more detailed discussion). Hereafter, we will assume that both $\epsilon_{\rm e}$\ and $\eta_\gamma$\ do not vary by a significant amount, but we explicitly write how our estimates depends on these two parameters, so that the effects of a different assumption can be easily computed.
\begin{figure}
\vskip -1.3truecm
\hskip -2.9truecm
\includegraphics[scale=0.37]{lc.ps}
\vskip -3.3truecm
\caption{Examples of synchrotron afterglow lightcurves at a frequency $\nu>\max(\nu_{\rm c},\nu_{\rm m})$ for a constant density profile of the surrounding medium. The afterglow parameters are the same in both cases, except for the initial Lorentz factor $\Gamma_0$. At large $\Gamma_0\sim10^3$, the light curve peaks at early times (see equation~\ref{eq:peaktime_s=0}), while the peak is shifted at much later times when $\Gamma_0\sim10^2$. }
\label{fig:lc}
\end{figure}
While during the deceleration phase the value of $\Gamma_0$\ does not affect the flux (which is rather determined by the blast wave energy), $\Gamma_0$\ plays an important role during the coasting phase and in determining the deceleration time, i.e. the time of the transition from a constant to a decreasing Lorentz factor.
For small $\Gamma_0$, the deceleration occurs at late times and the peak flux is smaller.
To clarify this point, Fig.~\ref{fig:lc} illustrates the afterglow lightcurves of two GRBs that have the same parameters except for the initial Lorentz factors $\Gamma_0$.
Even though they have the same energy $E_{\rm \gamma,iso}$\ (and hence same afterglow luminosity after deceleration), the chances to detect emission are very different in the two cases. Depending on the temporal window of observation as compared to the time of the peak, the afterglow of the low-$\Gamma_0$\ GRB might be completely missed.
For a GRB observed within the first few hundred seconds, chances of detection are larger for high-$\Gamma_0$\ events.
When observations extend to times longer than the peak time, where the luminosity is proportional to $E_{\rm \gamma,iso}$, the chances are dominated by the GRB energetics, and are larger for GRBs with a large $E_{\rm \gamma,iso}$.
From this example it is clear that three quantities play a fundamental role: the prompt energy $E_{\rm \gamma,iso}$, the observation time and $\Gamma_0$. When the first two quantities are known, a limit on $\Gamma_0$\ can be inferred from the non detection of the expected radiation.
\subsection{Synchrotron fluence at $\nu>\max(\nu_{\rm c},\nu_{\rm m})$}\label{sec:estimates}
Since the LAT is a photon-limited instrument, for a fixed spectral index $\alpha$ the detection capability is directly related to the fluence.
We then estimate the synchrotron afterglow fluence $S^{\rm aft}_{\rm [0.1-10]}$\ in the energy range $0.1-10\,$GeV (observer frame) under the assumption $\max(h\nu_{\rm c},h\nu_{\rm m})<0.1\,$GeV. In this spectral range, the spectral slope $\alpha$ (in the notation $F_{\nu}\propto\nu^{\alpha}$) is $\alpha=-p/2$.
We model the external shock dynamics starting from the coasting phase, following \cite{nava13}, and the radiation output following \cite{sari98} and \cite{nappo14}.
The choice of computing the afterglow fluence in the range 0.1-10 GeV is motivated by the fact that available estimates of LAT flux upper limits have been computed in this energy range \citep{latcatalogul12}. Moreover, this is also the energy range chosen in the First {\it Fermi}-LAT GRB catalog \citep{latcatalog} to quote fluxes and fluences of LAT detected GRBs, that can be directly compared to the estimates provided in the following. We also note that, if extended up to higher energies ($>10\,$GeV), the estimates of the expected afterglow flux might significantly depend on the possible presence of a spectral cutoff, caused for example, by the maximum synchrotron energy. Limiting the estimates at energies smaller than $10\,$GeV reduces these uncertainties (see a discussion in section \ref{sec:suppression}).
We consider two different radial density profiles characterized as $n \propto r^{-s}$: a constant ($s=0$) and a decreasing density ($s=2$).
While in both cases the afterglow flux after the deceleration time decreases with a temporal index $\beta_2=-(3p-2)/4$, before the deceleration the temporal indices are $\beta_1=2$ and $\beta_1=(2-p)/2$, for $s=0$ and $s=2$ respectively, where we used the notation $F(t)\propto t^\beta$.
If observations start at $t_{\rm i}$ and end at $t_{\rm f}$ the fluence is:
\begin{eqnarray}
S^{\rm aft}_{[0.1-10]}=\int^{t_{\rm f}}_{t_{\rm i}}F^{\rm aft}_{[0.1-10]}\, dt~,
\label{eq:integral}
\end{eqnarray}
where the flux $F^{\rm aft}_{[0.1-10]}$ is given by:
\begin{eqnarray}
F^{\rm aft}_{[0.1-10]}\!=\!A
\left\{ \!
\begin{array}{l}
t^{\beta_1} \quad {\rm for}\ t\ll t_{\rm dec}\\
\\
t^{\beta_1}_{\rm dec} \left( \frac{t}{t_{\rm dec}}\right)^{-\frac{3p-2}{4}}\quad {\rm for}\ t\gg t_{\rm dec}\\
\end{array} \right.
\end{eqnarray}
Here $t_{\rm dec}$ is the deceleration time in the observer frame.
If observations are characterised by temporal gaps, the integration in eq.~\ref{eq:integral} should be performed separately in each time interval where observations are available. The total expected afterglow fluence will be the sum of the contributions from each time interval.
In what follows, we give analytic approximations of the numerical results for the computation of $S^{\rm aft}_{\rm [0.1-10]}$\ (equation~\ref{eq:integral}), for different orders of the times $t_{\rm i}$, $t_{\rm f}$, and $t_{\rm dec}$.
We consider the general case $t_{\rm i}\ne0$, to account for cases where the GRB enters the LAT field-of-view (FoV) after the trigger time.
\subsubsection{Homogeneous medium: $n=constant$}
\begin{figure}
\vskip -0.25 truecm
\hskip -0.4truecm
\includegraphics[scale=0.51]{fluence_s=0.eps}
\vskip -0.25 truecm
\caption{Synchrotron afterglow fluence integrated from $t_i=0$ to $t_{\rm f}$ in the range $0.1-10\,$GeV (observer frame). The three different stripes correspond to three different integration times $t_{\rm f}=20, 300, 5\times10^3\,$seconds (from right to left). For each stripe, the solid lines correspond to different redshifts: $z=0.5$ (upper boundary), $z=2$ (central thick line), and $z=4$ (lower boundary). The filled dots show the Lorentz factor for which the lightcurve peak time is equal to the integration time: $t_{\rm dec}=t_{\rm f}$. All the curves have been derived assuming $\epsilon_{\rm e}$=0.1, $\epsilon_{\rm B}$=0.01, $\eta_\gamma$=0.2, $n$=1$\,$cm$^{-3}$, and $S_{\rm \gamma,iso}$=$10^{-4}$erg/cm$^2$. Different values of $S_{\rm \gamma,iso}$\ and $n_0$ significantly affect the curves, as indicated by the vertical arrows: in the first regime $S^{\rm aft}_{\rm [0.1-10]}$\ depends almost linearly on the external density (see equation~\ref{eq:S_LAT_early_s=0}), while in the second regime the LAT fluence $S^{\rm aft}_{\rm [0.1-10]}$\ depends linearly on the prompt fluence $S_{\rm \gamma,iso}$\ (see equation~\ref{eq:S_LAT_late_s=0}). }
\label{fig:fluence_s=0}
\end{figure}
The transition from the coasting to the deceleration regime occurs around the deceleration time, which is also the time at which the lightcurve peaks:
\begin{equation}
t_{\rm dec}=3\,(1+z_2)^{2/3}\left[ \frac{S_{\rm \gamma,iso,-4}(1-\eta_{\gamma})\,d^2_{\rm L,2}}{\Gamma_{0,3}^8\,n_0\,\eta_{\gamma}} \right]^{1/3} \rm s~,
\label{eq:peaktime_s=0}
\end{equation}
where $S_{\rm \gamma,iso,-4}$ is the bolometric prompt fluence in units of $10^{-4}\,$erg/cm$^2$, and $n_0$ is the density in cm$^{-3}$. We use the notation $Q_x=Q/10^x$, except for the redshift (where $z_2$ means that the numerical factor has been estimated for a typical redshift $z=2$) and the luminosity distance $d_{\rm L,2}=d_{\rm L}/d_{\rm L,z=2}$.
We estimate the integral in equation~\ref{eq:integral} for three different cases: $t_{\rm dec} > t_{\rm f}$ (relevant for short observing times and/or for small values of $\Gamma_0$), $t_{\rm i}<t_{\rm dec}<t_{\rm f}$ (relevant for longer observing time and/or larger values of the Lorentz factor), and $t_{\rm dec}<t_{\rm i}$ (relevant when the GRB enters the FoV at late times, when the fireball is already decelerating).
\begin{itemize}
\item $t_{\rm dec} > t_{\rm f}$:
\begin{eqnarray}
\begin{array}{r}
S^{\rm aft}_{\rm [0.1-10]}=2.5\times 10^{-7}t^3_{\rm f,3}\Gamma_{0,2}^{(2p+4)}\epsilon_{\rm B,-2}^{\frac{p-2}{4}}n^{\frac{p+2}{4}}_0\epsilon_{\rm e,-1}^{p-1}\times\\
\\
\times(1+z_2)^{-\frac{p+2}{2}} d^{-2}_{\rm L,2} \left[1-\left(\frac{t_{\rm i}}{t_{\rm f}}\right)^3\right]\,\rm erg/cm^2~.
\end{array}
\label{eq:S_LAT_early_s=0}
\end{eqnarray}
In this first regime the dependence on $\Gamma_0$\ is very strong and there is no dependence on $S_{\rm \gamma,iso}$. Moreover the fluence depends nearly linearly on $n_0$.\\
\item $t_{\rm i} < t_{\rm dec}<t_{\rm f}$:
\begin{eqnarray}
\begin{array}{c}
\label{eq:S_LAT_late_s=0}
S^{\rm aft}_{\rm [0.1-10]}=10^{-5} \rm erg/cm^2\, S_{\rm \gamma,iso,-4}\Gamma_{0,3}^{2(p-2)}\epsilon_{\rm B,-2}^{\frac{p-2}{4}}\,\times\\
\\
n^{\frac{p-2}{4}}_0\epsilon_{\rm e,-1}^{p-1}\frac{1-\eta_{\gamma}}{\eta_{\gamma}}(1+z_2)^{\frac{2-p}{2}}\times\\
\\
\left\{\left[1-\frac{4}{p+2}\left(\frac{t_{\rm f}}{t_{\rm dec}}\right)^{-\frac{3}{4}(p-2)}\right]- \frac{3(p-2)}{(p+2)}\left(\frac{t_{\rm i}}{t_{\rm f}}\right)^{-\frac{p-4}{2}}\right\} \,.
\end{array}
\end{eqnarray}
The dependences on $n_0$, $\epsilon_{\rm B}$, and $z$ are very weak and can be neglected.
Also, according to observations of GRBs with temporally extended emission, the term $\epsilon_{\rm e}^{p-1}\left[\frac{1-\eta}{\eta}\right]^{\frac{p+2}{4}}$ has a similar value for all GRBs \citep{nava14}.
The main parameters determining the afterglow fluence are then $S_{\rm \gamma,iso}$\ and (depending on the value of $p$) $\Gamma_0$.\\
\item $t_{\rm dec}<t_{\rm i}$:
\begin{eqnarray}
\begin{array}{c}
\label{eq:S_LAT_late_late_s=0}
S^{\rm aft}_{\rm [0.1-10]}=3\times10^{-5} S^{\frac{p+2}{4}}_{\rm \gamma,iso,-4}\epsilon_{\rm B,-2}^{\frac{p-2}{4}}\epsilon_{\rm e,-1}^{p-1}\left[\frac{1-\eta_{\gamma}}{\eta_{\gamma}}\right]^{\frac{p+2}{4}}\times\\
\\
\times\,d^{\frac{p-2}{2}}_{\rm L,2} t_{\rm i,3}^{-\frac{3(p-2)}{4}}\left[ 1-\left( \frac{t_{\rm f}}{t_{\rm i}} \right)^{-\frac{3(p-2)}{4}} \right]\rm erg/cm^2~.
\end{array}
\end{eqnarray}
In this last regime the synchrotron fluence is proportional to $S_{\rm \gamma,iso}$\ but, contrary to the previous regime, it is independent of $\Gamma_0$.
\end{itemize}
The results are summarised in Fig.~\ref{fig:fluence_s=0}, that shows curves of $S^{\rm aft}_{\rm [0.1-10]}$\ as a function of $\Gamma_0$. These have been derived for $t_{\rm i}=0$, but they hold as long as $t_{\rm i}<\min(t_{\rm dec},t_{\rm f})$, since for $n=const$ most of the emission is radiated at $t\gtrsim t_{\rm dec}$, and the initial integration time does not significantly affect the fluence estimates.
Each shaded stripe corresponds to a different value of the final integration time $t_{\rm f}$ (from left to right: $t_{\rm f}=5\times10^3, 300, 20$ seconds).
We chose $t_{\rm f}=5\times10^3$ as maximum value because this roughly corresponds to the maximum timescale over which observations can be performed without temporal gaps.
For each stripe, three different curves (corresponding to three different values of the redshift) are marked with a solid line: $z=0.5$ (upper boundary), $z=2$ (central thick line), and $z=4$ (lower boundary).
All curves have been derived for $S_{\rm \gamma,iso}=10^{-4}\,$erg cm$^{-2}$, $\epsilon_{\rm e}$=0.1, $\epsilon_{\rm B}$=0.01, $\eta_\gamma=0.2$, and $n=1\,$cm$^{-3}$.
Low values of $\Gamma_0$ correspond to late peak times. In this first regime, $S^{\rm aft}_{\rm [0.1-10]}$\ strongly depends on $\Gamma_0$\ and on the redshift (see equation~\ref{eq:S_LAT_early_s=0}). Moreover, it depends nearly linearly on the density: the curves should be moved up/down for increasing/decreasing density, as indicated by the arrows.
The prompt fluence plays no role in this regime.
For increasing $\Gamma_0$\ the peak time decreases. For each curve, the $\Gamma_0$\ at which $t_{\rm dec}=t_{\rm f}$ is marked by a filled dot. At larger $\Gamma_0$
we switch to the regime $t_{\rm dec}<t_{\rm f}$. In this second regime the afterglow fluence depends very weakly on all the unknown parameters, except $S_{\rm \gamma,iso}$. All the curves (for different $t_{\rm f}$ and redshifts) flatten (i.e. the dependence on $\Gamma_0$\ is weaker) and converge to a similar value, as predicted by equation~\ref{eq:S_LAT_late_s=0}. This value is proportional to $S_{\rm \gamma,iso}$: the curves should be moved up/down for increasing/decreasing prompt fluence, as indicated by the arrows.
If a LAT observation results in a non-detection, and the upper limit on the LAT average flux is estimated on a time [$t_{\rm i},t_{\rm f}$], these plots and equations \ref{eq:S_LAT_early_s=0} to~\ref{eq:S_LAT_late_late_s=0} can be used to set an upper limit on $\Gamma_0$.
Under favourable observing conditions, the most stringent limits that LAT can place on the 0.1-10 GeV fluence are around a few$\times 10^{-7}\,$erg/cm$^2$ \citep{latcatalogul12,latcatalog}. Our calculations show that strong limits ($\lesssim 200$) on $\Gamma_0$\ can hence be placed only if the GRB is observed for at least several hundred seconds (green stripe in Fig.~\ref{fig:fluence_s=0}).
While the curves in Fig.~\ref{fig:fluence_s=0} have been derived under the assumptions that LAT observations start at the trigger time and that there are no temporal gaps in the observations, eqs.~\ref{eq:S_LAT_early_s=0} to~\ref{eq:S_LAT_late_late_s=0} can also be used in the more general case where $t_{\rm i}\ne0$ and/or in case of gaps during observations, for example caused by Earth occultation. In this latter case, the equations should be applied to each time interval where observations are performed, and the total fluence can then be estimated as the sum of contributions from each interval.
\subsubsection{Wind-shaped environment: $n\propto r^{-2}$}
We derive the synchrotron fluence at $\nu > \max(\nu_{\rm c}, \nu_{\rm m})$ for a density $n=3\times10^{35}A_{\star} r^{-2}$, where $A_{\star}$ is defined such that $A_\star=1$ corresponds to the case of a typical wind from a Wolf-Rayet star \citep{chevalier00}.
The deceleration occurs around the time:
\begin{equation}
t_{\rm dec}=350\,\,\frac{S_{\rm \gamma,iso,-4}(1-\eta_{\gamma})\,d^2_{\rm L,2}}{\Gamma_{0,2}^4\,A_{\rm \star}\,\eta_{\gamma}}\,\, \rm s~.
\label{eq:peaktime_s=2}
\end{equation}
Also in this case, we consider all three possibilities for the order of $t_{\rm i}$, $t_{\rm f}$, and $t_{\rm dec}$.
Similar considerations to the case $s=0$ can be derived.\\
\begin{itemize}
\item $t_{\rm dec} > t_{\rm f}$:
\begin{eqnarray}
\begin{array}{r}
S^{\rm aft}_{\rm [0.1-10]}=3.7\times 10^{-6}t^{\frac{4-p}{2}}_{\rm f,3}\Gamma_{0,2}^{(p+2)}\epsilon_{\rm B,-2}^{(p-2)/4}A_{\rm \star}^{(p+2)/4}\times\\
\\
\times\epsilon_{\rm e,-1}^{p-1}d^{-2}_{\rm L,2}\left[ 1-\frac{t_{\rm i}}{t_{\rm f}}\right]^{\frac{4-p}{2}}\,\rm erg/cm^2~.
\end{array}
\label{eq:S_LAT_early_s=2}
\end{eqnarray}
\\
\item $t_{\rm i} < t_{\rm dec} < t_{\rm f}$:
\begin{eqnarray}
\begin{array}{r}
S^{\rm aft}_{\rm [0.1-10]}=10^{-5} \rm{erg/cm^2}\, S^{\frac{4-p}{2}}_{\rm \gamma,iso,-4}\Gamma_{0,2}^{3(p-2)}\times\\
\\
\times\,\epsilon_{\rm B,-2}^{\frac{p-2}{4}}A_{\rm \star}^{\frac{3(p-2)}{4}}\epsilon_{\rm e,-1}^{p-1}\left[\frac{1-\eta_{\gamma}}{\eta_{\gamma}}\right]^{\frac{4-p}{2}}d^{2-p}_{\rm L,2}\times\\
\\
\left\{\left[1-\frac{2(4-p)}{p+2}\left(\frac{t_{\rm f}}{t_{\rm dec}}\right)^{-\frac{3}{4}(p-2)}\right]- \frac{3(p-2)}{(p+2)}\left(\frac{t_{\rm i}}{t_{\rm f}}\right)^{-\frac{p-4}{2}}\right\}~,
\end{array}
\label{eq:S_LAT_late_s=2}
\end{eqnarray}
\\
\item $t_{\rm dec}<t_{\rm i}$:
\begin{eqnarray}
\begin{array}{r}
S^{\rm aft}_{\rm [0.1-10]}=3\times10^{-5} S^{\frac{p+2}{4}}_{\rm \gamma,iso,-4}\epsilon_{\rm B,-2}^{\frac{p-2}{4}}\epsilon_{\rm e,-1}^{p-1}\left[\frac{1-\eta_{\gamma}}{\eta_{\gamma}}\right]^{\frac{p+2}{4}}\times\\
\\
\times\,d^{\frac{p-2}{2}}_{\rm L,2} t_{\rm i,3}^{-\frac{3(p-2)}{4}}\left[ 1-\left( \frac{t_{\rm f}}{t_{\rm i}} \right)^{-\frac{3(p-2)}{4}} \right]\rm erg/cm^2~.
\end{array}
\label{eq:S_LAT_late_late_s=2}
\end{eqnarray}
\end{itemize}
The results are summarised in Fig.~\ref{fig:fluence_s=2}, that shows $S^{\rm aft}_{\rm [0.1-10]}$\ as a function of $\Gamma_0$, for the case $t_{\rm i}=0$. Each shaded stripe corresponds to a different value of the final integration time $t_{\rm f}$. Since the dependence on $t_{\rm f}$ is weaker as compared to the case $n=n_0$, only two cases are shown: $t_{\rm f}=5\times10^3, 10$ seconds (from left to right).
All curves have been derived for $S_{\rm \gamma,iso}=10^{-4}\,$erg cm$^{-2}$, $\epsilon_{\rm e}$=0.1, $\epsilon_{\rm B}$=0.01, $\eta_\gamma=0.2$, and $A_{\rm \star}=1$.
As in the constant density case, in the first regime ($t_{\rm f}<t_{\rm dec}$) the afterglow fluence depends on $\Gamma_0$\ and $z$, although the dependence on $\Gamma_0$\ is weaker (see equation~\ref{eq:S_LAT_early_s=2}).
Moreover, it depends nearly linearly on the density: the curves should be moved up/down for increasing/decreasing density.
The prompt fluence plays no role in this regime. For increasing $\Gamma_0$\ the deceleration time decreases and we switch to the regime $t_{\rm dec}<t_{\rm f}$. For each curve, the $\Gamma_0$\ at which $t_{\rm dec}=t_{\rm f}$ is marked by a filled circle. In the second regime the fluence depends very weakly on all the unknown parameters, except $S_{\rm \gamma,iso}$. All the curves converge to a similar value, as predicted by equation~\ref{eq:S_LAT_late_s=2}. This value is roughly proportional to $S_{\rm \gamma,iso}$.
\begin{figure}
\vskip -0.25 truecm
\hskip -0.35truecm
\includegraphics[scale=0.51]{fluence_s=2.eps}
\vskip -0.45 truecm
\caption{Same as in Fig.~\ref{fig:fluence_s=0} but for a wind circumburst density profile with $A_{\star}=1$ (see eqs.~\ref{eq:peaktime_s=2} to \ref{eq:S_LAT_late_s=2}). The two shaded stripes correspond to two different integration times: $t_{\rm f}=10, 5\times10^3\,$seconds (from right to left).}
\label{fig:fluence_s=2}
\end{figure}
In the wind density scenario, LAT upper limits as deep as a few$\times10^{-7}\,$erg/cm$^{-2}$ lead to place stronger limits on $\Gamma_0$, as compared to the constant density case, even in the case of relatively short observation times $t_{\rm f}$.
\subsection{Caveats}\label{sec:suppression}
The estimates presented in the previous section neglect possible physical processes that might decrease the expected flux.
The high-energy synchrotron afterglow emission might indeed be affected by:
\begin{itemize}
\item Inverse Compton scattering: in this case the synchrotron luminosity at frequencies larger than $\max(\nu_{\rm c},\nu_{\rm m})$ is suppressed by a factor (1+$Y$), where $Y$ is the Compton parameter. This can be relevant for small values of $\epsilon_{\rm B}$, a very uncertain parameter in GRB studies. However, at high-energies, the Compton scattering is in Klein-Nishina regime, and the relevance of inverse Compton effects is strongly reduced. \citet{beniamini15} have shown that $Y$ at $0.1-10\,$ GeV is of order unity, even for very small values ($<10^{-5}$) of $\epsilon_{\rm B}$;
\item The maximum synchrotron photon energy: this is limited by the maximal energy up to which electrons can be shock-accelerated.
The limit is estimated to be around $\Gamma\times70\,$MeV \citep{dejager92,piran10}. This means that the maximum photon energy is constant during the coasting phase and then it decreases. For $\Gamma<150$, this limit is then expected to produce a cutoff in the afterglow synchrotron spectrum around the energies relevant for this study.
The extrapolation of the synchrotron spectrum with index $\alpha=-p/2$ up to $10\,$GeV, might then be incorrect. In this case the flux is smaller then what estimated before, especially at late times, when $\Gamma$ has significantly decreased. Since we will apply our estimates to early time observations ($t_{\rm f}=100\,$s, see section~\ref{sec:UL}) and since $\alpha<-2$, for the application presented here, this effect, if present, introduces a flux suppression at most of a factor of 2-3;
\item $\gamma$-$\gamma$ absorption: for LAT observations performed simultaneously to the prompt emission, it might be relevant to include $\gamma$-$\gamma$ absorption of GeV photons passing the shell of lower energy, prompt photons. Even though afterglow photons are produced at much larger radii as compared to prompt photons, \citet{zou11} have demonstrated that opacity might still arise and partially suppress the GeV flux.
\end{itemize}
All these processes, if relevant, lower the expected synchrotron fluence, as compared to estimates presented in the previous section. This would lead to higher upper limits on $\Gamma_0$\ (i.e., if the expected flux is smaller, non-detections are consistent with theoretical expectations also for higher $\Gamma_0$, leading to less stringent upper limits on $\Gamma_0$). This might be regarded as a weakness of the method.
On the other hand, this can be used to check consistency by comparing
the upper limits derived with this method with lower limits and direct estimates derived with different methods. If the comparison does not outline any inconsistency, the assumption that the GeV afterglow flux is not strongly suppressed is well supported. On the other hand, an inconsistency between this and other methods would reveal the need for at least one of the mentioned processes to be at work.
As we will show later, inconsistencies are not found.
\section{Upper limits on $\Gamma_0$}\label{sec:UL}
As of January 2016 the GBM has detected prompt emission from almost $1800$\footnote{http://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html} GRBs. Around 105 have been detected also by the LAT\footnote{http://fermi.gsfc.nasa.gov/ssc/observations/types/grbs/lat\_grbs/}, corresponding to around 13\% of the GRBs falling within the nominal LAT FoV, i.e. at an angle of $65^\circ$ from the LAT boresight (see also \citealt{vianello15}).
\begin{figure*}
\includegraphics[scale=0.68]{g0_limit.eps}
\caption{Upper limits on $\Gamma_0$ for bursts with no LAT detection, as a function of the redshift, for a constant density medium (left panel) and a wind shaped medium (right panel). Only upper limits smaller than $\Gamma_0=2000$ are shown. Different colours of the curves refer to different values of the prompt GBM fluence: lighter colours are used for brighter bursts (see the color bar). Red arrows: upper limits for GRBs with measured redshift. Star symbols: GRBs for which $\Gamma_0$ has been estimated from the peak of the early optical lightcurve (green stars) and GeV light curve (yellow stars), taken from \citet{ghirlanda12}. }
\label{fig:G0limits}
\end{figure*}
\cite{latcatalogul12} have considered all GRBs with no evidence of emission above 100 MeV, that fell within the LAT FoV during the first 2.5 years (288 events).
The upper limits on the average flux in the range 0.1-10~GeV have been estimated on three different integration times: during the prompt emission, and for fixed 30 s and 100 s integration times, starting from the trigger time (i.e. $t_{\rm i}=0$).
We consider here the upper limits estimated for $t_{\rm f}=100\,$s.
For each burst in this sample, we have computed the prompt fluence $S_{\rm \gamma,iso}$\ in the energy range $1-10^4\,$keV using the best fit model reported in the {\it Fermi} GBM burst online catalog\footnote{http://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html} (\citealt{bhat16}).
The fit models used in the catalog include a simple power-law (PL), a power-law with an exponential cutoff (CPL), a smoothly broken PL (SBPL), and the so-called Band function. We have considered only those GRBs for which the best fit model is a peaked (in $\nu F_\nu$) function (i.e. either the CPL, SBPL, or Band models), otherwise a model extrapolation down to $1\,$keV and up to $10\,$MeV would be unsafe.
The final sample includes 190 GRBs.
In this sample we find that the limits on the LAT fluence $S^{UL}_{\rm [0.1-10]}$ in the first 100 seconds range from $5\times10^{-7}\,$erg/cm$^2$ to $8\times10^{-5}\,$erg/cm$^2$.
For this sample, $t_{\rm i}=0$, and $t_{\rm f}$, $S_{\rm \gamma,iso}$, and $S^{UL}_{\rm [0.1-10]}$ are known. Imposing $S^{\rm aft}_{\rm [0.1-10]}$$<S^{UL}_{\rm [0.1-10]}$, equations~\ref{eq:S_LAT_early_s=0} and \ref{eq:S_LAT_late_s=0} (or \ref{eq:S_LAT_early_s=2} and \ref{eq:S_LAT_late_s=2} for the wind case) can then be inverted to find the upper limit on $\Gamma_0$.
We assume $\epsilon_{\rm e}$=0.1, $\eta_\gamma$=0.2, $n_0=1$ (or $A_\star=1$ for the wind density case), $\epsilon_{\rm B}$=0.01 and $p=2.2$ (but $\epsilon_{\rm B}$\ and $p$ do not affect the estimates, and the density is important only at small values of $\Gamma_0$).
Since the redshift is known only for a small fraction of the sample, we derive the upper limits on $\Gamma_0$\ as a function of $z$.
The results are shown in Fig.~\ref{fig:G0limits} both for a constant density medium (left panel) and a wind medium (right panel), for $z$ in the range 0.1-10 (blue and light-blue curves). Red arrows mark those bursts for which the redshift is known.
Only cases resulting in upper limits smaller than 2000 are shown.
To emphasise the role of the prompt fluence, we use different colours for different values of $S_{\rm \gamma,iso}$: brighter (in the GBM range) bursts are marked with lighter colours. It is evident that stringent limits on $\Gamma_0$\ can be derived only for the brightest GRBs.
For a typical redshift $z\sim2$, the limits on $\Gamma_0$ lie above $200$ and in the range 100-400 for a constant and wind-like medium, respectively.
These limits can be compared with limits and direct estimates available in the literature and computed with different methods.
\cite{latcatalogul12} derived upper limits for 6 bright GRBs for which a high-energy cutoff in the prompt spectrum at energies $<100\,$MeV is implied by the LAT non detection.
Their upper limits on $\Gamma_0$\ as a function of $z$ are shown in their figure 11. The curves are similar to those derived here, with limiting values around $\sim$150 at $z=0.5$ and $\sim$500 at $z=5$.
Upper limits on $\Gamma_0$\ have been computed also from early time X-ray observations, resulting in maximum values around several hundreds, by \cite{zou10}. They have also shown that when these are combined with lower limits required to avoid the compactness problem, values of $\Gamma_0$\ are in the range $10^2-10^3$.
Concerning direct estimates (rather then limits) of $\Gamma_0$, a spectral break in the prompt component has been observed only in a few cases \citep{090926ALAT,tang15}.
Most of the available estimates of the value of $\Gamma_0$\ have been inferred from the detection of an early peak in the afterglow lightcurve.
\cite{ghirlanda12} collected all GRBs with known redshift and with an early peak in the optical light curve, and inferred $\Gamma_0$\ under the assumption that the peak corresponds to the blast wave deceleration time.
The $\Gamma_0$\ values have been derived both for a constant and wind-like medium, and are shown in Fig.~\ref{fig:G0limits} as star symbols (the green colour refers to optical lightcurves, while the yellow colour refers to a similar analysis applied to GeV lightcurves of LAT GRBs with temporally extended GeV emission).
The most stringent limits derived in this work lie above most of the values inferred from GRBs with an optical peak.
This implies that the non-detection of synchrotron afterglow radiation is consistent with the simplest model, and there is no evidence that mechanisms producing a suppression of the GeV flux (see section~\ref{sec:suppression}) are at work. The possibility to test the relevance of these processes is however limited by the instrument sensitivity. We can conclude that present instrument capabilities are not pointing to the need for a relevant suppression of the high-energy afterglow synchrotron flux.
On the other hand, the upper limits lie not far from (and sometimes below) the estimated values of $\Gamma_0$. This suggests that the LAT should be able to detect the synchrotron afterglow component for those GRBs with the largest bulk Lorentz factors and largest energetics.
A fraction of the LAT detected GRBs are indeed characterised by the presence of an emission above 100$\,$MeV lasting much longer than the prompt radiation, whose flux decays in time as a power-law \citep{latcatalog}. These are the brightest GBM GRBs, and a large Lorentz factor $\Gamma_0>500$ has been inferred for them. An association with synchrotron afterglow radiation has been claimed to be consistent also with their spectral and temporal properties \citep{kumar09,kumar10,ghisellini10,ghirlanda10,depasquale10,lemoine13,nava14,beniamini15}, although photons with particularly large energies ($>10$\,GeV) detected at late times ($>10^2\,$s) are in excess of the synchrotron limit and require a different explanation \citep{piran10,wang13,130427LAT}.
Finally, we comment on the dependence of these results on the unknown parameters $\epsilon_{\rm e}$\ and $\eta_\gamma$, with reference to a homogeneous density medium (similar considerations hold also for a wind-shaped density medium). In the first regime, where observations stop before the lightcurve reaches the peak (equation~\ref{eq:S_LAT_early_s=0}), our estimates of $\Gamma_0$\ do not depend on $\eta_\gamma$, and they depend very weakly on $\epsilon_{\rm e}$\ ($\Gamma_0$$\propto\epsilon_{\rm e}^{0.1}$). In the second regime (equation~\ref{eq:S_LAT_late_s=0}), the Lorentz factor appears also in the definition of $t_{\rm dec}$, and it is then less obvious to understand how different assumptions on $\eta_\gamma$\ and $\epsilon_{\rm e}$\ affect the results. From numerical estimates, we find that if the value of $\epsilon_{\rm e}(1-\eta_\gamma)/\eta_\gamma$ increases by a factor of 5 compared to our fiducial value of 0.4, the upper limits on $\Gamma_0$\ are smaller, by a factor of 1.5. They lie closer to the direct values estimated from the peak of optical lightcurves (green star symbols in Fig.~\ref{fig:G0limits}). Conversely, if the value of $\epsilon_{\rm e}(1-\eta_\gamma)/\eta_\gamma$ is decreased by a factor of 10, the upper limits increase by a factor of $\lesssim3$, and the most stringent values are now at the level of the direct estimates derived from GeV lightcurves of GRBs with LAT temporally extended emission (yellow star symbols in Fig.~\ref{fig:G0limits}).
\section{Implications for GRBs with detected GeV temporally extended emission}
\label{sec:detected}
\begin{figure*}
\hskip -0.4truecm \includegraphics[scale=0.83]{obs-angle.eps}
\caption{Prompt fluence (left panel) and peak flux (right panel) in the energy range 1-$10^4\,$keV vs. the angle $\theta$ to the LAT boresight. Grey dots represent {\it Fermi} GRBs detected only by the GBM. Square symbols represent GRBs detected also by the LAT: filled symbols refer to those with temporally extended emission, empty symbols refer to those with no evidence for extended emission, empty symbols with a cross inside refer to cases for which the classification is uncertain.}
\label{fig:prompt-angle}
\end{figure*}
In order to detect synchrotron afterglow radiation with the LAT, $S^{\rm aft}_{\rm [0.1-10]}$\ must be larger than the instrument threshold:
\begin{equation}
S^{\rm aft}_{\rm [0.1-10]}>S_{\rm th}[\rm bkg,\theta, t_{\rm f}-t_{\rm i}, \alpha].
\label{eq:condition}
\end{equation}
This threshold does not have the same value for all GRBs, because it strongly depends on the specific observing conditions. It is then impossible to identify a unique condition that all GRBs must satisfy in order to have a detectable GeV afterglow radiation.
More precisely, the minimum value of the fluence $S_{\rm th}$ required for detection will in general depend on the level of background, the angle $\theta$ between the burst location and the LAT boresight (which might also change during observations),
how long the GRB is inside the LAT FoV ($\Delta t=t_{\rm f}-t_{\rm i}$), and the spectral index $\alpha$.
The level of background depends on contamination from earth-albedo events CR-background, on the geomagnetic latitude, and on the location of the Earth limb.
$S_{\rm th}$ can then considerably vary from burst to burst, and two events with similar intrinsic properties and located at similar distances can result in a detection or non-detection due to different observing conditions.
As discussed in section~\ref{sec:estimates}, also the theoretical estimate of $S^{\rm aft}_{\rm [0.1-10]}$\ cannot be fully determined, because it depends on a few unknown parameters, such as $\Gamma_0$, $z$ (which is not measured in most cases) and possibly $n$ (depending on the interval time $t_{\rm i}-t_{\rm f}$ during which the event is observed).
However, for a typical $t_{\rm f}$ (of at least few hundred seconds) and for reasonably large Lorentz factors ($\Gamma_0>100$), the main parameter determining the afterglow fluence in the LAT range is the prompt fluence $S_{\rm \gamma,iso}$\ (see Figs.~\ref{fig:fluence_s=0} and \ref{fig:fluence_s=2}, and equations~\ref{eq:S_LAT_late_s=0} and \ref{eq:S_LAT_late_s=2}), if $\epsilon_{\rm e}$\ and $\eta_\gamma$\ do not vary significantly.
The condition for having a detectable afterglow fluence can then be roughly translated into a condition on the prompt fluence.
Keeping in mind that this is true only in the regime $t_{\rm f}>t_{\rm dec}>t_{\rm i}$ and that also $\Gamma_0$\ plays a role in determining the afterglow fluence, the prediction is that, when the emission detected by LAT is indeed afterglow radiation, these events should also be the ones with the largest prompt fluences.
A correlation between the prompt sub-MeV fluence and the GeV fluence arises also if both emissions are related to the prompt component, but in this case it is not trivial to explain why the GeV radiation extends in time significantly beyond the prompt phase.
Following these considerations we collect all GRBs detected by {\it Fermi} up to January 2016 and plot their distribution in the plane $S_{\rm \gamma,iso}$-$\theta$ (Fig.~\ref{fig:prompt-angle}, left panel), to verify if LAT detected GRBs with temporally extended emission show indeed a tendency to have larger prompt fluences.
For each burst in this sample, the prompt fluence $S_{\rm \gamma,iso}$\ has been estimated in the energy range $1-10^4\,$keV using the best fit model reported in the {\it Fermi} GBM burst online catalog\footnote{http://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html}.
Grey empty circles are GRBs detected by the GBM but with no emission detected by the LAT.
GRBs detected also by the LAT\footnote{http://fermi.gsfc.nasa.gov/ssc/observations/types/grbs/lat\_grbs/} are instead marked with a square symbol.
Note that, when GeV radiation is detected, there is a possibility that this radiation is not synchrotron afterglow emission, i.e. there are cases where the afterglow fluence is too faint, and photons of a different origin (for example, the high-energy extension of the prompt spectrum) are responsible for the LAT detection.
To account for this possible contamination, we classify LAT GRBs according to the duration of the LAT emission as compared to the duration of the prompt detected by the GBM.
According to information derived either from the GRB LAT catalog \citep{latcatalog}, the GCN archive, or literature, we divide the sample into three categories: i) GRBs with temporally extended emission (filled squares), ii) GRBs with no LAT emission after the end of the prompt emission (empty squares) and iii) GRBs for which the classification is uncertain, since a few photons have been detected by the LAT after the end of the prompt emission, but on timescales comparable to the prompt duration (squares with a plus symbol inside).
LAT GRBs with temporally extended emission\ and non-LAT GRBs clearly populate two different regions of the plane, with LAT GRBs to clustered in the high-$S_{\rm \gamma,iso}$/low-$\theta$ region (Fig.~\ref{fig:prompt-angle}, left panel).
We check if this tendency is present also when the prompt fluence is replaced with the prompt peak flux $F_{\rm\gamma,iso}$. The right panel in Fig.~\ref{fig:prompt-angle} shows, for the same sample, the prompt peak flux as a function of $\theta$.
GRBs detected by the LAT now span almost all the range of peak fluxes, and no clear separation is present between LAT and GBM-only GRBs, indicating that the prompt peak flux does not influence the possibility of having a bright long-lasting high-energy component.
The separation between LAT and non-LAT bursts is instead evident in terms of prompt fluence, consistently with the afterglow model.
\section{Discussion and Conclusions}\label{sec:conclusions}
The luminosity of the early afterglow emission strongly depends on the value of the initial Lorentz factor $\Gamma_0$. This parameter indeed affects the expected emission in two ways: (i) it is the main parameter determining the deceleration time, i.e. the transition between the coasting phase (where the Lorentz factor is constant) and the deceleration phase, and (ii) it is the main parameter determining the luminosity of the radiation during the initial coasting phase.
Large values of $\Gamma_0$\ imply a short deceleration time and a large peak flux. Afterglows of high-$\Gamma_0$\ GRBs are then easier to detect (Fig.~\ref{fig:lc}).
Early time flux upper limits can then be translated into upper limits on the afterglow luminosity, and in turn on the value of $\Gamma_0$.
In principle this method can be applied to optical and X-ray observations. The optical band, however, likely lies below the cooling frequency, where the flux depends on very uncertain parameters, especially $\epsilon_{\rm B}$.
Recent afterglow modelings on different samples selected in different energy bands (radio, optical, X-ray, and GeV) have showed that $\epsilon_{\rm B}$\ probably spans a large range of values, covering at least 4-5 orders of magnitude \citep{barniolduran14,santana14,zhang15,lemoine13,beniamini15}, making the predictions of the optical flux very uncertain.
The X-ray band instead, lies most likely above the cooling frequency, but, for small values of $\epsilon_{\rm B}$, this part of the synchrotron spectrum is strongly affected by inverse Compton scattering \citep{beniamini15,beniamini16}. Again, the very uncertain value of $\epsilon_{\rm B}$\ would reflect on a large uncertainty on the expected X-ray flux, and then in not very robust limits on $\Gamma_0$. Higher ($\sim$GeV) energies are less affected by these issues: first, we can safely assume that the LAT energy range is above the cooling frequency, and second, the Klein-Nishina cross section strongly limits the effects of the inverse Compton scattering on this part of the synchrotron spectrum.
We have modeled $\sim$GeV synchrotron afterglow emission during the coasting and deceleration phases, and compared model expectations with LAT observations. Since the LAT is a photon limited instrument, for a fixed photon index the relevant quantity for the detection is the fluence. We have presented equations to estimate the synchrotron afterglow fluence in the range 0.1-10~GeV (observer frame) as a function of all afterglow parameters, prompt fluence, redshift, and initial ($t_{\rm i}$) and final ($t_{\rm f}$) observation times (see equations~\ref{eq:peaktime_s=0} to \ref{eq:S_LAT_late_late_s=0} for a homogenous density medium, and equations~\ref{eq:peaktime_s=2} to \ref{eq:S_LAT_late_late_s=2} for a wind-like density medium).
For the case $t_{\rm i}=0$ (i.e., for GRBs that are inside the LAT FoV at the trigger time) the results are summarized in Fig.~\ref{fig:fluence_s=0} and \ref{fig:fluence_s=2} (for a constant and a wind-like density profile, respectively).
The fluence is shown as a function of $\Gamma_0$\ for different observing times $t_{\rm f}$ and for fixed $\epsilon_{\rm e}$=0.1, $\epsilon_{\rm B}$=0.01 and $p=2.2$ (the last two parameters however play a very little role in modifying the estimates), while the dependence on $n$, $z$, and $S_{\rm \gamma,iso}$\ are shown in the figures.
These curves and the equations provided in section \ref{sec:estimates} can be used to set a limit on $\Gamma_0$, if the upper limit on the LAT average flux from $t_{\rm i}$ to $t_{\rm f}$ is known.
We have applied these equations to a sample of 190 GRBs with no evidence for GeV emission \citep{latcatalogul12}.
We have used the upper limits on the average LAT flux (estimated in the first 100 seconds after the GRB trigger) to place upper limits on $\Gamma_0$\ (Fig.~\ref{fig:G0limits}).
For a typical redshift $z=2$, the inferred values are above 200 for a homogeneous medium, and in the range 100-400 for a wind-like density medium.
These values are consistent with estimates (and lower limits) available in literature and inferred with different methods.
These estimates rely on the assumption that processes such as the existence of a limit on the maximal synchrotron photon energy, $\gamma-\gamma$ absorption with lower energy, prompt photons, and inverse Compton scattering, do not significant lower the expected high-energy synchrotron flux (see section \ref{sec:suppression} for a discussion).
The lack of conflict between our results inferred from high-energy observations and estimates inferred (with other methods) from observations at lower frequencies implies there is no need to invoke a suppression of the high-energy afterglow flux.
An improved instrument sensitivity is required to probe the presence and relevance of the mentioned processes.
On the other hand, the fact that most of the inferred upper limits lie very close to $\Gamma_0$\ values (and lower limits) estimated with different methods (see Fig.~\ref{fig:G0limits}, star symbols) implies that the synchrotron afterglow radiation from GRBs with the highest $\Gamma_0$\ should be bright enough to be detected by the LAT.
For very large $\Gamma_0$, the deceleration time is small, and the afterglow luminosity is a good proxy for the blastwave energy. In turn, for a fixed value of the prompt efficiency $\eta_\gamma$, the blastwave energy is a proxy for the energy radiated during the prompt, $E_{\rm \gamma,iso}$, implying that the GeV synchrotron radiation should be detectable for the GRBs with the highest prompt fluences.
This scenario is consistent with the detection, in a considerable fraction of the LAT GRBs, of a slowly fading GeV radiation on timescales much longer than the prompt emission, whose luminosity is tightly correlated with the prompt energy \citep{nava14}.
The synchrotron afterglow scenario is then consistent not only with detections, but also with non-detections of GRBs by the LAT, and offers a method to place upper limits on $\Gamma_0$. This method, in combination with other estimates (mostly lower limits), provides a tool to restrict the acceptable range of values for the still uncertain parameter $\Gamma_0$.
\section*{Acknowledgements}
LN was partially supported by a Marie Curie Intra-European Fellowship of the European Community's 7th Framework Programme (PIEF-GA-2013-627715).
LN and TP were supported by the ISF-CHE I-Core center for Excellence for research in Astrophysics (1829/12),
and by the China-NSF-Israel-ISF grant.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,308
|
package org.fastcatsearch.db;
import java.io.IOException;
import java.net.URL;
import java.util.ArrayList;
import java.util.List;
import org.apache.ibatis.io.Resources;
import org.apache.ibatis.session.SqlSession;
import org.fastcatsearch.alert.ClusterAlertService;
import org.fastcatsearch.db.InternalDBModule.MapperSession;
import org.fastcatsearch.db.mapper.ManagedMapper;
import org.fastcatsearch.env.Environment;
import org.fastcatsearch.exception.FastcatSearchException;
import org.fastcatsearch.module.ModuleException;
import org.fastcatsearch.service.AbstractService;
import org.fastcatsearch.service.ServiceManager;
import org.fastcatsearch.settings.Settings;
public abstract class AbstractDBService extends AbstractService {
protected InternalDBModule internalDBModule;
private Class<?>[] mapperList;
public AbstractDBService(String dbPath, Class<?>[] mapperList, Environment environment, Settings settings, ServiceManager serviceManager) {
super(environment, settings, serviceManager);
this.mapperList = mapperList;
String absoluteDbPath = environment.filePaths().file(dbPath).getAbsolutePath();
// system관련 mapper설정.
List<URL> mapperFileList = new ArrayList<URL>();
for (Class<?> mapperDAO : mapperList) {
try {
String mapperFilePath = mapperDAO.getName().replace('.', '/') + ".xml";
URL mapperFile = Resources.getResourceURL(mapperFilePath);
mapperFileList.add(mapperFile);
} catch (IOException e) {
logger.error("error load MapperFile", e);
}
}
internalDBModule = new InternalDBModule(absoluteDbPath, mapperFileList, environment, settings);
}
public InternalDBModule internalDBModule() {
return internalDBModule;
}
public <T> MapperSession<T> getMapperSession(Class<T> type) {
SqlSession session = internalDBModule.openSession();
return new MapperSession<T>(session, session.getMapper(type));
}
@Override
protected boolean doStart() throws FastcatSearchException {
internalDBModule.load();
for (Class<?> mapperDAO : mapperList) {
Class<? extends ManagedMapper> clazz = (Class<? extends ManagedMapper>) mapperDAO;
MapperSession<? extends ManagedMapper> mapperSession = (MapperSession<? extends ManagedMapper>) getMapperSession(clazz);
try {
ManagedMapper managedMapper = mapperSession.getMapper();
try {
logger.debug("valiadte {}", clazz.getSimpleName());
managedMapper.validateTable();
} catch (Exception e) {
try {
logger.debug("drop {}", clazz.getSimpleName());
managedMapper.dropTable();
mapperSession.commit();
} catch (Exception ignore) {
}
try {
logger.debug("create table {}", clazz.getSimpleName());
managedMapper.createTable();
mapperSession.commit();
logger.debug("create index {}", clazz.getSimpleName());
managedMapper.createIndex();
mapperSession.commit();
initMapper(managedMapper);
} catch (Exception e2) {
logger.error("", e2);
ClusterAlertService.getInstance().alert(e);
}
}
} finally {
if (mapperSession != null) {
mapperSession.closeSession();
}
}
}
return true;
}
protected abstract void initMapper(ManagedMapper managedMapper) throws Exception;
@Override
protected boolean doStop() throws FastcatSearchException {
try {
internalDBModule.unload();
} catch (ModuleException e) {
logger.error(e.getMessage(), e);
return false;
}
return true;
}
@Override
protected boolean doClose() throws FastcatSearchException {
internalDBModule = null;
return true;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,333
|
{"url":"https:\/\/www.physicsforums.com\/threads\/galoi-group.441918\/","text":"# Galoi group\n\nAnybody can help me show that Gal(E\/Q) is isomorphic to Z4? E is the splitting field for X^5-1 over Q. Thanks.\n\nRelated Linear and Abstract Algebra News on Phys.org\nHallsofIvy\nHomework Helper\n$x^5- 1= (x- 1)(x^4+ x^3+ x^2+ x+ 1)$ has the single real root, x= 1, and 4 complex roots, $e^{2\\pi i\/5}$, $e^{4\\pi i\/5}$, $e^{6\\pi i\/5}$, and $e^{8\\pi i\/5}$. Can you construct the Galois group from that? What does Z4 look like?\n\nZ4 is {0,1,2,3} I can tell that their orders are all four. Just not sure about what's the rest needed to show isomorphic.\n\nOffice_Shredder\nStaff Emeritus","date":"2020-12-01 03:26:45","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\": 0.32556819915771484, \"perplexity\": 1407.7316037729051}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141542358.71\/warc\/CC-MAIN-20201201013119-20201201043119-00304.warc.gz\"}"}
| null | null |
\section{\label{sec:intro}Introduction}
Arrays of densely packed self-assembled Ge quantum dots (QD) on the Si(001) surface (Fig.~\ref{fig:example}) \cite{Smagina,classification} due to the phenomenon of quantum confinement of carriers are currently considered as a basis for development of prospective devices of photoelectronics \cite{Wang-properties,
Pchel_Review}.
Extensive investigations carried out for the last two decades
(see, e.g., Refs.~\onlinecite{Mo,Chem_Rev,Nucleation,Kastner,Island_growth,Fujikawa,Facet-105,Ge_QD_crystal})
resulted in the
technological achievements of the recent years that enabled the controllable formation of Ge QD arrays with the desired cluster densities (up to $10^{12}$~cm$^{-2}$, Refs.~\onlinecite{Smagina,classification}).
However, the problems of uniformity of cluster types in the arrays and the dispersion of cluster sizes are still far from solution. That is why the intensive investigations of the cluster morphology and growth process with view of reproducible formation of uniform and defectless QD arrays are strongly required. This is an issue of special importance for the ordered QD arrays \cite{Ge_QD_crystal} taking into account extremely exacting restrictions imposed on the uniformity by the aim of development of such arrays. Non-uniform ordered array composed of clusters of different types and sizes would not operate as 3D crystal of artificial atoms, or even would not reproduce regularity in successive QD layers if containing defects such as large and extended clusters or depleted regions \cite{defects_ICDS-25}.
\begin{figure}[b]
\includegraphics[scale=.6]{Fig_1a_new}(a)
\includegraphics[scale=.65]{Fig_1b}(b)
\includegraphics[scale=.7]{Fig_1c}(c)
\includegraphics[scale=.7]{Fig_1d_new}(d)
\caption{\label{fig:example}STM images of Ge pyramidal (a) and wedge-like (b) clusters, Ge QD dense array ($h_{\rm Ge}=10$\,\r{A}) on the Si(001) surface (c), and a fraction of wedges ($\blacksquare$) and pyramids ($\square$) in the arrays (d) {\it vs} Ge coverage ($T_{\rm gr}=360^{\circ}$C).}
\end{figure}
\begin{figure*}
\includegraphics[scale=1]{Fig_2a}(a)
\includegraphics[scale=1]{Fig_2b}(b)
\includegraphics[scale=1]{Fig_2c}(c)
\caption{\label{fig:arrays}STM empty state image of Ge QD array ($h_{\rm Ge}=6$~\r{A}, $T_{\rm gr}=360^{\circ}$C) on the Si(001) surface (a); $p(2\times 2)$ structure within the WL block, upper Ge atoms of the tilted dimers are resolved in the rows (b); pyramid (left) and wedge nuclei (1\,ML) on the neighboring WL blocks (c), both nuclei reconstruct the WL surface, a nucleus never exceeds the bounds of a single WL block. }
\end{figure*}
\begin{figure}
\includegraphics[scale=.75]{Fig_3a}(a)
\includegraphics[scale=.75]{Fig_3b}(b)
\includegraphics[scale=.85]{Fig_3c}(c)~~~~~\,
\includegraphics[scale=.8]{Fig_3d}(d)
\caption{\label{fig:nuclei}Nuclei of Ge hut clusters: STM empty state images (a,\,c) and atomic structures (b,\,d) of the pyramid (a,\,b)
and wedge (c,\,d) nuclei, 1 is WL.}
\end{figure}
Recently we showed that the \{105\} faceted clusters usually referred to as hut clusters \cite{Mo} are subdivided into two main {\it morphologically different} species---pyramids and wedges (Fig.~\ref{fig:example}) \cite{classification}. In the literature, both species of hut clusters are traditionally considered as structurally identical and genetically connected types \cite{Mo,Facet-105}. Explanations of transitions from square shaped to elongated islands (from pyramids to wedges in our terminology) are discussed \cite{Nucleation,Kastner,Island_growth} although no clear observations of such phenomenon have been described anywhere. Different models from simple coalescence of neighboring square shaped clusters \cite{Nucleation} to more sophisticated kinetic model of growth \cite{Kastner} have been brought forward which are in satisfactory agreement with observations. We found that at moderate growth temperatures the densities of clusters of both species are equal at the initial stage of the array formation (Fig.~\ref{fig:example}(d)). Then, as the Ge coverage is increased, the wedges become dominating in the arrays whereas the pyramids exponentially rapidly disappear \cite{classification,endnote_1}.
Lately we investigated by STM the structure of the $\{105\}$ cluster facets together with the structure of apexes (ridges and vertices) of the clusters and built structural models of both species of huts \cite{atomic_structure}. We found the structure of the ridges of the wedge-like clusters to be different from the structure of the vertices of the pyramidal ones, therefore a wedge-like cluster cannot arise from a pyramidal one and vice versa \cite{classification,atomic_structure}. Transitions between the shapes of the hut clusters are prohibited \cite{endnote_3}. One can find additional evidences of the above strong statement investigating the cluster nucleation and the initial stage of its growth by {\it in situ} STM with high enough resolution.
At present, nucleation of Ge clusters on the Si(001) surface is still very little-studied. Probably only two direct observations of this phenomenon were reported by Goldfarb {\it et~al.}~\cite{Nucleation,Goldfarb_JVST-A} and Vailionis {\it et~al.}~\cite{Vailionis}. Those comprehensive {\it in situ} STM studies explored gas-source-molecular-beam-epitaxy (GS-MBE) growth of Ge on Si(001) in the atmosphere of GeH$_4$ \cite{Nucleation,Goldfarb_JVST-A} or Ge$_2$H$_6$ \cite{Vailionis}. The chemistry of GS-MBE is obviously strongly different from that of ultrahigh vacuum (UHV) MBE which is usually employed for Ge deposition on Si substrates \cite{Pchel_Review}. Unfortunately, experimental and especially direct high resolution UHV STM investigations of Ge cluster nucleation and early stages of the cluster growth on Si(001) by UHV MBE have not been described in the literature thus far. No data are available on the morphology of nuclei and the beginning of cluster growth. Now we shall try to fill up this gap.
\begin{figure*}
\includegraphics[scale=1]{Fig_4a}(a)~~~~\,
\includegraphics[scale=1]{Fig_4b}(b)~~~~
\includegraphics[scale=1]{Fig_4c}(c)
\\
\includegraphics[scale=1]{Fig_4d}(d)
\includegraphics[scale=1]{Fig_4e-def}(e)
\includegraphics[scale=1]{Fig_4f}(f)
\caption{\label{fig:dots} STM empty state micrograph (a) of the 5-ML Ge pyramid ($h_{\rm Ge}=6$~\r{A}, $T_{\rm gr}=360^{\circ}$C), a top view of the pyramidal QD (b) and contrasted image of its vertex (c); STM empty state topographs ($h_{\rm Ge}=6$~\r{A}, $T_{\rm gr}=360^{\circ}$C) of the 2-ML Ge wedge-like cluster (d), a top view of the wedge-like QD (e) and an empty state image of the ridge of the 3-ML Ge wedge-like cluster (f); 1, 2 and 3 designate WL, the first and the second layers of QD respectively, d marks a defect arisen because of one translation uncertainty of the left dimer pair position. }
\end{figure*}
In this article, we investigate the nucleation and very beginning of growth of Ge hut clusters composing dense QD arrays formed by UHV MBE at moderate temperatures.
The atomic structure of cluster nuclei as well as the structures of very little clusters---as small as a few monolayers (ML) high over the wetting layer (WL)---are the issues of this study \cite{sabelnik}.
The results reported in the article evidence that there are two different types of nuclei on Ge wetting layer which evolve in the process of Ge deposition to pyramidal and wedge-like hut clusters. It might seem that solid proofs of this statement can be only obtained from STM
measurements during growth \cite{Nucleation, Kastner}. Unfortunately, such experiment is hardly possible now. STM operating at the growth temperatures cannot assure atomic resolution which is necessary to reveal an atomic structure of clusters and smaller objects on WL. We have made a different experiment. Having assumed that nuclei emerge on WL as combinations of dimer pairs and/or longer chains of dimers in epitaxial configuration \cite{epinucleation} and correspond to the known structure of apexes specific for each hut species \cite{classification, atomic_structure}
we have investigated WL patches, 1\,ML high formations on them and clusters of different heights (number of steps) over WL. This approach exactly simulates the above experiment ensuring the required high resolution. As a result, we succeeded to select two types of formations different in symmetry and satisfying the above requirements, which first appear at a coverage of $\sim 5$~\r{A} ($T_{\rm gr}=360^{\circ}$C) and then arise on WL during the array growth. We have interpreted them as hut nuclei, despite their sizes are much less than those predicted by the first principle calculations \cite{hut_stability}, and
traced their evolution to
huts \cite{endnote_7}.
\begin{figure}
\includegraphics[scale=.85]{Fig_5a}(a)
\includegraphics[scale=.85]{Fig_5b}(b)
\caption{\label{fig:phase_transition}
Rearrangement of the first layer (a) of a forming wedge during addition of dimer pairs of the second layer (b); labels are the same as in Fig.~\ref{fig:dots}.}
\end{figure}
The experiments were carried out using an ultra high vacuum instrument consisting of the UHV MBE chamber coupled with high resolution STM which enables the sample study at any stage of processing sequentially investigating the surface and giving additional treatments to the specimen; the samples never leave UHV ambient during experiments.
Silicon substrates ($p$-type, $\rho = 12~\Omega$\,cm) were completely deoxidized as a result of short annealing at the temperature of $\sim 925^\circ$C \cite{our_Si(001)_en}. Germanium was deposited directly on the atomically clean Si(001) surface from the source with the electron beam evaporation \cite{classification}.
The rate of Ge deposition was $\sim 0.1$~\r{A}/s and the Ge coverage ($h_{\rm Ge}$) \cite{endnote_4} was
varied from 3 to 14~\r{A}.
The substrate temperature $T_{\rm gr}$ was $360^\circ$C during Ge deposition. The rate of the sample cooling down to the room temperature was $\sim 0.4^\circ$C/s after the deposition.
The temperature was monitored with tungsten-rhenium thermocouple mounted in vacuum near the rear side of the samples and {\it in situ} graduated beforehand against the IMPAC~IS\,12-Si pyrometer which measured the sample temperature through the chamber window.
Specimens were scanned at room temperature in the constant tunneling current ($I_{\rm t}$) mode. The STM tip was zero-biased while a sample was positively or negatively biased ($U_{\rm s}$). The details of the sample preparation as well as the experimental techniques can be found elsewhere \cite{classification, our_Si(001)_en, WSxM}.
Fig.~\ref{fig:arrays}(a) presents an STM image of an array of small Ge clusters grown at $T_{\rm gr}=360^{\circ}$C and $h_{\rm Ge}=6$~\r{A}. WL is seen to have a block ($M \times N$ patched) structure. The blocks are usually $p(2\times 2)$ reconstructed (Fig.~\ref{fig:arrays}(b)) \cite{endnote_5}. We suppose that the process of the cluster nucleation consists in formation of new structures on the WL blocks. These 1\,ML high structures are well resolved in Fig.~\ref{fig:arrays}(c) on the neighboring WL blocks: The left feature is assumed to be a nucleus of the pyramid whereas the right one is considered as a nucleus of the wedge-like cluster. A good few of such structures are observed in the long shot of the array (Fig.~\ref{fig:arrays}(a)). STM images of the nuclei and their schematic plots are given in Fig.~\ref{fig:nuclei}. The further growth of the clusters is shown in Fig.~\ref{fig:dots}. Fig.~\ref{fig:dots}(a) presents an STM image of the 5\,ML high pyramid. It is commonly adopted that the hut clusters grow by successive filling the (001) terraces of the $\{105\}$ faces by the dimer rows \cite{Kastner}.
A schematic plot of the 2-ML pyramid based on this assumption (Fig.~\ref{fig:dots}(b)) demonstrates its atomic structure (even number of layers is shown in both (a) and (b) pictures, so the diagram reproduces the entire structure of the dot except for its height). It is seen comparing Figs.~\ref{fig:nuclei}(a) and \ref{fig:dots}(c) that the vertex repeats the structure of the nucleus drown in Fig.~\ref{fig:nuclei}(b)\cite{fig1}.
The characteristic distances exactly match.
The $<$100$>$ direction of the base sides is predetermined by the nucleus structure, thus the pyramids grow without phase transition when the second and subsequent layers are added. Only nucleus-like structures of their apexes are rotated $90^{\circ}$ with respect to the rows on previous terraces to form the correct epitaxial configuration when the heights are increased by 1\,ML, but this rotation does not violate the symmetry of the previous layers of the cluster.
\begin{figure}[t]
\includegraphics[scale=1.2]{Fig_6a}
\caption{\label{fig:face}
Schematic drawing of the $\{105\}$ facet superimposed on its STM image ($4.3\times 4.4$~nm, $U_{\rm s}=+3.0$~V, $I_{\rm t}=100$~pA), the cluster base side is parallel to the [100] direction, the steps rise from the lower right to the upper left corner.}
\end{figure}
A different scenario of growth of the wedge-like clusters have been observed. Figs.~\ref{fig:dots}(d,\,e) show an image and a schematic diagram of the 2-ML wedge-like cluster. The ridge structure is seen to be different from the nucleus structure presented in Figs.~\ref{fig:nuclei}(c,\,d). The structure of the ridge is well resolved in the image of the 3-ML cluster (Fig.~\ref{fig:dots}(f)) filtered to contrast the uppermost layer of atoms. In this image, the dimer pairs of the ridge are $90^{\circ}$ rotated compared to the 2-ML wedge that is in full agreement with the proposed atomic model\cite{ridge}. This structure of the wedge-like cluster arise due to rearrangement of rows of the first layer in the process of the second layer formation (Fig.~\ref{fig:phase_transition}). The phase transition in the first layer generates the base with all sides directed along the $<$100$>$ axes which is necessary to give rise to the $\{105\}$ faceted cluster (it is seen from Figs.~\ref{fig:nuclei}(c,\,d) that only one pair of sides of a wedge nucleus runs along the $<$100$>$ direction). After the transition, the elongation of the elementary structure is possible only along a single axis which is determined by the symmetry and clearly seen when comparing Figs.~\ref{fig:dots}(e) and \ref{fig:phase_transition}(b) (along the arrows in Fig.~\ref{fig:phase_transition}(b)).
This preferential growth direction determines the rapid growth on the triangular facets (short edges).
The growth on these facets does not change the orientation of the dimer pairs forming the ridge. It is obviously also that it cannot increase the cluster height but only its length. The increase of the cluster height is governed by the completion of the trapezoidal facet \cite{facet_growth}. The latter process is accompanied by the change of direction of the dimer pairs on the ridge when the apex terrace is completed. Note that the phenomenon of the wedge height limitation described in Ref.~\onlinecite{classification} differs from the process of its length self limitation. The former is mainly controlled by the growth temperature and the later is governed by either the area of the trapezoidal faces or the number and/or sizes of the WL blocks covered by the elongating cluster, as well as the competition of the processes of the in-height and longitudinal growth. In general, the cause of the wedge elongation is still unclear now.
It is necessary to remark here that the nuclei are always observed to arise on sufficiently large WL patches. There must be enough room for a nucleus on a single patch. A nucleus cannot be housed on more than one patch. So, cluster nucleation is impossible on little (too narrow or short) patches (Fig.~\ref{fig:arrays}(a)).
It should be noted also that according to the proposed model the wedge-like clusters always contain point defects on the triangular (short) facets. The defects are located in the upper corners of the facets and caused by uncertainty of one translation in the position a dimer pair which forms the penultimate terrace of the triangular facet (Figs.~\ref{fig:dots}(d--f)). The predicted presence of these defects removes the degeneracy of the facets and hence
an issue of the symmetry violation which occur if the pyramid-to-wedge transition is assumed (this issue was discussed in detail in Ref.~\onlinecite{classification}). These defects are absent on the facets of the pyramidal huts.
Their triangular facets are degenerate. Therefore, as it follows from our model, the trapezoidal and triangular facets of the wedge
are not degenerate with respect to one another even at very beginning of cluster growth. The wedges can easily elongate by growing on the
triangular facets faster than on trapezoidal ones, whereas pyramids, having degenerate facets, cannot elongate and grow
only in height outrunning wedges. This explains greater heights of pyramids \cite{classification}.
The proposed models being applied to draw the clusters by filling terrace by terrace (like it is done in Fig.~\ref{fig:dots}) allowed us to
deduce a model of the $\{105\}$ facets. This model resulting from the above simple crystallographic consideration corresponds to the PD \cite{Mo} (paired dimers) rather than more recent RS (rebonded step) model \cite{Fujikawa,Facet-105} which is now believed to improve the previous PD model by Mo {\it et al}. Being superposed with the empty state STM image of the cluster $\{105\}$ facet it demonstrates an excellent agreement with the experiment (Fig.~\ref{fig:face}). Dangling bonds of the derived in such a way $\{105\}$-PD facets in reality
may stimulate Ge atom addition and cluster growth.
Less stability of the $\{105\}$-PD facets compared to the Ge(105)/Si(105)-RS plane may cause fast completion of hut terraces during epitaxy.
It should be noticed also that, as it follows from the reported models, the growth of the wedge second layer requires reconstruction of the buried previous layer.
This phenomenon has been discussed theoretically before as ``critical
epinucleation'' on reconstructed surface \cite{epinucleation}. In particular, the atomic
models drawn in Figs.~\ref{fig:dots}(e) and~\ref{fig:phase_transition} show the ad-dimer rows
un-reconstructing the surface layer that can only happen beyond
a critical number of ad-dimers defined as ``epinucleus''. So, the presented data
could be one of the first experimental evidence of the
epinucleus \cite{thanks}. The critical epinucleation appears to be a basic phenomenon for hut formation on $(M\times N)$.
In conclusion,
we have reported the direct observation of nucleation of Ge hut clusters formed by UHV MBE on the Si surface.
The nuclei of the pyramidal and wedge-like clusters have been observed on the wetting layer $(M\times N)$ patches and found to have different structures. The atomic models of nuclei of both species of the hut clusters have been built as well as the models of the clusters at the early stage of growth.
The growth of the clusters of each species has been demonstrated to follow generic scenarios. The formation of the second atomic layer of the wedge-like cluster results in rearrangement of its first layer. Its ridge structure does not repeat the structure of the nucleus.
The pyramidal cluster grows without phase transitions. The structure of its vertex copies the structure of the nucleus. The cluster of one species cannot turn into the cluster of the other species.
The wedge-like clusters contain point defects in the upper corners of the triangular faces and have preferential directions of growth along the ridges.
The derived structure of the $\{105\}$ facet corresponds to the PD model. The critical epinucleation phenomenon may be responsible for hut formation on $(M\times N)$ patched WL.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 2,233
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Q: How to render the value of variable from a HTML page? suppose, If I have dictionary
game = [{'points': '534', 'Team': 'Ireland', 'Rating': '36', 'Matches': '15'}, {'points': '5146', 'Team': 'England Women', 'Rating': '129', 'Matches': '40'}, {'points': '5898', 'Team': 'Australia Women', 'Rating': '128', 'Matches': '46'},{'date-updated': '05 February 2018', 'match-type': "ICC Women's Championship"}]
If I wan't to use key's 'points','Team','Rating','Matches', I can use for loop like
{% for team in game %}
<tr class="table table-bordered">
<td>{{ team["Team"]}}</td>
<td>{{ team['Matches'] }}</td>
<td>{{ team['points'] }}</td>
<td>{{ team['Rating'] }}</td>
{% endif%}
and I can print these key's value.
How can I render the key value of date-updated and match-type without using for loop since there is only one set of dictionary present.
I tried to call it as bellow but it didn't print anything
<tr>
<th class="bg-danger text-lg-left">{{game['date-updated']}} </th>
<th class="bg-danger text-lg-right">{{game['match-type']}}</th>
</tr>
A: Without for loop you need to know the index of the array,
Following your example
<tr>
<th class="bg-danger text-lg-left">{{game[3]['date-updated']}} </th>
<th class="bg-danger text-lg-right">{{game[3]['match-type']}}</th>
</tr>
A: You can use negative indexing if the position of the dictionary is always in the last.
EX:
game = [{'points': '534', 'Team': 'Ireland', 'Rating': '36', 'Matches': '15'}, {'points': '5146', 'Team': 'England Women', 'Rating': '129', 'Matches': '40'}, {'points': '5898', 'Team': 'Australia Women', 'Rating': '128', 'Matches': '46'},{'date-updated': '05 February 2018', 'match-type': "ICC Women's Championship"}]
print game[-1]['date-updated']
print game[-1]['match-type']
Output:
05 February 2018
ICC Women's Championship
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 8
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The hidden millions in Australia's political finance system
follow the money trail
OUR SYSTEM is broken
Australian law requires all payments to politicians over $13,200 to be publicly declared - an important public transparency measure to stop corruption.
But right now there are some gaping legal loopholes that see tens of millions of dollars funnelled into the pockets of our politicians with no oversight, no accountability.
By piecing together fragments of publicly available data, our research reveals the full extent of hidden 'Dark Money' flooding our political system.
Welcome to the world of Dark Money -
where only 15% of payments to major parties are transparently disclosed.
UNDER THE SPOTLIGHT: HOW DO THE PARTIES STACK UP?
THE LIBERAL PARTY
Last election the Liberal Party transparently declared only 13% of their total private income.
The Liberal Party declared $8.98 million transparently, funnelled a further $5.5 million of donations through "affiliated entities", and listed $8.97 million as "other receipts". A full $45.9 million of their income was undisclosed Dark Money.
66% of the Liberal Party's private income last election came from undisclosed Dark Money.
Over the past decade, Dark Money has increased as a proportion of the Liberal Party's total income.
THE Labor PArty
Last election the Labor Party transparently declared 21% of their total private income.
Australian law requires all payments to politicians over $13,200 to be publicly declared. The Labor Party discloses more than they are required to, having made a public commitment to declare all payments over $1,000.
Labor declared $10.4 million transparently and listed $15 million as "other receipts" (note: Labor listed all income from affiliated entities as "other receipts"). A full $24.4 million of their income was entirely undisclosed Dark Money.
49% of the Labor Party's private income last election came from undisclosed Dark Money.
Over the past decade, the proportion of Labor Party income from Dark Money has remained about steady.
The Nationals
The Nationals run multiple "affiliated entities" and list a range of different income as "other receipts". The proportion of undisclosed income as a share of total income is higher than for the major parties - but the total sums involved are far smaller.
In the 2013-14 election year, the Nationals declared $0.5 million transparently, listed $0.9 million as "other receipts". $4 million of their income was undisclosed Dark Money.
70% of the Nationals private income in 2013-14 came from undisclosed Dark Money.
The total income of the Nationals - including Dark Money - is tiny compared to the income of the major parties.
The Greens' financial returns are different to the other parties - they don't run "affiliated entities" and declare only a tiny fraction of their income as "other receipts". Their proportion of undisclosed income is the highest of all the parties, however the total sums involved are far lower.
Australian law requires all payments to politicians over $13,200 to be publicly declared. The Greens disclose more than they are required to, having made a public commitment to declare all payments over $1,000. The Greens also publish this information on their website.
In the 2013-14 election year, the Greens declared $1.4 million transparently. $8.9 million income was undisclosed Dark Money.
85% of the Greens private income in 2013-14 came from undisclosed Dark Money.
The total income of the Greens - including Dark Money - is tiny compared to the income of the major parties.
HIDING THE MONEY TRAIL - HOW DO THEY DO IT?
Although there are some legitimate reasons why Dark Money may not be declared, it is a loophole that can be exploited.
Dark Money is hidden in three main ways...
Political parties have set up a range of "affiliated entities" or fundraising clubs - such as The Free Enterprise Foundation (Liberal) and Labor Holdings (ALP) - to obscure the origins of payments they receive.
other receipts
Political parties use creative accounting techniques to classify payments as "other receipts" instead of "donations" - meaning they get mixed in with all other income like share dividends and service fees for reporting purposes.
Donation Splitting
Donations over $13,200 need to be disclosed, but by splitting payments into smaller chunks donors can avoid detection. These might be distributed between party state and territory branches, or staggered over different days. This is the biggest loophole for Dark Money.
Using just donation splitting, it's possible to funnel half a million dollars per year to major political parties without disclosing a cent.
HOW CAN WE FIX THE SYSTEM?
Together we can put people back at the heart of our democracy - here's what we need to do:
End the shroud of secrecy: require all donations above $500 to be publicly disclosed on the internet in real-time, including donations to and from affiliated entities.
Stop the money game: cap the amount any individual or corporation can donate at $1000 per financial year, and introduce expenditure caps on election campaigns.
Stop offshore entities buying political influence: prohibit any corporation not registered in Australia, or any individual who doesn't have citizenship or residency, from making donations.
A corruption watchdog with teeth: create an independent federal corruption watchdog with broad investigative powers.
Close the revolving door: prevent Members of Parliament from engaging in lobbying work for a period of three years after they leave office.
To make our message heard over the flood of Dark Money, we need to speak together with one voice. Add your name today!
This website was made possible by the 3,300 incredible GetUp members that chipped in to fund groundbreaking research into Big Money in politics. Please note that all figures in this report exclude public funding and payments made between different branches of the same political party.
Follow this link to see the full Dark Money report: https://www.getup.org.au/dark-money-full-report
Authorised by Paul Oosting, 14/338 Pitt Street, Sydney NSW 2000
|
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| 2,664
|
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